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[ [ "Life after eruption - I. Spectroscopic observations of ten nova\n candidates" ], [ "Abstract We have started a project to investigate the connection of post-novae with the population of cataclysmic variables.", "Our first steps in this concern improving the sample of known post-novae and their properties.", "Here we present the recovery and/or confirmation of the old novae MT Cen, V812 Cen, V655 CrA, IL Nor, V2109 Oph, V909 Sgr, V2572 Sgr, and V728 Sco.", "Principal photometric and spectroscopic properties of these systems are discussed.", "We find that V909 Sgr is a probable magnetic CV, and that V728 Sco is a high-inclination system.", "We furthermore suggest that the two candidate novae V734 Sco and V1310 Sgr have been misclassified and instead are Mira variables." ], [ "Introduction", "A nova eruption in a cataclysmic binary star (CV) occurs as a thermonuclear explosion on the surface of the white dwarf primary star once it has accumulated a critical mass from its late-type, usually main-sequence, companion.", "In the process of the eruption, material is ejected into the interstellar medium.", "The mass of the shell typically amounts to $10^{-5}$ to $10^{-4}~\\mathrm {M_\\odot }$ [60].", "There is recent evidence that the mass of the white dwarf increases in the course of the secular evolution of CVs [62].", "Consequently, in nova eruptions less than the accumulated mass is ejected.", "With typical mass-transfer rates of a few $10^{-8}$ to $10^{-9}~\\mathrm {M_\\odot ~yr^{-1}}$ [50] the typical recurrence time of a nova eruption $>10^{3}~\\mathrm {yr}$ .", "This distinguishes the classical novae from the recurrent novae that have much shorter recurrence times in the order of decades, and in most cases have a giant donor.", "In between nova eruptions the binary is supposed to appear as a “normal\" CV, i.e.", "its behaviour is dominated by its current mass-transfer rate and the magnetic field strength of the white dwarf [51].", "Furthermore, the hibernation model predicts that most of the time between eruptions the system passes as a detached binary [43], [30].", "The proposed scenario here is that the binary separation increases due to the mass lost from the white dwarf during the eruption.", "The primary is heated by the thermonuclear runaway, thus in return heats the companion star, driving it far out of thermal equilibrium, and so sustaining a high mass-transfer rate in spite of the increased separation.", "The mass-transfer rate gradually decreases and eventually the donor can relax to thermal equilibrium, losing contact to its Roche lobe, and the system becomes detached.", "While there is still no observational evidence for the latter part of this scenario, i.e.", "the state of actual “hibernation\" [26], it is already well established that old novae are part of the CV class.", "For example, the system DQ Her (Nova Her 1934) is known as the prototype intermediate polar, while RR Pic (Nova Pic 1925) shows the characteristics of an SW Sex system [37].", "GK Per (Nova Per 1901) seems to be an intermediate polar, revealing, decades after its nova eruption, the typical behaviour of a dwarf nova, with semi-periodic outbursts [45].", "Furthermore, the discovery of a nova shell around the dwarf nova Z Cam [44] proves that (at least some) systems discovered as CVs are also old novae.", "In spite of these, more or less isolated and exotic, cases, there is still only a fragmentary knowledge of the generally valid long-term behaviour of classical novae before and after their eruption.", "A study based on 97 relatively well observed galactic novae by [52] revealed a decrease in brightness with a mean slope $21 \\pm 6~\\mathrm {mmag}$ per year during the first 130 years after the eruption.", "Similar results were obtained by [12] who derived, from a sample of nine very well covered cases, a mean decline rate of $10 \\pm 3~\\mathrm {mmag}$ per year, half a century after outburst.", "This is all our knowledge, at present.", "Most classical novae have orbital periods between 3 and 6 hours [6].", "In this period range we find several other classes of CVs, in particular dwarf novae, magnetic CVs, and nova-like variables like SW Sex and UX UMa stars, most of them characterised by high mass-transfer rates.", "Do all these CVs suffer nova eruptions?", "Is the magnetic field of the white dwarf of any importance for the eruption recurrence time?", "And do post-novae really go into hibernation, i.e.", "do CVs become detached for a time as a consequence of the nova eruption?", "We do not know, although answering these questions would be rather important for our understanding of CVs.", "The only way to progress here would be a more exhaustive investigation of the detailed behaviour of old novae decades and centuries after their outbursts.", "As a first step towards this goal, it is necessary to identify the remnants of past novae, especially those which had erupted long time ago.", "Our knowledge, in this respect, is largely incomplete.", "There are about 200 confirmed or suspected novae with eruptions before 1980, but for less than half of them a spectrum of the post-nova has been obtained.", "Only for 39 of these, a value for the orbital period is listed.", "Moreover, eight of these can be regarded as uncertain since they were obtained photometrically without the light curve presenting definite orbital features like eclipses or ellipsoidal variations, two more (DY Pup and DI Lac) are based on unpublished data [9], and the 5.714 d period of V1017 Sgr is marked as “preliminary\" [41].", "This leaves a mere 28 old novae with a well-established orbital period.", "To improve the current situation on systematic research on old novae we have started a program to identify nova candidates with $U\\!BV\\!R$ photometry via their specific colours and to confirm them with low-resolution spectroscopy [38], [40].", "For sufficiently bright systems, we furthermore plan to determine their orbital period by measuring radial velocities obtained with time-series spectroscopy.", "Since our main interest regards the underlying CV we intend to limit our research to novae where the nova shell is already sufficiently faint to provide only a negligible contribution at least to the optical spectrum.", "The time scale for the fading of the nova shell will be different for individual systems.", "However, a literature research on well-known novae shows that after about 30 years the nebular lines have disappeared from the spectra in almost all systemsFor a counter-example see the nova HR Del that still presents a significant contribution of nebular lines 40 years after the eruption [15].", "In order to facilitate the selection, we therefore limit our study to novae that erupted before 1980.", "We here present the results for ten candidate old novae." ], [ "Observations and reduction", "The data were obtained during three observing runs in May 2009, February and June 2011, at the ESO-NTT, La Silla, Chile, using EFOSC2 [14].", "Only during the first run the weather conditions allowed for calibrated photometry.", "The respective data were taken as $\\ge $ 3 exposures in the $U$ , $B$ , $V$ , and $R$ passbands.", "Between each exposure the telescope was moved slightly so that pixel deficiencies would average out when combining them to a single frame.", "The individual frames were bias-corrected, but not flatfielded, because EFOSC2 flats suffer significantly from a central light concentration problem.", "Subsequently, aperture photometry was performed using IRAF's phot package.", "These served as input for the standalone daomatch routine [47] to determine the offsets between the individual frames.", "Correcting for these offsets, the frames were combined to a single one using a 3$\\sigma $ clipping algorithm for the averaging.", "Since all photometric fields are located in crowded regions the stellar magnitudes were determined by fitting the point spread function (PSF).", "Finally, the magnitudes were calibrated using photometric data of standard stars [18], [19] that were taken in the same night.", "The spectroscopic data were collected using grism 4 and, in one case, grism 11, in combination with a 1.0$^{\\prime \\prime }$ slit.", "In another case, the 1.5$^{\\prime \\prime }$ slit was employed with grism 4.", "Similar to the photometry, the data consisted of three individual spectra that were later combined prior to the extraction process.", "The data were corrected for bias, divided by normalised flats, and afterwards combined and extracted using IRAF routines.", "Wavelength calibration was performed using Helium-Argon lamps.", "The resulting typical spectral ranges and resolutions are 3490–7470 Å at 13 Å (FWHM, i.e.", "Full Width at Half Maximum for an arc line) for grism 11, and 4050–7440 Å at 11.5 Å (34 Å in case of the 1.5$^{\\prime \\prime }$ slit) for grism 4.", "The spectra were corrected for the instrumental response function using standard star spectra obtained during the May 2009 run for grism 4, and the February 2011 run for grism 11.", "Since the conditions were not photometric during these nights, these spectra can not be considered as calibrated in absolute flux, but only regarding the relative spectral energy distribution (SED).", "In order to determine the coordinates reported in Table REF , we have performed an astrometric correction of the combined $R$ -band frames (or, for those objects without calibrated photometry, of the $R$ -band acquisition frame) using the routines embedded in Starlink's GAIAhttp://astro.dur.ac.uk/$\\sim $ pdraper/gaia/gaia.html tool (version 4.4.1) with the UCAC3 [61] catalogue.", "Prior to the final fit, saturated stars and ambiguous positions (close visual binaries) were manually deleted.", "The typical RMS of the fit amounted to significantly less than 1 pixel, i.e.", "less than 0.24$^{\\prime \\prime }$ ." ], [ "MT Cen = Nova Cen 1931", "MT Cen was classified as a fast nova ($t_3 \\sim 10~\\mathrm {d}$ ) that reached a photographic brightness of 8.4 mag on May 12, 1931 [10].", "Due to its faintness already the later stages of the eruption are only sparsely covered, and first attempts to observe the post-nova or its shell were unsuccessful [25], [16], Later, [57] identified a likely candidate for the post-nova by detecting flickering-like variability in their high-speed photometry.", "Our colour-colour diagram (Fig.", "REF , top) points out this same object as having colours different from the majority of the stars in the field.", "This becomes more pronounced if only the central part of the CCD image is considered (black squares in Fig.", "REF ).", "Our spectrum of the candidate has very low S/N (Fig.", "REF ), but the detection of weak H$\\alpha $ emission confirms the post-nova.", "In agreement with the not particularly blue colour derived photometrically (Table REF ), the continuum flux increases toward longer wavelengths.", "One possibility is that the late-type donor contributes significantly to the optical flux, which would indicate either a low-mass-transfer system, or a bright donor, the latter implying a long orbital period.", "However, this part of the sky suffers from considerable reddening (Table REF ).", "In order to examine the influence of the extinction we have dereddened the spectrum using the corresponding IRAF task, which is based on the relations derived by [2].", "Since the absolute extinction $A(V)$ is not known, we have used the standard value for the ratio $R(V) = A(V)/E(B\\!-\\!V) = 3.1$ .", "The resulting dereddened spectrum is shown in Fig.", "REF .", "It proves that the red slope of the continuum was entirely due to interstellar reddening, which even masked the presence of the emission components of H$\\beta $ and the Bowen blend.", "The corrected spectrum shows an SED that is typical for a high mass-transfer system." ], [ "V812 Cen = Nova Cen 1973", "This object was a late discovery, reported five years after its eruption by [23] who detected a typical spectrum of a nova in its nebular phase on objective prism plates.", "The authors determined the continuum brightness at this stage to $V = 11~\\mathrm {mag}$ .", "[11] identified a candidate for the post-nova, but attempts to confirm it spectroscopically [63] or by detecting the nova shell [8] remained unsuccessful.", "In our photometry we find this candidate as a blue object about 10 mag fainter than the nova at the time of its detection (Fig.", "REF , Table REF ).", "The spectroscopy shows a flat continuum with strong H$\\alpha $ emission, much weaker H$\\beta $ , as well as a couple of HeI lines (Fig.", "REF , Table REF ).", "The Balmer decrement is much stronger than usual in CVs [56].", "Since even after the dereddening the continuum slope is not particularly steep (Table REF ), it does not appear that either of the underlying stellar components of the binary is affecting the strength of the emission lines.", "Rather it seems more likely that an additional, non-stellar, source is contributing to the H$\\alpha $ emission.", "Although [8] did not detect a nova shell, such a shell still represents the most promising candidate for the additional H$\\alpha $ source.", "This is also the youngest nova in our sample, although three other systems with “normal\" values are not far behind (Table REF ).", "[33] found that the contribution by the nebula does not only depend on the time that has passed since the eruption, but also on the speed class, with the nebular contribution being more important for slower novae.", "Unfortunately there is no corresponding information available for V812 Cen, and we can thus only speculate that this nova has not been a particularly fast one.", "A similar, even slightly stronger, Balmer decrement was found by [33] for FH Ser ($t_3 = 62~\\mathrm {d}$ ) in a spectrum taken 20 years after the eruption." ], [ "V655 CrA = Nova CrA 1967", "Similar to V812 Cen, this nova was discovered some time after its original eruption on objective prism plates [34].", "The author determined the magnitude at the time of the discovery to 13 mag.", "Based on the characteristics of the spectrum he estimated the maximum brightness to 8 mag.", "He furthermore identifies a likely progenitor with a red magnitude of 17 mag, a value that was later corrected to $J = 17.6~\\mathrm {mag}$ by [11].", "However, the pre-nova candidate turned out to be a close visual binary, and the correct identification remained unclear.", "[13] took a spectrum, but it was of too low quality to even unambiguously detect the presence of emission lines.", "Finally, [58] identified a likely candidate with a mean magnitude of $V = 17.6$ based on flickering-type variability detected in their high-speed photometry.", "Our spectrum (Fig.", "REF ) confirms this object as the post-nova.", "It shows a steep blue continuum with comparatively narrow emission of H$\\alpha $ and H$\\beta $ , as well as a broader Bowen/HeII 4686 component (Table REF ).", "This system likely still sustains a high mass-transfer rate." ], [ "IL Nor = Nova Nor 1893", "This is the second oldest nova in this sample.", "It was discovered by M. Fleming on objective prism plates [28].", "Investigations of earlier photographic plates yielded a maximum magnitude of 7 [29].", "[11] classifies it as a moderately fast nova.", "The post nova is a member of a close visual triple system that for some time could not be resolved, although [13] reported a spectrum showing Balmer emission lines.", "An attempt by [16] to image the nova shell was unsuccessful.", "In a recent paper, [59] identified the post-nova in their high-speed photometry.", "The authors present two short light curves about one year apart (March 2003 and February 2004).", "Apart from some shorter term variability the second run showed the object on average 0.5 mag fainter than the first one.", "Our photometry (Table REF , Fig.", "REF ) reveals this candidate as a very blue object with $V = 19.0~\\mathrm {mag}$ , at the same brightness as in the second photometric run presented by [59].", "The spectrum consists of a blue continuum, with only weak Balmer and HeI emission.", "A Bowen/HeII component is also clearly visible.", "These characteristics suggest that more than a hundred years after its eruption this system still drives a comparatively high mass-transfer rate, which is in agreement with the results by [33] that generally the spectroscopic properties of novae after the initial decline phase (20–30 yr) change very little over the next decades." ], [ "V2109 Oph = Nova Oph 1969", "Another late discovery, this nova was reported eight years after its eruption when [22] detected an object with a red magnitude of 10.8 and an emission line spectrum on archival objective prism plates.", "A later study on Sonneberg plates by [55] revealed that the discovery plate missed maximum brightness by at least twelve days, when the object presented a blue magnitude of 8.9.", "[11] identified a likely candidate for the post-nova, that was later observed by [48] as a comparatively red star with $V = 19.4~\\mathrm {mag}$ , $B\\!-\\!V > 1.4$ and $V\\!-\\!R = 1.2$ .", "Our spectroscopy finds the object at $R \\sim 19.7~\\mathrm {mag}$ , and thus much fainter than during the [48] observations that took place 20 years after the eruption ($R = 18.2~\\mathrm {mag}$ ).", "The spectrum shows the red continuum that could be expected from Szkody's photometry.", "Like in the case of MT Cen (Sec.", "REF ) this is in large part due to the increased extinction in this area of the sky (Table REF ).", "Correcting for this effect yields a considerably bluer spectrum (Fig.", "REF ), although the SED is still rather unusual for a post-nova (see discussion in Sec. ).", "We furthermore find moderately strong emission lines of the Balmer and HeI series, as well as HeII 4686 (Fig.", "REF , Table REF ).", "Contrary to most of the other systems in this sample, the Bowen blend cannot be detected in V2109 Oph, indicating a comparatively low mass-transfer rate." ], [ "V909 Sgr = Nova Sgr 1941", "[11] summarises the discovery history of this nova that erupted in 1941, reached a photographic maximum brightness of 6.8 mag, and showed a very fast decline rate of $t_3 = 7~\\mathrm {d}$ .", "[6] report the post-nova as being an eclipsing system with an orbital period of 3.36 h, but emphasise the need for confirmation.", "Since they do not provide a finding chart, the correct identification of the post-nova remained ambiguous.", "In our colour-colour diagram (Fig.", "REF ) we detect a blue object very close to the coordinates listed in the [9] catalogue.", "The spectroscopy shows a blue continuum superposed with hydrogen and helium emission lines (Fig.", "REF ).", "Certainly the most remarkable feature is the strength of HeII in the system, evidenced in the presence of the $\\lambda $ 5412 Å line, and the $\\lambda $ 4686 Å line rivalling H$\\alpha $ in strength, while the Bowen blend appears absent (Table REF ).", "The spectrum – at least at the current resolution and S/N – bears a strong resemblance to that of the old nova and asynchronous polar V1500 Cyg [33], whose 3.35 h orbital period [27], [42] coincidentally is very close to the one reported by [6] for V909 Sgr." ], [ "V1310 Sgr = Nova Sgr 1935", "This star was flagged as a nova by A. D. Fokker in 1951 [11] based on photometric variability.", "Duerbeck classifies it as a slow nova with a “steep rise and [an] extremely slow, fairly smooth decline\".", "The time to drop by 3 magnitudes from the photographic maximum brightness of 11.7 mag was measured as $t_3 = 390~\\mathrm {d}$ .", "There is no spectroscopic information available, neither near maximum brightness, nor at minimum.", "[9] mark a fairly bright star [5] as the post-nova, but remark that the object is possibly a Mira variable instead of a nova.", "Our spectroscopy finds the star at $R \\sim 13.5~\\mathrm {mag}$ .", "The spectrum (Fig.", "REF ) looks like a late K or early M star.", "No emission lines are observed.", "We used the TiO5 index defined and calibrated by [32] to calculate a spectral type of M0$\\pm $ 0.5.", "This is probably a Mira type star.", "This poses the question if V1310 Sgr has to be qualified as a misidentification (i.e.", "the post-nova is some other object in the field) or as a misclassification (i.e.", "the original discovery report mistook the Mira variability for a nova eruption).", "Since both the shape and the time scale of the photometric variability as well as the observed difference in magnitude ($\\sim $ 2.5 mag) are consistent with a Mira type light curve [20], we favour the latter possibility, and suggest to remove V1310 Sgr from the list of potential classical novae." ], [ "V2572 Sgr = Nova Sgr 1969", "The nova was discovered due to its photometric variability that reached a maximum photographic brightness of 6.5 mag as reported by [1].", "[11] classifies it as a moderately fast nova with $t_3 = 44~\\mathrm {d}$ .", "The eruption light curve is given by [17], who furthermore reports a “fairly constant\" brightness of $V \\sim 13~\\mathrm {mag}$ for the pre-nova in the years 1957–1968, as well as for the post-nova in 1970–1972.", "Consistent with these findings, [31] present $V \\sim 12.5 \\pm 0.5~\\mathrm {mag}$ for the pre-nova in the years 1966–1967.", "However, in the same time range the authors also find an unusually red and strongly variable colour for this object to $B\\!-\\!V = 0.8-2.0~\\mathrm {mag}$ .", "Since modern finding charts [9] show a fairly crowded region of the sky at the given position, it is thus likely that above values for the pre- and post-nova do not represent the nova itself, but rather the combined brightness of the nova and its close visual companions.", "Supporting this interpretation we find the nova to have a steep blue continuum, and a much fainter brightness than reported of $R \\sim 17.8~\\mathrm {mag}$ .", "Emission lines of the Balmer and HeI series are clearly discernible, albeit comparatively weak.", "A Bowen/HeII component is also present.", "Overall, the spectroscopic characteristics give the impression of a high mass-transfer system.", "On our acquisition frame (Fig.", "REF ) we note that the vicinity of the post-nova appears “smudgy\", which could indicate the presence of a nova shell." ], [ "V728 Sco = Nova Sco 1862 = Nova Ara 1862", "Close to the border between constellations Scorpius and Ara (which is why it was originally assigned to the latter) a bright star of 5$^\\mathrm {th}$ magnitude was reported by [49] to have been observed visually on October 5–9, 1862.", "Only four days later he found it to have declined to below 11$^\\mathrm {th}$ mag.", "[11] identified two faint candidates ($j \\sim 20-21~\\mathrm {mag}$ ) for the post-nova based on Tebbutt's coordinates.", "[7] list the object as a potential magnetic nova due to its high eruption amplitude and fast decline.", "However, [36] note that this candidate presents colours that are more consistent with a main-sequence star than a CV.", "Based on their colour-colour diagram they instead suggest two new candidates that are within 1$^\\prime $ of the original coordinates.", "Our present photometric observations cover a larger area on the sky.", "The corresponding colour-colour diagram is presented in Fig.", "REF .", "The two candidates from [36] can be found as the bluest object in the field with $U\\!-\\!B = -0.88$ , $B\\!-\\!V = 0.02$ , and as the black square at $U\\!-\\!B = 0.38$ , $B\\!-\\!V = 0.24$ .", "Spectroscopic observations of the latter, which are not presented here in detail, showed it to be a blue star with Balmer absorption lines.", "The other candidate has not been observed spectroscopically.", "Instead we find the post-nova to be an object slightly more than 2$^\\prime $ northwest of the position given in the [9] catalogue (see Table REF and Fig.", "REF for the corrected position).", "The spectroscopy reveals a blue continuum and (for a nova) comparatively strong Balmer and HeI emission lines (Fig.", "REF , Table REF ).", "This system is the one in our sample that looks most like a dwarf nova [56] but for the blue slope and the presence of a broad Bowen/HeII component.", "The broad and structured Balmer emission lines (Fig.", "REF ) make it an interesting object for further studies, because they suggest a high-inclination, possibly eclipsing, CV." ], [ "V734 Sco", "As summarised by [11] this object was reported as a possible long-period variable or nova by L. Plaut who discovered it to be brighter than 15$^\\mathrm {th}$ mag for 10 days on photographic plates from 1937.", "The star is bright in the infrared: $J = 8.7~\\mathrm {mag}$ , $H = 7.6~\\mathrm {mag}$ , $K = 6.9~\\mathrm {mag}$ [5], and listed as a probable Mira variable in the [9] catalogue.", "Our spectrum in Fig.", "REF shows TiO bands of a cool star and Balmer emission lines.", "The strongest emission is observed in H$\\delta $ (equivalent width $W_\\lambda $ = $-$ 37 Å), followed by H$\\gamma $ ($W_\\lambda $ = $-$ 22 Å) and probably H$\\beta $ .", "This unusual Balmer decrement is typical for oxygen-rich (M-type) Mira stars [3].", "We used the TiO5 index defined by [32] and calibrated by [4] to calculate a spectral type of M8.7$\\pm $ 0.5.", "The overall shape around 6830 Å indicates a giant rather than a dwarf star, consistent with a Mira classification.", "Like in the case of V1310 Sgr we therefore conclude that V734 Sco is not a classical nova and should not be listed as such." ], [ "Discussion", "While the present work is mainly aimed at increasing the sample of confirmed old novae for later studies, and the data quality is mostly not suited for a thorough analysis, it still allows to remark on one or the other detail.", "In Table REF we list several photometric and spectroscopic properties of the confirmed post-novae.", "We have used the previously reported maximum brightness $m_\\mathrm {max}$ (in column 2) and the current brightness as derived from our observations $m_\\mathrm {min}$ (column 3) to derive the eruption amplitude $\\Delta m = m_\\mathrm {max}-m_\\mathrm {min}$ (column 4).", "This calculation ignores brightness differences between filters.", "Taking into account that the listed magnitudes are close to the visual range, and that colour differences within the $B-R$ range rarely exceed 1.0 mag (see Table REF ), we can estimate a typical uncertainty of $\\sim $ 0.5 mag.", "Column 5 presents the ”age\" of the post-nova, i.e.", "the time that has passed from the eruption to the current observations, and column 6 summarises the time $t_3$ in which the nova has declined by 3 magnitudes as taken from [11].", "In order to examine the SED of the confirmed novae, we have fitted a power law $F \\propto \\lambda ^{-\\alpha }$ to the continuum of the dereddened spectra.", "We restricted the fit to wavelengths $5000 - 7200~\\mathrm {Å}$ because the continuum bluewards of $5000~\\mathrm {Å}$ in many systems is less well defined due to the presence of several emission lines, and also the noise in general increases towards the blue end of the spectrum.", "The such derived values for the power-law exponent are given in column 7 of Table REF .", "Note that the corresponding uncertainty listed there corresponds to the standard deviation of the fit, and does not take into account additional uncertainties concerning the instrumental response function and the dereddening process.", "Finally, in column 8 we list the Full Width at Half Maximum of a Gaussian fit to the H$\\beta $ line.", "We have used this line instead of the stronger and usually better defined H$\\alpha $ line in order to avoid potential contamination of the shell material.", "Starting with the SED, we find that $\\alpha $ for most systems falls well below the value of 2.33 for a steady-state accretion disc [21].", "This is in agreement with [53] who places nova-like CVs in this range, but differs from the results of [33] who derive an average value of $\\alpha = 2.68$ with a standard deviation of $0.82$ for their sample of dereddened post-novae.", "Within our sample, MT Cen stands out by presenting with $\\alpha = 4.45$ by far the largest value.", "In principle this could mean that this system is dominated by a still hot dwarf, but considering that it was observed 78 years after the eruption, this appears unlikely.", "We furthermore note that the largest correction for interstellar extinction had to be used for MT Cen.", "This might be coincidental, but there is also the possibility that the uncertainties in the dereddening process are the reason behind MT Cen's exalted position, and perhaps also behind the difference between our results and [33].", "Another system with an SED significantly different from the others in our sample is V2109 Oph, whose continuum cannot be fitted by a single power-law, but instead follows different laws for the range redwards of 5900 Å ($\\alpha = 1.82$ ) and for the range 5000–5900 Å ($\\alpha = -0.21$ ).", "A similar phenomenon has been observed by [40] for the old novae V630 Sgr and V842 Cen, of which the former is more similar to our case, since it also presents the less steep continuum slope for the blue part of its SED.", "The authors interpret their findings as the signature of a disrupted inner accretion disc, and, encouraged by the strong HeII emission in the system, suggest that V630 Sgr is a magnetic CV.", "In V2109 Oph, however, the HeII line is not particularly strong for an old nova, and with the present data there is no reason to suspect magnetic accretion.", "Instead, Fig.", "REF shows that the continuum slope bluewards of 5000 Å again increases and appears similar to that redwards of 5900 Å.", "Such “bumpy\" continuum could in principle be the signature of an SED that is dominated by the stellar components instead of the accretion disc.", "Further evidence for this, e.g.", "in the form of stellar absorption lines, however is missing.", "Without further data it therefore remains unclear if the shape of the continuum in V2109 Oph is intrinsic or an artifact due to a problem in the extraction or dereddening process.", "Looking at the eruption amplitudes we find that two systems, V909 Sgr and V728 Sco, match the criterion of [39] for a Tremendous Outburst Nova (TON), $\\Delta m > 13~\\mathrm {mag}$ .", "Based on the assumption that the absolute eruption magnitude is very similar for all novae the authors [40] suggest that a large eruption amplitude indicates a faint post-nova, either because it is seen at high inclination [54], or because it is intrinsically faint due to a low mass-transfer rate.", "V728 Sco could fit into the former category.", "It is the only system in our sample whose emission lines are broad enough to be well-resolved (last column in Table REF ), and additionally these lines show a distinctive structure (Fig.", "REF ).", "Following [7] a large eruption amplitude and a fast decline can also indicate a magnetic nova, since discless systems, or those with a disrupted disc, should be intrinsically fainter than disc CVs.", "Since V909 Sgr additionally presents a particularly strong HeII emission, it is a good candidate for this category." ], [ "Summary", "We have conducted a study on ten candidate old novae.", "Among them we have found two probable Mira stars whose variability likely was confused with a nova eruption, and we propose to delete these stars, V1310 Sgr and V734 Sco, from the list of potential novae.", "We have furthermore spectroscopically confirmed seven previously selected candidate post-novae, as well as recovered one system, V728 Sco, that was found to be $\\sim $ 2$^\\prime $ away from the reported coordinates.", "Within the confirmed old novae we find a group of four objects that – after correction for interstellar reddening – share very similar spectral properties: MT Cen, V655 CrA, IL Nor, and V2572 Sgr, all have the blue continuum and the weak emission lines that are typical for high mass-transfer systems.", "The remaining four novae instead present at least one pecularity that distinguishes them from the other systems in our sample.", "V2109 Oph presents a ”bumpy\" continuum for which with the current data we do not find an explanation, and V909 Sgr shows several trademarks of a magnetic systems.", "V812 Cen has an unususally strong Balmer decrement, and one can assume that the ejected material still contributes to the H$\\alpha $ emission.", "Last, not least, V728 Sco is the only system in our sample where we find convincing evidence that it has a comparatively high inclination, and it should therefore be possible to derive its orbital period with time-series photometry." ], [ "Acknowledgments", "In fond memory of Hilmar W. Duerbeck (1948–2012) We would like to thank the referee, Mike Shara, for giving us the thumbs up.", "This research was supported by FONDECYT Regular grant 1120338 (CT and NV).", "AE acknowledges support by the Spanish Plan Nacional de Astrononomía y Astrofísica under grant AYA2011-29517-C03-01.", "REM acknowledges support by the Chilean Center for Astrophysics FONDAP 15010003 and from the BASAL Centro de Astrofisica y Tecnologias Afines (CATA) PFB–06/2007.", "We gratefully acknowledge ample use of the SIMBAD database, operated at CDS, Strasbourg, France, and of NASA's Astrophysics Data System Bibliographic Services.", "The Digitized Sky Surveys were produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166, based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope.", "We have furthermore made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.", "IRAF is distributed by the National Optical Astronomy Observatories." ], [ "Finding charts", "Here we present finding charts for the eight confirmed old novae.", "Where available the combined EFOSC2 $R$ band photometric images were used, otherwise the EFOSC2 acquisition frames were employed.", "The two discarded nova candidates, V1310 Sgr and V734 Sco, can be unambiguously identified on finding charts in the [9] catalogue.", "Figure: Finding charts for the confirmed old novae MT Cen, V812 Cen,V655 CrA, and IL Nor.", "The size of a chart is1.5 ' ^\\prime ×\\times 1.5 ' ^\\prime , and the orientation is such that Northis up and East is to the left.", "The images were taken in the RR band.Figure: Finding charts for the confirmed old novae V2109 Oph, V909 Sgr,V2572 Sgr, and V728 Sco.", "The size of a chart is1.5 ' ^\\prime ×\\times 1.5 ' ^\\prime , and the orientation is such that Northis up and East is to the left.", "The images were taken in the RR band." ] ]

[ [ "Dirac tachyons and antitachyons in many-particle system" ], [ "Abstract A consistent description of charged many-tachyon Fermi system is developed.", "Tachyons and antitachyons have the same chemical potential \\mu+=\\mu- because the axial coupling constant g+=g- is invariant under the charge conjugation, in contrast to reversion of the electric charge e+=-e-.", "The axial density n5=<\\psi^ \\gamma 5 \\psi> is incorporated in the thermodynamical functions instead of <\\psi^ \\psi> which is not associated with any conserved quantity.", "The number of tachyons and antitachyons are undefined but it is possible to estimate their difference and establish a link between the total electric charge density $en$ and $n5$." ], [ "Introduction", "Tachyon is a substance that moves faster than light.", "Its energy spectrum satisfies dispersion relation $\\varepsilon _p^2=p^2-m^2 $ and its group velocity $v=\\frac{d\\varepsilon _p}{dp}=\\frac{p}{\\sqrt{p^2-m^2}} $ exceeds the speed of light in vacuum $c=1$ , relative to any reference frame.", "Tachyons are commonly known in the field theory [1] and nonlinear optics [2].", "The Lagrangian of a free fermionic tachyon [3], [4]$L_0=\\bar{\\psi }\\left( i\\gamma _5\\gamma ^\\mu \\partial _\\mu -m\\right) \\psi $ corresponds to the equation of motion $\\left( i\\gamma ^\\mu \\partial _\\mu -\\gamma _5m\\right) \\psi =0 $ whose plane wave solution $\\psi =\\left(\\begin{array}{c}\\phi \\\\\\chi \\end{array}\\right) \\exp \\left( i\\vec{p}\\cdot \\vec{r}-i\\varepsilon _{p\\,}t\\right)$ results in the single-particle energy spectrum (REF ).", "The motion of tachyon in the presence of electromagnetic field $A_\\mu $ is described by equation [5] $\\left( i\\gamma ^\\mu \\left( \\partial _\\mu +ieA_\\mu \\right) -\\gamma _5m\\right)\\psi =0 $ that corresponds to appearance of interaction term $-e\\bar{\\psi }\\gamma _5\\gamma ^\\mu A_\\mu \\psi $ in the free tachyonic Lagrangian.", "The equation of motion $\\left( \\gamma ^\\mu \\left( i\\partial _\\mu -eA_\\mu \\right) -\\gamma _5m-g\\gamma _5\\gamma ^\\mu \\omega _\\mu \\right) \\psi =0 $ is written when the relevant Lagrangian $L=L_0-e\\bar{\\psi }\\gamma _5\\gamma ^\\mu A_\\mu \\psi -g\\bar{\\psi }\\gamma ^\\mu \\omega _\\mu \\psi $ includes the coupling term associated with vector field $\\omega _\\mu $ .", "When we consider an ensemble of particle and antiparticles in finite volume $V=\\int d^3r$ , we use standard definitions of the particle number density $n\\equiv \\left\\langle \\bar{\\psi }\\gamma ^0\\psi \\right\\rangle =\\frac{\\gamma }{\\left( 2\\pi \\right) ^3}\\int \\bar{\\psi }\\gamma ^0\\psi \\,d^3p $ and axial density $n_5\\equiv \\left\\langle \\bar{\\psi }\\gamma ^0\\gamma _5\\psi \\right\\rangle =\\frac{\\gamma }{\\left( 2\\pi \\right) ^3}\\int \\bar{\\psi }\\gamma ^0\\gamma _5\\psi \\,d^3p$ The relevant total charges $\\left\\langle Q_e\\right\\rangle \\equiv \\left( e_{+}n_{+}+e_{-}n_{-}\\right) V$ $\\left\\langle Q_g\\right\\rangle \\equiv \\left( g_{+}n_{5+}+g_{-}n_{5-}\\right) V$ include contributions from the two species that depend on the sign of elementary charges of particles $e_{+}$ , $g_{+}$ and antiparticles $e_{-}$ , $g_{-}$ .", "The electric charge conjugation [5] $e_{+}=-e_{-}\\equiv e$ implies that the total electric charge of tachyons and antitachyons $Q_e\\equiv e\\left( n_{+}-n_{-}\\right) V $ coincides with the relevant expression for ordinary fermions and antifermions [6].", "Most natural assumption $g_{-}=-g_{+}$ would lead to formula similar to (REF ) but relation between $g_{+}$ and $g_{-}$ is not evident beforehand.", "It is necessary to check the charge conjugation of tachyonic Dirac equation (REF ) and establish what consequences it implies to the thermodynamics of a many-tachyon system." ], [ "Charge conjugation", "Equation (REF ) is equivalent to $\\left( \\gamma _5\\gamma ^\\mu \\left( i\\partial _\\mu -eA_\\mu \\right) -m-g\\gamma ^\\mu \\omega _\\mu \\right) \\psi =0 $ The difference between $e$ and $g$ is evident from the corresponding bilinear transforms.", "Let us demonstrate it directly, following the previous analysis [5] and considering the charge conjugation of equation (REF ) or (REF ).", "The action of Hermite operator $\\dagger $ (transposition and complex conjugation) results in $\\psi ^{\\dagger }\\left( -i\\left[ \\gamma _5\\gamma ^\\mu \\right] ^{\\dagger }\\partial _\\mu -e\\left[ \\gamma _5\\gamma ^\\mu \\right] ^{\\dagger }A_\\mu -m-g\\gamma ^{\\mu \\dagger }\\omega _\\mu \\right) =0 $ Presenting $\\psi ^{\\dagger }\\equiv \\psi ^{\\dagger }\\gamma ^0\\gamma ^0=\\bar{\\psi }\\gamma ^0$ and multiplying equation (REF ) by $\\gamma ^0$ from the right, we have $\\bar{\\psi }\\gamma ^0\\left( -i\\left[ \\gamma _5\\gamma ^\\mu \\right] ^{\\dagger }\\partial _\\mu -e\\left[ \\gamma _5\\gamma ^\\mu \\right] ^{\\dagger }A_\\mu -m-g\\gamma ^{\\mu \\dagger }\\omega _\\mu \\right) \\times \\gamma ^0=0 $ Using identities $\\gamma ^0\\gamma ^{\\mu \\dagger }\\gamma ^0=\\gamma ^\\mu \\qquad \\gamma ^\\mu \\gamma _5=-\\gamma _5\\gamma ^\\mu \\qquad \\gamma _5=\\gamma _5^{\\dagger }\\qquad \\gamma ^0\\gamma _5\\gamma ^0=-\\gamma _5 $ and $\\gamma ^0\\left[ \\gamma _5\\gamma ^\\mu \\right] ^{\\dagger }\\gamma ^0=\\gamma ^0\\gamma ^{\\mu \\dagger }\\gamma _5^{\\dagger }\\gamma ^0=\\gamma ^0\\gamma ^{\\mu \\dagger }\\gamma ^0\\gamma ^0\\gamma _5\\gamma ^0=\\gamma _5\\gamma ^\\mu $ we obtain $\\bar{\\psi }\\left( -i\\gamma _5\\gamma ^\\mu \\partial _\\mu -e\\gamma _5\\gamma ^\\mu A_\\mu -m-g\\gamma ^\\mu \\omega _\\mu \\right) \\gamma ^0=0 $ Transposition of (REF ) yields $\\left( i\\left[ \\gamma _5\\gamma ^\\mu \\right] ^T\\partial _\\mu +e\\left[ \\gamma _5\\gamma ^\\mu \\right] ^TA_\\mu +m+g\\gamma ^{\\mu T}\\omega _\\mu \\right) \\bar{\\psi }^T=0 $ that can be also written in the equivalent form $\\left( i\\gamma ^{\\mu T}\\partial _\\mu +e\\gamma ^{\\mu T}A_\\mu -\\gamma _5m+g\\left[ \\gamma _5\\gamma ^\\mu \\right] ^T\\omega _\\mu \\right) \\bar{\\psi }^T=0$ Let us introduce the charge conjugation matrix $C$ with properties $C\\gamma ^{\\mu T}C^{-1}=-\\gamma ^\\mu $ that performs reversal $Ce\\gamma ^{\\mu T}A_\\mu C^{-1}=-e\\gamma ^\\mu A_\\mu \\qquad Cg\\gamma ^{\\mu T}\\omega _\\mu C^{-1}=-g\\gamma ^\\mu \\omega _\\mu $ Identity (REF ) immediately implies $C\\left[ \\gamma _5\\gamma ^\\mu \\right] ^TC^{-1}=C\\gamma ^{\\mu T}CC^{-1}\\gamma _5C^{-1}=-\\gamma ^\\mu \\gamma _5=\\gamma _5\\gamma ^\\mu $ and $&C\\gamma _5C^{-1} =iC\\left[ \\gamma ^0\\gamma ^1\\gamma ^2\\gamma ^3\\right]^TC^{-1}=iC\\gamma ^{3T}\\gamma ^{2T}\\gamma ^{1T}\\gamma ^{0T}C^{-1} =&\\nonumber \\\\&=i\\left(C\\gamma ^{3T}C\\right) \\left( C^{-1}\\gamma ^{2T}C\\right) \\left(C^{-1}\\gamma ^{1T}C\\right) \\left( C\\gamma ^{0T}C^{-1}\\right) = & \\nonumber \\\\&=i\\left( -\\gamma ^3\\right) \\left( -\\gamma ^2\\right) \\left( -\\gamma ^1\\right) \\left( -\\gamma ^0\\right) =i\\gamma ^3\\gamma ^2\\gamma ^1\\gamma ^0=\\gamma _5& $ Then, multiplying equation (REF ) by $C$ from the left, we have $C\\times \\left( i\\gamma ^{\\mu T}\\partial _\\mu +e\\gamma ^{\\mu T}A_\\mu -\\gamma _5m+g\\left[ \\gamma _5\\gamma ^\\mu \\right] ^T\\omega _\\mu \\right) C^{-1}C\\bar{\\psi }^T=0 $ hence, $\\left( i\\gamma ^\\mu \\partial _\\mu +e\\gamma ^\\mu A_\\mu +\\gamma _5m-g\\gamma _5\\gamma ^\\mu \\omega _\\mu \\right) \\bar{\\psi }^C=0 $ where $\\psi ^C=C\\bar{\\psi }^T$ is the charge conjugated wave function.", "Alternatively, multiplying equation (REF ) by $C$ from the left, we have $C\\times \\left( i\\left[ \\gamma _5\\gamma ^\\mu \\right] ^T\\partial _\\mu +e\\left[\\gamma _5\\gamma ^\\mu \\right] ^TA_\\mu +m+g\\gamma ^{\\mu T}\\omega _\\mu \\right)C^{-1}C\\bar{\\psi }^T=0 $ hence $\\left( i\\gamma _5\\gamma ^\\mu \\partial _\\mu +e\\gamma _5\\gamma ^\\mu A_\\mu +m-g\\gamma ^\\mu \\omega _\\mu \\right) \\psi ^C=0 $ that is equivalent to (REF ).", "It is not a problem that the tachyonic Dirac equation (REF ) is not invariant under the charge conjugation on account of wrong sign of the mass term in (REF ) because equation (REF ) is still CP and T invariant [5].", "The important fact we have seen now is that the charge conjugation implies no more than electric charge reversion $e\\leftrightarrow -e$ and does not concern the axial coupling constant $g$ .", "Thus, tachyons and antitachyons have opposite $e_{+}=-e_{-}$ but the same $g_{+}=g_{-}$ .", "Hence, in contrast to the total electric charge (REF ), the total axial charge (REF ) is calculated by formula $\\left\\langle Q_g\\right\\rangle \\equiv g\\left( n_{5+}+n_{5-}\\right) V=\\mathrm {const} $" ], [ "Thermodynamical functions", "According to the Noether theorem, a conserved current corresponds to each continuous symmetry of the Lagrangian.", "The relevant partition function is constructed so that its argument $L+\\mu O$ includes the chemical potential $\\mu $ as a free multiplier and the relevant conserved quantity $O$ [6].", "The tachyonic equation of motion (REF ) results in the continuity equation [7], [8] $\\partial _{\\mu \\,}j_5^\\mu =0$ of the axial current $j_5^\\mu =(j_5^0,\\vec{j}_5)=\\bar{\\psi }\\gamma ^\\mu \\gamma _5\\psi $ .", "Integration $\\int \\left( \\partial _{0\\,}j_5^0\\,+\\mathrm {div\\,}\\vec{j}_5\\right) d^3r=\\int \\partial _{0\\,}j_5^0\\,\\,d^3r $ implies conservation of quantity $N_5=\\int j_5^0d^3r=\\int \\bar{\\psi }\\gamma ^0\\gamma _5\\psi \\,d^3r $ called as the axial charge or axial particle number.", "The vector current $j^\\mu =\\bar{\\psi }\\gamma ^\\mu \\psi $ obeys equation [8] $\\partial _{\\mu \\,}j^\\mu =-2im\\bar{\\psi }\\gamma _5\\psi $ and quantity $N=\\int j^0d^3r=\\int \\bar{\\psi }\\gamma ^0\\psi \\,d^3r $ is not conserved (except the only chiral limit $m\\rightarrow 0$ ).", "Therefore, it is the axial charge (REF ) which is incorporated in the partition function of free tachyon Fermi gas $Z=\\int \\left[ d\\bar{\\psi }_{+}\\right] \\left[ d\\psi _{-}\\right] \\left[ d\\bar{\\psi }_{-}\\right] \\left[ d\\psi _{-}\\right] \\exp \\left\\lbrace \\int \\limits _0^{1/T}d\\tau \\left( \\int L_{+}d^3r+\\int L_{-}d^3r+\\mu _{+}N_{5+}+\\mu _{-}N_{5-}\\right) \\right\\rbrace $ where $\\tau =it$ .", "We emphasize that the number of particles (REF ) is not included in the partition function but the axial charge (REF ) is the main characteristic of tachyon Fermi gas.", "Replacement $N\\leftrightarrow N_5$ brings no formal change to the thermodynamical laws, however, the two quantities should not be mixed when we deal with the relevant charges (REF ) and (REF ): the quantity associated with the total electric charge is not conserved.", "The tachyonic Dirac equation is also time invariant [5].", "If time reversal is a good symmetry, a detailed balance must occur among all possible reactions in equilibrium and the Gibbs free energy will remain constant $G=\\mu _{+}n_{5+}V+\\mu _{-}n_{5-}V=\\mathrm {const} $ In the light of (REF ) it implies that tachyons and antitachyons must have the same chemical potential $\\mu \\equiv \\mu _{+}=\\mu _{-} $ This allows to simplify (REF ) in the form$\\ Z=Z_{\\pm }^2$ where $Z_{\\pm }\\equiv \\int \\left[ d\\bar{\\psi }\\right] \\left[ d\\psi \\right] \\exp \\left\\lbrace \\int \\limits _0^{1/T}d\\tau \\left( \\int Ld^3r+\\mu N_5\\right) \\right\\rbrace $ Then, the pressure, energy density and entropy are determined by standard formulas $P=\\frac{T}{V}\\ln Z=\\frac{\\gamma }{2\\pi ^2}T\\int \\limits _0^\\infty \\ln \\left(1+\\exp \\frac{\\mu -\\varepsilon _p}{T}\\right) p^2dp=\\frac{\\gamma }{6\\pi ^2}\\int \\limits _0^\\infty \\,f_\\varepsilon \\,\\frac{d\\varepsilon _p}{dp}p^2dp$ $E=\\frac{T^2}{V}\\left( \\frac{\\partial \\ln Z}{\\partial T}\\right) _{V,\\mu }+\\mu n_5=\\frac{\\gamma }{2\\pi ^2}\\int \\limits _0^\\infty \\,f_\\varepsilon \\,\\varepsilon _pp^2dp $ $S=V\\frac{E+P-\\mu n_5}{T} $ where $f_\\varepsilon =\\frac{1}{\\exp \\left[ \\left( \\varepsilon _p-\\mu \\right)/T\\right] +1} $ is the Fermi-Dirac distribution function, and the axial density satisfies identity $n_5\\equiv T\\left( \\frac{\\partial \\ln Z}{\\partial \\mu }\\right) _{V,T}=\\frac{\\gamma }{2\\pi ^2}\\int \\limits _0^\\infty f_\\varepsilon \\,p^2dp $ Since the thermodynamical functions of free tachyons and antitachyons are indistinguishable, the proper degeneracy factor of tachyons $\\gamma =1$ is doubled so that $\\gamma =2$ in all thermodynamical formulas (or we can write $2\\gamma $ implying that $\\gamma =1$ ).", "Note that the thermodynamical functions of ordinary baryonic matter are given by formulas [6] $P=\\frac{\\gamma }{6\\pi ^2}\\int \\limits _0^\\infty \\,\\left( f_{\\varepsilon +}+f_{\\varepsilon -}\\right) \\,\\frac{d\\varepsilon _p}{dp}p^2dp $ $E=\\frac{\\gamma }{2\\pi ^2}\\int \\limits _0^\\infty \\,\\left( f_{\\varepsilon +}+f_{\\varepsilon -}\\right) \\,\\varepsilon _pp^2dp $ $S=V\\frac{E+P-\\mu n}{T} $ where $n=\\frac{\\gamma }{2\\pi ^2}\\int \\limits _0^\\infty \\,\\left( f_{\\varepsilon +}-f_{\\varepsilon -}\\right) \\,p^2dp $ is the baryon number density, while distribution functions of particles and antiparticles are $f_{\\varepsilon +}=\\frac{1}{\\exp \\left[ \\left( \\varepsilon _p-\\mu \\right)/T\\right] +1}\\qquad f_{\\varepsilon -}=\\frac{1}{\\exp \\left[ \\left( \\varepsilon _p+\\mu \\right) /T\\right] +1} $ Indeed, the tachyonic formulas (REF )-(REF ) do immediately follow from the relevant formulas of hot baryonic matter (REF )-(REF ) if particles and antiparticles have the same chemical potential and $f_{\\varepsilon +}=f_{\\varepsilon -}$ .", "For a cold tachyon Fermi gas with Fermi momentum $p_F>m $ the distribution function (REF ) is reduced to the Heaviside step $f_\\varepsilon =\\Theta \\left( \\varepsilon _p -\\varepsilon _F\\right) =\\Theta \\left( p-p_F\\right) $ where  $\\varepsilon _F=\\sqrt{p_F^2-m^2} $ is the tachyon Fermi energy.", "The axial charge density (REF ) is immediately calculated $n_5=\\frac{\\gamma p_F^3}{6\\pi ^2} $ while formula (REF ) is reduced to $E+P=\\mu n_5 $ instead of ordinary $E+P=\\mu n$ .", "As we have emphasized above, the axial charge density $n_5$ appears in all thermodynamical formulas instead of the particle number density of the ordinary Fermi gas.", "Formula (REF ) is valid under condition (REF ) when the axial charge density exceeds $n_5>n_{\\star }=\\frac{\\gamma m^3}{6\\pi ^2} $ Of course, it does not imply that the number of tachyons must also exceed some finite bottom level.", "For thermodynamical relations (REF ), (REF )-(REF ) does not provide us any information about quantity $\\left\\langle \\bar{\\psi }\\gamma ^0\\psi \\right\\rangle \\, $ and it is not clear whether it is finite of has any physical meaning because the number of tachyons (REF ) is not conserved." ], [ "Scalar and particle number density", "Substituting plane-wave solution (REF ) in the tachyonic equation (REF ), we get a linear system for bispinors $\\begin{array}{c}\\left( \\vec{\\sigma }\\cdot \\vec{p}-m\\right) \\phi =\\varepsilon _p\\chi \\\\\\left( \\vec{\\sigma }\\cdot \\vec{p}+m\\right) \\chi =\\varepsilon _p\\phi \\end{array}$ Hence, $\\chi =\\frac{\\varepsilon _p}{hp+m}\\phi =\\frac{hp-m}{\\varepsilon _p}\\phi $ where $h=\\frac{\\vec{\\sigma }\\cdot \\vec{p}}{p}=\\pm 1 $ is helicity of tachyon (or antitachyon).", "It is clear that the sign of helicity remains the same regardless of the point of view of external observer moving at arbitrary subluminal velocity.", "Hence, if $h=+1$ , say for tachyons, it is always $h=-1$ for antitachyons and v.v.", "In the light of bispinor representation (REF ), we define the following quantities $j^0=\\bar{\\psi }\\gamma ^0\\psi =\\left\\Vert \\phi \\right\\Vert ^2+\\left\\Vert \\chi \\right\\Vert ^2=\\left[ 1+\\frac{\\left| \\varepsilon _p\\right| ^2}{\\left( hp+m\\right) ^2}\\right] \\left\\Vert \\phi \\right\\Vert ^2=\\left\\lbrace \\begin{array}{cc}\\cfrac{2hp\\left\\Vert \\phi \\right\\Vert ^2 }{hp+m} & p\\ge m \\\\\\cfrac{2m\\left\\Vert \\phi \\right\\Vert ^2 }{hp+m} & p<m\\end{array}\\right.", "$ $j_s=\\bar{\\psi }\\psi =\\left\\Vert \\phi \\right\\Vert ^2-\\left\\Vert \\chi \\right\\Vert ^2=\\left[1-\\frac{\\left| \\varepsilon _p\\right| ^2}{\\left( hp+m\\right) ^2}\\right]\\left\\Vert \\phi \\right\\Vert ^2=\\left\\lbrace \\begin{array}{cc}\\cfrac{2m\\left\\Vert \\phi \\right\\Vert ^2 }{hp+m} & p\\ge m \\\\\\cfrac{2hp\\left\\Vert \\phi \\right\\Vert ^2 }{hp+m} & p<m\\end{array}\\right.", "$ $j_5^0=\\bar{\\psi }\\gamma ^0\\gamma _5\\psi =\\phi ^{*}\\chi +\\chi ^{*}\\phi =\\frac{2\\mathrm {Re}\\varepsilon _p}{hp+m}\\left\\Vert \\phi \\right\\Vert ^2 $ The axial charge density $n_5=\\left\\langle j_5^0\\right\\rangle $ is determined by formula (REF ).", "Taking also into account formulas (REF ), (REF ), (REF ) and (REF ), we find general expression for the particle number density $\\left\\langle j^0\\right\\rangle \\equiv \\left\\langle \\bar{\\psi }\\gamma ^0\\psi \\right\\rangle =\\frac{\\gamma }{2\\pi ^2}\\int \\limits _m^\\infty f_\\varepsilon \\frac{h}{\\left| \\varepsilon _p\\right| }p^3dp+\\frac{\\gamma }{2\\pi ^2}\\int \\limits _0^m\\frac{m}{\\mathrm {Re}\\varepsilon _p}\\mathrm {\\ }p^2dp$ and, in the light of (REF ) and (REF ) we find general expression for the scalar density $\\left\\langle j_s\\right\\rangle \\equiv \\left\\langle \\bar{\\psi }\\psi \\right\\rangle =\\frac{\\gamma }{2\\pi ^2}\\int \\limits _m^\\infty f_\\varepsilon \\frac{m}{\\left| \\varepsilon _p\\right| }p^2dp+\\frac{\\gamma }{2\\pi ^2}\\int \\limits _0^mf_{\\varepsilon \\,}\\frac{\\mathrm {\\ }h}{\\mathrm {Re}\\varepsilon _p}p^3dp $ Since the particles and antiparticles have opposite helicity, we immediately write expressions each number density $n_{+}=\\left\\langle j_{+}^0\\right\\rangle =\\frac{\\gamma }{2\\pi ^2}\\int \\limits _m^\\infty f_{\\varepsilon +}\\frac{h}{\\left| \\varepsilon _p\\right| }p^3dp+\\frac{\\gamma }{2\\pi ^2}\\int \\limits _0^mf_{\\varepsilon -}\\frac{m}{\\mathrm {Re}\\varepsilon _p}\\mathrm {\\ }p^2dp $ $n_{-}=\\left\\langle j_{-}^0\\right\\rangle =\\frac{\\gamma }{2\\pi ^2}\\int \\limits _m^\\infty f_{\\varepsilon +}\\frac{\\left( -h\\right) }{\\left|\\varepsilon _p\\right| }p^3dp+\\frac{\\gamma }{2\\pi ^2}\\int \\limits _0^mf_{\\varepsilon -}\\frac{m}{\\mathrm {Re}\\varepsilon _p}\\mathrm {\\ }p^2dp\\,\\, $ and for the scalar density of particles and antiparticles $\\left\\langle j_{s+}\\right\\rangle =\\frac{\\gamma }{2\\pi ^2}\\int \\limits _m^\\infty f_{\\varepsilon +}\\frac{m}{\\left| \\varepsilon _p\\right| }p^2dp+\\frac{\\gamma }{2\\pi ^2}\\int \\limits _0^mf_{\\varepsilon -}\\frac{\\mathrm {\\ }h}{\\mathrm {Re}\\varepsilon _p}p^3dp $ $\\left\\langle j_{s-}\\right\\rangle =\\frac{\\gamma }{2\\pi ^2}\\int \\limits _m^\\infty f_{\\varepsilon +}\\frac{m}{\\left| \\varepsilon _p\\right| }p^2dp+\\frac{\\gamma }{2\\pi ^2}\\int \\limits _0^mf_{\\varepsilon -}\\frac{\\mathrm {\\ }\\left( -h\\right) }{\\mathrm {Re}\\varepsilon _p}p^3dp $ Taken into account that $n_{5+}=n_{5-}$ and $f_{\\varepsilon +}=f_{\\varepsilon -}\\equiv f_\\varepsilon $ because particles and antiparticles have the same chemical potential (REF ), we find by means of (REF ) the total electric charge $&Q_e =\\left( e_{+}n_{+}+e_{-}n_{-}\\right) V=&e\\left\\langle \\left( \\frac{hp}{\\left| \\varepsilon _p\\right| }+\\frac{m}{\\mathrm {Re}\\varepsilon _p}\\mathrm {\\ }\\right) j_5^0\\right\\rangle V-e\\left\\langle \\left( \\frac{-hp}{\\left|\\varepsilon _p\\right| }+\\frac{m}{\\mathrm {Re}\\varepsilon _p}\\mathrm {\\ }\\right)j_5^0\\right\\rangle V= \\nonumber \\\\&&=\\frac{\\gamma Veh}{2\\pi ^2}\\int \\limits _m^\\infty f_\\varepsilon \\frac{p^3dp}{\\sqrt{p^2-m^2}} \\quad \\quad \\quad $ and determine $n=n_{+}-n_{-}=h\\tilde{n}=\\frac{\\gamma h}{2\\pi ^2}\\int \\limits _m^\\infty f_\\varepsilon \\frac{p^3dp}{\\sqrt{p^2-m^2}}\\,\\, $ as the tachyonic ”particle number density”.", "It is finite in spite of the fact that each contribution of tachyons (REF ) and antitachyons (REF ) are separately divergent.", "We deliberately leave helicity $h$ in (REF ) because it implies correct definition of $n\\equiv \\left\\langle \\bar{\\psi }\\gamma ^0\\psi \\right\\rangle _{+}-\\left\\langle \\bar{\\psi }\\gamma ^0\\psi \\right\\rangle _{-}$ .", "The total electric charge $Q_e=en=eh\\tilde{n}$ depends on the handedness and the sign of tachyonic charge: for example, if left-handed tachyon ($h=-1$ ) has electric charge $e$ equal to 1 electron charge, then, the total electric charge of a many-tachyon system $Q_e$ has always opposite sign to the charge of electron.", "This can be compared with a hot nuclear matter where the number of protons always exceeds the number of antiprotons so that the total electric charge of nucleon is always positive (and counterbalanced by the negative charge of electron gas).", "Quantity $\\tilde{n}=\\frac{\\gamma }{2\\pi ^2}\\int \\limits _m^\\infty f_\\varepsilon \\frac{p^3dp}{\\sqrt{p^2-m^2}}\\,\\, $ is always positive and it can be called as effective particle number density, bearing in mind how it is incorporated in (REF )-(REF ).", "At zero temperature (REF ) it is reduced to $\\tilde{n}=\\left| n\\right| =\\frac{\\gamma }{6\\pi ^2}\\varepsilon _F\\left(\\varepsilon _F^2+3m^2\\right) \\, $ and the total electric charge (REF ) is easily expressed $Q_e=\\frac{eh\\gamma V}{6\\pi ^2}\\sqrt{p_F^2-m^2}\\left( p_F^2+2m^2\\right) \\,=\\frac{eh\\gamma V}{6\\pi ^2}\\sqrt{\\left( \\frac{6\\pi ^2n_5}{\\gamma }\\right)^{2/3}-m^2}\\left[ \\left( \\frac{6\\pi ^2n_5}{\\gamma }\\right) ^{2/3}+2m^2\\right]\\, $ in terms of the axial density $n_5$ (REF ).", "The total scalar density $n_s=\\left\\langle j_{s+}\\right\\rangle +\\left\\langle j_{s-}\\right\\rangle =\\left\\langle \\left( \\frac{m}{\\left| \\varepsilon _p\\right| }+\\frac{hp}{\\mathrm {Re}\\varepsilon _p}\\right) j_5^0\\right\\rangle +\\left\\langle \\left(\\frac{m}{\\left| \\varepsilon _p\\right| }+\\frac{-hp}{\\mathrm {Re}\\varepsilon _p}\\right) j_5^0\\right\\rangle =\\frac{\\gamma m}{2\\pi ^2}\\int \\limits _m^\\infty f_\\varepsilon \\frac{p^2dp}{\\sqrt{p^2-m^2}}\\, $ is also finite while each contribution of particles (REF ) and antiparticles (REF ) are divergent.", "The scalar density (REF ) at zero temperature is determined by formula $n_s=\\frac{\\gamma m}{4\\pi ^2}\\left( p_{F\\,}\\varepsilon _F+m^2\\ln \\frac{p_F+\\varepsilon _F}{m}\\right) $ that coincides with expression derived in the earlier work [9] in by means of intuitive analysis but without strict proof.", "Note that the scalar density of an ordinary fermionic matter $n_s=\\frac{\\gamma m}{4\\pi ^2}\\left( p_{F\\,}\\varepsilon _F-m^2\\ln \\frac{p_F+\\varepsilon _F}{m}\\right) $ differs from (REF ) by the sign before $m^2$ .", "The knowledge of (REF ), (REF ) and (REF ) is necessary for calculation the of thermodynamical functions of interacting tachyon Fermi gas.", "Comparing formulas (REF ), (REF ) and (REF ), one notes that $n>n_s$ at any $p_F$ , and $n_s>n_5$ when $1.08m<p_F<1.80m$ , while $n>n_5$ when $p_F>1.06m$ , see Fig.", "REF .", "Maximum ratio $n/n_5\\simeq 1.41$ is achieved at $p_F\\simeq 1.41$ .", "At large $p_F\\gg m$ and $\\varepsilon _F\\rightarrow p_F$ the tachyon matter behaves as an ordinary massless Fermi gas, and $n\\rightarrow n_5$ .", "The limit of low density $n\\rightarrow 0$ corresponds to $\\varepsilon _F\\rightarrow 0$ and $p_F\\rightarrow m$ while the minimum axial density (REF ) is achieved at the vanishing particle number density $n=0$ .", "However, this limit is not achieved in practice because a tachyon Fermi gas is unstable with respect to hydrodynamical perturbations at such small density since their causal propagation takes place only at [9] $p_F\\ge \\sqrt{\\frac{3}{2}}m\\, $ that, in the light of (REF ), corresponds to $n\\ge n_c=\\frac{5\\sqrt{2}\\gamma }{24\\pi ^2}m^3\\, $ The tachyon medium can exist only at finite material density $\\rho \\ge mn_c=\\frac{\\gamma 5\\sqrt{2}}{24\\pi ^2}m^4\\, $ while a rarefied tachyon Fermi gas will be unstable, perhaps, decaying into an aggregate of dense droplets.", "Of course, we could avoid divergences in (REF )-(REF ), considering the only sector $p>m$ and excluding the low-momentum states $p<m$ as unphysical ones.", "However, in spite of attractiveness of this approach, it results in serious contradictions so that statistical description of a many-tachyon system becomes senseless [10]." ], [ "Conclusion", "A many-tachyon system is looking strange and contrasting to an ordinary system of particles and antiparticles.", "The main invariant is the axial charge (REF ) while the number of tachyons (REF ) is not conserved (like the number of photons in black body radiation, or the number of thermal excitation in solids).", "The axial charge density $n_5$ determined by formula (REF ) is incorporated in the thermodynamical equations (REF )-(REF ) of a tachyon Fermi gas instead of the particle number density $n$ (REF ).", "The charge conjugation (REF ) changes the signs of electric charge while the axial charge remains the same implying that the tachyons and antitachyons have equal chemical potential $\\mu _{+}=\\mu _{-}$ (REF ).", "This fact is crucial in the thermodynamics of tachyon Fermi gas because negative $\\mu _{-}=-\\mu _{+}<0$ would not allow us to operate with unambiguous distribution function (REF ) at small momentum $p<m$ , implying impossibility of regularization of the total electric charge (REF ).", "The number of tachyons $N_{+}$ and antitachyons $N_{-}$ as well as their summary number $N_{+}+N_{-}$ cannot be defined or presented as a function on temperature.", "Nevertheless, the difference between the particles and antiparticles $N_{+}-N_{-}$ is reflected in the total electric charge (REF ) and we managed to estimate it in terms of the thermodynamical functions of tachyon gas (REF ).", "The scalar density is also estimated (REF ) and at zero temperature it coincides with expression found in the previous analysis [9].", "At zero temperature the axial density, particle number density, total electric charge and scalar density are given by formulas (REF ), (REF ), (REF ) and (REF ), respectively, see Fig.", "REF .", "As for the alternative Dirac equation with imaginary mass $\\left( i\\gamma ^\\mu \\partial _\\mu -im\\right) \\psi =0 $ it yields the same tachyonic dispersion relation (REF ).", "However, it is not associated with any conserved current because [7], [8] $\\partial _{\\mu \\,}j^\\mu =2m\\bar{\\psi }\\psi \\ne 0$ and $\\partial _{\\mu \\,}j_5^\\mu \\rightarrow \\infty $ .", "The relevant partition function will be equivalent to (REF ) at zero chemical potential, i.e.", "when tachyonic thermal excitations are considered.", "When a may-tachyon system is put in external filed, the single-particle energy spectrum of particles $\\varepsilon _{p+}$ and antiparticles $\\varepsilon _{p-}$ will be different.", "The thermodynamical functions of the whole system will be calculated by formulas (REF )-(REF ) and the distribution functions of the tachyons $f_{\\varepsilon +}$ and antitachyons $f_{\\varepsilon -}$ may not coincide.", "Then, the total electric charge (REF ) should be carefully estimated because the divergent terms in (REF ) and (REF ) may occur finite and will not be mutually eliminated if ${\\rm Re}\\varepsilon _{p \\pm } \\ne 0$ in the presence of external field that may imply request for production of other sorts of charged particles.", "Of course, for an electrically neutral system (for example, a mixture of positive-charged tachyons and electrons), this effect may play visible role only beyond the classical or mean-field level when the quantum exchange and correlation corrections to interaction are taken into account.", "This question requires development in the further research.", "The author is grateful to Konstantin Stepanyantz for inspiring discussions." ] ]

[ [ "Entanglement control in hybrid optomechanical systems" ], [ "Abstract We demonstrate the control of entanglement in a hybrid optomechanical system comprising an optical cavity with a mechanical end-mirror and an intracavity Bose-Einstein condensate (BEC).", "Pulsed laser light (tuned within realistic experimental conditions) is shown to induce an almost sixfold increase of the atom-mirror entanglement and to be responsible for interesting dynamics between such mesoscopic systems.", "In order to assess the advantages offered by the proposed control technique, we compare the time-dependent dynamics of the system under constant pumping with the evolution due to the modulated laser light." ], [ "The Physical Model", "We start describing the hybrid optomechanical setup at hand [11], [12], which consists of a Fabry-Perot cavity with a vibrating end-mirror.", "The cavity is pumped by a laser that couples to the mechanical mirror and an intracavity BEC.", "The Hamiltonian of the system (in a frame rotating at the frequency $\\omega _L$ of the pump field) reads $\\hat{\\cal H} = \\hat{\\cal H}_C + \\hat{\\cal H}_A + \\hat{\\cal H}_M + \\hat{\\cal H}_{AC} + \\hat{\\cal H}_{MC}.$ The Hamiltonian of the mirror is $\\hat{\\cal H}_M = \\frac{1}{2} m \\omega _m^2 \\hat{q}^2 +\\frac{\\hat{p}^2}{2m},$ where $m$ is the effective mechanical mass, $\\omega _m$ is the free mechanical frequency and $\\hat{q}$ ($\\hat{p}$ ) is the position (momentum) operator of the mirror.", "The Hamiltonian of the driven cavity is $\\hat{\\cal H}_C = \\hbar (\\omega _C{-}\\omega _L)\\hat{a}^\\dagger \\hat{a}{-}i \\hbar \\eta (\\hat{a}-\\hat{a}^\\dagger ),$ where $\\omega _C$ is the cavity frequency, $\\hat{a}$ is the cavity field's annihilation operator, and $\\eta = \\sqrt{2\\kappa {\\cal R}/\\hbar \\omega _L}$ accounts for the laser pumping [${\\cal R}$ is the laser power and $\\kappa $ is the cavity decay rate].", "The BEC is taken to be weakly interacting, allowing the separation of the atomic field operator into a classical component (the condensate wave function) and a quantum one (the fluctuations), expressed in terms of Bogoliubov modes.", "The cavity is strongly coupled to the mode with wavelength $\\lambda _c/2$ where $\\lambda _c$ is the cavity-mode wavelength [9], [11].", "We call $\\omega _b$ the frequency of the Bogoliubov mode and $\\hat{c}$ ($\\hat{c}^\\dagger $ ) the corresponding annihilation (creation) operator.", "As the condensate is at low temperature, any thermal fluctuation of the atoms is negligible.", "The free Hamiltonian of the Bogoliubov mode is given by $\\hat{\\cal H}_A = \\hbar \\omega _b\\hat{c}^\\dagger \\hat{c}$ , while the atom-cavity coupling is [9], [11] $\\hat{\\cal H}_{AC}=\\frac{\\hbar g^2 N_0}{2\\Delta _a}\\hat{a}^\\dagger \\hat{a} + \\hbar \\sqrt{2}\\zeta \\hat{Q}\\hat{a}^\\dagger \\hat{a},$ where $g$ is the atom-cavity coupling strength, $N_0$ is the condensate population, and $\\Delta _a$ is the atom-cavity detuning.", "In the second term, the atom-cavity coupling rate is denoted by $\\zeta $ .", "The complete derivation of its form can be found in Ref. [11].", "The position and momentum quadratures of the Bogoliubov mode are $\\hat{Q}{=}(\\hat{c}+\\hat{c}^\\dag )/\\sqrt{2}$ and $\\hat{P}{=}i(\\hat{c}^\\dag -\\hat{c})/\\sqrt{2}$ , respectively.", "Finally, the mirror-cavity interaction is given by ${\\hat{\\cal H}_{MC}{=}-\\hbar \\chi \\hat{q}\\hat{a}^\\dagger \\hat{a}}$ with $\\chi =\\omega _C/L$ the mirror-cavity coupling rate ($L$ is the length of the cavity).", "As can be seen by comparing $\\hat{\\cal H}_{MC}$ with the second term of Eq.", "(REF ), the BEC dynamics is analogous to a mechanical oscillator under the action of radiation pressure.", "As no direct coupling term ${\\cal H}_{AM}$ is present in Eq.", "(REF ), the interaction between the atomic mode and the mirror is mediated by the cavity.", "The relevant degrees of freedom of the system are grouped in the vector $\\hat{\\cal \\phi }^T = (\\hat{x},\\hat{y},\\hat{q},\\hat{p},\\hat{Q},\\hat{P})$ , where the cavity position and momentum-like quadrature operators are $\\hat{x}=(\\hat{a}+\\hat{a}^{\\dag })/\\sqrt{2}$ and $\\hat{y}=i(\\hat{a}^{\\dag }-\\hat{a})/\\sqrt{2}$ , respectively.", "Under intense laser pumping the operators can be linearised and expanded around their respective classical mean values $\\phi _{s,i}$ such that $\\hat{\\phi _i} \\rightarrow \\phi _{s,i} + \\delta \\hat{\\phi }_i$ , where $\\delta \\hat{\\cal \\phi }^T = (\\delta \\hat{x},\\delta \\hat{y},\\delta \\hat{\\tilde{q}},\\delta \\hat{\\tilde{p}},\\delta \\hat{Q},\\delta \\hat{P})$ is the vector of zero-mean quantum fluctuations for each operator in $\\hat{\\cal \\phi }$ .", "Here, the mirror position and momentum operators have been rescaled to dimensionless quantities as $\\hat{q}=\\sqrt{\\hbar /m\\omega _m}\\hat{\\tilde{q}}$ , and $\\hat{p}=\\sqrt{\\hbar m\\omega _m}\\hat{\\tilde{p}}$ .", "In the hybrid optomechanical system considered here, two sources of noise should be taken into account.", "The first comes from photons leaking out of the cavity, while the second is due to the mechanical Brownian motion performed by the mirror, which is typically in contact with a thermal bath at temperature $T$ .", "The open-system nature of the problem at hand allows for the establishment of a stationary state.", "In fact, the classical values $\\phi _{s,i}$ can be calculated by solving the steady-state Langevin equations [17], which leads to the new equilibrium positions for the mechanical mirror and the harmonic oscillator embodied by the Bogoliubov mode ${q_s =\\frac{\\hbar \\chi |\\alpha _s|^2}{m\\omega _m^2}}~~\\textrm {and},{Q_s = -\\frac{\\zeta |\\alpha _s|^2}{\\omega _b}}$ .", "Here we have introduced the mean intracavity field amplitude $|\\alpha _s|^2 ={\\eta ^2}/({\\Delta ^2+\\kappa ^2})$ and the total cavity-pump detuning (modified by the radiation pressure mechanisms and the shift of the cavity frequency due to its off-resonant coupling with the atomic medium) $\\Delta = \\omega _C - \\omega _L + \\frac{g^2N_0}{2\\Delta _a} - \\chi ^{\\prime } q_s + \\zeta Q_s.$ As the last three terms in $\\Delta $ are typically very small compared to the bare detuning $\\omega _C - \\omega _L$ , we neglect any bistability effect and assume $\\Delta $ to be an independent control parameter.", "As for the fluctuations, their dynamics can be described by the following vector equation $\\partial _t \\delta {\\hat{\\mathbf {\\phi }} }={{\\cal K}}\\delta \\hat{\\mathbf {\\phi }}{+}\\hat{\\cal {\\mathbf {N}}},$ where we have introduced the input-noise vector $\\hat{\\cal {\\mathbf {N}}}^T =(\\sqrt{2\\kappa }\\delta \\hat{x}_{in},\\sqrt{2\\kappa }\\delta \\hat{y}_{in},0,{\\hat{\\xi }}/{\\sqrt{\\hbar m \\omega _m}},0,0)$ .", "The drift matrix $\\cal K$ , given below, depends on the scaled coupling parameter $\\chi {=}\\chi ^{\\prime }\\sqrt{\\hbar /m\\omega _m}$ and the mirror dissipation rate $\\gamma {=}\\omega _m/Q$ [where $Q$ is the mechanical quality factor], $\\cal {K} = \\left(\\begin{array}{cccccc}-\\kappa & \\Delta & 0 & 0 & 0 & 0 \\\\-\\Delta & -\\kappa & \\sqrt{2}\\chi ^{\\prime }\\alpha _S & 0 & -\\sqrt{2}\\zeta \\alpha _S & 0 \\\\0 & 0 & 0 & \\omega _m & 0 & 0 \\\\\\sqrt{2}\\chi ^{\\prime }\\alpha _S & 0 & -\\omega _m & -\\gamma & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & \\Omega \\\\-\\sqrt{2}\\zeta \\alpha _S & 0 & 0 & 0 & -\\Omega & 0 \\\\\\end{array}\\right).$ The photon-leakage from the cavity is accounted for by the input-noise operators $\\delta \\hat{x}_{in}=\\frac{\\delta \\hat{a}^{\\dag }_{in}+\\delta \\hat{a}_{in}}{\\sqrt{2}}~~\\textrm {and}~~\\delta \\hat{y}_{in}=i\\frac{\\delta \\hat{a}^{\\dag }_{in}-\\delta \\hat{a}_{in}}{\\sqrt{2}}$ with $\\langle \\delta \\hat{a}_{in}\\rangle =\\langle \\delta \\hat{a}^\\dag _{in}\\rangle =0$ and $\\langle \\delta \\hat{a}_{in}(t)\\delta \\hat{a}^{\\dag }_{in}(t^{\\prime })\\rangle =\\delta (t-t^{\\prime })$ .", "The Langevin force operator $\\hat{\\xi }$ in $\\hat{\\cal {\\mathbf {N}}}$ models the effects of the mechanical Brownian motion.", "In the limit of a high mechanical quality factor, such noise can be faithfully considered as Markovian, as entailed by the asymptotic form of the auto-correlation function $\\langle \\xi (t) \\xi (t^{\\prime }) \\rangle {\\simeq }\\hbar \\gamma {m/\\beta _B}\\delta (t{-}t^{\\prime })$ , where $\\beta _B{=}\\hbar /{2k_B T}$ and $k_B$ is the Boltzmann constant.", "We consider a viable detection scheme to observe entanglement between the various bi-partitions.", "Our proposal is linked to the method put forward in Ref.", "[12], i.e.", "on the mapping of atom-field or mirror-field entanglement (and thus the atom-field one) into fully accessible all-optical quantum correlations by means of two extra fields that interact (locally) with the abovementioned subsystems.", "Both the operations are within reach of state-of-the-art experiments and, in fact, have been recently implemented [18], [19] with high efficiency.", "Our revelation scheme will thus be affected by the limitations of such methods, which embody the forefront of weakly disruptive detection schemes in such mesoscopic scenarios.", "Figure: (Color online) (a) Entanglement E MA (t)E_{MA}(t) between mirror and atoms against κt\\kappa t and Δ/ω m \\Delta /\\omega _m.", "(b) Same as panel (a) for E CM (t)=E CA (t)E_{CM}(t)=E_{CA}(t).Parameters: ω m /2π=3×10 6 s -1 \\omega _m/2\\pi =3\\times 10^6~s^{-1}; T=10μKT=10\\mu K; Q=3×10 4 Q=3\\times 10^4; m=50 ng m=50~\\mathrm {ng}; ℛ=50 mW {\\cal R}=50~\\mathrm {mW}, cavity finesse F=10 4 F=10^4, and ζ=χ\\zeta =\\chi ; cavity length is L=1 mm L=1~\\mathrm {mm} from which κ=πc/2LF\\kappa =\\pi c/2LF (cc is the speed of light).", "All plotted units are dimensionless." ], [ "Entanglement dynamics", "In Ref.", "[12], the stationary entanglement within the hybrid optomechanical system has been considered.", "Here we focus on the dynamical regime where the evolution of the entanglement is resolved in time.", "We consider the fully symmetric regime encompassed by equal frequencies for the Bogoliubov and mechanical modes (i.e.", "$\\omega _b=\\omega _m$ ) and identical coupling strengths in the bipartite cavity-mirror and cavity-atom subsystems (that is, we take $\\zeta =\\chi $ ).", "We are particularly interested in the emergence of atom-mirror entanglement at short interaction times.", "The analysis conducted in Ref.", "[12] has shown it to be absent for $t\\rightarrow \\infty $ .", "The linear nature of Eq.", "(REF ) preserves the Gaussian nature of any initial state of the overall system.", "This allows us to fully characterise the entanglement evolution through the covariance matrix ${\\cal V}_{ij} = \\frac{1}{2}\\langle \\lbrace \\delta \\hat{\\phi }_i,\\delta \\hat{\\phi }_j\\rbrace \\rangle {-}\\langle \\delta \\hat{\\phi }_i\\rangle \\langle \\delta \\hat{\\phi }_j\\rangle $ .", "Using this definition and Eq.", "(REF ), the dynamical equation regulating the evolution of the covariance matrix can be written as $\\dot{{\\cal V}} = {\\cal KV}+{\\cal VK}^T + {\\cal D},$ where we have introduced the noise matrix ${\\cal D} = \\mathrm {diag}[\\kappa , \\kappa , 0, \\gamma (2\\bar{n}{+}1), 0, 0]$ with $\\bar{n}=[\\mathrm {exp}(2\\beta _B\\omega _m){-}1]^{-1}$ as the thermal mean occupation number of the mechanical mode.", "In Eq.", "(REF ) we assume that the mean values of the mechanical and optical quadratures reach their stationary values much faster than the fluctuation dynamics (this is always verified in our calculations).", "Such an inhomogeneous first-order differential equation is solved assuming the initial conditions ${\\cal V}(0) = \\mathrm {diag}[{1},1,2\\bar{n}{+}{1},2\\bar{n}{+}{1},1,1]/2$ , which describe the vacuum state of both the cavity field and the BEC mode and the thermal state of the mechanical system.", "Physicality of the covariance matrix has been thoroughly checked by considering the fulfillment of the Heisenberg-Robinson uncertainty principle and checking that the minimum symplectic eigenvalue $\\nu =\\min \\mathrm {eig}(i\\omega {\\cal V})$ is such that $|\\nu |\\ge \\frac{1}{2}$ .", "Here used the $6 \\times 6$ symplectic matrix $\\omega =\\oplus ^3_{j=1}i\\sigma _y$ with $\\sigma _y$ the y-Pauli matrix.", "The entanglement measure that we use here to quantify entanglement between any two modes $\\alpha $ and $\\beta $ is the logarithmic negativity [20], defined as $E_{\\alpha \\beta } = -\\mathrm {log}|2\\nu _{min}|$ , where $\\nu _{min}$ is the smallest symplectic eigenvalue of the matrix ${\\cal V}_{\\alpha \\beta }^{T_\\beta }=P{\\cal V}_{\\alpha \\beta }P$ , for $\\alpha ,\\beta = C,A,M$ [the latter being the labels for the cavity, atomic and mirror modes, respectively].", "The reduced covariance matrix ${\\cal V}_{\\alpha \\beta }$ contains the entries of ${\\cal V}$ associated with modes $\\alpha $ and $\\beta $ while, by inverting the momentum quadrature of $\\beta $ , matrix $P=\\mathrm {diag}(1,1,1,-1)$ performs the partial transposition in phase space.", "The atom-mirror entanglement $E_{MA}(t)$ , whose time evolution is shown in Fig.", "REF(a) for different values of the effective detuning $\\Delta $ , gradually develops and reaches its peak value as the cavity-atom and cavity-mirror entanglement drop to a quasi-stationary value.", "As no direct atom-mirror interaction exists in this system, mediation through the cavity mode essentially results in a delay: quantum correlations between the atoms (the mirror) and the cavity mode must build up before any atom-mirror correlation can appear.", "This is clearly shown in Fig.", "REF(b), where $E_{CM}(t)=E_{CA}(t)$ reach their maximum well before $E_{MA}(t)$ starts to grow.", "The atom-mirror entanglement is non-zero only within a very short time window, signalling the fragility of quantum correlations resulting from only a second-order interaction between the BEC and the cavity end-mirror.", "These results go far beyond the limitations of the steady-state analysis conducted in [12] and prove the existence of a regime where all the various reductions obtained by tracing out one of the modes from the overall system are inseparable, a situation that, within the range of parameters considered in our investigation, is typical only of a time-resolved picture." ], [ "Optimal control of the early-time entanglement", "We now consider the effects of time-modulating the external pump power ${\\cal R}$ , which is now considered a function of time.", "In turn, this implies that we now take $\\eta \\rightarrow \\eta (t)$ and study the time behavior of the entanglement $E_{MA}$ set between the atoms and the mirror.", "We will show that a properly optimised $\\eta (t)$ can increase the maximum value of $E_{MA}(t)$ for values of $t$ within the same time interval $\\tau $ where atom-mirror entanglement has been shown to emerge in the unmodulated case.", "We assume to vary $\\eta (t)$ slowly in time, so that the classical mean values $\\phi _s$ adiabatically follow the change in $\\eta (t)$ .", "This approximation is valid as long as the number of intra-cavity photons is large enough to retain the validity of the linearization procedure and the time-variation of $\\eta (t)$ is slow compared to the time taken by the mean values to reach their stationary values.", "For all cases considered here we have verified the validity of such assumptions.", "The dynamics of the covariance matrix is thus still governed by Eq.", "(REF ) with the replacement ${\\cal K}\\rightarrow {\\cal K}(t)$ .", "In the following, we use the value of $\\Delta =2.7\\omega _m$ which maximises the short time entanglement $E_{MA}$ .", "Inspired by the techniques for dynamical optimization proposed in [16], we call $\\eta _0$ the unmodulated value of $\\eta $ and take $\\eta (t) = \\eta _0 + \\sum _{j=1}^{j_{max}}\\left[ A_j \\cos (\\omega _j t)+B_j\\sin (\\omega _j t) \\right],$ where $\\omega _j = 2\\pi j/\\tau +\\delta _j$ are the harmonics and $\\delta _j$ is a small random shift.", "The coefficients $A_j$ and $B_j$ are chosen in a way that the total energy brought about by the time-modulated field is the same as the one associated with the unmodulated case.", "We then set the time-window so that $\\tau =3.4\\kappa ^{-1}$ , when we observe the maximum value of $E_{MA}$ in the unmodulated instance.", "The other parameters are as in Fig.", "REF .", "We then look for the parameters $A_j$ and $B_j$ that maximise the value of $E_{MA}(\\tau )$ for a given set of random shifts $\\delta _j$ .", "We use standard optimisation routines to find a (local) maximum of $E_{MA}(\\tau )$ .", "We repeat the search of the optimal coefficients for different values of $\\delta _j$ and take the overall maximum.", "The corresponding results are presented in Fig.", "REF , where we show the optimal modulation $\\eta (t)$ and the optimal $E_{MA}(t)$ .", "These findings are also compared to the case without modulation.", "The maximum value attained in the interval $[0;\\tau ]$ is $E_{MA}(t)\\simeq 0.05$ which is 2.5 times larger than the case without modulation, thus demonstrating the effectiveness of our approach.", "Figure: (Color online) Entanglement dynamics E MA (t)E_{MA}(t) (dashed line) with the optimal laser intensity modulation η(t)\\eta (t) (solid line).", "The entanglement E MA (t)E_{MA}(t) for constant η(t)=η 0 \\eta (t)=\\eta _0 is also shown (dashed-dotted line).", "All plotted units are dimensionless." ], [ "Periodic modulation: long time entanglement", "In the two situations analyzed so far [constant laser intensity $\\eta $ and optimally modulated $\\eta (t)$ ], the atom-mirror entanglement $E_{MA}(t)$ is destined to disappear at long times.", "A complementary approach based on a periodic modulation of the laser intensity $\\eta (t)$ was used in the pure optomechanical setting [13] to increase the long-time light-mirror entanglement.", "Here we use a similar approach by assuming the monochromatic modulation of the laser-cavity coupling $\\eta (t) =\\eta ^{\\prime \\prime }_0+{\\eta ^{\\prime }_0}\\left[1 - \\sin (\\Sigma t)\\right]$ , where $\\Sigma $ is the frequency of the harmonic modulation, $\\eta ^{\\prime }_0=4\\eta ^{\\prime \\prime }_0={\\eta _0}/2$ , $\\eta _0$ being the same constant coupling parameter taken before.", "These choices ensure that the approximations used in the dynamical analysis are valid.", "After the transient dynamics, the covariance matrix and, in turn, $E_{MA}(t)$ become periodic functions of time.", "In order to achieve the best possible performance at long times, we compute the maximum of $E_{MA}(t)$ after the transient behavior, scanning the values of $\\Sigma $ .", "The result is shown in Fig.", "REF (a) (inset) revealing a sharp resonance with a maximum value of $E_{MA}\\simeq 0.12$ for $\\Sigma =\\bar{\\Sigma }\\sim 0.79 \\kappa $ (no further peak appears beyond this interval).", "This arises as a result of the effective interaction between atoms and mirror mediated by the cavity field.", "As shown in Fig.", "REF(a) the entanglement dynamics strongly depend on the effective detuning giving rise to such optimal behavior.", "Similar results have also been observed in Ref [13].", "Figure: (Color online) (a) Dynamics of the cavity-mirror and cavity-atoms entanglement E CM,CA E_{CM,CA} (solid line) and atoms-mirror entanglement E MA E_{MA} (dashed line) for Σ=Σ ¯∼0.79κ\\Sigma =\\bar{\\Sigma }\\sim 0.79 \\kappa .", "Inset: Maximum E MA E_{MA} for long times with a periodic modulation as a function of the frequency Σ\\Sigma .", "(b) Optimal periodic modulation η(t)\\eta (t) (solid line) for one period of time 2π/Σ ¯2\\pi /\\bar{\\Sigma } compared to the monochromatic modulation (dashed line).", "All plotted units are dimensionless.The analysis of the evolution of $E_{CM(CA)}$ and $E_{MA}$ , shown in the main panel of Fig.", "REF (a), reveals that while $E_{CM(CA)}$ develops very quickly due to the direct cavity-atoms and cavity-mirror couplings, $E_{MA}$ grows in a longer time lapse, during which the cavity disentangles from the dynamics.", "The quasi-asymptotic value achieved by $E_{MA}$ reveals a sixfold increase with respect to the unmodulated case.", "This behavior is worth commenting as it strengthens our intuition that any atom-mirror entanglement has to result from a process that effectively couples such subsystems, bypassing any mechanism giving rise to multipartite entanglement within the overall system.", "Finally, we discuss the results achieved in the long time case by adopting an optimal-control technique similar to the one used for enhancing the short time entanglement.", "We have considered the periodic modulation of the intensity at the frequency $\\bar{\\Sigma }$ given by $\\eta (t) = {\\eta ^{\\prime \\prime }_0}+{\\eta ^{\\prime }_0}\\left[1 - \\sum _{n=1}^{n_{max}} A_n\\sin (n\\bar{\\Sigma }t)+B_n\\cos (n\\bar{\\Sigma }t)\\right],$ and looked for the coefficients $\\lbrace A_n,B_n\\rbrace $ optimizing $E_{MA}(t)$ at long times with the constraint: $\\sum _{n=1}^{n_{max}}( A_n^2+B_n^2)\\le 1$ , ensuring that no instability is introduced in the dynamics of the overall system.", "In our simulation, $n_{max}=8$ has been taken to limit the complexity of the modulated signal.", "The resulting optimal coefficients give the periodic modulation shown in Fig.", "REF (b) and a maximum entanglement $E_{MA}\\simeq 0.17$ which is about $30\\%$ larger than the results obtained for the monochromatic modulation.", "This demonstrates the powerful nature of our scheme.", "Our extensive multiple-harmonic analysis is able to outperform quite significantly the single-frequency driving scheme addressed above and discussed in Ref.", "[13], proving strikingly the sub-optimality of the monochromatic modulation, both at short and, surprisingly, at long times of the system dynamics.", "This legitimates experimental efforts directed towards the use of time-modulated driving signals for the optimal control of the hybrid mesoscopic systems addressed here and similar ones based, for instance, on the use of a vibrating membrane [21] or a levitated nano-sphere [22] instead of the BEC.", "It should be noted that an adiabatic approach is used in Ref.", "[13] to find an effective Hamiltonian.", "When performed in our hybrid optomechnical scheme, such technique would remove the dynamics of the Bogoliubov modes of the atomic subsystem.", "We consider the robustness of our protocol with respect to inaccuracy in the control of the value $\\chi =\\xi $ as follows: after finding the best periodic modulation assuming $\\chi =\\xi $ , we ran again the simulations with the same modulation with $\\chi =1.1 \\xi $ and $\\chi =0.9 \\xi $ .", "These values correspond to a 10% inaccuracy in the nominal values of $\\chi $ and $\\xi $ .", "We found that the maximum entanglement is at most only 3% less than the original value thus confirming the stability of our result.", "Notice also that if the imbalance between $\\chi $ and $\\xi $ is known, for example by a calibration measurement, we can in principle run the optimization including the actual values of $\\chi $ and $\\xi $ therefore aiming at a larger entanglement value." ], [ "Conclusions", "We have demonstrated that the modulation-assisted driving of a hybrid optomechanical device gives rise to interesting and rich entanglement dynamics, surpassing the limitations associated with a steady-state analysis and a constant pump.", "A significant improvement in the genuinely mesoscopic entanglement between the mode embodied by the atomic system and the mechanical one can be achieved by pumping the cavity with a modulated driving field, in both the short-time case and the long-time case.", "We have shown the existence of modulations that are able to beat the performance of simple monochromatic driving in terms of the maximum entanglement created between the mirror and the atomic system, thus favoring the creation of entanglement at long times.", "Our study strengthens the idea that important advantages are in order when optimal control techniques are implemented in the open-system dynamics of mesoscopic devices.", "This contributes to the current quest for the grounding of such approaches as valuable instruments for the control of mesoscopic (and multipartite) systems of various realizations, including interesting configurations of current experimental interest [21], [22] to which our framework can be fully applied.", "We thank M. Genoni, V. Giovannetti, and A. Xuereb for fruitful discussions.", "This work is supported by the UK EPSRC through a Career Acceleration Fellowship and a grant under the “New Directions for EPSRC Research Leaders\" initiative (EP/G004759/1)." ] ]

[ [ "On the de Rham complex of mixed twistor D-modules" ], [ "Abstract Given a complex manifold S, we introduce for each complex manifold X a t-structure on the bounded derived category of C-constructible complexes of O_S-modules on X x S. We prove that the de Rham complex of a holonomic D_{XxS/S}-module which is O_S-flat as well as its dual object is perverse relatively to this t-structure.", "This result applies to mixed twistor D-modules." ], [ "Introduction", "Given a vector bundle $V$ of rank $d\\geqslant 1$ with an integrable connection $\\nabla :V\\rightarrow \\Omega ^1_X\\otimes V$ on a complex manifold $X$ of complex dimension $n$ , the sheaf of horizontal sections $V^\\nabla =\\ker \\nabla $ is a locally constant sheaf of $d$ -dimensional $-vector spaces, and is the only nonzero cohomology sheaf of the de~Rham complex $ DRX(V,)=(XV,)$.", "Assume moreover that $ (V,)$ is equipped with a harmonic metric in the sense of \\cite [p.~16]{Simpson92}.", "The twistor construction of \\cite {Simpson97} produces then a holomorphic bundle $ V$ on the product space $ X=X, where the factor $ has coordinate $ z$, together with a holomorphic flat $ z$-connection.", "By restricting to $ X*:=X*$, giving such a $ z$-connection on $ V*:=V|X*$ is equivalent to giving a flat relative connection~$$ with respect to the projection $ p:X**$.", "Similarly, the relative de~Rham complex $ DRX*/*(V*,)$ has cohomology in degree zero at most, and $ (V*):=$ is a locally constant sheaf of locally free $ p-1O*$-modules of rank $ d$.$ Holonomic ${D}_X$ -modules generalize the notion of a holomorphic bundle with flat connection to objects having (possibly wild) singularities, and a well-known theorem of Kashiwara [1] shows that the solution complex of such a holonomic ${D}_X$ -module has $-constructible cohomology, from which one can deduce that the de~Rham complex is of the same kind and more precisely that both are $ -perverse sheaves on $X$ up to a shift by $\\dim X$ .", "The notion of a holonomic ${D}_X$ -module with a harmonic metric has been formalized in [13] and [9] under the name of pure twistor ${D}$ -module (this generalizes holonomic ${D}_X$ -modules with regular singularities), and then in [14] and [10] under the name of wild twistor ${D}$ -modules (this takes into account arbitrary irregular singularities).", "More recently, Mochizuki [11] has fully developed the notion of a mixed (possibly wild) twistor ${D}$ -module.", "When restricted to ${X}^*$ , such an object contains in its definition two holonomic ${D}_{{X}^*/*}$ -modules, and we say that both underlie a mixed twistor ${D}$ -module The main result of this article concerns the de Rham complex and the solution complex of such objects.", "Theorem 1.1 The de Rham complex and the solution complex of a ${D}_{{X}^*/*}$ -module underlying a mixed twistor ${D}$ -module are perverse sheaves of $p^{-1}{O}_{*}$ -modules (up to a shift by $\\dim X$ ).", "In Section , we define the notion of relative constructibility and perversity.", "This applies to the more general setting where $p:{X}^*\\rightarrow *$ is replaced by a projection $p_X:{X}=X\\times S\\rightarrow S$ , where $S$ is any complex manifold.", "We usually set $p=p_X$ when $X$ is fixed.", "On the other hand, we call holonomic any coherent ${D}_{X\\times S/S}$ -module whose relative characteristic variety in $T^*(X\\times S/S)=(T^*X)\\times S$ is contained in a variety $\\Lambda \\times S$ , where $\\Lambda $ is a conic Lagrangian variety in $T^*X$ .", "We say that a ${D}_{X\\times S/S}$ -module is strict if it is $p^{-1}{O}_S$ -flat.", "Theorem 1.2 The de Rham complex and the solution complex of a strict holonomic ${D}_{X\\times S/S}$ -module whose dual is also strict are perverse sheaves of $p^{-1}{O}_S$ -modules (up to a shift by $\\dim X$ ).", "A ${D}_{{X}^*/*}$ -module ${M}$ underlying a mixed twistor ${D}$ -module is strict and holonomic (see [11]).", "Moreover, Mochizuki has defined a duality functor on the category of mixed twistor ${D}$ -modules, proving in particular that the dual of ${M}$ as a ${D}_{{X}^*/*}$ -module is also strict holonomic.", "Therefore, these results together with Theorem REF imply Theorem REF .", "Note that, while our definition of perverse objects in the bounded derived category $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p^{-1}{O}_S)$ intends to supply a notion of holomorphic family of perverse sheaves, we are not able, in the case of twistor ${D}$ -modules, to extend this notion to the case when the parameter $z\\in *=S$ also achieves the value zero, and to define a perversity property in the Dolbeault setting of [18] for the associated Higgs module." ], [ "Relative constructibility in the case of a projection", "We keep the setting as above, but $X$ is only assumed to be a real analytic manifold.", "Given a real analytic map $f:Y\\rightarrow X$ between real analytic manifolds, we will denote by $f_S$ (or $f$ if the context is clear) the map $f\\times \\operatorname{id}_S:Y\\times S\\rightarrow X\\times S$ ." ], [ "Sheaves of $-vector spaces and of $ p-1OS{{formula:25032082-7fcc-4cb2-a81d-7670ec1f6fc2}}", "Let $f:Y\\!\\rightarrow \\!", "X$ be such a map.", "There are functors $f^{-1},f^!,Rf_*,Rf_!$ between $\\operatorname{\\mathsf {D}}^\\mathrm {b}({X\\times S})$ and $\\operatorname{\\mathsf {D}}^\\mathrm {b}({Y\\times S})$ , and functors $f_S^{-1},f_S^!,Rf_{S,*},Rf_{S,!", "}$ between $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p_X^{-1}{O}_S)$ and $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p_Y^{-1}{O}_S)$ .", "These functors correspond pairwise through the forgetful functor $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p_X^{-1}{O}_S)\\rightarrow \\operatorname{\\mathsf {D}}^\\mathrm {b}({X\\times S})$ .", "Indeed, this is clear except for $f_S^!$ and $f^!$ .", "To check it, one decomposes $f$ as a closed immersion and a projection.", "In the first case, the compatibility follows from the fact that both are equal to $f^{-1}R\\Gamma _{f(X)}$ (see [4]) and for the case of a projection one uses [4].", "We note also that the Poincaré-Verdier duality theorem [4] holds on $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p^{-1}{O}_S)$ (see [4]).", "From now on, we will write $f^{-1}$ , etc.", "instead of $f_S^{-1}$ , etc.", "The ring $p_X^{-1}{O}_S$ is Noetherian, hence coherent (see [2]).", "For each $s_o\\in S$ let us denote by $\\mathfrak {m}_{s_o}$ the ideal of sections of ${O}_S$ vanishing at $s_o$ and by $i^\\star _{s_o}$ the functor $\\text{Mod}(p^{-1}_X {O}_{S})& \\longmapsto \\text{Mod}(X)\\\\F&\\longmapsto F\\otimes _{p_X^{-1}{O}_{S}} p_X^{-1}({O}_{S}/\\mathfrak {m}_{s_o}).$ This functor will be useful for getting properties of $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p_X^{-1}{O}_S)$ from well-known properties of $\\operatorname{\\mathsf {D}}^\\mathrm {b}(X)$ .", "Proposition 2.1 Let $F$ and $F^{\\prime }$ belong to $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p_X^{-1}{O}_{S})$ .", "Then, for each $s_o\\in S$ there is a well-defined natural morphism $Li^*_{s_o}(R{H}\\!om_{p^{-1}({O}_{S})}(F, F^{\\prime })) \\rightarrow R{H}\\!om_{{X}}(Li^*_{s_o}(F), Li^*_{s_o}(F^{\\prime }))$ which is an isomorphism in $\\operatorname{\\mathsf {D}}^\\mathrm {b}(X)$ .", "Let us fix $s_o\\in S$ .", "The existence of the morphism follows from [2].", "Moreover, since $p^{-1}_X{O}_S$ is a coherent ring as remarked above and $p_X^{-1}({O}_{S}/\\mathfrak {m}_{s_o})$ is $p^{-1}_X{O}_S$ -coherent, we can apply the argument given after (A.10) in loc. cit.", "to show that it is an isomorphism.", "Proposition 2.2 Let $F$ and $F^{\\prime }$ belong to $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p^{-1}_X{O}_S)$ and let $\\phi : F\\rightarrow F^{\\prime }$ be a morphism.", "Assume the following conditions: for all $j\\in \\mathbb {Z}$ and $(x,s)\\in X\\times S$ , ${H}^j(F)_{(x,s)}$ and ${H}^j(F^{\\prime })_{(x,s)}$ are of finite type over ${O}_{S,s}$ , for all $s_o\\in S$ , the natural morphism $Li^*_{s_o}(\\phi ): Li^*_{s_o}(F)\\rightarrow Li^*_{s_o}(F^{\\prime })$ is an isomorphism in $\\operatorname{\\mathsf {D}}^\\mathrm {b}(X)$ .", "Then $\\phi $ is an isomorphism.", "It is enough to prove that the mapping cone of $\\phi $ is quasi-isomorphic to 0.", "So we are led to proving that for $F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}(p^{-1}{O}_S)$ , if ${H}^j(F)_{(x,s)}$ are of finite type over ${O}_{S,s}$ for all $(x,s)\\in X\\times S$ , and $Li^*_{s_o}(F)$ is quasi-isomorphic to 0 for each $s_o\\in S$ , then $F$ is quasi-isomorphic to 0.", "We may assume that $S$ is an open subset of $n$ with coordinates $s^1,\\dots ,s^n$ and we will argue by induction on $n$ .", "Assume $n=1$ .", "For such an $F$ , for each $s_o\\in S$ and any $j\\in \\mathbb {Z}$ the morphism $(s^1-s^1_o):{H}^j(F)\\rightarrow {H}^j(F)$ is an isomorphism, hence ${H}^j(F)/(s^1-s^1_o){H}^j(F)=0$ and by Nakayama's Lemma, for any $x\\in X$ , ${H}^j(F)_{(x,s^1_o)}=0$ and the result follows.", "For $n\\geqslant 2$ , the cone $F^{\\prime }$ of the morphism $(s^n-s^n_o):F\\rightarrow F$ also satisfies $Li^*_{s^{\\prime }_o}F^{\\prime }=0$ for any $s^{\\prime }_o=(s^1_o,\\dots ,s^{n-1}_o)$ , hence is zero by induction, so we can argue as in the case $n=1$ ." ], [ "$S$ -locally constant sheaves", "We say that a sheaf $F$ of $-vector spaces (resp.~$ pX-1OS$-modules) on $ XS$ is \\emph {$ S$-locally constant} if, for each point $ (x,s)XS$, there exists a neighbourhood $ U=VxTs$ of $ (x,s)$ and a sheaf $ G(x,s)$ of $ -vector spaces (resp.", "${O}_S$ -modules) on $T_s$ , such that $F_{|U}\\simeq p_U^{-1}G^{(x,s)}$ .", "The category of $S$ -locally constant sheaves is an abelian full subcategory of that of sheaves of ${X\\times S}$ -vector spaces (resp.", "$p^{-1}{O}_S$ -modules), which is stable by extensions in the respective categories, by ${H}\\!om$ and tensor products.", "Moreover, if $\\pi :Y\\times X\\times S\\rightarrow Y\\times S$ is the projection, with $X$ contractible, then, if $F^{\\prime }$ is $S$ -locally constant on $Y\\times X\\times S$ , $\\pi _*F^{\\prime }$ is $S$ -locally constant on $Y\\times S$ , $R^k\\pi _*F^{\\prime }=0$ if $k>0$ , $F^{\\prime }\\simeq \\pi ^{-1}\\pi _*F^{\\prime }$ .", "Applying this to $Y=\\lbrace \\textup {pt}\\rbrace $ , we find that, if $F$ is $S$ -locally constant, then for each $x\\in X$ there exists a connected neighbourhood $V_x$ of $x$ and a $S$ -module (resp.", "${O}_S$ -module) $G^{(x)}$ such that $F=p_{V_x}^{-1}G^{(x)}$ , and one has $G^{(x)}=p_{V_x,*}F_{|V_x\\times S}=F_{|\\lbrace x\\rbrace \\times S}$ .", "We shall also denote by $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {lc}(p_X^{-1}{S})$ (resp.", "$\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {lc}(p_X^{-1}{O}_{S})$ ) the bounded triangulated category whose objects are the complexes having $S$ -locally constant cohomology sheaves.", "Similarly, for such a complex $F$ we have $F_{|V_x\\times S}\\simeq p_{V_x}^{-1}Rp_{V_x,*}F_{|V_x\\times S}\\simeq p_{V_x}^{-1}F_{|\\lbrace x\\rbrace \\times S}$ .", "We conclude from the previous remarks, by using the natural forgetful functor $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p_X^{-1}{O}_{S})\\rightarrow \\operatorname{\\mathsf {D}}^\\mathrm {b}({X\\times S})$ : Lemma 2.3 An object $F$ of $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p_X^{-1}{O}_{S})$ belongs to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {lc}(p_X^{-1}{O}_{S})$ if and only if, when regarded as an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}({X\\times S})$ , it belongs to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {lc}(p_X^{-1}{S})$ .", "For any object $F$ of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {lc}(p_X^{-1}{O}_{S})$ and for any $s_o\\in S$ , $Li^*_{s_o}F$ belongs to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {lc}(X)$ ." ], [ "$S$ -weakly {{formula:f417befc-7762-42dd-9ace-f2d00072cbfc}} -constructible sheaves", "As long as the manifold $X$ is fixed, we shall write $p$ instead of $p_X$ .", "Definition 2.4 Let $F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}({X\\times S})$ (resp.", "$F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}(p^{-1}{O}_{S})$ ).", "We shall say that $F$ is $S$ -weakly $\\mathbb {R}$ -constructible if there exists a subanalytic $\\mu $ -stratification $(X_\\alpha )$ of $X$ (see [4]) such that, for all $j\\in \\mathbb {Z}$ , ${H}^j(F)|_{X_\\alpha \\times S}$ is $S$ -locally constant.", "This condition is independent of the choice of the $\\mu $ -stratification and characterizes a full triangulated subcategory $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p^{-1}S)$ (resp.", "$\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p^{-1}{O}_{S})$ ) of $\\operatorname{\\mathsf {D}}^\\mathrm {b}({X\\times S})$ (resp.", "$\\operatorname{\\mathsf {D}}^\\mathrm {b}(p^{-1}{O}_{S})$ ).", "Due to Lemma REF , an object $F$ of $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p^{-1}{O}_{S})$ is in $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p^{-1}{O}_{S})$ if and only if it belongs to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p^{-1}S)$ when considered as an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}({X\\times S})$ .", "By mimicking for $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p^{-1}S)$ the proof of [4] and according to the previous remark for $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p^{-1}{O}_{S})$ , we obtain: Proposition 2.5 Let $F$ be $S$ -weakly $\\mathbb {R}$ -constructible on $X$ and let $X=\\bigsqcup _\\alpha X_\\alpha $ be a $\\mu $ -stratification of $X$ adapted to $F$ .", "Then the following conditions are equivalent: for all $j\\in \\mathbb {Z}$ and for all $\\alpha $ , ${H}^j(F)|_{X_\\alpha \\times S}$ is $S$ -locally constant.", "$SS(F)\\subset (\\bigsqcup _\\alpha T^*_{X_\\alpha } X)\\times T^* S$ .", "There exists a closed conic subanalytic Lagrangian subset $\\Lambda $ of $T^*X$ such that $SS(F)\\subset \\Lambda \\times T^*S$ .", "Proposition 2.6 Let $F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_{X}^{-1}{O}_{S})$ and let $s_o\\in S$ .", "Then $Li_{s_o}^*(F)\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(X)$ .", "Let $i_\\alpha :X_\\alpha \\hookrightarrow X$ denote the locally closed inclusion of a stratum of an adapted stratification $(X_\\alpha )$ .", "It is enough to observe that, for each $\\alpha $ , we have $i^{-1}_\\alpha Li^*_{s_o}(F)\\simeq Li^*_{s_o}(i_\\alpha ^{-1}F)$ , and to apply Lemma REF (REF ).", "Let now $Y$ be another real analytic manifold and consider a real analytic map $f: Y\\rightarrow X$ .", "The following statements for objects of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p^{-1}S)$ are easily deduced from Proposition REF similarly to the absolute case treated in [4], as consequences of Theorem $8.3.17$ , Proposition $8.3.11$ , Corollary $6.4.4$ and Proposition $5.4.4$ of loc. cit.", "In order to get the same statements for objects of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p^{-1}{O}_S)$ , one uses Lemma REF (REF ) together with §REF .", "We will not distinguish between $f$ and $f_S$ .", "Proposition 2.7 If $F$ is $S$ -weakly $\\mathbb {R}$ -constructible on $X$ , then so are $f^{-1}(F)$ and $f^!", "(F)$ .", "Assume that $F^{\\prime }$ is $S$ -weakly $\\mathbb {R}$ -constructible on $Y$ and that $f$ is proper on $\\operatorname{Supp}(F^{\\prime })$ .", "Then $Rf_*(F^{\\prime })$ is $S$ -weakly $\\mathbb {R}$ -constructible on $X$ .", "Given a closed subanalytic subset $Y\\subset X$ , we will denote by $i:Y\\times S\\hookrightarrow X\\times S$ the closed inclusion and by $j$ the complementary open inclusion.", "Corollary 2.8 Assume that $F^*$ is $S$ -weakly $\\mathbb {R}$ -constructible on $X\\setminus Y$ .", "Then the objects $Rj_!F^*$ and $Rj_*F^*$ are also $S$ -weakly $\\mathbb {R}$ -constructible on $X$ .", "The statement for $Rj_!F^*$ is obvious.", "Then Proposition REF implies that $i^!Rj_!F^*$ is $S$ -weakly $\\mathbb {R}$ -constructible.", "Conclude by using the distinguished triangle $Ri_*i^!Rj_!F^*\\rightarrow Rj_!F^*\\rightarrow Rj_*F^*\\xrightarrow{}$ and the $S$ -weak $\\mathbb {R}$ -constructibility of the first two terms.", "Proposition 2.9 An object $F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}({X\\times S})$ (resp.", "$F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}(p^{-1}({O}_S))$ ) is $S$ -weakly $\\mathbb {R}$ -constructible with respect to a $\\mu $ -stratification $(X_\\alpha )$ if and only if, for each $\\alpha $ , $i_\\alpha ^!F$ has $S$ -locally constant cohomology on $X_\\alpha $ .", "Assume that $F$ is $S$ -weakly $\\mathbb {R}$ -constructible with respect to a $\\mu $ -stratification $(X_\\alpha )$ of $X$ .", "Then $i_\\alpha ^!F$ has $S$ -locally constant cohomology on $X_\\alpha $ .", "Indeed the estimation of the micro-support of [4] implies that $SS(i_\\alpha ^!F)$ (like $SS(i_\\alpha ^*F)$ ) is contained in $T^*_{X_\\alpha }{X_\\alpha }\\times T^*S$ , so $i_\\alpha ^!F$ has locally constant cohomology on $X_\\alpha $ for each $\\alpha $ , according to Proposition REF .", "Conversely, if $i_\\alpha ^!F$ is locally constant for each $\\alpha $ , then $F$ is $S$ -weakly $\\mathbb {R}$ -constructible.", "Indeed, we argue by induction and we denote by $X_k$ the union of strata of codimension $\\leqslant k$ in $X$ .", "Assume we have proved that $F_{|X_{k-1}\\times S}$ is $S$ -weakly $\\mathbb {R}$ -constructible with respect to the stratification $(X_\\alpha )$ with $\\operatorname{codim}X_\\alpha \\leqslant k-1$ .", "We denote by $j_k:X_{k-1}\\hookrightarrow X_k$ the open inclusion and by $i_k$ the complementary closed inclusion.", "According to Corollary REF , $Rj_{k,*}j_k^{-1}F$ is $S$ -weakly $\\mathbb {R}$ -constructible with respect to $(X_\\alpha )_{|X_k}$ .", "Now, by using the exact triangle $i_k^!F\\rightarrow i_k^{-1}F\\rightarrow i_k^{-1}Rj_{k,*}j_k^{-1}F\\xrightarrow{}$ , we conclude that $i_k^{-1}F$ is locally constant, hence $F_{|X_k\\times S}$ is $S$ -weakly $\\mathbb {R}$ -constructible.", "Corollary 2.10 Let $F,F^{\\prime }\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ .", "Then $R{H}\\!om_{p_X^{-1}{O}_S}(F,F^{\\prime })$ also belongs to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ .", "In view of Proposition REF , it is sufficient to prove that for each $\\alpha $ , $i_\\alpha ^!R{H}\\!om_{p_X^{-1}{O}_S}(F,F^{\\prime })$ belongs to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {lc}(p_X^{-1}{O}_S)$ .", "We have: $i_\\alpha ^!R{H}\\!om_{p^{-1}{O}_S}(F,F^{\\prime })\\simeq R{H}\\!om_{p_\\alpha ^{-1}{O}_S}(i^{-1}_\\alpha F,i_\\alpha ^!F^{\\prime }).$ Since both $i^{-1}_\\alpha F$ and $i_\\alpha ^!F^{\\prime }$ belong to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {lc}(p_X^{-1}{O}_S)$ , according to Proposition REF , we have locally on $X_\\alpha $ isomorphisms $i^{-1}_\\alpha F=p_\\alpha ^{-1}G_\\alpha $ and $i_\\alpha ^!F^{\\prime }=p_\\alpha ^{-1}G^{\\prime }_\\alpha =p_\\alpha ^!G^{\\prime }_\\alpha [-\\dim _\\mathbb {R}X_\\alpha ]$ for some ${O}_S$ -modules $G_\\alpha $ and $G^{\\prime }_\\alpha $ .", "Then $R{H}\\!om_{p_\\alpha ^{-1}{O}_S}(i^{-1}_\\alpha F,i_\\alpha ^!F^{\\prime })&=R{H}\\!om_{p_\\alpha ^{-1}{O}_S}(p^{-1}_\\alpha G_\\alpha ,p_\\alpha ^!G^{\\prime }_\\alpha [-\\dim _\\mathbb {R}X_\\alpha ])\\\\&\\simeq p_\\alpha ^!R{H}\\!om_{{O}_S}(G_\\alpha ,G^{\\prime }_\\alpha )[-\\dim _\\mathbb {R}X_\\alpha ]\\\\&=p_\\alpha ^{-1}R{H}\\!om_{{O}_S}(G_\\alpha ,G^{\\prime }_\\alpha ).$ The following lemma will be useful in the next section.", "Assume that $X=Y\\times Z$ and that the $\\mu $ -stratification $(X_\\alpha )$ of $X$ takes the form $X_\\alpha =Y\\times Z_\\alpha $ , where $(Z_\\alpha )$ is a $\\mu $ -stratification of $Z$ .", "We denote by $q:X\\rightarrow Y$ the projection.", "Let $z_o\\in Z$ , let $U\\ni z_o$ be a coordinate neighbourhood of $z_o$ in $Z$ and, for each $\\varepsilon >0$ small enough, let $B_\\varepsilon \\subset U$ be the open ball of radius $\\varepsilon $ centered at $z_o$ and let $\\overline{B}_\\varepsilon $ be the closed ball and $S_\\varepsilon $ its boundary.", "For the sake of simplicity, we denote by $q_\\varepsilon ,q_{\\overline{\\varepsilon }},q_{\\partial \\varepsilon }$ the corresponding projections.", "We set $Z^*=Z\\setminus \\lbrace z_o\\rbrace $ and $X^*=Y\\times Z^*$ .", "We denote by $i:Y\\times \\lbrace z_o\\rbrace \\hookrightarrow Y\\times Z$ and by $j:Y\\times Z^*\\hookrightarrow Y\\times Z$ the complementary closed and open inclusions.", "Lemma 2.11 Let $F^*\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_{X^*}^{-1}S)$ (resp.", "$F^*\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_{X^*}^{-1}{O}_{S})$ ) be adapted to the previous stratification.", "Then there exists $\\varepsilon _o>0$ such that, for each $\\varepsilon \\in (0,\\varepsilon _o)$ , the natural morphisms $Rq_{\\partial \\varepsilon ,*}F^*_{|Y\\times S_\\varepsilon \\times S}\\longleftarrow Rq_{\\overline{\\varepsilon },*}Rj_*F^*\\longrightarrow Rq_{\\varepsilon ,*}Rj_*F^*\\longrightarrow i^{-1}Rj_*F^*$ are isomorphisms.", "We note that, according to Corollary REF , $F:=Rj_*F^*$ is $S$ -weakly $\\mathbb {R}$ -constructible, and is adapted to the stratification $(Y\\times Z_\\alpha )$ .", "On the other hand, according to §REF , it is enough to consider the case where $F^*$ is an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_{X^*}^{-1}S)$ .", "Let us start with the right morphisms.", "We can argue with any object $F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}S)$ , not necessarily of the form $Rj_*F^*$ .", "Recall that we have an adjunction morphism $q_{\\varepsilon }^{-1}Rq_{\\varepsilon ,*}\\rightarrow \\operatorname{id}$ and thus $i^{-1}q_{\\varepsilon }^{-1}Rq_{\\varepsilon ,*}\\rightarrow i^{-1}$ .", "Since $q_{\\varepsilon }\\circ i=\\operatorname{id}_{Y\\times S}$ , we get the second right morphism.", "The first one is the restriction morphism.", "According to [4], there exists $\\varepsilon _o>0$ such that, for $\\varepsilon ^{\\prime }<\\varepsilon $ in $(0,\\varepsilon _o)$ , the restriction morphisms $Rq_{\\overline{\\varepsilon },*}F\\rightarrow Rq_{\\varepsilon ,*}F\\rightarrow Rq_{\\overline{\\varepsilon }^{\\prime },*}F\\rightarrow Rq_{\\varepsilon ^{\\prime },*}F$ are isomorphisms.", "In particular, the first right morphism is an isomorphism.", "Let us take a $q$ -soft representative of $F$ , that we still denote by $F$ .", "The inductive system $q_{\\varepsilon ,*}F$ ($\\varepsilon \\rightarrow 0$ ) has limit $i^{-1}F$ and all morphisms of this system are quasi-isomorphisms.", "Hence the second right morphism is a quasi-isomorphism.", "Remark 2.12 A similar argument gives an isomorphism $i^!F\\xrightarrow{}Rq_{\\varepsilon ,!", "}F$ , by using [4].", "For the left morphism, we take a $q$ -soft representative of $F^*$ that we still denote by $F^*$ .", "For $\\varepsilon _-<\\varepsilon <\\varepsilon _+<\\varepsilon _o$ , we denote by $B_{\\varepsilon _-,\\varepsilon _+}$ the open set $B_{\\varepsilon _+}\\setminus \\overline{B}_{\\varepsilon _-}$ and by $q_{\\varepsilon _-,\\varepsilon _+}$ the corresponding projection.", "We have $q_{\\partial \\varepsilon ,*}F^*=\\varinjlim _{|\\varepsilon _+-\\varepsilon _-|\\rightarrow 0}q_{\\varepsilon _-,\\varepsilon _+,*}F^*$ .", "On the other hand, the morphisms of this inductive system are all quasi-isomorphisms, according to [4].", "Fixing $\\varepsilon ^{\\prime }\\in (\\varepsilon ,\\varepsilon _o)$ we find a quasi-isomorphism $q_{\\varepsilon ^{\\prime },*}F^*\\rightarrow q_{\\partial \\varepsilon ,*}F^*$ .", "On the other hand, from the first part we have $q_{\\varepsilon ^{\\prime },*}F^*\\xrightarrow{}q_{\\overline{\\varepsilon },*}F^*$ , hence the result." ], [ "$S$ -coherent local systems and {{formula:29b9757a-a9a3-4f09-b56a-6870c06a5a72}} -{{formula:97c12bfe-ddb8-4d02-a9b8-9de08a13f2d3}} -constructible sheaves", "Notation 2.13 We shall denote by $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathrm {lc\\,coh}}(p_X^{-1}{O}_S)$ the full triangulated subcategory of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {lc}(p_X^{-1}{O}_S)$ whose objects satisfy, locally on $X$ , $F\\simeq p_X^{-1}G$ with $G\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({O}_S))$ .", "Equivalently, for each $x\\in X$ , $F_{|\\lbrace x\\rbrace \\times S}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({O}_S)$ (see the remarks before Lemma REF ).", "Definition 2.14 Given $F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ , we say that $F$ is $\\mathbb {R}$ -constructible if, for some $\\mu $ -stratification of $X$ , $X=\\bigsqcup _\\alpha X_\\alpha $ , for all $j\\in \\mathbb {Z}$ , ${H}^j(F)|_{X_\\alpha \\times S}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {lc\\,coh}(p_{X_\\alpha }^{-1}{O}_S)$ .", "This condition characterizes a full triangulated subcategory of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_{S})$ which we denote by $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_{S})$ .", "Similarly to Proposition REF we have: Proposition 2.15 Let $F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_{X}^{-1}{O}_{S})$ and let $s_o\\in S$ .", "Then $Li_{s_o}^*(F)\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(X)$ .", "Remark 2.16 An object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ is in $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ if and only if, for any $x\\in X$ , $F_{|\\lbrace x\\rbrace \\times S}$ belongs to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({O}_S)$ .", "A straightforward adaptation of [4] gives: Proposition 2.17 Let $f: Y\\rightarrow X$ be a a morphism of manifolds and let $F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_Y^{-1}{O}_{S})$ .", "Assume that $f_S$ is proper on $\\operatorname{Supp}(F)$ .", "Then $Rf_{S,*}F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_{S}).$ We can also characterize $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ as in Corollary REF .", "Corollary 2.18 An object $F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}(p_X^{-1}{O}_S)$ is in $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ if and only if, for some subanalytic Whitney stratification $(X_\\alpha )$ of $X$ , the complexes $i_\\alpha ^!F$ belong to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathrm {lc\\,coh}}(p_\\alpha ^{-1}{O}_S)$ .", "Assume $F$ is in $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ .", "We need to prove the coherence of $i_\\alpha ^!F$ .", "We argue by induction as in Corollary REF , with the same notation.", "Since the question is local on $X_k$ , by the Whitney property of the stratification $(X_\\alpha )$ we can assume that $X_{k-1}=Z\\times Y_k$ and there exists a Whitney stratification $(Z_\\alpha )$ of $Z$ such that $X_\\alpha =Z_\\alpha \\times Y_k$ for each $\\alpha $ such that $X_\\alpha \\subset X_{k-1}$ (see e.g.", ").", "Proving that $i_k^!F$ is $p^{-1}{O}_S$ -coherent is equivalent to proving that $i_k^{-1}Rj_{k,*}j_k^{-1}F$ is so, since we already know that $i_k^{-1}F$ is so.", "According to Lemma REF , $i_k^{-1}Rj_{k,*}j_k^{-1}F$ is computed as $Rq_{\\partial \\varepsilon ,*}j_k^{-1}F$ , and since $q_{\\partial \\varepsilon }$ is proper, we can apply Proposition REF to get the coherence.", "Conversely, Corollary REF already implies that $F$ is an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ .", "We argue then as above: since we know by assumption that $i_k^!F$ is coherent, it suffices to prove that $i_k^{-1}Rj_{k,*}j_k^{-1}F$ is so, and the previous argument applies." ], [ "$S$ -weakly {{formula:c7efdd92-62b0-4bb2-8d0a-d4d3b76b36f0}} S{{formula:62db9ccd-82b9-4256-bf89-160246e712ab}} -constructible sheaves", "Let now assume that $X$ is a complex analytic manifold.", "Definition 2.19 Let $F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{S})$ (resp.", "$F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_{S})$ ).", "We shall say that $F$ is $S$ -weakly $-constructible if $ SS(F)$ is $ *$-conic.", "The corresponding categories are denoted by $ Dbw--c(pX-1S)$ (resp.~$ FDbw--c(pX-1OS)$).$ If $F$ belongs to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}{-c}}(p_X^{-1}{O}_{S})$ , we say that $F$ is $S$ -$-constructible if $ FDbR-c(pX-1OS)$, and we denote by $ Db-c(pX-1OS)$ the corresponding category, which is full triangulated sub-category of $ Db(pX-1OS)$.$ The following properties are obtained in a straightforward way, by using [4] in a way similar to [4].", "Properties 2.20 An object $F$ of $\\operatorname{\\mathsf {D}}^\\mathrm {b}(p_X^{-1}{O}_{S})$ belongs to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}{-c}}(p_X^{-1}{O}_{S})$ if and only if it belongs to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}{-c}}(p_X^{-1}{S})$ .", "Remark REF applies to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}{-c}}(p_X^{-1}{O}_S)$ and $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p_X^{-1}{O}_S)$ .", "Proposition REF applies to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}{-c}}$ .", "Propositions REF , REF , and Corollary REF apply to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p_X^{-1}{O}_S)$ .", "Corollary REF applies to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}{-c}}$ , $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}$ and $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}$ .", "Duality According to the syzygy theorem for the regular local ring ${O}_{S,s}$ (for any $s\\in S$ ) and e.g.", "[5] (for the opposite category), any object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({O}_S)$ is locally quasi-isomorphic to a bounded complex of locally free ${O}_S$ -modules of finite rank $L^{\\scriptscriptstyle \\bullet }$ .", "As a consequence, the local duality functor $D: \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({O}_S)\\rightarrow \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({O}_S),\\quad D({F}):=R{H}\\!om_{{O}_S}({F},{O}_S)$ is seen to be an involution, i.e., the natural morphism $\\operatorname{id}\\rightarrow D\\circ D$ is an isomorphism.", "However, the standard t-structure $\\big (\\operatorname{\\mathsf {D}}^{\\mathrm {b},\\leqslant 0}_\\mathrm {coh}({O}_S),\\operatorname{\\mathsf {D}}^{\\mathrm {b},\\geqslant 0}_\\mathrm {coh}({O}_S)\\big )$ defined by ${H}^jG=0$ for $j>0$ (resp.", "for $j<0$ ) is not interchanged by duality when $\\dim S\\geqslant 1$ (see e.g., [3] in the algebraic setting).", "Nevertheless, we have: Lemma 2.21 Let $G$ be an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({O}_S)$ .", "Assume that $DG$ belongs to $\\operatorname{\\mathsf {D}}^{\\mathrm {b},\\leqslant 0}_\\mathrm {coh}({O}_S)$ .", "Then $G$ belongs to $\\operatorname{\\mathsf {D}}^{\\mathrm {b},\\geqslant 0}_\\mathrm {coh}({O}_S)$ .", "Setting $G^{\\prime }=DG$ , the biduality isomorphism makes it equivalent to proving that $DG^{\\prime }$ belongs to $\\operatorname{\\mathsf {D}}^{\\mathrm {b},\\geqslant 0}_\\mathrm {coh}({O}_S)$ .", "The question is local on $S$ and we may therefore replace $G^{\\prime }$ with a bounded complex $L^{\\scriptscriptstyle \\bullet }$ as above.", "Moreover, $L^{\\scriptscriptstyle \\bullet }$ is quasi-isomorphic to such a bounded complex, still denoted by $L^{\\scriptscriptstyle \\bullet }$ , such that $L^k=0$ for $k>0$ .", "Indeed, note first that the kernel $K$ of a surjective morphism of locally free ${O}_S$ -modules of finite rank is also locally free of finite rank (being ${O}_S$ -coherent and having all its germs $K_s$ free over ${O}_{S,s}$ , because they are projective and ${O}_{S,s}$ is a regular local ring).", "By assumption, we have ${H}^j(L^{\\scriptscriptstyle \\bullet })=0$ for $j>0$ .", "Let $k>0$ be such that $L^k\\ne 0$ and $L^\\ell =0$ for $\\ell >k$ , and let $L^{\\prime k-1}=\\ker [L^{k-1}\\rightarrow L^k]$ .", "Then $L^{\\scriptscriptstyle \\bullet }$ is quasi-isomorphic to $L^{\\prime {\\scriptscriptstyle \\bullet }}$ defined by $L^{\\prime j}=L^j$ for $j<k-1$ and $L^{\\prime j}=0$ for $j\\geqslant k$ .", "We conclude by induction on $k$ .", "Now it is clear that $DG^{\\prime }\\simeq DL^{\\scriptscriptstyle \\bullet }$ is a bounded complex having terms in nonnegative degrees at most, and thus is an object of $\\operatorname{\\mathsf {D}}^{\\mathrm {b},\\geqslant 0}_\\mathrm {coh}({O}_S)$ .", "Remark 2.22 Let $G$ be an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({O}_S)$ .", "Assume that $G$ and $DG$ belong to $\\operatorname{\\mathsf {D}}^{\\mathrm {b},\\leqslant 0}_\\mathrm {coh}({O}_S)$ .", "Then $G$ and $DG$ are ${O}_S$ -coherent sheaves, hence $G$ and $DG$ are ${O}_S$ -locally free.", "We now set $\\omega _{X,S}=p_X^{-1}{O}_S[2\\dim X]=p_X^!", "{O}_S$ .", "Proposition 2.23 The functor $D:\\operatorname{\\mathsf {D}}^\\mathrm {b}(p_X^{-1}{O}_S)\\rightarrow \\operatorname{\\mathsf {D}}^+(p_X^{-1}{O}_S)$ defined by $DF=R{H}\\!om_{p_X^{-1}{O}_S}(F,\\omega _{X,S})$ induces an involution $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)\\rightarrow \\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ and $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p_X^{-1}{O}_S)\\rightarrow \\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p_X^{-1}{O}_S)$ .", "We will also set $D^{\\prime }F=R{H}\\!om_{p_X^{-1}{O}_S}(F,p_X^{-1}{O}_S)$ .", "Let us first show that, for $F$ in $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_{S})$ , the dual $DF$ also belongs to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_{S})$ .", "let $(X_\\alpha )$ be a $\\mu $ -stratification adapted to $F$ .", "According to Corollary REF , it is enough to show that $i_\\alpha ^!DF$ has locally constant cohomology for each $\\alpha $ .", "One can use [4] in our setting and get $i_\\alpha ^!DF=R{H}\\!om_{p_\\alpha ^{-1}{O}_S}(i_\\alpha ^{-1}F,\\omega _{X_\\alpha ,S}).$ Locally on $X_\\alpha $ , $i_\\alpha ^{-1}F=p_\\alpha ^{-1}G$ for some $G$ in $\\operatorname{\\mathsf {D}}^\\mathrm {b}(S)$ or $\\operatorname{\\mathsf {D}}^\\mathrm {b}({O}_S)$ .", "Then, locally on $X_\\alpha $ , $i_\\alpha ^!DF\\simeq R{H}\\!om_{p_\\alpha ^{-1}{O}_S}(p_\\alpha ^{-1}G,p_\\alpha ^!", "{O}_S)&=p_\\alpha ^!R{H}\\!om_{{O}_S}(G,{O}_S)\\\\&=p_\\alpha ^{-1}(DG)[2\\dim X_\\alpha ].$ The proof for $F$ in $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}{-c}}(p_X^{-1}{O}_{S})$ is similar.", "Moreover, by using Corollary REF instead of Corollary REF one shows that $D$ sends $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_{S})$ to itself and, according to Properties REF (REF ), $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p_X^{-1}{O}_{S})$ to itself.", "Let us prove the involution property.", "We have a natural morphism of functors $\\operatorname{id}\\rightarrow DD$ .", "It is enough to prove the isomorphism property after applying $Li^*_{s_o}$ for each $s_o\\in S$ , according to Proposition REF .", "On the other hand, Proposition REF implies that $Li^*_{s_o}$ commutes with $D$ , so we are reduced to applying the involution property on $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{{-c}}}(X)$ , according to the ${{-c}}$ -analogue of Proposition REF , which is known to be true (see e.g. [4]).", "Remark 2.24 By using the biduality isomorphism and the isomorphism $i_x^!DF\\simeq Di_x^{-1}F$ for $F$ in $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ or $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p_X^{-1}{O}_S)$ , where $i_x:\\lbrace x\\rbrace \\times S\\hookrightarrow X\\times S$ denotes the inclusion, we find a functorial isomorphism $i_x^{-1}DF\\simeq Di_x^!F$ .", "Perversity We will now restrict to the case of $S$ -$-constructible complexes, which is the only case which will be of interest for us, although one could consider the case of $ S$-$ R$-constructible complexes as in \\cite [§10.2]{K-S90}.$ We define the category $\\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\leqslant 0}_{{-c}}(p_X^{-1}{O}_S)$ as the full subcategory of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p_X^{-1}{O}_S)$ whose objects are the $S$ -$-constructible bounded complexes~$ F$ such that, for some adapted $$-stratification $ (X)$ ($ ix$ is as above),\\begin{equation}\\forall \\,\\alpha ,\\;\\forall \\,x\\in X_\\alpha ,\\;\\forall \\,j>-\\dim X_\\alpha ,\\quad {H}^ji_x^{-1}F=0.\\end{equation}Similarly, \\qquad \\mathrm {(Supp)}$ pD0-c(pX-1OS)$ consists of objects $ F$ such that\\begin{equation}\\forall \\,\\alpha ,\\;\\forall \\,x\\in X_\\alpha ,\\;\\forall \\,j<\\dim X_\\alpha ,\\quad {H}^ji_x^!F=0.\\end{equation}$ In the preceding situation in view of Corollary REF we have, similarly to [4]: Lemma 2.25 $F\\in \\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\leqslant 0}_{{-c}}(p_X^{-1}{O}_S)$ if and only if for any $\\alpha $ and $j>-\\dim (X_\\alpha )$ , ${H}^j(i^{-1}_\\alpha F)=0.$ $F\\in \\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\geqslant 0}_{{-c}}(p_X^{-1}{O}_S)$ if and only if for any $\\alpha $ and $j<-\\dim (X_\\alpha )$ , ${H}^j(i^{!", "}_\\alpha F)=0.$ Namely, if $F\\in \\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\leqslant 0}_{{-c}}(p_X^{-1}{O}_S)$ and $Z$ is a closed analytic subset of $X$ such that $\\dim Z=k$ , then $i_{Z\\times S}^{-1} F$ is concentrated in degrees $\\leqslant -k$ , and if $F^{\\prime }\\in \\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\geqslant 0}_{{-c}}(p_X^{-1}{O}_S)$ , then $i^!_{Z\\times S} F^{\\prime }$ is concentrated in degrees $\\geqslant -k$ .", "We have the following variant of [4]: Proposition 2.26 Let $F$ be an object of $\\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\leqslant 0}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ and $F^{\\prime }$ an object of $\\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\geqslant 0}_{\\textup {w-}\\mathbb {R}\\textup {-c}}(p_X^{-1}{O}_S)$ .", "Then ${H}^jR{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })=0,\\quad \\text{for } j<0.$ Let $(X_\\alpha )$ be a $\\mu $ -stratification of $X$ adapted to $F$ and $F^{\\prime }$ .", "By assumption, for each $\\alpha $ , $i_\\alpha ^{-1}{H}^jF={H}^ji_\\alpha ^{-1}F=0$ for $j>-\\dim X_\\alpha $ .", "Similarly, ${H}^ji_\\alpha ^!F^{\\prime }=0$ for $j<-\\dim X_\\alpha $ .", "Let $X_\\alpha $ be a stratum of maximal dimension such that $i_\\alpha ^{-1}{H}^jR{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })\\ne 0\\quad \\text{for some $j<0$}.$ Let $V$ be an open neighbourhood of $X_\\alpha $ in $X$ such that $V\\setminus X_\\alpha $ intersects only strata of dimension $>\\dim X_\\alpha $ , and let $j_\\alpha :(V\\setminus X_\\alpha )\\times S\\hookrightarrow V\\times S$ be the inclusion.", "Then the complex $i_\\alpha ^{-1}Rj_{\\alpha ,*}j_\\alpha ^{-1}R{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })$ has nonzero cohomology in nonnegative degrees only: indeed, by the definition of $X_\\alpha $ , this property holds for $j_\\alpha ^{-1}R{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })$ , hence it holds for $Rj_{\\alpha ,*}j_\\alpha ^{-1}R{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })$ , and then clearly for the complex $i_\\alpha ^{-1}Rj_{\\alpha ,*}j_\\alpha ^{-1}R{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })$ .", "From the distinguished triangle $i_\\alpha ^!R{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })\\rightarrow i_\\alpha ^{-1}R{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })\\\\\\rightarrow i_\\alpha ^{-1}Rj_{\\alpha ,*}j_\\alpha ^{-1}R{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })\\xrightarrow{}$ we conclude that ${H}^ji_\\alpha ^!R{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })\\rightarrow {H}^j i_\\alpha ^{-1}R{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })= i_\\alpha ^{-1}{H}^jR{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })$ is an isomorphism for all $j<0$ .", "Therefore, we obtain, for this stratum $X_\\alpha $ and for any $j<0$ , $i_\\alpha ^{-1}{H}^jR{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })&\\simeq {H}^ji_\\alpha ^!R{H}\\!om_{p^{-1}_X{O}_S}(F,F^{\\prime })\\\\&\\simeq {H}^jR{H}\\!om_{p^{-1}_X{O}_S}(i_\\alpha ^{-1}F,i_\\alpha ^!F^{\\prime }).$ Since $i_\\alpha ^{-1}F$ has nonzero cohomology in degrees $\\leqslant -\\dim X_\\alpha $ at most and $i_\\alpha ^!F^{\\prime }$ in degrees $\\geqslant -\\dim X_\\alpha $ at most, ${H}^jR{H}\\!om_{p^{-1}_X{O}_S}(i_\\alpha ^{-1}F,i_\\alpha ^!F^{\\prime })=0$ for $j<0$ , a contradiction with the definition of $X_\\alpha $ .", "Theorem 2.27 $\\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\leqslant 0}_{{-c}}(p_X^{-1}{O}_S)$ and $\\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\geqslant 0}_{{-c}}(p_X^{-1}{O}_S)$ form a t-structure of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p_X^{-1}{O}_S)$ , whose heart is denoted by $\\mathrm {Perv}(p_X^{-1}{O}_S)$ .", "We have to prove: $\\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\leqslant 0}_{{-c}}\\subset \\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\leqslant 1}_{{-c}}$ and $\\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\geqslant 0}_{{-c}}\\supset \\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\geqslant 1}_{{-c}}$ .", "For $F\\in \\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\leqslant 0}_{{-c}}(p_X^{-1}{O}_S)$ and $F^{\\prime }\\in \\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\geqslant 1}_{{-c}}(p_X^{-1}{O}_S)$ , $\\operatorname{Hom}_{\\operatorname{\\mathsf {D}}^\\mathrm {b}(p_X^{-1}{O}_S)}(F,F^{\\prime })=0.$ For any $F\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p_X^{-1}{O}_S)$ there exist $F^{\\prime }\\in \\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\leqslant 0}_{{-c}}(p_X^{-1}{O}_S)$ and $F^{\\prime \\prime }\\in \\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\geqslant 1}_{{-c}}(p_X^{-1}{O}_S)$ , giving rise to a distinguished triangle $F^{\\prime }\\rightarrow F\\rightarrow F^{\\prime \\prime }\\overset{+1}{\\rightarrow }$ .", "Then, following the line of the proof of [4], we observe that (REF ) is obvious and (REF ) follows from Proposition REF .", "Now, (REF ) is deduced by mimicking stepwise the proof of (c) in [4].", "According to the preliminary remarks before Lemma REF , one cannot expect that the previous t-structure is interchanged by duality when $\\dim S\\geqslant 1$ .", "However we have: Proposition 2.28 Let $F$ be an object of $\\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\leqslant 0}_{{-c}}(p_X^{-1}{O}_S)$ such that $DF$ also belongs to $\\operatorname{{}^\\mathrm {p}\\mathsf {D}}^{\\leqslant 0}_{{-c}}(p_X^{-1}{O}_S)$ .", "Then $F$ and $DF$ are objects of $\\mathrm {Perv}(p_X^{-1}{O}_S)$ .", "Let us fix $x\\!\\in X_\\alpha $ .", "We have $i_x^!F\\simeq D(i_x^{-1}DF)$ , as already observed in Remark REF .", "By assumption $G:=i_x^{-1}DF$ belongs to $\\operatorname{\\mathsf {D}}^{\\mathrm {b},\\leqslant -\\dim X_\\alpha }_\\mathrm {coh}({O}_S)$ , and Lemma REF suitably shifted and applied to $DG$ implies that $DG$ belongs to $\\operatorname{\\mathsf {D}}^{\\mathrm {b},\\geqslant \\dim X_\\alpha }_\\mathrm {coh}({O}_S)$ , which is the cosupport condition (Cosupp) for $F$ .", "Assume $F\\in \\mathrm {Perv}(p_X^{-1}{O}_S)$ .", "The description of the dual standard t-structure on $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({O}_S)$ given in [3] supplies the following refinement to () and () when $DF$ is also perverse.", "Corollary 2.29 Let $F\\in \\mathrm {Perv}(p_X^{-1}{O}_S)$ and assume that $DF\\in \\mathrm {Perv}(p_X^{-1}{O}_S)$ .", "Let $(X_\\alpha )$ be a stratification adapted to $F$ .", "Then for each $\\alpha $ , each $x\\in X_\\alpha $ and each closed analytic subset $Z\\subset S$ , we have ${H}^k(i^!_{Z\\times \\lbrace x\\rbrace }F)=0,\\quad \\forall \\,k<\\operatorname{codim}_SZ+\\dim X_\\alpha .\\qquad \\mathrm {(Cosupp+)}$ (The perversity of $F$ only gives the previous property when $Z=S$ .)", "The de Rham complex of a holonomic ${D}_{X\\times S/S}$ -module In what follows $X$ and $S$ denote complex manifolds and we set $n=\\dim X$ , $\\ell =\\dim S$ .", "We shall keep the notation of the preceding section.", "Let $\\pi :T^*(X\\times S)\\rightarrow T^*X\\times S$ denote the projection and let ${D}_{X\\times S/S}$ denote the subsheaf of ${D}_{X\\times S}$ of relative differential operators with respect to $p_X$ (see [17]).", "Recall that $p^{-1}_X{O}_{S}$ is contained in the center of ${D}_{X\\times S/S}$ .", "With the same proof as for Proposition REF we obtain: Proposition 3.1 Let $s_o\\in S$ be given.", "Let ${M}$ and ${N}$ be objects of $\\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S})$ .", "Then, there is a well-defined natural morphism $Li^*_{s_o}(R{H}\\!om_{{D}_{X\\times S/S}}({M}, {N}))\\rightarrow R{H}\\!om_{i^*_{s_o}({D}_{X\\times S/S})}(Li^*_{s_o}({M}), Li^*_{s_o}({N}))$ which is an isomorphism in $\\operatorname{\\mathsf {D}}^\\mathrm {b}(X)$ .", "Duality for coherent ${D}_{X\\times S/S}$ -modules We refer for instance to [2] for the coherence properties of the ring ${D}_{X\\times S/S}$ .", "The classical methods used in the absolute case, i.e, for coherent ${D}_X$ -objects (see for instance [7], [8]) apply here: Proposition 3.2 Let ${M}$ be a coherent ${D}_{X\\times S/S}$ -module.", "Then ${M}$ locally admits a resolution of length at most $2n+\\ell $ by free ${D}_{X\\times S/S}$ -modules of finite rank.", "Proposition REF and [5] (for the opposite category) imply: Corollary 3.3 Let ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ .", "Let us assume that ${M}$ is concentrated in degrees $[a,b]$ .", "Then, in a neighborhhod of each $(x,z)\\in X\\times S$ , there exist a complex ${L}^{\\scriptscriptstyle \\bullet }$ of free ${D}_{X\\times S/S}$ -modules of finite rank concentrated in degrees $[a-2n-\\ell , b]$ and a quasi-isomorphism $ {L}^{\\scriptscriptstyle \\bullet }\\rightarrow {M}$ .", "We set $\\Omega _{X\\times S/S}=\\Omega ^{n}_{X\\times S/S}$ , where $\\Omega ^{n}_{X\\times S/S}$ denotes the sheaf of relative differential forms of degree $n=\\dim X$ .", "Definition 3.4 The duality functor $D(\\cdot ): \\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S}) \\rightarrow \\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S})$ is defined as: ${M}\\mapsto D{M}=R{H}\\!om_{{D}_{X\\times S/S}}({M}, {D}_{X\\times S/S}\\otimes _{{O}_{X\\times S}}\\Omega _{X\\times S/S}^{\\otimes ^{-1}})[n].$ We also set $D^{\\prime }{M}:=R{H}om_{{D}_{X\\times S/S}}({M}, {D}_{X\\times S/S})\\in \\operatorname{\\mathsf {D}}^{\\mathrm {b}}({D}_{X\\times S/S}^\\mathrm {opp})$ .", "By Proposition REF , ${D}_{X\\times S/S}$ has finite cohomological dimension, so [2] gives a natural morphism in $\\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S})$ : ${M}\\rightarrow D^{\\prime }D^{\\prime }{M}\\simeq DD{M}.$ Moreover, in view of Corollary REF , if ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ , then $D^{\\prime }{M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S}^\\mathrm {opp}).$ Indeed, we may choose a local free finite resolution ${L}^{\\scriptscriptstyle \\bullet }$ of ${M}$ , so that $D^{\\prime }{M}$ is quasi isomorphic to the transposed complex $({L}^{\\scriptscriptstyle \\bullet })^t$ whose entries are free.", "By the same argument we deduce that (REF ) is an isomorphism whenever ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ .", "Again by Proposition REF , ${D}_{X\\times S/S}$ has finite flat dimension so we are in conditions to apply [2]: given ${M}, {N}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S})$ there is a natural morphism: $D^{\\prime }{M}\\overset{L}{\\otimes }_{{D}_{X\\times S/S}}{N}\\rightarrow R{H}\\!om_{{D}_{X\\times S/S}}({M},{N})$ which an isomorphism provided that ${M}$ or ${N}$ belong to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ .", "When ${M}, {N}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ , composing (REF ) with the biduality isomorphism (REF ) gives a natural isomorphism $R{H}\\!om_{{D}_{X\\times S/S}}({M},{N})\\simeq R{H}\\!om_{{D}_{X\\times S/S}}(D{N}, D{M}).$ Characteristic variety Recall (see [16]) that the characteristic variety $\\operatorname{Char}{M}$ of a coherent ${D}_{X\\times S/S}$ -module ${M}$ is the support in $T^*X\\times S$ of its graded module with respect to any (local) good filtration.", "One has (see [16]) $\\begin{split}\\operatorname{Char}({D}_{X\\times S}\\otimes _{{D}_{X\\times S/S}} {M})&=\\pi ^{-1}\\operatorname{Char}{M},\\\\\\operatorname{Char}{M}&=\\pi \\big (\\operatorname{Char}({D}_{X\\times S}\\otimes _{{D}_{X\\times S/S}} {M})\\big ).\\end{split}$ One may as well define the characteristic variety of an object ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ as the union of the characteristic varieties of its cohomology modules.", "By the flatness of ${D}_{X\\times S}$ over ${D}_{X\\times S/S}$ , (REF ) holds for any object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ .", "Proposition 3.5 ([17]) For ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ we have $\\operatorname{Char}({M})=\\operatorname{Char}(D{M}).$ The de Rham and solution complexes For an object ${M}$ of $\\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S})$ we define the functors $\\operatorname{DR}{M}&:=R{H}\\!om_{{D}_{X\\times S/S}}({O}_{X\\times S},{M}),\\\\\\operatorname{Sol}{M}&:=R{H}\\!om_{{D}_{X\\times S/S}}({M},{O}_{X\\times S})$ which take values in $\\operatorname{\\mathsf {D}}^{\\mathrm {b}}(p_X^{-1}{O}_S)$ .", "If ${M}$ is a ${D}_{X\\times S/S}$ -module, that is, a ${O}_{X\\times S}$ -module equipped with an integrable relative connection $\\nabla :{M}\\rightarrow \\Omega ^1_{X\\times S/S}\\otimes {M}$ , the object $\\operatorname{DR}{M}$ is represented by the complex $(\\Omega ^{\\scriptscriptstyle \\bullet }_{X\\times S/S}\\otimes _{{O}_{X\\times S}}{M},\\nabla )$ .", "Noting that $R{H}\\!om_{{D}_{X\\times S/S}}({O}_{X\\times S},{D}_{X\\times S/S})\\simeq \\Omega _{X\\times S/S}[-\\dim X]$ we get $D{O}_{X\\times S}\\simeq {O}_{X\\times S}.$ For ${N}={O}_{X\\times S}$ , (REF ) implies a natural isomorphism, for ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ : $\\operatorname{Sol}{M}\\simeq \\operatorname{DR}D{M}.$ Holonomic ${D}_{X\\times S/S}$ -modules Let ${M}$ be a coherent ${D}_{X\\times S/S}$ -module.", "We say that it is holonomic if its characteristic variety $\\operatorname{Char}{M}\\subset T^*X\\times S$ is contained in $\\Lambda \\times S$ for some closed conic Lagrangian complex analytic subset of $T^*X$ .", "We will say that a complex $\\mu $ -stratification $(X_\\alpha )$ is adapted to ${M}$ if $\\Lambda \\subset \\bigcup _\\alpha T^*_{X_\\alpha }X$ .", "Similar definitions hold for objects of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ .", "An object ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ is said to be holonomic if its cohomology modules are holonomic.", "We denote the full triangulated category of holonomic complexes by $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ .", "Corollary 3.6 (of Prop.", "REF ) If ${M}$ is an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ , then so is $D{M}$ .", "Theorem 3.7 Let ${M}$ be an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ .", "Then $\\operatorname{DR}({M})$ and $\\operatorname{Sol}{M}$ belong to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p^{-1}_X{O}_S)$ .", "Firstly, it follows [4], that $\\operatorname{Sol}({M})$ and $\\operatorname{DR}({M})$ have their micro-support contained in $\\Lambda \\times T^*S$ (see [17]) and, according to Proposition REF , these complexes are objects of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}{-c}}(p^{-1}_X{O}_S)$ .", "Let $x\\in X$ .", "In order to prove that $i_x^{-1}\\operatorname{DR}{M}$ has ${O}_S$ -coherent cohomology, we can assume that $x$ is a stratum of a stratification adapted to $\\operatorname{DR}{M}$ and we use Lemma REF to get $i_x^{-1}\\operatorname{DR}{M}\\simeq Rp_{\\overline{\\varepsilon },*}({\\overline{B}_\\varepsilon \\times S}\\otimes _{M})$ for $\\varepsilon $ small enough, where $\\overline{B}_\\varepsilon $ is a closed ball of radius $\\varepsilon $ centered at $x$ .", "One then remarks that $({\\overline{B}_\\varepsilon \\times S},{M})$ forms a relative elliptic pair in the sense of [17], and Proposition 4.1 of loc. cit.", "gives the desired coherence.", "The statement for $\\operatorname{Sol}{M}$ is proved similarly.", "Lemma 3.8 (see [13]) For ${M}$ in $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ with adapted stratification $(X_\\alpha )$ and for any $s_o\\in S$ , $Li^*_{s_o}{M}$ is ${D}_X$ -holonomic and $(X_\\alpha )$ is adapted to it.", "Corollary 3.9 For ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ , there is a natural isomorphism $D^{\\prime }\\operatorname{Sol}{M}\\simeq \\operatorname{DR}{M}$ .", "We consider the canonical pairing $\\operatorname{DR}{M}\\overset{L}{\\otimes }_{p^{-1}_X{O}_S}\\operatorname{Sol}{M}\\rightarrow p_X^{-1}{O}_S$ which gives a natural morphism $\\operatorname{DR}{M}\\rightarrow D^{\\prime }\\operatorname{Sol}{M}$ in $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{{-c}}}(p_X^{-1}{O}_S)$ .", "We have for each $s_o\\in S$ , by Proposition REF $Li^*_{s_o}(\\operatorname{DR}{M})&\\simeq \\operatorname{DR}Li^*_{s_o}({M}),\\\\Li^*_{s_o}(\\operatorname{Sol}{M})&\\simeq \\operatorname{Sol}Li^*_{s_o}({M}) .$ Since $Li^*_{s_o}({M})\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_X)$ by Lemma REF , we have $\\operatorname{DR}Li^*_{s_o}({M})\\simeq D^{\\prime }\\operatorname{Sol}Li^*_{s_o}({M}),$ so by Proposition REF and Proposition REF $D^{\\prime }\\operatorname{Sol}Li^*_{s_o}({M})\\simeq D^{\\prime }Li^*_{s_o}(\\operatorname{Sol}{M})\\simeq Li^*_{s_o}(D^{\\prime }\\operatorname{Sol}{M}).$ The assertion then follows by Proposition REF .", "In the following proposition, the main argument is that of strictness, which is essential.", "We will set $\\operatorname{{}^\\mathrm {p}DR}{M}:=\\operatorname{DR}{M}[\\dim X]$ and $\\operatorname{{}^\\mathrm {p}Sol}{M}=\\operatorname{Sol}{M}[\\dim X]$ .", "Proposition 3.10 Let ${M}$ be a holonomic ${D}_{X\\times S/S}$ -module which is strict, i.e., which is $p^{-1}{O}_S$ -flat.", "Then $\\operatorname{{}^\\mathrm {p}DR}{M}$ satisfies the support condition () with respect to a $\\mu $ -stratification adapted to ${M}$ .", "We prove the result by induction on $\\dim S$ .", "Since it is local on $S$ , we consider a local coordinate $s$ on $S$ and we set $S^{\\prime }=\\lbrace s=0\\rbrace $ .", "The strictness property implies that we have an exact sequence $0\\rightarrow {M}\\xrightarrow{}{M}\\rightarrow i_{S^{\\prime }}^*{M}\\rightarrow 0,$ and $i_{S^{\\prime }}^*{M}$ is ${D}_{X\\times S^{\\prime }/S^{\\prime }}$ -holonomic and $p^{-1}{O}_{S^{\\prime }}$ -flat.", "We deduce an exact sequence of complexes $0\\rightarrow \\operatorname{{}^\\mathrm {p}DR}{M}\\xrightarrow{}\\operatorname{{}^\\mathrm {p}DR}{M}\\rightarrow \\operatorname{{}^\\mathrm {p}DR}i_{S^{\\prime }}^*{M}\\rightarrow 0$ .", "Let $X_\\alpha $ be a stratum of a $\\mu $ -stratification of $X$ adapted to ${M}$ (hence to $i_{S^{\\prime }}^*{M}$ , after Lemma REF ).", "For $x\\in X_\\alpha $ , let $k$ be the maximum of the indices $j$ such that ${H}^ji_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}\\ne 0$ .", "For any $S^{\\prime }$ as above, we have a long exact sequence $\\cdots \\rightarrow {H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}\\xrightarrow{}{H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}\\rightarrow {H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}i_{S^{\\prime }}^*{M}\\rightarrow 0.$ If $k>-\\dim X_\\alpha $ , we have ${H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}i_{S^{\\prime }}^*{M}=0$ , according to the support condition for $i_{S^{\\prime }}^*{M}$ (inductive assumption), since $(X_\\alpha )$ is adapted to it.", "Therefore, $s:{H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}\\rightarrow {H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}$ is onto.", "On the other hand, by Theorem REF , ${H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}$ is ${O}_S$ -coherent.", "Then Nakayama's lemma implies that ${H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}=0$ in some neighbourhood of $S^{\\prime }$ .", "Since $S^{\\prime }$ was arbitrary, this holds all over $S$ , hence the assertion.", "It is a direct consequence of the following.", "Theorem 3.11 Let ${M}$ be an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ and let $D{M}$ be the dual object.", "Then there is an isomorphism $\\operatorname{{}^\\mathrm {p}DR}D{M}\\simeq D\\operatorname{{}^\\mathrm {p}DR}{M}$ .", "Indeed, with the assumptions of Theorem REF , $D{M}$ is holonomic since ${M}$ is so (see Corollary REF ), and both ${M}$ and $D{M}$ are strict.", "Then both $\\operatorname{{}^\\mathrm {p}DR}{M}$ and $\\operatorname{{}^\\mathrm {p}DR}D{M}$ satisfy the support condition, according to Proposition REF .", "Hence, according to Theorem REF and Proposition REF , $\\operatorname{{}^\\mathrm {p}DR}{M}$ satisfies the cosupport condition.", "Similarly, $\\operatorname{{}^\\mathrm {p}Sol}{M}\\simeq D\\operatorname{{}^\\mathrm {p}DR}{M}$ and $D(\\operatorname{{}^\\mathrm {p}Sol}{M})\\simeq \\operatorname{{}^\\mathrm {p}DR}{M}$ both satisfy the support condition, hence $\\operatorname{Sol}{M}[\\dim X]$ is a perverse object.", "Combining (REF ) with [4] (with $f=\\operatorname{id}$ , ${A}={D}_{X\\times S/S}$ and ${B}=p_X^{-1}{O}_S$ ) entails, for any ${N}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ , a natural morphism $R{H}\\!om_{{D}_{X\\times S/S}}({N},{M})\\rightarrow R{H}\\!om_{p_X^{-1}{O}_S}(\\operatorname{DR}D{M}, \\operatorname{DR}D{N}).$ When ${N}={O}_{X\\times S}$ , we obtain a natural morphism $ \\operatorname{DR}{M}\\rightarrow D^{\\prime }\\operatorname{DR}D{M},\\quad \\text{that is,}\\quad \\operatorname{{}^\\mathrm {p}DR}{M}\\rightarrow D\\operatorname{{}^\\mathrm {p}DR}D{M}.$ Suppose now that ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ .", "Recall that $D{M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ , so $\\operatorname{{}^\\mathrm {p}DR}D{M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p^{-1}_X{O}_S)$ .", "Hence, by biduality, we get a morphism $D\\operatorname{{}^\\mathrm {p}DR}{M}\\leftarrow \\operatorname{{}^\\mathrm {p}DR}D{M}.$ On the other hand, since $Li^*_{s_o}({M})\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_X)$ for each $s_o\\in S$ , the morphisms above induce isomorphisms $Li^*_{s_o}(D\\operatorname{{}^\\mathrm {p}DR}{M})\\simeq \\operatorname{{}^\\mathrm {p}DR}DLi^*_{s_o}({M})$ according to Proposition REF and Proposition REF , where in the right hand side we consider the duality for holonomic ${D}_X$ -modules.", "Thus (REF ) is an isomorphism by Proposition REF and the local duality theorem for holonomic ${D}_X$ -modules (see [12] and the references given there).", "Example 3.12 Let $X$ be the open unit disc in $ with coordinate $ x$ and let~$ S$ be a connected open set of $ with coordinate $s$ .", "Let $\\varphi :S\\rightarrow be a non constant holomorphic function on $ S$ and consider the holonomic $ DXS/S$-module $ M=DXS/S/DXS/SP$, with $ P=xx-(s)$.", "It is easy to check that $ M$ has no $ OS$-torsion and admits the resolution $ 0DXS/SPDXS/SM0$, so that the dual module $ DM$ has a similar presentation and is also $ OS$-flat.", "The complex $ pSolM$ is represented by $ 0OXSPOXS0$ (terms in degrees $ -1$ and $ 0$).", "Consider the stratification $ X1=X{0}$ and $ X0={0}$ of $ X$.", "Then $ H-1pSolM|X1$ is a locally constant sheaf of free $ p-1XOS$-modules generated by a local determination of~$ x(s)$, and $ H0pSolM|X1=0$.", "On the other hand, $ H-1pSolM|X0=0$ and $ H0pSolM|X0$ is a skyscraper sheaf on $ X0S$ supported on $ {sS(s)Z}$.$ For each $x_0$ we have ${{i_{x_0}^{!}}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})\\\\\\simeq i^{-1}_{\\lbrace x_0\\rbrace \\times S}R{H}\\!om_{{D}_{X\\times S}}({D}_{X\\times S}\\otimes _{{D}_{X\\times S/S}}{M}, R\\Gamma _{\\lbrace x_0\\rbrace \\times S|X\\times S}{O}_{X\\times S})[\\dim X]\\\\\\simeq i^{-1}_{\\lbrace x_0\\rbrace \\times S}R{H}\\!om_{{D}_{X\\times S}}({D}_{X\\times S}\\otimes _{{D}_{X\\times S/S}}{M}, B_{\\lbrace x_0\\rbrace \\times S|X\\times S})$ where $B_{\\lbrace x_0\\rbrace \\times S|X\\times S}:={H}^1_{[\\lbrace x_0\\rbrace \\times S]}({O}_{X\\times S})$ denotes the sheaf of holomorphic hyperfunctions (of finite order) along $x=x_0$ (cf. [15]).", "The second isomorphism follows from the fact that ${D}_{X\\times S}\\otimes _{{D}_{X\\times S/S}}{M}$ is regular specializable along the submanifold $x=x_0$ (cf. [6]).", "Recall that the sheaves $B_{\\lbrace x_0\\rbrace \\times S|X\\times S}$ are flat over $p_X^{-1}{O}_S$ because locally they are inductive limits of free $p_X^{-1}{O}_S$ -modules of finite rank.", "Since ${i_{x_0}^{!}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})$ is quasi isomorphic to the complex $0\\rightarrow B_{\\lbrace x_0\\rbrace \\times S|X\\times S}|_{\\lbrace x_0\\rbrace \\times S}\\xrightarrow{}B_{\\lbrace x_0\\rbrace \\times S|X\\times S}|_{\\lbrace x_0\\rbrace \\times S}\\rightarrow 0$ it follows that the flat dimension over ${O}_S$ of ${{i_{x_0}^{!}}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})$ in the sense of [3] is $\\leqslant 0$ for any $x_0$ .", "Moreover, ${H}^0{i_{x_0}^{!}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})=0$ and, if $x_0\\ne 0$ , ${H}^1{i_{x_0}^{!}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})$ is a locally free ${O}_S$ -module of rank 1.", "Hence the flat dimension of ${i_{x_0}^{!}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})$ is $\\leqslant 1$ .", "This shows explicitly that $\\operatorname{{}^\\mathrm {p}Sol}{M}$ satisfies the condition (REF ) of Corollary REF .", "Application to mixed twistor ${D}$ -modules Let ${R}_{X\\times be the sheaf on X\\times of z-differential operators, locally generated by {O}_{X\\times and the z-vector fields z\\partial _{x_i} in local coordinates (x_1,\\dots ,x_n) on~X.", "When restricted to X\\times *, the sheaf {R}_{X\\times *} is isomorphic to {D}_{X\\times */*}.", "}A mixed twistor {D}-module on X (see \\cite {Mochizuki11}) is a triple {T}=({M}^{\\prime },{M}^{\\prime \\prime },C), where {M}^{\\prime },{M}^{\\prime \\prime } are holonomic {R}_{X\\times -modules and C is a certain pairing with values in distributions, that we will not need to make precise here.", "Such a triple is subject to various conditions.", "We say that a {D}_{X\\times */*}-module {M} underlies a mixed twistor {D}-module {T} if {M} is the restriction to X\\times * of~{M}^{\\prime } or {M}^{\\prime \\prime }.", "}Theorem \\ref {th:main} is now a direct consequence of the following properties of mixed twistor {D}-modules, since they imply that {M} satisfies the assumptions of Theorem \\ref {th:main2}.", "If {M} underlies a mixed twistor {D}-module, then\\begin{itemize}\\item there exists a locally finite filtration W_{\\scriptscriptstyle \\bullet }{M} indexed by \\mathbb {Z} by {R}_{X\\times -submodules such that each graded module underlies a pure polarizable twistor {D}-module; then each \\mathrm {gr}_\\ell ^W{M} is strict and holonomic (see \\cite [Prop.\\,4.1.3]{Sabbah05} and \\cite [§17.1.1]{Mochizuki08}), and thus so is {M};\\item the dual of {M} as a {R}_{X\\times *}-module also underlies a mixed twistor {D}-module, hence is also strict holonomic (see \\cite [Th.\\,12.9]{Mochizuki11}); using the isomorphism {R}_{X\\times *}\\simeq {D}_{X\\times */*}, we see that the dual D{M} as a {D}_{X\\times */*}-module is strict and holonomic.\\Box }\\end{itemize}}\\begin{thebibliography}{10}\\end{thebibliography}\\bibitem {G-M88}M.~Goresky and {R.D}.~MacPherson, \\emph {{Stratified Morse theory}},Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge 3 Band 14,Springer-Verlag, Berlin, Heidelberg, New York, 1988.$ M. Kashiwara, On the maximally overdetermined systems of differential equations, Publ.", "RIMS, Kyoto Univ.", "10 (1975), 563–579.", ", $D$ -modules and microlocal calculus, Translations of Mathematical Monographs, vol.", "217, American Mathematical Society, Providence, RI, 2003. , t-Structure on the derived categories of ${D}$ -modules and ${O}$ -modules, Moscow Math. J.", "4 (2004), no.", "4, 847–868.", "M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren der mathematischen Wissenschaften, vol.", "292, Springer-Verlag, 1990. , Categories and sheaves, Grundlehren der mathematischen Wissenschaften, vol.", "332, Springer-Verlag, 2006.", "Y. Laurent and T. Monteiro Fernandes, Systèmes différentiels fuchsiens le long d'une sous-variété, Publ.", "RIMS, Kyoto Univ.", "24 (1988), no.", "3, 397–431.", "Z. Mebkhout, Le formalisme des six opérations de Grothendieck pour les ${D}$ -modules cohérents, Travaux en cours, vol.", "35, Hermann, Paris, 1989. , Le théorème de positivité, le théorème de comparaison et le théorème d'existence de Riemann, Éléments de la théorie des systèmes différentiels géométriques, Séminaires & Congrès, vol.", "8, Société Mathématique de France, Paris, 2004, pp. 165–310.", "T. Mochizuki, Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$ -modules, vol.", "185, Mem.", "Amer.", "Math.", "Soc., no.", "869-870, American Mathematical Society, Providence, RI, 2007, arXiv: math.DG/0312230 & math.DG/0402122.", ", 1Wild harmonic bundles and wild pure twistor $D$ -modules, Astérisque, vol.", "340, Société Mathématique de France, Paris, 2011. , 2Mixed twistor D-Module, arXiv: 1104.3366, 2011.", "L. Narváez Macarro, The local duality theorem in ${D}$ -module theory, Éléments de la théorie des systèmes différentiels géométriques, Séminaires & Congrès, vol.", "8, Société Mathématique de France, Paris, 2004, pp. 59–87.", "C. Sabbah, Polarizable twistor ${D}$ -modules, Astérisque, vol.", "300, Société Mathématique de France, Paris, 2005. , Wild twistor $D$ -modules, Algebraic Analysis and Around: In Honor of Professor M. Kashiwara's 60th Birthday (Kyoto, June 2007), Advanced Studies in Pure Math., vol.", "54, Math.", "Soc.", "Japan, Tokyo, 2009, pp.", "293–353, arXiv: 0803.0287.", "M. Sato, T. Kawai, and M. Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (Katata, 1971), Lect.", "Notes in Math., vol.", "287, Springer-Verlag, 1973, pp. 265–529.", "P. Schapira, Microdifferential systems in the complex domain, Grundlehren der mathematischen Wissenschaften, vol.", "269, Springer-Verlag, 1985.", "P. Schapira and J.-P. Schneiders, Index theorem for elliptic pairs, Astérisque, vol.", "224, Société Mathématique de France, Paris, 1994.", "C. Simpson, Higgs bundles and local systems, Publ.", "Math.", "Inst.", "Hautes Études Sci.", "75 (1992), 5–95.", ", Mixed twistor structures, Prépublication Université de Toulouse & arXiv: math.AG/9705006, 1997." ], [ "The de Rham complex of a holonomic ${D}_{X\\times S/S}$ -module", "In what follows $X$ and $S$ denote complex manifolds and we set $n=\\dim X$ , $\\ell =\\dim S$ .", "We shall keep the notation of the preceding section.", "Let $\\pi :T^*(X\\times S)\\rightarrow T^*X\\times S$ denote the projection and let ${D}_{X\\times S/S}$ denote the subsheaf of ${D}_{X\\times S}$ of relative differential operators with respect to $p_X$ (see [17]).", "Recall that $p^{-1}_X{O}_{S}$ is contained in the center of ${D}_{X\\times S/S}$ .", "With the same proof as for Proposition REF we obtain: Proposition 3.1 Let $s_o\\in S$ be given.", "Let ${M}$ and ${N}$ be objects of $\\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S})$ .", "Then, there is a well-defined natural morphism $Li^*_{s_o}(R{H}\\!om_{{D}_{X\\times S/S}}({M}, {N}))\\rightarrow R{H}\\!om_{i^*_{s_o}({D}_{X\\times S/S})}(Li^*_{s_o}({M}), Li^*_{s_o}({N}))$ which is an isomorphism in $\\operatorname{\\mathsf {D}}^\\mathrm {b}(X)$ ." ], [ "Duality for coherent ${D}_{X\\times S/S}$ -modules", "We refer for instance to [2] for the coherence properties of the ring ${D}_{X\\times S/S}$ .", "The classical methods used in the absolute case, i.e, for coherent ${D}_X$ -objects (see for instance [7], [8]) apply here: Proposition 3.2 Let ${M}$ be a coherent ${D}_{X\\times S/S}$ -module.", "Then ${M}$ locally admits a resolution of length at most $2n+\\ell $ by free ${D}_{X\\times S/S}$ -modules of finite rank.", "Proposition REF and [5] (for the opposite category) imply: Corollary 3.3 Let ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ .", "Let us assume that ${M}$ is concentrated in degrees $[a,b]$ .", "Then, in a neighborhhod of each $(x,z)\\in X\\times S$ , there exist a complex ${L}^{\\scriptscriptstyle \\bullet }$ of free ${D}_{X\\times S/S}$ -modules of finite rank concentrated in degrees $[a-2n-\\ell , b]$ and a quasi-isomorphism $ {L}^{\\scriptscriptstyle \\bullet }\\rightarrow {M}$ .", "We set $\\Omega _{X\\times S/S}=\\Omega ^{n}_{X\\times S/S}$ , where $\\Omega ^{n}_{X\\times S/S}$ denotes the sheaf of relative differential forms of degree $n=\\dim X$ .", "Definition 3.4 The duality functor $D(\\cdot ): \\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S}) \\rightarrow \\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S})$ is defined as: ${M}\\mapsto D{M}=R{H}\\!om_{{D}_{X\\times S/S}}({M}, {D}_{X\\times S/S}\\otimes _{{O}_{X\\times S}}\\Omega _{X\\times S/S}^{\\otimes ^{-1}})[n].$ We also set $D^{\\prime }{M}:=R{H}om_{{D}_{X\\times S/S}}({M}, {D}_{X\\times S/S})\\in \\operatorname{\\mathsf {D}}^{\\mathrm {b}}({D}_{X\\times S/S}^\\mathrm {opp})$ .", "By Proposition REF , ${D}_{X\\times S/S}$ has finite cohomological dimension, so [2] gives a natural morphism in $\\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S})$ : ${M}\\rightarrow D^{\\prime }D^{\\prime }{M}\\simeq DD{M}.$ Moreover, in view of Corollary REF , if ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ , then $D^{\\prime }{M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S}^\\mathrm {opp}).$ Indeed, we may choose a local free finite resolution ${L}^{\\scriptscriptstyle \\bullet }$ of ${M}$ , so that $D^{\\prime }{M}$ is quasi isomorphic to the transposed complex $({L}^{\\scriptscriptstyle \\bullet })^t$ whose entries are free.", "By the same argument we deduce that (REF ) is an isomorphism whenever ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ .", "Again by Proposition REF , ${D}_{X\\times S/S}$ has finite flat dimension so we are in conditions to apply [2]: given ${M}, {N}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S})$ there is a natural morphism: $D^{\\prime }{M}\\overset{L}{\\otimes }_{{D}_{X\\times S/S}}{N}\\rightarrow R{H}\\!om_{{D}_{X\\times S/S}}({M},{N})$ which an isomorphism provided that ${M}$ or ${N}$ belong to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ .", "When ${M}, {N}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ , composing (REF ) with the biduality isomorphism (REF ) gives a natural isomorphism $R{H}\\!om_{{D}_{X\\times S/S}}({M},{N})\\simeq R{H}\\!om_{{D}_{X\\times S/S}}(D{N}, D{M}).$" ], [ "Characteristic variety", "Recall (see [16]) that the characteristic variety $\\operatorname{Char}{M}$ of a coherent ${D}_{X\\times S/S}$ -module ${M}$ is the support in $T^*X\\times S$ of its graded module with respect to any (local) good filtration.", "One has (see [16]) $\\begin{split}\\operatorname{Char}({D}_{X\\times S}\\otimes _{{D}_{X\\times S/S}} {M})&=\\pi ^{-1}\\operatorname{Char}{M},\\\\\\operatorname{Char}{M}&=\\pi \\big (\\operatorname{Char}({D}_{X\\times S}\\otimes _{{D}_{X\\times S/S}} {M})\\big ).\\end{split}$ One may as well define the characteristic variety of an object ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ as the union of the characteristic varieties of its cohomology modules.", "By the flatness of ${D}_{X\\times S}$ over ${D}_{X\\times S/S}$ , (REF ) holds for any object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ .", "Proposition 3.5 ([17]) For ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ we have $\\operatorname{Char}({M})=\\operatorname{Char}(D{M}).$" ], [ "The de Rham and solution complexes", "For an object ${M}$ of $\\operatorname{\\mathsf {D}}^\\mathrm {b}({D}_{X\\times S/S})$ we define the functors $\\operatorname{DR}{M}&:=R{H}\\!om_{{D}_{X\\times S/S}}({O}_{X\\times S},{M}),\\\\\\operatorname{Sol}{M}&:=R{H}\\!om_{{D}_{X\\times S/S}}({M},{O}_{X\\times S})$ which take values in $\\operatorname{\\mathsf {D}}^{\\mathrm {b}}(p_X^{-1}{O}_S)$ .", "If ${M}$ is a ${D}_{X\\times S/S}$ -module, that is, a ${O}_{X\\times S}$ -module equipped with an integrable relative connection $\\nabla :{M}\\rightarrow \\Omega ^1_{X\\times S/S}\\otimes {M}$ , the object $\\operatorname{DR}{M}$ is represented by the complex $(\\Omega ^{\\scriptscriptstyle \\bullet }_{X\\times S/S}\\otimes _{{O}_{X\\times S}}{M},\\nabla )$ .", "Noting that $R{H}\\!om_{{D}_{X\\times S/S}}({O}_{X\\times S},{D}_{X\\times S/S})\\simeq \\Omega _{X\\times S/S}[-\\dim X]$ we get $D{O}_{X\\times S}\\simeq {O}_{X\\times S}.$ For ${N}={O}_{X\\times S}$ , (REF ) implies a natural isomorphism, for ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ : $\\operatorname{Sol}{M}\\simeq \\operatorname{DR}D{M}.$" ], [ "Holonomic ${D}_{X\\times S/S}$ -modules", "Let ${M}$ be a coherent ${D}_{X\\times S/S}$ -module.", "We say that it is holonomic if its characteristic variety $\\operatorname{Char}{M}\\subset T^*X\\times S$ is contained in $\\Lambda \\times S$ for some closed conic Lagrangian complex analytic subset of $T^*X$ .", "We will say that a complex $\\mu $ -stratification $(X_\\alpha )$ is adapted to ${M}$ if $\\Lambda \\subset \\bigcup _\\alpha T^*_{X_\\alpha }X$ .", "Similar definitions hold for objects of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ .", "An object ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ is said to be holonomic if its cohomology modules are holonomic.", "We denote the full triangulated category of holonomic complexes by $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ .", "Corollary 3.6 (of Prop.", "REF ) If ${M}$ is an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ , then so is $D{M}$ .", "Theorem 3.7 Let ${M}$ be an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ .", "Then $\\operatorname{DR}({M})$ and $\\operatorname{Sol}{M}$ belong to $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p^{-1}_X{O}_S)$ .", "Firstly, it follows [4], that $\\operatorname{Sol}({M})$ and $\\operatorname{DR}({M})$ have their micro-support contained in $\\Lambda \\times T^*S$ (see [17]) and, according to Proposition REF , these complexes are objects of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{\\textup {w-}{-c}}(p^{-1}_X{O}_S)$ .", "Let $x\\in X$ .", "In order to prove that $i_x^{-1}\\operatorname{DR}{M}$ has ${O}_S$ -coherent cohomology, we can assume that $x$ is a stratum of a stratification adapted to $\\operatorname{DR}{M}$ and we use Lemma REF to get $i_x^{-1}\\operatorname{DR}{M}\\simeq Rp_{\\overline{\\varepsilon },*}({\\overline{B}_\\varepsilon \\times S}\\otimes _{M})$ for $\\varepsilon $ small enough, where $\\overline{B}_\\varepsilon $ is a closed ball of radius $\\varepsilon $ centered at $x$ .", "One then remarks that $({\\overline{B}_\\varepsilon \\times S},{M})$ forms a relative elliptic pair in the sense of [17], and Proposition 4.1 of loc. cit.", "gives the desired coherence.", "The statement for $\\operatorname{Sol}{M}$ is proved similarly.", "Lemma 3.8 (see [13]) For ${M}$ in $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ with adapted stratification $(X_\\alpha )$ and for any $s_o\\in S$ , $Li^*_{s_o}{M}$ is ${D}_X$ -holonomic and $(X_\\alpha )$ is adapted to it.", "Corollary 3.9 For ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ , there is a natural isomorphism $D^{\\prime }\\operatorname{Sol}{M}\\simeq \\operatorname{DR}{M}$ .", "We consider the canonical pairing $\\operatorname{DR}{M}\\overset{L}{\\otimes }_{p^{-1}_X{O}_S}\\operatorname{Sol}{M}\\rightarrow p_X^{-1}{O}_S$ which gives a natural morphism $\\operatorname{DR}{M}\\rightarrow D^{\\prime }\\operatorname{Sol}{M}$ in $\\operatorname{\\mathsf {D}}^\\mathrm {b}_{{{-c}}}(p_X^{-1}{O}_S)$ .", "We have for each $s_o\\in S$ , by Proposition REF $Li^*_{s_o}(\\operatorname{DR}{M})&\\simeq \\operatorname{DR}Li^*_{s_o}({M}),\\\\Li^*_{s_o}(\\operatorname{Sol}{M})&\\simeq \\operatorname{Sol}Li^*_{s_o}({M}) .$ Since $Li^*_{s_o}({M})\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_X)$ by Lemma REF , we have $\\operatorname{DR}Li^*_{s_o}({M})\\simeq D^{\\prime }\\operatorname{Sol}Li^*_{s_o}({M}),$ so by Proposition REF and Proposition REF $D^{\\prime }\\operatorname{Sol}Li^*_{s_o}({M})\\simeq D^{\\prime }Li^*_{s_o}(\\operatorname{Sol}{M})\\simeq Li^*_{s_o}(D^{\\prime }\\operatorname{Sol}{M}).$ The assertion then follows by Proposition REF .", "In the following proposition, the main argument is that of strictness, which is essential.", "We will set $\\operatorname{{}^\\mathrm {p}DR}{M}:=\\operatorname{DR}{M}[\\dim X]$ and $\\operatorname{{}^\\mathrm {p}Sol}{M}=\\operatorname{Sol}{M}[\\dim X]$ .", "Proposition 3.10 Let ${M}$ be a holonomic ${D}_{X\\times S/S}$ -module which is strict, i.e., which is $p^{-1}{O}_S$ -flat.", "Then $\\operatorname{{}^\\mathrm {p}DR}{M}$ satisfies the support condition () with respect to a $\\mu $ -stratification adapted to ${M}$ .", "We prove the result by induction on $\\dim S$ .", "Since it is local on $S$ , we consider a local coordinate $s$ on $S$ and we set $S^{\\prime }=\\lbrace s=0\\rbrace $ .", "The strictness property implies that we have an exact sequence $0\\rightarrow {M}\\xrightarrow{}{M}\\rightarrow i_{S^{\\prime }}^*{M}\\rightarrow 0,$ and $i_{S^{\\prime }}^*{M}$ is ${D}_{X\\times S^{\\prime }/S^{\\prime }}$ -holonomic and $p^{-1}{O}_{S^{\\prime }}$ -flat.", "We deduce an exact sequence of complexes $0\\rightarrow \\operatorname{{}^\\mathrm {p}DR}{M}\\xrightarrow{}\\operatorname{{}^\\mathrm {p}DR}{M}\\rightarrow \\operatorname{{}^\\mathrm {p}DR}i_{S^{\\prime }}^*{M}\\rightarrow 0$ .", "Let $X_\\alpha $ be a stratum of a $\\mu $ -stratification of $X$ adapted to ${M}$ (hence to $i_{S^{\\prime }}^*{M}$ , after Lemma REF ).", "For $x\\in X_\\alpha $ , let $k$ be the maximum of the indices $j$ such that ${H}^ji_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}\\ne 0$ .", "For any $S^{\\prime }$ as above, we have a long exact sequence $\\cdots \\rightarrow {H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}\\xrightarrow{}{H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}\\rightarrow {H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}i_{S^{\\prime }}^*{M}\\rightarrow 0.$ If $k>-\\dim X_\\alpha $ , we have ${H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}i_{S^{\\prime }}^*{M}=0$ , according to the support condition for $i_{S^{\\prime }}^*{M}$ (inductive assumption), since $(X_\\alpha )$ is adapted to it.", "Therefore, $s:{H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}\\rightarrow {H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}$ is onto.", "On the other hand, by Theorem REF , ${H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}$ is ${O}_S$ -coherent.", "Then Nakayama's lemma implies that ${H}^ki_x^{-1}\\operatorname{{}^\\mathrm {p}DR}{M}=0$ in some neighbourhood of $S^{\\prime }$ .", "Since $S^{\\prime }$ was arbitrary, this holds all over $S$ , hence the assertion.", "It is a direct consequence of the following.", "Theorem 3.11 Let ${M}$ be an object of $\\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ and let $D{M}$ be the dual object.", "Then there is an isomorphism $\\operatorname{{}^\\mathrm {p}DR}D{M}\\simeq D\\operatorname{{}^\\mathrm {p}DR}{M}$ .", "Indeed, with the assumptions of Theorem REF , $D{M}$ is holonomic since ${M}$ is so (see Corollary REF ), and both ${M}$ and $D{M}$ are strict.", "Then both $\\operatorname{{}^\\mathrm {p}DR}{M}$ and $\\operatorname{{}^\\mathrm {p}DR}D{M}$ satisfy the support condition, according to Proposition REF .", "Hence, according to Theorem REF and Proposition REF , $\\operatorname{{}^\\mathrm {p}DR}{M}$ satisfies the cosupport condition.", "Similarly, $\\operatorname{{}^\\mathrm {p}Sol}{M}\\simeq D\\operatorname{{}^\\mathrm {p}DR}{M}$ and $D(\\operatorname{{}^\\mathrm {p}Sol}{M})\\simeq \\operatorname{{}^\\mathrm {p}DR}{M}$ both satisfy the support condition, hence $\\operatorname{Sol}{M}[\\dim X]$ is a perverse object.", "Combining (REF ) with [4] (with $f=\\operatorname{id}$ , ${A}={D}_{X\\times S/S}$ and ${B}=p_X^{-1}{O}_S$ ) entails, for any ${N}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {coh}({D}_{X\\times S/S})$ , a natural morphism $R{H}\\!om_{{D}_{X\\times S/S}}({N},{M})\\rightarrow R{H}\\!om_{p_X^{-1}{O}_S}(\\operatorname{DR}D{M}, \\operatorname{DR}D{N}).$ When ${N}={O}_{X\\times S}$ , we obtain a natural morphism $ \\operatorname{DR}{M}\\rightarrow D^{\\prime }\\operatorname{DR}D{M},\\quad \\text{that is,}\\quad \\operatorname{{}^\\mathrm {p}DR}{M}\\rightarrow D\\operatorname{{}^\\mathrm {p}DR}D{M}.$ Suppose now that ${M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ .", "Recall that $D{M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_{X\\times S/S})$ , so $\\operatorname{{}^\\mathrm {p}DR}D{M}\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_{{-c}}(p^{-1}_X{O}_S)$ .", "Hence, by biduality, we get a morphism $D\\operatorname{{}^\\mathrm {p}DR}{M}\\leftarrow \\operatorname{{}^\\mathrm {p}DR}D{M}.$ On the other hand, since $Li^*_{s_o}({M})\\in \\operatorname{\\mathsf {D}}^\\mathrm {b}_\\mathrm {hol}({D}_X)$ for each $s_o\\in S$ , the morphisms above induce isomorphisms $Li^*_{s_o}(D\\operatorname{{}^\\mathrm {p}DR}{M})\\simeq \\operatorname{{}^\\mathrm {p}DR}DLi^*_{s_o}({M})$ according to Proposition REF and Proposition REF , where in the right hand side we consider the duality for holonomic ${D}_X$ -modules.", "Thus (REF ) is an isomorphism by Proposition REF and the local duality theorem for holonomic ${D}_X$ -modules (see [12] and the references given there).", "Example 3.12 Let $X$ be the open unit disc in $ with coordinate $ x$ and let~$ S$ be a connected open set of $ with coordinate $s$ .", "Let $\\varphi :S\\rightarrow be a non constant holomorphic function on $ S$ and consider the holonomic $ DXS/S$-module $ M=DXS/S/DXS/SP$, with $ P=xx-(s)$.", "It is easy to check that $ M$ has no $ OS$-torsion and admits the resolution $ 0DXS/SPDXS/SM0$, so that the dual module $ DM$ has a similar presentation and is also $ OS$-flat.", "The complex $ pSolM$ is represented by $ 0OXSPOXS0$ (terms in degrees $ -1$ and $ 0$).", "Consider the stratification $ X1=X{0}$ and $ X0={0}$ of $ X$.", "Then $ H-1pSolM|X1$ is a locally constant sheaf of free $ p-1XOS$-modules generated by a local determination of~$ x(s)$, and $ H0pSolM|X1=0$.", "On the other hand, $ H-1pSolM|X0=0$ and $ H0pSolM|X0$ is a skyscraper sheaf on $ X0S$ supported on $ {sS(s)Z}$.$ For each $x_0$ we have ${{i_{x_0}^{!}}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})\\\\\\simeq i^{-1}_{\\lbrace x_0\\rbrace \\times S}R{H}\\!om_{{D}_{X\\times S}}({D}_{X\\times S}\\otimes _{{D}_{X\\times S/S}}{M}, R\\Gamma _{\\lbrace x_0\\rbrace \\times S|X\\times S}{O}_{X\\times S})[\\dim X]\\\\\\simeq i^{-1}_{\\lbrace x_0\\rbrace \\times S}R{H}\\!om_{{D}_{X\\times S}}({D}_{X\\times S}\\otimes _{{D}_{X\\times S/S}}{M}, B_{\\lbrace x_0\\rbrace \\times S|X\\times S})$ where $B_{\\lbrace x_0\\rbrace \\times S|X\\times S}:={H}^1_{[\\lbrace x_0\\rbrace \\times S]}({O}_{X\\times S})$ denotes the sheaf of holomorphic hyperfunctions (of finite order) along $x=x_0$ (cf. [15]).", "The second isomorphism follows from the fact that ${D}_{X\\times S}\\otimes _{{D}_{X\\times S/S}}{M}$ is regular specializable along the submanifold $x=x_0$ (cf. [6]).", "Recall that the sheaves $B_{\\lbrace x_0\\rbrace \\times S|X\\times S}$ are flat over $p_X^{-1}{O}_S$ because locally they are inductive limits of free $p_X^{-1}{O}_S$ -modules of finite rank.", "Since ${i_{x_0}^{!}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})$ is quasi isomorphic to the complex $0\\rightarrow B_{\\lbrace x_0\\rbrace \\times S|X\\times S}|_{\\lbrace x_0\\rbrace \\times S}\\xrightarrow{}B_{\\lbrace x_0\\rbrace \\times S|X\\times S}|_{\\lbrace x_0\\rbrace \\times S}\\rightarrow 0$ it follows that the flat dimension over ${O}_S$ of ${{i_{x_0}^{!}}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})$ in the sense of [3] is $\\leqslant 0$ for any $x_0$ .", "Moreover, ${H}^0{i_{x_0}^{!}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})=0$ and, if $x_0\\ne 0$ , ${H}^1{i_{x_0}^{!}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})$ is a locally free ${O}_S$ -module of rank 1.", "Hence the flat dimension of ${i_{x_0}^{!}}", "(\\operatorname{{}^\\mathrm {p}Sol}{M})$ is $\\leqslant 1$ .", "This shows explicitly that $\\operatorname{{}^\\mathrm {p}Sol}{M}$ satisfies the condition (REF ) of Corollary REF ." ], [ "Application to mixed twistor ${D}$ -modules", "Let ${R}_{X\\times be the sheaf on X\\times of z-differential operators, locally generated by {O}_{X\\times and the z-vector fields z\\partial _{x_i} in local coordinates (x_1,\\dots ,x_n) on~X.", "When restricted to X\\times *, the sheaf {R}_{X\\times *} is isomorphic to {D}_{X\\times */*}.", "}A mixed twistor {D}-module on X (see \\cite {Mochizuki11}) is a triple {T}=({M}^{\\prime },{M}^{\\prime \\prime },C), where {M}^{\\prime },{M}^{\\prime \\prime } are holonomic {R}_{X\\times -modules and C is a certain pairing with values in distributions, that we will not need to make precise here.", "Such a triple is subject to various conditions.", "We say that a {D}_{X\\times */*}-module {M} underlies a mixed twistor {D}-module {T} if {M} is the restriction to X\\times * of~{M}^{\\prime } or {M}^{\\prime \\prime }.", "}Theorem \\ref {th:main} is now a direct consequence of the following properties of mixed twistor {D}-modules, since they imply that {M} satisfies the assumptions of Theorem \\ref {th:main2}.", "If {M} underlies a mixed twistor {D}-module, then\\begin{itemize}\\item there exists a locally finite filtration W_{\\scriptscriptstyle \\bullet }{M} indexed by \\mathbb {Z} by {R}_{X\\times -submodules such that each graded module underlies a pure polarizable twistor {D}-module; then each \\mathrm {gr}_\\ell ^W{M} is strict and holonomic (see \\cite [Prop.\\,4.1.3]{Sabbah05} and \\cite [§17.1.1]{Mochizuki08}), and thus so is {M};\\item the dual of {M} as a {R}_{X\\times *}-module also underlies a mixed twistor {D}-module, hence is also strict holonomic (see \\cite [Th.\\,12.9]{Mochizuki11}); using the isomorphism {R}_{X\\times *}\\simeq {D}_{X\\times */*}, we see that the dual D{M} as a {D}_{X\\times */*}-module is strict and holonomic.\\Box }\\end{itemize}}\\begin{thebibliography}{10}\\end{thebibliography}\\bibitem {G-M88}M.~Goresky and {R.D}.~MacPherson, \\emph {{Stratified Morse theory}},Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge 3 Band 14,Springer-Verlag, Berlin, Heidelberg, New York, 1988.$ M. Kashiwara, On the maximally overdetermined systems of differential equations, Publ.", "RIMS, Kyoto Univ.", "10 (1975), 563–579.", ", $D$ -modules and microlocal calculus, Translations of Mathematical Monographs, vol.", "217, American Mathematical Society, Providence, RI, 2003. , t-Structure on the derived categories of ${D}$ -modules and ${O}$ -modules, Moscow Math. J.", "4 (2004), no.", "4, 847–868.", "M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren der mathematischen Wissenschaften, vol.", "292, Springer-Verlag, 1990. , Categories and sheaves, Grundlehren der mathematischen Wissenschaften, vol.", "332, Springer-Verlag, 2006.", "Y. Laurent and T. Monteiro Fernandes, Systèmes différentiels fuchsiens le long d'une sous-variété, Publ.", "RIMS, Kyoto Univ.", "24 (1988), no.", "3, 397–431.", "Z. Mebkhout, Le formalisme des six opérations de Grothendieck pour les ${D}$ -modules cohérents, Travaux en cours, vol.", "35, Hermann, Paris, 1989. , Le théorème de positivité, le théorème de comparaison et le théorème d'existence de Riemann, Éléments de la théorie des systèmes différentiels géométriques, Séminaires & Congrès, vol.", "8, Société Mathématique de France, Paris, 2004, pp. 165–310.", "T. Mochizuki, Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$ -modules, vol.", "185, Mem.", "Amer.", "Math.", "Soc., no.", "869-870, American Mathematical Society, Providence, RI, 2007, arXiv: math.DG/0312230 & math.DG/0402122.", ", 1Wild harmonic bundles and wild pure twistor $D$ -modules, Astérisque, vol.", "340, Société Mathématique de France, Paris, 2011. , 2Mixed twistor D-Module, arXiv: 1104.3366, 2011.", "L. Narváez Macarro, The local duality theorem in ${D}$ -module theory, Éléments de la théorie des systèmes différentiels géométriques, Séminaires & Congrès, vol.", "8, Société Mathématique de 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[ [ "Improved collision strengths and line ratios for forbidden [O III]\n far-infrared and optical lines" ], [ "Abstract Far-infrared and optical [O III] lines are useful temeprature-density diagnostics of nebular as well as dust obscured astrophysical sources.", "Fine structure transitions among the ground state levels 1s^22s^22p^3 \\ ^3P_{0,1,2} give rise to the 52 and 88 micron lines, whereas transitions among the $^3P_{0,1,2}, ,^1D_2, ^1S_0$ levels yield the well-known optical lines 4363, 4959 and 5007 Angstroms.", "These lines are excited primarily by electron impact excitation.", "But despite their importance in nebular diagnostics collision strengths for the associated fine structure transitions have not been computed taking full account of relativistic effects.", "We present Breit-Pauli R-matrix calculations for the collision strengths with highly resolved resonance structures.", "We find significant differences of up to 20% in the Maxwellian averaged rate coefficients from previous works.", "We also tabulate these to lower temperatures down to 100 K to enable determination of physical conditions in cold dusty environments such photo-dissociation regions and ultra-luminous infrared galaxies observed with the Herschel space observatory.", "We also examine the effect of improved collision strengths on temperature and density sensitive line ratios." ], [ "Introduction", "[O iii] optical lines have long been standard nebular temperature diagnostics with wavelengths almost in the middle of the optical spectrum at $\\lambda \\lambda $ 4363, 4959, 5007 (viz.", "Aller 1956, Dopita and Sutherland 2003, Pradhan and Nahar 2011).", "In recent years, owing to the advent of far-infrared (FIR) space observatories and instruments such as the Intfrared Space Observatory - Long Wavelength Spectrograph (ISO-LWS), the Spitzer Infrared Spectrograph, and the Herschel Photodetector Array Camera and Spectrometer (PACS) the [O iii] FIR lines have proven to have great potential in providing diagnostics of physical conditions in a variety of astrophysical objects that are generally obscured by dust extinction at optical or shorter wavelengths.", "These range from Galactic H ii regions (Martin-Hernandez et al.", "2002, Morisset et al.", "2002) to star-forming galaxies at intermediate redhift (Liu et al.", "2008) and ultra-luminous infrared galaxies (ULIRGs).", "For example, the [O iii] FIR lines at $\\lambda \\lambda $ 88 and 52 $\\mu $ m are observed from dusty ULIRGs, which are copious IR emitters and become more prominent with increasing redshift (Nagao et al.", "2011).", "They may be valuable indicators of the metallicity evolution from otherwise inaccessible star-forming regions buried deep within the galaxies (Houck et al.", "2004, 2005).", "The forbidden FIR lines arise from very low-energy excitations within the fine structure levels of the ground state of atomic ions, such as the [O iii] $^3P_o\\rightarrow ^3P_1$ transition at 88.36 $\\mu $ m and the $^3P_1\\rightarrow ^3P_2$ transition at 51.81 $\\mu $ m .", "As such they can be excited by electron impact at low temperatures, even T$_e$$\\sim $ 1000 K or less.", "That also accounts for their utility since the FIR lines can be formed in (and therefore probe) not only H ii regions but also photo-dissociation regions (PDRs) where the temperaure-density gradients are large (Nagao et al.", "2011).", "However, excitation of levels lying very close to each other implies that the associated cross sections need to be computed with great accuracy at very low energies in order to yield reliable rate coefficients.", "The Maxwellian electron distribution at low temperatures samples only the near-threshold energies above the small excitation energy of the fine structure transition.", "Relativistic fine structure separations therefore assume special importance even for low-Z atomic ions in determining not only the energy separation but also the interaction of the incident electron with the target levels.", "Owing to its prominence in astrophysical spectra, a large number of previous studies have been carried out on electron impact excitation of O iii (viz.", "compilation of evaluated data by Pradhan and Zhang 2001).", "Among the recent ones, whose collision strengths are employed in astrophysical models, are Burke et al.", "(1989) and Aggarwal and Keenan (1999).", "But these calculations are basically in LS coupling (Burke et al.", "1989), or with intermediate coupling effects introduced perturbatively via an algebraic transformation from the LS to LSJ scheme (Aggarwal and Keenan 1999).", "Although the earlier calculations employed the coupled channel R-matrix method which takes account of the extensive resonance structures, the fine structure separations are not considered.", "In this report we take account of both the resonances and fine structure in an ab initio manner.", "Another recent development in relativistic R-matrix codes is the inclusion of the two-body fine structure Breit interaction terms in the Breit-Pauli hamiltonian (Eissner and Chen 2012, Nahar et al.", "2011).", "A relativistic calculation of collision strengths can therefore be carried out, including fine structure explicitly and more accurately than in previous works.", "Relativistic effects are likely to be insignificant for optical transitions compared to the FIR transitions since the former involve relatively larger energy separations and relatvistic corrections are small.", "Nevertheless, we consider all 10 forbidden transitions among the levels dominated by the ground configuration of O iii." ], [ "Theory and computations", "A brief theoretical description of the calculations is given.", "In particular, we describe relatvistic effects and the representation of the (e + ion) system." ], [ "Relativistic fine structure", "The relativistic Hamiltonian (Rydberg units) in the Breit-Pauli R-matrix (BPRM) approximation is given by $\\begin{array}{l}H_{N+1}^{\\rm BP} = \\\\ \\sum _{i=1}^{N+1}\\left\\lbrace -\\nabla _i^2 -\\frac{2Z}{r_i}+ \\sum _{j>i}^{N+1} \\frac{2}{r_{ij}}\\right\\rbrace +H_{N+1}^{\\rm mass} +H_{N+1}^{\\rm Dar} + H_{N+1}^{\\rm so}.\\end{array}$ where the last three terms are relativistic corrections, respectively: $\\begin{array}{l}{\\rm the~mass~correction~term},~H^{\\rm mass} =-{\\alpha ^2\\over 4}\\sum _i{p_i^4},\\\\{\\rm the~Darwin~term},~H^{\\rm Dar} = {Z\\alpha ^2 \\over 4}\\sum _i{\\nabla ^2({1\\over r_i})}, \\\\{\\rm the~spin-orbit~interaction~term},~H^{\\rm so}= Z\\alpha ^2\\sum _i{1\\over r_i^3} {\\bf l_i.s_i}.\\end{array}$ Eq.", "(2) representes the one-body terms of the Breit interaction.", "In addition, another version of BPRM codes including the two-body terms of the Breit-interaction (Nahar et al.", "2011; W. Eissner and G. X. Chen, in preparation) has been developed, and is employed in the present work." ], [ "Effective collision strengths", "Cross sections or collision strengths at very low energies may be inordinately influenced by near-threshold resonances.", "Those, in turn, affect the effective collision strengths or rate coefficients computed by convolving the collision strengths over a Maxwellian function at a given temperature T as $\\Upsilon _{ij}(T_e) = \\int _0^{\\infty } \\Omega _{ij} (\\epsilon )\\exp (-\\epsilon /kT_e) d(\\epsilon /kT_e), $ where $E_{ij}$ is the energy difference and $\\Omega _{ij}$ is the collision strength for the transition $i \\rightarrow j$ .", "The exponentially decaying Maxwellian factor implies that at low temperatures only the very low energy $\\Omega _{ij}$ (E) would determine the $\\Upsilon $ (T).", "Figure: Collision strengths for the [O iii] IR fine structure transitions2p 2 ( 3 P 0 - 3 P 1 , 3 P 1 - 3 P 2 2p^2 (^3P_0-^3P_1, ^3P_1-^3P_2) at λλ\\lambda \\lambda a) 88μ\\mu m and b) 52 μ\\mu m respectively.High resolution at near-thresholdenergies is necessary for accuracy in rate coefficients at lowtemperatures.", "The top panel shows an expanded view in the region E≤\\le 1 Rydberg; both transitions have similar resonancestructures.Figure: Maxwellian averaged effective collision strengths Υ(T)\\Upsilon (T)(Eq.", "1) for the transitions 0 P - 3 P 1 ^P_0-^3P_1 at 88 μ\\mu m and 3 P 1 - 3 P 2 ^3P_1-^3P_2 at52 μ\\mu m (solid lines, c.f.", "Fig.", ").Previous results without relativistic effects (Aggarwal and Keenan 1999)are also shown (dashed lines) in the temperature range available T e _e≥\\ge 2500 K." ], [ "Wavefunction representation and calculations", "Based on the coupled channel approximaton, the R-matrix method (Burke et al.", "1971) entails a wavefunction expansion of the (e + ion) system in terms of the eigenfuctions for the target ion.", "In the present case we are interested in low-lying O/FIR transitions of the ground configuration $2s^22p^2$ .", "Therefore we confine ourselves to an accurate wavefunction representation for the first 19 levels dominated by the spectroscopic configurations $[1s^2] 2s^22p^2, 2s2p^3, 2s^22p3s$ .", "A much larger set of correlation configurations is included for configuration interaction with the spectroscopic terms using the atomic structure code SUPERSTRUCTURE (Eissner et al.", "1974, Nahar et al.", "2003): $[1s^2] 2p^4, 2s^22p3p, 2s^22p3d,2s^22p4s, 2s^22p4p, 2s2p^23s,\\\\ 2s2p^23p, 2s2p^23d, 2s^23s^2,2s^23p^2,2s^23d^2, 2s^24s^2, 2s^24p^2,\\\\ 2s^23s3p, 2s^23s4s, 2s^23p3d, 2p^33s,2p^33p, 2p^33d$ .", "We note here that the crucial fine structure separations between the ground state $^3P_{0,1.2}$ levels reproduced theoretically agree with experimentally measured values to $\\sim $ 3% (Nahar et al.", "2011, see Pradhan and Nahar (2011) for a general description of atomic processes and calculations).", "The observed energies were substituted for theoretical ones in order to reproduce the threshold energies more accurately.", "This is of particular importance for excitation at low temperatures dominated by near-threshold resonances.", "Even though the observed and experimental values are close, a small difference in resonance positions relative to threshold can introuce a significant uncertainty in the effective collision strengths.", "The collision strengths were computed employing the extended Breit-Pauli R-matrix (BPRM) codes (Eissner and Chen 2012).", "Particular care is taken to test and ensure convergence of collision strengths with respect to partial waves and energy resolution.", "Total (e + ion) symmetries up to (LS)J$\\pi $ with J $\\le 19.5$ were included in the calculations, though it was found that the collision strengths for all forbidden transition transitions converged for J $\\le 9.5$ .", "An energy mesh of $\\Delta E <10^{-4}$ Rydbergs was used to resolve the near-thresold resonances.", "The resonances were delineated in detail prior to averaging over the Maxwellian distribution.", "We describe the two main sets of results for the FIR and the optical transitions, as well as diagnostics line ratios." ], [ "Far-infrared transitions", "The BPRM collision strengths for the two FIR fine structure transitions $\\lambda \\lambda $ 88, 52 $\\mu $ m are shown in Fig.", "1a,b.", "Although the resonance structures look similar the magnitude and energy variations are not the same.", "The Maxwellian averaged effective collision strengths $\\Upsilon $ (T) are quite different, as shown in Fig.", "2.", "While $\\Upsilon (T;^3P_o-^3P_1)$ for the 88 $\\mu $ m transition is relatively constant over three orders of magnitude in temperature, the $\\Upsilon (T;^3P_1-^3P_2)$ for the 52 $\\mu $ m transition varies by about a factor of 1.5 from the low-temperature limit of 100 K to temperatures T $>$ 10,000 K. A comparison with the earlier work by Aggarwal and Keenan (1999) is shown as dashed lines, which range down to their lowest tabulated temperature 2500K.", "It can be noted that if the Aggarwal and Keenan values are extrapolated linearly to lower temperatures then one would obtain fairly constant effective collision strengths.", "But the present results show marked difference owing to resonance structures as in Fig. 1.", "Such temperature sensitivity of the otherwise density sensitive 52/88 line ratio is illustrated in Fig. 3a,b.", "In Fig.", "3a the solid lines are ratios with the present collision strengths, and the dashed lines are using previous results (Aggarwal and Keenan 1999).", "We find very good agreement, implying that at all temperatures down to 2500K the differences in line ratios would be negligble.", "Fig.", "3b shows the 52/88 ratio at various temperatures between 100K and 10,000K.", "Whereas the ratio is relatively constant with density at 100K, its dependence on density varies significantly with increasing temperature.", "The density-temperature dianostic value of the 52/88 ratio is apparent from these curves.", "Therefore, care must be exercised to establish a temperature regime for the emitting region.", "Fig.", "3a shows line ratios computed at 2500 K and 10,000K, and variations with electron density.", "It is also found that the values of line ratios at 2500K and 1000K are very close together, implyling covergence for T $\\sim $ 1000K.", "Fig.", "3b clearly demonstrates that the line ratio decreases rapidly for T $<$ 1000K to almost flat at 100K.", "The low-temperature regime 100-1000K is therefore indicated by the curves shown in Fig.", "3b, as well as the limit where the 52/88 ratio is temperature invariant.", "So the 52/88 ratio is excellent density diagnostics in the typical density range Log N$_e$$\\sim $ 3-4 for T $>$ 1000K without much dependence on temperature (Fig.", "3a).", "However, at lower temepratures the ratio may differ by up to a factor of ten (Fig.", "3b).", "Whereas the primary variations are owing to the exponential factors in $\\Upsilon (T)$ , (Eq.", "3), we emphasize the role of relativistic fine structure splitting between the $^3P_{0,1,2}$ levels and near-threshold resonances lying in between.", "Figure: The density and temperature dependence of the λλ\\lambda \\lambda 52/88 μ\\mu m lineratios; a) the solid curves are present results at 2500K and 30000 K andthe dashed lines are using the Aggarwal Keenan (1999) effective collisionstrengths; b) 52/88 ratio at temperatures 100≤T(K)≤10000100 \\le T(K) \\le 10000." ], [ "Forbidden optical transitions", "Fig.", "4 shows the collision strengths for the optical transitions $^3P_1-^1D_2$ , $^3P_2-^1D_2,^1S_0-^1D_2$ at $\\lambda \\lambda $ 4959, 5007, 4363 respectively.", "The effective collision strengths for the [O iii] optical lines are shown in Fig. 5.", "These also differ significantly from previously available ones, by up to 15%.", "The new results are also obtained down to 100K; their limiting values at low temperatures tend to 0.4:0.6:1.0.", "Since $\\lambda \\lambda $ 4959, 5007 are often blended, it is common to plot the bleneded line ratio (4959+5007)/4363 shown in Fig. 6.", "This ratio varies over orders of magnitude since the upper-most $^1S_0$ level is far less excited at low energies than the $^1D_2$ , and therefore the level populations and line intensities depend drastically on temperature.", "A comparison is made with fine structure collision strengths derived from the LS term values of Aggarwal and Keenan (1999) divided according to statistical weights, again shown as dashed lines in Figs.", "5 and 6.", "However, similar to Fig.", "3a), the differences in effective collision strengths does not translate into any significant differences in line ratios even at Log T = 4.5 ($\\approx $ 30000K) wwhere the values differ most.", "Figure: Collision strengths of the [O iii] optical transitions.The two transitions 1 D 2 - 3 P 1 , 1 D 2 - 3 P 2 ^1D_2-^3P_1, ^1D_2-^3P_2 in a) and b)have similar resonancestructures; the top panel a) presents an expanded view below E ≤\\le 1 Rydberg.Figure: Effective collision strengths of the [O iii] optical transitions 1 D 2 - 3 P 1 , 1 D 2 - 3 P 2 , 1 S 0 - 1 D 2 ^1D_2-^3P_1, ^1D_2-^3P_2, ^1S_0-^1D_2 atλλ\\lambda \\lambda 4959, 5007 and 4363 respectively (c.f.", "Fig.", "4).Figure: Blended [O iii] line ratio (4959 + 5007)/4363 vs. temperature,at N e _e= 10 3 ^3 cm -3 ^{-3}.", "The dashed line using earlier data (Aggarwaland Keenan 1999) is plotted down to 2500K." ], [ "Maxwellian averaged collision strengths", "In Table 1 we present the effective collision strengths (Eq.", "3) for the 10 transitions among the ground configuration levels and their wavelengths.", "The tabulation is carried out at a range of temperatures typical of nebular environments, including the low temperature range $T \\le 1000$ K not heretofore considered.", "Table: Effective Maxwellian averaged collision strengths" ], [ "Conclusion", "Improved collision strengths including fine structure with relativistic effects are computed.", "Owing to the diagnostic importance of the [O iii] forbidden FIR and optical lines, the relatively small but significant differences of up to 20% should provide more accurate line ratios.", "Particular attention is paid to the resolution of resonances in the very small energy region above threshold(s), enabling the study of low temperature behavior.", "The line emissivities and ratios computed in this work demonstrate the temperature-density behaviour at low temperatures and at typical nebular temperatures.", "However, depending on the astrophysical sources a complete model of line emissivities may also need to take into consideration the Bowen fluorescence mechanism: the radiative excitation of $2p^2 \\ ^3P_2 - 2p3d^3P^o_2$ by He ii Ly$\\alpha $ at 304$Å$ and cascades into the upper levels of the forbidden transitions considered herein (viz.", "Saraph and Seaton 1980, Pradhan and Nahar 2011).", "In addition, for higher temperatures T $>$ 20,000K proton impact excitation of the ground state fine structure levels $^3P_{0,1,2}$ needs to be taken into account; at lower temperatures the excitation rate coefficent due to electrons far exceeds that due to protons (Ryans et al.", "1999).", "Finally, there may be some contribution from (e + ion) recombination from O iv to [O iii], since recombination rate coefficients increase sharply towards lower temperatures while collisional excitation rates decrease (level-specific and total recombination rate coefficients may be obtained from S. N. Nahar's database NORAD at: www.astronomy.ohio-state.edu$sim$nahar$ ).", "Recombination contribution depends on the O iv/[O iii] ionization fraction, which at low temperatures would be small." ], [ "Acknowledgments", "The computational work was carried out at the Ohio Supercomputer Center in Columbus Ohio.", "This work was partially supported by a grant from the NASA Astrophysical Research and Analysis program.", "EP would like to gratefully acknowledge a Summer Undergradute Research Program grant from the Ohio State University." ] ]

[ [ "AdS and Lifshitz Black Holes in Conformal and Einstein-Weyl Gravities" ], [ "Abstract We study black hole solutions in extended gravities with higher-order curvature terms, including conformal and Einstein-Weyl gravities.", "In addition to the usual AdS vacuum, the theories admit Lifshitz and Schr\\\"odinger vacua.", "The AdS black hole in conformal gravity contains an additional parameter over and above the mass, which may be interpreted as a massive spin-2 hair.", "By considering the first law of thermodynamics, we find that it is necessary to introduce an associated additional intensive/extensive pair of thermodynamic quantities.", "We also obtain new Liftshitz black holes in conformal gravity and study their thermodynamics.", "We use a numerical approach to demonstrate that AdS black holes beyond the Schwarzschild-AdS solution exist in Einstein-Weyl gravity.", "We also demonstrate the existence of asymptotically Lifshitz black holes in Einstein-Weyl gravity.", "The Lifshitz black holes arise at the boundary of the parameter ranges for the AdS black holes.", "Outside the range, the solutions develop naked singularities.", "The asymptotically AdS and Lifshitz black holes provide an interesting phase transition, in the corresponding boundary field theory, from a relativistic Lorentzian system to a non-relativistic Lifshitz system." ], [ "Introduction", "Theories of gravity extended by the addition of higher-order curvature terms are of interest for a number of reasons.", "One motivation is to investigate whether suitably extended four-dimensional gravity can be quantized in its own right.", "It has been shown that Einstein gravity extended by the inclusion of quadratic curvature terms is perturbatively renormalizable [1].", "However, the price to be paid for achieving renormalizability is that the theory then contains massive spin-2 modes as well as the massless graviton and, furthermore, that the massive modes are ghostlike (i.e.", "their kinetic term has the wrong sign).", "Three-dimensional models of gravity, for which the UV divergence problems are inherently less severe, have also been studied extensively.", "While Einstein gravity itself is essentially trivial in three dimensions, extensions to include higher-order derivative terms lead to interesting toy models with dynamical content and the possibility of well-controlled UV behaviour.", "Such extensions in three dimensions include topologically massive gravity [2], and more recently, “New Massive Gravity” [3].", "The theory can be rendered ghost free, and equivalent to a theory with a standard Einstein-Hilbert action, after truncating out modes with logarithmic fall-off by imposing an appropriate boundary condition of AdS$_3$ .", "(See, for example, [4].)", "Supersymmetric extensions were considered in [5]-[9].", "The situation is rather more subtle in four dimensions.", "An analogous “critical gravity” in four dimensions was considered in [10].", "The Lagrangian consists of the Einstein-Hilbert term with a cosmological constant $\\Lambda $ and an additional higher-order term proportional to the square of the Weyl tensor, with a coupling constant $\\alpha $ .", "It was shown that there is a critical relation between $\\alpha $ and $\\Lambda $ for which the generically-present massive spin-2 modes disappear, and are instead replaced by modes with a logarithmic fall-off [11] (see also [12], [13]).", "These log modes are ghostlike in nature [14], [15], but since they fall off more slowly than do the massless spin-2 modes, they can be truncated out by imposing an appropriate AdS boundary condition.", "The resulting theory is then somewhat trivial, in the sense that the remaining massless graviton has zero on-shell energy.", "Furthermore, the mass and the entropy of standard Schwarzschild-AdS black holes are both zero in the critical theory.", "Analogous critical theories exist also in higher-dimensional extended gravities with curvature-squared terms [16].", "In fact, it was observed in [17] that one can generalise critical gravity to a wider class of Weyl-squared extensions to cosmological Einstein gravity, where $\\alpha $ does not take the critical value.", "For a certain range of values for $\\alpha $ , the mass-squared of the massive spin-2 mode in the AdS$_4$ background is negative, but not sufficiently negative to imply tachyonic behaviour.", "However, this mode, which is again ghostlike, has a slower fall-off than the massless graviton and so it can be truncated by imposing appropriate AdS boundary conditions.", "This extended class of models has been investigated further in [18]-[34].", "One possible approach would be to begin with the conformally-invariant theory described by a pure Weyl-squared action.", "Being the local gauge theory of the conformal group, this theory of “conformal gravity” has the virtue of yielding a convergent Euclidean functional integral, and also of renormalizability and asymptotic freedom [35].", "One might then argue [36], [37] that quantum fluctuations would break the scale invariance, and thereby generate an Einstein-Hilbert term in the low-energy effective action.", "Thus, the Einstein-Weyl extensions of critical gravity described above effectively describe the emergence [38] of Einstein gravity from conformal gravity.", "It then becomes of interest to investigate the classical solutions of conformal gravity and Einstein-Weyl gravity.", "Any solution of Einstein gravity with a cosmological constant is also a solution of Einstein-Weyl gravity.", "However, the Weyl-squared term gives rise to fourth-order equations of motion, which are highly non-linear, and so it is in general rather difficult to find the further new solutions that exist over and above the pure Einstein solutions.", "One of the main purposes of the present paper is to search for such new solutions under the simplifying assumption of spherical symmetry (and certain generalisations of this).", "The investigation of solutions in higher-derivative extensions of Einstein gravity is also of interest from the AdS/CFT viewpoint, not least because it is known that such higher-order terms arise in string theory.", "Furthermore, although originally the AdS/CFT correspondence was conceived as a duality between a conformal field theory and a string theory, the idea of holography has been generalized to broader classes of gauge/gravity duality outside the string theoretical context.", "Recently, holographic techniques have been used to study non-relativistic systems, such as atomic gases at ultra-low temperature.", "This entails two types of gravitational backgrounds: those which correspond to Lifshitz-like fixed points [39] and Schrödinger-like fixed points [40], [41].", "In the context of condensed matter theory, various systems exhibit a dynamical scaling near fixed points: $t\\rightarrow \\lambda ^z t,\\qquad x_i\\rightarrow \\lambda x_i,\\qquad z\\ne 1\\,.$ In other words, rather than obeying the conformal scale invariance $t\\rightarrow \\lambda t$ , $x_i\\rightarrow \\lambda x_i$ , the temporal and the spatial coordinates scale anisotropically.", "Requiring also time and space translation invariance, spatial rotational symmetry, spatial parity and time reversal invariance, the authors of [39] were led to consider $D$ dimensional geometries of the form $ds^2=\\ell ^2\\biggl (-r^{2z}dt^2+r^2dx_idx^i+\\frac{dr^2}{r^2}\\biggr )\\,.$ This metric obeys the scaling relation (REF ) if one also scales $r\\rightarrow \\lambda ^{-1}r$ .", "If $z=1$ , the metric reduces to the usual AdS metric in Poincaré coordinates with AdS radius $\\ell $ .", "Metrics of the form (REF ) can be obtained as solutions in general relativity with a negative cosmological constant if appropriate matter is included.", "For example, solutions were found by introducing 1-form and 2-form gauge fields [39]; a massive vector field [42]; in an abelian Higgs model [43]; and with a charged perfect fluid [44].", "A class of Lifshitz black hole solutions with non-planar horizons was found in [45], [46].", "String theory and supergravity embeddings have been found in [47]-[55].", "In a similar vein, $D$ -dimensional geometries which exhibit Schrödinger symmetry are described by a metric of the form [40], [41] $ds^2=\\ell ^2\\biggl (-r^{2z}dt^2+r^2(-dtd\\xi +dx_idx^i)+\\frac{dr^2}{r^2}\\biggr )\\,,$ which is conformally related to a pp-wave spacetime.", "This metric obeys the scaling relation $t\\rightarrow \\lambda ^z t,\\qquad x_i\\rightarrow \\lambda x_i,\\qquad r\\rightarrow \\lambda r,\\qquad \\xi \\rightarrow \\lambda ^{2-z} \\xi ,\\qquad z\\ne 1\\,.", "$ If momentum along the $\\xi $ direction is interpreted as rest mass, then this metric describes a system which exhibits time and space translation invariance, spatial rotational symmetry, and invariance under the combined operations of time reversal and charge conjugation.", "These geometries have been embedded in string theory [56], [57].", "The organization of this paper is as follows.", "Section 2 contains a brief description of four-dimensional Einstein-Weyl gravity, including the equations of motion.", "In section 3, we summarise some salient features of the AdS$_4$ solution of Einstein-Weyl gravity and the nature of the linearised fluctuations around the AdS$_4$ background.", "We also discuss the Lifshitz solutions of Einstein-Weyl gravity, deriving the relation between the coupling strength $\\alpha $ of the Weyl-squared term and the value of the Lifshitz anisotropy parameter $z$ .", "We also find Schrödinger-type solutions.", "In section 4, we consider a black hole type ansatz for spherically-symmetric solutions of Einstein-Weyl gravity, and generalisations where the spatial sections are flat or hyperbolic instead of spherical.", "We show that the fourth-order equations of motion can be reduced to second-order equations for the metric functions.", "In the special case of flat spatial sections, we also derive a conserved Noether charge, which for the standard black hole solution is related to the mass.", "In section 5, we consider spherically-symmetric black holes in the specific case of pure conformal gravity.", "They can have either AdS or Lifshitz asymptotic behaviour.", "The AdS black holes in conformal gravity have an additional parameter over and above the mass, and this leads to interesting consequences when one considers their thermodynamics.", "We discuss how one may generalise the first law of thermodynamics to include the additional parameter.", "We also consider the asymptotically-Lifshitz black holes in conformal gravity, which can have either $z=4$ or $z=0$ .", "In section 6, we extend this discussion to Einstein-Weyl gravities.", "Now, it appears that the equations governing the metric functions for the spherically-symmetric ansatz are too complicated to be solvable in general, and so we have to resort to numerical methods in order to go beyond the known Schwarzschild-AdS metrics.", "To do this, we first give a discussion of the forms of the solutions in the near-horizon and the asymptotic regions.", "Then upon performing numerical integrations outwards from the near-horizon region, we find indications that more general black hole solutions do indeed exist, at least when the $\\alpha $ coupling parameter for the Weyl-squared term lies in an appropriate range.", "Section 7 contains our conclusions.", "We present further solutions of conformal gravity and general extended gravities with quadratic curvature-squared terms in appendices A and B, respectively.", "We summarise some results on the calculation of the mass for black holes in the critical theory and in conformal gravity in Appendices C and D, respectively." ], [ "Extended and Critical Gravity", "We begin by considering the action $I = {\\frac{1}{2\\kappa ^2}}\\,\\int \\sqrt{-g}\\,d^4x(R -2\\Lambda + \\alpha R^{\\mu \\nu } R_{\\mu \\nu } +\\beta R^2 + \\gamma E_{\\rm GB})\\,,$ where $\\kappa ^2 = 8\\pi G$ and $E_{GB}$ is the Gauss-Bonnet term $E_{GB}=R^2 - 4 R^{\\mu \\nu }R_{\\mu \\nu } +R^{\\mu \\nu \\rho \\sigma }R_{\\mu \\nu \\rho \\sigma }\\,.$ Although this term does not contribute to the equations of motion in four dimensions, it can have non-trivial consequences for thermodynamics in the higher-derivative theory, and so we shall include it in our discussion.", "The equations of motion that follow from the action (REF ) are ${{\\cal G}}_{\\mu \\nu } + E_{\\mu \\nu }=0\\,,$ where ${{\\cal G}}_{\\mu \\nu } &=& R_{\\mu \\nu } -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } R_{\\mu \\nu } + \\Lambda \\, g_{\\mu \\nu }\\,,\\\\E_{\\mu \\nu } &=& 2\\alpha (R_{\\mu \\rho }\\, R_\\nu {}^\\rho -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } } R^{\\rho \\sigma }R_{\\rho \\sigma }\\, g_{\\mu \\nu }) + 2\\beta R\\, (R_{\\mu \\nu } -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } } R\\, g_{\\mu \\nu })\\nonumber \\\\&&+\\alpha \\, ( \\square R_{\\mu \\nu } + \\nabla _\\rho \\nabla _\\sigma R^{\\rho \\sigma }\\,g_{\\mu \\nu } - 2\\nabla _\\rho \\nabla _{(\\mu } R_{\\nu )}{}^\\rho ) +2\\beta \\, (g_{\\mu \\nu }\\, \\square R -\\nabla _\\mu \\nabla _\\nu R)\\,.$ When $\\beta =-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\alpha $ and $\\gamma ={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\alpha $ , the theory describes what we shall call Einstein-Weyl gravity, with the action $I = {\\frac{1}{2\\kappa ^2}}\\,\\int \\sqrt{-g}\\,d^4x(R -2\\Lambda + {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\alpha |{\\rm Weyl}|^2)\\,,$ where $|{\\rm Weyl}|^2 = R^{\\mu \\nu \\rho \\sigma } R_{\\mu \\nu \\rho \\sigma } - 2R^{\\mu \\nu }R_{\\mu \\nu } + {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } } R^2\\,.$ Note that the equations of motion following from this action can be written asIn deriving this result it is helpful to note that, in four dimensions, the Weyl tensor satisfies the identity $C_{\\mu \\rho \\sigma \\lambda } C_\\nu {}^{\\rho \\sigma \\lambda } = {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }C_{\\rho \\sigma \\lambda \\tau } C^{\\rho \\sigma \\lambda \\tau }\\, g_{\\mu \\nu }$ .", "This can be seen easily by employing 2-component spinor notation.", "$R_{\\mu \\nu } -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } R g_{\\mu \\nu } + \\Lambda g_{\\mu \\nu }-2\\alpha (2\\nabla ^\\rho \\nabla ^\\sigma + R^{\\rho \\sigma })C_{\\mu \\rho \\sigma \\nu }=0 \\,.", "$ We shall also sometimes consider the limit of conformal gravity, where only the Weyl-squared term in the action is retained.", "This can be described as the $\\alpha \\longrightarrow \\infty $ limit of the Einstein-Weyl action (REF ).", "The action of conformal gravity is conformally invariant, which implies that the equations of motion determine the metric only up to an arbitrary conformal factor.", "A special feature of four-dimensional Einstein gravity with curvature-squared terms is that any solution of the pure Einstein theory continues to be a solution of the theory with the quadratic modifications.", "Thus, in particular, the Schwarzschild-AdS black hole solution $ds^2 = -\\Big (k - {\\frac{2m}{r}} - {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\Lambda r^2\\Big )\\, dt^2 +\\Big (k - {\\frac{2m}{r}} - {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\Lambda r^2\\Big )^{-1}\\, dr^2 +r^2 d\\Omega _{2,k}^2 $ of Einstein gravity is also a solution in the higher-derivative theory.", "Here $d\\Omega _{2,k}^2$ denotes the metric on a unit 2-sphere ($k=1$ ), unit hyperbolic plane ($k=-1$ ) or 2-torus ($k=0$ ), and may be written as $d\\Omega _{2,k}^2 ={\\frac{dx^2}{1-k\\,x^2}} + (1-k\\,x^2) dy^2\\,.$" ], [ "AdS, Lifshitz and Schrödinger Vacua", "Unlike the $D=3$ or $D\\ge 5$ cases, for $D=4$ the cosmological constant of the (A)dS vacuum is not modified by the quadratic curvature terms, and hence we have only one such vacuum with cosmological constant $\\Lambda $ .", "In this paper we shall consider only negative $\\Lambda $ , and furthermore, without loss of generality, we shall from now on set $\\Lambda =-3$ .", "The linearised fluctuations around the AdS$_4$ vacuum in Einstein-Weyl gravity were analyzed in [10].", "It turns out that the scalar trace mode decouples from the spectrum, which then contains just massless and massive spin-2 modes, satisfying $\\alpha (\\Box +2)(\\Box +2 -m^2)h_{\\mu \\nu }=0\\,,$ where $h_{\\mu \\nu }$ is transverse traceless and $m^2 =-2 -{\\frac{1}{\\alpha }}\\,.$ The characteristics of the linearised theory depend upon the value of the parameter $\\alpha $ , and are summarised in the table below.", "Table: NO_CAPTION Table 1: The characteristics of the massive and massless spin-2 modes in Einstein-Weyl gravity for finite values of the parameter $\\alpha $ .", "When not indicated to the contrary, the modes are non-ghostlike.", "Owing to the fact that the background is AdS rather than Minkowski spacetime, there is an allowed range of negative mass-squared values for the massive mode, $-{\\textstyle {\\frac{\\scriptstyle 9}{\\scriptstyle 4} } }\\le m^2<0$ , for which it is still non-tachyonic.", "In this range, the massive mode actually falls off less rapidly at infinity than the massless mode, and so it can be truncated from the theory by imposing a suitable boundary condition at infinity.", "In the critical theory, which occurs when $\\alpha =-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ , the massive mode becomes massless and in fact a new type of mode with logarithmic fall off arises.", "The usual massless mode has zero norm in the critical theory, and the logarithmic mode is ghostlike.", "The logarithmic mode could be truncated by imposing appropriate boundary conditions, but this would leave only the zero-norm massless graviton [14], [15].", "The case $\\alpha =0$ corresponds to ordinary Einstein gravity, in which case there is of course no massive mode.", "Not depicted in the table is the case $\\alpha =\\pm \\infty $ , which corresponds to the pure Weyl-squared conformal theory.", "In the conformal theory the massive mode has $m^2=-2$ , and so although negative, it is not tachyonic.", "In addition to the AdS vacuum, the theory (REF ) also admits Lifshitz solutions, for which the metric is given by $ds^2={\\frac{dr^2}{\\sigma \\,r^2}} - r^{2z} dt^2 + r^2 (dx^2 + dy^2)\\,,$ where $(z^2+2)\\alpha + 2(z^2 + 2z + 3)\\beta ={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 12} } } (z^2 + 2z+3)\\,,\\qquad \\sigma = {\\frac{6}{z^2+2z + 3}}\\,.$ For the special case of Einstein-Weyl gravity, where $\\beta =-\\alpha /3$ , we have $\\alpha ={\\frac{z^2+2z+3}{4z(z-4)}}\\,.$ For conformal gravity, corresponding to $\\alpha =\\infty $ , equation (REF ) implies that the Lifshitz scaling parameter $z$ can take the values $z=4$ or $z=0$ .", "At the critical point, on the other hand, where $\\alpha =-{\\frac{1}{2}}$ , both roots of (REF ) give $z=1$ .", "For general values of $\\alpha $ we have $z={\\frac{8\\alpha +1\\pm \\sqrt{2(1+2\\alpha )(16\\alpha -1)}}{4\\alpha -1}}\\,.$ Thus, the reality of $z$ requires that $\\alpha \\ge {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 16} } }$ or $\\alpha \\le {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ .", "For conformal gravity, we find that there are also Lifshitz-like solutions with $S^2$ or $H^2$ spatial topologies as well as $T^2$ .", "The metrics for all three cases can be written as $ds^2 = -r^{2z} (1+{\\frac{k}{r^2}}) dt^2+ {\\frac{4dr^2}{r^2(1+{\\frac{k}{r^2}})}} + r^2d\\Omega _{2,k}^2\\,,$ with $z=0$ and 4.", "Finally, we consider Schrödinger vacua, whose metric takes the form $ds^2=-r^{2z} dt^2 + {\\frac{dr^2}{r^2}} + r^2 (-2dt dx + dy^2)\\,.$ For $z=1$ and $z=-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ , the metrics are Einstein with $\\Lambda =-3$ , and hence they are solutions for all $\\alpha $ and $\\beta $ .", "In particular, the $z=1$ case is simply the AdS metric, whilst if $z=-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ it is the Kaigorodov metric describing a pp-wave propagating in AdS [58], [59].", "In general, $z$ satisfies [12] $1 - 24 \\beta + \\alpha (4z^2 - 2z - 8)=0\\,.$ In Einstein-Weyl gravity, we have $\\alpha ={\\frac{1}{2z(1-2z)}}\\,.$ In conformal gravity, $z$ can take values 1, ${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ , 0 or $-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ .", "Some asymptotic Schrödinger solutions are presented in appendix A.1." ], [ "General equations for $k=1$ , 0 or {{formula:1321538e-d07f-457f-9538-57ede53f3e53}}", "In this paper, we focus on the construction of static, spherically-symmetric (or $H^2$ or $T^2$ symmetric) black hole solutions that are asymptotic to either the AdS or the Lifshitz vacua discussed in the previous section.", "We may therefore, without loss of generality, consider the ansatz $ds^2=-a(r)\\, dt^2 + {\\frac{dr^2}{f(r)}} + r^2d\\Omega _{2,k}^2\\,.$ The equations of motion for $a$ and $f$ may be derived from the Lagrangian obtained by substituting this ansatz into the action (REF ).", "Since we are interested specifically in the case of Einstein-Weyl gravity, where $\\beta =-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\alpha $ , and since the equations of motion are greatly simplified in this case,The reason for the simplification is that the trace of the Weyl-squared contribution to the equations of motion vanishes (see equation (REF )), and so this projection is identical in Einstein gravity and Einstein-Weyl gravity.", "we shall present the results under this specialisation.", "We then find that the equations can be reduced to the second-order system $a^{\\prime \\prime }={\\frac{r^2f a^{\\prime 2} + 4a^2(k+6r^2-f-rf^{\\prime }) - ra a^{\\prime }(4f+rf^{\\prime })}{2r^2af}}\\,,$ and $f^{\\prime \\prime } &=& {\\frac{1}{2 r^2 a^2 f (r a^{\\prime }-2a)}}\\Big ({\\frac{4r^2a^2}{\\alpha }}(a(k+3r^2-f) - r f a^{\\prime }) +r^3 f^2 a^{\\prime 3} + 2r^2 a^2 fa^{\\prime }(8r-f^{\\prime })\\cr &&-r^2afa^{\\prime 2}(3f+r f^{\\prime }) - a^3 (48r^4-16r^2f + 8f^2-24r^3f^{\\prime } + 4rff^{\\prime }+3r^2f^{\\prime 2})\\cr &&-4ka^3(4r^2-2f-r f^{\\prime })\\Big )\\,.$" ], [ "Conserved Noether charge for $k=0$ case", "In the case of a toroidal spatial geometry (i.e.", "$k=0$ ), the system of equations has an additional global symmetry, and hence there is an associated conserved Noether charge.", "In order to discuss this, it is helpful temporarily to choose a different parameterisation of the metric ansatz (REF ), using a new radial coordinate $\\rho $ such that $r^2=b(\\rho )$ and $dr^2/f=ab^2h d\\rho ^2$ , so that the metric is now written as $ds^2 = - a(\\rho ) dt^2 + a(\\rho ) b(\\rho )^2\\, h(\\rho ) d\\rho ^2 + b(\\rho )\\,dx^i dx^i\\,.$ Since the additional global symmetry arises regardless of whether or not we choose the Weyl-squared combination $\\beta =-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\alpha $ , we shall keep these two parameters arbitrary in the following discussion.", "Substituting into the action (REF ) yields an effective Lagrangian for $a$ , $b$ and $h$ .", "The function $h(\\rho )$ can be viewed as parameterising general coordinate transformations of the radial variable, and its equation of motion yields the Hamiltonian constraint $H=0$ .", "Having obtained this equation, we can impose $h(\\rho )=1$ as a coordinate gauge condition.", "In this case $H$ , which must vanish, is given by $H&=&{\\frac{a^{\\prime }b^{\\prime }}{ab}} + {\\frac{b^{\\prime 2}}{2b^2}} - 6 ab^2 - 2 k a b \\cr &&-{\\frac{\\alpha }{4a^5b^6}}\\Big (10a^4a^{\\prime 4} + 20 ab^3 a^{\\prime 3}b^{\\prime } + 22a^2 b^2a^{\\prime 2} b^{\\prime 2} + 36 a^3 b a^{\\prime } b^{\\prime 3} + 47 a^4b^{\\prime 4}\\Big )\\cr && + {\\frac{\\alpha }{a^4b^5}}\\Big (b(ba^{\\prime }+ a b^{\\prime })(3ba^{\\prime }+2ab)a^{\\prime \\prime } + a(b a^{\\prime } + 3a b^{\\prime })(b a^{\\prime } + 4 a b^{\\prime }) b^{\\prime \\prime }\\Big )\\cr && +{\\frac{\\alpha }{a^3b^4}}\\Big (b^2a^{\\prime \\prime 2} + 2 a b a^{\\prime \\prime } b^{\\prime \\prime } + 3a^2 b^{\\prime \\prime 2}\\Big )-{\\frac{2\\alpha k(k a b^3 + b^{\\prime 2})}{b^3}}\\cr &&-{\\frac{\\beta }{4a^5b^6}}\\Big (20b^4 a^{\\prime 4} + 64 a b^3 a^{\\prime 3} b^{\\prime } + 72a^2b^2 a^{\\prime 2} b^{\\prime 2} + 124 a^3 b a^{\\prime } b^{\\prime 3} + 125a^4b^{\\prime 4}\\Big )\\cr &&+{\\frac{2\\beta }{a^4b^5}}\\Big (3b(b^2a^{\\prime 2}+3aba^{\\prime }b^{\\prime }+a^2b^{\\prime 2})a^{\\prime \\prime } +a(b^2a^{\\prime 2}+13ab a^{\\prime } b^{\\prime } + 16a^2 b^{\\prime 2})b^{\\prime \\prime }\\Big )\\cr &&+{\\frac{\\beta (ba^{\\prime \\prime }+2ab^{\\prime \\prime })^2}{a^3b^4}}-{\\frac{2\\beta (ba^{\\prime }+2ab^{\\prime })(ba^{\\prime \\prime \\prime }+2ab^{\\prime \\prime \\prime })}{a^3b^4}}- {\\frac{2\\beta k(2k a b^3 + 3 b^{\\prime 2})}{b^3}}\\,,$ where a prime here denotes a derivative with respect to $\\rho $ .", "We may then also set $h=1$ in the effective Lagrangian, so that the remaining equations, for $a$ and $b$ , can be obtained from ${\\cal L}&=&{\\frac{a^{\\prime }b^{\\prime }}{a b}} + {\\frac{b^{\\prime 2}}{2b^2}} + 6 ab^2 + 2 k ab\\cr && + {\\frac{\\alpha }{4a^5b^6}} \\Big (2b^4a^{\\prime 4} + 5ab^3 a^{\\prime 3}b^{\\prime } + 10 a^2 b^2a^{\\prime 2} b^{\\prime 2} +12ab^3a^{\\prime }b^{\\prime 3}+ 11a^4b^{\\prime }4\\cr &&-2ab(2b^3a^{\\prime 2}a^{\\prime \\prime } + 2ab^2 a^{\\prime }b^{\\prime }a^{\\prime \\prime } + 3a^2bb^{\\prime 2} a^{\\prime \\prime } +2ab^2a^{\\prime 2}b^{\\prime \\prime } + 4a^2ba^{\\prime }b^{\\prime }b^{\\prime \\prime } + 8a^3b^{\\prime 2}b^{\\prime \\prime })\\cr &&+2a^2b^2(b^2a^{\\prime \\prime 2} + 2 a b a^{\\prime \\prime } b^{\\prime \\prime } +3a2b^{\\prime \\prime 2})\\Big ) +{\\frac{2k\\,\\alpha }{b^3}}(kab^3 + b^{\\prime 2} - b b^{\\prime \\prime })\\cr &&+{\\frac{\\beta }{4a^5b^6}}\\Big (4ka^3b^3 + 2b^2a^{\\prime 2} + 2aba^{\\prime }b^{\\prime } +5 a^2 b^{\\prime 2} -2ab(ba^{\\prime \\prime } + 2ab^{\\prime \\prime })\\Big )^2\\,.$ For the $k=0$ case, corresponding to a black brane solution, the Lagrangian (REF ) and Hamiltonian (REF ) have a global scaling symmetry with $a\\rightarrow \\xi ^2\\,a\\,,\\qquad b\\rightarrow \\xi ^{-1}\\, b\\,.$ This enables us to derive a conserved Noether charge, $\\lambda $ .", "Having done this, it is more convenient now to revert to the original radial coordinate $r$ and the metric functions $a$ and $f$ in (REF ).", "The Noether charge is then given by $\\lambda &=& {\\frac{\\sqrt{f}}{ra^{5/2}}}\\Big [2r^2a^2(ra^{\\prime }-2a)-\\alpha \\Big ( 8a^3 f - 2 r a^2 f a^{\\prime } - 4r^2 a f a^{\\prime 2} + 3r^3 f a^{\\prime 3} +6r a^3 f^{\\prime }\\cr &&- 3r^3 a a^{\\prime 2} f^{\\prime } + 3r^2a (2af-2rfa^{\\prime }+raf^{\\prime })a^{\\prime \\prime } +r^2a^2(ra^{\\prime }-2a) +2r^3a^2ff^{\\prime \\prime \\prime }\\Big )\\cr &&+2\\beta (ra^{\\prime }-2a)(4a^2f +4rafa^{\\prime }-r^2fa^{\\prime 2}+4ra^2f^{\\prime }+r^2aa^{\\prime }f^{\\prime }+2r^2afa^{\\prime \\prime })\\Big ]\\,.", "$ It should be emphasized that this quantity is conserved only for the case $k=0$ .", "The analogous Noether charge was studied in [60] for Lifshitz black holes (with $T^2$ horizon topology) in Einstein gravity coupled to a massive vector field.", "It was shown [61] that it is related to the energy of the black branes: $E=-{\\frac{\\lambda \\omega _2}{16\\pi \\,(z+2)}}={\\frac{2}{(z+2)}} T\\,S\\,,$ where $T$ and $S$ are the temperature and the entropy of the black brane, and $\\omega _2$ is its area.", "We can test this formula with the $k=0$ Schwarzschild-AdS black holes, corresponding to $a=f=r^2 - r_+^3/r$ in the metric ansatz (REF ).", "This solution exists for all values of the $\\alpha $ and $\\beta $ parameters.", "The temperature and the entropy are given by $T={\\frac{3r_+}{4\\pi }}\\,,\\qquad S={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\omega _2 r_+^2[1 - 6(\\alpha + 4\\beta )]\\,.$ Note that the Gauss-Bonnet term does not contribute to the entropy in this case.", "The energy is given by $E={\\frac{1}{8\\pi }} [1 - 6(\\alpha + 4\\beta )] r_+^3 = {\\textstyle {\\frac{\\scriptstyle 2}{\\scriptstyle 3} } } TS\\,.$ The Noether charge $\\lambda $ in this case is given by $\\lambda =-6 [1 - 6 (\\alpha + 4\\beta )] r_+^3\\,.$ Thus, we find that the relation (REF ) holds for this $z=1$ case.", "In general we find that the second equality in (REF ) always holds, whilst the definition of energy in terms of the Noether charge does not apply for solutions of higher-derivative gravity when massive spin-2 modes are excited.", "For the case of Einstein-Weyl gravity, i.e.", "when $\\beta =-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\alpha $ , the Noether charge (REF ) simplifies considerably, and becomes $\\lambda &=&{\\frac{1}{\\sqrt{a^3f}\\,(ra^{\\prime }-2a)}}\\Big (2ra(18ra^2-10a^2f-2rafa^{\\prime }-r^2fa^{\\prime 2})\\cr &&-\\alpha (4ra-fa^{\\prime }-af^{\\prime })(36r^2a^2-8a^2f - rafa^{\\prime } - 2r^2 fa^{\\prime 2}-9ra^2f^{\\prime })\\Big )\\,.$" ], [ "AdS and Lifshitz Black Holes in Conformal Gravity", "In this section, we focus on the special case of conformal gravity, i.e.", "the limiting case of Einstein-Weyl gravity when $\\alpha $ goes to infinity.", "The equations of motion are given by $(2\\nabla ^\\rho \\nabla ^\\sigma + R^{\\rho \\sigma })C_{\\mu \\rho \\sigma \\nu }=0\\,.$ Note that for the metric ansatz (REF ), the $\\alpha $ -independent trace equation (REF ) does not apply in conformal gravity.", "Thus, the equation of motion is not simply (REF ) with $\\alpha $ sent to $\\infty $ ." ], [ "AdS black holes", "The most general spherically-symmetric solution in conformal gravity was found in [62], [63], [64].", "(See also, [65], [31].)", "The solution can easily be generalized to the other horizon topologies $T^2$ and $H^2$ .", "The solution for all three cases is given by $ds^2=-f dt^2 + {\\frac{dr^2}{f}} + r^2 d\\Omega _{2,k}^2\\,,\\qquad f=b r^2+ {\\frac{c^2-k^2}{3d}}\\, r + c + {\\frac{d}{r}}\\,,$ where $b$ , $c$ and $d$ are arbitrary constants.", "The coefficients of $r^2$ and $1/r$ are related to the excitations of the massless graviton, while the coefficient of $r$ and the constant $c$ are related to the massive spin-2 mode.", "If $c$ is chosen so that $c=k$ , the solution reduces to the usual AdS black hole for each of the cases $k=1$ , $k=-1$ and $k=0$ .", "Of course, since the equations of motion for conformal gravity leave an overall conformal factor undetermined, it follows that $d\\tilde{s}^2 = \\Omega ^2\\, ds^2$ is also a spherically-symmetric static solution, where $ds^2$ is given by (REF ) and $\\Omega $ is an arbitrary function of $r$ .", "In fact, the solution (REF ) can easily be derived by starting from the Schwarzschild-AdS solution $d\\tilde{s}^2= -\\Big (k -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\Lambda \\rho ^2 -{\\frac{2M}{\\rho }}\\Big )\\,dt^2 + \\Big (k -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\Lambda \\rho ^2 -{\\frac{2M}{\\rho }}\\Big )^{-1}\\,d\\rho ^2 + \\rho ^2\\,d\\Omega _{2,k}^2\\,,$ noting that not only this, but also $ds^2=\\Omega (\\rho )^{-2}\\,d\\tilde{s}^2$ , is therefore a solution of conformal gravity, and then defining a new radial coordinate via $r=\\rho \\, \\Omega (\\rho )^{-1}$ .", "Requiring that the resulting metric have the form $ds^2=-h dt^2 +h^{-1} dr^2 + r^2 d\\Omega _{2,k}^2$ implies that $\\Omega =1 + q \\rho $ where $q$ is an arbitrary constant, and hence $r=\\rho /(1+q\\rho )$ .", "The function $h$ is therefore given by $h= (2M q^3 + k q^2-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\Lambda ) r^2 -2q(k+3Mq)r + (k+ 6M q)-{\\frac{2M}{r}}\\,,$ which precisely reproduces the function $f$ in (REF ) with $b= 2M q^3 + k q^2-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\Lambda \\,,\\qquad c= k+ 6Mq\\,,\\qquad d= -2M\\,.$ The fact that the solution (REF ) is related to the usual Schwarzschild-AdS black hole does not imply that these solutions are completely equivalent.", "The scaling of the Schwarzschild-AdS metric leaves the thermodynamic properties of the black hole unchanged only if the conformal factor is finite and non-vanishing in the regions between the horizon and asymptotic infinity.", "However, the conformal factor $\\Omega =1+q\\rho $ that relates (REF ) to the usual Schwarzschild-AdS black hole metric is in fact singular at $\\rho =\\infty $ , and so it alters the global structure.", "In turn, this affects the thermodynamic properties, as we shall discuss below." ], [ "Thermodynamics of AdS black holes in conformal gravity", "We begin by reviewing the thermodynamic properties of the standard Schwarzschild-AdS black hole (REF ) in the context of conformal gravity.", "Letting $\\rho _+$ denote the radius of the outer horizon, we can solve for $M$ to get $M={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 6} } } \\rho _+ (3k - \\Lambda \\rho _+^2)\\,.$ The Hawking temperature can be obtained from a calculation of the surface gravity in the standard way.", "The entropy can be derived from the Wald formula [66], giving $S=-\\frac{\\alpha }{8}\\int C^{\\mu \\nu \\rho \\sigma }\\epsilon _{\\mu \\nu }\\epsilon _{\\rho \\sigma }d\\Sigma \\,.", "$ Thus we have $T={\\frac{k-\\Lambda \\rho _+^2}{4\\pi \\rho _+}}\\,,\\qquad S={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 6} } } \\alpha (3k-\\Lambda \\rho _+^2)\\omega _2\\,,$ where $\\omega _2$ denotes the volume of $d\\Omega _{2,k}^2$ .", "It is worth remarking that at first sight the entropy of the black hole in conformal gravity is not simply proportional to the area of the horizon, but now it is given by $S={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\alpha \\Big (k\\,\\omega _2 + {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }(-\\Lambda )\\,A\\Big )\\,,$ where $A=\\rho _+^2 \\omega _2^2$ is the area of the horizon.", "However the first term in the above is a pure constant, independent of the parameters in the solution, and can be removed by introducing a Gauss-Bonnet term in the Lagrangian.", "In fact if we use the action (REF ) with $\\beta =-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\alpha $ and $\\gamma =0$ , the first term in (REF ) vanishes and hence the entropy is then proportional to the area of the horizon.", "The free energy $F$ can be obtained from the Euclidean action $I_E$ of conformal gravity, using the relation $F=I_E\\, T$ .", "The action converges for the black hole solution, leading to $F=-{\\frac{\\alpha \\omega _2}{32\\pi }} \\int _{r_+}^\\infty r^2 dr\\,|\\hbox{Weyl}|^2 =-{\\frac{\\alpha (3k - \\Lambda \\rho _+^2)^2 \\omega _2}{72\\pi \\rho _+}}\\,.$ The energy can be determined by integrating the first law, $dE=T dS$ , assuming that $\\Lambda $ is held fixed, giving $E={\\frac{\\alpha \\Lambda \\rho _+ (-3k + \\Lambda \\rho _+^2)\\omega _2}{36\\pi }} = {\\frac{\\alpha (-\\Lambda ) \\omega _2}{6\\pi }}\\, M\\,.$ This expression for the energy can also be confirmed independently by using either the Deser-Tekin [67] or the AMD method [68], [69], [70], [71].", "In conformal gravity the cosmological constant $\\Lambda $ is a parameter of the solution, rather than of the theory, and hence we may treat $\\Lambda $ as a further thermodynamic variable, leading to the more general thermodynamic relations $&&dE = T dS + \\Theta \\, d\\Lambda \\,,\\qquad F = E - T S\\,,\\qquad \\Theta = -{\\frac{\\alpha \\rho _+ (3k - \\Lambda \\rho _+^2)\\omega _2}{72\\pi }}\\,.$ Thus, treating the cosmological constant as a thermodynamic variable does not affect the relationship between $F$ and $E$ .", "We can simply start by assuming that $\\Lambda $ is constant and obtain the first law of thermodynamics.", "The first law can then be straightforwardly modified by treating $\\Lambda $ as a variable, thus determining the corresponding conjugate variable $\\Theta $ , whilst the other thermodynamic quantities remain unchanged.", "Treating the cosmological constant as a thermodynamic variable has been considered previously.", "See, for example, [72], [73], [74].", "In Einstein gravity, where the entropy is simply one quarter of the horizon area, without explicit dependence on the cosmological constant $\\Lambda $ , the quantity $\\Theta \\sim \\rho _+^3$ is proportional to the volume conjugate to the cosmological constant, which can then be interpreted as a pressure [74].", "In conformal gravity, on the other hand, the entropy has a manifest dependence on $\\Lambda $ , and hence the quantity $\\Theta $ given in (REF ) is not simply proportional to the volume, but has a linear $\\rho _+$ dependence as well, for non-vanishing $k$ .", "It is also worth remarking that the Smarr formula $E=2TS -2\\Theta \\Lambda $ in Einstein gravity [74] now becomes $E=2\\Theta \\Lambda $ in conformal gravity.", "We shall discuss this further in section 5.1.5.", "We are now in a position to discuss the more general AdS black holes in conformal gravity.", "We shall begin by taking the cosmological constant to be fixed,When we refer to the “cosmological constant” in the context of conformal gravity, where of course there is no cosmological term in the action, we always mean the cosmological constant of the asymptotic AdS space.", "by setting $b=1$ in (REF ), corresponding to setting the cosmological constant of the asymptotic AdS space to be $\\Lambda =-3$ .", "Letting $r_+$ be the radius of the outer horizon and writing $d=-r_+ \\tilde{d}$ , we find that $r_+^2 = -c + {\\frac{c^2-k^2}{3\\tilde{d}}} + \\tilde{d} >0\\,.$ The temperature and entropy can then be straightforwardly calculated; they are given by $T={\\frac{(3\\tilde{d}-c)^2 - k^2}{12\\pi r_+ \\tilde{d}}}\\,,\\qquad S={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 6} } }\\alpha (k+3 \\tilde{d}-c) \\omega _2\\,.$ The free energy $F$ can also be obtained from the Euclidean action, giving $F=-{\\frac{\\alpha \\omega _2 \\Big ((c-k)^2 - 3 (c-k) \\tilde{d} + 3 \\tilde{d}^2\\Big )}{24\\pi r_+}}\\,.$ When $c=k$ , the system reduces to the previous Schwarzschild-AdS black hole with cosmological constant fixed at $\\Lambda = -3$ .", "Thus, the general solution with $c\\ne k$ contains an additional independent parameter.", "As we have remarked in the previous subsection, the general solution can be obtained by performing a conformal transformation of the Schwarzschild-AdS black hole, whose cosmological constant $\\Lambda $ can be promoted to being a parameter of the solution.", "In the new solution, we have chosen to set $b=1$ .", "Thus as a local solution, our new variables $(c,d)$ are related to the $(M,\\Lambda )$ variables in the Schwarzschild-AdS solution (REF ).", "It is natural to ask whether the thermodynamics of the new solution are simply the same as (REF ), but expressed in terms of new variables.", "In order to address this issue, we note that the transformation described in subsection 5.1 amounts to $q={\\frac{c-k}{6M}}\\,,\\qquad M={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 6} } } \\rho _+(3k - \\Lambda \\rho _+^2)\\,,$ with $\\rho _+={\\frac{3r_+ \\tilde{d}}{k-c + 3\\tilde{d}}}\\,,\\qquad \\Lambda ={\\frac{(2k+c) r_+ - 3k\\rho _+}{(r_+ - \\rho _+) \\rho _+^2}}={\\frac{(c-k - 3\\tilde{d})^2 (c+2k - 3 \\tilde{d})}{9 r_+^2 \\tilde{d}^2}}\\,.$ It is easy to see that when $c=k$ , we have $\\rho _+=r_+$ and $\\Lambda = -3$ , as we should have expected.", "It is straightforward to verify that the temperature and entropy in (REF ) are indeed mapped into those in (REF ).", "However, the free energy in (REF ) becomes $F\\rightarrow \\widetilde{F} = -{\\frac{\\alpha (3 \\tilde{d}-c+k)^3\\omega _2}{216 \\pi r_+ \\tilde{d}}}\\,,$ which is different from the free energy given in (REF ).", "The reason for this can be easily understood as follows.", "The $r$ and $\\rho $ coordinates are related to each other by $r={\\frac{\\rho }{1 + q \\rho }}\\,,\\qquad \\rho = {\\frac{r}{1-q r}}\\,.$ The temperature and entropy are, in a sense, “local” properties, evaluated on the horizon $r=r_+$ or $\\rho =\\rho _+$ , and related by the above equation.", "Since the theory is conformal, the temperature and entropy are not modified by the conformal transformation.", "On the other hand, the free energy, and hence the energy, are evaluated by an integration over the regions $[r_+, \\infty )$ of the general black hole or $[\\rho _+,\\infty )$ of the Schwarzschild-AdS black hole.", "From the relationship (REF ), we find $r_+\\le r <\\infty &\\longrightarrow & (-\\infty <\\rho \\le -{\\frac{1}{q}})\\quad \\cup \\quad (\\rho _+\\le \\rho <\\infty )\\,,\\cr \\rho _+ \\le \\rho <\\infty & \\longrightarrow & r_+ \\le r <{\\frac{1}{q}}\\,.$ Thus, we see that the outer region $\\rho \\ge \\rho _+$ of the Schwarzschild-AdS black hole covers only part of the outer region $r\\ge r_+$ of the general black hole (REF ).", "The exterior of the general black hole maps into disconnected regions of the Schwarzschild-AdS solution.", "Thus, we see that although the conformal transformation does not affect the location of the horizon or the expression for $\\sqrt{-g} |\\hbox{Weyl}|^2$ , the structure of the asymptotic region is altered by the transformation.", "Therefore, the Euclidean actions are different for the two solutions.", "Analogously, the energy of the two solutions, which are typically evaluated at asymptotic infinity, are also different.", "The upshot is that the two solutions cannot be viewed as equivalent.", "Having established that the new solutions are globally inequivalent to the Schwarzschild-AdS black hole with $(M,\\Lambda )$ parameters, we shall now proceed to investigate the thermodynamics of the general black holes in conformal gravity.", "We should not expect the usual first law $dE=T dS$ still to be satisfied, since the general solutions are now described by two independent parameters, $c$ and $d$ , rather than just one.", "As we shall see, it is necessary now to introduce an additional pair of intensive and extensive thermodynamic variables, which we shall call $\\Psi $ and $\\Xi $ , and the first law will be modified to $dE=TdS+\\Psi d\\Xi $ .", "Once the additional parameter of the AdS black holes in conformal gravity is turned on, by taking $c\\ne k$ , we find that neither the Deser-Tekin nor the AMD methods gives a finite result for the mass.", "In appendix D, we describe a new procedure for calculating the mass in conformal gravity.", "It is instructive first to look at the solution where the parameter $d$ is set to zero.", "In the parameterisation in (REF ), this can be done by first writing $c^2-k^2= 3 \\Xi d$ , and then sending $d$ to zero, giving $f=r^2 + \\Xi \\, r + k\\,.$ This solution has a curvature singularity at $r=0$ , which can be shielded by an horizon at $r=r_0$ provided that $\\Xi $ is chosen so that $\\Xi ^2\\ge 4k$ .", "The temperature is given by $T_0={\\frac{r_0^2-k}{4\\pi r_0}}\\,.$ However, we find that the entropy and free energy both vanish, suggesting that the energy should vanish also.", "Thus the solution can be viewed as a “thermalized vacuum.” (This is analogous to the Schwarzschild black hole in critical gravity, where all thermodynamic quantities except for temperature vanish [10].)", "In a Deser-Tekin or AMD calculation, this thermalized vacuum will generate a divergence in the evaluation of the mass, and it should be subtracted.", "In fact it is easy to verify that this thermalized vacuum is locally conformal to a de Sitter background.", "To see this, we define $d\\hat{s}^2=\\Omega ^2 ds^2$ , with $\\Omega = {\\frac{1}{\\Xi r + 2k}}\\,,$ and introduce the new radial coordinate $\\rho =r\\Omega $ .", "We then have $d\\hat{s}^2 = -{\\frac{\\hat{f}}{4k}} dt^2 + {\\frac{d\\rho ^2}{k\\, \\hat{f}}} +\\rho ^2 d\\Omega _{2,k}^2\\,, $ where $\\hat{f} = 1 -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } } \\Lambda \\rho ^2\\,,\\qquad \\Lambda = 3(\\Xi ^2-4k)\\,.$ The condition for the solution (REF ) to have real roots defining the horizons is $\\Xi ^2-4k\\ge 0$ , and so this means the conformally-related metric $d\\hat{s}^2$ in (REF ) is de Sitter spacetime, with positive cosmological constant.", "The horizon in the metric with $f$ given by (REF ) maps into the de Sitter horizon in (REF ).", "From appendix D, if we take the conserved quantity $Q$ to furnish a definition of energy, we have $E={\\frac{\\alpha \\,\\omega _2}{4\\pi }} (-d + m)\\,,$ where $m\\equiv {\\frac{(c-k)(c^2-k^2)}{18d}}\\,.$ Note that when $c=k$ , it reproduces the energy for the Schwarzschild-AdS black hole.", "When $d=0$ , it is necessary that $c\\rightarrow k$ with $\\Xi =(c^2-k^2)/(3d)$ held fixed.", "In this limit, the quantity $m$ vanishes, and hence we see that the thermalized vacuum indeed has zero energy.", "It turns out that with this definition of energy for the general AdS black holes in conformal gravity we have $F=E - TS\\,.$ We find that, as mentioned earlier, the standard first law $dE=TdS$ is not satisfied for the general AdS black holes, since the solutions are characterised by two independent parameters.", "If we first consider the situation where the quantity $\\Xi =(c^2-k^2)/(3d)$ is held fixed, then the first law $dE=TdS$ does hold.", "This corresponds to allowing only variations that keep the thermalized vacuum fixed.", "If instead we allow general variations of the two parameters in the solution, by allowing $\\Xi $ to vary also, then we find that we should add another term in the first law, which now becomes $dE=T dS + \\Psi d\\Xi \\,,\\qquad \\Psi ={\\frac{\\alpha \\omega _2 (c-k)}{24\\pi }}\\,.$ The quantity $\\Psi $ here is a new thermodynamic variable conjugate to $\\Xi $ , which is determined from the requirement of integrability of the generalized first law." ], [ "The Noether charge of the $k=0$ solution", "As discussed in section 4.2, for $k=0$ , the system has an additional conserved Noether charge.", "For conformal gravity, the Noether charge for the ansatz (REF ) is given by $\\lambda &=& {\\frac{\\alpha }{12ra^2\\sqrt{a f} (ra^{\\prime } - 2a)}}(4a^2 f-10 r af a^{\\prime }+ 7r^2f a^{\\prime 2} + 6ra^2 f^{\\prime } - 3r^2a a^{\\prime } f^{\\prime } - 6r^2 a f a^{\\prime \\prime })\\cr &&\\qquad \\times (4a^2 f - 2a f r a^{\\prime } - f r^2 a^{\\prime 2} -2a^2 r f^{\\prime } + ar^2a^{\\prime }f^{\\prime } + 2 a fr^2a^{\\prime \\prime })\\,.$ Thus for the black hole (REF ) with $k=0$ , we have $\\lambda = {\\frac{4\\alpha (27d^2-c^3)}{9d}}\\,.$ For the Schwarzschild-AdS black hole (REF ), we have $\\lambda = 8\\Lambda \\alpha M\\,.$ Note that in both cases we have $\\lambda \\omega _2 = -32\\pi T S$ .", "In other words, the second equality of (REF ) always holds.", "Indeed, these two Noether charges for the general and the Schwarzschild-AdS solutions can map to each other by the conformal transformation discussed in the previous subsection.", "Let us define $\\widetilde{E}$ as $\\widetilde{E} = -{\\frac{\\lambda \\, \\omega _2}{48\\pi }}\\,.$ For the Schwarzschild-AdS black brane, $\\widetilde{E}$ is precisely the mass of the solution.", "It follows from the argument presented in the previous subsection that $\\widetilde{E}$ cannot be the energy of the more general solution that has an additional parameter $c$ .", "Thus now we have two conserved quantities; one is the true energy $E$ given in (REF ) and the other is $\\widetilde{E}$ .", "The difference is $E-\\widetilde{E} = {\\frac{c^3}{216 \\pi d}}= {\\frac{m}{12\\pi }}\\,,$ where $m$ is given in (REF )." ], [ "Conformal boundary term", "It is possible to write a conformally-invariant boundary term in four dimensions.", "Thus for completeness, this boundary term should be included in conformal gravity.", "The conformal boundary term is given by $I_c=\\eta \\alpha \\int d^3x \\sqrt{-\\tilde{g}} C^{\\mu \\nu \\rho \\sigma }n_\\mu n_\\rho \\nabla _\\nu n_\\sigma \\,,$ where $n_\\mu $ is the unit outward normal to the boundary, and $\\eta $ is an arbitrary pure numerical constant.", "This boundary term does not contribute to the equations of motion, and so it has no effect on the local solutions, but it can contribute to the thermodynamics.", "For example, it yields a non-trivial contribution to the Euclidean action, implying that the free energy is now modified, and is given by $F=-{\\frac{\\alpha \\omega _2[(c-k)^2 - 3(c-k)\\tilde{d} + 3 \\tilde{d}^2]}{24\\pi r_+}} - {\\frac{\\eta \\alpha \\, \\omega _2\\,m}{16\\pi }}\\,,$ where $m$ is given by (REF ).", "It is of interest to note that the contribution of the conformal boundary term to the free energy is of the same form as the $m$ term appearing in the expression (REF ) for the energy in conformal gravity without the boundary term." ], [ "Extremal limit", "Since the general AdS black hole (REF ) has the parameter $c-k$ in addition to the usual $d$ parameter of the Schwarzschild-AdS black hole, it is possible to find an extremal limit for which the temperature vanishes and the near-horizon geometry has an AdS$_2$ factor.", "For both $k=\\pm 1$ , the extremal solution takes the same form, given by $f={\\frac{(r-r_+)^2 (r r_+ - r_+^2 + 1)}{r r_+}}\\,.$ For $k=1$ , the near horizon geometry is AdS$_2\\times S^2$ , with vanishing temperature and entropy.", "The energy, free energy and $\\Psi $ are given by $E=F=-{\\frac{\\alpha (r_+^2-1)^2\\omega _2}{8\\pi r_+}}\\,,\\qquad \\Psi ={\\frac{\\alpha \\omega _2 (r_+^2-1)}{8\\pi }}\\,,$ which all vanish for $r_+=1$ .", "Thus the $r_+=1$ solution may also be stable vacuum of the theory.", "For $k=0$ , it turns out that there is no extremal limit, since $f(r)$ has either a single root or a triple root." ], [ "Thermodynamics with varying $\\Lambda $", "Finally we consider the thermodynamics of the general AdS black holes in conformal gravity when $\\Lambda $ , the cosmological constant of the asymptotically AdS region, is treated as a thermodynamic variable also.", "This is natural in conformal gravity since the cosmological constant arises as a parameter of the solution rather than as a fixed parameter of the theory.", "The solution is given by $f=-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\Lambda r^2 + \\Xi \\, r + c + {\\frac{d}{r}}\\,, \\qquad {\\rm with}\\qquad 3\\Xi \\, d = c^2 - k^2\\,.$ Letting $r_+$ be the radius of the outer horizon, and defining $d=-r_+ \\tilde{d}$ , we have $&& T={\\frac{(3\\tilde{d}-c)^2 - k^2}{12\\pi r_+ \\tilde{d}}}\\,,\\qquad S={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 6} } } \\alpha \\omega _2 (k+3 \\tilde{d}-c) \\,,\\cr &&\\Psi ={\\frac{\\alpha \\omega _2 (c-k)}{24\\pi }}\\,,\\qquad \\Theta = {\\frac{\\alpha \\omega _2 d}{24\\pi }}\\,,\\cr &&F=-{\\frac{\\alpha \\omega _2 \\Big ((c-k)^2 - 3 (c-k) \\tilde{d} + 3 \\tilde{d}^2\\Big )}{24\\pi r_+}}\\,,\\qquad E= 2\\Theta \\, \\Lambda + \\Psi \\, \\Xi \\,.$ These thermodynamic quantities satisfy the relations $dE=T dS + \\Theta \\, d\\Lambda + \\Psi \\, d\\Xi \\,,\\qquad F=E - T\\, S\\,.$ Note that the last equation in (REF ) is the Smarr formula for the general black holes in conformal gravity.", "Its rather unusual form can be understood by considering the following scaling argument.", "Since the parameter $\\alpha $ has dimensions of length-squared, $L^2$ , and it is treated as a fixed parameter of the theory (which may be set, without loss of generality, to $\\alpha =1$ ), it follows that the effective scaling dimensions for the thermodynamic quantities are given by $[E]={\\frac{1}{L}}\\,,\\quad [T]={\\frac{1}{L}}\\,,\\quad [S]=1\\,,\\quad [\\Theta ]=L\\,,\\quad [\\Lambda ]={\\frac{1}{L^2}}\\,,\\quad [\\Psi ]=1\\,,\\quad [\\Xi ]={\\frac{1}{L}}\\,.$ Thus if $E$ is viewed as a function of $S$ , $\\Lambda $ and $\\Xi $ , namely $E=h(S,\\Lambda ,\\Xi )$ , then under scaling we shall have $E= \\mu h(S,\\mu ^{-2}\\Lambda , \\mu ^{-1} \\Xi )$ .", "Differentiating with respect to $\\mu $ , setting $\\mu =1$ , and using the first law in (REF ) then gives the Smarr relation $E=2\\Theta \\Lambda + \\Psi \\Xi $ we found in (REF ).A Smarr formula with more conventional coefficients would arise if we were to view the coupling constant $\\alpha $ as another thermodynamic variable, so that the thermodynamic quantities would all have their standard “engineering” scaling dimensions.", "We would then have a generalised first law $dE=TdS+\\Theta d\\Lambda + \\Psi d\\Xi + \\sigma d\\alpha $ , where $\\sigma $ is a new thermodynamic variable conjugate to $\\alpha $ .", "The Smarr formula would then be $E=2TS -2\\Theta \\Lambda -\\Psi \\Xi +2\\sigma \\, \\alpha $ .", "However, since $\\alpha $ is an overall parameter in conformal gravity, including it as an additional variable represents an over-parameterisation of the system.", "This is reflected in the fact that there is then a 1-parameter family of possible Smarr relations, with $E=\\lambda TS +2(1-\\lambda ) \\Theta \\Lambda + (1-\\lambda ) \\Psi \\Xi + \\lambda \\sigma \\alpha $ , where $\\lambda $ is an arbitrary constant.", "The entropy of the general black hole can be decomposed as $S={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } \\alpha w_2\\, k + {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 6} } }\\alpha \\, (-\\Lambda ) A + 8\\pi \\Psi +{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\alpha \\omega _2 \\Xi r_+\\,,$ where $A=r_+^2 \\omega _2$ is the area of the horizon, and the first pure numerical term is the contribution from the Gauss-Bonnet term in the action (REF ) with $\\beta =-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\alpha $ and $\\gamma ={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\alpha $ .", "We see from the constraint (REF ) on the parameters that in the limit $c\\rightarrow k$ , we can either set $d=0$ with $\\Xi $ fixed, or set $\\Xi =0$ with $d$ fixed.", "The former leads to the thermalized vacuum (REF ) and the latter leads to the Schwarzschild-AdS black hole." ], [ "$z=4$ Lifshitz black holes", "We find that conformal gravity admits static asymptotically-Lifshitz black hole solutions also, both for $z=4$ and $z=0$ .", "We shall first discuss the case with $z=4$ .", "The solution is given by $ds^2 = -r^8 f dt^2+ {\\frac{4 dr^2}{r^2f}} + r^2d\\Omega _{2,k}^2\\,,\\qquad f=1 + {\\frac{c}{r^2}} + {\\frac{c^2-k^2}{3r^4}} +{\\frac{d}{r^6}}\\,.$ This solution for Lifshitz black holes is locally equivalent to the AdS black hole solution (REF ) up to an overall conformal factor.", "Specifically, it can be seen that the metric $d\\hat{s}^2 = \\Omega ^2 ds^2$ with $\\Omega = {\\frac{q}{r(c+ 3 r^2-k)}}\\,, $ becomes, after transforming to the new radial coordinate $\\rho = r\\Omega $ , $d\\hat{s}^2= -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 9} } } q^2\\, \\hat{f} dt^2 + \\hat{f}^{-1}\\, d\\rho ^2 +\\rho ^2\\, d\\Omega _{2,k}^2\\,,$ where $\\hat{f} = k + {\\frac{q}{3\\rho }} -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\Lambda \\, \\rho ^2\\,,\\qquad \\Lambda = {\\frac{(c^3-27d-3ck^2+2k^3)}{q^2}}\\,.$ The conformal factor (REF ) is non-singular on the horizon $r=r_+$ of the Lifshitz black hole (except, as we shall see below, in the case of the $k=1$ extremal limit), and the horizon is mapped to that of the conformally-related (A)dS black hole (REF ).", "However, since the conformal factor becomes singular at $r=\\infty $ , the asymptotic regions, and hence the global structure, are very different for the two metrics.", "The equation $f(r)=0$ determines the locations of the horizons.", "This yields a cubic equation for $r^2$ , which will have either three real roots or one real root, according to whether the discriminant $\\Delta = -{\\frac{1}{27}}(c^3-27d-3ck^2-2k^3)(c^3-27d-3ck^2+2k^3)\\,,$ is positive or negative.", "In particular, in the case that $\\Delta >0$ , the cosmological constant of the conformally-related metric (REF ) will be positive, and it describes a de Sitter black hole.", "Using $r_+$ as usual to denote the radius of the outer horizon of the Lifshitz black hole, we have $d=-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } } r_+^2(c^2-k^2+3c r_+^2 +3r_+^4)\\,.$ We find that the temperature and the entropy are given by $T={\\frac{(c+3r_+^2-k)(c+3r_+^2 +k)}{12\\pi }}\\,,\\qquad S=-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 6} } }\\alpha \\omega _2(c+3r_+^2-k)\\,.$ The above expressions suggest that the constant $c$ might be spurious, since it always arises in the combination $c+3r_+^2$ .", "Indeed we can, locally, remove it by first making the coordinate transformation $r^2=\\tilde{r}^2 -c/3$ , and then scaling the metric by the factor $\\tilde{r}^2/r^2$ .", "However, if $c$ is negative, this transformation can be singular, if $c + 3 r_+^2 <0$ , and so we cannot simply use the above transformation to set $c=0$ .", "Indeed, one can see from (REF ) that if $c=0$ then $T$ and $S$ cannot both be positive (if $\\alpha >0$ ).", "On the other hand, if $c$ is sufficiently negative then we can arrange the parameters so that $T$ and $S$ are both positive.", "There is no obvious way to calculate the energy of an asymptotically-Lifshitz black hole directly (for example, the conserved charge given by (REF ) diverges).", "We can, however, integrate the first law, $dE=TdS$ , to obtain a thermodynamic definition of the energy, up to an undetermined additive constant.", "From (REF ) we find $E=-{\\frac{\\alpha \\omega _2 (c+3r_+^2-k)^2 (c+3 r_+^2 + 2k)}{216\\pi }}\\,,$ where we have made a choice for the additive constant that is convenient for the cases $k=1$ or $k=0$ .", "The energy definition for $k=-1$ will be given presently.", "The Euclidean action for the $z=4$ Lifshitz black hole diverges for large $r$ .", "We can instead define the free energy from the thermodynamic relation $F=E-TS$ , yielding $F={\\frac{\\alpha \\omega _2 (c+3 r_+^2 -k)^2(2c+6 r_+^2+k)}{216\\pi }}\\,.$ We now examine the three cases $k=0$ , 1 or $-1$ in more detail.", "For $k=0$ , there exists the Noether charge (REF ), giving $\\lambda ={\\textstyle {\\frac{\\scriptstyle 4}{\\scriptstyle 9} } } \\alpha (c^3-27d)\\,.$ Therefore, we see that (REF ) holds for this solution.", "In particular, we have $E={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } } TS\\,.$ The $k=0$ solution has no extremal limit, since then if the function $f$ has a double real root then it necessarily has a triple real root.", "For $c+3r_+^2>0$ we can set $c=0$ without loss of generality, since the conformal factor $\\tilde{r}^2/r^2$ is non-singular, as discussed earlier.", "In this case, the positivity of both the entropy and energy would require that $\\alpha <0$ .", "On the other hand, when we have $c+3r_+^2<0$ , the constant $c$ cannot be set to zero, since now the conformal factor $\\tilde{r}^2/r^2$ runs from a negative value to 1 when $r$ goes from the horizon to infinity.", "The positivity of both the entropy and energy now requires that $\\alpha >0$ .", "When $c+3r_+^2=0$ , which would be the extremal limit for the $k=0$ black holes, the solution instead has a naked singularity at $r=r_+$ .", "Thus for a given $\\alpha $ , only one of the two branches ($c+3r_+^2>0$ or $c+3r_+^2<0$ ) is well defined, since the entropy of one branch is positive at the price that in the other branch it is negative.", "For $k=1$ , then again if $c+3r_+^2>0$ we can set the parameter $c=0$ without loss of generality.", "In this case, the solution has an extremal limit with $r_+^2=1/3$ , for which both the entropy and energy vanish.", "For this branch of solutions, $r_+^2\\ge 1/3$ and the non-negativeness of the entropy and the energy defined by (REF ) is guaranteed as long as $\\alpha $ is negative.", "If $c+3r_+^2<0$ , then the parameter $c$ becomes non-trivial.", "The range where $-2<c+3r_+^2<0$ in fact cannot arise, since then the function $f$ actually has a third positive root that is larger than the putative largest root $r_+$ , and so $r=r_+$ is not the outer horizon.", "If $c+3r_+^2 < -2$ , the entropy and energy are non-negative provided that $\\alpha >0$ .", "There is an extremal limit at $c+3r_+^2=1$ , but, since $c+3r_+^2>0$ we can reduce this to the $c=0$ , $r_+^2=1/3$ extremal case discussed previously.", "Although the function $f$ also has a double root, at $r=r_0$ if $c+3r_0^2=-1$ , there is a larger positive root at $r=\\sqrt{r_0^2+1}$ , so this does not describe an extremal black hole.", "Note that for a given sign of $\\alpha $ , only one of the above two branches of solutions is well defined, and only the branch with $c+ 3 r_+^2>0$ has an extremal limit.", "The near-horizon geometry of the extremal limit is AdS$_2\\times S^2$ .", "The behavior of the metric functions is the same for the $k=-1$ solution as for the $k=1$ solution.", "Thus for the $c+3r_+^2\\ge 0$ branch we can again set $c=0$ , and extremality occurs at $r_+^2=1/3$ .", "The near horizon limit of the extremal black hole is AdS$_2\\times H^2$ .", "For solutions to have positive energy, we shift the previous energy (REF ) by a different additive constant, and define $\\widetilde{E}=-{\\frac{\\alpha \\omega _2 (c+3r_+^2-1)^2 (c+3 r_+^2 +2)}{216\\pi }}=E- {\\frac{\\alpha \\omega _2}{54\\pi }}\\,.$ The solution has non-negative energy and entropy provided that $\\alpha <0$ .", "For the $c+3r_+^2< -2$ branch, the energy and entropy are non-negative provided that $\\alpha >0$ .", "The solution is extremal at $c+3r_+^2=1$ , but this reduces to the $c=0$ , $r_+^2=1/3$ extremal case discussed above.", "For a given $\\alpha $ , only one of the two branches of solutions can be well defined." ], [ "$z=0$ Lifshitz black holes", "We now turn our attention to the $z=0$ Lifshitz black hole, for which the solution is given by $ds^2=-f dt^2 + {\\frac{4dr^2}{r^2 f}} + r^2 d\\Omega _{2,k}^2\\,,\\qquad f=1 + {\\frac{c}{r^2}} + {\\frac{c^2-k^2}{3r^4}}\\,.$ The solution has a power-law curvature singularity at $r=0$ .", "For $k=0$ , the singularity is naked.", "The Noether charge is given by $\\lambda =-4\\alpha c/3$ .", "Since the $k=0$ solution is not a black hole, we cannot use this example to test the validity of (REF ).", "For $k=\\pm 1$ , there is an horizon at the largest root of $f$ , given by $r_+^2={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 6} } }(\\sqrt{3(4-c^2)} - 3c)\\,.$ The requirement that $r_+^2>0$ implies that $-2\\le c<1$ .", "It follows that the temperature and entropy are given by $T={\\frac{1}{\\pi (2-{\\frac{2\\sqrt{3}\\,c}{\\sqrt{4-c^2}}})}}\\,,\\qquad S={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 12} } }\\alpha \\,\\omega _2 (c+2k + \\sqrt{3(4-c^2)})\\,.$ As in the case of the $z=4$ Lifshitz black hole, here too we can define the energy, up to an undetermined additive constant, by integrating the first law $dE=TdS$ .", "Here, we find $E={\\frac{1}{24\\pi }}\\alpha \\, \\omega _2\\,(c+2)\\,,$ where we have chosen the (parameter-independent) additive constant so that the energy is positive for $c> -2$ .", "Note that when $c=-2$ , the solution becomes extremal, with $f$ given by $f={\\frac{(r^2-1)^2}{r^4}}\\,,$ and the energy defined in (REF ) vanishes.", "The solution has a double root at $r=1$ , with the near-horizon geometry being AdS$_2\\times S^2$ or AdS$_2\\times H^2$ .", "For $k=1$ , the entropy vanishes in the extremal limit.", "The metric (REF ) is conformal to (A)dS.", "Defining $d\\hat{s}^2=\\Omega ^2 ds^2$ with $\\Omega = {\\frac{q r}{c+k+r^2}}\\,,$ we find after defining a new radial coordinate $\\rho = r\\Omega $ that $d\\hat{s}^2 = -{\\frac{q^2}{(c+k)^2}}\\, \\hat{f} dt^2 + \\hat{f}^{-1}\\, d\\rho ^2+ \\rho ^2\\, d\\Omega _{2,k}^2\\,,$ where $\\hat{f} = k + {\\frac{(c-k)q}{3\\rho }} -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\Lambda \\rho ^2\\,,\\qquad \\Lambda = {\\frac{c+2k}{q^2}}\\,.$ Thus the conformally-related metric describes an AdS black hole if $c+2k<0$ and a dS black hole if $c+2k>0$ .", "The condition for having real roots for $r^2$ in the $z=0$ Lifshitz black hole is that $4k^2-c^2\\ge 0$ .", "In particular, if $k=+1$ then the conformally-related metric will describe a de Sitter black hole.", "As with the $z=4$ Lifshitz black hole discussed previously, here too the conformal factor is non-singular on the horizon (except in the extremal limit), and so the horizon of the non-extremal $z=0$ Lifshitz black hole maps to the horizon of the (A)dS black hole.", "Once again, however, the conformal factor becomes singular at infinity, and the asymptotic regions of the two conformally-related metrics are very different." ], [ "AdS and Lifshitz Black Holes in Einstein-Weyl Gravity", "The existence of asymptotically AdS black holes in conformal gravity over and above the standard Schwarzschild-AdS black holes suggests that analogous more general solutions should exist also in Einstein-Weyl gravity, possibly including at the critical point.", "Furthermore, the existence of Lifshitz vacua in these theories and their generalisations to Lifshitz black holes in conformal gravity suggests that such Lifshitz black holes may also exist in Einstein-Weyl gravity.", "However, no exact solutions with either type of asymptotic behaviour have been found, beyond the usual Schwarzschild-AdS black holeThere exists a degenerate case with $\\alpha =0$ and $8\\beta \\Lambda +1=0$ , in which the Lagrangian is simply $\\sqrt{-g} (R-R_0)^2$ .", "This degenerate case allows any metric with constant scalar curvature $R_0$ to be a solution, including some Lifshitz black holes [75], [76].", "Since the equations of motion in this case are reduced to a scalar rather than a tensor equation, the system has no linear massive spin-2 excitations..", "In this section, we establish their existence by using a numerical approach.", "For $k=0$ AdS and Lifshitz black holes, by studying the horizon expansion, we find the following general relation between the Noether charge and the temperature and entropy: $\\lambda \\omega _2 = -32\\pi T S\\,.$ By examining the asymptotic behaviour at infinity, we find examples for which the energy is given by $E=-\\lambda \\omega _2/(16\\pi (z+2))$ , and hence the relation (REF ) appears to hold in these cases.", "However, as we have seen in conformal gravity discussed in the previous section, the first equality of (REF ) obtained in [60] does not hold in general in higher-derivative gravity, when massive spin-2 hair is involved." ], [ "Horizon expansion", "The equations of motion for Einstein-Weyl gravity which follow from (REF ) or from () with $\\beta =-\\alpha /3$ , appear not to be explicitly solvable for the most general static, spherically-symmetric solutions.", "We shall again consider the ansatz (REF ), and so the equations of motion for the metric functions $a(r)$ and $f(r)$ are again given by (REF ) and (REF ).", "As remarked previously, the Schwarzschild-AdS metrics (REF ) are solutions of these equations, but now we shall have to resort to numerical methods in order to investigate the most general static, spherically-symmetric solutions.", "In order to do this, we first construct Taylor expansions for the metric functions $a(r)$ and $f(r)$ in the vicinity of a black hole horizon.", "These will then be used to set the initial conditions for $a$ , $a^{\\prime }$ , $f$ and $f^{\\prime }$ just outside the horizon, so that the equations (REF ) and (REF ) for $a^{\\prime \\prime }$ and $f^{\\prime \\prime }$ can be numerically integrated out to large distances.", "It is instructive first to look at the near-horizon expansions of $a$ and $f$ for the Schwarzschild-AdS black hole (REF ).", "If we set $\\Lambda =-3$ as usual, and define the horizon radius $r_0$ by $k-2m/r_0 +r_0^2=0$ , then we have $a=f = r^2 + k -{\\frac{r_0(k+r_0^2)}{r}}\\,,$ and so the expansions are of the form $a(r)=f(r) &=& \\Big (3 r_0 + {\\frac{k}{r_0}}\\Big )\\, (r-r_0) -{\\frac{k}{r_0^2}}\\, (r-r_0)^2 +\\Big ( {\\frac{k}{r_0^3}} + {\\frac{1}{r_0}}\\Big )\\, (r-r_0)^3+\\cdots \\,.$ Since an overall constant factor in $a(r)$ can be absorbed into a rescaling of the time coordinate, for the general solutions we can consider a near-horizon expansion of the form $a(r) &=& (r-r_0) + a_2\\, (r-r_0)^2 + a_3\\, (r-r_0)^3 + a_4\\, (r-r_0)^4 +\\cdots \\,,\\\\f(r) &=& f_1(r-r_0) + f_2\\, (r-r_0)^2 + f_3\\, (r-r_0)^3+ f_4 \\, (r-r_0)^4 +\\cdots \\,.$ (Note that these expansions are for non-extremal black holes.", "The discussion for extremal black holes will be given presently.)", "Substituting these expansions into (REF ) and (REF ), we may then solve order by order in powers of $(r-r_0)$ , thus obtaining expressions for $a_n$ and $f_n$ with $n\\ge 2$ in terms of $f_1$ , $r_0$ , $k$ and $\\alpha $ .", "For example, we find $a_2 &=& {\\frac{3 r_0^3 + 5 f_1\\, r_0^2 - 2 f_1^2\\, r_0 + k\\, r_0 + k\\, f_1}{f_1^2\\, r_0^2}} - {\\frac{(3 r_0^2 - f_1\\, r_0 + k)}{4\\alpha \\, f_1^2\\, r_0}}\\,,\\nonumber \\\\f_2 &=& {\\frac{(f_1-3 r_0)(3 r_0^2 -2 f_1\\, r_0 + k)}{f_1\\, r_0^2}}+ {\\frac{3(3 r_0^2 - f_1\\, r_0 + k)}{4\\alpha \\, f_1\\, r_0}}\\,.$ The expressions for the coefficients with higher $n$ become rapidly quite complicated, and we shall not present them here.", "They are easily found, up to any desired order, using algebraic computing methods.", "Since we have fixed the cosmological constant, by setting $\\Lambda =-3$ , we see that $r_0$ and $f_1$ are non-trivial parameters characterising the solutions for each choice of $k=0$ , 1 or $-1$ .", "The case $f_1=3 r_0 + k/r_0$ corresponds to the Schwarzschild-AdS solution, for which the series expansions can be found from (REF ).", "The temperature and the entropy are given by $T={\\frac{\\sqrt{a_1 f_1}}{4\\pi }}\\,,\\qquad S={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\omega _2 r_0\\Big (r_0 +2\\alpha (f_1 - 2r_0)\\Big )\\,.$ The entropy is calculated with respect to the action of Einstein-Weyl gravity.", "When $k=0$ , the Noether charge is given by $\\lambda =-2 \\sqrt{a_1f_1} r_0 \\Big (r_0-2\\alpha (2 r_0-f_0)\\Big )\\,.$ It follows that the relation (REF ) indeed holds in general.", "In the above entropy calculation, we used the action (REF ) with the Gauss-Bonnet term set to zero ($\\gamma =0$ ).", "The Gauss-Bonnet term contributes $S_{\\rm GB}=\\gamma \\,k$ , which is a purely numerical constant, independent of the metric modulus parameters.", "In the above consideration, the functions $a$ and $f$ have the same single root $r=r_0$ , giving rise to non-extremal black holes.", "In the extremal limit, these functions have a double root, so that the near-horizon geometry has an AdS$_2$ factor.", "The Taylor expansion is given by $a(r) &=& (r-r_0)^2 + a_3\\, (r-r_0)^3 + a_4 \\, (r-r_0)^4+\\cdots \\,,\\cr f(r) &=& f_2\\, (r-r_0)^2 + f_3\\, (r-r_0)^3 + f_4 \\, (r-r_0)^4+\\cdots \\,.$ We find that the leading-order expansion of the equations of motion when $r\\rightarrow r_0$ implies that $(4\\alpha -1) (3r_0^2 + k)=0\\,.$ Thus we see that for $k=0,1$ , extremal black holes do not exist except for $\\alpha ={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }$ , on which we shall focus.", "Taking this $\\alpha $ value, we find that $&&a_3=-{\\frac{2 (4 k + 15 r_0^2)}{3 r_0 (k + 6 r_0^2)}}\\,,\\qquad a_4={\\frac{41 k^2 + 358 k r_0^2 + 759 r_0^4}{9 r_0^2 (k + 6r_0^2)^2}}\\,,\\cr &&f_2={\\frac{k + 6 r_0^2}{r_0^2}}\\,,\\qquad f_3=-{\\frac{2 (2 k + 9r_0^2)}{3 r_0^3}}\\,,\\qquad f_4={\\frac{15 k^2 + 148 k r_0^2 + 363r_0^4}{9 r_0^4 (k + 6 r_0^2)}}\\,.$ As one would have expected, the Noether charge of the $k=0$ solution vanishes in the extremal limit.", "Note that in the extremal limit, there is only one non-trivial parameter $r_0>0$ .", "As we shall discuss presently, these near-horizon geometries can extend smoothly to the asymptotic AdS or Lifshitz infinities.", "When $k=-1$ , the constraint (REF ) can be solved with $r_0^2=1/3$ for arbitrary $\\alpha $ .", "However, the resulting solution is simply the Schwarzschild-AdS solution whose $f$ can have a double zero for $k=-1$ ." ], [ "Asymptotically AdS solutions", "For asymptotically AdS solutions, the asymptotic regions behave roughly as follows: $a&\\sim & r^2(1+c_0) + k - {\\frac{2M}{r}} + c_1 r^{n+1} +{\\frac{c_2}{r^n}}\\,,\\cr f&\\sim & r^2(1+c_0) + k - {\\frac{2M}{r}} -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }c_1(n-1)r^{n+1} + {\\frac{c_2(n+2)}{3r^n}}\\,,$ where we have parameterized $\\alpha $ by $\\alpha = -{\\frac{1}{n(n+1)}}\\qquad \\Rightarrow \\qquad n=-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } \\pm {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\sqrt{1-{\\frac{4}{\\alpha }}}\\,.$ Note that there are a total of four parameters in (REF ), corresponding to four excitations.", "The coefficients $(c_0,M)$ correspond to the massless spin-2 modes, whilst the $(c_1,c_2)$ correspond to the massive spin-2 modes.", "For $0<\\alpha <4$ , the constant $n$ is complex, implying that the excitation takes the form $\\sqrt{r}\\Big (c_1\\, \\cos ({\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\sqrt{4/\\alpha -1}\\, \\log r) +c_2\\,\\sin ({\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\sqrt{4/\\alpha -1}\\, \\log r)\\Big )\\,.$ Let us present some explicit examples.", "The first is $n=-1/2$ , corresponding to $\\alpha =4$ .", "The functions $a$ and $f$ at asymptotic infinity are given by $a&=& r^2 +m \\sqrt{r}+ k -{\\frac{2M}{r}} +{\\frac{5k\\,m}{16r^{3/2}}} -{\\frac{m(m^2+48M)}{96r^{5/2}}} + \\cdots \\,,\\cr f&=& r^2 +{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }m\\sqrt{r} + k- {\\frac{32M+7m^2}{r}}-{\\frac{15k\\,m}{32r^{3/2}}} -{\\frac{5m(16M+m^2)}{64r^{5/2}}} + \\cdots \\,.$ When $k=0$ , we have the Noether charge $\\lambda =-27(8M+m^2)/2$ .", "In this case, the usual Deser-Tekin and AMD methods of energy calculation lead to divergent results, and hence we do not have an independent method of calculating $E$ to verify whether the first equality of (REF ) holds.", "The second example is $n=1/2$ , corresponding to $\\alpha =-4/3$ .", "We have $a&=& r^2 + k +{\\frac{m}{r^{{\\frac{1}{2}}}}} -{\\frac{2M}{r}}+{\\frac{11k\\,m}{96r^{{\\frac{5}{2}}}}} + {\\frac{m^2}{12r^3}} -{\\frac{Mm}{6r^{{\\frac{7}{2}}}}}+\\cdots \\,,\\cr f&=& r^2 + k +{\\frac{5m}{6r^{{\\frac{1}{2}}}}} -{\\frac{2M}{r}}+{\\frac{65k\\,m}{192r^{{\\frac{5}{2}}}}} + {\\frac{25m^2}{144r^3}} -{\\frac{Mm}{4r^{{\\frac{7}{2}}}}}+\\cdots \\,.$ For $k=0$ , the Noether charge is $\\lambda =20M$ , which is independent of $m$ .", "In principle, the asymptotic behavior could have the $r^{3/2}$ series as well, but it does not appear to be the case.", "The third example is $n=2$ , corresponding to $\\alpha =-1/6$ , for which we find $a&=&r^2 + k -{\\frac{2M}{r}} + {\\frac{m}{r^2}} -{\\frac{5k\\,m}{21r^4}} +{\\frac{Mm}{3r^5}}+\\cdots \\,,\\cr f&=&r^2 + k -{\\frac{2M}{r}} + {\\frac{4m}{3r^2}}-{\\frac{10k\\,m}{21r^4}} + {\\frac{5Mm}{9r^5}}+\\cdots \\,.$ For $k=0$ , the Noether charge is $\\lambda =-8M$ .", "The energy density can be calculated using the Deser-Tekin or AMD methods, giving $E=M/(6\\pi )$ .", "Thus for this case, the first equality in (REF ) holds.", "However, the numerical results indicate that only the $m=0$ case, i.e.", "the Schwarzschild-AdS solution, describes a black hole with an horizon.", "The final example is the critical point, namely $\\alpha =-1/2$ , corresponding to $n=1$ .", "We find that $a &=& r^2 + k - {\\frac{2\\tilde{M}}{r}} - {\\frac{7k\\,m}{15r^3}} +{\\frac{2m\\,\\tilde{M}}{3r^4}} + \\cdots \\,,\\cr f &=& r^2 + k - {\\frac{2\\tilde{M} -2m/3}{r}} - {\\frac{k\\,m}{r^3}} +{\\frac{9(18\\tilde{M} + 7m)}{18 r^4}} + \\cdots \\,, $ where $\\tilde{M}=m \\log r + M\\,.$ For $k=0$ , the Noether charge is $\\lambda =-18m$ .", "In appendix C, we derive the mass formula for this case, and we find the energy density is $E=3m/(8\\pi )$ .", "Thus, in this case, the first equality of (REF ) holds." ], [ "Asymptotic Lifshitz behavior", "In this case, we are primarily concerned with the $k=0$ case.", "We find that the large $r$ expansion is given by $a &\\sim & r^{2z}\\Big (1+{\\frac{(z^2+2)m}{z(z+2)r^{z+2}}} + \\tilde{c}_+r^{-1-{\\frac{1}{2}}z + {\\frac{1}{2}}\\Delta } + \\tilde{c}_- r^{-1-{\\frac{1}{2}} z -{\\frac{1}{2}}\\Delta }\\Big )\\,,\\cr f&\\sim & \\sigma r^2\\Big (1 + {\\frac{m}{r^{z+2}}} + c_+ r^{-1-{\\frac{1}{2}}z +{\\frac{1}{2}}\\Delta } + c_- r^{-1-{\\frac{1}{2}} z - {\\frac{1}{2}}\\Delta }\\Big )\\,,\\cr \\tilde{c}_+&=&{\\frac{4-11z+z^2-3z^3 + (1-3z-z^2)\\Delta }{2(z-1)^2(3z-1)}}c_+\\,,\\qquad \\Delta = \\sqrt{3(4-4z+3z^2)}\\,,\\cr \\tilde{c}_-&=&{\\frac{4-11z+z^2-3z^3 - (1-3z-z^2)\\Delta }{2(z-1)^2(3z-1)}} c_-\\,.$ The Noether charge is given by $\\lambda =-{\\frac{6\\sqrt{6} (2-3z+z^3)m}{z^2(z-4)\\sqrt{z^2+2z+3}}}\\,.$ Our numerical results suggest that there are Lifshitz-like black holes with $S^2$ and $H^2$ topology that have the same leading-order behavior as the above." ], [ "Numerical analysis", "We have carried out a numerical analysis for a variety of choices for the coefficient $\\alpha $ that multiplies the Weyl-squared term in the action.", "The choice of the horizon radius $r_0$ is a non-trivial parameter, given that we have fixed the cosmological constant ($\\Lambda =-3$ ).", "The value of the expansion coefficient $f_1$ is also a non-trivial parameter in the solutions.", "The deviation of $f_1$ from the value $3 r_0 + k/r_0$ determines the deviation of the black hole from the usual Schwarzschild-AdS solution.", "The Schwarzschild-AdS black hole can be thought of as a solution where only the massless spin-2 modes are excited.", "Deviating from $f_1= 3 r_0 + k/r_0$ corresponds to setting initial conditions near the horizon that cause the massive spin-2 modes to be excited also.", "Our numerical investigations suggest that solutions of this type exist, in the sense that the numerical routines give a reasonably stable result with the metric functions showing no sign of runaway behaviour, provided that the linearised spin-2 massive mode falls off less rapidly than the spin-2 massless mode.", "This fall-off rate is governed by the mass $m$ of the linearised fluctuation, and in turn, this is related to the value of the parameter $\\alpha $ in the Lagrangian.", "Specifically, the condition of less rapid fall-off is achieved if the massive mode has negative mass-squared.", "Solutions with stable behaviour appear to exist regardless of whether the negative $m^2$ lies in the non-tachyonic region $-{\\textstyle {\\frac{\\scriptstyle 9}{\\scriptstyle 4} } }\\le m^2<0$ or the tachyonic region $m^2<-{\\textstyle {\\frac{\\scriptstyle 9}{\\scriptstyle 4} } }$ .", "Of course in the latter case one would expect the solutions to exhibit time-dependent runaway behaviour, but this will not show up with the static metric ansatz that we are considering here.", "In terms of the constant $\\alpha $ that characterises the coefficient of Weyl-squared in the action, the condition that the massive linearised mode have $m^2<0$ corresponds to $\\alpha <-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ or $\\alpha >0$ .", "We find that if $\\alpha $ lies in the region $-\\infty <\\alpha <-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ , then defining $f_1= 3 r_0 + k/r_0 + \\delta \\,,$ there is a range for $\\delta $ , with $\\delta _- <\\delta <\\delta _+$ , for which the numerical solutions indicate the occurrence of asymptotically AdS black holes.", "The lower limit $\\delta _-$ is negative, while the upper limit $\\delta _+$ is positive.", "If the value of $\\delta $ is fine-tuned to be equal to $\\delta _-$ or $\\delta _+$ , then the asymptotic behaviour of the black hole changes from AdS to Lifshitz.", "The value of $z$ in the asymptotically Lifshitz case is given by the larger root in (REF ).", "If the parameter $\\delta $ is chosen to lie outside the range $\\delta _-\\le \\delta \\le \\delta _+$ , then the numerical analysis indicates that the solution becomes singular.", "As an example, let us consider $\\alpha = -{\\frac{11}{16}}$ , which from (REF ) implies that there should exist asymptotically Lifshitz solutions with $z=2$ .", "Taking $k=0$ and choosing $r_0=10$ , we find that the limiting values for $\\delta $ in (REF ) are $\\delta _-\\approx -11.596956988\\,,\\qquad \\delta _+\\approx 62.826397763\\,.$ In our numerical routine, we set initial conditions just outside the horizon at $r=r_0 + 0.0001$ , and run out to $r=100000$ .", "For the asymptotically Lifshitz black hole with $\\delta =\\delta _-$ , we obtained plots of $a(r)$ , $f(r)$ , given in Figure 1, and $a(r)/r^4$ and $f(r)/r^2$ , given in Figure 2.", "Note that although we integrated out to $r=100000$ , we only plot the functions out to $r=100$ in order to be able to generate more illustrative displays.", "The asymptotic value of the ratio $f(r)/r^2$ reaches about $0.545454545452$ as $r$ approaches 100000, which is indeed close to the expected ratio $6/11$ (see equation (REF )).", "Figure: The metricfunctions a(r)a(r) and f(r)f(r) for the asymptotically Lifshitz black hole.Figure: The asymptotic forms for a(r)/r 4 a(r)/r^4 and f(r)/r 2 f(r)/r^2,illustrating the z=2z=2 Lifshitz behaviour.The solution with $\\delta =\\delta _+$ exhibits very similar Lifshitz behaviour.", "If we choose a value of $\\delta $ that lies in between the two Lifshitz extremes, we obtain an asymptotically AdS black hole.", "Figures 3 and 4 below illustrate this, again for $\\alpha =-{\\textstyle {\\frac{\\scriptstyle 11}{\\scriptstyle 16} } }$ , $k=0$ and $r_0=10$ , in the case that $\\delta =20$ .", "Figure 3 shows the functions $a$ and $f$ , while Figure 4 shows the functions $a/r^2$ and $f/r^2$ .", "Figure: The metricfunctions a(r)a(r) and f(r)f(r) for the asymptotically AdS black hole.Figure: The asymptotic forms for a(r)/r 2 a(r)/r^2 and f(r)/r 2 f(r)/r^2,illustrating the AdS behaviour.The story is very similar for solutions with $k=1$ or $-1$ .", "For example, if we consider $k=1$ solutions, again with $\\alpha =-{\\textstyle {\\frac{\\scriptstyle 11}{\\scriptstyle 16} } }$ and $r_0=10$ , we find that the upper and lower limits on the range of $\\delta $ in (REF ) is now $\\delta _-\\approx -11.596956988\\,,\\qquad \\delta _+\\approx 62.826397763\\,.$ We find solutions exhibiting asymptotically Lifshitz type of behaviour, again with $z=2$ , if $\\delta $ is taken to be either of the extreme values.", "If, on the other hand, $\\delta $ lies in between the limiting values $\\delta _-$ and $\\delta _+$ , then we find solutions with asymptotically AdS behaviour.", "The forms of the metric functions $a$ and $f$ are qualitatively similar to those illustrated in the $k=0$ examples above.", "When $\\alpha =-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ , corresponding to the case of critical gravity, numerical analysis indicates that asymptotically AdS black hole solutions again exist, within some range of values for the $\\delta $ parameter in (REF ).", "However, $\\alpha =-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ is on the borderline for stability of the solutions, with $-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } <\\alpha <0$ seemingly being unstable, and so it is not easy to extract meaningful quantitative results in the critical case.", "We also perform the numerical analysis for the extremal case with the parameter $r_0$ , whose horizon behavior is given by (REF ).", "For $S^2$ or $T^2$ topology, such a solution exists only for $\\alpha =1/4$ .", "For $k=1$ , the numerical results indicate that the horizon can smoothly extend to the asymptotic AdS$_4$ infinity for all parameters $r_0>0.251976578$ .", "When $r_0=0.251976578$ , the asymptotic behavior becomes Lifshitz-like with exponent $z=-1/2$ .", "For $r_0<0.251976578$ , the solution becomes singular.", "For $k=0$ , the horizon can extend smoothly to AdS in the asymptotic region for any $r_0>0$ .", "For $k=-1$ , we must have $r_0>1/\\sqrt{6}$ .", "When $r_0=1/\\sqrt{3}$ , the usual Schwarzschild-AdS solution emerges.", "There is no indication of Lifshitz behavior for $k=-1$ ." ], [ "Conclusions", "In this paper, we have considered four-dimensional Einstein gravity extended by the addition of general quadratic-curvature terms.", "In addition to the usual AdS vacuum, the theory contains Lifshitz and Schrödinger vacuum solutions.", "Our primary purpose was to construct black holes obeying asymptotically AdS or Lifshitz boundary conditions, with spherical, 2-torus or hyperbolic $H^2$ spatial symmetry.", "We focused on conformal gravity, with a purely Weyl-squared action, as well as Einstein-Weyl gravity, for which the standard Einstein action with cosmological constant is augmented with a Weyl-squared term.", "The general spherically-symmetric local solution in conformal gravity was known previously.", "It involves two nontrivial parameters, one of which is associated with the mass of the black hole while the other, which we call $\\Xi $ , may be thought of as characterising massive spin-2 hair.", "Owing to the presence of the second non-trivial parameter in the general AdS black hole solutions, one can expect that the usual first law of thermodynamics, $dE=TdS$ , will need to be augmented by an additional term involving a new pair of intensive and extensive thermodynamic variables.", "We studied this in detail in the case of AdS black holes in conformal gravity, showing how the first law becomes $dE=TdS+\\Psi d\\Xi $ , where the variable $\\Psi $ , conjugate to $\\Xi $ , is determined by requiring the integrability of the equation.", "We also needed to find a satisfactory definition of energy for the black holes in conformal gravity.", "Its derivation, as a conserved charge evaluated at infinity, is described in appendix D. In conformal gravity, the cosmological constant $\\Lambda $ of the AdS black holes is a parameter of the solution rather than a parameter in the action; it characterises the “AdS radius” of the asymptotically AdS region.", "It is therefore natural to promote $\\Lambda $ to being another thermodynamic quantity that can be varied in the first law.", "We showed that this indeed gives a consistent extension of the thermodynamic phase space.", "We then constructed Lifshitz black holes in conformal gravity, with a temporal/spatial anisotropic scaling parameter $z=4$ .", "These solutions involve only a single non-trivial parameter, and hence the thermodynamic quantities can be easily evaluated.", "We showed that, since the Lifshitz black hole has $T^2$ spatial sections, there exists a conserved Noether charge $\\lambda $ .", "Moreover, $\\lambda $ is related to the energy of the black hole, and to the product of temperature and entropy (REF ), in the same way as has previously been observed in [60] for certain two-derivative theories.", "However, for the more general AdS black holes (with $T^2$ spatial sections) involving massive spin-2 hair, characterised by $\\Xi $ , the Noether charge no longer seems to provide a natural definition for the energy, although the second equality of (REF ) always holds.", "By contrast, in the case of Schwarzschild-AdS black holes with $T^2$ spatial sections, the relation (REF ) always holds.", "We also obtained Lifshitz-like black holes with $S^2$ and $H^2$ spatial sections, with Lifshitz exponent $z=4$ and 0, and we found that the thermodynamic relations are obeyed in these cases.", "The existence of well-defined AdS and Lifshitz black holes in conformal gravity with additional massive spin-2 hair prompted us to seek similar solutions in Einstein-Weyl gravities.", "It does not appear to be possible to obtain closed-form expressions for such solutions, and so we resorted to numerical integration of the equations of motion.", "The procedure is to first obtain both the horizon and asymptotic expansions, and then use the horizon expansion as the initial boundary conditions for numerical analysis and compare the resulting solution for large radial values with the asymptotic expansions.", "We find that the horizon geometry involves an extra parameter over and above that of the usual Schwarzschild-AdS solution which is an Einstein metric.", "The numerical analysis suggests that asymptotically AdS black holes exist within a continuous range of values for the additional parameter.", "At the boundary of this parameter region, the asymptotic behavior changes to that of Lifshitz solutions, giving rise to corresponding asymptotically Lifshitz black holes.", "Beyond these parameter boundaries, the solutions develop naked curvature singularities.", "For a black brane with $k=0$ , for which there is an additional Noether charge, we find that the second equality in (REF ) always holds, whereas the first equality does not.", "These solutions provide an interesting phase transition of the corresponding boundary field theory from a relativistic Lorentzian system to a non-relativistic Lifshitz system.", "We further examine the existence of extremal solutions whose near-horizon geometry has an AdS$_2$ factor.", "It turns out that non-trivial AdS extremal solutions arise only for $\\alpha ={\\frac{1}{4}}$ .", "In the case of $k=1$ , there exists an extremal Lifshitz-like black hole with exponent $z=-{\\frac{1}{2}}$ .", "It would be interesting to explore the possibility of embedding extended gravity within string theory, given that string theory contains higher derivative corrections due to stringy or quantum effects.", "In fact, other than some special cases for which there is maximal supersymmetry, not much is known regarding the forms of these higher derivative corrections.", "In light of the vast string landscape, one expects that there are generic corrections, which include the higher-order curvature terms discussed in this paper.", "One might then invoke holographic techniques, in which case the black hole solutions discussed in this paper could be used to describe three-dimensional field theories or condensed matter systems.", "For the AdS black holes, the extra parameter of the AdS black hole solution would be mapped to a parameter in the dual field theory associated with finite coupling corrections.", "The additional global symmetry exhibited by the AdS black brane solutions would then be associated with a particular scaling symmetry in which space and time are rescaled differently, which is present at the conformal fixed point as well as away from it." ], [ "Acknowledgements", "H.L.", "is grateful to the George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy for hospitality during the course of this work.", "The research of H.L.", "is supported in part by NSFC grant 11175269.", "The research of C.N.P.", "is supported in part by DOE grant DE-FG03-95ER40917.", "The research of J.F.V.P.", "is supported in part by NSF grant PHY-0969482 and a PSC-CUNY Award.", "C.N.P.", "gratefully acknowledges the hospitality of the Mitchell Family Foundation at Cook's Branch Conservancy during the completion of this work." ], [ "Asymptotically Schrödinger solutions", "Here we construct solutions that are asymptotic to the Schrödinger solutions discussed in [41].", "We consider the metric ansatz $ds^2=-r^{2z} f dt^2 + {\\frac{dr^2}{r^2 f}} + r^2 (-2dt dx + dy^2)\\,,$ for $z=(1,{\\frac{1}{2}},0,-{\\frac{1}{2}})$ .", "We find that the equations are reduced to the fourth-order differential equation $0&=&f^{\\prime \\prime \\prime \\prime } + {\\frac{(6f+18zf + 7rf^{\\prime })f^{\\prime \\prime \\prime }}{2rf}} + {\\frac{f^{\\prime \\prime }}{2r^2f^2}} \\Big (4r^2ff^{\\prime \\prime }+ 2r^2 f^{\\prime 2} + (53z + 5) r f f^{\\prime }\\cr && + 4(16z^2 + 3 z - 1) f^2\\Big )+{\\frac{f^{\\prime }}{2r^3 f^2}} \\Big (2(3z - 1)r^2 f^{\\prime 2} + (78 z^2- 11z-9)r f f^{\\prime }\\cr && + 4(28 z^2 - 10 z^2 - 7z + 1) f^2\\Big ) +{\\frac{4z(z-1)(2z-1)(2z+1)f}{r^4}}\\,.$ For $z=1$ , we find a solution $f=1-M/r$ , giving $ds^2=-r^{2} f dt^2 + {\\frac{dr^2}{r^2 f}} + r^2 (-2dt dx +dy^2)\\,,\\qquad f=1-{\\frac{M}{r}}\\,.$" ], [ "Generalized Plebanski metric", "Using the Plebanski metric ansatz [77], we find solutions in conformal gravity given by $ds^2=-{\\frac{\\Delta _x}{x^2 + y^2}} (dt + y^2 d\\psi )^2 +{\\frac{\\Delta _y}{x^2+y^2}} (dt-x^2 d\\psi )^2 + {\\frac{x^2+y^2}{\\Delta _x}}dx^2 + {\\frac{x^2 + y^2}{\\Delta _y}} dy^2\\,,$ where $\\Delta _x &=& c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4 x^4\\,,\\cr \\Delta _y &=& c_0 + d_1 y - c_2 y^2 + {\\frac{c_1c_3}{d_1}} y^3 + c_4 y^4\\,.$ This metric is conformal to the Plebanski-Demianski [78] metric.", "Namely, the metric $d\\hat{s}^2=\\Omega ^2 ds^2$ with $\\Omega =(1+c_3 x\\,y/d_1)^{-1}$ is Einstein and satisfies $\\hat{R}_{\\mu \\nu }=\\Lambda \\hat{g}_{\\mu \\nu }$ with $\\Lambda =-{\\frac{3(c_0 c_3^2+c_4 d_1^2)}{d_1^2}}\\,.$" ], [ "Time-dependent metrics", "For the general quadratic action (REF ) with arbitrary $\\alpha $ and $\\beta $ , we will consider time-dependent and spatially flat isotropic solutions described by the metric $ds^2=-dt^2+f(t)^2 \\sum _{i=1}^3 dx_i^2\\,.$ Applying the trace condition reduces the equations to the single third-order equation $3f^2 f^{\\prime 2}-\\Lambda f^4-6(\\alpha +3\\beta ) \\left( 3f^{\\prime 4}-2ff^{\\prime 2}f^{\\prime \\prime }+f^2 f^{\\prime \\prime 2}-2f^2f^{\\prime }f^{\\prime \\prime \\prime }\\right) =0\\,.", "$ For Einstein-Weyl gravity, $\\alpha +3\\beta =0$ and, for a positive cosmological constant, the only solution is de Sitter spacetime.", "We will now consider non-isotropic time-dependent solutions for extended gravity with zero cosmological constant, described by the Kasner metric $ds^2=-dt^2+\\sum _{i=1}^3 t^{2p_i} dx_i^2\\,.", "$ It can be shown that a metric of this form must satisfy the conditions $ \\sum _{i=1}^3 p_i=\\sum _{i=1}^3 p_i^2=1\\,.", "$ In other words, the quadratic terms in the action (REF ) do not modify the Kasner conditions that arise in Einstein gravity.", "This disallows isotropic expansion and, in particular, one exponent must be negative.", "However, in conformal gravity the exponents need only satisfy the condition $2\\sum _{i=1}^3 p_i +2\\sum _{i=1}^3 p_i^2-\\left( \\sum _{i=1}^3p_i\\right)^2=3\\,.", "$ This solution includes the Kasner metric for which both conditions in (REF ) are obeyed." ], [ "pp-wave metrics", "A general class of pp-wave solutions for the general quadratic action (REF ) with arbitrary $\\alpha $ and $\\beta $ has the metric $ds^2=H dx^2+{\\frac{dr^2}{r^2}}+r^2 (-2dt dx+dy^2)\\,,$ where $H=f_1r^2+{\\frac{f_2}{r}}+f_3r^{2z_+}+f_4r^{2z_-}+g_1(1+y^2r^2)\\,,$ the $f_i$ and $g_i$ are functions of $x$ only and $z=z_\\pm $ satisfy the equation $1-24\\beta +\\alpha (4z_{\\pm }^2-2z_{\\pm }-8)=0\\,.$ For conformal gravity and critical gravity, the $H$ function can have additional terms.", "Namely, for conformal gravity $H$ has the form $H=f_1 r^2+{\\frac{f_2}{r}}+f_3r+f_4+g_1y^2r^2+g_2y^3r^2\\,,$ while for critical gravity it is given by $H=f_1r^2+{\\frac{f_2}{r}}+f_3r^2\\log r+f_4{\\frac{\\log r}{r}}+g_1(1+y^2r^2)\\,.$ For $g_i=0$ these metrics all reduce to ones presented in [12], [13], for which $H$ is a function only of $x$ and $r$ .", "Metrics for which the $H$ function involves sinusoidal dependence on the $y$ coordinate are also discussed in [13].", "For $f_1=f_3=f_4=g_i=0$ all of these metrics reduce to the Kaigorodov [58] metric." ], [ "Energy of Logarithmic Black Hole in Critical Gravity", "In this appendix, we derive the mass of the logarithmic black hole using the Abbott-Deser-Tekin (ADT) and the Ashtekar-Magnon-Das (AMD) procedures.", "The main idea of the ADT method is to write the asymptotic AdS black hole metric in the form $g_{\\mu \\nu }=\\bar{g}_{\\mu \\nu } +h_{\\mu \\nu }$ , where $\\bar{g}_{\\mu \\nu }$ is the metric on AdS, and then interpret the linearised variation of the field equation, given in our case by (REF ), as an effective gravitational energy-momentum tensor $T_{\\mu \\nu }$ for the black hole field.", "One then writes the conserved current $J^\\mu =T^{\\mu \\nu }\\,\\xi _\\mu $ , where $\\xi ^\\mu $ is a Killing vector that is timelike at infinity, as the divergence of a 2-form ${\\cal F}_{\\mu \\nu }$ ; i.e.", "$J^\\mu =\\nabla _\\nu {\\cal F}^{\\mu \\nu }$ .", "From this, one obtains the ADT mass for the Lagrangian corresponding to (REF ): $8\\pi GE =(1+8\\Lambda \\beta +2\\Lambda \\alpha )\\, \\int _{S_\\infty }\\,dS_{i}\\, {\\cal F}_{(0)}^{0i} +(2\\beta +\\alpha ) \\int _{S_\\infty }\\,dS_{i}\\, {\\cal F}_{(1)}^{0i}+\\alpha \\int _{S_\\infty }\\, dS_{\\mu \\nu }\\,{\\cal F}_{(2)}^{0i}\\,, $ where $dS_{i}$ is the area of the sphere at infinity.", "The definition of ${\\cal F}^{\\mu \\nu }$ associated with the various terms in the equations of motion have been calculated in [67].", "One may verify that upon defining ${\\cal F}_{(0)}^{\\mu \\nu } &=& \\xi _\\alpha \\nabla ^{[\\mu }h^{\\nu ]\\alpha }+ \\xi ^{[\\mu }\\nabla ^{\\nu ]}\\, h + h^{\\alpha [\\mu }\\nabla ^{\\nu ]}\\xi _\\alpha -\\xi ^{[\\mu }\\, \\nabla _\\alpha h^{\\nu ]\\alpha } + {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } h \\nabla ^\\mu \\xi ^\\nu \\,,\\nonumber \\\\{\\cal F}_{(1)}^{\\mu \\nu } &=& 2\\xi ^{[\\mu }\\, \\nabla ^{\\nu ]}\\, R^L +R^L\\, \\nabla ^\\mu \\xi ^\\nu \\,,\\nonumber \\\\{\\cal F}_{(2)}^{\\mu \\nu } &=& -2\\xi _\\alpha \\, \\nabla ^{[\\mu }\\,{{\\cal G}}_L^{\\nu ]\\alpha }-2 {{\\cal G}}_L^{\\alpha [\\mu }\\ \\nabla ^{\\nu ]}\\, \\xi _\\alpha \\,,$ it follows that $\\nabla _\\nu \\, {\\cal F}_{(0)}^{\\mu \\nu } &=& {{\\cal G}}_L^{\\mu \\nu }\\, \\xi _\\nu \\,,\\nonumber \\\\\\nabla _\\nu \\, {\\cal F}_{(1)}^{\\mu \\nu } &=&\\Big [(-\\nabla _\\mu \\nabla _\\nu +g^{\\mu \\nu }\\square + \\Lambda \\, g^{\\mu \\nu }) R^L\\Big ] \\xi _\\nu \\,,\\nonumber \\\\\\nabla _\\nu \\, {\\cal F}_{(2)}^{\\mu \\nu } &=&\\Big [(\\square - {\\frac{2\\Lambda }{3}}){{\\cal G}}_L^{\\mu \\nu } -{\\frac{2\\Lambda }{3}}\\, R^L\\, g^{\\mu \\nu }\\Big ]\\, \\xi _\\nu \\,.$ At the critical point $\\Lambda \\alpha =-3\\Lambda \\beta ={\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ , the first term in (REF ) vanishes, and the contributions to the mass of the logarithmic black hole from the two remaining terms is $E_{\\rm {log}}^{\\rm ADT}=\\frac{3m}{8\\pi G}\\,.$ Since the logarithmic black hole is asymptotically AdS, one can also try to apply the AMD method to this case.", "The derivation of AMD conserved quantities relies on a detailed analysis of the fall-off rate of the curvature near the boundary, which is weighted by a smooth function $\\Omega $ (the conformal boundary is defined at $\\Omega =0$ ).", "For details on the requirement for $\\Omega $ , the reader is referred to [68], [69].", "For $n$ -dimensional asymptotic AdS spacetime, for generic cases in which the leading fall off of the Weyl tensor goes as $\\Omega ^{n-5}$ , the AMD formula for conserved quantities in quadratic curvature theories were explored in [70], [71].", "However, in the case of AdS logarithmic black holes, the leading fall off of the Weyl tensor near the boundary is modified to be $C_{abcd}\\rightarrow \\Omega ^{n-5}K_{abcd}+\\Omega ^{n-5}\\log (\\Omega )L_{abcd}\\,.$ Here $a$ and $b$ are indices related to a new coordinate system which adopts $\\Omega $ as the radial coordinate.", "It is found that, at critical points where a logarithmic term can appear, the fall-off behavior of the energy-momentum tensor is still at the order of $\\Omega ^{n-3}$ .", "Thus, the flux across the boundary is finite.", "This implies that the AMD conserved quantities for the logarithmic black hole may be well defined.", "In [71], the AMD conserved quantities corresponding to the logarithmic black hole are found to be given by $Q_{\\xi }[C] = \\frac{\\alpha R_0}{8\\pi G_{(n)}n(n-3)} \\int _C dx^{n-2}\\sqrt{\\hat{\\sigma }} \\hat{\\cal L}_{ab} \\xi ^{a} \\hat{N}^{b}\\,,$ with $R_0 =-n(n-1)\\,,$ where the AdS radius has been set to 1 and $\\hat{\\cal L}_{ab}\\equiv \\ell ^2L_{eafb}{\\hat{n}}^{e}{\\hat{n}}^{f}$ .", "Specifically for the four-dimensional AdS-logarithmic black hole solutions with the asymptotic expansion given by (REF ), one finds that $E_{\\rm {log}}^{\\rm AMD}=\\frac{3m}{8\\pi G}\\,.$" ], [ "Energy of AdS Black Holes in Conformal Gravity", "In this appendix, we present details of the proposal for calculating the mass of AdS black holes in conformal gravity that we discussed in section 5.1.", "The Lagrangian for conformal gravity is given by $e^{-1}{\\cal L}=\\frac{\\alpha }{2} C^{\\mu \\nu \\rho \\lambda }C_{\\mu \\nu \\rho \\lambda }\\,.$ The solutions of conformal gravity discussed in section 5 can be written as $ds^2=-f dt^2 + {\\frac{dr^2}{f}} + r^2 d\\Omega _{2,k}^2\\,,\\qquad f= r^2 +b r + c + {\\frac{d}{r}},\\qquad 3bd-c^2+k^2=0\\,,$ To apply the ADT method to this solution, a background subtraction is necessary.", "It turns out that if we simply choose the static AdS metric as the background, the energy calculated is divergent.", "On the other hand, the background-independent AMD method will also give a infinite result since the leading fall-off of the Weyl tensor of the solution (REF ) is slower than that of the usual AdS black hole.", "Motivated by finding a proper definition of energy for black hole solutions (REF ) in conformal gravity, we adapt the standard Noether method to the Lagrangian of conformal gravity.", "In the following, we briefly review the Noether procedure for deriving a conserved current associated with symmetry generated by the Killing vector $\\xi $ .", "The first variation of the Lagrangian 4-form generated by the vector $\\xi $ can always be expressed as ${\\cal L}_{\\xi }L=E{\\cal L}_{\\xi }\\phi +d\\Theta (\\phi ,{\\cal L}_{\\xi }\\phi ),$ where $\\phi $ represents a collection of tensorial fields, $E$ denotes their equations of motion, and ${\\cal L}_\\xi $ denotes the Lie derivative.", "Using the identity ${\\cal L}_{\\xi }=di_{\\xi }+i_{\\xi }d,$ for the Lie derivative of a differential form, we find a conserved current defined by $J=\\Theta -i_{\\xi }L.$ On shell, we have $dJ=0\\Rightarrow J=dQ,$ where $Q$ is the conserved charge density associated with the symmetry generated by $\\xi $ .", "Applying this procedure to conformal gravity, we find that $Q=\\frac{1}{4}\\epsilon _{\\mu \\nu \\rho \\sigma }Q^{\\rho \\sigma }dx^{\\mu }\\wedge dx^{\\nu },$ with $Q^{\\rho \\sigma } = -\\frac{\\alpha }{8\\pi G}(C^{\\rho \\sigma \\mu \\nu }\\nabla _{\\mu }\\xi _{\\nu }-2\\xi _{\\nu }\\nabla _{\\mu }C^{\\rho \\sigma \\mu \\nu }).$ It is well known that the conserved charge $Q$ derived from the Einstein-Hilbert action only accounts for one half of the true ADM mass.", "(The other half can be understood as coming from a total derivative term added to the Einstein-Hilbert action [79].)", "The validity of the proposal to take (REF ) as the definition of energy for black holes in conformal gravity can be tested by applying it to the known examples of the Schwarzschild-AdS and Kerr-AdS black holes.", "We find that the results using (REF ) coincide with those obtained from the AMD method and in particular, by setting $\\alpha ={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ , we recover the result presented in [72] and [80], thus confirming the tree level equivalence between Einstein gravity and Weyl gravity that was proposed in [38].", "Finally, we calculate the conserved charge for the metric (REF ) associated with the timelike Killing vector $\\partial /\\partial t$ .", "It is given by $\\int Q=-\\frac{\\alpha \\omega _2}{16\\pi G}[4d-{\\textstyle {\\frac{\\scriptstyle 2}{\\scriptstyle 3} } }b(c-k)]\\,.$ As we discuss in section 5, we can use this conserved quantity to provide a definition of energy, which turns out to be consistent with the first law of thermodynamics for the general AdS black holes in conformal gravity." ] ]

[ [ "An Effective Information Retrieval for Ambiguous Query" ], [ "Abstract Search engine returns thousands of web pages for a single user query, in which most of them are not relevant.", "In this context, effective information retrieval from the expanding web is a challenging task, in particular, if the query is ambiguous.", "The major question arises here is that how to get the relevant pages for an ambiguous query.", "We propose an approach for the effective result of an ambiguous query by forming community vector based on association concept of data minning using vector space model and the freedictionary.", "We develop clusters by computing the similarity between community vectors and document vectors formed from the extracted web pages by the search engine.", "We use Gensim package to implement the algorithm because of its simplicity and robust nature.", "Analysis shows that our approach is an effective way to form clusters for an ambiguous query." ], [ "Introduction", "On the web, search engines are key for the information retrieval (IR) for any user query.", "However, resolving ambiguous query is a challenging task, hence a vibrant area of research.", "Due to short and ambiguity in the user query, retrieving the information as per the intention of user in large volume of web is not straight forward.", "The ambiguities in queries is due to the short query length, which is on an average is 2.33 times on a popular search engine [1].", "In this context, Sanderson [2] reports that 7%-23% of the queries frequently occur in two search engines are ambiguous with the average length one.", "For e.g.", "the familiar word Java which is ambiguous as it has multiple senses viz.", "Java coffee, Java Island and Java programming language etc.", "In the user query, ambiguities can also exists which do not appear in surface.", "Because of such ambiguities, search engine generally does not understand in what context user is looking for the information.", "Hence, it returns huge amount of information, in which most of the retrieved pages are irrelevant to the user.", "These huge amount of heterogeneous information retrieve not only increases the burden for search engine but also decreases its performance.", "In this paper we propose an approach to improve the effectiveness of search engine by making clusters of word sense based on association concept of data mining, using vector space model of Gensim [6] and the freedictionary [13].", "The association concept on which the clusters has formed can be describe as follows.", "Suppose, if user queried for the word Apple, which is associated in multiple context viz.", "computer, fruit, company etc.", "Each of this context associated with Apple is again associate with different word senses viz.", "computer is associated with the keyboard, mouse, monitor etc.", "Hence computer can be taken as community vector or cluster whose components/elements are the associated words keyboard, mouse, monitor, etc.", "Here, each element in the cluster represent the sense of computer vector for apple.", "So, if a user looking for apple as a computer, s'he may look for `apple keyboard' or `apple mouse' or `apple monitor' etc.", "We use Minipar [16] to transform a complete sentence into a dependency tree and for the classification of words and phrases into lexical categories.", "The paper is organized as follows.", "In section 2 we examine the related work on the information retrieval based on clustering technique.", "In section 3 we briefly discuss the Gensim package for the implementation of our approach.", "In section 4 we present our approach for the effective information retrieval in the context of user query.", "Section 5 contains analysis of the algorithm.", "Finally Section 6 is the conclusion of the paper." ], [ "Related Work", "Ranking and Clustering are the two most popular methods for information retrieval on the web.", "In ranking, a model is designed using training data, such that model can sort new objects according to their relevance's.", "There are many ranking models [14] which can be roughly categorized as query-dependent and query-independent models.", "In the other method i.e.", "clustering, an unstructured set of objects form a group, based on the similarity among each other.", "One of the most popular algorithms on clustering is k-means algorithm.", "However, the problem of this algorithm is that an inappropriate choice of clusters (k) may yield poor results.", "In case of an ambiguous query, word sense discovery is one of the useful method for IR in which documents are clustered in corpus.", "Discovering word senses by clustering the words according to their distributional similarity is done by Patrick et al, 2002.", "The main drawback of this approach is that they require large training data to make proper cluster and its performance is based on cluster centroid, which changes whenever a new web page is added to it.", "Hence identifying relevant cluster will be a tedious work.", "Herrera et al., 2010 gave an approach, which uses several features extracted from the document collection and query logs for automatically identifying the user’s goal behind their queries.", "This approach success to classifies the queries into different categories like navigational, informational and transactional (B. J. Jansen et al., 2008) but fails to classify the ambiguous query.", "As query logs has been used, it may raise privacy concerns as long sessions are recorded and may led to ethical issues surrounding the users data collections.", "Lilyaa et.al [15] uses statistical relational learning (SRL) for the short ambiguous query based only on a short glimpse of user search activity, captured in a brief search session.", "Many research has been done to map user queries to a set of categories (Powell et al., 2003; Dolin et al., 1998; Yu et al., 2001).", "But all of the above techniques fails to identify the user intention behind the user query.", "The Word Sense Induction (Roberto Navigli et.al, 2010) method is a graph based clustering algorithm, in which snippets are clustered based on dynamic and finer grained notion of sense.", "The approach (Ahmed Sameh et al, 2010) with the help of modified Lingo algorithm, identifying frequent phrases as a candidate cluster label, the snippets are assigned to those labels.", "In this approach semantic recognition is identified by WordNet which enables recognition of synonyms in snippets.", "Clusters formation by the above two approaches not contain all the relevant pages of user choice.", "Our work uses free dictionary and association concept of data mining has been added to our approach to form clusters.", "Secondly it can handle the dynamic nature of the web as Gensim has been used.", "Hence the user intention behind the ambiguous query can be identified in simple and efficient manner.", "In 2008, Jiyang Chen et.", "al.", "purposed an unsupervised approach to cluster results by word sense communities.", "Clusters are made based on dependency based keywords which are extracted for large corpus and manual label are assigned to each cluster.", "In this paper we form the community vector and eliminate the problem of manual assignment of the cluster lable.", "We use Gensim package to avoid the dependency of the large training corpus size [5], and its ease of implementing vector space model (e.g.", "LSI, LDA)." ], [ "Gensim", "Gensim package is a python library for vector space modeling, aims to process raw, unstructured digital texts (“plain text\").", "It can automatically extract semantic topics from documents, used basically for the Natural Language Processing (NLP) community.", "Its memory (RAM) independent feature with respect to the corpus size allows to process large web based corpora.", "In Gensism one can easily plugin his own input corpus and data stream and other vector space algorithms can be trivially incorporated in it.", "In Gensim, many unsupervised algorithms are based on word co-occurrence patterns within a corpus of training documents.", "Once these statistical patterns are found, any plain text documents can be succinctly expressed in the new semantic representation and can be queried for the topical similarity against other documents and so on.", "In addition it has following salient features Straightforward interfaces, scalable software framework, low API learning curve and prototyping.", "Efficient implementations of several popular vector space algorithms, calculation of TF-IDF (term frequency-inverse document frequency), distributed incremental Latent Semantic Analysis, distributed incremental incremental Latent Dirichlet Allocation(LDA).", "I/O wrappers and converters around several popular data formats.", "Vector Space Model: In vector space model, each document is defined as a multidimensional vector of keywords in euclidean space whose axis correspond to the keyword i.e., each dimension corresponds to a separate keyword [4].", "The keywords are extracted from the document and weight associated with each keyword determines the importance of the keyword in the document.", "Thus, a document is represented as, $ D_j = (w_{1j}, w_{2j}, w_{3j}, w_{4j}, ..........w_{nj})$ where $w_{ij}$ is the weight of term $i$ in document $j$ indicating the relevance and importance of the keyword.", "TF-IDF Concept: TF is the measure of how often a word appears in a document and IDF is the measure of the rarity of a word within the search index.", "Combining TF-IDF is used to measure the statistical strength of the given word in reference to the query.", "Mathematically, $ {\\text{TF}_i} = \\frac{n_i}{\\sum _k n_k}$ where, $n_i$ is the number of occurrences of the considered terms and $n_k$ is the number of occurrences of all terms in the given document ${\\text{IDF}_i}= \\log \\frac{N}{df_i}$ where, $N$ is the number of occurrences of the considered terms and $df_i$ is the number of documents that contain term $i$ .", "${\\text{TF-IDF}} = {\\text{TF}}_i \\times \\log \\frac{N}{df_i}$ Cosine Similarity Measure: It is a technique to measure the similarity between the document and the query.", "The angle ($\\theta $ ) between the document vector and the query vector determines the similarity between the document and the query and it is written as $\\cos \\theta = \\frac{\\sum w_{q,j}w_{ij}}{\\sqrt{\\sum w^2_{q,j}} \\sqrt{\\sum w^2_{i,j}}}$ $\\sqrt{\\sum w^2_{q,j}}$ and $\\sqrt{\\sum w^2_{i,j}}$ is the length of the query and document vector respectively.", "If $\\theta = 0^\\circ $ then the document and query is similar.", "As $\\theta $ changes from 0o to 90o, the similarity between the document and query decreases i.e.", "${\\mathbf {D}_2}$ will be more similar to query than ${\\mathbf {D}_1}$ , if the angle between ${\\mathbf {D}_2}$ and query is smaller than the angel between ${\\mathbf {D}_1}$ and query." ], [ "Our Approach ", "Our approach for an ambiguous query is described below in five steps and depicted in the flow chart (Fig.", "1).", "Web page extraction and preprocessing: Submit the ambiguous query to a search engine and extract top $\\mathbf {n}$ pages.", "Preprocess the retrieve corpus as follows: Remove the stop and unwanted words.", "Select noun as the keywords from the corpus using Minipar [16] and ignore other categories, such as verbs, adjectives, adverbs and pronounce.", "Do stemming using porter algorithm [12].", "Save each processed $\\mathbf {n}$ pages as documents $\\mathbf {D_k}$ , where $k = 1,2,3,.....n$ .", "Document vectors: Compute TF and IDF score for all the keywords of each $\\mathbf {D_k}$ using Gensim and make document vectors of all the retrieved pages.", "Cluster formation: We use the freedictionary with the option start with to form the community vector of the queried word as follows Submit the ambiguous query (say apple) to the freedictionary, preprocess the retrieved data i.e.", "remove the queried, stop & unwanted words.", "After stemming, save all the noun as keywords ($\\mathbf {W_j}$ ) in a file $\\mathbf {F_c}$ , where $j = 1,2,3, ......m$ Now submit each $\\mathbf {W_j}$ again to the freedictionary, preprocess the retrieved data and save the noun as keywords along with the queried word in a community file $F_\\mathbf {W_j}$ .", "Search all the words of $F_\\mathbf {W_j}$ in $\\mathbf {D_k}$ using regular expression search technique.", "Delete those words in $F_\\mathbf {W_j}$ which are not present in $\\mathbf {D_k}$ .", "$\\mathbf {W_j}$ is the formed community vectors (clusters) whose elements are the words saved in the file $F_\\mathbf {W_j}$ Compute TF-IDF for each word in $F_\\mathbf {W_j}$ in compare with $\\mathbf {D_k}$ to form community vectors.", "Similarity check: Compute the cosine similarities between the formed documents and community vectors using eq.", "1.", "Assignment of Documents to the Clusters: Assign the documents to that cluster which has maximum similarity.", "Figure: An effective IR for an ambiguous query" ], [ "Test Results", "To illustrate our approach we took four sample documents as shown in Table 1.", "We preprocess the documents and extracted ten keywords (apple, computer, tree, keyboard, mouse, juice, country, vegetables, fruit, monitor) from the sample (Table 2).", "After assigning a token ID to each selected keyword (Table 3) TF & IDF are computed which is shown in Table 4.", "In Table 5 computed weight (TF-IDF) of all the four sample documents are given.With the calculated weight and respective token IDs, document vectors are generated (Table 6).", "The community vectors are formed as described in the section 4 (Table 7) and the corresponding TF-IDF and weights are calculated (Table 8).", "Cosine similarity are calculated defined by the eq.", "1.", "Now the similarity between each community vector (C1, C2) and the set of document vectors (D1, D2, D3 and D4) are computed and maximum values of the similarity between community and document vectors form the cluster.", "From our experimental result, we found that (D1, D3 ) and (D2, D4) associated with C1 and C2 respectively i.e.", "two clusters are generated (Table 9 and 10).", "As an example, from the Table 10 we say that if the user search the ambiguous word apple, s'he will get two clusters C1 and C2, containing most relevant documents." ], [ "Conclusion", "For an ambiguous query, we propose an effective approach for the IR by forming the clusters of relevant web pages.", "For cluster formation we use standard vector space model and the freedictionary.", "From our approach we find that user intention behind ambiguous query can be identify significantly.", "This unsupervised approach not only handles the corpus by extracting and analyzing significant terms, but also form desire clusters for real time query.", "Further we would extend our work for the multi word query and improving these clusters using ranking techniques." ], [ "Acknowledgment", "We are thankful to Bharat Deshpande and our colleague Aruna Govada and K.V.", "Santhilata for their useful discussions and valuable suggestions." ] ]

[ [ "Isospin violating decay of $\\psi(3770)\\rightarrow J/\\psi + \\pi^0$" ], [ "Abstract The strong-isospin violation in $\\psi(3770)\\rightarrow J/\\psi + \\pi^0$ via intermediate $D$ meson loops is investigated in an effective Lagrangian approach.", "In this process, there is only one $D$-meson loop contributing to the absorptive part, and the uncertainties due to the introduction of form factors can be minimized.", "With the help of QCD spectral sum rules (QSSR), we extract the $J/\\psi DD^*$ form factor as an implement from the first principle of QCD.", "The $DD^*\\pi^0$ form factor can be well determined from the experimental data for $D\\rightarrow\\pi l\\nu$.", "The exploration of the dispersion relation suggests the dominance of the dispersive part via the intermediate $D$ meson loops even below the open charm threshold.", "This investigation could provide further insights into the puzzling question on the mechanisms for $\\psi(3770)\\to$ non-$D\\bar{D}$ transitions." ], [ "Introduction", "The non-$D\\bar{D}$ decays of $\\psi (3770)$ have attracted a lot attention during the past decades.", "As $\\psi (3770)$ is the first state above the open charm $D\\bar{D}$ threshold, its decay was believed to be saturated by the $D\\bar{D}$ channel via the Okubo-Zweig-Iizuka (OZI) connected diagram.", "Such an anticipation was supported by early experimental data which showed that exclusive decays of $\\psi (3770)\\rightarrow $ non-$D\\bar{D}$ were negligibly small.", "Theoretical calculations of the perturbative QCD (pQCD) leading order contributions also suggested rather small non-$D\\bar{D}$ branching ratios for $\\psi (3770)$  [1], [2], [3], [4], [5], [6].", "Interestingly, recent studies of the $\\psi (3770)$ non-$D\\bar{D}$ decays in experiment and theory have exposed unexpected results which complicated the situation.", "In experiment, the $D\\bar{D}$ cross section measurement by the CLEO collaboration suggests that the non-$D\\bar{D}$ branching ratio is consistent with zero with an upper limit of about 6.8% [7], [8], [9].", "Rather contradicting the CLEO results, the BES collaboration finds much larger non-$D\\bar{D}$ branching ratios of $\\sim 15\\%$ in the direct measurement of non-$D\\bar{D}$ inclusive cross section [10].", "Recently a next-to-leading-order (NLO) nonrelativistic QCD (NRQCD) calculation of the $c\\bar{c}$ annihilation width for $\\psi (3770)$ suggests that the higher order contributions can account for about $5\\%$ of the $\\psi (3770)$ non-$D\\bar{D}$ decay branching ratios at most [11].", "In Refs.", "[12], [13], it was proposed that the open-charm threshold effects via intermediate meson loops (IML) could serve as an important nonperturbative mechanism to produce sizable non-$D\\bar{D}$ branching ratios.", "Note that $\\psi (3770)$ is close to the $D\\bar{D}$ open threshold.", "A natural conjecture is that the $D\\bar{D}$ threshold would play an important role in its production and decay.", "This mechanism turns out to be successful in the explanation of the decay of $\\psi (3770)\\rightarrow \\text{vector}+\\text{pseudoscalar}$ as one of the non-$D\\bar{D}$ decay channel of $\\psi (3770)$  [12], [13].", "During the past few years, there have been observations of a large number of heavy quarkonium states [14] at the B-factories (Belle and BaBar) and electron storage-rings (CLEO).", "Some of those states have masses close to open thresholds and cannot be easily accommodated in the framework of potential quark models.", "For instance, the well-established $X(3872)$ is located in the vicinity of $D^*\\bar{D}$ threshold and its mass as a $1^{++}$ state is nearly 100 MeV lower than the first radial excitation of $\\chi _{c1}$ in potential models.", "Such observations, on the one hand, have raised serious questions on the constituent degrees of freedom within heavy quarkonia, and on the other hand, raised questions on the role played by the open decay thresholds via the IML as an important nonperturbative mechanism in the understanding of the properties of those newly observed states.", "Such a mechanism symbolizes a general dynamical feature in the charmonium mass region, thus should be explored broadly in various processes.", "To gain further insights into the underlying dynamics and understand better the properties of the IML, we are motivated to study the decays of $\\psi (3770)\\rightarrow J/\\psi +\\eta $ and $J/\\psi +\\pi ^0$ .", "First, we note that the decay of $\\psi (3770)\\rightarrow J/\\psi +\\eta $ is one of few measured non-$D\\bar{D}$ decay channels in experiment with $BR(\\psi (3770)\\rightarrow J/\\psi +\\eta ) = (9\\pm 4)\\times 10^{-4}$  [15].", "One can estimate the branching ratio of $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ via $\\eta $ -$\\pi ^0$ mixing based on the leading-order chiral perturbation theory.", "The mixing intensity can be expressed as [16] $\\epsilon _0=\\frac{\\sqrt{3}}{4}\\frac{m_d-m_u}{m_s-(m_u+m_d)/2}.$ Using Dashen's theorem [17], one obtains $\\epsilon _0=\\frac{1}{\\sqrt{3}}\\frac{M_{K^0}^2-M_{K^+}^2+M_{\\pi ^+}^2-M_{\\pi ^0}^2}{M_\\eta ^2-M_{\\pi ^0}^2}=0.01\\ .$ Taking the $\\eta -\\eta ^\\prime $ mixing into account, the mixing intensity is slightly enhanced [18] $\\hat{\\epsilon }=\\epsilon _0\\sqrt{3}\\cos \\phi ,$ where $\\sqrt{3}\\cos \\phi =1.34$ would be unity if $\\phi $ is the ideal mixing angle.", "With the $\\eta $ -$\\pi ^0$ mixing intensity in a range of $0.01\\sim 0.02$ , the branching ratio of $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ from $\\eta $ -$\\pi ^0$ mixing is at most the order of $10^{-6}$ .", "This result actually sets up a limit for the $\\eta $ -$\\pi ^0$ mixing contributions in $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ .", "In contrast, the Particle Data Group (PDG2010) [15] gives an experimental upper limit $BR(\\psi (3770)\\rightarrow J/\\psi +\\pi ^0)< 2.8\\times 10^{-4}$ .", "A recent investigation of $\\psi ^{\\prime }\\rightarrow J/\\psi +\\pi ^0$  [19], [20] based on a nonrelativistic effective field theory (NREFT) suggests that the strong-isospin violation via the IML is relatively enhanced by $1/v$ in comparison with the tree-level contribution where the pion is emitted directly from the charmonium through soft gluon exchanges, where $v\\simeq 0.5$ is the velocity of the intermediate charmed meson.", "Since $\\psi (3770)$ is close to the $D\\bar{D}$ threshold, we expect that such a strong-isospin mechanism would also play a role.", "As a consequence, the IML mechanism may lead to a sizable branching ratio of $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ which might be significantly larger than that given by the $\\eta $ -$\\pi ^0$ mixing.", "In an early study [21], [22], [23], the absorptive contribution from the intermediate $D\\bar{D}$ in the decay of $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0(\\eta )$ was calculated with an exponential form factor determined by the characteristic mass scale in the exchange channel.", "It was also argued that the real part contribution from $D\\bar{D}$ and other heavier $D$ meson loops should cancel each other in order to obey the OZI rule successfully in $J/\\psi $ decay.", "However, it is found [24] that the IML effects may still be important in the decay of charmonium close to the open charm threshold.", "This is because the quark-hadron duality turns out to have been broken locally.", "As a consequence, the decay of a charmonium state can still experience the open threshold effects significantly if its mass is close to the open threshold.", "Such a scenario may imply that the real parts of the exclusive $\\psi (3770)$ decays could not be neglected and could explain the observed sizeable non-$D\\bar{D}$ branching ratios of $\\psi (3770)$  [10].", "In this work, we shall apply an effective Lagrangian approach (ELA) to investigate the IML effects in $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ , and demonstrate that the IML transitions have dominant contributions to this isospin-violating decay channel.", "As an important improvement of this approach, we shall implement form factors from QCD spectral sum rules (QSSR) for the off-shell $J/\\psi D D^*$ coupling vertex, while the $D^*D\\pi ^0$ form factor can be extracted from the semileptonic decay of $D\\rightarrow \\pi ^0 l\\nu $ .", "We mention in advance that this elaborate treatment will allow a reliable estimate of the absorptive amplitude of $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ .", "Meanwhile, we also explore the real part of the IML contributions with the help of dispersion relation, within which the effective threshold is determined by the experimental data of $\\psi (3770)\\rightarrow J/\\psi +\\eta $ .", "This paper is organized as follows: In Sec.", "the ELA for the IML transitions is formulated.", "In Sec.", "the form factors from QCDSR and $D$ meson semileptonic decays are investigated.", "The numerical results for the isospin-violating decay $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ are presented in Sec.", ", and a brief summary is given in Sec.", ".", "As illustrated by Fig.", "REF , only the $DD(D^*)$ loop (the meson in the parenthesis denotes the exchanged particle between $J/\\psi $ and $\\pi ^0$ ) contributes to the imaginary part.", "The effective Lagrangians are as follows [25], [26], $\\mathcal {L}_{\\psi ^{\\prime \\prime } DD}&=&-ig_{\\psi ^{\\prime \\prime }DD}\\psi ^{\\prime \\prime \\mu }D_i^\\dag \\stackrel{\\leftrightarrow }{\\partial _\\mu }D_i \\ , \\\\\\mathcal {L}_{\\psi DD^*}&=&g_{\\psi DD^*}\\epsilon _{\\mu \\nu \\alpha \\beta }\\partial ^\\mu \\psi _n^\\nu \\lbrace D_i^{*\\beta \\dag }\\stackrel{\\leftrightarrow }{\\partial ^\\alpha }D_i-D_i^\\dag \\stackrel{\\leftrightarrow }{\\partial ^\\alpha }D_i^{*\\beta }\\rbrace \\ ,\\\\\\mathcal {L}_{D^*D\\pi }&=&-ig_{D^*DP}\\left(D^i\\partial ^\\mu P_{ij}D_\\mu ^{*j\\dag }-D_\\mu ^{*i}\\partial ^\\mu P_{ij}D^{j\\dag }\\right) \\ ,$ where the coupling $g_{\\psi DD^*}\\equiv M_\\psi /(f_\\psi \\sqrt{M_{D^*}M_D})\\ \\mathrm {GeV}^{-1}$ .", "The same convention has also been adopted in Ref. [24].", "The decay amplitude via the $D\\bar{D}(D^*)$ loop is $\\mathcal {M}_{fi}&=&\\sum _{Polarization}\\int \\frac{\\mathrm {d}^4p_5}{(2\\pi )^4}(2g_{\\psi ^{\\prime \\prime }DD}p_3\\cdot \\epsilon _1)(-2g_{\\psi DD^*}\\epsilon _{\\mu \\nu \\alpha \\beta }q_2^\\mu \\epsilon _2^{*\\nu }p_5^\\alpha \\epsilon _5^\\beta )(\\frac{-g_{D^*DP}}{\\sqrt{2}}p\\cdot \\epsilon _5^*)\\nonumber \\\\&&\\times \\frac{i}{p_3^2-M_D^2}\\frac{i}{p_4^2-M_D^2}\\frac{i}{p_5^2-M_{D^*}^2}F(p_i^2) \\ .$ At each vertex of the loop diagram, the off-shell effect or the finite size effect should be taken into account by introducing the form factor $F(p_i^2)$ , which can be regarded as the extended version of the local couplings in the original effective Lagrangian.", "The form factor is also necessary for cutting off the ultraviolet divergence in the loop integrals.", "Figure: The DD-loop diagram contributing to the absorptive part.Applying the Cutkosky rule, the discontinuity of the decay amplitude is $\\emph {Disc.", "}\\mathcal {M}_{fi}&=&-i(-2\\pi i)^2\\int \\frac{\\mathrm {d}^4p_5}{(2\\pi )^4}(2g_{\\psi ^{\\prime \\prime }DD})(p_3\\cdot \\epsilon _1)\\epsilon _{q_2\\epsilon _2^*p_5\\mu }p_\\nu \\left(-g^{\\mu \\nu }+\\frac{p_5^\\mu p_5^\\nu }{M_{D^*}^2}\\right)\\nonumber \\\\&&\\times \\delta (p_3^2-M_D^2)\\times \\delta (p_4^2-M_D^2)\\times \\frac{F_{\\psi DD^*}(p_5^2)F_{D^*D\\pi ^0}(p_5^2)}{p_5^2-M_{D^*}^2}$ with $\\epsilon _{q_2\\epsilon _2^*p_5\\mu }\\equiv \\epsilon _{\\alpha \\beta \\gamma \\mu }q_2^\\alpha \\epsilon _2^{*\\beta }p_5^\\gamma $ for a short notation.", "Then, there are two form factors depending only on the virtuality of $D^*$ left in our calculation.", "In practice, the product of these form factors can be parameterized as the product of the local couplings and an empirical form factor: $F_{\\psi _nDD^*}(p_5^2)F_{D^*D\\pi ^0}(p_5^2)\\equiv -2g_{\\psi _nDD^*}\\frac{-g_{D^*DP}}{\\sqrt{2}}F(p_5^2) \\ ,$ where a dipole form factor is adopted, $F(p_5^2)=\\left(\\frac{\\Lambda ^2-M_{D^*}^2}{\\Lambda ^2-p_5^2}\\right)^2,$ with $\\Lambda =M_{D^*}+\\alpha \\Lambda _{QCD}$ and $\\Lambda _{QCD}=0.22 \\ \\mathrm {GeV}$ .", "Further deduction gives the discontinuity of the decay amplitude $\\emph {Disc.", "}\\mathcal {M}_{fi}&=&2ig_{\\psi ^{\\prime \\prime }DD}\\int \\frac{\\mathrm {d}^3\\vec{p}_3}{(2\\pi )^32E_{p_3}2E_{p_4}}\\frac{F_{\\psi DD^*}(p_5^2)F_{D^*D\\pi ^0}(p_5^2)}{p_5^2-M_{D^*}^2}\\nonumber \\\\&&\\times 2\\pi \\delta (p_3^0+p_4^0-q_1^0)\\epsilon _{q_2\\epsilon _2^*p_3p}p_3\\cdot \\epsilon _1|_{constraints} \\ ,$ where $E_{p_i}=\\sqrt{|\\vec{p}_i|^2+M_i^2}$ , and $constraints\\equiv (p_3^0>0,p_4^0>0,p_3^2=M_D^2,p_4^2=M_D^2,p_5=q_2-p_3,\\vec{p}_4=\\vec{p}_5+\\vec{p})$ .", "Extracting the antisymmetric Lorentz structure $\\epsilon _{q_2\\epsilon _2^*\\mu p}\\epsilon _{1\\nu }$ , we get the tensor amplitude $\\emph {Disc.", "}\\mathcal {M}_{fi}^{\\mu \\nu }&=&2ig_{\\psi ^{\\prime \\prime }DD}\\int \\frac{\\mathrm {d}^3\\vec{p}_3}{(2\\pi )^32E_{p_3}2E_{p_4}}\\frac{F_{\\psi DD^*}(p_5^2)F_{D^*D\\pi ^0}(p_5^2)}{p_5^2-M_{D^*}^2}\\nonumber \\\\&&\\times 2\\pi \\delta (p_3^0+p_4^0-q_1^0)p_3^\\mu p_3^\\nu |_{constraints}$ From Lorentz invariance, this tensor structure can be decomposed into terms built out of the external momenta and metric tensor: $\\emph {Disc.", "}\\mathcal {M}_{fi}^{\\mu \\nu }&=&g^{\\mu \\nu }\\mathcal {M}_A+p^\\mu p^\\nu \\mathcal {M}_B+q_2^\\mu q_2^\\nu \\mathcal {M}_C+(p^\\mu q_2^\\nu +p^\\nu q_2^\\mu )\\mathcal {M}_D,$ where only $\\mathcal {M}_A$ will contribute to the final result when we contract the tensor amplitude with the extracted antisymmetric Lorentz structure.", "Contracting the tensor amplitude with the metric tensor, we obtain $g_{\\mu \\nu }\\emph {Disc.", "}\\mathcal {M}_{fi}^{\\mu \\nu }&=&2ig_{\\psi ^{\\prime \\prime }DD}\\int \\frac{\\mathrm {d}^3\\vec{p}_3}{(2\\pi )^32E_{p_3}2E_{p_4}}\\frac{F_{\\psi DD^*}(p_5^2)F_{D^*D\\pi ^0}(p_5^2)}{p_5^2-M_{D^*}^2}\\nonumber \\\\&&\\times 2\\pi \\delta (p_3^0+p_4^0-q_1^0)p_3^2|_{constraints}\\nonumber \\\\&=&2ig_{\\psi ^{\\prime \\prime }DD}\\int \\frac{|\\vec{p}_3|^2\\mathrm {d}|\\vec{p}_3|\\sin \\theta \\mathrm {d}\\theta \\mathrm {d}\\varphi }{(2\\pi )^32E_{p_3}2E_{p_4}}\\frac{F_{\\psi DD^*}(p_5^2)F_{D^*D\\pi ^0}(p_5^2)}{p_5^2-M_{D^*}^2}\\nonumber \\\\&&\\times 2\\pi \\delta (p_3^0+p_4^0-q_1^0)p_3^2|_{constraints}\\nonumber \\\\$ In the rest frame of $\\psi (3770)$ and setting the direction of $\\vec{q}_2$ as the $z$ -axis, the dynamic variables can be expressed as $h&\\equiv &|\\vec{q}_2|=\\sqrt{\\left(\\frac{M_\\psi ^2+M_{\\psi ^{\\prime \\prime }}^2-M_\\pi ^2}{2M_{\\psi ^{\\prime \\prime }}}\\right)^2-M_\\psi ^2}\\nonumber \\\\\\frac{r}{2}&\\equiv &|\\vec{p}_3|=\\frac{\\sqrt{M_{\\psi ^{\\prime \\prime }}^2-4M_D^2}}{2}\\nonumber \\\\v&\\equiv &q_2^0=\\frac{M_{\\psi ^{\\prime \\prime }}^2+M_\\psi ^2-M_\\pi ^2}{2M_{\\psi ^{\\prime \\prime }}}\\nonumber \\\\w&\\equiv &p^0=\\frac{M_{\\psi ^{\\prime \\prime }}^2-M_\\psi ^2+M_\\pi ^2}{2M_{\\psi ^{\\prime \\prime }}}\\nonumber \\\\q_1^0&=&v+w=M_{\\psi ^{\\prime \\prime }}\\nonumber \\\\p_3^0&=&\\frac{v+w}{2}$ Then, the Lorentz invariant amplitude is $g_{\\mu \\nu }\\emph {Disc.", "}\\mathcal {M}_{fi}^{\\mu \\nu }&=&ig_{\\psi ^{\\prime \\prime }DD}\\times \\frac{r}{8\\pi (v+w)}\\times \\int _{-1}^1\\mathrm {d}x\\,\\frac{M_D^2F_{\\psi DD^*}(x)F_{D^*D\\pi ^0}(x)}{G(x)}\\nonumber \\\\$ where the propagator of $D^*$ is $G(x)=M_\\psi ^2+M_D^2-2[v(v+w)/2-h rx/2]-M_{D^*}^2$ .", "Similarly, we can get other three Lorentz invariant amplitudes: $p_\\mu p_\\nu \\emph {Disc.", "}\\mathcal {M}_{fi}^{\\mu \\nu }&=&ig_{\\psi ^{\\prime \\prime }DD}\\times \\frac{r}{8\\pi (v+w)}\\nonumber \\\\&&\\times \\int _{-1}^1\\mathrm {d}x\\,\\frac{[w(v+w)/2+h r x/2]^2 F_{\\psi DD^*}(x)F_{D^*D\\pi ^0}(x)}{G(x)}\\nonumber \\\\$ $q_{2\\mu }q_{2\\nu }\\emph {Disc.", "}\\mathcal {M}_{fi}^{\\mu \\nu }&=&ig_{\\psi ^{\\prime \\prime }DD}\\times \\frac{r}{8\\pi (v+w)}\\nonumber \\\\&&\\times \\int _{-1}^1\\mathrm {d}x\\,\\frac{[v(v+w)/2-h r x/2]^2 F_{\\psi DD^*}(x)F_{D^*D\\pi ^0}(x)}{G(x)}\\nonumber \\\\$ $p_\\mu q_{2\\nu }\\emph {Disc.", "}\\mathcal {M}_{fi}^{\\mu \\nu }&=&ig_{\\psi ^{\\prime \\prime }DD}\\times \\frac{r}{8\\pi (v+w)}\\nonumber \\\\&&\\times \\int _{-1}^1\\mathrm {d}x\\,\\frac{[v(v+w)/2-h r x/2][w(v+w)/2+hr x/2] F_{\\psi DD^*}(x)F_{D^*D\\pi ^0}(x)}{G(x)}\\nonumber \\\\$ Solving these four equations simultaneously, we can get the complicated expression of the invariant amplitude $\\mathcal {M}_A$ and the absorptive part of the decay amplitude $\\emph {Disc.", "}\\mathcal {M}_{fi}$ .", "The charge conjugate contribution gives the same result.", "Because of the mass of $\\psi (3770)$ being above the charmed meson pair, the coupling constant $g_{\\psi ^{\\prime \\prime }DD}$ and the isospin difference may be difficult to get from theory because of the rescattering mechanism.", "So we will extract this coupling directly from the experimental data: $\\Gamma _{\\psi ^{\\prime \\prime }\\rightarrow D\\bar{D}}&=&\\frac{4g_{\\psi ^{\\prime \\prime }DD}^2|\\vec{p}_3|}{8\\pi M_{\\psi ^{\\prime \\prime }}^2}\\times \\frac{1}{3}\\sum _{\\epsilon _1}(p_3\\cdot \\epsilon _1)(p_3\\cdot \\epsilon _1^*)\\nonumber \\\\&=&\\frac{g_{\\psi ^{\\prime \\prime }DD}^2|\\vec{p}_3|}{6\\pi M_{\\psi ^{\\prime \\prime }}^2}\\left(-M_D^2+\\frac{M_{\\psi ^{\\prime \\prime }}^2}{4}\\right) \\ .$" ], [ "Dispersive part", "In principle, all the meson loops of which the thresholds are above the $\\psi (3770)$ mass would contribute to the dispersive part (i.e.", "the real part) of the transition amplitude.", "Because of the introduction of form factors in the loop integrals, some model dependence seems inevitable in the evaluation of the real part.", "Given that the imaginary part of the amplitude can be reliably determined as in the previous subsection, we shall apply the dispersion relation to obtain the real part of the decay amplitude.", "Taking the assumption that the spectral density can be approximated by the extrapolation $\\mathcal {M}_A(M_{\\psi ^{\\prime \\prime }}^2)\\rightarrow \\mathcal {M}_A(s_1)$ , we have the unsubtracted dispersion relation: $Re[\\mathcal {M}_{fi}^{Tot}]=\\frac{1}{2\\pi i}\\left(\\mathcal {P}\\int _{(2M_{D^+})^2}^{{th_C}}\\frac{2\\mathcal {M}_A^{C}(s_1)}{s_1-M_{\\psi ^{\\prime \\prime }}^2}\\mathrm {d}s_1+\\mathcal {P}\\int _{(2M_{D^0})^2}^{{th_N}}\\frac{2\\mathcal {M}_A^{N}(s_1)}{s_1-M_{\\psi ^{\\prime \\prime }}^2}\\mathrm {d}s_1\\right)\\ , $ where $\\mathcal {M}_A^{C/N}$ corresponds to the charged or neutral $D$ meson loop's contribution, and the factor 2 in front of $\\mathcal {M}_A(s_1)$ refers to the charge conjugate contribution.", "Then the total decay width is $\\Gamma &=&\\frac{h}{8\\pi M_{\\psi ^{\\prime \\prime }}^2}\\int \\frac{\\mathrm {d}\\Omega _{cm}}{4\\pi }\\left[Re[\\mathcal {M}_{fi}^{Tot}]^2+\\left(\\frac{2\\mathcal {M}_A^{Tot}}{2i}\\right)^2\\right]\\times \\frac{1}{3}\\sum _{\\epsilon _1,\\epsilon _2}\\epsilon _{q_2\\epsilon _2^*\\epsilon _1p}\\epsilon _{q_2\\epsilon _2\\epsilon _1^*p}\\nonumber \\\\&=&-\\frac{h}{12\\pi M_{\\psi ^{\\prime \\prime }}^2}\\left[Re[\\mathcal {M}_{fi}^{Tot}]^2+\\left(\\frac{2\\mathcal {M}_A^{Tot}}{2i}\\right)^2\\right]\\nonumber \\\\&&\\times \\left[M_\\pi ^2\\left(v-w+\\frac{M_\\pi ^2}{v+w}\\right)(v+w)-(vw+h^2)^2\\right] \\ ,$ where $\\mathcal {M}_A^{Tot}=\\mathcal {M}_A^{C}+\\mathcal {M}_A^{N}$ .", "Two points should be stressed: one is the upper limit of the dispersive integral, and the other is the virtuality dependence of the coupling $g_{\\psi ^{\\prime \\prime }DD}$ .", "Generally speaking, the upper limit of the dispersive integral should be infinity from the mathematical viewpoint.", "But in practice, we only take a finite effective threshold ${th}$ because the spectral density is only an approximation.", "It is presented that the form factor of $D+J/\\psi (virtual)\\rightarrow D$ is harder than that of $D+\\rho (virtual)\\rightarrow D$ because $D$ can “see” the size of smaller $J/\\psi $  [27].", "We expect that the heavier $\\psi (3770)$ also gives a harder $g_{\\psi ^{\\prime \\prime }DD}(s_1)$ form factor at large $s_1$ so that in a limited $s_1$ region the $s_1$ -dependence can be neglected.", "There is also literature [13] to take this $s_1$ -dependence into account by adding a suppression factor $\\exp (-I|\\vec{p}_3|^2)$ into the integrand of Eq.", "(REF ), where $I$ is the square of the interaction length [28].", "We will discuss both points in detail in the following numerical analysis." ], [ "QSSR reanalysis of the form factor $F_{\\psi DD^*}(p_5^2)$", "Since the mass of $J/\\psi $ is below the lowest threshold of open charm $D\\bar{D}$ , it is not possible to measure the form factor $F_{\\psi DD^*}(p_5^2)$ in experiment directly.", "There has been a systematic investigation of the charmonium to open charmed meson form factors in the framework of QSSR [29], [30].", "As a crucial criterion of QSSR, the pole contribution should take a dominant part in the dispersion integral.", "To our surprise, it seems not possible to satisfy this condition with the parameters given in the literature.", "This stimulate us to reinvestigate the $F_{\\psi DD^*}(p_5^2)$ with the improved QSSR approach, and crosscheck the result with finite energy sum rules (FESR).", "We shall be concerned with the three-point correlation function: $\\Gamma _{\\mu \\nu }(q_2,p_3)=\\int \\mathrm {d}^4x\\,\\mathrm {d}^4ye^{ip_3\\cdot x}e^{-i(p_3-q_2)\\cdot y}\\langle 0|T\\left\\lbrace J^3(x)J_\\mu ^{2\\dag }(y)J_\\nu ^{1\\dag }(0)\\right\\rbrace |0\\rangle \\ ,$ where $J_\\nu ^1=\\bar{c}\\gamma _\\nu c$ , $J_\\mu ^2=\\bar{q}\\gamma _\\mu c$ and $J^3=i\\bar{q}\\gamma _5 c$ denote the interpolating currents for the incoming $J/\\psi (q_2,\\epsilon _2)$ , incoming $D^*(p_5,\\epsilon _5)$ and outgoing $D$ , respectively.", "Taking the advantage of the unique Lorentz structure for the $VVP$ coupling, we can decompose $\\Gamma _{\\mu \\nu }$ simply as: $\\Gamma _{\\mu \\nu }(q_2,p_3)\\equiv \\Lambda (q_2^2,p_3^2,p_5^2)\\epsilon _{\\mu \\nu \\alpha \\beta }q_2^\\alpha p_5^\\beta \\ ,$ where $p_3=p_5+q_2$ .", "The above expression has an arbitrary sign compared with that the preceding section.", "Using a double dispersion relation, one can express the invariant amplitude as: $\\Lambda (q_2^2,p_3^2)=\\frac{-1}{4\\pi ^2}\\int \\mathrm {d}s\\,\\mathrm {d}u\\frac{\\rho (s,u,p_5^2)}{(s-q_2^2)(u-p_3^2)}\\ .$ For the $D^*$ -meson off-shell, the spectral density can be obtained from the Cutkosky rule presented in the previous section.", "On the phenomenological side, the three-point correlation function can be approximated by the lowest resonance plus the “QCD continuum” contributions, where the latter come from the discontinuity of the QCD diagrams from a threshold: $\\sqrt{u_0}(\\sqrt{s_0})\\equiv M_D (M_\\psi ) +\\Delta \\ ,$ and smears the contributions of all higher resonance contributions.", "In this way, the phenomenological part of the three-point function reads: $\\Lambda ^{phen}=\\frac{\\delta _c F_{\\psi DD^*}(p_5^2)\\epsilon _{\\mu \\nu q_2p_5}}{(p_5^2-M_{D^*}^2)(q_2^2-M_\\psi ^2)(p_3^2-M_D^2)}+\\textrm {``QCD\\,\\,continuum^{\\prime \\prime }}$ where $\\delta _c\\equiv {M_D^2M_{D^*}M_\\psi f_D f_{D^*}f_\\psi }/{m_c}$ , the form factor with virtual $D^*$ is defined as $\\langle D(p_3)|J_\\mu ^{2\\dag }|J/\\psi (q_2)\\rangle &=& \\frac{ \\langle D^*(p_5)|J_\\mu ^{2\\dag }|0\\rangle \\langle D(p_3)|D^*(p_5)J/\\psi (q_2)\\rangle }{p_5^2-M_{D^*}^2}\\ ,$ with $\\langle D(p_3)|D^*(p_5)J/\\psi (q_2)\\rangle &\\equiv & F_{\\psi DD^*}(p_5^2)\\epsilon _{\\epsilon _2\\epsilon _5p_3p_5} \\ ,$ and the decay constants are normalized as: $\\langle D^*(p_5)|J_\\mu ^{2\\dag }|0\\rangle &=&M_{D^*}f_{D^*}\\epsilon _\\mu ^* \\ ,\\nonumber \\\\\\langle 0|J^3|D(p_3)\\rangle &=&\\frac{M_D^2 f_D}{m_c} \\ ,\\nonumber \\\\\\langle J/\\psi (q_2)|J_\\nu ^1|0\\rangle &=&M_\\psi f_\\psi \\epsilon _\\nu ^*\\ .$ Matching the two sides of correlation function, and performing the Borel transformation (Laplace SR), the lowest perturbative diagram gives $F_{\\psi DD^*}(p_5^2)&=&-\\frac{1}{4\\pi ^2}\\frac{p_5^2-M_{D^*}^2}{\\delta _c}\\int _{4m_c^2}^{s_0}\\int _{u_{min}}^{u_0}\\mathrm {d}s\\,\\mathrm {d}u \\,\\,\\rho (u,s,t) \\nonumber \\\\&&\\times e^{-(s-M_\\psi ^2)\\tau _1}e^{-(u-M_D^2)\\tau _2}\\theta (u_{max}-u) \\ ,$ with $\\rho (s,t,u)&=&\\frac{3m_c}{\\sqrt{\\lambda }}\\left(1+\\frac{s\\lambda _2}{\\lambda }\\right) \\ ,\\nonumber \\\\u_{min}^{max}&=&\\frac{1}{2m_c}\\left[-s t+m_c^2(s+2t)\\pm \\sqrt{s(s-4m_c^2)(t-m_c^2)^2}\\right] \\ ,$ where $t=p_5^2$ , $\\lambda \\equiv (u+s-t)^2-4u s$ , $\\lambda _2\\equiv u+t-s+2m_c^2$ , and $\\tau _{1,2}$ are the inverse squares of the corresponding Borel masses.", "Taking the limits $\\tau _1\\rightarrow 0$ and $\\tau _2\\rightarrow 0$ , we obtain the FESR.", "Here, we neglect the numerically small gluon condensate contribution [30].", "To the leading order approximation where the three-point correlation function is evaluated, it is consistent to extract the decay constants $f_D$ and $f_\\psi $ from the corresponding two-point functions at the lowest order other than the value extracted from the experiment directly, e.g.", "$f_\\psi =0.405\\pm 0.015 \\ \\mathrm {GeV}$ .", "The QCD expressions of the pseudoscalar and vector two-point functions are well known [31].", "We show our analysis for $f_D$ and $f_\\psi $ in Fig.", "REF and Fig.", "REF , respectively.", "Stabilities in both the two-point sum rule variables $\\tau _{\\psi ,D}$ and variation of the continuum threshold $\\Delta $ are observable.", "We show in Fig.", "REF the $\\Delta $ behavior of the ratio $\\tau _\\psi /\\tau _D$ which is rather stable, especially for $m_c=1.26\\ \\mathrm {GeV}$ .", "The obtained optimal ratios (stable with $\\Delta $ ) are: $\\frac{\\tau _\\psi }{\\tau _D} \\simeq {\\left\\lbrace \\begin{array}{ll}0.30& \\mathrm {for}\\ m_c=1.26 \\ \\mathrm {GeV},\\\\0.28& \\mathrm {for}\\ m_c=1.47 \\ \\mathrm {GeV},\\end{array}\\right.", "}$ while the ad-hoc phenomenological choice used in the literature is: $\\frac{\\tau _\\psi }{\\tau _D}=\\frac{M_D^2}{M_\\psi ^2}=0.364 \\ .$ The relations between the two-point parameters $\\tau _{\\psi ,D}$ and the corresponding three-point parameters $\\tau _{1,2}$ are [32]: $\\tau _1\\simeq \\frac{\\tau _\\psi }{2}, \\,\\,\\tau _2\\simeq \\frac{\\tau _D}{2} \\ .$ Figure: (color online).", "(a) The two-point SR of f D f_D versusτ D \\tau _D with Δ D =0.8 GeV \\Delta _D=0.8 \\ \\mathrm {GeV}.", "The red dashed line isfor m c =1.26 GeV m_c=1.26 \\ \\mathrm {GeV} and the green solid line for m c =1.47 GeV m_c=1.47\\ \\mathrm {GeV}.", "(b) f D f_D versus Δ D \\Delta _D with the minimum ofτ D \\tau _D adopted.Figure: (color online).", "(a) The two-point SR of f ψ f_\\psi versusτ ψ \\tau _\\psi with Δ ψ =0.8 GeV \\Delta _\\psi =0.8 \\ \\mathrm {GeV}.", "The red dashedline is for m c =1.26 GeV m_c=1.26 \\ \\mathrm {GeV} and the green solid line form c =1.47 GeV m_c=1.47 \\ \\mathrm {GeV}.", "(b) f ψ f_\\psi versus Δ ψ \\Delta _\\psi withthe minimum of τ ψ \\tau _\\psi adopted.Figure: (color online).", "Behavior of the ratio of SR variablesτ ψ /τ D \\tau _\\psi /\\tau _D versus Δ\\Delta for two values of m c =1.26 GeV m_c=1.26 \\ \\mathrm {GeV} (red dashed line) and 1.47 GeV 1.47 \\ \\mathrm {GeV} (solidgreen line).As follows, we adopt $m_c=1.26 \\ \\mathrm {GeV}$ and $\\Delta _D=\\Delta _\\psi \\equiv \\Delta $ as inputs since it is difficult to find a global maximum with $m_c=1.47 \\ \\mathrm {GeV}$ in our three-point SR and the variation of $\\tau $ ratio is more stable with $m_c=1.26 \\ \\mathrm {GeV}$   [33].", "To obtain more concrete information about the form factor, we consider a large virtuality interval $0\\le -p_5^2\\le 5 \\ \\mathrm {GeV^2}$ , which is the same as in Ref. [29].", "As an illustration, we show in Fig.", "REF the form factor $F_{\\psi DD^*}(p_5^2)$ at $p_5^2=-3 \\ \\mathrm {GeV^2}$ with both $\\Delta \\ge 0.4 \\ \\mathrm {GeV}$ and ${\\tau _1}/{\\tau _2}=0.3$ .", "For simplicity, $f_{D^*}=0.24 \\ \\mathrm {GeV}$ is the same as in Ref. [29].", "The ratios of the pole contribution versus the whole dispersion integral are also depicted in Fig.", "REF and parameterized as $\\mathrm {R}&\\equiv &\\mathrm {\\frac{PI}{WI}},\\\\\\mathrm {PI}&=&\\int _{4m_c^2}^{s_0} \\mathrm {d} s\\int _{m_c^2}^{u_0}\\mathrm {d} u\\,\\rho (s,u,Q^2)\\nonumber \\\\&&\\times \\theta (u_\\mathrm {max}-u)\\theta (u-u_\\mathrm {min})e^{-s \\tau _1-u \\tau _2},\\\\\\mathrm {WI}&=&\\int _{4m_c^2}^{\\infty } \\mathrm {d}s\\int _{u_\\mathrm {min}}^{u_\\mathrm {max}} \\mathrm {d} u\\,\\rho (s,u,Q^2)\\nonumber \\\\&&\\times \\theta (u_\\mathrm {max}-u)\\theta (u-u_\\mathrm {min})e^{-s\\tau _1-u \\tau _2}.", "$ Figure: (color online).", "(a) The τ 1 \\tau _1 windows of the form factorand (b) the contribution from the corresponding pole at p 5 2 =-3 GeV 2 p_5^2=-3 \\ \\mathrm {GeV^2}, where m c =1.26 GeV m_c=1.26 \\ \\mathrm {GeV},τ 1 /τ 2 =0.3\\tau _1/\\tau _2=0.3, and Δ\\Delta as a variable.With the increase of $\\Delta $ , the pole contributions will become larger.", "It is obvious to see that the pole contributions are less than one half at the maximum $\\tau _1=0.05 \\ \\mathrm {GeV}$ even with $\\Delta $ as large as $1 \\ \\mathrm {GeV}$ .", "The situation will be worse with larger $D^*$ virtuality.", "This phenomenon seems to be a common problem for the form factors of charmonium to open charmed mesons.", "To avoid this difficulty of the SR criterion, we deduce the form factors from Laplace SR varying with different $\\Delta $ , and also show the predictions from FESR in terms of $\\Delta $ .", "In principle, these two SRs should give the same solution, which means that the result at the intersection point is the reliable one from QSSR, see Fig.", "REF .", "The form factor from the above method is shown in Fig.", "REF , and we use three different parameterizations to extend the form factor to broader regions of the $D^*$ virtuality: $F_{\\psi DD^*}(p_5^2)={\\left\\lbrace \\begin{array}{ll}10.58 \\exp \\left[-\\genfrac{}{}{}0{(-p_5^2+21.30)^2}{422.62}\\right]&\\text{Gaussian}\\\\[10pt]\\genfrac{}{}{}0{332.61}{p_5^4-8.79 p_5^2+91.98} & \\text{Dipole}\\\\[10pt]\\genfrac{}{}{}0{-25.83}{p_5^2-7.03} & \\textrm {Monopole}\\end{array}\\right.", "}$ The form factor obtained in Ref.", "[29] with fixed $f_\\psi =0.405 \\ \\mathrm {GeV}$ , $f_D=0.17 \\ \\mathrm {GeV}$ , $m_c=1.3 \\ \\mathrm {GeV}$ , $\\Delta _\\psi =\\Delta _D=0.5 \\ \\mathrm {GeV}$ and $0.09<\\tau _1<0.14 \\ \\mathrm {GeV^{-2}}$ are parameterized by the Gaussian formula [29] $F_{\\psi DD^*}(p_5^2)=19.9 \\exp \\left[\\frac{-(-p_5^2+27)^2}{345}\\right]\\,\\, .$ Below the $D^*$ threshold, our improved form factors are slightly larger and decline slower than the one in Ref. [29].", "In fact, the form factors used in our following calculation are usually restricted to a small region $-5<p_5^2<2 \\ \\mathrm {GeV^2}$ with on-shell $D$ mesons.", "Therefore, those different parameterizations would not bring noticeable differences to the calculation results, although in a broader momentum region they turn out to be different from each other especially in the timelike region.", "Usually, pQCD predicts the power falloff of the form factors, we will use the dipole fit in the following calculations.", "Notice that there is no real roots, i.e.", "the unphysical state, in the denominator of our dipole fit (we label it as power fit hereafter to distinguish it from the empirical dipole form factor).", "Figure: (color online).", "(a) The intersection point of Laplace SRand FESR at p 5 2 =0 GeV 2 p_5^2=0\\mathrm {GeV}^2.", "(b) The J/ψDD * J/\\psi DD^* formfactor derived from our method compared with the Gaussian fit .", "The blue dash-dotted line is ourmonopole fit, the green solid line is our dipole fit, the red dottedone is our Gaussian fit, and the black dashed one is the Gaussianfit  .", "The dots are our SR result." ], [ "The Form factor of $F_{D^*D\\pi ^0}(p_5^2)$", "As mentioned earlier, the form factor $F_{D^*D\\pi ^0}(p_5^2)$ is determined by the $D$ meson semileptonic decays, i.e.", "$D\\rightarrow \\pi ^0l\\nu $ .", "In the momentum transfer region $p_5^2<2 \\ \\mathrm {GeV}^2$ , we expect that the $D^*$ pole has the dominant contribution.", "Thus, the transition matrix element can be expressed as $&&\\langle \\pi ^0(p)|\\bar{d}\\gamma _\\mu c|D^+(p_4)\\rangle \\nonumber \\\\&\\sim &\\sum _{\\epsilon _5} \\langle \\pi ^0(p)D^{*+}(p_5)|D^+(p_4)\\rangle \\langle 0|\\bar{d}\\gamma _\\mu c|D^{*+}(p_5)\\rangle \\frac{1}{p_5^2-M_{D^{*+}}^2}\\nonumber \\\\&=&\\frac{F_{D^*D\\pi ^0}(p_5^2)M_{D^{*+}}f_{D^{*+}}}{p_5^2-M_{D^{*+}}^2}\\left(-p_\\mu +\\frac{p_{5\\mu } p\\cdot p_5}{M_{D^{*+}}^2}\\right) \\ ,$ where we have used the following definitions consistent with the effective Lagrangian: $\\langle \\pi ^0(p)D^{*+}(p_5)|D^+(p_4)\\rangle &=&F_{D^*D\\pi ^0}(p_5^2)(p\\cdot \\epsilon _5^*) \\ ,\\nonumber \\\\\\langle 0|\\bar{d}\\gamma _\\mu c|D^{*+}(p_5)\\rangle &=&M_{D^{*+}}f_{D^{*+}}\\epsilon _{5\\mu } \\ .$ The transition matrix element of the weak decay can be defined as [34] $\\langle \\pi ^0(p)|\\bar{d}\\gamma _\\mu c|D^+(p+p_5)\\rangle &=&\\frac{1}{\\sqrt{2}}[(2p+p_5)_\\mu f_+(p_5^2)+p_{5\\mu }f_-(p_5^2)] \\ ,$ where the form factor $f_+(p_5^2)$ has been measured with high accuracy, and can be parameterized as a modified pole formula [35] $f_+(p_5^2)=\\frac{-f_+(0)M_{D^*}^2}{(p_5^2-M_{D^*}^2)\\left(1-\\alpha _0\\frac{p_5^2}{M_{D^{*}}^2}\\right)}\\ .$ Compared with the $p^\\mu $ part of the weak decay form factor definition, we obtain the needed form factor: $F_{D^*D\\pi ^0}(p_5^2)=\\frac{\\sqrt{2}f_+(0)M_{D^*}}{\\left(1-\\alpha _0\\frac{p_5^2}{M_{D^{*}}^2}\\right)f_{D^{*}}} \\ ,$ where $\\alpha _0=0.21$ for $D^0$ , $\\alpha _0=0.24$ for $D^+$  [35], and $f_+(0)=0.64$ from the lattice QCD simulations [36], [37] for our numerical calculation.", "Note that $f_+(0)$ from QSSR [38] are consistent with the lattice result very well.", "The local coupling $g_{D^{*+}D^+\\pi ^0}$ can thus be extracted from the form factor at $p_5^2=M_{D^*}^2$ with $f_{D^{*}}=0.24\\mathrm {GeV}$ , i.e.", "$g_{D^*D\\pi ^0}(p_5^2)\\equiv F_{D^*D\\pi ^0}(p_5^2=M_{D^*}^2)\\simeq 9.97,$ which is slightly different from the value extracted from the decay of $D^*\\rightarrow D+\\pi $ , i.e.", "$g_{D^*D\\pi ^0}=g_{D^*DP}/\\sqrt{2}=17.9/\\sqrt{2}=12.7$  [39].", "One can also extract this coupling from QSSR or QCD light-cone SR.", "However, both SRs suffer from their inherent uncertainties and the corresponding couplings from most SRs are nearly the same as what we derived from the weak decay form factor.", "One can refer to Ref.", "[40] for a review on this issue.", "Another reason for the discrepancy of the coupling values is that in the momentum transfer region $0<p_5^2<3 \\ \\mathrm {GeV}^2$ which corresponds to the experimental kinematics, the form factor may vary drastically near the pole position of $D^*$ .", "One could of course calculate the dispersive part with empirical form factors, but we must emphasize that in some diagrams containing more $D^*$ mesons, e.g.", "$D^*D(D^*)$ loop in the vector charmonium decay to a $VP$ final state, the empirical dipole form factor used by most of the references is not enough to suppress the ultraviolet divergence in the loops so that we need other more complicated form factors such as the Gaussian form factor.", "Thus, we leave the direct calculation with empirical form factor aside.", "We present the empirical dipole form factor (Eq.", "(REF )) with different cutoff in Fig.", "REF and compare it with our QCD-motivated form factors.", "It is obvious that our form factors favor $\\alpha >2$ , which is consistent with the value used in the study of $X(3872)$ decays [41].", "Figure: (color online).", "The QCD induced form factor F ψDD * (p 5 2 )F D * Dπ 0 (p 5 2 )F_{\\psi DD^*}(p_5^2)F_{D^*D\\pi ^0}(p_5^2) (charged DD) compared with theempirical dipole form factors.", "The red dotted line is our power fit,the black dashed line is the form factor obtained inRef.", ", the green solid line, theorange dash-dot-dotted line and the blue dash-dotted line correspondto the empirical dipole form factor with α=4\\alpha =4, α=2\\alpha =2 andα=1\\alpha =1, respectively." ], [ "Numerical results and discussion ", "The determination of the effective threshold ${th}$ of the dispersive part is not a trivial task.", "If we know the full information of the spectral density, ${th}$ should be extended to be infinity.", "Usually, ${th}\\equiv (M_D+M_{D^*})^2$ is taken in the literatures as a natural cutoff [41] on the assumption that the spectral density can be approximated by the extrapolation of the imaginary part.", "As an improvement, it is assumed that the corresponding effective threshold of $\\psi (3770)\\rightarrow J/\\psi +\\eta $ containing $u,d$ components should be the same as that of $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ .", "So, we can determine the effective threshold from the decay width of $\\psi (3770)\\rightarrow J/\\psi +\\eta $ .", "The flavor mixing scheme is taken in our calculation, and the mixing angle between $|\\bar{n}n\\rangle \\equiv (|\\bar{u}u\\rangle +|\\bar{d}d\\rangle )/\\sqrt{2}$ and $|\\bar{s}s\\rangle $ is $\\alpha _P\\equiv \\theta _P+\\arctan \\sqrt{2}\\simeq 38^\\circ $ , where $\\theta _P$ is the mixing angle between the flavor singlet and octet.", "Note that, as mentioned in the Introduction that the IML contributions are relatively enhanced by $1/v$ in comparison with the tree-level contribution, it leads to the dominance of the IML contributions in $\\psi (3770)\\rightarrow J/\\psi +\\eta $ which is consistent with the study of Ref. [12].", "It can be understood that the isospin violation from the $\\eta $ -$\\pi ^0$ mixing is different from the IML transitions.", "In the latter, there is no $\\eta $ pole contributions to the strong isospin violations.", "In the following calculation of the dispersive part, we will take the SU(3) flavor symmetry for the production of $s\\bar{s}$ and $q\\bar{q}$ component within the light pseudoscalar mesons.", "It means the dispersive part of $s\\bar{s}$ is approximated by the average of the dispersive integrals of $q\\bar{q}$ .", "The decay constants of $g_{\\psi ^{\\prime \\prime }DD}$ in our numerical simulation are listed in Table REF , where $g_{\\psi ^{\\prime \\prime }DD}^{low,cen,up}$ correspond to the lower bound, central value, and upper bound allowed by the experimental data [15].", "From the experimental branching ratio $Br(\\psi (3770)\\rightarrow J/\\psi +\\eta )=9\\pm 4\\times 10^{-4}$[15], we obtain the corresponding $\\delta {th}\\equiv {th}-(M_D+M_{D^*})^2$ in Tables REF and  REF .", "One can see that the frequently used natural cutoff $\\delta {th}=0$ is not supported by our calculation.", "Notice that we distinguish the charged and neutral $D(D^*)$ masses in our numerical calculation.", "To estimate the uncertainty from $J/\\psi D D^*$ form factor, we choose our power parametrization and the Gaussian form in Ref.", "[29] for comparison.", "The $s_1$ -dependence of $g_{\\psi ^{\\prime \\prime }DD}$ is also taken into account by adding the suppression factor $\\exp (-I|\\vec{p}_3|^2)$ into the dispersive integral, where $I=0.4 \\ \\mathrm {GeV}^{-2}$ is extracted from the charmonium mass shift [28].", "It should be stressed that the dispersive integral with the suppression factor is not considered priority than the original one with a lower effective threshold from the phenomenological viewpoint.", "Moreover, both imaginary parts numerically decrease faster than $1/s_1$ , so the unsubtracted dispersion relation used here is self-contained.", "In most of the parameter space, the dispersive part of the branching ratio is dominant over the absorptive part for $J/\\psi \\eta $ , while for $J/\\psi \\pi ^0$ both absorptive and dispersive parts are important.", "The difference between different $J/\\psi D D^*$ form factor is small, despite that the absorptive part of our power fit is systematically larger than the Gaussian fit in Ref.", "[29] as expected.", "As shown in Table REF , the branching ratios of $J/\\psi \\pi ^0$ obtained from the corresponding effective thresholds are in good agreement with the experimental upper limit $2.8\\times 10^{-4}$  [15].", "As an interesting investigation, we take the threshold asymptotic to infinity with the suppression factor, and the corresponding branching ratios of $J/\\psi \\pi ^0$ are still below the upper limit except for $g_{\\psi ^{\\prime \\prime }DD}^{low}$ .", "In contrast, the asymptotic limits of $J/\\psi \\eta $ are far beyond the upper limit of the experiment $13\\times 10^{-4}$ .", "Then, even taking the suppression factor into account, the spectral information from other resonances and continuum are still ambiguous, so that the asymptotic limit is questionable and the effective threshold is still necessary.", "It is essential to recognize that the isospin symmetry breaking with the vertex couplings is also an important dynamic source apart from effects caused by the mass differences between the charged and neutral $D^{*}$ mesons.", "In our formulation, the coupling $g_{\\psi ^{\\prime \\prime }DD}$ and form factor $F_{D^*D\\pi ^0}(p_5^2)$ are extracted from experimental data which suggest different values for the charged and neutral couplings, respectively.", "Since the form factor $F_{D^*D\\pi ^0}(p_5^2)$ can be better fixed by the experimental data, the results listed in Tables REF , REF and REF also reflect the effects from the isospin breakings of $g_{\\psi ^{\\prime \\prime }DD}$ .", "It is interesting to note that the larger absorptive contributions actually favor smaller difference between the charged and neutral $g_{\\psi ^{\\prime \\prime }DD}$ couplings, which is also observed in Refs.", "[22], [23] considering the theoretical Coulomb correction for $g_{\\psi ^{\\prime \\prime }DD}$ .", "Taking into account the dispersive part, the central values of $g_{\\psi ^{\\prime \\prime }DD}$ give relatively small branching ratios for $\\psi ^{\\prime \\prime }\\rightarrow J/\\psi +\\pi ^0$ , while deviations from the central values can produce larger branching ratios for $\\psi ^{\\prime \\prime }\\rightarrow J/\\psi +\\pi ^0$ .", "Within the present experimental uncertainty bounds [15], the predicted branching ratios for $\\psi ^{\\prime \\prime }\\rightarrow J/\\psi +\\pi ^0$ are at the order of $10^{-5}\\sim 10^{-4}$ .", "Confirmation of this decay branching ratio would be a strong evidence for the open charm threshold effects in $\\psi ^{\\prime \\prime }\\rightarrow J/\\psi +\\pi ^0$ .", "Note that our prediction of the absorptive part is also close to the prediction of Ref.", "[23] with isospin $I=0$ for $\\psi (3770)$ , i.e.", "$Br_\\eta (\\mathrm {Abs})=8\\times 10^{-5}$ and $Br_\\pi (\\mathrm {Abs})=2\\times 10^{-5}$ , which is a consequence of the similar values of the form factors in both approaches.", "Table: Different g ψ '' DD g_{\\psi ^{\\prime \\prime }DD} from experimentaldata .Table: The branching ratio without suppression factor.“Abs” denotes the absorptive part, and “Tot” is for the sum ofthe absorptive and dispersive part.", "The flavor mixing angle isα P =38 ∘ \\alpha _P=38^\\circ .Table: The branching ratio with suppression factor.", "“Tot” is forthe sum of the absorptive and dispersive part, and “Asym” means wetake the asymptotic limit δth→∞\\delta th\\rightarrow \\infty .", "Flavormixing angle is 38 ∘ 38^\\circ ." ], [ "Conclusion", "The ELA is very useful to investigate the nature of the near threshold charmonia and charmoniumlike resonances.", "The largest uncertainty of the ELA comes from the determination of the off-shell effect, i.e.", "the form factors.", "In this paper, we investigate the isospin violating decay of $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ .", "In this process, there is only one $D$ -meson loop contributing to the absorptive part, and the form factors encountered in the loop calculation can be relatively well controlled.", "With the help of QSSR, we extract the $J/\\psi DD^*$ form factor as an implement from the first principle of QCD.", "The $DD^*\\pi ^0$ form factor can be well determined from the experimental data of $D\\rightarrow \\pi l\\nu $ , which has been measured with high accuracy.", "We also explore the dispersion relation to evaluate the dispersive part of $\\psi (3770)$ non-$D\\bar{D}$ decays, and find they take an important part in most of the parameter space.", "It means that the IML effects below the open charmed meson threshold cannot be neglected in general.", "Different from the traditional natural cutoff of the effective threshold in the dispersive integral, we extract them from the experimental data of $\\psi (3770)\\rightarrow J/\\psi +\\eta $ .", "Our prediction of the branching ratio of $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ is less than $3\\times 10^{-5}$ with the couplings $g_{\\psi ^{\\prime \\prime }DD}$ extracted from the central values of the data.", "Within the experimental uncertainty bounds for the extracted $g_{\\psi ^{\\prime \\prime }D^0\\bar{D^0}}$ and $g_{\\psi ^{\\prime \\prime }D^+D^-}$ , the branching ratio of $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ can reach the order of $10^{-4}$ .", "Notice that the understanding of the isospin violation of $g_{\\psi ^{\\prime \\prime }D\\bar{D}}$ is not a trivial task despite the Coulomb correction favors the experimental central value [42].", "It is also suggested that a small admixture of isovector four-quark component of $\\psi (3770)$ may also give a measurable decay rate to $J/\\psi \\pi ^0$  [6].", "In Ref.", "[4] the four-quark component is viewed as a reannihilation effect of $D\\bar{D}$ .", "To some extent, the nature of $\\psi (3770)$ hides in the coupling $g_{\\psi ^{\\prime \\prime }DD}$ .", "Meanwhile, the forthcoming BESIII measurement of $\\psi (3770)\\rightarrow J/\\psi +\\pi ^0$ will be able to provide useful information about the QCD motivated form factors and clarify the role played by the IML.", "We plan to discuss the isospin violations with the charged and neutral couplings $g_{\\psi ^{\\prime \\prime }D^+D^-}$ and $g_{\\psi ^{\\prime \\prime }D^0\\bar{D^0}}$ elsewhere.", "This work is supported, in part, by the France–China Particle Physics Laboratory, National Natural Science Foundation of China (Grant No.", "11035006), Chinese Academy of Sciences (KJCX2-EW-N01), and Ministry of Science and Technology of China (2009CB825200)." ] ]

[ [ "On Rogers-Ramanujan functions, binary quadratic forms and eta-quotients" ], [ "Abstract In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions.", "We observe that the functions that appear in Ramanujan's identities can be obtained from a Hecke action on a certain family of eta products.", "We establish further Hecke-type relations for these functions involving binary quadratic forms.", "Our observations enable us to find new identities for the Rogers-Ramanujan functions and also to use such identities in turn to find identities involving binary quadratic forms." ], [ "Introduction", "The Rogers–Ramanujan functions are defined for $ |q| < 1 $ by $G(q) := \\sum _{n=0}^{\\infty }\\dfrac{q^{n^2}}{(q;q)_n} \\qquad \\text{and} \\qquad H(q) := \\sum _{n=0}^{\\infty }\\dfrac{q^{n(n+1)}}{(q;q)_n},$ where $(a;q)_0 :=1$ and, for $n \\ge 1$ , $(a;q)_n := \\prod _{k=0}^{n-1}(1-aq^k).$ These functions satisfy the famous Rogers–Ramanujan identities [11] $G(q) = \\dfrac{1}{(q;q^5)_{\\infty }(q^4;q^5)_{\\infty }} \\qquad \\text{and} \\qquad H(q) = \\dfrac{1}{(q^2;q^5)_{\\infty }(q^3;q^5)_{\\infty }},$ where $(a;q)_{\\infty } := \\lim _{n\\rightarrow \\infty }(a;q)_n, \\qquad |q| <1.$ In a handwritten manuscript published with his lost notebook [10], Ramanujan stated without proofs forty identities for the Rogers–Ramanujan functions.", "These identities were established in a series of papers by L. J. Rogers [12] in 1921, G. N. Watson [13] in 1933, D. Bressoud [6] in 1977, A. J. F. Biagioli [5] in 1989, and by the second author [14] in 2012.", "A detailed history of Ramanujan's forty identities can be found in [4].", "Ramanujan's identities mainly involve the function $U(r,s,q):=U(r,s)={\\left\\lbrace \\begin{array}{ll}G(q^r)G(q^s)+q^{(s+r)/5}H(q^r)H(q^s) & \\text{if } s+r\\equiv 0 \\pmod {5},\\\\H(q^r)G(q^s)-q^{(s-r)/5}G(q^r)H(q^s) & \\text{if } s-r\\equiv 0 \\pmod {5}.\\end{array}\\right.", "}$ The modular properties of the function $U(r,s)$ were first established by Biagioli [5].", "M.  Koike [9] observed that for certain values of $r$ and $s$ the function $U(r,s)$ could be written in terms of Thompson series.", "His observations were later proved by K. Bringmann and H. Swisher [7] by using the theory of modular forms.", "However, it was not realized that this function occurs naturally as a Hecke action on eta products, as given by the following theorem.", "Theorem 1.1 Let $r$ and $s$ be two positive integers with $r+s \\equiv 0 \\pmod {24}$ .", "If $r+s \\equiv 0 \\pmod {5}$ , then $T_5(\\eta (r\\tau )\\eta (s\\tau ))=\\eta (r\\tau )\\eta (s\\tau )\\left(q^{-(r+s)/60}U(r,s)\\right)^2,$ and if $r-s \\equiv 0 \\pmod {5}$ , then $T_5(\\eta (r\\tau )\\eta (s\\tau ))=-\\eta (r\\tau )\\eta (s\\tau )\\left(q^{(11r-s)/60}U(r,s)\\right)^2,$ where in either case $\\eta (\\tau ):=\\sum _{n=-\\infty }^\\infty (-1)^n q^{(6n-1)^2/24}$ with $ q = \\exp (2\\pi i\\tau ) $ and $ {\\textrm {Im} } \\, \\tau > 0$ .", "Here and later in this manuscript, for any function to which we apply the Hecke action with prime 5 we have the reduction formula $T_5\\left(\\sum _{n=0}^{\\infty }a(n)q^n\\right)=\\sum _{n=0}^{\\infty }\\left(a(5n)+a(n/5)\\right)q^n,$ where we employ the convention that $a(k) = 0$ whenever $k$ is not an integer.", "We will also make use of the notation $U_5\\left(\\sum _{n=0}^{\\infty }a(n)q^n\\right) := \\sum _{n=0}^{\\infty }a(5n)q^n.$ During our investigations, we also observed the following two theorems, where $(q) := \\sum _{n=-\\infty }^{\\infty }q^{n^2}$ .", "Theorem 1.2 If $r+s\\equiv 0 \\pmod {5}$ , then the following two identities hold: $2(q^r)(q^s)+T_5((q^r)(q^s)) &= 4E(q^{2r})E(q^{2s})U(r,s)U(4r,4s)\\\\2(q^r)(q^s)-T_5((q^r)(q^s)) &= 4q^rE(q^{2r})E(q^{2s})U(r,4s)U(4r,s).$ Theorem 1.3 If $r-s\\equiv 0 \\pmod {5}$ , then the following two identities hold: $2(q^r)(q^s)+T_5((q^r)(q^s)) &= 4E(q^{2r})E(q^{2s})U(r,4s)U(s,4r),\\\\2(q^r)(q^s)-T_5((q^r)(q^s)) &= 4q^rE(q^{2r})E(q^{2s})U(r,s)U(4r,4s).$ Let $(a,b,c)$ denote the positive definite quadratic form $an^2+bnm+cn^2$ with discriminant $D=b^2-4ac<0$ .", "For simplicity, we will not distinguish the quadratic form $(a,b,c)$ and its generating function $\\sum _{n,m=-\\infty }^{\\infty }q^{an^2+bnm+cm^2}$ .", "It is well known that a Hecke action on a binary quadratic form of a given discriminant can be written as a linear combination of the quadratic forms of this discriminant [8].", "In applications of Theorem REF , we first express the eta product as a linear combination of the relevant quadratic forms.", "In this format, Theorems REF –REF are easier to apply.", "These theorems enable us to find new identities for the Rogers–Ramanujan functions and also to use such identities in return to find identities involving binary quadratic forms.", "Among the many such results presented in this paper we give two examples (see (REF ) and (REF )): letting $\\chi (q):=(-q;q^2)_\\infty $ , they are $2qU(1,71,q^2)=-2q^3+\\chi (q)\\chi (q^{71})-\\chi (-q)\\chi (-q^{71})-2q^9\\dfrac{1}{\\chi (-q^2)\\chi (-q^{142})}$ and $\\dfrac{(1,1,10)+(2,1,5)-(1,0,39)-(5,2,8)}{(3,0,13)+(2,1,5)-(3,3,4)-(5,2,8)}=\\dfrac{(-q^6)(-q^{26})}{(-q^2)(-q^{78})}.$ We proceed by collecting the necessary definitions and formulas in the next section.", "In Section , we give proofs of Theorems REF –REF .", "In Section , we present several applications.", "We conclude in the last section with a brief description of the prospects for future work." ], [ "Definitions and Preliminary Results", "We first recall Ramanujan's definitions for a general theta function and some of its important special cases.", "Set $f(a,b) := \\sum _{n=-\\infty }^{\\infty }a^{n(n+1)/2}b^{n(n-1)/2}, \\qquad |ab| <1.$ The function $ f(a,b) $ satisfies the well-known Jacobi triple product identity [2] $f(a,b) = (-a;ab)_{\\infty }(-b;ab)_{\\infty }(ab;ab)_{\\infty }.$ The three most important special cases of (REF ) are $\\varphi (q) := f(q,q) = \\sum _{n=-\\infty }^{\\infty }q^{n^2} =(-q;q^2)_{\\infty }^2(q^2;q^2)_{\\infty },$ $\\psi (q) := f(q,q^3) = \\sum _{n=0}^{\\infty }q^{n(n+1)/2} =\\dfrac{(q^2;q^2)_{\\infty }}{(q;q^2)_{\\infty }},$ and $E(q) := f(-q,-q^2) = \\sum _{n=-\\infty }^\\infty (-1)^n q^{n(3n-1)/2}= (q;q)_\\infty = q^{-1/24}\\eta (\\tau ).$ The product representations in (REF )–(REF ) are special cases of (REF ).", "The function $ f(a,b) $ also satisfies a useful addition formula.", "For each integer $n$ , let $U_n := a^{n(n+1)/2}b^{n(n-1)/2} \\qquad \\text{and} \\qquad V_n :=a^{n(n-1)/2}b^{n(n+1)/2}.$ Then [2] $f(U_1,V_1) =\\sum _{r=0}^{n-1}U_rf\\left(\\dfrac{U_{n+r}}{U_r},\\dfrac{V_{n-r}}{U_r}\\right).$ With $a=b=q$ and $n=2$ , we find from (REF ) that $(q)=(q^4)+2q\\psi (q^8).$ Similarly, with $a=q$ , $b=q^3$ , and $n=2$ , we find that $\\psi (q)=f(q^6,q^{10})+qf(q^2,q^{14}).$ By (REF ) and (REF ), we see that $G(q)=\\dfrac{f(-q^2,-q^3)}{E(q)} \\quad \\text{and} \\quad H(q)=\\dfrac{f(-q,-q^4)}{E(q)}.$ A useful consequence of (REF ) in conjunction with the Jacobi triple product identity (REF ) is $G(q)H(q) = \\dfrac{E(q^5)}{E(q)}.$ The odd-even dissections of $G$ and $H$ were given by Watson [13]: $\\begin{split}G(q)&=\\dfrac{E(q^8)}{E(q^2)}\\bigr (G(q^{16})+qH(-q^4)\\bigl ),\\\\H(q)&=\\dfrac{E(q^8)}{E(q^2)}\\bigr (q^3H(q^{16})+G(-q^4)\\bigl ).\\end{split}$ Recall that the general theta function $f$ is defined by (REF ).", "For convenience, we also define $f_k(a,b) := {\\left\\lbrace \\begin{array}{ll}f(a,b) & \\text{if } k \\equiv 0 \\pmod {2},\\\\f(-a,-b) & \\text{if } k \\equiv 1 \\pmod {2}.\\end{array}\\right.", "}$ Let $m$ be an integer and let $\\alpha $ , $\\beta $ , $p$ , and $\\lambda $ be positive integers such that $\\alpha m^2+\\beta =p\\lambda .$ Let ${\\delta }$ and $\\epsilon $ be integers, and let $t$ and $l$ be reals.", "With the parameters defined this way, we set $\\begin{split}R(\\epsilon ,{\\delta },l,t,\\alpha ,\\beta ,m,p,\\lambda ) :=\\sum _{\\begin{array}{c}k=0\\\\n:=2k+t\\end{array}}^{p-1}\\Bigl (&(-1)^{\\epsilon k}q^{\\lbrace \\lambda n^2+p\\alpha l^2+2\\alpha nml\\rbrace /8}\\\\&\\times f_{\\delta }(q^{(1+l)p\\alpha /2+\\alpha nm/2},q^{(1-l)p\\alpha /2-\\alpha nm/2})\\\\&\\times f_{\\epsilon p/2+m{\\delta }/2}(q^{p\\beta /2+\\beta n/2},q^{p\\beta /2 - \\beta n/2})\\Bigr ).\\end{split}$ Then, by [15] with $x=y=1$ and $q$ replaced by $q^{1/2}$ , we have $R(\\epsilon ,{\\delta },l,t,\\alpha ,\\beta ,m,p,\\lambda )=\\sum _{u,v=-\\infty }^{\\infty }(-1)^{{\\delta }v+ \\epsilon u}q^{(\\lambda U^2 +2\\alpha m UV +p\\alpha V^2)/8},$ where $U:=2u+t$ and $V:=2v+l$ .", "From this representation it follows that [15] $R(\\epsilon ,{\\delta },l,t,\\alpha ,\\beta ,m,p,\\lambda )=R({\\delta },\\epsilon ,t,l,1,\\alpha \\beta , \\alpha m, \\lambda ,p\\alpha ).$ Moreover, we have the following lemma.", "Lemma 2.1 [15] Let $\\alpha _1$ , $\\beta _1$ , $m_1$ , $p_1$ be another set of parameters such that $\\alpha _1m_1^2+\\beta _1=p_1\\lambda $ , $\\alpha \\beta =\\alpha _1\\beta _1$ , and $\\lambda \\mid (\\alpha m-\\alpha _1 m_1)$ .", "Set $a:=\\dfrac{\\alpha m-\\alpha _1 m_1}{\\lambda }.$ Then, $R(\\epsilon ,{\\delta },l,t,\\alpha ,\\beta ,m,p,\\lambda )=R(\\epsilon ,{\\delta }+a\\epsilon ,l,t+al,\\alpha _1,\\beta _1,m_1,p_1,\\lambda ).$ From (REF ) we then conclude that $(a,b,c)=R(0,0,0,0,1,-D,b,2c,2a).$" ], [ "Hecke-Type Relations", "In this section we presents proofs of Theorems REF –REF ." ], [ "Proof of Theorem ", "The proofs of (REF ) and (REF ) are essentially same, so we will only prove (REF ).", "For simplicity we set $Q:=q^5$ .", "We start with the well-known 5-dissection of $E(q)$ as found in [2]: $E(q)E(Q)=f^2(-Q^2,-Q^3)-q^2f^2(-Q,-Q^4)-qE(Q)E(Q^5).$ Using (REF ), we can write (REF ) in its equivalent form $\\dfrac{E(q)}{E(Q)}=G^2(Q)-q^2H^2(Q)-q\\dfrac{E(Q^5)}{E(Q)}.$ From (REF ), we find that $\\dfrac{E(q^r)E(q^s)}{E(Q^{r})E(Q^{s})}=\\left(G^2(Q^r)-q^{2r}H^2(Q^r)-q^r\\dfrac{E(Q^{5r})}{E(Q^r)}\\right)\\left(G^2(Q^s)-q^{2s}H^2(Q^s)-q^s\\dfrac{E(Q^{5s})}{E(Q^s)}\\right),$ from which we deduce that $\\dfrac{U_5(E(q^r)E(q^s))}{E(q^r)E(q^s)}&=G^2(q^r)G^2(q^s)+q^{2(r+s)/5}H^2(q^r)H^2(q^s)+q^{(r+s)/5}\\dfrac{E(Q^{r})E(Q^s)}{E(q^r)E(q^s)}\\\\&=U(r,s)^2-2q^{(r+s)/5}G(q^r)G(q^s)H(q^r)H(q^s)+q^{(r+s)/5}\\dfrac{E(Q^{r})E(Q^s)}{E(q^r)E(q^s)}\\\\&=U(r,s)^2-q^{(r+s)/5}\\dfrac{E(Q^{r})E(Q^s)}{E(q^r)E(q^s)},$ where in the last step we used (REF ).", "Therefore, we have that $U_5(E(q^r)E(q^s))+q^{(r+s)/5}E(Q^{r})E(Q^s)=E(q^r)E(q^s)U(r,s)^2.$ Finally, we multiply both sides of (REF ) by $q^{(r+s)/120}$ and use the fact that $\\eta (\\tau )=q^{1/24}E(q)$ to arrive at $U_5(\\eta (r\\tau )\\eta (s\\tau ))+\\eta (5r\\tau )\\eta (5s\\tau )=\\eta (r\\tau )\\eta (s\\tau )\\left(q^{-(r+s)/60}U(r,s)\\right)^2,$ which is clearly equivalent to (REF ).", "$\\Box $ The proofs of the Theorems REF and REF are identical, so we will only prove Theorem REF .", "For convenience, we again set $Q:=q^5$ .", "By (REF ), with $a=b=q$ and $n=5$ , and by (REF ), we get $(q)=(Q^5)+2qf(Q^3,Q^7)+2q^4f(Q,Q^9).$ From (REF ), with simple product manipulations we find that $A(q):=f(q^3,q^7)=E(q^2)H(q)G(q^4) \\quad \\text{and} \\quad B(q):=f(q,q^9)=E(q^2)G(q)H(q^4).$ We have from (REF ) and (REF ) that $(q^r)(q^s)=\\left((Q^{5r})+2qA(Q^r)+2q^4B(Q^r)\\right)\\left((Q^{5s})+2qA(Q^s)+2q^4B(Q^s)\\right).$ From (REF ), together with the fact that $r+s\\equiv 0 \\pmod {5}$ , we conclude that $U_5\\left((q^r)(q^s)\\right)=(Q^r)(Q^s)+4q^{(r+s)/5}A(q^r)A(q^s)+4q^{4(r+s)/5}B(q^r)B(q^s).$ The following two identities of Ramanujan [4] will be employed in our proofs: $G(q)G(q^4)+qH(q)H(q^4) = \\dfrac{\\varphi (q)}{E(q^2)}.$ and $G(q)G(q^4)-qH(q)H(q^4) = \\dfrac{\\varphi (q^5)}{E(q^2)}.$ From (REF ), we have $\\begin{split}4q^{(r+s)/5}A(q^r)A(q^s)+4&q^{4(r+s)/5}B(q^r)B(q^s)\\\\&= 4E(q^{2r})E(q^{2s})\\left(q^{(r+s)/5}H(q^r)G(q^{4r})H(q^s)G(q^{4s})\\right.\\\\&\\quad \\left.+q^{4(r+s)/5}G(q^r)H(q^{4r})G(q^s)H(q^{4s})\\right)\\\\&= 4E(q^{2r})E(q^{2s})\\left(G(q^r)G(q^s)+q^{(r+s)/5}H(q^r)H(q^s)\\right)\\\\&\\quad \\times \\left(G(q^{4r})G(q^{4s})+q^{4(r+s)/5}H(q^{4r})H(q^{4s})\\right) - 4E(q^{2r})E(q^{2s})\\\\&\\quad \\times \\left(G(q^r)G(q^s)G(q^{4r})G(q^{4s})+q^{r+s}H(q^r)H(q^s)H(q^{4r})H(q^{4s})\\right)\\\\&= 4E(q^{2r})E(q^{2s})U(r,s)U(4r,4s) - \\left((q^r)+(Q^r)\\right)\\left((q^s)+(Q^s)\\right)\\\\&\\quad -\\left((q^r)-(Q^r)\\right)\\left((q^s)-(Q^s)\\right)\\\\&= 4E(q^{2r})E(q^{2s})U(r,s)U(4r,4s)-2\\left((q^r)(q^s)+(Q^r)(Q^s)\\right),\\end{split}$ where in the next to last step we use (REF ) and (REF ).", "We now return to (REF ) and use (REF ) to find that $2(q^r)(q^s)+U_5\\left((q^r)(q^s)\\right)+(Q^r)(Q^s)=4E(q^{2r})E(q^{2s})U(r,s)U(4r,4s),$ which is clearly equivalent to (REF ).", "While we can prove () exactly the same way we proved (REF ) by simply grouping terms differently in (REF ), we can also give a direct proof by showing that $(q^r)(q^s)=E(q^{2r})E(q^{2s})\\left(U(r,s)U(4r,4s)+q^rU(r,4s)U(4r,s)\\right).$ To prove (REF ), we consider the system of equations $U(r,s) &= G(q^r)G(q^{s})+q^{(r+s)/5}H(q^r)H(q^{s}),\\\\U(r,4s) &= -q^{(4s-r)/5}G(q^r)H(q^{4s})+H(q^r)G(q^{4s}),\\\\\\dfrac{(q^r)}{E(q^{2r})} &= G(q^{r})G(q^{4r})+q^rH(q^r)H(q^{4r}),$ where the last equation is simply (REF ).", "It follows that $\\left|\\begin{matrix}U(r,s) & G(q^{s}) & q^{(r+s)/5} H(q^s) \\\\U(r,4s) & -q^{(4s-r)/5}H(q^{4s}) & G(q^{4s}) \\\\\\dfrac{(q^r)}{E(q^{2r})} & G(q^{4r}) & q^rH(q^{4r})\\end{matrix}\\right|=0.$ By expanding this determinant we discover that $0 &= U(r,s)\\left(-q^{(r+s)/5}H(q^{4s})H(q^{4r})-G(q^{4r})G(q^{4s})\\right)\\\\&\\quad -U(r,4s)\\left(q^rG(q^s)H(q^{4r})-q^{(r+s)/5}H(q^s)H(q^{4r})\\right)\\\\&\\quad +\\dfrac{(q^r)}{E(q^{2r})}\\left(G(q^s)G(q^{4s})+q^sH(q^s)H(q^{4s})\\right)\\\\&= -U(r,s)U(4r,4s)-q^rU(r,4s)U(4r,s)+\\dfrac{(q^r)(q^s)}{E(q^{2r})E(q^{2s})},$ which is ().", "$\\Box $" ], [ "Applications", "The first set of identities we will prove involves the quadratic forms $(1,1,10)$ , $(2,1,5)$ , and $(3,3,4)$ of discriminant $-39$ and the quadratic forms $(1,0,39)$ , $(3,0,13)$ , and $(5,2,8)$ of discriminant $-156$ .", "From (REF ), we observe that $R(0,0,0,0,1,39,1,4,10)=R(0,0,0,0,3,13,-3,4,10)$ .", "By (REF ), we also have $(2,1,5)=(5,1,2)=R(0,0,0,0,1,39,1,4,10)$ .", "For simplicity we now set $Q:=q^{13}$ .", "From (REF ), we find that $R(0,0,0,0,1,39,1,4,10)&=f(q^2,q^2)f(Q^6,Q^6)+2q^5f(q,q^3)f(Q^3,Q^9)+q^{20}f(1,q^4)f(1,Q^{12})\\\\&=(q^2)(Q^6)+2q^5\\psi (q)\\psi (Q^3)+4q^{20}\\psi (q^4)\\psi (Q^{12}).$ Similarly, $R(0,0,0,0,3,13,-3,4,10)&=f(q^6,q^6)f(Q^2,Q^2)+2q^2f(q^3,q^9)f(Q,Q^3)+q^8f(1,q^{12})f(1,Q^4)\\\\&=(q^6)(Q^2)+2q^2\\psi (q^3)\\psi (Q)+4q^{8}\\psi (q^{12})\\psi (Q^{4}).$ Hence, we conclude that $\\begin{split}(2,1,5) &= (q^2)(Q^6)+2q^5\\psi (q)\\psi (Q^3)+4q^{20}\\psi (q^4)\\psi (Q^{12})\\\\&= (q^6)(Q^2)+2q^2\\psi (q^3)\\psi (Q)+4q^{8}\\psi (q^{12})\\psi (Q^{4}).\\end{split}$ By similar considerations one can also obtain $(1,1,10)=(q)(Q^3)+4q^{10}\\psi (q^2)\\psi (Q^6),$ $(3,3,4)=(q^3)(Q)+4q^4\\psi (q^6)\\psi (Q^2),$ and $\\begin{split}(5,2,8) &= (q^{24})(Q^8)+2{q}^{8}\\psi (q^{12})\\psi (Q^4) +4q^{32}\\psi (q^{48})\\psi (Q^{16})\\\\&\\quad +2{q}^{5}f({q}^{6},{q}^{42}) f({Q}^{6},{Q}^{10}) +2{q}^{15}f({q}^{18},{q}^{30}) f({Q}^{2},{Q}^{14}) \\\\&= (q^{8})(Q^{24})+2{q}^{20}\\psi (q^{4})\\psi (Q^{12}) +4q^{80}\\psi (q^{16})\\psi (Q^{48})\\\\&\\quad +2{q}^{5}f({Q}^{18},{Q}^{30}) f({q}^{6},{q}^{10}) +2{q}^{45}f({Q}^{6},{Q}^{42}) f({q}^{2},{q}^{14}).\\end{split}$ Theorem 4.1 The following four facts are true (with $Q:=q^{13}$ ): $2q^2E(q^2)E(Q^6)U(1,39)U(1,39,-q) = (1,1,10)+(2,1,5)-(1,0,39)-(5,2,8),$ $2q^2E(q^6)E(Q^2)U(3,13)U(3,13,-q) = (3,0,13)+(2,1,5)-(3,3,4)-(5,2,8),$ $\\dfrac{(3,0,13)-(5,2,8)}{(1,0,39)+(5,2,8)} = q^3\\dfrac{\\psi (-q)\\psi (-Q^3)}{\\psi (-q^3)\\psi (-Q)},$ $\\dfrac{(1,1,10)+(2,1,5)-(1,0,39)-(5,2,8)}{(3,0,13)+(2,1,5)-(3,3,4)-(5,2,8)} = \\dfrac{(-q^6)(-Q^2)}{(-q^2)(-Q^6)}.$ We start by proving (REF ).", "Using Theorem REF , we see that $4E(q^2)E(Q^6)U(1,39)U(1,39,q^4)=2(1,0,39)+T_5(1,0,39)=2(1,0,39)+2(5,2,8).$ From the odd-even dissections of the functions $G(q)$ and $H(q)$ , i.e.", "(REF ), we have $\\dfrac{E(q^2)E(Q^6)}{E(q^8)E(Q^{24})}U(1,39)=U(1,39,q^{16})+q^{8}U(1,39,-q^4)+qU(1,156,-q^4)+q^{39}U(39,4,-q^4).$ From (REF ) we get $(q^r)(q^s)=(q^{4r})(q^{4s})+4q^{r+s}\\psi (q^{8r})\\psi (q^{8s})+2q^s(q^{4r})\\psi (q^{8s})+2q^r(q^{4s})\\psi (q^{8r}).$ Examining (REF ) and (REF ), we observe that the even part of $(1,0,39)$ is $(1,1,10,q^4)$ .", "From (REF ) and (REF ), we immediately see that the even part of $(5,2,8)$ is $(1,1,10,q^4)$ .", "Therefore, by equating the even parts in both sides of (REF ), we conclude that $2E(q^8)E(Q^{24})\\left(U(1,39,q^{16})+q^{8}U(1,39,-q^4)\\right)U(1,39,q^4)=(1,1,10,q^4)+(2,1,5,q^4).$ Next, we replace $q^4$ by $q$ and use (REF ) to arrive at (REF ).", "To prove (REF ), we use Theorem REF and start with the identity $4q^3E(q^6)E(Q^2)U(3,13)U(3,13,q^4)=2(3,0,13)-T_5(3,0,13)=2(3,0,13)-2(5,2,8).$ By looking at the even parts in both sides of this equation and arguing as before, we arrive at (REF ).", "Ramanujan observed that [4] $E(q)E(Q^3)U(1,39)=E(q^3)E(Q)U(3,13).$ By (REF ), (REF ), and (REF ), and using some elementary product manipulations, we deduce (REF ).", "Similarly, (REF ) follows from (REF ), (REF ), and (REF ).", "Remark It can be easily verified by appealing to theory of modular forms that $2E(q^3)E(Q)U(1,39)=(1,1,10)+(2,1,5)$ and $2q^2E(q)E(Q^3)U(3,13)=(2,1,5)-(3,3,4).$ It is also easy to prove (see, for example, the proof of (REF )) that $2qE(q)E(Q^3)U(1,39)=2qE(q^3)E(Q)U(3,13)= (1,1,10)-(3,3,4).$ From these last three equations we easily observe that $\\sqrt{\\dfrac{ (2,1,5)-(3,3,4)}{(1,1,10)+(2,1,5)}}=\\dfrac{(2,1,5)-(3,3,4)}{(1,1,10)-(3,3,4)}=\\dfrac{(1,1,10)-(3,3,4)}{(1,1,10)+(2,1,5)}=q\\dfrac{E(q)E(Q^3)}{E(q^3)E(Q)}.$ Next, we treat identities involving the quadratic forms $(2,1,44)$ , $(8,1,11)$ , $(4,1,22)$ , $(10,7,10)$ , $(5,3,18)$ , $(1,1,88)$ , and $(9,3,10)$ of discriminant $-351$ .", "Theorem 4.2 The following five facts are true (with $Q := q^{13}$ ): $(2,1,44)-(8,1,11)=2q^2E(q^9)E(Q^3),$ $(5,3,18)-(8,1,11)=2q^5E(q^3)E(Q^9),$ $(4,1,22)-(10,7,10)=2q^4E(q^9)E(Q^3)U(3,13,q^3)^2,$ $(1,1,88)-(9,3,10)=2qE(q^3)E(Q^9)U(1,39,q^3)^2,$ $\\dfrac{(4,1,22)-(10,7,10)}{(1,1,88)-(9,3,10)}=\\dfrac{(5,3,18)-(8,1,11)}{(2,1,44)-(8,1,11)}=q^3\\dfrac{E(q^3)E(Q^9)}{E(q^9)E(Q^3)}.$ Using (REF ) and (REF ), we get $\\begin{split}(2,1,44) &= (44,1,2) = R(0,0,0,0,1,351,1,4,88)\\\\&= (q^2)(Q^{54})+2q^{44}\\psi (q)\\psi (Q^{27})+4q^{176}\\psi (q^4)\\psi (Q^{108})\\end{split}$ as well as $\\begin{split}(8,1,11) &= (11,1,8) = R(0,0,0,0,1,351,1,16,22)\\\\&= (q^8)(Q^{216})+2q^{11}f(q^7,q^{9})f(Q^{189},Q^{243})+2q^{44}f(q^6,q^{10})f(Q^{162},Q^{270})\\\\&\\quad +2q^{99}f(q^5,q^{11})f(Q^{135},Q^{297})+2q^{176}\\psi (q^4)\\psi (Q^{108})+2q^{275}f(q^3,q^{13})f(Q^{81},Q^{351})\\\\&\\quad +2q^{396}f(q^2,q^{14})f(Q^{54},Q^{378})+2q^{539}f(q,q^{15})f(q^{27},q^{405})+4q^{704}\\psi (q^{16})\\psi (Q^{432}).\\end{split}$ We take (REF ) and we expand $(q^2)(Q^{54})$ by using (REF ) and we similarly expand $\\psi (q)\\psi (Q^{27})$ by using (REF ).", "After this expansion we subtract (REF ) from the expanded (REF ) and arrive at $\\begin{split}(2,1,44)-(8,1,11) &= 2{q}^{2}\\psi ({q}^{16})( {Q}^{216}) -2{q}^{11}f({q}^{7},{q}^{9}) f({Q}^{189},{Q}^{243})\\\\&\\quad +2{q}^{45}f({q}^{2},{q}^{14}) f({Q}^{162},{Q}^{270}) -2{q}^{99}f({q}^{5},{q}^{11}) f({Q}^{135},{Q}^{297})\\\\&\\quad +2{q}^{176}f({q}^{4},{q}^{12}) f({Q}^{108},{Q}^{324})-2{q}^{275}f({q}^{3},{q}^{13}) f({Q}^{81},{Q}^{351})\\\\&\\quad +2{q}^{395}f({q}^{6},{q}^{10}) f({Q}^{54},{Q}^{378}) -2{q}^{539}f(q,{q}^{15}) f({Q}^{27},{Q}^{405})\\\\&\\quad +2{q}^{702}({q}^{8})\\psi ({Q}^{432}).\\end{split}$ By (REF ), we have $R(0,1,0,1,9,39,1,3,16)=2q^2E(q^9)E(Q^3).$ Next, we employ (REF ) and find that $R(0,1,0,1,9,39,1,3,16)=R(1,0,1,0,1,351,9,16,27)$ .", "By employing (REF ) one more time we observe that $R(1,0,1,0,1,351,9,16,27)$ equals exactly the right side of (REF ), which completes the proof of (REF ).", "We now observe that $2T_5(2,1,44)=(9,3,10)+(10,7,10) \\quad \\text{and} \\quad 2T_5(8,1,11)=(4,1,22)+(9,3,10).$ Therefore, by (REF ), we find that $T_5(2q^2E(q^9)E(Q^3))=(10,7,10)-(4,1,22),$ from which by way of (REF ) we immediately arrive at (REF ).", "The proofs of (REF ) and (REF ) go along the same lines as the proofs of (REF ) and (REF ), respectively, so we omit them.", "In fact one can go from one identity to the other via the map $\\tau \\mapsto -351/\\tau $ .", "Finally, (REF ) follows from (REF ).", "Remark By a straightforward but quite lengthy argument one can eliminate $q^3$ from either (REF ) or (REF ), resulting in $\\begin{split}\\left(E(q^3)E(Q)U(3,13)\\right)^2 &= \\left(E(q)E(Q^3)U(1,39)\\right)^2\\\\&=(-q^2)(-Q^6)\\psi (-Q)\\psi (-q^3)-q^3(-Q^2)(-q^6)\\psi (-q)\\psi (-Q^3).\\end{split}$ From (REF ) we may then deduce the following identity which is similar to those found by Ramanujan: $U(1,39)U(3,13)=\\dfrac{\\chi (q)\\chi (Q^3)}{\\chi (-q^6)\\chi (-Q^2)}-q^3\\dfrac{\\chi (Q)\\chi (q^3)}{\\chi (-Q^6)\\chi (-q^2)}.$ The next theorem concerns relations for the quadratic forms $(1,1,18)$ , $(2,1,9)$ , $(4,3,5)$ , and $(3,1,6)$ of discriminant $-71$ .", "Here we now set $Q:=q^{71}$ .", "Theorem 4.3 The following five facts are true (with $Q := q^{71}$ ): $2q^3E(q)E(Q)=(3,1,6)-(4,3,5),$ $2qE(q)E(Q)U(1,71)^2=(3,1,6)-(4,3,5)-(2,1,9)+(1,1,18),$ $2q^2E(q)E(Q)U(1,71)=(2,1,9)-(3,1,6),$ $2qU(1,71,q^2)=-2q^3+\\chi (q)\\chi (Q)-\\chi (-q)\\chi (-Q)-2q^9\\dfrac{1}{\\chi (-q^2)\\chi (-Q^2)},$ $\\left((3,1,6)-(4,3,5)-(2,1,9)+(1,1,18)\\right)\\left((3,1,6)-(4,3,5)\\right)=\\left((2,1,9)-(3,1,6)\\right)^2.$ The proof of (REF ) is similar to that of (REF ) so we omit the details.", "The identity (REF ) follows from (REF ) once we observe that $T_5(3,1,6)=(4,3,5)+(2,1,9) \\quad \\text{and} \\quad T_5(4,3,5)=(3,1,6)+(1,1,18).$ Using (REF ) and (REF ), we get $\\begin{split}R(0,1,0,1,1,71,3,5,16) &= 2{q}^{2}f(-q,-{q}^{4}) f(-{Q}^{2},-{Q}^{3}) -2{q}^{16}f(-{q}^{2},-{q}^{3}) f(-Q,-{Q}^{4})\\\\&= 2q^2E(q)E(Q)U(1,71).\\end{split}$ From (REF ) and (REF ), we also find that $\\begin{split}R(0,1,0,1,1&,71,3,5,16)\\\\&= R(1,0,1,0,1,71,3,16,5)\\\\&= 2{q}^{2}\\psi ({q}^{16})(Q^8) -2{q}^{3}f({q}^{3},{q}^{13}) f({Q}^{7},{Q}^{9}) +2{q}^{9}f({q}^{6},{q}^{10}) f({Q}^{6},{Q}^{10})\\\\&\\quad -2{q}^{20}f({q}^{7},{q}^{9}) f({Q}^{5},{Q}^{11}) +2{q}^{36}\\psi ({q}^{4}) \\psi ({Q}^{4}) -2{q}^{57}f(q,{q}^{15}) f({Q}^{3},{Q}^{13})\\\\&\\quad +2{q}^{81}f({q}^{2},{q}^{14}) f({Q}^{2},{Q}^{14}) -2{q}^{109} f({q}^{5},{q}^{11}) f(Q,{Q}^{15}) +2{q}^{142}({q}^{8})\\psi ({Q}^{16}).\\end{split}$ By (REF ) and (REF ), we have $\\begin{split}(2,1,9)=(9,2,1)&=R(0,0,0,0,1,71,1,4,18)\\\\&=(q^2)(Q^2)+2q^9\\psi (q)\\psi (Q)+4q^{36}\\psi (q^4)\\psi (Q^4).\\end{split}$ From (REF ), (REF ), and (REF ), we deduce that $\\begin{split}(3,1,6)&= R(0,0,0,0,1,71,1,12,6)\\\\&= R(0,0,0,0,1,71,-5,16,6)\\\\&= (q^8)(Q^8)+2q^3f(q^3,q^{13})f(Q^7,Q^9)+2q^{10}f(q^2,q^{14})f(Q^6,Q^{10})\\\\&\\quad +2q^{20}f(q^7,q^9)f(Q^5,Q^{11})+2q^{36}\\psi (q^4)\\psi (Q^4)+2q^{57}f(q,q^{15})f(Q^3,Q^{13})\\\\&\\quad +2q^{80}f(q^6,q^{10})f(Q^2,Q^{14})+2q^{109}f(q^5,q^{11})f(Q,Q^{15})+4q^{144}\\psi (q^{16})\\psi (Q^{16}).\\end{split}$ We then take (REF ), expand $(q^2)(Q^{54})$ as per (REF ) and expand $\\psi (q)\\psi (Q^{27})$ as per (REF ), and then subtract (REF ) to obtain the right-hand side of (REF ).", "Then by (REF ) and (REF ), the proof of (REF ) is complete.", "Next observe that by (REF ) and (REF ) we have $(4,3,5)&= (5,4,3)\\\\&= R(0,0,0,0,1,71,3,8,10)\\\\&= (q^4)(Q^4)+2q^5f(q,q^7)f(Q^3,Q^5)+2q^{18}\\psi (q^2)\\psi (Q^2)\\\\&\\quad +2q^{40}f(q^3,q^5)f(Q,Q^7)+4q^{72}\\psi (q^8)\\psi (Q^8)\\\\&= \\left((q)(Q)+(-q)(-Q)\\right)/2+2q^{18}\\psi (q^2)\\psi (Q^2)\\\\&\\quad +2q^5f(q,q^7)f(Q^3,Q^5)+2q^{40}f(q^3,q^5)f(Q,Q^7),$ where in the last step we used (REF ).", "We then replace $q$ by $q^2$ and use (REF ) to conclude that $\\begin{split}2(4,3,5,q^2) &= (q^2)(Q^2)+(-q^2)(-Q^2)+4q^{36}\\psi (q^4)\\psi (Q^4)\\\\&\\quad +2q^9\\psi (q)\\psi (Q)-2q^9\\psi (-q)\\psi (-Q).\\end{split}$ In (REF ), we use (REF ), then we replace $q$ by $q^2$ and subtract (REF ) from the resulting identity to obtain $\\begin{split}2(2,1,9,q^2)-2(4,3,5,q^2) &= (q)(Q)+(-q)(-Q)+4q^{18}\\psi (q^2)\\psi (Q^2)\\\\&\\quad -(q^2)(Q^2)-(-q)(-Q)-4q^{36}\\psi (q^4)\\psi (Q^4)\\\\&\\quad -2q^9\\psi (q)\\psi (Q)+2q^9\\psi (-q)\\psi (-Q).\\end{split}$ From the identities established in [2] it now follows that $(2,1,9,q^2)-(4,3,5,q^2)=q^3E(-q)E(-Q)-q^3E(q)E(Q)-2q^{12}E(q^4)E(Q^4).$ By (REF ) and (REF ), with $q$ replaced by $q^2$ in each, and by (REF ), we conclude that $2q^4E(q^2)E(Q^2)U(1,71,q^2)=q^3E(-q)E(-Q)-q^3E(q)E(Q)-2q^{12}E(q^4)E(Q^4)-2q^6E(q^2)E(Q^2),$ which is clearly equivalent to (REF ).", "Lastly, the identity (REF ) follows trivially from (REF )–(REF ).", "Remark Ramanujan almost always expressed the function $U(r,s)$ in terms of the function $\\chi (q)$ at related arguments, but he did not have an identity for the modulus 71.", "The following identity is for the quadratic forms of discriminant $-56$ .", "This identity was stated in [1] without a proof.", "Note that we now set $Q:=q^7$ .", "Theorem 4.4 Let $Q:=q^7$ .", "Then, $\\dfrac{(1,0,14)-(3,2,5)}{(2,0,7)+(3,2,5)}=q\\dfrac{E^2(Q^4)E^2(q^2)}{E^2(Q^2)E^2(q^4)}.$ From (REF ) we have $4qE(q^2)E(Q^4)U(1,56)U(4,14)=2(1,0,14)-T_5(1,0,14)=2(1,0,14)-2(3,2,5).$ Employing (REF ), we also find that $4E(q^4)E(Q^2)U(2,28)U(7,8)=2(2,0,7)+T_5(2,0,7)=2(2,0,7)+2(3,2,5).$ Ramanujan observed that [12] $\\dfrac{U(1,14)}{U(2,7)}=\\dfrac{E^2(q^2)E^2(Q)}{E^2(q)E^2(Q^2)}.$ From [3] we get $\\dfrac{U(1,56)}{U(7,8)}=\\dfrac{E(q^4)E(Q^2)}{E(q^2)E(Q^4)}.$ The identity (REF ) now follows from (REF )–(REF ), where (REF ) is used with $q$ replaced by $q^2$ .", "The next identity concerns quadratic forms of discriminant $-224$ .", "Theorem 4.5 Let $Q:=q^7$ .", "Then, $\\dfrac{(1,0,56)-(5,4,12)}{(7,0,8)+(3,2,19)}=q\\dfrac{E(q^4)E(Q^8)}{E(q^8)E(Q^4)}.$ The proof of (REF ) is very similar to that of (REF ), so we omit the details.", "Identities similar to (REF ) and (REF ) are established for $U(1,56)U(1,56,q^4)$ and $U(7,8)U(7,8,q^4)$ by using (REF ) and (REF ), and then the identity (REF ) is used twice with $q$ replaced by $q^4$ in its second application.", "By using Ramanujan's identities [6] $\\dfrac{U(1,54)}{U(2,27)}=\\dfrac{E(q^{27})E(q^{18})E(q^3)E(q^2)}{E(q^{54})E(q^9)E(q^6)E(q)}, \\quad \\dfrac{U(1,34)}{U(2,17)}=\\dfrac{\\chi (-q^{17})}{\\chi (-q)}, \\quad \\text{and} \\quad U(2,13)=U(1,26),$ and his other identities [5] $\\dfrac{U(1,66)}{U(2,33)}=\\dfrac{E(q^{11})E(q^6)}{E(q^{22})E(q^3)} \\quad \\text{and} \\quad \\dfrac{U(3,22)}{U(6,11)}=\\dfrac{E(q^2)E(q^{33})}{E(q)E(q^{66})},$ and by following exactly the same arguments as in the previous proof, we can easily establish the following identities, in corresponding order to the identities in (REF ) and (REF ), involving quadratic forms of discriminants $-216$ , $-136$ , $-104$ , and $-264$ .", "Theorem 4.6 The following five facts are true : $\\dfrac{(2,0,27)-(7,6,9)}{(1,0,54)+(5,2,11)}=q^2\\dfrac{\\psi (-q)\\psi (-q^6)\\psi (-q^9)\\psi (-q^{54})}{E(q^3)E(q^{72})\\psi (-q^2)\\psi (-q^{27})},$ $\\dfrac{(2,0,17)-(5,2,7)}{(1,0,34)+(5,2,7)}=q^2\\dfrac{(-q^4)\\psi (q^{17})}{(-q^{68})\\psi (q)},$ $\\dfrac{(1,0,26)-(5,4,6)}{(2,0,13)+(3,2,9)}=q\\dfrac{E(q^2)E(q^{52})}{E(q^4)E(q^{26})},$ $\\dfrac{(6,0,11)-(7,4,10)}{(3,0,22)+(5,4,14)}=q^6\\dfrac{E(q^{12})E(q^{22})\\psi (-q)\\psi (-q^{66})}{E(q^6)E(q^8)E(q^{33})E(q^{44})},$ $\\dfrac{(1,0,66)-(5,4,14)}{(2,0,33)+(7,4,10)}=q\\dfrac{E(q^{132})E(q^{44})E(q^{24})E(q^{11})E(q^6)E(q^2)}{E(q^{88})E(q^{66})E(q^{22})E(q^{12})E(q^{4})E(q^3)}.$ The following relations are for quadratic forms of discriminant $-1664$ .", "Here we set $Q:=q^{13}$ .", "Theorem 4.7 The following facts are true (with $Q:=q^{13}$ ): $2q^3E(q^{16})E(Q^8)U(2,13,q^8)=(3,2,139)-(12,4,35),$ $2q^{7}E(q^8)E(Q^{16})U(1,26,q^8)=(7,4,60)-(15,4,28),\\\\$ $2q^{9}E(q^8)E(Q^{16})=(9,8,48)-(17,6,25),\\\\$ $2q^5E(q^{16})E(Q^8)=(5,4,84)-(20,4,21),\\\\$ $2q^5E(q^8)E(Q^{16})U(1,26,q^8)^2 &= (5,4,84)+(21,10,21)-(13,0,32)-(20,4,21)\\\\&=2q^5E(q^{16})E(Q^{8})-2q^{13}\\psi (Q^8)(-q^8),$ $2qE(q^{16})E(Q^8)U(2,13,q^8)^2 &= (1,0,416)+(17,6,25)-(4,4,105)-(9,8,48)\\\\&=2q\\psi (q^8)(-Q^8)-2q^9E(q^8)E(Q^{16}),$ $\\begin{split}\\dfrac{(5,4,84)+(21,10,21)-(13,0,32)-(20,4,21)}{(1,0,416)+(17,6,25)-(4,4,105)-(9,8,48)} &= \\dfrac{(9,8,48)-(17,6,25)}{(5,4,84)-(20,4,21)}\\\\&= \\dfrac{(7,4,60)-(15,4,28)}{(3,2,139)-(12,4,35)}\\\\&= q^4\\dfrac{E(q^8)E(Q^{16})}{E(q^{16})E(Q^8)}.\\end{split}$ The derivations of (REF ), (REF ), (REF ), and (REF ) are similar to that of (REF ).", "The identity (REF ) is obtained from (REF ) by arguing as in the proof of (REF ).", "This reasoning also applies to obtaining (REF ) from (REF ).", "The last identity, (REF ), follows from (REF ), (REF ), (REF ), and (REF ), together with the right-most identity in (REF ) with $q$ replaced by $q^8$ .", "By (REF ), the identity () reduces to $(1,0,416)-(4,4,105)=2q\\psi (q^8)(-Q^8)$ .", "From (REF ), (REF ), and (REF ), we have $(4,4,105)&= (105,4,4)\\\\&= R(0,0,0,0,2,832,2,4,210)\\\\&= R(0,0,0,0,4,416,1,2,210)\\\\&= (q^4)(Q^{32})+4q^{105}\\psi (q^8)\\psi (Q^{64}).$ If we apply (REF ) twice, with $q$ replaced by $-Q^8$ in the second application, we can conclude that $(1,0,416)-(4,4,105)&= (q)(Q^{32})-(q^4)(Q^{32})-4q^{105}\\psi (q^8)\\psi (Q^{64})\\\\&= (Q^{32})\\left((q)-(q^4)\\right)-4q^{105}\\psi (q^8)\\psi (Q^{64})\\\\&= 2q(Q^{32})\\psi (q^8)-4q^{105}\\psi (q^8)\\psi (Q^{64})\\\\&= 2q\\psi (q^8)\\left((Q^{32})-2q^{104}\\psi (Q^{64})\\right)\\\\&= 2q\\psi (q^8)(-Q^8),$ thus proving ().", "The proof of () is similar and so we skip its details.", "Remark Replacing $q^8$ with $q$ in () and () yields Ramanujan's identity [5] $U(1,26)=U(2,13)=\\sqrt{\\dfrac{\\chi (-Q)}{\\chi (-q)}-q\\dfrac{\\chi (-q)}{\\chi (-Q)}}.$ Next, we obtain relations for the quadratic forms $(3,0,7)$ , $(1,0,21)$ , $(5,4,5)$ , and $(2,2,11)$ of discriminant $-84$ .", "Here we set $Q:=q^7$ .", "Theorem 4.8 The following two facts are true (with $Q:=q^7$ ): $\\dfrac{(1,0,21)-(5,4,5)}{(3,0,7)+(2,2,11)}=q\\dfrac{E(q^2)E(Q^6)}{E(q^6)E(Q^2)},$ $\\dfrac{(3,0,7)-(2,2,11)}{(1,0,21)+(5,4,5)}=-q^2\\dfrac{E(q^6)E(Q^2)\\psi ^2(-q)\\psi ^2(-Q^3)}{E(q^2)E(Q^6)\\psi ^2(-q^3)\\psi ^2(-Q)}.$ In [3], we obtained identities relating the functions $U(7,12)$ , $U(4,21)$ , $U(3,28)$ , and $U(1,84)$ .", "While it was not stated there it trivially follows from [3] that $\\dfrac{U(7,12)}{U(4,21)}=\\dfrac{\\psi (-q)\\psi (-Q^3)}{\\psi (-q^3)\\psi (-Q)}.$ Also from [3] we find that $\\dfrac{U(3,28)}{U(1,84)}=\\dfrac{\\psi (-q)\\psi (-Q^3)}{\\psi (-q^3)\\psi (-Q)}.$ Together that is $\\dfrac{U(7,12)}{U(4,21)}=\\dfrac{U(3,28)}{U(1,84)}=\\dfrac{\\psi (-q)\\psi (-Q^3)}{\\psi (-q^3)\\psi (-Q)}.$ Next, we have the following from (REF ) and (REF ): $4E(q^6)E(Q^2)U(3,7)U(3,7,q^4)=2(3,0,7)+T_5(3,0,7)=2(3,0,7)+2(2,2,11),$ $\\begin{split}4q^3E(q^6)E(Q^2)U(12,7)U(3,28) &= -4q^2E(q^6)E(Q^2)U(7,12)U(3,28)\\\\&= 2(3,0,7)-T_5(3,0,7)=2(3,0,7)-2(2,2,11),\\end{split}$ $4qE(q^2)E(Q^6)U(1,21)U(1,21,q^4)=2(1,0,21)-T_5(1,0,21)=2(1,0,21)-2(5,4,5),$ $4E(q^2)E(Q^6)U(4,21)U(1,84)=2(1,0,21)+T_5(1,0,21)=2(1,0,21)+2(5,4,5).$ Finally, (REF ) and (REF ) follow from (REF ), (REF )–(REF ), and Ramanujan's identity [5]: $U(3,7)=U(1,21).$ For our last application, we treat discriminant $-76$ .", "Here we set $Q:=q^{19}$ and we provide identities similar to those that Ramanujan gave for his functions.", "Theorem 4.9 The following two facts are true (with $Q:=q^{19}$ ): $4U(1,19,q)U(1,19,q^4)=3\\chi (q)^2\\chi (Q)^2+\\chi (-q)^2\\chi (-Q)^2+4q^5\\dfrac{1}{\\chi (-q^2)^2\\chi (-Q^2)^2},$ $4qU(1,19)U(1,19,-q)=\\chi (q)^2\\chi (Q)^2-\\chi (-q)^2\\chi (-Q)^2+12q^5\\dfrac{1}{\\chi (-q^2)^2\\chi (-Q^2)^2}.$ Using (REF ), we find that $4E(q^2)E(Q^2)U(1,19)U(1,19,q^4)=2(1,0,19)+T_5(1,0,19)=2(1,0,19)+2(4,2,5),\\\\\\multicolumn{2}{l}{\\text{and}}\\\\4qE(q^2)E(Q^2)U(4,19)U(1,76)=2(1,0,19)-T_5(1,0,19)=2(1,0,19)-2(4,2,5).$ From [3] we know that $U(4,19)U(1,76)=U(1,19,q^2).$ Ramanujan observed that [6] $4qU(1,19,q^2)&=\\dfrac{(q)(Q)-(-q)(-Q)-4q^5\\psi (q^2)\\psi (Q^2)}{E(q^2)E(Q^2)}\\\\&=\\chi (q)^2\\chi (Q)^2-\\chi (-q)^2\\chi (-Q)^2-4q^5\\dfrac{1}{\\chi (-q^2)^2\\chi (-Q^2)^2}.$ By adding the identities in (REF ) and () and by using (REF ) and () we arrive at (REF ).", "From (REF ), (REF ), and (REF ), we conclude that $2(4,2,5)=(q)(Q)+(-q)(-Q)+4q^5\\psi (q^2)\\psi (Q^2).$ If we use (REF ) and equate the even parts in both sides of (REF ), then by arguing as in the proof of (REF ), we will arrive at (REF )." ], [ "conclusion", "There are similar identities for many other discriminants that can be proved by establishing identities for the relevant Rogers–Ramanujan functions.", "For example, for quadratic forms of discriminant $-111$ , we find that $\\dfrac{(4,1,7)-(5,3,6)}{(1,1,28)-(4,1,7)}=\\dfrac{(3,3,10)-(4,1,7)}{(2,1,14)+(4,1,7)}=q^3\\dfrac{E(q)E(q^{111})}{E(q^3)E(q^{37})}.$ Another set of examples is for quadratic forms of discriminant $-119$ : $\\dfrac{(4,3,8)-(6,5,6)}{(1,1,30)+(5,1,6)}=q^4\\dfrac{E(q)E(q^{119})}{E(q^7)E(q^{17})}$ $\\left((4,3,8)-(2,1,15)\\right)\\left((3,1,10)-(5,1,6)\\right)=\\left((1,1,30)+(5,1,6)\\right)\\left((6,5,6)-(5,1,6)\\right).$ There are further identities for quadratic forms not related to Rogers–Ramanujan functions.", "As an example, for quadratic forms of discriminant $-80$ we have $\\dfrac{(1,0,20)-(3,2,7)}{(3,2,7)+(4,0,5)}=q\\dfrac{E(q^{40})E(q^2)}{E(q^{10})E(q^8)}.$ The identities (REF )— (REF ), along with similar types of identities, will be discussed elsewhere." ], [ "Acknowledgment", "We would like to thank Rainer Schulze-Pillot and Keith Grizzell for their interest and helpful comments." ] ]

[ [ "The Relationship Between Beam Power and Radio Power for Classical Double\n Radio Sources" ], [ "Abstract Beam power is a fundamental parameter that describes, in part, the state of a supermassive black hole system.", "Determining the beam powers of powerful classical double radio sources requires substantial observing time, so it would be useful to determine the relationship between beam power and radio power so that radio power could be used as a proxy for beam power.", "A sample of 31 powerful classical double radio sources with previously determined beam and radio powers are studied; the sources have redshifts between about 0.056 and 1.8.", "It is found that the relationship between beam power, Lj, and radio power, P, is well described by Log(Lj) = 0.84 Log(P) + 2.15, where both L_j and P are in units of 10^(44) erg/s.", "This indicates that beam power is converted to radio power with an efficiency of about 0.7%.", "The ratio of beam power to radio power is studied as a function of redshift; there is no significant evidence for redshift evolution of this ratio over the redshift range studied.", "The relationship is consistent with empirical results obtained by Cavagnolo et al.", "(2010) for radio sources in gas rich environments, which are primarily FRI sources, and with the theoretical predictions of Willott et al.", "(1999)." ], [ "INTRODUCTION", "A powerful classical double radio source, also known as an FRII source (Fanaroff & Riley 1974), is powered by large scale outflows from a supermassive black hole system that resides at the center of a galaxy (e.g.", "Blandford & Rees 1974; Scheuer 1974).", "The energy per unit time, or beam power, carried from the vicinity of the supermassive black hole to the large scale radio source is a fundamental physical parameter that, in part, describes the physical state of the black hole system.", "The beam power can be used to study many aspects of the source, the source population, interactions of the source with its environment, and the role of feedback between the black hole system, its environment, and the host galaxy (e.g.", "Silk & Rees 1998; Willott et al.", "1999; Eilek & Owen 2002; Birzan et al.", "2004; Croston et al.", "2005; Dunn et al.", "2005; Dunn & Fabian 2006; Rafferty et al.", "2006; Belsole et al.", "2007; Jetha et al.", "2007; Birzan et al.", "2008; Cavagnolo et al.", "2008; Daly 2009, 2011; Cavagnolo et al.", "2010; Martínez-Sansigre & Rawlings 2011).", "It is possible to determine the beam power of a classical double (FRII) radio source using multifrequency radio data that covers most of the radio emitting region and resolves several different regions of each radio lobe (e.g.", "Rawlings & Saunders 1991; Daly 1994; Kharb et al.", "2008; O'Dea et al.", "2009; Daly et al.", "2010); see O'Dea et al.", "(2009) for a detailed discussion of the method.", "Obtaining these data requires a considerable amount of observing time.", "If a relationship between beam power and radio power can be established, it can be applied to other FRII sources so that their beam power can be estimated from their radio power.", "The relationship between beam power and radio power has been studied for samples of radio sources that reside in gas rich environments.", "These sources are primarily FRI sources (Fanaroff & Riley 1974) at relatively low redshift (Cavagnolo et al.", "2010; Birzan et al.", "2008; Rafferty et al.", "2006; Dunn et al.", "2005; Birzan et al.", "2004).", "It has also been studied for a small sample of 14 FRII galaxies in the context of an open cosmological model (Wan & Daly 1998).", "Here, the relationship between beam power and radio power is determined for a sample of 31 FRII radio galaxies with a broad range of redshift.", "The sample is described in section 2.", "The results are presented in section 3, discussed and compared with other results and model predictions in section 4, and summarized in section 5.", "A standard spatially flat cosmological model with normalized mean mass density $\\Omega _m = 0.3$ , cosmological constant $\\Omega _{\\Lambda } = 0.7$ , and Hubble constant $\\rm {H_0} = 70~ \\rm { km~s}^{-1}\\rm { Mpc}^{-1}$ is assumed throughout." ], [ "SAMPLE", "The sample studied consists of 31 very powerful FRII radio galaxies with redshifts ranging from about 0.056 to 1.79 and core-hot spot sizes ranging from about 30 to 400 kpc drawn from the sample studied by O'Dea et al.", "(2009).", "The sources have both beam powers and radio powers determined, and are 3CRR sources (Laing, Riley, & Longair 1983).", "The sources and source properties are listed in Table 1.", "The intrinsic 178 MHz energy per unit time per unit frequency, $P_{\\nu }(178~\\rm {MHz})$ , is listed in column 3, and is determined from the 178 MHz flux density, $f_{\\nu }$ , and the radio spectral index, $\\alpha $ , defined as $f_{\\nu } \\propto \\nu ^{-\\alpha }$ .", "Radio flux densities and spectral indices are obtained from Jackson & Rawlings (1997) and Willott et al.", "(1999).", "The 178 MHz radio power, $P_{178}$ , is obtained by multiplying $P_{\\nu }(178~\\rm {MHz})$ by 178 MHz, and the integrated radio power, $P_{int}$ is obtained by integrating $P_{\\nu }$ over the frequency range from 200 to 400 MHz using the radio spectral index listed in column 9.", "This frequency range is selected to match that studied by Cavagnolo et al.", "(2010) so that the results can be easily compared.", "Values of $P_{178}$ and $P_{int}$ are listed in columns 4 and 5, respectively, of Table 1.", "Beam powers, $L_j$ , are the weighted sum of the beam power from each side of a given source obtained from O'Dea et al.", "(2009) and are listed in column 6.", "The ratio of the beam power to the radio power is listed in columns 7 and 8 for $P_{178}$ and $P_{int}$ , respectively.", "The classical double radio galaxies studied here are among the most powerful extended radio sources known in the cosmos.", "These powerful radio sources are often much larger than their parent galaxies and have cigar-like radio lobe structure indicating that the forward region of the radio emitting lobe is moving into the ambient gas supersonically (e.g.", "Alexander & Leahy 1987; Leahy, Muxlow, & Stephens 1989; Liu, Pooley, & Riley 1992).", "Since the structure of the sources indicates that each is growing at a supersonic rate, the equations of strong shock physics may be applied to the sources (Leahy, Muxlow, & Stephens 1989).", "As discussed in detail by Leahy (1990), Rawlings & Saunders (1991), Daly (1990, 1994), Wan & Daly (1998), Wan et al.", "(2000), and O'Dea et al.", "(2009), for example, this means that the beam power $L_j = \\kappa _L v P^{\\prime } a^2$ , where $\\kappa _L$ is a constant, $v$ is the rate of growth of the source, $a$ is the radius of the cross-sectional area of the forward region of the shock, and $P^{\\prime }$ is the postshock pressure.", "Each of these parameters has been determined empirically.", "O'Dea et al.", "(2009) provide a detailed description of the method and related uncertainties and a comprehensive list of beam powers obtained using this method; these are the sources studied here.", "Remarkably, O'Dea et al.", "(2009) found that the beam power is independent of offsets from minimum energy conditions due to a cancellation of the way the offset enters into the determinations of $P^{\\prime }$ and $v$ , so the beam power is insensitive to assumptions regarding minimum energy conditions.", "The beam power studied here has been obtained using the equation $L_j = \\kappa _L v P^{\\prime } a^2$ , where $\\kappa _L = 4 \\pi $ (e.g.", "Leahy 1990; Wan et al.", "2000).", "The rate of growth of the source $v$ is obtained using the equation $v = \\Delta x/\\Delta t$ .", "The change in the radio spectrum across a region $\\Delta x$ along the symmetry axis of the source is used to obtain the time $\\Delta t$ that has elapsed for the source length to change by an amount $\\Delta x$ .", "The spectral aging method of Meyers & Spangler (1985) was applied to the Kardashev-Pacholczyk and Jaffe-Perola models (Jaffe & Perola 1973) to obtain $\\Delta t$ , and similar results were obtained with both models.", "Details of the method used to obtain the quantities described above and the computations of their uncertainties are given in sections 2.1, 2.2, 3.1, 3.2, and 3.3 of O'Dea et al.", "(2009).", "In brief, uncertainties in the injection spectral index, and the flux density at each frequency studied are included in the uncertainty of $\\Delta t$ obtained for each value of $\\Delta x$ .", "Several different values of $\\Delta x$ were considered for each source, and the value of $\\Delta x/\\Delta t$ obtained was found to be independent of the value of $\\Delta x$ .", "In the computation of the rate of growth of the source, offsets of the true magnetic field strength $B$ from minimum energy conditions were parameterized by $B = b B_{min}$ where $B_{min}$ is the minimum energy magnetic field, and the geometric mean of the minimum energy fields 10 and 25 kpc from the hot spot along the symmetry axis of the source were used to obtain $B_{min}$ over the spectral aging region.", "The postshock pressure $P^{\\prime }$ was taken to be the pressure in the region 10 kpc behind the hot spot moving toward the core of the host galaxy along the symmetry axis of the radio source; the pressure can be written $P^{\\prime } = (1.33b^{-1.5}+b^2)B^2_{min}/(24\\pi )$ (e.g.", "Wan et al.", "2000), where $B_{min}$ here is obtained 10 kpc from the hot spot.", "It is shown in section 3.3 of O'Dea et al.", "(2009) that the offset from minimum energy conditions entering through $v$ cancels that entering through $P^{\\prime }$ so that, for the sources studied here, $L_j$ is independent of offsets from minimum energy conditions.", "The uncertainty of the beam power is dominated by the uncertainties of the source rate of growth and pressure.", "The half-width of the source $a$ is taken to be the deconvolved half-width of the source, which is equal to $3^{-1/2}$ times the deconvolved FWHM of the source (see, for example, section 5.1 of Wellman et al.", "1997).", "Widths were extracted at 1.4 GHz a distance 10 kpc behind the hot spot.", "Detailed studies of these sources show that the source width is independent of the frequency of observation for the range of frequencies studied (Daly et al.", "2010).", "For the new sources studied by O'Dea et al.", "(2009), the widths were obtained using a cross-sectional slice taken perpendicular to the symmetry axis of the source, as described by Daly et al.", "(2010).", "The deconvolved FWHM, $w_t$ , is given by $w_t = (w_G^2 -w_b^2)$ , where $w_G$ is the FWHM of the best fit Gaussian to the slice and $w_b$ is the observing beamwidth.", "The widths obtained were found to be in good agreement with those obtained earlier by Wellman et al.", "(1997).", "The uncertainty of the deconvolved FWHM, $\\delta w_t$ , was obtained using the expression $(\\delta w_t)^2 = (w_G/w_t)^2(\\delta w_G)^2+(w_b/w_t)^2(\\delta w_b)^2$ , where $\\delta w_G$ is the uncertainty of the FWHM of the best fit Gaussian profile and $\\delta w_b$ is the uncertainty of the observing beam.", "In practice, $\\delta w_t$ was always very close to $\\delta w_G$ ; only widths very close to the hot spot had a larger uncertainty, and these were at most about 35 % larger than $\\delta w_G$ ." ], [ "RESULTS", "The beam power was studied as a function of radio power for the full sample of 31 sources and for a sample of 30 galaxies obtained by excluding the lone low redshift source in the sample, Cygnus A.", "Very similar results are obtained using the radio power defined by $P_{178}$ and $P_{int}$ , and considering the samples of 30 and 31 sources.", "The results are illustrated in Figure 1, and best fit parameters are listed in Table 2.", "The uncertainties on the best fit parameters have been adjusted to bring the reduced chi-square of the fit to unity.", "Overall, the data are described by the equation $\\log (L_{44}) = 0.84 (\\pm 0.14)\\log (P_{44})+ 2.15 (\\pm 0.07)~,$ where $L_{44} \\equiv L_j/(10^{44} \\rm {erg~s}^{-1})$ , and $P_{44} \\equiv P/(10^{44} \\rm {erg~s}^{-1})$ .", "For powerful FRII sources, most of the radio power is produced by the outer half of the radio lobe (near the radio hot spots) (e.g.", "Leahy, Muxlow, & Stephens 1989).", "The result obtained here, $L_{44} \\approx (140 \\pm 20) P_{44}^{0.84 \\pm 0.14}$ , is consistent with $L_j \\propto P$ at about one sigma.", "These results are consistent with those obtained with a sample of 14 radio galaxies (Wan & Daly 1998).", "The efficiency with which beam power is converted to radio power is $\\epsilon = P/L_j \\approx 0.007 P_{44}^{0.16 \\pm 0.14}$ , and the constant of proportionality suggests the beam power may be converted to radio power with an overall efficiency of about 0.7 %.", "The radio power and the beam power each increase with redshift, so there is a concern that the intrinsic relationship between beam power and radio power may be masked because each is correlated with redshift.", "This effect can be circumvented by studying the ratio $L_j/P$ rather than studying each quantity separately.", "To explore this, the ratio of the beam power to the radio power, $L_j/P$ , is studied as a function of redshift with and without Cygnus A.", "The results are illustrated in Figure 2, and best fit parameters are listed in Table 2.", "Without Cygnus A, the ratio $L_j/P$ is independent of redshift, indicating that $L_j \\propto P$ is intrinsic to the sources.", "When Cygnus A is included, there is a tendency for the ratio to increase with redshift though the significance of the result is only about one sigma.", "This suggests that the relationship between beam power and radio power obtained for powerful classical double radio sources is intrinsic to the sources and is independent of redshift for redshifts ranging from about 0.4 to 2, and possibly for redshifts from about zero to 2.", "The expression used to obtain the beam power can be compared with that used to obtain the radio power of a given source.", "The radio power is obtained by integrating the radio surface brightness of the the source to obtain the flux density, multiplying this by the luminosity distance squared and appropriate redshift factors, and multiplying by the rest frame radio frequency or integrating over radio frequency to obtain $P_{178}$ or $P_{int}$ , respectively.", "The beam power depends on $L_j \\propto \\Delta x \\nu _T^{1/2} a_{10}^{17/14} a_{25}^{-3/14}S_{10}^{11/14} S_{25}^{3/14} (1+z)^{(3+ \\alpha )}$ , where $a_{10}$ and $a_{25}$ are the widths and $S_{10}$ and $S_{25}$ are the mean surface brightnesses of cross-sectional slices of the radio lobe 10 and 25 kpc, respectively, from the hot spot along the symmetry axis of the source and $\\nu _T$ is the spectral aging break frequency.", "This expression assumes that synchrotron cooling dominates over inverse Compton cooling and that $b \\le 1$ , both of which apply to the sources studied here (O'Dea et al.", "2009).", "As noted above, the length $\\Delta x$ is measured from the hot spot to the location along the symmetry axis of the lobe where the break frequency $\\nu _T$ is measured, and the product $\\Delta x \\nu _T^{1/2}$ was found to be independent of $\\Delta x$ , thus, the beam power is also independent of $\\Delta x$ (O'Dea et al.", "2009).", "The beam power depends upon several factors that do not enter into the calculation of the radio power, and the radio power depends upon factors that do not enter the beam power, so the correlation obtained here between these two parameters is likely to characterize an intrinsic physical relationship between beam power and radio power.", "In addition, the results obtained here are consistent with those obtained by Cavagnolo et al.", "(2010) who used a completely different and independent method of determining the beam power, as discussed in section 4.", "Figure: Beam power as a function of radio power for the 31 radio galaxiesin the sample.The radio power shown is obtained at 178 MHz as described in the text.The best fit line to this data set is shown, andparameter values are listed in Table 2.Figure: The ratio of the beam power to the radio power obtained at178 MHz as a function of redshift.", "Thirty of the radio galaxies areindicated as filled circles; Cygnus A is shown as an open square.The best fit line to 30 radio galaxies is indicated as a solid line andthat for all 31 sources is shown as a dotted line.", "The best fitparameters are listed in Table 2." ], [ "DISCUSSION", "The results obtained here can be compared with those obtained by Cavagnolo et al.", "(2010) who studied radio sources in gas rich clusters of galaxies and other gas rich environments.", "Cavagnolo et al.", "(2010) expanded upon the sample studied by Birzan et al.", "(2008) by adding very lower power radio sources to the sample.", "The sources studied by these groups are primarily FRI sources.", "The method used to obtain the beam power for these sources is significantly different from that described in section 3.", "For the radio sources in gas rich environments studied by Birzan et al.", "(2008) and Cavagnolo et al.", "(2010), the P dV work done by the relativistic plasma on the intracluster medium is obtained and combined with the buoyancy timescale of the radio source to obtain the beam power (e.g.", "Fabian et al.", "2006; Rafferty et al.", "2006).", "Cavagnolo et al.", "(2010) obtain $L_{44} \\approx 0.58 (10^4~P_{44})^{0.7}$ , or $L_{44} \\approx 366 P_{44}^{0.7}$ , and the coefficient may be approximated as $370 \\pm 130$ .", "As noted by these authors, the buoyancy timescale is likely to be an overestimate of the true source age and some of the sources may be affected by multiple outflows; these uncertainties would increase the uncertainties on parameters stated above.", "The slope of the relationship obtained here, $0.84 \\pm 0.14$ , is within one sigma of that obtained by Cavagnolo et al.", "(2010), $0.70 \\pm 0.12$ .", "The normalization of the relationship obtained by Cavagnolo et al.", "(2010), $370 \\pm 130$ , differs from that obtained here, $140 \\pm 20$ by less than 2 sigma.", "Thus, the results obtained are remarkably similar even though different types of radio sources at different redshifts are studied.", "The small differences that do exist may be accounted for by uncertainties in fitted parameters, or by intrinsic differences between the sources.", "More data will be needed to compare results obtained with different methods in more detail.", "The agreement between the results obtained here and those of Cavagnolo et al.", "(2010) is particularly remarkable because the interaction of the sources with their environments is quite different: the forward region of FRII sources is moving supersonically with respect to the ambient gas resulting in a strong shock whereas FRI sources are not moving supersonically and in any case the method used by Cavagnolo et al.", "(2010) does not include energy that has gone into shocks.", "The results of Cavagnolo et al.", "(2010) described above imply an efficiency of conversion of beam power to radio power of $\\epsilon = P/L_j \\approx 0.003P_{44}^{0.3 \\pm 0.12}$ ; the slope is consistent with zero at about 2.5 sigma and the coefficient suggests a conversion efficiency of about 0.3%.", "If the slope is taken to be non-zero, it would suggest that the conversion efficiency increases with radio power and is about 0.3% at a power of $10^{44} \\rm {erg/s}$ .", "Thus, both types of source appear to convert beam power to radio power with similar efficiencies: about 0.7% for FRII sources and about 0.3% for FRI sources.", "This provides a hint that the physics of the conversion of beam power to relativistic electrons and magnetic fields may be similar for FRI and FRII sources, perhaps with the efficiency of conversion increasing with radio power for FRI sources.", "Alternatively, it is possible that different physical processes for the conversion of beam power to radio power have similar efficiencies.", "The results obtained here can be compared with the theoretical model described by Willott et al.", "(1999), who found that $L_j \\approx 1.7 \\times 10^{45}f^{3/2}(P_{44})^{6/7} \\rm {erg ~s}^{-1}$ , where the term $f$ is expected to lie between 1 and 20 and is a combination of factors including the filling factor of the lobes, departure from minimum energy conditions, and the fraction of energy in non-radiating particles.", "This model prediction is obtained in the context of a detailed model based on the work of Falle (1991) and Kaiser & Alexander (1997) for extended radio sources.", "The dependence of beam power on radio power predicted by the model, which has an exponent of $0.86$ , is remarkably close to the value of $0.84$ obtained here.", "The normalization of the relationship obtained here, $L_{44} \\approx 140 P_{44}^{0.84}$ , indicates a value of $f \\approx 4$ .", "The fact that both the results obtained here and those obtained by Cavagnolo et al.", "(2010) are in fairly good agreement with the model predictions suggests that the fundamental model provides a good description of the sources." ], [ "SUMMARY", "A sample of 31 powerful FRII radio galaxies is studied to determine the relationship between beam power and radio power for sources of this type.", "The data indicated that $\\log (L_{44}) = 0.84 (\\pm 0.14)\\log (P_{44})+ 2.15 (\\pm 0.07)$ , where $L_{44} \\equiv L_j/(10^{44} \\rm {erg~s}^{-1})$ , and $P_{44} \\equiv P/(10^{44} \\rm {erg~s}^{-1})$ .", "The results listed in Table 2 may be applied to samples of FRII sources with known radio power to obtain estimates of the beam power.", "To determine whether the relationship between beam power and radio power evolves with redshift, the ratio of the beam power to radio power was studied as a function of redshift with and without the one low redshift source, Cygnus A.", "Excluding Cygnus A, there is no redshift evolution of the ratio of beam power to radio power.", "Including Cygnus A, there is a tendency for the ratio to increase with redshift, but the significance of the result is marginal.", "Thus, there is no conclusive evidence at this time that the ratio evolves with redshift.", "The results obtained here are broadly consistent with those obtained by Cavagnolo et al.", "(2010) for samples of radio sources in gas rich environments, which consist primarily of low redshift FRI sources.", "Interestingly, the results obtained here are consistent with the radio power being proportional to the beam power, with an efficiency of conversion of beam power to radio power of about 0.7% that is independent of or very weakly dependent upon radio power.", "This is comparable to the efficiency of about 0.3% indicated by FRI sources, which may be weakly dependent upon radio power in the sense that the efficiency of conversion of beam power to radio power may increase weakly with radio power.", "The relationship between beam power and radio power obtained here may be compared with theoretical predictions obtained in the context of a specific model for the sources.", "The agreement between the results obtained here and the detailed model described by Willott et al.", "(1999) is quite impressive." ], [ "Acknowledgments", "We would like to thank the referee for very helpful comments and suggestions.", "This work is supported in part by Penn State University (R.A.D.", "and T.B.S.)", "and the Radcliffe Institute for Advanced Study at Harvard University (S.A.B.).", "Table: Radio and Beam Powers of FRII SourcesTable: Best Fit Parameters" ] ]

[ [ "Binary properties of subdwarfs selected in the GALEX survey" ], [ "Abstract We describe our programme to identify and analyse binary stars among the bright subdwarfs selected in the Galaxy Evolution Explorer (GALEX) survey.", "Radial velocity time-series helped us identify subdwarfs with low-mass or compact stellar companions: We describe work conducted on the bright binaries GALEX J0321+4727 and GALEX J2349+3844, and we present a radial velocity study of several objects that include three new likely binaries.", "We also carried out photometric observations that allowed us to detect long period pulsations in the subdwarf components in two of the close binaries." ], [ "Introduction", "Hot subdwarf stars are found at the blue end of the horizontal branch and are thought to evolve with a thin hydrogen envelope as a result of binary interaction.", "Approximately half of sdB stars are in close binary systems [11], [12], [4], either with a white dwarf or a cool main-sequence companion.", "This fraction of subdwarfs in close binaries can be explained by population synthesis studies such as those of [7], [6].", "Their models evolve binary stars through either the common envelope or Roche-lobe overflow.", "On the other hand, single sdB stars could be the result of mergers of two low-mass helium white dwarfs.", "Since the discovery of pulsations in sdB stars [10], the internal structure of sdB stars has become accessible to inquiries.", "Using asteroseismology the mass and hydrogen envelope thickness have been determined for a number of hot subdwarfs [13], [1].", "Such studies have shown that most sdB stars have masses between 0.4 and 0.5 M$_\\odot $ and thin hydrogen envelopes ($\\log {M_{\\rm H}/M_*} \\sim -4$ ).", "We have identified a sample of bright subdwarf candidates using ultraviolet photometry from the GALEX all-sky survey and photographic magnitudes from the Guide Star Catalog (GSC2.3.2).", "Details of the selection criteria are described in [14].", "We have now obtained high-quality spectroscopic observations for $\\sim $ 170 hot subdwarfs and conducted model atmosphere analyses.", "[14] presented the results of their analysis of 52 subdwarfs from this sample that included measurements of their effective temperature, surface gravity and helium abundance.", "An analysis of the complete sample will be presented in Németh, Kawka, & Vennes (in preparation; see also these proceedings).", "We have initiated a follow-up programme aimed at identifying new binary systems among confirmed hot subdwarfs in the GALEX sample.", "Our first results on two of these binaries, GALEX J0321$+$ 4727 and GALEX J2349$+$ 3844, were presented in [9].", "Here we report our progress in identifying and analysing the properties of subdwarfs in close binaries." ], [ "Observations", "We observed several bright GALEX targets using the spectrograph at the coudé focus of the 2 m telescope at Ondřejov.", "We used the 830.77 lines/mm grating with a SITe $2030\\times 800$ CCD providing a resolution of $R=13\\,000$ and a spectral range from 6254 Å to 6763 Å [15].", "We also obtained spectra centred on H$\\alpha $ with the coudé spectrograph attached to the 2 m telescope at Rozhen Observatory.", "We used the 632 lines/mm grating with a SITe $1024\\times 1024$ CCD providing a resolution of $R=28\\,000$ .", "We have also started a similar programme in the southern hemisphere using the Wide Field Spectrograph [2] attached to the 2.3 m telescope at Siding Spring Observatory (SSO).", "The first set of data was obtained between UT 2010 July 14 and 18.", "We used the B3000 and R7000 gratings in the blue and red, respectively.", "We have obtained follow-up photometry for several of our targets with the 0.6 m telescope at Białków Observatory (7 objects) and 0.6 m telescope at Perth Observatory (46 objects).", "Here, we present a summary of observations of GALEX J0321$+$ 4727 and GALEX J2349$+$ 3844 that were obtained at Białków Observatory.", "GALEX J0321$+$ 4727 was observed during 6 nights between UT 2010 October 9 and November 14 and GALEX J2349$+$ 3844 was observed during 6 nights between UT 2010 October 9 and December 4.", "Figure: Radial velocity measurements of GALEX J0321++4727 and GALEX J2349++3844 phased on theorbital period.", "Radial velocity residuals in GALEX J2349++3844reveal the eccentricity of the orbit." ], [ "Binary Parameters", "We determined the radial velocities by measuring the centre of the H$\\alpha $ core.", "The velocities were adjusted to the solar system barycentre.", "Table REF updates the orbital parameters for GALEX J0321$+$ 4727 and GALEX J2349$+$ 3844 that were originally presented in [9].", "Figure REF shows the new radial velocity curves and residuals to the best-fitting sinusoidal curve.", "The residuals in the radial velocity curve of GALEX J2349$+$ 3844 show systematic deviations revealing an orbital eccentricity $e = 0.06\\pm 0.02$ .", "The short orbital period of the binary implies that it must have evolved through at least one common envelope (CE) phase [7].", "The CE phase is expected to circularize the orbit and therefore a measurable eccentricity is not expected for post-CE binaries.", "[3] reported similar periodic residual patterns for a small number of close post-CE binaries containing a subdwarf also interpreted as orbital eccentricity.", "The eccentricity may be the result of past interaction with a third body or anisotropic mass-loss.", "Table: Binary parametersWe determined the atmospheric parameters of the two subdwarfs by fitting them with synthetic spectra that were calculated using TLUSTY/SYNSPEC codes.", "For details of the fitting see Németh, Kawka & Vennes (these proceedings).", "The temperature and surface gravity for GALEX J2349$+$ 3844 have been remeasured using the complete Balmer line series.", "The temperature is 4700 K cooler and the gravity is 0.5 dex lower than originally estimated by [9] who relied on H$\\alpha $ only.", "We have obtained spectroscopic observations of several other bright GALEX subdwarfs using the Ondřejov 2 m telescope and the 2.3 m telescope at SSO.", "Most of the objects do not show significant velocity variations, however there are a few interesting cases that deserve further observations.", "GALEX J2038$-$ 2657 is a hot sdO star with a G type subgiant (III-IV) (Németh et al.", "in preparation).", "We obtained two spectra, a day apart, that show variable H$\\alpha $ emission.", "Table REF lists the number of spectra obtained for each star, as well as the average and dispersion of the velocity measurements.", "This sample also includes two subdwarf candidates (TYC4000-216-1 and TYC4406-285-1) from [8].", "We added to our sample the “constant” velocity star Feige 66 and adopted its velocity dispersion as representative of 1$\\sigma $ measurement uncertainty at Ondřejov.", "Table: Summary of radial velocities for a sample of GALEX subdwarfsThree objects show a dispersion in velocity measurements larger than 3$\\sigma $ including GALEX J0639$+$ 5156, which is also a V361 Hya pulsating subdwarf (Vučković et al.", "these proceedings).", "The most promising object that shows variable velocity measurements is GALEX J1526$+$ 7941.", "Although we have only obtained five exposures so far, the difference in velocity between two consecutive nights was $\\sim $ 60 km s$^{-1}$ .", "GALEX J1911$-$ 1406 is a hot sdO star; we used HeII$\\lambda $ 6560.09Å line to measure the radial velocities which do not appear to vary.", "GALEX J1736$+$ 2806 is a composite sdB plus main-sequence G system [14], and it does not show variability.", "This is one of the objects for which we obtained a reasonably large number of spectra over a period of two years.", "Therefore, either the binary is at a low-inclination, or more likely the system passed through a Roche-lobe overflow phase resulting in a wide binary with a long period $\\sim $ 1000 days [7]." ], [ "Photometric variations", "Based on an analysis of NSVS light curves GALEX J0321$+$ 4727 was reported to be photometrically variable [9].", "The NSVS photometric measurements have relatively large uncertainties and were obtained sporadically, and, therefore, timeseries photometry seems preferable.", "Photometric time series of GALEX J0321$+$ 4727 were obtained in $BVRI$ .", "In allowing us to accurately phase the data with our new ephemeris, these time series confirmed the reflection effect of the subdwarf by the cool companion.", "Figure REF shows the photometric measurements of GALEX J0321$+$ 4727 in the four bands folded on the best orbital period.", "As expected for the reflection effect, the highest amplitude variations are observed in the $I$ band ($\\Delta I \\sim 0.08$ mag) and the lowest in the $B$ band ($\\Delta B \\sim 0.04$ mag).", "The photometric observations also revealed weaker variations indicating that GALEX J0321$+$ 4727 is a pulsating subdwarf.", "Figure REF shows the pulsation lightcurve after removing the variations due to the reflection effect.", "We were able to detect two frequencies in our data: $\\sim $ 28.0 d$^{-1}$ and $\\sim $ 6.0 d$^{-1}$ with the respective amplitudes of $\\sim $ 1.3 mmag and $\\sim $ 1.4 mmag.", "Figure REF shows the photometric $V$ measurements of GALEX J2349$+$ 3844 revealing it to be a pulsating subdwarf.", "We detected three frequencies in our data: $\\sim $ 16.5 d$^{-1}$ , $\\sim $ 12.2 d$^{-1}$ and $\\sim $ 9.1 d$^{-1}$ with the respective amplitudes of $\\sim $ 2.9, $\\sim $ 2.8 and $\\sim $ 2.2 mmag.", "The low frequencies place GALEX J0321$+$ 4727 and GALEX J2349$+$ 3844 among the group of slowly pulsating V2093 Her type subdwarfs [5].", "GALEX J2349$+$ 3844 is at the red edge of the instability strip of V2093 Her pulsators, while GALEX J0321$+$ 4727 is closer to the blue edge.", "Figure: The photometry of GALEX J0321++4727 is phased on the orbitalperiod determined from radial velocity measurements and shows the reflectioneffect.", "The photometric bands plotted from top to bottom are BB, VV, RR andII.Figure: Photometry of GALEX J0321++4727 (top) with the reflectioneffect removed and the power spectrum (bottom) showing the main pulsationfrequencies.Figure: Photometry of GALEX J2349++3844 (top) and the powerspectrum (bottom) showing the main pulsation frequencies." ], [ "Summary", "We are currently conducting a radial velocity and photometric survey of the brightest hot subdwarf stars selected in the GALEX survey.", "With only two previously confirmed close binary systems, GALEX J0321$+$ 4727 and GALEX J2349$+$ 3844, our sample appears to have only a $\\sim $ 10% success rate of detection.", "However, with the inclusion of the three potentially variable stars, this rate increases to $\\sim $ 30%.", "The aims of the photometric survey are to identify new candidates for pulsation studies and for binarity via the detection of the reflection effect.", "We report the detection of pulsation periods of $\\sim $ 1-1.5 hrs in the subdwarf stars GALEX J0321$+$ 4727 and GALEX J2349$+$ 3844.", "Therefore, the two stars belong to the V2093 Her class of pulsators [5].", "A.K., P.N., and S.V.", "are supported by GA AV grant numbers IAA300030908 and IAA301630901, and by GA ČR grant number P209/10/0967.", "A.P.", "acknowledges Polish MNiSzW grant N N203 302635.", "We thank Dawid Moździerski for sharing a night with us." ] ]

[ [ "Microlocal limits of plane waves and Eisenstein functions" ], [ "Abstract We study microlocal limits of plane waves on noncompact Riemannian manifolds (M,g) which are either Euclidean or asymptotically hyperbolic with curvature -1 near infinity.", "The plane waves E(z,\\xi) are functions on M parametrized by the square root of energy z and the direction of the wave, \\xi, interpreted as a point at infinity.", "If the trapped set K for the geodesic flow has Liouville measure zero, we show that, as z\\to +\\infty, E(z,\\xi) microlocally converges to a measure \\mu_\\xi, in average on energy intervals of fixed size, [z,z+1], and in \\xi.", "We express the rate of convergence to the limit in terms of the classical escape rate of the geodesic flow and its maximal expansion rate - when the flow is Axiom A on the trapped set, this yields a negative power of z.", "As an application, we obtain Weyl type asymptotic expansions for local traces of spectral projectors with a remainder controlled in terms of the classical escape rate." ], [ "Outline of the proofs", "In this section, we explain the ideas of the proofs of Theorems REF and REF , in the case of manifolds Euclidean near infinity.", "We also describe the structure of the paper.", "We start with Theorem REF .", "Take $t>0$ ; we will use $\\lim _{t\\rightarrow +\\infty }\\lim _{h\\rightarrow 0}$ limits, therefore remainders that decay in $h$ with constants depending on $t$ will be negligible.", "Since $E_h$ is a generalized eigenfunction of the Laplacian (REF ), we have for $\\lambda >0$ $E_h(\\lambda ,\\xi )=e^{-it\\lambda ^2/(2h)}U(t)E_h(\\lambda ,\\xi ).$ Here $U(t)=e^{ith\\Delta /2}$ is the semiclassical Schrödinger propagator, quantizing the geodesic flow $g^t$ .", "Since $E_h$ does not lie in $L^2(M)$ , we cannot apply the operator $U(t)$ to it; however, (REF ) can be made rigorous, with an $\\mathcal {O}(h^\\infty )$ error, by using appropriate cutoffs – see Lemma REF .", "We will not write these cutoffs here for the sake of brevity.", "Take a compactly supported and compactly microlocalized semiclassical pseudodifferential operator $A$ on $M$ ; then by (REF ), $\\langle AE_h,E_h\\rangle =\\langle AU(t)E_h,U(t)E_h\\rangle =\\langle A^{-t} E_h,E_h\\rangle ,$ where $A^{-t}:=U(-t)AU(t)$ is a pseudodifferential operator with principal symbol $\\sigma (A)\\circ g^{-t}$ .", "(It is not compactly supported, but we ignore this issue here.)", "We now use the following decomposition of plane waves (see (REF )): for a fixed $\\lambda >0$ , $E_h=\\chi _0 E_h^0+E_h^1,\\quad E_h^0(m)=e^{{i\\lambda \\over h}\\xi \\cdot m},\\ E_h^1=-R_h(\\lambda )F_h,\\ F_h:=(h^2\\Delta -\\lambda ^2)\\chi _0E_h^0.$ Here $E_h^0$ is the outgoing part of the plane wave, defined in a certain neighborhood of infinity and solving (REF ) there, while $\\chi _0$ is a cutoff function equal to 1 near infinity and supported inside the domain of $E_h^0$ ; then $F_h=[h^2\\Delta ,\\chi _0]E_h^0$ is compactly supported and we can apply to it the semiclassical incoming resolvent $R_h(\\lambda )$ defined by $R_h(\\lambda ):=\\lim _{\\varepsilon \\rightarrow 0+}(h^2\\Delta -(\\lambda -i\\varepsilon )^2)^{-1}$ when acting on compactly supported functions, where $(h^2\\Delta -z)^{-1}$ is bounded on $L^2$ for $z\\notin [0,\\infty )$  – see (REF ) for the definition in the Euclidean case and (REF ) for a similar description in the hyperbolic case.", "For $\\lambda =1+\\mathcal {O}(h)$ , the function $F_h$ is microlocalized inside the set $W_\\xi :=\\lbrace (m,\\nu )\\mid m\\in \\operatorname{supp}(d\\chi _0),\\ \\nu =\\xi \\rbrace \\subset S^*M.$ In general, we cannot expect the resolvent $R_h(\\lambda )$ to be polynomially bounded in $h$ , and thus cannot determine the wavefront set of $E_h^1$ .", "However, we will show the following weaker propagation of singularities statement: the function $\\widetilde{E}_h^1(\\lambda ,\\xi ):={E_h^1(\\lambda ,\\xi )\\over 1+\\Vert E_h(\\lambda ,\\xi )\\Vert _{L^2(K_0)}},$ where $K_0\\subset M$ is a sufficiently large compact set, is polynomially bounded in $h$ and for each $(m,\\nu )\\in \\operatorname{WF}_h(\\widetilde{E}_h^1)$ , the geodesic $g^t(m,\\nu )$ is either trapped as $t\\rightarrow +\\infty $ or passes through $W_\\xi $ for some $t\\ge 0$ .", "For the case of manifolds Euclidean near infinity, this follows directly from the explicit formula for the scattering resolvent on the free Euclidean space; for manifolds hyperbolic near infinity, we use the microlocal properties of the resolvent established in [55].", "See assumption (A6) in Section REF , Section REF , and Proposition REF for details.", "If $A$ and $1-\\chi _0$ are both supported in the ball of radius $R$ , let $\\varphi \\in C_0^\\infty (M)$ be independent of $t$ and equal to 1 in the ball of radius $R+1$ .", "Then we write $A^{-t}=A_0^{-t}+A_1^{-t},\\ A_0^{-t}:=A^{-t}\\varphi ,\\ A_1^{-t}:=A^{-t}(1-\\varphi ).$ Now, each $(m,\\nu )\\in \\operatorname{WF}_h(A_1^{-t})$ has the following properties: $|m|\\ge R+1$ , and for $(m^{\\prime },\\nu ^{\\prime })=g^{-t}(m,\\nu )$ , $|m^{\\prime }|\\le R$ .", "(See Figure REF .)", "Therefore, the geodesic $g^s(m,\\nu )$ escapes to infinity for $s\\ge 0$ and never passes through $W_\\xi $ ; it follows from the discussion of the wavefront set of $\\widetilde{E}_h^1$ in the previous paragraph that $\\Vert A^{-t}_1E^1_h\\Vert _{L^2}=\\mathcal {O}(h^\\infty (1+\\Vert E_h\\Vert _{L^2(K_0)})).$ Therefore, we can write $\\langle AE_h,E_h\\rangle =\\langle A_1^{-t}\\chi _0E_h^0,\\chi _0E_h^0\\rangle +\\langle A^{-t}_0E_h,E_h\\rangle +\\mathcal {O}(h^\\infty (1+\\Vert E_h\\Vert _{L^2(K_0)}^2)).$ The first term on the right-hand side is explicit, as we have a formula for $E_h^0$ ; we can calculate for Lebesgue almost every $\\xi $ and $\\lambda =1+\\mathcal {O}(h)$ , $\\lim _{t\\rightarrow +\\infty }\\lim _{h\\rightarrow 0}\\langle A_1^{-t}\\chi _0 E_h^0(\\lambda ,\\xi ),\\chi _0 E_h^0(\\lambda ,\\xi )\\rangle =\\int _{S^*M} a\\,d\\mu _\\xi .$ Figure: A phase space picture of the main argument.The right side of each picture represents infinity;χ 0 =1\\chi _0=1 in the lighter shaded region and dχ 0 d\\chi _0is supported in the darker shaded region, while ϕ=1\\varphi =1to the left of the vertical dashed line.", "The horizontal dashedlines on the right represent the wavefront set of E ˜ h 1 \\widetilde{E}^1_h;they terminate at the solid arrows, which denote the set W ξ W_\\xi .It then remains to estimate the second and third terms on average in $\\lambda $ and $\\xi $ .", "For this, we use the relation (REF ) of distorted plane waves to the spectral measure of the Laplacian to get for any bounded compactly supported pseudodifferential operator $B$ , $h^{-1}\\Vert BE_h(\\lambda ,\\xi )\\Vert ^2_{L^2_{m,\\xi ,\\lambda }(M\\times \\partial \\overline{M}\\times [1,1+h])}\\le Ch^n\\Vert B\\operatorname{1\\hspace{-2.75pt}l}_{[1,(1+h)^2]}(h^2\\Delta )\\Vert ^2_{\\operatorname{HS}}.$ Here $\\operatorname{HS}$ denotes the Hilbert–Schmidt norm.", "One can estimate the right-hand side of (REF ) uniformly in $h$  – see Lemma REF and the proof of Proposition REF .", "Then $h^{-1}\\Vert E_h\\Vert ^2_{L^2(K_0)}$ , when integrated over $\\lambda \\in [1,1+h]$ and $\\xi $ , is bounded uniformly in $h$ ; this removes the third term on the right-hand side of (REF ).", "Finally, the average in $\\lambda ,\\xi $ of the second term on the right-hand side of (REF ) can be bounded, modulo an $\\mathcal {O}_t(h)$ remainder, by the $L^2$ norm $\\Vert \\sigma (A^{-t}_0)\\Vert _{L^2(S^*M)}$ of the restriction of the principal symbol of $A^{-t}_0$ to the energy surface $S^*M$ , with respect to the Liouville measure.", "Now, $\\sigma (A^{-t}_0)=(\\sigma (A)\\circ g^{-t})\\varphi $ converges to zero as $t\\rightarrow +\\infty $ at any point which is not trapped in the backwards direction.", "Since we assumed $\\mu _L(K)=0$ , by the dominated convergence theorem $\\Vert \\sigma (A^{-t}_0)\\Vert _{L^2(S^*M)}$ converges to zero as $t\\rightarrow +\\infty $ ; this finishes the proof of Theorem REF .", "For the estimate (REF ) in Theorem REF , we need to take $t$ up to the Ehrenfest time: $t=t_e:=\\Lambda _0^{-1}\\log (1/h)/2,$ replacing the $\\lim _{t\\rightarrow +\\infty } \\lim _{h\\rightarrow 0}$ limit in the argument of Theorem REF by just the $\\lim _{h\\rightarrow 0}$ limit, but with $t$ depending on $h$ .", "The operator $A^{-t}$ is then still pseudodifferential, though in a mildly exotic class.", "To avoid a quantization procedure uniform at infinity, we give an iterative argument, propagating $A$ for a fixed time for $\\sim \\log (1/h)$ steps, applying $t$ -independent cutoffs and removing the microlocally negligible terms at each step.", "The proof then works as before, with the term $\\langle A^{-t}_0 E_h,E_h\\rangle $ bounded by the Liouville measure of the microsupport of $A^{-t}_0$ (see Definition REF ), which depends on $h$ and is contained in $g^t(\\mathcal {T}(t))$ , where $\\mathcal {T}(t)$ is defined in (REF ); this proves (REF ).", "The interpolated quantity $r(h,\\Lambda )$ from (REF ) appears because of the subprincipal terms in (REF ).", "For (), we have to propagate to twice the Ehrenfest time: $t=2t_e$ .", "The operator $A^{-t}$ is not pseudodifferential, but we can use (REF ) to write $\\langle A^{-t}_0 E_h,E_h\\rangle =\\langle U(-t/2)AU(t/2)\\cdot U(t/2)\\varphi U(-t/2)E_h,E_h\\rangle .$ The operators $U(-t/2)AU(t/2)$ and $U(t/2)\\varphi U(-t/2)$ are both pseudodifferential in a mildly exotic class; multiplying them, we get a pseudodifferential operator whose full symbol is supported inside $g^{t/2}(\\mathcal {T}(t))$ , and thus (REF ) can be estimated by the Liouville measure of this set, giving the remainder ().", "A problem arises when trying to get a rate of convergence in (REF ) for $t$ up to twice the Ehrenfest time.", "We are unable to propagate the Lagrangian state $E_h^0(\\lambda ,\\xi )$ pointwise in $\\xi $ and $\\lambda $ for time $t$ , therefore we do not get an $L^1_\\xi $ estimate in ().", "However, for $f\\in C^\\infty (\\partial \\overline{M})$ we can still approximate the integral $\\int _{\\partial \\overline{M}} f(\\xi )\\langle A^{-t}_1\\chi _0 E_h^0, \\chi _0 E_h^0\\rangle \\,d\\xi $ as follows.", "Define the operator $\\Pi ^0_f(\\lambda ):=\\int _{\\partial \\overline{M}} f(\\xi ) (\\chi _0 E_h^0(\\lambda ,\\xi ))\\otimes (\\chi _0 E_h^0(\\lambda ,\\xi ))\\,d\\xi .$ Here $\\otimes $ denotes the Hilbert tensor product; that is, if $u,v\\in C^\\infty (M)$ , then $u\\otimes v$ is the operator with the Schwartz kernel $K_{u\\otimes v}(m,m^{\\prime })=u(m)\\overline{v(m^{\\prime })}.$ We can show that if $\\widetilde{X}$ is a pseudodifferential operator with compactly supported Schwartz kernel and microlocalized in a compact subset of $T^*M$ , satisfying certain conditions, then $\\widetilde{X} \\Pi ^0_f\\widetilde{X}^*$ is a Fourier integral operator associated to the canonical relation $\\lbrace (m,\\nu ;m^{\\prime },\\nu ^{\\prime })\\mid (m,\\nu )\\in S^*M,\\ (m^{\\prime },\\nu ^{\\prime })=g^s(m,\\nu )\\text{ for some }s\\in (-T_0,T_0)\\rbrace ,$ for a fixed $T_0>0$ depending on $\\widetilde{X}$ .", "(For comparison, for the spectral measure of $h^2\\Delta $ we would have to formally take all possible values of $s$ , which would destroy any hope on microlocally approximating it when the geodesic flow is chaotic.", "The use of $E_h^0$ instead of $E_h$ here puts us in a `nontrapping' situation $M=\\mathbb {R}^n$ , where the cutoff $\\widetilde{X}$ restricts the range of times $s$ we have to consider.)", "We can then write $\\widetilde{X} \\Pi ^0_f(\\lambda )\\widetilde{X}^*=(2\\pi h)^n\\int _{-T_0}^{T_0} e^{-i\\lambda ^2 s/(2h)}U(s)B_s\\,ds,$ where $B_s$ is a smooth family of pseudodifferential operators, compactly supported in $s\\in (-T_0,T_0)$  – see Lemma REF .", "We then write the integral (REF ) as ${\\begin{array}{c}\\operatorname{Tr}(U(-t)AU(t)(1-\\varphi ) \\Pi ^0_f(\\lambda ))=\\operatorname{Tr}\\int _{-T_0}^{T_0} e^{-i\\lambda ^2 s/(2h)} U(-t)AU(t)(1-\\varphi )U(s)B_s\\,ds\\\\=\\operatorname{Tr}\\int _{-T_0}^{T_0}e^{-i\\lambda ^2 s/(2h)} U(-t/2)AU(t/2)\\cdot U(t/2)(1-\\varphi )U(s)B_sU(-s-t/2)\\cdot U(s)\\,ds.\\end{array}}$ The operators $U(-t/2)AU(t/2)$ and $U(t/2)(1-\\varphi )U(s)B_sU(-s-t/2)$ are pseudodifferential in a mildly exotic class; thus their product is also pseudodifferential and (bearing in mind that $s$ varies in a bounded set) one gets a microlocal expansion for (REF ) through a local trace formula for Schrödinger propagators – see Lemma REF and Proposition REF ." ], [ "Other possible geometric assumptions", "Our results should be true for asymptotically hyperbolic manifolds without the constant curvature assumption near infinity.", "The main difficulty here is constructing a good semiclassical parametrix for the Eisenstein function $E_h(\\lambda ,\\xi )$ near $\\xi \\in \\partial \\overline{M}$ ; this can be done by WKB approximation, and the phase is a Busemann function $\\phi _\\xi (m)$ near $\\xi $ , however one would need a good understanding of the regularity of $\\phi _\\xi (m)$ as $m\\rightarrow \\xi $ .", "This is in a way related to the high-frequency parametrix of [36] in the non-trapping setting.", "For asymptotically Euclidean or asymptotically conic ends, this might be more complicated as we would need a parametrix of $E_h(\\lambda ,\\xi )$ in a large neighbourhood of $\\xi \\in \\partial \\overline{M}$ , essentially in a region with closure containing a ball of radius $\\pi /2$ in $\\partial \\overline{M}$ .", "In particular, the Lagrangian supporting the semiclassical parametrix of $E_h(\\lambda ,\\xi )$ would not a priori be projectable far from $\\xi $ , which would make the construction more technical.", "We leave these questions for future research.", "The convergence result in Theorem REF should be true in the case where $M$ has a boundary, for instance $M=\\mathbb {R}^{n+1}\\setminus \\Omega $ with $\\Omega $ a piecewise smooth obstacle.", "In fact, it should be straightforward to check that the method of proof applies when combined with the idea of [61], based on the fact that the region in phase space near the boundary where the dynamics is complicated is of Liouville measure 0 (since we assume $\\mu _L(K)=0$ ).", "To get a good remainder in that setting would be more involved since one would need to care about the amount of mass of plane waves staying in the regions near the boundary where the dynamics is complicated, as we propagate up to Ehrenfest time.", "A reasonable case to start with is that of strictly convex obstacles." ], [ "Structure of the paper", "In Section , we review certain notions of semiclassical analysis and derive several technical lemmata; in particular, in Section REF , we review the local theory of semiclassical Lagrangian distributions and Fourier integral operators and in Section REF we study microlocal properties of Schrödinger propagators, including the Hilbert–Schmidt norm bound (Lemma REF ).", "In Section , we formulate the general assumptions on the studied manifolds and derive some immediate corollaries; Section REF contains the geometric assumptions and the definition of the trapped set and Section REF contains the analytic assumptions on distorted plane waves.", "In Section REF we construct the limiting measures $\\mu _\\xi $ and in Section REF we prove averaged estimates on Eisenstein functions.", "In Section , we give the proofs of our main theorems.", "Section REF contains the proof of Theorem REF , Section REF contains the proof of the estimate (REF ) in Theorem REF , while Section REF contains the proof of the estimate () in Theorem REF .", "Section REF also contains the Tauberian argument proving an expansion of the local trace of a spectral projector (Theorem REF ).", "Sections  and  study the Euclidean and hyperbolic near infinity manifolds, respectively, and show that the general assumptions of Section  are satisfied in these cases.", "Appendix  provides a formula for the limiting measures in the case of a convex co-compact hyperbolic quotient, which generalizes the limiting measure of [20] to the case $\\delta \\ge n/2$ .", "Appendix  discusses the classical escape rate, in particular explaining (REF ).", "Appendix  gives a self-contained proof of Egorov's theorem up to the Ehrenfest time (Proposition REF ).", "Finally, Appendix  contains a short proof of (a special case of) quantum ergodicity in the semiclassical setting, which is simpler than that of [25] because it does not rely on [9], [43]." ], [ "Semiclassical preliminaries", "In this section, we review the methods of semiclassical analysis needed for our argument.", "Most of the constructions listed below are standard: pseudodifferential operators, wavefront sets, local theory of Fourier integral operators, and Egorov's theorem.", "However, Section REF contains the propagation result for generalized eigenfunctions (Lemma REF ) and a Hilbert–Schmidt norm estimate in an $\\mathcal {O}(h)$ spectral window (Lemma REF ), which the authors were unable to find in previous literature." ], [ "Notation", "In this subsection, we briefly review certain notation used in semiclassical analysis.", "The reader is referred to [62] (especially Chapter 14 on semiclassical calculus on manifolds) or [8] for a detailed introduction to the subject.", "The phase space.", "Let $M$ be a $d$ -dimensional manifold without boundary.", "We denote elements of the cotangent bundle $T^*M$ by $(m,\\nu )$ , where $\\nu \\in T^*_mM$ .", "Following [55], consider the fiber-radial compactification $\\overline{T}^*M$ of $T^*M$ , with the boundary definining function given by $\\langle \\nu \\rangle ^{-1}$ for any smooth inner product on the fibers of $T^*M$ .", "The boundary $\\partial \\overline{T}^*M$ , called the fiber infinity, is diffeomorphic to the cosphere bundle $S^*M$ over $M$ .Unlike [55], we do not use the notation $S^*M$ for fiber infinity — we reserve it for the unit cotangent bundle $\\lbrace |\\nu |_g=1\\rbrace \\subset T^*M$ .", "Except in Propositions REF and REF , we will use compactly microlocalized operators, for which the fiber-radial compactification is not necessary.", "Symbol classes.", "For $k\\in \\mathbb {R}$ and $\\rho \\in [0,1/2)$ , consider the symbol class $S^k_\\rho (M)$ defined as follows: a smooth function $a(m,\\nu ;h)$ on $T^*M\\times [0,h_0)$ lies in $S^k_\\rho (M)$ if and only if for each compact set $K\\subset M$ and each multiindices $\\alpha ,\\beta $ , there exists a constant $C_{\\alpha \\beta K}$ such that $\\sup _{m\\in K,\\ \\nu \\in T^*_m M}|\\partial ^\\alpha _m \\partial ^\\beta _\\nu a(m,\\nu ;h)|\\le C_{\\alpha \\beta K} h^{-\\rho (|\\alpha |+|\\beta |)}\\langle \\nu \\rangle ^{k-|\\beta |}.$ These classes are independent of the choice of coordinates on $M$ .", "Note that we do not fix the behaviour of the symbols as $m\\rightarrow \\infty $ .", "The important special case is $\\rho =0$ , which includes the classical symbols studied in [55].", "The class $S^k_0(M)$ , denoted simply by $S^k(M)$ , would be sufficient for the convergence Theorem REF .", "The classes $S^k_\\rho $ with $\\rho >0$ will be important for obtaining the remainder estimate of Theorem REF ; these classes arise when propagating symbols in $S^k_0$ for short logarithmic times, as in Proposition REF .", "Since plane waves are microlocalized on the cosphere bundle, away from the fiber infinity, we will most often work with the classes $S^{\\operatorname{comp}}_\\rho $ , consisting of compactly supported functions satisfying (REF ); we have $S^{\\operatorname{comp}}_\\rho \\subset S^k_\\rho $ for all $k$ .", "Pseudodifferential operators.", "Following [62], we can define the algebra $\\Psi ^k_\\rho (M)$ of pseudodifferential operators with symbols in $S^k_\\rho (M)$ .", "(The properties of the symbol classes $S^k_\\rho $ required for the construction of [62] are derived as in [62]; see also [27] or [18].)", "As before, denote $\\Psi ^k=\\Psi ^k_0$ .", "Since our symbols can grow arbitrarily fast as $m\\rightarrow \\infty $ , we do not make any a priori assumptions on the behavior of elements of $\\Psi ^k_\\rho $ near the infinity in $M$ .", "However, we require that all operators $A\\in \\Psi ^k(M)$ be properly supported; namely, the restriction of each of the projection maps $\\pi _m,\\pi _{m^{\\prime }}:M\\times M\\rightarrow M$ to the support of the Schwartz kernel $K_A(m,m^{\\prime })$ of $A$ is a proper map.", "See for example [27] for how to obtain properly supported quantizations on noncompact manifolds.", "Then each element of $\\Psi ^k(M)$ acts $H^s_{h,\\operatorname{loc}}(M)\\rightarrow H^{s-k}_{h,\\operatorname{loc}}(M)$ , where $H^s_{h,\\operatorname{loc}}(M)$ denotes the space of distributions locally in the semiclassical Sobolev space $H_h^s$ (see for example [62] for the definition of semiclassical Sobolev spaces).", "We also include properly supported operators that are $\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }}$ into all considered pseudodifferential classes, see for example [27].", "We have the semiclassical principal symbol map $\\sigma :\\Psi ^k_\\rho (M)\\rightarrow S^k_\\rho (M)/h^{1-2\\rho }S^{k-1}_\\rho (M)$ and its right inverse, a non-canonical quantization map $\\operatorname{Op}_h:S^k_\\rho (M)\\rightarrow \\Psi ^k_\\rho (M).$ For $A\\in \\Psi ^k_\\rho (M)$ , we often use $\\sigma (A)$ to denote any representative of the corresponding equivalence class, hence the remainder terms below.", "The standard operations of pseudodifferential calculus with symbols in $S^k_\\rho $ have an $\\mathcal {O}(h^{1-2\\rho })$ remainder instead of the $\\mathcal {O}(h)$ remainder valid for the class $S^k_0$ .", "More precisely, we have for $A\\in \\Psi ^k_\\rho (M)$ and $B\\in \\Psi ^{k^{\\prime }}_\\rho (M)$ , ${\\begin{array}{c}\\sigma (A^*)=\\overline{\\sigma (A)}+\\mathcal {O}(h^{1-2\\rho })_{S^{k-1}_\\rho (M)},\\\\\\sigma (AB)=\\sigma (A)\\sigma (B)+\\mathcal {O}(h^{1-2\\rho })_{S^{k+k^{\\prime }-1}_\\rho (M)},\\\\\\sigma ([A,B])=-ih \\lbrace \\sigma (A),\\sigma (B)\\rbrace +\\mathcal {O}(h^{2(1-2\\rho )})_{S^{k+k^{\\prime }-2}_{\\rho }(M)}.\\end{array}}$ Here $\\lbrace \\cdot ,\\cdot \\rbrace $ stands for the Poisson bracket and the adjoint is with respect to $L^2(M)$ .", "The $\\mathcal {O}(\\cdot )$ notation is used in the present paper in the following way: we write $u=\\mathcal {O}_z(F)_{\\mathcal {X}}$ if the norm of the function, or the operator, $u$ in the functional space $\\mathcal {X}$ is bounded by the expression $F$ times a constant depending on the parameter $z$ .", "Wavefront sets.", "If $A:C^\\infty (M)\\rightarrow C^\\infty (M)$ is a properly supported operator, we say that $A=\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }}$ if $A$ is smoothing and each of the $C^\\infty (M\\times M)$ seminorms of its Schwartz kernel is $\\mathcal {O}(h^\\infty )$ .", "For each $A\\in \\Psi ^k_\\rho (M)$ , we have $A=\\operatorname{Op}_h(a)+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }}$ for some $a\\in S^k_\\rho (M)$ .", "Define the semiclassical wavefront set $\\operatorname{WF}_h(A)\\subset \\overline{T}^*M$ of $A$ as follows: a point $(m,\\nu )\\in \\overline{T}^*M$ does not lie in $\\operatorname{WF}_h(A)$ , if there exists a neighborhood $U$ of $(m,\\nu )$ in $\\overline{T}^*M$ such that each $(m,\\nu )$ -derivative of $a$ is $\\mathcal {O}(h^\\infty \\langle \\nu \\rangle ^{-\\infty })$ in $U\\cap T^*M$ .", "Operators with compact wavefront sets in $T^*M$ are called compactly microlocalized; those are exactly operators of the form $\\operatorname{Op}_h(a)+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }}$ for some $a\\in S^{\\operatorname{comp}}_\\rho $ .", "We denote by $\\Psi ^{\\operatorname{comp}}_\\rho (M)$ the class of all compactly microlocalized elements of $\\Psi ^k_\\rho (M)$ ; as before, we put $\\Psi ^{\\operatorname{comp}}(M)=\\Psi ^{\\operatorname{comp}}_0(M)$ .", "Compactly microlocalized operators should not be confused with compactly supported operators (operators whose Schwartz kernels are compactly supported).", "We will need a finer notion of microsupport on $h$ -dependent sets, used in the proofs in Sections REF and REF , for example in Proposition REF : Definition 3.1 An operator $A\\in \\Psi ^{\\operatorname{comp}}_\\rho (M)$ is said to be microsupported on an $h$ -dependent family of sets $V(h)\\subset T^*M$ , if we can write $A=\\operatorname{Op}_h(a)+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }}$ , where for each compact set $K\\subset T^*M$ , each differential operator $\\partial ^\\alpha $ on $T^*M$ , and each $N$ , there exists a constant $C_{\\alpha N K}$ such that for $h$ small enough, $\\sup _{(m,\\nu )\\in K\\setminus V(h)}|\\partial ^\\alpha a(m,\\nu ;h)|\\le C_{\\alpha N K}h^N.$ Since the change of variables formula for the full symbol of a pseudodifferential operator [62] contains an asymptotic expansion in powers of $h$ , consisting of derivatives of the original symbol, Definition REF does not depend on the choice of the quantization procedure $\\operatorname{Op}_h$ .", "Moreover, if $A\\in \\Psi ^{\\operatorname{comp}}_\\rho $ is microsupported inside some $V(h)$ and $B\\in \\Psi ^k_\\rho $ , then $AB$ , $BA$ , and $A^*$ are also microsupported inside $V(h)$ .", "It follows from the definition of the wavefront set that $(m,\\nu )\\in T^*M$ does not lie in $\\operatorname{WF}_h(A)$ for some $A\\in \\Psi ^{\\operatorname{comp}}_\\rho $ , if and only if there exists an $h$ -independent neighborhood $U$ of $(m,\\nu )$ such that $A$ is microsupported on the complement of $U$ .", "Note however that $A$ need not be microsupported on $\\operatorname{WF}_h(A)$ , though it will be microsupported on any $h$ -independent neighborhood of $\\operatorname{WF}_h(A)$ .", "Finally, it can be seen by Taylor's formula that if $A\\in \\Psi ^{\\operatorname{comp}}_\\rho (M)$ is microsupported in $V(h)$ and $\\rho ^{\\prime }>\\rho $ , then $A$ is also microsupported on the set of all points in $V(h)$ which are at least $h^{\\rho ^{\\prime }}$ away from the complement of $V(h)$ .", "Ellipticity.", "For $A\\in \\Psi ^k_\\rho (M)$ , define its elliptic set $\\operatorname{ell}(A)\\subset \\overline{T}^*M$ as follows: $(m,\\nu )\\in \\operatorname{ell}(A)$ if and only if there exists a neighborhood $U$ of $(m,\\nu )$ in $\\overline{T}^*M$ and a constant $C$ such that $|\\sigma (A)|\\ge C^{-1}\\langle \\nu \\rangle ^k$ in $U\\cap T^*M$ .", "The following statement is the standard semiclassical elliptic estimate; see [27] for the closely related microlocal case and for example [10] for the semiclassical case.", "Proposition 3.2 Assume that $P\\in \\Psi _\\rho ^k(M)$ , $A\\in \\Psi _\\rho ^{k^{\\prime }}(M)$ , and $\\operatorname{WF}_h(A)\\subset \\operatorname{ell}(P)$ .", "Assume moreover that $A$ is compactly supported.", "Then there exists a constant $C$ and a function $\\chi \\in C_0^\\infty (M)$ such that for each $s\\in \\mathbb {R}$ , each $u\\in H^{s+k^{\\prime }}_{h,\\operatorname{loc}}(M)$ and each $N$ , we have $\\Vert Au\\Vert _{H^s_h}\\le C\\Vert \\chi Pu\\Vert _{H^{s+k^{\\prime }-k}_h}+\\mathcal {O}(h^\\infty )\\Vert \\chi u\\Vert _{H^{-N}}.$ Moreover, if $P$ is a differential operator, then we can take any $\\chi $ such that the Schwartz kernel of $A$ is supported in $\\lbrace \\chi \\ne 0\\rbrace \\times \\lbrace \\chi \\ne 0\\rbrace $ .", "Semi-classical wave-front sets of distributions.", "An $h$ -dependent family $u(h)\\in \\mathcal {D}^{\\prime }(M)$ is called h-tempered, if for each open $U$ compactly contained in $M$ , there exist constants $C$ and $N$ such that $\\Vert u(h)\\Vert _{H^{-N}_{h}(U)}\\le Ch^{-N}.$ For a tempered distribution $u$ , we say that $(m_0,\\nu _0)\\in \\overline{T}^*M$ does not lie in the wavefront set $\\operatorname{WF}_h(u)$ , if there exists a neighborhood $V(m_0,\\nu _0)$ in $\\overline{T}^*M$ such that for each $A\\in \\Psi ^0(M)$ with $\\operatorname{WF}_h(A)\\subset V$ , we have $Au=\\mathcal {O}(h^\\infty )_{C^\\infty }$ .", "By Proposition REF , $(m_0,\\nu _0)\\notin \\operatorname{WF}_h(u)$ if and only if there exists compactly supported $A\\in \\Psi ^0(M)$ elliptic at $(m_0,\\nu _0)$ such that $Au=\\mathcal {O}(h^\\infty )_{C^\\infty }$ .", "The wavefront set of $u$ is a closed subset of $\\overline{T}^*M$ ; it is empty if and only if $u=\\mathcal {O}(h^\\infty )_{C^\\infty (M)}$ .", "We can also verify that for $u$ tempered and $A\\in \\Psi ^k_\\rho (M)$ , $\\operatorname{WF}_h(Au)\\subset \\operatorname{WF}_h(A)\\cap \\operatorname{WF}_h(u)$ ." ], [ "Semiclassical Lagrangian distributions", "In this subsection, we review some facts from the theory of semiclassical Lagrangian distributions.", "See [21] or [56] for a detailed account, and [28] or [18] for the closely related microlocal case.", "However, note that we do not attempt to define the principal symbols as global invariant geometric objects; this makes the resulting local theory considerably simpler.", "Phase functions.", "Let $M$ be a manifold without boundary.", "We denote its dimension by $d$ ; in the convention used in the present paper, $d=n+1$ .", "As before, we denote elements of $T^*M$ by $(m,\\nu )$ , $m\\in M$ , $\\nu \\in T^*_m M$ .", "Let $\\varphi (m,\\theta )$ be a smooth real-valued function on some open subset $U_\\varphi $ of $M\\times \\mathbb {R}^L$ , for some $L$ ; we call $m$ base variables and $\\theta $ oscillatory variables.", "We say that $\\varphi $ is a (nondegenerate) phase function, if the differentials $d(\\partial _{\\theta _1}\\varphi ),\\dots ,d(\\partial _{\\theta _L}\\varphi )$ are linearly independent on the critical set $C_\\varphi :=\\lbrace (m,\\theta )\\mid \\partial _\\theta \\varphi =0\\rbrace \\subset U_\\varphi .$ In this case $\\Lambda _\\varphi :=\\lbrace (m,\\partial _m\\varphi (m,\\theta ))\\mid (m,\\theta )\\in C_\\varphi \\rbrace \\subset T^*M$ is an (immersed, and we will shrink the domain of $\\varphi $ to make it embedded) Lagrangian submanifold.", "We say that $\\varphi $ generates $\\Lambda _\\varphi $ .", "Symbols.", "Let $\\rho \\in [0,1/2)$ .", "A smooth function $a(m,\\theta ;h)$ is called a compactly supported symbol of type $\\rho $ on $U_\\varphi $ , if it is supported in some compact $h$ -independent subset of $U_\\varphi $ , and for each differential operator $\\partial ^\\alpha $ on $M\\times \\mathbb {R}^L$ , there exists a constant $C_\\alpha $ such that $\\sup _{U_\\varphi }|\\partial ^\\alpha a|\\le C_\\alpha h^{-\\rho |\\alpha |}.$ Similarly to Section REF , we write $a\\in S^{\\operatorname{comp}}_\\rho (U_\\varphi )$ and denote $S^{\\operatorname{comp}}:=S^{\\operatorname{comp}}_0$ .", "Lagrangian distributions.", "Given a phase function $\\varphi $ and a symbol $a\\in S^{\\operatorname{comp}}_\\rho (U_\\varphi )$ , consider the $h$ -dependent family of functions $u(m;h)=h^{-L/2}\\int _{\\mathbb {R}^L} e^{i\\varphi (m,\\theta )/h}a(m,\\theta ;h)\\,d\\theta .$ We call $u$ a Lagrangian distribution of type $\\rho $ generated by $\\varphi $ .", "By the method of non-stationary phase, if $\\operatorname{supp}a$ is contained in some $h$ -independent compact set $K\\subset U_\\varphi $ , then $\\operatorname{WF}_h(u)\\subset \\lbrace (m,\\partial _m\\varphi (m,\\theta ))\\mid (m,\\theta )\\in C_\\varphi \\cap K\\rbrace \\subset \\Lambda _\\varphi .$ The principal symbol $\\sigma _\\varphi (u)\\in S^{\\operatorname{comp}}_\\rho (\\Lambda _\\varphi )$ of $u$ is defined modulo $\\mathcal {O}(h^{1-2\\rho })$ by the expression $\\sigma _\\varphi (u)(m,\\partial _m\\varphi (m,\\theta );h)= a(m,\\theta ;h),\\ (m,\\theta )\\in C_\\varphi .$ That $\\sigma _\\varphi (u)$ does not depend (modulo $\\mathcal {O}(h^{1-2\\rho })$ ) on the choice of $a$ producing $u$ will follow from Proposition REF and (REF ).", "Following [18], we introduce a certain (local) canonical form for Lagrangian distributions.", "Fix some local system of coordinates on $M$ (shrinking $M$ to the domain of this coordinate system and identifying it with a subset of $\\mathbb {R}^d$ ) and consider $\\Lambda _F=\\lbrace (m,\\nu )\\mid m=-\\partial _\\nu F(\\nu ),\\ \\nu \\in U_F\\rbrace \\subset T^*M,$ where $F$ is a smooth real-valued function on some open set $U_F\\subset \\mathbb {R}^d$ , such that the image of $-\\partial _\\nu F$ is contained in $M$ .", "Then $\\Lambda _F$ is Lagrangian; in fact, it is generated by the phase function $m\\cdot \\nu +F(\\nu )$ , with $\\nu $ the oscillatory variable.", "One can also prove that any Lagrangian submanifold not intersecting the zero section (which is always the case for the Lagrangians considered in this paper) can be locally brought under the form (REF ) for an appropriate choice of the coordinate system on $M$  – see for example [18].", "If $b(\\nu ;h)\\in S^{\\operatorname{comp}}_\\rho (U_F)$ and $\\chi \\in C_0^\\infty (M)$ is equal to 1 near $-\\partial _\\nu F(\\operatorname{supp}b)$ , then we can define a Lagrangian distribution by the following special case of (REF ): $v(m;h)=\\chi (m)h^{-d/2}\\int _{U_F} e^{i(m\\cdot \\nu +F(\\nu ))/h}b(\\nu ;h)\\,d\\nu .$ We need $\\chi $ to make $v\\in C_0^\\infty (M)$ ; however, by (REF ) (or directly by the method of nonstationary phase), if we choose $\\chi $ differently, then $v$ will change by $\\mathcal {O}(h^\\infty )_{C_0^\\infty }$ .", "If $v$ is given by (REF ), then we can recover the symbol $b$ by the Fourier inversion formula: $e^{i F(\\nu )/h}b(\\nu ;h)=(2\\pi )^{-d}h^{-d/2}\\int _Me^{-im\\cdot \\nu /h} v(m;h)\\,dm+\\mathcal {O}(h^\\infty )_{S(\\mathbb {R}^d)}.$ Note that if $v=\\mathcal {O}(h^\\infty )_{C_0^\\infty }$ , then $b(\\nu ;h)=\\mathcal {O}(h^\\infty )_{C_0^\\infty }$ .", "Moreover, if $v\\in C_0^\\infty (M)$ satisfies (REF ) for some $b\\in S^{\\operatorname{comp}}_\\rho (U_F)$ , then $v$ is given by (REF ) modulo $\\mathcal {O}(h^\\infty )_{C_0^\\infty }$ .", "Following [18], by the method of stationary phase each Lagrangian distribution can be brought locally into the form (REF ): Proposition 3.3 Assume that $\\varphi $ is a phase function, and the corresponding Lagrangian $\\Lambda =\\Lambda _\\varphi $ can be written in the form (REF ).", "For $a(m,\\theta ;h)\\in S^{\\operatorname{comp}}_\\rho (U_\\varphi )$ and $b(\\nu ;h)\\in S^{\\operatorname{comp}}_\\rho (U_F)$ , denote by $u_a$ and $v_b$ the functions given by (REF ) and (REF ), respectively.", "Then: 1.", "For each $a\\in S^{\\operatorname{comp}}_\\rho (U_\\varphi )$ , there exists $b\\in S^{\\operatorname{comp}}_\\rho (U_F)$ such that $u_a=v_b+\\mathcal {O}(h^\\infty )_{C_0^\\infty }$ .", "Moreover, we have the following asymptotic decomposition for $b$ : $b(\\nu ;h)=\\sum _{0\\le j<N} h^j L_j a(m,\\theta ;h)+\\mathcal {O}(h^{N(1-2\\rho )})_{S^{\\operatorname{comp}}_\\rho (U_F)},$ where each $L_j$ is a differential operator of order $2j$ on $U_\\varphi $ , and $(m,\\theta )\\in C_\\varphi $ is the solution to the equation $(m,\\partial _m\\varphi (m,\\theta ))=(-\\partial _\\nu F(\\nu ),\\nu )$ .", "In particular, if $\\sigma _\\varphi (u)$ is given by (REF ), then $\\sigma _\\varphi (u)(-\\partial _\\nu F(\\nu ),\\nu ;h)=f_{\\varphi F} b(\\nu ;h)+\\mathcal {O}(h^{1-2\\rho })_{S^{\\operatorname{comp}}_\\rho (U_F)},$ where $f_{\\varphi F}$ is some nonvanishing function depending on $\\varphi $ and the coordinate system on $M$ .", "Adding a certain constant to the function $F$ , we can make $f_{\\varphi F}$ independent of $h$ .", "2.", "For each $b\\in S^{\\operatorname{comp}}_\\rho (U_F)$ , there exists $a\\in S^{\\operatorname{comp}}_\\rho (U_\\varphi )$ such that $v_b=u_a+\\mathcal {O}(h^\\infty )_{C_0^\\infty }$ .", "Definition 3.4 Let $\\Lambda \\subset T^*M$ be an embedded Lagrangian submanifold.", "We say that an $h$ -dependent family of functions $u(m;h)\\in C_0^\\infty (M)$ is a (compactly supported and compactly microlocalized) Lagrangian distribution of type $\\rho $ associated to $\\Lambda $ , if it can be written as a sum of finitely many functions of the form (REF ), for different phase functions $\\varphi $ parametrizing open subsets of $\\Lambda $ , plus an $\\mathcal {O}(h^\\infty )_{C_0^\\infty }$ remainder.", "Denote by $I^{\\operatorname{comp}}_\\rho (\\Lambda )$ the space of all such distributions, and put $I^{\\operatorname{comp}}(\\Lambda ):=I^{\\operatorname{comp}}_0(\\Lambda )$ .", "By Proposition REF , if $\\varphi $ is a phase function and $u\\in I^{\\operatorname{comp}}_\\rho (\\Lambda _\\varphi )$ , then $u$ can be written in the form (REF ) for some symbol $a$ , plus an $\\mathcal {O}(h^\\infty )_{C_0^\\infty }$ remainder.", "The symbol $\\sigma _\\varphi (u)$ , given by (REF ), is well-defined modulo $\\mathcal {O}(h^{1-2\\rho })$ .", "The action of a pseudodifferential operator on a Lagrangian distribution is given by the following proposition, following from Proposition REF and the method of stationary phase: Proposition 3.5 Let $u\\in I^{\\operatorname{comp}}_\\rho (\\Lambda )$ and $P\\in \\Psi ^k_\\rho (M)$ .", "Then $Pu\\in I^{\\operatorname{comp}}_\\rho (\\Lambda )$ .", "Moreover, 1.", "If $\\Lambda =\\Lambda _\\varphi $ for some phase function $\\varphi $ , then $\\sigma _\\varphi (Pu)=\\sigma (P)|_{\\Lambda _\\varphi }\\cdot \\sigma _\\varphi (u)+\\mathcal {O}(h^{1-2\\rho })_{S^{\\operatorname{comp}}_\\rho (\\Lambda )}.$ 2.", "Assume that $\\Lambda =\\Lambda _F$ is given by (REF ) in some coordinate system on $M$ .", "Let $b(\\nu ;h)$ and $b^P(\\nu ;h)$ be the symbols corresponding to $u$ and $Pu$ , respectively, via (REF ).", "Let also $P=\\operatorname{Op}_h(p)$ for some quantization procedure $\\operatorname{Op}_h$ .", "Then $b^P(\\nu ;h)=\\sum _{0\\le j<N}h^jL_j (p(m,\\nu ^{\\prime };h)b(\\nu ;h))|_{\\nu ^{\\prime }=\\nu ,\\,m=-\\partial _\\nu F(\\nu )}+\\mathcal {O}(h^{N(1-2\\rho )})_{S^{\\operatorname{comp}}_\\rho (U_F)},$ where each $L_j$ is a differential operator of order $2j$ on $M\\times U_F\\times U_F$ .", "Finally, we give the following estimate of the $L^2$ norm of a Lagrangian distribution, following from the boundedness of the Fourier transform on $L^2$ : Proposition 3.6 Assume that $u\\in I^{\\operatorname{comp}}_\\rho (\\Lambda _F)$ , where $\\Lambda _F$ is given by (REF ).", "Let $u$ be given by (REF ), with $b(\\nu ;h)$ the corresponding symbol.", "Then for some constant $C$ independent of $h$ , $\\Vert u(m;h)\\Vert _{L^2}\\le C\\Vert b(\\nu ;h)\\Vert _{L^2(U_F)}.$ Fourier integral operators.", "A special case of Lagrangian distributions are Fourier integral operators associated to canonical transformations.", "Let $M,M^{\\prime }$ be two manifolds of the same dimension $d$ , and let $\\kappa $ be a symplectomorphism from an open subset of $T^*M$ to an open subset of $T^*M^{\\prime }$ .", "Consider the Lagrangian $\\Lambda _\\kappa =\\lbrace (m,\\nu ;m^{\\prime },-\\nu ^{\\prime }) \\mid \\kappa (m,\\nu )=(m^{\\prime },\\nu ^{\\prime })\\rbrace \\subset T^*M\\times T^*M^{\\prime }=T^*(M\\times M^{\\prime }).$ A compactly supported operator $U:\\mathcal {D}^{\\prime }(M^{\\prime })\\rightarrow C_0^\\infty (M)$ is called a (semiclassical) Fourier integral operator of type $\\rho $ associated to $\\kappa $ , if its Schwartz kernel $K_U(m,m^{\\prime })$ lies in $h^{-d/2}I^{\\operatorname{comp}}_\\rho (\\Lambda _\\kappa )$ .", "We write $U\\in I^{\\operatorname{comp}}_\\rho (\\kappa )$ .", "Note that we quantize a canonical transformation $T^*M\\rightarrow T^*M^{\\prime }$ as an operator $\\mathcal {D}^{\\prime }(M^{\\prime })\\rightarrow C_0^\\infty (M)$ , in contrast with the standard convention, which would quantize it as an operator $\\mathcal {D}^{\\prime }(M)\\rightarrow C_0^\\infty (M^{\\prime })$ .", "The $h^{-d/2}$ factor is explained as follows: the normalization for Lagrangian distributions is chosen so that $\\Vert u\\Vert _{L^2}\\sim 1$ , while the normalization for Fourier integral operators is chosen so that $\\Vert U\\Vert _{L^2(M^{\\prime })\\rightarrow L^2(M)}\\sim 1$ .", "After sufficiently shrinking the domain of $\\kappa $ and choosing an appropriate coordinate system on $M^{\\prime }$ (which is possible for all $\\kappa $ whose graph does not intersect the zero section of $T^*M^{\\prime }$ , see the remark following (REF )), we can find a generating function $S(m,\\nu ^{\\prime })$ for $\\kappa $ ; that is, $\\kappa (m,\\nu )=(m^{\\prime },\\nu ^{\\prime })\\iff \\partial _m S(m,\\nu ^{\\prime })=\\nu ,\\ \\partial _{\\nu ^{\\prime }} S(m,\\nu ^{\\prime })=m^{\\prime }.$ Here $(m,\\nu ^{\\prime })$ vary in some open set $U_S\\subset M\\times \\mathbb {R}^d$ .", "The phase function $S(m,\\nu ^{\\prime })-m^{\\prime }\\cdot \\nu ^{\\prime }$ , with $\\nu ^{\\prime }$ the oscillatory variable, parametrizes $\\Lambda _\\kappa $ and for $U\\in I^{\\operatorname{comp}}_\\rho (\\kappa )$ , we write similarly to (REF ), $K_U(m,m^{\\prime })=h^{-d}\\chi (m^{\\prime })\\int _{\\mathbb {R}^d} e^{{i\\over h}(S(m,\\nu ^{\\prime })-m^{\\prime }\\cdot \\nu ^{\\prime })}b(m,\\nu ^{\\prime };h)\\,d\\nu ^{\\prime }+\\mathcal {O}(h^\\infty )_{C_0^\\infty },$ for some symbol $b\\in S^{\\operatorname{comp}}_\\rho (U_S)$ and any $\\chi \\in C_0^\\infty (M^{\\prime })$ such that $\\chi =1$ near the set $\\partial _{\\nu ^{\\prime }}S(\\operatorname{supp}b)$ .", "The function $b$ is determined uniquely by $U$ modulo $\\mathcal {O}(h^\\infty )_{S^{\\operatorname{comp}}_\\rho (U_S)}$ , similarly to (REF ).", "Note that if $\\kappa $ is the identity map, then $S(m,\\nu ^{\\prime })=m\\cdot \\nu ^{\\prime }$ and we arrive to the quantization formula for a semiclassical pseudodifferential operator.", "Similarly to Proposition REF , we have Proposition 3.7 Assume that $U\\in I^{\\operatorname{comp}}_\\rho (\\kappa )$ and $P\\in \\Psi ^k_\\rho (M^{\\prime })$ .", "Then $UP\\in I^{\\operatorname{comp}}_\\rho (\\kappa )$ .", "If moreover $\\kappa $ is given by (REF ), $b(m,\\nu ^{\\prime };h)$ and $b^P(m,\\nu ^{\\prime };h)$ are the symbols corresponding to $U$ and $UP$ , respectively, via (REF ), and $P=\\operatorname{Op}_h(p)$ for some quantization procedure $\\operatorname{Op}_h$ , then we have the following asymptotic decomposition for $b^P$ : $b^P(m,\\nu ^{\\prime };h)=\\sum _{0\\le j<N}h^j L_j(p(m^{\\prime },\\tilde{\\nu })b(m,\\nu ^{\\prime }))|_{\\tilde{\\nu }=\\nu ^{\\prime },\\,m^{\\prime }=\\partial _{\\nu ^{\\prime }} S(m,\\nu ^{\\prime })}+\\mathcal {O}(h^{N(1-2\\rho )})_{S^{\\operatorname{comp}}_\\rho (U_S)}.$ Here each $L_j$ is a differential operator of order $2j$ on $M^{\\prime }\\times \\mathbb {R}^d\\times U_S$ .", "In particular, $b^P(m,\\nu ^{\\prime };h)=p(\\partial _{\\nu ^{\\prime }} S(m,\\nu ^{\\prime }),\\nu ^{\\prime };h)b(m,\\nu ^{\\prime };h)+\\mathcal {O}(h^{1-2\\rho })_{S^{\\operatorname{comp}}_\\rho (U_S)}.$ A similar statement is true for an operator of the form $PU$ , where $P\\in \\Psi ^k_\\rho (M)$ and the terms of the asymptotic decomposition have the form $h^j L_j(p(\\widetilde{m},\\nu )b(m,\\nu ^{\\prime }))|_{\\widetilde{m}=m,\\,\\nu =\\partial _m S(m,\\nu ^{\\prime })}$ ." ], [ "Schrödinger propagators", "Let $(M,g)$ be a complete Riemannian manifold, $\\Delta =\\Delta _g$ the corresponding (nonnegative) Laplace–Beltrami operator, and $p(m,\\nu )=|\\nu |_g^2$ the semiclassical principal symbol of $h^2\\Delta \\in \\Psi ^2(M)$ .", "We use the notation $S^*M=p^{-1}(1)\\subset T^*M$ for the unit cotangent bundle.", "The geodesic flow $g^t$ on $T^*M$ is related to the Hamiltonian flow $e^{tH_p}$ of $p$ by the formula $g^t=e^{tH_p/2}$ .", "The operator $\\Delta $ is essentially self-adjoint on $L^2(M)$ by [6] and its domain is given by the Friedrichs extension.", "Let $U(t)=e^{ith\\Delta /2}=e^{{it\\over h}(h^2\\Delta /2)}$ be the semiclassical Schrödinger propagator; it is a unitary operator on $L^2(M)$ .", "The basic microlocal properties of $U(t)$ are given by the following Proposition 3.8 For each $t\\in \\mathbb {R}$ , 1.", "(Egorov's Theorem) For each compactly supported $A\\in \\Psi ^{\\operatorname{comp}}_\\rho (M)$ , there exists compactly supported $A^t\\in \\Psi ^{\\operatorname{comp}}_\\rho (M)$ such that $U(t)AU(-t)=A^t+\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}.$ Moreover, $\\operatorname{WF}_h(A^t)\\subset g^{-t}(\\operatorname{WF}_h(A))$ and $\\sigma (A^t)=\\sigma (A)\\circ g^t+\\mathcal {O}(h^{1-2\\rho })$ .", "2.", "(Microlocalization) $U(t)$ is microlocalized on the graph of $g^{-t}$ , namely if $A,B\\in \\Psi ^k_\\rho (M)$ are compactly supported and at least one of them is compactly microlocalized, then $g^t(\\operatorname{WF}_h(A))\\cap \\operatorname{WF}_h(B)=\\emptyset \\ \\Longrightarrow \\ AU(t)B=\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}.$ 3.", "(Parametrix) If $A\\in \\Psi ^{\\operatorname{comp}}(M)$ is compactly supported, then $U(t)A$ is the sum of a compactly microlocalized Fourier integral operator (of type 0) associated to $g^t$ , as defined in Section REF , and an $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ remainder.", "The proofs are standard; part 1 can be found in [62] (with the mildly exotic classes $\\Psi ^{\\operatorname{comp}}_\\rho $ handled as in Appendix ), part 2 follows directly from part 1, and part 3 is proved similarly to [62].", "The operator $U(t)A$ quantizes $g^t$ , not $g^{-t}$ , because of the convention adopted in Section REF that a canonical transformation $T^*M\\rightarrow T^*M^{\\prime }$ is quantized as an operator $\\mathcal {D}^{\\prime }(M^{\\prime })\\rightarrow C_0^\\infty (M)$ .", "Egorov's theorem until the Ehrenfest time.", "Proposition REF is valid for bounded times $t$ ; as $t\\rightarrow \\infty $ , the constants in the estimates for the corresponding symbols will blow up.", "However, it is still possible to prove Egorov's Theorem for $t$ bounded by a certain multiple of $\\log (1/h)$ , called the Ehrenfest time.", "To define this time, we fix an open bounded set $U\\subset M$ with geodesically convex closure in the sense of (REF ) and define the maximal expansion rate $\\Lambda _{\\max }:=\\limsup _{|t|\\rightarrow \\infty }{1\\over |t|}\\log \\sup _{m\\in U,\\,|\\nu |_g=1,\\atop g^t(m,\\nu )\\in U}\\Vert dg^t(m,\\nu )\\Vert .$ Here $\\Vert dg^t(m,\\nu )\\Vert $ is the operator norm of the differential $dg^t(m,\\nu ):T_{(m,\\nu )}T^*M\\rightarrow T_{g^t(m,\\nu )}T^*M$ with respect to any given smooth norm on the fibers of $T(T^*M)$ .", "Since we will work on a noncompact manifold, we introduce cutoffs into the corresponding propagators: Proposition 3.9 Assume that $X_1,X_2\\in \\Psi ^0(M)$ satisfy $\\Vert X_j\\Vert _{L^2\\rightarrow L^2}\\le 1+\\mathcal {O}(h)$ and are compactly supported inside $U$ .", "Let $\\varepsilon _e>0$ and take $\\Lambda _0,\\Lambda ^{\\prime }_0>0$ such that $\\Lambda _0>\\Lambda ^{\\prime }_0>(1+2\\varepsilon _e)\\Lambda _{\\max }$ .", "Fix $t_0\\in \\mathbb {R}$ .", "Then for each integer $l\\in [0, \\log (1/h)/(2|t_0|\\Lambda _0)],$ and each compactly supported $A\\in \\Psi ^{\\operatorname{comp}}(M)$ with $\\operatorname{WF}_h(A)\\subset \\mathcal {E}_{\\varepsilon _e}:=\\lbrace 1-\\varepsilon _e\\le |\\nu |_g\\le 1+\\varepsilon _e\\rbrace $ , the compactly supported operator $A^{(l)}:=(X_2 U(t_0))^l A (U(-t_0)X_1)^l$ lies in $\\Psi ^{\\operatorname{comp}}_{\\rho _l}(M)$ , modulo an $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ remainder, with $\\rho _l=l|t_0|\\Lambda ^{\\prime }_0/\\log (1/h)<1/2.$ Moreover, the $S^{\\operatorname{comp}}_{\\rho _l}$ seminorms of the full symbol of $A^{(l)}$ are bounded uniformly in $l$ , in the following sense: the order $k$ derivatives of this symbol are bounded by $Ch^{-k\\rho _l}$ , where $C$ is a constant independent of $h$ and $l$ .", "The principal symbol of $A^{(l)}$ is $\\sigma (A^{(l)})=(\\sigma (A)\\circ g^{lt_0})\\prod _{j=0}^{l-1}(\\sigma (X_1)\\sigma (X_2))\\circ g^{jt_0}+\\mathcal {O}(h^{1-2\\rho _l}).$ The wavefront set of $A^{(l)}$ , for $l>0$ , is contained in $\\operatorname{WF}_h(X_1)\\cap \\operatorname{WF}_h(X_2)\\cap \\mathcal {E}_{\\varepsilon _e}$ .", "Finally, if $U_A$ and $U_X$ are open sets such that $\\operatorname{WF}_h(A)\\subset U_A$ and $\\operatorname{WF}_h(X_1)\\cap \\operatorname{WF}_h(X_2)\\subset U_X$ , then $A^{(l)}$ is microsupported, in the sense of Definition REF , inside the set $V^{(l)}:=g^{-lt_0}(U_A)\\cap \\bigcap _{j=0}^{l-1} g^{-jt_0}(U_X).$ The set $V^{(l)}$ does not depend on $h$ directly, however it depends on $l$ , which is allowed to depend on $h$ , and our microlocal vanishing statement is uniform in $l$ .", "Proposition REF is the main technical tool of obtaining the polynomial remainder bound of Theorem REF ; it is also the reason why the classes $\\Psi ^{\\operatorname{comp}}_\\rho $ appear.", "Its proof, following the methods of [1] and [62], is given in Appendix .", "See also [44] for a more refined estimate in the more restrictive setting of two-dimensional manifolds with hyperbolic geodesic flows.", "We do not impose any restrictions on the set $U$ at this point, however in our actual argument it will have to contain a neighborhood of the trapped set – see the beginning of Section REF .", "Propagating generalized eigenfunctions.", "The following fact, similar to [11], will be used to propagate the Eisenstein functions by the group $U(t)$ : Lemma 3.10 Assume that $u\\in C^\\infty (M)$ solves the equation $(h^2\\Delta -z)u=0,\\ |1-z|\\le Ch.$ Let $\\chi \\in C_0^\\infty (M)$ ; take $t\\in \\mathbb {R}$ and assume that $\\chi _t\\in C_0^\\infty (M)$ is supported in the interior of a compact set $K_t\\subset M$ and satisfies $d_g(\\operatorname{supp}\\chi ,\\operatorname{supp}(1-\\chi _t))>|t|.$ Here $d_g$ denotes Riemannian distance on $M$ .", "Then $\\chi u=\\chi e^{-itz/(2h)}U(t)\\chi _t u+\\mathcal {O}(h^\\infty \\Vert u\\Vert _{L^2(K_t)})_{L^2(M)}.$ Without loss of generality, we assume that $t\\ge 0$ .", "For $0\\le s\\le t$ , define $u_s=\\chi (u-e^{-isz/(2h)}U(s)\\chi _t u)$ .", "We need to prove that $\\Vert u_t\\Vert _{L^2}=\\mathcal {O}(h^\\infty )\\Vert u\\Vert _{L^2(K_t)}.$ Since $\\chi =\\chi \\chi _t$ , we have $u_0=0$ ; next, ${\\begin{array}{c}2hD_s u_s= - \\chi e^{-isz/(2h)}U(s)(h^2\\Delta -z)\\chi _t u= - e^{-isz/(2h)}\\chi U(s) [h^2\\Delta ,\\chi _t]u.\\end{array}}$ Let $B\\in \\Psi ^{\\operatorname{comp}}$ be compactly supported inside $K_t\\times K_t$ , equal to the identity microlocally near $\\operatorname{supp}\\chi _t\\cap S^*M$ , but microlocalized in a small neighborhood of $S^*M$ so that by (REF ), $g^s(\\operatorname{supp}\\chi )\\cap \\operatorname{WF}_h(B)\\cap \\operatorname{supp}(1-\\chi _t)=\\emptyset .$ Note that $\\operatorname{WF}_h([h^2\\Delta ,\\chi _t])\\subset \\operatorname{supp}(1-\\chi _t)$ .", "Then by part 2 of Proposition REF , $\\Vert \\chi U(s)[h^2\\Delta ,\\chi _t]Bu\\Vert _{L^2}=\\mathcal {O}(h^\\infty )\\Vert u\\Vert _{L^2(K_t)},\\ 0\\le s\\le t.$ Moreover, by Proposition  REF $\\Vert \\chi U(s)[h^2\\Delta ,\\chi _t](1-B)u\\Vert _{L^2}=\\mathcal {O}(h^\\infty )\\Vert u\\Vert _{L^2(K_t)}.$ Combining (REF ) and (REF ), we get $\\Vert \\partial _s u_s\\Vert _{L^2}=\\mathcal {O}(h^\\infty )\\Vert u\\Vert _{L^2(K_t)}$ ; it remains to integrate in $s$ to get (REF ).", "Hilbert–Schmidt norm estimates.", "We now prove Hilbert–Schmidt norm estimates for the product of a pseudodifferential operator with a spectral projector.", "(See [27] for the properties of Hilbert–Schmidt and trace class operators.)", "To simplify notation, we consider a spectral interval of size $h$ centered at $\\lambda =1$ ; similar statement is true for the interval $[\\lambda +c_1h,\\lambda +c_2h]$ with $\\lambda >0$ , replacing $S^*M$ by $\\lambda S^*M$ .", "Lemma 3.11 Fix $c_1,c_2\\in \\mathbb {R}$ and let $\\operatorname{1\\hspace{-2.75pt}l}_{[1+c_1h,1+c_2h]}(h^2\\Delta )$ be defined by means of spectral theory.", "Assume that $A\\in \\Psi ^{\\operatorname{comp}}_\\rho (M)$ is compactly supported.", "Then $h^{(d-1)/2}\\Vert \\operatorname{1\\hspace{-2.75pt}l}_{[1+c_1h,1+c_2h]}(h^2\\Delta )A\\Vert _{\\operatorname{HS}}\\le C\\Vert \\sigma (A)\\Vert _{L^2(S^*M)}+\\mathcal {O}(h^{1-2\\rho }).$ Here $C$ is a constant independent of $A$ (if $\\operatorname{WF}_h(A)$ is contained in a fixed compact set), however the constant in $\\mathcal {O}(h^{1-2\\rho })$ depends on $A$ .", "We take the $L^2$ norm of $\\sigma (A)$ on the energy surface $S^*M$ with respect to the Liouville measure $\\mu _L$ .", "Moreover, if $\\operatorname{WF}_h(A)$ is microsupported, in the sense of Definition REF , in some $h$ -dependent family of sets $V(h)\\subset T^*M$ , then $h^{(d-1)/2}\\Vert \\operatorname{1\\hspace{-2.75pt}l}_{[1+c_1h,1+c_2h]}(h^2\\Delta )A\\Vert _{\\operatorname{HS}}\\le C \\mu _L(V(h)\\cap S^*M)^{1/2}+\\mathcal {O}(h^\\infty ).$ Here $\\mu _L(V(h)\\cap S^*M)$ denotes the volume of $V(h)\\cap S^*M$ with respect to the Liouville measure on $S^*M$ and the constant $C$ depends on a certain $S^{\\operatorname{comp}}_\\rho $ -seminorm of the full symbol of $A$ .", "Take a function $\\chi \\in S(\\mathbb {R})$ such that $\\hat{\\chi }$ is compactly supported in some interval $(-T,T)$ and $\\chi $ does not vanish on $[c_1,c_2]$ (for example, take nonzero $\\psi \\in C_0^\\infty (\\mathbb {R})$ with $\\psi \\ge 0$ , then $|\\hat{\\psi }|>0$ in an interval $[c_1\\varepsilon ,c_2\\varepsilon ]$ ; set $\\chi (x):=\\hat{\\psi }(\\varepsilon x)$ ).", "Then $\\operatorname{1\\hspace{-2.75pt}l}_{[1+c_1h,1+c_2h]}(h^2\\Delta )=Z\\chi ((h^2\\Delta -1)/h),\\ $ where $Z$ is a certain function of $h^2\\Delta $ and it is bounded on $L^2(M)$ uniformly in $h$ .", "It then suffices to estimate the Hilbert–Schmidt norm of $B = h^{(d-1)/2}\\chi ((h^2\\Delta -1)/h)A= (2\\pi )^{-1}h^{(d-1)/2}\\int _{-T}^T\\hat{\\chi }(t) e^{-it/h}U(2t)A\\,dt.$ Let $A_0\\in \\Psi ^{\\operatorname{comp}}_0(M)$ be compactly supported and equal to the identity microlocally near $\\operatorname{WF}_h(A)$ .", "By part 3 of Proposition REF , for each $t$ we have $U(2t)A_0=U_{2t}+R_{2t}$ , where $U_{2t}\\in I^{\\operatorname{comp}}(g^{2t})$ and $R_{2t}=\\mathcal {O}(h^{\\infty })_{L^2\\rightarrow L^2}$ .", "Then $(U(2t)-U_{2t})A=\\mathcal {O}(h^\\infty )_{\\operatorname{HS}}.$ Indeed, we can write the left-hand side of (REF ) as the sum of $R_{2t}A$ and $U(2t)(1-A_0)A$ ; it remains to note that $R_{2t}=\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ , $\\Vert A\\Vert _{\\operatorname{HS}}$ is polynomially bounded in $h$ , and $\\Vert (1-A_0)A\\Vert _{\\operatorname{HS}}=\\mathcal {O}(h^\\infty )$ .", "By (REF ), we can replace $U(2t)$ by $U_{2t}$ in the definition of $B$ .", "Now, $\\Vert B\\Vert _{\\operatorname{HS}}$ is equal to the $L^2(M\\times M)$ norm of the Schwartz kernel $K_B$ .", "Using the local normal form (REF ), we can write $K_B$ , up to an $\\mathcal {O}(h^\\infty )_{C_0^\\infty }$ remainder and an appropriate cutoff in the $m^{\\prime }$ variable, as a finite sum of expressions of the form (in a fixed coordinate system on $M$ ) $h^{-(d+1)/2}\\int _{-T}^T \\int _{\\mathbb {R}^d}e^{i(S(m,\\nu ^{\\prime };2t)-m^{\\prime }\\cdot \\nu ^{\\prime }-t)/h}b(m,\\nu ^{\\prime },t;h)\\,d\\nu ^{\\prime } dt.$ Here $S(m,\\nu ^{\\prime };2t)$ is a generating function for $g^{2t}$ and $b$ is a certain symbol in $S^{\\operatorname{comp}}_\\rho $ .", "Moreover, $b$ admits an asymptotic expansion in terms of the full symbol of $A$ , by Proposition REF .", "The fact that $S$ and $b$ can be choosen to depend smoothly on $t$ follows from the proof of part 3 of Proposition REF .", "By [62], $S$ satisfies the Hamilton–Jakobi equation $g^{2t}(m,\\nu )=(m^{\\prime },\\nu ^{\\prime })\\Longrightarrow \\partial _t(S(m,\\nu ^{\\prime };2t))=p(m,\\partial _m S(m,\\nu ^{\\prime };2t)).$ It follows that $\\Phi (m,m^{\\prime },\\nu ^{\\prime },t)=S(m,\\nu ^{\\prime };2t)-m^{\\prime }\\cdot \\nu ^{\\prime }-t$ is a nondegenerate phase function (with $m,m^{\\prime }$ as base variables and $\\nu ^{\\prime },t$ as the oscillatory variables) and generates the (immersed) Lagrangian $\\Lambda =\\lbrace (m,\\nu ;m^{\\prime },-\\nu ^{\\prime })\\mid p(m,\\nu )=1,\\ \\exists t\\in (-T,T):g^{2t}(m,\\nu )=(m^{\\prime },\\nu ^{\\prime })\\rbrace .$ Then (REF ) lies in $I^{\\operatorname{comp}}_\\rho (\\Lambda )$ .", "By the local normal form (REF ) of a Lagrangian distribution, we can write (REF ), up to an $\\mathcal {O}(h^\\infty )_{C_0^\\infty }$ remainder and an appropriate cutoff in the $(m,m^{\\prime })$ variables, as the sum of finitely many expressions of the form $h^{-d}\\int _{\\mathbb {R}^{2d}} e^{i(m\\cdot \\nu +m^{\\prime }\\cdot \\nu ^{\\prime }+F(\\nu ,\\nu ^{\\prime }))/h}\\tilde{b}(\\nu ,\\nu ^{\\prime };h)\\,d\\nu d\\nu ^{\\prime },$ where $F$ parametrizes some open subset of $\\Lambda $ by (REF ) and $\\tilde{b}$ is a symbol in $S^{\\operatorname{comp}}_\\rho $ .", "By Proposition REF and Proposition REF , we see that the symbol $\\tilde{b}$ has the following asymptotic expansion in terms of the full symbol $a$ of $A$ : $\\tilde{b}(\\nu ,\\nu ^{\\prime };h)=\\sum _{0\\le j<N} h^jL_ja(m^{\\prime },\\nu ^{\\prime };h)+\\mathcal {O}(h^{N(1-2\\rho )})_{S^{\\operatorname{comp}}_\\rho },$ where each $L_j$ is a differential operator of order $2j$ and $m,m^{\\prime }$ are given by the relation $(m,\\nu ,m^{\\prime },-\\nu ^{\\prime })\\in \\Lambda $ ; in particular, $(m^{\\prime },\\nu ^{\\prime })\\in S^*M$ .", "We now use Proposition REF to estimate the $L^2$ norm of (REF ); as $B$ is, modulo $\\mathcal {O}(h^\\infty )_{\\operatorname{HS}}$ , a sum of operators with Schwartz kernels of the form (REF ), this would give an estimate on the Hilbert–Schmidt norm of $B$ .", "For (REF ), we can write $\\tilde{b}(\\nu ,\\nu ^{\\prime };h)$ as a multiple of $a(m^{\\prime },\\nu ^{\\prime })$ plus an $\\mathcal {O}(h^{1-2\\rho })$ remainder and note that $(m^{\\prime },\\nu ^{\\prime })$ always lies in $S^*M$ .", "For (REF ), we use that $\\tilde{b}=\\mathcal {O}(h^\\infty )$ outside of the preimage of $V(h)$ under the map $(\\nu ,\\nu ^{\\prime })\\mapsto (m^{\\prime },\\nu ^{\\prime })$ , and that $\\sup |\\tilde{b}|$ can be estimated by a certain $S^{\\operatorname{comp}}_\\rho $ -seminorm of $a$ .", "Local traces of integrated Schrödinger propagators.", "We give the following version of the Schrödinger propagator trace formula when there are no contributions from closed geodesics: Lemma 3.12 Assume that $M$ is a $d$ -dimensional complete Riemannian manifold and $B_s$ is a family of compactly supported pseudodifferential operators in $\\Psi ^{\\operatorname{comp}}_\\rho (M)$ , smooth and compactly supported in $s\\in (-T_0,T_0)$ , where $T_0>0$ is fixed.", "Assume also that all $B_s$ are microsupported, in the sense of Definition REF , in some $h$ -dependent family of bounded sets $V(h)\\subset T^*M$ , and the following nonreturning condition holds: $(m,\\nu )\\in V(h),\\ |s|< T_0\\Longrightarrow d((m,\\nu ),g^s(m,\\nu ))\\ge C^{-1}|s|h^\\rho .$ Here $C$ is some constant and $d$ denotes some smooth distance function on $T^*M$ .", "Let $B_s=\\operatorname{Op}_h(b(s))+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }}$ for some family of symbols $b(s,m,\\nu )$ and some quantization procedure $\\operatorname{Op}_h$ .", "Then for each $N$ and each $\\lambda >0$ , we have the trace expansion ${\\begin{array}{c}(2\\pi h)^{d-1}\\int _{-T_0}^{T_0}e^{-i\\lambda ^2s/(2h)}\\operatorname{Tr}(U(s)B_s)\\,ds\\\\=\\sum _{0\\le j<N} h^j\\int _{S^*M} L_jb(0,m,\\lambda \\nu )\\,d\\mu _L(m,\\nu )+\\mathcal {O}(h^{N(1-2\\rho )})_{C^\\infty _\\lambda },\\end{array}}$ where $\\mu _L$ is the Liouville measure and each $L_j$ is a differential operator of order $2j$ on $T^*M_{(m,\\nu )}\\times (-T_0,T_0)_s$ , independent of $B_s$ and smooth in $\\lambda $ .", "In particular, $L_0=\\lambda ^{d-2}$ .", "As in the proof of Lemma REF , we can reduce to computing the trace of the operator with the Schwartz kernel (in some fixed local coordinates) $K(m,m^{\\prime })=(2\\pi h)^{-1}\\int _{-T_0}^{T_0}\\int _{\\mathbb {R}^d}e^{{i\\over h}(S(m,\\nu ^{\\prime };s)-m^{\\prime }\\cdot \\nu ^{\\prime }-\\lambda ^2s/2)}\\tilde{b}(m,\\nu ^{\\prime },s;h)\\,d\\nu ^{\\prime }ds,$ where $S(m,\\nu ^{\\prime };s)$ is a local generating function for $g^s$ in the sense of (REF ) and $\\tilde{b}(m,\\nu ^{\\prime },s;h)$ is a certain symbol in $S_\\rho $ having an asymptotic expansion in terms of the jet of $b_s$ at the point $(\\partial _{\\nu ^{\\prime }} S(m,\\nu ^{\\prime };s),\\nu ^{\\prime })$ .", "The trace of the corresponding operator is $\\int \\limits _M K(m,m)\\,dm=(2\\pi h)^{-1}\\int \\limits _{-T_0}^{T_0}\\int \\limits _{M\\times \\mathbb {R}^d} e^{{i\\over h}(S(m,\\nu ^{\\prime };s)-m\\cdot \\nu ^{\\prime }-\\lambda ^2s/2)}\\tilde{b}(m,\\nu ^{\\prime },s;h)\\,dmd\\nu ^{\\prime }ds.$ We now use the method of stationary phase.", "The stationary points of the phase are solutions to the equations $g^s(m,\\nu ^{\\prime })=(m,\\nu ^{\\prime })$ and $|\\nu ^{\\prime }|_g=\\lambda $ ; they occur at $s=0$ and may also occur for $\\lambda |s|\\ge r_i$ , where $r_i>0$ is the injectivity radius of $M$ .", "For $\\lambda |s|\\ge r_i/2$ , we see by (REF ) that the expression under the integral can be split into two pieces, on one of which the symbol is $\\mathcal {O}(h^\\infty )$ and on the other, the differential of the phase function has length at least $C^{-1}h^{\\rho }$ ; by repeated integration by parts, the latter integral is $\\mathcal {O}(h^\\infty )$ .", "It remains to evaluate the contribution of the stationary set $\\lbrace s=0\\rbrace \\cap \\lambda S^*M$ .", "The phase function is degenerate on these points; however, one can pass to polar coordinates $\\nu ^{\\prime }=r\\omega $ , with $|\\omega |_g=1$ and $r>0$ , and apply the method of stationary phase in the $(r,s)$ variables, resulting in the expansion (REF ).", "See for example the proofs of [45] or [46] for details of the computation." ], [ "General assumptions", "In this section, we list general geometric assumptions on the manifold $M$ and analytic assumptions on its Laplacian required for our results to hold.", "As noted in the introduction, they are satisfied in particular if outside of a compact set, $M$ is isometric to either the Euclidean space (studied in Section ) or an asymptotically hyperbolic space of constant curvature (studied in Section ).", "We also derive some direct consequences of the general assumptions, including averaged estimates on plane waves and the existence of limiting measures $\\mu _\\xi $ ." ], [ "Geometric assumptions", "In this subsection, we specify the geometry of the manifold $M$ at infinity.", "Let us introduce some notation and terminology first.", "On a complete Riemannian manifold $(M,g)$ we denote by $g^t$ the geodesic flow of the metric $g$ , considered as a map on the cotangent bundle $T^*M$ .", "Any smooth function $f$ on $M$ can be lifted to a function on $T^*M$ ; denote by $\\dot{f},\\ddot{f}\\in C^\\infty (T^*M)$ the derivatives of $f$ with respect to the geodesic flow: $\\dot{f}(m,\\nu ):=d_tf(g^t(m,\\nu ))|_{t=0},\\ \\ddot{f}(m,\\nu ):=d_t^2f(g^t(m,\\nu ))|_{t=0}.$ We denote by $S^*M$ the unit cotangent bundle $\\lbrace (m,\\nu )\\mid |\\nu |_g=1\\rbrace \\subset T^*M$ .", "A boundary defining function on a smooth compact manifold $\\overline{M}$ with boundary is a smooth function $x:\\overline{M}\\rightarrow [0,\\infty )$ such that $x>0$ on $M$ and $x$ vanishes to first order on $\\partial \\overline{M}$ .", "We make the following assumptions: $(M,g)$ is a complete Riemannian manifold of dimension $d=n+1$ .", "Moreover, there exists a compactification of $M$ , namely a compact manifold with boundary $\\overline{M}$ such that $M$ is diffeomorphic to the interior of $\\overline{M}$ .", "The boundary $\\partial \\overline{M}$ is called the boundary at infinity; There exists a boundary defining function $x$ on $M$ and a constant $\\varepsilon _0>0$ such that for any point $(m,\\nu )\\in S^*M$ , $\\text{if }x(m,\\nu )\\le \\varepsilon _0\\text{ and }\\dot{x}(m,\\nu )=0,\\text{ then }\\ddot{x}(m,\\nu )<0;$ For each $(m,\\nu )\\in S^*M$ such that $x(m)\\le \\varepsilon _0$ and $\\dot{x}(m,\\nu )\\le 0$ , the geodesic $g^t(m,\\nu )$ (projected onto the base space $M$ ) converges as $t\\rightarrow +\\infty $ , in the topology of $\\overline{M}$ , to some point $\\xi _{+\\infty }(m,\\nu )\\in \\partial \\overline{M}$ .", "The function $\\xi _{+\\infty }$ depends smoothly on $(m,\\nu )$ , and we extend it naturally (as the limit of the corresponding geodesic) to a smooth function on $S^*M\\setminus \\Gamma _-$ , with $\\Gamma _-$ given in Definition REF below; There exists an open set $U_\\infty \\subset M\\times \\partial \\overline{M}$ such that $\\overline{U}_\\infty $ contains a neighbourhood of $\\lbrace (\\xi ,\\xi )\\in \\overline{M}\\times \\partial \\overline{M}\\mid \\xi \\in \\partial \\overline{M}\\rbrace $ and a smooth real-valued function $\\phi (m,\\xi )=\\phi _\\xi (m)$ on $U_\\infty $ such that $|\\partial _m\\phi _\\xi (m)|_g=1$ everywhere and the function $\\tau (m,\\xi ):=(m,\\partial _m\\phi _\\xi (m))\\in S^*M,\\ (m,\\xi )\\in U_\\infty ,$ is a diffeomorphism from $U^+_\\infty $ onto $V^+_\\infty $ with inverse given by $\\tau ^{-1}(m,\\nu )=(m,\\xi _{+\\infty }(m,\\nu )),\\ (m,\\nu )\\in V^+_\\infty $ where the sets $U^+_\\infty $ and $V^+_\\infty $ are defined by ${\\begin{array}{c}U^+_\\infty :=\\lbrace (m,\\xi )\\in U_\\infty \\mid x(m)\\le \\varepsilon _0,\\,\\, \\dot{x}(\\tau (m,\\xi ))\\le 0\\rbrace ,\\\\V^+_\\infty :=\\lbrace (m,\\nu )\\in S^*M\\mid x(m)\\le \\varepsilon _0,\\, \\dot{x}(m,\\nu )\\le 0,\\,\\, (m,\\xi _{+\\infty }(m,\\nu ))\\in U_\\infty \\rbrace ;\\end{array}}$ if $(m,\\nu )\\in V^+_\\infty $ , then $g^t(m,\\nu )\\in V^+_\\infty $ for all $t\\ge 0$ ; if $\\xi \\in \\partial \\overline{M}$ and $m,m^{\\prime }\\in M$ are such that $(m,\\xi ),(m^{\\prime },\\xi )\\in U_\\infty ^+$ , then $\\partial _\\xi \\phi _\\xi (m)=\\partial _\\xi \\phi _\\xi (m^{\\prime })$ if and only if $\\tau (m,\\xi )$ and $\\tau (m^{\\prime },\\xi )$ lie on the same geodesic.", "Moreover, the matrix $\\partial _m \\partial _\\xi \\phi _\\xi (m)$ has rank $n$ .", "Figure: Illustrations for (G2) and (G3).", "Left: (G2) is not satisfied.", "Right: (G2) is satisfied.The point (m,ν)(m,\\nu ) does not escape directly in the forward direction,but the point (m ' ,ν ' )(m^{\\prime },\\nu ^{\\prime }) does, illustrating (G3).Escaping trajectories and the trapped set.. We now define the incoming/outgoing tails $\\Gamma _\\pm $ and the trapped set $K$ : Definition 4.1 Let $\\gamma (t)$ be a unit speed geodesic.", "We say that it escapes in the forward, respectively backward, direction, if $\\gamma (t)$ goes to infinity in $M$ as $t\\rightarrow +\\infty $ , respectively $t\\rightarrow -\\infty $ .", "If $\\gamma (t)$ does not escape in some direction, we call it trapped in this direction.", "Denote by $\\Gamma _+\\subset S^*M$ the union of all geodesics trapped in the backward direction, by $\\Gamma _-$ the union of all geodesics trapped in the forward direction, and put $K=\\Gamma _+\\cap \\Gamma _-$ ; we call $K$ the trapped set.", "An escaping geodesic could potentially spend a long time in the compact part of the manifold.", "It is helpful to consider geodesics that escape in a straightforward way (with the boundary defining function $x$ decreasing along them); they appeared in assumption (G3) for instance.", "Definition 4.2 We say that $(m,\\nu )\\in S^*M$ directly escapes in the forward, respectively backward, direction, if $x(m)\\le \\varepsilon _0$ and $\\dot{x}(m,\\nu )\\le 0$ , respectively $\\dot{x}(m,\\nu )\\ge 0$ .", "Here $\\varepsilon _0$ is the constant from (G2).", "Denote by $\\mathcal {DE}_+$ , respectively $\\mathcal {DE}_-$ , the set of all points directly escaping in the forward, respectively backward, direction.", "One can verify that $\\Gamma _\\pm $ are closed sets and the trapped set $K$ is compact (see [15]); in fact, since $S^*M\\cap \\lbrace x\\le \\varepsilon _0\\rbrace \\subset \\mathcal {DE}_+\\cup \\mathcal {DE}_-$ , we have $K\\subset \\lbrace x>\\varepsilon _0\\rbrace $ .", "For the example of $M=\\mathbb {R}^{n+1}$ discussed below, we have $\\Gamma _\\pm =\\emptyset $ .", "The point $(m,\\nu )$ lies in $\\mathcal {DE}_+$ if and only if $x(m)\\le \\varepsilon _0$ and $m\\cdot \\nu \\ge 0$ .", "Comments on the geometric assumptions.", "A basic example to have in mind for a manifold satisfying our assumptions is $M=\\mathbb {R}^{n+1}$ with the radial compactification $\\overline{M}$ being a closed ball and the boundary at infinity $\\partial \\overline{M}$ equal to the sphere $\\mathbb {S}^n$ .", "We will often use this example to illustrate the somewhat abstract assumptions of this section.", "(A more general version will be considered in Section .)", "An important corollary of the assumption (G2) is that for $\\varepsilon \\le \\varepsilon _0$ , the compact set $\\lbrace x\\ge \\varepsilon \\rbrace \\subset M$ is geodesically convex in the sense of (REF ).", "For the example of $M=\\mathbb {R}^{n+1}$ , we can take $x=(1+|m|^{-2})^{-1/2}$ , where $|m|$ is the Euclidean length of $m\\in \\mathbb {R}^d$ ; the corresponding sets $\\lbrace x\\ge \\varepsilon \\rbrace $ are balls centered at zero.", "It also follows from (G2) that for $(m,\\nu )\\in \\mathcal {DE}_+$ , the function $x(g^t(m,\\nu ))$ is decreasing for $t\\ge 0$ .", "One can show that $x(g^t(m,\\nu ))\\rightarrow 0$ as $t\\rightarrow +\\infty $ and thus $g^t(m,\\nu )$ escapes in the forward direction; we do not give a proof of this fact as it follows from the more restrictive assumption (G3).", "Also, if a geodesic $\\gamma (t)$ escapes in the forward direction, then for $t$ large enough we have $\\gamma (t)\\in \\mathcal {DE}_+$ .", "For $M=\\mathbb {R}^{n+1}$ , we have $\\xi _{+\\infty }(m,\\nu )=\\nu \\in \\mathbb {S}^n$ .", "Assumption (G4) means that for $m$ sufficiently close to the infinity, the covectors $\\nu $ such that $(m,\\nu )\\in \\mathcal {DE}_+$ are in one-to-one correspondence with the limit points $\\xi _{+\\infty }(m,\\nu )$ , and the inverse correspondence can be described by a phase function.", "It follows in particular from (G4) that for a fixed $\\xi \\in \\partial \\overline{M}$ , the set of directly escaping points $(m,\\nu )$ such that $\\xi _{+\\infty }(m,\\nu )=\\xi $ and $(m,\\xi )\\in U_\\infty $ is the intersection of $\\mathcal {DE}_+$ with the Lagrangian $\\Lambda _\\xi :=\\lbrace (m,\\partial _m\\phi _\\xi (m))\\mid (m,\\xi )\\in U_\\infty \\rbrace .$ In the model case $M=\\mathbb {R}^{n+1}$ we can put for any $R>0$ , $U_\\infty =\\lbrace (m,\\xi )\\mid |m|> R\\rbrace $ , and $\\phi _\\xi (m)=m\\cdot \\xi $ , so that $\\tau $ is the canonical map from $\\mathbb {R}^{n+1}\\times \\mathbb {S}^n$ to $S^* \\mathbb {R}^{n+1}$ .", "Then $U^+_\\infty =\\lbrace (m,\\xi )\\mid |m|>R,\\ m\\cdot \\xi \\ge 0\\rbrace $ and $V^+_\\infty =\\lbrace (m,\\nu )\\mid |m|\\ge R,\\ m\\cdot \\nu \\ge 0\\rbrace $ ; the difference is that $U^+_\\infty $ is considered as a subset of $\\mathbb {R}^{n+1}\\times \\mathbb {S}^n$ , while $V^+_\\infty $ is considered as a subset of $S^* \\mathbb {R}^{n+1}$ .", "The condition (G5) can also be viewed as a condition on $U_\\infty ^+$ , saying that for any $(m,\\xi )\\in U_\\infty ^+$ , the whole geodesic `segment' $\\gamma (m,\\xi )$ relating $m$ and the point $\\xi \\in \\partial \\overline{M}$ is such that $\\gamma (m,\\xi )\\times \\lbrace \\xi \\rbrace $ is contained in $U_\\infty ^+$ .", "The condition (G6) is required in Proposition REF .", "To explain it, note that under the assumption (G4), if $(m,\\xi )\\in U^+_\\infty $ and $(m(t),\\nu (t))=g^t(\\tau (m,\\xi ))$ , then $\\partial _t\\phi _\\xi (m(t))|_{t=0}=\\partial _m\\phi _\\xi (m)\\cdot \\partial _t m(t)|_{t=0}=g(\\partial _m\\phi _\\xi (m),\\partial _m\\phi _\\xi (m))=1.$ Therefore, $\\partial _\\xi \\phi _\\xi (m)$ is constant on the geodesic passing through $\\tau (m,\\xi )$ ." ], [ "Analytic assumptions", "In this subsection, we formulate the analytic assumptions on plane waves.", "We will prove that they are satisfied in the Euclidean near infinity setting (in Section REF ) and in the hyperbolic near infinity setting (in Section REF ).", "Let $M$ be as in the previous subsection, $\\Delta $ be the (nonnegative definite) Laplace–Beltrami operator on $M$ , and $h>0$ be the semiclassical parameter.", "We make the following assumptions: There exists $c_0\\ge 0$ (equal to 0 for the Euclidean and to $n^2/4$ for the hyperbolic case), such that for each $\\lambda >0$ , $h>0$ and $\\xi \\in \\partial \\overline{M}$ , there exists a function, called distorted plane wave, $E_h(\\lambda ,\\xi ;m)=E(\\lambda /h,\\xi ;m)$ , smooth in all variables and solving on $M$ the differential equation (REF ) in $m$ : $(h^2\\Delta -c_0h^2-\\lambda ^2)E_h(\\lambda ,\\xi ;\\cdot )=0.$ Here $\\xi $ gives the direction of the plane wave, while $\\lambda $ corresponds to its semiclassical energy; for each $0< \\lambda _1\\le \\lambda _2$ , the Schwartz kernel of the semiclassical spectral projector $\\Pi _{[\\lambda _1,\\lambda _2]}:=\\operatorname{1\\hspace{-2.75pt}l}_{[\\lambda _1^2+c_0h^2,\\lambda _2^2+c_0h^2]}(h^2\\Delta )$ can be written in the form $\\Pi _{[\\lambda _1,\\lambda _2]}(m,m^{\\prime })=(2\\pi h)^{-n-1}\\int _{\\lambda _1}^{\\lambda _2}\\lambda ^n f_\\Pi (\\lambda /h) \\int _{\\partial \\overline{M}}E_h(\\lambda ,\\xi ;m)\\overline{E_h(\\lambda ,\\xi ;m^{\\prime })}\\,d\\xi d\\lambda .$ Here integration in $\\xi $ is carried with respect to a certain given volume form $d\\xi $ on $\\partial \\overline{M}$ and $f_\\Pi (z)>0$ is a smooth function of $z$ such that $|\\partial _z^kf_\\Pi (z)|\\le C_k\\langle z\\rangle ^{-k}$ for each $k$ and $f_\\Pi (z)\\rightarrow 1$ as $z\\rightarrow \\infty $ .", "We now assume that plane waves admit the decomposition $E_h(\\lambda ,\\xi ;m)=\\chi _0(m;\\xi )E^0_h(\\lambda ,\\xi ;m)+E^1_h(\\lambda ,\\xi ;m),$ where $\\chi _0,E^0_h,E_h^1$ are respectively a cutoff function, an explicit `outgoing' part of the wave, and the `incoming' part, satisfying more precisely the following properties: $\\chi _0(m;\\xi )$ is a function smooth in $m\\in \\overline{M}$ and $\\xi \\in \\partial \\overline{M}$ , supported inside the set $U_\\infty $ from (G4) and $\\chi _0(m,\\xi )=1$ for $m$ sufficiently close to $\\xi $ ; $E^0_h(\\lambda ,\\xi ;m)$ is a smooth function of $\\lambda /h \\in \\mathbb {R}^*$ and $(m,\\xi )\\in U_\\infty $ , of the form $E^0_h(\\lambda ,\\xi ;m)=e^{{i\\lambda \\over h}\\phi _\\xi (m)}b^0(\\lambda ,\\xi ,m;h),$ where $U_\\infty $ and $\\phi _\\xi $ are defined in (G4) and $b^0(\\lambda ,\\xi ,m;h)=b^0(\\lambda /h,\\xi ,m)$ is a classical symbol in $h$ for $(m,\\xi )\\in U_\\infty $ , $\\lambda \\in \\mathbb {R}^*$ ; that is, $b^0(\\lambda ,\\xi ,m;h)$ is smooth in all variables, including $h$ , up to $h=0$ .", "The limit $b^0(\\lambda ,\\xi ,m;0)=\\lim _{h\\rightarrow 0}b^0(\\lambda /h,\\xi ,m)$ for $\\lambda >0$ is independent of $\\lambda $ ; for $\\lambda $ in a fixed compact subset of $(0,\\infty )$ and $\\varepsilon _0$ defined in (G2), the function $\\widetilde{E}^1_h(\\lambda ,\\xi ;m):={E^1_h(\\lambda ,\\xi ;m)\\over 1+\\Vert E_h(\\lambda ,\\xi ;m)\\Vert _{L^2(\\lbrace x\\ge \\varepsilon _0\\rbrace )}}$ is $h$ -tempered in the sense of (REF ); for $\\lambda $ in a fixed compact subset of $(0,\\infty )$ , each $\\xi \\in \\partial \\overline{M}$ , and each $(m,\\lambda \\nu )\\in \\operatorname{WF}_h(\\widetilde{E}^1_h(\\lambda ,\\xi ))$ , we have $(m,\\nu )\\in S^*M$ and either the geodesic $\\gamma (t)=g^t(m,\\nu )$ does not escape in the forward direction (i.e.", "$(m,\\nu )\\in \\Gamma _-$ ) or there exists $t\\ge 0$ such that $\\gamma (t)$ lies in the set $W_\\xi :=\\lbrace (m,\\partial _m\\phi _\\xi (m))\\mid m\\in \\operatorname{supp}(\\partial _m \\chi _0)\\rbrace .$ The constants in the corresponding estimates (in the definition of the wave front set of a distribution given in Section REF ) are uniform in $\\lambda $ and $\\xi $ ; there exists $\\varepsilon _1\\in (0,\\varepsilon _0)$ such that for $(m,\\nu )\\in S^*M$ directly escaping in the forward direction and $x(m)\\le \\varepsilon _1$ , the point $(m,\\xi _{+\\infty }(m,\\nu ))$ lies in the set $U_\\infty $ defined in (G4) and $\\chi _0=1$ near this point; Let $\\tau :U^+_\\infty \\rightarrow V^+_\\infty $ be the diffeomorphism from (G4).", "Then its Jacobian with respect to the volume measure $\\operatorname{dvol}(m)d\\xi $ on $U^+_\\infty $ and the Liouville measure on $V^+_\\infty $ , is equal to $|b^0(1,\\xi ,m;0)|^2$ , with $b^0$ defined in (A4).", "For example, for $M=\\mathbb {R}^{n+1}$ we put $c_0=0$ , $E_h(\\lambda ,\\xi ;m)=e^{i\\lambda \\xi \\cdot m/h}$ and use the standard volume form on the sphere $\\partial \\overline{M}=\\mathbb {S}^{n}$ .", "The equation (REF ) then follows from the Fourier inversion formula.", "Let us informally explain how the decomposition (REF ) is constructed and provide a justification for assumptions (A3)–(A6), putting for simplicity $\\lambda =1$ .", "First of all, (A4) implies that for any $\\chi \\in C_0^\\infty (M)$ , $\\chi \\chi _0 E^0_h$ , as a function of $m$ , is a Lagrangian distribution associated to the Lagrangian $\\Lambda _\\xi $ from (REF ).", "In fact, in the cases considered in the present paper, $E^0_h$ solves on its domain the equation (REF ); however, we do not make this assumption here, as in more complicated cases (such as asymptotically hyperbolic manifolds of variable curvature) $E^0_h$ might only be an approximate solution to (REF ) in a certain sense.", "If we assume that $E^0_h$ solves (REF ) on its domain, then the function $F_h(\\lambda ,\\xi ;m)=(h^2\\Delta -\\lambda ^2-c_0h^2)(\\chi _0(m)E^0_h(\\lambda ,\\xi ;m))$ is equal to $[h^2\\Delta ,\\chi _0]E^0_h$ .", "Since $E^0_h$ is a Lagrangian distribution associated to $\\Lambda _\\xi $ , the wavefront set of $F_h$ is contained in $W_\\xi $ .", "We will now take $E^1_h=-R_h(\\lambda )F_h$ , where $R_h(\\lambda )$ is the incoming scattering resolvent, a certain right inverse of $h^2\\Delta -\\lambda ^2-c_0h^2$ .", "Moreover, in our cases $R_h(\\lambda )$ will be microlocally incoming in the weak sense: if we multiply $F_h$ by a (possibly small) constant to make $R_h(\\lambda )F_h$ bounded polynomially in $h$ , then each point in the wavefront set of $R_h(\\lambda )F_h$ , when propagated forward by the geodesic flow, will either converge to the trapped set or pass through $\\operatorname{WF}_h(F_h)$ .", "Thus, the assumption (A6) should be viewed as a direct consequence of the fact that the scattering resolvent is microlocally incoming and of propagation of singularities.", "The assumption (A7) looks less natural, but will play an essential role in our proofs, in Propositions REF and REF .", "It holds for both Euclidean and hyperbolic infinities, but for different reasons.", "For the hyperbolic infinity, $\\chi _0(\\cdot ;\\xi )$ is equal to 1 in a small neighborhood of $\\xi $ and one can see that for $(m,\\nu )$ directly escaping in the forward direction and converging to $\\xi $ , the distance from $m$ to $\\xi $ in $\\overline{M}$ is $\\mathcal {O}(x(m))$ .", "This is not true in the Euclidean case; however, in that case $\\chi _0$ is equal to 1 outside of a compact subset of $M$ (that is, near the whole boundary $\\partial \\overline{M}$ , not just near $\\xi $ ).", "The assumption (A8) is required to relate the natural measure arising from the function $E_h^0$ to the Liouville measure.", "If $E_h$ were equal to $E^0_h$ , then this assumption would simply follow by taking the trace in (REF ) with a compactly supported pseudodifferential operator and a smooth cutoff function in $\\lambda $ ." ], [ "Limiting measures", "We now define the family of limiting measures $\\mu _\\xi $ .", "These measures result from propagating the natural measure arising from the `outgoing' part $E^0_h$ of the plane wave, which is supported on the Lagrangian $\\Lambda _\\xi $ from (REF ), backwards along the geodesic flow.", "In contrast with [11], where the exponential decay of the measure along the flow ensured its convergence, our measures will only be defined for almost every $\\xi $ .", "We first define the measure $\\tilde{\\mu }_\\xi $ on $S^*M$ , corresponding to $E^0_h$ , as follows: for each compactly supported continuous function $a$ on $S^*M$ , put $\\int _{S^*M} a\\,d\\tilde{\\mu }_\\xi =\\int _{(m,\\xi )\\in U_\\infty ^+}|b^0(1,\\xi ,m;0)|^2 a(\\tau (m,\\xi ))\\,\\operatorname{dvol}(m).$ The support of $\\tilde{\\mu }_\\xi $ is contained in the Lagrangian $\\Lambda _\\xi $ from (REF ) and the integral (REF ) depends continuously on $\\xi $ .", "We see from (A8) that for any continuous function $f$ on $\\partial \\overline{M}$ , $\\int _{\\partial \\overline{M}} f(\\xi )\\int _{S^*M} a(m,\\nu )\\,d\\tilde{\\mu }_\\xi (m,\\nu ) d\\xi =\\int _{V_\\infty ^+} f(\\xi _{+\\infty }(m,\\nu ))a(m,\\nu )\\,d\\mu _L(m,\\nu ).$ We now want to define the measure $\\mu _\\xi $ by $\\int _{S^*M} a\\,d\\mu _\\xi =\\lim _{t\\rightarrow +\\infty }\\int _{S^*M} a\\circ g^{-t}\\,d\\tilde{\\mu }_\\xi ,$ valid for all compactly supported continuous functions $a$ .", "To show that the limit exists for almost every $\\xi $ (chosen independently of $a$ ) and for every $a$ , we will use monotonicity.", "By (REF ), (G5), and using the invariance of the function $\\xi _{+\\infty }$ and the Liouville measure $\\mu _L$ under the geodesic flow, we see that if $a$ and $f$ are nonnegative, then $\\int _{\\partial \\overline{M}} f(\\xi )\\int _{S^*M}(a\\circ g^{-t})\\,d\\tilde{\\mu }_\\xi d\\xi =\\int _{g^{-t}(V^+_\\infty )} f(\\xi _{+\\infty }(m,\\nu ))a(m,\\nu )\\,d\\mu _L(m,\\nu )$ is increasing with $t$ .", "Therefore, for each $\\xi $ the integral $I_{a,t}(\\xi )=\\int _{S^*M} (a\\circ g^{-t})\\,d\\tilde{\\mu }_\\xi $ is increasing in $t$ for any nonnegative $a$ .", "Moreover, the integral of $I_{a,t}(\\xi )$ in $\\xi $ is bounded by a $t$ -independent constant, namely by the integral of $a$ by the Liouville measure.", "Taking $a$ to be an approximation of the characteristic function of each member of a countable family of compact sets exhausting $S^*M$ , and using the monotone convergence theorem, we see that there exists a measure zero set $\\mathcal {X}\\subset \\partial \\overline{M}$ such that for $\\xi \\notin \\mathcal {X}$ , we have for any compactly supported continuous function $a$ , $\\lim _{t\\rightarrow +\\infty }\\int _{S^*M} (a\\circ g^{-t})\\,d\\tilde{\\mu }_\\xi <\\infty .$ This limit is a continuous functional on the space of continuous compactly supported functions on $S^*M$ ; therefore, there exists unique Borel measure $\\mu _\\xi $ such that (REF ) holds.", "Moreover, we see that the limit (REF ) is uniform in $a$ , as soon as we fix a compact set containing $\\operatorname{supp}a$ and impose a bound on $\\sup _{S^*M}|a|$ .", "One also sees immediately (REF ), namely that for compactly supported continuous $a$ , $\\operatorname{supp}a\\cap \\overline{\\xi _{+\\infty }^{-1}(\\xi )}=\\emptyset \\Longrightarrow \\int _{S^*M}a\\,d\\mu _\\xi =0,$ as $\\int _{S^*M}(a\\circ g^{-t})\\,d\\tilde{\\mu }_\\xi =0$ for all $t$ .", "We can integrate the measure $\\mu _\\xi $ in $\\xi $ , getting back the Liouville measure: Proposition 4.3 For each $f\\in C^\\infty (\\partial \\overline{M})$ and each $a\\in C_0^\\infty (S^*M)$ we have $\\int _{\\partial \\overline{M}}f(\\xi )\\int _{S^*M} a(m,\\nu )\\,d\\mu _\\xi (m,\\nu )d\\xi =\\int _{S^*M\\setminus \\Gamma _-} f(\\xi _{+\\infty }(m,\\nu ))a(m,\\nu )\\,d\\mu _L(m,\\nu ).$ In particular, if $\\mu _L(\\Gamma _-)=0$ (which will always be the case in our theorems, see (REF )), then $\\int \\mu _\\xi \\,d\\xi $ is the Liouville measure.", "The left-hand side can be written as $\\lim _{t\\rightarrow +\\infty } \\int _{g^{-t}(V^+_\\infty )} f(\\xi _{+\\infty }(m,\\nu ))a(m,\\nu )\\,d\\mu _L(m,\\nu ).$ It remains to use the dominated convergence theorem; indeed, the function under the integral is bounded and compactly supported, we have $g^{-t_1}(V^+_\\infty )\\subset g^{-t_2}(V^+_\\infty )$ for $t_1<t_2$ , and the union of $g^{-t}(V^+_\\infty )$ over all $t\\in \\mathbb {R}$ is exactly $S^*M\\setminus \\Gamma _-$ , as for every geodesic $\\gamma (t)$ escaping in the forward direction and for $t$ large enough, the point $\\gamma (t)$ is directly escaping in the forward direction and $(\\gamma (t),\\xi _{+\\infty }(\\gamma (t)))\\in U_\\infty $ .", "Finally, the following lemma will be useful to relate our measure $\\mu _\\xi $ to the one obtained from $E^0_h$ in the proofs of Theorems REF and REF : Lemma 4.4 Let $\\xi \\notin \\mathcal {X}$ , so that $\\mu _\\xi $ is well-defined.", "Let $a$ be a compactly supported continuous function on $S^*M$ .", "1.", "$\\mu _\\xi $ is invariant under the geodesic flow: for each $t\\in \\mathbb {R}$ , $\\int _{S^*M} a\\circ g^t\\,d\\mu _\\xi =\\int _{S^*M} a\\,d\\mu _\\xi .$ 2.", "If $\\operatorname{supp}a\\subset \\mathcal {DE}_+\\cap \\lbrace x\\le \\varepsilon _1\\rbrace $ , where $\\mathcal {DE}_+$ is given by Definition REF and $\\varepsilon _1$ is defined in (A7), then $\\int _{(m,\\xi )\\in U_\\infty } |b^0(1,\\xi ,m;0)\\chi _0(m;\\xi )|^2a(m,\\partial _m\\phi _\\xi (m))\\,\\operatorname{dvol}(m)=\\int _{S^*M} a\\,d\\mu _\\xi .$ 1.", "Follows immediately from the definition (REF ).", "2.", "First of all, note that for $m$ in the support of the function $a(m,\\partial _m\\phi _\\xi (m))$ , we have $(m,\\xi )\\in U^+_\\infty $ and $\\chi _0(m;\\xi )=1$ by (A7); therefore, the left-hand side of (REF ) becomes the integral of $a$ over the measure $\\tilde{\\mu }_\\xi $ defined in (REF ).", "By (REF ), it is enough to show that for $t\\ge 0$ , $\\int _{S^*M} a\\circ g^{-t}\\,d\\tilde{\\mu }_\\xi =\\int _{S^*M} a\\,d\\tilde{\\mu }_\\xi .$ For that, it is enough to show that for each $f\\in C_0^\\infty (\\partial \\overline{M})$ , $\\int _{\\partial M}f(\\xi )\\int _{S^*M} a\\circ g^{-t}\\,d\\tilde{\\mu }_\\xi =\\int _{\\partial M}\\int _{S^*M} a\\,d\\tilde{\\mu }_\\xi .$ Using (REF ), we rewrite this as $\\int _{g^{-t}(V^+_\\infty )} f(\\xi _{+\\infty }) a\\,d\\mu _L=\\int _{V^+_\\infty } f(\\xi _{+\\infty }) a\\,d\\mu _L.$ This is true as $\\operatorname{supp}a\\subset V^+_\\infty \\subset g^{-t}(V^+_\\infty )$ ." ], [ "Averaged estimates on plane waves", "One of the principal tools of the present paper are microlocal estimates on the plane waves $E_h(\\lambda ,\\xi )$ on average in $\\lambda ,\\xi $ , where $\\lambda $ takes values in a size $h$ interval.", "They are direct consequences of (REF ) and the Hilbert–Schmidt norm estimate (REF ).", "More precisely, restricting to the case $\\lambda =1+\\mathcal {O}(h)$ for simplicity, we have the following Proposition 4.5 Let $\\chi \\in C_0^\\infty (M)$ .", "Then: 1.", "$\\chi \\Pi _{[1,1+h]}$ is a Hilbert–Schmidt operator and there exists a global constant $C$ such that for each bounded operator $A:L^2(M)\\rightarrow L^2(M)$ , we have $h^{-1}\\Vert A\\chi (m) E_h(\\lambda ,\\xi ;m)\\Vert _{L^2_{m,\\xi ,\\lambda }(M\\times \\partial \\overline{M}\\times [1,1+h])}^2\\le C h^{n}\\Vert A\\chi \\Pi _{[1,1+h]}\\Vert _{\\operatorname{HS}}^2.$ 2.", "The functions $\\chi E_h$ are bounded in $L^2$ on average in the following sense: there exists a constant $C(\\chi )$ such that for any $h$ , $h^{-1}\\Vert \\chi (m) E_h(\\lambda ,\\xi ;m)\\Vert ^2_{L^2_{m,\\xi ,\\lambda }(M\\times \\partial \\overline{M}\\times [1,1+h])}\\le C(\\chi ).$ The $h^{-1}$ prefactor in both cases is due to the fact that we are integrating over an interval of size $h$ in $\\lambda $ .", "1.", "It follows immediately from (REF ) that $h^{-1}\\int _1^{1+h} f_\\Pi (\\lambda /h)\\lambda ^n \\int _{\\partial \\overline{M}}(\\chi E_h(\\lambda ,\\xi ))\\otimes (\\chi E_h(\\lambda ,\\xi ))\\,d\\xi d\\lambda =(2\\pi )^{n+1}h^n\\chi \\Pi _{[1,1+h]}\\bar{\\chi }.$ Here $\\otimes $ denotes the Hilbert tensor product, defined in (REF ).", "The integral on the left-hand side converges in the trace class norm, as the Schwartz kernels of the integrated operators are smooth and compactly supported.", "Therefore, $\\chi \\Pi _{[1,1+h]}\\bar{\\chi }$ is trace class.", "Since $\\chi \\Pi _{[1,1+h]}\\bar{\\chi }=(\\chi \\Pi _{[1,1+h]})(\\chi \\Pi _{[1,1+h]})^*,$ we see that $\\chi \\Pi _{[1,1+h]}$ is a Hilbert–Schmidt operator.", "Now, multiplying both sides of (REF ) by $A$ on the left and $A^*$ on the right and taking the trace, we get ${\\begin{array}{c}h^{-1}\\Vert \\lambda ^{n/2}f_\\Pi (\\lambda /h)^{1/2}A\\chi (m)E_h(\\lambda ,\\xi ;m)\\Vert _{L^2_{m,\\xi ,\\lambda }(M\\times \\partial \\overline{M}\\times [1,1+h])}^2\\\\=(2\\pi )^{n+1}h^n\\operatorname{Tr}((A\\chi \\Pi _{[1,1+h]})(A\\chi \\Pi _{[1,1+h]})^*)\\\\=(2\\pi )^{n+1}h^n\\Vert A\\chi \\Pi _{[1,1+h]}\\Vert _{\\operatorname{HS}}^2.\\end{array}}$ 2.", "We would like to use Lemma REF to estimate $\\Vert \\chi \\Pi _{[1,1+h]}\\Vert _{\\operatorname{HS}}$ (we can put $\\chi $ on the other side of the projector in (REF ) by taking the adjoint), however this is not directly possible as $\\chi $ is not compactly microlocalized.", "We thus use that $E_h$ solve the equation (REF ), writing by the elliptic parametrix construction (same as for the proof of Proposition REF ) $\\chi =B+Q_\\lambda (h^2\\Delta -\\lambda ^2-c_0h^2)+R_\\lambda $ for $\\lambda \\in [1,1+h]$ , where $B\\in \\Psi ^{\\operatorname{comp}}(M)$ , $Q_\\lambda \\in \\Psi ^{-2}(M)$ , and $R_\\lambda \\in h^\\infty \\Psi ^{-\\infty }(M)$ are compactly supported and $B$ is independent of $\\lambda $ and equal to $\\chi $ microlocally near $S^*M$ .", "We can also assume that $Q_\\lambda $ and $R_\\lambda $ are smooth in $\\lambda $ .", "Now, we substitute (REF ) into the left-hand side of (REF ) and use the triangle inequality.", "By (REF ), the term featuring $B$ is bounded by a constant times $h^n\\Vert B\\Pi _{[1,1+h]}\\Vert _{\\operatorname{HS}}^2$ , which is bounded uniformly in $h$ by Lemma REF .", "The term featuring $Q_\\lambda $ is zero by (REF ).", "Finally, we show that the term featuring $R_\\lambda $ is $\\mathcal {O}(h^\\infty )$ .", "This does not follow immediately from (REF ), as the operator $R_\\lambda $ depends on $\\lambda $ .", "We use the following variant of (REF ): for $\\tilde{\\lambda }\\in [1,1+h]$ , $h^{-1}\\Vert \\lambda ^{n/2}f_\\Pi (\\lambda /h)^{1/2}R_{\\tilde{\\lambda }}E_h(\\lambda )\\Vert ^2_{L^2_{m,\\xi ,\\lambda }(M\\times \\partial \\overline{M}\\times [1,\\tilde{\\lambda }])}=(2\\pi )^{n+1}h^n\\Vert R_{\\tilde{\\lambda }}\\Pi _{[1,\\tilde{\\lambda }]}\\Vert _{\\operatorname{HS}}^2.$ Differentiating in $\\tilde{\\lambda }$ , we get ${\\begin{array}{c}(2\\pi )^{n+1}h^n\\partial _{\\tilde{\\lambda }}\\Vert R_{\\tilde{\\lambda }}\\Pi _{[1,\\tilde{\\lambda }]}\\Vert _{\\operatorname{HS}}^2=h^{-1}\\Vert \\tilde{\\lambda }^{n/2}f_\\Pi (\\tilde{\\lambda }/h)^{1/2}R_{\\tilde{\\lambda }}E_h(\\tilde{\\lambda })\\Vert ^2_{L^2(m,\\xi )(M\\times \\partial M)}+\\\\2h^{-1}\\operatorname{Re}\\langle \\lambda ^{n/2}f_\\Pi (\\lambda /h)^{1/2}(\\partial _{\\tilde{\\lambda }} R_{\\tilde{\\lambda }})E_h(\\lambda ),\\lambda ^{n/2}f_\\Pi (\\lambda /h)^{1/2}R_{\\tilde{\\lambda }}E_h(\\lambda )\\rangle _{L^2_{m,\\xi ,\\lambda }(M\\times \\partial M\\times [1,\\tilde{\\lambda }])}.\\end{array}}$ We now integrate in $\\tilde{\\lambda }$ from 1 to $1+h$ .", "The integral of the left-hand side is bounded by a constant times $h^n\\Vert R_{1+h}\\Vert _{\\operatorname{HS}}^2=\\mathcal {O}(h^\\infty )$ .", "The integral of the first term on the right-hand side is the quantity we are estimating.", "Finally, the second term on the right-hand side is bounded by a constant times $h^n|\\operatorname{Tr}((\\partial _{\\tilde{\\lambda }}R_{\\tilde{\\lambda }})\\Pi _{[1,1+h]}R_{\\tilde{\\lambda }}^*)|$ , which is $\\mathcal {O}(h^\\infty )$ uniformly in $\\tilde{\\lambda }$ , as the Hilbert–Schmidt norms of both $R_{\\tilde{\\lambda }}$ and $\\partial _{\\tilde{\\lambda }}R_{\\tilde{\\lambda }}$ are $\\mathcal {O}(h^\\infty )$ ." ], [ "Proof of Theorem ", "In this section, we prove the convergence Theorem REF under the following assumption: $\\mu _L(K)=0,$ where $\\mu _L$ denotes the Liouville measure on $S^*M$ and $K$ is the trapped set.", "First of all, note that (REF ) implies $\\mu _L(\\Gamma _\\pm )=0.$ Indeed, fix $\\varepsilon \\in (0,\\varepsilon _0)$ , where $\\varepsilon _0$ is defined in (G2), and take the set $\\Gamma _+^\\varepsilon =\\Gamma _+\\cap \\lbrace x\\ge \\varepsilon \\rbrace $ .", "For $(m,\\nu )\\in \\Gamma _+\\cap \\lbrace x=\\varepsilon \\rbrace $ , we have $\\dot{x}(m,\\nu )<0$ ; indeed, otherwise $(m,\\nu )$ directly escapes in the backward direction and thus cannot lie in $\\Gamma _+$ .", "It follows that $g^{-t}(\\Gamma _+^\\varepsilon )\\subset \\Gamma _+^\\varepsilon $ for $t\\ge 0$ .", "Since $\\Gamma _+^\\varepsilon $ is bounded, and $\\mu _L$ is invariant under the geodesic flow, we have $\\mu _L(\\Gamma _+^\\varepsilon )=\\lim _{t\\rightarrow +\\infty }\\mu _L(g^{-t}(\\Gamma _+^\\varepsilon ))=\\mu _L\\bigg (\\bigcap _{t\\ge 0}g^{-t}(\\Gamma _+^\\varepsilon )\\bigg )=\\mu _L(K)=0.$ Letting $\\varepsilon \\rightarrow 0$ , we get (REF ).", "We next note that the averaged $L^2$ bound (REF ) on $E_h$ on compact sets, together with (REF ) and the elliptic Proposition REF , give the following Proposition 5.1 Assume that $A\\in \\Psi ^0(M)$ is compactly supported and $\\operatorname{WF}_h(A)\\cap S^*M=\\emptyset $ .", "Then $h^{-1}\\Vert \\langle A E_h(\\lambda ,\\xi ),E_h(\\lambda ,\\xi )\\rangle \\Vert _{L^1_{\\xi ,\\lambda }(\\partial \\overline{M}\\times [1,1+h])}=\\mathcal {O}(h^\\infty ).$ Therefore, it is enough to prove (REF ) for a compactly supported $A\\in \\Psi ^{\\operatorname{comp}}(M)$ microlocalized in an arbitrarily small neighborhood of $S^*M$ .", "Take $t>0$ ; we will calculate limits of the form $\\lim _{t\\rightarrow +\\infty }\\lim _{h\\rightarrow 0}$ , thus $\\mathcal {O}_t(h^\\infty )$ expressions (that is, expressions that are $\\mathcal {O}(h^\\infty )$ with the constants depending on $t$ ) will be negligible.", "Take $\\chi \\in C_0^\\infty (M)$ indepedendent of $t$ and such that $A=\\chi A\\chi $ .", "We first use that $E_h$ is a generalized eigenfunction of the Laplacian (REF ) and apply Lemma REF : for each $\\lambda \\in [1,1+h]$ and each $\\xi \\in \\partial M$ , $\\chi E_h=\\chi e^{-it(\\lambda ^2+c_0h^2)/2h}U(t)\\chi _t E_h+\\mathcal {O}_t(h^\\infty \\Vert E_h\\Vert _{L^2(K_t)})_{L^2}.$ Here $U(t)=e^{ith\\Delta /2}$ is the semiclassical Schrödinger propagator and $\\chi _t\\in C_0^\\infty (M)$ is supported in the interior of the compact set $K_t\\subset M$ and satisfies $d_g(\\operatorname{supp}\\chi ,\\operatorname{supp}(1-\\chi _t))>t$ .", "We also assume that $|\\chi _t|\\le 1$ everywhere and $K_t$ contains $\\lbrace x\\ge \\varepsilon _0\\rbrace $ , where $\\varepsilon _0$ is defined in (G2).", "By Proposition REF , we can write $U(-t)AU(t)=A^{-t}+\\mathcal {O}_t(h^\\infty )_{L^2\\rightarrow L^2}$ , where $A^{-t}\\in \\Psi ^{\\operatorname{comp}}$ is compactly supported.", "Then $\\langle AE_h,E_h\\rangle =\\langle A^{-t}\\chi _t E_h,\\chi _t E_h\\rangle +\\mathcal {O}_t(h^\\infty )\\Vert E_h\\Vert _{L^2(K_t)}^2.$ We will now write $A^{-t}=A^{-t}_0+A^{-t}_1,\\ A^{-t}_0:=A^{-t}\\varphi ,\\ A^{-t}_1:=A^{-t}(1-\\varphi ),$ where the $L^2$ norm of the principal symbol of $A_0^{-t}$ will decay with $t$ and the operator $A^{-t}_1$ will be negligible on $E^1_h$ .", "The function $\\varphi \\in C_0^\\infty (M)$ is taken independent of $t$ and such that $\\operatorname{supp}\\chi \\subset \\lbrace x>\\varepsilon _\\chi \\rbrace $ for some $\\varepsilon _\\chi $ and $\\varphi =1$ near $\\lbrace x\\ge \\varepsilon _\\chi \\rbrace $ .", "We also require that $\\varphi =1$ near $\\lbrace x\\ge \\varepsilon _1\\rbrace $ , where $\\varepsilon _1$ comes from the assumption (A7).", "We first show that the terms in (REF ) featuring both $A^{-t}_1$ and $E_h^1$ are $\\mathcal {O}(h^\\infty )$ .", "For that, we need to show that the trajectories in $\\operatorname{WF}_h(A^{-t}_1)\\subset \\operatorname{supp}(1-\\varphi )\\cap g^t(\\operatorname{supp}\\chi )$ satisfy the geometric property shown on Figure REF : Lemma 5.2 Let $t\\ge 0$ .", "Assume that $(m,\\nu )\\in S^*M$ satisfies $m\\in \\operatorname{supp}(1-\\varphi )$ , but $g^{-t}(m,\\nu )\\in \\operatorname{supp}\\chi $ .", "Then: $(m,\\nu )$ directly escapes in the forward direction, in the sense of Definition REF ; for each $s\\ge 0$ , $g^s(m,\\nu )$ does not lie in the set $W_\\xi $ defined in (REF ), for any $\\xi \\in \\partial \\overline{M}$ .", "(1) We have $x(m)<\\varepsilon _1\\le \\varepsilon _0$ ; therefore, if $(m,\\nu )$ does not directly escape in the forward direction, then it directly escapes in the backward direction; this would imply that $x(g^{-t}(m,\\nu ))$ is decreasing in $t\\ge 0$ , which is impossible as $x(m)<\\varepsilon _\\chi <x(g^{-t}(m,\\nu ))$ .", "(2) The point $g^s(m,\\nu )$ directly escapes in the forward direction and $x(g^s(m,\\nu ))<\\varepsilon _1$ .", "If $g^s(m,\\nu )\\in W_\\xi $ , then by (G4), $\\xi =\\xi _{+\\infty }(m,\\nu )$ , but this is impossible as $\\chi _0=1$ near $(g^s(m,\\nu ),\\xi _{+\\infty }(m,\\nu ))$ by (A7).", "Combining Lemma REF with the microlocal information we have on $E^1_h$ , we get Proposition 5.3 If $E_h=\\chi _0E^0_h+E^1_h$ is the decomposition (REF ), then for each $t\\ge 0$ , ${\\begin{array}{c}\\langle AE_h,E_h\\rangle =\\langle A^{-t}_1\\chi _t \\chi _0E^0_h,\\chi _t \\chi _0E^0_h\\rangle +\\langle A^{-t}_0 \\chi _t E_h,\\chi _t E_h\\rangle +\\mathcal {O}_t(h^\\infty (1+\\Vert E_h\\Vert _{L^2(K_t)}^2)).\\end{array}}$ where $A_0^{-t},A_1^{-t}$ are defined in (REF ).", "By (REF ), it is enough to show that $\\langle A^{-t}_1\\chi _t E_h,\\chi _t E_h\\rangle -\\langle A^{-t}_1\\chi _t\\chi _0 E_h^0,\\chi _t \\chi _0E_h^0\\rangle =\\mathcal {O}_t(h^\\infty (1+\\Vert E_h\\Vert _{L^2(K_t)}^2)).$ Given that $\\Vert \\chi _0 E_h^0\\Vert _{L^2(K_t)}=\\mathcal {O}(1)$ , it suffices to prove $\\Vert B\\chi _t E^1_h\\Vert _{L^2}=\\mathcal {O}_t(h^\\infty (1+\\Vert E_h\\Vert _{L^2(K_t)})),$ where $B$ is equal to either $A^{-t}_1$ or its adjoint.", "This in turn follows from $\\Vert B\\chi _t \\widetilde{E}^1_h\\Vert _{L^2}=\\mathcal {O}_t(h^\\infty ),$ with $\\widetilde{E}^1_h$ defined in (REF ).", "Take $(m,\\nu )\\in \\operatorname{WF}_h(B\\chi _t\\widetilde{E}^1_h)\\subset S^*M$ .", "Then by Proposition REF , $(m,\\nu )\\in \\operatorname{WF}_h(B)\\subset \\operatorname{WF}_h(A^{-t})\\cap \\operatorname{supp}(1-\\varphi )\\subset g^t(\\operatorname{WF}_h(A))\\cap \\operatorname{supp}(1-\\varphi ).$ Since $\\operatorname{WF}_h(A)\\subset \\operatorname{supp}\\chi $ , we see that $m\\in \\operatorname{supp}(1-\\varphi )$ and $g^{-t}(m,\\nu )\\in \\operatorname{supp}\\chi $ ; therefore, by Lemma REF , the geodesic $g^s(m,\\nu )$ escapes in the forward direction and does not pass through $W_\\xi $ for $s\\ge 0$ .", "But then by (A6) the point $(m,\\nu )$ cannot lie in $\\operatorname{WF}_h(\\widetilde{E}^1_h)$ , a contradiction.", "We showed that the wavefront set of $B\\chi _t\\widetilde{E}^1_h$ is empty, which implies (REF ).", "We now use the averaged estimate (REF ) and the Hilbert–Schmidt norm estimates from Section REF , to estimate the second term on the right-hand side of (REF ): Proposition 5.4 There exists a constant $C$ independent of $t$ such that $h^{-1}\\Vert \\langle A^{-t}_0\\chi _t E_h,\\chi _t E_h\\rangle \\Vert _{L^1_{\\xi ,\\lambda }(\\partial \\overline{M}\\times [1,1+h])}\\le C\\Vert (\\sigma (A)\\circ g^{-t})\\varphi \\Vert _{L^2(S^*M)}+\\mathcal {O}_t(h).$ Here $\\Vert a\\Vert _{L^2(S^*M)}$ is the $L^2$ norm of the restriction of $a$ to $S^*M$ with respect to the Liouville measure.", "Take a real-valued function $\\varphi _1\\in C_0^\\infty (M)$ independent of $t$ such that $\\varphi _1=1$ near $\\operatorname{supp}\\varphi $ .", "Then the left-hand side of (REF ) is bounded by $h^{-1}\\Vert \\langle A^{-t}_0 \\chi _t E_h,\\varphi _1 \\chi _t E_h\\rangle \\Vert _{L^1_{\\xi ,\\lambda }}+h^{-1}\\Vert \\langle (1-\\varphi _1)A^{-t}_0 \\chi _t E_h, \\chi _t E_h\\rangle \\Vert _{L^1_{\\xi ,\\lambda }},$ where the $L^1$ , and later $L^2$ , norms in $\\xi ,\\lambda $ are taken over $\\partial \\overline{M}\\times [1,1+h]$ .", "The second term here is $\\mathcal {O}_t(h^\\infty )$ by the bound (REF ) and since $(1-\\varphi _1)A^{-t}_0=\\mathcal {O}_t(h^\\infty )_{L^2\\rightarrow L^2}$ is compactly supported.", "The first term can be estimated by applying the Cauchy–Schwarz inequality first in $m$ and then in $(\\lambda ,\\xi )$ : ${\\begin{array}{c}h^{-1}\\Vert \\langle A^{-t}_0 \\chi _t E_h,\\varphi _1 \\chi _t E_h\\rangle \\Vert _{L^1_{\\xi ,\\lambda }}\\le h^{-1}\\Vert \\,\\Vert A^{-t}_0 \\chi _t E_h\\Vert _{L^2(M)}\\cdot \\Vert \\varphi _1 \\chi _t E_h\\Vert _{L^2(M)}\\Vert _{L^1_{\\xi ,\\lambda }}\\\\\\le h^{-1/2}\\Vert A^{-t}_0\\chi _t E_h\\Vert _{L^2_{m,\\xi ,\\lambda }}\\cdot h^{-1/2} \\Vert \\varphi _1 \\chi _t E_h\\Vert _{L^2_{m,\\xi ,\\lambda }}.\\end{array}}$ Now, $h^{-1/2}\\Vert \\varphi _1 \\chi _t E_h\\Vert _{L^2_{m,\\xi ,\\lambda }}$ is bounded (independently of $t$ ) uniformly in $h$ by (REF ).", "As for $h^{-1/2}\\Vert A^{-t}_0\\chi _t E_h\\Vert _{L^2_{m,\\xi ,\\lambda }}$ , we can estimate it using (REF ) by a constant times $h^{n/2}\\Vert A^{-t}_0\\chi _t\\Pi _{[1,1+h]}\\Vert _{\\operatorname{HS}}.$ Note that the operator $A^{-t}_0\\chi _t\\in \\Psi ^{\\operatorname{comp}}$ is compactly supported and it is compactly microlocalized independently of $t$ .", "It then remains to apply (REF ) (to the adjoint of our operator); by Proposition REF , the principal symbol of $A^{-t}_0\\chi _t$ is given by $(\\sigma (A)\\circ g^{-t})\\varphi $ .", "We now use the dynamical assumption that $\\mu _L(K)=0$ .", "The function $(\\sigma (A)\\circ g^{-t})\\varphi $ is supported in a $t$ -independent compact set and bounded uniformly in $t$ .", "Moreover, it converges to zero pointwise on $S^*M\\setminus \\Gamma _+$ as $t\\rightarrow +\\infty $ .", "Therefore, by (REF ) and the dominated convergence theorem we have $(\\sigma (A)\\circ g^{-t})\\varphi \\rightarrow 0$ in $L^2(S^*M)$ , as $t\\rightarrow +\\infty $ .", "It then follows from (REF ) together with the bound (REF ) and from (REF ) that $\\lim _{t\\rightarrow +\\infty }\\limsup _{h\\rightarrow 0}h^{-1}\\Vert \\langle AE_h,E_h\\rangle -\\langle A^{-t}_1\\chi _t\\chi _0E_h^0,\\chi _t\\chi _0E_h^0\\rangle \\Vert _{L^1_{\\xi ,\\lambda }(\\partial \\overline{M}\\times [1,1+h])}=0.$ To prove Theorem REF , it now remains to show that $\\lim _{t\\rightarrow +\\infty }\\limsup _{h\\rightarrow 0}h^{-1}\\bigg \\Vert \\langle A^{-t}_1\\chi _t\\chi _0E_h^0,\\chi _t\\chi _0E_h^0\\rangle -\\int _{S^*M}\\sigma (A)\\,d\\mu _\\xi \\bigg \\Vert _{L^1_{\\xi }(\\partial \\overline{M})}=0$ uniformly in $\\lambda =1+\\mathcal {O}(h)$ .", "We first note that by (REF ) the function $\\chi _t\\chi _0E^0_h(\\lambda ,\\xi ;m)=e^{{i\\lambda \\over h}\\phi _\\xi (m)}\\chi _t(m)\\chi _0(m,\\xi )b^0(1,\\xi ,m;0)+\\mathcal {O}_t(h)_{L^2}$ is a compactly supported Lagrangian distribution associated to the Lagrangian $\\Lambda _\\xi $ from (REF ).", "Therefore, by Proposition REF , we find $A^{-t}_1\\chi _t\\chi _0E^0_h(\\lambda ,\\xi )=e^{{i\\lambda \\over h}\\phi _\\xi }\\chi _t\\chi _0b^0(1,\\xi ,m;0)\\sigma (A^{-t}_1)(m,\\partial _m\\phi _\\xi (m))+\\mathcal {O}_t(h)_{L^2}.$ Therefore, ${\\begin{array}{c}\\langle A^{-t}_1\\chi _t\\chi _0 E_h^0,\\chi _t\\chi _0E_h^0\\rangle =\\int _M \\sigma (A^{-t}_1)(m,\\partial _m\\phi _\\xi (m))|\\chi _t\\chi _0b^0(1,\\xi ,m;0)|^2\\,\\operatorname{dvol}(m)+\\mathcal {O}_t(h).\\end{array}}$ Now, by Proposition REF , $\\sigma (A^{-t}_1)=(\\sigma (A)\\circ g^{-t})(1-\\varphi )$ .", "By Lemma REF , this function is supported in $\\mathcal {DE}_+\\cap \\lbrace x<\\varepsilon _1\\rbrace $ , with $\\mathcal {DE}_+$ from Definition REF .", "Also, $\\chi _t=1$ near $\\operatorname{supp}\\sigma (A^{-t}_1)$ .", "Then by part 2 of Lemma REF , $\\langle A^{-t}_1\\chi _t\\chi _0 E_h^0,\\chi _t\\chi _0 E_h^0\\rangle =\\int _{S^*M} (\\sigma (A)\\circ g^{-t})(1-\\varphi )\\,d\\mu _\\xi +\\mathcal {O}_t(h).$ Therefore, (REF ) reduces to $\\lim _{t\\rightarrow +\\infty }\\bigg \\Vert \\int _{S^*M}(\\sigma (A)\\circ g^{-t})(1-\\varphi )\\,d\\mu _\\xi -\\int _{S^*M}\\sigma (A)\\,d\\mu _\\xi \\bigg \\Vert _{L^1_\\xi (\\partial \\overline{M})}=0.$ By part 1 of Lemma REF and (REF ), we write the norm on the left-hand side of (REF ) as $\\bigg \\Vert \\int _{S^*M}\\sigma (A)(\\varphi \\circ g^t)\\,d\\mu _\\xi \\bigg \\Vert _{L^1_\\xi (\\partial \\overline{M})}\\le \\int _{S^*M}|\\sigma (A)(\\varphi \\circ g^t)|\\,d\\mu _L.$ The expression under the integral on the right-hand side is bounded and compactly supported uniformly in $t$ and converges to zero pointwise on $S^*M\\setminus \\Gamma _-$ ; by (REF ) and the dominated convergence theorem, we get (REF ).", "This finishes the proof of Theorem REF .", "The nontrapped case.", "We briefly discuss the situation when $\\operatorname{WF}_h(A)\\cap \\Gamma _-=\\emptyset $ .", "In this case, for $t$ large enough (depending on $A$ ), for any $(m,\\nu )\\in \\operatorname{WF}_h(A)$ we have $g^t(m,\\nu )\\notin \\operatorname{supp}\\varphi $ and thus $A_0^{-t}=\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}.$ Then by (REF ) and the bound (REF ), $\\langle AE_h,E_h\\rangle =\\langle A^{-t}_1\\chi _t\\chi _0E^0_h,\\chi _t\\chi _0E^0_h\\rangle +\\mathcal {O}(h^\\infty )_{L^1_{\\xi ,\\lambda }(\\partial \\overline{M}\\times [1,1+h])}.$ The quantity $\\langle A^{-t}_1\\chi _t\\chi _0 E^0_h,\\chi _t\\chi _0E^0_h\\rangle $ is calculated in (REF ) up to $\\mathcal {O}(h)$ .", "However, since $E^0_h$ is a Lagrangian distribution, one can get by Proposition REF a full expansion of this quantity in powers of $h$ ; this yields $\\langle AE_h(\\lambda ,\\xi ),E_h(\\lambda ,\\xi )\\rangle =\\sum _{0\\le j<N}h^j\\int _{S^*M}L_ja\\,d\\mu _\\xi +\\mathcal {O}(h^{N+1})_{L^1_{\\xi ,\\lambda }(\\partial \\overline{M}\\times [1,1+h])},$ where $A=\\operatorname{Op}_h(a)$ for some symbol $a$ and some quantization procedure $\\operatorname{Op}_h$ and each $L_j$ is a differential operator of order $2j$ on $T^*M$ , with $L_0=1$ ." ], [ "Estimates on the remainder", "In this subsection, we prove (REF ) and establish an approximation fact (Proposition REF ) used in the proofs of () and Theorem REF .", "Classical escape rate and Ehrenfest time.", "Let $K_0\\subset M$ be a compact geodesically convex set (in the sense of (REF )) containing a neighborhood of the projection of the trapped set $K$ onto $M$ .", "As in (REF ), define the set $\\mathcal {T}(t)=\\lbrace (m,\\nu )\\in S^*M\\mid m\\in K_0,\\ g^t(m,\\nu )\\in K_0\\rbrace .$ The choice of $K_0$ does not matter here: if $K^{\\prime }_0\\subset M$ is another set with same properties and $\\mathcal {T}^{\\prime }(t)$ is defined using $K^{\\prime }_0$ in place of $K_0$ , then there exists a constant $T_0>0$ such that for each $T\\ge T_0$ and $t\\ge 0$ , $g^T(\\mathcal {T}^{\\prime }(t+2T))\\subset \\mathcal {T}(t).$ Indeed, assume that (REF ) were false.", "Then there exists sequences $T_j\\rightarrow +\\infty $ , $t_j\\ge 0$ , and $(m_j,\\nu _j)\\in S^*M$ such that $g^{-T_j}(m_j,\\nu _j)$ and $g^{t_j+T_j}(m_j,\\nu _j)$ both lie in $K^{\\prime }_0$ , but for each $j$ , either (1) $(m_j,\\nu _j)\\notin K_0$ or (2) $g^{t_j}(m_j,\\nu _j)\\notin K_0$ .", "We may assume that case (1) holds for all $j$ ; case (2) is handled similarly, reversing the direction of the flow and taking $g^{t_j}(m_j,\\nu _j)$ in place of $(m_j,\\nu _j)$ .", "Take $\\varepsilon >0$ such that $K^{\\prime }_0\\subset \\lbrace x\\ge \\varepsilon \\rbrace $ ; since $\\lbrace x\\ge \\varepsilon \\rbrace $ is geodesically convex (in the sense of (REF )) for $\\varepsilon $ small enough, we have $(m_j,\\nu _j)\\in \\lbrace x\\ge \\varepsilon \\rbrace $ .", "Passing to a subsequence, we can assume that $(m_j,\\nu _j)\\rightarrow (m,\\nu )\\in S^*M$ as $j\\rightarrow +\\infty $ .", "Now, since $g^{-T_j}(m_j,\\nu _j)\\in K^{\\prime }_0$ and $T_j\\rightarrow +\\infty $ , we have $(m,\\nu )\\in \\Gamma _+$ (indeed, otherwise there would exist $s>0$ such that $g^{-s}(m,\\nu )\\in \\lbrace x<\\varepsilon \\rbrace $ and this would also hold in a neighborhood of $(m,\\nu )$ ).", "Similarly, since $g^{t_j+T_j}(m_j,\\nu _j)\\in K^{\\prime }_0$ and $t_j+T_j\\rightarrow +\\infty $ , we have $(m,\\nu )\\in \\Gamma _-$ .", "It follows that $(m,\\nu )\\in K$ , which is impossible, as each $(m_j,\\nu _j)$ does not lie in $K_0$ , which contains a neighborhood of $K$ .", "By changing $\\Lambda _0$ slightly and using (REF ), we see that the choice of $K_0$ does not matter for the validity of (REF ) and (); more precisely, if $\\Lambda _0>\\Lambda ^{\\prime }_0$ , then $r^{\\prime }(h,\\Lambda ^{\\prime }_0)\\le Cr(h,\\Lambda _0)$ , where $r^{\\prime }$ is defined by (REF ) using $\\mathcal {T}^{\\prime }$ in place of $\\mathcal {T}$ .", "Also, the maximal expansion rate $\\Lambda _{\\max }$ defined in (REF ) does not depend on the choice of $K_0$ .", "We now choose a geodesically convex $K_0$ in the sense of (REF ) such that its interior contains the supports of all cutoff functions and compactly supported operators used in the argument below.", "We will rely on Proposition REF (with $U$ equal to the interior of $K_0$ ); we let $\\Lambda _0>\\Lambda _{\\max }$ and fix $\\varepsilon _e>0$ and $\\Lambda ^{\\prime }_0$ such that $\\Lambda _0>\\Lambda ^{\\prime }_0>(1+2\\varepsilon _e)\\Lambda _{\\max }$ .", "Define the Ehrenfest time $t_e:=\\log (1/h)/(2\\Lambda _0).$ Then when propagating an operator in $\\Psi ^{\\operatorname{comp}}$ microlocalized inside $\\mathcal {E}_{\\varepsilon _e}:=\\lbrace 1-\\varepsilon _e\\le |\\nu |_g\\le 1+\\varepsilon _e\\rbrace $ with cutoffs supported inside $K_0$ , as in Proposition REF , for time $t=lt_0\\in [-t_e,t_e]$ , we get a mildly exotic pseudodifferential operator in $\\Psi ^{\\operatorname{comp}}_{\\rho _e}$ , where $\\rho _e:=t_e\\Lambda ^{\\prime }_0/\\log (1/h)=\\Lambda ^{\\prime }_0/(2\\Lambda _0)<1/2.$ First decomposition of $\\langle AE_h,E_h\\rangle $.", "By Proposition REF , we may assume that $A\\in \\Psi ^{\\operatorname{comp}}(M)$ is compactly supported and microlocalized inside the set $\\mathcal {E}_{\\varepsilon _e}$ defined in (REF ).", "We first establish the following decomposition similar to (REF ): ${\\begin{array}{c}\\langle AE_h,E_h\\rangle =e^{il\\beta } \\langle A(\\varphi U(t_0))^l\\varphi E_h,E_h\\rangle \\\\+\\sum _{j=1}^l e^{ij\\beta }\\langle A(\\varphi U(t_0))^j(1-\\varphi )\\varphi _{t_0}\\chi _0E^0_h,E_h\\rangle +\\mathcal {O}(h^\\infty \\mathcal {N}(E_h)^2),\\end{array}}$ uniformly in $\\xi \\in \\partial \\overline{M}$ and $\\lambda \\in [1,1+h]$ ; comparing with (REF ), the first term in the right hand side of (REF ) corresponds to the $A_0^{-t}$ term in (REF ), the sum over $j$ corresponds to the $A_1^{-t}$ term in (REF ).", "Here $l=\\mathcal {O}(\\log (1/h))$ is a nonnegative integer and $t_0>0$ and $\\varphi ,\\varphi _{t_0}\\in C_0^\\infty (M)$ , specified below, are independent of $j$ .", "The quantity $\\mathcal {N}(E_h)$ , defined in (REF ), is related to the $L^2$ norm of $E_h$ on a certain compact set, and is bounded on average by (REF ).", "The real-valued parameter $\\beta $ is equal to $\\beta =-t_0(\\lambda ^2+c_0h^2)/(2h)$ and will not play a big role in our argument.", "To show (REF ), we start by considering the functions $\\varphi ,\\varphi _1,\\varphi _2\\in C_0^\\infty (M)$ such that: $0\\le \\varphi ,\\varphi _1,\\varphi _2\\le 1$ everywhere, $\\varphi =1$ near $\\operatorname{supp}\\varphi _2$ and $\\varphi _1=1$ near $\\operatorname{supp}\\varphi $ , and $\\varphi _2=1$ both near the support of $A$ and near the set $\\lbrace x\\ge \\varepsilon _1\\rbrace $ , with $\\varepsilon _1$ defined in (A7).", "The proof of (REF ) only uses the function $\\varphi $ , however the other two functions will be required for the more precise decomposition (REF ) below.", "We now have the following analogue of Lemma REF : Lemma 5.5 There exists $t_0\\ge 0$ such that if $(m,\\nu )\\in S^*M$ satisfies $m\\in \\operatorname{supp}(1-\\varphi _2)\\text{ and }g^{-t}(m,\\nu )\\in \\operatorname{supp}\\varphi _1\\text{ for some }t\\ge t_0,$ then: $(m,\\nu )$ directly escapes in the forward direction; for each $s\\ge 0$ , $g^s(m,\\nu )$ does not lie in the set $W_\\xi $ defined in (REF ) for any $\\xi \\in \\partial \\overline{M}$ ; and for each $s\\ge t_0$ , $g^s(m,\\nu )\\notin \\operatorname{supp}\\varphi _1$ .", "Figure: An illustration of Lemma .", "The functionsϕ,ϕ 1 ,ϕ 2 \\varphi ,\\varphi _1,\\varphi _2 are supported to the left of the correspondingdashed lines; the right side of the figure represents infinity.", "(1) Let $\\operatorname{supp}\\varphi _1\\subset \\lbrace x\\ge \\varepsilon _\\varphi \\rbrace $ .", "The set $\\mathcal {DE}_-\\cap \\lbrace x\\ge \\varepsilon _\\varphi \\rbrace $ , where $\\mathcal {DE}_-$ is specified in Definition REF , is compact; therefore, there exists $t_0>0$ such that for $t\\ge t_0$ and $(m,\\nu )\\in \\mathcal {DE}_-\\cap \\lbrace x\\ge \\varepsilon _\\varphi \\rbrace $ , we have $g^{-t}(m,\\nu )\\notin \\operatorname{supp}\\varphi _1$ .", "Now, assume that $(m,\\nu )$ satisfies (REF ), but it does not directly escape in the forward direction.", "Since $(m,\\nu )\\in \\operatorname{supp}(1-\\varphi _2)$ , we have $x(m)\\le \\varepsilon _0$ ; therefore, $(m,\\nu )\\in \\mathcal {DE}_-$ .", "Then $x(m)\\ge x(g^{-t}(m,\\nu ))\\ge \\varepsilon _\\varphi $ ; therefore, $(m,\\nu )\\in \\mathcal {DE}_-\\cap \\lbrace x\\ge \\varepsilon _\\varphi \\rbrace $ , a contradiction with the fact that $g^{-t}(m,\\nu )\\in \\operatorname{supp}\\varphi _1$ and $t\\ge t_0$ .", "(2) This is proved exactly as part 2 of Lemma REF .", "(3) It is enough to use part (1), take $t_0$ large enough, and use that the set $\\mathcal {DE}_+\\cap \\lbrace x\\ge \\varepsilon _\\varphi \\rbrace $ is compact.", "Take $t_0$ from Lemma REF .", "Let $\\varphi _{t_0}\\in C_0^\\infty (M)$ be real-valued and satisfy $d_g(\\operatorname{supp}\\varphi _1,\\operatorname{supp}(1-\\varphi _{t_0}))>t_0$ .", "Take a compact set $K_{t_0}\\subset M$ whose interior contains $\\operatorname{supp}\\varphi _{t_0}$ .", "Put $\\mathcal {N}(E_h):=1+\\Vert E_h\\Vert _{L^2(K_{t_0})};$ this quantity depends on $\\lambda $ and $\\xi $ and we know by (REF ) that $h^{-1}\\Vert \\mathcal {N}(E_h)\\Vert ^2_{L^2_{\\xi ,\\lambda }(\\partial \\overline{M}\\times [1,1+h])}=\\mathcal {O}(1).$ By (REF ) and Lemma REF , we have similarly to (REF ), $\\varphi E_h=e^{i\\beta }\\varphi U(t_0)\\varphi _{t_0}E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}.$ Here $\\beta $ is given by (REF ).", "Iterating (REF ) by writing $\\varphi _{t_0}=\\varphi +(1-\\varphi )\\varphi _{t_0}$ , we get for $l=\\mathcal {O}(\\log (1/h))$ (or even for $l$ polynomially bounded in $h$ ) $\\varphi E_h=e^{il\\beta } (\\varphi U(t_0))^l \\varphi E_h+\\sum _{j=1}^l e^{ij\\beta } (\\varphi U(t_0))^j (1-\\varphi )\\varphi _{t_0}E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2},$ uniformly in $\\xi \\in \\partial \\overline{M}$ and $\\lambda \\in [1,1+h]$ .", "Same is true if $\\varphi $ is replaced by any function $\\varphi ^{\\prime }\\in C_0^\\infty (M)$ such that $d_g(\\operatorname{supp}\\varphi ^{\\prime },\\operatorname{supp}(1-\\varphi _{t_0}))>t_0$ .", "One can also replace $U(t_0)$ by $U(-t_0)$ .", "We now use our knowledge of the wavefront set of $\\widetilde{E}^1_h$ to prove the following analogue of Proposition REF : Proposition 5.6 If $E_h=\\chi _0E^0_h+E^1_h$ is the decomposition (REF ), then $\\Vert \\varphi U(t_0)(1-\\varphi )\\varphi _{t_0}E^1_h\\Vert _{L^2}=\\mathcal {O}(h^\\infty \\mathcal {N}(E_h)),$ uniformly in $\\xi \\in \\partial \\overline{M}$ and $\\lambda \\in [1,1+h]$ .", "Same is true if we replace each instance of $\\varphi $ by any function in the set $\\lbrace \\varphi ,\\varphi _1,\\varphi _2\\rbrace $ .", "Recalling the definition (REF ) of $\\widetilde{E}^1_h$ , we see that (REF ) follows from $\\Vert \\varphi U(t_0)(1-\\varphi )\\varphi _{t_0}\\widetilde{E}^1_h\\Vert _{L^2}=\\mathcal {O}(h^\\infty ).$ We now make the following observation: a point $(m,\\nu )\\in S^*M$ in the wavefront set of $\\widetilde{E}^1_h$ will make an $\\mathcal {O}(h^\\infty )$ contribution to (REF ) unless $m\\in \\operatorname{supp}(1-\\varphi )$ , but $g^{-t_0}(m,\\nu )\\in \\operatorname{supp}\\varphi $ ; however, by (A6) and Lemma REF , in this case $(m,\\nu )\\notin \\operatorname{WF}_h(\\widetilde{E}^1_h)$ .", "To make this argument rigorous, we can write (bearing in mind that $\\widetilde{E}^1_h$ is polynomially bounded) $\\varphi _{t_0}\\widetilde{E}^1_h=B \\widetilde{E}^1_h+\\mathcal {O}(h^\\infty )_{L^2},$ where $B\\in \\Psi ^{\\operatorname{comp}}$ is compactly supported and such that $(m,\\nu )\\in \\operatorname{WF}_h(B)\\cap \\operatorname{supp}(1-\\varphi ) \\Longrightarrow g^{-t_0}(m,\\nu )\\notin \\operatorname{supp}\\varphi .$ Then the operator $\\varphi U(t_0)(1-\\varphi )B$ is $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ by part 2 of Proposition REF , which proves (REF ).", "Using (REF ), we can replace $E_h$ by $\\chi _0 E^0_h$ in each term of the sum (REF ): ${\\begin{array}{c}\\varphi E_h=e^{il\\beta }(\\varphi U(t_0))^l\\varphi E_h+\\sum _{j=1}^l e^{ij\\beta }(\\varphi U(t_0))^j(1-\\varphi )\\varphi _{t_0}\\chi _0E^0_h\\\\+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}.\\end{array}}$ Applying the operator $A=\\varphi A\\varphi $ , we get (REF ).", "Properties of propagators up to Ehrenfest time.", "We will now establish certain properties of the cut off and iterated propagators up to the Ehrenfest time $t_e$ defined in (REF ), or, in certain cases, up to twice the Ehrenfest time.", "The need for these properties arises mostly because of the cutoffs present in the argument.", "Define the Ehrenfest index $l_e := \\lfloor t_e/t_0\\rfloor + 1\\sim \\log (1/h).$ Lemma 5.7 Assume that $\\varphi ^{\\prime },\\varphi ^{\\prime \\prime }\\in C_0^\\infty (M)$ satisfy $|\\varphi ^{\\prime }|,|\\varphi ^{\\prime \\prime }|\\le 1$ everywhere.", "Let $B\\in \\Psi ^{\\operatorname{comp}}$ be compactly supported and microlocalized inside the set $\\mathcal {E}_{\\varepsilon _e}$ defined in (REF ).", "Then: 1.", "If $\\varphi ^{\\prime \\prime }=1$ near $\\operatorname{supp}\\varphi ^{\\prime }$ , then for $0\\le j\\le l_e$ , $(\\varphi ^{\\prime } U(\\pm t_0))^jBU(\\mp jt_0)=(\\varphi ^{\\prime } U(\\pm t_0))^jB(U(\\mp t_0)\\varphi ^{\\prime \\prime })^j+\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2},\\\\(U(\\pm t_0)\\varphi ^{\\prime })^j BU(\\mp jt_0)=(U(\\pm t_0)\\varphi ^{\\prime })^j B(\\varphi ^{\\prime \\prime }U(\\mp t_0))^j+\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}.$ 2.", "If $B_1,B_2\\in \\Psi ^{\\operatorname{comp}}$ satisfy same conditions as $B$ and moreover $\\operatorname{WF}_h(B_1)\\cap \\operatorname{WF}_h(B_2)=\\emptyset $ , then for $0\\le j\\le 2l_e$ (that is, up to twice the Ehrenfest time) $B_1(\\varphi ^{\\prime } U(\\pm t_0))^j B(U(\\mp t_0)\\varphi ^{\\prime \\prime })^j B_2=\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}.$ Same is true if we replace $\\varphi ^{\\prime } U(\\pm t_0)$ by $U(\\pm t_0)\\varphi ^{\\prime }$ and/or replace $U(\\mp t_0)\\varphi ^{\\prime \\prime }$ by $\\varphi ^{\\prime \\prime } U(\\mp t_0)$ .", "3.", "If $\\varphi ^{\\prime \\prime }=1$ near $\\operatorname{supp}\\varphi ^{\\prime }$ and both $\\varphi ^{\\prime \\prime }$ and $B$ are supported at distance more than $t_0$ from $\\operatorname{supp}(1-\\varphi _{t_0})$ , then for $0\\le j\\le l_e$ ($\\beta $ is defined in (REF )) $e^{\\pm ij\\beta }(\\varphi ^{\\prime } U(\\pm t_0))^j B E_h=(\\varphi ^{\\prime } U(\\pm t_0))^j B (U(\\mp t_0)\\varphi ^{\\prime \\prime })^j E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2},\\\\e^{\\pm ij\\beta }(U(\\pm t_0)\\varphi ^{\\prime })^j B E_h=(U(\\pm t_0)\\varphi ^{\\prime })^j B (\\varphi ^{\\prime \\prime } U(\\mp t_0))^j\\varphi _{t_0} E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}.$ We will repeatedly use Propositions REF and REF and omit the $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ remainders present there.", "1.", "We prove (REF ); () is proved similarly.", "Assume that the signs are chosen so that (REF ) features $\\varphi ^{\\prime }U(t_0)$ .", "We argue by induction in $j$ .", "The case $j=0$ is obvious.", "Now, assume that (REF ) is true for $j-1$ in place of $j$ .", "Then $(\\varphi ^{\\prime } U(t_0))^j BU(- j t_0)=\\varphi ^{\\prime } B^{\\prime }+\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2},$ where $B^{\\prime }=U(t_0)(\\varphi ^{\\prime }U(t_0))^{j-1}B(U(-t_0)\\varphi ^{\\prime \\prime })^{j-1}U(-t_0)$ is a compactly supported operator in $\\Psi _{\\rho _e}^{\\operatorname{comp}}$ (modulo the $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ remainder from Proposition REF , which we henceforth omit), with $\\rho _e$ defined in (REF ).", "Since $\\operatorname{supp}\\varphi ^{\\prime }\\cap \\operatorname{supp}(1-\\varphi ^{\\prime \\prime })=\\emptyset $ , we have $\\varphi ^{\\prime } B^{\\prime }=\\varphi ^{\\prime }B^{\\prime }\\varphi ^{\\prime \\prime }+\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ and (REF ) follows.", "2.", "We again assume that the signs are chosen so that (REF ) features $\\varphi ^{\\prime }U(t_0)$ .", "Write $j=j_1+j_2$ , where $0\\le j_1,j_2\\le l_e$ , and write the left-hand side of (REF ) as $U(j_1t_0)\\widetilde{B}_1\\widetilde{B} \\widetilde{B}_2U(-j_1t_0)$ , where ${\\begin{array}{c}\\widetilde{B}=(\\varphi ^{\\prime }U(t_0))^{j_2}B(U(-t_0)\\varphi ^{\\prime \\prime })^{j_2},\\\\\\widetilde{B}_1=U(-j_1 t_0)B_1(\\varphi ^{\\prime }U(t_0))^{j_1},\\ \\widetilde{B}_2=(U(-t_0)\\varphi ^{\\prime \\prime })^{j_1}B_2 U(j_1 t_0).\\end{array}}$ Now, $\\widetilde{B}$ is a compactly supported member of $\\Psi _{\\rho _e}^{\\operatorname{comp}}$ .", "Same can be said about $\\widetilde{B}_1$ and $\\widetilde{B}_2$ , by applying () and its adjoint (where the role of $\\varphi ^{\\prime }$ is played by either $\\varphi ^{\\prime }$ or $\\varphi ^{\\prime \\prime }$ and the role of $\\varphi ^{\\prime \\prime }$ , by a suitably chosen cutoff function).", "Moreover, if $U_1,U_2$ are bounded open subsets of $T^*M$ such that $\\operatorname{WF}_h(B_k)\\subset U_k$ and $U_1\\cap U_2=\\emptyset $ , then by Proposition REF , $\\widetilde{B}_k$ is microsupported, in the sense of Definition REF , on the set $g^{j_1t_0}(U_k)$ ; since these two sets do not intersect, we see that $\\widetilde{B}_1\\widetilde{B}\\widetilde{B}_2=\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ as needed.", "3.", "We once again fix the sign so that $U(t_0)$ stands next to $\\varphi ^{\\prime }$ .", "Formally, (REF ) and () follow by applying (REF ) and (), respectively, to the identity $e^{ij\\beta }E_h=U(- jt_0)E_h$ .", "To make this observation rigorous, we write by Lemma  REF ${\\begin{array}{c}e^{i\\beta } BE_h=BU(- t_0)\\varphi _{t_0}E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2},\\\\e^{i\\beta }\\varphi ^{\\prime \\prime } E_h= \\varphi ^{\\prime \\prime } U(-t_0)\\varphi _{t_0}E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}.\\end{array}}$ We now use induction in $j$ .", "For $j=0$ , both (REF ) and () are trivial.", "Now, assume that they both hold for $j-1$ in place of $j$ .", "We then write ${\\begin{array}{c}e^{ij\\beta }(\\varphi ^{\\prime }U(t_0))^j BE_h\\\\=e^{i\\beta } (\\varphi ^{\\prime }U(t_0))^j B(U(-t_0)\\varphi ^{\\prime \\prime })^{j-1}E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}\\\\=(\\varphi ^{\\prime } U(t_0))^j B(U(-t_0)\\varphi ^{\\prime \\prime })^{j-1}U(-t_0)\\varphi _{t_0}E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}.\\end{array}}$ The operator $(\\varphi ^{\\prime } U(t_0))^jB(U(-t_0)\\varphi ^{\\prime \\prime })^{j-1}U(-t_0)$ is a compactly supported element of $\\Psi ^{\\operatorname{comp}}_\\rho $ ; moreover, as $j\\ge 1$ , the wavefront set of this operator is contained in $\\operatorname{supp}\\varphi ^{\\prime }$ .", "Since $\\varphi ^{\\prime \\prime }=1$ near $\\operatorname{supp}\\varphi ^{\\prime }$ , we can replace $\\varphi _{t_0}$ by $\\varphi ^{\\prime \\prime }$ in the last formula, proving (REF ).", "We next write $e^{ij\\beta }(U(t_0)\\varphi ^{\\prime })^jBE_h=e^{i\\beta }(U(t_0)\\varphi ^{\\prime })^jB(\\varphi ^{\\prime \\prime }U(-t_0))^{j-1}\\varphi _{t_0} E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}.$ However, $\\varphi ^{\\prime } (U(t_0)\\varphi ^{\\prime })^{j-1}B(\\varphi ^{\\prime \\prime } U(-t_0))^{j-1}$ is a compactly supported element of $\\Psi ^{\\operatorname{comp}}_\\rho $ and its wavefront set is contained in $\\operatorname{supp}\\varphi ^{\\prime }$ .", "Since $\\varphi ^{\\prime \\prime }=1$ near $\\operatorname{supp}\\varphi ^{\\prime }$ , we can replace $\\varphi _{t_0}$ by $\\varphi ^{\\prime \\prime }$ , obtaining (): ${\\begin{array}{c}e^{ij\\beta }(U(t_0)\\varphi ^{\\prime })^jBE_h\\\\=e^{i\\beta }(U(t_0)\\varphi ^{\\prime })^jB(\\varphi ^{\\prime \\prime }U(-t_0))^{j-1}\\varphi ^{\\prime \\prime } E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}\\\\=(U(t_0)\\varphi ^{\\prime })^jB(\\varphi ^{\\prime \\prime } U(-t_0))^{j-1}\\varphi ^{\\prime \\prime } U(-t_0)\\varphi _{t_0}E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}.\\end{array}}$ Second decomposition of $\\langle AE_h,E_h\\rangle $.", "We now analyse the terms of (REF ), reducing $\\langle AE_h,E_h\\rangle $ to an expression depending on the `outgoing' part $E^0_h$ of the plane wave (see (REF )), with remainder estimated by the classical escape rate for up to twice the Ehrenfest time.", "We will use Lemma REF ; since it only applies to pseudodifferential operators microlocalized inside the set $\\mathcal {E}_{\\varepsilon _e}$ from (REF ), we take an operator $X_0\\in \\Psi ^{\\operatorname{comp}}(M),\\ \\operatorname{WF}_h(X_0)\\subset \\mathcal {E}_{\\varepsilon _e},\\ X_0=1\\text{ near }S^*M\\cap \\operatorname{supp}\\varphi _{t_0},$ compactly supported inside $K_{t_0}$ .", "By (REF ) and the elliptic estimate (Proposition REF ), we have $\\varphi _{t_0} E_h=X_0\\varphi _{t_0}E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}=\\varphi _{t_0}X_0E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}.$ Same is true if we replace $E_h$ by $\\chi _0E_h^0$ , as by (A4) and the fact that $|\\partial _m\\phi _\\xi |_g=1$ , we have $\\operatorname{WF}_h(\\chi _0E_h^0)\\subset S^*M$ .", "We also recall that $\\operatorname{WF}_h(A)\\subset \\mathcal {E}_{\\varepsilon _e}$ .", "We start by estimating the first term on the right-hand side of (REF ) for $l$ up to twice the Ehrenfest time, in terms of the classical escape rate: Proposition 5.8 There exists a constant $C$ such that for $0\\le l\\le 2l_e$ , we have $h^{-1}\\Vert \\langle A(\\varphi U(t_0))^l\\varphi E_h,E_h\\rangle \\Vert _{L^1_{\\xi ,\\lambda }(\\partial \\overline{M}\\times [1,1+h])}\\le C\\mu _L(\\mathcal {T}(lt_0))+\\mathcal {O}(h^\\infty ).$ We write $l=l_1+l_2$ , where $0\\le l_1,l_2\\le l_e$ ; then $\\langle A(\\varphi U(t_0))^l\\varphi E_h,E_h\\rangle =\\langle (\\varphi U(t_0))^{l_1}\\varphi E_h,(U(-t_0)\\varphi )^{l_2}A^* E_h\\rangle .$ Now, by (REF ) ${\\begin{array}{c}e^{il_1\\beta }(\\varphi U(t_0))^{l_1}\\varphi E_h=e^{il_1\\beta }(\\varphi U(t_0))^{l_1}X_0\\varphi E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}\\\\=B_l^1 E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2},\\end{array}}$ where $B_l^1=(\\varphi U(t_0))^{l_1}X_0\\varphi (U(-t_0)\\varphi _1)^{l_1}.$ Similarly, by () (recalling that $A=\\varphi A\\varphi $ ) $e^{-i l_2\\beta }(U(-t_0)\\varphi )^{l_2}A^* E_h=B_l^2E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2},$ where $B_l^2=(U(-t_0)\\varphi )^{l_2}A^* (\\varphi _1U(t_0))^{l_2}\\varphi _{t_0}.$ Put $B_l=(B_l^2)^*B_l^1$ ; recalling (REF ), it is then enough to show that $h^{-1}\\Vert \\langle B_l E_h,E_h\\rangle \\Vert _{L^1_{\\xi ,\\lambda }(\\partial \\overline{M}\\times [1,1+h])}\\le C\\mu _L(\\mathcal {T}(lt_0))+\\mathcal {O}(h^\\infty ).$ Now, by Proposition REF , the operator $B_l^1$ is a compactly supported element of $\\Psi ^{\\operatorname{comp}}_{\\rho _e}$ (modulo an $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ remainder which we will omit), and it is microsupported, in the sense of Definition REF , inside the set $g^{-l_1t_0}(\\lbrace \\varphi _1\\ne 0\\rbrace )$ (here we only use that $\\operatorname{supp}\\varphi \\subset \\lbrace \\varphi _1\\ne 0\\rbrace $ ).", "Similarly, $B_l^2\\in \\Psi ^{\\operatorname{comp}}_{\\rho _e}$ is microsupported inside $g^{l_2 t_0}(\\lbrace \\varphi _1\\ne 0\\rbrace )$ .", "Therefore, $B_l$ is microsupported on the set $\\mathcal {S}_l=g^{-l_1t_0}(\\lbrace \\varphi _1\\ne 0\\rbrace )\\cap g^{l_2 t_0}(\\lbrace \\varphi _1\\ne 0\\rbrace ).$ Note also that $B_l$ is compactly supported independently of $l$ .", "Now, by taking the convolution of the indicator function of an $h^{\\rho _e}$ sized neighborhood of $\\mathcal {S}_l$ with an appropriately rescaled cutoff function, we can construct a compactly supported operator $\\widetilde{B}_l\\in \\Psi ^{\\operatorname{comp}}_{\\rho _e}$ such that $B_l=\\widetilde{B}_l^* B_l+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }}$ and $\\widetilde{B}_l$ is microsupported inside an $\\mathcal {O}(h^{\\rho _e})$ sized neighborhood $\\widetilde{\\mathcal {S}}_l$ of $\\mathcal {S}_l$ .", "Using (REF ), (REF ), and the estimate on the Lipschitz constant of the flow given by (REF ), we see that for $(\\widetilde{m},\\tilde{\\nu })\\in \\widetilde{\\mathcal {S}}_l\\cap S^*M$ , there exists $(m,\\nu )\\in \\mathcal {S}_l\\cap S^*M$ such that $d((\\widetilde{m},\\tilde{\\nu }),(m,\\nu ))\\le Ch^{\\rho _e}$ and for $\\Lambda ^{\\prime }_0>\\Lambda ^{\\prime \\prime }_0>(1+2\\varepsilon _e)\\Lambda _{\\max }$ , $d(g^{l_1t_0}(\\widetilde{m},\\tilde{\\nu }),g^{l_1t_0}(m,\\nu ))\\le Ce^{l_1t_0\\Lambda ^{\\prime \\prime }_0}h^{\\rho _e}\\le Ce^{t_e\\Lambda ^{\\prime \\prime }_0}h^{\\rho _e}\\le Ce^{-t_e(\\Lambda ^{\\prime }_0-\\Lambda ^{\\prime \\prime }_0)}$ is bounded by some positive power of $h$ .", "Here $d$ denotes some smooth distance function on $T^*M$ .", "Same is true if we replace $g^{l_1t_0}$ with $g^{-l_2t_0}$ ; therefore, if the compact set $K_0$ used in the definition (REF ) of $\\mathcal {T}(t)$ is chosen large enough, we have $\\widetilde{\\mathcal {S}_l}\\cap S^*M\\subset g^{l_2t_0}(\\mathcal {T}(lt_0)).$ Using the Cauchy–Schwartz inequality and (REF ), we bound the left-hand side of (REF ) by ${\\begin{array}{c}h^{-1}\\Vert \\langle B_l E_h,E_h\\rangle \\Vert _{L^1_{\\xi ,\\lambda }}\\le h^{-1}\\Vert \\langle B_l E_h,\\widetilde{B}_l E_h\\rangle \\Vert _{L^1_{\\xi ,\\lambda }}+\\mathcal {O}(h^\\infty )\\\\\\le h^{-1}\\Vert B_lE_h\\Vert _{L^2(M)L^2_{\\xi ,\\lambda }}\\cdot \\Vert \\widetilde{B}_lE_h\\Vert _{L^2(M)L^2_{\\xi ,\\lambda }}+\\mathcal {O}(h^\\infty )\\\\\\le C(h^{n/2}\\Vert B_l\\Pi _{[1,1+h]}\\Vert _{\\operatorname{HS}})(h^{n/2}\\Vert \\widetilde{B}_l\\Pi _{[1,1+h]}\\Vert _{\\operatorname{HS}})+\\mathcal {O}(h^\\infty ).\\end{array}}$ It remains to use (REF ) (or rather its adjoint).", "Indeed, both $B_l$ and $\\widetilde{B}_l$ are bounded in $\\Psi ^{\\operatorname{comp}}_{\\rho _e}$ uniformly in $l$ , and they are microsupported in $\\widetilde{\\mathcal {S}}_l$ ; therefore, by (REF ) $h^{-1}\\Vert \\langle B_l E_h,E_h\\rangle \\Vert _{L^1_{\\xi ,\\lambda }}\\le C\\mu _L(\\widetilde{\\mathcal {S}}_l\\cap S^*M)+\\mathcal {O}(h^\\infty )\\le C\\mu _L(\\mathcal {T}(lt_0))+\\mathcal {O}(h^\\infty ).$ As for the sum in (REF ), we have the following Proposition 5.9 For $1\\le j\\le 2 l_e$ , we have ${\\begin{array}{c}e^{ij\\beta }\\langle A(\\varphi U(t_0))^j(1-\\varphi )\\varphi _{t_0}\\chi _0 E^0_h,E_h\\rangle =\\langle \\widetilde{A}^j \\chi _0E_h^0,\\chi _0E_h^0\\rangle +\\mathcal {O}(h^\\infty \\mathcal {N}(E_h)^2),\\\\\\widetilde{A}^{j}:=\\varphi _{t_0}(1-\\varphi _2)(U(-t_0)\\varphi _1)^jA(\\varphi U(t_0))^j(1-\\varphi )\\varphi _{t_0},\\end{array}}$ uniformly in $\\xi \\in \\partial \\overline{M}$ and $\\lambda \\in [1,1+h]$ .", "Since $A=\\varphi A\\varphi $ , we can replace $E_h$ by $\\varphi _1 E_h$ on the left-hand side of (REF ).", "Writing down (REF ) for $\\varphi _1$ in place of $\\varphi $ and using $\\varphi _2$ in place of $\\varphi _1$ in the splitting $\\varphi _{t_0}=\\varphi _1+(1-\\varphi _1)\\varphi _{t_0}$ in the last step, we get ${\\begin{array}{c}\\varphi _1 E_h=e^{ij\\beta }(\\varphi _1 U(t_0))^j\\varphi _2 E_h+e^{ij\\beta }(\\varphi _1 U(t_0))^j (1-\\varphi _2)\\varphi _{t_0}E_h\\\\+\\sum _{k=1}^{j-1}e^{ik\\beta }(\\varphi _1 U(t_0))^k(1-\\varphi _1)\\varphi _{t_0}E_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2}.\\end{array}}$ We now substitute (REF ) into the left-hand side of (REF ).", "The first term gives, after using (REF ) to replace $\\varphi _{t_0}\\chi _0E^0_h$ by $X_0\\varphi _{t_0}\\chi _0 E^0_h$ and $\\varphi E_h$ by $X_0\\varphi E_h$ $\\langle A(\\varphi U(t_0))^j (1-\\varphi )\\varphi _{t_0}\\chi _0 E^0_h,(\\varphi _1 U(t_0))^j\\varphi _2 E_h\\rangle =\\langle B_0 \\chi _0 E^0_h,E_h\\rangle +\\mathcal {O}(h^\\infty \\mathcal {N}(E_h)^2),$ where $B_0=\\varphi _2X_0^*(U(-t_0)\\varphi _1)^j A(\\varphi U(t_0))^j (1-\\varphi )X_0\\varphi _{t_0}=\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ by (REF ), as $\\operatorname{supp}\\varphi _2\\cap \\operatorname{supp}(1-\\varphi )=\\emptyset $ .", "Next, we use Proposition REF to write the second term of (REF ) as $e^{ij\\beta }(\\varphi _1 U(t_0))^j(1-\\varphi _2)\\varphi _{t_0}\\chi _0E^0_h+\\mathcal {O}(h^\\infty \\mathcal {N}(E_h))_{L^2};$ therefore, this term gives the right-hand side of (REF ).", "It remains to estimate the contribution of each term of the sum in (REF ), which we can write, using (REF ), as $e^{i(j-k)\\beta }\\langle B_k\\chi _0 E^0_h,\\chi _0E^0_h\\rangle +\\mathcal {O}(h^\\infty \\mathcal {N}(E_h)^2)$ , with $B_k=\\varphi _{t_0}X_0^*(1-\\varphi _1)(U(-t_0)\\varphi _1)^k A(\\varphi U(t_0))^j(1-\\varphi )X_0\\varphi _{t_0}.$ We need to show that $\\Vert B_k\\Vert _{L^2\\rightarrow L^2}=\\mathcal {O}(h^\\infty )$ for $1\\le k<j$ .", "For that, we consider two cases.", "First, assume that $k\\le l_e$ .", "Then we have $\\varphi _{t_0} X_0^*(1-\\varphi _1)(U(-t_0)\\varphi _1)^kA(\\varphi U(t_0))^k\\varphi =\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2},$ as the supports of $1-\\varphi _1$ and $\\varphi $ do not intersect and the operator in between them is a compactly supported element of $\\Psi ^{\\operatorname{comp}}_{\\rho _e}$ (modulo an $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ remainder which we will omit).", "Since $B_k$ is obtained by multiplying the left-hand side of (REF ) on the right by $U(t_0)(\\varphi U(t_0))^{j-1-k}(1-\\varphi )X_0\\varphi _{t_0}$ , it is also $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ .", "Now, assume that $k\\ge l_e$ .", "Take $\\widetilde{\\varphi }_1\\in C_0^\\infty (M)$ equal to 1 near $\\operatorname{supp}\\varphi _1$ and such that $|\\widetilde{\\varphi }_1|\\le 1$ everywhere.", "We write by (REF ) and its adjoint, ${\\begin{array}{c}U((k-l_e)t_0)B_k U(-(j-l_e)t_0)=B_k^1B_k^2B_k^3+\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2},\\\\B_k^1=(\\widetilde{\\varphi }_1 U(t_0))^{k-l_e}\\varphi _{t_0}X_0^*(1-\\varphi _1)(U(-t_0)\\varphi _1)^{k-l_e},\\\\B_k^2=(U(-t_0)\\varphi _1)^{l_e}A(\\varphi U(t_0))^{l_e},\\\\B_k^3=(\\varphi U(t_0))^{j-l_e}(1-\\varphi )X_0\\varphi _{t_0}(U(-t_0)\\varphi _1)^{j-l_e}.\\end{array}}$ Now all $B_k^i$ , $i=1,2,3$ , are compactly supported members of $\\Psi ^{\\operatorname{comp}}_{\\rho _e}$ .", "Let $U_1,U_2$ be two bounded open sets such that $\\operatorname{supp}(\\varphi _{t_0}(1-\\varphi _1))\\subset U_1$ , $\\operatorname{supp}\\varphi \\subset U_2$ , and $U_1\\cap U_2=\\emptyset $ .", "Since $k-l_e>j-l_e$ and by Proposition REF , the operator $B_k^1$ is microsupported, in the sense of Definition REF , on the set $g^{-(k-l_e)t_0}(U_1)$ , while $B_k^3$ is microsupported on the set $g^{-(k-l_e)t_0}(U_2)$ ; since these two sets do not intersect, we get $B_k=\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ , finishing the proof.", "Combining (REF ) with (REF ) and (REF ), we have finally proved Proposition 5.10 For $0\\le l\\le 2 l_e$ and $A\\in \\Psi ^{\\operatorname{comp}}$ microlocalized inside the set $\\mathcal {E}_{\\varepsilon _e}$ defined in (REF ), we have ${\\begin{array}{c}\\langle A E_h,E_h\\rangle =\\sum _{j=1}^l\\langle \\widetilde{A}^j\\chi _0 E^0_h,\\chi _0 E^0_h\\rangle +\\mathcal {O}\\big (h\\mu _L(\\mathcal {T}(lt_0))+h^\\infty \\big )_{L^1_{\\xi ,\\lambda }(\\partial \\overline{M}\\times [1,1+h])},\\\\\\widetilde{A}^{j}:=\\varphi _{t_0}(1-\\varphi _2)(U(-t_0)\\varphi _1)^jA(\\varphi U(t_0))^j(1-\\varphi )\\varphi _{t_0}.\\end{array}}$ Here $l_e$ is defined in (REF ) and $\\mathcal {T}(t)$ in (REF ).", "Note that the sum on the right-hand side corresponds to the $A_1^{-t}$ term in (REF ), while the remainder contains both the $A_0^{-t}$ term (dealt with in Proposition REF ) and the remainder in (REF ).", "We have kept the $\\mathcal {O}(h^\\infty )$ remainder to include the nontrapping case.", "Proof of (REF ).", "Put $l$ equal to the number $l_e$ defined in (REF ).", "By (REF ), it is enough to approximate the terms $\\langle \\widetilde{A}^j\\chi _0 E^0_h,\\chi _0E^0_h\\rangle $ .", "This is done by the following improvement of (REF ), relying on the Lagrangian structure of $E^0_h$ and featuring the interpolated escape rate $r(h,\\Lambda )$ from (REF ): Proposition 5.11 Put $l=l_e$ given by (REF ), and $r(h,\\Lambda )$ defined in (REF ).", "Then the sum on the right-hand side of (REF ) is approximated as follows: $\\sum _{j=1}^l \\langle \\widetilde{A}^j\\chi _0 E^0_h,\\chi _0E^0_h\\rangle =\\int _{S^*M} \\sigma (A)\\,d\\mu _\\xi +\\mathcal {O}(hr(h,2\\Lambda _0))_{L^1_{\\xi ,\\lambda }(\\partial \\overline{M}\\times [1,1+h])}.$ By Proposition REF , the operator $\\widetilde{A}^j$ is compactly supported and lies in $\\Psi _{\\rho _j}^{\\operatorname{comp}}$ , modulo an $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ remainder, where $\\rho _j={jt_0\\over t_e}\\rho _e, \\quad \\textrm { with }\\, jt_0/t_e\\le 1+o(1)$ with $t_e$ and $\\rho _e$ defined in (REF ) and (REF ), respectively.", "Next, $\\widetilde{A}^j$ is microsupported, in the sense of Definition REF , in the set $\\mathcal {Q}_j:=g^{t_0}(\\lbrace \\varphi _1\\ne 0\\rbrace )\\cap g^{jt_0}(\\lbrace \\varphi _1\\ne 0\\rbrace ).$ If the set $K_0$ from the definition (REF ) of $\\mathcal {T}(t)$ is large enough, then $\\mathcal {Q}_j\\cap S^*M\\subset g^{jt_0}(\\mathcal {T}(jt_0))$ ; by the definition (REF ) of $r(h,\\Lambda )$ , we find $h^{1-jt_0/t_e} \\mu _L(\\mathcal {Q}_j\\cap S^*M)\\le r(h,2\\Lambda _0).$ By (A4) and Proposition REF , we have the following analogue of (REF ): $\\widetilde{A}^j\\chi _0E^0_h=e^{{i\\lambda \\over h}\\phi _\\xi }\\chi _0 b^0(1,\\xi ,m;0)\\sigma (\\widetilde{A}^j)(m,\\partial _m\\phi _\\xi (m))+\\mathcal {O}(h^{1-2\\rho _j})_{L^\\infty }.$ Here and below, we also use the fact that the seminorms of $\\widetilde{A}^j$ are uniform with respect to $j$ in the sense stated in Proposition REF , in order to control the remainders.", "Moreover, by part 2 of the same proposition, we see that $\\widetilde{A}^j\\chi _0E^0_h$ is $\\mathcal {O}(h^\\infty )$ outside of the set of points $m\\in U_\\xi $ such that $(m,\\partial _m\\phi _\\xi (m))\\in \\mathcal {Q}_j$ .", "By Lemma REF , $\\sigma (\\widetilde{A}^j)$ is supported in $\\operatorname{supp}(1-\\varphi )\\cap g^{t_0}(\\operatorname{supp}\\varphi _1)\\subset \\mathcal {DE}_+\\cap \\lbrace x<\\varepsilon _1\\rbrace $ , with $\\mathcal {DE}_+$ from Definition REF .", "Using part 2 of Lemma REF , we then get $\\langle \\widetilde{A}^j \\chi _0 E^0_h,\\chi _0 E^0_h\\rangle =\\int _{S^*M} \\sigma (\\widetilde{A}^j)\\,d\\mu _\\xi +\\mathcal {O}(h^{1-2\\rho _j}\\mu _\\xi (\\mathcal {Q}_j))+\\mathcal {O}(h^\\infty ),$ uniformly in $\\xi \\in \\partial \\overline{M}$ and $\\lambda \\in [1,1+h]$ .", "Now, we write by (REF ) and Proposition REF , ${\\begin{array}{c}\\sum _{j=1}^l h^{1-2\\rho _j}\\Vert \\mu _\\xi (\\mathcal {Q}_j)\\Vert _{L^1_\\xi }=\\sum _{j=1}^l h^{1-2\\rho _j}\\mu _L(\\mathcal {Q}_j\\cap S^*M)\\\\\\le r(h,2\\Lambda _0)\\sum _{j=1}^l h^{(1-2\\rho _e)jt_0/t_e}=r(h,2\\Lambda _0)\\sum _{j=1}^l e^{-2\\Lambda _0(1-2\\rho _e)jt_0}\\le Cr(h,2\\Lambda _0).\\end{array}}$ It remains to sum up the integrals in (REF ).", "We have by Proposition REF , bearing in mind that $\\varphi \\varphi _1=\\varphi $ , $(1-\\varphi )(1-\\varphi _2)=1-\\varphi $ , $A=\\varphi A\\varphi $ , and $d_g(\\operatorname{supp}(1-\\varphi _{t_0}),\\operatorname{supp}\\varphi )>t_0$ , $\\sigma (\\widetilde{A}^j)=(\\sigma (A)\\circ g^{-jt_0})(1-\\varphi )\\prod _{k=1}^{j-1}\\varphi \\circ g^{-kt_0}.$ By part 1 of Lemma REF , we write ${\\begin{array}{c}\\sum _{j=1}^l\\int _{S^*M}\\sigma (\\widetilde{A}^j)\\,d\\mu _\\xi =\\int _{S^*M} \\sigma (A)\\sum _{j=1}^l (1-\\varphi \\circ g^{jt_0})\\prod _{k=1}^{j-1}\\varphi \\circ g^{kt_0}\\,d\\mu _\\xi \\\\=\\int _{S^*M}\\sigma (A)\\Big (1-\\prod _{k=1}^l\\varphi \\circ g^{kt_0}\\Big )\\,d\\mu _\\xi .\\end{array}}$ It remains to note that by Proposition REF , $\\int _{\\partial \\overline{M}}\\int _{S^*M}|\\sigma (A)|\\prod _{k=1}^l\\varphi \\circ g^{kt_0}\\,d\\mu _\\xi d\\xi =\\int _{S^*M}|\\sigma (A)|\\prod _{k=1}^l\\varphi \\circ g^{kt_0}\\,d\\mu _L=\\mathcal {O}(\\mu _L(\\mathcal {T}(lt_0)))$ since the expression under the last integral is supported in $\\mathcal {T}(lt_0)$ ." ], [ "Trace estimates", "In this subsection, we prove a stronger remainder bound () for the case when $\\langle AE_h,E_h\\rangle $ is paired with a test function in $\\xi $ and obtain an expansion of the trace of spectral projectors with a fractal remainder – Theorem REF .", "Expressing $E^0_h\\otimes E^0_h$ via Schrödinger propagators.", "Our argument will be based on the decomposition (REF ).", "The remainder in this decomposition is already controlled by the escape rate at twice the Ehrenfest time $t_e$ defined in (REF ).", "However, in the previous subsection (see Proposition REF ), we were only able to estimate the sum in (REF ) for $l$ up to the Ehrenfest index $l_e\\sim t_e/t_0$ defined in (REF ).", "We therefore need a better way of writing down the Lagrangian states $E^0_h$ , when coupled with a test function in $\\xi $ , and such a way is provided by (with the cutoffs allowing for representation of the spectral measure in terms of a finite time integral) Lemma 5.12 Let $f(\\xi )\\in C^\\infty (\\partial \\overline{M})$ and define for $\\lambda \\in (1/2,2)$ , $\\Pi ^0_f(\\lambda ):=\\int _{\\partial \\overline{M}}f(\\xi )(\\chi _0 E^0_h(\\lambda ,\\xi ))\\otimes (\\chi _0 E^0_h(\\lambda ,\\xi ))\\,d\\xi .$ Here $\\otimes $ denotes the Hilbert tensor product, see (REF ).", "Assume that $\\widetilde{X}_1,\\widetilde{X}_2\\in \\Psi ^{\\operatorname{comp}}(M)$ are compactly supported and the projections $\\pi _S(\\operatorname{WF}_h(\\widetilde{X}_j))$ of $\\operatorname{WF}_h(\\widetilde{X}_j)$ onto $S^*M$ along the radial rays in the fibers of $T^*M$ lie inside $\\mathcal {DE}_+\\cap \\lbrace x\\le \\varepsilon _1\\rbrace $ , with $\\mathcal {DE}_+$ defined in (REF ) and $\\varepsilon _1$ from (A7).", "Then $\\widetilde{X}_1\\Pi ^0_f(\\lambda )\\widetilde{X}_2^*=(2\\pi h)^n\\int _{-T_0}^{T_0} e^{-i\\lambda ^2 s/(2h)} U(s)B_s(\\lambda )\\,ds+\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2},$ where $T_0>0$ depends on the support of the Schwartz kernels of $\\widetilde{X}_j$ but not on $h$ , $B_s(\\lambda )\\in \\Psi ^{\\operatorname{comp}}(M)$ is compactly supported on $M$ , smooth and compactly supported in $s\\in (-T_0,T_0)$ , depending smoothly on $\\lambda $ .", "Moreover, if $\\xi _{+\\infty }$ is the function defined in (G3), then $\\sigma (B_0(1))|_{S^*M}=f(\\xi _{+\\infty })\\sigma (\\widetilde{X}_1\\widetilde{X}_2^*).$ We write $\\widetilde{X}_1\\Pi ^0_f(\\lambda )\\widetilde{X}_2^*=\\int _{\\partial \\overline{M}} f(\\xi )(\\widetilde{X}_1 \\chi _0E^0_h(\\lambda ,\\xi ))\\otimes (\\widetilde{X}_2\\chi _0 E^0_h(\\lambda ,\\xi ))\\,d\\xi .$ By (A4), $\\chi _0E^0_h(\\lambda ,\\xi )$ is a Lagrangian distribution associated to $\\lambda $ times the Lagrangian $\\Lambda _\\xi $ from (REF ).", "By Proposition REF , we can write $\\widetilde{X}_j \\chi _0E^0_h(\\lambda ,\\xi )(m)=e^{{i\\lambda \\over h}\\phi _\\xi (m)}b_j(\\lambda ,\\xi ,m;h)+\\mathcal {O}(h^\\infty )_{C_0^\\infty },$ where $\\phi _\\xi $ is defined in (G4) and $b_j$ is a classical symbol in $h$ smooth in $\\lambda ,\\xi ,m$ and compactly supported in $m$ .", "The symbol $b_j$ depends on the operator $\\widetilde{X}_j$ ; in fact, we can make $\\operatorname{supp}b_j\\subset \\tau ^{-1}(\\pi _S(\\operatorname{WF}_h(\\widetilde{X}_j)))$ , with $\\tau $ defined in (REF ).", "We then write the Schwartz kernel of $\\widetilde{X}_1\\Pi ^0_f(\\lambda )\\widetilde{X}_2^*$ , modulo an $\\mathcal {O}(h^\\infty )_{C^\\infty _0}$ remainder, as $\\widetilde{\\Pi }(m,m^{\\prime };\\lambda ,h)=\\int _{\\partial \\overline{M}} e^{{i\\lambda \\over h}(\\phi _\\xi (m)-\\phi _\\xi (m^{\\prime }))}f(\\xi )b_1(\\lambda ,\\xi ,m;h)\\overline{b_2(\\lambda ,\\xi ;m^{\\prime },h)}\\,d\\xi .$ Now, the support of each $b_j$ in the $(m,\\xi )$ variables lies in the set $U^+_\\infty $ defined in (G4).", "The critical points of the phase $\\lambda (\\phi _\\xi (m)-\\phi _\\xi (m^{\\prime }))$ are given by $\\partial _\\xi \\phi _\\xi (m)=\\partial _\\xi \\phi _\\xi (m^{\\prime })$ ; using (G6), we see that $h^{-n/2}\\widetilde{\\Pi }(m,m^{\\prime };\\lambda ,h)$ is a Lagrangian distribution associated to the Lagrangian $\\widetilde{\\Lambda }_\\lambda :=\\lbrace (m,\\nu ;m^{\\prime },\\nu ^{\\prime })\\mid |\\nu |_g=\\lambda ,\\ \\exists s\\in (-T_0,T_0): g^s(m,\\nu )=(m^{\\prime },\\nu ^{\\prime })\\rbrace .$ Here $T_0>0$ is large, but fixed.", "Now, take some family $B_s(\\lambda )\\in \\Psi ^{\\operatorname{comp}}(M)$ smooth and compactly supported in $s\\in (-T_0,T_0)$ and define the operator $\\Pi _B(\\lambda ):=(2\\pi h)^n\\int _{-T_0}^{T_0}e^{-i\\lambda ^2 s/(2h)}U(s)B_s(\\lambda )\\,ds.$ Following the proof of Lemma REF , we see that $h^{-n/2}$ times the Schwartz kernel $\\Pi _B(m,m^{\\prime };\\lambda ,h)$ of $\\Pi _B(\\lambda )$ is, up to an $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ remainder, a compactly supported and compactly microlocalized Lagrangian distribution associated to the Lagrangian $\\widetilde{\\Lambda }_\\lambda $ .", "Moreover, the principal symbol of $h^{-n/2}\\Pi _B(m,m^{\\prime };\\lambda ,h)$ at $(m,\\nu ,m^{\\prime },\\nu ^{\\prime })$ such that $g^s(m,\\nu )=(m^{\\prime },\\nu ^{\\prime })$ is a nonvanishing factor times $\\sigma (B_s)(m^{\\prime },\\nu ^{\\prime })$ .", "Arguing as in the proof of part 2 of Proposition REF , we see that we can find a family of operators $B_s(\\lambda )$ such that $\\widetilde{\\Pi }(m,m^{\\prime };\\lambda ,h)=\\Pi _B(m,m^{\\prime };\\lambda ,h)+\\mathcal {O}(h^\\infty )_{C^\\infty _0}.$ It remains to check that the family $B_s(\\lambda )$ can be chosen to depend smoothly on $\\lambda $ uniformly in $h$ (this is not automatic, as multiplication by $e^{{i\\over h}\\psi (\\lambda )}$ for some function $\\psi $ destroys this property, but does not change the Lagrangians where our kernels are microlocalized for each $\\lambda $ ).", "For that, it is enough to note (by Proposition REF ) that if we consider $h^{-n/2}\\widetilde{\\Pi }$ and $h^{-n/2}\\Pi _B$ as Lagrangian distributions in $m,m^{\\prime },\\lambda $ , they are associated to the Lagrangian $\\lbrace (m,\\nu ,m^{\\prime },\\nu ^{\\prime },\\lambda ,q_\\lambda )\\mid |\\nu |_g=\\lambda ,\\ \\exists s\\in (-T_0,T_0):g^s(m,\\nu )=(m^{\\prime },\\nu ^{\\prime }),\\ q_\\lambda =-\\lambda s\\rbrace ,$ where $q_\\lambda $ is the momentum corresponding to $\\lambda $ .", "For $\\widetilde{\\Pi }$ , this is true as when $\\tau (m^{\\prime },\\xi )=g^{\\lambda s}(\\tau (m,\\xi ))$ , we have $\\phi _\\xi (m)-\\phi _\\xi (m^{\\prime })=-\\lambda s$ by (REF ); for $\\Pi _B$ , this is seen directly from the parametrization (REF ), keeping in mind the factor $e^{-i\\lambda ^2 s/(2h)}$ in the definition of $\\Pi _B$ .", "Finally, to show the formula (REF ), put $\\lambda =1$ , take an arbitrary $Z\\in \\Psi ^{\\operatorname{comp}}$ , and compute the trace $\\operatorname{Tr}(\\widetilde{X}_1\\Pi ^0_f(1)\\widetilde{X}_2^*Z)=(2\\pi h)^n\\int _{-T_0}^{T_0}e^{-is/(2h)}\\operatorname{Tr}(U(s)B_s(1)Z)\\,ds+\\mathcal {O}(h^\\infty ).$ The left-hand side of (REF ) can be computed as at the end of Section REF , using the limiting measure $\\mu _\\xi $ ; by Proposition REF , it is equal to the integral of $f(\\xi _{+\\infty })\\sigma (\\widetilde{X}_2^*Z\\widetilde{X}_1)$ over the Liouville measure on $S^*M$ , plus an $\\mathcal {O}(h)$ remainder.", "The right-hand side of (REF ) can be computed by the trace formula (REF ), and is equal to the integral of $B_0(1)Z$ over the Liouville measure on $S^*M$ , plus an $\\mathcal {O}(h)$ remainder.", "Therefore, $\\int _{S^*M}\\sigma (Z)f(\\xi _{+\\infty })\\sigma (\\widetilde{X}_1\\widetilde{X}_2^*)\\,d\\mu _L=\\int _{S^*M}\\sigma (Z)\\sigma (B_0(1))\\,d\\mu _L$ for any $Z$ ; (REF ) follows.", "Proof of ().", "By (REF ), it is enough to approximate the sum in this formula up to twice the Ehrenfest time: Proposition 5.13 Fix $f\\in C^\\infty (\\partial \\overline{M})$ and $A\\in \\Psi ^{\\operatorname{comp}}(M)$ is microlocalized inside the set $\\mathcal {E}_{\\varepsilon _e}$ defined in (REF ).", "Put $l=2l_e$ , where $l_e$ is defined in (REF ), and consider the following function $S_f(\\lambda ):=\\sum _{j=1}^l \\int _{\\partial \\overline{M}}f(\\xi )\\langle \\widetilde{A}^j\\chi _0E^0_h(\\lambda ,\\xi ),\\chi _0E^0_h(\\lambda ,\\xi )\\rangle \\,d\\xi .$ If $\\xi _{+\\infty }(m,\\nu )$ is the limit of $g^t(m,\\nu )$ as $t\\rightarrow +\\infty $ , for $(m,\\nu )\\in S^*M\\setminus \\Gamma _-$ (see (G3)), then for $\\lambda \\in [1,1+h]$ , $S_f(\\lambda )=\\int _{S^*M} f(\\xi _{+\\infty })\\sigma (A)\\,d\\mu _L+\\mathcal {O}(r(h,\\Lambda _0)).$ Here $r(h,\\Lambda )$ is defined in (REF ).", "Moreover, for each $k$ $\\sup _{\\lambda \\in [1,1+h]}|\\partial _\\lambda ^k S_f(\\lambda )|\\le C_k h^{-k\\rho _e},$ where $\\rho _e$ is defined in (REF ).", "First of all, take a compactly supported operator $\\widetilde{X}\\in \\Psi ^{\\operatorname{comp}}$ such that $\\operatorname{WF}_h(\\widetilde{X})\\cap S^*M$ lies inside the set $\\mathcal {DE}_+\\cap \\lbrace x\\le \\varepsilon _1\\rbrace $ and for $X_0$ defined in (REF ), ${\\begin{array}{c}\\varphi U(t_0)(1-\\varphi )\\varphi _{t_0}X_0(1-\\widetilde{X})=\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2},\\\\\\varphi _1 U(t_0)(1-\\varphi _2)\\varphi _{t_0}X_0(1-\\widetilde{X})=\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}.\\end{array}}$ Such an operator exists by Lemma REF (it can be easily seen that in part 1 of this lemma, $(m,\\nu )$ actually lies in the interior of $\\mathcal {DE}_+$ ).", "Then by (REF ), the definition (REF ) of $\\widetilde{A}^j$ , and Lemma REF , ${\\begin{array}{c}S_f(\\lambda )=\\sum _{j=1}^l\\int _{\\partial \\overline{M}} f(\\xi )\\langle \\widetilde{A}^jX_0\\widetilde{X}\\chi _0 E_h^0(\\lambda ,\\xi ),\\widetilde{X}\\chi _0 E_h^0(\\lambda ,\\xi )\\rangle \\,d\\xi +\\mathcal {O}(h^\\infty )\\\\=\\sum _{j=1}^l\\operatorname{Tr}(\\widetilde{A}^jX_0\\widetilde{X}\\Pi ^0_f(\\lambda )\\widetilde{X}^*)+\\mathcal {O}(h^\\infty )\\\\=(2\\pi h)^n\\sum _{j=1}^l\\int _{-T_0}^{T_0} e^{-i\\lambda ^2s/(2h)}\\operatorname{Tr}(\\widetilde{A}^j X_0U(s)B_s(\\lambda ))\\,ds+\\mathcal {O}(h^\\infty )\\end{array}}$ for some fixed $T_0>0$ and some family $B_s(\\lambda )\\in \\Psi ^{\\operatorname{comp}}$ smooth in $s$ and $\\lambda $ and compactly supported in $s$ ; we can make $B_s$ microlocalized inside the set $\\mathcal {E}_{\\varepsilon _e}$ defined in (REF ).", "We will henceforth ignore the $\\mathcal {O}(h^\\infty )$ term.", "Now, take $1\\le j\\le l$ and put $j=j_1+j_2$ , where $0\\le j_1,j_2\\le l_e$ , $j_2\\ge 1$ , and $|j_1-j_2|\\le 1$ .", "Using the cyclicity of the trace, we find ${\\begin{array}{c}\\operatorname{Tr}(\\widetilde{A}^j X_0U(s)B_s(\\lambda ))= \\operatorname{Tr}(U(s)B_1^j B_{2,s}^j(\\lambda )),\\\\B_1^j := (U(-t_0)\\varphi _1)^{j_1}A(\\varphi U(t_0))^{j_1},\\\\B_{2,s}^j(\\lambda ) := (\\varphi U(t_0))^{j_2}(1-\\varphi )\\varphi _{t_0}X_0U(s)B_s(\\lambda )\\varphi _{t_0}(1-\\varphi _2)(U(-t_0)\\varphi _1)^{j_2}U(-s).\\end{array}}$ Put $\\rho _j=(jt_0/t_e)\\rho _e$ ; since $j_1,j_2\\le j/2+1$ , by Proposition REF the operator $B_1^j$ is a compactly supported element of $\\Psi ^{\\operatorname{comp}}_{\\rho _j/2}$ (modulo an $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ remainder which we will ignore).", "Same can be said about the operator $B_{2,s}^j(\\lambda )=(\\varphi U(t_0))^{j_2}\\cdot (1-\\varphi )\\varphi _{t_0}X_0 U(s)B_s(\\lambda )\\varphi _{t_0}(1-\\varphi _2)U(-s)\\cdot (U(-t_0)\\cdot U(s)\\varphi _1U(-s))^{j_2}.$ (The operator $U(s)\\varphi _1 U(-s)$ is not pseudodifferential because $\\varphi _1$ is not compactly microlocalized, but this can be easily fixed by taking $\\widetilde{X}_0\\in \\Psi ^{\\operatorname{comp}}$ equal to the identity on a sufficiently large compact set and replacing $U(s)\\varphi _1U(-s)$ by $U(s)\\varphi _1\\widetilde{X}_0 U(-s)$ in $B_{2,s}(\\lambda )$ , with an $\\mathcal {O}(h^\\infty )$ error.)", "Therefore, $B_1^jB^j_{2,s}(\\lambda )$ also lies in $\\Psi ^{\\operatorname{comp}}_{\\rho _j/2}$ ; moreover, it depends smoothly on $s$ and $\\lambda $ , uniformly in this operator class.", "(In principle, we get powers of $l\\sim \\log (1/h)$ when differentiating in $s$ , due to the $(U(-t_0)\\cdot U(s)\\varphi _1 U(s))^{j_2}$ term, but they can be absorbed into the powers of $h$ in the expansion (REF ).)", "We now use the trace formula of Lemma REF , writing $S_f(\\lambda )=(2\\pi h)^n\\sum _{j=1}^l\\int _{-T_0}^{T_0}e^{-i\\lambda ^2 s/(2h)}\\operatorname{Tr}(U(s)B_1^jB_{2,s}^j(\\lambda ))\\,ds.$ The operator $B_{2,s}(\\lambda )^j$ is microsupported, in the sense of Definition REF , inside the set $g^{-j_2t_0}(\\lbrace \\varphi _2\\ne 1\\rbrace )\\cap g^{-(j_2-1)t_0}(\\lbrace \\varphi _1\\ne 0\\rbrace )$ ; by Lemma REF , this set lies inside $g^{-j_2t_0}(\\mathcal {DE}_+)$ and in particular does not intersect any closed geodesics, therefore (REF ) holds.", "The estimate (REF ) now follows immediately from (REF ).", "The power $h^{-k\\rho _e}$ arises because we integrate over the energy surface $\\lbrace |\\nu |_g=\\lambda \\rbrace $ depending on $\\lambda $ ; therefore, $\\partial ^k_\\lambda S_{f}(\\lambda )$ will involve $k$ th derivatives of the full symbol of $B_1^jB^j_{2,s}(\\lambda )$ in the direction transversal to the energy surface, which are bounded by $h^{-k\\rho _j}$ .", "The sum (REF ) has $l\\sim \\log (1/h)$ terms; however, our estimate is not multiplied by $\\log (1/h)$ because one can see that the sum of Liouville measures of the sets where these terms are microsupported is bounded.", "As for the approximation (REF ), we write (note that we take $s=0$ in $B_{2,s}$ ) ${\\begin{array}{c}\\sigma (B_1^j)=(\\sigma (A)\\circ g^{-j_1t_0})\\prod _{k=1}^{j_1-1} \\varphi \\circ g^{-kt_0},\\\\\\sigma (B_{2,0}^j(\\lambda ))|_{S^*M}=\\big ((1-\\varphi )\\sigma (B_0(\\lambda ))\\big )\\circ g^{j_2t_0}\\prod _{k=0}^{j_2-1}\\varphi \\circ g^{kt_0}.\\end{array}}$ Since the Liouville measure is invariant under the geodesic flow, the contribution of the principal term of (REF ) to $S_f(\\lambda )$ for $\\lambda =1$ is $\\int _{S^*M}\\sigma (A)\\sum _{j=1}^l\\big ((1-\\varphi )\\sigma (B_0(\\lambda ))\\big )\\circ g^{jt_0}\\prod _{k=1}^{j-1} \\varphi \\circ g^{kt_0}\\,d\\mu _L.$ Now, by (REF ), $\\sigma (B_0(1))=f(\\xi _{+\\infty })$ on the support of the integrated expression; recombining the terms as in the proof of Proposition REF , we get the right-hand side of (REF ), with an $\\mu _L(\\mathcal {T}(lt_0))$ remainder.", "The subprincipal terms (and also the difference $S_f(\\lambda )-S_f(1)$ for $\\lambda \\in [1,1+h]$ ) are estimated using the bound on the Liouville measure of the set where $B_1^jB^j_{2,s}$ is microsupported; arguing as in the proof of Proposition REF , we see that they are bounded by a constant times $r(h,\\Lambda _0)$ .", "Expansion of the trace of spectral projectors in powers of $h$.", "We now use the results obtained so far to derive an asymptotic expansion for the trace of the product of the spectral projector $\\operatorname{1\\hspace{-2.75pt}l}_{[0,\\lambda ^2]}(P(h))$ with a compactly supported pseudodifferential operator, with the remainder depending on the classical escape rate for up to twice the Ehrenfest time.", "Here we denote $P(h):=h^2(\\Delta -c_0),$ with the constant $c_0$ from (A1).", "It will also be more convenient for us to use the spectral parameter $s=\\lambda ^2$ in the following corollary and theorem (not to be confused with the time variable $s$ used in Lemma REF ).", "We start with the following consequence of the decomposition (REF ), the bound (REF ), and the spectral formula (REF ): Corollary 5.14 Take $\\Lambda _0>\\Lambda _{\\max }$ , with $\\Lambda _{\\max }$ defined in (REF ), and let $\\mathcal {T}(t)$ be defined in (REF ).", "For $\\varepsilon >0$ , let $\\varphi \\in C_0^\\infty ((1-\\varepsilon ,1+\\varepsilon ))$ equal to 1 near $[1-\\varepsilon /2,1+\\varepsilon /2]$ and for $s\\in \\mathbb {R}$ , define $\\varphi _s:=\\varphi \\cdot \\operatorname{1\\hspace{-2.75pt}l}_{(-\\infty ,s]}$ .", "If $\\varepsilon >0$ is small enough, then for each compactly supported $A \\in \\Psi ^0(M)$ , there exist some functions $S_h(s),Q_h(s)$ and some constants $C>0$ , $C_k>0$ such that for all $s\\in \\mathbb {R}$ and all $k\\in \\mathbb {N}$ ${\\begin{array}{c}\\operatorname{Tr}\\big (A\\varphi _s(P(h))\\big )=S_h(s)+Q_h(s),\\ |\\partial _{s}^kS_h(s)|\\le C_kh^{-n-1-k/2},\\\\|Q_h(s+u)-Q_h(s)|\\le Ch^{-n}\\mu _L\\bigg (\\mathcal {T}\\bigg (\\frac{|\\log h|}{\\Lambda _0}\\bigg )\\bigg )+\\mathcal {O}(h^\\infty )\\text{ for }u\\in [0,h].\\end{array}}$ By (REF ), $\\operatorname{Tr}\\big (A\\varphi _s(P(h))\\big )=(2\\pi h)^{-n-1}\\int \\limits _{\\sqrt{1-\\varepsilon }}^{\\sqrt{1+\\varepsilon }} \\lambda ^n f_\\Pi (\\lambda /h)\\varphi _s(\\lambda ^2)\\int \\limits _{\\partial \\overline{M}}\\langle A E_h(\\lambda ,\\xi ),E_h(\\lambda ,\\xi )\\rangle \\,d\\xi d\\lambda .$ By Proposition REF , we may assume that $A\\in \\Psi ^{\\operatorname{comp}}$ .", "Now, note that the decomposition (REF ) (with $l=2l_e$ ) is still valid in any $\\mathcal {O}(h)$ sized interval inside $(\\sqrt{1-\\varepsilon },\\sqrt{1+\\varepsilon })$ , if $\\varepsilon $ is small enough.", "More precisely, if we write $S_h(s):=(2\\pi h)^{-n-1}\\int \\limits _{\\sqrt{1-\\varepsilon }}^{\\sqrt{1+\\varepsilon }}\\lambda ^nf_\\Pi (\\lambda /h)\\varphi _s(\\lambda ^2)S_1(\\lambda )\\,d\\lambda ,$ where $S_1(\\lambda )$ is defined by (REF ) with $f(\\xi )\\equiv 1$ , then we have the expansion (REF ) with $Q_h(s)$ satisfying the required bound.", "To estimate the derivatives of $S_h(s)$ , it now suffices to use the bound (REF ), noting that it is valid for $|\\lambda ^2-1|<\\varepsilon $ if $\\varepsilon $ is small enough.", "Using the last corollary, we can show the following trace decomposition with a fractal remainder, the proof of which is based on a Tauberian argument: Theorem 4 Let $P(h)$ be defined in (REF ), let $a\\in S^{0}(M)$ be compactly supported and $A={\\rm Op}_h(a)\\in \\Psi ^{0}(M)$ a compactly supported quantization.", "Then there exist some smooth differential operators $L_j$ of order $2j$ on $T^*M$ , depending on the quantization procedure ${\\rm Op}_h$ , with $L_0=1$ , such that for any compact interval $I\\subset (0,\\infty )$ , all $s\\in I$ , all $h>0$ small, and all $N\\in \\mathbb {N}$ $\\begin{split}{\\rm Tr}\\big (A\\operatorname{1\\hspace{-2.75pt}l}_{[0,s]}(P(h))\\big )= &(2\\pi h)^{-n-1}\\sum _{j=0}^{N} h^j\\int \\limits _{|\\nu |_g^2\\le s}L_ja\\, d\\mu _\\omega +h^{-n}\\mathcal {O}\\big (\\mu _L(\\mathcal {T}(\\Lambda _0^{-1}|\\log h|))+h^{N}\\big )\\end{split}$ where the remainder is uniform in $s$ .", "Here $\\mu _\\omega $ is the standard volume form on $T^*M$ ; we have $\\mu _\\omega =\\omega _S^{n+1}/(n+1)!$ , where $\\omega _S$ is the symplectic form.", "By rescaling $h$ , it suffices to prove the result for $|s-1|\\le \\varepsilon /2$ where $\\varepsilon >0$ is obtained in Corollary REF , we can thus assume $|s-1|\\le \\varepsilon /2$ .Ê Let $\\varphi _s$ be defined as in Corollary REF , and $\\psi \\in C_0^\\infty ((-1+\\varepsilon /2,1-\\varepsilon /2))$ such that $\\psi +\\varphi =1$ on $[0,1+\\varepsilon /2]$ .", "For $s\\in (1-\\varepsilon /2,1+\\varepsilon /2)$ , one has $\\operatorname{1\\hspace{-2.75pt}l}_{[0,s]}(P(h))=\\varphi _s(P(h))+\\psi (P(h))$ and it suffices to study the expansion in $h$ of $ \\sigma _{A,h}(s)$ and $\\operatorname{Tr}(A\\psi (P(h)))$ where $\\sigma _{A,h}(s):=\\operatorname{Tr}\\big (A\\varphi _s(P(h))\\big )=\\operatorname{Tr}\\big (A\\varphi _s(P(h))\\chi \\big ).$ if $\\chi \\in C_0^\\infty (M)$ is such that $A=\\chi A\\chi $ .", "Since $A$ is compactly supported, one can use the functional calculus of Helffer–Sjöstrand [8] to deduce that $A\\psi (P(h))\\chi \\in \\Psi ^{\\operatorname{comp}}(M)$ is a compactly supported and microsupported pseudodifferential operatorAn alternative method in the Euclidean near infinity setting is the functional calculus of Helffer–Robert [26].. Its trace has a complete expansion in powers of $h$ (see [8]): $\\operatorname{Tr}(A\\psi (P(h))\\chi )=(2\\pi h)^{-n-1}\\sum _{j=0}^Nh^j\\int _{T^*M}L^{\\prime \\prime }_ja \\,d\\mu _\\omega +\\mathcal {O}(h^{-n+N})$ where $L^{\\prime \\prime }_j$ are some differential operators of order $2j$ and $L^{\\prime \\prime }_0a(m,\\nu )=a(m,\\nu )\\psi (|\\nu |_g^2)$ .", "Let us now analyze the function $\\sigma _{A,h}$ .", "This is a smooth function of $s>0$ by the smoothness assumption on the $E_h(\\lambda ,\\xi )$ in $\\lambda $ , it is constant in $s$ for $|1-s|>\\varepsilon $ , and we know that $\\sigma _{A,h}(s)=\\mathcal {O}(h^{-n-1})$ uniformly in $s$ by Lemma REF .", "Let $\\theta (s)\\in {S}(\\mathbb {R})$ be a Schwartz function such that $\\hat{\\theta }\\in C_0^\\infty (-\\eta ,\\eta )$ for some small $\\eta >0$ and $\\hat{\\theta }(t)=1$ near $t=0$ , and let $\\theta _h(s)=h^{-1}\\theta (s/h)$ .", "We write $\\sigma ^{\\prime }_{A,h}(s):=\\partial _s \\sigma _{A,h}(s)=\\operatorname{Tr}(A\\varphi (P(h))d\\Pi _{s}(P(h))\\chi )\\in C_0^\\infty ((0,\\infty ))$ where $d\\Pi _s(P(h))$ is the spectral measure of $P(h)$ .", "The operator $A\\varphi (P(h))d\\Pi _{s}(P(h))\\chi $ has a smooth compactly supported kernel and is trace class.", "We clearly have $\\sigma ^{\\prime }_{A,h}\\star \\theta _h\\in {S}(\\mathbb {R})$ and by a simple computation, its semi-classical Fourier transform is given by $\\int _{\\mathbb {R}}e^{-i\\frac{t}{h}s}\\sigma ^{\\prime }_{A,h}\\star \\theta _h(s)ds=\\operatorname{Tr}(A\\varphi (P(h))e^{-i\\frac{t}{h}P(h)})\\hat{\\theta }(t)$ and thus $\\sigma ^{\\prime }_{A,h}\\star \\theta _h(s)=\\frac{1}{2\\pi h}\\int _{\\mathbb {R}}e^{i\\frac{s}{h}t} \\operatorname{Tr}(A\\varphi (P(h))e^{-i\\frac{t}{h}P(h)})\\hat{\\theta }(t)dt.$ Now we can apply Lemma REF with $B_s=\\frac{1}{2}e^{ic_0hs/2}\\hat{\\theta }(-s/2)A\\varphi (P(h))$ ; the condition (REF ) is satisfied because $\\hat{\\theta }$ is supported in a small neighborhood of zero.", "This shows that, as $h\\rightarrow 0$ , we have the expansion (locally uniformly in $s$ ) $\\sigma ^{\\prime }_{A,h}\\star \\theta _h(s)=(2\\pi h)^{-n-1}\\Big (\\sum _{j=0}^{N} h^j\\int _{S^*M}\\widetilde{L}_jb(m,\\sqrt{s}\\nu )d\\mu _L(m,\\nu )+\\mathcal {O}(h^{N+1})\\Big )$ for all $N\\in \\mathbb {N}$ , where $b$ is a symbol such that ${\\rm Op}_h(b)=\\frac{1}{2}A\\varphi (P(h))+\\mathcal {O}(h^\\infty )$ , $\\widetilde{L}_j$ are differential operators of order $2j$ on $T^*M$ , smooth in $\\sqrt{s}$ , with $\\widetilde{L}_0=s^{\\frac{n-1}{2}}$ .", "In particular, one has $\\widetilde{L}_0b(m,\\sqrt{s}\\nu )=\\frac{1}{2}s^{\\frac{n-1}{2}}a(m,\\sqrt{s}\\nu )\\varphi (s|\\nu |_g^2)+\\mathcal {O}(h)$ .", "Notice that $b$ is supported in $\\lbrace (m,\\nu )\\in T^*M\\mid |\\nu |_g^2\\in \\operatorname{supp}\\varphi \\rbrace $ thus $\\widetilde{L}_jb(m,\\sqrt{s}\\nu )$ is smooth in $s\\in \\mathbb {R}$ when $(m,\\nu )\\in S^*M$ .", "Since $\\sigma _{A,h}(s)$ is bounded in $s$ , the convolution $\\sigma _{A,h}\\star \\theta _h(s)$ is well defined (as an element in $L^\\infty (\\mathbb {R})$ ) and we have for all $N$ and for all $s\\le 2$ $\\begin{split}\\sigma _{A,h}\\star \\theta _h(s)= (2\\pi h)^{-n-1}\\Big (\\sum _{j=0}^{N} h^j\\int _{0}^s\\int _{S^*M}\\widetilde{L}_jb(m,\\sqrt{u}\\nu )d\\mu _L(m,\\nu )du+\\mathcal {O}(h^{N+1})\\Big ).\\end{split}$ We are going to show that, uniformly in $s\\in \\mathbb {R}$ , $\\sigma _{A,h}(s)-\\sigma _{A,h}\\star \\theta _h(s)=\\mathcal {O}(h^\\infty )+\\mathcal {O}\\big (h^{-n}\\mu _L(\\mathcal {T}(\\Lambda _0^{-1}|\\log h|))\\big ),$ using the decomposition $\\sigma _{A,h}(s)=S_h(s)+Q_h(s)$ defined in (REF ).", "Since $\\partial _s S_h(s)$ is a compactly supported symbol we get by integrating by parts $N$ times $(1-\\hat{\\theta }(t)) \\int e^{-i\\frac{t}{h}s}\\partial _s S_{h}(s)ds=(1-\\hat{\\theta }(t))\\int e^{-i\\frac{t}{h}s}\\bigg (\\frac{h}{it}\\bigg )^N\\partial ^{N+1}_{s}(S_{h}(s))ds=\\mathcal {O}\\bigg (\\frac{h^{N/4}}{(1+|t|)^{N}}\\bigg )$ for all $t\\in \\mathbb {R}$ and all $N\\gg 1$ .", "Thus, taking the Fourier transform we deduce that $S_h(s)-S_h\\star \\theta _h(s)=\\mathcal {O}(h^{\\infty })$ uniformly in $s$ .", "From (REF ), we obtain by induction that for all $s,u$ $|Q_h(s+u)-Q_h(s)|\\le Ch^{-n}\\bigg (1+\\frac{|u|}{h}\\bigg )\\mu _L(\\mathcal {T}(\\Lambda _0^{-1}|\\log h|))+\\mathcal {O}(h^\\infty )$ then multiplying by $\\theta _h(-u)$ and integrating in $u$ , we obtain (REF ).", "Given (REF ), we have ${\\begin{array}{c}\\sigma _{A,h}(s)=(2\\pi h)^{-n-1}\\sum _{j=0}^{N} h^j\\int _{0}^s\\int _{S^*M}\\widetilde{L}_jb(m,\\sqrt{u}\\nu )d\\mu _L(m,\\nu )du\\\\+\\mathcal {O}\\big (h^{-n}\\mu _L(\\mathcal {T}(\\Lambda _0^{-1}|\\log h|))\\big )+\\mathcal {O}(h^{-n+N}).\\end{array}}$ Since the symbol of $b$ is explicitly obtained from $a$ using Moyal product, we can rewrite this expression with $a$ instead of $b$ and with some new differential operators with the same properties as $\\widetilde{L}_j$ but supported in $\\lbrace |\\nu |^2_g\\in \\operatorname{supp}\\varphi \\rbrace $ ; using polar coordinates $S^*M\\times \\mathbb {R}^+_{\\sqrt{u}}$ on $T^*M$ , we deduce that there exist some differential operators $L_j^{\\prime }$ of order $2j$ on $T^*M$ such that $\\int _{0}^s \\int _{S^*M}\\widetilde{L}_jb(m,\\sqrt{u}\\nu )d\\mu _L(m,\\nu )du=\\int _{|\\nu |^2_g\\le s}L^{\\prime }_ja(m,\\nu )d\\mu _\\omega (m,\\nu )$ and $L^{\\prime }_0a(m,\\nu )= \\varphi (|\\nu |^2_g)a(m,\\nu )$ .", "Combining this with (REF ) and (REF ), we obtain the desired result where $L_j$ in the statement of the Theorem corresponds now to $L^{\\prime }_j+L^{\\prime \\prime }_j$ ." ], [ "Euclidean near infinity manifolds", "In this section, we assume that $(M,g)$ is a complete Riemannian manifold such that there exists a compact set $K_0\\subset M$ such that for $\\mathcal {E}:=M\\setminus K_0$ , $(\\mathcal {E},g) \\textrm { is isometric to }(\\mathbb {R}^{n+1}\\setminus B(0,R), g_{\\rm eucl})$ where $R>0$ , $B(0,R)$ is the Euclidean ball of center 0 and radius $R$ and $g_{\\rm eucl}$ is the Euclidean metric.", "We will check that all the assumptions of Section  are satisfied." ], [ "Geometric assumptions", "We let $x\\in C^\\infty (M)$ be an everywhere positive function equal to $x(m)=|m|^{-1}$ in $\\mathcal {E}$ identified with $\\mathbb {R}^{n+1}\\setminus B(0,R)$ , and such that $x\\ge R^{-1}$ in $K_0$ .", "(We take it instead of the function $(1+|m|^2)^{-1/2}$ used in Section  for the model case of $\\mathbb {R}^{n+1}$ , to simplify the calculations and since we no longer need smoothness at zero.)", "We shall use the polar coordinates $m=\\omega / x$ in $\\mathcal {E}$ , where $\\omega \\in \\mathbb {S}^{n}$ .", "Assumption (G1) is satisfied by taking the radial compactification of $M$ , i.e.", "adding the sphere at infinity: the map $\\psi :\\mathbb {R}^{n+1}\\setminus B(0,R)\\rightarrow (0,1/R)\\times \\mathbb {S}^n$ defined by $\\psi (m)=(x(m),x(m)m)$ is a diffeomorphism and the radial compactification of $M$ is obtained by setting $\\overline{M}=M\\sqcup \\partial \\overline{M}$ where $\\partial \\overline{M}:=\\mathbb {S}^n$ , the smooth structure on $\\overline{M}$ is the same as before on $M$ but we extend it to $\\overline{M}$ by asking that $\\psi $ extends smoothly to the boundary $\\partial \\overline{M}$ and $\\psi (\\xi )=(0,\\xi )$ if $\\xi \\in \\partial \\overline{M}=\\mathbb {S}^n$ (see for instance [35] for more details).", "In other words, smooth functions on $\\overline{M}$ are smooth functions on $M$ with an asymptotic expansion in integer powers of $1/|m|$ to any order near infinity.", "Assumption (G2) is clearly satisfied for $\\varepsilon _0:=1/2R$ since the trajectories of the geodesic flow in $x\\le \\varepsilon _0$ are simply $g^t(m,\\nu )=(m+t\\nu ,\\nu )$ .", "A point $(m,\\nu )\\in S^*M$ is directly escaping in the forward direction in the sense of Definition REF if and only if $x(m)\\le \\varepsilon _0$ and $m\\cdot \\nu \\ge 0$ .", "Now, (G3) is satisfied with $\\xi _{+\\infty }(m,\\nu )=\\nu $ for $(m,\\nu )\\in \\mathcal {DE}_+$ .", "For the assumption (G4), we define ${\\begin{array}{c}\\widetilde{U}_\\infty =\\lbrace x<\\varepsilon _0\\rbrace \\times \\partial \\overline{M}\\subset \\overline{M}\\times \\partial \\overline{M},\\\\\\phi _\\xi (m)=m\\cdot \\xi ,\\ (m,\\xi )\\in U_\\infty .\\end{array}}$ Then $\\tau :U_\\infty \\rightarrow S^*M$ from (REF ) maps each $(m,\\xi )\\in (\\mathbb {R}^n\\setminus B(0,2R))\\times \\mathbb {S}^n$ to itself as an element of $S^*(\\mathbb {R}^n\\setminus B(0,2R))$ .", "Assumptions (G4) and (G5) follow immediately.", "To see assumption (G6), we note that for $x(m),x(m^{\\prime })<\\varepsilon _0$ and some $\\xi \\in \\mathbb {S}^n$ , we have $\\partial _\\xi \\phi _\\xi (m)=\\partial _\\xi \\phi _\\xi (m^{\\prime })$ if and only if $m-m^{\\prime }$ is a multiple of $\\xi $ ." ], [ "Distorted plane waves and analytic assumptions", "We recall a few well-known facts about scattering theory on perturbations of $\\mathbb {R}^n$ , we refer to [35] for a geometric approach and to [37], [23] in a more general setting (asymptotically Euclidean case).", "A plane wave for the flat Laplacian on $\\mathbb {R}^{n+1}$ is the function, for $\\lambda \\in (1/2,2)$ , $u(\\lambda , \\xi ;m):=ce^{{i\\lambda \\over h} m\\cdot \\xi } , \\quad \\xi \\in \\mathbb {S}^{n},\\ m\\in \\mathbb {R}^{n+1},\\ c\\in \\mathbb {C}.$ This is a semiclassical Lagrangian distribution, its oscillating phase has level sets given by planes orthogonal to $\\xi $ .", "The continuous spectrum of the Laplacian $\\Delta $ associated to the metric $g$ is the half-line $[0,\\infty )$ .", "We will take the resolvent of $h^2\\Delta $ to be the $L^2$ -bounded operator $R_h(\\lambda ):=(h^2\\Delta -\\lambda ^2)^{-1} \\textrm { in } {\\rm Im}(\\lambda )<0.$ This admits a continuous extension to $\\lbrace \\lambda \\ne 0, {\\rm Im}(\\lambda )\\le 0\\rbrace $ as a bounded operator from $L^2_{\\rm comp}$ to $L^2_{\\rm loc}$ .", "For $\\lambda >0$ we call $R_h(\\lambda )$ the incoming resolvent and $R_h(-\\lambda )$ the outgoing resolvent.", "For $\\lambda >0$ , $h>0$ , and $m\\in M$ fixed, the Schwartz kernel $R_h(\\lambda ;m,m^{\\prime })$ of the incoming resolvent $R_h(\\lambda )$ has an asymptotic expansion along the lines $m^{\\prime }\\in m^{\\prime }_0+\\mathbb {R}\\xi $ directed by $\\xi \\in S^n$ given by $R_h(\\lambda ;m,\\xi /x^{\\prime })\\sim _{x^{\\prime }\\rightarrow 0} (x^{\\prime })^{\\frac{n}{2}}e^{-\\frac{i\\lambda }{hx^{\\prime }}}f(\\lambda ,\\xi ;m)+\\mathcal {O}((x^{\\prime })^{\\frac{n}{2}+1})$ for some smooth function $f$ and the remainder is uniform for $m$ in compact sets (see for example [37], [23]).", "Using this expansion, we define the distorted plane wave by $E_h(\\lambda ,\\xi ;m):={2i\\lambda h}\\Big (\\frac{2\\pi h}{i\\lambda }\\Big )^\\frac{n}{2}\\lim _{x^{\\prime }\\rightarrow 0}[(x^{\\prime })^{-n/2}e^{{i\\lambda \\over hx^{\\prime }}}R_h(\\lambda ; m,\\xi /x^{\\prime })],$ with $\\xi \\in S^n$ and $\\xi /x^{\\prime }\\in \\mathcal {E}$ .", "This is a smooth function of $(m,\\xi )\\in M\\times \\mathbb {S}^{n}$ , and in the case of $M=\\mathbb {R}^{n+1}$ it is given by (REF ) with $c=1$ (see [35]).", "We shall use the notation $E_h(\\lambda ,\\xi )$ for the $C^\\infty (M)$ function defined by $m\\mapsto E_h(\\lambda ,\\xi ;m)$ and we notice that $(h^2\\Delta -\\lambda ^2)E_h(\\lambda ,\\xi )=0$ in $M$ .", "One has $\\overline{E_h(\\lambda ,\\xi ;m)}=E_h(-\\lambda ,\\xi ;m)$ since $R_h(\\lambda )^*=R_h(-\\lambda )$ for $\\lambda \\in \\mathbb {R}$ , and the decomposition of the spectral measure in terms of these functions in given as follows: by Stone's formula, the semiclassical spectral measure is given by $d\\Pi _h(\\lambda )=\\frac{i\\lambda }{\\pi }(R_h(\\lambda )-R_h(-\\lambda ))\\,d\\lambda \\quad \\textrm { for }\\lambda \\in (0,\\infty )$ in the sense that $F(h^2\\Delta )=\\int _0^\\infty F(\\lambda ^2)d\\Pi _h(\\lambda )$ for any bounded function $F$ ; by combining this with the Green's type formula of [23], we deduce that $d\\Pi _h(\\lambda ;m,m^{\\prime })=\\lambda ^n(2\\pi h)^{-n-1}\\int _{\\mathbb {S}^n}E_h(\\lambda ,\\xi ;m)\\overline{E_h(\\lambda ,\\xi ;m^{\\prime })}\\, d\\xi d\\lambda .$ Here $d\\xi $ corresponds to the standard volume form on the sphere $\\mathbb {S}^n$ .", "The assumptions (A1) and (A2) are then satisfied.", "In fact, using [23], one can define distorted plane waves and verify assumptions (A1) and (A2) for the more general case of scattering manifolds.", "Outgoing/incoming decomposition.", "We now construct the decomposition (REF ) of $E_h$ into the outgoing and incoming parts and verify assumptions (A3)–(A8).", "Take $\\chi _0\\in C^\\infty (\\overline{M})$ (thus constant in $\\xi $ ) supported in $\\lbrace x<\\varepsilon _0\\rbrace $ and equal to 1 near $\\lbrace x\\le \\varepsilon _0/2\\rbrace $ , so that assumptions (A3) and (A7) hold, where we put $\\varepsilon _1:=\\varepsilon _0/2$ .", "We next put $E_h^0(\\lambda ,\\xi ;m):=e^{{i\\lambda \\over h}m\\cdot \\xi },\\ x(m)<\\varepsilon _0,$ so that (A4) holds with $b^0\\equiv 1$ and (A8) follows.", "We then claim that $E_h=\\chi _0 E_h^0+E_h^1,$ where $E_h^1:=-R_h(\\lambda )F_h,\\ F_h(\\lambda ,\\xi )=(h^2\\Delta -\\lambda ^2)(\\chi _0E_h^0(\\lambda ,\\xi ))=[h^2\\Delta ,\\chi _0]E_h^0(\\lambda ,\\xi ).$ We can apply $R_h(\\lambda )$ to $F_h(\\lambda ,\\xi )$ as the latter lies in $C_0^\\infty (M)$ ; in fact, $\\operatorname{supp}F_h\\subset \\lbrace \\varepsilon _0/2<x<\\varepsilon _0\\rbrace $ .", "To show (REF ), note that the incoming resolvent $R_h(\\lambda )$ satisfies $R_h(\\lambda )\\chi _1=\\chi _0R_h^0(\\lambda )\\chi _1-R_h(\\lambda )[h^2\\Delta ,\\chi _0]R_h^0(\\lambda )\\chi _1$ if $\\chi _1\\in C^\\infty (\\overline{M})$ is such that $\\chi _0=1$ on $\\operatorname{supp}(\\chi _1)$ and $R_h^0(\\lambda )$ is the incoming scattering resolvent of the free semiclassical Laplacian $h^2\\Delta $ on $\\mathbb {R}^{n+1}$ (we use again the isometry $\\mathcal {E}\\simeq \\mathbb {R}^{n+1}\\setminus B(0,R)$ ).", "To obtain $E_h(\\lambda ,\\xi )$ from (REF ), we shall use definition (REF ); consider the Schwartz kernels of the operators in (REF ) and multiply them by $(x^{\\prime })^{-\\frac{n}{2}}e^{i\\lambda \\over hx^{\\prime }}$ in the right variable; since $\\chi _1=1$ near infinity, one has by the remark following (REF ) $E^0_h(\\lambda ,\\xi ;m)={2i\\lambda h}(\\frac{2\\pi h}{i\\lambda })^\\frac{n}{2}\\lim _{x^{\\prime }\\rightarrow 0}[(x^{\\prime })^{-n/2}e^{{i\\lambda \\over hx^{\\prime }}}R^0_h(\\lambda ; m,\\xi /x^{\\prime })\\chi _1(\\xi /x^{\\prime })].$ Now the Schwartz kernel $\\kappa (m^{\\prime \\prime },m^{\\prime })$ of $[h^2\\Delta ,\\chi _0]R_h^0(\\lambda )\\chi _1$ is smooth in $M\\times M$ by ellipticity and compactly supported in the first variable, moreover $R_h(m,m^{\\prime \\prime })$ is in $L^1_{\\rm loc}$ in the $m^{\\prime \\prime }$ variable for $m\\in M$ fixed, thus we get by dominated convergence for fixed $m,\\xi $ ${\\begin{array}{c}\\lim _{x^{\\prime }\\rightarrow 0}\\Big ({x^{\\prime }}^{-\\frac{n}{2}}e^{\\frac{i\\lambda }{hx^{\\prime }}}\\int _MR_h(m,m^{\\prime \\prime })\\kappa (m^{\\prime \\prime },\\xi /x^{\\prime }){\\rm dvol}_M(m^{\\prime \\prime })\\Big )=\\\\\\int _{M}R_h(m,m^{\\prime \\prime })[h^2\\Delta ,\\chi _0]E_h^0(\\lambda ,\\xi ; m^{\\prime \\prime }){\\rm dvol}_M(m^{\\prime \\prime })\\end{array}}$ and by combining this with (REF ), this proves (REF ).", "Microlocalization of $E^1_h$.", "It remains to verify assumptions (A5) and (A6).", "By rescaling $h$ and using that $E_h(\\lambda ,\\cdot )$ depends only on $\\lambda /h$ , we may assume that $\\lambda =1$ .", "Fix $\\xi $ and take $\\chi _2\\in C^\\infty (\\overline{M})$ equal to 1 near $\\lbrace x\\le \\varepsilon _0\\rbrace $ , but supported inside $\\mathcal {E}$ .", "Then $\\chi _2 E^1_h=R^0_h(\\lambda )F^0_h,\\quad F^0_h:=(h^2\\Delta -\\lambda ^2)(\\chi _2E^1_h)=-F_h+[h^2\\Delta ,\\chi _2]E^1_h.$ The function $F^0_h$ is supported inside $\\lbrace x>\\varepsilon _0/2\\rbrace $ and $\\Vert F^0_h\\Vert _{H^{-1}_h}\\le Ch(1+\\Vert E_h\\Vert _{L^2(\\lbrace x\\ge \\varepsilon _0\\rbrace )}).$ The free resolvent $R^0_h(\\lambda )$ is bounded $H^{-1}_{h,\\operatorname{comp}}\\rightarrow L^2_{\\operatorname{loc}}$ with norm $\\mathcal {O}(h^{-1})$ by [4]; therefore, for each compact set $K\\subset M$ , there exists a constant $C_K$ such that $\\Vert E^1_h\\Vert _{L^2(K)}\\le C_K (1+\\Vert E_h\\Vert _{L^2(\\lbrace x\\ge \\varepsilon _0\\rbrace )}).$ This shows (A5), namely that the function $\\widetilde{E}^1_h:={E^1_h\\over 1+\\Vert E_h\\Vert _{L^2(\\lbrace x\\ge \\varepsilon _0\\rbrace )}}$ is $h$ -tempered.", "To prove (A6), we use semiclassical elliptic estimate and propagation of singularities (see for example [55]).", "We have $(h^2\\Delta -\\lambda ^2)\\widetilde{E}^1_h=-\\widetilde{F}_h,\\ \\widetilde{F}_h:={F_h\\over 1+\\Vert E_h\\Vert _{L^2(\\lbrace x\\ge \\varepsilon _0\\rbrace )}}.$ Now, $F_h$ is a Lagrangian distribution associated to $\\lbrace (m,\\xi )\\mid m\\in \\operatorname{supp}(d\\chi _0)\\rbrace $ ; therefore, $\\operatorname{WF}_h(\\widetilde{F}_h)\\subset \\operatorname{WF}_h(F_h)\\subset W_\\xi ,$ with $W_\\xi \\subset S^*M$ defined in (REF ).", "Take $(m,\\nu )\\in \\operatorname{WF}_h(\\widetilde{E}^1_h)$ .", "By the elliptic estimate, $(m,\\nu )\\in S^*M$ .", "Next, if $\\gamma (t)=g^t(m,\\nu )$ , then by propagation of singularities, either $\\gamma (t)\\in \\operatorname{WF}_h(\\widetilde{F}_h)\\subset W_\\xi $ for some $t\\ge 0$ or $\\gamma (t)\\in \\operatorname{WF}_h(\\widetilde{E}^1_h)$ for all $t\\ge 0$ .", "Now, the free resolvent $R^0_h(\\lambda )$ is semiclassically incoming in the following sense: if $f$ is a compactly supported $h$ -tempered family of distributions, then for each $(m^{\\prime },\\nu ^{\\prime })\\in \\operatorname{WF}_h(R^0_h(\\lambda )f)$ , there exists $t\\ge 0$ such that $g^t(m^{\\prime },\\nu ^{\\prime })\\in \\operatorname{supp}f$ .", "This can be seen for example from the explicit formulas for $R^0_h(\\lambda )$ , see [35].", "By (REF ) and since $\\operatorname{supp}(F^0_h)\\subset \\lbrace x>\\varepsilon _0/2\\rbrace $ , we see that for $(m^{\\prime },\\nu ^{\\prime })\\in \\operatorname{WF}_h(\\widetilde{E}^1_h)$ , we cannot have $x(m^{\\prime })<\\varepsilon _0/2$ and $m^{\\prime }\\cdot \\nu ^{\\prime }\\ge 0$ .", "Therefore, if $\\gamma (t)\\notin W_\\xi $ for all $t\\ge 0$ , then $\\gamma (t)$ is trapped as $t\\rightarrow +\\infty $ ; this proves (A6)." ], [ "Hyperbolic near infinity manifolds", "In this section, we verify the assumptions of Section  for certain asymptotically hyperbolic manifolds.", "Let $(M,g)$ be an $(n+1)$ -dimensional asymptotically hyperbolic manifold as defined in the introduction.", "It has a compactification $\\overline{M}=M\\cup \\partial \\overline{M}$ and the metric can be written in the product form (REF ): $g=\\frac{dx^2+h(x)}{x^2}$ where $x$ is a boundary defining function and $h(x)$ a smooth family of metrics on $\\partial \\overline{M}$ defined near $x=0$ .", "The function $x$ putting the metric in the form (REF ) is not unique, and those functions (thus satisfying $|d\\log (x)|_g=1$ near $\\partial \\overline{M}$ ) are called geodesic boundary defining functions.", "The set of such functions parametrizes the conformal class of $h(0)$ , as shown in [17].", "The metric is called even if $h(x)$ is an even function of $x$ , this condition is independent of the choice of geodesic boundary defining function.", "A choice of geodesic boundary defining function induces a metric on $\\partial \\overline{M}$ by taking $h_0=h(0)=x^2g|_{T\\partial \\overline{M}}$ , and therefore one has a Riemannian volume form, denoted $d\\xi $ , on $\\partial \\overline{M}$ induced by the choice of $x$ .", "Any other choice $\\hat{x}=e^{\\omega }x$ of boundary defining function induces a volume form $\\widehat{d\\xi }=e^{n\\omega _0}d\\xi \\quad \\textrm { where }\\omega _0=\\omega |_{\\partial \\overline{M}}.$ We will further assume that $M$ has constant sectional curvature $-1$ outside of some compact set, even though some of the assumptions of Section  hold for general asymptotically hyperbolic manifolds with no simplification provided by the additional assumption on curvature – we will give the proofs in higher generality where appropriate." ], [ "Geometric assumptions", "Let $(M,g)$ be an asymptotically hyperbolic manifold.", "The assumption (G1) is satisfied.", "We are now going to prove a Lemma which implies directly that the assumptions (G2) and (G3) are satisfied, except that this only proves continuous dependence of $\\xi _{+\\infty }$ in $(m,\\nu )$ in (G3).", "To prove $C^1$ dependence in a general setting, a bit more analysis would be required, but we shall later concentrate only on cases with constant curvature near infinity, in which case the dependence is smooth (see below).", "Lemma 7.1 Let $(M,g)$ an asymptotically hyperbolic manifold.", "Then there exists $\\varepsilon _0>0$ such that the function $x$ satisfies (REF ) and for any unit speed geodesic $\\gamma (t)=(m(t),\\nu (t))$ with $x(m(0))\\le \\varepsilon _0$ and $\\partial _t x(m(t))|_{t=0}\\le 0$ , one has the following: $\\partial _t x(m(t)) \\le 0$ for all $t\\ge 0$ and $m(t)$ converges in the topology of $\\overline{M}$ to some point $\\xi _{+\\infty }\\in \\partial \\overline{M}$ .", "More precisely, the distance with respect to the compactified metric $\\bar{g}=x^2g$ between $m(t)$ and $\\xi _{+\\infty }$ is bounded by $d_{\\bar{g}}( m(t), \\xi _{+\\infty })\\le Ct^{-1}.$ Consider coordinates $(m,\\nu )=(x,y;\\rho dx+\\theta \\cdot dy)$ on $T^*M$ near the boundary $\\partial \\overline{M}=\\lbrace x=0\\rbrace $ .", "The geodesic flow is the Hamiltonian flow of $p/2$ , where $p(m,\\nu )=x^2(\\rho ^2+|\\theta |^2_{h_m})$ ; if dots denote time derivatives with respect to the geodesic flow, we get $\\dot{x}=\\rho x^2, \\quad \\dot{\\rho }=-x^{-1} p(m,\\nu )-x^2\\partial _xh_{(x,y)}(\\theta ,\\theta )/2.$ Since $\\partial _x h_{(x,y)}$ is smooth up to $x=0$ , there exists a constant $C$ such that $|x^2\\partial _x h_{(x,y)}(\\theta ,\\theta )/2|\\le C x^2 h_{(x,y)}(\\theta ,\\theta )\\le C p(m,\\nu ).$ Therefore, there exists $\\varepsilon _0>0$ such that along any unit speed geodesic, we have $x\\le \\varepsilon _0\\Longrightarrow \\dot{\\rho }=-x^{-1}+\\mathcal {O}(1)\\le -x^{-1}/2<0.$ This in particular implies (REF ).", "Now, let $\\gamma (t)=(x(t),y(t);\\rho (t),\\theta (t))$ be a unit speed geodesic and assume that $x(0)\\le \\varepsilon _0$ and $\\dot{x}(0)\\le 0$ .", "It follows from (REF ) that for $t\\ge 0$ , we have $\\dot{x}(t)\\le 0$ and thus $x(t)\\le \\varepsilon _0$ .", "(Indeed, for each $s\\ge 0$ the minimal value of $x(t)$ on the interval $[0,s]$ has to be achieved at $t=s$ .)", "It remains to show that as $t\\rightarrow +\\infty $ , $x(t)$ converges to 0 and $y(t)$ converges to some $\\xi _{+\\infty }\\in \\partial M$ .", "For that, note that by (REF ), $\\dot{\\rho }(t)\\le -\\varepsilon _0^{-1}/2$ for $t\\ge 0$ ; since $\\dot{x}(0)\\le 0$ , we have $\\rho (0)\\le 0$ and thus $\\rho (t)\\le -{\\varepsilon _0^{-1}\\over 2}t.$ Setting $u(t):=x(t)^{-1}$ , we find $\\dot{u}(t)=-\\rho (t)\\ge (\\varepsilon _0^{-1}/2)t$ ; therefore, $x(t)\\le {\\varepsilon _0\\over 1+t^2/4}.$ In particular, $x(t)\\rightarrow 0$ as $t\\rightarrow +\\infty $ .", "Now the equation for $\\dot{y}(s)$ tells us that $\\dot{y}_i(t)=x\\sum _{k=1}^nh^{ki}_{(x,y)}x\\theta _k=\\mathcal {O}(x(t))=\\mathcal {O}(t^{-2})$ and therefore $|y(t)-y(t^{\\prime })|\\le C/t^{\\prime }$ for any $t>t^{\\prime }>0$ , which implies $\\lim _{t\\rightarrow \\infty }m(t)= \\xi _{\\infty }$ for some $\\xi _\\infty \\in \\partial \\overline{M}$ and $|m(t)-\\xi _\\infty |=\\mathcal {O}(1/t)$ .", "The geometric assumption (G4) is a more complicated one, and we will restrict ourselves to asymptotically hyperbolic manifolds with constant curvature $-1$ in a neighbourhood of $\\partial M$ and $x$ a geodesic boundary defining function.", "Let $\\xi \\in \\partial \\overline{M}$ , then there exists a neighborhood $V_{\\xi }$ of $\\xi $ in $\\overline{M}$ , and an isometric diffeomorphism $\\psi _{\\xi }$ from $V_{\\xi }\\cap M$ into the following neighbourhood $V_{q_0}$ of the north pole $q_0$ in the unit ball $\\mathbb {B}:=\\lbrace m\\in \\mathbb {R}^{n+1}; |m|<1\\rbrace $ equipped with the hyperbolic metric $g_{0}$ $V_{q_0}:=\\lbrace q\\in \\mathbb {B}\\mid |q-q_0|< 1/4\\rbrace , \\quad g_{0}=4\\frac{dq^2}{(1-|q|^2)^2}$ where $\\psi _{\\xi }(\\xi )=q_0$ and $|\\cdot |$ denotes the Euclidean length.", "This statement is proved for instance in [22].", "We shall choose the boundary defining function on the ball $\\mathbb {B}$ to be $x_0=2\\frac{1-|q|}{1+|q|}.$ and the induced metric $x_0^2g_0|_{\\mathbb {S}^n}$ on $\\mathbb {S}^n=\\partial \\overline{\\mathbb {B}}$ is the canonical one with curvature $+1$ .", "The function $x_0$ can be viewed locally as a boundary defining function (through the chart $\\psi _\\xi $ ) near a point $\\xi \\in \\partial \\overline{M}$ but in general there does not exist a global geodesic boundary defining function $x$ on $M$ so that $x=\\psi _\\xi ^*x_0$ in a whole family of charts $V_\\xi $ covering a neighbourhood of $\\partial \\overline{M}$ .", "We define for each $p\\in \\mathbb {S}^n$ the Busemann function on $\\mathbb {B}$ $\\phi ^{\\mathbb {B}}_{p}(q)=\\log \\Big (\\frac{1-|q|^2}{|q-p|^2}\\Big ).$ The geodesic trajectory $g^t(q,d \\phi ^{\\mathbb {B}}_{p}(q))$ generated by the differential $d\\phi ^{\\mathbb {B}}_{p}$ converges (in the Euclidean ball topology) to $p$ and the Lagrangian manifold $\\Lambda ^{\\mathbb {B}}_{p}:=\\lbrace (q,d\\phi ^{\\mathbb {B}}_{p}(q))\\in S^* {\\mathbb {H}^{n+1}}\\mid q\\in {\\mathbb {B}}\\rbrace $ is the stable manifold of the geodesic flow associated to $p$ on ${\\mathbb {B}}$ .", "The level sets of $\\phi ^{\\mathbb {B}}_p$ are horospheres based at $p$ .", "We cover a neighbourhood of $\\partial \\overline{M}$ by finitely many $V_{\\xi _j}$ for some $\\xi _j\\in \\partial \\overline{M}$ and take a partition of unity $\\chi _j\\in C^\\infty (\\partial \\overline{M})$ on $\\partial \\overline{M}$ with $\\chi _j$ supported in $V_{\\xi _j}\\cap \\partial \\overline{M}$ .", "Then there exists $\\varepsilon >0$ such that for all $j$ and all $\\xi \\in \\operatorname{supp}\\chi _j$ , the set $U_{\\xi }:=\\lbrace m\\in \\overline{M}\\mid d_{\\bar{g}}(m,\\xi )<\\varepsilon \\rbrace $ lies inside $V_{\\xi _j}$ , where $\\bar{g}=x^2g$ is the compactified metric.", "Put $U_\\infty :=\\lbrace (m,\\xi )\\in M\\times \\partial \\overline{M}\\mid m\\in U_\\xi \\rbrace .$ Define the function $\\phi _\\xi (m):=\\sum _j \\chi _j(\\xi ) \\phi ^{\\mathbb {B}}_{\\psi _{\\xi _j}(\\xi )}(\\psi _{\\xi _j}(m)),\\ (m,\\xi )\\in U_\\infty .$ Since $\\psi _{\\xi _j}$ are isometries, each function $\\phi ^j_\\xi (m):=\\phi ^{\\mathbb {B}}_{\\psi _{\\xi _j}(\\xi )}(\\psi _{\\xi _j}(m))$ is such that $d\\phi ^j_\\xi (m)$ is the unit covector which generates the unique geodesic in $M$ starting at $m$ , staying in $U_\\xi $ for positive times, and converging to $\\xi $ (therefore, the difference of any two functions $\\phi ^j_\\xi $ for different $j$ is a function of $\\xi $ only).", "Therefore $\\partial _m\\phi _\\xi (m)=\\sum _j\\chi _j(\\xi )\\partial _m\\phi ^j_\\xi (m)$ is also equal to this unit covector; (G4) and (G5) follow.", "The dependence of all objects in $m,\\xi $ is smooth here.", "Finally, (G6) can be reduced via $\\psi _{\\xi _j}$ to the following statement that can be verified by a direct computation: if $p\\in \\mathbb {S}^n$ and $q,q^{\\prime }\\in \\mathbb {B}$ , then $\\partial _p\\phi ^{\\mathbb {B}}_p(q)=\\partial _p\\phi ^{\\mathbb {B}}_p(q^{\\prime })$ if and only if $q$ and $q^{\\prime }$ lie on a geodesic converging to $p$ , and the matrix $\\partial ^2_{pq}\\phi ^{\\mathbb {B}}_p(q)$ has rank $n$ ." ], [ "Eisenstein functions and analytic assumptions", "Let $(M,g)$ be asymptotically hyperbolic.", "The Laplacian $\\Delta $ on $(M,g)$ has absolutely continuous spectrum on $[n^2/4,\\infty )$ and a possibly non-empty finite set of eigenvalues in $(0,n^2/4)$ .", "By [34], [19], if $g$ is an even metricThere is a simpler proof by Guillopé-Zworski [22] when the curvature is constant outside a compact set., the resolvent of the Laplacian $R(s):=(\\Delta -s(n-s))^{-1} \\quad \\textrm { defined in the half plane }{\\rm Re}(s)>n/2$ admits a meromorphic continuation to the whole complex plane $\\mathbb {C}$ , with poles of finite rank (i.e.", "the Laurent expansion at each pole consists of finite rank operators), as a family of bounded operators $R(s): x^NL^2(M)\\rightarrow x^{-N}L^2(M), \\quad \\textrm { if }\\,\\,\\, \\operatorname{Re}(s)-n/2+N>0,$ moreover it has no poles on the line ${\\rm Re}(s)=\\frac{n}{2}$ except possibly $s=\\frac{n}{2}$ , as shown by Mazzeo [33].", "The continuous spectrum for the spectral parameter $s$ correspond to $\\operatorname{Re}(s)=n/2$ and we write $s=\\frac{n}{2}+i\\lambda /h$ with $\\lambda >0$ bounded and $h>0$ small for the high-frequency regime.", "Let us fix a geodesic boundary defining function $x$ on $\\overline{M}$ .", "By [34], the resolvent integral kernel $R(s;m,m^{\\prime })$ near the boundary $\\partial \\overline{M}$ has an asymptotic expansion given as follows: for any $m\\in M$ fixed $m^{\\prime }\\mapsto R(s;m ,m^{\\prime } )x(m^{\\prime })^{-s} \\in C^\\infty (\\overline{M})$ and similarly for $m^{\\prime }\\in M$ fixed and $m\\rightarrow \\partial \\overline{M}$ .", "Since we are interested in high frequency asymptotics, we will consider the semiclassical rescaled versions $R_h(\\lambda ):=h^{-2}R(n/2+i\\lambda / h).$ Note that the physical region $\\operatorname{Re}s>n/2$ , in which the resolvent is bounded on $L^2$ , corresponds to $\\operatorname{Im}\\lambda <0$ , which agrees with our convention for Euclidean case, see (REF ).", "Definition 7.2 Let $1/2\\le |\\lambda |\\le 2$ and $h>0$ , then Eisenstein functions are the functions in $C^\\infty (M\\times \\partial \\overline{M})$ defined for any fixed $\\xi \\in \\partial \\overline{M}$ by the following limit of the resolvent kernel at infinity ${\\begin{array}{c}E_h(\\lambda ,\\xi ;m):=\\frac{2i \\lambda h}{C(\\lambda /h)}\\lim _{m^{\\prime }\\rightarrow \\xi }x(m^{\\prime })^{-n/2-i\\lambda /h}R_h(\\lambda ;m,m^{\\prime }), \\\\C(z):=2^{-iz}(2\\pi )^{-\\frac{n}{2}}\\frac{\\Gamma (\\frac{n}{2}+iz)}{\\Gamma (iz)}.\\end{array}}$ The normalisation constant in (REF ) is like the constant in (REF ) so that in $\\mathbb {B}$ , $E_h(\\lambda )$ is a horospherical wave as described below in (REF ).", "For any $\\xi \\in \\partial \\overline{M}$ , we will denote by $E_h(\\lambda ,\\xi )$ the function $m\\mapsto E_h(\\lambda ,\\xi ;m)$ , and we observe that they solve (REF ): $(h^2(\\Delta -n^2/4)-\\lambda ^2)E_h(\\lambda ,\\xi )=0.$ One also has $\\overline{E_h(\\lambda ,\\xi ;m)}=E_h(-\\lambda ,\\xi ;m)$ as an easy consequence of $R_h(\\lambda )^*=R_h(-\\lambda )$ for $\\lambda \\in \\mathbb {R}$ .", "From its definition, $E_h(\\lambda ,\\xi )$ depends on the choice of the boundary defining function $x$ , but considering such a change we easily see from (REF ) that the density on $\\partial \\overline{M}$ $\\langle AE_h(\\lambda ,\\xi ),E_h(\\lambda ,\\xi )\\rangle _{L^2(M)}\\,d\\xi \\quad \\textrm { for }\\, A\\in \\Psi ^{\\operatorname{comp}}(M)$ is independent of $x$ .", "Let us recall the decomposition of the spectral measure in terms of these functions.", "By Stone's formula, the semiclassical spectral measure is given by $d\\Pi _h(\\lambda )=\\frac{i\\lambda }{\\pi }(R_h(\\lambda )-R_h(-\\lambda ))\\,d\\lambda \\quad \\textrm { for }\\lambda \\in (0,\\infty )$ in the sense that $F(h^2(\\Delta -n^2/4))=\\int _0^\\infty F(\\lambda ^2)d\\Pi _h(\\lambda )$ for any bounded function $F$ supported in $(0,\\infty )$ .", "Now we can write (see [19]) for any $m,m^{\\prime }$ $d\\Pi _h(\\lambda ;m,m^{\\prime })=\\frac{|C(\\lambda /h)|^2}{2\\pi h}\\int _{\\partial \\overline{M}}E_h(\\lambda ,\\xi ;m)E_h(-\\lambda ,\\xi ;m^{\\prime })d\\xi \\,\\, d\\lambda .$ where $(2\\pi h)^n|C(\\lambda /h)|^2\\rightarrow \\lambda ^n$ as $h\\rightarrow 0$ uniformly in $\\lambda \\in [1/2,2]$ .", "The assumptions (A1) and (A2) are then satisfied in the general asymptotically hyperbolic case (without asking the constant curvature near infinity).", "Outgoing/incoming decomposition.", "To check assumptions (A3)–(A8), we give a representation of the Eisenstein functions as sums of the `outgoing' part $E^0_h$ and the `incoming' part $E^1_h$ .", "We assume constant curvature near infinity in what follows.", "The expression for $E_h^{\\mathbb {B}}(\\lambda )$ on hyperbolic space $\\mathbb {H}^{n+1}$ viewed as a unit ball $\\mathbb {B}$ , defined using the boundary defining function $x_0$ of (REF ), is given by [20] ${\\begin{array}{c}E^{\\mathbb {B}}_h(\\lambda ,p; q)=\\Big (\\frac{1-|q|^2}{|q-p|^2}\\Big )^{n/2+i\\lambda / h},\\ p\\in \\mathbb {S}^n,\\ q\\in \\mathbb {B}\\end{array}}$ We thus set $E_h^0(\\lambda ,\\xi ;m)$ to be $E_h^0(\\lambda ,\\xi ;m):= e^{(n/2+i\\lambda /h)\\phi _\\xi (m)},$ where $\\phi _\\xi $ is the Busemann function defined in (REF ).", "Viewing the neighbourhood $U_{\\xi }$ as a subset of one of the $V_{\\xi _j}\\simeq _{\\psi _{\\xi _j}} V_{q_0}$ where $V_{q_0}\\subset \\mathbb {B}$ is defined in (REF ), the Laplacian in this hyperbolic chart pulls back to $\\Delta _{\\mathbb {H}^{n+1}}$ .", "Since $\\phi _\\xi (m)=\\phi ^{\\mathbb {B}}_{\\psi _{\\xi _j}(\\xi )}(\\psi _{\\xi _j}(m))+c_j(\\xi )$ for some function $c_j(\\xi )$ independent of $m$ , we directly have in $U_\\xi $ ($U_\\xi $ is defined in (REF )) $(h^2(\\Delta -n^2/4)-\\lambda ^2)E_h^0(\\lambda ;m,\\xi )=0.$ We let $\\chi _0\\in C^\\infty (\\partial \\overline{M}\\times \\overline{M})$ be a function such that $\\chi _0(\\xi ,\\cdot )$ is supported in $U_\\xi $ , equal to 1 near $\\xi $ and smooth in $x^2$ .", "Therefore we obtain $\\begin{split}F_h(\\lambda , \\xi ):=(h^2(\\Delta -n^2/4)-\\lambda ^2)\\chi _0E^0_h(\\lambda ,\\xi )=[h^2\\Delta ,\\chi _0]E^0_h(\\lambda ,\\xi )\\,\\,\\end{split}$ and we claim that $\\,F_h(\\lambda , \\xi )\\in x^{\\frac{n}{2}+2+\\frac{i\\lambda }{h}}C^\\infty (\\overline{M})\\textrm { and } \\Vert x^{-1}F_h(\\lambda , \\xi )\\Vert _{L^2(M)}=\\mathcal {O}(h)$ uniformly in $\\xi $ .", "Indeed, this is an elementary calculation since from (REF ) we see that $E^0_h(\\lambda ,\\xi )\\in x^{\\frac{n}{2}+i\\frac{\\lambda }{h}}C^\\infty (\\overline{M}\\setminus \\lbrace \\xi \\rbrace )$ and in geodesic normal coordinates near the boundary $[\\Delta ,\\chi _0] = -x^2(\\partial ^2_x\\chi _0)-2x(\\partial _x\\chi _0) x\\partial _x +x^2[\\Delta _{h(x)},\\chi _0]+n(x\\partial _x\\chi _0)-\\frac{1}{2}\\operatorname{Tr}_{h(x)}(\\partial _xh(x))x^2(\\partial _x\\chi _0)$ is a first order operator with coefficients vanishing in a neighbourhood of $\\xi $ .", "We thus correct the error by the incoming resolvent $R_h(\\lambda )$ by setting $E_h(\\lambda ,\\xi ):=\\chi _0E^0_h(\\lambda ,\\xi )+E^1_h(\\lambda ,\\xi ),\\,\\,\\,\\, \\textrm { with }\\,\\,E^1_h(\\lambda ,\\xi ):=-R_h(\\lambda )F_h(\\lambda ,\\xi )$ and this makes sense since for $\\lambda \\in \\mathbb {R}$ , $R_h(\\lambda ):x^{\\alpha }L^2(M)\\rightarrow x^{-\\alpha }L^2(M)$ for any $\\alpha >0$ and $F_h\\in xL^2(M)$ .", "We claim that Proposition 7.3 The function $E_h(\\lambda ,\\xi )$ of (REF ) is the Eisenstein function defined in (REF ) for a certain boundary defining function $x$ .", "Let $R_h^{\\mathbb {B}}(\\lambda )$ be the resolvent of the hyperbolic space (that is, the incoming right inverse to $h^2(\\Delta _{\\mathbb {H}^{n+1}}-n^2/4)-\\lambda ^2$ ) in the ball model and let $\\chi _1\\in C^\\infty (\\partial \\overline{M}\\times \\overline{M})$ be such that $\\chi _1(\\xi ,\\cdot )$ is supported in $U_\\xi $ and $\\chi _1\\chi _0=\\chi _1$ .", "Through the pull-back by $\\psi _{\\xi _j}$ (for each $j$ ), the operator $R_h^{\\mathbb {B}}(\\lambda )$ induces an operator $R_h^j(\\lambda )$ on $V_{\\xi _j}$ ; if $U_\\xi \\subset V_{\\xi _j}$ , then we have the resolvent identity $R_h(\\lambda )\\chi _1=\\chi _0 R_h^j(\\lambda )\\chi _1-R_h(\\lambda )[h^2\\Delta ,\\chi _0]R_h^j(\\lambda )\\chi _1$ for $\\lambda \\in \\mathbb {R}$ , the composition makes sense as a map $x^{\\alpha }L^2\\rightarrow x^{-\\alpha }L^2$ for any $\\alpha >0$ .", "Let $x$ be a boundary defining function, so in $V_{\\xi _j}$ , one has $\\psi _{\\xi _j}^*x_0=xe^{\\omega _j}$ for some function $\\omega _j\\in C^\\infty (\\partial \\overline{M}\\cap V_{\\xi _j})$ .", "Then multiplying (REF )Ê by $x^{-n/2-i\\lambda /h}$ on the right, and taking the restriction of the Schwartz kernels on $M\\times \\partial \\overline{M}$ , we have $E_h(\\lambda ,\\xi )=\\chi _0\\widetilde{E}_h^0(\\lambda ,\\xi )-R_h(\\lambda )[h^2\\Delta ,\\chi _0]\\widetilde{E}_h^0(\\lambda ,\\xi )$ with $\\widetilde{E}_h^0(\\lambda ;m,\\xi )= \\frac{2i\\lambda h}{C(\\lambda /h)}\\lim _{m^{\\prime }\\rightarrow \\xi }(x(m^{\\prime })^{-\\frac{n}{2}-i\\frac{\\lambda }{h}}R_h^j(\\lambda ;m,m^{\\prime }))$ a smooth function of $m\\in U_\\xi $ and $C(\\lambda /h)$ the constant in (REF ).", "Note that the Schwartz kernels of $R_h^j$ and $R_h^k$ are the same on the intersection of their domains, therefore $\\widetilde{E}_h^0$ does not depend on the choice of $j$ .", "Now, since $E^{\\mathbb {B}}_h(\\lambda ;m,\\xi )$ in (REF ) is the Eisenstein function on $\\mathbb {B}$ for the defining function $x_0$ , we deduce that in $U_\\xi \\subset V_{\\xi _j}$ , one has $\\widetilde{E}_h^0(\\lambda ,\\xi ;m)=E_h^0(\\lambda ,\\xi ;m)e^{(\\frac{n}{2}+i\\frac{\\lambda }{h})(\\omega _j(\\xi )-c_j(\\xi ))}$ .", "Here $c_j(\\xi )=\\phi _\\xi (m)-\\phi ^{\\mathbb {B}}_{\\psi _{\\xi _j}(\\xi )}(\\psi _{\\xi _j}(m))$ .", "Since $E_h^0(\\lambda ;m,\\xi )$ does not vanish, this shows that on any intersection $\\partial \\overline{M}\\cap V_{\\xi _j}\\cap V_{\\xi _k}$ of the cover of $\\partial \\overline{M}$ by the open sets $V_{\\xi _j}\\cap \\partial \\overline{M}$ , we get $\\omega _j(\\xi )-c_j(\\xi )=\\omega _k(\\xi )-c_k(\\xi )$ and therefore this defines a global smooth function $\\theta $ on $\\partial \\overline{M}$ .", "In its definition, $E_h(\\lambda ,\\xi )$ only depends on the first jet of $x$ at $\\partial \\overline{M}$ and thus modifying $x$ to be $xe^{\\theta }$ , this shows the claim.", "It follows that (A3) and (A4) are satisfied, with $b^0=e^{{n\\over 2}\\phi _\\xi (m)}$ .", "Assumption (A8) is then checked by a direct calculation, with the measure $d\\xi $ on $\\partial \\overline{M}$ corresponding to the choice of the function $x$ in Proposition REF .", "Assumption (A7) can be reduced, using the isometries $\\psi _{\\xi _j}$ , to the following statement: if $(q,\\nu )\\in S^* \\mathbb {H}^{n+1}$ is directly escaping in the forward direction and converging to some $p\\in \\mathbb {S}^n$ , then $|q-p|\\le C x_0(q)$ for some global constant $C$ ; the latter statement is verified directly, see Figure REF .", "Figure: Illustration of (A7) for the half-plane model of ℍ n+1 \\mathbb {H}^{n+1}: the set of points on trajectories convergingto ξ∈∂ℍ n+1 \\xi \\in \\partial \\mathbb {H}^{n+1} with x ˙<0\\dot{x}<0 andx<εx<\\varepsilon is the triangle formed by dashed lines, lyingO(ε)O(\\varepsilon ) close to ξ\\xi .", "For ε\\varepsilon small enough, thistriangle lies inside the lighter shaded region, denoting the set {χ 0 =1}\\lbrace \\chi _0=1\\rbrace .Microlocalization of $E^1_h$.", "Finally, assumptions (A5) and (A6) follow, by rescaling $h$ and using that $E_h(\\lambda ,\\cdot )$ is a function of $\\lambda /h$ , from Proposition 7.4 Let $K_0\\subset M$ be a compact set containing a neighborhood of the trapped set.", "Assume that $\\lambda =1$ and define $\\widetilde{E}^1_h(\\lambda ,\\xi )=\\frac{E_h^1(\\lambda ,\\xi )}{1+\\Vert E_h(\\lambda ,\\xi )\\Vert _{L^2(K_0)}}.$ Then: 1.", "$\\widetilde{E}^1_h(\\lambda ,\\xi )$ is $h$ -tempered in the sense of (REF ).", "2.", "The wavefront set $\\operatorname{WF}_h(\\widetilde{E}^1_h)$ is contained in $S^*M$ .", "3.", "If $(m,\\nu )\\in S^*M$ and $g^t(m,\\nu )$ escapes to infinity as $t\\rightarrow +\\infty $ and never passes through the set $W_\\xi :=\\lbrace (m,\\partial _m\\phi _\\xi (m))\\mid m\\in \\operatorname{supp}(\\partial _m\\chi _0)\\rbrace $ for $t\\ge 0$ , then $(m,\\nu )\\notin \\operatorname{WF}_h(\\widetilde{E}^1_h)$ .", "Moreover, the corresponding estimates are uniform in $\\lambda \\in [1/2,2]$ and $\\xi \\in \\partial M$ .", "We will use the construction of [55].", "(See also [54]; note however that in that paper $L_+$ and $L_-$ switch places compared to the notation of [55] that we are using.)", "Let $\\overline{M}_{\\operatorname{even}}$ (called $X_{0,\\operatorname{even}}$ in [55]) be the space $\\overline{M}$ with the smooth structure at the boundary $\\partial \\overline{M}$ changed so that $x^2$ is the new boundary defining function.", "As in [55], introduce the modified Laplacian $P_1(\\lambda ):=x^{-2}x^{-s}(1+x^2)^{s/4-n/8}(h^2\\Delta -s(n-s))(1+x^2)^{n/8-s/4}x^{s},\\ s:=n/2+i\\lambda /h.$ (The conjugation by $(1+x^2)^{s/4-n/8}$ is irrelevant in our case, as $s/4-n/8=i\\lambda /(4h)$ is purely imaginary.", "In [55], it is needed to show estimates far away in the physical plane, that is for $\\operatorname{Re}s\\gg 1$ .)", "Note that we change the sign of $\\lambda $ in the conjugation (in the notation of [55], $P_1(\\lambda )=P_\\sigma $ with $\\sigma =-\\lambda /h$ ); therefore, our resolvent will be semiclassically incoming, instead of semiclassically outgoing, for $\\lambda >0$ .", "The operator $P_1$ is smooth up to the boundary of $\\overline{M}_{\\operatorname{even}}$ ; as in [55], we embed $\\overline{M}_{\\operatorname{even}}$ as an open set in a certain compact manifold without boundary $X$ , and extend $P_1$ to a differential operator in $\\Psi ^2(X)$ .", "We also consider the semiclassical complex absorbing operator $Q(\\lambda )\\in \\Psi ^2(X)$ satisfying the assumptions of [55]; in particular, $Q(\\lambda )$ is supported outside of $\\overline{M}_{\\operatorname{even}}\\subset X$ .", "Then $(P_1(\\lambda )-iQ(\\lambda ))^{-1}:C^\\infty (X)\\rightarrow C^\\infty (X)$ is a meromorphic family of operators in $\\lambda $ , and for $f\\in C^\\infty (X)$ , we have (see the proof of [55]) $x^s(1+x^2)^{n/8-s/4}(P_1(\\lambda )-iQ(\\lambda ))^{-1}f|_M=R_h(\\lambda )(1+x^2)^{n/8-s/4}x^s x^2 (f|_M).$ Here $R_h(\\lambda )$ is the incoming scattering resolvent on $M$ .", "In principle, depending on the choice of $Q(\\lambda )$ , the operator $(P_1(\\lambda )-iQ(\\lambda ))^{-1}$ could have a pole at $\\lambda $ .", "However, as $R(\\lambda )$ does not have a pole for $\\lambda \\in [1/2,2]$ , the terms in the Laurent expansion of $(P_1(\\lambda )-iQ(\\lambda ))^{-1}$ have to be supported outside of $\\overline{M}_{\\operatorname{even}}$ and we can ignore them in the analysis.", "Figure: Left: physical space picture of geodesics convergingto ξ\\xi .", "The darker shaded region is the support of dχ 0 d\\chi _0, andthus of F h F_h.", "In the lighter shaded region, χ 0 =1\\chi _0=1.", "Right: phasespace picture near ξ\\xi after the conjugation of .", "L - L_- isthe sink consisting of radial points, QQ is the complex absorbingoperator, and the shaded region corresponds to the wavefront set ofF ^ h \\widehat{F}_h.", "The vertical line hitting L - L_- is the boundary ofM ¯ even \\overline{M}_{\\operatorname{even}}, while the horizontal line is the fiberinfinity.", "In both pictures, we mark two points (m,ν)(m,\\nu ) satisfyingthe assumption of part 3 of Proposition  and the forwardgeodesics starting at these points.Let $\\widehat{F}_h\\in C^\\infty (X)$ be any function such that $\\widehat{F}_h=\\mathcal {O}(h)_{H^N_{h}}$ for all $N$ , and $F_h=(1+x^2)^{n/8-s/4}x^{s+2}(\\widehat{F}_h|_M).$ Such a function exists as $x^{-s}\\chi _0E^0_h\\in C^\\infty (\\overline{M}_{\\operatorname{even}}\\setminus \\xi )$ , $\\chi _0\\in C^\\infty (\\overline{M}_{\\operatorname{even}})$ , and $F_h=x^{2+s}(1+x^2)^{n/8-s/4}[P_1(s),\\chi _0](1+x^2)^{s/4-n/8}x^{-s}E^0_h$ is supported away from $\\xi $ .", "Define the function $\\widehat{E}^1_h\\in C^\\infty (X)$ by $\\widehat{E}^1_h=-{(P_1(\\lambda )-iQ(\\lambda ))^{-1}\\widehat{F}_h\\over 1+\\Vert E_h(\\lambda ,\\xi )\\Vert _{L^2(K_0)}}.$ Then $\\widetilde{E}^1_h=x^{s}(1+x^2)^{n/8-s/4}\\widehat{E}^1_h|_M.$ Consider the map $\\iota : T^*M\\rightarrow T^*X$ given by $\\iota ( m,\\nu )=\\bigg (m,\\nu -d\\bigg (\\ln x(m)-{1\\over 4}\\ln (1+x(m)^2)\\bigg )\\bigg ),\\ m\\in M,\\ \\nu \\in T^*_{m}M;$ then for an $h$ -tempered $u\\in C^\\infty (X)$ , $\\operatorname{WF}_h(x^s(1+x^2)^{n/8-s/4} u|_M)=\\iota ^{-1}(\\operatorname{WF}_h(u)).$ Then $\\operatorname{WF}_h((P_1(\\lambda )-iQ(\\lambda ))\\widehat{E}^1_h)\\cap T^*M\\subset \\iota (\\operatorname{WF}_h(F_h))\\subset \\iota (W_\\xi ).$ Now, as $\\Vert E^0_h\\Vert _{L^2(K_0)}\\le C$ and thus $\\Vert E^1_h\\Vert _{L^2(K_0)}\\le C+\\Vert E_h\\Vert _{L^2(K_0)}$ , we have $\\Vert \\widehat{E}^1_h\\Vert _{L^2(K_0)}\\le C.$ Consider an operator $Q_K\\in \\Psi ^{\\operatorname{comp}}(X)$ supported in $K_0$ such that $\\sigma (Q_K)\\le 0$ everywhere and each unit speed geodesic $\\gamma (t)$ either escapes as $t\\rightarrow +\\infty $ or passes through the region $\\lbrace \\sigma (Q_K)<0\\rbrace $ at some positive time.", "This is possible since $K_0$ contains a neighborhood of the trapped set.", "Then the operator $P_1(\\lambda )-iQ(\\lambda )-iQ_K$ satisfies the semiclassical nontrapping assumptions [55]; therefore, by the nontrapping estimate [55], ${\\begin{array}{c}\\Vert \\widehat{E}^1_h\\Vert _{L^2(X)}\\le Ch^{-1}\\Vert (P_1(\\lambda )-iQ(\\lambda )-iQ_K)\\widehat{E}^1_h\\Vert _{L^2(X)}\\\\\\le Ch^{-1} \\Vert \\widehat{F}_h\\Vert _{L^2(X)}+Ch^{-1}\\Vert Q_K \\widehat{E}^1_h\\Vert _{L^2(X)}.\\end{array}}$ However, $\\Vert Q_K \\widehat{E}^1_h\\Vert $ is bounded by $\\Vert \\widehat{E}^1_h\\Vert _{L^2(K_0)}$ ; therefore, $\\Vert \\widehat{E}^1_h\\Vert _{L^2(X)}=\\mathcal {O}(h^{-1})$ and in particular $\\widehat{E}^1_h$ is tempered; it follows that $\\widetilde{E}^1_h$ is also tempered.", "This proves part 1 of the proposition; part 2 follows by ellipticity (note that $\\operatorname{WF}_h(F_h)\\subset W_\\xi \\subset S^*M$ ).", "Now, assume that $(m,\\nu )\\in S^*M$ satisfies the assumption of part 3 of this proposition.", "Then it follows directly from (REF ), the analysis of [55], and the definition of $\\iota $ , that the Hamiltonian flow line of $\\sigma (P_1)$ starting at $\\iota (m,\\nu )$ converges to the set $L_-$ of radial points as $t\\rightarrow +\\infty $ and does not intersect $\\operatorname{WF}_h((P_1(\\lambda )-iQ(\\lambda ))\\widehat{E}^1_h)$ for $t\\ge 0$ .", "In a fashion similar to the global argument of [55] (see also a similar semiclassical outgoing property of [55]), we combine elliptic regularity and propagation of singularities (see [55]) with the radial points lemma [55] for $L_-$ , to get $\\iota (m,\\nu )\\notin \\operatorname{WF}_h(\\widehat{E}^1_h)$ .", "Therefore, $(m,\\nu )\\notin \\operatorname{WF}_h(\\widetilde{E}^1_h)$ as required." ], [ "Limiting measures for hyperbolic quotients", "In this appendix, we give an explicit description of the limiting measures $\\mu _\\xi $ in case when $M$ is a hyperbolic quotient $\\Gamma \\backslash \\mathbb {H}^{n+1}$ , in terms of the group $\\Gamma $ .", "This is a particular case of asymptotically hyperbolic manifolds discussed in Section ." ], [ "Convex co-compact groups", "Let $\\mathbb {B}$ be the unit ball in $\\mathbb {R}^{n+1}$ , and $\\mathbb {H}^{n+1}$ the $(n+1)$ -dimensional hyperbolic space, which we view as $\\mathbb {B}$ equipped with the constant negative curvature metric $g_{\\mathbb {H}^{n+1}}:=4|dm|^2/(1-|m|^2)^2$ .", "The boundary $\\mathbb {S}^n=\\partial \\overline{\\mathbb {B}}$ is the sphere of radius 1, which is also the conformal boundary of $\\mathbb {H}^{n+1}$ .", "A convex co-compact group $\\Gamma $ of isometries of $\\mathbb {H}^{n+1}$ is a discrete group of hyperbolic transformations (i.e., transformations having 2 disjoint fixed points on $\\overline{\\mathbb {B}}$ ) with a compact convex core, and $\\Gamma $ is not co-compact.", "The convex core is the smallest convex subset in $\\Gamma \\backslash \\mathbb {H}^{n+1}$ , which can be obtained as follows.", "The limit set $\\Lambda _\\Gamma $ of the group and the discontinuity set $\\Omega _\\Gamma $ are defined by $\\Lambda _\\Gamma :=\\overline{\\lbrace \\gamma (m)\\in \\mathbb {B}; \\gamma \\in \\Gamma \\rbrace }\\cap \\mathbb {S}^n\\,, \\quad \\Omega _\\Gamma :=\\mathbb {S}^n\\setminus \\Lambda _\\Gamma $ where the closure is taken in the closed unit ball $\\overline{\\mathbb {B}}$ and $m\\in \\mathbb {B}$ is any point (the set $\\Lambda _\\Gamma $ does not depend on the choice of $m$ ).", "The group $\\Gamma $ acts on the convex hull of $\\Lambda _\\Gamma $ (with respect to hyperbolic geodesics) and the convex core is the quotient space.", "An important quantity is the Hausdorff dimension of $\\Lambda _\\Gamma $ $\\delta :=\\dim _{H}\\Lambda _{\\Gamma } < n$ which in turn is, by Patterson [41] and Sullivan [50], the exponent of convergence of Poincaré series: for any $m\\in \\mathbb {B}$ , $\\sum _{\\gamma \\in \\Gamma } e^{-s d(m,\\gamma m)}<\\infty \\iff s>\\delta ;$ we henceforth denote by $d(\\cdot ,\\cdot )$ the distance function of the hyperbolic metric on $\\mathbb {B}$ .", "Notice that the series (REF ) is locally uniformly bounded in $m\\in \\mathbb {B}$ .", "The group $\\Gamma $ acts properly discontinuously on $\\Omega _\\Gamma $ as conformal transformations of the sphere and the quotient space $\\Gamma \\backslash \\Omega _\\Gamma $ is a smooth compact manifold of dimension $n$ .", "The quotient $M=\\Gamma \\backslash \\mathbb {H}^{n+1}$ is a smooth non-compact manifold equipped with the hyperbolic metric $g$ induced by $g_{\\mathbb {H}^{n+1}}$ , and it admits a smooth compactification by setting $\\overline{M}=M\\cup (\\Gamma \\backslash \\Omega _\\Gamma )$ , i.e.", "with $\\partial \\overline{M}=\\Gamma \\backslash \\Omega _\\Gamma $ .", "Then $M$ is an asymptotically hyperbolic manifold in the sense of Section , of constant sectional curvature $-1$ .", "We shall denote the covering map by $\\pi : \\mathbb {B} \\cup \\Omega _{\\Gamma } \\rightarrow \\overline{M}.$ We refer the reader to [38] for more details and properties of convex co-compact groups." ], [ "Limiting measures in this setting", "In constant curvature, it turns out that the limiting measure $\\mu _\\xi $ exists for all $\\xi $ (rather than for Lebesgue almost every $\\xi $ as in Section REF ), and can be described as a converging sum over the group.", "We give an expression below, which is the same as the one obtained in [20] when $\\delta <n/2$ .", "For $\\xi \\in \\mathbb {S}^n$ , we let $\\phi _\\xi $ be the Busemann functionIn Section , we used the coordinate $q\\in \\mathbb {B}$ , $p\\in \\mathbb {S}^n$ for certain charts near infinity of $M$ , and the notation $\\phi ^{\\mathbb {B}}_p(q)$ for the Busemann function on the ball.", "This was to avoid confusion with the coordinate $m,\\xi $ on $M,\\partial \\overline{M}$ .", "We keep in this appendix the notation $\\phi _\\xi (m)$ to match the notation of the general setting of the article.", "on the unit ball $\\mathbb {B}$ defined by $\\phi _\\xi (m)=\\log \\Big (\\frac{1-|m|^2}{|m-\\xi |^2}\\Big ).$ The map $\\Phi $ defined by $\\Phi : \\mathbb {B}\\times \\mathbb {S}^n\\rightarrow S^*\\mathbb {H}^{n+1}, \\quad \\Phi : (m,\\xi )\\mapsto (m,\\partial _m\\phi _\\xi (m))$ gives a diffeomorphism between the unit cotangent bundle $S^*\\mathbb {H}^{n+1}$ and $\\mathbb {B}\\times \\mathbb {S}^n$ , and satisfies $\\Phi ^*d\\mu _L=e^{n\\phi _{\\xi }(m)}\\operatorname{dvol}_{\\mathbb {H}^{n+1}}(m)\\wedge d\\xi , \\quad \\textrm { with }\\,\\, e^{n\\phi _{\\xi }(m)}=\\Big (\\frac{1-|m|^2}{|m-\\xi |^2}\\Big )^n,$ if $d\\mu _L$ is the Liouville measure (viewed as a volume form on the unit cotangent bundle) and $d\\xi $ the canonical measure on $\\mathbb {S}^n$ .", "(This is a more general version of (A8) for the considered case.)", "Any isometry $\\gamma $ of $\\mathbb {H}^{n+1}$ acts on both spaces by ${\\begin{array}{c}\\gamma .", "(m,\\nu )=(\\gamma m, (d\\gamma (m)\\nu ^*)^*), \\textrm { for }(m,\\nu )\\in S^*\\mathbb {H}^{n+1};\\\\\\quad \\gamma .", "(m,\\xi )=(\\gamma m,\\gamma \\xi ),\\,\\,\\textrm { for }(m,\\xi )\\in \\mathbb {B}\\times \\mathbb {S}^n, \\end{array}}$ where $^*$ denotes the map identifying $T^*\\mathbb {H}^{n+1}$ with $T \\mathbb {H}^{n+1}$ through the metric.", "We have $\\Phi (\\gamma .", "(m,\\xi ))=\\gamma .\\Phi (m,\\xi )$ and thus $\\Phi $ descends to a map $\\Gamma \\backslash (\\mathbb {H}^{n+1}\\times \\mathbb {S}^n)\\rightarrow S^*(\\Gamma \\backslash \\mathbb {H}^{n+1})$ , which we also denote by $\\Phi $ .", "The limiting measure $\\mu _\\xi $ in the considered case is given by Lemma 1.1 Let $M=\\Gamma \\backslash \\mathbb {H}^{n+1}$ be a quotient of $\\mathbb {H}^{n+1}$ by a convex co-compact group $\\Gamma $ of isometries, let $\\mathcal {F}$ be a fundamental domain.", "Then the measure $\\mu _{\\pi (\\xi )}$ of (REF ) exists for all $\\xi \\in \\Omega _\\Gamma $ and is described as a converging series by the following expression: if $\\xi \\in \\Omega _\\Gamma \\cap \\overline{\\mathcal {F}}$ and $a\\in C_0^\\infty (S^*M)$ , then $\\int _{M} a\\, d\\mu _{\\pi (\\xi )}= \\int _{\\mathcal {F}}\\sum _{\\gamma \\in \\Gamma }a(m,d\\phi _{\\gamma \\xi }(m))e^{n(\\phi _{\\gamma \\xi }(m)+\\log |d\\gamma (\\xi )|)} \\operatorname{dvol}_{\\mathbb {H}^{n+1}}(m)$ where $\\phi _\\xi (m)$ is the Busemann function on $\\mathbb {B}$ associated to $\\xi \\in \\mathbb {S}^n$ and $|d\\gamma (\\xi )|$ is the Euclidean norm of $d\\gamma (\\xi )$ .", "We can view $a$ as a compactly supported function on the unit cotangent bundle $S^*\\mathcal {F}$ over a fundamental domain $\\mathcal {F}\\subset \\mathbb {B}$ and we extend $a$ by 0 in $S^*\\mathbb {H}^{n+1}\\backslash S^*\\mathcal {F}$ (the resulting function might not be smooth, but it does not matter here).", "The flow $g^t$ on $S^*M$ is obtained by projecting down the geodesic flow $\\widetilde{g}^t$ of the cover $S^*\\mathbb {H}^{n+1}$ .", "Let $\\xi \\in \\Omega _\\Gamma \\cap \\overline{\\mathcal {F}}$ , then small neighbourhoods of $\\pi (\\xi )$ in $M$ are isometric through $\\pi $ to small neighbourhoods of $\\xi $ in the unit ball $\\mathbb {B}$ .", "By the construction of the decomposition (REF ) for the asymptotically hyperbolic case in Section REF , the function $E_h^0(\\lambda ,\\pi (\\xi );\\pi (m))$ is equal to $e^{(n/2+i\\lambda /h)\\phi _\\xi (m)}$ for $m$ near $\\xi $ ($\\xi $ being fixed) and thus $|b^0|^2=e^{n\\phi _\\xi (m)}$ .", "One has $\\int _{M}a(g^{-t}(m,d\\phi _\\xi (m)))e^{n\\phi _{\\xi }(m)}\\operatorname{dvol}_M(m)=\\int _{\\mathcal {F}}\\widetilde{a}(\\widetilde{g}^{-t}\\Phi (m,\\xi ))e^{n\\phi _{\\xi }(m)}\\operatorname{dvol}_{\\mathbb {H}^{n+1}}(m)$ where $\\widetilde{a}(m,\\nu ):=\\sum _{\\gamma \\in \\Gamma }a(\\gamma .", "(m,\\nu ))$ is the lift to $S^*\\mathbb {H}^{n+1}$ of the function $a$ on $S^*M$ and $\\operatorname{dvol}_{\\mathbb {H}^{n+1}}(m)$ is the Riemannian measure on $\\mathbb {H}^{n+1}$ .", "Using the map $\\Phi $ of (REF ), one can define a map $\\widetilde{g}^t_\\xi :\\mathbb {B}\\rightarrow \\mathbb {B}$ by $\\widetilde{g}^t\\Phi (m,\\xi )=\\Phi (\\widetilde{g}_\\xi ^t(m),\\xi ),$ this is a diffeomorphism which preserves the measure $e^{n\\phi _\\xi (m)} \\operatorname{dvol}_{\\mathbb {H}^{n+1}}$ .", "By [20], we have $e^{n\\phi _\\xi (\\gamma ^{-1}m)}=e^{n\\phi _{\\gamma \\xi }(m)}|d\\gamma (\\xi )|^{n}$ , but we also have $\\gamma .\\Phi (m,\\xi )=\\Phi (\\gamma m,\\gamma \\xi )$ .", "Let $U^+_\\infty $ be defined in (G4) and put $U:=\\lbrace m\\mid (m,\\pi (\\xi ))\\in U^+_\\infty \\rbrace $ , then $U$ lies in a small neighborhood of $\\pi (\\xi )$ in $M$ .", "We can identify $U$ with a small neighbourhood $\\widetilde{U}$ of $\\xi $ in $\\mathcal {F}$ and we get for $\\tilde{\\mu }_{\\pi (\\xi )}$ defined in (REF ), $\\begin{split}\\int _{S^*M} (a\\circ g^{-t})\\,d\\tilde{\\mu }_{\\pi (\\xi )}=&\\int _{U}a(g^{-t}(m,d\\phi _\\xi (m)))e^{n\\phi _{\\xi }(m)}\\operatorname{dvol}_M(m)\\\\=&\\int _{\\widetilde{U}}\\sum _{\\gamma \\in \\Gamma }a(\\gamma .\\widetilde{g}^{-t}\\Phi (m,\\xi ))e^{n\\phi _{\\xi }( m)}\\operatorname{dvol}_{\\mathbb {H}^{n+1}}(m)\\\\=&\\sum _{\\gamma \\in \\Gamma }\\int _{\\widetilde{U}}a(\\Phi (\\gamma \\widetilde{g}_\\xi ^{-t}m,\\gamma \\xi ))e^{n\\phi _{\\xi }(m)}\\operatorname{dvol}_{\\mathbb {H}^{n+1}}(m)\\\\=&\\sum _{\\gamma \\in \\Gamma }\\int _{\\gamma \\widetilde{g}_\\xi ^{-t}(\\widetilde{U})} a(m,d\\phi _{\\gamma \\xi }(m))e^{n\\phi _{\\xi }(\\gamma ^{-1}m)}\\operatorname{dvol}_{\\mathbb {H}^{n+1}}(m).\\end{split}$ We now observe that for all $\\gamma \\in \\Gamma $ , $\\lim _{t\\rightarrow +\\infty }\\operatorname{1\\hspace{-2.75pt}l}_{\\gamma \\widetilde{g}_\\xi ^{-t}\\widetilde{U}}=1$ , since $\\widetilde{U}$ is a neighbourhood of $\\xi $ in $\\mathbb {B}$ containing all points directly escaping to $\\xi $ .", "This achieves the proof by recalling the definition (REF ) of $\\mu _{\\pi (\\xi )}$ and taking the limit in (REF ) and using the dominated convergence theorem, as there exists $C,C^{\\prime }>0$ such that for all $m$ in the compact set $\\operatorname{supp}(a)$ $\\sum _{\\gamma \\in \\Gamma } e^{n\\phi _{\\xi }(\\gamma ^{-1}m)} = \\sum _{\\gamma \\in \\Gamma }\\Big (\\frac{1-|\\gamma ^{-1} m|}{|\\gamma ^{-1}m-\\xi |^2}\\Big )^n\\le C\\sup _{m\\in \\operatorname{supp}(a)} e^{-n\\, d(\\gamma ^{-1}m,0)}\\le C^{\\prime }$ by locally uniform (in $m$ ) convergence of Poincaré series (REF ) at $s=n$ ." ], [ "The escape rate", "Let us discuss the classical escape rate in some particular cases, following the work of Bowen-Ruelle [3], Young [57], and Kifer [30]." ], [ "Escape rate and the pressure of the unstable Jacobian", "We consider $(M,g)$ a complete non-compact Riemannian manifold and say that a compact set $K_0\\subset S^*M$ is geodesically convex if any geodesic trajectory in $S^*M$ which leaves $K_0$ never comes back: $\\exists t_1, \\exists t_0<t_1,\\,\\, g^{t_0}(m,\\nu )\\in K_0 \\textrm { and } g^{t_1}(m,\\nu )\\in M\\setminus K_0\\Longrightarrow \\forall t\\ge t_1,\\,\\, g^t(m,\\nu )\\in M\\setminus K_0.$ A compact set $K_0\\subset M$ is said geodesically convex if $\\pi ^{-1}(K_0)$ is geodesically convex where $\\pi :S^*M\\rightarrow M$ is the natural projection.", "Let $K_0\\subset S^*M$ be a geodesically convex compact set containing a neighborhood of the trapped set $K$ .", "The examples we consider are $(M,g)$ which are hyperbolic or Euclidean near infinity, and $K_0=S^*M\\cap \\lbrace x\\ge \\varepsilon _0\\rbrace $ with $x,\\varepsilon _0$ given in (G2).", "The trapped set from Definition REF can be written as $K=\\bigcap _{t\\in \\mathbb {R}}g^t(K_0)=\\bigcap _{j\\in \\mathbb {Z}}g^j(K_0)$ This is a compact maximal invariant set for the flow $g^t$ .", "We define the escape rate as in [57], [30] by $Q:=\\limsup _{t\\rightarrow \\infty }\\frac{1}{t}\\log \\mu _L(\\mathcal {T}(t)),$ with $\\mu _L$ the Liouville measure and $\\mathcal {T}(t)$ defined in (REF ).", "Note that, since $K_0$ is geodesically convex, we have $\\mathcal {T}(t_2)\\subset \\mathcal {T}(t_1)$ for $0\\le t_1\\le t_2$ .", "The escape rate is clearly non-positive.", "In this section, we assume that $\\mu _L(K)=0$ and write $Q$ in terms of the topological pressure of the flow, under certain dynamical assumptions.", "More precisely, we assume that the trapped set $K$ is uniformly partially hyperbolic, in the following sense: there exists $\\varepsilon _f>0$ and a splitting of $T(S^*M)$ over $K$ into continuous subbundles invariant under the flow $T_{z}S^*M=E^{cs}_z\\oplus E^u_z, \\quad \\forall z\\in K$ such that the dimensions of $E^u$ and $E^{cs}$ are constant on $K$ and for all $\\varepsilon >0$ , there is $t_0\\in \\mathbb {R}$ such that $\\forall z\\in K, \\,\\, \\forall t\\ge t_0,\\left\\lbrace \\begin{array}{ll}\\forall v\\in E^{u}_z,\\,\\, |dg_z^t v|\\ge e^{\\varepsilon _f t}|v|,\\\\\\forall v\\in E^{cs}_z, \\,\\, |dg^t_zv|\\le e^{\\varepsilon t}|v|.\\end{array}\\right.$ Let $J^u$ be the unstable Jacobian of the flow, defined by $J^u(z):=-\\partial _{t}(\\det dg^t_z|_{E^u_z})|_{t=0}$ where $dg^t: E^u_z\\rightarrow E^u_{g^t_z}$ and the determinant is defined using the Sasaki metric for choosing orthornormal bases in $E^u$ .", "If $\\mu $ is a $g^t$ -invariant measure on $K$ , one has $\\int _{K}J^u d\\mu =-\\int _{K}\\sum _j \\Lambda _j^+ d\\mu $ where $\\Lambda _j^+(z)$ are the positive Lyapunov exponents at a regular point $z\\in K$ counted with multiplicity (regular points are points where the exponents are well defined, and this is set of full $\\mu $ -measure by the Oseledec theorem).", "It is also direct to see that $\\int _{K}J^u d\\mu =-\\int _{K}\\log \\det (dg^1|_{E^u})d\\mu $ .", "The topological pressure of a continuous function $\\varphi :K\\rightarrow \\mathbb {R}$ with respect to the flow can be defined by the variational formula $P(\\varphi ):=\\sup _{\\mu \\in \\mathcal {M}(K)}\\bigg (h_{\\mu }(g^1)+\\int _K \\varphi \\, d\\mu \\bigg )$ where $\\mathcal {M}(K)$ is the set of $g^t$ -invariant Borel probability measures and $h_\\mu (g^1)$ is the measure theoretic entropy of the flow at time 1 with respect to $\\mu $ .", "In particular $P(0)$ is the topological entropy of the flow.", "A particular case of uniformly partially hyperbolic dynamics is when $K$ is uniformly hyperbolic, that is when there is a continuous $g^t$ -invariant splitting $E^{cs}=\\mathbb {R}H_p\\oplus E^s$ into flow direction ($H_p$ is the vector field generating the geodesic flow) and stable directions $E^s$ where for $t\\ge t_0$ $\\forall v\\in E^{s}_z,\\,\\, |dg_z^t v|\\le e^{-\\varepsilon _f t}|v|.$ The flow is said to be Axiom A when the trapped set $K$ is a uniformly hyperbolic set such that the periodic orbits of $g^t$ on $K$ are dense in $K$ .", "It is proved by Young [57] that if $K$ is uniformly partially hyperbolic, then $Q=\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log \\mu _L(\\mathcal {T}(t))=P(J^u).$ In the Axiom A case, the same formula was essentially contained in the work of Bowen-Ruelle (using the volume lemma [3]).", "Moreover by [3], if the incoming tail $\\Gamma _-$ (which is the union of stable manifolds over the trapped set) has Liouville measure 0, then $P(J^u)<0$ .", "Thus we deduce by (REF ) $\\mu _L(K)=0 \\textrm { and } g^t \\textrm { is Axiom A } \\Longrightarrow P(J^u)<0.$ Young [57] gives a lower bound $Q\\ge P(-\\sum _j \\Lambda _j^+)$ which applies without any assumption on $K$ (but we are more interested in an upper bound)." ], [ "Relation with fractal dimensions in particular cases", "Assume first that the metric has constant curvature $-1$ in a small neighbourhood of the trapped set $K$ (this includes the case of convex co-compact hyperbolic quotients studied in Appendix ).", "Then the geodesic flow on $S^*M$ is uniformly hyperbolic on $K$ and has Lyapunov exponents 0 (with multiplicity 1) and $\\pm 1$ (each with multiplicity $n$ ).", "Therefore, the maximal expansion rate $\\Lambda _{\\max }$ from (REF ) is equal to 1, one has $J^u(z)=-n$ for all $z\\in K$ , and (see for example [14]) $P(J^u)=h_{\\rm top}(K)-n=(\\dim _{H}(K)-1)/2-n$ where $h_{\\rm top}$ is the topological entropy of the flow on $K$ , and $\\dim _H(K)\\in (0,n)$ is the Hausdorff dimension of $K$ (which is equal to the Minkowski box dimension in this case).", "For convex co-compact hyperbolic quotients $\\Gamma \\backslash \\mathbb {H}^{n+1}$ (see Section  for definition), one has by Sullivan [51] $\\delta :=\\dim _{H}(\\Lambda _\\Gamma )=h_{\\rm top}(K)$ where $\\Lambda _\\Gamma $ is the limit set of the group $\\Gamma $ defined in (REF ).", "If $g$ has negative pinched curvature near the trapped set, then one still has upper and lower bounds on $P(J^u)$ in terms of $h_{\\rm top}(K)$ and the pinching constant.", "If the trapped set $K$ is uniformly hyperbolic, it is also shown in [14] that $\\dim _H(K)\\le 1+2h_{\\rm top}(K)/\\Lambda _{\\max }$ .", "In dimension 2 there is an explicit relation between the Hausdorff dimension $\\dim _{H}(K)$ and pressures for Axiom A cases: if $a^{u}(z)=\\lim _{t\\rightarrow 0}\\frac{1}{t}\\log \\Vert dg^t|_{E^u}\\Vert >0, \\quad a^{s}(z)=\\lim _{t\\rightarrow 0}\\frac{1}{t}\\log \\Vert Dg^t|_{E^u}\\Vert <0$ then Pesin–Sadovskaya [42] show the following formula $\\dim _{H}(K)=1+t^u+t^s, \\quad \\textrm { with } P(-t^u a^u)=P(-t^sa^s)=0.$" ], [ "Egorov's theorem until Ehrenfest time", "In this section, we prove Proposition REF , following the methods of [2], [1], and [62].", "See also [44].", "Without lack of generality, we assume that $t_0>0$ ." ], [ "Estimating higher derivatives of the flow", "First of all, we need to estimate the derivatives of symbols under propagation for long times.", "Consider the open set $U_1=\\lbrace (m,\\nu )\\in T^*M\\mid m\\in U,\\ 1-2\\varepsilon _e<|\\nu |_g<1+2\\varepsilon _e\\rbrace .$ For each $k$ , we fix a norm $\\Vert \\cdot \\Vert _{C^k(\\overline{U_1})}$ for the space $C^k(\\overline{U_1})$ of $k$ times differentiable functions on $\\overline{U_1}$ .", "(The particular choice of the norm does not matter, as long as it does not depend on $t$ .)", "The following estimate is an analogue of [1]; we include the proof for the case of manifolds for the reader's convenience.", "Lemma 3.1 Take $\\Lambda _1>(1+2\\varepsilon _e)\\Lambda _{\\max }$ .", "Then for each $k$ , there exists a constant $C(k)$ such that for each $a\\in C_0^\\infty (\\overline{U_1})$ and each $t\\in \\mathbb {R}$ , $\\Vert a\\circ g^t\\Vert _{C^k(\\overline{U_1})}\\le C(k) e^{k\\Lambda _1|t|}\\Vert a\\Vert _{C^k(\\overline{U_1})}.$ Without loss of generality, we assume that $t>0$ .", "We first recall the formula for derivatives of the composition $b\\circ \\psi $ of a function $b\\in C^\\infty (\\mathbb {R}^d)$ with a map $\\psi :\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ : $\\partial ^\\alpha (b\\circ \\psi )=\\sum _{\\alpha ,j}c_{\\alpha ,j}(\\partial _{j_1\\dots j_m}b)\\circ \\psi \\cdot \\prod _{l=1}^m \\partial ^{\\alpha _l}\\psi _{j_l},$ where $c_{\\alpha ,j}$ are constants, $j_1,\\dots ,j_m\\in \\lbrace 1,\\dots ,d\\rbrace $ , and $\\alpha _1,\\dots ,\\alpha _m$ are nonzero multiindices whose sum equals $\\alpha $ .", "We see from (REF ) that (REF ) is implied by the following estimate on the derivatives of the flow $g^t$ (required to hold in any coordinate system): $|\\alpha |\\le k\\Longrightarrow \\sup _{U_1\\cap g^{-t}(U_1)}|\\partial ^\\alpha g^t|\\le C_\\alpha e^{|\\alpha |\\Lambda _1t}.$ The converse is also true, which can be seen by substituting cooordinate functions in place of $a$ in (REF ).", "To estimate higher derivatives of the flow, we will need several definitions from differential geometry.", "For a vector field $X$ on $\\overline{U_1}$ , define its pushforward $g^t_*X$ by $X(a\\circ g^t)=((g^t_*X) a)\\circ g^t,\\ a\\in C^\\infty (g^t(\\overline{U_1})).$ Then $g^t_* X$ is a vector field on $g^t(\\overline{U_1})$ .", "In local coordinates, we have $(g_*^tX)^j=\\sum _l (X^l\\partial _l g^t_j)\\circ g^{-t}.$ Note that since $g^t=\\exp (tH_p/2)$ and $g^t_* H_p=H_p$ , we have $\\partial _t g^t_* X=-{1\\over 2}[H_p,g^t_* X]=-{1\\over 2}g^t_*[H_p,X].$ We fix a symmetric affine connection $\\nabla $ on $T^*M$ .", "For vector fields $X$ and $Y$ , consider the differential operator $\\nabla ^2_{XY}$ , acting on functions or on vector fields, defined as follows: for a function $f$ and a vector field $Z$ , $\\nabla ^2_{XY}f=XYf-(\\nabla _XY)f,\\quad \\nabla ^2_{XY}Z=\\nabla _X\\nabla _Y Z-\\nabla _{\\nabla _XY}Z.$ In local coordinates, we have (using Einstein's summation convention) ${\\begin{array}{c}\\nabla ^2_{XY}f=X^iY^j(\\partial ^2_{ij}f-\\Gamma _{ij}^l \\partial _l f),\\\\(\\nabla ^2_{XY}Z)^m=X^iY^j(\\partial ^2_{ij}Z^m+\\Gamma ^m_{j\\alpha }\\partial _iZ^\\alpha +\\Gamma ^m_{i\\alpha }\\partial _j Z^\\alpha -\\Gamma _{ij}^\\alpha \\partial _\\alpha Z^m\\\\+(\\partial _i \\Gamma _{j\\alpha }^m+\\Gamma _{i\\beta }^m\\Gamma _{j\\alpha }^\\beta -\\Gamma _{ij}^\\beta \\Gamma _{\\alpha \\beta }^m)Z^\\alpha ).\\end{array}}$ Here $\\Gamma _{ij}^l$ are the Christoffel symbols of the connection $\\nabla $ .", "The advantage of $\\nabla ^2_{XY}$ over $XY$ is that the coefficients of this differential operator at any point depend (bilinearly) only on the values of $X$ and $Y$ at this point, but not on their derivatives.", "We now return to the proof of (REF ).", "The estimate (REF ) for $k=1$ follows directly from the definition (REF ) of $\\Lambda _{\\max }$ .", "It is then enough to assume that (REF ) holds for some $k\\ge 1$ and prove the estimate (REF ) for $k+1$ .", "It suffices to show that for any two vector fields $X,Y$ on $T^*M$ and any $a\\in C_0^\\infty (U_1)$ , we have the estimate $\\Vert XY(a\\circ g^t)\\Vert _{C^{k-1}(\\overline{U_1})}\\le C e^{(k+1)\\Lambda _1 t}\\Vert a\\Vert _{C^{k+1}(\\overline{U_1})}.$ The left-hand side of (REF ) is equal to $\\Vert (g^t_*Xg^t_*Y a)\\circ g^t\\Vert _{C^{k-1}(\\overline{U}_1)}$ .", "We first claim that $\\Vert (\\nabla ^2_{g^t_*Xg^t_*Y}a)\\circ g^t\\Vert _{C^{k-1}(\\overline{U_1})}\\le C e^{(k+1)\\Lambda _1 t}\\Vert a\\Vert _{C^{k+1}(\\overline{U_1})}.$ Indeed, in local coordinates $(\\nabla ^2_{g^t_*Xg^t_*Y}a)\\circ g^t=(X^\\alpha \\partial _\\alpha g_i^t)(Y^\\beta \\partial _\\beta g_j^t)\\big ((\\partial ^2_{ij}a-\\Gamma _{ij}^l \\partial _l a)\\circ g^t\\big ).$ We can now apply (REF ) to get an expression for any derivative of order no more than $k-1$ of (REF ).", "The result will involve derivatives of orders $1,\\dots ,k$ of $g^t$ , but not its $k+1$ 'st derivative; therefore, we can apply (REF ) to get (REF ).", "Given (REF ) and (REF ), it is enough to show $\\Vert ((\\nabla _{g^t_*X}g^t_*Y)a)\\circ g^t\\Vert _{C^{k-1}(\\overline{U_1})}\\le Ce^{(k+1)\\Lambda _1 t}\\Vert a\\Vert _{C^k(\\overline{U_1})}.$ The vector field $\\nabla _{g^t_*X}g^t_*Y$ involves the second derivatives of $g^t$ , therefore the left-hand side of (REF ) depends on the $k+1$ 'st derivatives of $g^t$ and we cannot apply (REF ) directly.", "We will instead use the method of the proof of [2], computing by (REF ) ${\\begin{array}{c}\\partial _t(g_*^{-t}(\\nabla _{g_*^tX}g^t_*Y))={1\\over 2}g_*^{-t}([H_p,\\nabla _{g_*^tX}g^t_*Y]-\\nabla _{[H_p,g_*^tX]}g_*^tY-\\nabla _{g_*^tX}[H_p,g^t_*Y])={1\\over 2}g^{-t}_*Z_t,\\end{array}}$ where $Z_t$ is the vector field given by $Z_t= \\nabla ^2_{g_*^tXg_*^tY}H_p+R_\\nabla (H_p,g_*^tX)(g_*^tY)$ .", "Here $R_\\nabla $ is the curvature tensor of the connection $\\nabla $ .", "Then $\\nabla _{g^t_*X}g^t_* Y=g^t_*(\\nabla _XY)+{1\\over 2}\\int _0^tg_*^{t-s}Z_s\\,ds.$ We have ${\\begin{array}{c}\\Vert (g^t_*(\\nabla _XY)a)\\circ g^t\\Vert _{C^{k-1}(\\overline{U_1})}=\\Vert \\nabla _XY(a\\circ g^t)\\Vert _{C^{k-1}(\\overline{U_1})}\\\\\\le C\\Vert a\\circ g^t\\Vert _{C^k(\\overline{U_1})}\\le Ce^{k\\Lambda _1 t}\\Vert a\\Vert _{C^k(\\overline{U_1})}.\\end{array}}$ It is then enough to handle the integral part of (REF ).", "The field $Z_s$ depends quadratically on the first derivatives of $g^s$ , but does not depend on its higher derivatives; therefore, writing an expression for $Z_s$ in local coordinates similar to (REF ), we get for $a\\in C_0^\\infty (U_1)$ , $\\Vert (Z_s a)\\circ g^s\\Vert _{C^{k-1}(\\overline{U_1})}\\le Ce^{(k+1)\\Lambda _1s}\\Vert a\\Vert _{C^k(\\overline{U_1})}.$ Applying (REF ) for the $C^k$ norm (given by the induction hypothesis) and using the geodesic convexity of $U$ , we get ${\\begin{array}{c}\\int _0^t \\Vert ((g_*^{t-s}Z_s) a)\\circ g^t\\Vert _{C^{k-1}(\\overline{U_1})}\\,ds=\\int _0^t\\Vert (Z_s(a\\circ g^{t-s}))\\circ g^s\\Vert _{C^{k-1}(\\overline{U_1})}\\,ds\\\\\\le C\\int _0^t e^{(k+1)\\Lambda _1 s}\\Vert a\\circ g^{t-s}\\Vert _{C^k(\\overline{U_1})}\\,ds\\le C\\int _0^t e^{(k+1)\\Lambda _1 s}e^{k\\Lambda _1(t-s)}\\Vert a\\Vert _{C^k(\\overline{U_1})}\\,ds\\\\\\le Ce^{(k+1)\\Lambda _1t}\\Vert a\\Vert _{C^k(\\overline{U_1})}\\end{array}}$ and the proof is finished." ], [ "Proof of Proposition ", "The proof of Proposition REF is based on repeatedly applying the following corollary of Lemma REF .", "The functions $b^{(j)}$ below will be the remainders in the formula for the commutator $[h^2\\Delta ,A^{(j)}(t)]$ , while the functions $c^{(j)}$ will be the errors arising from multiplying our operators by $X_1$ and $X_2$ .", "Proposition 3.2 Take $\\Lambda _1>(1+2\\varepsilon _e)\\Lambda _{\\max }$ .", "Fix $t_0>0$ and let $\\varphi \\in C_0^\\infty (U_1)$ satisfy $|\\varphi |\\le 1$ .", "Assume that $a_0\\in C^\\infty (T^*M)$ and for each $j\\ge 0$ , $b^{(j)}(t)\\in C^\\infty ([0,t_0]\\times T^*M)$ , and $c^{(j)}\\in C^\\infty (T^*M)$ , with support contained in some $j$ -independent compact set.", "For $j\\ge 0$ , define $a^{(j)}\\in C^\\infty ([0,t_0]\\times T^*M)$ inductively as the solutions to the equations ${\\begin{array}{c}a^{(0)}(0)=a_0,\\ a^{(j+1)}(0)=\\varphi \\cdot a^{(j)}(t_0)+c^{(j+1)};\\\\\\partial _t a^{(j)}(t)={1\\over 2}H_p a^{(j)}(t)+b^{(j)}(t).\\end{array}}$ Then for each $k$ , and each $j$ , we have (bearing in mind that each $a^{(j)}$ is supported inside some $j$ -independent compact set and thus its $C^k$ norm is well-defined up to a constant) ${\\begin{array}{c}\\sup _{t\\in [0,t_0]}\\Vert a^{(j)}(t)\\Vert _{C^k(T^*M)}\\le C(k) \\big (e^{jk\\Lambda _1t_0}\\Vert a_0\\Vert _{C^k}\\\\+\\max _{0\\le i\\le j}e^{(j-i)k\\Lambda _1t_0}(\\sup _{t\\in [0,t_0]}\\Vert b^{(i)}(t)\\Vert _{C^k}+\\Vert c^{(i)}\\Vert _{C^k})\\big ),\\end{array}}$ where $C(k)$ is a constant independent of $j$ .", "We can write $a^{(j)}(t)=a^{(j)}(0)\\circ g^t+\\int _0^t b^{(j)}(s)\\circ g^{t-s}\\,ds.$ Since $t_0$ is fixed, it is enough to estimate the derivatives of $a^{(j)}(0)$ .", "Define $\\varphi ^{(j)}=\\prod _{0\\le m<j} (\\varphi \\circ g^{mt_0});$ applying the Leibniz rule to $\\varphi ^{(j)}$ , estimating each nontrivial derivative of $\\varphi \\circ g^{mt_0}$ by Lemma REF , using that $|\\varphi |\\le 1$ and absorbing the (polynomial in $l$ ) number of different terms in the Leibniz formula into the exponential by increasing $\\Lambda _1$ slightly, we get $\\Vert \\varphi ^{(j)}\\Vert _{C^k}=\\mathcal {O}(e^{jk\\Lambda _1t_0})$ .", "Now, ${\\begin{array}{c}a^{(j)}(0)=\\varphi ^{(j)}\\cdot (a_0\\circ g^{jt_0})+\\sum _{i=0}^{j-1}\\varphi ^{(j-i)}\\int _0^{t_0} b^{(i)}(s)\\circ g^{(j-i)t_0-s}\\,ds\\\\+\\sum _{i=1}^j \\varphi ^{(j-i)}\\cdot (c^{(i)}\\circ g^{(j-i)t_0}).\\end{array}}$ Here we put $\\varphi ^{(0)}=1$ .", "We can now apply Lemma REF again to get the required estimate.", "We are now ready to prove Proposition REF .", "Fix a quantization procedure $\\operatorname{Op}_h$ on $M$ ; our symbols will be supported in a certain compact set (in fact, no more than distance $t_0$ to the set $U$ ) and we require that the corresponding operators be compactly supported.", "Put $\\Lambda _1=\\Lambda ^{\\prime }_0$ .", "Let $l$ satisfy (REF ).", "We will construct the operators $A^{(j)}_m(t)=\\operatorname{Op}_h\\bigg (\\sum _{0\\le m^{\\prime }\\le m}a^{(j)}_{m^{\\prime }}(t)\\bigg ),\\ 0\\le t\\le t_0,\\ 0\\le j\\le l,\\ m\\ge 0,$ Here the symbols $a^{(j)}_m$ will be supported in a fixed compact subset of $T^*M$ and satisfy the derivative bounds $\\sup _{t\\in [0,t_0]}\\Vert a^{(j)}_m(t)\\Vert _{C^k}\\le C(k,m)h^{(1-2\\rho _j)m-\\rho _j k}.$ with the constants $C(k,m)$ independent on $j$ and $\\rho _j$ defined by (REF ).", "The operators $A^{(j)}_m(t)$ will satisfy the relations ${\\begin{array}{c}A^{(0)}_m(0)=A+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }},\\\\A^{(j+1)}_m(0)=X_2 A^{(j)}_m(t_0)X_1+\\operatorname{Op}_h(c^{(j)}_m)+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }},\\\\hD_t A^{(j)}_m(t)={1\\over 2}[h^2\\Delta ,A^{(j)}_m(t)]+{h\\over i}\\operatorname{Op}_h(b^{(j)}_m(t))+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }},\\end{array}}$ where the symbols $b^{(j)}_m(t)$ and $c^{(j)}_m$ are supported in some fixed compact set and satisfy bounds $\\sup _{t\\in [0,t_0]}\\Vert b^{(j)}_m(t)\\Vert _{C^k},\\Vert c^{(j)}_m\\Vert _{C^k}\\le C(k,m)h^{(1-2\\rho _j)(m+1)-\\rho _j k},$ with the constants $C(k,m)$ again independent on $j$ .", "We construct the symbols $a^{(j)}_m$ iteratively, by requiring that they solve the equations ${\\begin{array}{c}a^{(0)}_m(0)=\\delta _{m0}\\cdot a_0,\\ a^{(j+1)}_m(0)=\\varphi a^{(j)}_m(t_0)-c^{(j)}_{m-1},\\\\\\partial _t a^{(j)}_m(t)={1\\over 2}H_p a^{(j)}_m(t)-b^{(j)}_{m-1}(t).\\end{array}}$ Here $A=\\operatorname{Op}_h(a_0)+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }}$ and we put $b^{(j)}_{-1}=c^{(j)}_{-1}=0$ .", "The function $\\varphi \\in C_0^\\infty (U_1)$ is equal to $\\sigma (X_1)\\sigma (X_2)\\psi (|\\nu |)$ , where $\\psi \\in C_0^\\infty (1-2\\varepsilon _e,1+2\\varepsilon _e)$ is such that $\\psi (|\\nu |)=1$ near $\\operatorname{WF}_h(A)$ .", "We use the fact that the function $|\\nu |$ is invariant under the geodesic flow.", "The estimate (REF ) follows immediately from (REF ) and Proposition REF .", "As for the equations (REF ) and the bounds (REF ), they follow from (REF ) and the following commutator formula: $[h^2\\Delta ,\\operatorname{Op}_h(a)]={h\\over i}\\operatorname{Op}_h(H_pa)+\\operatorname{Op}_h(b)+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }},\\ b=\\mathcal {O}(h^{2-2\\rho }\\Vert a\\Vert _{S_\\rho })_{S_\\rho },$ true for any $\\rho <1/2$ and any $a\\in S^{\\operatorname{comp}}_\\rho $ .", "Now, consider the asymptotic sums $a^{(j)}(t)\\sim \\sum _{m\\ge 0} a^{(j)}_m(t)$ and define the operators $A^{(j)}(t)=\\operatorname{Op}_h(a^{(j)}(t))$ .", "By (REF ), these operators satisfy ${\\begin{array}{c}A^{(0)}(0)=A+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }},\\ A^{(j+1)}(0)=X_2 A^{(j)}(t_0)X_1+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }},\\\\hD_t A^{(j)}(t)={1\\over 2}[h^2\\Delta ,A^{(j)}(t)]+\\mathcal {O}(h^\\infty )_{\\Psi ^{-\\infty }}.\\end{array}}$ We then have $(X_2U(t_0))^lA(U(-t_0)X_1)^l=A^{(l)}(0)+\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}.$ It remains to recall that $a^{(l)}(0)\\in S^{\\operatorname{comp}}_{\\rho _l}$ uniformly in $l$ .", "The principal symbol and microlocal vanishing statements follow directly from the procedure used to construct the symbols $a^{(j)}_m$ ." ], [ "Proof of quantum ergodicity in the semiclassical setting", "In this section, we illustrate how our methods yield a proof of the following integrated quantum ergodicity statement in the semiclassical setting: Theorem 5 Let $(M,g)$ be a compact Riemannian manifold of dimension $d$ and assume that the geodesic flow $g^t$ on $M$ is ergodic with respect to the Liouville measure $\\mu _L$ on the unit cotangent bundle $S^*M$ .", "For each $h>0$ , let $(e_j)_{j\\in \\mathbb {N}}$ be an orthonormal basis of eigenfunctions of $h^2\\Delta $ with eigenvalues $\\lambda _j^2$ .", "Then for each semiclassical pseudodifferential operator $A\\in \\Psi ^0(M)$ , we have $h^{d-1}\\sum _{\\lambda _j\\in [1,1+h]} \\bigg |\\langle A e_j,e_j\\rangle _{L^2(M)}-{1\\over \\mu _L(S^*M)}\\int _{S^*M} \\sigma (A)\\,d\\mu _L\\bigg |\\rightarrow 0\\text{ as }h\\rightarrow 0.$ A more general version of Theorem REF was proved in [25], in particular relying on the result of [9], [43] on $o(h)$ remainders for the Weyl law when the closed geodesics form a set of measure zero.", "The purpose of this Appendix is to provide a shorter proof.", "Theorem REF is formulated here for the semiclassical Laplacian for simplicity of notation, but it applies to any self-adjoint semiclassical pseudodifferential operator $P(h)$ with compact resolvent on a compact manifold, if the Hamiltonian flow of the principal symbol $p$ of $P(h)$ has no fixed points and is ergodic on the energy surface $p^{-1}(0)$ and we take eigenvalues in the interval $[0,h]$ .", "The key component of our proof is the following estimate: Lemma 4.1 Let $M$ be as in Theorem REF .", "Then for each $A\\in \\Psi ^0(M)$ , we have $h^{d-1}\\sum _{\\lambda _j\\in [1,1+h]}\\Vert Ae_j\\Vert _{L^2(M)}^2\\le (C\\Vert \\sigma (A)\\Vert _{L^2(S^*M)}+\\mathcal {O}(h))^2.$ Here $\\Vert \\sigma (A)\\Vert _{L^2(S^*M)}$ is the $L^2$ norm of the restriction of $\\sigma (A)$ to $S^*M$ with respect to the Liouville measure.", "The constant in $\\mathcal {O}(h)$ depends on $A$ , but the constant $C$ does not.", "Assume first that $A$ is compactly microlocalized.", "We can rewrite the left-hand side of (REF ) as the square of the Hilbert–Schmidt norm of $h^{(d-1)/2}A\\Pi _{[1,1+h]}$ , where $\\Pi _{[1,1+h]}=\\operatorname{1\\hspace{-2.75pt}l}_{[1,(1+h)^2]}(h^2\\Delta )$ is a spectral projector.", "It can then be estimated using the local theory of semiclassical Fourier integral operators, by (REF ) (applied to the adjoint of the operator in interest).", "To handle the case of a general $A$ , it remains to note that if $\\operatorname{WF}_h(A)\\cap S^*M=\\emptyset $ , then the left-hand side of (REF ) is $\\mathcal {O}(h^\\infty )$ , as each $Ae_j$ is $\\mathcal {O}(h^\\infty )$ by the elliptic estimate (Proposition REF ; see also the proof of Proposition REF ).", "Putting $A$ equal to the identity in (REF ), we get the following upper Weyl bound: $\\#\\lbrace j\\mid \\lambda _j\\in [1,1+h]\\rbrace \\le Ch^{1-d}.$ We can now prove Theorem REF .", "Take $A\\in \\Psi ^0(M)$ ; by subtracting a multiple of the identity operator and applying the ellipticity estimate, we may assume that $A$ is compactly microlocalized and $\\int _{S^*M}\\sigma (A)\\,d\\mu _L=0.$ Define the quantum average $\\langle A\\rangle _T={1\\over T}\\int _0^T U(t)AU(-t)\\,dt.$ Here $U(t)=e^{ith\\Delta /2}$ is the semiclassical Schrödinger propagator.", "By Egorov's theorem (Proposition REF ), for any fixed $T$ the operator $\\langle A\\rangle _T$ lies in $\\Psi ^0$ , modulo an $\\mathcal {O}(h^\\infty )_{L^2\\rightarrow L^2}$ remainder, and its principal symbol is $\\sigma (\\langle A\\rangle _T)=\\langle \\sigma (A)\\rangle _T:={1\\over T}\\int _0^T \\sigma (A)\\circ g^t\\,dt.$ Note that for each $j$ , we have $U(t)e_j=e^{it\\lambda _j/(2h)}$ and thus $\\langle \\langle A\\rangle _T e_j,e_j\\rangle =\\langle Ae_j,e_j\\rangle $ .", "Using Cauchy–Schwarz inequality in $j$ and the bounds (REF ) and (REF ), we get ${\\begin{array}{c}h^{d-1}\\sum _{\\lambda _j\\in [1,1+h]}|\\langle Ae_j,e_j\\rangle |=h^{d-1}\\sum _{\\lambda _j\\in [1,1+h]}|\\langle \\langle A\\rangle _Te_j,e_j\\rangle |\\\\\\le h^{d-1}\\sum _{\\lambda _j\\in [1,1+h]}\\Vert \\langle A\\rangle _T e_j\\Vert _{L^2}\\le C\\bigg (h^{d-1}\\sum _{\\lambda _j\\in [1,1+h]}\\Vert \\langle A\\rangle _T e_j\\Vert _{L^2}^2\\bigg )^{1/2}\\\\\\le C\\Vert \\langle \\sigma (A)\\rangle _T\\Vert _{L^2(S^*M)}+\\mathcal {O}_T(h).\\end{array}}$ However, by (REF ) and the von Neumann ergodic theorem [62], we have $\\Vert \\langle \\sigma (A)\\rangle _T\\Vert _{L^2(S^*M)}\\rightarrow 0$ as $T\\rightarrow \\infty $ .", "Therefore, for each $\\varepsilon >0$ we can choose $T$ large enough so that the left-hand side of (REF ) is bounded by $\\varepsilon /2+\\mathcal {O}(h)$ .", "Then for $h$ small enough, it is bounded by $\\varepsilon $ ; since the latter was chosen arbitrarily small, we get (REF ).", "Acknowledgements.", "We would like to thank Viviane Baladi, Dima Jakobson, Frederic Naud, Stéphane Nonnenmacher, Steve Zelditch, and Maciej Zworski for useful discussions and providing references on the subject.", "We additionally thank Stéphane Nonnenmacher for explaining some estimates on higher derivatives of the flow (forming the basis for Lemma REF ), and the anonymous referees for their interest in this work and suggesting many improvements.", "S.D.", "would also like to thank the DMA of Ecole Normale Supérieure where part of this work was done.", "S.D.", "was partially supported by NSF grant DMS-0654436.", "C.G.", "is supported by ANR grant ANR-09-JCJC-0099-01." ] ]

[ [ "Complex Variability of the H$\\alpha$ Emission Line Profile of the T\n Tauri Binary System KH 15D: The Influence of Orbital Phase, Occultation by\n the Circumbinary Disk, and Accretion Phenomenae" ], [ "Abstract We have obtained 48 high resolution echelle spectra of the pre-main sequence eclipsing binary system KH~15D (V582 Mon, P = 48.37 d, $e$ $\\sim$ 0.6, M$_{A}$ = 0.6 M$_{\\odot}$, M$_{B}$ = 0.7 M$_{\\odot}$).", "The eclipses are caused by a circumbinary disk seen nearly edge on, which at the epoch of these observations completely obscured the orbit of star B and a large portion of the orbit of star A.", "The spectra were obtained over five contiguous observing seasons from 2001/2002 to 2005/2006 while star A was fully visible, fully occulted, and during several ingress and egress events.", "The H$\\alpha$ line profile shows dramatic changes in these time series data over timescales ranging from days to years.", "A fraction of the variations are due to \"edge effects\" and depend only on the height of star A above or below the razor sharp edge of the occulting disk.", "Other observed variations depend on the orbital phase: the H$\\alpha$ emission line profile changes from an inverse P Cygni type profile during ingress to an enhanced double-peaked profile, with both a blue and red emission component, during egress.", "Each of these interpreted variations are complicated by the fact that there is also a chaotic, irregular component present in these profiles.", "We find that the complex data set can be largely understood in the context of accretion onto the stars from a circumbinary disk with gas flows as predicted by the models of eccentric T Tauri binaries put forward by Artymowicz & Lubow, G\\\"{u}nther & Kley, and de Val-Borro et al.", "In particular, our data provide strong support for the pulsed accretion phenomenon, in which enhanced accretion occurs during and after perihelion passage." ], [ "Introduction", "KH 15D, first observed at Van Vleck Observatory in 1995, gained its notoriety due to the regular and rather large amplitude variability it displayed (Kearns & Herbst 1998).", "At that time, the system's light was observed to diminish by nearly 3 magnitudes every 48.4 days, and remain in this diminished or eclipsed state for approximately 16 days, with an unusual return to near-normal brightness close to mid-eclipse.", "By 2000, the central reversal in the light curve had faded and the length of time that the star spent in the eclipsed state had grown slightly ($\\sim $ 1 day).", "Spectra taken of the object in and out of eclipse had shown that there was effectively no color change between states, implying that the source of obscuration was either an optically thick disk or rather large particles (Hamilton et al.", "2001).", "Several models were proposed at this time (Herbst et al.", "2002; Barge & Viton 2003; Agol et al.", "2004) that involved some sort of warp or swarm of particles in a circumstellar disk around a single star.", "KH 15D was also discovered to be the source of a bipolar jet revealed in H$\\alpha $ and [OI] (Hamilton et al.", "2003) whose possible launching mechanism has been discussed in detail by Mundt et al.", "(2010).", "The system is associated with a shocked H$_{2}$ emission filament (Deming, Charbonneau, & Harrington 2004; Tokunaga et al.", "2004).", "During eclipse, the spectrum of KH 15D often exhibits extended wings of H$\\alpha $ emission, up to several hundreds of km s$^{-1}$ , characteristic of actively accreting classical T Tauri stars (CTTSs).", "Out of eclipse, the EW of H$\\alpha $ is only a few angstroms, which would normally lead to a classification as a weak-lined T Tauri star (WTTS) (Hamilton et al.", "2003).", "It is perhaps best described as a weakly accreting T Tauri star.", "Its age, estimated from membership in NGC 2264, is $\\sim $ 3 Myr.", "In 2004, it was discovered that KH 15D is a binary system.", "Johnson et al.", "(2004) conducted a radial velocity survey of the system over the course of two observing seasons (2002/2003 and 2003/2004) during which KH 15D underwent significant radial velocity variations consistent with a binary companion with an orbital period equal to that of the photometric period of 48 days.", "Almost simultaneously, two models were put forward to explain the changing light curve as the result of the progressive occultation of a pre-main sequence binary orbit by a precessing circumbinary inner disk or ring (Winn et al.", "2004; Chiang & Murray-Clay 2004).", "This type of model had the advantage of also being consistent with the historical light curves obtained from a study of archival photographic plates (Winn et al.", "2003; Johnson & Winn 2004).", "Winn et al.", "(2006; hereafter W06) further refined this model by including a wealth of additional photometric data available from the literature (Johnson et al.", "2005; Maffei et al.", "2005; Hamilton et al.", "2005; Barsunova et al.", "2005; Kusakabe et al.", "2005).", "Our current working model of the system (W06) has the distinguishing aspect that the system is fortuitously observed nearly along the plane of an opaque circumbinary ring that is, itself, somewhat tilted with respect to the binary's orbital plane.", "To orient the reader, a schematic of the KH 15D system is shown in Figure 1.", "The ring is presumably the inner part ($<$ 5 AU) of a more extensive circumbinary disk (CBD) that provides gas for continuing accretion.", "The outer disk has recently been detected at mm wavelengths with the SMA (Herbst & Wilner, in preparation).", "Precession of the ring has, over the past 50 years, gradually occulted the orbit of the binary as projected on the sky.", "Various studies have shown (e.g., Herbst et al.", "2002; Herbst et al.", "2010) that the edge of this occulting ring is surprisingly sharp (much less than one stellar radius) and behaves like a knife edge.", "For a more general discussion of the current properties of the KH 15D light curve and its interpretation, the reader is referred to Herbst et al.", "(2010).", "When first noticed in 1995 (Kearns & Herbst 1998) the occulting edge had just about completely covered the orbit of the more massive star (designated star B following the nomenclature of W06).", "From 1995 to 2009 the knife edge moved progressively across the orbit of star A, producing the observed light curve (Hamilton et al.", "2005; Herbst et al.", "2010).", "During this time the system exhibited dramatic photometric variations on the orbital cycle as star A regularly rose and set with respect to the ring horizon.", "Throughout part of this phase of its evolution (2001-2006) we were able to obtain a substantial number of spectra (N=48) of the system at a variety of orbital phases and heights of star A above and below the ring edge.", "This has allowed us to do a kind of “occultation mapping\" of gas flows in the magnetosphere of star A and perhaps elsewhere within the inner cavity of the CBD.", "By far the most useful spectral feature available to us for this work is H$\\alpha $ .", "Hamilton et al.", "(2003) used 3 spectra obtained during one cycle in 2001 to infer the presence of a jet and a magnetospheric accretion flow.", "At that time, the system was believed to be only a single star with a circumstellar disk containing a warp.", "In this paper, we present the analysis of additional high resolution spectroscopic data of the KH 15D system, specifically, the H$\\alpha $ emission line profiles, obtained while in its bright state, during ingress/egress, and during eclipse over the course of a 5 year observing period (2001-2006).", "Section 2 discusses the observations and reductions.", "The H$\\alpha $ emission line profiles are presented and characterized in Section 3.", "Profile decompositions based on gaussian fitting and the velocity behavior of the absorption component are discussed in Section 4.", "The high velocity emission component of the H$\\alpha $ line profile is presented in Section 5, and constraints on the size of the magnetosphere of star A are given in Section 6.", "General conclusions are drawn in Section 7." ], [ "Observations and Data Reduction", "The high resolution echelle spectra analyzed here were obtained during five observing seasons from 2001/2002 through 2005/2006 at various observing facilities.", "Table 1 gives the UT dates of the observations, telescope/instrument used, wavelength coverage and resolution.", "We attempted to exploit the “natural coronagraph\" that this system represents by focusing primarily on phases during or near ingress and egress, however, spectra were also obtained near mid-eclipse, and when star A was well out of eclipse.", "Figure 2 shows the Cousins $I$ -band ($I_{C}$ ) data from Hamilton et al.", "(2005) or Herbst et al.", "(2010) phased with the orbital period for each of the five observing seasons overlain with the dates on which spectra were obtained.", "It is clear that the width of the eclipse increased significantly between the beginning and end of the data acquisition period so a particular orbital phase that represented an “out-of-eclipse\" observation in 2001/2002 might be “in eclipse\" in 2005/2006.", "This is demonstrated in Figure 3.", "Each spectrum is characterized, therefore, by two numbers: orbital phase, ranging from –0.5 to +0.5 with 0.0 representing the time of mid-eclipse, and the position of star A with respect to the ring edge, ranging from $\\Delta $ X = –4 (4 stellar radii above the edge) to $\\Delta $ X = +10 (10 stellar radii below the ring edge).", "The parameter $\\Delta $ X used here comes from Model 3 of W06Model 3 employs the astrophysical constraint that star A be less massive than star B because it is less luminous.", "This constraint produces a model that is in best agreement with the observed photometry through 2006. and is the elevation of the center of star A above (–) or below (+) the edge of the occulting disk in units of the radius of star A (R$_{A}$ = 1.3 R$_{\\odot }$ = 9 x 10$^{8}$ m).", "As is evident from Table 1, there were five different instruments employed to obtain the observational material analyzed here.", "These are: the Keck 10-m telescope and High Resolution Echelle Spectrometer (HIRES; solid red lines in Fig.", "2), the European Southern Observatory's 8.2-m Very Large Telescope and UV-Visual Echelle Spectrograph (UVES; long-dashed blue lines), the 2.1-m Otto Struve Telescope at McDonald Observatory and Sandiford Cassegrain Echelle Spectrometer (CE; dash-dot green lines), the 8-m Hobby-Ebberly Telescope at McDonald Observatory and High Resolution Spectrograph (HRS; dashed pink lines) and the 6.5-m Magellan II (Clay) Telescope and Magellan Inamori Kyocera Echelle Spectrograph (MIKE; dash-dot-dot-dot orange lines).", "Brief descriptions of the data and reduction process are now given for each instrument." ], [ "The VLT/UVES Spectra", "The data obtained with the VLT and UVES used in this analysis were taken during the 2001/2002 and 2004/2005 observing seasons.", "The data from 2001/2002 were obtained in queue mode and have already been published.", "The reader is referred to Hamilton et al.", "(2003) for more information regarding their reduction.", "The data taken during the 2004/2005 observing season were reduced with a set of custom echelle reduction routines written in IDL.", "The data reduction procedure is described by Valenti (1994) and Hinkle et al.", "(2000) and includes bias subtraction, flat fielding by a normalized flat spectrum, scattered light subtraction, and optimal extraction of the spectrum.", "Due to nebular H$\\alpha $ emission near KH 15D, it is important for this study to perform a sky subtraction when reducing the spectra.", "The VLT/UVES spectral format contains enough room between the orders that background sky spectra are recorded both above and below the stellar spectrum in the slit.", "The spectral traces defining the order locations for the orders in the stellar spectrum were offset in each direction along the slit and sky spectra were extracted.", "The two sky spectra were averaged and used to subtract the sky background from the spectra of KH 15D.", "The wavelength solution was determined by fitting a two-dimensional polynomial to $n\\lambda $ as function of pixel and order number, $n$ , for several hundred extracted thorium lines observed from an internal lamp assembly." ], [ "The Keck/HIRES Spectra", "Some of the data that were obtained with the Keck I telescope and HIRES for this project were also used as a part of the radial velocity study Johnson et al.", "(2004).", "A full description of the reduction procedure used for the Keck data obtained in 2003/2004 is given in that paper.", "Additional Keck data obtained in Feb 2005 were reduced using the same custom IDL echelle reduction routines referenced above in §2.1.", "Similar to the VLT/UVES data, sky subtraction was also performed for the Keck/HIRES data." ], [ "The McDonald Observatory 2.1-m/CE Spectra", "The January 2004 data obtained with the McDonald Observatory 2.1-m Otto Struve Telescope and CE spectrometer were also used as a part of the radial velocity survey presented by Johnson et al.", "(2004).", "The reader is again directed to that paper for a full description of the reduction procedures for these spectra." ], [ "2.1-m Sky Subtraction", "The spectral format of the CE is quite compact, requiring a relatively short slit (in this case 2.$^{^{\\prime \\prime }}$ 5) in order to keep the orders well separated on the CCD.", "As a result, traditional sky subtraction with this system is not possible.", "As mentioned above, there is an H$\\alpha $ component in the line profiles of KH 15D due to the nebular emission from NGC 2264, the cluster in which KH 15D is located.", "In order to study the intrinsic variations of the H$\\alpha $ profile of KH 15D, it is helpful to remove this nebular component.", "To do so, we utilized a sky spectrum obtained with one of the HET/HRS spectra of KH 15D (see below) as a proxy for sky subtracting the McDonald 2.1-m CE data.", "After continuum normalizing the H$\\alpha $ order in the 2.1-m CE spectra, we scaled the HET sky spectrum so that when it is subtracted from the 2.1-m CE spectra the nearby [NII] $\\lambda $ 6583 emission line is entirely removed.", "This line is a pure nebular emission line, so the underlying assumption in this procedure is that the ratio of this line to the nebular H$\\alpha $ line stays constant throughout the nebula surrounding KH 15D.", "As a check on this assumption, we examined the spectra taken with Keck/HIRES which has a long enough slit to separately extract sky spectra on either side of the stellar spectrum.", "We scaled the sky spectrum on one side of the stellar specrum so that the flux in the [NII] line matched that from the sky spectrum taken on the other side of the star.", "We then differenced the two sky spectra.", "In each case, no residual H$\\alpha $ flux is detected, and the 3$\\sigma $ upper limit on the H$\\alpha $ flux in the difference spectrum is $\\sim 2$ % of the nebular H$\\alpha $ emission.", "It thus appears that this is a fairly robust way to remove the sky$+$ nebular emission from the 2.1-m CE spectra." ], [ "The Magellan/MIKE Spectra", "The MIKE spectra were reduced with the “MIKE redux\" code written by S. Burles, X. Prochaska, and R. Bernsteini (see http://web.mit.edu/$\\sim $ burles/www/MIKE/).", "This reduction package performs bias subtraction, flat fielding, order-edge tracing, sky subtraction, and optimal extraction.", "Wavelength calibration is also performed as part of the package using spectra of a Thorium-Argon arc lamp obtained just prior to each observation of KH 15D.", "The wavelength solution for the MIKE data was performed with the custom IDL software described above and referenced in Valenti (1994) and Hinkle et al.", "(2000)." ], [ "The HET/HRS Spectra", "The HET/HRS spectra used here were all reduced with the custom IDL software described above in §2.1 and referenced in Valenti (1994) and Hinkle et al.", "(2000).", "The HRS instrument at the HET is a fiber fed spectrometer.", "For each observation of KH 15D, sky fibers were placed on either side of the star fiber.", "As a result of the spectra from these sky fibers appearing on the CCD, the optimal extraction routines in the reduction package were not used.", "Instead, the counts normal to the dispersion direction were summed at each wavelength in order to produce the extracted spectrum.", "The observations used here come from two programs looking at KH 15D.", "As a result, about half the observations used 2$^{\\prime \\prime }$ fibers while the other half used 3$^{\\prime \\prime }$ fibers.", "The throughput of the fibers used to feed the stellar and sky spectra vary somewhat.", "As a result, the two sky spectra were summed to increase the signal to noise and the result was then scaled to match the [NII] $\\lambda $ 6583 flux in the stellar spectrum before the sky subtraction was performed.", "Spectra of a Thorium-Argon lamp were also extracted for each night and again a wavelength solution was created by fitting a two-dimensional polynomial to $n\\lambda $ as function of pixel and order number, $n$ , for several hundred thorium lines." ], [ "The H$\\alpha $ Emission Line Profiles", "In order to interpret the wealth of complex information produced by the occultation mapping of the H$\\alpha $ line profile we required the height of star A above or below the occulting screen, which is taken from Model 3 of W06.", "Table 2 lists the Julian Date of observation, telescope/instrument, orbital phase, the height of Star A above the disk, the radial velocity (RV) of star A as predicted by W06, the radial velocity of star B as predicted by W06, the calculated or measured (when available) Cousins $R$ magnitude ($R_{C}$ ), the measured $I_{C}$ (from Hamilton et al.", "2005 or Herbst et al.", "2010), the flux ratio used to scale the normalized spectrum, and the barycentric correction.", "All profiles have been continuum-fit and normalized to 1.0.", "Their flux was then scaled in reference to the 2002 December 10 out-of-eclipse spectrum obtained with the HET at McDonald Observatory.", "The 2002 December 10 spectrum was chosen as the reference spectrum to which all others would be scaled because star A had the largest height above the occulting screen as predicted by the W06 model on that date and presumably represents the H$\\alpha $ profile most intrinsic to star A.", "The scale factor was calculated by computing the difference between the $R_{C}$ magnitude on the particular date of interest and the $R_{C}$ magnitude observed on 2002 December 10.", "The flux on 2002 December 10 was assumed to be 1.00.", "The Cousins $R$ filter encompasses the wavelength region corresponding to H$\\alpha $ , making it the best available proxy for measurement of the flux in the continuum near H$\\alpha $ .", "Actual observed $R_{C}$ magnitudes were used when available, however, the majority of the photometric monitoring during these seasons was only done in the Cousins $I$ -band.", "When $R_{C}$ magnitudes were not available in the photometric database, they were estimated from measured $I_{C}$ values using a color-magnitude relation from Hamilton et al.", "(2005) based on USNO 2002/2003 data.", "Typical photometric errors are $\\sim $ 0.01 out of eclipse and $\\sim $ 0.1 in eclipse (Hamilton et al.", "2005).", "The scaled profiles are shown in Figures 4 and 5.", "The profiles are plotted from left to right in these figures according to distance above or below the projection of the edge of the occulting disk as calculated by W06.", "They are shown, in reference to Figure 2, from out of eclipse, through ingress, during mid-eclipse, through egress, and out of eclipse again.", "The profiles are plotted this way to assist in identifying whether or not there are systematic differences in the profiles given the specific location in the orbit, not just the height above or below the occulting disk.", "All profiles have been corrected for the Earth's motion, and are shown in the rest frame of the center of mass of the system." ], [ "Characterizing the Variability", "When trying to understand the complex variations in the H$\\alpha $ line profiles presented here, one should consider three more or less independent factors influencing the variability: (1) the “edge effect\" - occultation by the CBD as measured by the distance $\\Delta $ X with respect to the occulting edge; (2) the “orbital phase effect\" - potential variability due to increased accretion on both stars or other changes during certain phases of the eccentric binary orbit (see de Val-Borro et al.", "2011 and references therein); and (3) an irregular/chaotic component that is best seen when comparing the profiles taken at similar heights and orbital phase but at different times.", "We have several multi-day spectral sequences (some with coverage on consecutive days) obtained during ingress, egress, and while the observable star is out of eclipse.", "These observations occurred over the course of five different observing seasons so we can assess both the short-term and long-term variability of the H$\\alpha $ emission line profile.", "In the following section, we will begin by demonstrating the changes that are seen in the H$\\alpha $ emission line profile as the stars move throughout their orbits, followed by changes that are produced as star A is occulted by the CBD.", "Finally, we will discuss variations observed which can be attributed to the irregular nature of the T Tauri accretion process.", "By looking at Figures 4 and 5, it is evident that the KH 15D system exhibits highly variable H$\\alpha $ profiles.", "We begin our analysis by taking the appropriately scaled (see §3) profiles from Figures 4 and 5 and averaging them according to the predicted distances of star A with respect to the edge of the occulting disk at the time of observation.", "The spectra were split up into two sets of 7 different bins ($\\Delta $ X = –4 to –2, $\\Delta $ X = –2 to –1, $\\Delta $ X = –1 to 0, $\\Delta $ X = 0 to +1, $\\Delta $ X = +1 to +2, $\\Delta $ X = +2 to +4, and $\\Delta $ X = +7 to +10; see Table 3 for details), with the first set of bins for the spectra taken during ingress (approaching mid-eclipse, phases –0.5 to 0 according to Figure 2), and the second set of bins for those taken during egress (following mid-eclipse, phases 0 to +0.5 according to Figure 2).", "These sets of averaged profiles are displayed in Figure 6.", "To orient the reader, “time\" in Figure 6 runs down the left hand side of panels and then up the right hand side of panels back to the top.", "As one examines the averaged, scaled emission line profiles presented in Figure 6, three distinct “features\" can be discerned and are indicated by arrows in the appropriate panels.", "They are: The absorption component seen mostly out of eclipse and during ingress/egress.", "The broad component mainly seen once star A is fully occulted.", "The double peak emission profile primarily observable during full eclipse.", "The top four panels in Figure 6 (a-d) represent the out-of-eclipse spectra obtained when $\\Delta $ X ranges from –4 to –1.", "The next four panels (e-h) show the ingress and egress spectra obtained when $\\Delta $ X ranges from –1 to +1.", "In these eight panels, a “central\" absorption feature near velocity = 0 is predominant, and sometimes extends well below the stellar continuum.", "The bottom six panels in Figure 6 (i-n) show the averaged H$\\alpha $ profile after the star is completely obscured and at a distance of $\\Delta $ X = +1 or greater.", "Here, we see a generally double-peaked emission line profile, with an underlying broad component.", "The underlying broad component ranges in velocity from approximately -300 to +300 km s$^{-1}$ .", "To emphasize the presence of this faint broad component, which is difficult to see in the regularly scaled plots, the profiles plotted in the bottom 6 panels of Figure 6 have been multiplied by a factor of 10 and over-plotted on the same graph.", "These modified profiles are shown as a dash-dot-dot-dot line in red.", "Note that the bottom two panels in Figure 6 (m and n) are identical.", "The largest variation in the profiles examined here naturally occurs during the eclipse itself as the star and its magnetosphere sink below, and alternatively rise from behind, the occulting screen.", "To focus on these phases, we show in Figures 7 and 8, two ingress sequences; one from 2003 February and one from 2004 December.", "The ingresses are plotted two ways: 1) The top panel shows the fluxed profile with an offset, so that each individual profile is visible.", "The zero point for each spectrum is indicated by the small horizontal line along the y-axis.", "2) The bottom panel shows each spectrum properly fluxed with its continuum adjusted back up to 1.0 so that they can be plotted on top of one another allowing one to see the systematic decline of the flux in the far wings.", "Figures 7 and 8 demonstrate that a “typical\" ingress spectrum may be characterized as an inverse P Cygni type profile, consisting of red-shifted H$\\alpha $ absorption, superimposed upon blue-shifted H$\\alpha $ emission.", "We note that the absorption feature starts out extending well below the stellar continuum.", "As star A sinks below the occulting edge during ingress, the overall photospheric spectrum and all of the absorption that is seen in the H$\\alpha $ profile drops to near zero flux levels.", "The profile that is left once the star is fully occulted and several stellar radii below the occulting edge primarily consists of a rather narrow double-peaked emission feature with peaks at $\\sim $ $\\pm $ 30 km s$^{-1}$ , as well as an underlying faint broad component ranging in velocity from approximately -300 to 300 km s$^{-1}$ , as mentioned earlier.", "This double-peaked emission feature was first noted by Hamilton et al.", "(2003), and its interpretation in terms of a bipolar jet has been discussed in great detail by Mundt et al.", "(2010), and will not be further discussed here.", "After mid-eclipse, or periastron, star A emerges again, rising from behind the occulting screen during egress.", "During this phase, we see a more generally enhanced, roughly symmetric, double-peaked H$\\alpha $ emission profile with a slightly blue-shifted absorption feature, as shown in Figure 9.", "Additionally, Figure 9 shows that during this particular egress, there is a reduction of peak emission flux as more of the star is exposed (i.e., 2005 February 28 vs. 2005 March 01).", "This seems to be contrary to what one would expect; as more of the star is exposed, more of the H$\\alpha $ emitting region, i.e., the magnetosphere, should be exposed, and therefore a greater amount of H$\\alpha $ flux would be expected.", "We return to this issue later.", "In summary, the disappearance/reappearance of both the “central\" absorption feature as well as the overall photospheric spectrum observed in the KH 15D system can be attributed to the effect that the edge of the CBD and its occultation of star A and its close stellar environment has on the observed H$\\alpha $ emission line spectrum.", "Additionally, the variation from inverse P Cygni profile to double-peaked profile must be dependent on where star A is in its orbit, since the inverse P Cygni profile is mostly observed out of eclipse and into ingress.", "The significance of these observations will be discussed in §5." ], [ "Additional Variability", "T Tauri stars (TTSs) are generally characterized by their irregular variability due to accretion and other effects.", "As a way of assessing the baseline variability of the visible TTS in this system, high-resolution spectra were obtained during the out-of-eclipse phases with the HET during the 2002/2003 (Hessman, PI) and 2005/2006 (Herbst, PI) observing seasons.", "These data have been scaled by the appropriate flux ratio, had their continuua adjusted back up to 1 for ease of comparison, and are shown in Figure 10.", "The left-hand side of Figure 10 shows the H$\\alpha $ emission line profiles obtained out-of-eclipse in 2002 December and 2003 January/February.", "The bottom panel shows the 2002/2003 light curve and the vertical lines indicate when the spectra were obtained.", "The phases are listed in each plot as are the model predicted heights above the edge of the CBD on that given date.", "On the right-hand side of Figure 10, we show the out-of-eclipse H$\\alpha $ emission line profiles obtained during 2005 December and 2006 February.", "The bottom panel shows where the spectra were obtained in reference to the 2005/2006 light curve.", "Focusing our attention first on the 2002/2003 season, we note that in this season, the spectra were obtained when star A was located anywhere between $\\sim $ 3 and 4 stellar radii above the edge of the CBD (see Figure 3 and Table 3).", "The phases also range from +0.42 to –0.37, passing through phase $\\pm $ 0.5.", "We remind the reader that positive phases are those following mid-eclipse, representing the egress side of the orbit, while negative phases are those preceding mid-eclipse, representing the ingress side of the orbit.", "A phase of $\\pm $ 0.5 is exactly opposite mid-eclipse, which is denoted as phase = 0.", "A comparison of the blue-shifted peaks observed in 2002 December and 2003 Jan/Feb does not show a great deal of variation in the maximum flux within the line.", "In both cases, the relative line flux is $\\sim $ 2.1-2.2 at the peak.", "The maximum flux occurs when the star is at its greatest height ($\\Delta $ X = –4.02, phase = –0.48 for 2002 December, and $\\Delta $ X = –3.95, phase = +0.49 for 2003 January).", "What is highly variable, however, is the emission, and alternatively, the absorption, in the red-shifted portion of the H$\\alpha $ line profile.", "The interpretation of this absorption feature and the cause of its variability is discussed in §4.", "When comparing the observed profiles obtained during 2005/2006 season, one can see that the maximum flux in the blue-shifted peak in 2005 December is greater than that observed in 2006 February, one full orbital cycle apart, suggesting that this may be the result of the intrinsic variability of TTSs.", "However, it is important to note that the 2005 December spectra were taken as star A was rising up from the edge of the CBD, as denoted by its X position (phases ranging from 0.38 to 0.46).", "We see in this set of profiles that as the star continues to rise from behind the edge, the red-shifted emission increases in flux as the blue-shifted emission becomes somewhat narrower and lower in flux.", "During this time, we also see an increase in the depth of the “central\" absorption feature, eventually extending below the continuum.", "Comparatively, the profiles obtained in 2006 February (phases ranging from 0.39 to 0.5 and then to –0.47) are similar in peak blue-emission flux to the profiles shown on the left-hand side of Figure 10 from 2002/2003.", "The maximum flux occurs around phase 0.5 when the star is at its greatest predicted height above the occulting edge (2006 February 12).", "Here we see both blue and red emission with a narrow “central\" absorption that extends below the stellar continuum for the first three profiles (2006 February 9, 10, and 12).", "As the star begins its “descent\" back toward the obscuring edge (phases have gone from + to –), the red-shifted emission feature disappears and is replaced by a broad absorption feature, apparently separate from the “central\" absorption, and centered around +100 km s$^{-1}$ .", "This type of change in profile shape is also seen in the top left panel of Figure 10 (2002 December).", "As star A moves through phase 0.5 and begins its descent to ingress, the profile loses its red-ward emission as it is replaced by a broad red-shifted absorption feature.", "Note also that the peak blue-emission flux in 2006 February is similar in nature to the peak blue-emission flux seen in the out-of-eclipse spectra observed in 2002/2003, and shown on the left-hand side of Figure 10.", "In summary, the observations show that the H$\\alpha $ emission line profile in the KH 15D system varies systematically with the position of star A in its orbit and as a result of obscuration of the H$\\alpha $ emitting region by the CBD.", "We see a characteristic inverse P Cygni profile during ingress and a double-peaked, enhanced emission profile during egress.", "Our observations also reveal that the “central\" absorption feature is primarily red-shifted with respect to the systemic velocity during ingress and blue-shifted during egress.", "Additionally, it appears as if the profile makes its transition from double-peak to inverse P Cygni just as star A passes through phase = $\\pm $ 0.5, or apastron.", "In the next section, we explore the various components that make up the H$\\alpha $ emission line profile." ], [ "The Absorption Component", "It is clear from Figs.", "4 and 5 that there is a substantial, but highly variable absorption component to the H$\\alpha $ profiles when star A is above the occulting edge.", "Figure 6 shows that this feature depends on the orbital phase of star A.", "During ingress the central absorption is broad and red-shifted relative to the systemic velocity (see panels a, c, e, and g of Figure 6).", "During egress, the absorption feature tends to be narrower and more central or slightly blue-shifted (Figure 6, panels b, d, f, and h).", "Whenever star A is visible above the CBD, the absorption feature extends below the continuum, indicating that the gas responsible for it is at least partly projected onto the stellar photosphere.", "To quantify the velocity behavior of the absorption component we have fit the profiles when the star is not fully eclipsed with three gaussians.", "Normally these were chosen to fit the red and blue emission wings and the central absorption.", "In one or two cases the red-shifted absorption was so strong that there was no blue emission wing visible and we fit two absorption components.", "Note that the intention here is to simply help quantify the mean velocity of the absorbing material clearly visible in the profiles.", "Examples of this fitting are shown in Figure 11, with the combined fit shown as the solid curve in red.", "Note that in this figure, the left hand panels represents the ingress portion of the orbit, while the right hand panels represents the egress portion of the orbit.", "Additionally, we draw attention to the fact that the top panels show out of eclipse spectra ($\\Delta $ X values $\\sim $ +4), while the bottom panels represent spectra when the star was mostly eclipsed ($\\Delta $ X $\\sim $ +0.5).", "Plotting the profiles in this way helps to identify the differences between ingress and egress.", "There is a substantial amount of red-shifted absorption that comes and goes on the orbital cycle, being strongest during ingress and weaker or perhaps even absent during egress.", "It is not clear whether there is a separate narrow absorption component centered near the systemic velocity or the velocity of star A.", "In at least one case shown in Figure 11, where we used two absorption components to model the line, this appears to be the case.", "The phase-dependent red-shifted component seen during ingress is, however, always present.", "Figure 12 shows the velocity of the red-shifted absorption component plotted against binary phase.", "The top panel shows the velocity of the gas relative to the center of mass of the system, while the bottom panel shows it in the frame of reference of star A.", "Again, the feature is only visible when at least some portion of star A is above the occulting edge, so there is a gap during perihelion passage.", "It is quite clear from the figure that as star A becomes visible on the far side of the disk hole from us, the central absorption is redshifted by only a small amount ($\\sim $ 10 km s$^{-1}$ ) with respect to star A.", "But once the star “turns the corner\" and starts to move toward the observer on its way to the near side of the gap where it will soon begin to be occulted, the highly red-shifted absorption component appears, with velocities reaching up to +100 km s$^{-1}$ or more.", "We believe that this relatively high velocity absorption component is a regular feature of the system that varies with orbital phase, and is not merely a stochastic feature attributable to unsteady accretion.", "The evidence is that the feature is present in every spectrum during the appropriate orbital phase, even those taken many cycles (and years) apart.", "It is not always precisely the same strength or velocity, but it is always identifiable as a feature of the H$\\alpha $ profile.", "This suggests that it is a stable accretion feature (or “stream\") associated with star A.", "It is interesting that models of accretion in eccentric binary systems (e.g., de Val-Borro et al.", "2011) predict precisely the sort of stable accretion streams that these observations require.", "In these models, the disk and binary are co-planar, however one can expect the general flow pattern to be rather similar for a slightly inclined system (10-20$^{\\circ }$ ).", "Furthermore, one can expect stronger accretion streams in the inclined system, which may help to understand why this WTTS has such strong accretion signatures.", "A schematic representation of how such an accretion stream would need to be configured to explain our observations was introduced in Figure 1.", "Detailed modeling will be required to determine whether such a model satisfies the observations, but an accretion stream does seem promising as a way to understand this absorption component and its variation with orbital phase.", "Note that the maximum velocity observed, 100 km s$^{-1}$ , is close to the free-fall velocity for gas falling from the inner edge of the CBD near 0.6 AU to $\\sim $ 10 stellar radii.", "In this interpretation, then, we would identify the high velocity gas observed during ingress to be material in the accreting stream just outside the magnetosphere.", "It is quite possible that the gas accretes onto a disk at this location where it awaits a more rapid transfer to the star (or ejection into a jet) during perihelion passage; see next section).", "The lower velocity, but still accreting, gas seen during egress and perhaps at all times, may be part of a remnant disk of gas that did not accrete during perihelion passage or the outer parts of the accretion stream.", "Additionally, since the streams are in co-rotation, when star A is at apastron, gas will be seen approaching with much larger velocities than the star, which would produce a blue-shifted wing.", "Again, a detailed analysis of these events requires a dynamical model tuned to the properties of the KH 15D system and that is beyond the scope of this work." ], [ "The High Velocity Emission Component", "Figure 6 shows clearly that the high velocity wings of the H-alpha profiles vary substantially and systematically with orbital phase.", "For single CTTSs, this emission is believed to arise within the magnetospheres of stars as they accrete material from a circumstellar disk and its strength is a proxy for the accretion rate.", "During phases when star A is only partly occulted or entirely visible the high velocity component is associated with its magnetosphere.", "At other phases there may be a contribution from star B or from the combined magnetospheres.", "During these fully occulted phases much, if not all, of the light reaches us by reflection from the back wall of the CBD (Herbst et al.", "2008), allowing us to observe phenomena occurring during perihelion passage that would otherwise be invisible.", "The data show an interesting and clear trend that repeats in all cycles observed.", "Namely, we observe that the high velocity emission is more intense following perihelion passage (i.e.", "during egress) than it is prior to perihelion passage (i.e.", "during ingress).", "This may be seen by comparing the left hand side of Figure 6, corresponding to ingress, and the right hand side, corresponding to egress.", "The effects are particularly noticeable in panels near the bottom that apply to the fully occulted system just before and just after perihelion passage.", "On the left hand panels, one clearly sees the high velocity emission but it is nowhere near as intense as it is on the right hand panels.", "To make this clearer we have multiplied the relevant profiles by a factor of ten in the bottom six panels.", "Another way to demonstrate this point is to compare individual profiles obtained at the same locations with respect to the occulting disk and determine whether the ingress profiles look different from the egress profiles.", "This direct comparison is shown in Figure 13 where the spectra have been properly fluxed and had their continuum adjusted back up to 1.0, and for two different values of $\\Delta $ X where we happened to have both ingress and egress spectra.", "It is clear that at both locations relative to the occulting disk (partly visible, $\\Delta $ X $\\sim $ +0.5, top panel, and fully occulted, $\\Delta $ X = +1.09, bottom panel) the magnetospheric emission is more intense during egress than ingress.", "This will be quantified further in the next section where we discuss the equivalent widths of the blue emission wing in more detail.", "We note that in addition to the phase-dependent variations discussed above there is clearly a time-dependent variation as well.", "It is not the case that every egress spectrum looks identical in its profile to others obtained at the same orbital phase.", "This time variability is illustrated in Figure 14 where we show some egress spectra obtained near the same orbital phase and at times when star A was fully occulted, or nearly so.", "These spectra are of somewhat lower resolution than many but serve to make the main point: there is substantial variation in the emission line wings that is independent of orbital phase or location relative to the occulting disk edge.", "The 2001 Dec 20 spectrum is particularly notable, and was discussed in Hamilton et al.", "(2003) in the context of magnetospheric accretion.", "Here we see a very extended and substantial wing indicating that during this perihelion passage there was a substantial amount of activity.", "In Figure 15 we show a sequence of spectra taken during a single egress while the system was fully occulted.", "Here, also, we see substantial variability in the high velocity emission, now on a daily timescale.", "There is some evidence for a high velocity “feature\" that shows up as a small bump in the blue wing on 2005 Dec 12 and migrates to lower velocity and higher intensity by 2005 Dec 12.", "Whether this is related to the changing aspect of the star caused by stellar rotation and orbital motion or to some feature of the accretion is unknown.", "It is clear, however, that there is variability in the magnetospheric emission even on timescales as short as a day.", "Our observations appear to confirm a general prediction of models of the binary accretion process in eccentric systems (Artymowicz & Lubow 1996; Günther & Kley 2002; and de Val-Borro et al.", "2011) that accretion onto the stars occurs predominantly during periastron.", "It is during egress, i.e.", "after perihelion passage, that we see the most activity in the high velocity wings of the line and the greatest intensity overall.", "Similar results have been reported for DQ Tau (whose binary parameters are very similar to those for KH 15D; see Table 4) by Mathieu et al.", "(1997) and Basri et al.", "(1997), for V4046 Sgr by Stempels & Gahm (2004), for UZ Tau E by Jensen et al.", "(2007), and for V773 Tau A by Boden et al.", "(2007).", "In some cases mentioned above, the evidence for pulsed accretion includes phase-dependent continuum variations detected by broad band photometric studies.", "It has long been known that KH 15D shows a brightening near perihelion passage (see Hamilton et al.", "2001; Hamilton et al.", "2005; Herbst et al.", "2010) at $I_{C}$ and some variations in color.", "The general phenomenon has been interpreted as the influence of star B, which is inferred to be the brighter of the pair, reaching its greatest extension toward the edge of the CBD at perihelion.", "At these orbital phases it, therefore, becomes the dominant source of the reflected light.", "It could, of course, be that some of this variation is caused by an increased brightness of one or both stars during perihelion passage.", "If the accretion is primarily onto the lower mass star (star A), we may not be able to witness the more dramatic brightness enhancements that occur, because star A is at its lowest point with respect to the disk edge then and perhaps completely invisible to us, even by reflected light.", "This would, however, be in contrast to the current models, which predict that mass is preferentially channeled to the primary.", "Although the photometry cannot directly confirm the case for increased accretion during perihelion passage in the case of KH 15D, they are certainly not inconsistent with the spectroscopic evidence reported here." ], [ "Constraints on the Size of the Magnetosphere of Star A", "Our data taken during eclipses can, at most, be used to constrain the size of the magnetosphere of star A in the vertical direction.", "As outlined in previous papers (e.g., Herbst et al.", "2002) the occulting edge of the circumbinary disk/ring is extremely “sharp,\" i.e., the transition region from optically thin to optically thick is less than $\\sim $ 0.1 R$_{A}$ (and perhaps much less) so that the eclipses can be modeled with a “knife-edge.\"", "The magnetospheric emission model profiles of Hartmann and collaborators (Hartmann et al.", "1994; Muzerolle et al.", "1998; Muzerolle et al.", "2001) show that the blue wing of these profiles is an emission component, while the red wing has contributions from both emission and absorption.", "We have calculated the emission flux and equivalent width of the blue-shifted wing of the H$\\alpha $ profile between –280 km s$^{-1}$ and –85 km s$^{-1}$ , ignoring the velocity range from 0 to –85 km s$^{-1}$ because it is likely affected by emission from the jet.", "For velocities less than -280 km s$^{-1}$ , there is very little emission.", "In Figure 16 we show the fluxes (top panel) and equivalent width (lower panel) for two eclipse sequences obtained during ingress (2003 Feb and 2004 Dec; red squares and blue triangles, respectively) and two eclipse sequences obtained during egress (2005 Feb and 2005 Dec; green stars and pink circles, respectively).", "Each ingress or egress is indicated in the upper left corner of the figure, and the change in position from the center of mass of the binary of star A (r) is indicated for each ingress/egress shown.", "For reference, the binary semi-major axis is 0.25 AU = 41.4 R$_{A}$ .", "Note that for the ingress observations, time increases from left to right, while for the egress observations it is just the opposite.", "To interpret the data displayed in Figure 16, we make use of the “knife-edge\" model, and assume that the emission region is spherically symmetric and homogeneously emitting as an initial thought experiment.", "We begin with the top panel of Figure 16, which shows the blue wing emission flux.", "Within the knife-edge model, the somewhat rapid decrease of the relative line fluxes during ingress can be most easily explained by a decreasing geometrical size of the emission region (i.e., less of the emission region is exposed as star A “sets\" below the edge of the occulting disk), provided this region is not highly variable on time scales of a few days.", "In principal, most of the flux data for the two egress phases can be explained in the same way.", "However, for the 2005 December egress, there is also a strong time variable component superimposed, which causes a strong decline in the line flux for the end of the egress phase (recall that for egress, increasing time runs from right to left).", "As already outlined elsewhere in this paper (see §5) the line fluxes during egress are typically a factor 3 higher than during ingress, which has been attributed to increased accretion from the circumbinary disk/ring onto the two stars at the time near or after periastron passage.", "To constrain the size of the H$\\alpha $ emission region relative to the size of the stellar disk, we use the equivalent width (EW) of the emission contained within the velocity range of –85 to –280 km s$^{-1}$ , because the EW is a measure of the ratio between line flux and continuum flux.", "To interpret the EW measurements within the context of the above mentioned “knife-edge\" model, assuming a spherically symmetric emission line region, we consider two extreme cases: 1) the case of an emission region that is at least a few times larger than the stellar disk, and 2) the other extreme where the emission region is not extended and is roughly the same size as the stellar disk.", "In the first case, the EW should increase between $\\Delta $ X = 0 and $\\Delta $ X = +1, because at $\\Delta $ X = +1, all of the stellar disk is effectively covered and the only continuum emission visible would be that due to the scattered component.", "If the emission region were indeed larger than the stellar disk, say with a radius of 2R$_{A}$ , only a fraction of that region would be covered.", "Therefore, the ratio between emission line flux and continuum flux should be highly increased.", "We do, in fact, see a slight increase in the EW between $\\Delta $ X = 0 and $\\Delta $ X = +1 for both ingresses, however, it is more pronounced for the 2003 Feb ingress.", "If instead, we examine the second case where the emitting region is roughly the same size as the stellar disk, we would not expect a change in the EW at all.", "It is difficult to interpret the observed EW measurements in exactly this way, though the relatively small change in EW suggests a relatively small magnetosphere.", "The egress data displayed in Figure 16 show that during egress the EW of H$\\alpha $ is about a factor of 2-4 greater than during ingress.", "This means that within the simplified spherically symmetric model considered here either the radius or the surface brightness of the line emission region has increased.", "Unfortunately, the egress data are not well sampled between $\\Delta $ X = 0 and $\\Delta $ X = +1 in order to check whether the radius of the emission line region has indeed increased.", "The 2005 Feb egress data suggest that there is a factor of 2 decrease in EW from $\\Delta $ X = +1 to +0.5, but this rests on only one data point.", "In 2005 Dec, the same sort of trend is seen (decreasing EW as the star moves closer to the occulting edge) but the trend is not as steep for that egress, and the EW is on average greater.", "The EW data in Figure 16 can only constrain the extended emission perpendicular to the knife edge since it is believed that star A moves roughly up and down with respect to the occulting edge.", "We cannot exclude a disk-like emission region around the star with a disk of e.g.", "several stellar radii in size.", "In fact, in the models of Hartmann et al.", "(1994), the vertical extent of their magnetospheric accretion flow is barely bigger than the star, however, the magnetosphere extends out to 2-3 R$_{\\star }$ .", "At most, we can argue that this disk like structure could be roughly 10-15% thicker than the stellar diameter (in the direction perpendicular to the knife edge).", "If the H$\\alpha $ emitting region were uniformly bright, we would expect that the EW measurements for both ingress and egress to be the same.", "We show in Figure 17, the flux (top panel) and EW (bottom panel) of the H$\\alpha $ emission contained within the velocity range of –85 to –280 km s$^{-1}$ for all observed profiles.", "Open symbols represent ingress measurements, while filled symbols represent egress measurements.", "Here we see a generally increasing EW for ingress from $\\Delta $ X = 0 to $\\Delta $ X = +1, and that all egress measurements are a factor of 2-4 times greater than ingress measurements.", "If we take the EW of the H$\\alpha $ emitting region as a proxy for accretion rate, we can conclude that the accretion rate onto star A has increased significantly after periastron (phase = 0).", "This has been modeled for eccentric binaries and is shown in Figure 2 of Artymowicz & Lubow (1996), Figure 12 of Günther & Kley (2002), and de Val-Borro et al.", "(2011).", "Both Günther & Kley (2002) and de Val-Borro et al.", "(2011) model the DQ Tau system, which can be used as a proxy for the KH 15D system (see Table 4).", "All models demonstrate that the accretion rates are greater during periastron passage, however, both Artymowicz & Lubow (1996) and Günther & Kley (2002) show that the peak accretion occurs just prior to periastron passage.", "To compare our results to those presented in Figure 2 of Artymowicz & Lubow (1996), we plot our EW measurements as a function of phase and approximated their model for gas accretion rate as a simple gaussian.", "Figure 18 shows that if we scale the gaussian to match our data on ingress (solid red line), we cannot match the egress accretion rate.", "If instead, we scale the gaussian to match our egress data, we cannot match the ingress observations (dashed blue line).", "This suggests that the simulation does not quite accurately predict the timing of the increase in accretion, or there may be something significant within the KH 15D system that is not included within the model, like a differential occultation of the magnetosphere.", "The early calculations by Artymowicz & Lubow (1996) used here were based on smooth particle hydrodynamics and rather coarse.", "A new theoretical approach to this problem is warranted." ], [ "Conclusions", "Our spectroscopic studies have revealed intrinsic variability and amazing structure in the H$\\alpha $ line profiles caused by high velocity gas flows in the vicinity of the stars within the KH 15D system.", "The profiles observed during ingress and egress are distinctly different, changing from an inverse P-Cygni profile during ingress to an enhanced double-peaked profile with broad extended wings during egress.", "The differences in these profiles can be understood in terms of models of accretion flows within an eccentric binary (Artymowicz & Lubow 1996; Günther & Kley 2002; and de Val-Borro et al.", "2011).", "Measurements of the flux and EW of the blue-shifted emission located in the wings of H$\\alpha $ (velocity ranging from –280 km s$^{-1}$ to –85 km s$^{-1}$ ) give us an indication that the H$\\alpha $ emitting region is compact and variable in brightness on the time scale of the orbital period.", "There is potentially a great deal more information locked in these spectra than we have been able to extract.", "The present study will hopefully serve as an incentive and guide to more detailed modeling.", "The dynamics of the binary system are very well understood (W06) and it should be quite possible to model with definiteness the expected gas flows and accretion dynamics using codes that have already been applied to similar systems.", "These data could potentially serve as a critical test of such models.", "Some subtleties of the profile variations have undoubtedly escaped us and comparison with models of this particular system may reveal them.", "We would like to thank the anonymous referee for the suggested improvements to this paper.", "We also thank Dr. Frederic V. Hessman for graciously allowing us to use the data he collected with the HET.", "C.M.H would like to thank Dr. Eric Mamajek and Dr. David James for helpful advice and discussions regarding the creation of Figure 1.", "C.M.H.", "acknowledges partial support by the American Association of University Women through an American Fellowship.", "C.M.J.-K. acknowledges partial support by the NASA Origins of Solar Systems program through the following grants to Rice University: NNX08AH86G and NNX10AI53G.", "W.H.", "acknowledges partial support by the NASA Origins of Solar Systems program.", "Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration.", "The Observatory was made possible by the generous financial support of the W.M.", "Keck Foundation.", "The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community.", "We are most fortunate to have the opportunity to conduct observations from this mountain.", "The Hobby-Eberly Telescope (HET) is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen.", "The HET is named in honor of its principal benefactors, William P. Hobby and Robert E. Eberly.", "This paper is based in part on observations collected at the European Southern Observatory (Program 074.C- 0604A).", "C.M.H.", "would also like to thank Darla and Steve McKee, as well as the Masci family, for graciously supporting her and her daughter throughout the beginning, and final production, of this work, respectively.", "This paper is dedicated to Robert Masci.", "lccc 0pt Spectroscopic Observations of KH 15D UT Date Telescope/Instrument Wavelength Coverage ($\\mbox{Å} $ ) ${\\lambda /\\Delta \\lambda }$ 2001 Nov 29 VLT/UVES 4800-6800 44,000 2001 Dec 14 VLT/UVES 4800-6800 44,000 2001 Dec 20 VLT/UVES 4800-6800 44,000 2002 Dec 06 HET/HRS 6380-7330a 15,000 2002 Dec 10 HET/HRS 6380-7330a 15,000 2002 Dec 13 HET/HRS 6380-7330a 15,000 2003 Jan 23 HET/HRS 6380-7330a 15,000 2003 Jan 26 HET/HRS 6380-7330a 15,000 2003 Feb 02 HET/HRS 6380-7330a 15,000 2003 Feb 05 HET/HRS 6380-7330a 15,000 2003 Feb 08 Keck/HIRES 4200-6600 70,000 2003 Feb 09 Keck/HIRES 4200-6600 70,000 2003 Feb 10 Keck/HIRES 4200-6600 70,000 2003 Mar 06 HET/HRS 6380-7330a 15,000 2003 Mar 23 HET/HRS 6380-7330a 15,000 2003 Dec 16 Keck/HIRES 4400-6800 40,000 2003 Dec 18 Keck/HIRES 4400-6800 40,000 2004 Jan 04 McD/CE 5600-6900 30,000 2004 Jan 05 McD/CE 5600-6900 30,000 2004 Jan 10 McD/CE 5600-6900 30,000 2004 Feb 05 Keck/HIRES 4400-6800 39,000 2004 Mar 10 Keck/HIRES 4700-7100 40,000 2004 Mar 12 Keck/HIRES 4800-7100 40,000 2004 Dec 10 McD/CE 5600-6900 30,000 2004 Dec 13 VLT/UVES 5800-7300 44,000 2004 Dec 14 VLT/UVES 5900-7300 44,000 2004 Dec 15 VLT/UVES 5800-7300 44,000 2004 Dec 16 VLT/UVES 5800-7300 55,000 2004 Dec 17 VLT/UVES 5800-7300 55,000 2004 Dec 18 VLT/UVES 5800-7300 55,000 2005 Feb 27 Keck/HIRES 4800-7100 42,000 2005 Feb 28 Keck/HIRES 4800-7100 42,000 2005 Mar 01 Keck/HIRES 4800-7100 42,000 2005 Dec 12 Mag/MIKE 5800-6800 25,000 2005 Dec 13 Mag/MIKE 5800-6800 25,000 2005 Dec 14 Mag/MIKE 5800-6800 25,000 2005 Dec 15 Mag/MIKE 5800-6800 25,000 2005 Dec 20 HET/HRS 5880-6770b 30,000 2005 Dec 21 HET/HRS 5880-6770b 30,000 2005 Dec 22 HET/HRS 5880-6770b 30,000 2005 Dec 23 HET/HRS 5880-6770b 30,000 2005 Dec 24 HET/HRS 5880-6770b 30,000 2006 Feb 07 HET/HRS 5880-6770b 30,000 2006 Feb 09 HET/HRS 5880-6770b 30,000 2006 Feb 10 HET/HRS 5880-6770b 30,000 2006 Feb 12 HET/HRS 5880-6770b 30,000 2006 Feb 13 HET/HRS 5880-6770b 30,000 2006 Feb 14 HET/HRS 5880-6770b 30,000 aThese wavelengths were covered on the blue chip.", "In addition, the wavelength region 7480-8320 $\\mbox{Å} $ was covered on the red chip.", "bThese wavelengths were covered on the blue chip.", "In addition, the wavelength region 6880-7800 $\\mbox{Å} $ was covered on the red chip.", "lccccccccc 0pt Parameters Used to Plot and Scale the H$\\alpha $ Emission Line Profiles Shown in Figures 4 and 5.", "Julian Date Telescope/Instrument Orbital Phasea Position of Star Ab $RV_{A}$ b $RV_{B}$ b $R$ c $I$ d Flux Ratioe Barycentric Correction km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ 2452242.7446 VLT/UVES -0.26 -1.46 -9.95 4.52 15.28 14.49 0.990 16.23 2452257.8271 VLT/UVES 0.05 8.74 59.86 -27.21 18.86 18.18 0.036 9.10 2452263.7026 VLT/UVES 0.17 1.95 -2.19 0.99 18.31 17.61 0.060 6.35 2452614.9333 HET/HRS 0.44 -3.90 -18.89 8.58 15.25 14.48 1.020 12.80 2452618.9292 HET/HRS -0.48 -4.02 -18.43 8.61 15.27 14.50 1.000 10.89 2452621.7431 HET/HRS -0.43 -3.65 -17.34 7.88 15.26 14.49 1.010 9.90 2452662.6451 HET/HRS 0.42 -3.68 -18.83 8.55 15.25 14.48 1.020 -10.68 2452665.8014 HET/HRS 0.49 -3.95 -18.76 8.52 15.25 14.48 1.020 -12.51 2452672.6090 HET/HRS -0.37 -2.91 -15.82 7.18 15.29 14.52 0.980 -15.23 2452675.7688 HET/HRS -0.31 -1.71 -12.99 5.90 15.29 14.52 0.980 -16.92 2452678.8193 Keck/HIRES -0.26 -0.12 -8.94 4.06 15.57 14.80 0.760 -18.29 2452679.8276 Keck/HIRES -0.24 0.50 -7.20 3.27 16.20 15.46 0.420 -18.71 2452680.8356 Keck/HIRES -0.22 1.17 -5.15 2.42 18.09 17.39 0.070 -19.13 2452704.6771 HET/HRS 0.29 -1.39 -15.35 6.97 15.33 14.56 0.950 -26.23 2452721.6361 HET/HRS -0.36 -2.59 -15.33 6.96 15.28 14.46 0.990 -28.70 2452990.0431 Keck/HIRES 0.18 2.72 -4.83 2.19 18.29$^{\\ast }$ 17.62 0.060 7.84 2452992.0413 Keck/HIRES 0.22 1.09 -10.63 4.83 18.11$^{\\ast }$ 17.42 0.070 6.84 2453008.7960 McD/CE -0.42 -2.71 -17.33 7.87 15.25 14.47 1.020 -1.33 2453009.8729 McD/CE -0.40 -2.48 -16.74 7.60 15.22 14.44 1.050 -2.06 2453014.7920 McD/CE -0.30 -0.77 -12.58 5.71 15.34 14.57 0.940 -4.42 2453040.8025 Keck/HIRES 0.22 0.92 -11.45 5.20 18.24 17.56 0.060 -16.88 2453074.7882 Keck/HIRES -0.07 7.61 23.23 -10.56 18.85$^{\\ast }$ 18.25 0.040 -27.41 2453076.8125 Keck/HIRES -0.03 9.36 42.73 -19.43 18.51$^{\\ast }$ 17.81 0.050 -27.79 2453349.8684 McD/CE -0.38 -1.31 -15.86 7.20 15.28 14.51 0.990 10.85 2453352.7976 VLT/UVES -0.31 -0.28 -13.29 6.03 15.35 14.59 0.930 9.56 2453353.6774 VLT/UVES -0.29 0.11 -12.31 5.59 15.51 14.74 0.800 9.28 2453354.7199 VLT/UVES -0.27 0.60 -11.01 5.00 16.35 15.60 0.370 8.73 2453355.6510 VLT/UVES -0.25 1.09 -9.69 4.40 17.55 16.85 0.120 8.32 2453356.6724 VLT/UVES -0.23 1.66 -8.04 3.65 17.89 17.20 0.090 7.80 2453357.6852 VLT/UVES -0.21 2.28 -6.16 2.79 18.20 17.50 0.067 7.28 2453428.8268 Keck/HIRES 0.26 1.09 -13.25 6.02 17.31 16.60 0.150 -24.91 2453429.8250 Keck/HIRES 0.28 0.52 -14.70 6.67 15.86 15.10 0.580 -25.18 2453430.8201 Keck/HIRES 0.30 0.02 -15.84 7.19 15.47 14.70 0.830 -25.45 2453716.7708 Mag/MIKE 0.21 3.31 -8.20 3.72 18.80 18.13 0.040 9.98 2453717.7708 Mag/MIKE 0.23 2.58 -10.71 4.86 18.27 17.59 0.060 9.49 2453718.7708 Mag/MIKE 0.25 1.93 -12.69 5.76 18.14 17.45 0.070 9.00 2453719.7708 Mag/MIKE 0.27 1.35 -14.25 6.47 17.60 16.90 0.120 8.51 2453724.7264 HET/HRS 0.38 -0.63 -18.26 8.30 15.38 14.61 0.900 6.35 2453725.7340 HET/HRS 0.40 -0.87 -18.59 8.44 15.31 14.52 0.960 5.83 2453726.7222 HET/HRS 0.42 -1.05 -18.79 8.53 15.28 14.51 0.990 5.35 2453727.7181 HET/HRS 0.44 -1.19 -18.89 8.58 15.27 14.50 1.000 4.84 2453728.7090 HET/HRS 0.46 -1.28 -18.90 8.58 15.22 14.45 1.050 4.35 2453773.7556 HET/HRS 0.39 -0.67 -18.49 8.40 15.49 14.73 0.820 -17.81 2453775.7458 HET/HRS 0.43 -1.03 -18.86 8.57 15.34 14.57 0.940 -18.59 2453776.5862 HET/HRS 0.45 -1.12 -18.91 8.59 15.28 14.50 0.990 -18.59 2453778.7409 HET/HRS 0.50 -1.21 -18.72 8.51 15.23 14.46 1.040 -19.74 2453779.5889 HET/HRS -0.49 -1.19 -18.55 8.43 15.25 14.48 1.020 -19.74 2453780.7305 HET/HRS -0.47 -1.12 -18.22 8.28 15.26 14.49 1.010 -20.45 aThe orbital phase was calculated using the following ephemeris for mideclipse: JD (mideclipse) = 2,453,077.59 + 48.37$E$ .", "bThese values come from Model 3 of Winn et al.", "(2006).", "The height above the disk is given in units of the radius of star A.", "The model predicted systemic velocity of 18.676 km s$^{-1}$ has been subtracted out of the radial velocities to put them in the reference frame of the system.", "cThe $R$ magnitudes were either measured directly (marked with an asterisk) or calculated using the following relation derived from the 2002/2003 color data from Hamilton et al.", "(2005): $R$ -$I$ = 1.18301 - 0.0283473*$I$ .", "dThe $I$ magnitudes were taken from Hamilton et al.", "(2005) except for the 2005/2006 observing season, which were taken from Herbst et al.", "(2010).", "eThe flux ratios were calculated relative to the 2002 December 10 (JD = 2452721.6361) observation because it has the highest calculated position above the disk as predicted by Model 3 of Winn et al.", "(2006), and therefore, is presumably the least obscured observation.", "cccc Dates and predicted heights for spectra included in each distance bin presented in Figure 6.", "0pt Distance Bin UT Date Height (R$_{\\star }$ ) Obs/Instr $\\Delta $ X = -4 to -2 2002 Dec 10 -4.02 HET/HRS (ingress) 2002 Dec 13 -3.65 HET/HRS 2003 Feb 02 -2.91 HET/HRS 2004 Jan 04 -2.71 McD/CE 2003 Mar 23 -2.59 HET/HRS 2004 Jan 05 -2.48 McD/CE $\\Delta $ X = -2 to -1 2003 Feb 05 -1.71 HET/HRS (ingress) 2001 Nov 29 -1.46 VLT/UVES 2004 Dec 10 -1.31 McD/CE 2006 Feb 13 -1.19 HET/HRS 2006 Feb 14 -1.12 HET/HRS $\\Delta $ X = -1 to 0 2004 Jan 10 -0.77 McD/CE (ingress) 2004 Dec 13 -0.28 VLT/UVES 2003 Feb 08 -0.12 Keck/HIRES $\\Delta $ X = 0 to 1 2004 Dec 14 0.11 VLT/UVES (ingress) 2003 Feb 09 0.50 Keck/HIRES 2004 Dec 15 0.60 VLT/UVES $\\Delta $ X = 1 to 2 2004 Dec 16 1.09 VLT/UVES (ingress) 2003 Feb 10 1.17 Keck/HIRES 2004 Dec 17 1.66 VLT/UVES $\\Delta $ X = 2 to 4 2004 Dec 18 2.28 VLT/UVES $\\Delta $ X = 7 to 10 2004 Mar 10 7.61 Keck/HIRES (mid-eclipse) 2001 Dec 14 8.74 VLT/UVES 2004 Mar 12 9.36 Keck/HIRES $\\Delta $ X = 4 to 2 2005 Dec 12 3.31 Magellan/MIKE (egress) 2003 Dec 16 2.72 Keck/HIRES 2005 Dec 13 2.58 Magellan/MIKE $\\Delta $ X = 2 to 1 2001 Dec 20 1.95 VLT/UVES (egress) 2005 Dec 14 1.93 Magellan/MIKE 2005 Dec 15 1.35 Magellan/MIKE 2005 Feb 27 1.10 Keck/HIRES 2003 Dec 18 1.09 Keck/HIRES $\\Delta $ X = 1 to 0 2004 Feb 05 0.92 Keck/HIRES (egress) 2005 Feb 28 0.52 Keck/HIRES 2005 Mar 01 0.02 Keck/HIRES $\\Delta $ X = 0 to -1 2005 Dec 20 -0.63 HET/HRS (egress) 2006 Feb 07 -0.67 HET/HRS 2005 Dec 21 -0.87 HET/HRS $\\Delta $ X = -1 to -2 2006 Feb 09 -1.03 HET/HRS (egress) 2005 Dec 22 -1.05 HET/HRS 2006 Feb 10 -1.12 HET/HRS 2005 Dec 23 -1.19 HET/HRS 2006 Feb 12 -1.21 HET/HRS 2005 Dec 24 -1.28 HET/HRS 2003 Mar 06 -1.39 HET/HRS $\\Delta $ X = -2 to -4 2003 Jan 23 -3.68 HET/HRS (egress) 2002 Dec 06 -3.90 HET/HRS 2003 Jan 26 -3.95 HET/HRS cccccc Orbital Parameters for DQ Tau and KH 15D.", "0pt Period $a$ Binary System (days) $M_{2}/M_{1}$ $e$ (AU) $i$ DQ Taua 15.804 0.97 0.56 0.13 23$^{\\circ }$ KH 15Db 48.38 0.83 0.57 0.25 92.5$^{\\circ }$ aParameters for DQ Tau come from Mathieu et al.", "1997. bParameters for KH 15D come from Model 3 of W06." ] ]

[ [ "Modelling rogue waves through exact dynamical lump soliton controlled by\n ocean currents" ], [ "Abstract Rogue waves are extraordinarily high and steep isolated waves, which appear suddenly in a calm sea and disappear equally fast.", "However, though the Rogue waves are localized surface waves, their theoretical models and experimental observations are available mostly in one dimension(1D) with the majority of them admitting only limited and fixed amplitude and modular inclination of the wave.", "We propose a two-dimensional(2D), exactly solvable Nonlinear Schr\\\"odinger equation(NLS), derivable from the basic hydrodynamic equations and endowed with integrable structures.", "The proposed 2D equation exhibits modulation instability and frequency correction induced by the nonlinear effect, with a directional preference, all of which can be determined through precise analytic result.", "The 2D NLS equation allows also an exact lump solution which can model a full grown surface Rogue wave with adjustable height and modular inclination.", "The lump soliton under the influence of an ocean current appear and disappear preceded by a hole state, with its dynamics controlled by the current term.These desirable properties make our exact model promising for describing ocean rogue waves." ], [ "Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents A. Kundu , A. Mukherjee & T. Naskar Theory Division, Saha Institute of Nuclear Physics Calcutta, INDIA The ocean rogue wave, one of the mysteries of nature, has not yet been understood or modelled satisfactorily, in spite of being in the intense lime-light in recent years and the concept spreading fast to other disciplines [1], [2], [3], [4], [5], [6], [7], [8], [9].", "Rogue waves are extraordinarily high and steep surface waves.", "However, most of their theoretical models and experimental observations, excluding a few [3], [4], [10], [11] are one-dimensional, admitting limited high intensity and steepness.", "We propose here a novel two-dimensional integrable nonlinear Schrödinger equation allowing an exact lump-soliton with special asymmetry and directional preference.", "The soliton can appear on surface waves making a hole just before surging up high, with adjustable height and steepness and disappear again followed by the hole.", "The dynamics, speed and the duration of the soliton is controlled by ocean currents.", "These desirable properties make our exact model promising for describing deep sea large rogue waves.", "RWs are reported to being observed in oceans, which apparently appear on a relatively calm sea from nowhere, make a sudden hole in the sea just before attaining surprisingly high amplitude and disappear again without a trace [12], [8], [13].", "This elusive freak wave caught the imagination of the broad scientific community quite recently, triggering off an upsurge in theoretical [14], [11], [10], [15] and experimental [1], [2], [8], [3], [4], [5], [6], [7], [9] studies of this unique phenomenon.", "For identifying the class of such extreme waves the suggested signature of this rare event is a deviation of the probability distribution function (PDF) of the wave amplitude from its usual random Gaussian distribution (GD), by having a long-tail, indicating that the appearance of high intensity pulses are probable more often than that predicted by the GD [16].", "In conformity with this definition RWs were detected in erbium doped fiber laser [9] in optical lasers [1], [8] , in nonlinear optical cavity [4] , in acoustic turbulence in He II [2] and other set ups [6].", "As for the theoretical studies on the ocean RW, apart from few nonlinear models based on the modulation instability, four-wave mixing and conformal mapping [17], [6], [15] or on some other effects [4], [10] , the nonlinear Schrödinger (NLS) equation $i \\partial _tq= \\partial ^2_{x^2}q+2|q|^2q, $ is the most accepted one.", "Equation (REF ) is a well known integrable evolution equation in one space dimension (1D) admitting the Lax pair and exact soliton solutions [18].", "Some models of RW generalise the NLS equation by adding extra terms on physical reasons like ocean current [13] , nonlinear dispersion [14], [19] etc., which however makes the system non-integrable, allowing only numerical simulations.", "The most popular RW model used in many studies is a unique analytic solution of the original equation (REF ), given by the Peregin breather (PB) or its trigonometric variant [5], [11], [7], [20], [21].", "The conventional soliton solution: $ q_s= {\\rm sech}\\kappa (x-vt)e^{i(kx+\\omega t)} $ of the NLS equation (REF ), representing a localised translational wave behaves like a stable particle and unlike a RW propagates with unchanged shape and amplitude.", "In contrast, the exact PB solution of (REF ) $q_P(x,t)= e^{-2it}(u+iv), \\ u=G-1, \\ v=-4tG, $ where $ G= 1 /F, \\ F(x,t)=x^2+4t^2+\\frac{1}{4},$ represents a breather mode ${\\rm cos} 2t $ with unit intensity at both distant past and future, with its amplitude rising suddenly at $t=0 $ to its maximum at $x=0 $ , but subsiding again with time to the same breathing state.", "This intriguing RW like behaviour makes the PB a popular candidate for the rogue wave [5], [11], [7], [21].", "Notice however that, the NLS equation (REF ) together with its different generalisations are equations in $(1+1) $ -dimensions and therefore all of their solutions, including the PB, can describe the time evolution of a wave only along a line.", "Looking more closely into the PB we also realise that the maximum amplitude attained by the RW described by this solution is fixed, and just three times more than that of the background waves.", "The steepness of this wave as well as the fastness of its appearance are also fixed, since solution (REF ) admits no free parameter.", "Therefore, though the well accepted PB or other solutions of the generalised NLS equation could be fitted into the working definition of the ocean RW, saying any wave with height more than twice the nearby significant height (average height among one-third of the highest waves) could be treated as the RW [16], they perhaps, with their severely restricted characteristics, can only explain moderately intense RW-like events in 1D, as observed in water channels [7], optical fibers [5], [9] or optical lasers [1], [8], but are grossly unfit as a model for the ocean RWs.", "First, ocean RWs are two-dimensional (2D) lumps appearing on the sea surface, which are not possible to describe by a 1D equation like the NLS (REF ).", "Second, the ocean RWs, as reported, might be as high as 17-30 meters in a relatively calm sea [8], [13], which is way above the value allowed by 1D solutions like the PB.", "Third, the steepness of the oceanic RW and the speed of their appearance may vary from event to event, whereas they are fixed in the PB (REF ) admitting no free parameter, and are of values much below than those reported for the deep sea RWs.", "Note that in 2D water basin experiments as well as in the related simulations the amplitude and steepness of the RWs were found to be higher [3], [6], [10], [11] than those predicted and observed in 1D [7], [11].", "The above arguments should be convincing enough to reject the PB together with other solutions of the generalised 1D NLS equation and look for a more realistic model for the ocean RWs.", "A straightforward 2D extension of the NLS equation: $i \\partial _tq= d_1 \\partial ^2_{x^2}q -d_2 \\partial ^2_{y^y}q +2|q|^2q, $ was proposed in some studies as a possible RW model [10], [11].", "However the 2D NLS equation (REF ) is not an integrable system and gives only approximate solutions with no stable soliton.", "Nevertheless, this unlikely candidate is found to exhibit RW like structures numerically, with higher intensity and steepness and with an intriguing directional preference [10], [3] and broken special symmetry [11], [4].", "In the light of not so satisfactory present state in modelling the deep sea RWs, we propose a new integrable extension of the 2D NLS equation: $i \\partial _tq= d_1 \\partial ^2_{x^2}q -d_2 \\partial ^2_{y^2}q +2iq(j_x-j_y), \\ j_a=q\\partial _aq^*-q^*\\partial _aq $ together with its exact lump-soliton as a suitable RW model.", "We find that, when the conventional amplitude-like nonlinear term in the non-integrable equation (REF ) is replaced by a current-like nonlinear term, the resulting equation (REF ) miraculously becomes a completely integrable system with all its characteristic properties, which is much rarer in 2D than in 1D.", "More surprisingly equation (REF ) admits an exact 2D generalisation of the PB with the desired properties of a real RW.", "Before proceeding further we notice that, our 2D NLS equation (REF ) can be simplified through a rotation on the plane to $i \\partial _tq+\\partial ^2_{\\bar{x} \\bar{y}}q +2iq(q\\partial _{\\bar{y}}q^*-q^*\\partial _{\\bar{y}}q), $ where the $bar $ over the coordinates will be omitted in what follows.", "Encouragingly, we can find an exact stable soliton solution of the 2D equation (REF ) as $ q_{s(2d)}(x,y,t)= {\\rm sech}\\kappa (y+\\rho x-vt)e^{k_1x+k_2y+\\omega t} $ together with the associated Lax pair, infinite set of conserved charges and higher soliton solutions proving its integrability (see supplementary Note).", "However, the most important finding relevant to our present problem, is to discover an exact solution of (REF ) as a 2D extension of the PB.", "Before presenting the dynamical solution we consider first its static 2D lump-like structure localised in both space directions, describing a fully developed RW: $q_{P(2d)}(x,y)&=& e^{4ix}(u+iv), \\ u=G-1, \\ v=-4xG, \\ \\nonumber \\\\\\mbox{ where} \\ G=\\frac{1}{F}, & & \\ F(x,y)=\\alpha y^2+4x^2+ c. $ One can check by direct insertion that (REF ) is an exact static solution of the 2D nonlinear equation (REF ) with two arbitrary parameters $\\alpha $ and $\\ c $ .", "It marks an important difference of solution (REF ) from the well known PB (REF ), which has no free parameter in spite of its close resemblance with our solution.", "Looking closely into our 2D lump soliton (REF ), as shown in Fig.", "1d, we find that the wave attains its maximum amplitude: $A_{rog}=(1-c )/c=N-1 $ , at the centre $x =0 , \\ y =0 $ , where parameter $c=1/N $ is chosen through an arbitrary integer $N $ .", "On the other hand, at large distances: $|x| \\rightarrow \\infty , \\ |y| \\rightarrow \\infty $ the wave goes to the background plane wave modulating as $ {\\rm cos} (4x)$ , with its amplitude decreasing to a constant: $A_{\\infty }=1, $ (Fig.", "1a).", "The maximum amplitude relative to the background: $\\frac{A_{rog}}{A_{\\infty }}=N-1 $ , reachable by our RW solution, can be adjusted by choosing $N $ to fit different observed heights of the RWs, and particularly the deep sea RWs having significantly higher amplitudes [9], [13] than usually assumed [16].", "Similarly, the steepness of the RW solution (REF ) at both sides as observed from the front: $|\\partial _yq_{P(2d)}| $ , linked to another free parameter $\\alpha $ , can also be changed to fit different observations.", "Note that the amplitude of the wave falls to its minimum: $A_{0}=0 $ , at $x=0, \\ y= \\pm y_0 $ with $y_0 =\\sqrt{\\frac{1}{\\alpha }(1-c)} $ , which is relevant for the hole-wave formation as we see below.", "Our next aim is to construct a dynamical lump soliton out of the static solution (REF ), to create a true picture of a RW which can appear and disappear fast with time.", "For constructing such a solution however we have to clarify first, whether it is possible in principle for our lump soliton to disappear, i.e.", "whether the soliton is free from all topological restrictions, which otherwise would prevent such a vanishing without a trace.", "The reason for such suspicion is due an interesting lesson from topology stating that, when a complex field $q(x,y) $ is defined on a 2D space with non-vanishing boundary condition $|q| \\rightarrow 1 $ at large distances, but having vanishing values $q \\rightarrow 0 $ close to the centre, we can define a unit vector $\\hat{\\phi }= \\frac{q}{|q|} $ on an 1-sphere $S^1 $ .", "However, this vector $\\hat{\\phi }=(\\phi _1,\\phi _2)$ is well defined only at space boundaries: $\\partial {\\rm R}^2\\sim S^1 $ (since $q=0 $ at inner points), realising a smooth map: $S^1 \\rightarrow S^1 $ with possible nontrivial topological charge $Q=n $ .", "This charge with integer values $n=0,1,2,\\ldots $ , labels the distinct homotopy classes and is defined as the degree of the map, which unlike a Nöther charge is conserved irrespective of the dynamics of the system.", "Such a situation occurs for example in type II superconductors with the charge linked to the quantised flux of vortices for the magnetic field ${\\bf B} (x,y) $ [22] : ${2 \\pi } Q= \\int d {\\bf S} \\cdot {\\bf B}=\\int _C {\\bf dl} \\cdot {\\bf A}, $ where $ \\ {\\bf B}= curl {\\bf A}= \\partial _x\\phi _1 \\partial _y \\phi _2-\\partial _x\\phi _2 \\partial _y \\phi _1.$ Notice that, our complex field solution $q_{P(2d)}(x,y) $ possesses clearly the features of $\\hat{\\phi }$ discussed above, since (REF ) goes to a constant modulation $-e^{4ix} $ at large distances and vanishes at points $(0,\\pm y_0) $ .", "Note that, such a solution related to a sphere to sphere map can not go to a trivial configuration , if it belongs to a homotopy class with nontrivial topological charge: $Q=N, N=1,2,3,... $ , due to conservation of the charge, with the only exception for the class with zero charge $Q=0 $ .", "Therefore, for confirming the possible appearance/disappearance property of a RW for solution (REF ), we have to establish first that in spite of defining a nontrivial topological map, it belongs nevertheless to the sector with topological charge: $Q=0 $ , i.e.", "our lump soliton is indeed shrinkable to the vacuum solution.", "For this we calculate explicitly the topological charge (REF ) associated with (REF ) as ${2 \\pi } Q= \\int _C {\\bf dl} \\cdot {\\bf A}=\\int (dx { A}_x+ dy { A}_x),, $ with $ \\ A_a=\\phi _1\\partial _a \\phi _2, \\ \\phi _1= Re q/|q|, \\ \\phi _2= Im q/|q|, $ where the contour integral along $x $ and $y$ are taken along a closed square ${} $ at the boundaries of the plane.", "Substituting explicit form of solution $q(x,y) $ from (REF ) and arguing about the oddness and evenness of the integrants with respect to $x,y $ or checking directly by Mathematica one can show that the related charge is indeed $Q=0 $ and therefore the solution belongs to the trivial topological sector as we wanted.", "The intriguing reason behind this fact is that, the two holes appearing here have in fact opposite charges resulting to the combined charge being zero.", "For constructing a dynamical extension of the 2D RW solution (REF ) we realise that, the sudden change of amplitude with time, as necessary to mimic the RW behaviour, would result to a non-conservation of energy.", "Therefore it can not be described by an integrable equation alone, which insists on a strict conservation of all charges and therefore our integrable equation (REF ) needs certain modification.", "On the other hand, the importance of ocean currents in the formation of RWs, which is not considered in (REF ), is documented and repeatedly emphasised [13], [17] .", "This motivates us to solve both these problems in one go, by modifying our equation (REF ) with the inclusion of the effect of ocean currents, by adding a term in the form $-iU_cq_y $ , as in [13] .", "For obtaining an exact dynamical RW solution to our modified 2D NLS equation, we choose the currents flowing along transverse directions and changing with the time and location as $U_c(y,t)=\\frac{\\mu t}{\\alpha y}$ .", "As is apparent, the currents would flow to the centre of formation of the RW from both the transverse directions, with the speed increasing as they approach, stopping however completely at the moment of the full surge, at $t=0 $ .", "The picture gets reverted after the event with currents flowing back quickly, away from the centre.", "Though $U_c $ looks ill-defined, the multiplicative factor $q_y $ makes the current term well-behaved and the modified 2D NLS equation ((REF ) with the inclusion of the current term) admits now an exact dynamical 2D Peregin soliton, which interestingly has the same form as (REF ), only with the function $F $ becoming dynamical with the inclusion of time variable as $F(x,y,t)= F(x,y)+ \\mu t^2, \\ \\ F(x,y)=4x^2+\\alpha y^2+ c .$ The arbitrary parameter $\\mu $ appearing in solution (REF ) is related to the ocean current and can control how fast the RW appears and how long it stays.", "It is convincingly demonstrated in Fig.", "1 , how this exact dynamical 2D lump-soliton evolves from a background plane wave with a slight depression appearing already at the centre $x=0,y=0 $ (Fig.", "1a).", "As the time passes a sudden hole is formed right in the centre (Fig.", "1b) at the moment $t_h=-\\sqrt{\\frac{1}{\\mu }(1-c)} $ , as told in marine-lore [11], [15], which splits into two and shift apart in the transverse direction (Fig.", "1c) to make space for a high steep upsurge of the lump forming the actual RW (Fig.", "1d) at time $ t=0$ .", "The RW disappears fast into the background waves with the holes merging at the centre and vanishing again, thus describing vividly well the reported picture of the ocean RWs [9], [13], [17] as well as those found in large scale experiments [3].", "The surface waves modelled by our solution (REF ), as visible also from Fig.", "1, show a distict directional preference and an asymmetry between the two space variables.", "a                                   b Figure: NO_CAPTIONFigure: NO_CAPTION c Figure: NO_CAPTION d Figure: NO_CAPTION Figure 1: Formation of 2D rogue wave and holes Panels a-c show the modulus of the dynamical 2D lump-solution and panel d its real part at time $t=0.$ Figures correspond to parameter values $c=1/6,\\ \\alpha =1.2, \\ \\mu =1.2 $ .", "a, Almost constant amplitude background wave with a slight depression appearing at the centre (at time $t=-30.00$ ).", "b, Creation of a hole at the centre at time $t_h=-0.83$ .", "c, The hole splitted into two are drifting away from the centre (at $t=-0.40 $ ).", "d, Full grown RW formed over the background modulation.", "The amplitude attained by the RW is five-times that of the background waves (linked to the choice of $c$ ).", "We conclude by listing a few distinguishing features of our proposed dynamical lump soliton ((REF ) with (REF )), which are important for a ocean RW model.", "1) This is the first 2D dynamical RW model given in an analytic form.", "2) It is an exact Peregin like lump soliton solution of a novel $1+2$ -dimensional nonlinear integrable equation.", "3) The dynamics of the solution is controlled by the ocean currents.", "4) The height and steepness of the RW can be adjusted by independent free parameters.", "5) The fastness of its appearance and the duration of stay can also be regulated by another parameter linked to the ocean current, which becomes the key factor in creating the dynamical RW.", "6) The proposed solution exhibits a clear broken special symmetry in variables $x$ and $y $ as well as a directional preference, which are suspected to be crucial features in the formation of 2D RWs [11], [4], [3], [10].", "6) Strange appearance (and disappearance) of a hole just before (and after) the formation of the rogue wave [11], [17] is also confirmed in our model.", "We hope that, this significant breakthrough in describing large ocean RWs through a dynamical analytic lump-soliton would also be valuable for experimental findings of 2D RWs in other systems like capillary fluid waves [6] optical cavity waves [4] and basin water waves [3].", "Acknowledgement: A.K.", "thanks P. Mitra for valuable discussions.", "Supplementary Note: Integrable properties of the Novel 2D NLS Equation (equation (5) in the main text) I. Lax pair representation: The integrable nonlinear equations can be linked to a linear system $\\Phi _x=U \\Phi , \\ \\Phi _t=V \\Phi , $ associated with a pair of Lax martices $ U,V$ .", "The existence of a Lax pair is considered to be a guarantee for the integrability of a nonlinear partial differential equation.", "The Lax pair dependent on an additional parameter $\\lambda $ , called spectral parameter, are constructed in such a way that their flatness condition : $U_t-V_x+ [U,V]=0 , $ which is the compatibily condition of the overdetermined linear system, yields the given nonlinear equation [18].", "For our nonlinear evolution equation (equation (5) in the main text) in 2D space we can find the Lax pair given through the standard $2 \\times 2 $ Pauli matrices $ \\sigma ^a, a=+,-,3$ as $U= i ( U_{11} \\sigma ^3+ U_{12}\\sigma ^+ +U_{21}\\sigma ^-), $ where $ \\ \\ U_{11}=2\\lambda ^2 +|q|^2 ,U_{12}=U^*_{21} =2\\lambda q-iq_y, $ and $V=i ( V_{11} \\sigma ^3+ V_{12}\\sigma ^+ +V_{21}\\sigma ^-) $ where $ \\ \\ V_{11}=-4\\lambda ^3 +2 \\lambda |q|^2+i(q q^*_y-q^*q_y),$ $V_{12}=V^*_{21} =-4\\lambda ^2 q+2i\\lambda q_{y}+|q|^2q+q_{yy},$ where the notation $q_y=\\partial _y q $ etc.", "as popular in the soliton commulity has been introduced.", "It can be checked directly that our integrable equation (5) (in the main text) is obtained from the flatness condition of the Lax pair (REF , REF ).", "II.", "Conserved charges & soliton solutions : The infinite set of conserved charges associated with our integrable system can be derived systematically from the Riccati equation obtained from the first equation in (REF ) with the Lax operator (REF ) [23].", "Exact higher soliton solutions of equation (5) (in the main text) can be obtained by the Hirota's bilinearization method, similar to the NLS solitons [24].", "III.", "2D rogue waves as exact solutions : The static 2D rogue wave (6) (in the main text) as an exact solution of the 2D NLS equation (5) (in the main text) can be checked by direct insertion.", "The The dynamical 2D rogue wave solution (equation (6) with (9) in the main text) can be checked as an exact solution, by inserting it in the modified 2D NLS equation obtained by adding the ocean current term $-iU_cq_y $ to equation (5) (see the main text)." ] ]

[ [ "HST/ACS Photometry of Old Stars in NGC 1569: The Star Formation History\n of a Nearby Starburst" ], [ "Abstract (abridged) We used HST/ACS to obtain deep V- and I-band images of NGC 1569, one of the closest and strongest starburst galaxies in the Universe.", "These data allowed us to study the underlying old stellar population, aimed at understanding NGC 1569's evolution over a full Hubble time.", "We focus on the less-crowded outer region of the galaxy, for which the color-magnitude diagram (CMD) shows predominantly a red giant branch (RGB) that reaches down to the red clump/horizontal branch feature (RC/HB).", "A simple stellar population analysis gives clear evidence for a more complicated star formation history (SFH) in the outer region.", "We derive the full SFH using a newly developed code, SFHMATRIX, which fits the CMD Hess diagram by solving a non-negative least squares problem.", "Our analysis shows that the relative brightnesses of the RGB tip and RC/HB, along with the curvature and color of the RGB, provide enough information to ameliorate the age-metallicity-extinction degeneracy.", "The distance/reddening combination that best fits the data is E(B-V) = 0.58 +/- 0.03 and D = 3.06 +/- 0.18 Mpc.", "Star formation began ~ 13 Gyr ago, and this accounts for the majority of the mass in the outer region.", "However, the initial burst was followed by a relatively low, but constant, rate of star formation until ~ 0.5-0.7 Gyr ago when there may have been a short, low intensity burst of star formation." ], [ "Introduction", "At high redshift, the star formation rate density is considerably higher than what we observe in the local Universe (e.g., [34]).", "Many galaxies, such as Lyman break galaxies, are thought to be undergoing massive bursts of star formation.", "These starbursts are important to understand since they drive the evolution of galaxies.", "Energy generated in a starburst provides thermal and mechanical heating to the host galaxy, and supernova winds spread chemically enriched material throughout the ISM.", "In some cases, starbursts have enough energy to create a galactic wind that escapes the gravitational potential of the host galaxy and thereby disperses metals throughout the intergalactic medium (e.g., [73]).", "Given the unresolved nature of high redshift galaxies, the information we can glean from them has its limits.", "One approach to understand the physical processes at work during the starburst phase of high redshift galaxies is to turn to studies of nearby starburst galaxies (although these may represent quite a different mass and evolutionary stage).", "Although they are rare, the proximity of starburst galaxies in the local Universe allows us to resolve them into individual stars and thereby study their star formation histories (SFHs) in great detail over the lifetime of the galaxy.", "A starburst is loosely defined as an intense period of star formation that is unsustainable over a Hubble time due to the limited supply of gas available in the galaxy.", "Starbursts in the local Universe are typically found in dwarf irregular (dIrr) and blue compact dwarf (BCD) galaxies, systems which are characterized by their relatively blue colors due to young stellar populations, high gas content, and low metallicities.", "In some cases, their metal abundances are so low that these galaxies have been targeted as possible primeval galaxies that may have only recently begun forming stars [40].", "However, in all dIrrs and BCDs that have been observed to a sufficient photometric depth, red giant branch stars (RGB; i.e., evolved stars older than $\\sim $ 1 Gyr) have been detected, proving that these galaxies have been forming stars for an extended period of time (e.g., I Zw 18, [1]; SBS 1415+437, [2]; I Zw 36, [62]).", "The dIrr galaxy NGC 1569 is one of the closest and most extreme starbursts.", "NGC 1569 has a total dynamical mass of $M = 3.3 \\times 10^8$ M$_{\\odot }$ , 1/3 of which is in HI [37].", "With a mean oxygen abundance of 12 + log($O/H$ ) $= 8.2 \\pm 0.2$ (or $Z =0.25$ Z$_{\\odot }$ , assuming [O/Fe] = 0.0; [28] and references therein), NGC 1569 is thus a gas-rich system with an SMC-like chemical composition.", "Long assumed to lie at a distance of 2.2 Mpc [37], in [30] we unequivocally identified for the first time the tip of the red giant branch (TRGB) and found that its true distance is $\\sim $ 3 Mpc (see also §5.1).", "This makes it a likely member of the IC 342 group of galaxies.", "In the past $\\sim $ 25 Myr, NGC 1569 has formed a large number ($\\sim $ 150) of star clusters with masses similar to Milky Way open clusters ([4]; see also [51], [36]), as well as three super star clusters.", "Based on our improved distance, the three super star clusters, NGC 1569-A1 and NGC 1569-A2 ([16], [24]) and NGC 1569-B ([44]) have masses of (6-7) $\\times 10^5$ M$_{\\odot }$ .", "These clusters, which are more massive than any of the young clusters in the Milky Way or Large Magellanic Cloud, will likely survive for the entire lifetime of the galaxy, thus making them possible precursors to globular clusters.", "Many authors have studied the recent star formation in NGC 1569 using various tracers of the ISM, including: CO ([74]; [29]; [68]); HI ([38]; [66]); H$\\alpha $ ([35]; [18]); and the X-ray emission from hot gas ([33]; [17]; [48]).", "All of these indicators point towards an ISM heavily impacted by a starburst driven galactic superwind.", "Star formation in the central region of NGC 1569 has been studied by a number of authors using HST photometry (e.g., [71], [28], [5]).", "The most straightforward way to determine a galaxy's SFH is by comparing its observed color-magnitude diagram (CMD) to synthetic CMDs, created from stellar evolution models, with star formation rates that vary over time.", "Both [28] and [5] used this method to determine the SFH of NGC 1569 and found that in the past $\\sim $ 1 Gyr, NGC 1569's star formation rate per unit area is 2-3 times higher than in other strong starbursts and 2-3 orders of magnitude higher than what is seen in Local Group irregulars and the solar neighborhood.", "At such a rate, the star formation in NGC 1569 would have exhausted its gas supply in $\\sim $ 1 Gyr.", "Thus, to have sustained star formation for an extended period of time, NGC 1569 would need to have accreted gas, possibly from a nearby HI cloud ([50], [66]).", "However, since neither program's data reached the faint magnitudes needed to sample the RGB, they were not able to constrain NGC 1569's SFH prior to 1 Gyr ago.", "While the existence of an RGB indicates stars older than 1 Gyr, the age-metallicity degeneracy of these stars makes it difficult to resolve the age distribution of stars from the RGB alone.", "The most accurate way to determine the SFH of any stellar population is by resolving the main sequence turn off (MSTO) for even the oldest stars.", "However, at $M_I\\sim 4$ for 13 Gyr old stars, the MSTO is too faint to be observed in a reasonable amount of telescope time for galaxies outside of about 1 Mpc.", "So instead, we must turn to brighter CMD features, such as the core helium burning stars of the red clump/horizontal branch (RC/HB; $M_I\\sim -0.2$ ) to help lift the age-metallicity degeneracy.", "[53] and [54] analyzed a CMD of the elliptical galaxy Cen A (NGC 5128) that reached 0.5 mag below the RC/HB and showed that photometry reaching these depths can provide enough information to determine a galaxy's SFH over a Hubble time.", "Therefore, to determine the SFH of NGC 1569's oldest stars, we obtained deep HST ACS/WFC $V$ - and $I$ -band images that reach down to the RC/HB and cover both the crowded inner and relatively sparsely populated outer regions (HST GO-10885, PI: A. Aloisi).", "Our observational program was designed to reach $\\sim $ 0.5 mag below the RC/HB.", "However, since NGC 1569 is roughly 50% farther away than previously believed, our photometry only barely reaches the RC/HB and not well below it.", "Recently, as a part of a study of 18 nearby dwarf starburst galaxies, [49] analyzed a subset of our images.", "Using the CMD fitting program, MATCH ([19]), they determined the full SFH of NGC 1569.", "Although they focused primarily on the starburst properties, they also determine a relatively coarse age distribution for stars older than 1 Gyr.", "We will discuss their results further in §5.3.", "Since the SFH of young stars in NGC 1569 has been well studied by previous authors, in this paper we present an analysis focused on the SFH of stars older than 1 Gyr.", "In [30] we showed that the recent star formation is concentrated in the core of NGC 1569, while the outer region shows no signs of a significant young population.", "In [30] we adopted the convention of referring to the bottom half of our NGC 1569 field (Fig.", "REF ) as the core and the top half as the halo.", "Recently, in [58], we used four WFPC2 fields to study the number density profile of RGB stars in NGC 1569 out to 8 scale radii and found that NGC 1569 does appear to transition from an exponential disk to a halo population.", "However, as this occurs outside of our ACS field, herein we refer to the upper half of our ACS field as the outer region of NGC 1569.", "Due to the fact that the outer region is much less crowded than the core, these data necessarily go deeper, allowing us to more accurately detect and measure the brightness of stars on the RC/HB.", "This is illustrated by the luminosity functions (LFs) in Fig.", "4 of [30], where the peak due to the RC/HB is visible in the outer region, but not in the core.", "Therefore, herein we focus solely on our observations of NGC 1569's outer region.", "In §2 we present our data and detail the process used to determine the photometry and errors.", "The resulting CMDs for the core and outer region are discussed in §3.", "In §4 we test whether or not NGC 1569's outer region can be treated as a simple stellar population (i.e., if it can be represented by a single age, single metallicity population) while in §5 we use the method of synthetic CMD fitting to determine the full SFH of NGC 1569 and compare our results to the work of [49].", "In §6 we discuss NGC 1569's SFH in the context of interactions with the IC 342 group of galaxies.", "Conclusions are summarized in §7.", "Appendix A provides an overview of our new code SFHMATRIX for determining the SFH a galaxy.", "We observed NGC 1569 with the HST ACS/WFC in November 2006 and January 2007 as a part of HST program GO-10885 (PI: Aloisi).", "Images were taken in the F606W ($V$ ) and F814W ($I$ ) broad-band filters as well as the F658N (H$\\alpha $ ) narrow-band filter.", "Total exposure times were: 61716 s in the $V$ -band, composed of 54 individual images; 24088 s in the $I$ -band, composed of 22 images, and 4620 s in the H$\\alpha $ filter, composed of 4 images.", "While all of the images were centered on the galaxy, telescope rotation between the two observation dates caused both the $V$ - and $I$ -band images to be split between two different orientation angles.", "NGC 1569 was only imaged in H$\\alpha $ during the second visit and, therefore, these data are only at one orientation.", "WFPC2 was used in parallel to image regions in the outskirts of NGC 1569, $\\sim 6$ from the galaxy's center, also in the $V$ - and $I$ -bands.", "These data are discussed in [58].", "Our ACS images were dithered using a standard sub-pixel plus integer dither pattern.", "The sub-pixel pattern is used to improve the sampling of the point spread function (PSF) and aid in the removal of bad/hot pixels and cosmic rays.", "The integer pixel step is necessary to fill in the gap between the ACS chips.", "While these are the same data that were presented in [30], we have reprocessed all of the images using the most up-to-date versions of the ACS pipeline (CALACS) and calibration frames.", "Images in each filter were then combined into a single image using the MULTIDRIZZLE software package ([23]).", "The MULTIDRIZZLE software fine-tunes the image alignment, corrects for small shifts, rotations, and geometric distortions between the images, and removes cosmic rays and bad pixels.", "We experimented with the drizzle parameters within MULTIDRIZZLE and found that resampling the images to 0.7 times the original ACS/WFC pixel scale provides the best resolution and PSF sampling for our data.", "Our final combined images have a pixel size of $0.035$ and cover roughly $3.5\\times 3.5$ .", "Figure REF shows our 3-color image of NGC 1569.", "The H$\\alpha $ image was not used for photometry and is thus excluded from further discussion in this paper." ], [ "Photometry and Calibration", "Using the stand-alone versions of DAOPHOT and ALLSTAR ([65]), we performed photometry on the $V$ - and $I$ -band images in the following manner.", "We created a rough PSF model from $\\sim $ 300 bright, uncrowded stars in each image.", "This rough model was then used to remove neighbors from around the full set of $\\sim $ 1000 PSF stars in each image, thereby allowing us to create a more robust PSF.", "Our PSF stars were chosen to have excellent spatial coverage across the entire image so as to accurately model variations in the PSF shape as a function of position.", "We note that, due to crowding effects, we have avoided using any stars in the crowded inner region of NGC 1569.", "Next, we used ALLSTAR to fit the improved PSF model to independent source detection lists for the $V$ - and $I$ -band images.", "In an effort to find and measure faint stars, we performed a single iteration of subtracting from the images all sources measured in the first ALLSTAR run, searching for previously undetected objects, adding those objects to the original source detection list, and then re-running the PSF fitting on the original images, this time using the newly updated catalog.", "We matched the $V$ and $I$ photometry lists, requiring that detections were within 0.5 pixel to be considered a match.", "The resulting photometric catalog contains over 400,000 sources.", "To further clean our catalog of false detections or background galaxies, we made cuts by both position and photometric quality.", "Positional cuts were used to trim detections from around overexposed stars, where diffraction spikes can lead to false detections, as well as from the edges of the images and the ACS chip gap, which had lower exposure time due to dithering and therefore a much higher incidence of noise peaks that were coincident between the $V$ and $I$ images.", "ALLSTAR provides three estimates of the quality of the PSF fit to any given source: $\\sigma $ , the error in the PSF magnitude; $\\chi $ , the goodness-of-fit for the PSF fit to each star; and $sharpness$ , a measure of the intrinsic size of the source relative to the PSF.", "Assuming that noise peaks follow a Gaussian distribution around the mean background level, we expect on the order of one noise peak that meets our matching criteria and lies 3.5$\\sigma $ above the mean in both bands.", "Since a signal-to-noise ratio of 3.5 translates to an error of $\\sim $ 0.3 mag, we keep only those stars with $\\sigma _{V} \\le 0.3$ and $\\sigma _{I} \\le 0.3$ .", "Sources with a $sharpness$ $\\ll $ 0 are considerably more narrow than the PSF and are likely to be false detections, while those with $sharpness$ $\\gg $ 0 are more extended and likely to be background galaxies or stellar blends.", "Thus, we cut from our catalog any stars with $|sharpness| > 1$ in either band.", "We have chosen to not make any cuts based on $\\chi $ since most of the faint objects with large $\\chi $ values have already been cut based on their $\\sigma $ and $sharpness$ values.", "Bright sources with large $\\chi $ values are generally real stars, with the large $\\chi $ due to inadequacies in the PSF (which appear more significant when the star is bright and the noise is low).", "After applying all of the above cuts, our photometric catalog contains over 370,000 stars, with approximately 31,000 of those stars residing in the outer region.", "Finally, we apply the necessary zeropoints and corrections to place our photometry on the Johnson-Cousins magnitude system following the prescriptions in [63].", "First, we converted instrumental magnitudes to the HST VEGAMAG system according to the equation $m_{vegamag} = m_i + C_{ap} + ZP - C_{inf} + C_{sky} + C_{cte}.$ $m_i$ is the PSF-fitting magnitude [$-2.5 \\times log(flux)$ ] within a radius of 10 pixels (0.35) and $C_{ap}$ is the correction necessary to convert that magnitude to the conventional 0.5 radius aperture.", "ZP is the HST VEGAMAG zeropointhttp://www.stsci.edu/hst/acs/analysis/zeropoints for a given filter and $C_{inf}$ is an offset to convert magnitudes from the 0.5 radius aperture to an infinite aperture [63].", "$C_{sky}$ corrects for the fact that our aperture correction, $C_{ap}$ , uses an annulus that is a finite distance from the star to estimate the sky background.", "Since a small amount of star light is included in the annulus, the sky background is overestimated in the aperture correction step.", "The ACS CCDs suffer from losses in charge transfer efficiency (CTE) as a result of their prolonged exposure to radiation in space.", "We corrected for CTE losses following the prescription in [56], namely, $C_{cte} = 10^A \\times SKY^B \\times FLUX^C \\times \\frac{Y}{2048} \\times \\frac{MJD - 52333}{365},$ where $SKY$ is the sky background counts per pixel, $FLUX$ is the star counts per exposure within our PSF fitting radius, $Y$ is the number of charge transfers, and $MJD$ is the Modified Julian Date.", "The coefficients were taken from [56] for an aperture radius of $r = 7$ pixels on the native pixel scale (or 10 pixels on our resampled drizzled image), and are $A = -0.7$ , $B = -0.34$ , and $C =-0.36$ .", "Due to the fact that our images were both dithered and taken at two different orientations, each star has 54 different charge transfer values, $Y$ , in the $V$ -band and 22 different $Y$ values in the $I$ -band.", "We tested two approaches for dealing with this in calculating $C_{cte}$ for each star.", "First, we calculated the CTE corrections for each star in each individual image and averaged those together to get $C_{cte}$ for each star.", "In the second approach, we determine the average number of charge transfers for each star and then calculate a single CTE correction for each star.", "We found that these two approaches give similar $C_{cte}$ values for each star.", "Our average CTE correction is $\\sim $ 0.01 mag, but can be as large as $\\sim $ 0.2 mag for the faintest stars.", "After applying all of the zeropoints and corrections, the last step in our calibration was to convert photometry on the VEGAMAG system to the Johnson-Cousins system by following the procedure outlined in [63].", "Note that we have not applied any reddening corrections to our photometry since these are included as part of our models (see § and §)." ], [ "Completeness and Errors", "To properly evaluate the role of incompleteness and photometric errors in the analysis of our data, we performed artificial star tests on our $V$ - and $I$ -band images.", "We added artificial stars to our images using the ADDSTAR routine within DAOPHOT.", "This routine simulates real stars covering a user supplied range of positions and magnitudes by adding the appropriate Poisson noise to the previously generated PSFs.", "In an effort to compromise between efficient computing of the completeness and not changing the crowding on the images, we divided the images into a grid of boxes 30 pixels $\\times $ 30 pixels in size and placed one artificial star in each box, with the added restriction that no two artificial stars lie within two PSF fitting radii (20 pixels) of each other.", "To fully sample the images, we varied both the starting position of the grid as well as the position of the artificial star within each box, allowing for shifts as small as 0.01 pixels.", "We chose the range of magnitudes covered by the artificial stars to extend $\\sim $ 1 mag brighter and fainter than the observed data.", "To better mimic the observed luminosity function, we generated twice as many artificial stars in the faint half of the magnitude range as in the bright half.", "After placing the artificial stars on the image, we followed the same procedure as outlined in the previous section and performed PSF fitting on the entire image.", "We then cross-correlated the input artificial star list with the output photometric catalog, again using a 0.5 pixel matching radius.", "An artificial star was considered to be “lost\" if it was not matched in the output catalog, or if its PSF fitting magnitude differs from its input magnitude by more than 0.75 mag, it has $\\sigma >0.3$ or $|sharpness| > 1$ .", "We note that we also trimmed the artificial stars by position in the same way as for the observed catalog, but these stars were not counted in the lost/recovered statistics.", "In the outer region, we simulated over 170,000 stars in each band, or more than 5 times the number of stars observed in the outer region.", "In Figs.", "REF and REF we present the resulting CMDs for the core and outer region of NGC 1569.", "Previous authors have shown that NGC 1569 has recently undergone massive bursts of star formation (e.g., [28], [5]).", "This is evident in the CMD of the core (Fig.", "REF ), which is dominated by features that are due to young stellar populations.", "Readily visible above $I \\sim 24$ and with $0.6 (V-I) 1.1$ is the blue plume, which contains both young ($$ 10 Myr) main sequence stars and massive evolved stars ($$ 9 M$_{\\odot }$ ) on the blue part of their core helium-burning phase.", "The red plume of supergiants at $1.9 (V-I)2.5$ and the blue loop stars (between the red and blue plumes) indicate the existence of evolved stars with masses $$ 5 M$_{\\odot }$ .", "Also visible at $I \\sim 24$ and $(V-I) 2.5$ are the intermediate-mass ($\\sim $ 1.2 - 6 M$_{\\odot }$ ) carbon stars of the thermally pulsating asymptotic giant branch (AGB), as well as M-type AGB stars.", "Unlike the core, the CMD of the outer region (Fig.", "REF ) shows no signs of a significant young stellar population, indicating that the recent bursts of star formation in NGC 1569 were restricted to the core.", "In the outer region, the only outstanding feature is the upper $\\sim $ 4 mag of the red giant branch (RGB), which is the result of stars older than $\\sim 1$ Gyr that have evolved off of the main sequence and are expanding as their He cores increase in mass, contract, and work their way toward becoming fully degenerate.", "RGB stars are also present in the core, though heavily blended in the CMD with younger evolutionary features, indicating a global star formation history that began at least 1 Gyr ago.", "Given NGC 1569's location ($\\ell = 14368213$ , $b = 1124174$ ) near the Galactic Plane, we expect our CMDs to suffer from some foreground contamination due to the Galaxy.", "This contamination can be seen in Fig.", "REF as the swath of stars stretching from $I\\sim 19$ , $(V-I) \\sim 1$ to $I \\sim 25$ , $(V-I) \\sim 4$ .", "In Fig.", "REF , we confirm that this “feature\" is due to the Milky Way by plotting a CMD of the expected foreground stars in our ACS field based on the Besançon model of the Galaxy [57].", "The Besançon model includes four populations: the thin disk, thick disk, bulge, and spheroid.", "The plume of faint stars around $V-I \\sim $ 1.3 is due to white dwarfs in the disk, and the stars in top right part of the CMD are primarily M-dwarfs in the disk.", "These foreground stars are not considered further in the remainder of this paper.", "We note that after taking into account the foreground stars, there is a sparse blue ($V-I\\sim 0.4$ in Fig.", "REF ) plume of stars in the outer region that are not related to the old RGB stars.", "These stars are spread throughout the outer region, but with a higher concentration toward the core of the galaxy, and are possibly young stars that have migrated out of the core (§5.3)." ], [ "Luminosity Function Analysis with Simple Stellar Population Models", "As we have shown above, the outer region of NGC 1569 appears to be a purely old population and thus provides the opportunity to study the SFH of stars older than $\\sim $ 1 Gyr.", "Since we have no a priori reason to assume a specific SFH for the outer region (e.g., that it formed in a single burst or experienced extended episodes of star formation), we begin with the most straightforward approach of treating NGC 1569's outer region as a simple stellar population (SSP), i.e., one that can be fit by a theoretical isochrone with a single age and metallicity.", "Models with more complicated SFHs are addressed in §.", "For both the SSP analysis and full SFH analysis, we use stellar evolution models from the Padova grouphttp://stev.oapd.inaf.it/cgi-bin/cmd (e.g., [46]) as they provide comprehensive coverage of the stages of stellar evolution and are readily available for a large range of ages, metallicities, and photometric systems.", "The metallicity of an isochrone is $Z$ .", "Throughout this paper we will often use instead the approximation $log(Z/Z_{\\odot }) =$ [Fe/H], where the equality holds for a solar composition mix.", "Like the Padova models, we adopt $Z_{\\odot } = 0.019$ from work by [3]There are many indications for a lower value of $Z_{\\odot } = 0.0134$ as advocated by [6], but it has so far not been possible to bring such a lower number in agreement with Solar sound speed measurements from helioseismology.", "Recent 3D modeling of the solar photosphere from [12] gives $Z_{\\odot } = 0.0153$ , which moves in the direction of reconciling solar photosphere and helioseismology measurements." ], [ "Theoretical RGB Luminosity Function", "The RGB contains a plethora of information, with age, metallicity, distance, and reddening all playing a role in the luminosity, shape and color of the RGB.", "For a large range of ages and abundances, the $I$ -band luminosity of the tip of the RGB (TRGB) is approximately constant at $M_I \\sim -4.0$ [7], thus acting as a standard candle.", "Unfortunately, while the shape of the RGB can place some constraints on [Fe/H], the RGB color suffers from a well known degeneracy, where different combinations of age, metallicity, and reddening can create RGBs that are similar in appearance.", "Rather than dealing with the RGB as a whole, we instead turn to three features that may be visible in the LF of the RGB: the HB/RC, RGB bump, and AGB bump.", "The HB/RC feature on the CMD is a collection of evolved stars that are in their core helium burning phase.", "The exact luminosity of the RC/HB is dependent on both the age and metallicity of the stars (e.g., [31], [26]), but is roughly $3.5-4$ mag fainter than the TRGB in the $I-$ band (Fig.", "REF ).", "Both the RGB bump and AGB bump are more subtle features that are typically not identifiable in the CMD of a stellar population and are only sometimes visible as discrete “bumps\" in the LF.", "The RGB bump feature occurs when the H burning shell in RGB stars moves outward and crosses the chemical discontinuity produced during the first dredge up.", "Since the luminosity of the H burning shell, $L_H$ , is related to the mean molecular weight, $\\mu $ , as $L_H \\propto \\mu ^7$ , the sudden drop in $\\mu $ at the discontinuity (due to an increase in the H abundance) causes a temporary drop in the luminosity of an RGB star.", "RGB stars spend $\\sim $ 20% of their total RGB lifetime in the bump phase.", "Due to the fact that the depth of the chemical discontinuity is dependent on both the mass and composition of the star, the luminosity of the RGB bump acts as a tracer of the age and metallicity (Fig.", "REF ).", "Following central He exhaustion, RC/HB stars evolove up the AGB.", "The transition from core helium burning to a thick He shell burning configuration causes a temporary drop in the luminosity of AGB stars, which results in the formation of the AGB bump.", "Fig.", "REF shows that the luminosity of the AGB bump is primarily dependent on the age of the population, with [Fe/H] playing a role only for the most metal rich systems.", "Based on theoretical models, the RC/HB is visible in all stellar populations with log(age) $$ 8.8 (630 Myr), regardless of [Fe/H].", "On the other hand, while both the RGB bump and AGB bump features form in stellar populations with log(age) $$ 9.1 (1.25 Gyr), these features cannot always be identified in a LF.", "Aside from situations where the number of stars is so low that few stars exist on the RGB (e.g., open clusters), the significant range in brightness of the RGB bump ($>$ 2.5 mag) can cause it to overlap with either the AGB bump or the RC/HB.", "In contrast, the AGB bump varies in brightness by less than 0.5 mag and is never less than 1.0 mag brighter than the RC/HB.", "This gives rise to stellar populations that can have either two or three distinct bumps in their LFs, assuming they are well populated.", "Although there is still some age/metallicity degeneracy with the RC/HB, RGB bump and AGB bump, by analyzing the brightness of each of these features relative to each other and relative to the TRGB, we can eliminate any dependence on distance and reddening and place considerable constraints on the age and metallicity of the stellar population.", "Then, with age and metallicity in hand, we can use our synthetic CMDs to compare the predicted intrinsic color of the RGB with the observed color and thereby determine the reddening of the stellar population.", "At known reddening, the TRGB magnitude provides the galactic distance." ], [ "Age and Metallicity", "In Fig.", "REF we plot the observed LF of NGC 1569's outer region as the solid black histogram with the completeness corrected LF plotted in red, where we have used our artificial star tests to determine the correction factor for each magnitude bin.", "Fainter than the TRGB (dashed vertical line), the LF of NGC 1569's outer region shows a number of peaks, the most obvious of which is the RC/HB feature at $I \\sim $ 28.", "A comparison with the completeness corrected LF shows that we have only reached the bright end of the RC/HB, a result of the fact that NGC 1569 is 50% farther away than previously believed (§5.1).", "To determine which of the other peaks may be real features, in Fig.", "REF we fit the slope of the RGB between $I$ = 24.8 and 26.8 with a solid line.", "We also show error bars with the Poisson error for each magnitude bin.", "Although none of the other peaks along our LF are as conspicuous as seen by other authors (e.g., [53]), there are two small bumps, marked with the arrows, that stick up above the RGB and are in the correct locations to possibly be the AGB and RGB bumps.", "Having identified the RC/HB, as well as AGB and RGB bump candidates in NGC 1569's LF, we can now compare the positions of these three features, plus that of the TRGB, to the predictions of theoretical isochrones.", "To measure the magnitudes of both bumps and the RC/HB in the observed LF of NGC 1569, we first subtract the slope of the RGB, leaving just the features in which we are interested.", "We fit gaussians to both bump features and take the center of the gaussian as the $I$ -band magnitude of each feature.", "For the two bumps we find: $I_{bump1} = 26.94 \\pm 0.02$ and $I_{bump2} = 27.12 \\pm 0.03$ , where the errors listed are the errors in the fit.", "Due to completeness issues near the RC/HB, which result in a steep drop off in the LF, fitting a gaussian to the RC/HB feature may yield a biased measurement.", "Instead, we simply take the magnitude of the LF bin with the most stars as our measurement of the RC/HB.", "For the RC/HB, which is $\\sim $ 0.4 mag wide (full width, zero intensity), or about eight times the LF bin size, we measure $I_{RC/HB}= 27.93 \\pm 0.025$ .", "This is an upper limit on the brightness of the RC/HB due to completeness effects.", "The quoted error is half the width of the magnitude bins in the LF.", "In §REF we discuss the measurement of the TRGB, for which we find $I_{TRGB} = 24.47 \\pm 0.04$ .", "Due to the fact that our observations only reach down to the RC/HB branch feature, and not well below it, completeness and photometric errors may play a role in the magnitudes that we measure for the RC/HB and RGB and AGB bumps.", "Therefore, we cannot simply compare our numbers to those predicted by stellar evolution models.", "Instead, we use the same code as described below in Appendix A to create synthetic CMDs from each individual isochrone.", "In this, we make use of our artificial star test results to apply the appropriate errors and incompleteness.", "We have assumed a reddening of $E(B-V)$ = 0.56 [37], which, when combined with our measured TRGB value, gives a distance of $\\sim $ 3 Mpc.", "Then, in the same way as for NGC 1569's LF, we measure the magnitudes of the RGB bump, AGB bump, and RC/HB.", "At the brightness of the TRGB, completeness and errors are not expected to significantly affect our measurements.", "We therefore use the TRGB magnitude as reported in the isochrones.", "Since then we are confident in our identification and measurement of the RC/HB, we use this feature to make the first cut in determining which age and metallicity best fits NGC 1569's outer region.", "In Fig.", "REF we plot the values of $I_{TRGB} - I_{RC/HB}$ , as predicted by our synthetic CMDs, as a function of age for a range of abundances.", "Overplotted as the dashed line is our measured value for NGC 1569, with the dotted lines representing the 3$\\sigma $ error bars.", "Only those synthetic CMDs that fall within 3$\\sigma $ of our observed value are considered further.", "Because we have no a priori knowledge of the order of the AGB and RGB bumps, or if the bumps we have identified are actually real features, we consider all combinations of these two features in determining our best fit LF.", "We find that all of the best fit models for the RC/HB show only the AGB bump, with the RGB bump `lost' in the RC/HB feature.", "The isochrone that provides the best fit to the positions of the TRGB, AGB bump, and RC/HB to within the measurement errors has log(age) = 10 (10.0 Gyr) and [Fe/H] = -1.4 (Z = 0.0008).", "Its LF is plotted in Fig.", "REF as the red histogram along with NGC 1569's outer region (black histogram).", "While this is the best fit model, there are two major differences between the observational data and model.", "First, the AGB bump and RC/HB features in the model are much more pronounced than what we observe.", "Second, the slope of the synthetic LF is similar to what is seen in star clusters ($N(L) \\propto L^{-\\beta }$ with $\\beta \\approx 0.32$ ; e.g., [75]) whereas the slope of NGC 1569's RGB is much steeper ($\\beta = 0.37$ ).", "The poor fit to the bump features opens the possibility that the bumps we have identified in the LF in Fig.", "REF may not be associated with the actual RGB and AGB bumps in NGC 1569.", "That is not to say that they do not exist in NGC 1569, but rather that they have been smoothed out due to a superposition of mixed stellar populations.", "Combined with the difference between predicted and observed LF slopes, this suggests that NGC 1569's LF as a whole is not represented by any single age, single metallicity population; it must have had a more complex SFH, which we derive explicitly in §." ], [ "Distance and Reddening", "As discussed in [30], our data represent the first unequivocal detection of the TRGB in NGC 1569 and thereby allow us to determine its distance via the luminosity of the TRGB.", "Although we presented the distance to NGC 1569 in [30], since we have reprocessed our data using the most recent HST ACS calibration data we revisit our distance calculation, which also depends on the reddening.", "We calculate the distance and reddening here, still assuming NGC 1569 can be represented by a SSP, and discuss the results from more complicated SFHs in §5.2.", "In the LF of a stellar population the TRGB acts as a discontinuity, the position of which is easily measured.", "Using the software developed by one of us (R. P. v. d. M.) and detailed in [14], we find that $I_{TRGB} = 24.47 \\pm 0.04.$ The location of the TRGB is marked by the vertical dashed line in Fig.", "REF .", "We note that this value is slightly fainter than reported in [30] due to the updated processing and photometry.", "For the reddening appropriate for NGC 1569, we can either use values from the literature (as we did in [30]), or we can estimate it from our data.", "Literature values for the foreground extinction toward, and intrinsic reddening in, NGC 1569 span a wide range of values.", "The most commonly adopted value is $E(B-V)$ = 0.56 [37], which was calculated using integrated UV photometry of the core of NGC 1569.", "Since the recent star formation, and therefore the gas and dust, in NGC 1569 is concentrated in the core, we expect the reddening in the outer region to be lower.", "Thus, in [30] we adopted the foreground extinction due to the Milky Way provided by the [11] reddening map, $E(B-V)$ = 0.50.", "We note that the reddening map of [61] gives a much higher value, $E(B-V)$ = 0.68, for NGC 1569.", "However, as discussed by [55], this estimate of the foreground extinction is higher than the total extinction in NGC 1569 derived by many authors (e.g.", "[37], [51]) suggesting that it is likely an overestimate of the reddening at the position of NGC 1569.", "In the previous section we used the relative brightnesses of the TRGB, AGB bump, and RC/HB features to determine an estimate for the age and [Fe/H] of the dominant population in NGC 1569.", "Since the RGB spans a very small range in color, this approach gives an age and metallicity that is independent of the reddening of the stellar population.", "Thus, to determine $E(B-V)$ from our data we can compare the observed RGB color of NGC 1569 with that predicted for a synthetic population that has an age of 10 Gyr and [Fe/H] = -1.4.", "For RGB stars on the synthetic CMD $\\sim $ 0.5 mag below the TRGB, we find an intrinsic color of $(V-I)_0 = 1.29$ .", "Over the same magnitude range, RGB stars in NGC 1569's outer region have an average apparent color $(V-I) = 1.98$ .", "This implies a reddening of $E(B-V)$ = 0.58 (adopting the reddening law of [13], where $A_V = 3.1E(B-V) $ and $A_I = 0.614 A_V$ for photometry on the Johnson-Cousins system).", "The absolute $I$ -band magnitude of the TRGB ($M_I^{TRGB}$ ) is well known to be roughly constant around $\\sim $ -4.0 mag in SSPs with age $\\mathrel {>\\hspace{-10.0pt}\\sim }$$ 2 Gyr and [Fe/H] $ <$\\sim $$ -0.5.", "\\cite {barkeretal2004} usedsynthetic CMDs to study the reliability of the TRGB as a standard candlefor resolved stellar populations with complex SFHs.", "They found that$ MITRGB = -4.0 0.1$ independent of the SFH, provided that themedian dereddened color of RGB stars $$ 0.5 mag below the TRGB is$ (V-I)0 1.9$.", "It is therefore appropriate to assume $ MITRGB = -4.0 0.1$ for our distance calculations.", "Combined with $ ITRGB = 24.47 0.04$ and $$E(B-V)$ = 0.50 0.05$ (from \\cite {bh1982}), thisyields $$ {(m-M)}_{0} $ = 27.52 0.14$ or $ D = 3.19 0.21$ Mpc.", "If weinstead use the reddening estimated from our RGB analysis, $$E(B-V)$ = 0.58 0.06$ (we assume a 10\\% error), we find $$ {(m-M)}_{0} $ = 27.36 0.16$ or$ D = 2.96 0.22$ Mpc.", "Both of these distances are much farther thanthe typically assumed distance of 2.2 Mpc (e.g., \\cite {israel1988})and place NGC 1569 on the near edge of the IC 342 group of galaxies.$" ], [ "Methodology", "We showed in § that the LF of the NGC 1569 outer region is not well fit by an SSP and must have had a more complex SFH.", "To infer this SFH we use the new SFHMATRIX code (van der Marel & Grocholski in prep.)", "described in Appendix A.", "Starting from a set of isochrones, and for any assumed distance and extinction, the code finds the SFH as function of age and metallicity that best matches the Hess diagram of the observed CMD in a $\\chi ^2$ sense.", "We note that the full SFH calculations are independent of our SSP analysis and that the accuracy of our SFH is ultimately tied to the accuracy of the stellar evolution models we have adopted.", "For a given assumed extinction, we find the best-fitting distance as follows.", "We start with a trial distance and find the best fitting SFH.", "From this we create a synthetic CMD realization (i.e., a Monte-Carlo realization that has the same number of stars on the CMD, taking incompleteness and photometric errors into account, as the observed CMD).", "We then analyze both the observed and the synthetic CMD with the TRGB analysis software described in Section REF .", "Based on the inferred difference between the observed and synthetic TRGB magnitudes we adjust the trial distance, and iterate this procedure till convergence (agreement better than $0.01$ mag).", "The iteration is necessary because not all SFHs yield the same absolute TRGB magnitude; the distance depends on the SFH.", "There generally is agreement at the $\\pm 0.1$ mag level [7], but this is not sufficient for the most accurate results.", "To infer the extinction, we run the procedure for a range of trial extinction values and plot the $\\chi ^2$ of the CMD fit as a function of extinction (see Fig.", "REF ).", "The data vary smoothly and are well fit by a fourth-order polynomial (dashed curve).", "This has a minimum at $E(B-V)$ = 0.58, which provides the best-fitting extinction.", "The corresponding best-fitting distance modulus is $ {(m-M)}_{0} $ = 27.43, yielding a distance of 3.06 Mpc.", "These results agree very well with published reddenings as well as the results from our SSP analysis of the LF and RGB color in Section REF .", "In principle, the random errors on the extinction and distance can be robustly determined using a repeated analysis of pseudo-data created from the observed CMD data set using bootstrapping (see Appendix A).", "In practice, systematic errors of various kinds are probably of more importance.", "Based on the combined insights from literature studies of the extinction, and our own SSP and SFH analyses, we adopt as our final estimates $E(B-V) = 0.58 \\pm 0.03$ and $D = 3.06 \\pm 0.18$ Mpc.", "The distance error is dominated by the systematic uncertainty in the knowledge/calibration of the absolute magnitude of the TRGB feature.", "It is of interest to note that the SFH analysis yields extinction values that are consistent with those inferred through independent techniques.", "This means that the CMD data has enough information content to break the age-metallicity-extinction degeneracy for old populations.", "The color and curvature of the RGB is driven primarily by metallicity with only a small contribution from age.", "Metal-poor isochrones are relatively blue with little curvature, while metal-rich isochrones are red and have a significant amount of curvature.", "Therefore, if we use too low of a value for the extinction, the SFH code must use the metal-rich isochrones to match the observed color of the RGB, resulting in a synthetic CMD with too much curvature on the RGB.", "Conversely, high values of extinction lead to metal-poor synthetic CMDs with too little RGB curvature.", "Since our data reach $\\sim 4$ mag below the TRGB, the curvature is observationally well constrained.", "We found that the high $\\chi ^2$ values in Fig.", "REF for models with relatively low or high extinction values are driven in part by the fact the models for these extinctions cannot fit the observed RGB curvature at the given (observed) color." ], [ "The CMD Fit", "Fig.", "REF compares the observed CMD (right) to a synthetic CMD realization (left) from the best-fit SFH.", "The area below the solid line was excluded from the $\\chi ^2$ minimization because the completeness is below 20% there, making the artificial star corrections unreliable.", "The area above and to the right of the dashed box was also excluded, for two reasons.", "First, this is where foreground stars are found (see Fig.", "REF ), which are not explicitly accounted for in our models.", "Second, this is where thermally pulsating AGB (TP-AGB) stars are located.", "Near the end of their AGB lifetimes, these stars undergo a series of He shell flashes due to the He- and H-burning shells turning on and off.", "While the updated Padova models (see [27]) provide a much improved treatment of the TP-AGB stars, this is still a complex and highly uncertain phase of stellar evolution.", "Moreover, a significant but unknown fraction of TP-AGB stars may be obscured by dust shells [10] making it difficult to use such stars to constrain the SFH.", "In the region of the CMD that was fitted, the agreement between the synthetic and observed CMDs is excellent.", "The $\\chi ^2$ of the fit is 2668, for $N_{\\rm DF} = N_{\\rm pix} - N_{\\rm basis} = 898$ degrees of freedom.", "Here $N_{\\rm pix} = 1773$ is the number of pixels in the CMD Hess diagram that was fitted, and $N_{\\rm basis} = 875$ is the number of different (isochrone) basis functions that was used to build the fit.", "The ratio $\\chi ^2/N_{\\rm DF} = 2.97$ is somewhat higher than expected purely from random errors, but such ratios are not atypical for studies of this kind [49].", "The only visually obvious discrepancies between the synthetic and observed CMDs occur in the regions that were masked in the fit.", "These discrepancies are well-understood: shortcomings in the TP-AGB evolutionary models at bright magnitudes and shortcomings in the correcting of very incomplete data with artificial start tests at faint magnitudes.", "These do not impact the inferred SFH, because of the exclusion of these regions from the fit.", "Fig.", "REF compares the LF of NGC 1569 (black) to the model LF for the best-fit SFH (red).", "As described above, there are well-understood discrepancies brighter than the TRGB (arrow) and fainter than $I \\gtrsim 28$ .", "However, at the intermediate magnitudes that were actually included in the $\\chi ^2$ minimization, the fit is excellent.", "This is particularly true when compared to the predictions of the best-fit SSP in Fig.", "REF .", "The best-fit SFH shows excellent agreement with the observed slope of the RGB, the observed magnitude and prominence of the RC/HB, and the lack of other prominent features (e.g., the RGB and AGB bumps).", "Fig.", "REF compares the LF of NGC 1569 (black) to the model LF for the best-fit SFH (red).", "As described above, there are well-understood discrepancies brighter than the TRGB (arrow) and fainter than $I \\gtrsim 28$ .", "However, at the intermediate magnitudes that were actually included in the $\\chi ^2$ minimization, the fit is excellent.", "The best-fit SFH matches the observed slope of the RGB, the observed magnitude and prominence of the RC/HB, and the lack of other prominent features (e.g., the RGB and AGB bumps).", "This is in sharp contrast to our best fit SSP in Fig.", "REF , and confirms our assessment in § REF that NGC 1569 is not well fit by any SSP and must have had a more complex SFH." ], [ "Star Formation History", "Fig.", "REF shows the full SFH of NGC 1569's outer region as a color-contour plot as a function of log (age) and [Fe/H].", "The quantity that is plotted is the predicted number of stars on the unmasked portion of the CMD.", "Thus, this provides a direct assessment of which (age,metallicity) combinations contribute most to the observed CMD.", "The SFH shows a number of salient features, which we will discuss in turn.", "(1) The peak of the SFH is found at old ages (log(age) $\\gtrsim 10$ ) and metallicities from [Fe/H]$ = -1$ to $-2$ , but centered around [Fe/H]$ = -1.25 \\pm 0.1$ .", "This is very similar to the properties of the best-fit SSP inferred in §REF , which has log(age) $= 10.0$ and [Fe/H]$ = -1.4$ (open star in Fig.", "REF ).", "There is a well-known age-metallicity degeneracy in the modeling of old stellar populations.", "Before accepting a SFH result, one must therefore ask if the age-metallicity dependence of the inferred SFH is uniquely implied by the data, and if the data did indeed have the information content necessary to ameliorate the degeneracy.", "In this case, the answer is yes.", "We have tried to match our observations of NGC 1569 using basis functions with restricted parameter space [e.g., log (age) $\\le $ 9.9 (8 Gyr)], but find that with these restrictions we cannot match the observed LF.", "Thus, while the precise age and metallicity of stellar populations can only be determined through a combination of spectroscopy and MSTO photometry, the information available on the RGB (brightness of the TRGB and RC/HB, color and curvature of the RGB) is sufficient to prefer a predominantly old population with a spread in metallicity.", "(2) The SFH shows a general trend of younger stars having higher metallicities.", "The metallicity of the gas in the central regions of NGC 1569 has been measured to be [Fe/H]$ = -0.6$ (dashed horizontal line in Fig.", "REF ).", "There are few stars in the SFH model at metallicities higher than this, and the youngest stars inferred with significance (log(age) $\\approx 8.7$ ) have a metallicity that approaches this value.", "This is all as generically expected in scenarios of chemical evolution.", "Younger stars formed from gas that was pre-enriched by the ejecta of older stars, and the youngest stars should have metallicities similar to that of the gas.", "These results therefore also provide additional credibility for the SFH results.", "(3) Even though the CMD only shows a predominant RGB, the SFH actually has significant star formation in the outer region for the age range from approximately $0.5-2.0$ Gyr ago (log(age) between $8.7$ and $9.3$ ).", "Since this may appear counter-intuitive, it is important to understand why this occurs in the models.", "The luminosity at the TRGB, i.e.", "the point of central He ignition, is virtually constant with increasing stellar mass, up to close to the RGB phase transition mass.", "Stars that are less massive than the phase transition mass have fully degenerate cores when they begin He fusion, while He ignition occurs in more massive stars under non-degenerate conditions [67].", "As stars approach this transition mass, the luminosity of the TRGB rapidly decreases by $\\sim $ 2 mag in the $I$ -band.", "In terms of age, the RGB phase transition occurs in the Padova isochrones at $\\sim $ 1-2 Gyr, with the younger, more massive stars populating a short RGB, while the older, less massive stars form the fully extended RGB.", "Our observed RGB luminosity function can only be reproduced with an intermediate age component, which steepens the LF by populating the magnitude range $27 I 28$ , on top of the extended RGB component provided by the older stellar populations.", "This argument is illustrated in Fig.", "REF where we plot the synthetic CMD realization from the best-fit SFH, broken down into its age (left) and metallicity (right) components.", "(4) The observed stars in the outer region span a range of ages, from $\\sim 0.5$ Gyr to a Hubble time (Fig.", "REF ).", "However, the mass in the outer region resides almost exclusively in the oldest stars.", "As a population ages, the fraction of the stars that is observable decreases.", "Therefore, a single observed star that is old hints at a much larger underlying reservoir of unseen stars than does a star that is younger.", "For the best-fit SFH, 93% of the total outer region mass resides in stars with $\\log (age) \\ge 9.7$ (age $>5$ Gyr).", "(5) Even the oldest stars have a significant spread in metallicity, ranging between [Fe/H]$ = -1$ and $-2$ .", "This is evident from Fig.", "REF , which shows the distribution of stellar mass vs. [Fe/H], for only those stars with $\\log (age) \\ge 9.7$ .", "The error bars indicate that the individual peaks and valleys seen as a function of metallicity in this range are probably not statistically significant, with the possible exception of a dip around -1.8 dex.", "However, the spread in metallicity is a persistent feature of the various SFH fits we have explored.", "This is due to the observed RGB width, which exceeds the photometric errors, and therefore requires a metallicity spread to be reproduced by the SFH code.", "Note that the total mass in the field under study is small, because we are looking at a low density field in the outer region.", "The integral under the histogram in Fig.", "REF corresponds to only $5.0 \\times 10^7 M_{\\odot }$ .", "The spread in [Fe/H] in NGC 1569's outer region is similar to that of the Milky Way halo; using SDSS data, [39] find that the Milky Way halo metallicity distribution is well fit by a gaussian with [Fe/H] = -1.46 and an intrinsic width, $\\sigma $ = 0.30.", "(6) After most of the outer region stars formed early in the Universe ($\\log (age) \\ge 9.7$ ), NGC 1569 experienced a relatively constant rate of star formation until $\\sim $ 0.5 Gyr ago.", "Fig.", "REF shows the star formation rate (SFR; expressed in $M_{\\odot }/yr$ ) as function of log(age) (integrated over all metallicities).", "The error bars indicate that the increase in SFR some $\\sim 0.5$ Gyr ago ($\\log (age) \\ge 8.7$ ) may be statistically significant, but not necessarily so.", "The oldest epoch of star formation in NGC 1569's outer region is congruent to that of the Milky Way globular clusters, which formed from the earliest times up until about 7 Gyr ago [47].", "This is also similar to the $\\Lambda $ CDM simulations of a Milky Way-like halo by [22], which showed that 80% of the mass in the inner halo (R $<$ 20 kpc) was accreted by $\\sim $ 9 Gyr ago, with almost no mass ($\\ll 1\\%$ ) assembled in the last 5 Gyr.", "For reference, at the distance of NGC 1569, the full ACS field-of-view is $\\sim $ 3.1 kpc $\\times $ 3.1 kpc.", "In contrast to the oldest stars, the extended formation of stars in NGC 1569's outer region until 0.5 Gyr ago is inconsistent with the formation scenarios of the Milky Way halo, indicating that NGC 1569 likely had atypical environmental influences (see §).", "(7) The outer region of NGC 1569 does have a few young stars.", "The sparsely populated CMD plume at $V-I \\approx 0.4$ (see Fig.", "REF ) is fit by the SFH code as a low level of star formation at ages $\\lesssim 30$ Myr.", "It is possible that these stars may have formed closer to the core of the galaxy, followed by outward migration through dynamical processes.", "To travel from the galaxy center to the outer region in 30 Myr requires a transverse velocity of $\\sim 50$ km/s, which is not out of the question." ], [ "Core-Outer Region SFH Comparison", "We have focused here on a study of the SFH of the NGC 1569 outer region.", "It is of interest to see how our results compare to those previously derived by other authors for the core of the galaxy.", "The most recent analysis of this was performed by [49].", "They used a subset of our ACS images, performed their own photometry and determined the SFH of NGC 1569 via CMD fitting with MATCH.", "Similar to us, they assumed a Salpeter IMF and used the Padova stellar evolution models.", "Because of uncertainties due to the extreme crowding and differential reddening in the central region of the galaxy, [49] required that [Fe/H] increases with time.", "While our analysis allows more general chemical evolution, the inferred SFH for the outer region is in fact generally consistent with an increasing [Fe/H] with time.", "The published [49] SFH refers to the entire ACS field.", "[49] divided the galaxy into low and high surface brightness regions, which is likely similar to our core/outer region distinction.", "Their published SFH is the sum of the individual SFRs for the low- and high-surface brightness regions.", "Since the stars and mass of NGC 1569 are heavily concentrated towards the galaxy core, that region carries almost all the weight in their SFH.", "For simplicity, we'll refer to their result as the SFH of the core.", "In Fig.", "REF we compare our outer region SFR as a function of age (top panel) to the core SFR from [49].", "Of course, the core SFR is significantly enhanced at young ages compared to what is seen in the outer region.", "But for the more ancient populations, our studies are in good overall agreement: most of the old stars formed $\\sim $ 10 Gyr ago; a low but significant amount of star formation persisted between 1 and 10 Gyr ago; and there was a SF peak/burst at $\\sim 0.3$ –$0.7$ Gyr ago.", "The similarity between the outer region and core SFHs for old stars implies that there are no strong radial population gradients for the old stars.", "This is also consistent with our recent study in [58] of HST/WFPC2 parallel fields much further out in the outer region.", "Out to 8 scale radii, we detected no variation in either RGB color or carbon-to-RGB star count ratio in the outer region.", "In view of these results, the results for our outer region can be taken to be representative of the old stellar populations throughout the galaxy.", "As with SFHMATRIX, MATCH can be used to determine the best fit distance and reddening of the galaxy.", "For the high surface brightness region, [49] find a best fit distance of 3.2 $\\pm $ 0.1 Mpc and $E(B-V)$ = 0.58 $\\pm $ 0.03, similar to what we find for the outer region (see §5.1).", "However, for the low surface brightness region, which should be similar to our outer region, they find a larger distance, 3.5 $\\pm $ 0.1 Mpc, and lower reddening, $E(B-V)$ = 0.48 $\\pm $ 0.04, than we calculate for the outer region.", "The source of this discrepancy is unclear, but is possibly due to a difference in photometric depth.", "As we discussed in §4.3, reaching fainter features on the RGB LF helps to ameliorate the degeneracy between age, metallicity, and reddening.", "Our outer region LF (Fig.", "REF ) shows that we barely reach the RC/HB in NGC 1569.", "[49], on the other hand, use only a subset of our data.", "A comparison of their published CMD of the entire NGC 1569 field (their Fig.", "3) with our CMD of the core (see Fig.", "REF ) shows that our photometry reaches 0.5 - 0.75 mag deeper.", "Thus, their calculation of the reddening and distance does not benefit from the added constraints provided by the RC/HB.", "It is interesting to note, however, that even though [49] do not reach the same photometric depth as we do, their SFH is very similar to ours.", "This suggests that even without the RC/HB feature a significant amount of information can still be gleaned from the CMD." ], [ "Possible Star Formation Triggers", "The central region of NGC 1569 is currently undergoing a strong starburst.", "Our CMD analysis of its outer region indicates that a constant level of star formation has been maintained there for much of the Hubble time.", "This prompts the question of what may have triggered this star formation activity.", "Its projection on the sky places NGC 1569 near the IC 342 group.", "The IC 342 group is comprised of nine known galaxies and has a mean distance of $3.35\\pm 0.09$ Mpc, with a line-of-sight depth of 0.25 Mpc ($1\\sigma $ ; [42]).", "NGC 1569 was long assumed to lie only 2.2 Mpc away (e.g., [37]), placing it in a relatively isolated position between the Local Group and the IC 342 group.", "However, the TRGB distance inferred from our HST data, $D = 3.06\\pm 0.18$ Mpc, places NGC 1569 inside the IC 342 group, raising the possibility that interactions with the group or its galaxies may have triggered the star formation in NGC 1569 [30].", "In the rest frame of the Local Group, the IC 342 group (excluding NGC 1569 and UGCA 92, see below) has a line-of-sight velocity $\\langle V_{LG} \\rangle $ = 226$\\pm $ 18 km s$^{-1}$ , with a dispersion of 54 km s$^{-1}$ ([42]).", "NGC 1569 has a lower velocity, $V_{LG}$ = 88 km s$^{-1}$ , that is 2.5$\\sigma $ from the group average.", "So NGC 1569 may reside in the tail of the velocity distribution of the IC 342 group, or it may not be bound to it at all (depending on its unknown transverse velocity, $v_{\\rm trans}$ ).", "Since NGC 1569 lies at the front side of the IC 342 group, NGC 1569 is now moving away from it.", "Given the known line-of-sight velocities and distances, we can calculate that it was at the same distance as the IC 342 group $\\sim 2.1$ Gyr ago, and entered the far side of the IC 342 group $\\sim 3.8$ Gyr ago.", "Fig.", "REF shows that the NGC 1569 outer region had a relatively constant non-zero star formation rate in this period.", "Thus, interactions with the IC 342 group, whether from tidal forces between galaxies or through ram pressure compression by IGM gas may have played a role in its SFH.", "It is possible that NGC 1569 may have interacted strongly with a particular galaxy in the IC 342 group.", "One candidate is the dwarf galaxy UGCA 92.", "Like NGC 1569, UGCA 92 has a distance of $\\sim $ 3 Mpc and $V_{LG} = 89$ km s$^{-1}$ [43].", "They are separated on the sky by only 123, which gives a physical distance of $\\sim $ 65 kpc.", "A recent study by [41] shows that NGC 1569 is sitting in a large, cold HI cloud, with tidal tails stretching out toward UGCA 92.", "The combination of small spatial separation, similar line-of-sight velocity, and the HI gas possibly connecting the two galaxies suggests that NGC 1569 and UGCA 92 may be interacting with each other.", "It is therefore possible that tidal forces between these galaxies have driven the long term SFH of both galaxies.", "Dynamical models have shown that close encounters between two dwarf galaxies, such as the Magellanic Clouds, can lead to widespread star formation (e.g., [8]; Besla et al.", "in prep.).", "However, the distance between NGC 1569 and UGCA 92 is about three times the current separation between the Magellanic Clouds.", "In absence of knowledge about $v_{\\rm trans}$ , it is not possible to know whether NGC 1569 may have interacted directly with IC 342 itself.", "Also, the distance to IC 342 is not particularly well known, with many widely separated values quoted in the literature (e.g., [69], [59] and references therein)." ], [ "Summary and Conclusions", "NGC 1569 is one of the closest and strongest starburst galaxies in the Universe.", "While its ISM and recent star formation have been thoroughly studied, until recently, little was know about NGC 1569's old stellar populations.", "This has made it difficult to determine the exact duration of the starburst, to unravel its triggers, and to understand NGC 1569's evolution over a full Hubble time.", "For this reason, we used HST/ACS to obtain deep $V$ - and $I$ -band images that reach down to $M_V \\sim $ -0.5.", "These data allowed us to show definitively for the first time that NGC 1569 has a significant population of stars older than 1 Gyr [30].", "Here we have used the same data to derive the properties and full SFH of this old population.", "We focused our analysis on the outer region of the galaxy, which is largely devoid of young stars and is much less crowded than the core.", "The improved photometric depth provides access to the RC/HB, which helps to ameliorate the age-metallicity degeneracy that plagues SFH studies that focus only on the RGB.", "We have used first an approach based on simple stellar populations.", "We then presented a newly developed synthetic CMD fitting code, SFHMATRIX, and used it to determine the full SFH.", "By combining the results from the different approaches, we were able to derive the following coherent picture for NGC 1569's old stellar populations.", "(1) By treating NGC 1569's outer region as an SSP, we find that the relative brightnesses of observed LF features (TRGB, RC/HB, and AGB bump) are best matched by a SSP with age = 10 Gyr and [Fe/H] = -1.4.", "However, the model SSP LF slope is flatter and the features more pronounced that what we observe in NGC 1569.", "This discrepancy suggests that NGC 1569 cannot be treated as an SSP as it must have had a more complex SFH.", "(2) The distance/reddening combination that best fits the data with a full SFH analysis is $E(B-V)$ = 0.58 $\\pm $ 0.03 and $D = 3.06 \\pm 0.18$ Mpc.", "The reddening is in agreement with values published using other techniques, but the distance is $\\sim $ 50% farther than what has been typically assumed.", "(3) Star formation in the outer region of NGC 1569 began $\\sim $ 13 Gyr ago and lasted until $\\sim $ 0.5 Gyr ago.", "The majority of star formation in our observed field occurred early on, with 93% of the stars, by mass, having formed more than 5 Gyr ago.", "This initial burst was followed by a relatively low, but constant, rate of star formation until $\\sim $ 0.5-0.7 Gyr ago when there may have been a short, low intensity burst of star formation.", "(4) The SFH for the old population in the NGC 1569 outer region follows a trend of increasing metallicity with time.", "The youngest significant population of $\\sim $ 0.5-0.7 Gyr age has a metallicity similar to that of the ionized gas in NGC 1569.", "These results are consistent with the basic expectations of chemical evolution scenarios.", "(5) NGC 1569's dominant old population (age $$ 10 Gyr) shows a considerable spread in metallicity, ranging from [Fe/H] = -1 to -2, with a peak around [Fe/H] = -1.25.", "The mean of this distribution is very similar to what we derived in our SSP analysis.", "The metallicity-spread of the dominant old population is similar to that for the Milky Way halo.", "However, the star formation in NGC 1569's outer region extended for much longer than in the Milky Way, indicating that NGC 1569 likely had atypical environmental influences.", "(6) The distance and line-of-sight velocity of NGC 1569 indicate that it moved through the IC 342 group of galaxies, at least in the past few Gyr.", "This may be the reason for the extended low-level star formation seen in its outer region.", "By contrast, its recent starburst may be related to interactions with the companion UGCA 92.", "(7) Comparison with recent work from [49] and [58], which was more heavily weighted towards smaller and larger radii in the galaxy, respectively, provides no evidence for radial population gradients in the old population of NGC 1569.", "This suggests that our results for the outer region are representative for the old stellar population throughout the galaxy.", "We would like to thank Jay Anderson for helpful discussions regarding early versions of the photometry, Leo Girardi for answering many questions about the Padova models, and Kristen McQuinn for kindly providing an electronic version of her SFH for NGC 1569.", "The authors would also like to thank the anonymous referee for comments that helped to improve the clarity of the paper.", "We are grateful to Jennifer Mack, Luca Angeretti, Enrico Held, Donatella Romano and Marco Sirianni for collaboration on earlier stages of this project.", "Support for proposal GO-10885 was provided by NASA through a grant from STScI, which is operated by AURA, Inc., under NASA contract NAS 5-26555.", "F.A.", "and M.T.", "acknowledge partial financial support from ASI through contracts ASI-INAF I/016/07/0 and I/009/10/0.", "Facility: HST (ACS)." ], [ "Star Formation History Analysis with SFHMATRIX", "The Star Formation History (SFH) of a galaxy can be described mathematically as a sum of delta functions.", "Each delta function has a fixed combination of age and metallicity, and a weight corresponding to the mass of stars that formed with that combination.", "This implies that the CMD of a galaxy can be described as the weighted linear sum of a set of “basis functions”, where each basis function corresponds to the synthetic CMD for an isochrone with fixed age and metallicity [70].", "The basis functions must reflect the characteristics of the problem at hand, namely: (1) observational details, such as the completeness and photometric errors inferred from artificial star tests; (2) galaxy properties, such as the distance and total (foreground+internal) extinction; and (3) stellar population properties, such as the Initial Mass Function (IMF) with which an isochrone is populated, and the binary fraction.", "Framed in this manner, the problem of inferring the SFH of a galaxy from its CMD reduces to the problem of finding the weighted combination of synthetic basis function CMDs that best reproduces the observed CMD.", "Since the data consist of a discrete set of points, this can be expressed mathematically as a maximum likelihood problem (e.g., [19]).", "However, when the number of basis functions is large, such problems can be complicated to solve numerically.", "It is therefore advantageous to consider instead the density of points in the CMD, i.e., the Hess diagram, rather than the CMD itself (both for the observations and the basis functions).", "By pixelating the CMD and assigning an appropriate error bar to the observed density in each pixel, the problem reduces to a linear $\\chi ^2$ minimization problem (e.g., [32]).", "For large numbers of stars, these approaches become mathematically equivalent due to the central limit theorem.", "Many software implementations have been developed to infer the SFH from an observed CMD, mostly using approaches similar to those outlined above (see, e.g., the compilation and comparison of various methods described in [64]).", "Here we use a new code that we developed, called SFHMATRIX.", "We describe here the salient features, with a more detail description planned for a separate paper (van der Marel & Grocholski in prep.).", "Our code resembles the STARFISH code of [32].", "Our method of constructing a synthetic CMD from an isochrone is largely similar to theirs, and like STARFISH, we describe the problem mathematically as a $\\chi ^2$ minimization.", "Our code differs primarily from the STARFISH implementation in how it finds the $\\chi ^2$ minimum, i.e., the best-fitting SFH.", "STARFISH finds the minimum using a brute-force minimization in $N$ -dimensional space, where $N$ is the number of basis functions (i.e., the number of different isochrone weights to be optimized).", "By contrast, we phrase the problem as the solution of a matrix equation (hence the name “SFHMATRIX”) $\\sum _{j=1}^N A_{ij} m_j = \\rho _i \\pm \\Delta \\rho _i , \\qquad \\forall i=1,\\ldots ,M \\qquad .$ Here $\\rho _i$ is the density of the stars in CMD space, $\\Delta \\rho _i$ is the associated Poisson uncertainty, and the index $i$ counts the $M$ pixels of the Hess diagram.", "The density $\\rho _i$ equals the integer number of stars $L_i \\ge 0$ that is detected in Hess diagram pixel $i$ , divided by the area of that pixel.", "We adopt $max(1,\\sqrt{L_i})$ as the Poisson error on the detected number of stars.", "Hence, $\\Delta \\rho _i / \\rho _i = max(1,\\sqrt{L_i}) /L_i$ .", "The vector of weights $m_j \\ge 0$ gives the mass (in $M_{\\odot }$ ) of stars associated with the $N$ basis functions (i.e., isochrones) $j$ .", "The matrix $A_{ij}$ gives the density of stars in Hess diagram pixel $i$ for a $1 M_{\\odot }$ population of stars on synthetic isochrone $j$ .", "Finding the solution of this matrix equation is a non-negative least squares (NNLS) problem in linear algebra.", "This can be solved with efficient general purpose subroutines (e.g., [45]) that have been well tested in other areas of astronomy (e.g., the dynamical modeling of galaxies using Schwarzschild's orbit superposition technique; e.g., [72]).", "Our approach has several advantages over the STARFISH approach.", "First, the NNLS matrix routines are guaranteed to converge on a global minimum [45], and cannot inadvertently find local minima as do brute-force searching routines.", "Second, the speed of NNLS routines makes it easy to use many basis function (large $N$ ), so that it is possible to effectively search and characterize the space of relevant models on a fine grid (both in age and metallicity, as well as in other parameters such as extinction and distance).", "Third, the speed of NNLS routines makes it easy to run the code in Monte-Carlo sense on many simulated datasets, so that the uncertainty on the final SFH can be robustly characterized.", "Our code can create basis functions from any set of isochrones.", "It is important that the basis functions fully span the range of ages and metallicities that are relevant for the galaxy under study.", "It is also important that the spacing between adjacent isochrones is relatively fine.", "If it is too coarse, then a fitted CMD superposition looks choppy and discontinuous in those regions of CMD space where the observational errors are small.", "On the other hand, the computational effort of the NNLS solution scales with the number of basis functions as $N^3$ .", "So it is best not to choose more basis functions than can realistically be resolved by the data given the observational errors.", "Here we use the isochrones from the same Padova models as discussed in § and we note that these are the same isochrones used in the paper by [49].", "We use metallicities ranging from -2.28 dex to +0.20 dex and log(age) values from 6.60 to 10.12.", "We obtained isochrones spaced by 0.05 dex in metallicity and 0.01 dex in log(age).", "Most of these isochrones are obtained from interpolation between a much coarser set of isochrones (in particular in metallicity) for which actual stellar evolutionary calculations were performed (e.g., [20], [21]).", "We created synthetic CMDs from each basis function by randomly drawing many stars from the isochrone, given an assumed IMF, and then rescaling the corresponding Hess diagram density to correspond to $1 M_{\\odot }$ .", "Stars are assigned a simulated photometric error and are marked as either detected or not (thus accounting for incompleteness) based on a randomly drawn artificial star from the artificial star test results.", "To construct basis functions that are not sampled quite as finely, we co-add the synthetic CMD results within bins of size 0.1 dex in metallicity and 0.1 dex in log(age).This results in each basis function corresponding to a two-dimensional comb function that approximates a two-dimensional boxcar.", "The resulting SFH is therefore more akin to a two-dimensional histogram, than a sum of delta-functions.", "Finding the best-fitting SFH from an observed CMD is mathematically an inverse problem.", "Methods for solving such problems have the well-known tendency for amplifying noise, leading to solutions that appear unphysically spiky.", "This can be counteracted by enforcing smoothness on the solution.", "Regularization is one popular technique for doing this (e.g., [52]).", "In the context of NNLS solutions, linear regularization constraints can be enforced by adding $K$ additional rows of the following form to the matrix equation $\\sum _{j=1}^N B_{kj} m_j = 0 \\pm \\Delta , \\qquad \\forall k=1,\\ldots ,K \\qquad .$ (see [15]).", "The $B_{kj}$ are chosen such that for given $k$ , the expression $\\sum B_{kj} m_j$ is a second order divided difference of basis functions that are adjacent in age at fixed metallicity, or vice versa.", "The divided differences are zero if the SFH is locally well approximated by a linear function.", "In this manner, smoothness is enforced in (age,metallicity) space, with the exact amount depending on the parameter $\\Delta $ .", "If $\\Delta =\\infty $ , then the regularization constraints are ignored.", "If $\\Delta =0$ , then the data are ignored, and the code cares only about returning a smooth solution.", "For intermediate values, the NNLS solutions tries to fit the data as best as possible, while keeping the noise in the SFH in check.", "In practice we applied a small amount of regularization to obtain the results in this paper, but none of our main results depend sensitively on this.", "Calculating error bars on the best-fitting SFH returned by the code is straightforward.", "For this we create many realizations of pseudo datasets with properties similar to the real data.", "We analyze each of these pseudo datasets in Monte-Carlo fashion as we do the real data.", "The RMS scatter in the SFH results at a given (age,metallicity) combination is the error bar.", "The pseudo data can be created with either of two possible approaches.", "The first approach is to use bootstrapping [52].", "In this approach one obtains a new dataset of $S$ stars from the existing dataset of $S$ stars, by random drawing with replacement.", "The second approach is to use the best-fitting SFH already inferred from the real data, and to draw many Monte-Carlo pseudo dataset realizations from this SFH.", "In the present paper we use the bootstrapping approach to calculate SFH error bars.", "To test the accuracy of the code we explored two approaches.", "In the first approach we used pseudo dataset realizations drawn from a known SFH, and then used the code to verify that the inferred SFH agrees with the input SFH to within the errors.", "In the second approach we used our code and the publicly available STARFISH code on the same input data, and verified that the inferred SFHs agreed to within the errors.", "Both tests were passed successfully.", "We also explored the sensitivity of the results to the small changes in the stellar evolutionary models using two approaches.", "In the first approach we used our code with either the isochrones from [9] or [25] to infer the SFH from the same data.", "In the second approach we used our code with basis functions that were created either from the [9] isochrones, or directly from the [20], [21] evolutionary tracks from which these isochrones were derived, and then derived the SFH from the same data.", "The approaches all yielded results in satisfactory agreement.", "However, more generally this does depend on both the actual differences in the underlying evolutionary calculations, and the characteristics and quality of the available data.", "Different stellar evolutionary tracks can yield different implied SFHs when used to analyze the same data (e.g., Skillman & Gallart 2002).", "We have assumed a Salpeter IMF [60] from $0.1-100$ M$_{\\odot }$ in the calculations presented in this paper, similar to prior studies of the SFH in NGC 1569 ([49], [5]).", "The exact IMF choice does not affect the shape of the inferred SFH for the outer region of NGC 1569.", "This is because all observed stars have evolved off the main sequence, and have approximately the same mass.", "However, the IMF choice does affect the normalization of the SFH, since different IMF slopes predict different amounts of (unobserved) lower-mass stars for a given (observed) population of giants.", "The assumed binary fraction also affects the SFH normalization, because some single observed sources may have the mass of two stars.", "For the calculations in the paper the SFH shape was not found to depend sensitively on the assumed binary fraction, and all results reported herein assume a zero binary fraction.", "Our code has the option to assign different reddening to different stars in the basis functions.", "We did not use this feature for the outer region of NGC 1569.", "Our H$\\alpha $ image showed very little emission there, so that differential reddening is not expected to be an issue.", "Figure: ACS/WFC 3-color image of NGC 1569, where the Hα\\alpha -, II-and VV-bands have been colored red, orange, and blue, respectively.", "(The faint ellipse near the center of the field is the ghost image of abright star).Figure: CMD composed of all stars in the core of NGC 1569 (bottom halfof Fig. ).", "Featuresresulting from young, massive, main sequence and evolved stars dominatethe CMD (see §3 for details).", "The 20% completeness limit is shown asthe solid line.", "The remainder of this paper focusessolely on the outer region of NGC 1569.Figure: CMD of all stars in NGC 1569's outer region (top halfof Fig. ).", "Unlike the core CMD,only the old stars on the RGB are readily visible in the outer region,illustrating that the recent star formation in NGC 1569 is concentratedin the core.", "For visibility, we have increased the size of the pointsfor stars that are off of the RGB.", "As in the previous figure, thesolid line represents the 20% completeness limit.Figure: Besançon model of the expected MW foreground contaminationin the direction of NGC 1569 , for a field size equalto that of the outer region (i.e., half of the total HST/ACS field).Figure: Absolute II-band magnitude of the RC/HB feature plotted as afunction of age for a range of metallicities.", "ForFigs.", "- we have used the theoretical values fromthe Padova isochrones.Figure: Absolute II-band magnitude of the RGB bump plotted as afunction of age for a range of metallicities.Figure: Absolute II-band magnitude of the AGB bump plotted as afunction of age for a range of metallicities.Figure: Observed (black line) and completeness corrected(thin red line) LFs of NGC1569's outer region.", "Brighter than I ∼\\sim 27.3, our photometry of NGC1569's outer region is ∼\\sim 100% complete.", "At I=28.4I = 28.4 thecompletenessis only 20%, and the completeness correction is not very reliablebeyond this magnitude.", "The vertical dashed line marks the position ofthe TRGB and the solid (blue) line follows the slope of the RGB.", "Threefeatures of interest, the RC/HB (labeled) and the possible AGB bumpand RGB bump features (arrows) are seen to extend above the expectedRGB slope.", "Error bars overplotted on the observed LF show the Poissonnoise in each magnitude bin and suggest that the possible AGB bump andRGB bump features may be real.", "The bottom panel shows the RGBslope-subtracted LF, with gaussian fits to the three features overplotted.Figure: Difference in II-band magnitude of the TRGB and RC/HB as afunction of age.", "Theoretical values for a range of metallicities(shown as symbols of different colors) were calculated by creatinga synthetic CMD for each isochrone using our SFH code, which takes intoaccount errors and incompleteness, and measuring the brightness of thefeatures in the same way as for NGC 1569.", "The dashed line is ourmeasured TRGB-RC/HB value forstars in NGC 1569's outer region while the dotted lines mark the3σ\\sigma error in our measurement.", "Only those isochrones with TRGB-RC/HB valueswithin 3σ\\sigma of our measured value are considered as possiblematches to NGC 1569.Figure: Theoretical SSP LF (thin red line) compared to NGC 1569's outerregionLF (black line).", "The synthetic LF shown, with an age of 10 Gyr and[Fe/H] = -1.4, provides the best simultaneous match to the positions ofthe AGB bump and RC/HB relative to the TRGB, to within our measurementerrors.", "At this age and metallicity, the RGB bump is faint enough thatit blends in with the RC/HB.", "While this synthetic population providesthe best match, its LF features are much more pronounced and its LFslope flatter than what we observe.", "This difference suggests that NGC1569's outer region is not an SSP.", "Instead, it must have undergone manyepochsof star formation, which served to both smooth the LF features and makeit steeper.Figure: χ 2 \\chi ^2 of our SFH model fits to the CMD as a function ofreddening.", "The χ 2 \\chi ^2 values vary smoothly with E(B-V)E(B-V) and are wellfit with a 4th order polynomial (dashed curve).", "The fit has a minimumat E(B-V)E(B-V) = 0.58.", "At this best-fit reddening, the galaxy distanceimplied by the TRGB magnitude is 3.063.06 Mpc.Figure: Comparison of the observed CMD (right) to a synthetic CMDrealization (left) from the best-fit SFH.", "The areas below the solidline and above and to the right of the dashed box were excluded fromthe fit.", "At the faint magnitudes the completeness is too low (<20<20%)to yield an accurate fit.", "At the bright magnitudes there areforeground stars (see Fig. )", "and TP-AGB stars, a complexstate of stellar evolution that has historically been difficult tomodel.", "The model matches the data well in the RGB region thatwas fit.Figure: Luminosity function of NGC 1569 (black line) compared to themodel LF for the best-fit SFH (thin red line).", "The TRGB is marked withthe arrow.", "Theisochrones slightly overpredict the number of TP-AGB stars brighter thanI∼24I \\sim 24.", "Below the TRGB, the model provides excellent agreementwith the data.", "Discrepancies at I≳28I \\gtrsim 28 are due to thelimitations of artificial star test corrections when the incompletenessgets very low.Figure: Contour plot of the SFH of the NGC 1569 outer region,interpolatedonto a grid with a step size of 0.01 in both log (age) and [Fe/H].", "Thecolor coding indicates the predicted number of stars on the parts ofthe CMD that are not masked in Fig. .", "The open star marksthe position of the best-fitting SSP model as derived in§.", "The dashed line indicates the abundance of the gas inNGC 1569.", "The properties and interpretation of this SFH are discussedin §.Figure: A synthetic CMD realization from the best-fit SFH.", "Differentcolors indicate a break-down into separate age (left) andmetallicity (right) components.", "Stars in yellow correspond to ayoung and metal-rich component that steepens the RGB LF at the faintend, in accordance with the observations.Figure: Stellar mass in the best-fit SFH vs. [Fe/H], for those starswith log(age)≥9.7\\log (age) \\ge 9.7.", "Error bars were calculated usingbootstrapping as described in Appendix A.", "The old stars shown in thisfigure show asignificant spread in metallicity, ranging between [Fe/H]=-1 = -1 and-2-2.Figure: SFR as function of log(age), integrated over allmetallicities.", "(top) NGC 1569 outer region, as derived in thepresentpaper.", "(bottom) NGC 1569 total, heavily dominated by the core,from .", "The core shows much elevated starformation at young ages.", "But for the more ancient populations, ourstudies are in good overall agreement: most of the old stars formed∼\\sim 10 Gyr ago; a low but significant amount of star formationpersisted between 1 and 10 Gyr ago; and there was a SF peak/burst at∼0.3\\sim 0.3–0.70.7 Gyr ago." ] ]

[ [ "Top quark pair production via (un)polarized photon collisions in the\n littlest Higgs model with T-parity at the ILC" ], [ "Abstract We study the top-quark pair production via polarized and unpolarized photon collisions at the International Linear Collider in the context of the littlest Higgs model with T-parity.", "We calculate the production cross section of the process $\\gamma\\gamma\\rightarrow t\\bar{t}$ and find the effects can be more significant in the $- -$ polarized photon collision mode than in other collision modes, and the relative correction can be expected to reach about -1% in the favorable parameter space." ], [ " Introduction", "The detailed analysis of the dynamics of top-quark production and decay is a major objective of experiments at the Tevatron, the Large Hadron Collider(LHC), and a possible International Linear Collider(ILC).", "Top quarks can be produced at hadron colliders via top-antitop pair [1] and single top [2] production channels.", "For top-pair production, the leading-order (LO) partonic processes are $q\\bar{q}\\rightarrow t\\bar{t}$ , which is dominant at Tevatron energies, and $gg\\rightarrow t\\bar{t}$ , dominant at LHC energies.", "At the ILC, one of the most important reactions will be top-pair production well above the threshold.", "Because of the small statistics, the top-quark properties have not been precisely measured at the Tevatron.", "Several tens of millions of top pair signals per year will be produced at the LHC, this large number of top quarks allow very precise measurements in the top-quark properties.", "Compared to the LHC, the $t\\bar{t}$ production cross section is less at the ILC[3], but it will be an ideal place for further investigating the top quark duo to its clean background.", "The little Higgs theory constructs the Higgs as a Pseudogoldstone boson to solve the hierarchy problem of the Standard Model(SM)[4].", "The littlest Higgs (LH) model [5] is an economical approach to implement this idea, but electroweak precision tests give it strong constraints [6] so that the fine-tuning problem in the Higgs potential would be reintroduced[7].", "Then, a discrete symmetry called T-parity is proposed to tackle this problem[8], this resulting model is referred to as the littlest Higgs model with T-parity (LHT).", "The LHT model predicts new particles, such as T-odd gauge bosons and T-odd fermions.", "In the LHT model, there are interactions between the SM fermions and the mirror fermions mediated by the new T-odd gauge bosons or T-odd Goldstone bosons.", "These interactions can contribute to the $\\gamma t\\bar{t}$ coupling and the production cross section of the process $\\gamma \\gamma \\rightarrow t\\bar{t}$ .", "Furthermore, an additional heavy quark $T^{+}$ and its partner $T^{-}$ can also contribute to the $\\gamma t\\bar{t}$ coupling.", "In this paper, we will study the polarized and unpolarized $\\gamma \\gamma $ collisions in the LHT model at the ILC.", "This paper is organized as follows.", "In Sec.II we give a brief review of the LHT model.", "In Sec.III and IV we respectively calculate the one-loop contributions of the LHT model to the $\\gamma \\gamma \\rightarrow t\\bar{t}$ in polarized and unpolarized photon-photon collision modes and show some numerical results at the ILC.", "Finally, we give our conclusions in Sec.V." ], [ " A brief review of the LHT model", "The LHT model [8] is based on an $SU(5)/SO(5)$ non-linear sigma model, where with the global group $SU(5)$ being spontaneously broken into $SO(5)$ by a $5\\times 5$ symmetric tensor at the scale $f\\sim \\mathcal {O}(TeV)$ , the gauged subgroup $[SU(2)\\times U(1)]_{1}\\times [SU(2)\\times U(1)]_{2}$ of $SU(5)$ is broken to the diagonal subgroup $SU(2)_{L}\\times U(1)_{Y}$ of $SO(5)$ .", "From the symmetry breaking, there arise 4 new heavy gauge bosons $W_{H}^{\\pm },Z_{H},A_{H}$ whose masses up to $\\mathcal {O}(\\upsilon ^{2}/f^{2})$ are given by $M_{W_{H}}=M_{Z_{H}}=gf(1-\\frac{\\upsilon ^{2}}{8f^{2}}),M_{A_{H}}=\\frac{g^{\\prime }f}{\\sqrt{5}}(1-\\frac{5\\upsilon ^{2}}{8f^{2}})$ with $g$ and $g^{\\prime }$ being the SM $SU(2)$ and $U(1)$ gauge couplings, respectively.", "A consistent viable implementation of T-parity in the fermion sector requires the introduction of mirror fermions.", "For each SM quark, a copy of mirror quark with T-odd quantum number is added.", "We denote up-type and down-type mirror quarks by $u_{H}^{i},d_{H}^{i}$ respectively, where i= 1, 2, 3 are the generation index, whose masses up to $\\mathcal {O}(\\upsilon ^{2}/f^{2})$ are given by $m_{d_{H}^{i}}=\\sqrt{2}\\kappa _if, m_{u_{H}^{i}}=m_{d_{H}^{i}}(1-\\frac{\\upsilon ^2}{8f^2})$ where $\\kappa _i$ are the diagonalized Yukawa couplings of the mirror quarks.", "An additional heavy quark $T^{+}$ is introduced to cancel the large contributions to the Higgs mass from one-loop quadratic divergences.", "The implementation of T-parity then requires also a T-odd partner $T^{-}$ , which is an exact singlet under $SU(2)_{1}\\times SU(2)_{2}$ .", "Their masses up to $\\mathcal {O}(\\upsilon ^{2}/f^{2})$ are given by $m_{T^{+}}&=&\\frac{f}{v}\\frac{m_{t}}{\\sqrt{x_{L}(1-x_{L})}}[1+\\frac{v^{2}}{f^{2}}(\\frac{1}{3}-x_{L}(1-x_{L}))]\\\\m_{T^{-}}&=&\\frac{f}{v}\\frac{m_{t}}{\\sqrt{x_{L}}}[1+\\frac{v^{2}}{f^{2}}(\\frac{1}{3}-\\frac{1}{2}x_{L}(1-x_{L}))]$ where $x_{L}$ is the mixing parameter between the SM top-quark $t$ and the heavy quark $T^{+}$ .", "In the LHT model, one of the important ingredients of the mirror sector is the existence of four CKM-like unitary mixing matrices, two for mirror quarks and two for mirror leptons: $V_{Hu},V_{Hd},V_{Hl},V_{H\\nu }$ where $V_{Hu}$ and $V_{Hd}$ are for the mirror quarks which are present in our analysis.", "They satisfy the relation $V_{Hu}^{\\dag }V_{Hd}=V_{CKM}$ .", "We follow Ref.", "[9] to parameterize $V_{Hd}$ with three angles $\\theta ^d_{12},\\theta ^d_{23},\\theta ^d_{13}$ and three phases $\\delta ^d_{12},\\delta ^d_{23},\\delta ^d_{13}$ $V_{Hd}=\\begin{pmatrix}c^d_{12}c^d_{13}&s^d_{12}c^d_{13}e^{-i\\delta ^d_{12}}&s^d_{13}e^{-i\\delta ^d_{13}}\\\\-s^d_{12}c^d_{23}e^{i\\delta ^d_{12}}-c^d_{12}s^d_{23}s^d_{13}e^{i(\\delta ^d_{13}-\\delta ^d_{23})}&c^d_{12}c^d_{23}-s^d_{12}s^d_{23}s^d_{13}e^{i(\\delta ^d_{13}-\\delta ^d_{12}-\\delta ^d_{23})}&s^d_{23}c^d_{13}e^{-i\\delta ^d_{23}}\\\\s^d_{12}s^d_{23}e^{i(\\delta ^d_{12}+\\delta ^d_{23})}-c^d_{12}c^d_{23}s^d_{13}e^{i\\delta ^d_{13}}&-c^d_{12}s^d_{23}e^{i\\delta ^d_{23}}-s^d_{12}c^d_{23}s^d_{13}e^{i(\\delta ^d_{13}-\\delta ^d_{12})}&c^d_{23}c^d_{13}\\end{pmatrix}$" ], [ "Top quark pair production via $\\gamma \\gamma $ collision in the LHT model", "In the context of the LHT model, the relevant Feynman diagrams of the one-loop correction to the process $\\gamma \\gamma \\rightarrow t\\bar{t}$ are shown in Fig.1, where the black dot represents the effective $\\gamma t\\bar{t}$ vertex which is shown in Fig.2 and the black diamond represents the fermion propagator.", "In our calculation, the higher order couplings between the scalar triplet $\\Phi $ and top quark and the high order $\\mathcal {O}(\\upsilon ^{2}/f^{2})$ terms in the masses of new particles and in the Feynman rules are neglected.", "The relevant Feynman rules can be found in Ref.[10].", "We use the 't Hooft-Feynman gauge, so the masses of the Goldstone bosons and the ghost fields are the same as their corresponding gauge bosons.", "The ultraviolet divergences have been regulated by the dimensional regularization scheme and the divergences have been canceled according to the on-shell renormalization scheme.", "Figure: Feynman diagrams ofthe one-loop correction to the process γγ→tt ¯\\gamma \\gamma \\rightarrow t\\bar{t} in the LHT model.Figure: The effective γtt ¯\\gamma t\\bar{t} vertex diagrams at one-loop level in the LHT model.For the $\\gamma \\gamma $ collision at the ILC, the photon beams are generated by the backward Compton scattering of incident electron- and laser-beams just before the interaction point.", "The total cross section $\\sigma (s)$ for the top-pair production can be obtained by folding the elementary cross section $\\hat{\\sigma }(\\hat{s})$ for the subprocess $\\gamma \\gamma \\rightarrow t\\bar{t}$ with the photon luminosity at the $e^+e^-$ colliders given in Ref.", "[11] $\\sigma (s)=\\int _{2m_t/\\sqrt{s}}^{x_{\\rm max}} {\\rm d}z \\frac{{\\rm d}L_{\\gamma \\gamma }}{{\\rm d}z}\\hat{\\sigma }(\\hat{s}) \\ \\ (\\gamma \\gamma \\rightarrow t\\bar{t} \\ {\\rm at} \\ \\hat{s}=z^2 s),$ where $\\sqrt{s}$ and $\\sqrt{\\hat{s}}$ are the $e^+e^-$ and $\\gamma \\gamma $ center-of-mass energies respectively, and ${\\rm d}L_{\\gamma \\gamma }/{\\rm d}z$ is the photon luminosity, which can be expressed as $\\frac{{\\rm d}L_{\\gamma \\gamma }}{{\\rm d}z}=2z\\int _{z^2/x_{\\rm max}}^{x_{\\rm max}}\\frac{{\\rm d}x}{x} F_{\\gamma /e}(x)F_{\\gamma /e}(z^2/x).$ For unpolarized initial electron and laser beams, the energy spectrum of the backscattered photon is given by $F_{\\gamma /e}(x)&=&\\frac{1}{D(\\xi )} \\left( 1-x+\\frac{1}{1-x}-\\frac{4x}{\\xi (1-x)}+ \\frac{4 x^2}{\\xi ^2 (1-x)^2} \\right) .$ where $\\xi =4E_e E_0/m_e^2$ ,$m_e$ and $E_e$ are respectively the incident electron mass and energy, $E_0$ is the initial laser photon energy.", "In our numerical calculation, we choose $\\xi =4.8$ , $D(\\xi )=1.83$ and $x_{max}=0.83$ .", "The $\\gamma \\gamma $ collisions have five polarization modes as follows: $++, --, +-,-+$ and unpolarized collision modes, where the notation $+$ and $-$ represent the helicities of the two incoming photons being $\\lambda _{1}=1$ and $\\lambda _{1}=-1$ , respectively." ], [ "Numerical results", "In our numerical calculations, we take the SM parameters as[12] $\\nonumber G_{F}&=&1.16637\\times 10^{-5}GeV^{-2},~~~S_{W}^{2}=\\sin ^{2}\\theta _{W}=0.231,\\\\\\alpha _{e}&=&1/128,~M_{Z_{L}}=91.2GeV,~m_{t}=172.4GeV,~m_{h}=120GeV.$ The LHT parameters relevant to our study are the scale $f$ , the mixing parameter $x_{L}$ , the mirror quark masses and the parameters in the matrices $V_{Hu},V_{Hd}$ .", "For the mirror quark masses, we get $m_{u_{H}^{i}}=m_{d_{H}^{i}}$ at $\\mathcal {O}(\\upsilon /f)$ and further assume $m_{u_{H}^{1}}=m_{u_{H}^{2}}=m_{d_{H}^{1}}=m_{d_{H}^{2}}=M_{12},m_{u_{H}^{3}}=m_{d_{H}^{3}}=M_{3}$ From the couplings between the mirror quarks and the heavy gauge bosons or the heavy Goldstone bosons, we can see the main contribution comes from the third family couplings.", "In order to show the largest correction, for the matrices $V_{Hu},V_{Hd}$ , we follow Ref.", "[13] to choose the following scenario: $V_{Hu}=1,V_{Hd}=V_{CKM}$ .", "In this scenario, the contribution of the LHT model comes entirely from the third family mirror quarks and the additional heavy quarks $T^{+},T^{-}$ .", "Figure: The relative correction of the top-quark pair productioncross section δσ/σ\\delta \\sigma /\\sigma as functions of thecenter-of-mass energy s\\sqrt{s} for M 3 =1000GeVM_{3}=1000GeV(a) and themirror quark mass M 3 M_{3} for s=500GeV\\sqrt{s}=500GeV(b) in unpolarizedphoton collision mode, respectively.In Fig.3(a), we discuss the dependance of the relative correction $\\delta \\sigma /\\sigma $ on the center-of-mass energy $\\sqrt{s}$ .", "To see the influence of the scale $f$ on the $\\delta \\sigma /\\sigma $ , we take $f = 500, 2000GeV$ , respectively.", "We can see $\\delta \\sigma /\\sigma $ firstly decreases then increases with $\\sqrt{s}$ .", "The higher the scale $f$ is taken, the shallower the curve becomes.", "To see the influence of the $x_{L}$ on the $\\delta \\sigma /\\sigma $ , we take $x_{L} = 0.1, 0.7$ , respectively.", "We can see that the larger the $x_{L}$ is taken, the less the negative relative correction is generated.", "Furthermore, the larger the scale $f$ is taken, the less influence the $x_{L}$ can exert, which means the contributions of the $T^{+},T^{-}$ are supressed by the high scale $f$ .", "Considering the constraints of Ref.", "[14], the maximum of the relative correction can reach about $-0.65\\%$ .", "In Fig.3(b), we discuss the dependance of $\\delta \\sigma /\\sigma $ on the third family mirror quark mass $M_{3}$ .", "We can see $\\delta \\sigma /\\sigma $ is negative and becomes larger with the $M_{3}$ increasing.", "Same as above, we take $f = 500,2000GeV$ respectively to see the influence of the scale $f$ on the $\\delta \\sigma /\\sigma $ and take $x_{L} = 0.1,0.7$ respectively to see the influence of the $x_{L}$ on the $\\delta \\sigma /\\sigma $ .", "The larger the scale $f$ is taken, the shallower the curve becomes, which means the contributions of the $T^{+},T^{-}$ and the third family mirror quark are all supressed by the high scale $f$ .", "For the same $x_{L}$ , the overall trend is that the higher the scale $f$ is taken, the smaller the relative correction $\\delta \\sigma /\\sigma $ is generated.", "For the same scale $f$ , the larger the $x_{L}$ is taken, the less the relative correction $\\delta \\sigma /\\sigma $ is generated, which means $\\delta \\sigma /\\sigma $ becomes less with the $M_{T^{+}},M_{T^{-}}$ increasing.", "Furthermore, we find the contribution of the third family mirror quark is negative while the contributions of the $T^{+},T^{-}$ change from negative to positive with the $x_{L}$ from 0.1 to 0.7.", "For the case $f=500GeV,x_L=0.7$ , the contributions of the $T^{+},T^{-}$ are positive so that they counteract the contribution of the third family mirror quark very strongly.", "As a result, the $\\delta \\sigma /\\sigma $ for $f=500GeV,x_L=0.7$ is smaller than the case for $f=2000GeV, x_L=0.7$ when $M_{3}<750GeV$ .", "For the case $f=500GeV,x_L=0.1$ , the contributions of the $T^{+},T^{-}$ are negative so that the contribution of the third family mirror quark is enhanced obviously.", "The maximum of the relative correction can reach about $-0.7\\%$ .", "Figure: The relative correction of the top-quark pair productioncross section δσ/σ\\delta \\sigma /\\sigma as functions of thecenter-of-mass energy s\\sqrt{s} for M 3 =1000GeV,f=500GeV,x L =0.1M_{3}=1000GeV, f=500GeV,x_{L}=0.1(a) and the mirror quark mass M 3 M_{3} fors=500GeV,f=500GeV,x L =0.1\\sqrt{s}=500GeV, f=500GeV, x_{L}=0.1(b), respectively.To see the maximum of the relative correction in the LHT model, we take $f=500GeV, x_{L}=0.1$ for the process $\\gamma \\gamma \\rightarrow t\\bar{t}$ in polarized photon collision mode.", "We show the dependance of the relative correction $\\delta \\sigma /\\sigma $ on the center-of-mass energy $\\sqrt{s}$ and the third family mirror quark mass $M_{3}$ for the process $\\gamma \\gamma \\rightarrow t\\bar{t}$ with unpolarized and completely $+ -, + +, - -,- +$ polarized photon beams in Fig4.", "(a) and Fig4.", "(b), respectively.", "From Fig.4(a) and Fig.4(b) we can see clearly that the relative correction $\\delta \\sigma /\\sigma $ in $- -$ photon polarization collision mode are larger than in other photon collision modes.", "In $- -$ photon polarization collision mode, the maximum of the relative correction can be expected to reach about $-1\\%$ in the favorable parameter space." ], [ "Conclusions", "In the framework of the LHT model, we studied the one-loop contributions of the T-odd particles to the top-quark pair production cross section in unpolarized and polarized photon collision modes.", "Because the contributions of the $T^{+},T^{-}$ change from negative to positive with the $x_{L}$ increasing, in some cases the contribution of the third family mirror quark to the relative correction $\\delta \\sigma /\\sigma $ was enhanced, and in other cases the contribution was counteracted.", "In all collision modes, we found that the relative correction $\\delta \\sigma /\\sigma $ can be more significant in the $- -$ polarized photon collision mode than in other collision modes.", "In the favorable parameter space, the relative correction can be expected to reach about $-1\\%$ .", "Acknowledgments We thank Cao Jun-jie for providing the calculation programs and thank Wu Lei for useful discussions.", "This work is supported by the National Natural Science Foundation of China under Grant Nos.10775039, 11075045, by Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20094104110001 and by HASTIT under Grant No.2009HASTIT004.", "Appendix: The expression of the renormalization vertex $\\hat{\\Gamma }^{\\mu }_{\\gamma t\\bar{t}}$ and the renormalization propagator $-i\\hat{\\Sigma }^f(p)$ [15] (I)Renormalization vertex Figure: NO_CAPTION$\\hat{\\Gamma }^{\\mu }_{\\gamma t\\bar{t}}&=&\\Gamma ^{\\mu }_{\\gamma t\\bar{t}}-ieQ_{t}\\gamma ^{\\mu }(\\delta Z_{V}^{t}-\\gamma _{5}\\delta Z_{A}^{t}-\\frac{S_{W}}{2C_{W}}\\delta Z_{ZA})+ie\\gamma ^{\\mu }(v_{t}-a_{t}\\gamma _{5})\\frac{1}{2}\\delta Z_{ZA}\\nonumber $ where $v_{t}\\equiv \\frac{I_{t}^{3}-2Q_{t}S_{W}^{2}}{2C_{W}S_{W}},\\quad a_{t}\\equiv \\frac{I_{t}^{3}}{2C_{W}S_{W}},\\quad I_{t}^{3}=\\frac{1}{2},\\quad Q_{t}=\\frac{2}{3}~~~~~~~~~~~\\\\\\quad \\delta Z_{ZA}=2\\frac{\\Sigma _{T}^{AZ}(0)}{M_{Z_{L}}^{2}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\\delta Z_{L}^{t}=Re\\Sigma _{L}^{t}(m_{t}^{2})+m_{t}^{2}\\frac{\\partial }{\\partial P_{t}^{2}}Re[\\Sigma _{L}^{t}(P_{t}^{2})+\\Sigma _{R}^{t}(P_{t}^{2})+2\\Sigma _{S}^{t}(P_{t}^{2})]|_{P_{t}^{2}=m_{t}^{2}}\\\\\\delta Z_{R}^{t}=Re\\Sigma _{R}^{t}(m_{t}^{2})+m_{t}^{2}\\frac{\\partial }{\\partial P_{t}^{2}}Re[\\Sigma _{L}^{t}(P_{t}^{2})+\\Sigma _{R}^{t}(P_{t}^{2})+2\\Sigma _{S}^{t}(P_{t}^{2})]|_{P_{t}^{2}=m_{t}^{2}}\\\\\\delta Z_{V}^{t}=\\frac{1}{2}(\\delta Z_{L}^{t}+\\delta Z_{R}^{t}),\\delta Z_{A}^{t}=\\frac{1}{2}(\\delta Z_{L}^{t}-\\delta Z_{R}^{t})~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $\\hat{\\Gamma }^{LHT,\\mu }_{\\gamma t\\bar{t}}&=&\\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(\\eta )+ \\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(\\omega ^{0})+\\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(\\omega ^{\\pm })+\\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(\\pi ^{0})+\\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(h)\\\\&+&\\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(A_{H})+\\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(Z_{H})+\\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(W_{H}^{\\pm })+\\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(Z)+\\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(\\omega ^{\\pm },W_{H}^{\\pm })\\\\&+&\\delta \\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(\\eta )+ \\delta \\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(\\omega ^{0})+\\delta \\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(\\omega ^{\\pm })+ \\delta \\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(\\pi ^{0})+\\delta \\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(h)\\\\&+&\\delta \\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(A_{H})+\\delta \\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(Z_{H})+\\delta \\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(W_{H}^{\\pm })+\\delta \\Gamma ^{\\mu }_{\\gamma t\\bar{t}}(Z)$ (II)Renormalization propagator Figure: NO_CAPTION$-i\\hat{\\Sigma }^{f}(p)=-i\\Sigma ^{f}(p)+(-i\\delta \\Sigma ^{f}(p))$ where $\\Sigma ^{f}(p)&=&m_{f}\\Sigma ^{f}_{S}(p^{2})+\\unknown.", "{\\hspace{0.56917pt}/}{p}P_{L}\\Sigma ^{f}_{L}(p^{2})+\\unknown.", "{\\hspace{0.56917pt}/}{p}P_{R}\\Sigma ^{f}_{R}(p^{2})\\\\\\delta \\Sigma ^{f}(p)&=&\\delta m_{f}+m_{f}\\frac{1}{2}\\delta Z_{L}^{f}+m_{f}\\frac{1}{2}\\delta Z_{R}^{f}-\\unknown.", "{\\hspace{0.56917pt}/}{p}P_{L}\\delta Z_{L}^{f}-\\unknown.", "{\\hspace{0.56917pt}/}{p}P_{R}\\delta Z_{R}^{f}\\\\\\delta m_{f}&=&-m_{f}Re[\\Sigma ^{f}_{S}(m_{f}^{2})+\\frac{1}{2}\\Sigma ^{f}_{L}(m_{f}^{2})+\\frac{1}{2}\\Sigma ^{f}_{R}(m_{f}^{2})]\\\\\\delta Z_{L}^{f}&=&Re\\Sigma ^{f}_{L}(m_{f}^{2})+m_{f}^{2}\\frac{\\partial }{\\partial p^{2}}Re[\\Sigma ^{f}_{L}(p^{2})+\\Sigma ^{f}_{R}(p^{2})+2\\Sigma ^{f}_{S}(p^{2})]|_{p^{2}=m_{f}^{2}}\\\\\\delta Z_{R}^{f}&=&Re\\Sigma ^{f}_{R}(m_{f}^{2})+m_{f}^{2}\\frac{\\partial }{\\partial p^{2}}Re[\\Sigma ^{f}_{L}(p^{2})+\\Sigma ^{f}_{R}(p^{2})+2\\Sigma ^{f}_{S}(p^{2})]|_{p^{2}=m_{f}^{2}}$" ] ]

[ [ "Effect of Position-dependent Mass on Dynamical Breaking of Type B and\n Type X_2 N-fold Supersymmetry" ], [ "Abstract We investigate effect of position-dependent mass profiles on dynamical breaking of N-fold supersymmetry in several type B and type X_2 models.", "We find that N-fold supersymmetry in rational potentials in the constant-mass background are steady against the variation of mass profiles.", "On the other hand, some physically relevant mass profiles can change the pattern of dynamical N-fold supersymmetry breaking in trigonometric, hyperbolic, and exponential potentials of both type B and type X_2.", "The latter results open the possibility of detecting experimentally phase transition of N-fold as well as ordinary supersymmetry at a realistic energy scale." ], [ "Introduction", "In recent years, the study of quantum mechanical systems with a position-dependent mass (PDM) have attracted a lot of interest due to their relevance in describing the physics of many microstructures of current interests, such as compositionally graded crystals [1], semiconductor heterostructure [2], quantum dots [3], $^3$ He clusters [4], metal clusters [5] etc.", "The concept of PDM comes from the effective-mass approximation [6], [7] which is a useful tool for studying the motion of carrier electrons in pure crystals and also for the virtual-crystal approximation in the treatment of homogeneous alloys (where the actual potential is approximated by a periodic potential) as well as in graded mixed semiconductors (where the potential is not periodic).", "Recent interest in this field stems from extraordinary development in crystal-growth techniques like molecular beam epitaxy, which allow the production of nonuniform semiconductor specimen with abrupt heterojunctions [8].", "In these mesoscopic materials, the effective mass of the charge carrier are position dependent.", "Consequently, the study of the position-dependent mass Schrödinger equation (PDMSE) becomes relevant for deeper understanding of the non-trivial quantum effects observed on these nanostructures.", "It has also been found that such equations appear in many different areas.", "For example, it has been shown that constant mass Schrödinger equations in curved space and those based on deformed commutation relations can be interpreted in terms of PDMSE [9].", "The PDM also appear in nonlinear oscillator [10], [11] and $\\mathcal {PT}$ -symmetric cubic anharmonic oscillator [12].", "The most general form of the PDM Hamiltonian proposed by von Roos [13] is defined by H = -14(m(q)ddqm(q) ddqm(q)+m(q)ddq m(q)ddqm(q)) + V(q), where the ambiguity parameters $\\alpha $ , $\\beta $ , $\\gamma $ are related by $\\alpha +\\beta +\\gamma =-1$ .", "The above Hamiltonian always has the following form: H=-12m(q)d2dq2 + m'(q)2m(q)2 ddq + U(q), where the effective potential $U(q)$ is given by U(q) = V(q) -(+) m”(q)4 m(q)2 + (+ +) m'(q)22m(q)3.", "It is quite natural that physical interests just described above have also enhanced the studies on exact solutions to PDMSE [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] by employing various methods e.g.", "supersymmetric (SUSY) quantum mechanics [31] and point canonical transformation [32] to mention a few.", "Later, PDM quantum systems were successfully formulated in the framework of $\\mathcal {N}$ -fold SUSY in Ref.", "[33], which has provided until now the most general tool for constructing a PDM system which admits exact solutions because of its equivalence to weak quasi-solvability.", "To avoid confusion, we here note that $\\mathcal {N}$ -fold SUSY is different from nonlinear SUSY which has been long employed since the work by Samuel and Wess [34] in 1983 to indicate the nonlinearly realized SUSY originated from the work by Akulov and Volkov [35] in 1972.", "For a review of $\\mathcal {N}$ -fold SUSY see Ref.", "[36], while for recent works on nonlinear SUSY see, e.g., Ref.", "[37] and references cited therein.", "Very recently, new classes of exactly solvable PDM quantum systems whose eigenfunctions are expressible in terms of so-called $X_{1}$ polynomials were constructed in Ref. [28].", "The new findings of $X_{n}$ polynomials ($n\\ge 1$ ) were associated with the more fundamental mathematical concept of exceptional polynomial subspaces of codimension $n$ introduced in Refs.", "[38], [39], [40], whose origin can be traced back to the pioneering work on the classification of monomial spaces preserved by second-order linear differential operators [41].", "The purpose of the present paper is two-fold.", "The first one is to bring the purely mathematical concept of exceptional polynomial subspaces into more physical settings by allowing the position dependence of mass (in a spirit similar to Ref.", "[28]) in the framework of $\\mathcal {N}$ -fold SUSY.", "In the constant-mass case, form of potentials related to exceptional polynomial systems is very limited.", "Thus, we can enlarge the physical applicability of the mathematical concept by introducing PDM to quantum systems.", "On the other hand, the framework of $\\mathcal {N}$ -fold SUSY enables us to talk about the physical phenomenon of dynamical $\\mathcal {N}$ -fold SUSY breaking.", "The second purpose is actually to examine effect of PDM profiles on dynamical breaking of $\\mathcal {N}$ -fold SUSY.", "In this respect, it is rather surprising that there have been few papers, like Ref.", "[11], where broken as well as unbroken SUSY is described in PDM backgrounds depending on the mass profiles.", "One of the main reasons would be that SUSY has been mostly used just as a technique to obtain exact solutions.", "The true significance of the Witten's SUSY quantum mechanics [31], however, rather resides in the nonperturbative aspects of dynamical SUSY breaking.", "Hence, one of our main purposes is, in other words, to examine change of nonperturbative nature of quantum systems caused by variations of mass profiles in view of dynamical $\\mathcal {N}$ -fold SUSY breaking.", "The paper is organized as follows.", "In Section , we provide a self-contained review of $\\mathcal {N}$ -fold SUSY in a PDM background, especially for those who are not familiar with the subject.", "We also summarize mathematical structure of type B and type $X_{2}$ $\\mathcal {N}$ -fold SUSY.", "In Section , we construct several $\\mathcal {N}$ -fold SUSY PDM quantum systems and examine dynamical $\\mathcal {N}$ -fold SUSY breaking in different PDM backgrounds.", "The first three models of type B $\\mathcal {N}$ -fold SUSY have rational, trigonometric, and exponential potentials in the constant mass case.", "We show in particular that the models whose bound state eigenfunctions were shown to be expressed in terms of $X_{1}$ polynomials in Ref.", "[42] for the constant mass case and in Ref.", "[28] for the PDM cases can be obtained as type B systems.", "The last three models of type $X_{2}$ $\\mathcal {N}$ -fold SUSY have rational, hyperbolic, and exponential potentials in the constant mass case.", "For both types of $\\mathcal {N}$ -fold SUSY, we find that the rational potentials have steady $\\mathcal {N}$ -fold SUSY against variation of mass profile while all the other types of potentials can receive effect of PDM on their dynamical breaking of $\\mathcal {N}$ -fold SUSY.", "Finally, we summarize the results and discuss their implications and prospects in Section ." ], [ "Review of $\\mathcal {N}$ -fold Supersymmetry in a PDM background", "An $\\mathcal {N}$ -fold SUSY one-body quantum mechanical system with PDM is composed of a pair of PDM Hamiltonians H=-12m(q)d2dq2+m'(q)2m(q)2 ddq+U(q), and an $\\mathcal {N}$ th-order linear differential operator PN-=m(q)-N/2dNdqN +k=0N-1wk[N](q)dkdqk, which satisfy the following intertwining relations PN-H-=H+PN-,      PN+H+=H-PN+.", "In the above, $P_{\\mathcal {N}}^{+}$ is the transposition [43] of $P_{\\mathcal {N}}^{-}$ given by PN+=(PN-)T=(-ddq )Nm(q)-N/2+k=0N-1( -ddq)kwk[N](q).", "Actually, the two relations in () are not independent; the first implies the second and vice versa, since the PDM Hamiltonians () are invariant under the transposition $(H^{\\pm })^{\\mathrm {T}}=H^{\\pm }$ .", "One of the significant consequences of the intertwining relations () is weak quasi-sovability, that is, $H^{\\pm }$ preserves a finite-dimensional linear space $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ spanned by the kernel of the operator $P_{\\mathcal {N}}^{\\pm }$ HVNVN,      VN=PN.", "Each space $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ is called a solvable sector of $H^{\\pm }$ .", "Except for the $\\mathcal {N}=2$ case (cf., Refs.", "[36], [44]), virtually all the $\\mathcal {N}$ -fold SUSY systems so far found admit analytic expression of $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ in closed form, and thus are quasi-solvable.", "In addition, it sometimes happens when either $H^{-}$ or $H^{+}$ does not depend essentially on $\\mathcal {N}$ and preserves an infinite flag of the solvable sectors V1-/+V2-/+VN-/+ .", "In this case, it is said to be solvable, which is a necessary condition for exact solvability.", "We note that $H^{-}$ and $H^{+}$ are usually simultaneously solvable due to the intertwining relations ().", "A set of an $\\mathcal {N}$ -fold SUSY system $H^{\\pm }$ and $P_{\\mathcal {N}}^{\\pm }$ provides a representation of $\\mathcal {N}$ -fold superalgebra defined by [QN,H]={QN,QN }=0,      {QN-,QN+}=2N PN(H), where $\\mathsf {P}_{\\mathcal {N}}(x)$ is a monic polynomial of degree $\\mathcal {N}$ in $x$ .", "Indeed, it is realized by defining ${H}$ and ${Q}_{\\mathcal {N}}^{\\pm }$ as H=H--++H++-,   QN+=PN-+,   QN-=PN+-, where $\\psi ^{\\pm }$ is a pair of fermionic variables satisfying $\\lbrace \\psi ^{\\pm },\\psi ^{\\pm }\\rbrace =0$ and $\\lbrace \\psi ^{-},\\psi ^{+}\\rbrace =1$ .", "It is easy to check that the above ${H}$ and ${Q}_{\\mathcal {N}}^{\\pm }$ satisfy the first part of algebra ().", "In particular, the intertwining relations in () guarantee the commutativity of ${H}$ and ${Q}_{\\mathcal {N}}^{\\pm }$ .", "Regarding the second part of algebra, the monic polynomial $\\mathsf {P}_{\\mathcal {N}}$ is given, in the above representation, by [43], [33] PN(H)=(H-H|VN), namely, the characteristic polynomial for $H^{\\pm }$ restricted to the solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ .", "Whether $\\mathcal {N}$ -fold SUSY of the system under consideration is dynamically broken is determined by a property of the solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ since they characterize $\\mathcal {N}$ -fold SUSY states, namely, states annihilated by the pair of $\\mathcal {N}$ -fold supercharges ${Q}_{\\mathcal {N}}^{\\pm }$ .", "Let $|0\\rangle $ and $|1\\rangle $ be the fermionic vacuum and the one fermion state, respectively, which satisfy -|0=0,      |1=+|0.", "Then, superstates $|\\Psi _{0}^{-}\\rangle =\\Psi _{0}^{-}(q)|0\\rangle $ and $|\\Psi _{0}^{+}\\rangle =\\Psi _{0}^{+}(q)|1\\rangle $ , respectively, are annihilated by both of ${Q}_{\\mathcal {N}}^{\\pm }$ QN|0-=0,      QN|0+ =0, if and only if $\\Psi _{0}^{-}(q)\\in \\mathcal {V}_{\\mathcal {N}}^{-}$ and $\\Psi _{0}^{+}(q)\\in \\mathcal {V}_{\\mathcal {N}}^{+}$ , respectively.", "However, such states do not necessarily satisfy physical requirements.", "Suppose $S\\subset \\mathbb {C}$ is a domain where both of the Hamiltonians $H^{\\pm }$ have no singularities and are thus naturally defined, and $\\mathfrak {F}(S)$ is a linear space of complex functions in which both of $H^{\\pm }$ act.", "In a usual physical application, the domain $S$ is the real line $\\mathbb {R}$ or a real half-line $\\mathbb {R}_{+}=(0,\\infty )$ , and the linear space $\\mathfrak {F}$ is a Hilbert space $L^{2}$ , so that $\\mathfrak {F}(S)=L^{2}(\\mathbb {R})$ , or $L^{2}(\\mathbb {R}_{+})$ .", "In the latter cases, the physical requirement is the normalizability (square integrability) on $S$ .", "Then, there exists physical (normalizable) $\\mathcal {N}$ -fold SUSY states $|\\Psi _{0}^{-}\\rangle $ and/or $|\\Psi _{0}^{+}\\rangle $ which satisfies () if $\\mathcal {V}_{\\mathcal {N}}^{-}(S)\\subset L^{2}(S)$ and/or $\\mathcal {V}_{\\mathcal {N}}^{+}(S)\\subset L^{2}(S)$ , in other words, if $H^{-}$ and/or $H^{+}$ is quasi-exactly solvable.", "If there are no such physical $\\mathcal {N}$ -fold SUSY states in the Hilbert space $L^2(S)$ exists then $\\mathcal {N}$ -fold SUSY of the system is said to be dynamically broken.", "It was first shown correctly in Ref.", "[45] that the generalized Witten index characterizes $\\mathcal {N}$ -fold SUSY breaking, which corrected the wrong statement made earlier in Ref. [46].", "For $\\mathcal {N}>1$ , we can have an intriguing situation where not the whole of, but a subspace of the solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{-}(S)$ and/or $\\mathcal {V}_{\\mathcal {N}}^{+}(S)$ belong to the Hilbert space $L^{2}(S)$ .", "In this case, $\\mathcal {N}$ -fold SUSY of the system is said to be partially broken.", "Partial breaking of $\\mathcal {N}$ -fold SUSY was first discovered in Ref. [47].", "We note that it is different in nature from the partial breaking of (nonlinear) SUSY [48], [49].", "Construction of an $\\mathcal {N}$ -fold SUSY system is in general quite difficult, especially for a larger value of $\\mathcal {N}$ , since the intertwining relations () compose of coupled nonlinear differential equations for $U^{\\pm }(q)$ and $w_{k}^{[\\mathcal {N}]}(q)$ ($k=0,\\dots ,\\mathcal {N}-1$ ).", "For the direct calculations of intertwining relations in a PDM background in the cases of $\\mathcal {N}=1$ and 2, see Ref. [24].", "To circumvent the difficulty, a systematic algorithm for constructing an $\\mathcal {N}$ -fold SUSY system was developed in Ref.", "[47] for constant-mass quantum mechanics and was later generalized to PDM systems in Ref. [33].", "The significant feature which is common in both constant-mass and PDM systems is that an $\\mathcal {N}$ -dimensional linear space of functions VN-=1(z),...,N(z), preserved by a second-order linear differential operator $\\tilde{H}^{-}$ can determine whole of an $\\mathcal {N}$ -fold SUSY system.", "Indeed, we can construct a pair of $\\mathcal {N}$ th-order linear differential operators $\\bar{\\tilde{P}}_{\\mathcal {N}}^{\\pm }$ and another $\\mathcal {N}$ -dimensional vector space $\\bar{\\mathcal {V}}_{\\mathcal {N}}^{+}$ such that $\\bar{\\tilde{\\mathcal {V}}}_{\\mathcal {N}}^{\\pm }=\\ker \\bar{\\tilde{P}}_{\\mathcal {N}}^{\\pm }$ .", "Then, we can show that a pair of second-order linear differential operators given by H=-A(z)d2dz2+[N-22 A'(z)Q(z)]ddz-C(z) -(11)[N-12Q'(z)-12A'(z)wN-1[N] (z)-A(z)wN-1[N](z)], is weakly quasi-solvable with respect to the spaces $\\bar{\\tilde{\\mathcal {V}}}_{\\mathcal {N}}^{\\pm }$ , namely, $\\bar{\\tilde{H}}^{\\pm }\\bar{\\tilde{\\mathcal {V}}}_{\\mathcal {N}}^{\\pm }\\subset \\bar{\\tilde{\\mathcal {V}}}_{\\mathcal {N}}^{\\pm }$ .", "With the choice of the change of variable $z=z(q)$ and the gauge potential $\\mathcal {W}_{\\mathcal {N}}^{\\pm }$ determined by z'(q)2=2m(q)A(z),   WN=-14|m(q)|+N-14 |2A(z)|dz m(q)Q(z)2A(z), we can obtain an $\\mathcal {N}$ -fold SUSY system by H=e-WNHeWN |z=z(q),      PN=e-WN PNeWN|z=z(q).", "With the change of variable and the gauge transformation, both of $H^{\\pm }$ get the form of PDM Hamiltonian () and their effective potentials $U^{\\pm }(q)$ are given by U(q)=12m(q)[(dWN-dq )2-d2WN-dq2+m'(q)m(q) dWN-dq]-C(z(q)) -(11)[N-12Q'(z)-12A'(z)wN-1[N](z) -A(z)wN-1[N](z)]z=z(q).", "The solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ of $H^{\\pm }$ are evidently given by VN=PN=e-WN VN|z=z(q).", "In principle, we can construct a pair of $\\mathcal {N}$ -fold SUSY PDM Hamiltonians $H^{\\pm }$ and its solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ by using the formulas () and ().", "However, there is an easier way to obtain such a system when we have already had an ordinary $\\mathcal {N}$ -fold SUSY constant-mass quantum system at hand.", "Suppose the latter system is such that its pair of potentials $V^{(0)\\pm }(q)$ , its gauge potentials $\\mathcal {W}_{\\mathcal {N}}^{(0)\\pm }(q)$ , its solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{(0)\\pm }[q]$ are all known.", "Then, an $\\mathcal {N}$ -fold SUSY PDM system having a pair of effective potentials $U^{\\pm }(q)$ , gauge potentials $\\mathcal {W}_{\\mathcal {N}}^{\\pm }(q)$ , and solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{\\pm }[q]$ can be constructed immediately via the following prescription: U(q)=V(0)(u(q))+m”(q)8m(q)2-7m'(q)2 32m(q)3, WN(q)=-14|m(q)|+WN(0)(u(q)), VN[q]=m(q)1/4VN(0)[u(q)], where the function $u(q)$ is given by u(q)=dqm(q).", "Actually, the above relations are consistent with the formulas obtained by the point canonical transformation, see, e.g., equations (2.7) and (2.8) in Ref.", "[22], equation (7) of [21] and equations (10), (13), and (14) in Ref. [19].", "The above relations () have also been verified in Ref.", "[33] where type A $\\mathcal {N}$ -fold SUSY has been constructed in PDM background.", "One of the most salient features unveiled by the algorithmic construction is that both constant-mass and PDM quantum systems with $\\mathcal {N}$ -fold SUSY have totally the same structure in the gauged $z$ -space.", "That is, the functional forms of the gauged operators such as $\\bar{\\tilde{P}}_{\\mathcal {N}}^{\\pm }$ and $\\bar{\\tilde{H}}^{\\pm }$ given by () are identical in both the cases.", "It means in particular that the starting vector space $\\tilde{\\mathcal {V}}_{\\mathcal {N}}^{-}$ determines all in the algorithm regardless of whether mass is constant or not.", "Hence, different types of $\\mathcal {N}$ -fold SUSY are characterized by different types of vector spaces $\\tilde{\\mathcal {V}}_{\\mathcal {N}}^{-}$ and vice versa.", "Until now, four different types have been discovered, namely, type A [50], [44], type B [51], type C [47], and type $X_{2}$  [52].", "We note that almost all the models having essentially the same symmetry as $\\mathcal {N}$ -fold SUSY but called with other terminologies in the literature, such as Pöschl–Teller and Lamé potentials, are actually particular cases of type A $\\mathcal {N}$ -fold SUSY.", "In this article, we focus on constructing PDM quantum systems with type B and type $X_{2}$ $\\mathcal {N}$ -fold SUSY since the other types (type A and type C) are not related to exceptional polynomial subspaces.", "In what follows, we shall review the general structure of these two types of $\\mathcal {N}$ -fold SUSY." ], [ "Type B $\\mathcal {N}$ -fold Supersymmetry", "Type B $\\mathcal {N}$ -fold SUSY was first discovered in Ref.", "[51] and was found to be associated with the following monomial space VN-=VN(B):=1,z,...,zN-2, zN, called type B, which was considered in Ref.", "[41] in the context of the classification of monomial spaces preserved by second-order linear ordinary differential operators.", "Applying the algorithm to the type B monomial space, we obtain [36] the gauged $\\mathcal {N}$ -fold supercharge components PN-=z'(q)N(ddz-1z) dN-1dzN-1,      PN+=z'(q)N dN-1dzN-1(ddz+1z ), and the functions which characterize the gauged Hamiltonians () are given by A(z)=a4z4+a3z3+a2z2+a1z+a0, 2Q(z)=-Na3z2+2b1z-Na1, C(z)=N(N-3)a4z2+N(N-2)a3z+c0, and $\\tilde{w}_{\\mathcal {N}-1}^{[\\mathcal {N}]}(z)=-z^{-1}$ .", "The other linear space $\\bar{\\mathcal {V}}_{\\mathcal {N}}^{+}$ preserved by $\\bar{H}^{+}$ is given by VN+=z-11,z2,...,zN.", "We note that both the monomial spaces (REF ) and (REF ) are actually exceptional polynomial subspaces of codimension 1, see Ref. [40].", "We can easily check that the type B Hamiltonian $H^{+}$ preserves an infinite flag of the following spaces V1+e-WN+V2+ e-WN+VN+ e-WN+, where $\\bar{\\mathcal {V}}_{\\mathcal {N}}^{+}$ and $\\mathcal {W}_{\\mathcal {N}}^{+}$ are given by (REF ) and (), respectively, and thus $H^{+}$ is solvable if and only if $a_{3}=a_{4}=0$ .", "On the other hand, the partner type B Hamiltonian $H^{-}$ does not appear to be solvable for any parameter value since the type B monomial space (REF ) does not constitutes an infinite flag due to the fact that $\\tilde{\\mathcal {V}}_{\\mathcal {N}}^{(\\mathrm {B})}\\lnot \\subset \\tilde{\\mathcal {V}}_{\\mathcal {N}+1}^{(\\mathrm {B})}$ for all $\\mathcal {N}=1,2,\\ldots $ .", "However, it turns out [36] that, when $a_{3}=a_{4}=0$ and $H^{+}$ gets solvable, the partner Hamiltonian $H^{-}$ does preserve an infinite flag of linear spaces given by V1(A)e-WN-V2(A) e-WN-VN(A) e-WN-, where $\\mathcal {W}_{\\mathcal {N}}^{-}$ is given by () and $\\tilde{\\mathcal {V}}_{\\mathcal {N}}^{(\\mathrm {A})}$ is the type A monomial space defined by VN(A)=1,z,...,zN-1.", "That is, $H^{-}$ and $H^{+}$ can be solvable simultaneously.", "In this paper, all the type B models we will consider later satisfy the solvability condition $a_{3}=a_{4}=0$ .", "Thus, all the pairs of type B Hamiltonians $H^{\\pm }$ preserve the infinite-dimensional solvable sectors $\\mathcal {V}^{\\pm }$ given by $\\mathcal {V}^{-}&=\\bigl \\langle {1,z(q),z(q)^{2},\\ldots }\\bigr \\rangle \\,\\mathrm {e}^{-\\mathcal {W}_{\\mathcal {N}}^{-}(q)},\\\\\\mathcal {V}^{+}&=\\bigl \\langle {1,z(q)^{2},z(q)^{3},\\ldots }\\bigr \\rangle z(q)^{-1}\\mathrm {e}^{-\\mathcal {W}_{\\mathcal {N}}^{+}(q)}.$ An interesting consequence of the fact that $H^{-}$ and $H^{+}$ preserve different types of infinite flag of spaces in the solvable case is that the eigenfunctions of $H^{-}$ are expressed in terms of a classical polynomial system while those of $H^{+}$ are in terms of an $X_{1}$ polynomial system.", "It is exactly the underlying reason why some of the Hamiltonians whose eigenfunctions are expressed in terms of the $X_{1}$ Laguerre or Jacobi polynomials were obtained by those whose bound state eigenfunctions are expressed in terms of the classical Laguerre or Jacobi polynomials using an intertwining or SUSY techniques in Refs.", "[53], [54], [27]." ], [ "Type $X_{2}$ {{formula:15393db5-b079-4b73-b5ea-640b61062f76}} -fold Supersymmetry", "Type $X_{2}$ $\\mathcal {N}$ -fold SUSY constructed in Ref.", "[52] is associated with the following exceptional polynomial subspace of codimension 2 VN-=1(z;),...,N(z;), where $\\tilde{\\varphi }_{n}(z;\\alpha )$ is a polynomial of degree $n+1$ in $z$ with a parameter $\\alpha (\\ne 0,1)$ defined by n(z;)=(+n-2)zn+1+2(+n-1)(-1)zn +(+n)(-1)zn-1.", "Applying the algorithm to the $X_{2}$ space (REF ), we obtain [52] the gauged $\\mathcal {N}$ -fold supercharge components $\\tilde{P}_{\\mathcal {N}}^{-}&=z^{\\prime }(q)^{\\mathcal {N}}\\frac{f(z;\\alpha )}{f(z;\\alpha +\\mathcal {N})}\\prod _{k=0}^{\\mathcal {N}-1}\\frac{f(z;\\alpha +k+1)}{f(z;\\alpha +k)}\\left(\\frac{\\mathrm {d}}{\\mathrm {d}z}-\\frac{f^{\\prime }(z;\\alpha +k+1)}{f(z;\\alpha +k+1)}\\right),\\\\\\bar{P}_{\\mathcal {N}}^{+}&=z^{\\prime }(q)^{\\mathcal {N}}\\left[\\prod _{k=0}^{\\mathcal {N}-1}\\left(\\frac{\\mathrm {d}}{\\mathrm {d}z}+\\frac{f^{\\prime }(z;\\alpha +\\mathcal {N}-k)}{f(z;\\alpha +\\mathcal {N}-k)}\\right)\\frac{f(z;\\alpha +\\mathcal {N}-k)}{f(z;\\alpha +\\mathcal {N}-k-1)}\\right]\\frac{f(z;\\alpha )}{f(z;\\alpha +\\mathcal {N})},$ where $\\prod _{k=0}^{\\mathcal {N}-1}A_{k}:=A_{\\mathcal {N}-1}\\dots A_{1}A_{0}$ , and the functions $f(z;\\alpha )$ and $\\tilde{w}_{\\mathcal {N}-1}^{[\\mathcal {N}]}(z)$ are given by f(z;)=z2+2(-1)z+(-1), wN-1[N](z)=-(N-1)f'(z;)f(z;) -f'(z;+N)f(z;+N).", "The most general forms of the functions $A(z)$ , $Q(z)$ , and $C(z)$ appeared in $\\bar{\\tilde{H}}^{\\pm }$ depend on four parameters $a_{i}$ ($i=1,\\dots ,4$ ), but in this paper we only consider models with $a_{4}=a_{3}=0$ .", "In the latter case, they read as A(z)=a2z2+a1z+(-1)(+N-1)a2, Q(z)=-a2z2-(3a2+a1)z-(-1)(3+3N-7)a2 +2+N-82a1+4(-1)D(z)f(z;), C(z)=a2z+c0-4(-1)D(z)f(z;), where $D(z)$ is given by D(z)=-[(2+N-3)a2-a1]z-(-1)(2+N-1)a2 +a1.", "For their most general forms, please refer to Ref. [52].", "The other linear space $\\bar{\\mathcal {V}}_{\\mathcal {N}}^{+}$ preserved by $\\bar{H}^{+}$ is given by VN+=1(z;+N),..., N(z;+N)f(z;)-1f(z;+N)-1, where $\\bar{\\chi }_{n}(z;\\alpha )$ is a polynomial of degree $n+1$ in $z$ defined by n(z;)=(-n)(-n+1)zn+1+2(-n-1) (-n+1)(-1)zn +(-n-1)(-n)(-1)zn-1.", "The solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ of the constant-mass Hamiltonians $H^{\\pm }$ are VN-=1(z(q);),...,N(z(q); ) e-WN-(q), VN+=1(z(q);+N),..., N(z(q);+N)f(z(q);)f(z(q);+N)  e-WN+(q).", "Finally, the type $X_{2}$ Hamiltonians $H^{\\pm }$ preserve the infinite flag of the spaces $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ ($\\mathcal {N}=1,2,\\ldots $ ) and are simultaneously solvable if and only if $a_{2}=(a_{3}=a_{4}=)0$ ." ], [ "Type B and Type $X_2$ {{formula:ee6990db-ccd5-4a50-ad93-f40832f808aa}} -fold Supersymmetry for position-dependent\nmass", "In this section, we shall consider some models which belong to type B and type $X_{2}$ $\\mathcal {N}$ -fold supersymmetry.", "In order to study effect of PDM in these models, we need to consider simultaneously the corresponding constant-mass type B and type $X_{2}$ models as well.", "In particular, we shall address ourselves to the following question: Does position dependent mass have any effect on dynamical breaking of type B and type $X_{2}$ $\\mathcal {N}$ -fold SUSY?", "By comparing the solvable sectors of both constant and position-dependent mass cases, we shall see below that the answer is in the affirmative in some cases for particular choices of physically interesting mass functions.", "In order to explore in detail the impact of mass functions on symmetry breaking or restoration, it will be appropriate to consider more than one mass function in a few examples.", "Also it will be shown that the bound state wavefunctions of one of the partner potentials obtained in type B $\\mathcal {N}$ -fold SUSY are associated with exceptional $X_{1}$ Laguerre and Jacobi polynomials while those of the other partner are associated with classical Laguerre and Jacobi polynomials." ], [ "Effects of PDM on Dynamical Symmetry Breaking of Type B\n$\\mathcal {N}$ -fold SUSY", "Here we shall consider three examples of type B $\\mathcal {N}$ -fold SUSY corresponding to three different choices of $A(z)$ .", "In each of the examples, we first show the results in the constant mass case, followed by the corresponding results in the PDM case.", "As we referred to before, all the type B models constructed below satisfy the solvability condition $a_{3}=a_{4}=0$ and thus their solvable sectors in the constant-mass case are given by (REF ).", "Example 3.1.", "$A(z)=k(z-z_{0})\\quad (k\\ne 0)$ Potentials: V(0)-(q)=b1 28q2+4(z0b1-Nk)2 -k28 k2q2+Nb12+V0, V(0)+(q)=b1 28q2+4z0 2b1 2 -k28 k2q2+2kk q2+2 z0 -8k z0(k q2+2 z0)2+V0, where $V_{0}$ is an irrelevant constant given by V0=(z0b1-Nk)b12k+b1N-R. Solvable sectors: V(0)-=1,z(q),z(q)2,... q(2z0b1-2Nk+k)/(2k)eb1q2/4, V(0)+=1,z(q)2,z(q)3,...z(q)-1 q-(2z0b1-k)/(2k)e-b1q2/4.", "We assume $k>0$ and $z_{0}>0$ so that the pair of potentials $V^{\\pm }(q)$ has no singularities except for at $q=0$ .", "Thus, the system is naturally defined in $L^{2}(\\mathbb {R}_{+})$ , $\\mathbb {R}_{+}=(0,\\infty )$ .", "In the latter Hilbert space, $\\mathcal {V}^{(0)-}(\\mathbb {R}_{+})\\subset L^{2}(\\mathbb {R}_{+})$ if and only if b1<0   and   z0b1>(N-1)k, which cannot be satisfied by any $b_{1}\\in \\mathbb {R}$ .", "On the other hand, $\\mathcal {V}^{(0)+}(\\mathbb {R}_{+})\\subset L^{2}(\\mathbb {R}_{+})$ if and only if b1>0   and   k>z0b1.", "Hence, $\\mathcal {N}$ -fold SUSY of the system is unbroken if and only if $0<b_{1}<k/z_{0}$ on the constant-mass background.", "Now, the relevant expressions for partner potentials, gauge potentials and corresponding solvable sectors of type B PDM systems can be obtained using Eqs.", "(REF )–(REF ) and relations ().", "Since our main objective in this section is to study effect of mass function on dynamical breaking of $\\mathcal {N}$ -fold SUSY, we give below only the solvable sectors $\\mathcal {V}^{\\pm }$ for an arbitrary mass function $m(q)$ : V-=1,z(u(q)),z(u(q))2,... m(q)1/4u(q)(2z0b1-2Nk+k)/(2k)eb1u(q)2/4, V+=1,z(u(q))2,z(u(q))3,...z(u(q))-1 m(q)1/4u(q)-(2z0b1-k)/(2k)e-b1u(q)2/4, where $u(q)$ is given by ().", "At this point, we are in a position to choose a particular mass function.", "Let the mass function be m(q)=e-bq,   b>0,   q(-,), which was considered in Ref.", "[28] where the PDM potentials were associated with $X_{1}$ -Laguerre polynomials.", "This exponentially behaved mass function has been often used in the study of confined energy states for carriers in semiconductor quantum well [19], [28].", "It has also been used to compute transmission probabilities for scattering in abrupt heterostructures [25] which may be useful in the design of semiconductor devices [55].", "For the mass function, the change of variable is given by u(q)=-2be-bq/2, and the pair of potentials $U^{\\pm }(q)$ reads from () as U-(q)=b1 22b2e-bq+b2[(z0b1 -Nk)2-k2]8k2ebq+Nb12+V0, U+(q)=b1 22b2e-bq+b2(z0 2 b1 2-k2)8k2ebq+k b22ke-bq +z0b2-2kz0b4(2ke-bq+z0b2)2+V0, respectively.", "It is worth mentioning here that the potential $U^{+}(q)$ given in (REF ) is identical with the potential $V_{eff}(q)$ associated with exceptional $X_{1}$ Laguerre polynomials [e.g., Eq.", "(12) of Ref.", "[28]], if one takes $k=1/2$ , $b_{1}=b^{2}/2$ , and $z_{0}=\\alpha /b^{2}$ .", "On the other hand, for the same choices of parameters the other potential $U^{-}(q)$ coincides with the potential [after making a translation $\\alpha \\rightarrow \\alpha -\\mathcal {N}$ ] previously obtained in Ref.", "[27] corresponding to classical Laguerre polynomials.", "The solvable sectors of the potentials (REF ) and (REF ) are respectively given by V-=1,e-bq+z0, (e-bq+z0)2,... [-(z0b1k-N+1)b2q +b1b2e-bq], V+=1,(e-bq+z0)2, (e-bq+z0)3,... (e-bq+z0)-1[(z0b1k-1 )b2q-b1b2e-bq], where $\\bar{z}_{0}=z_{0}b^{2}/(2k)$ .", "Here the potentials have no singularities in the finite part of the real line, so the domain is $\\mathbb {R}$ .", "Since $b>0$ , so $\\mathcal {V}^{-}(\\mathbb {R})\\subset L^{2}(\\mathbb {R})$ if and only if $b_{1}<0$ .", "On the other hand, $\\mathcal {V}^{+}(\\mathbb {R})\\subset L^{2}(\\mathbb {R})$ if and only if $b_{1}>0$ .", "Hence, the $\\mathcal {N}$ -fold SUSY of the PDM system is unbroken unless $b_{1}=0$ .", "Comparing the solvable sectors of both the constant and position-dependent mass scenarios, it can be observed that it is not possible to break $\\mathcal {N}$ -fold SUSY dynamically for the particular choice of mass function $m(q)=\\mathrm {e}^{-bq}$ .", "In addition, we have checked that many physically interesting mass functions also have no effect on symmetry breaking.", "Example 3.2.", "$A(z)=a^{2}[1-(z-z_{0})^{2}]/2\\quad (a>0)$ Potentials: V(0)-(q)=(4b1 2-N 2a4)z04a2 aq2aq+(2b1-Na2)2z0 2 +(2b1+Na2)2-a48 a22aq +b1N2+V0, V(0)+(q)=(2b1-Na2)2z04 a2 aq2aq+(2b1-Na2)2(z0 2 +1)-a48a22aq +a2z0aq+z0-a2(z0 2-1) (aq+z0)2-b1N2+V0, where $V_{0}$ is an irrelevant constant given by V0=b1N+a2(N2-7)12 +(2b1z0-Nz0a2)28a2-R. Solvable sectors: V(0)-=1,z(q),z(q)2,...|aq|b1a2 +N-12(1+aq1-aq )-(2b1-Na2)z04a2, V(0)+=1,z(q)2,z(q)3,...z(q)-1 |aq|-b1a2+N-12( 1+aq1-aq)(2b1-Na2)z04a2.", "It is worth mentioning here that the potential $V^{(0)+}(q)$ coincides with the potential whose bound state wave functions are given in terms of exceptional $X_{1}$ Jacobi polynomial [42] for $a=1$ , $b_{1}=B+\\mathcal {N}/2$ , $z_{0}=-(2A-1)/(2B)$ whereas potential $V^{(0)-}(q)$ coincides with the Scarf I potential [54] [after making an change $B\\rightarrow B+\\mathcal {N}$ ] whose bound state wave functions are given in terms of classical Jacobi polynomials.", "We choose here a domain of the system as $S=(-\\frac{\\pi }{2a},\\frac{\\pi }{2a})$ and assume $z_{0}>1$ so that the pair of potentials $V^{(0)\\pm }(q)$ has no singularities except for at the boundary $\\partial S=\\lbrace -\\frac{\\pi }{2a},\\frac{\\pi }{2a}\\rbrace $ .", "Thus, the Hilbert space for the system is $L^{2}(S)$ .", "Then, $\\mathcal {V}^{(0)-}(S)\\subset L^{2}(S)$ if and only if b1a2+N-12(2b1-Na2)z02a2 >-12, that is, Na22z0-1z0+1<b1<Na22 z0+1z0-1   for   z0>1.", "Similarly, $\\mathcal {V}^{(0)+}(S)\\subset L^{2}(S)$ if and only if -b1a2+N-12(2b1-Na2)z02 a2>-12, that is, b1>Na22   and   z0>1.", "Hence, $\\mathcal {N}$ -fold SUSY of the system is broken for the constant mass case if and only if $z_{0}>1$ and b1Na22z0-1z0+1   or   b1Na22z0+1z0-1.", "In a PDM case, the solvable sectors $\\mathcal {V}^{\\pm }$ of the type B PDM $\\mathcal {N}$ -fold SUSY partner Hamiltonians $H^{\\pm }$ for an arbitrary mass function $m(q)$ are deformed according to () as V-=1,z(u(q)),z(u(q))2,...m(q)14 |a u(q)|b1a2+N-12( 1+a u(q)1-a u(q))-(2b1-Na2)z04a2, V+=1,z(u(q))2,z(u(q))3,... m(q)14 |a u(q)|-b1a2+N-12a u(q)+z0(1+a u(q)1-a u(q))(2b1 -Na2)z04a2.", "where $u(q)$ is given by ().", "In this case, the choice of mass function and the corresponding change of variable are given by m(q)=2e-2q2,      u(q)=Erfq,       q(-,).", "Consequently, the partner potentials $U^{\\pm }(q)$ read as U-(q)=(4b1 2-N 2a4)z04a2(aErfq)2(aErfq)-(3q2+1)e2q24 +b1N2+V0 +(2b1-Na2)2z0 2+(2b1+Na2)2-a48 a22(aErfq), U+(q)=(2b1-Na2)2z04a2(aErfq)2(aErfq)-(3q2+1)e2q24 -b1N2+V0 +a2z0(aErfq)+z0-a2(z0 2-1) [(aErfq)+z0]2 +(2b1-Na2)2(z0 2+1)-a48a22(aErfq).", "The solvable sectors of the potentials (REF ) and (REF ) are given by V-=1,z(u(q)),z(u(q))2,... e-q2/4 |(aErfq)|b1a2+N-12 (1+(aErfq)1-(aErfq)) -(2b1-Na2)z04a2, V+=1,z(u(q))2,z(u(q))3,... e-q2/4 |(aErfq)|-b1a2+N-12 (aErfq)+z0(1+(aErfq)1-(aErfq) )(2b1-Na2)z04a2.", "The potentials $U^{\\pm }(q)$ as well as the mass function are well behaved in $q\\in (-\\infty ,\\infty )$ .", "So, we can take the domain as the whole real line $\\mathbb {R}$ .", "Since $\\operatorname{Erf}q\\rightarrow \\pm 1$ as $q\\rightarrow \\pm \\infty $ , so both the solvable sectors $\\mathcal {V}^{\\pm }(\\mathbb {R})$ belong to $L^{2}(\\mathbb {R})$ , irrespective of the parameter values of $b_{1}$ and $z_{0}$ .", "Hence, it manifests unbroken SUSY.", "So, in this case position-dependent mass affects the symmetry breaking scenario.", "But the mass profile $m(q)=\\operatorname{sech}^{2}aq$ , $q\\in (-\\infty ,\\infty )$ has no effect on dynamical breaking of $\\mathcal {N}$ -fold SUSY which can be observed by considering the leading behavior of the solvable sectors (REF ) and (REF ).", "We have found that same is true for many other mass functions.", "Also associated to this mass profile, one of the partner potentials given in equation (REF ) is identical with the $V_{eff}(q)$ whose bound state wave functions are given by exceptional $X_{1}$ Jacobi polynomials [e.g., Eq.", "(18) of Ref.", "[28]], for the choice of parameters $b_{1}=(\\alpha -\\beta +\\mathcal {N})a^{2}/2$ , $z_{0}=(\\alpha +\\beta )/(\\alpha -\\beta )$ .", "The simplified form of the other partner potential $U^{-}(q)$ matches with the potential previously obtained in [27] corresponding to classical Jacobi polynomials.", "It is worth mentioning that this mass profile $m(q)=\\operatorname{sech}^{2}aq$ has been previously used in PDM Hamiltonians of BenDaniel–Duke [56] and Zhu–Kroemer [57] type and interesting connection was shown [58] between the discrete eigenvalues of such Hamiltonians and the stationary 1-soliton and 2-soliton solutions of the Korteweg-de Vries (KdV) equation.", "For the latter choice of the mass function, the change of variable is given by $u(q)=\\tan ^{-1}(\\sinh aq)/a$ and corresponding pair of potentials $U^{\\pm }(q)$ read as U(q)=[2b1(z0+1)-Na2z0(N-2)a2] [2b1(z0+1)-Na2z0(N+2)a2]32a2e2aq +[2b1(z0-1)-Na2z0(N-2)a2] [2b1(z0-1)-Na2z0(N+2)a2]32a2e-2aq +112a2z0+1[1-2(z0-2)z0-1 +(z0+1)e2aq-4(z0-1)(z0-1+(z0+1)e2aq)2 ] Nb14 + V0.", "Example 3.3.", "$A(z)=(z-z_{0})^{2}/2$ Potentials: V(0)-(q)=(2b1+N)2z0 28e-2q +(4b1 2-N 2)z04e-q+V0, V(0)+(q)=(2b1+N)2z0 28e-2q +(2b1+N)2z04e-q -z0e-q(1+z0e-q)2+V0, where $V_{0}$ is an irrelevant constant given by V0=b1 22+b1N+N 2+1124-R. Solvable sectors: V(0)-=1,z(q),z(q)2,...[ -(2b1+N)z02e-q-N-1-2b12q], V(0)+=1,z(q)2,z(q)3,...z(q)-1[ (2b1+N)z02e-q-N-1+2b12q].", "We assume $z_{0}>0$ so that the pair of potentials $V^{(0)\\pm }(q)$ has no singularities in $(-\\infty ,\\infty )$ .", "As we will show in what follows, the $\\mathcal {N}$ -fold SUSY in this case can be partially broken.", "To see this, we first introduce a pair of $k$ -dimensional subspaces $\\mathcal {V}_{k}^{(0)\\pm }$ of the solvable sectors $\\mathcal {V}^{(0)\\pm }$ as Vk(0)-=1,z(q),...,z(q)k-1[ -(2b1+N)z02e-q-N-1-2b12q], Vk(0)+=1,z(q)2,...,z(q)kz(q)-1[ (2b1+N)z02e-q-N-1+2b12q].", "Then, for a fixed $k\\in \\mathbb {N}$ , we have Vk(0)-(R)L2(R)      -N<2b1<N+1-2k, Vk(0)+(R)L2(R)      2k-N-1<2b1<-N. From these conditions, it is easy to observe that $\\mathcal {V}_{k}^{(0)-}(\\mathbb {R})\\subset L^{2}(\\mathbb {R})$ if and only if $-\\mathcal {N}/2<b_{1}<(\\mathcal {N}+1-2k)/2$ for a $k\\in \\mathbb {N}$ satisfying $k<\\mathcal {N}+1/2$ , while there is no $k\\in \\mathbb {N}$ which satisfy the condition (REF ) and thus $\\mathcal {V}^{(0)+}(\\mathbb {R})\\lnot \\subset L^{2}(\\mathbb {R})$ $\\forall b_{1}\\in \\mathbb {R}$ .", "Hence, the $\\mathcal {N}$ -fold SUSY in the constant-mass background is partially broken if there is a positive integer $k\\le \\mathcal {N}$ for which the parameter $b_{1}$ satisfies -N2<b1<N+1-2k2, and fully broken otherwise.", "The solvable sectors $\\mathcal {V}^{\\pm }$ of the corresponding PDM Hamiltonians $H^{\\pm }$ are written as V-=1,z(u(q)),z(u(q))2,...m(q)1/4 [-(2b1+N)z02e-u(q) -N-1-2b12u(q)], V+=1,z(u(q))2,z(u(q))3,...z(u(q))-1 m(q)1/4 [(2b1+N)z02e-u(q) -N-1+2b12u(q)].", "and the potentials $U^{\\pm }(q)$ can be obtained using Eqs.", "(), (REF a) and (REF b).", "We have checked the normalizability of the solvable sectors (REF ) with the following two mass functions.", "(i) $m(q)=(1-q^{2})^{-1}$ , $q\\in (-1,1)$ for which the change of variable is $u(q)=\\sin ^{-1}q$ .", "This mass profile has been used in Refs.", "[11], [10] while considering the effective-mass quantum nonlinear oscillator.", "This mass function has effect on dynamical symmetry breaking because it manifests broken SUSY [i.e., neither $\\mathcal {V}^{-}$ nor $\\mathcal {V}^{+}$ belongs to $ L^{2}(-1,1)$ ], which is clear from the following expressions of $\\mathcal {V}^{-}$ and $\\mathcal {V}^{+}$ : V-=1,z(u(q)),z(u(q))2,... 1(1-q2)1/4 [-(2b1+N)z02e--1q -N-1-2b12-1q], V+=1,z(u(q))2,z(u(q))3,... 1(1-q2)1/4(e-1q+z0) [(2b1+N)z02e--1q -N-1+2b12-1q].", "(ii) $m(q)=2\\mathrm {e}^{-2q^{2}}/\\pi $ for which the solvable sectors (REF ) reduce to V-=1,z(u(q)),z(u(q))2,... [-q24-(2b1+N)z02e-Erfq -N-1-2b12Erfq], V+=1,z(u(q))2,...,z(u(q))Nz(u(q))-1 [-q24+(2b1+N)z02 e-Erfq-N-1+2b12Erfq].", "From the above solvable sectors, we observe that both $\\mathcal {V}^{-}(\\mathbb {R})$ and $\\mathcal {V}^{+}(\\mathbb {R})$ belong to $L^{2}(\\mathbb {R})$ , irrespective of the parameter value $b_{1}$ , which means unbroken $\\mathcal {N}$ -fold SUSY.", "Hence, the mass function $m(q)=2\\mathrm {e}^{-2q^{2}}/\\pi $ affects dynamical breaking of the $\\mathcal {N}$ -fold SUSY.", "Hence, comparing the normalizability conditions in both the constant and position dependent mass cases, we conclude that both the mass functions change the behaviours of symmetry breaking." ], [ "Effects of PDM on Dynamical Symmetry Breaking of Type $X_{2}$ \n{{formula:76cf54fa-6552-45e2-9d6e-8dba31884b77}} -fold SUSY", "In this section, we examine three different models of type $X_{2}$ $\\mathcal {N}$ -fold SUSY characterized by different choices of the two parameters $a_{1}$ and $a_{2}$ ; $a_{1}\\ne 0$ and $a_{2}=0$ for the first model, $a_{1}=0$ and $a_{2}\\ne 0$ for the second, and $a_{1}a_{2}\\ne 0$ for the third.", "The first two choices lead to the rational- and hyperbolic-type potential pairs already shown in Ref.", "[52], while the last choice to an exponential-type potential pair which is new and has not been investigated in the literature.", "Example 3.4.", "$A(z)=2z$ $[a_{1}=2]$ .", "Potentials: V(0)-(q)=q22+42-18q2 +4[q2-+1f(q2;) -4(-1)q2f(q2;)2]-N+V0, V(0)+(q)=q22+4(+N)2-18q2 +4[q2--N+1f(q2;+N) -4(+N-1)q2f(q2;+N)2]+V0, where $V_{0}=\\mathcal {N}-\\alpha +3-c_{0}$ is an irrelevant constant.", "Solvable sectors: VN(0)-=1(q2;),...,N (q2;)q+1/2e-q2/2f(q2;), VN(0)+=1(q2;+N),..., N(q2;+N)q--N+1/2 eq2/2f(q2;+N).", "In this case, the solvability condition $a_{2}(=a_{3}=a_{4})=0$ for type $X_{2}$ is satisfied and thus the corresponding constant-mass Hamiltonians $H^{(0)\\pm }$ are simultaneously solvable.", "For $\\alpha >1$ , a natural choice for the domain of these potentials is a real half-line $S=\\mathbb {R}_{+}$ .", "On this domain $\\mathbb {R}_{+}$ , it is evident from (REF ) that $\\mathcal {V}_{\\mathcal {N}}^{(0)-}(\\mathbb {R}_{+})\\subset L^{2}(\\mathbb {R}_{+})$ and $\\mathcal {V}_{\\mathcal {N}}^{(0)-}(\\mathbb {R}_{+})\\lnot \\subset L^{2}(\\mathbb {R}_{+})$ .", "Therefore, it manifests unbroken $\\mathcal {N}$ -fold SUSY of the system in the constant-mass background.", "According to (), the solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ of the corresponding PDM Hamiltonians $H^{\\pm }$ for an arbitrary mass function $m(q)$ read as VN-=1(u(q)2;),...,N (u(q)2;)m(q)1/4u(q)+1/2 e-u(q)2/2f(u(q)2;), VN+=1(u(q)2;+N),..., N(u(q)2;+N)m(q)1/4 u(q)--N+1/2eu(q)2/2f(u(q)2;+N), where $u(q)$ is given by () and the PDM potentials $U^{\\pm }(q)$ can be obtained using Eqs.", "() and (REF ).", "In this case, we have not been able to find out any realistic mass function which could break the $\\mathcal {N}$ -fold SUSY.", "In other words, we can say that the $\\mathcal {N}$ -fold SUSY in this case is steady against many variations of mass functions [e.g., $m(q)=\\mathrm {e}^{-q}$ , $\\operatorname{sech}^{2}q$ ].", "Example 3.5.", "$A(z)=(z^{2}+\\zeta ^{2})/2$ , $[a_{2}=1/2$ , $\\zeta ^{2}=(\\alpha -1)(\\alpha +\\mathcal {N}-1)>0]$ .", "Potentials: V(0)-(q)=282q+N-14q+V0 +182q[ 4(N-1)q+42+4(N-2)-N 2-2N+4] -2(-1)[q--N+3f(q; )-2(-1)2q-N+1f(q; )2], V(0)+(q)=282q+3N-14q+V0         -182q[ 4(N+1)q-42-4(N-2)+N 2+6N-4] -2(+N-1)[q-+3f(q; +N)-2(+N-1)2q+N+1 f(q;+N)2], where $V_{0}$ is an irrelevant constant given by V0=42+4(N-4)+N 2+168-c0.", "Solvable sectors: VN(0)-=1(q;),..., N(q;)e-(q)/2 -gdq(q)N/2-1f(q;), VN(0)+=1(q;+N),..., N(q;+N) e(q)/2 +gdq(q)N/2f(q;+N), where $\\operatorname{gd}q=\\tan ^{-1}(\\sinh q)$ is the Gudermann function.", "The solvability condition is not satisfied in this case and both of the Hamiltonians are only quasi-solvable.", "For $\\alpha >1$ , the potentials $V^{\\pm }(q)$ given in (REF ) are defined on the whole real line $\\mathbb {R}$ .", "From the solvable sectors (REF ), it is clear that neither $\\mathcal {V}_{\\mathcal {N}}^{(0)-}(\\mathbb {R})$ nor $\\mathcal {V}_{\\mathcal {N}}^{(0)+}(\\mathbb {R})$ belongs to $L^{2}(\\mathbb {R})$ , so the $\\mathcal {N}$ -fold SUSY is dynamically broken in the constant-mass background.", "Now, the PDM potentials $U^{\\pm }(q)$ can be obtained with help of Eqs.", "(), (REF ), and (REF ), and the solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ of the corresponding PDM Hamiltonians $H^{\\pm }$ for an arbitrary mass function $m(q)$ read from () as VN-=1(u(q);),..., N(u(q);) m(q)1/4e-(u(q))/2 -gdu(q)(u(q))N/2-1f(u(q);), VN+=1(u(q);+N),..., N(u(q);+N) m(q)1/4e(u(q))/2 +gdu(q)(u(q))N/2f(u(q);+N), where $u(q)$ is given by ().", "Let us now consider two cases: (i) $m(q)=\\operatorname{sech}^{2}q$ , $q\\in (-\\infty ,\\infty ),$ for which the change of variable is $u(q)=\\operatorname{gd}q$ .", "Then, the solvable sectors of $U^{\\pm }(q)$ are given by VN-=1(u(q);),..., N(u(q);) sechq e-(gdq)/2 -gd(gdq)[(gdq)]N/2-1f(u(q);), VN+=1(u(q);+N),...,N(u(q);+N) sechq e(gdq)/2 +gd(gdq)[(gdq)]N/2f(u(q);+N).", "In this case, the mass function as well as the potentials $U^{\\pm }(q)$ are well behaved on $(-\\infty ,\\infty )$ , so we can consider the whole real line $\\mathbb {R}$ as a domain of the potentials.", "From the solvable sectors (REF ), it is clear that both $\\mathcal {V}_{\\mathcal {N}}^{\\pm }(\\mathbb {R})$ belong to $L^{2}(\\mathbb {R})$ , which means unbroken $\\mathcal {N}$ -fold SUSY, i.e., the mass profile affects symmetry restoration.", "(ii) $m(q)=2\\mathrm {e}^{-2q^{2}}/\\pi $ .", "In this case, the solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ reduce to VN-=1(u(q);),..., N(u(q);) [-q2/4-(Erfq)/2-gd(Erfq)] [(Erfq)]N/2-1f(u(q);), VN+=1(u(q);+N),..., N(u(q);+N) [-q2/4+(Erfq)/2+gd(Erfq)] [(Erfq)]N/2f(u(q);+N).", "From the above solvable sectors $(\\ref {e74})$ , it is clear that both $\\mathcal {V}_{\\mathcal {N}}^{\\pm }(\\mathbb {R})$ belong to $L^{2}(\\mathbb {R})$ .", "That is, in this case we again have unbroken $\\mathcal {N}$ -fold SUSY.", "We note that there are other mass functions, e.g., $m(q)=(\\beta +q^{2})^{2}/(1+q^{2})^{2}$ , which have no effect on the dynamical breaking of $\\mathcal {N}$ -fold SUSY, i.e., it is also possible to construct PDM systems which maintain the broken $\\mathcal {N}$ -fold SUSY.", "Example 3.6.", "$A(z)=(z+\\zeta )^{2}/2$ , $[a_{2}=1/2,a_{1}=\\zeta =\\sqrt{(\\alpha -1)(\\alpha +\\mathcal {N}-1)}]$ .", "Potentials: V(0)-(q)=18e2q-N+14eq -(N-1)(N+2-2-1)4e-q +2[N 2+2N(4-2-3) +4(2-2-3)+4+5]8e-2q -2[(--1)eqf(eq-;) +2(-1)e2qf(eq-;)2]+V0, V(0)+(q)=18e2q+N-14eq +(N+1)(N+2-2-1)4e-q +2[N 2+2N(4-2-3) +4(2-2-3)+4+5]8e-2q -2[(+N--1)eqf(eq-;+N)+2(+N-1)e2qf(eq-;+N)2]+V0, where $V_{0}$ is an irrelevant constant given by V0=(N+2)2+2(N-2)+2(7-8+8)8 -c0.", "Solvable sectors: VN(0)-=1(eq-;), ...,N(eq-;)f(eq-; ) [-eq2+2-2-N+12e-q-N-22q], VN(0)+=1(eq-;+N),...,N(eq-;+N) f(eq-;+N) [eq2-2-2-N+12e-q-N2q].", "This system is new and presented in this paper for the first time.", "The exponential-type $V_{\\mathcal {N}}^{\\pm }(q)$ are naturally defined on the whole real line $\\mathbb {R}$ since they have no singularity on it, so the Hilbert space is $L^{2}(\\mathbb {R})$ .", "Noting that $2\\zeta -2\\alpha -\\mathcal {N}+1<0$ for $\\alpha >1$ , since 42-(2+N-1)2=-4-(N-1)(N+3)<-(N+1)2<0, we see that $\\mathcal {V}_{\\mathcal {N}}^{(0)-}(\\mathbb {R})\\subset L^{2}(\\mathbb {R})$ and $\\mathcal {V}_{\\mathcal {N}}^{(0)+}(\\mathbb {R})\\lnot \\subset L^{2}(\\mathbb {R})$ for $\\zeta >0$ .", "Hence, it manifests unbroken $\\mathcal {N}$ -fold SUSY.", "For $\\zeta <0$ , on the other hand, neither $\\mathcal {V}_{\\mathcal {N}}^{(0)-}(\\mathbb {R})$ nor $\\mathcal {V}_{\\mathcal {N}}^{(0)+}(\\mathbb {R})$ belongs to $L^{2}(\\mathbb {R})$ , so the $\\mathcal {N}$ -fold SUSY is broken in the constant-mass background.", "In a PDM background, the solvable sectors $\\mathcal {V}_{\\mathcal {N}}^{\\pm }$ of the type $X_{2}$ PDM Hamiltonians $H^{\\pm }$ are deformed as [cf., Eq.", "()] VN-=1(eu(q)-;), ...,N(eu(q)-;)f(eu(q) -;) m(q)1/4[-eu(q)2+2-2-N+12e-u(q)-N-22u(q)], VN+=1(eu(q)-; +N),...,N(eu(q)-;+N) f(eu(q)-;+N) m(q)1/4[eu(q)2-2-2-N+12e-u(q)-N2u(q)], and the potentials $U^{\\pm }(q)$ can be obtained using Eqs.", "(), (REF ), and (REF ).", "In this case, the choice of mass functions are as follows: (i) $m(q)=(1-q^{2})^{-1}$ , $q\\in (-1,1)$ , for which the solvable sectors of the PDM Hamiltonians $H^{\\pm }$ are given by VN-=1(eu(q)-;), ...,N(eu(q)-;)(1-q2)1/4 f(eu(q)-;) [-e-1q2+2-2-N+12 e--1q-N-22-1q], VN+=1(eu(q)-;+N),...,N(eu(q)-;+N) (1-q2)1/4f(eu(q)-;+N) [e-1q2-2-2-N+12 e--1q-N2-1q].", "From the above solvable sectors, it is clear that both $\\mathcal {V}_{\\mathcal {N}}^{\\pm }(-1,1)$ do not belong to $L^{2}(-1,1)$ , so it manifests broken $\\mathcal {N}$ -fold SUSY irrespective of the sign of $\\zeta $ .", "Hence, comparing the normalizability conditions in both the constant and position-dependent mass cases, we conclude that the mass function $m(q)=(1-q^{2})^{-1}$ affects dynamical breaking of $\\mathcal {N}$ -fold SUSY for $\\zeta >0$ .", "(ii) $m(q)=2\\mathrm {e}^{-2q^{2}}/\\pi $ , $q\\in (-\\infty ,\\infty )$ , for which the $\\mathcal {N}$ -fold SUSY remains unbroken, which is evident from the corresponding solvable sectors given by VN-=1(eu(q)-;), ...,N(eu(q)-;)f(eu(q) -;)[-q24.", ".-eErfq2+2-2-N+12e-Erfq-N-22Erfq], VN+=1(eu(q)-; +N),...,N(eu(q)-;+N) f(eu(q)-;+N) [-q24+eErfq2 -2-2-N+12e-Erfq-N2Erfq].", "From the normalizability conditions in the constant and position-dependent mass cases, we see that the mass function $m(q)=2\\mathrm {e}^{-2q^{2}}/\\pi $ affects the dynamical breaking of $\\mathcal {N}$ -fold SUSY for $\\zeta <0$ ." ], [ "Summary and Perspectives", "In this paper, we have investigated effect of position-dependent mass background on dynamical breaking of type B and type $X_{2}$ $\\mathcal {N}$ -fold SUSY.", "We have selected three different models in the constant mass background for each type, and then examined whether some of the physically relevant effective mass profiles can affect the pattern of $\\mathcal {N}$ -fold SUSY breaking in each model.", "We summarize the results in Table REF .", "We can easily see from Table REF that, except for the rational potentials, some of the PDM profiles can actually affect and change the patterns of dynamical $\\mathcal {N}$ -fold SUSY breaking in all the trigonometric, hyperbolic, and exponential potentials.", "Although we have selected the specific types of $\\mathcal {N}$ -fold SUSY to develop physical applicability of the new mathematical concept of exceptional polynomial subspaces, we can of course make a similar analysis on other types of $\\mathcal {N}$ -fold SUSY such as type A and type C to find out positive effect of PDM on SUSY breaking in some models.", "Hence, it would be possible to observe experimentally transition between a broken and an unbroken phases if an effective mass can be controlled experimentally such that the constant mass limit can be also realized in an experimental setting.", "The physical meanings of a position-dependent mass depend on each physical system under consideration, for instance, the curvature of the local band structure of an alloy in the momentum space for electrons in a crystal with graded composition [1], the particle densities of 3He and 4He in pure and mixed helium clusters with doping atoms or molecules [4], the effective electron mass for electrons confined in a quantum dot [3] and for dipole excitations of sodium clusters [5], and so on.", "Thus, if we can prepare such an atomic, molecular, or condensed matter system which is described by a certain PDM quantum model subjected to an $\\mathcal {N}$ -fold SUSY potential with mass profiles, e.g., $m(q)=\\mathrm {e}^{-\\nu ^{2}q^{2}}$ or $(1-\\nu ^{2}q^{2})^{-1}$ where $\\nu $ is an experimentally adjustable parameter such that $\\nu \\rightarrow 0$ is realizable, then the spectral change of the system could be observed at $\\nu =0$ due to the phase transition.", "The essence and novelty of our idea rely on the observation that the physically controllable PDM can cause the phase transition by changing the normalizability of the solvable sector although the latter is superficially a simple mathematical aspect.", "Hence, it is quite important to note that the normalizability of wave functions can play much more roles than the quantization of energy spectrum which is referred to by any standard textbook on quantum mechanics.", "Table: The effects of PDM profiles on dynamical breaking of 𝒩\\mathcal {N}-fold SUSYin various type B and type X 2 X_{2} models.We note that this experimental observability might have impact not only on some atomic, molecular, and condensed matter problems from which PDM quantum theory originated, but also on high-energy physics.", "Until now many high-energy physicists have believed that SUSY is realized at the GUT or Planck scale as a resolution of the naturalness and the hierarchy problem but is broken at least at the electroweak scale.", "Unfortunately, however, theoretical analysis on dynamical SUSY breaking in field theoretical models are extremely difficult on the one hand, and it is virtually impossible to make a GUT scale experiment on the other hand.", "The aforementioned experimental observability suggests that we might extract some clues to understand dynamical SUSY breaking in high-energy physics from realistic eV scale experiments in atomic, molecular, and condensed matter physics.", "It is because the Witten's work [31] has indicated that the mechanism of dynamical SUSY breaking in quantum field theory and quantum mechanics is essentially the same.", "We also note that the careful non-perturbative analyses in Refs.", "[59], [60] have shown that the mechanism of dynamical breaking of ordinary and $\\mathcal {N}$ -fold SUSY is also the same.", "Hence, dynamical aspects of SUSY quantum field theoretical models would be mimicked in $\\mathcal {N}$ -fold SUSY quantum mechanical toy models, regardless of whether or not $\\mathcal {N}$ -fold SUSY can be realized in higher dimensions.", "Therefore, we believe that further studies in this direction are worth pursuing both theoretically and experimentally.", "From a theoretical point of view it is a challenging issue to investigate both a perturbation theory and the non-renormalization theorem in PDM quantum systems.", "This work (T. T.) was partially supported by the National Cheng Kung University under the grant No.", "HUA:98-03-02-227." ] ]

[ [ "Quantum Statistics of Surface Plasmon Polaritons in Metallic Stripe\n Waveguides" ], [ "Abstract Single surface plasmon polaritons are excited using photons generated via spontaneous parametric down-conversion.", "The mean excitation rates, intensity correlations and Fock state populations are studied.", "The observed dependence of the second order coherence in our experiment is consistent with a linear uncorrelated Markovian environment in the quantum regime.", "Our results provide important information about the effect of loss for assessing the potential of plasmonic waveguides for future nanophotonic circuitry in the quantum regime." ], [ "1. Second-order quantum coherence function $g^{(2)}(\\tau )$", "For quantized electromagnetic fields propagating in the $x$ -direction with an arbitrary lateral beam profile, and represented by the electric field operator $\\hat{E}^+(x,t)$ , we have at a fixed position, $x=0$ , the following definition [22] $g^{(2)}(\\tau )=\\frac{\\langle \\hat{E}^-(0)\\hat{E}^-(\\tau )\\hat{E}^+(\\tau )\\hat{E}^+(0)\\rangle }{\\langle \\hat{E}^-(0)\\hat{E}^+(0)\\rangle ^2}.$ Here $\\langle \\hat{X}\\rangle $ represents the expectation value of the operator $\\hat{X}$ with respect to the initial state of the field, i.e.", "an averaging over ensembles.", "The average of the intensity of the field is assumed to be constant over time, $\\langle \\hat{E}^-(\\tau )\\hat{E}^+(\\tau )\\rangle =\\langle \\hat{E}^-(0)\\hat{E}^+(0)\\rangle $ .", "Throughout we will suppress the position dependence of $\\hat{E}^+(x,t)$ , as $x$ is fixed at zero.", "At zero time delay, $\\tau =0$ , for $n$ -excitation states $\\left|{n}\\right\\rangle $ , we have that $\\left\\langle {n}\\right| \\hat{E}^-(0)\\hat{E}^-(0)\\hat{E}^+(0)\\hat{E}^+(0)\\left|{n}\\right\\rangle =n(n-1)$ and $\\left\\langle {n}\\right| \\hat{E}^-(0)\\hat{E}^+(0)\\left|{n}\\right\\rangle =n$ , leading to the relation $g^{(2)}(0)=1-1/n$ .", "In particular, for $n=1$ (single excitations), we have $g^{(2)}(0)=0$ .", "Similarly, for $n=2$ , $g^{(2)}(0)=0.5$ : a measured value of $g^{(2)}(0)$ between 0 and 0.5 is a confirmation that we are dealing with single excitations.", "On the other hand, for attenuated laser light described by a weak coherent state $\\left|{\\alpha }\\right\\rangle =\\sum _{n=0}^{\\infty }e^{-|\\alpha |^2}\\frac{|\\alpha |^{2n}}{n!", "}\\left|{n}\\right\\rangle $ , where $|\\alpha |^2=\\langle n \\rangle $ is the mean excitation number, we have $g^{(2)}(0)=1$ .", "Moreover, it can be shown using the Cauchy-Schwartz inequality that for any classical electromagnetic field, due to the absence of operators and their commutation relations for the classical electric field $E^+(x,t)$ , the numerator in Eq.", "REF factorizes to give the inequality $g^{(2)}(0)\\ge 1$ .", "Thus by measuring $g^{(2)}(0)$ for a given field, we can determine whether or not it is in the nonclassical regime ($g^{(2)}(0)<1$ ).", "The interaction Hamiltonian for type-I spontaneous parametric down-conversion is given by [4], [16] $\\hat{H}_I = \\hbar ~\\xi ~ \\hat{a}^\\dag _A\\hat{a}^\\dag _B + H.c.$ Here, $\\xi \\propto \\chi ^{(2)}{\\cal E}_p$ , where $\\chi ^{(2)}$ is the second-order nonlinear susceptibility of the BBO crystal in our setup and ${\\cal E}_p$ is the amplitude of the classical coherent laser pump field.", "In addition, $\\hat{a}^\\dag _A$ ($\\hat{a}^\\dag _B$ ) is a creation operator for a photon in mode $A$ ($B$ ) and $H.c.$ represents the Hermitian conjugate.", "Taking the initial state of modes $A$ and $B$ to be the vacuum $\\left|{\\psi (0)}\\right\\rangle =\\left|{0}\\right\\rangle _{AB}=\\left|{0}\\right\\rangle _{A}\\left|{0}\\right\\rangle _{B}$ and evolving it according to the Schrödinger equation as $\\left|{\\psi (t)}\\right\\rangle =e^{-it\\hat{H}_I/\\hbar }\\left|{\\psi (0)}\\right\\rangle ,$ we obtain, up to first order in time, the state $\\left|{\\psi (t)}\\right\\rangle =(1-\\mu ^2/2)\\left|{0}\\right\\rangle _{A}\\left|{0}\\right\\rangle _{B}-i\\mu \\left|{1}\\right\\rangle _{A}\\left|{1}\\right\\rangle _{B},$ where $\\mu =\\xi t$ .", "By detecting a photon in mode $A$ we remove the first (vacuum) term and `herald' the presence of a single-photon state $\\left|{1}\\right\\rangle _B$ in mode $B$ , up to first order.", "By tuning the pump laser intensity appropriately, higher order terms can be made negligible in mode $B$ , even if the detection in mode $A$ is not photon number resolving.", "Thus, we can use type-I SPDC to produce high-quality single photon states, $\\left|{1}\\right\\rangle $ , with larger generation rates than currently achieved with emitter-type sources, such as quantum dots [24]." ], [ "b) Characterization of our single photon source", "In the down conversion process, the phase matching conditions are not perfect in the experiment.", "For this reason, the down-converted light is not monochromatic, but presents a spectrum with finite width [31]: this effect is observed with our source as shown in Fig.", "REF (a), where we present its spectral properties.", "In Fig.", "REF (b) the unconditioned coincidence rates $R_{B_1B_2}=N_{B_1B_2}/T$ (where $T$ is the integration time) are shown.", "No correlation between those two arms arises.", "On the other hand, when we plot the conditioned rates $R_{AB_1}=N_{AB_1}/T$ (Fig.", "REF (c)) it is apparent that there is a very strong correlation between arm $B_1$ and the reference arm $A$ at zero delay.", "The same occurs for $B_2$ .", "This figure shows as well that the configuration used allows for a single photon generation rate of about 10$^6$  s$^{-1}$ .", "In addition, the triple coincidence rate $R_{AB_1B_2}=N_{AB_1B_2}/T$ is shown in Fig.", "REF (d).", "On first thought, one would expect to see a value of zero at zero delay, as a coincidence between $A$ and $B_1$ should indicate the presence of a single photon in the system, and thus forbidding any simultaneous detection on $B_2$ .", "However, there is a peak in $R_{AB_1B_2}$ at zero delay: this is due to the fact that we count coincidences within a finite time window $\\Delta t$ .", "In our case, $\\Delta t=2\\text{~ns}$ : two \"clicks\" detected within a 2 ns-wide time window are considered as a coincidence, even if they are not exactly simultaneous.", "For this reason, accidental coincidences are measured, at a rate determined solely by the count rates on each detector, the integration time $T$ and the time window $\\Delta t$ .", "One can show that [23], if $R_{B_1}$ and $R_{B_2}$ are the single count rates at $B_1$ and $B_2$ respectively, the accidental coincidence rate at zero delay for three detectors $R_{acc}(0)$ is: $R_{acc}(0)=\\Delta t R_{AB_1}R_{B_2}+\\Delta t R_{AB_2}R_{B_1}.$ The rate of triples observed at zero delay agree very well with the value expected for solely accidental coincidences.", "Additionally, these accidental coincidences lead to a value of $g^{(2)}(0)$ higher than zero.", "One can show [23] that the offset on $g^{(2)}(0)$ due to accidental coincidences is: $g^{(2)}_{acc}(0)=\\Delta t R_A \\left(\\frac{R_{B_1}}{R_{AB_1}}+\\frac{R_{B_2}}{R_{AB_2}}\\right).$ By using the values observed in Fig.", "REF and the count rates at each detector, one finds the value of $g^{(2)}(0)=0.23$ for our single photon source for $\\Delta t=2\\text{~ns}$ , which is the value observed in Fig.", "2(b) in the main text: the non-zero value of $g^{(2)}(0)$ is solely due to accidental coincidences." ], [ "3. Fock state population tomography", "Here we provide details of the tomographic method used to reconstruct the populations.", "We used the technique of Zambra et al.", "[30] to measure the photon statistics based on on/off detection.", "An arbitrary quantum state can be written in the number basis as $\\rho =\\sum _{nm} \\rho _{nm }\\left|{n}\\right\\rangle \\left\\langle {m}\\right|$ , where the diagonal elements $\\rho _{nn}=\\rho _{n}=\\left\\langle {n}\\right|\\rho \\left|{n}\\right\\rangle $ give the photon number distribution of the state $\\rho $ .", "Let $\\eta $ be the efficiency of a given detector, so that $\\eta $ is the probability for a single photon to be revealed and $(1-\\eta )$ is the probability for it not to be revealed.", "Thus, the total probability of the detector not giving a `click' is $p(\\eta )=\\sum _n(1-\\eta )^n\\rho _n$ .", "Consider now a set of such detectors with different efficiencies, $\\eta _\\nu $ .", "We then have $p(\\eta _\\nu )=\\sum _n(1-\\eta _\\nu )^n\\rho _n$ , or more compactly written $p_\\nu =\\sum _nA_{\\nu n} \\rho _n.$ Here, $p_\\nu $ can be obtained experimentally and $A_{\\nu n}$ can be set by artificially changing the efficiency of the detector, leaving $\\rho _n$ as the unknown parameter.", "Then, by assuming the $\\rho _n$ 's are negligible for $n>n_t$ , where $n_t$ is a truncation number and we have at least $N>n_t$ detector efficiencies, Eq.", "REF is a LINPOS problem [30] and one can use the expectation maximization (EM) algorithm, which converges to the maximum likelihood solution.", "By imposing the physical constraint $\\sum _n \\rho _n=1$ we have the iterative solution $\\rho _n^{(i+1)}=\\rho _n^{(i)}\\sum _{\\nu }^N\\frac{A_{\\nu n}}{\\sum _\\lambda A_{\\lambda n}}\\frac{f_\\nu }{p_\\nu [\\lbrace \\rho _n^{(i)}\\rbrace ]},$ where $\\rho _n^{(i)}$ is the value of $\\rho _n$ evaluated at the $i$ -th iteration, $f_\\nu $ are the experimental frequencies of the `no-click' events for $\\eta =\\eta _\\nu $ (whose ideal values are $p_\\nu $ ) and $p_\\nu [\\lbrace \\rho _n^{(i)}\\rbrace ]$ are the probabilities $p_\\nu $ calculated using the reconstructed distribution $\\lbrace \\rho _n^{(i)}\\rbrace $ at the $i$ -th iteration, i.e.", "$p_\\nu [\\lbrace \\rho _n^{(i)}\\rbrace ]=\\sum _n^{n_t}(1-\\eta _\\nu )^n\\rho _n^{(i)}$ .", "We start the iterations with the unbiased distribution $\\rho _n^{(0)}=1/(1+n_t)$ and use the experimental results $f_\\nu =n_{0,\\nu }/n_\\nu $ , where for a given detector efficiency $\\eta _\\nu $ , $n_\\nu $ is the total number of runs (state preparation and measurement) and $n_{0,\\nu }$ is the number of no-click events for these runs.", "The EM algorithm is then carried out until the changes in the population numbers $\\rho _n$ between iterations reduce below a given threshold, $\\epsilon $ .", "For the field out-coupled from the waveguides for the single photon source, we have $n_\\nu =N_A$ and $n_{0,\\nu }=N_A-N_{AB_{1,\\nu }}\\eta _x$ .", "Here, $\\eta _x$ is a loss scaling factor given by $\\eta _x=\\eta _d/\\eta _0$ , which allows us to consider the tomography being performed on the state that enters the detection and analysis stage in our setup (rather than a tomography of the initial state generated, as carried out by Zambra et al. [30]).", "The loss $\\eta _d=0.55/2$ corresponds to detector $B_1$ 's intrinsic efficiency around the operating wavelength of the field used ($\\lambda =$ 808nm), combined with that of the beamsplitter in front of it.", "The loss $\\eta _0=N_{AB_{1,0}}/N_A$ is the total loss from initial state generation to detection at $B_1$ .", "To measure the coincidences $N_{AB_{1,\\nu }}$ we have set the coincidence window to $\\Delta t=2$ ns.", "A set of efficiencies are then introduced using an ND filter wheel.", "Here, the efficiencies $\\eta _\\nu =\\eta _dN_{B_1,\\nu }/N_{B_{1,0}}$ .", "For the attenuated laser source, as it is not based on conditional measurements at detector $A$ , we set a window of 500 ns for a measurement duration every 10 $\\mu $ s and carry out 10,000 runs.", "Thus $n_\\nu =10,000$ and $n_{0,\\nu }=10,000-N_{AB_{1,\\nu }}$ , where $N_{AB_{1,\\nu }}$ is the total number of clicks from the 10,000 runs.", "The efficiencies $\\eta _\\nu =\\eta _dN_{B_1,\\nu }/N_{B_{1,0}}$ , where $\\eta _d=0.55/2$ is used as before.", "In Fig.", "REF we show the dependence of the no-click frequencies $f_\\nu $ with the detector efficiencies $\\eta _\\nu $ measured in our experiment for the light out-coupled from the waveguides for the single-photon source and that for the attenuated laser.", "In both plots, background counts were subtracted from the singles ($N_A$ and $N_{B_1}$ ) and doubles ($N_{AB}$ and $N_{AB_1}$ ) at the detectors.", "Using these plots, the reconstructed populations from the EM algorithm are found and shown in the main text in Fig. 4.", "To calculate the errors in the populations we used a Monte-Carlo approach, propagating the errors from the measured data shown in Fig.", "REF through the reconstruction algorithm, with Gaussian distributions placed on the values." ], [ "4. Grating coupling", "Here we provide a theoretical model to describe the transfer of single photons to single SPPs via the grating coupling method used in our experiment.", "At the single-photon level only small intensities of the photon field are involved and therefore any nonlinear terms in the photon-SPP coupling can be effectively neglected [32], leading to the following linear coupling Hamiltonian [19] $\\hat{H}&=& \\int _{0}^{\\infty }{\\mathrm {d}} \\omega \\hbar \\omega \\hat{a}^{\\dag }(\\omega )\\hat{a}(\\omega )+\\int _{0}^{\\infty }{\\mathrm {d}} \\omega \\hbar \\omega \\hat{b}^{\\dag }(\\omega )\\hat{b}(\\omega ) \\\\& & + i \\hbar \\int _{0}^{\\infty }{\\mathrm {d}} \\omega [g(\\omega )\\hat{a}^{\\dag }(\\omega )\\hat{b}(\\omega )-g^*(\\omega )\\hat{b}^{\\dag }(\\omega )\\hat{a}(\\omega )].", "\\nonumber $ Here, the $\\hat{a}(\\omega )$ 's ($\\hat{a}^{\\dag }(\\omega )$ 's) correspond to annihilation (creation) operators for the photons which obey bosonic commutation relations $[\\hat{a}(\\omega ),\\hat{a}^{\\dag }(\\omega ^{\\prime })]=\\delta (\\omega -\\omega ^{\\prime })$ .", "Due to the collective nature of the electron charge density waves and the frequency regime of our experiment, a macroscopic approach for the resulting electromagnetic field is appropriate for the SPPs [33].", "Upon quantization, they are therefore assumed to behave as bosonic modes.", "Thus, the $\\hat{b}(\\omega )$ 's ($\\hat{b}^{\\dag }(\\omega )$ 's) correspond to annihilation (creation) operators for the SPPs which should, in principle, obey bosonic commutation relations $[\\hat{b}(\\omega ),\\hat{b}^{\\dag }(\\omega ^{\\prime })]=\\delta (\\omega -\\omega ^{\\prime })$ .", "In the Hamiltonian given in Eq.", "(9), the first and second terms are the photon and SPP fields' free-energy respectively.", "The last term describes interactions between the two fields, where the coupling $g(\\omega )$ is proportional to the overlap of the field scattered by the grating with the SPP mode on the waveguide stripe.", "Carrying out the overlap integrals for the fields in a particular scenario gives the phase-matching conditions for the coupling and the coupling value itself.", "For perfect coupling, i.e.", "a single photon injected into the grating scatters and couples to a single SPP with unit efficiency, we have $g(\\omega )=\\pi /2$  [19].", "Thus, the geometry of the grating should be optimised to achieve a coupling as close as possible to $\\pi /2$ , with negligible deviation over the bandwidth, $\\Delta \\omega $ , of the incoming photon - in order to avoid significant wavepacket distortion and loss during the transfer process.", "A first approximation for the phase-matching condition for photons injected normal to the surface and with TM polarization (with respect to the surface plane) is given by [29] $k_{sp}=k_g$ , where $k_{sp}$ is the magnitude of the SPP wavevector in the direction of propagation along the waveguide and $k_g=\\frac{2 \\pi }{\\Lambda }m$ is the grating momentum ($\\Lambda $ is the grating period and $m$ is an integer).", "For the metallic strip in our experiment, we use the approximation [20], $k_{sp}=\\sqrt{\\frac{\\omega ^2}{c^2}\\frac{\\epsilon _m}{1+\\epsilon _m}-\\frac{\\pi ^2}{W^2}}$ , where $\\epsilon _m$ is the permittivity of gold, $W=3\\mu $ m is the width of the waveguide and $\\omega $ corresponds to a free space wavelength of $\\lambda _0=808$  nm.", "Taking the $m=+1$ grating momentum, one finds a grating period of $\\Lambda =802$ nm.", "We use this as our starting point and perform FEM simulations to optimise the coupling by modifying the height, width, period and number of grooves for the grating on the waveguide.", "We find the optimal period of $\\Lambda =680$  nm.", "Such a large deviation from the approximate result can be explained by the use of deep grooves in our gratings, as the phase-matching condition $k_{sp}=k_g$ is only approximate for shallow gratings [29] (weak perturbations)." ] ]

[ [ "Fundamental $C^*$-algebras associated to automata groups" ], [ "Abstract We propose to study some properties of the $C^*$-algebra naturally built out of the fundamental action that an automaton group $G$ admits on a regular rooted trees $\\tree$." ], [ "Introduction", "This paper aims to study some $C^*$ -algebras associated to groups generated by automata.", "We will mainly focus on the fundamental $C^*$ -algebra of such a group $G$ , that is the one naturally built out of the action of $G$ on a regular rooted tree $\\mathcal {T}_d$ that defines it as an automaton group.", "The paper is organised as follows.", "The first section aims to recall some basic definitions and properties of regular rooted trees and their group of automorphisms, and to fix some notation which will be used in the sequel.", "In Section 2, we introduce the fundamental $C^*$ -algebra of a automaton group and study some general properties of it.", "For instance, we will be interested in spectrum approximation for non-normal elements in the von Neumann algebra generated by this fundamental $C^*$ -algebra.", "This problem is the one that motivated I. Marin and the author to write the Appendix.", "Section 3 aims to extend the study initiated in [9] of the relation between the fundamental and the regular representation for an automaton group.", "The last one focus on groups generated by reversible automata.", "In this case, the self-similar structure of $G$ allows us to get results on its reduced $C^*$ -algebra $C^*_{\\lambda }\\left(G\\right)$ , and on certain $C^*$ -algebras considered by V. Nekrashevych in [8]." ], [ "Groups generated by automata", "We start by recalling some definitions on rooted trees and groups generated by automata.", "For a general presentation on the subject, we refer to [12], [7].", "Let $d\\ge 1$ be an integer and $\\mathcal {T}_d$ be the $d$ -regular rooted tree.", "That is, $\\mathcal {T}_d$ is the tree in which all vertices have degree $d+1$ , except one which has degree $d$ .", "This last vertex is called the root of $\\mathcal {T}_d$ and is denoted by $\\varnothing $ .", "If we let $X=\\left\\lbrace 1,\\dots ,d \\right\\rbrace $ , then $\\mathcal {T}_d$ is the Cayley graph of the free monoid $X^*$ generated by $X$ .", "Therefore, the vertex set $\\mathcal {T}_d^0$ can be identified with $X^*$ .", "In particular, each vertex $v\\in \\mathcal {T}_d^0$ induces an isometry on $\\mathcal {T}_d$ by left translation.", "We denote by $v\\mathcal {T}_d$ the image of $\\mathcal {T}_d$ under $v$ and call it the subtree rooted at $v$: Notation 2.1 $v\\mathcal {T}_d:= \\left\\lbrace x \\in \\mathcal {T}_d\\ | \\ v \\in \\left[ \\varnothing ,x\\right] \\right\\rbrace $ where $\\left[ a,b\\right] $ is the segment between $a$ and $b$ .", "Let us also introduce, for all non-negative integer $n$ , the $n$ -th level $L_n$ which consists of all vertices whose distance to the root is $n$ : Notation 2.2 $L_n := \\left\\lbrace v\\in \\mathcal {T}_d^0 \\ | \\ \\textrm {dist}\\left(\\varnothing ,v \\right)=n \\right\\rbrace $ We refer to Fig.", "REF for a less rigourous but more visual presentation of these definitions.", "Figure: dd-regular rooted tree 𝒯 d \\mathcal {T}_dThe boundary $\\partial \\mathcal {T}_d$ of $\\mathcal {T}_d$ is the set of infinite geodesic paths starting from the root $\\varnothing $ .", "This set is a Cantor set; if we denote by $\\partial \\left(v\\mathcal {T}_d\\right)$ the subset of geodesics going through the vertex $v\\in \\mathcal {T}_d^0$ , then $F :=\\left\\lbrace \\partial \\left(v\\mathcal {T}_d\\right) \\ | \\ v\\in \\mathcal {T}_d^0\\right\\rbrace $ is a fundamental system of open sets in $\\partial \\mathcal {T}_d$ .", "Let us denote by $\\mu _n$ the uniform probability measure on the $n$ -th level $L_n$ .", "The limit $\\mu $ of the these measures is a probability measure on $\\partial \\mathcal {T}_d$ characterized by $\\mu \\left( \\partial \\left(v\\mathcal {T}_d\\right)\\right)= \\frac{1}{d^{\\textrm {L}\\left(v\\right)}}$ where $\\textrm {L}\\left(v\\right) $ is the level of the vertex $v$ , i.e.", "$\\textrm {L}\\left(v\\right)=\\textrm {dist}\\left( v,\\varnothing \\right)$ .", "Now, let $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ be the group of the automorphisms of $\\mathcal {T}_d$ .", "Each element $g$ in $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ fixes the root and for all $n$ , $g$ preserves the level $L_n$ , as well as the finite set $\\left\\lbrace v\\mathcal {T}_d\\ | \\ v \\in L_n\\right\\rbrace $ of subtrees rooted at its vertices.", "In particular, $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ acts faithfully on $\\partial \\mathcal {T}_d$ by measure preserving homeomorphisms.", "The self-similar structure of $\\mathcal {T}_d$ implies that the group $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ admits a natural decomposition in terms of the automorphisms group of subtrees.", "More precisely, $g$ induces a permutation $g_1$ on the set $L_1$ , as well as an isomorphism $\\varphi _{g(v)}(g)$ from $v\\mathcal {T}_d$ onto $g(v)\\mathcal {T}_d$ , for every vertex $v$ in $L_1$ .", "These two subtrees are canonically isomorphic to $\\mathcal {T}_d$ ; therefore, $\\varphi _{g(v)}(g)$ can be seen as an element of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ .", "It is easy to see that this data completely determines the action of $g$ on $\\mathcal {T}_d$ .", "In fact, we have the following decomposition: $\\begin{array}{ccc}\\text{Aut}\\left(\\mathcal {T}_d\\right)& \\overset{\\Phi }{\\longrightarrow } & \\left( \\text{Aut}\\left(\\mathcal {T}_d\\right)\\times \\dots \\times \\text{Aut}\\left(\\mathcal {T}_d\\right)\\right) \\rtimes \\mathfrak {S}_{d} \\\\g & \\longrightarrow & ( \\varphi _{1}(g), \\dots , \\varphi _{d}(g)) .", "g_1\\end{array}$ where $\\mathfrak {S}_{d}$ denotes the symetric group on the set of $d$ elements $L_1$ .", "Its action on $\\left( \\text{Aut}\\left(\\mathcal {T}_d\\right) \\times \\dots \\times \\text{Aut}\\left(\\mathcal {T}_d\\right) \\right) $ is the permutation of the coordinates.", "Definition 2.3 The isomorphism $\\Phi $ is called the recursion isomorphism.", "Generalizing further, we denote by $\\Phi ^{(n)}$ the decomposition of $\\text{Aut}\\left(\\mathcal {T}_d\\right) $ with respect to the level $L_n$ : $\\begin{array}{ccc}\\text{Aut}\\left(\\mathcal {T}_d\\right) & \\overset{\\Phi ^{(n)}}{\\longrightarrow } & \\left(\\prod _{w \\in L_n} \\text{Aut}\\left(\\mathcal {T}_d\\right)\\right) \\rtimes \\text{Aut}\\left(\\rm T_{d,n}\\right) \\\\g & \\longrightarrow & \\left( \\varphi _w \\right)_{w\\in L_n} .\\ g_n\\end{array}$ where $\\rm T_{d,n} $ is the restriction of $\\mathcal {T}_d$ to its $n$ first levels, and $\\text{Aut}(\\rm T_{d,n} )$ is the restriction of $\\text{Aut}\\left(\\mathcal {T}_d\\right) $ to this stable subgraph $\\rm T_{d,n} $ .", "We remark that an element $g\\in \\text{Aut}\\left(\\mathcal {T}_d\\right) $ fixes the restriction $\\rm T_{d,n} $ if and only if $g_n$ equals 1, which is also equivalent to $g$ fixing the $n$ -th level $L_n$ .", "Following [8], we can write the relation between the actions of $\\mathcal {T}_d^0$ and $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ by: $\\forall g\\in \\text{Aut}\\left(\\mathcal {T}_d\\right), \\ \\forall w \\in \\mathcal {T}_d^0, \\ g \\circ w = g(w)\\circ \\varphi _{g(w)}(g).$ We can now define the class of automata groups (note that in this article, we suppose every automaton to be finite): Definition 2.4 A subgroup $G$ of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ is said to be generated by a automaton (on $d$ letters) if $G$ admits a finite generating set $S$ which fulfills $\\forall g\\in S, \\ \\forall v\\in L_1, \\ \\varphi _v \\left( g\\right) \\in S.$ Let us finish this preliminary section with a definition and some general notation: Definition 2.5 A subgroup $G$ of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ will be said to act spherically transitively on $\\mathcal {T}_d$ if and only if its action on each level is transitive.", "Notation 2.6 Let $G$ be a group acting on a set $X$ .", "Let $g\\in G$ and $A\\subset X$ .", "Then, $\\textrm {Stab}_G \\left(A\\right) :=\\left\\lbrace g\\in G \\ | \\ \\forall a \\in A, \\ g\\left(a\\right)=a \\right\\rbrace ,$ $\\text{Fix}_X \\left(g \\right) := \\left\\lbrace x\\in X \\ | \\ g\\left(x\\right)=x\\right\\rbrace .$" ], [ "The fundamental $C^*$ -algebras associated to an automaton group", "Let $d\\ge 2$ be an integer and $G$ a subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ ." ], [ "Definitions", "Let $\\rho $ be the fundamental representation of $G$ on $\\ell ^2 \\left( \\mathcal {T}_d^0 \\right)$ induced from its action on $\\mathcal {T}_d$ : $\\begin{array}{ccc}G & \\overset{\\rho }{\\longrightarrow } & \\mathcal {U} \\left( \\ell ^2 \\left( \\mathcal {T}_d^0 \\right)\\right)\\\\g & \\longrightarrow & \\xi \\overset{\\rho (g)}{\\rightarrow } \\xi (g^{-1}.", ")\\end{array}$ where $\\mathcal {U} \\left( \\ell ^2 \\left( \\mathcal {T}_d^0 \\right)\\right)$ is the group of unitary operators of the Hilbert space $ \\ell ^2 \\left( \\mathcal {T}_d^0 \\right)$ .", "The map $\\rho $ can be extended by linearity to a $\\ast $ -homomorphism from $G̏$ into the bounded operators $\\mathcal {B} \\left( \\ell ^2 \\left( \\mathcal {T}_d^0 \\right)\\right)$ .", "Definition 3.1 The fundamental $C^*$ -algebra of $G$ is $C^{*}_{\\rho }\\left( G \\right):= \\overline{\\rho \\left( \\mathbb {C}G \\right) }.$" ], [ "Approximation of norm and spectrum", "Let $\\ell ^2 \\left( L_n \\right)\\simeq {d^n}$ be the subspace of $\\ell ^2 \\left( \\mathcal {T}_d^0 \\right)$ of functions whose support is included in the $n$ -th level $L_n$ .", "We denote by $p_n$ the orthonormal projection onto $\\ell ^2 \\left( L_n \\right)$ , i.e.", "$p_n: \\ell ^2 \\left( \\mathcal {T}_d^0 \\right)\\twoheadrightarrow \\ell ^2 \\left( L_n \\right), \\ p_n=p_{n}^*=p_{n}^2.$ For all $n\\in \\mathbb {N}$ and $x\\in C^{*}_{\\rho }\\left( G \\right)$ , let $\\mathcal {R}_n \\left( G \\right) := p_n C^{*}_{\\rho }\\left( G \\right)p_n , \\ x_n := p_n x p_n.$ The group $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ preserves the level $L_n$ , and thus all its subgroups admit a representation $\\rho _n$ on $\\ell ^2 \\left( L_n \\right)$ .", "Clearly, $\\rho _n$ is contained in $\\rho $ and $\\begin{array}{ccc}C^{*}_{\\rho }\\left( G \\right)& \\overset{\\rho _n}{\\longrightarrow } & \\mathcal {R}_n \\left( G \\right) \\\\x & \\longrightarrow & x_n\\end{array}$ is a surjective $\\ast $ -homomorphism.", "Moreover, it is easy to see that $\\rho _n$ is contained in $\\rho _{n+1}$ .", "Indeed, for all $v\\in L_n$ , let $C(v)$ be the set of children of $v$ , i.e.", "the set of vertices in $L_{n+1}$ whose distance to $v$ is 1.", "The group $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ preserves the partition $L_{n+1}=\\bigsqcup _{v\\in L_n} C(v)$ and thus subspace $\\mathrm {Sh}\\left(n,n+1 \\right) := \\left\\lbrace f\\in \\ell ^2 \\left( L_{n+1} \\right) \\ | \\ f \\textrm { depends only on }C(v), \\ v\\in L_n \\right\\rbrace .$ Then, denoting by $\\mathbf {1}_{A} $ the characteristic function of a subset $A$ , the map $\\begin{array}{ccc}\\ell ^2(L_{n}) & \\overset{Q_n}{\\longrightarrow } & \\mathrm {Sh}\\left(n,n+1 \\right)\\\\\\mathbf {1} _{ \\lbrace z \\rbrace } & \\longrightarrow & \\frac{1}{\\sqrt{d}} \\mathbf {1} _{C(z)}\\end{array}$ is a $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ -equivariant unitary, i.e.", "it intertwines the representations $\\rho _n$ and $\\rho _{n+1}\\mid _{\\mathrm {Sh}\\left(n,n+1 \\right)}$ .", "Therefore, we have surjective $\\ast $ -homorphisms $q_n : \\mathcal {R} _n \\left( G \\right) \\twoheadrightarrow \\mathcal {R} _{n-1} \\left( G \\right), \\ n\\in \\mathbb {N}$ which make the following diagram commute : ${& & &C^{*}_{\\rho }\\left( G \\right)@{->>}[dlll]_{\\rho _0}@{->>}[dll]^{\\rho _1}@{->>}[d]_{\\rho _n}@{-->>}[dr]\\\\\\mathcal {R} _0 \\left( G \\right) & \\mathcal {R} _1 \\left(G \\right) @{->>}[l]^{q_1} & \\dots @{->>}[l] & \\mathcal {R} _n \\left(G\\right) @{->>}[l]^{q_n} & \\dots @{->>}[l] }$ The next proposition is a direct consequence of this diagram and the fact that $\\left( \\sum _{k=0}^{n}p_n \\right)_{n\\ge 0}$ is a sequence of projections converging strongly to the identity $Id$ in $ \\mathcal {B}\\left( \\ell ^2 \\left( \\mathcal {T}_d\\right) \\right)$ .", "Property 3.2 For all $x\\in C^{*}_{\\rho }\\left( G \\right)$ , $\\Vert x\\Vert =\\sup _{n \\in \\mathbb {N}} \\Vert x_n\\Vert =\\lim _{n \\rightarrow +\\infty } \\Vert x_n\\Vert .$ In particular, $C^{*}_{\\rho }\\left( G \\right)$ is a residually finite dimensional $C^*$ -algebra.", "Remark 3.3 If for all $n \\ge 1$ we remove $\\rho _{n-1} $ to $\\rho _n$ , we get a representation $\\tilde{\\rho }$ weakly isomorphic to $\\rho $ .", "It is easy to see that $\\tilde{\\rho }$ is isomorphic to the unitary representation $\\rho _{mes}$ that $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ admits on $L^2 \\left( \\partial \\mathcal {T}_d\\right)$ (see [10] §3.7).", "In particular, for every countable subgroup $G$ of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ , the $C^*$ -algebra $C^{*}_{\\rho }\\left( G \\right)$ is isomorphic to $C^*_{\\rho _{mes}} \\left( G \\right) := \\overline{\\rho _{mes} \\left( \\mathbb {C}G \\right) }$ .", "Now, we consider: $\\mathcal {S}\\left(G \\right) :=\\left\\lbrace \\left(x_0,x_1,\\dots \\right) \\ | \\ x_n \\in \\mathcal {R} _n \\left( G \\right),\\ q_{n+1}\\left(x_{n+1}\\right)=x_n \\textrm { and } \\sup _n \\Vert x_n\\Vert <\\infty \\right\\rbrace $ Each $x=\\left(x_0,\\dots ,x_n,\\dots \\right)$ in $\\mathcal {S} \\left( G \\right)$ acts on $\\ell ^2 \\left( \\mathcal {T}_d^0 \\right)$ via the operator $\\tilde{x}=\\sum _{n=0}^{+\\infty } p_n x_n p_n,$ where the convergence is understood for the strong topology.", "Here again, $\\Vert \\tilde{x}\\Vert =\\lim _{n \\rightarrow +\\infty } \\Vert x_n\\Vert .$ Then, under this identification $\\mathcal {S} \\left( G \\right) \\subset \\mathcal {B} \\left(\\ell ^2 \\left( \\mathcal {T}_d^0 \\right)\\right)$ , we have Proposition 3.4 Let $G$ be a subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ .", "Then, the von Neumann algebra generated by $C^{*}_{\\rho }\\left( G \\right)$ in $\\mathcal {B}\\left( \\ell ^2 \\left( \\mathcal {T}_d\\right) \\right)$ is $\\mathcal {S}\\left( G \\right)$ .", "$\\mathcal {S} (G)$ is a unital, involutive subalgebra of $\\mathcal {B} (\\ell ^2 \\left( \\mathcal {T}_d^0 \\right))$ .", "Let us first show that it is weakly closed.", "Let $\\left( x(k) \\right)_{k\\ge 0}$ be a sequence of elements in $\\mathcal {S} (G)$ that converges weakly to $x \\in \\mathcal {B} (\\ell ^2 \\left( \\mathcal {T}_d^0 \\right))$ : $\\forall \\xi , \\eta \\in \\ell ^2 \\left( \\mathcal {T}_d^0 \\right), \\ \\left< x(k)(\\xi ) \\vert \\eta \\right> \\underset{k \\rightarrow \\infty }{\\longrightarrow } \\left< x(\\xi ) \\vert \\eta \\right> .$ For all $n\\ne p$ and $\\xi _n \\in \\ell ^2 (L_n), \\ \\xi _p \\in \\ell ^2 (L_p)$ , we have $\\left< x(\\xi _n) \\vert \\xi _p \\right> =0$ since $x(k)$ preserves $\\ell ^2 (L_n)$ .", "Thus, $x$ leaves these subspaces invariant; we denote by $x_i$ the operator that $x$ induces on $\\ell ^2 (L_i)$ .", "For all $i\\in \\mathbb {N}$ and $\\xi , \\eta $ in $\\ell ^2 (L_i) $ , one has $\\left< x_i(\\xi ) \\vert \\eta \\right> = \\left< x(\\xi ) \\vert \\eta \\right> = \\lim _{k \\rightarrow \\infty }\\left< x(k)(\\xi ) \\vert \\eta \\right> = \\lim _{k \\rightarrow \\infty }\\left< x(k)_i(\\xi ) \\vert \\eta \\right>$ which shows that $x(k)_i$ converges weakly to $x_i$ .", "Now, since the algebras $\\mathcal {R} _i (G)$ have finite dimension, this convergence holds for the norm topology: letting $k\\rightarrow +\\infty $ , one sees that $x_i $ belongs to $ \\mathcal {R} _i (G)$ and that $Q_i (x_i)$ equals $x_{i-1}$ .", "Therefore, $x=\\left( x_0,\\dots ,x_i,\\dots \\right) \\in \\mathcal {S} (G)$ .", "Let us show that $C^{*}_{\\rho }\\left( G \\right)$ is weakly dense in $\\mathcal {S} (G)$ .", "Let $x=\\left( x_0,\\dots ,x_i,\\dots \\right) $ in $ \\mathcal {S} (G) $ .", "By construction, for each $n\\in \\mathbb {N}$ , there exists $x(n)\\in C^{*}_{\\rho }\\left( G \\right)$ such that for all $ i\\le n, \\ x_i = x(n)_i$ .", "This last equality is equivalent to $x_n$ equals $x(n)_n $ .", "It is a standard fact that if $f$ is a $\\ast $ -homomorphism between two $C^*$ -algebras, then any element in the image of $f$ admits a preimage of same norm.", "Thus, applying the last remark to $x(n) \\mapsto x(n)_n$ , one sees that it is possible to choose the $x(n)$ ’s uniformly bounded ($\\Vert x(n)\\Vert =\\Vert x_n\\Vert \\le \\Vert x\\Vert $ ).", "Then, $\\forall \\xi ,\\eta \\in \\ell ^2 \\left( \\mathcal {T}_d^0 \\right),$ $\\vert \\left< (x-x(n))\\xi \\vert \\eta \\right>\\vert &=& \\vert \\sum _{i>n}\\left< (x_i-x(n)_i) p_i (\\xi ) \\vert p_i (\\eta ) \\right> \\vert \\\\& \\le & \\underbrace{\\left( \\Vert x\\Vert +\\sup _{n} \\Vert x(n)\\Vert \\right) \\Vert (1-p_n)\\xi \\Vert \\Vert (1-p_n)\\eta \\Vert }_{\\underset{n \\rightarrow \\infty }{\\longrightarrow }0}$ which finishes the proof.", "Concerning the spectrum of elements in $\\mathcal {S} \\left( G \\right)$ , one has the following result.", "Proposition 3.5 If $x\\in \\mathcal {S} \\left( G \\right)$ is normal, then $\\textrm {sp}(x)=\\overline{\\bigcup _{n=0}^{+\\infty }\\textrm {sp}(x_n)}.$ Suppose $G$ is finitely generated.", "Then, the previous equality holds for each element $x\\in \\mathcal {S} \\left( G \\right)$ if and only if $G$ is virtually abelian.", "Suppose that $\\textrm {sp}(x) \\setminus \\overline{\\bigcup _{n=0}^{+\\infty }\\textrm {sp}(x_n)} \\ne \\emptyset .$ This non-empty set is open in $\\textrm {sp}(x)$ : one can choose a continous function $f$ defined on $\\textrm {sp}(x)$ such that - $f\\ne 0$ , - $f(z)=0, \\ \\forall z \\in \\overline{\\bigcup _{n=0}^{+\\infty }\\textrm {sp}(x_n)}$ .", "The first property implies that $f(x)$ is non-zero.", "The second implies that for all $n \\in \\mathbb {N}$ , $0=f(x_n)=f(x)_n$ , the last equality coming from the fact that the continous functional calculus commutes with continous $\\ast $ -homomorphisms.", "Since $\\Vert f(x)\\Vert =\\lim _{n \\rightarrow +\\infty } \\Vert f(x)_n\\Vert $ , we get a contradiction.", "First, let us make some remarks on $\\mathcal {S} \\left( G \\right)$ .", "The algebras $\\mathcal {R} _n\\left( G \\right)$ have finite dimension.", "Thus, each of them is a sum of matrix algebras: $\\forall n \\in \\mathbb {N}, \\ \\exists d_{n,1},d_{n,2},\\dots ,d_{n,j_n} , \\ \\mathcal {R} _n\\left( G \\right) = \\bigoplus _{i=1}^{j_n} M_{d_{n,i}}\\left( )\\right.$ where the $d_{n,i}$ are non-negative integers corresponding to the dimension of the irreducible representations contained in $\\rho _n$ .", "Then, the homomorphism $q_n$ correspond to a projection onto certain factors of this decomposition.", "Therefore, the von Neumann algebra $\\mathcal {S} \\left( G \\right)$ is a $\\ell ^{\\infty }$ sum of matrix algebras: $\\mathcal {S} \\left( G \\right) &= \\int ^{\\oplus }_{\\mathbb {N}} M_{d_k} \\left( )\\\\&:= \\left\\lbrace \\left( a_0,a_1,\\dots ,a_k,\\dots \\right) \\ | \\ \\exists M, \\ \\forall k, \\ a_k \\in M_{d_k} \\left( ) \\textrm { and } \\Vert a_k\\Vert \\le M\\right\\rbrace .", "\\nonumber \\right.Moreover, even if \\right.a_k \\ne x_n in general (except a_0=x_0), one has\\bigcup _{k=0}^{+\\infty }\\textrm {sp}(a_k)=\\bigcup _{n=0}^{+\\infty }\\textrm {sp}(x_n).Since the representation \\rho is faithful, Theorem \\ref {theoIJF} in Appendix \\ref {apen} implies that the integers d_k are uniformly bounded if and only if G is virtually abelian.$ We are now ready to prove REF .REF .", "First, suppose that $G$ is virtually abelian and let $d=\\max _{k\\in \\mathbb {N}} d_k$ .", "Let $x\\in \\mathcal {S} \\left(G\\right)$ .", "It is obvious that $\\overline{\\bigcup _{n=0}^{+\\infty }\\textrm {sp}(x_n)}$ is contained in $\\textrm {sp}\\left( x \\right)$ so let us prove the other inclusion.", "We write the decomposition of $x$ given by REF by $x=\\left(a_0,a_1,\\dots \\right), \\ a_k \\in M_{d_k} \\left( ).\\right.$ Let $\\lambda \\notin \\overline{\\bigcup _{k=0}^{+\\infty }\\textrm {sp}(a_k)}$ and set $r:= \\textrm {dist}\\left( \\lambda , \\overline{\\bigcup _{k=0}^{+\\infty }\\textrm {sp}(a_k)} \\right) >0$ .", "For all $k \\in \\mathbb {N}$ , $a_k-\\lambda $ is invertible; we want to prove that $\\left( \\left(a_0-\\lambda \\right)^{-1}, \\left( a_1-\\lambda \\right)^{-1},\\dots \\right) \\in \\mathcal {S} \\left( G \\right)$ which is equivalent to the $\\left( a_k-\\lambda \\right)^{-1} $ ’s to be uniformly bounded.", "For each $k\\in \\mathbb {N}$ and $M\\in M_{d_k} \\left( ) \\right.$ , let $\\Vert M\\Vert _{_{\\infty }}=\\sup _{1 \\le i,j \\le d_k} \\left|M_{i,j}\\right|.$ Then, $\\Vert M\\Vert \\le \\sqrt{d_k}\\times \\Vert M\\Vert _{_{\\infty }}\\le \\sqrt{d_k}\\times \\Vert M\\Vert $ .", "Denoting by $\\textrm {Comat}\\left(M\\right)$ the cofactor matrix of a square matrix $M$ , one has: $\\Vert \\left( a_k-\\lambda \\right)^{-1}\\Vert &= \\left| \\det \\left( a_k-\\lambda \\right) \\right|^{-1} \\times \\Vert \\textrm {Comat}\\left( a_k-\\lambda \\right)^t\\Vert \\\\&\\le r^{-d}\\times \\sqrt{d} \\times \\Vert \\textrm {Comat}\\left( a_k-\\lambda \\right)\\Vert _{_{\\infty }} \\\\&\\le r^{-d}\\times \\sqrt{d} \\times \\left(d-1\\right)!\\times \\Vert a_k-\\lambda \\Vert _{_{\\infty }}^{d-1} \\\\&\\le r^{-d}\\times \\sqrt{d} \\times \\left(d-1\\right)!\\times \\left(\\Vert x\\Vert +\\left|\\lambda \\right| \\right)^{d-1}.$ Let us now prove the inverse implication: suppose $G$ is not virtually abelian so that $\\sup _{k\\in \\mathbb {N}} d_k = +\\infty $ .", "For all $k \\in \\mathbb {N}$ , set $a_k = \\left( \\begin{matrix}1 & -1 & 0 & \\cdots \\\\0 & 1 & -1 & \\\\0 & 0 & 1 & \\ddots \\\\\\vdots & \\vdots & &\\ddots \\end{matrix}\\right) \\in M_{d_k} \\left( ) .\\right.$ It is easy to see that $a_k^{-1} = \\left( \\begin{matrix}1 & 1 & 1 & \\cdots \\\\0 & 1 & 1 & \\cdots \\\\0 & 0 & 1 & \\cdots \\\\\\vdots & \\vdots & &\\ddots \\end{matrix}\\right)$ so that $\\Vert a_k\\Vert \\le 2$ , $\\Vert a_k^{-1}\\Vert \\ge \\sqrt{d_k}$ .", "The first property implies that there exists an element $x\\in \\mathcal {S} \\left( G \\right)$ whose decomposition is given by $x=\\left(a_0,a_1,\\dots \\right) $ and the second implies that $x$ is not invertible.", "However, $0\\notin \\overline{\\bigcup _{n=0}^{+\\infty }\\textrm {sp}(x_n)} = \\left\\lbrace 1 \\right\\rbrace $ .", "Remark 3.6 It is clear that REF .REF also holds for $x\\in C^{*}_{\\rho }\\left( G \\right)$ (if $x\\in \\mathcal {A} \\subset \\mathcal {B}$ , then $\\textrm {sp}_{\\mathcal {A}}\\left(x\\right) =\\textrm {sp}_{\\mathcal {B}}\\left(x\\right)$ for an inclusion of unital $C^*$ -algebras $\\mathcal {A} \\subset \\mathcal {B}$ ).", "However, the author does not know whether REF .REF holds for $C^{*}_{\\rho }\\left( G \\right)$ ." ], [ "Extension of the recursion morphism", "Recall that $\\mathcal {T}_d$ is the Cayley graph of the semi-group generated by $X=\\left\\lbrace 1,2,\\dots ,d \\right\\rbrace $ .", "In this section, we assume that $G$ is a self-similar subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ , i.e.", "$\\forall g \\in G, \\ \\forall v\\in \\mathcal {T}_d^0, \\ \\varphi _v \\left( g\\right) \\in G.$ For each $g$ in $G$ , we consider the following matrix $\\Phi (g)$ , of size $d \\times d$ and whose entries belong to the group $G$ : $\\Phi (g)=\\left(\\begin{matrix}\\varphi _1 (g)\\\\&\\varphi _2 (g)\\\\& & \\ddots \\\\& & &\\varphi _{d} (g)\\end{matrix} \\right) \\times M_{g_1}$ where $M_{g_1}$ is the matrix of the permutation $g_1 \\in \\mathfrak {S}_d$ that $g$ induces on $X$ .", "Now, let us consider the Hilbert subspace $(1-p_0)\\ell ^2 \\left( \\mathcal {T}_d^0 \\right)$ of the functions in $\\ell ^2 \\left( \\mathcal {T}_d^0 \\right)$ null at the root $\\varnothing $ of $\\mathcal {T}_d$ .", "This space is the sum indexed by $v\\in L_1 = X $ of the subspaces $\\ell ^2 \\left( v\\mathcal {T}_d^0 \\right)$ : $(1-p_0)\\ell ^2 \\left( \\mathcal {T}_d^0 \\right)=\\bigoplus _{v \\in L_1} \\ell ^2 \\left( v\\mathcal {T}_d^0 \\right)=\\bigoplus _{v=1}^{d} \\ell ^2 \\left( v\\mathcal {T}_d^0 \\right)$ where $\\ell ^2 \\left( v\\mathcal {T}_d^0 \\right)$ is the space of functions supported on the subtree $v\\mathcal {T}_d$ rooted at the vertex $v$ of the first level $L_1$ .", "First, the group $G$ preserves $(1-p_0)\\ell ^2 \\left( \\mathcal {T}_d^0 \\right)$ and therefore, $\\rho $ restricts to it.", "Moreover, $\\ell ^2 \\left( v\\mathcal {T}_d^0 \\right)$ is isometric to $\\ell ^2 \\left( \\mathcal {T}_d^0 \\right)$ via the isometric shift $\\mathcal {T}_d\\underset{v^{-1}}{\\overset{v}{\\rightleftarrows }} v\\mathcal {T}_d$ .", "Hence, there is a homorphism $C^{*}_{\\rho }\\left( G \\right)\\longrightarrow \\mathcal {B} \\left( \\bigoplus _{v=1}^{d} \\ell ^2 \\left( \\mathcal {T}_d^0 \\right)\\right)=M_{d}\\left( \\mathcal {B}\\left(\\ell ^2 \\left( \\mathcal {T}_d^0 \\right)\\right) \\right).$ Since $\\mathcal {R} _1 \\left( G \\right) \\twoheadrightarrow \\mathcal {R} _0 \\left( G \\right)$ (see diagram REF ), this homomorphism is injective.", "It is also easy to see that it coincides with $\\Phi $ on $G\\subset C^{*}_{\\rho }\\left( G \\right)$ .", "Summarizing everything, one has: Property 3.7 The recursion homomorphism $\\Phi $ gives rise to an isometric embedding $C^{*}_{\\rho }\\left( G \\right)\\overset{\\Phi }{\\longrightarrow } M_{d}\\left( C^{*}_{\\rho }\\left( G \\right)\\right)$ defined on $G\\subset C^{*}_{\\rho }\\left( G \\right)$ by REF ." ], [ "The trace $Tr$", "For all $n\\in \\mathbb {N}$ , let $tr_n$ be the usual normalized trace on the finite dimensional $C^*$ -algebra $\\mathcal {B} \\left( \\ell ^2 \\left( L_n\\right)\\right)=M_{d^n}\\left()\\right.$ .", "That is, $tr_n$ is defined for all $M\\in M_{d^n}\\left()\\right.$ by: $tr_n \\left( M \\right) := \\frac{1}{d^n} \\sum _{i=1}^{d^n} M_{i,i}.$ We remark that for an element $g$ in $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ , $tr_n \\left( g_n \\right)$ measures the fixed point set of $g_n$ acting on $L_n$ : $tr_n \\left( g_n \\right) = \\frac{1}{d^n} \\left| \\textrm {Fix}_{L_n}\\left(g_n\\right) \\right| = \\mu _n \\left(\\textrm {Fix}_{L_n}\\left(g_n\\right)\\right)$ where $\\mu _n$ is the uniform probability measure on the finite set $L_n$ .", "It is easy to see that $tr_n (g_n) = \\frac{1}{d}\\left( \\sum _{i \\in \\textrm {Fix}_{ L_1}(g)} tr_{n-1}(\\varphi _i (g)_{n-1})\\right).$ In particular, for all $g\\in \\text{Aut}\\left(\\mathcal {T}_d\\right)$ , the sequence $\\left(tr_n\\left(g_n\\right)\\right)_{n}$ is a decreasing sequence of reals in $\\left[ 0,1\\right]$ , hence its limit exists.", "More precisely, $\\lim _{n \\rightarrow +\\infty } tr_n \\left( g_n \\right)= \\mu \\left(\\textrm {Fix}_{\\partial \\mathcal {T}_d}\\left( g\\right) \\right).$ Definition 3.8 For all $g\\in \\text{Aut}\\left(\\mathcal {T}_d\\right)$ , we define $Tr\\left(g\\right)=\\lim _{n \\rightarrow +\\infty } tr_n \\left( g_n \\right)= \\mu \\left(\\textrm {Fix}_{\\partial \\mathcal {T}_d}\\left( g\\right) \\right)$ .", "Property 3.9 Let $G$ be a subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ .", "Then, the map $Tr$ extends to a continous and normalized trace on $C^{*}_{\\rho }\\left( G \\right)$ .", "If moreover $G$ is self-similar, then for all $x\\in C^{*}_{\\rho }\\left( G \\right)$ and $n\\in \\mathbb {N}$ , one has $Tr\\left( x\\right)=\\left(tr_n\\otimes Tr\\right)(\\Phi ^{(n)} \\left(x \\right)).$ First extend $Tr$ by linearity to $\\rho \\left( G̏ \\right)$ .", "Since the linear forms $tr_n$ defined on $\\mathcal {B} \\left(\\ell ^2\\left( L_n\\right) \\right)$ all have norm equal to 1, Property REF implies that $\\sup _{x\\in \\rho \\left( G̏\\right), \\Vert x\\Vert =1} Tr\\left( x \\right) \\le 1.$ Thus, the linear form $Tr$ can be extended by continuity to the Banach space $C^{*}_{\\rho }\\left( G \\right)$ in which $\\rho \\left( G̏\\right)$ is dense.", "It is obvious that $Tr$ is normalized and REF is obtained by letting $n$ go to $+\\infty $ in REF ." ], [ "Automata groups", "The trace we just defined is linked to the the fixed points set of elements in $G$ .", "If the latter is generated by an automaton, one can describe this set more precisely.", "Recall first the following lemma proven in [6]: Lemma 3.10 (Grigorchuk-Żuk) Let $G$ be a subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ generated by an automaton.", "Let $g\\in G$ be such that its fixed points set $\\textrm {Fix}_{\\partial \\mathcal {T}_d}\\left(g\\right)$ has an empty interior.", "Then, $\\textrm {Fix}_{\\partial \\mathcal {T}_d}\\left(g\\right)$ has measure 0 (equivalently $Tr\\left(g\\right)=0$ ).", "Here is a generalisation of this result: Lemma 3.11 Let $G$ be a subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ generated by an automaton and $g\\in G$ .", "Then, there exists a partition $\\partial \\mathcal {T}_d= F\\sqcup T \\sqcup N$ of the boundary $\\partial \\mathcal {T}_d$ such that: $F$ is open (i.e.", "is a union of boundary of subtrees) and for $\\mu $ -almost all $f\\in F$ , $g(f)\\ne f$ .", "$T$ is open and for all $t\\in T$ , $g(t)=t$ .", "$N$ has measure 0.", "Let us first define $F$ and $T$ .", "We set ${T}:=\\left\\lbrace t\\in \\mathcal {T}_d^0 \\ | \\ t\\mathcal {T}_d\\subseteq \\textrm {Fix}(g) \\right\\rbrace \\textrm { and } {F}:=\\left\\lbrace f \\in \\mathcal {T}_d^0 \\ | \\ \\forall z \\in f\\mathcal {T}_d, \\ z\\mathcal {T}_d\\nsubseteq \\textrm {Fix}(g) \\right\\rbrace $ These two sets are union of subtrees.", "Let us define $\\mathfrak {T}$ and $\\mathfrak {F}$ by the following property: ${T}=\\bigsqcup _{t\\in \\mathfrak {T}} t\\mathcal {T}_d& \\textrm { and } T=\\bigsqcup _{t\\in \\mathfrak {T}} \\partial \\left(t\\mathcal {T}_d\\right) \\nonumber \\\\{F}=\\bigsqcup _{f\\in \\mathfrak {F}} f\\mathcal {T}_d& \\textrm { and } F=\\bigsqcup _{f\\in \\mathfrak {F}} \\partial \\left(f\\mathcal {T}_d\\right) \\nonumber $ It is clear that $F$ and $T$ are open, that their intersection is empty and that for all $t\\in T$ , $g(t)=t$ .", "Let us prove that the action of $g$ on $F$ is almost free: $\\mu \\left(\\textrm {Fix}_{\\partial \\mathcal {T}_d} (g) \\cap F\\right) &= \\mu \\left(\\textrm {Fix}_{\\partial \\mathcal {T}_d} (g) \\cap \\bigsqcup _{f \\in \\mathfrak {F}} \\partial \\left( f\\mathcal {T}_d\\right) \\right) \\nonumber \\\\&= \\sum _{f\\in \\mathfrak {F} \\cap \\textrm {Fix}(g)} \\mu \\left(\\textrm {Fix}_{\\partial \\mathcal {T}_d} (g) \\cap \\partial \\left( f\\mathcal {T}_d\\right) \\right)\\nonumber \\\\&=\\sum _{f\\in \\mathfrak {F} \\cap \\textrm {Fix}(g)} \\frac{\\mu \\left(\\textrm {Fix}_{\\partial \\mathcal {T}_d} (\\varphi _{f} (g)) \\right)}{d^{\\textrm {L}(f)}} \\nonumber $ where $\\textrm {L}(f)$ still denotes the level of the vertex $f$ .", "Now, each term in this last sum is 0 thanks to REF .", "Indeed, by construction, $g$ fixes no subtree included in $T$ .", "The same holds a fortiori for $\\varphi _{f} (g)$ if $f\\in \\mathfrak {F} \\cap \\textrm {Fix}(g)$ , and this is equivalent to $\\textrm {Fix}_{\\partial \\mathcal {T}_d} (\\varphi _{f} (g)) $ having empty interior.", "We will now prove that $\\mu \\left(T\\right)+\\mu \\left(F\\right)=1$ , which will finish the proof.", "Let us define for all $n\\in \\mathbb {N}$ , $u_n= \\vert L_n \\cap \\left({T} \\cup {F}\\right)\\vert .$ We want to prove that $v_n:=\\frac{u_n}{d^n}\\underset{n \\rightarrow \\infty }{\\longrightarrow }1 $ .", "Fact 1: The sequence $\\left(v_n\\right)_n$ converges to a real $l\\in \\left[0,1\\right]$ .", "Indeed, it is an increasing sequence (${T}$ and $ {F}$ are both union of subtrees) of reals in $\\left[0,1\\right]$ .", "Fact 2: There exists $k\\in \\mathbb {N}$ such that for all $n\\in \\mathbb {N}$ , $v_{kn}\\ge v_{k(n-1)}+\\frac{1}{d^k}\\left(1-v_{k(n-1)}\\right) .", "$ To see this, let us denote by $\\vert .\\vert $ the word length on $G$ relative to the finite generating set which defines $G$ as an automaton group.", "In particular, for all $w\\in \\mathcal {T}_d^0, \\ \\vert \\varphi _w (g)\\vert \\le \\vert g\\vert $ .", "Then, let $k \\in \\mathbb {N}$ be such that for all $ h\\in G$ with $ \\vert h \\vert \\le \\vert g\\vert $ , the following holds: $\\exists w \\in \\mathcal {T}_d^0 \\textrm { s.t. }", "w\\mathcal {T}_d\\subset \\textrm {Fix}(h) \\Longrightarrow \\exists w \\in \\mathcal {T}_d^0 \\textrm { s.t. }", "w\\mathcal {T}_d\\subset \\textrm {Fix}(h) \\textrm { and } \\textrm {L}(w) \\le k .$ Such a positive integer exists because the set $\\left\\lbrace h \\ | \\ \\vert h\\vert \\le \\vert g\\vert \\right\\rbrace $ is finite.", "Let $N\\in \\mathbb {N}$ and $w\\in L_n$ which is not in ${T} \\cup {F} $ .", "This means that $g(w)=w, \\ \\varphi _w (g) \\ne 1$ and there exists $w^{\\prime } \\in w\\mathcal {T}_d$ such that $w^{\\prime }\\mathcal {T}_d\\subset \\textrm {Fix}(g)$ (i.e.", "$w^{\\prime } \\in {T}$ ).", "According to REF (applied to $\\varphi _w (g)$ ), $w^{\\prime }$ even exists with the additional property of having a level smaller than $N+k$ .", "Therefore, one gets the following inequality: $u_{N+k} \\ge \\underbrace{d^k u_N}_{\\textrm { contribution of } L_N\\cap \\left({T} \\cup {F}\\right)} + \\underbrace{d^N -u_N}_{\\textrm { contribution of } L_N \\setminus \\left({T} \\cup {F}\\right)} .$ If $N:=k(n-1)$ , then $u_{nk} \\ge d^k u_{k(n-1)} + d^{k(n-1)}-u_{k(n-1)}$ which, dividing by $d^{kn}$ , yields REF .", "Now, let $n$ tend to $\\infty $ in REF to obtain $\\frac{1}{d^k}(1-l)\\le 0$ which implies that $l=1$ .", "From this follows the unicity of $Tr$ : Corollary 3.12 If $G$ is a subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ generated by an automaton, then $Tr$ is the unique continous normalized trace $\\tau $ on $C^{*}_{\\rho }\\left( G \\right)$ for which $\\forall x\\in C^{*}_{\\rho }\\left( G \\right), \\ \\forall n\\in \\mathbb {N}, \\ \\tau \\left( x\\right)=\\left(tr_n\\otimes \\tau \\right)(\\Phi ^{(n)} \\left(x \\right)).$ Let $\\tau $ be such a trace.", "It is sufficient to prove that for each $g\\in G$ , $\\tau \\left(g\\right)=Tr\\left(g\\right)$ since $\\tau $ is continous and $\\rho \\left( G̏ \\right)$ is dense in $C^{*}_{\\rho }\\left( G \\right)$ .", "Let $g\\in G$ .", "Let $\\alpha _n$ (respectively $\\beta _n$ ) be the number of non-zero elements (respectively of 1’s) in the diagonal of $\\Phi ^{(n)}(g)$ .", "Lemma REF says that $Tr(g)=\\lim _{n\\rightarrow +\\infty } \\frac{\\alpha _n}{d^n}=\\lim _{n\\rightarrow +\\infty } \\frac{\\beta _n}{d^n}.$ (the first equality is just the definition of $Tr$ ).", "Moreover, the diagonal entries of $\\Phi ^{(n)}(g)$ are either 0 or elements in $G$ .", "In particular, $\\tau $ being continous, there exists $M>0$ such that for each $x$ in the diagonal of $\\Phi ^{(n)}(g)$ , $\\left|\\tau (x)\\right|\\le M$ .", "Therefore, $\\left| \\tau \\left(g\\right) - Tr(g)\\right|&=\\left| \\tau \\left(\\Phi ^{(n)}(g)\\right) - Tr(g)\\right|\\nonumber \\\\&\\le M\\frac{\\alpha _n-\\beta _n}{d^n}+\\left| \\tau (1)\\frac{\\beta _n}{d^n}-Tr(g)\\right| \\nonumber $ Since $\\tau $ is assumed normalized (i.e.", "$\\tau (1)=1$ ), REF implies that the left hand side tends to 0 as $n$ goes to $+\\infty $ ." ], [ "Generalities", "For a discrete group $G$ , we denote by $\\lambda $ the regular representation of $G$ , i.e.", "$\\lambda := G \\longrightarrow \\mathcal {U}\\left(\\ell ^2 \\left(G\\right)\\right).$ In [10], [9], the following general problem is studied: if $G$ is a group acting on a rooted tree $\\mathcal {T}$ , when does the induced representation $\\rho $ on $\\ell ^2 \\left( \\mathcal {T}\\right)$ weakly contain the regular representation $\\lambda $ ?", "The same relation can be investigated for the representations $\\rho ^{\\otimes n}$ coming from the diagonal action of $G$ on the product $\\mathcal {T} \\times \\dots \\times \\mathcal {T}$ of $n$ copies of the rooted tree.", "It turns out the answer is related to the size of the stabilizers of subtrees $\\textrm {Stab}_G \\left(v\\mathcal {T}\\right):= \\left\\lbrace g\\in G \\ | \\ g(x)=x, \\ \\forall x\\in v\\mathcal {T}_d\\right\\rbrace $ .", "More precisely, the following results are proved: Theorem 4.1 ([9]) Let $G$ be a countable group acting faithfully on a rooted tree $\\mathcal {T}$ .", "If for every vertex $v\\in \\mathcal {T}^0$ , the stabilizer $\\textrm {Stab}_G (v\\mathcal {T})$ is trivial, then $\\lambda \\prec \\rho $ .", "If the set $\\bigcup _{v\\in \\mathcal {T}^0} \\textrm {Stab}_G (v\\mathcal {T})$ is finite and has cardinality $n$ , then $\\lambda \\prec \\rho ^{\\otimes n}$ .", "One always has $\\lambda \\prec \\bigoplus _{n=1}^{+\\infty }\\rho ^{\\otimes n} $ .", "It is shown that the inverse implication of Theorem REF .REF is not true, unless one assumes a certain algebraic condition on $G$ : Notation 4.2 A countable group $G$ is said to satisfy (A) if the normalizer $N_G \\left(H \\right)$ of any non-central finite group $H$ has infinite index in $G$ .", "Theorem 4.3 ([9]) Let $G$ be a countable group satisfying (A).", "Suppose that $G$ acts spherically transitively on a rooted tree $\\mathcal {T}$ .", "If there exists a subtree $v\\mathcal {T}$ whose stabilizer $\\textrm {Stab}_G \\left( v\\mathcal {T}\\right)$ in $G$ is not trivial, then the $\\ast $ -homomorphism $\\rho $ defined on $G̏$ is not injective.", "These results obviously apply when $\\mathcal {T}$ is a regular rooted tree $\\mathcal {T}_d$ and $G$ is generated by an automaton.", "The two next paragraphs aim to show that the language of automata groups is relevant for giving partial answers to certain natural questions arising from [9]." ], [ "The inverse implication of Theorem ", "As suggested by Theorem REF , the general strategy to study the inverse implication of Theorem REF .REF is the following: if there exists a subtree $v\\mathcal {T}_d$ whose stabilizer in $G$ is not trivial, and the action of $G$ on $\\mathcal {T}_d$ is spherically transitive, then there are natural elements $M\\in G̏$ in the kernel of $\\rho $ .", "The difficulty is to prove that one of them is not 0.", "This cannot always be achieved, and in fact the sufficient condition in Theorem REF .REF is not necessary in general (see [9]).", "When $G$ is self-similar, the existence of a subtree whose stabilizer in $G$ is not trivial clearly implies that the same holds for a subtree rooted at the first level, $\\exists v \\in \\mathcal {T}_d^0, \\ \\textrm {Stab}_G \\left(v\\mathcal {T}_d\\right) \\ne \\left\\lbrace 1\\right\\rbrace \\Longrightarrow \\exists v \\in L_1, \\ \\textrm {Stab}_G \\left(v\\mathcal {T}_d\\right) \\ne \\left\\lbrace 1\\right\\rbrace .$ For small valences, this remark allows us to study more easily the non-nullity of these elements $M\\in G̏ \\cap \\text{ker} \\rho $ .", "More precisely, we can in these cases replace the algebraic condition (A) in Theorem REF by conditions involving only the action of $G$ on $\\mathcal {T}_d$ .", "Proposition 4.4 Let $G$ be a self-similar subgroup of $\\text{Aut}\\left(\\mathcal {T} _2 \\right)$ .", "Then $\\lambda \\prec \\rho $ if and only if $\\textrm {Stab}_G \\left(v\\mathcal {T} _2 \\right) =\\left\\lbrace 1\\right\\rbrace $ for all subtree $v\\mathcal {T} _2$ .", "The if part is implied by Theorem REF .REF .", "Let us now prove the only if part, assuming that $G$ is not trivial, since this case is trivial.", "Suppose $v\\in L_1=\\left\\lbrace 1,2\\right\\rbrace $ and $g$ is a non-trivial element in $G$ fixing $v\\mathcal {T} _2 $ .", "Since $G$ is self-similar and non-trivial, there exists in $G$ an element $h\\notin \\textrm {Stab}_G \\left(L_1\\right)$ .", "Then, there are non-trivial elements $x$ and $y$ in $G$ such that: $\\Phi \\left(\\rho \\left(g\\right)\\right)= \\left(\\begin{array}{cc} x & 0 \\\\ 0 & 1\\end{array}\\right) \\text{ and } \\Phi \\left(\\rho \\left(hgh^{-1}\\right)\\right)= \\left(\\begin{array}{cc} 1 & 0 \\\\ 0 & y\\end{array}\\right)$ or maybe the other way around.", "In both cases, if we let $M=\\left(1-g\\right)\\left(1-hgh^{-1}\\right),$ one sees that $\\Phi \\left(\\rho \\left(M\\right) \\right)=0$ which implies that $\\rho \\left(M\\right)=0$ since $\\Phi $ is injective on $C^{*}_{\\rho }\\left( G \\right)$ by Property REF .", "Moreover, $M\\ne 0$ .", "Indeed, $M=1+ghgh^{-1}-g-hgh^{-1} $ so that 1 cannot be cancelled since neither $g$ nor $hgh^{-1} $ is trivial.", "Proposition 4.5 Let $G$ be a self-similar subgroup of $ \\text{Aut}\\left(\\mathcal {T} _3\\right)$ acting spherically transitively.", "Suppose there is a vertex $v \\in L_1=\\left\\lbrace 1,2,3 \\right\\rbrace $ such that the group $\\textrm {Stab}_G\\left(v \\mathcal {T} _3\\right)\\cap \\textrm {Stab}_G\\left(L_1\\right)\\ne \\left\\lbrace 1\\right\\rbrace $ .", "Then, the $\\ast $ -homomorphism $\\rho $ defined on $G̏$ is not injective.", "We will need the following lemma.", "Lemma 4.6 Let $G$ be a self-similar subgroup of $ \\text{Aut}\\left(\\mathcal {T} _3\\right)$ acting spherically transitively.", "Let $H$ be a subgroup of $G$ in which all elements have order 2.", "Then, $H$ is not normal.", "Suppose that $H$ is normal.", "It is also abelian since all its elements have order 2.", "Therefore, $\\forall h\\in H, \\ \\forall g\\in G,\\ h \\text{ and } ghg^{-1} \\text{commute}.$ Let $h\\in H$ be non-trivial and $n$ the smallest non-negative integer such that the permutation $h_n$ acts non-trivially on $L_n$ (of course, $n>0$ ).", "This permutation can be decomposed into a non-trivial product of transpositions $\\tau _i$ with disjoints supports: $h_n=\\prod _{i=1 \\dots k} \\tau _i $ By the minimality of $n$ , there is for each $i$ a vertex $v_i\\in L_{n-1}$ such that the support of $\\tau _i$ is in the first level of $v_i \\mathcal {T} _3 $ : $\\forall i=1 \\dots k, \\ \\exists v_i \\in L_{n-1}, \\ x_0(i),x_1(i) \\in \\left\\lbrace 1,2,3 \\right\\rbrace \\textrm { s.t. }", "\\\\x_0(i)\\ne x_1(i) \\textrm { and } \\tau _i = \\left( v_i x_0(i), v_i x_1(i)\\right).$ Let us choose an index $i$ and let $x$ be the unique element in $\\left\\lbrace 1,2,3 \\right\\rbrace \\setminus \\left\\lbrace x_0(i),x_1(i) \\right\\rbrace $ .", "Let also $g\\in G$ such that $g\\left(v_i x_0(i)\\right)= v_i x$ .", "In particular, $g\\left(v_i\\right)=v_i$ .", "Thus $g$ preserves the first level of the subtree $v_i \\mathcal {T} _3 $ (in which the support of $\\tau _i$ is contained) which imply that the permutations $h_n$ and $g_n h_n g_n^{-1}$ can be restricted to it.", "By REF the permutations $ h_n$ and $g_n h_n g_n^{-1}$ commute, so do these restrictions.", "But by construction, the last are transpositions in $\\mathfrak {S}_3$ with distinct support, leading to a contradiction.", "For each subset $A$ of the first level $L_1$ of $\\mathcal {T} _3$ , we define $\\textrm {Stab}_G \\left(A\\mathcal {T} _3 ,1\\right):= \\bigcap _{v\\in A}\\textrm {Stab}_G\\left(v \\mathcal {T} _3\\right)\\cap \\textrm {Stab}_G\\left(L_1\\right).$ The assumption of the proposition implies that there exists $v\\in L_1$ such that $ \\textrm {Stab}_G \\left(\\left\\lbrace v \\right\\rbrace \\mathcal {T} _3 ,1\\right) \\ne \\left\\lbrace 1\\right\\rbrace $ .", "The group $G$ acts transitively on $L_1$ and therefore the same holds for all $v\\in L_1$ since these groups are conjugate.", "1st case: there exists $A \\subset L_1$ with $\\left| A \\right| =2$ and $\\textrm {Stab}_G \\left(A\\mathcal {T} _3 ,1\\right)\\ne \\left\\lbrace 1\\right\\rbrace $ .", "The construction of a non-zero element of $G̏$ in the kernel of $\\rho $ is here the same as in the proof of Proposition REF : we consider a non-trivial element $g\\in \\textrm {Stab}_G \\left(A\\mathcal {T} _3 ,1\\right)$ and $h\\in G$ such that $A\\cup h\\left(A\\right)=L_1$ (equivalently $h\\left(A\\right) \\nsubseteq A $ ; such an $h$ exists since $G$ acts transitively on $L_1$ ).", "Now, $M:=\\left(1-g\\right)\\left(1-hgh^{-1}\\right)$ is the desired element.", "2nd case: there exists $v_1\\ne v_2$ and non-trivial elements $g_1\\in \\textrm {Stab}_G \\left(\\left\\lbrace v_1 \\right\\rbrace \\mathcal {T} _3 ,1\\right)$ , $g_2\\in \\textrm {Stab}_G \\left(\\left\\lbrace v_2 \\right\\rbrace \\mathcal {T} _3 ,1\\right)$ which do not commute.", "In this case, the commutator $\\left[g_1,g_2\\right]$ yields a non-trivial element in $\\textrm {Stab}_G \\left(\\left\\lbrace v_1,v_2 \\right\\rbrace \\mathcal {T} _3 ,1\\right) $ and we conclude using the first case.", "3rd case: for all $i\\in \\left\\lbrace 1,2,3 \\right\\rbrace = L_1$ and $g_i \\in \\textrm {Stab}_G \\left(\\left\\lbrace i \\right\\rbrace \\mathcal {T} _3 ,1\\right)$ , the elements $g_i$ and $g_j$ commute as soon as $i\\ne j$ .", "Then, let $M\\left(g_1,g_2,g_3\\right):= \\left(1-g_1\\right)\\left(1-g_2\\right)\\left(1-g_3\\right).$ One has $\\rho \\left( M\\left(g_1,g_2,g_3\\right) \\right)=0$ and we want to find the $g_i$ ’s such that $M\\left(g_1,g_2,g_3\\right)$ is non-zero.", "If $M\\left(g_1,g_2,g_3\\right)=0$ , then $ 1-\\left(g_1+g_2+g_3\\right)+\\left(g_1 g_2 + g_2 g_3 + g_1 g_3\\right) - g_1 g_2 g_3 = 0.", "$ Since the $g_i$ ’s are all non-trivial, one has $g_1 g_2 g_3=1$ .", "The same argument shows that $g_1=g_2 g_3$ .", "Thus, $g_1$ has order 2.", "Using the fact that the $g_i$ ’s commute, one shows in the same way that $g_2$ and $g_3$ have order 2.", "Therefore, if $\\forall g_1,g_2,g_3 \\text{ s.t. }", "g_i \\in \\textrm {Stab}_G \\left(\\left\\lbrace i \\right\\rbrace \\mathcal {T} _3 ,1\\right), \\ M\\left(g_1,g_2,g_3\\right) = 0,$ then the subgroups $\\textrm {Stab}_G \\left(\\left\\lbrace i \\right\\rbrace \\mathcal {T} _3 ,1\\right) $ would consist only of order 2 elements.", "The same would hold for the group that their union generates since $\\left[g_i,g_j\\right]=1$ as soon as $i\\ne j$ .", "But this last group is normal in $G$ , hence Lemma REF prevents REF from holding." ], [ "The classes $C_p^{d}$", "Let $d>1$ be an integer and $G$ a subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ .", "Theorem REF motivates the following definition: Definition 4.7 For each positive integer $p$ , one says that $G$ belongs to the class $C_p ^d$ if the regular representation $\\lambda $ of $G$ is weakly contained in $p$ -th tensor power $\\rho ^{\\otimes p}$ of the fundamental representation $\\rho $ .", "The last section will give examples of automata groups in $C_1 ^d$ .", "Besides, it is shown in [9] that there are subgroups of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ which do not belong to $C_p^{d}$ , for all positive integer $p$ (these are the so-called weakly branched subgroups of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ ).", "Note that for all positive $p$ , $\\rho ^{\\otimes p} \\prec \\rho ^{\\otimes p+1}$ because the trivial representation is contained in $\\rho $ .", "Therefore $C_1^{d} \\subset C_2^{d}\\subset \\dots C_p^{d}\\subset \\dots .$ The following construction aims to exhibit examples of automata groups that distinguish some of the $C_p^{d}$ ’s.", "To simplify the exposition, we will restrict ourselves to the case $d=2$ , but the following construction could be extended to any valence.", "Let $G$ be a subgroup of $\\text{Aut}\\left( \\mathcal {T}_2\\right)$ generated by a finite set $S=\\left\\lbrace g_1,\\dots ,g_l \\right\\rbrace $ .", "Let then consider $\\mathfrak {T} \\left(G\\right)$ the subgroup of $\\text{Aut}\\left( \\mathcal {T}_2\\right) $ generated by $S$ and the elements $\\tilde{g_i}$ defined for all $i=1 \\dots l$ by $\\Phi (\\tilde{g_i}) = \\left(1,g_i \\right)$ , i.e.", "$\\mathfrak {T} (G)=\\left\\langle g_1,\\dots ,g_l,\\Phi ^{-1}\\left(\\left(1,g_1\\right)\\right),\\dots ,\\Phi ^{-1}\\left(\\left(1,g_l\\right)\\right) \\right\\rangle .$ It is easy to see that if $G$ is self-similar (resp.", "generated by an automaton), then $\\mathfrak {T} \\left(G\\right)$ is also self-similar (resp.", "generated by an automaton).", "Moreover, the group we get is independent of the choice of the generating set $S$ of $G$ .", "Indeed, $\\mathfrak {T} \\left(G\\right)$ is also the group generated by $G \\bigcup \\left\\lbrace \\left(1,g\\right)\\ | \\ g\\in G \\right\\rbrace $ .", "Thus, for any positive integer $p$ , one can consider without ambiguity the group $\\mathfrak {T}^{(p)} \\left(G\\right)$ that we get by applying this construction $p$ times.", "Now, let $G$ be a finitely generated subgroup of $\\text{Aut}\\left( \\mathcal {T}_2\\right)$ fulfilling moreover the following conditions: $G$ is self-similar, the action of $G$ on the boundary $\\partial \\mathcal {T} _2$ is essentially free, the action of $G$ on $ \\mathcal {T} _2$ is spherically transitive.", "Then, Proposition 4.8 For all positive integer $p$ , the group $\\mathfrak {T}^{(p)} \\left(G\\right)$ distinguishes the classes $C _{2^p}^2$ and $C _{2^p -1}^2$ .", "We will need the following lemma: Lemma 4.9 Let $G$ be a finitely generated subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ .", "Suppose there exists $n\\in \\mathbb {N}$ such that: - the group $G$ acts transitively on the $n$ -th level $L_n$ of $\\mathcal {T}_d$ , - there is a vertex $v_0 \\in L_n$ such that the group $\\text{Rist}_G \\left(v_0\\right) := \\bigcap _{v\\in L_n \\setminus \\left\\lbrace v_0 \\right\\rbrace } \\textrm {Stab}_G \\left(v\\mathcal {T}_d\\right) $ is not trivial.", "Then, $G$ does not belong to the class $C _{d^n-1}^d $ .", "Suppose there are $l$ elements $\\xi _1,\\xi _2,\\dots ,\\xi _l$ in the boundary $\\partial \\mathcal {T}_d$ such that $\\bigcap _{i=1\\dots l} \\textrm {Stab}_G(\\xi _i) =\\left\\lbrace 1 \\right\\rbrace $ Then, the group $G$ belongs to the class $C _l ^2$ .", "i.", "The strategy is the same that the one used in the previous subsection: we will find a non-zero element of $G̏$ in the kernel of $\\rho ^{\\otimes n}$ .", "Since $G$ acts transitively on $L_n$ , the groups $\\textrm {Rist}_G \\left(v\\right)$ for $v\\in L_n$ are all conjugate and thus, all non-trivial.", "Let us choose, for each $v\\in L_n$ , a non-trivial element $g_v$ in the group $\\textrm {Rist}_G \\left(v\\right)$ .", "We have $\\Phi ^{(n)}\\left(g_v \\right) = \\left(1,1,\\dots ,1,\\varphi _v \\left(g_v\\right),1,\\dots \\right)$ where the non-trivial element $\\varphi _v \\left( g_v \\right)$ appears in the position corresponding to $v$ .", "It is clear that the $g_v$ 's commute.", "Let $M=\\prod _{v\\in L_n} \\left( 1- g_v \\right) \\in G̏.$ Then, $M \\ne 0$ .", "Indeed, the nullity of $M$ would imply the existence of a subset $\\mathcal {A}$ of $L_n$ (of odd cardinality) such that $1=\\prod _{v\\in \\mathcal {A}} g_v,$ and this is impossible because, if $w$ is any vertex in $\\mathcal {A}$ , (REF ) implies that $\\varphi _w \\left(\\prod _{v\\in \\mathcal {A}} g_v \\right)=\\varphi _w \\left(g_w \\right)\\ne 1.$ Let us show that $\\rho ^{\\otimes d^n -1} \\left( M \\right)$ is 0.", "This is equivalent to proving that for every $d^n-1$ -tuple $\\left(z_1,\\dots ,z_{d^n -1} \\right)$ consisting of elements in $\\mathcal {T}_d^0$ , $\\rho ^{\\otimes d^n -1} \\left( M \\right) \\left(\\delta _{z_1}\\otimes \\dots \\otimes \\delta _{z_{d^n -1}} \\right)=0 $ where $\\delta _z \\in \\ell ^2\\left(\\mathcal {T}_d^0\\right)$ is the Dirac function over the vertex $z\\in \\mathcal {T}_d^0$ .", "There is necessarily a vertex $v_0$ among the $d^n$ in $L_n$ such that the subtree $v_0\\mathcal {T}_d$ does not contain any $z_i$ .", "By construction, $\\rho ^{\\otimes d^n -1} \\left(1-g_{v_0} \\right)\\left(\\delta _{z_1}\\otimes \\dots \\otimes \\delta _{z_{d^n -1}} \\right)=0.$ As the $g_v$ 's commute, $M=\\left( \\prod _{v \\in L_n \\setminus \\left\\lbrace v_0\\right\\rbrace } \\left( 1-g_v\\right) \\right) (1-g_{v_0})$ and this implies $\\rho ^{\\otimes d^n -1}\\left( M \\right) \\left(\\delta _{z_1}\\otimes \\dots \\otimes \\delta _{z_{d^n -1}} \\right) =0 $ .", "ii- we will use the following result [9]: Proposition Let $G$ be a countable group acting on a countable set $X$ .", "Let $\\rho $ be the permutational representation that $G$ then admits on $\\ell ^2 \\left( X \\right)$ : $\\rho (g) (\\xi ) (x)= \\xi (g^{-1}\\cdot x)$ (with $g\\in G, \\ \\xi \\in \\ell ^2\\left( X \\right) $ and $x \\in X$ ).", "Suppose that for every finite subset $F$ of $G$ which does not contain 1, there is an element $x$ in $X$ such that $\\textrm {Stab}_G (x) \\cap F $ is empty i.e.", "$\\forall f \\in F, f.x \\ne x .$ Then $\\lambda \\prec \\rho $ .", "Let us show that the action of $G$ on the product $\\mathcal {T}_d^0 \\times \\cdots \\times \\mathcal {T}_d^0$ of $l$ copies of $\\mathcal {T}_d^0$ satisfies the condition of this proposition.", "Let $F$ be a finite subset of $G$ which does not contain 1.", "For each $f\\in F$ , the set $F\\cap \\textrm {Stab}_G\\left(\\xi _1 \\times \\dots \\times \\xi _l\\right)=F \\cap \\bigcap _{i=1\\dots l} \\textrm {Stab}_G\\left(\\xi _i\\right)$ is empty, i.e.", "$\\forall f \\in F, f\\left(\\xi _1\\times \\dots \\times \\xi _l \\right) \\ne \\xi _1\\times \\dots \\times \\xi _l.$ Since $F$ is finite, there exists $n\\in \\mathbb {N}$ such that (denoting by $(\\xi _i)_n$ the restriction to the $n$ -th level of the path $\\xi _i$ ), one has: $\\forall f \\in F, f\\left((\\xi _1)_n\\times \\dots \\times (\\xi _l)_n \\right) \\ne (\\xi _1)_n\\times \\dots \\times (\\xi _l)_n,$ which completes the proof.", "Let $p$ be a positive integer.", "Since $G$ is a subgroup of $\\mathfrak {T}^{(p)} \\left(G\\right)$ , the last acts spherically transitively on $\\mathcal {T} _2$ .", "Moreover, an easy induction on $p$ implies that for each vertex $w \\in L_p$ : the group $\\textrm {Rist}_{\\mathfrak {T}^{(p)} \\left(G\\right)}\\left(w\\right)$ is not trivial, for all $\\alpha \\in \\mathfrak {T}^{(p)}(G)$ , $\\varphi _w (\\alpha ) \\in G$ .", "The first point (1) and Lemma REF .REF already imply that the group $\\mathfrak {T}^{(p)} \\left(G\\right) $ does not belong to the class $\\mathcal {C} _{2^p -1}^2$ .", "Let us now show that $\\mathfrak {T}^{(p)} \\left(G\\right) $ belongs to $\\mathcal {C} _{2^p}^2$ .", "For all non-trivial element $g \\in G$ , the set $\\textrm {Fix}_{\\partial \\mathcal {T}_2}\\left(g\\right)$ has measure 0.", "Since $G$ is countable, the union of these fixed point sets also has measure 0.", "Hence, we can choose for each $w\\in L_p$ an element $\\xi _w \\in \\partial \\left(w \\mathcal {T}_2\\right)$ in its complement, i.e.", "$\\xi _w \\in \\partial (w \\mathcal {T}_2) \\textrm { is such that } \\forall g \\in G, \\ \\left(g\\ne 1 \\Longrightarrow g(\\xi _w)\\ne \\xi _w\\right).$ Now, let $\\alpha \\in \\mathfrak {T}^{(p)} \\left(G\\right) $ and suppose that for all $w \\in L_p$ , $\\alpha \\left(\\xi _w\\right)=\\xi _w $ .", "Necessarly, $\\alpha $ fixes each element in the $p-$ level $L_p$ and thus, the decomposition of $\\alpha $ with respect to $L_n$ is: $\\Phi ^{(p)}\\left(\\alpha \\right)=\\left(g_{w_0}, \\dots , g_{w_{2^p-1}} \\right)$ with $g_{w_{i}} \\in G$ thanks to (2).", "By the definition of $\\alpha $ , $g_{w_{i}} $ fixes $\\xi _{w_i}$ for all $i=0 \\dots 2^p-1$ .", "By REF , all the $g_{w_i}$ ’s are trivial.", "Therefore, $\\alpha =1$ which implies that $\\mathfrak {T}^{(p)} \\left(G\\right) $ satisfies the conditions of Lemma REF .REF with $l=2^p$ .", "Hence, $\\mathfrak {T}^{(p)} \\left(G\\right) $ belongs to $\\mathcal {C} _{2^p}^2$ ." ], [ "The ¨free¨ case", "We are interested here in automata groups fulfilling the condition of Theorem REF .REF , that is acting topologically freely on the boundary $\\partial \\left(\\mathcal {T}_d\\right)$ .", "These are exactly the groups generated by reversible automata (see Chapter 2.7 in [3]).", "Note that this class contains a lot of interesting examples such as: the lamplighter group $\\left( \\oplus _{n\\in \\mathbb {Z}}\\mathbb {Z}/ 2\\mathbb {Z}\\right)\\rtimes \\mathbb {Z}$ [6], free groups [1], [11] and [5], lattices in $p$ -adic Lie groups [5] and $\\mathbb {Z}^n \\rtimes GL_n\\left( \\mathbb {Z}\\right)$ [4].", "Let $d>1$ be an integer and $G$ be a subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ generated by a reversible automaton.", "Lemma REF implies that the action of $G$ on $\\partial \\left( \\mathcal {T}_d\\right)$ is essentially free, i.e.", "$\\forall g\\in G, \\ g \\ne 1 \\Longrightarrow Tr(g)=0 .$ This means that $Tr$ coincides on $ G̏ $ with the usual trace $\\tau $ on $C^* _{\\lambda } \\left(G \\right)$ defined by $\\tau (g)=\\left\\langle g \\left(\\delta _1\\right) | \\delta _1\\right\\rangle $ , where $\\delta _1 \\in \\ell ^2 \\left( G \\right)$ is the Dirac function over the neutral element $1\\in G$ .", "Moreover, Theorem REF .REF implies that $\\lambda $ is weakly contained in $\\rho $ , i.e.", "$C^* _{\\lambda } \\left(G \\right)$ is a quotient of $C^{*}_{\\rho }\\left( G \\right)$ .", "Thus, we have the following commutative diagram.", "$ {C^{*}_{\\rho }\\left( G \\right)@{->>}[r]^{\\lambda } [dr]_{Tr} & C^*_{\\lambda }\\left( G\\right) [d]_{\\tau }\\\\&}$ Since the trace $\\tau $ is faithful, one has $\\ker \\lambda = \\left\\lbrace x \\in C^{*}_{\\rho }\\left( G \\right)\\ | \\ Tr(x^* x)=0 \\right\\rbrace .$ Remark 5.1 The representation $\\rho $ contains the trivial representation.", "In particular, REF implies that $Tr$ is a faithful trace on $C^{*}_{\\rho }\\left( G \\right)$ if and only if $G$ is amenable.", "The author does not know how to decide the faithfulness of $Tr$ when $G$ does not act essentially freely on $\\partial \\left(\\mathcal {T}_d\\right)$ .", "Here is the central result of this section.", "Theorem 5.2 Let $G$ be a subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ generated by a reversible automaton.", "Then: The recursion isomorphism $\\Phi $ defined on $G̏ $ extends to an isometry from $C^{*}_{\\lambda }(G) $ into $M_d(C^{*}_{\\lambda }(G))$ : $ \\Phi : C^{*}_{\\lambda }\\left(G\\right) \\hookrightarrow M_d\\left( C^{*}_{\\lambda }\\left(G\\right)\\right) .$ For all $n\\in \\mathbb {N}$ , there exists a conditional expectation $E_n$ from $M_{d^n}\\left(C^{*}_{\\lambda }\\left(G\\right)\\right)$ onto $C^{*}_{\\lambda }\\left(G\\right) $ , the last being identified with its image in $M_{d^n}\\left(C^{*}_{\\lambda }\\left(G\\right)\\right) $ via $\\Phi ^{(n)}$ : $E_n : M_{d^n}\\left(C^{*}_{\\lambda }\\left(G\\right)\\right) \\twoheadrightarrow \\Phi ^{(n)}\\left(C^{*}_{\\lambda }\\left(G\\right)\\right) .$ Example 5.3 Consider $\\mathbb {F}_3=\\left\\langle a,b,c\\right\\rangle $ the free group on three generators $a$ , $b$ and $c$ .", "Then, Theorem REF and the main result of [11] imply that the map $a \\rightarrow \\left(\\begin{array}{cc}0 & b \\\\ c & 0\\end{array}\\right), \\ b \\rightarrow \\left(\\begin{array}{cc}0 & c \\\\ b & 0\\end{array}\\right), \\ c \\rightarrow \\left(\\begin{array}{cc}a & 0 \\\\ 0 & a\\end{array}\\right)$ defines an isometry from $C^{*}_{\\lambda }\\left(\\mathbb {F}_3\\right) $ into $M_{2}\\left(C^{*}_{\\lambda }\\left(\\mathbb {F}_3\\right)\\right)$ .", "Moreover, there exists a conditional expectation from the last $C^*$ -algebra onto the image of this isometry.", "i. REF and Proposition REF yield the following commutative diagram: $ {@{=}[rr] && & C^{*}_{\\rho }\\left( G \\right)[lu]^{Tr} @{^{(}->}[rr]^{\\Phi } @{->>}[dl]^{\\lambda } && M_{d}\\left(C^{*}_{\\rho }\\left( G \\right)\\right) [lu]_{Tr} @{->>}[dl]^{Id \\otimes \\lambda }\\\\C^{*}_{\\lambda }\\left( G\\right) [uu]^{\\tau } @{.>}[rr]^{?}", "&& M_{d}\\left( C^{*}_{\\lambda }\\left(G\\right)\\right) [uu]^{\\tau } |!", "{[ul];[ur]}\\\\}$ Therefore, $\\Phi \\left( \\ker \\lambda \\right) & = & \\Phi \\left( \\left\\lbrace x \\in C^{*}_{\\rho }\\left( G \\right)\\ | \\ Tr(x^* x)=0\\right\\rbrace \\right)\\\\& = & \\left( \\textrm {Im }\\Phi \\right) \\cap \\left\\lbrace x \\in M_d\\left(C^{*}_{\\rho }\\left( G \\right)\\right) \\ | \\ Tr(x^*x)=0\\right\\rbrace \\\\& = & \\left(\\textrm {Im }\\Phi \\right) \\cap \\ker \\left(Id \\otimes \\lambda \\right) .$ Thus, the map $\\Phi $ defines an injective (hence isometric) $\\ast $ -homomorphism from $C^{*}_{\\lambda }\\left(G\\right) $ into $M_{d}\\left(C^{*}_{\\lambda }\\left(G\\right)\\right) $ .", "ii.", "Let us fix $n\\in \\mathbb {N}$ .", "To simplify notation, we set - $A := \\Phi ^{(n)}\\left( C^{*}_{\\lambda }\\left(G\\right)\\right)$ and $A ^{\\prime \\prime }$ the von Neumann algebra generated by $A $ in $\\mathcal {B} \\left(\\ell ^2\\left(G\\right)\\right) \\otimes M_{d^n}\\left()\\right.$ , - $B :=M_{d^n}\\left(C^{*}_{\\lambda }\\left(G\\right)\\right) $ and $B ^{\\prime \\prime }:=M_{d^n}\\left(L\\left(G\\right)\\right)$ the von Neumann algebra generated by $B $ in $\\mathcal {B} \\left(\\ell ^2\\left(G\\right)\\right) \\otimes M_{d^n}\\left()\\right.$ .", "One knows that there exists a conditional expectation $E_n$ from $B ^{\\prime \\prime }$ onto $A ^{\\prime \\prime }$ determined by the following equation: $\\forall a \\in A^{\\prime \\prime }, \\ \\forall b \\in B^{\\prime \\prime } , \\ \\tau (a^* b)=\\tau (a^* E_n(b)).$ The space $M_{d^n}(G̏)=M_{d^n}(\\otimes G̏ $ is generated by the elements $e_{i,j} \\otimes g$ where $g$ runs over $G$ and the ordered pair $(i,j)$ runs over $\\left\\lbrace 1,\\dots ,d^n\\right\\rbrace ^2 $ (i.e., the $(i,j)$ -th entry of the matrix $e_{i,j} \\otimes g$ is $g$ , the others are 0).", "We will show that $E_n \\left( e_{i,j} \\otimes g\\right) \\in \\Phi ^{(n)}\\left(G̏\\right)$ which will prove the second part of Theorem REF .", "Indeed, the linear map $E_n$ being bounded, REF will imply that $E_n\\left( B \\right)$ is a subspace of $ A$ , and a fortiori is $ A$ since by definition, $E_n$ is the identity restricted to this $C^*$ -algebra $A\\subset B $ .", "Let $g\\in G$ and $(i,j)\\in \\left\\lbrace 1,\\dots ,d^n\\right\\rbrace ^2 $ .", "If there exists $h\\in G$ such that the $(i,j)$ -th entry of the matrix $\\Phi ^{(n)}(h)$ is $g$ , then $h$ is unique; indeed: $\\exists (i,j) \\textrm { s.t.", "}\\left( \\Phi ^{(n)}(h) \\right)_{i,j} = \\left( \\Phi ^{(n)}(h^{\\prime }) \\right)_{i,j} &\\Longleftrightarrow & \\exists v\\in L_n \\textrm { s.t. }", "h^{-1}h^{\\prime } \\in \\textrm {Stab}_G(v \\mathcal {T}_d)\\\\&\\Longleftrightarrow &h=h^{\\prime }$ the last equivalence being the assumption of the theorem.", "Thus, we can define $\\tilde{E_n}(e_{i,j} \\otimes g) = \\left\\lbrace \\begin{array}{ll}\\frac{1}{d^n}\\Phi ^{(n)}(h) & \\textrm {if } \\exists h \\textrm { s.t.", "}\\left( \\Phi ^{(n)}(h) \\right)_{i,j}=g, \\\\0 & \\textrm {otherwise.", "}\\end{array} \\right.$ This defines a linear map $\\tilde{E_n}$ on $M_{d^n}(G̏)$ .", "In fact, $\\tilde{E_n} =E_n$ , which yields REF .", "To see this, it is sufficient to prove: $\\forall a\\in A ^{\\prime \\prime }, \\ \\tau \\left(a^* \\cdot (e_{i,j} \\otimes g)\\right)=\\tau \\left(a^* \\cdot \\tilde{E_n}(e_{i,j} \\otimes g)\\right)$ since $E_n$ is determined by REF and the $e_{i,j} \\otimes g $ ’s generate $B ^{\\prime \\prime }$ .", "For $h \\in G$ , the diagonal of the matrix $\\Phi ^{(n)}\\left(h\\right)^{*} \\cdot (e_{i,j} \\otimes g)$ consists only of 0’s except maybe at the $j$ -th position: - Either $\\left(\\Phi ^{(n)}(h)^{*} \\cdot (e_{i,j} \\otimes g)\\right)_{j,j} = 1$ .", "This is equivalent to $\\left( \\Phi ^{(n)}(h) \\right)_{i,j} =g$ and in this case, $\\tau \\left( \\Phi ^{(n)}\\left(h\\right)^{*} \\cdot (e_{i,j} \\otimes g)\\right) =\\frac{1}{d^n}= \\tau \\left( \\Phi ^{(n)}\\left(h\\right)^{*} \\cdot \\tilde{E_n}(e_{i,j} \\otimes g)\\right).$ - Or $\\left(\\Phi ^{(n)}\\left(h\\right)^{*} \\cdot (e_{i,j} \\otimes g)\\right)_{j,j} = 0$ or $ \\alpha $ , where $\\alpha \\ne 1$ belongs to $G$ .", "In this case, $\\tau \\left(\\Phi ^{(n)}\\left(h\\right)^{*}\\cdot (e_{i,j} \\otimes g)\\right) =0= \\tau \\left( \\Phi ^{(n)}\\left(h\\right)^{*} \\cdot \\tilde{E_n}(e_{i,j} \\otimes g)\\right).", "$ Thus, we observe that $\\tau \\left(a^* \\cdot (e_{i,j} \\otimes g)\\right)=\\tau \\left(a^* \\cdot \\tilde{E_n}(e_{i,j} \\otimes g)\\right) $ for each element $a$ in the set $\\left\\lbrace \\Phi ^{(n)}\\left(h\\right) \\ | \\ h \\in G\\right\\rbrace $ .", "Since $A ^{\\prime \\prime }$ is generated by this set, we have proved REF , hence the theorem.", "From Theorem REF .REF follows the existence of the direct system $ \\left\\lbrace M_{d^n}\\left(C^* _{\\lambda }\\left(G\\right)\\right),\\Phi \\right\\rbrace $ .", "Therefore, one can consider the $C^*$ -algebra $\\mathcal {L} _G$ limit of this system: $\\mathcal {L} _G = \\varinjlim _{\\Phi ,n} M_{d^n}\\left(C^* _{\\lambda }\\left(G\\right)\\right).$ Moreover, Theorem REF .REF gives Corollary 5.4 Let $G$ be a subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ generated by a reversible automaton.", "Then, there exists a conditional expectation $E$ from $\\mathcal {L} _G$ onto the image of $C^* _{\\lambda }\\left(G\\right) $ in $\\mathcal {L} _G$ .", "Let us first remark that, since the $\\Phi ^{(n)}$ ’s are injective, the image of $C^* _{\\lambda }\\left(G\\right) $ in $\\mathcal {L} _G$ is actually isomorphic to $C^* _{\\lambda }\\left(G\\right) $ .", "For all $n\\in \\mathbb {N}$ , we can define $F_n : M_{d^n} \\left(C^* _{\\lambda }\\left(G\\right)\\right) \\stackrel{E_n}{\\longrightarrow } \\Phi ^n \\left(C^* _{\\lambda }\\left(G\\right)\\right) \\stackrel{i}{\\longrightarrow } \\varinjlim _{n} \\Phi ^{(n)} \\left(C^* _{\\lambda }\\left(G\\right)\\right)$ where $E_n$ is the conditional expectation built in REF and $i$ is the injection of $M_{d^n}\\left(C^* _{\\lambda }\\left(G\\right)\\right)$ in $\\mathcal {L} _G$ .", "The map $E_n$ has norm 1 since it is a normalized conditional expectation.", "The same holds for $i$ and thus for $F_n$ .", "Moreover, by the definition of $E_n$ , the linear maps $F_n$ ’s are consistent with the direct system $ \\left\\lbrace M_{d^n}\\left(C^* _{\\lambda }\\left(G\\right)\\right),\\Phi \\right\\rbrace $ .", "Therefore, there exists a linear map $E$ from the algebraic direct limit $\\tilde{\\mathcal {L} _G}$ onto $\\varinjlim _{n} \\Phi ^n \\left(C^* _{\\lambda }\\left(G\\right)\\right) \\simeq C^* _{\\lambda }\\left(G\\right)$ .", "It has norm 1 and it is straightforward to check that it fulfills all the axioms of a positive linear map of the $C^* _{\\lambda }\\left(G\\right)$ -$ C^* _{\\lambda }\\left(G\\right)$ -bimodule $\\tilde{\\mathcal {L} _G}$ : its unique extension to the closure $\\mathcal {L} _G$ of $\\tilde{\\mathcal {L} _G}$ is the conditional expectation we wanted.", "In [8], V. Nekrashevych defines for each self-similar subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ its universal Cuntz-Pimsner $C^*$ -algebra $\\mathcal {O} _G$ : Definition ([8]) For $G$ a self-similar subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ and $X = \\left\\lbrace 1,\\dots ,d \\right\\rbrace $ (i.e.", "the first level $L_1$ of $\\mathcal {T}_d$ ), $\\mathcal {O} _G$ is the universal $C^*$ -algebra generated by a familly $\\left(u_g\\right)_{g\\in G}$ of unitaries and a finite set $\\left( S_v\\right)_{v\\in X}$ of partial isometries satisfying: for all $g,h \\in G$ , $u_g \\cdot u_h=u_{gh}$ , for all $x\\in X$ , $S_x^{*}S_x=1$ and $\\sum _{x\\in X}S_x S_x^{*} =1$ , for all $g \\in G$ and $x\\in X$ , $u_g \\cdot S_x = S_{g(x)} \\cdot u_{\\varphi _x (g)}$ .", "V. Nekrashevych proves in particular: Theorem ([8]) There exists a conditional expectation $\\mathfrak {E}$ from $\\mathcal {O} _G$ onto a subalgebra $\\mathcal {M} _G$ of $\\mathcal {O} _G$ .", "Moreover, $\\mathcal {M} _G$ isomorphic to the limit of the direct system $ \\left\\lbrace M_{d^n}\\left( C^* _{\\textrm {max}}\\left(G\\right)\\right),\\Phi \\right\\rbrace $ : $ \\mathcal {M} _G = \\varinjlim _{\\Phi ,n} M_{d^n}\\left(C^* _{\\textrm {max}}\\left(G\\right)\\right).$ The author also shows that under certain conditions, the algebras $\\mathcal {O}_G$ and $\\mathcal {M}_G$ are nuclear if $G$ is contractible (see [8]).", "In our case, the situation concerning the nuclearity of these algebras is the following.", "The $C^*$ -algebra $\\mathcal {M} _G$ clearly surjects onto $\\mathcal {L} _G$ .", "Thus, for a group generated by a reversible automaton, we have: $\\mathcal {O} _G \\stackrel{\\mathfrak {E}}{\\rightarrow }\\mathcal {M} _G \\twoheadrightarrow \\mathcal {L} _G \\stackrel{E}{\\rightarrow }C^* _{\\lambda }\\left(G\\right) .$ Corollary 5.5 Let $G$ be a subgroup of $\\text{Aut}\\left(\\mathcal {T}_d\\right)$ generated by a reversible automaton.", "Among the $C^*$ -algebras $\\mathcal {O} _G$ , $\\mathcal {M} _G$ , $\\mathcal {L} _G$ and $C^* _{\\lambda }\\left(G\\right)$ , one is nuclear if and only if all the others are, that is if and only if $G$ is amenable.", "It is essentially a combination of known results on nuclearity for $C^*$ -algebras.", "Let us quote them: nuclearity passes to quotient (see for instance [2]), to the image of a conditional expectation (see for instance [2]) and to direct limit (see for instance [2]).", "Moreover, it is classical that $C^* _{\\lambda }\\left(G\\right)$ is nuclear if and only if $G$ is amenable, that is if and only if $C^* _{\\lambda }\\left(G\\right)\\simeq C^* _{max}\\left(G\\right)$ .", "It is easy to see that these results give all the implications of Corollary REF , except $\\mathcal {M} _G $ nuclear $\\Rightarrow \\mathcal {O} _G$ nuclear.", "This is a application of the following isomorphism proved by V. Nekrashevych ([8]): $\\left(\\mathbb {K} \\otimes \\mathcal {M} _G \\right) \\rtimes \\mathbb {Z}\\simeq \\mathbb {K} \\otimes \\mathcal {O} _G$ where $\\mathbb {K}$ is the (nuclear) algebra of compact operators.", "If $\\mathcal {M} _G $ is nuclear, so is $\\left(\\mathbb {K} \\otimes \\mathcal {M} _G \\right) \\rtimes \\mathbb {Z}$ (see for instance [2]).", "Therefore, the algebra $\\mathbb {K} \\otimes \\mathcal {O} _G $ is nuclear, and $\\mathcal {O} _G $ as well (see for instance [2]).", "Remark 5.6 Note that it is still an open question whether all contractible groups are amenable.", "Corollary REF does not bring anything new to this question, since it is already known that contractible groups acting essentially freely on $\\partial \\mathcal {T}_d$ have polynomial growth (see [7])." ], [ "Acknowledgments.", "This work is part of my PhD-thesis.", "I want to express my gratitude to my adviser Andrzej Żuk.", "I also would like to thank Mikael de la Salle for usefull discussions, Suliman Albandik for having suggested the application REF of Lemma REF , and Ivan Marin for having accepted to write the following appendix with me." ], [ "Virtually abelian groups", "by Ivan MarinI.", "MARIN, Institut de Mathématiques de Jussieu, Université Paris 7. [email protected] and Jean-François PlanchatJ.-F. PLANCHAT, Mathematisches Institut, Georg-August Universität Göttingen.", "[email protected] Let $\\Gamma $ be a finitely generated and residually finite group.", "We consider a decreasing sequence $\\left(\\Gamma _i\\right)_i$ of finite index subgroups of $\\Gamma $ .", "It is easy to see that the following conditions are equivalent: The natural representation that $\\Gamma $ admits on $\\bigoplus _{i} \\ell ^2 \\left( \\Gamma /\\Gamma _i \\right)$ is faithful.", "The natural action that $\\Gamma $ admits on $\\bigsqcup _{i} \\Gamma /\\Gamma _i$ is faithful.", "One has $\\bigcap _{i} \\text{Core}\\left(\\Gamma _i\\right)=\\left\\lbrace 1\\right\\rbrace $ where $\\text{Core}\\left(\\Gamma _i\\right) := \\bigcap _{\\gamma \\in \\Gamma } \\gamma \\Gamma _i \\gamma ^{-1}$ is the biggest normal subgroup of $\\Gamma $ contained in $\\Gamma _i$ .", "Definition 1.1 An F-filtration of $\\Gamma $ is a decreasing sequence $\\left(\\Gamma _i\\right)_i$ of finite index subgroups fulfilling one of the previous equivalent conditions.", "Theorem 1.2 Let $\\Gamma $ be a finitely generated and residually finite group.", "The following are equivalent.", "$\\Gamma $ is virtually abelian.", "$\\Gamma $ is of type $I$ (see [13]).", "There exists $N\\in \\mathbb {N}$ and a F-filtration $\\left(\\Gamma _i\\right)_i$ of $\\Gamma $ , such that the natural representation of $\\Gamma $ on $\\ell ^2(\\Gamma /\\Gamma _i)$ can be decomposed in irreducible representations of dimension at most $N$ for all $i$ .", "There exists $N\\in \\mathbb {N}$ such that, for all F-filtrations $\\left(\\Gamma _i\\right)_i$ of $\\Gamma $ , the natural representation of $\\Gamma $ on $\\ell ^2(\\Gamma /\\Gamma _i)$ can be decomposed in irreducible representations of dimension at most $N$ for all $i$ .", "There exists $N \\in \\mathbb {N}$ such that every finite image finite-dimensional irreducible representation of $\\Gamma $ has dimension at most $N$ .", "There exists $N\\in \\mathbb {N}$ such that every finite-dimensional irreducible unitary representation of $\\Gamma $ has dimension at most $N$ .", "(1) is equivalent to (2) by [13].", "(6) implies (5) and (5) implies (4) trivially.", "(4) implies (3) since $\\Gamma $ is residually finite.", "We prove that (3) implies (1).", "By assumption, each $\\ell ^2(\\Gamma /\\Gamma _i)$ is the direct sum of irreducible representations of dimension at most $N$ , which are all unitary since the representation of $\\Gamma $ on $\\ell ^2(\\Gamma /\\Gamma _i)$ is.", "By considering all $i$ , we thus get an embedding $\\Gamma \\hookrightarrow \\bigoplus _{j \\in J} G_j$ with $G_j < U_n :=\\mathcal {U} \\left(N\\right)$ a finite subgroup of $U_N$ .", "By Jordan's theorem (see [15]) there exists $q \\ge 0$ such that every finite subgroup of $U_N$ has a normal abelian subgroup of index at most $q$ .", "For all $j\\in J$ , we let $H_j$ be such a subgroup for $G_j$ .", "We denote by $\\varphi _j : G_j \\twoheadrightarrow G_j/H_j$ the canonical projection and $\\tilde{\\varphi }_j : \\Gamma \\rightarrow G_j/H_j$ the induced morphism.", "By assumption $|G_j/H_j| \\le q$ hence $G_j/H_j < \\mathfrak {S}_q$ .", "The set of subgroups $\\mathrm {Ker}\\tilde{\\varphi }_j < \\Gamma $ is thus a subset of $\\lbrace \\mathrm {Ker}\\psi \\ | \\ \\psi \\in \\mathrm {Hom}(\\Gamma ,\\mathfrak {S}_q) \\rbrace $ , and $\\mathrm {Hom}(\\Gamma ,\\mathfrak {S}_q)$ is finite because $\\Gamma $ is finitely generated.", "It follows that $H = \\bigcap _j \\mathrm {Ker}\\tilde{\\varphi }_j$ is a finite index subgroup of $\\Gamma $ .", "Moreover, $H$ is abelian because its image in $\\bigoplus _j G_j$ is abelian.", "We now prove that (1) implies (6), and let $H \\vartriangleleft \\Gamma $ be an abelian normal subgroup such that $\\Gamma /H$ is finite.", "Let $\\rho : \\Gamma \\rightarrow \\mathcal {U} \\left( V \\right)$ with $V$ a finite-dimensional hermitian space be a unitary representation of $\\Gamma $ .", "The restriction of $\\rho $ to $H$ is a direct sum $\\bigoplus _{\\chi \\in \\mathrm {Irr}(H) } a_{\\chi } \\chi $ where $\\mathrm {Irr}(H)$ denotes the set of irreducible (1-dimensional) unitary representations of $H$ , and $a_{\\chi }\\in \\mathbb {N}$ denotes the multiplicity of $\\chi $ .", "Letting $V_r := \\bigoplus _{a_{\\chi } = r}r \\chi $ we have a canonical decomposition $V = \\bigoplus _{r \\ge 0} V_r.$ We first prove that $V = V_r$ for some $r$ .", "Since $V$ is irreducible, this is equivalent to saying that $\\Gamma .", "V_r \\subset V_r$ for all $r$ .", "Since $V$ is finite dimensional, this is equivalent (by descending induction on $r$ ) to saying that $\\Gamma .", "V_r \\subset \\bigoplus _{s \\ge r} V_s$ for all $r$ .", "We prove this last statement, and decompose $V_r = U_1 \\oplus \\dots \\oplus U_m$ with $\\dim U_i = r$ and $h(x) = \\chi _i(h) x$ for every $x \\in U_i, h \\in H$ , for the given $\\chi _i \\in \\mathrm {Irr}(H)$ associated to $i$ (by construction, $i \\ne j \\Leftrightarrow \\chi _i \\ne \\chi _j$ ).", "Let $g \\in \\Gamma $ , and $i \\in \\lbrace 1, \\dots ,m \\rbrace $ .", "Since $H $ is a normal subgroup of $ \\Gamma $ , the subspace $g.U_i$ is $H$ -stable, hence is included in some $V_s$ .", "Moreover, $H$ acts through $\\chi _i^g : h \\mapsto \\chi _i(g^{-1}hg)$ on $g .", "U_i$ , hence $\\chi _i^g \\in \\mathrm {Irr}(H)$ occurs in $V_s$ with multiplicity at least $r$ .", "It follows that $s \\ge r$ .", "Thus $V = V_r = U_1 \\oplus \\dots \\oplus U_m.$ Moreover, we proved $g.", "U_i = U_{\\pi (g)(i)}$ for some $\\pi (g) \\in \\mathfrak {S}_m$ , and clearly $\\pi : \\Gamma \\rightarrow \\mathfrak {S}_m$ is a group morphism, which factors through $\\Gamma /H$ .", "By irreducibility of $V$ , $\\pi (\\Gamma )= \\pi (\\Gamma /H)$ must be transitive, which implies $m \\le |\\Gamma /H|$ .", "We now introduce $St(U_1) = \\lbrace g \\in \\Gamma \\ | \\ g .U_1 = U_1 \\rbrace $ , and prove that $U_1$ is irreducible under $St(U_1)$ .", "Let $E \\subset U_1$ be non-zero $St(U_1)$ -stable subspace.", "By irreducibility of $V$ under $\\Gamma $ , we have $\\sum _{g \\in \\Gamma } g E = V$ .", "But $g E \\subset U_{\\pi (g)(1)}$ for all $g \\in \\Gamma $ , so this can be true only if $\\sum _{g \\in \\Gamma \\ | \\ \\pi (g)(1) = 1} g E = U_1 \\ \\Longleftrightarrow \\ E = \\sum _{g \\in St(U_1)} g E = U_1, $ which proves that $U_1$ is $St(U_1)$ -irreducible.", "Since $H$ acts by scalars on $U_1$ , this last subspace can be considered as an irreducible projective representation of the subgroup $St(U_1)/H$ of $\\Gamma /H$ .", "This last group is finite, hence it has a finite number of subgroups, and these subgroups have a finite number of irreducible projective representations (for instance by the theory of Schur covers, see [14]).", "Taking for $e$ the maximal degree of such representations, we get $r \\le e$ hence $\\dim V = m r \\le N = |\\Gamma /H| e$ , which proves the claim." ] ]

[ [ "Non-Abelian statistics of vortices with non-Abelian Dirac fermions" ], [ "Abstract We extend our previous analysis on the exchange statistics of vortices having a single Dirac fermion trapped in each core, to the case where vortices trap two Dirac fermions with U(2) symmetry.", "Such a system of vortices with non-Abelian Dirac fermions appears in color superconductors at extremely high densities, and in supersymmetric QCD.", "We show that the exchange of two vortices having doublet Dirac fermions in each core is expressed by non-Abelian representations of a braid group, which is explicitly verified in the matrix representation of the exchange operators when the number of vortices is up to four.", "We find that the result contains the matrices previously obtained for the vortices with a single Dirac fermion in each core as a special case.", "The whole braid group does not immediately imply non-Abelian statistics of identical particles because it also contains exchanges between vortices with different numbers of Dirac fermions.", "However, we find that it does contain, as its subgroup, a genuine non-Abelian statistics for the exchange of the identical particles, that is, vortices with the same number of Dirac fermions.", "This result is surprising compared with conventional understanding because all Dirac fermions are defined locally at each vortex, unlike the case of Majorana fermions for which Dirac fermions are defined non-locally by Majorana fermions located at two spatially separated vortices." ], [ "Introduction", "Topological insulators/superconductors have an attractive property that some of them possess quantum vortices which trap zero-energy, Majorana or Dirac, fermions in their cores [1], [2].", "The existence of such zero-energy fermions is topologically protected and is robust against small perturbations [3], [4].", "Thus, when we consider adiabatic manipulation of vortices such as interchanging the positions of two vortices, we can treat the vortices as objects that are always accompanied by zero-energy fermions.", "In particular, according to the recent discoveries, the exchange of such vortices can be represented by a non-trivial representation of a braid group, whose precise form is determined by the trapped zero-energy Majorana fermions [5], [6], [7], [8], [9], [10] and Dirac fermions [11].", "The exchange of vortices with Majorana fermions gives non-Abelian statistics because they are all identical particles.", "The statistics is called non-Abelian because quantum states of two vortices transform non-diagonally under the exchange of two vortices (i.e., the exchange operation is described by non-diagonal matrices acting on the quantum states), and two adjacent exchange operations (such as those for vortex pairs (1,2) and (2,3)) do not commute with each other.", "This is highly contrasted with the ordinary statistics where only a phase factor ${\\rm e}^{i\\theta }$ appears under the exchange of two particles ($\\theta =0$ for the Bose-Einstein, $\\theta =\\pi $ for the Fermi-Dirac, and others for the anyon statistics), and two adjacent operations are commutative.", "On the other hand, it is unclear whether the exchange of vortices with Dirac fermions gives a non-Abelian statistics or not because, in general, it exchanges different particles, that is, vortices with different numbers of Dirac fermions.", "For U(1) Dirac fermions, vortices can be distinguished by the occupancy of Dirac fermions.", "According to the classification of topological insulators/superconductors [1], [2] and its extension to the case with topological defects [12], [4], the vortices with the Majorana or Dirac fermions are categorized into different types: class D for the Majorana and classes C and DIII for the Dirac.", "However, the essential difference between the vortices with Majorana fermions and Dirac fermions is the parity of the number of zero-energy Majorana fermions trapped to a single vortex.", "Notice that a single Dirac fermion corresponds to two Majorana fermions.", "Thus, when a vortex traps an even number of the Majorana fermions, it should be regarded as the Dirac case, while an odd number of the Majorana fermions, just as the Majorana case [4].", "So far, non-Abelian unitary transformations (non-Abelian representations of a braid group) are found both in the Majorana and Dirac cases.", "In the Majorana case, the non-Abelian statistics was first discovered when a vortex has a single Majorana fermion [6] and later when a vortex can have multiple Majorana fermions with non-Abelian symmetry [8], [9], [10].", "In contrast, in the Dirac case, while non-Abelian representation of a braid group is analyzed when a vortex traps a single Dirac fermion [11], it is unclear if it gives genuine non-Abelian statistics.In the previous paper for the U(1) Dirac case [11], we used the word “non-Abelian statistics\" to imply non-Abelian representations of the braid group.", "However, in the present paper, we use the word “non-Abelian statistics\" only for exchanges of two identical states, which should be a subgroup of the whole representation.", "The present paper discusses the non-Abelian representation of a braid group for the exchange of vortices which have multiple Dirac fermions with non-Abelian symmetry, to complete the series of analyses.", "We then show that it indeed contains genuine non-Abelian statistics of the exchanges of identical particles, that is, vortices with the same numbers of Dirac fermions.", "This result is somewhat surprising compared with conventional understanding because non-Abelian statistics appears in a system with only locally-defined Dirac fermions.", "There is an important difference between the vortices with Majorana fermions and Dirac fermions.", "Consider a system of vortices each of which traps only a single Majorana fermion.", "Recall that the Majorana fermion has a unique property that there is no distinction between a particle and a hole (anti-particle) [13].", "Thus, in order to define the Fock vacuum, one has to introduce a Dirac fermion by using two Majorana fermions that belong to different vortices [5], [6].", "Therefore, the Dirac fermions thus constructed are non-local objects.", "When there are $2m$ vortices, the total Hilbert space of the zero-energy fermions has a dimension (degeneracy) of $2^m$ , where each zero-energy Dirac fermion has the dimension 2 (empty or occupied).", "On the other hand, in the system of vortices each of which traps a single Dirac fermion, we can immediately construct the Hilbert space without introducing `non-local' Dirac fermions [11], and find a non-Abelian representation of a braid group (but not non-Abelian statistics) for the exchange of two vortices.", "When there are $2m$ vortices, the dimension of the Hilbert space is $2^{2m}=4^m$ .", "In the Majorana case, increasing the number of Majorana fermions in a single vortex brings in an interesting nontrivial structure.", "When the multiple Majorana fermions are in the vector representations of SO(3) [8] and, in general, SO($2N-1$ ) [10], the exchange matrices of two vortices are given as tensor products of the matrices that appear in the single Majorana case (called the Ivanov matrices) and generators of the Coxeter group of the $A_{2m-1}$ type (for $2m$ vortices).", "Besides, the dimension of the Hilbert space of zero-energy fermions becomes larger than that of the single Majorana case by the internal degrees of freedom.", "For example, when three (triplet) Majorana fermions with SO(3) symmetry are trapped, the dimension is $2^{3m}$ which should be compared with $2^m$ for the case with a single Majorana fermion.", "It is not known if a similar structure exists and how the dimension of the Hilbert space is enlarged, when increasing the number of Dirac fermions at each vortex.", "The purpose of the present paper is to show the explicit forms of the non-Abelian representation of a braid group for vortices with multiple Dirac fermions, and that it contains genuine non-Abelian statistics in the sectors of the exchange of vortices with the same numbers of Dirac fermions.", "As the simplest but non-trivial example, we focus on the Dirac fermions with U(2) symmetry.", "Extension to general cases will become more complicated, but should be straightforward.", "It should be noticed that, since the vortices with zero-energy fermions are characterized by topology, they appear in many different quantum systems.", "For example, vortices with the Majorana fermions are realized in chiral $p$ -wave superconductors [3], such as Sr$_2$ RuO$_4$ [14], in chiral $p$ -wave superfluids, such as the A-phase of $^3$ He in 2+1 dimensions, and also in other systems [15], [16].", "Vortices with the Dirac fermions are identified with the integer (singular) vortices in the $^3$ He A-phase in 2+1 dimensions [17] the normal “$o$ \" vortices in the $^3$ He B-phase in 3+1 dimensions [18] and also with dislocation lines in topological insulators [19].", "In these examples, the vortex has only a single zero-energy Majorana or Dirac fermion at its core.", "So far, there is no condensed-matter example of vortices with more than two Majorana fermions.", "However, we know at least two examples in high-energy physics.", "In fact, the existence of zero-energy modes in the vortex-fermion system was discussed long time ago in the context of relativistic quantum field theory [20].", "The primary example is the color superconductor in QCD which could exist in extremely high density matter such as in the cores of neutron stars [21].", "In particular, vortices with non-Abelian symmetries appear in the color-flavor locked (CFL) phase where the original color SU(3) and flavor SU(3) symmetry in the vacuum breaks down to SU(3) with color and flavor degrees are locked [22].", "The SU(3) CFL symmetry is further broken down to a U(2) symmetry in the core of non-Abelian vortices [23].", "It has been recently shown [24] by using the index theorem that there appear two types of non-Abelian vortices: the vortex which traps a triplet of zero-energy Majorana fermions [9] and the vortex which traps an U(2) doublet of zero-energy Dirac fermions.", "While the former gives an example of non-Abelian statistics of vortices with multiple Majorana fermions [8], the latter gives an example of non-Abelian representation of a braid group for the exchange of vortices with doublet Dirac fermions focused in the present paper.", "The secondary example is the non-Abelian vortices in supersymmetric QCD with U($N$ ) gauge symmetry [25] (see Ref.", "[26] for reviews).", "In this case, the color U($N$ ) symmetry and the flavor SU($N$ ) symmetry are spontaneously broken down to the SU($N$ ) color-flavor locked symmetry by the scalar quark condensates in the vacuum, and it is further broken down to a U($N-1$ ) symmetry in the core of non-Abelian vortices.", "These vortices contain one singlet and one (${N-1}$ )-plet of zero-energy Dirac fermions in their core.", "Therefore, it gives an example of non-Abelian statistics of vortices with arbitrary number of Dirac fermions.", "Before finishing Introduction, let us briefly comment on the potential application to quantum computers [28], [27].", "As discussed above, the system of vortices with zero-energy fermions is robust against small perturbations from environment, and has the Hilbert space with a large dimension.", "These are desirable properties as quantum computers.", "Comparing the Majorana and Dirac cases, it should be noticed that the Dirac case is simpler because we do not have to introduce non-local Dirac fermions.", "Since vortex systems with non-Abelian symmetry have larger dimensions, it would be worth to consider the case with non-Abelian Dirac fermions, even though it has not been realized in laboratory.", "This paper is organized as follows.", "In Section , we briefly summarize the non-Abelian statistics for the vortices trapping a single Dirac fermion with U(1) symmetry at each core, as presented in Ref. [11].", "In Section , we discuss the non-Abelian representation of the braid group for the exchange of the vortices trapping doublet Dirac fermions with U(2) symmetry.", "In Section , we discuss the difference between non-Abelian representations of the braid group for the exchange of the Dirac fermion with U(1) symmetry and that with U(2) symmetry.", "In Section , we show that the whole braid group contains genuine non-Abelian statistics as its subgroup, in the sectors of the vortices with the same number of Dirac fermions.", "Section is devoted to a summary.", "In Appendices, we present detailed supplementary information.", "In Appendix , we give the transformation matrices for U(1) (singlet) Dirac fermions.", "In Appendix we give the Hilbert space and the exchange matrices for $n=4$ U(2) Dirac vortices.", "In Appendix , we discuss restricted Hilbert subspaces, in which two successive exchanges of vortices is equal to the identity." ], [ "Non-Abelian representation of braid group for exchange of\nU(1) Dirac vortices", "Let us first explain how the non-Abelian representation of the braid group appears in the system of vortices having a single Dirac fermion in each core, which corresponds to considering the Dirac fermion with the U(1) symmetry.", "This is a brief summary of the recent work done by three of us [11].", "We highlight the similarities to and differences from the case with Majorana fermions.", "Below, we call vortices with the Dirac fermions “the Dirac vortices”, while vortices with the Majorana fermions, “the Majorana vortices”.", "Consider $n$ Dirac vortices which are labelled by $k=1,\\cdots , n$ .", "The number of vortices, $n$ , can be arbitrary in contrast with the case of Majorana vortices where we define Dirac fermions by using two Majorana fermions, and thus the total fermion number is even.", "The operator $\\hat{\\psi }_k$ denotes the Dirac fermion of the $k$ -th vortex.", "Together with its hermitian conjugate $\\hat{\\psi }^\\dag _k$ , they satisfy the following algebra: $\\lbrace \\hat{\\psi }_{k}, \\hat{\\psi }_{\\ell }^{\\dag } \\rbrace =\\delta _{k\\ell }, \\quad \\lbrace \\hat{\\psi }_{k}, \\hat{\\psi }_{\\ell } \\rbrace =0, \\quad \\lbrace \\hat{\\psi }_{k}^{\\dag }, \\hat{\\psi }_{\\ell }^{\\dag } \\rbrace =0.$ We regard $\\hat{\\psi }_k$ and $\\hat{\\psi }_k^\\dag $ as the annihilation and creation operators, respectively.", "Exchange of the $k$ -th and $(k+1)$ -th vortices, which is denoted by $T_{k}$ , induces the following exchange of $\\hat{\\psi }_{k}$ and $\\hat{\\psi }_{k+1}$ , $T_k : \\left\\lbrace \\begin{array}{l}\\hat{\\psi }_{k} \\quad \\rightarrow \\, \\hat{\\psi }_{k+1} \\\\\\hat{\\psi }_{k+1} \\rightarrow -\\hat{\\psi }_{k}\\end{array}\\right.", ",$ with the rest $\\hat{\\psi }_{\\ell }$ ($\\ell \\ne k$ and $k+1$ ) unchanged.", "We note that $T_{k}$ 's satisfy the braid relations, $\\mbox{(i) } &&T_{k}T_{\\ell }T_{k}=T_{\\ell }T_{k}T_{\\ell }\\quad \\mbox{ for } \\quad |k-\\ell |=1\\, , \\\\\\mbox{(ii) } && T_{k}T_{\\ell }=T_{\\ell }T_{k}\\quad \\mbox{ for }\\quad |k-\\ell |>1\\, ,$ as a general rule of exchange operations.", "Notice that these relations and the transformation (REF ) are the same as in the case of the Majorana vortices, while the fermion operators $\\hat{\\psi }_k$ and $\\hat{\\psi }^\\dag _k$ now satisfy Eq.", "(REF ), instead of the Clifford algebra $\\lbrace \\hat{\\gamma }_k,\\hat{\\gamma }_\\ell \\rbrace =2\\delta _{k\\ell }$ satisfied by the Majorana fermion operators $\\hat{\\gamma }_k$ 's.", "Therefore, one has the same property of the operator $T_k$ , namely, four-time successive application of $T_k$ is equivalent to the identity, $(T_k)^4=1$ .", "This fact by itself suggests that the exchange of these vortices shows a representation of the braid group different from the ordinary Bose-Einstein, Fermi-Dirac or Abelian anyon one which shows Abelian representation of the braid group.", "In fact, the exchange statistics of the Majorana vortices showing the same property $(T_k)^4=1$ turned out to be non-Abelian [6], [8], [10], and we will discuss below if the same is true for the Dirac vortices.", "The transformation (REF ) can be represented by the following unitary operator $\\hat{\\tau }_{k}^{\\rm s} \\equiv 1+ \\hat{\\psi }^{}_{k+1} \\hat{\\psi }_{k}^{\\dag }+ \\hat{\\psi }_{k+1}^{\\dag } \\hat{\\psi }^{}_{k}- \\hat{\\psi }_{k+1}^{\\dag } \\hat{\\psi }^{}_{k+1}- \\hat{\\psi }_{k}^{\\dag } \\hat{\\psi }^{}_{k}+ 2 \\hat{\\psi }_{k+1}^{\\dag }\\hat{\\psi }_{k+1}^{}\\hat{\\psi }_{k}^{\\dag } \\hat{\\psi }_{k}^{}\\, ,$ so that $\\hat{\\tau }_{k}^{\\rm s} \\hat{\\psi }_{\\ell } (\\hat{\\tau }_{k}^{\\rm s})^{-1}$ ($\\ell =1, \\cdots , n$ ) reproduces the transformation law.", "The superscript “$\\mathrm {s}$ ” implies the singlet Dirac fermion.", "One can also confirm by a straightforward calculation that $\\hat{\\tau }_{k}$ 's satisfy the braid relations, $\\mbox{(i')} && \\hat{\\tau }_{k}^{\\rm s} \\hat{\\tau }_{\\ell }^{\\rm s}\\hat{\\tau }_{k}^{\\rm s} = \\hat{\\tau }_{\\ell }^{\\rm s} \\hat{\\tau }_{k}^{\\rm s}\\hat{\\tau }_{\\ell }^{\\rm s} \\quad \\mbox{ for } \\quad |k-\\ell |=1\\, ,\\\\\\mbox{(ii')} && \\hat{\\tau }_{k}^{\\rm s} \\hat{\\tau }_{\\ell }^{\\rm s} =\\hat{\\tau }_{\\ell }^{\\rm s} \\hat{\\tau }_{k}^{\\rm s}\\quad \\mbox{ for } \\quad |k-\\ell |>1\\, .$ Having the explicit form of the exchange operator (REF ), we are able to check the representation of the braid group for the exchange of the Dirac vortices.", "First of all, as we mentioned above, four successive exchanges of the $k$ -th and $(k+1)$ -th vortices indeed yield the identity: $(\\hat{\\tau }_{k}^{\\rm s})^{4}=1\\, , $ while two successive exchanges do not, $(\\hat{\\tau }_{k}^{\\rm s})^{2} =(1-2\\hat{\\psi }_{k}^{\\dag }\\hat{\\psi }^{}_{k})(1-2\\hat{\\psi }_{k+1}^{\\dag }\\hat{\\psi }^{}_{k+1})\\ne 1 \\, .$ Next, it should be noted that $\\hat{\\tau }_{k}^{\\rm s}$ and $\\hat{\\tau }_{k+1}^{\\rm s}$ are not commutative; $\\left[\\hat{\\tau }_{k}^{\\rm s}\\, ,\\hat{\\tau }_{k+1}^{\\rm s}\\right] \\ne 0.$ Therefore, one observes at the operator level that the exchange operation of two Dirac vortices is in general non-Abelian.", "However, this observation does not necessarily mean that the exchange is always non-Abelian.", "In order to confirm whether the exchange operation is indeed non-Abelian, we have to check if the commutator $\\left[\\hat{\\tau }_{k}^{\\rm s}\\, , \\hat{\\tau }_{k+1}^{\\rm s}\\right]$ does not vanish in the matrix representation.", "Let us consider the matrix representation of the operator $\\hat{\\tau }_{k}^{\\rm s}$ .", "As a basis of the Hilbert space, we choose the Fock states defined by the Dirac fermion operators $\\hat{\\psi }_{k}$ 's.", "One of the merits of the Dirac vortices is that we can construct the Hilbert space naturally by using locally-defined Dirac fermions, which is in clear contrast with the Majorana case.", "We first define the Fock vacuum state $|0\\rangle $ by $\\hat{\\psi }_{\\ell }|0\\rangle =0 \\quad \\mbox{ for all } \\ \\ell .$ Then, by acting successively the creation operators $\\hat{\\psi }_{\\ell }^{\\dag }$ 's on the vacuum, we obtain the $f$ -particle state ($0 \\le f \\le n$ ) [0] $&&|0\\cdots 0 1 \\cdots 1 \\cdots 1 0\\cdots 0\\rangle =\\hat{\\psi }^{\\dag }_{\\ell _{1}} \\dots \\hat{\\psi }^{\\dag }_{\\ell _{i}} \\dots \\hat{\\psi }^{\\dag }_{\\ell _{f}} |0\\rangle \\, , \\\\&&\\ \\check{1}\\qquad \\check{\\ell }_{1}\\quad \\, \\check{\\ell }_{i}\\quad \\,\\check{\\ell }_{f}\\quad \\ \\, \\check{n}\\nonumber $ in which the $\\ell _{i}$ -th ($i=1, \\cdots , f$ ) vortex is occupied by a Dirac fermion, while the other vortices are empty.", "When we have only one vortex, $n=1$ ($k$ -th vortex), there are two Fock states, $|0\\rangle $ and $|1\\rangle \\equiv \\hat{\\psi }_{k}^{\\dag }|0\\rangle $ .", "When we have two vortices, $n=2$ ($k$ -th and $(k+1)$ -th vortices), there are $2^2=4$ Fock states, $|00\\rangle \\equiv |0\\rangle $ , $|10\\rangle \\equiv \\hat{\\psi }_{k}^{\\dag }|0\\rangle $ , $|01\\rangle \\equiv \\hat{\\psi }_{k+1}^{\\dag }|0\\rangle $ and $|11\\rangle \\equiv \\hat{\\psi }_{k}^{\\dag } \\hat{\\psi }_{k+1}^{\\dag }|0\\rangle $ .", "We can similarly obtain the Fock states for any number of vortices $n$ .", "The basis of the whole Hilbert space for the $n$ -vortex system is given by a tensor product of the 4 Fock states constructed at each vortex.", "Because the fermion number operator $\\hat{f}^{\\mathrm {s}}\\equiv \\sum _{i=1}^{n} \\hat{\\psi }_{i}^{\\dag }\\hat{\\psi }^{}_{i}$ commutes with $\\hat{\\tau }_{\\ell }^{\\mathrm {s}}$ for $\\ell =1, \\cdots ,n-1$ , the fermion number $f$ (an eigenvalue of $\\hat{f}^{\\mathrm {s}}$ ) is a conserved quantity under the exchange of vortices.", "Thus, the whole Hilbert space $\\mathbb {H}^{(n)}$ for $n$ vortices (with the dimension $2^n$ ) can be decomposed into subspaces $\\mathbb {H}^{(n,f)}$ labelled further by the total fermion number $f$ : $\\mathbb {H}^{(n)} = \\oplus _{f=1}^{n} \\mathbb {H}^{(n,f)}$ .", "Then, in each subspace, the operators $\\hat{\\tau }_{\\ell }^{\\rm s}$ 's are represented by matrices, whose explicit expressions are shown up to $n=4$ in Appendix .", "One can confirm that the matrix representations of the exchange operators $\\hat{\\tau }_{\\ell }$ 's are indeed non-Abelian for $n\\ge 3$ ." ], [ "Non-Abelian representation of braid group for exchange of\nU(2) Dirac vortices", "Now let us turn to the case with vortices which trap two massless Dirac fermions having the “pseudo-spin” U(2) symmetry.", "In particular, we consider Dirac fermions in the doublet of U(2), which are denoted by $\\hat{\\psi }_{k}^{a}$ $(a=1,2)$ for the $k$ -th vortex.", "We also use the vector notation for the doublet $(\\hat{\\psi }_{k}^{1},\\hat{\\psi }_{k}^{2})^{\\rm t}$ .", "The exchange of the $k$ -th and $(k+1)$ -th vortices induces transformation of the Dirac fermions.", "Here we consider the simplest transformation similarly as the single Dirac fermion shown in Eq.", "(REF ): $T_k : \\left\\lbrace \\begin{array}{l}\\hat{\\psi }_{k}^{a} \\quad \\rightarrow \\, \\hat{\\psi }_{k+1}^{a} \\\\\\hat{\\psi }_{k+1}^{a} \\rightarrow -\\hat{\\psi }_{k}^{a}\\end{array}\\right.", ",$ for each $a=1,2$ , with the rest $\\hat{\\psi }_{\\ell }^{a}$ ($\\ell \\ne k$ and $k+1$ ) unchanged.", "This transformation is expressed by the unitary operator $\\hat{\\tau }_{k} &\\equiv & \\prod _{a=1,2} \\left(1 + \\hat{\\psi }_{k+1}^{a}\\hat{\\psi }_{k}^{a\\dag } + \\hat{\\psi }_{k+1}^{a\\dag }\\hat{\\psi }_{k}^{a} - \\hat{\\psi }_{k+1}^{a\\dag }\\hat{\\psi }_{k+1}^{a} - \\hat{\\psi }_{k}^{a\\dag }\\hat{\\psi }_{k}^{a} + 2\\hat{\\psi }_{k+1}^{a\\dag }\\hat{\\psi }_{k+1}^{a} \\hat{\\psi }_{k}^{a\\dag }\\hat{\\psi }_{k}^{a}\\right),$ which is invariant under the U(2) transformation, $(\\hat{\\psi }_{k}^{1},\\hat{\\psi }_{k}^{2})^{\\rm t} \\rightarrow \\exp (i\\varphi ) \\exp (i\\vec{\\theta } \\cdot \\vec{\\sigma }/2)(\\hat{\\psi }_{k}^{1},\\hat{\\psi }_{k}^{2})^{\\rm t}$ with $\\varphi $ and $\\vec{\\theta }$ being parameters and $\\vec{\\sigma }$ the Pauli matrices.", "We confirm that $\\hat{\\tau }^{}_{k} \\hat{\\psi }_{\\ell }^{a} \\hat{\\tau }_{k}^{-1}$ ($a=1$ , 2) reproduces the transformation (REF ), and that $\\hat{\\tau }_{k}$ 's satisfy the braid relations, (i') $\\hat{\\tau }_{k}\\hat{\\tau }_{\\ell }\\hat{\\tau }_{k}=\\hat{\\tau }_{\\ell }\\hat{\\tau }_{k}\\hat{\\tau }_{\\ell }$ for $|k-\\ell |=1$ and (ii') $\\hat{\\tau }_{k}\\hat{\\tau }_{\\ell }=\\hat{\\tau }_{\\ell }\\hat{\\tau }_{k}$ for $|k-\\ell |>1$ .", "Similarly as in the U(1) case (see Eqs.", "(REF ) and (REF )), the exchange operator $\\hat{\\tau }_k$ satisfies $(\\hat{\\tau }_{k})^{4} = 1\\, ,$ while two successive exchanges do not go back to the identity at the operator level: $(\\hat{\\tau }_{k})^{2} = (1-2\\hat{\\psi }_{k}^{1\\,\\dag }\\hat{\\psi }_{k}^{1}) (1-2\\hat{\\psi }_{k}^{2\\,\\dag }\\hat{\\psi }_{k}^{2}) (1-2\\hat{\\psi }_{k+1}^{1\\,\\dag }\\hat{\\psi }_{k+1}^{1}) (1-2\\hat{\\psi }_{k+1}^{2\\,\\dag }\\hat{\\psi }_{k+1}^{2}) \\ne 1\\, .$ We can also check that $\\hat{\\tau }_k$ and $\\hat{\\tau }_{k+1}$ are non-commutative, which suggests that the representation of the braid group for the exchange of U(2) Dirac vortices is non-Abelian.", "We define the number operator of the Dirac fermions $\\hat{f} = \\sum _{\\ell =1}^{n} \\sum _{a=1,2} \\hat{\\psi }_{\\ell }^{a\\,\\dag } \\hat{\\psi }_{\\ell }^{a},$ whose eigenvalues $f$ give the total number of zero-energy Dirac fermions in the vortex system ($0 \\le f \\le 2n$ ).", "We note that $\\sum _{a=1,2}\\psi _{k}^{a\\,\\dag } \\psi _{k}^{a}$ is U(2) invariant, hence $\\hat{f}$ is also U(2) invariant.", "We also note that $\\hat{f}$ commutes with $\\hat{\\tau }_{\\ell }$ for $\\ell =1,\\cdots ,n-1$ , hence $f$ is a conserved quantity under the exchange of two Dirac vortices.", "We can construct the Hilbert space by successively applying $\\hat{\\psi }_{\\ell }^{a\\dag }$ to the Fock vacuum $|0\\rangle $ defined by $\\hat{\\psi }_{\\ell }^{a}|0\\rangle =0\\ \\mbox{ for all \\ $\\ell $ \\ and \\ $a=1$, $2$}.$ Below, we explicitly construct the Hilbert spaces when the numbers of vortices are $n=1,2,3$ , and see the non-Abelian properties of the exchange operation in the matrix representations for $n=2$ and 3.", "The results for $n=4$ vortices are quite involved and are shown in Appendix ." ], [ "The case of $n=1$", "When we have only one vortex, we are not able to discuss the exchange of vortices.", "However, let us consider this case to demonstrate how to construct the Hilbert space for the U(2) Dirac vortex.", "We have two massless Dirac fermion operators $\\hat{\\psi }_k^1$ and $\\hat{\\psi }^2_k$ (to avoid notational confusion, we use the label $k$ to specify this single vortex).", "By applying the creation operators $\\hat{\\psi }^{1\\dag }_{k}$ and $\\hat{\\psi }^{2\\dag }_{k}$ to the Fock vacuum $| 0 \\rangle $ , we obtain $2\\times 2=4$ energetically degenerate states (2 for empty/occupied, and another 2 for $a=1,2$ ).", "Let us introduce the notation $\\vert {\\cal R}_f \\rangle $ where ${\\cal R}$ is the representation of U(2) group and $f$ is the total number of the Dirac fermions.", "Namely, $| {\\bf 1}_{0} \\rangle &\\equiv & | 0 \\rangle , \\nonumber \\\\| {\\bf 2}_{1} \\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\\\| {\\bf 1}_{2} \\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } | 0 \\rangle .\\nonumber $ The bold numbers 1, 2 imply the singlet and doublet representations of the U(2) group.", "Notice that the fully occupied state $\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } | 0 \\rangle $ is invariant under the U(2) transformation, thus it belongs to the singlet representation.", "Hence, there are two singlet states $| {\\bf 1}_{0} \\rangle $ (empty) and $| {\\bf 1}_{2} \\rangle $ (fully-occupied by two fermions), and one doublet state $| {\\bf 2}_{1} \\rangle $ (occupied by one fermion).", "We use these states as the basis to span the whole Hilbert space of the zero-energy states.", "Therefore, we have decomposed the representations of the U(2) pseudo-spin as ${\\bf 1}_{0} + {\\bf 2}_{1} + {\\bf 1}_{2}\\, , $ and correspondingly the whole Hilbert space of a single vortex $\\mathbb {H}^{\\lbrace n=1\\rbrace }$ into a direct sum $\\mathbb {H}^{\\lbrace n=1\\rbrace } = \\mathbb {H}^{{\\bf 1}_{[0]}} \\oplus \\mathbb {H}^{{\\bf 2}_{[1]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[2]}},$ where $\\mathbb {H}^{{\\bf 1}_{[0]}} \\equiv \\lbrace | {\\bf 1}_{0} \\rangle \\rbrace $ , $\\mathbb {H}^{{\\bf 2}_{[1]}} \\equiv \\lbrace | {\\bf 2}_{1} \\rangle \\rbrace $ and $\\mathbb {H}^{{\\bf 1}_{[2]}} \\equiv \\lbrace | {\\bf 1}_{2} \\rangle \\rbrace $ ." ], [ "The case of $n=2$", "Consider the case when we have two Dirac vortices which are respectively labelled by $k$ and $k+1$ .", "First of all, we can use the decomposition for a single Dirac vortex, Eq.", "(REF ), to decompose the whole Hilbert space of two Dirac vortices into representations.", "Since each vortex contains the representations $({\\bf 1}_{0} + {\\bf 2}_{1} + {\\bf 1}_{2})_{{\\rm vortex}\\,\\ell }$ ($\\ell =k$ , $k+1$ ), the total representation is obtained as a tensor product of them, which can be decomposed as $&&\\hspace{-56.9055pt}({\\bf 1}_{0} + {\\bf 2}_{1} + {\\bf 1}_{2})_{{\\rm vortex}\\, k} \\otimes ({\\bf 1}_{0} + {\\bf 2}_{1} + {\\bf 1}_{2})_{{\\rm vortex}\\, k+1} \\nonumber \\\\&=& {\\bf 1}_{00} + {\\bf 1}_{11} + {\\bf 1}_{20} + {\\bf 1}_{02} + {\\bf 1}_{22} \\nonumber \\\\&+& {\\bf 2}_{10} + {\\bf 2}_{01} + {\\bf 2}_{21} + {\\bf 2}_{12} \\nonumber \\\\&+& {\\bf 3}_{11}, $ where the bold numbers denote representations, and the subscript $n_{k}n_{k+1}$ ($n_{k}$ , $n_{k+1}=0$ , 1, 2) denotes the number of the Dirac fermions, $n_{k}$ and $n_{k+1}$ , at the $k$ -th and $(k+1)$ -th vortices, respectively.", "Next, one obtains the basis of the whole Hilbert space by applying $\\hat{\\psi }_{\\ell }^{a\\dag }$ ($\\ell =k$ , $k+1$ and $a=1$ , 2) successively to the Fock vacuum $| 0 \\rangle $ defined by $\\hat{\\psi }_{\\ell }^{a} | 0 \\rangle = 0$ for all $\\ell $ and $a=1$ , 2.", "Then, one reorganizes the Fock states according to the decomposition into representations.", "Below, we explicitly show the basis of the Hilbert space according to the decomposition in Eq.", "(REF ).", "We introduce the notations $\\vert {\\cal R}_{n_{k} n_{k+1}}\\rangle $ for the basis states and $\\mathbb {H}^{{\\cal R}_{[N_{1} N_{2}]}}$ for the subspaces of the Hilbert space.", "Here ${\\cal R}$ denotes the representation of pseudo-spin, and the subscript $n_{k}n_{k+1}$ ($n_{k}, n_{k+1}=0,1,2$ ) denotes the number of the Dirac fermions at the $k$ -th and $(k+1)$ -th vortices, respectively.", "In the subspaces, we do not distinguish $n_{k}$ and $n_{k+1}$ for the reasons mentioned later, and thus $N_1$ and $N_2$ are defined as $N_1=\\max \\lbrace n_{k},n_{k+1}\\rbrace $ and $N_2=\\min \\lbrace n_{k},n_{k+1}\\rbrace $ .", "For example, $(N_1,N_2)=(1,0)$ contains two cases $(n_{k},n_{k+1})=(1,0),(0,1)$ .", "Then, the whole Hilbert space of two Dirac vortices is decomposed into seven subspaces: $\\mathbb {H}^{\\lbrace n=2\\rbrace }&=& \\mathbb {H}^{{\\bf 1}_{[00]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[11]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[20]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[22]}} \\nonumber \\\\& \\oplus & \\mathbb {H}^{{\\bf 2}_{[10]}} \\oplus \\mathbb {H}^{{\\bf 2}_{[21]}}\\nonumber \\\\& \\oplus & \\mathbb {H}^{{\\bf 3}_{[11]}}\\, .", "$ This should be compared with the decomposition (REF ).", "The first line in Eq.", "(REF ) corresponds to the singlet subspaces.", "They are defined by the basis states as $\\mathbb {H}^{{\\bf 1}_{[00]}} \\equiv \\lbrace | {\\bf 1}_{00} \\rangle \\rbrace $ , $\\mathbb {H}^{{\\bf 1}_{[11]}} \\equiv \\lbrace | {\\bf 1}_{11} \\rangle \\rbrace $ , $\\mathbb {H}^{{\\bf 1}_{[20]}} \\equiv \\lbrace | {\\bf 1}_{20} \\rangle , | {\\bf 1}_{02} \\rangle \\rbrace $ , and $\\mathbb {H}^{{\\bf 1}_{[22]}} \\equiv \\lbrace | {\\bf 1}_{22} \\rangle \\rbrace $ .", "Explicit forms of the basis states are given as follows: $| {\\bf 1}_{00} \\rangle &\\equiv & | 0 \\rangle ,\\\\| {\\bf 1}_{11} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) | 0 \\rangle ,\\\\| {\\bf 1}_{20} \\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 1}_{02} \\rangle &\\equiv & \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle ,\\\\| {\\bf 1}_{22} \\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle .$ The second line in Eq.", "(REF ) corresponds to the doublet subspaces.", "They are defined by the basis states as $\\mathbb {H}^{{\\bf 2}_{[10]}} \\equiv \\lbrace | {\\bf 2}_{10} \\rangle , | {\\bf 2}_{01} \\rangle \\rbrace $ and $\\mathbb {H}^{{\\bf 2}_{[21]}} \\equiv \\lbrace | {\\bf 2}_{21} \\rangle , | {\\bf 2}_{12} \\rangle \\rbrace $ .", "Explicit forms of the basis states are given as follows: $| {\\bf 2}_{10} \\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\nonumber \\\\| {\\bf 2}_{01} \\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) | 0 \\rangle ,\\\\| {\\bf 2}_{21} \\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\nonumber \\\\| {\\bf 2}_{12} \\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle .$ Here, in the last two expressions, we have factored out the singlet part so that the vector structure becomes manifest.", "Lastly, the third line in Eq.", "(REF ) corresponds to the triplet subspace.", "It is defined by the basis states as $\\mathbb {H}^{{\\bf 3}_{[11]}} \\equiv \\lbrace | {\\bf 3}_{11} \\rangle \\rbrace $ with $| {\\bf 3}_{11} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } +\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) | 0 \\rangle .$ Now we have prepared to discuss the exchange of vortices.", "We recall that the exchange of $\\hat{\\psi }_{k}^{a}$ and $\\hat{\\psi }_{k+1}^{a}$ is expressed by the operator $\\hat{\\tau }_{k}$ defined in Eq.", "(REF ) as a unitary transformation $\\hat{\\tau }_{k} \\hat{\\psi }_{\\ell }^{a} \\hat{\\tau }_{k}^{-1}$ ($\\ell =k$ , $k+1$ and $a=1$ , 2).", "With the explicit forms of the basis states in Hilbert subspaces, we are able to express the operator $\\hat{\\tau }_{k}$ as matrices.", "In the singlet subspaces $\\mathbb {H}^{{\\bf 1}_{[00]}}$ , $\\mathbb {H}^{{\\bf 1}_{[11]}}$ , $\\mathbb {H}^{{\\bf 1}_{[20]}}$ and $\\mathbb {H}^{{\\bf 1}_{[22]}}$ , the corresponding matrices are $\\tau _{k}^{{\\bf 1}_{[00]}} &=& 1, \\nonumber \\\\\\tau _{k}^{{\\bf 1}_{[11]}} &=& -1, \\nonumber \\\\\\tau _{k}^{{\\bf 1}_{[20]}} &=&\\left(\\begin{array}{cc}0 & 1 \\\\1 & 0\\end{array}\\right), \\nonumber \\\\\\tau _{k}^{{\\bf 1}_{[22]}} &=& 1\\, .", "\\nonumber $ In the doublet subspaces $\\mathbb {H}^{{\\bf 2}_{[10]}}$ and $\\mathbb {H}^{{\\bf 2}_{[21]}}$ , the corresponding matrices are $\\tau _{k}^{{\\bf 2}_{[10]}} &=&\\left(\\begin{array}{cc}0 & -1 \\\\1 & 0\\end{array}\\right), \\nonumber \\\\\\tau _{k}^{{\\bf 2}_{[21]}} &=&\\left(\\begin{array}{cc}0 & 1 \\\\-1 & 0\\end{array}\\right)\\, .", "\\nonumber $ Lastly, in the triplet subspace $\\mathbb {H}^{{\\bf 3}_{[11]}}$ , the matrix is $\\tau _{k}^{{\\bf 3}_{[11]}} &=& 1\\, .", "\\nonumber $ Since $\\mathbb {H}^{{\\bf 1}_{[00]}}$ , $\\mathbb {H}^{{\\bf 1}_{[11]}}$ , $\\mathbb {H}^{{\\bf 1}_{[22]}}$ in the singlet subspace and $\\mathbb {H}^{{\\bf 3}_{[11]}}$ in the triplet subspace consist of only one basis state, we have one-dimensional representations.", "In contrast, the other subspaces $\\mathbb {H}^{{\\bf 1}_{[20]}}$ , $\\mathbb {H}^{{\\bf 2}_{[10]}}$ , and $\\mathbb {H}^{{\\bf 2}_{[21]}}$ have two basis states, thus yielding two-dimensional representations.", "Notice that the exchange matrices in these subspaces have off-diagonal elements.", "However, this does not mean that the representation of the braid group is non-Abelian.", "Rather, this simply implies that two basis states are mixed with each other by the exchange operation (this is the reason why we did not specify the order of $n_{k}$ and $n_{k+1}$ in defining the subspaces).", "In fact, one can choose appropriate basis states so that the exchange matrices are expressed as diagonal.", "For example, $\\tau _{k}^{{\\bf 1}_{[20]}}$ is diagonalized with eigenvalues $\\pm 1$ showing the Abelian representation of the braid group, while $\\tau _{k}^{{\\bf 2}_{[10]}}$ and $\\tau _{k}^{{\\bf 2}_{[21]}}$ are diagonalized with $\\pm i$ , showing anyon-like Abelian representation of the braid group." ], [ "The case of $n=3$ : emergence of non-Abelian representation", "Let us finally consider the case of three Dirac vortices which are respectively labelled by $k$ , $k+1$ and $k+2$ .", "Basically we will follow the same procedures presented in the case of two Dirac vortices, but as easily expected, the analysis becomes quite involved.", "Still, we present here all the information since this is the simplest case where the non-Abelian representation of the braid group appears.", "In fact, when we have only two vortices ($k$ -th and $(k+1)$ -th vortices), there is only one exchange operator $\\hat{\\tau }_k$ , thus we are not able to discuss the non-commutativity of two exchange operations.", "It makes sense only when we have three or more vortices.", "When we have three vortices ($k$ -th, $(k+1)$ -th and $(k+2)$ -th vortices), we can define two exchange operators $\\hat{\\tau }_k$ and $\\hat{\\tau }_{k+1}$ , and thus we can check if these two are commutative or not.", "First of all, by using Eq.", "(REF ) for a single Dirac vortex, we can decompose the pseudo-spin structure made of three Dirac vortices into representations of U(2).", "We again introduce the notation ${\\cal R}_{n_{k}n_{k+1}n_{k+2}}$ similar to the $n=2$ case.", "Here the subscript $n_{k}n_{k+1}n_{k+2}$ ($n_{k}$ , $n_{k+1}$ , $n_{k+2}=0$ , 1, 2) denotes the number of Dirac fermions at the $k$ -th, $(k+1)$ -th and $(k+2)$ -th vortices, respectively.", "Then, one finds $&&\\hspace{-28.45274pt}({\\bf 1}_{0} + {\\bf 2}_{1} + {\\bf 1}_{2})_{{\\rm vortex}\\, k} \\otimes ({\\bf 1}_{0} + {\\bf 2}_{1} + {\\bf 1}_{2})_{{\\rm vortex}\\, k+1} \\otimes ({\\bf 1}_{0} + {\\bf 2}_{1} + {\\bf 1}_{2})_{{\\rm vortex}\\, k+2}\\nonumber \\\\&=& {\\bf 1}_{000} + ({\\bf 1}_{020} + {\\bf 1}_{200} + {\\bf 1}_{002})+ ({\\bf 1}_{110} + {\\bf 1}_{011} + {\\bf 1}_{101})+ ({\\bf 1}_{220} + {\\bf 1}_{022} + {\\bf 1}_{202})+ ({\\bf 1}_{112} + {\\bf 1}_{211} + {\\bf 1}_{121}) + {\\bf 1}_{222} \\nonumber \\\\&+& ({\\bf 2}_{100} + {\\bf 2}_{010} + {\\bf 2}_{001})+ ({\\bf 2}_{210} + {\\bf 2}_{120} + {\\bf 2}_{012} + {\\bf 2}_{102}+ {\\bf 2}_{021} + {\\bf 2}_{201})+ ({\\bf 2}_{212} + {\\bf 2}_{122} + {\\bf 2}_{221}) + ({\\bf 2}_{{\\rm A}\\,\\underline{11}1} + {\\bf 2}_{{\\rm S}\\,\\underline{11}1})\\nonumber \\\\&+& ({\\bf 3}_{110} + {\\bf 3}_{011} + {\\bf 3}_{101}) +({\\bf 3}_{112} + {\\bf 3}_{211} + {\\bf 3}_{121}) \\nonumber \\\\&+& {\\bf 4}_{{\\rm S}\\,\\underline{11}1}\\, .$ Here we have also introduced new notations ${\\bf 2}_{{\\rm A}\\,\\underline{11}1}$ , ${\\bf 2}_{{\\rm S}\\,\\underline{11}1}$ and ${\\bf 4}_{{\\rm S}\\,\\underline{11}1}$ , meaning that the first two subscripts with underlines are made Asymmetric (Symmetric) with respect to the indices.", "We will note again when we present the explicit forms of the basis states.", "Each term corresponds to the basis state in the Hilbert subspace.", "Notice that we have already grouped the representations so as not to distinguish the ordering of $n_{k}$ , $n_{k+1}$ and $n_{k+2}$ .", "Thus, one can easily see that the whole Hilbert space $\\mathbb {H}^{\\lbrace n=3\\rbrace }$ of three U(2) Dirac vortices can be decomposed into a direct sum $\\mathbb {H}^{\\lbrace n=3\\rbrace }&=& \\mathbb {H}^{{\\bf 1}_{[000]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[200]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[110]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[220]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[211]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[222]}} \\nonumber \\\\& \\oplus & \\mathbb {H}^{{\\bf 2}_{[100]}} \\oplus \\mathbb {H}^{{\\bf 2}_{[210]}} \\oplus \\mathbb {H}^{{\\bf 2}_{[221]}} \\oplus \\mathbb {H}^{{\\bf 2}_{[111]}} \\nonumber \\\\& \\oplus & \\mathbb {H}^{{\\bf 3}_{[110]}} \\oplus \\mathbb {H}^{{\\bf 3}_{[211]}} \\nonumber \\\\& \\oplus & \\mathbb {H}^{{\\bf 4}_{[111]}}.$ Next, one can obtain the basis states of the Hilbert space by applying $\\hat{\\psi }_{\\ell }^{a\\dag }$ ($\\ell =k$ , $k+1$ , $k+2$ and $a=1$ , 2) successively to the vacuum $| 0 \\rangle $ defined by $\\hat{\\psi }_{\\ell }^{a} | 0 \\rangle = 0$ for all $\\ell $ and $a=1$ , 2.", "The first line in the decomposition (REF ) corresponds to the singlet subspaces.", "They are defined by the basis states as $\\mathbb {H}^{{\\bf 1}_{[000]}} \\equiv \\lbrace | {\\bf 1}_{000} \\rangle \\rbrace $ , $\\mathbb {H}^{{\\bf 1}_{[200]}} \\equiv \\lbrace | {\\bf 1}_{020} \\rangle , | {\\bf 1}_{200} \\rangle , | {\\bf 1}_{002} \\rangle \\rbrace $ , $\\mathbb {H}^{{\\bf 1}_{[110]}} \\equiv \\lbrace | {\\bf 1}_{110} \\rangle , | {\\bf 1}_{011} \\rangle , | {\\bf 1}_{101} \\rangle \\rbrace $ , $\\mathbb {H}^{{\\bf 1}_{[220]}} \\equiv \\lbrace | {\\bf 1}_{220} \\rangle , | {\\bf 1}_{022} \\rangle , | {\\bf 1}_{202} \\rangle \\rbrace $ , $\\mathbb {H}^{{\\bf 1}_{[211]}} \\equiv \\lbrace | {\\bf 1}_{112} \\rangle , | {\\bf 1}_{211} \\rangle , | {\\bf 1}_{121} \\rangle \\rbrace $ and $\\mathbb {H}^{{\\bf 1}_{[222]}} \\equiv \\lbrace | {\\bf 1}_{222} \\rangle \\rbrace $ .", "Notice that all these states have even numbers of Dirac fermions.", "Explicit forms of the basis states are given as follows: $\\mathbb {H}^{{\\bf 1}_{[000]}} :\\qquad && \\quad \\,| {\\bf 1}_{000} \\rangle \\equiv | 0 \\rangle ,\\\\\\mathbb {H}^{{\\bf 1}_{[200]}}: \\qquad &&\\left\\lbrace \\begin{array}{l}| {\\bf 1}_{020} \\rangle \\equiv \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 1}_{200} \\rangle \\equiv \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 1}_{002} \\rangle \\equiv \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle ,\\end{array}\\right.", "\\\\\\mathbb {H}^{{\\bf 1}_{[110]}}:\\qquad &&\\left\\lbrace \\begin{array}{l}| {\\bf 1}_{110} \\rangle \\equiv \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) | 0 \\rangle , \\\\| {\\bf 1}_{011} \\rangle \\equiv \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) | 0 \\rangle , \\\\| {\\bf 1}_{101} \\rangle \\equiv \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) | 0 \\rangle ,\\end{array}\\right.", "\\\\\\mathbb {H}^{{\\bf 1}_{[220]}}:\\qquad &&\\left\\lbrace \\begin{array}{l}| {\\bf 1}_{220} \\rangle \\equiv \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 1}_{022} \\rangle \\equiv \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 1}_{202} \\rangle \\equiv \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle ,\\end{array}\\right.\\\\\\mathbb {H}^{{\\bf 1}_{[211]}}: \\qquad &&\\left\\lbrace \\begin{array}{l}| {\\bf 1}_{112} \\rangle \\equiv \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 1}_{211} \\rangle \\equiv \\frac{1}{\\sqrt{2}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) | 0 \\rangle , \\\\| {\\bf 1}_{121} \\rangle \\equiv \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle ,\\end{array}\\right.\\\\\\mathbb {H}^{{\\bf 1}_{[222]}} : \\qquad && \\quad \\,| {\\bf 1}_{222} \\rangle \\equiv \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle .$ It is interesting to notice that the structures in $\\mathbb {H}^{{\\bf 1}_{[110]}}$ and $\\mathbb {H}^{{\\bf 1}_{[211]}}$ , or in $\\mathbb {H}^{{\\bf 1}_{[200]}}$ and $\\mathbb {H}^{{\\bf 1}_{[220]}}$ are similar.", "This is the symmetry between “occupied\" and “empty\" states (or “particles\" and “holes\"), reflecting the ambiguity in defining the creation operator (i.e., we could define $\\hat{\\psi }_k^a$ as a creation operator, instead of the annihilation operator).", "The second line of Eq.", "(REF ) corresponds to the doublet subspaces.", "These four subspaces are respectively defined by the basis states as $\\mathbb {H}^{{\\bf 2}_{[100]}} \\equiv \\lbrace | {\\bf 2}_{100}\\rangle , | {\\bf 2}_{010} \\rangle , | {\\bf 2}_{001} \\rangle \\rbrace $ , $\\mathbb {H}^{{\\bf 2}_{[210]}} \\equiv \\lbrace | {\\bf 2}_{210} \\rangle , | {\\bf 2}_{120} \\rangle , | {\\bf 2}_{012} \\rangle , | {\\bf 2}_{102} \\rangle , | {\\bf 2}_{021} \\rangle , | {\\bf 2}_{201} \\rangle \\rbrace $ , $\\mathbb {H}^{{\\bf 2}_{[221]}} \\equiv \\lbrace | {\\bf 2}_{212} \\rangle , | {\\bf 2}_{122} \\rangle , | {\\bf 2}_{221} \\rangle \\rbrace $ , and $\\mathbb {H}^{{\\bf 2}_{[111]}} \\equiv \\lbrace | {\\bf 2}_{{\\rm A}\\,\\underline{11}1} \\rangle , | {\\bf 2}_{{\\rm S}\\,\\underline{11}1} \\rangle \\rbrace $ .", "This time, the total numbers of Dirac fermions are odd in these states.", "Explicit forms of the basis states are given as follows: $\\mathbb {H}^{{\\bf 2}_{[100]}} : \\qquad &&\\left\\lbrace \\begin{array}{l}| {\\bf 2}_{100} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\\\| {\\bf 2}_{010} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\\\| {\\bf 2}_{001} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) | 0 \\rangle ,\\end{array}\\right.", "\\\\\\mathbb {H}^{{\\bf 2}_{[210]}}:\\qquad &&\\left\\lbrace \\begin{array}{l}| {\\bf 2}_{210} \\rangle \\equiv \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\\\| {\\bf 2}_{120} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 2}_{012} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 2}_{102} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 2}_{021} \\rangle \\equiv \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\\\| {\\bf 2}_{201} \\rangle \\equiv \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) | 0 \\rangle ,\\end{array}\\right.\\\\\\mathbb {H}^{{\\bf 2}_{[221]}}:\\qquad &&\\left\\lbrace \\begin{array}{l}| {\\bf 2}_{212} \\rangle \\equiv \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 2}_{122} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 2}_{221} \\rangle \\equiv \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) | 0 \\rangle ,\\end{array}\\right.\\\\\\mathbb {H}^{{\\bf 2}_{[111]}}:\\qquad &&\\left\\lbrace \\begin{array}{l}| {\\bf 2}_{{\\rm A}\\,\\underline{11}1} \\rangle \\equiv \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } )\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\\\| {\\bf 2}_{{\\rm S}\\,\\underline{11}1} \\rangle \\equiv \\frac{1}{\\sqrt{6}} \\left(\\begin{array}{c}2\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag }- ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\\\( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{2\\dag } - 2 \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag }\\end{array}\\right) | 0 \\rangle \\, .\\end{array}\\right.", "$ Now the meaning of the notations A and S is evident.", "For example, in $| {\\bf 2}_{{\\rm A}\\,\\underline{11}1} \\rangle $ the first ($k$ -th) and second (($k+1$ )-th) vortices form anti-symmetric combination with respect to the indices.", "The third line of the decomposition (REF ) corresponds to the triplet subspaces.", "They are defined by the basis states as $\\mathbb {H}^{{\\bf 3}_{[110]}} \\equiv \\lbrace | {\\bf 3}_{110} \\rangle ,| {\\bf 3}_{011} \\rangle ,| {\\bf 3}_{101} \\rangle \\rbrace $ and $\\mathbb {H}^{{\\bf 3}_{[211]}} \\equiv \\lbrace | {\\bf 3}_{112} \\rangle ,| {\\bf 3}_{211} \\rangle ,| {\\bf 3}_{121} \\rangle \\rbrace $ .", "Total fermion numbers are now even.", "Explicit forms of the basis states are given as follows: $\\mathbb {H}^{{\\bf 3}_{[110]}}:\\qquad &&\\left\\lbrace \\begin{array}{l}| {\\bf 3}_{110} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\\\| {\\bf 3}_{011} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\\\| {\\bf 3}_{101} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) | 0 \\rangle ,\\end{array}\\right.", "\\\\\\mathbb {H}^{{\\bf 3}_{[211]}}:\\qquad &&\\left\\lbrace \\begin{array}{l}| {\\bf 3}_{112} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\\\| {\\bf 3}_{211} \\rangle \\equiv \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\\\| {\\bf 3}_{121} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle .\\end{array}\\right.$ The last line of the decomposition (REF ) corresponds to the quartet subspace.", "It is defined by the basis states as $\\mathbb {H}^{{\\bf 4}_{[111]}} \\equiv \\lbrace | {\\bf 4}_{{\\rm S}\\,\\underline{11}1} \\rangle \\rbrace $ .", "Explicitly, $\\mathbb {H}^{{\\bf 4}_{[111]}}: \\qquad | {\\bf 4}_{{\\rm S}\\,\\underline{11}1} \\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag }\\hat{\\psi }_{k+2}^{2\\dag } + \\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag }\\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag }\\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } +\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{2\\dag }+ \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) | 0 \\rangle .", "\\nonumber $ Now we can finally discuss the exchange of U(2) Dirac vortices in the Hilbert space constructed for three Dirac vortices.", "As in the case of $n=2$ , we express the operators $\\hat{\\tau }_{k}$ and $\\hat{\\tau }_{k+1}$ as matrices with the basis states presented above.", "In the singlet subspaces $\\mathbb {H}^{{\\bf 1}_{[000]}}$ , $\\mathbb {H}^{{\\bf 1}_{[200]}}$ , $\\mathbb {H}^{{\\bf 1}_{[110]}}$ , $\\mathbb {H}^{{\\bf 1}_{[220]}}$ , $\\mathbb {H}^{{\\bf 1}_{[211]}}$ and $\\mathbb {H}^{{\\bf 1}_{[222]}}$ , the exchange matrices are $&& \\tau _{k}^{{\\bf 1}_{[000]}} = \\tau _{k+1}^{{\\bf 1}_{[000]}} = 1, \\nonumber \\\\&& \\tau _{k}^{{\\bf 1}_{[200]}} =\\left(\\begin{array}{ccc}0 & 1 & 0 \\\\1 & 0 & 0 \\\\0 & 0 & 1\\end{array}\\right), \\quad \\tau _{k+1}^{{\\bf 1}_{[200]}} =\\left(\\begin{array}{ccc}0 & 0 & 1 \\\\0 & 1 & 0 \\\\1 & 0 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k}^{{\\bf 1}_{[110]}} =\\left(\\begin{array}{ccc}-1 & 0 & 0 \\\\0 & 0 & 1 \\\\0 & -1 & 0\\end{array}\\right), \\quad \\tau _{k+1}^{{\\bf 1}_{[110]}} =\\left(\\begin{array}{ccc}0 & 0 & -1 \\\\0 & -1 & 0 \\\\1 & 0 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k}^{{\\bf 1}_{[220]}} =\\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & 0 & 1 \\\\0 & 1 & 0\\end{array}\\right), \\quad \\tau _{k+1}^{{\\bf 1}_{[220]}} =\\left(\\begin{array}{ccc}0 & 0 & 1 \\\\0 & 1 & 0 \\\\1 & 0 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k}^{{\\bf 1}_{[211]}} =\\left(\\begin{array}{ccc}-1 & 0 & 0 \\\\0& 0 & 1 \\\\0 & -1 & 0\\end{array}\\right), \\quad \\tau _{k+1}^{{\\bf 1}_{[211]}} =\\left(\\begin{array}{ccc}0 & 0 & -1 \\\\0 & -1 & 0 \\\\1 & 0 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k}^{{\\bf 1}_{[222]}} = \\tau _{k+1}^{{\\bf 1}_{[222]}} = 1\\, .$ In the doublet subspaces, $\\mathbb {H}^{{\\bf 2}_{[100]}}$ , $\\mathbb {H}^{{\\bf 2}_{[210]}}$ , $\\mathbb {H}^{{\\bf 2}_{[221]}}$ and $\\mathbb {H}^{{\\bf 2}_{[111]}}$ , the exchange matrices are $&& \\tau _{k}^{{\\bf 2}_{[100]}} =\\left(\\begin{array}{ccc}0 & -1 & 0 \\\\1 & 0 & 0 \\\\0 & 0 & 1\\end{array}\\right), \\quad \\tau _{k+1}^{{\\bf 2}_{[100]}} =\\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & 0 & -1 \\\\0 & 1 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k}^{{\\bf 2}_{[210]}} =\\left(\\begin{array}{cccccc}0 & 1 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & 1 & 0\\end{array}\\right), \\quad \\tau _{k+1}^{{\\bf 2}_{[210]}} =\\left(\\begin{array}{cccccc}0 & 0 & 0 & 0 & 0 & -1 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k}^{{\\bf 2}_{[221]}} =\\left(\\begin{array}{ccc}0 & 1 & 0 \\\\-1 & 0 & 0 \\\\0 & 0 & 1\\end{array}\\right), \\quad \\tau _{k+1}^{{\\bf 2}_{[221]}} =\\left(\\begin{array}{ccc}0 & 0 & -1 \\\\0 & 1 & 0 \\\\1 & 0 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k}^{{\\bf 2}_{[111]}} =\\left(\\begin{array}{cc}-1 & 0 \\\\0 & 1\\end{array}\\right), \\quad \\tau _{k+1}^{{\\bf 2}_{[111]}} =\\left(\\begin{array}{cc}\\frac{1}{2} & \\frac{\\sqrt{3}}{2} \\\\\\frac{\\sqrt{3}}{2} & -\\frac{1}{2}\\end{array}\\right)\\, .", "$ In the triplet subspaces, $\\mathbb {H}^{{\\bf 3}_{[110]}}$ and $\\mathbb {H}^{{\\bf 3}_{[211]}}$ , the exchange matrices are $&& \\tau _{k}^{{\\bf 3}_{[110]}} =\\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & 0 & 1 \\\\0 & -1 & 0\\end{array}\\right), \\quad \\tau _{k+1}^{{\\bf 3}_{[110]}} =\\left(\\begin{array}{ccc}0 & 0 & -1 \\\\0 & 1 & 0 \\\\1 & 0 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k}^{{\\bf 3}_{[211]}} =\\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & 0 & 1 \\\\0 & -1 & 0\\end{array}\\right), \\quad \\tau _{k+1}^{{\\bf 3}_{[211]}} =\\left(\\begin{array}{ccc}0 & 0 & -1 \\\\0 & 1 & 0 \\\\1 & 0 & 0\\end{array}\\right)\\, .$ Lastly, in the quartet subspace $\\mathbb {H}^{{\\bf 4}_{[111]}}$ , the exchange matrices are $&& \\tau _{k}^{{\\bf 4}_{[111]}} = \\tau _{k+1}^{{\\bf 4}_{[111]}} =1\\, .$ We now find that, except for the one-dimensional representations (in $\\mathbb {H}^{{\\bf 1}_{[000]}}$ , $\\mathbb {H}^{{\\bf 1}_{[222]}}$ and $\\mathbb {H}^{{\\bf 4}_{[111]}}$ ), all the exchange matrices are non-commutative; $[\\tau _{k}, \\tau _{k+1}] \\ne 0$ .", "Therefore, the exchange operations in these subspaces exhibit non-Abelian representation of the braid group.", "Recall that we have shown that the exchange operators $\\hat{\\tau }_{\\ell }$ 's satisfy $(\\hat{\\tau }_{\\ell })^4=1$ at the operator level (see Eq.", "(REF )).", "Of course this is satisfied by all the exchange matrices obtained above.", "However, interestingly, a stronger relation $(\\hat{\\tau }_{\\ell })^2=1$ is satisfied in some subspaces.", "In addition to the trivial one-dimensional representations in $\\mathbb {H}^{{\\bf 1}_{[000]}}$ , $\\mathbb {H}^{{\\bf 1}_{[222]}}$ and $\\mathbb {H}^{{\\bf 4}_{[111]}}$ , we observe that the exchange matrices in $\\mathbb {H}^{{\\bf 1}_{[200]}}$ , $\\mathbb {H}^{{\\bf 1}_{[220]}}$ and $\\mathbb {H}^{{\\bf 2}_{[111]}}$ satisfy this stronger relation.", "More details are discussed in Appendix .", "We can continue the construction of the Hilbert space when the number of vortices is more than three.", "The dimension of the total Hilbert space of $n$ -vortex system is $(2^2)^n$ .", "We present the results for $n=4$ vortices in Appendix , which also show non-Abelian representation of the braid group." ], [ "U(1) Dirac structure embedded in U(2) Dirac vortices", "Recall that we already found in the previous paper [11] that the system of U(1) Dirac vortices shows non-Abelian representation of the braid group.", "Then, it is natural to ask how the previous results are related to the present results.", "In this section, we discuss that we can indeed identify the same structure as the U(1) vortices by a simple reduction of the U(2) system.", "So far, we have not specified any detail of the interaction, but suppose that one can turn on an interaction which breaks the global U(2) symmetry so that only the U(1) symmetry is preserved.", "Then, in the presence of such an interaction, only one Dirac fermion would remain massless.", "We can realize such a case by simply ignoring the lower component $\\hat{\\psi }_{\\ell }^{2}$ ($\\ell =1, \\cdots , n$ ) of the U(2) Dirac fermions.", "Under this simple procedure, the Hilbert space $\\mathbb {H}^{\\lbrace n\\rbrace }$ of the U(2) Dirac vortices reduces to the Hilbert space $\\mathbb {H}^{(n)}$ of the U(1) Dirac vortices discussed in the previous paper [11] (and also in Sec.", "and Appendix ).", "Let us discuss the cases of $n=2$ and 3, separately.", "Consider first the case of $n=2$ (see Sec.", "REF ).", "One finds that only the following states survive after the reduction: $|{\\bf 1}_{00}\\rangle =|0\\rangle ,\\quad |{\\bf 2}_{10}\\rangle = \\left(\\begin{matrix}\\hat{\\psi }^{1\\dag }_k\\\\ 0\\end{matrix}\\right)|0\\rangle ,\\quad |{\\bf 2}_{01}\\rangle = \\left(\\begin{matrix}\\hat{\\psi }^{1\\dag }_{k+1}\\\\ 0\\end{matrix}\\right)|0\\rangle ,\\quad |{\\bf 3}_{11}\\rangle = \\left(\\begin{matrix}\\hat{\\psi }^{1\\dag }_k\\hat{\\psi }^{1\\dag }_{k+1}\\\\ 0\\\\ 0\\end{matrix}\\right)|0\\rangle .$ This means that the whole Hilbert space $\\mathbb {H}^{\\lbrace 2\\rbrace }$ shrinks into its subspaces $\\mathbb {H}^{{\\bf 1}_{[00]}}$ , $\\mathbb {H}^{{\\bf 2}_{[10]}}$ and $\\mathbb {H}^{{\\bf 3}_{[11]}}$ .", "Notice that these four states are equivalent to the basis states for $\\mathbb {H}^{(2,0)}$ , $\\mathbb {H}^{(2,1)}$ and $\\mathbb {H}^{(2,2)}$ in the U(1) Dirac vortices.", "Thus, it is not surprising that the exchange matrices in the reduced U(2) Dirac vortices are equivalent to those of the U(1) vortices: $\\tau _{k}^{{\\bf 1}_{[00]}} &=& \\tau _{k}^{(2,0)}, \\\\\\tau _{k}^{{\\bf 2}_{[10]}} &=& \\tau _{k}^{(2,1)}, \\\\\\tau _{k}^{{\\bf 3}_{[11]}} &=& \\tau _{k}^{(2,2)}.$ Similarly, in the case of $n=3$ , one finds that only the subspaces $\\mathbb {H}^{{\\bf 1}_{[000]}}$ , $\\mathbb {H}^{{\\bf 2}_{[100]}}$ , $\\mathbb {H}^{{\\bf 3}_{[110]}}$ and $\\mathbb {H}^{{\\bf 4}_{[111]}}$ survive after the reduction of the lower component.", "These Hilbert subspaces correspond, respectively, to $\\mathbb {H}^{(3,0)}$ , $\\mathbb {H}^{(3,1)}$ , $\\mathbb {H}^{(3,2)}$ and $\\mathbb {H}^{(3,3)}$ in the U(1) Dirac vortices.", "The matrices of these two cases are equivalent to each other: $\\tau _{\\ell }^{{\\bf 1}_{[000]}} &=& \\tau _{\\ell }^{(3,0)}, \\\\\\tau _{\\ell }^{{\\bf 2}_{[100]}} &=& \\tau _{\\ell }^{(3,1)}, \\\\\\tau _{\\ell }^{{\\bf 3}_{[110]}} &=& \\tau _{\\ell }^{(3,2)}, \\\\\\tau _{\\ell }^{{\\bf 4}_{[111]}} &=& \\tau _{\\ell }^{(3,3)},$ with $\\ell =k$ , $k+1$ .", "By repeating the same arguments for general $n$ , one can easily find that there will be a correspondence between the Hilbert subspaces $\\mathbb {H}^{{\\cal M}_{[1\\cdots 10\\cdots 0]}}$ and $\\mathbb {H}^{(n,f)}$ .", "The matrices in these two subspaces will be equivalent; $\\tau _{\\ell }^{{\\cal M}_{[1\\cdots 10\\cdots 0]}} = \\tau _{\\ell }^{(n,f)},$ with $\\ell = 1, \\cdots , n-1$ .", "Here, the U(2) representation ${\\cal M}_{[\\underbrace{1\\cdots 1}_{f}\\underbrace{0\\cdots 0}_{n-f}]}$ contains $f$ Dirac fermions, and the dimension of ${\\cal M}_{[1\\cdots 10\\cdots 0]}$ is equal to $f+1$ .", "Thus, the Hilbert space $\\mathbb {H}^{\\lbrace n\\rbrace }$ of the U(2) Dirac vortices covers $\\mathbb {H}^{(n)}$ of the U(1) Dirac vortices.", "This is confirmed also for $n=4$ vortices in Appendix ." ], [ "Sectors of exchanges of the identical vortices", "We have constructed the braid group made of exchanges of U(2) Dirac vortices.", "One may be able to regard it as a kind of exchange statistics since all the Dirac vortices to be exchanged are energetically the same.", "However, this is true only if we are allowed to neglect the difference between the occupation and absence of Dirac fermions in each vortex.", "Genuine non-Abelian statistics in a strict sense should appear in exchanges of identical particles, which implies that we should distinguish the number of Dirac fermions in each vortex.", "In the case of vortices with U(1) Dirac fermions, the whole braid group representation is non-Abelian, but does not have genuine non-Abelian statistics when restricted to a subspace of exchanges of identical states (i.e., between two occupied states or two un-occupied states).", "What is truly remarkable in the U(2) case is that the braid group which we have found indeed contains genuine non-Abelian statistics as its subgroup where we consider only the exchanges between the identical vortices, the vortices with the same numbers of Dirac fermions.", "Below, we explicitly show them in subspaces with $n=3$ and 4 vortices.", "The doublet Hilbert subspace $\\mathbb {H}^{{\\bf 2}_{[111]}}$ for three U(2) Dirac vortices ($n=3$ ; $k$ -th, $(k+1)$ -th and $(k+2)$ -th vortices) with single Dirac fermion occupations.", "The minimum genuine non-Abelian statistics appears in the doublet Hilbert subspace $\\mathbb {H}^{{\\bf 2}_{[111]}}$ in $n=3$ vortices.", "The two-dimensional Hilbert subspace $\\mathbb {H}^{{\\bf 2}_{[111]}}$ is spanned by the basis $| {\\bf 2}_{{\\rm A}\\,\\underline{11}1} \\rangle $ and $| {\\bf 2}_{{\\rm S}\\,\\underline{11}1} \\rangle $ defined in Eq.", "(REF ).", "The Dirac fermion numbers in each U(2) Dirac vortex are one so that they are all identical vortices.", "Two exchange matrices, i.e., $\\tau _{k}^{{\\bf 2}_{[111]}}$ for the exchange of the $k$ -th and $(k+1)$ -th vortices, and $\\tau _{k+1}^{{\\bf 2}_{[111]}}$ for the exchange of the $(k+1)$ -th and $(k+2)$ -th vortices, can be found in the last line in Eq.", "(REF ): $&& \\tau _{k}^{{\\bf 2}_{[111]}} =\\left(\\begin{array}{cc}-1 & 0 \\\\0 & 1\\end{array}\\right), \\quad \\tau _{k+1}^{{\\bf 2}_{[111]}} =\\left(\\begin{array}{cc}\\frac{1}{2} & \\frac{\\sqrt{3}}{2} \\\\\\frac{\\sqrt{3}}{2} & -\\frac{1}{2}\\end{array}\\right)\\, .", "$ They are non-commutative: $\\tau _{k}^{{\\bf 2}_{[111]}}\\tau _{k+1}^{{\\bf 2}_{[111]}} \\ne \\tau _{k+1}^{{\\bf 2}_{[111]}}\\tau _{k}^{{\\bf 2}_{[111]}}$ , showing genuine non-Abelian statistics.", "The non-Abelian transformation equivalent to Eq.", "(REF ) are embedded in a sector of four vortices.", "For instance, let us consider four vortices three of which contain one Dirac fermion at each vortex but the rest of which contains no Dirac fermion, then the exchange of the identical vortices with the one Dirac fermion occupation leads to the same result as in Eq.", "(REF ).", "To see this more explicitly, consider the doublet subspace with the basis $|{\\bf 2}_{\\mathrm {A}\\,01\\underline{11}}\\rangle $ and $|{\\bf 2}_{\\mathrm {S}\\,01\\underline{11}}\\rangle $ defined in Eq.", "(REF ).", "We can see that the submatrices made of the fifth and sixth columns and rows in $\\tau _{k+1}^{{\\bf 2}_{[1110]}}$ and $\\tau _{k+2}^{{\\bf 2}_{[1110]}}$ in Eq.", "(REF ) are nothing but the matrices in Eq.", "(REF ).", "The rest corresponding submatrix in $\\tau _{k}^{{\\bf 2}_{[1110]}}$ in Eq.", "(REF ) is consistently zero, because we are considering the exchange of only three identical vortices with one Dirac fermion occupations.", "The singlet Hilbert space $\\mathbb {H}^{{\\bf 1}_{[1111]}}$ for four U(2) Dirac vortices ($n=4$ ; $k$ -th, $(k+1)$ -th, $(k+2)$ -th and $(k+3)$ -th vortices) with single Dirac fermion occupations.", "We have now four identical vortices each of which contains one Dirac fermion.", "The singlet Hilbert space $\\mathbb {H}^{{\\bf 1}_{[1111]}}$ is a two-dimensional subspace with the basis $| {\\bf 1}_{\\mathrm {AA}\\,\\underline{11}\\,\\underline{11}} \\rangle $ and $| {\\bf 1}_{\\mathrm {SS}\\,\\underline{11}\\,\\underline{11}} \\rangle $ defined in Eq.", "(REF ).", "The non-Abelian exchange matrices can be found in Eq.", "(REF ) as $\\tau _{k}^{{\\bf 1}_{[1111]}} =\\left(\\begin{array}{cc}-1 & 0 \\\\0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 1}_{[1111]}} =\\left(\\begin{array}{cc}\\frac{1}{2} & \\frac{\\sqrt{3}}{2} \\\\\\frac{\\sqrt{3}}{2} & -\\frac{1}{2}\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 1}_{[1111]}} =\\left(\\begin{array}{cc}-1 & 0 \\\\0 & 1\\end{array}\\right),$ The exchange of the Dirac vortices are non-Abelian; $\\tau _{\\ell }^{{\\bf 1}_{[1111]}} \\tau _{\\ell +1}^{{\\bf 1}_{[1111]}} \\ne \\tau _{\\ell +1}^{{\\bf 1}_{[1111]}} \\tau _{\\ell }^{{\\bf 1}_{[1111]}}$ for $\\ell =k$ , $k+1$ .", "The triplet Hilbert space $\\mathbb {H}^{{\\bf 3}_{[1111]}}$ for four U(2) Dirac vortices ($n=4$ ; $k$ -th, $(k+1)$ -th, $(k+2)$ -th and $(k+3)$ -th vortices) with single Dirac fermion occupations.", "The four identical Dirac vortices with single Dirac fermion occupations allow for another sector having genuine non-Abelian statistics.", "The triplet Hilbert space $\\mathbb {H}^{{\\bf 3}_{[1111]}}$ is three dimensional with the basis $|{\\bf 3}_{\\mathrm {AS}\\,\\underline{11}\\,\\underline{11}}\\rangle $ , $|{\\bf 3}_{\\mathrm {SA}\\,\\underline{11}\\,\\underline{11}}\\rangle $ and $|{\\bf 3}_{\\mathrm {SS}\\,\\underline{11}\\,\\underline{11}}\\rangle $ in Eq.", "(REF ).", "The exchange matrices can be found in Eq.", "(REF ) as $\\tau _{k}^{{\\bf 3}_{[1111]}} =\\left(\\begin{array}{ccc}-1 & 0 & 0 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 3}_{[1111]}} =\\left(\\begin{array}{ccc}\\frac{1}{2} & -\\frac{1}{2} & \\frac{1}{\\sqrt{2}} \\\\-\\frac{1}{2} & \\frac{1}{2} & \\frac{1}{\\sqrt{2}} \\\\\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} & 0\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 3}_{[1111]}} =\\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & -1 & 0 \\\\0 & 0 & 1\\end{array}\\right),$ The exchange of these Dirac vortices are non-Abelian; $\\tau _{\\ell }^{{\\bf 3}_{[1111]}} \\tau _{\\ell +1}^{{\\bf 3}_{[1111]}} \\ne \\tau _{\\ell +1}^{{\\bf 3}_{[1111]}} \\tau _{\\ell }^{{\\bf 3}_{[1111]}}$ for $\\ell =k$ , $k+1$ .", "From the above examples, we find that the U(2) Dirac vortices with one Dirac fermion occupation give non-Abelian quantum statistics of identical particles." ], [ "Entanglement in the Hilbert space", "If one reminds of the case of Majorana vortices showing the genuine non-Abelian statistics, one may critically wonder why we have found the genuine non-Abelian statistics even though Dirac fermions are defined locally at each vortex.", "As is well known, for vortices with Majorana fermions, Dirac fermions have to be defined non-locally by using two Majorana fermions localized at spatially separated vortices, and, thus far, it has been commonly thought that emergence of non-Abelian statistics is attributed to such a non-locality of Dirac fermions.", "In contrast, in our case, Dirac fermions are introduced from the beginning, and are of course defined locally at each vortex.", "Then, it is natural to ask why we get non-Abelian statistics from local Dirac fermions, or to raise question if the non-locality is really essential for the emergence of non-Abelian statistics.", "Our answer is that the non-locality is indeed essential but it does not come from the definition of Dirac fermions but rather from the definition of the basis of the Hilbert space.", "Namely, the use of entangled states, which was absent in the previous U(1) Dirac case, is essential in the U(2) case.", "For instance, in the first example of the previous subsection, the basis $| {\\bf 2}_{{\\rm A}\\,\\underline{11}1} \\rangle $ and $| {\\bf 2}_{{\\rm S}\\,\\underline{11}1} \\rangle $ in Eq.", "(REF ) are entangled because of (anti-)symmetric combination of indices.", "Such entangled states are necessary because the basis of the Hilbert space have to belong to the irreducible representations of U(2) symmetry.", "Therefore, genuine non-Abelian statistics does not appear in U(1) Dirac vortices but it first appears in U(2) Dirac vortices because of non-Abelian U(2) symmetry group acting on the doublet Dirac fermions on the vortices.", "From these considerations, we conclude that some kind of non-locality is necessary to obtain the genuine non-Abelian statistics: It is the non-local definition of the Dirac fermions in the case of Majorana vortices, while it is the entanglement in the Hilbert space in the case of Dirac vortices." ], [ "Summary", "We have considered the simplest system of non-Abelian Dirac vortices, namely the system of vortices in which U(2) doublet zero-energy Dirac fermions are trapped.", "We have constructed the non-Abelian representation of the braid group for the exchange of vortices with U(2) Dirac fermions.", "This was confirmed both at the operator level and in matrix representations of the exchange operations.", "In particular, the whole Hilbert spaces for $n=2$ , 3, 4 vortices are decomposed into subspaces according to representations of U(2), and the matrix forms of the exchange operations were obtained in these subspaces.", "We have found they have off-diagonal elements in some subspaces.", "This analysis is an extension of the previous one for the U(1) (one component) Dirac vortices [11].", "By using a simple reduction of the U(2) results, we have indeed identified the same matrices as in the U(1) Dirac vortices.", "This way of identifying the U(1) structure is rather different from the one in the non-Abelian Majorana vortices where one can extract the U(1) part as a tensor product.", "It is not trivial at this point if a similar identification is possible in the Dirac vortices.", "Finally, we have found that the whole braid group contains a subgroup of genuine non-Abelian statistics for the exchange of the vortices with one Dirac fermion.", "The reason why non-Abelian statistics appears only from locally defined Dirac fermions is that the basis of the Hilbert space are entangled because of the representation of U(2) acting on the Dirac fermions.", "In this sense, the `spatial' non-locality of Dirac fermions is not needed to have non-Abelian statistics but rather the non-locality, namely entanglement, in representations is needed.", "Table: Summary of the results on non-Abelian representations of the braidgroup and non-Abelian statistics ofMajorana/Dirac vortices with Abelian/non-Abelian symmetry.Note that the braid group acts on all particles while the non-Abelian statistics concerns only on identical particles.Multiple zero-energy fermions are in the vector representations ofthe groups shown in the table.", "In the Majorana vortices, the exchangematrices for a single fermion are called Ivanov matrices, and thosefor multiple fermions are decomposed into the Ivanov matrices andgenerators of the Coxeter group.In Table I, we have summarized the results of Majorana/Dirac vortices with Abelian/non-Abelian symmetry.", "All of them exhibit non-Abelian representations of the braid group.", "The right most column corresponds to the present result.", "The exchange of Majorana fermions automatically gives non-Abelian statistics, while the exchange of Dirac fermions contains non-Abelian statistics when one exchanges identical particles, that is possible at least for U(2).", "So far, we know only high-energy-physics examples of vortices with multiple Dirac fermions , i.e., non-Abelian vortices in dense QCD and supersymmetric QCD.", "As future studies, examples in condensed matter physics and experimental observation of the non-Abelian statistics in laboratory will be important.", "Applications to quantum computing should also be explored." ], [ "Exchange matrices for U(1) Dirac vortices", "In this Appendix, we present the matrix representations of the exchange operators $\\tau _{\\ell }$ 's ($\\ell =1,\\cdots ,n-1$ ) for the U(1) Dirac vortices which was first obtained in Ref.", "[11] (below we omit the superscript “s\" for notational simplicity).", "The Hilbert space of $n$ vortices, $\\mathbb {H}^{(n)}$ , is decomposed into subspaces $\\mathbb {H}^{(n,f)}$ which are specified by the total fermion number $f$ .", "In the case of $n=2$ ($k$ -th and $(k+1)$ -th vortices), we have only one exchange operation $T_{k}$ .", "The Hilbert space $\\mathbb {H}^{(2)}$ is decomposed into a direct sum of three subspaces: $\\mathbb {H}^{(2,0)}\\equiv \\lbrace |00\\rangle \\rbrace $ , $\\mathbb {H}^{(2,1)}\\equiv \\lbrace |10\\rangle , |01\\rangle \\rbrace $ and $\\mathbb {H}^{(2,2)}\\equiv \\lbrace |11\\rangle \\rbrace $ .", "Thus, $\\mathbb {H}^{(2)} = \\mathbb {H}^{(2,0)} \\oplus \\mathbb {H}^{(2,1)} \\oplus \\mathbb {H}^{(2,2)}$ .", "Then, we obtain the matrix representations of $\\hat{\\tau }_{k}$ as $\\tau _{k}^{(2,0)} &=& 1, \\\\\\tau _{k}^{(2,1)} &=&\\left(\\begin{array}{cc}0 & -1 \\\\1 & 0\\end{array}\\right), \\\\\\tau _{k}^{(2,2)} &=& 1.$ In the case of $n=3$ ($k$ -th, $(k+1)$ -th and $(k+2)$ -th vortices), we have two operations $T_{k}$ and $T_{k+1}$ .", "The Hilbert space $\\mathbb {H}^{(3)}$ is decomposed into a direct sum of $\\mathbb {H}^{(3,0)}\\equiv \\lbrace |000\\rangle \\rbrace $ , $\\mathbb {H}^{(3,1)}\\equiv \\lbrace |100\\rangle , |010\\rangle , |001\\rangle \\rbrace $ , $\\mathbb {H}^{(3,2)}\\equiv \\lbrace |110\\rangle , |011\\rangle , |101\\rangle \\rbrace $ and $\\mathbb {H}^{(3,3)}\\equiv \\lbrace |111\\rangle \\rbrace $ .", "Namely, $\\mathbb {H}^{(3)} = \\mathbb {H}^{(3,0)} \\oplus \\mathbb {H}^{(3,1)} \\oplus \\mathbb {H}^{(3,2)} \\oplus \\mathbb {H}^{(3,3)}$ .", "We thus have the matrix representations of $\\hat{\\tau }_{k}$ and $\\hat{\\tau }_{k+1}$ as $\\tau _{k}^{(3,0)} &=& \\tau _{k+1}^{(3,0)} = 1, \\\\\\tau _{k}^{(3,1)} &=&\\left(\\begin{array}{ccc}0 & -1 & 0 \\\\1 & 0 & 0 \\\\0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{(3,1)} =\\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & 0 & -1 \\\\0 & 1 & 0\\end{array}\\right), \\\\\\tau _{k}^{(3,2)} &=&\\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & 0 & 1 \\\\0 & -1 & 0\\end{array}\\right), \\,\\tau _{k+1}^{(3,2)} =\\left(\\begin{array}{ccc}0 & 0 & -1 \\\\0 & 1 & 0 \\\\1 & 0 & 0\\end{array}\\right), \\\\\\tau _{k}^{(3,3)} &=& \\tau _{k+1}^{(3,3)} = 1.", "$ In the case of $n=4$ ($k$ -th, $(k+1)$ -th, $(k+2)$ -th and $(k+3)$ -th vortices), we have three operations $T_{k}$ , $T_{k+1}$ and $T_{k+2}$ .", "The Hilbert space $\\mathbb {H}^{(4)}$ is decomposed into a direct sum of five sectors: $\\mathbb {H}^{(4,0)}\\equiv \\lbrace |0000\\rangle \\rbrace ,$ $\\mathbb {H}^{(4,1)}$ $\\equiv \\lbrace |1000\\rangle ,|0100\\rangle , |0010\\rangle , |0001\\rangle \\rbrace $ , $\\mathbb {H}^{(4,2)}\\equiv \\lbrace |1100\\rangle $ , $|1010\\rangle , |1001\\rangle , |0110\\rangle , |0101\\rangle , |0011\\rangle \\rbrace $ , $\\mathbb {H}^{(4,3)}\\equiv \\lbrace |1110\\rangle $ , $|1101\\rangle , |1011\\rangle , |0111\\rangle \\rbrace $ and $\\mathbb {H}^{(4,4)}\\equiv \\lbrace |1111\\rangle \\rbrace $ .", "Namely, $\\mathbb {H}^{(4)} = \\mathbb {H}^{(4,0)} \\oplus \\mathbb {H}^{(4,1)} \\oplus \\mathbb {H}^{(4,2)} \\oplus \\mathbb {H}^{(4,3)} \\oplus \\mathbb {H}^{(4,4)}$ .", "The matrix representations of $\\hat{\\tau }_{k}$ , $\\hat{\\tau }_{k+1}$ and $\\hat{\\tau }_{k+2}$ are $\\tau _{k}^{(4,0)} &=& \\tau _{k+1}^{(4,0)} = \\tau _{k+2}^{(4,0)} = 1, \\\\\\tau _{k}^{(4,1)} &=&\\left(\\begin{array}{cccc}0 & -1 & 0 & 0 \\\\1 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{(4,1)} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+2}^{(4,1)} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & -1 \\\\0 & 0 & 1 & 0\\end{array}\\right), \\\\\\tau _{k}^{(4,2)} &=&\\left(\\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{(4,2)} =\\left(\\begin{array}{cccccc}0 & -1 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 \\\\0 & 0 & 0 & 0 & 1 & 0\\end{array}\\right), \\,\\tau _{k+2}^{(4,2)} =\\left(\\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right), \\\\\\tau _{k}^{(4,3)} &=&\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & -1 \\\\0 & 0 & 1 & 0\\end{array}\\right), \\,\\tau _{k+1}^{(4,3)} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+2}^{(4,3)} =\\left(\\begin{array}{cccc}0 & -1 & 0 & 0 \\\\1 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right), \\\\\\tau _{k}^{(4,4)} &=& \\tau _{k+1}^{(4,4)}= \\tau _{k+2}^{(4,4)} = 1.", "$" ], [ "The case of $n=4$ for the U(2) Dirac vortices", "We consider the system of $n=4$ vortices, which is an ensemble of $k$ -th, $(k+1)$ -th, $(k+2)$ -th, and $(k+3)$ -th vortices.", "The Hilbert space is constructed straightforwardly from the $n=2$ and $n=3$ cases.", "The representations of the U(2) symmetry are decomposed as $&&({\\bf 1}_{0} + {\\bf 2}_{1} + {\\bf 1}_{2})_{{\\rm vortex}\\, k} \\otimes ({\\bf 1}_{0} + {\\bf 2}_{1} + {\\bf 1}_{2})_{{\\rm vortex}\\, k+1} \\otimes ({\\bf 1}_{0} + {\\bf 2}_{1} + {\\bf 1}_{2})_{{\\rm vortex}\\, k+2} \\otimes ({\\bf 1}_{0} + {\\bf 2}_{1} + {\\bf 1}_{2})_{{\\rm vortex}\\, k+3} \\nonumber \\\\&=& {\\bf 1}_{0000} + {\\bf 1}_{0020} + {\\bf 1}_{0002} + {\\bf 1}_{2000} + {\\bf 1}_{0200} \\nonumber \\\\&+& {\\bf 1}_{0011} + {\\bf 1}_{1100} + {\\bf 1}_{1010} + {\\bf 1}_{1001} + {\\bf 1}_{0110} + {\\bf 1}_{0101} \\nonumber \\\\&+& {\\bf 1}_{0022} + {\\bf 1}_{2020} + {\\bf 1}_{2002} + {\\bf 1}_{0220} + {\\bf 1}_{0202} + {\\bf 1}_{2200} \\nonumber \\\\&+& {\\bf 1}_{1120} + {\\bf 1}_{1102} + {\\bf 1}_{2011} + {\\bf 1}_{0211} + {\\bf 1}_{1021} + {\\bf 1}_{1012} + {\\bf 1}_{0121} + {\\bf 1}_{0112} + {\\bf 1}_{2110} + {\\bf 1}_{2101} + {\\bf 1}_{1210} + {\\bf 1}_{1201} \\nonumber \\\\&+& {\\bf 1}_{2022} + {\\bf 1}_{0222} + {\\bf 1}_{2220} + {\\bf 1}_{2202} \\nonumber \\\\&+& {\\bf 1}_{\\mathrm {AA}\\;\\underline{11}\\,\\underline{11}} + {\\bf 1}_{\\mathrm {SS}\\;\\underline{11}\\,\\underline{11}} \\nonumber \\\\&+& {\\bf 1}_{1122} + {\\bf 1}_{2211} + {\\bf 1}_{2121} + {\\bf 1}_{2112} + {\\bf 1}_{1221} + {\\bf 1}_{1212} \\nonumber \\\\&+& {\\bf 1}_{2222} \\nonumber \\\\&+& {\\bf 2}_{1000} + {\\bf 2}_{0100} + {\\bf 2}_{0010} + {\\bf 2}_{0001} \\nonumber \\\\&+& {\\bf 2}_{0021} + {\\bf 2}_{0012} + {\\bf 2}_{2010} + {\\bf 2}_{2001} + {\\bf 2}_{0210} + {\\bf 2}_{0201} + {\\bf 2}_{1020} + {\\bf 2}_{1002} + {\\bf 2}_{0120} + {\\bf 2}_{0102} + {\\bf 2}_{2100} + {\\bf 2}_{1200} \\nonumber \\\\&+& {\\bf 2}_{\\mathrm {A}\\,\\underline{11}10} + {\\bf 2}_{\\mathrm {A}\\,\\underline{11}01} + {\\bf 2}_{\\mathrm {A}\\,10\\underline{11}} + {\\bf 2}_{\\mathrm {S}\\,10\\underline{11}} + {\\bf 2}_{\\mathrm {A}\\,01\\underline{11}} + {\\bf 2}_{\\mathrm {S}\\,01\\underline{11}} + {\\bf 2}_{\\mathrm {S}\\,\\underline{11}10} + {\\bf 2}_{\\mathrm {S}\\,\\underline{11}01} \\nonumber \\\\&+& {\\bf 2}_{2021} + {\\bf 2}_{2012} + {\\bf 2}_{0221} + {\\bf 2}_{0212} + {\\bf 2}_{2210} + {\\bf 2}_{2201} + {\\bf 2}_{1022} + {\\bf 2}_{0122} + {\\bf 2}_{2120} + {\\bf 2}_{2102} + {\\bf 2}_{1220} + {\\bf 2}_{1202} \\nonumber \\\\&+& {\\bf 2}_{\\mathrm {A}\\,\\underline{11}21} + {\\bf 2}_{\\mathrm {A}\\,\\underline{11}12} + {\\bf 2}_{\\mathrm {A}\\,21\\underline{11}} + {\\bf 2}_{\\mathrm {S}\\,21\\underline{11}} + {\\bf 2}_{\\mathrm {A}\\,12\\underline{11}} + {\\bf 2}_{\\mathrm {S}\\,12\\underline{11}} + {\\bf 2}_{\\mathrm {S}\\,\\underline{11}21} + {\\bf 2}_{\\mathrm {S}\\,\\underline{11}12} \\nonumber \\\\&+& {\\bf 2}_{2221} + {\\bf 2}_{2212} + {\\bf 2}_{2122} + {\\bf 2}_{1222} \\nonumber \\\\&+& {\\bf 3}_{1100} + {\\bf 3}_{1010} + {\\bf 3}_{1001} + {\\bf 3}_{0110} + {\\bf 3}_{0101} + {\\bf 3}_{0011} \\nonumber \\\\&+& {\\bf 3}_{2011} + {\\bf 3}_{0211} + {\\bf 3}_{1021} + {\\bf 3}_{1012} + {\\bf 3}_{0121} + {\\bf 3}_{0112} + {\\bf 3}_{2110} + {\\bf 3}_{2101} + {\\bf 3}_{1210} + {\\bf 3}_{1201} + {\\bf 3}_{1120} + {\\bf 3}_{1102} \\nonumber \\\\&+& {\\bf 3}_{\\mathrm {AS}\\,\\underline{11}\\,\\underline{11}} + {\\bf 3}_{\\mathrm {SA}\\,\\underline{11}\\,\\underline{11}} + {\\bf 3}_{\\mathrm {SS}\\,\\underline{11}\\,\\underline{11}} \\nonumber \\\\&+& {\\bf 3}_{2211} + {\\bf 3}_{2121} + {\\bf 3}_{2112} + {\\bf 3}_{1221} + {\\bf 3}_{1212} + {\\bf 3}_{1122} \\nonumber \\\\&+& {\\bf 4}_{\\mathrm {S}\\,\\underline{11}10} + {\\bf 4}_{\\mathrm {S}\\,\\underline{11}01} + {\\bf 4}_{\\mathrm {S}\\,10\\underline{11}} + {\\bf 4}_{\\mathrm {S}\\,01\\underline{11}} \\nonumber \\\\&+& {\\bf 4}_{\\mathrm {S}\\,21\\underline{11}} + {\\bf 4}_{\\mathrm {S}\\,12\\underline{11}} + {\\bf 4}_{\\mathrm {S}\\,\\underline{11}21} + {\\bf 4}_{\\mathrm {S}\\,\\underline{11}12} \\nonumber \\\\&+& {\\bf 5}_{\\mathrm {S}\\,\\underline{11}\\,\\underline{11}},$ where the subscript $n_{k}n_{k+1}n_{k+2}n_{k+3}$ ($n_{k}$ , $n_{k+1}$ , $n_{k+2}$ , $n_{k+3}=0$ , 1, 2) denotes the number of the Dirac fermions, $n_{k}$ , $n_{k+1}$ $n_{k+2}$ and $n_{k+3}$ , at the $k$ -th, $(k+1)$ -th, $(k+2)$ -th and $(k+3)$ -th vortices, respectively.", "One obtains the basis states of the Hilbert space by applying $\\hat{\\psi }_{\\ell }^{a\\dag }$ ($\\ell =k$ , $k+1$ , $k+2$ , $k+3$ and $a=1$ , 2) successively to the Fock vacuum $| 0 \\rangle $ defined by $\\hat{\\psi }_{\\ell }^{a} | 0 \\rangle = 0$ for all $\\ell $ and $a=1$ , 2.", "For singlet, there are nine Hilbert spaces, $\\mathbb {H}^{{\\bf 1}_{[0000]}}$ , $\\mathbb {H}^{{\\bf 1}_{[2000]}}$ , $\\mathbb {H}^{{\\bf 1}_{[1100]}}$ , $\\mathbb {H}^{{\\bf 1}_{[2200]}}$ , $\\mathbb {H}^{{\\bf 1}_{[2110]}}$ , $\\mathbb {H}^{{\\bf 1}_{[2220]}}$ , $\\mathbb {H}^{{\\bf 1}_{[1111]}}$ , $\\mathbb {H}^{{\\bf 1}_{[2211]}}$ and $\\mathbb {H}^{{\\bf 1}_{[2222]}}$ .", "They are defined as $\\mathbb {H}^{{\\bf 1}_{[0000]}} \\equiv \\lbrace | {\\bf 1}_{0000} \\rangle \\rbrace $ with $|{\\bf 1}_{0000} \\rangle \\equiv | 0 \\rangle ,$ $\\mathbb {H}^{{\\bf 1}_{[2000]}} \\equiv \\lbrace | {\\bf 1}_{0020} \\rangle , | {\\bf 1}_{0002} \\rangle , | {\\bf 1}_{2000} \\rangle , | {\\bf 1}_{0200} \\rangle \\rbrace $ with $| {\\bf 1}_{0020} \\rangle &\\equiv & \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{0002} \\rangle &\\equiv & \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2000} \\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{0200} \\rangle &\\equiv & \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle ,$ $\\mathbb {H}^{{\\bf 1}_{[1100]}} \\equiv \\lbrace | {\\bf 1}_{0011} \\rangle , | {\\bf 1}_{1100} \\rangle , | {\\bf 1}_{1010} \\rangle , | {\\bf 1}_{1001} \\rangle , | {\\bf 1}_{0110} \\rangle , | {\\bf 1}_{0101} \\rangle \\rbrace $ with $| {\\bf 1}_{0011} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{1100} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{1010} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{1001} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{0110} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{0101} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) | 0 \\rangle ,$ $\\mathbb {H}^{{\\bf 1}_{[2200]}} \\equiv \\lbrace | {\\bf 1}_{0022} \\rangle , | {\\bf 1}_{2020} \\rangle , | {\\bf 1}_{2002} \\rangle , | {\\bf 1}_{0220} \\rangle , | {\\bf 1}_{0202} \\rangle , | {\\bf 1}_{2200} \\rangle \\rbrace $ with $| {\\bf 1}_{0022} \\rangle &\\equiv & \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2020} \\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2002} \\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{0220} \\rangle &\\equiv & \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{0202} \\rangle &\\equiv & \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2200} \\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle ,$ $\\mathbb {H}^{{\\bf 1}_{[2110]}} \\equiv \\lbrace | {\\bf 1}_{1120} \\rangle , | {\\bf 1}_{1102} \\rangle , | {\\bf 1}_{2011} \\rangle , | {\\bf 1}_{0211} \\rangle , | {\\bf 1}_{1021} \\rangle , | {\\bf 1}_{1012} \\rangle , | {\\bf 1}_{0121}\\rangle , | {\\bf 1}_{0112} \\rangle , | {\\bf 1}_{2110} \\rangle , | {\\bf 1}_{2101} \\rangle , | {\\bf 1}_{1210} \\rangle , | {\\bf 1}_{1201} \\rangle \\rbrace $ with $| {\\bf 1}_{1120} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{1102} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2011} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{0211} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{1021} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{1012} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{0121} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{0112} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2110} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2101} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{1210} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{1201} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } | 0 \\rangle ,$ $\\mathbb {H}^{{\\bf 1}_{[2220]}} \\equiv \\lbrace | {\\bf 1}_{2022} \\rangle , | {\\bf 1}_{0222} \\rangle , | {\\bf 1}_{2220} \\rangle , | {\\bf 1}_{2202} \\rangle \\rbrace $ with $| {\\bf 1}_{2022} \\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{0222} \\rangle &\\equiv & \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2220} \\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2202} \\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle ,$ $\\mathbb {H}^{{\\bf 1}_{[1111]}} \\equiv \\lbrace | {\\bf 1}_{\\mathrm {AA}\\,\\underline{11}\\,\\underline{11}} \\rangle , | {\\bf 1}_{\\mathrm {SS}\\,\\underline{11}\\,\\underline{11}} \\rangle \\rbrace $ with $| {\\bf 1}_{\\mathrm {AA}\\,\\underline{11}\\,\\underline{11}} \\rangle &\\equiv & \\frac{1}{2} (\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } -\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) (\\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } -\\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) | 0 \\rangle , \\\\| {\\bf 1}_{\\mathrm {SS}\\,\\underline{11}\\,\\underline{11}} \\rangle &\\equiv &\\frac{1}{\\sqrt{3}} \\left\\lbrace \\hat{\\psi }_{k}^{1\\dag }\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\hat{\\psi }_{k+3}^{2\\dag }- \\frac{1}{2} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } +\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) (\\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } +\\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } )+ \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\right\\rbrace | 0 \\rangle ,\\nonumber $ with a notation A (S) for an antisymmetric (symmetric) combination in first and second pair of indices with the underline in the $k$ -th and $(k+1)$ -th vortices and in the $(k+2)$ -th and $(k+3)$ -th vortices.", "$\\mathbb {H}^{{\\bf 1}_{[2211]}} \\equiv \\lbrace | {\\bf 1}_{1122} \\rangle , | {\\bf 1}_{2211} \\rangle ,| {\\bf 1}_{2121} \\rangle , | {\\bf 1}_{2112} \\rangle ,|{\\bf 1}_{1221} \\rangle , | {\\bf 1}_{1212} \\rangle \\rbrace $ with $| {\\bf 1}_{1122} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag }\\hat{\\psi }_{k+1}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag }\\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag }\\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag }\\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2211} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} \\hat{\\psi }_{k}^{1\\dag }\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag }\\hat{\\psi }_{k+1}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag }\\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k+2}^{2\\dag }\\hat{\\psi }_{k+3}^{1\\dag } ) | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2121} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} \\hat{\\psi }_{k}^{1\\dag }\\hat{\\psi }_{k}^{2\\dag } ( \\hat{\\psi }_{k+1}^{1\\dag }\\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k+1}^{2\\dag }\\hat{\\psi }_{k+3}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag }\\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{2112} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} \\hat{\\psi }_{k}^{1\\dag }\\hat{\\psi }_{k}^{2\\dag } ( \\hat{\\psi }_{k+1}^{1\\dag }\\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k+1}^{2\\dag }\\hat{\\psi }_{k+2}^{1\\dag } ) \\hat{\\psi }_{k+3}^{1\\dag }\\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{1221} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag }\\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag }\\hat{\\psi }_{k+3}^{1\\dag } ) \\hat{\\psi }_{k+1}^{1\\dag }\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag }\\hat{\\psi }_{k+2}^{2\\dag } | 0 \\rangle , \\nonumber \\\\| {\\bf 1}_{1212} \\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag }\\hat{\\psi }_{k+2}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag }\\hat{\\psi }_{k+2}^{1\\dag } ) \\hat{\\psi }_{k+1}^{1\\dag }\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag }\\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle ,$ and $\\mathbb {H}^{{\\bf 1}_{[2222]}} \\equiv \\lbrace | {\\bf 1}_{2222} \\rangle \\rbrace $ with $| {\\bf 1}_{2222} \\rangle \\equiv \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } | 0 \\rangle .$ For doublet, there are six Hilbert spaces, $\\mathbb {H}^{{\\bf 2}_{[1000]}}$ , $\\mathbb {H}^{{\\bf 2}_{[2100]}}$ , $\\mathbb {H}^{{\\bf 2}_{[1110]}}$ , $\\mathbb {H}^{{\\bf 2}_{[2210]}}$ , $\\mathbb {H}^{{\\bf 2}_{[2111]}}$ and $\\mathbb {H}^{{\\bf 2}_{[2221]}}$ .", "They are defined as $\\mathbb {H}^{{\\bf 2}_{[1000]}} \\equiv \\lbrace |{\\bf 2}_{1000}\\rangle , |{\\bf 2}_{0100}\\rangle , |{\\bf 2}_{0010}\\rangle , |{\\bf 2}_{0001}\\rangle \\rbrace $ with $|{\\bf 2}_{1000}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\nonumber \\\\|{\\bf 2}_{0100}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\nonumber \\\\|{\\bf 2}_{0010}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) | 0 \\rangle , \\nonumber \\\\|{\\bf 2}_{0001}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) | 0 \\rangle ,$ $\\mathbb {H}^{{\\bf 2}_{[2100]}} \\equiv \\lbrace |{\\bf 2}_{0021}\\rangle , |{\\bf 2}_{0012}\\rangle , |{\\bf 2}_{2010}\\rangle , |{\\bf 2}_{2001}\\rangle , |{\\bf 2}_{0210}\\rangle , |{\\bf 2}_{0201}\\rangle , |{\\bf 2}_{1020}\\rangle , |{\\bf 2}_{1002}\\rangle , |{\\bf 2}_{0120}\\rangle , |{\\bf 2}_{0102}\\rangle , |{\\bf 2}_{2100}\\rangle , |{\\bf 2}_{1200}\\rangle \\rbrace $ with $|{\\bf 2}_{0021}\\rangle &\\equiv & \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{0012}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{2010}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{2001}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{0210}\\rangle &\\equiv & \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{0201}\\rangle &\\equiv & \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{1020}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{1002}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{0120}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{0102}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{2100}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{1200}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } |0\\rangle ,$ $\\mathbb {H}^{{\\bf 2}_{[1110]}} \\equiv \\lbrace |{\\bf 2}_{\\mathrm {A}\\,\\underline{11}10}\\rangle , |{\\bf 2}_{\\mathrm {A}\\,\\underline{11}01}\\rangle ,|{\\bf 2}_{\\mathrm {A}\\,10\\underline{11}}\\rangle , |{\\bf 2}_{\\mathrm {S}\\,10\\underline{11}}\\rangle , |{\\bf 2}_{\\mathrm {A}\\,01\\underline{11}}\\rangle ,|{\\bf 2}_{\\mathrm {S}\\,01\\underline{11}}\\rangle , |{\\bf 2}_{\\mathrm {S}\\,\\underline{11}10}\\rangle , |{\\bf 2}_{\\mathrm {S}\\,\\underline{11}01}\\rangle \\rbrace $ with $|{\\bf 2}_{\\mathrm {A}\\,\\underline{11}10}\\rangle &\\equiv & \\frac{1}{\\sqrt{2}} (\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } -\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } )\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {A}\\,\\underline{11}01}\\rangle &\\equiv & \\frac{1}{\\sqrt{2}} (\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } -\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } )\\left(\\begin{array}{c}\\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {A}\\,10\\underline{11}}\\rangle &\\equiv & \\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } -\\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) |0\\rangle ,\\nonumber \\\\|{\\bf 2}_{\\mathrm {S}\\,10\\underline{11}}\\rangle &\\equiv &\\left(\\begin{array}{c}\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag }\\hat{\\psi }_{k+3}^{1\\dag } - \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } (\\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } +\\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\frac{1}{\\sqrt{6}} \\hat{\\psi }_{k}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag }\\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag }\\hat{\\psi }_{k+3}^{1\\dag } ) - \\sqrt{\\frac{2}{3}}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {A}\\,01\\underline{11}}\\rangle &\\equiv & \\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } -\\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) |0\\rangle ,\\nonumber \\\\|{\\bf 2}_{\\mathrm {S}\\,01\\underline{11}}\\rangle &\\equiv &\\left(\\begin{array}{c}\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } - \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{1\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\frac{1}{\\sqrt{6}} \\hat{\\psi }_{k+1}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) - \\sqrt{\\frac{2}{3}} \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {S}\\,\\underline{11}10}\\rangle &\\equiv &\\left(\\begin{array}{c}\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{6}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{2\\dag } - \\sqrt{\\frac{2}{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {S}\\,\\underline{11}01}\\rangle &\\equiv &\\left(\\begin{array}{c}\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{6}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+3}^{2\\dag } - \\sqrt{\\frac{2}{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag }\\end{array}\\right) |0\\rangle , $ with a notation A (S) for an antisymmetric (symmetric) combination in a pair of indices with the underline in the $k$ -th and $(k+1)$ -th vortices, or in the $(k+2)$ -th and $(k+3)$ -th vortices.", "$\\mathbb {H}^{{\\bf 2}_{[2210]}} \\equiv \\lbrace |{\\bf 2}_{2021}\\rangle , |{\\bf 2}_{2012}\\rangle , |{\\bf 2}_{0221}\\rangle , |{\\bf 2}_{0212}\\rangle , |{\\bf 2}_{2210}\\rangle , |{\\bf 2}_{2201}\\rangle , |{\\bf 2}_{1022}\\rangle , |{\\bf 2}_{0122}\\rangle , |{\\bf 2}_{2120}\\rangle , |{\\bf 2}_{2102}\\rangle , |{\\bf 2}_{1220}\\rangle , |{\\bf 2}_{1202}\\rangle \\rbrace $ with $|{\\bf 2}_{2021}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{2012}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{0221}\\rangle &\\equiv & \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{0212}\\rangle &\\equiv & \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{2210}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{2201}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{1022}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{0122}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{2120}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{2102}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{1220}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{1202}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle ,$ $\\mathbb {H}^{{\\bf 2}_{[2111]}} \\equiv \\lbrace |{\\bf 2}_{\\mathrm {A}\\,\\underline{11}21}\\rangle , |{\\bf 2}_{\\mathrm {A}\\,\\underline{11}12}\\rangle ,|{\\bf 2}_{\\mathrm {A}\\,21\\underline{11}}\\rangle , |{\\bf 2}_{\\mathrm {S}\\,21\\underline{11}}\\rangle , |{\\bf 2}_{\\mathrm {A}\\,12\\underline{11}}\\rangle ,|{\\bf 2}_{\\mathrm {S}\\,12\\underline{11}}\\rangle , |{\\bf 2}_{\\mathrm {S}\\,\\underline{11}21}\\rangle , |{\\bf 2}_{\\mathrm {S}\\,\\underline{11}12}\\rangle \\rbrace $ with $|{\\bf 2}_{\\mathrm {A}\\,\\underline{11}21}\\rangle &\\equiv &\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {A}\\,\\underline{11}12}\\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } )\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {A}\\,21\\underline{11}}\\rangle &\\equiv & \\frac{1}{\\sqrt{2}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {S}\\,21\\underline{11}}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\left(\\begin{array}{c}\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } - \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{1\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\frac{1}{\\sqrt{6}} \\hat{\\psi }_{k+1}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) - \\sqrt{\\frac{2}{3}} \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {A}\\,12\\underline{11}}\\rangle &\\equiv & \\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {S}\\,12\\underline{11}}\\rangle &\\equiv & \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\left(\\begin{array}{c}\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } - \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\frac{1}{\\sqrt{6}} \\hat{\\psi }_{k}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) - \\sqrt{\\frac{2}{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {S}\\,\\underline{11}21}\\rangle &\\equiv &\\left(\\begin{array}{c}\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{6}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+3}^{2\\dag } - \\sqrt{\\frac{2}{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{\\mathrm {S}\\,\\underline{11}12}\\rangle &\\equiv &\\left(\\begin{array}{c}\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } - \\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{6}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{2\\dag } - \\sqrt{\\frac{2}{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle ,$ with a notation A (S) for an antisymmetric (symmetric) combination in a pair of indices with the underline in the $k$ -th and $(k+1)$ -th vortices, or in the $(k+2)$ -th and $(k+3)$ -th vortices.", "$\\mathbb {H}^{{\\bf 2}_{[2221]}} \\equiv \\lbrace |{\\bf 2}_{2221}\\rangle , |{\\bf 2}_{2212}\\rangle , |{\\bf 2}_{2122}\\rangle , |{\\bf 2}_{1222}\\rangle \\rbrace $ with $|{\\bf 2}_{2221}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 2}_{2212}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{2122}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 2}_{1222}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle .$ For triplet, there are four Hilbert spaces, $\\mathbb {H}^{{\\bf 3}_{[1100]}}$ , $\\mathbb {H}^{{\\bf 3}_{[2110]}}$ , $\\mathbb {H}^{{\\bf 3}_{[1111]}}$ and $\\mathbb {H}^{{\\bf 3}_{[2211]}}$ .", "They are defined as $\\mathbb {H}^{{\\bf 3}_{[1100]}} \\equiv \\lbrace |{\\bf 3}_{1100}\\rangle , |{\\bf 3}_{1010}\\rangle , |{\\bf 3}_{1001}\\rangle , |{\\bf 3}_{0110}\\rangle , |{\\bf 3}_{0101}\\rangle , |{\\bf 3}_{0011}\\rangle \\rbrace $ with $|{\\bf 3}_{1100}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{1010}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{1001}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{0110}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{0101}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{0011}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle ,$ $\\mathbb {H}^{{\\bf 3}_{[2110]}} \\equiv \\lbrace |{\\bf 3}_{2011}\\rangle , |{\\bf 3}_{0211}\\rangle , |{\\bf 3}_{1021}\\rangle , |{\\bf 3}_{1012}\\rangle , |{\\bf 3}_{0121}\\rangle , |{\\bf 3}_{0112}\\rangle , |{\\bf 3}_{2110}\\rangle , |{\\bf 3}_{2101}\\rangle , |{\\bf 3}_{1210}\\rangle , |{\\bf 3}_{1201}\\rangle , |{\\bf 3}_{1120}\\rangle , |{\\bf 3}_{1102}\\rangle \\rbrace $ with $|{\\bf 3}_{2011}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{0211}\\rangle &\\equiv & \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{1021}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 3}_{1012}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 3}_{0121}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 3}_{0112}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 3}_{2110}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{2101}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{1210}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 3}_{1201}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 3}_{1120}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 3}_{1102}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle ,$ $\\mathbb {H}^{{\\bf 3}_{[1111]}} \\equiv \\lbrace |{\\bf 3}_{\\mathrm {AS}\\,\\underline{11}\\,\\underline{11}}\\rangle , |{\\bf 3}_{\\mathrm {SA}\\,\\underline{11}\\,\\underline{11}}\\rangle , |{\\bf 3}_{\\mathrm {SS}\\,\\underline{11}\\,\\underline{11}}\\rangle \\rbrace $ with $|{\\bf 3}_{\\mathrm {AS}\\,\\underline{11}\\,\\underline{11}}\\rangle &\\equiv & \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } - \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } )\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{\\mathrm {SA}\\,\\underline{11}\\,\\underline{11}}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{\\mathrm {SS}\\,\\underline{11}\\,\\underline{11}}\\rangle &\\equiv &\\left(\\begin{array}{c}\\frac{1}{2} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) - \\frac{1}{2} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\frac{1}{\\sqrt{2}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{2} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag } - \\frac{1}{2} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } )\\end{array}\\right) |0\\rangle , $ with a notation A (S) for an antisymmetric (symmetric) combination in first or second pair of indices with the underline in the $k$ -th and $(k+1)$ -th vortices, and in the $(k+2)$ -th and $(k+3)$ -th vortices.", "$\\mathbb {H}^{{\\bf 3}_{[2211]}} \\equiv \\lbrace |{\\bf 3}_{2211}\\rangle , |{\\bf 3}_{2121}\\rangle , |{\\bf 3}_{2112}\\rangle , |{\\bf 3}_{1221}\\rangle , |{\\bf 3}_{1212}\\rangle , |{\\bf 3}_{1122}\\rangle \\rbrace $ with $|{\\bf 3}_{2211}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{2121}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{2112}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 3}_{1221}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 3}_{1212}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 3}_{1122}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\\\\\frac{1}{\\sqrt{2}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle .$ For quartet, there are two Hilbert spaces, $\\mathbb {H}^{{\\bf 4}_{[1110]}}$ and $\\mathbb {H}^{{\\bf 4}_{[2111]}}$ .", "They are defined as $\\mathbb {H}^{{\\bf 4}_{[1110]}} \\equiv \\lbrace |{\\bf 4}_{\\mathrm {S}\\,\\underline{11}10}\\rangle , |{\\bf 4}_{\\mathrm {S}\\,\\underline{11}01}\\rangle , |{\\bf 4}_{\\mathrm {S}\\,10\\underline{11}}\\rangle , |{\\bf 4}_{\\mathrm {S}\\,01\\underline{11}}\\rangle \\rbrace $ with $|{\\bf 4}_{\\mathrm {S}\\,\\underline{11}10}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{2\\dag } + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 4}_{\\mathrm {S}\\,\\underline{11}01}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+3}^{2\\dag } + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 4}_{\\mathrm {S}\\,10\\underline{11}}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 4}_{\\mathrm {S}\\,01\\underline{11}}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{1\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle ,$ and $\\mathbb {H}^{{\\bf 4}_{[2111]}} \\equiv \\lbrace |{\\bf 4}_{\\mathrm {S}\\,21\\underline{11}}\\rangle , |{\\bf 4}_{\\mathrm {S}\\,12\\underline{11}}\\rangle , |{\\bf 4}_{\\mathrm {S}\\,\\underline{11}21}\\rangle , |{\\bf 4}_{\\mathrm {S}\\,\\underline{11}12}\\rangle \\rbrace $ with $|{\\bf 4}_{\\mathrm {S}\\,21\\underline{11}}\\rangle &\\equiv & \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k}^{2\\dag }\\left(\\begin{array}{c}\\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{1\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag } \\\\\\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle , \\nonumber \\\\|{\\bf 4}_{\\mathrm {S}\\,12\\underline{11}}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 4}_{\\mathrm {S}\\,\\underline{11}21}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+3}^{2\\dag } + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } |0\\rangle , \\nonumber \\\\|{\\bf 4}_{\\mathrm {S}\\,\\underline{11}12}\\rangle &\\equiv &\\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } + \\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\frac{1}{\\sqrt{3}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{2\\dag } + \\frac{1}{\\sqrt{3}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag }\\end{array}\\right) \\hat{\\psi }_{k+3}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } |0\\rangle ,$ with a notation A (S) for an antisymmetric (symmetric) combination in a pair of indices with the underline in the $k$ -th and $(k+1)$ -th vortices, or in the $(k+2)$ -th and $(k+3)$ -th vortices.", "For quintet, there is one Hilbert space $\\mathbb {H}^{{\\bf 5}_{[1111]}}$ which is defined as $\\mathbb {H}^{{\\bf 5}_{[1111]}} \\equiv \\lbrace |{\\bf 5}_{\\mathrm {SS}\\,\\underline{11}\\,\\underline{11}}\\rangle \\rbrace $ with $|{\\bf 5}_{\\mathrm {SS}\\,\\underline{11}\\,\\underline{11}}\\rangle \\equiv \\left(\\begin{array}{c}\\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{2} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) + \\frac{1}{2} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } )\\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{\\sqrt{6}} \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{1\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\frac{1}{\\sqrt{6}} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) + \\frac{1}{\\sqrt{6}} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{1\\dag } \\\\\\frac{1}{2} ( \\hat{\\psi }_{k}^{1\\dag } \\hat{\\psi }_{k+1}^{2\\dag } + \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{1\\dag } ) \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\frac{1}{2} \\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } ( \\hat{\\psi }_{k+2}^{1\\dag } \\hat{\\psi }_{k+3}^{2\\dag } + \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{1\\dag } ) \\\\\\hat{\\psi }_{k}^{2\\dag } \\hat{\\psi }_{k+1}^{2\\dag } \\hat{\\psi }_{k+2}^{2\\dag } \\hat{\\psi }_{k+3}^{2\\dag }\\end{array}\\right) |0\\rangle ,$ with a notation S for a symmetric combination in first and second pair of indices with the underline in the $k$ -th and $(k+1)$ -th vortices, and in the $(k+2)$ -th and $(k+3)$ -th vortices.", "Therefore, the Hilbert space is totally given as a direct sum, $\\mathbb {H}^{\\lbrace n=4\\rbrace }&=& \\mathbb {H}^{{\\bf 1}_{[0000]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[2000]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[1100]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[2200]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[2110]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[2220]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[1111]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[2211]}} \\oplus \\mathbb {H}^{{\\bf 1}_{[2222]}} \\nonumber \\\\& \\oplus & \\mathbb {H}^{{\\bf 2}_{[1000]}} \\oplus \\mathbb {H}^{{\\bf 2}_{[2100]}} \\oplus \\mathbb {H}^{{\\bf 2}_{[1110]}} \\oplus \\mathbb {H}^{{\\bf 2}_{[2210]}} \\oplus \\mathbb {H}^{{\\bf 2}_{[2111]}} \\oplus \\mathbb {H}^{{\\bf 2}_{[2221]}} \\nonumber \\\\& \\oplus & \\mathbb {H}^{{\\bf 3}_{[1100]}} \\oplus \\mathbb {H}^{{\\bf 3}_{[2110]}} \\oplus \\mathbb {H}^{{\\bf 3}_{[1111]}} \\oplus \\mathbb {H}^{{\\bf 3}_{[2211]}} \\nonumber \\\\& \\oplus & \\mathbb {H}^{{\\bf 4}_{[1110]}} \\oplus \\mathbb {H}^{{\\bf 4}_{[2111]}} \\nonumber \\\\& \\oplus & \\mathbb {H}^{{\\bf 5}_{[1111]}}.$ With these basis states in the Hilbert spaces, the operators $\\hat{\\tau }_{k}$ , $\\hat{\\tau }_{k+1}$ and $\\hat{\\tau }_{k+2}$ are expressed as matrices.", "For singlet, the matrices are $\\tau _{k}^{{\\bf 1}_{[0000]}} = \\tau _{k+1}^{{\\bf 1}_{[0000]}} = \\tau _{k+2}^{{\\bf 1}_{[0000]}} = 1,$ for $\\mathbb {H}^{{\\bf 1}_{[0000]}}$ , $\\tau _{k}^{{\\bf 1}_{[2000]}} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 1 \\\\0 & 0 & 1 & 0\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 1}_{[2000]}} =\\left(\\begin{array}{cccc}0 & 0 & 0 & 1 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\1 & 0 & 0 & 0\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 1}_{[2000]}} =\\left(\\begin{array}{cccc}0 & 1 & 0 & 0 \\\\1 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 1}_{[2000]}}$ , $\\tau _{k}^{{\\bf 1}_{[1100]}} =\\left(\\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 1}_{[1100]}} =\\left(\\begin{array}{cccccc}0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 0\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 1}_{[1100]}} =\\left(\\begin{array}{cccccc}-1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 \\\\0 & 0 & 0 & 0 & 1 & 0\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 1}_{[1100]}}$ , $&& \\tau _{k}^{{\\bf 1}_{[2200]}} =\\left(\\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 1}_{[2200]}} =\\left(\\begin{array}{cccccc}0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 1}_{[2200]}} =\\left(\\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 1}_{[2200]}}$ , $&& \\tau _{k}^{{\\bf 1}_{[2110]}} =\\left(\\begin{array}{cccccccccccc}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\end{array}\\right),\\,\\tau _{k+1}^{{\\bf 1}_{[2110]}} =\\left(\\begin{array}{cccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k+2}^{{\\bf 1}_{[2110]}} =\\left(\\begin{array}{cccccccccccc}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 1}_{[2110]}}$ , $\\tau _{k}^{{\\bf 1}_{[2220]}} =\\left(\\begin{array}{cccc}0 & 1 & 0 & 0 \\\\1 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 1}_{[2220]}} =\\left(\\begin{array}{cccc}0 & 0 & 0 & 1 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\1 & 0 & 0 & 0\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 1}_{[2220]}} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 1 \\\\0 & 0 & 1 & 0\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 1}_{[2220]}}$ , $\\tau _{k}^{{\\bf 1}_{[1111]}} =\\left(\\begin{array}{cc}-1 & 0 \\\\0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 1}_{[1111]}} =\\left(\\begin{array}{cc}\\frac{1}{2} & \\frac{\\sqrt{3}}{2} \\\\\\frac{\\sqrt{3}}{2} & -\\frac{1}{2}\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 1}_{[1111]}} =\\left(\\begin{array}{cc}-1 & 0 \\\\0 & 1\\end{array}\\right), $ for $\\mathbb {H}^{{\\bf 1}_{[1111]}}$ , $\\tau _{k}^{{\\bf 1}_{[2211]}} =\\left(\\begin{array}{cccccc}-1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 1}_{[2211]}} =\\left(\\begin{array}{cccccc}0 & 0 & 0 & 0 & 0 & -1 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 1}_{[2211]}} =\\left(\\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & -1 & 0\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 1}_{[2211]}}$ , $\\tau _{k}^{{\\bf 1}_{[2222]}} = \\tau _{k+1}^{{\\bf 1}_{[2222]}} = \\tau _{k+2}^{{\\bf 1}_{[2222]}} = 1,$ for $\\mathbb {H}^{{\\bf 1}_{[2222]}}$ .", "For doublet, the matrices are $\\tau _{k}^{{\\bf 2}_{[1000]}} =\\left(\\begin{array}{cccc}0 & -1 & 0 & 0 \\\\1 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 2}_{[1000]}} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 2}_{[1000]}} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & -1 \\\\0 & 0 & 1 & 0\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 2}_{[1000]}}$ , $&& \\tau _{k}^{{\\bf 2}_{[2100]}} =\\left(\\begin{array}{cccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 2}_{[2100]}} =\\left(\\begin{array}{cccccccccccc}0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k+2}^{{\\bf 2}_{[2100]}} =\\left(\\begin{array}{cccccccccccc}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 2}_{[2100]}}$ , $&& \\tau _{k}^{{\\bf 2}_{[1110]}} =\\left(\\begin{array}{cccccccc}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right),\\,\\tau _{k+1}^{{\\bf 2}_{[1110]}} =\\left(\\begin{array}{cccccccc}\\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\0 & 0 & \\frac{1}{2} & \\frac{\\sqrt{3}}{2} & 0 & 0 & 0 & 0 \\\\0 & -\\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & \\frac{\\sqrt{3}}{2} \\\\0 & -\\frac{\\sqrt{3}}{2} & 0 & 0 & 0 & 0 & 0 & -\\frac{1}{2} \\\\0 & 0 & 0 & 0 & \\frac{1}{2} & \\frac{\\sqrt{3}}{2} & 0 & 0 \\\\0 & 0 & 0 & 0 & \\frac{\\sqrt{3}}{2} & -\\frac{1}{2} & 0 & 0 \\\\\\frac{\\sqrt{3}}{2} & 0 & 0 & 0 & 0 & 0 & -\\frac{1}{2} & 0 \\\\0 & 0 & -\\frac{\\sqrt{3}}{2} & \\frac{1}{2} & 0 & 0 & 0 & 0\\end{array}\\right),\\nonumber \\\\&& \\tau _{k+2}^{{\\bf 2}_{[1110]}} =\\left(\\begin{array}{cccccccc}0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\end{array}\\right), $ for $\\mathbb {H}^{{\\bf 2}_{[1110]}}$ , $&& \\tau _{k}^{{\\bf 2}_{[2210]}} =\\left(\\begin{array}{cccccccccccc}0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 2}_{[2210]}} =\\left(\\begin{array}{cccccccccccc}0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k+2}^{{\\bf 2}_{[2210]}} =\\left(\\begin{array}{cccccccccccc}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 2}_{[2210]}}$ , $&& \\tau _{k}^{{\\bf 2}_{[2111]}} =\\left(\\begin{array}{cccccccc}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right),\\,\\tau _{k+1}^{{\\bf 2}_{[2111]}} =\\left(\\begin{array}{cccccccc}0 & 0 & 0 & 0 & \\frac{1}{2} & \\frac{\\sqrt{3}}{2} & 0 & 0 \\\\0 & \\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & \\frac{\\sqrt{3}}{2} \\\\0 & 0 & \\frac{1}{2} & \\frac{\\sqrt{3}}{2} & 0 & 0 & 0 & 0 \\\\0 & 0 & \\frac{\\sqrt{3}}{2} & -\\frac{1}{2} & 0 & 0 & 0 & 0 \\\\-\\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\-\\frac{\\sqrt{3}}{2} & 0 & 0 & 0 & 0 & 0 & -\\frac{1}{2} & 0 \\\\0 & 0 & 0 & 0 & -\\frac{\\sqrt{3}}{2} & \\frac{1}{2} & 0 & 0 \\\\0 & \\frac{\\sqrt{3}}{2} & 0 & 0 & 0 & 0 & 0 & -\\frac{1}{2}\\end{array}\\right), \\nonumber \\\\&& \\tau _{k+2}^{{\\bf 2}_{[2111]}} =\\left(\\begin{array}{cccccccc}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 2}_{[2111]}}$ , $\\tau _{k}^{{\\bf 2}_{[2221]}} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 1 \\\\0 & 0 & -1 & 0\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 2}_{[2221]}} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 2}_{[2221]}} =\\left(\\begin{array}{cccc}0 & 1 & 0 & 0 \\\\-1 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 2}_{[2221]}}$ .", "For triplet, the matrices are $\\tau _{k}^{{\\bf 3}_{[1100]}} =\\left(\\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 3}_{[1100]}} =\\left(\\begin{array}{cccccc}0 & -1 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 \\\\0 & 0 & 0 & 0 & 1 & 0\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 3}_{[1100]}} =\\left(\\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 3}_{[1100]}}$ , $&& \\tau _{k}^{{\\bf 3}_{[2110]}} =\\left(\\begin{array}{cccccccccccc}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 3}_{[2110]}} =\\left(\\begin{array}{cccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right), \\nonumber \\\\&& \\tau _{k+2}^{{\\bf 3}_{[2110]}} =\\left(\\begin{array}{cccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 3}_{[2110]}}$ , $\\tau _{k}^{{\\bf 3}_{[1111]}} =\\left(\\begin{array}{ccc}-1 & 0 & 0 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 3}_{[1111]}} =\\left(\\begin{array}{ccc}\\frac{1}{2} & -\\frac{1}{2} & \\frac{1}{\\sqrt{2}} \\\\-\\frac{1}{2} & \\frac{1}{2} & \\frac{1}{\\sqrt{2}} \\\\\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} & 0\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 3}_{[1111]}} =\\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & -1 & 0 \\\\0 & 0 & 1\\end{array}\\right), $ for $\\mathbb {H}^{{\\bf 3}_{[1111]}}$ , $\\tau _{k}^{{\\bf 3}_{[2211]}} =\\left(\\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 \\\\0 & -1 & 0 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 3}_{[2211]}} =\\left(\\begin{array}{cccccc}0 & 1 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & -1 & 0\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 3}_{[2211]}} =\\left(\\begin{array}{cccccc}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 3}_{[2211]}}$ .", "For quartet, the matrices are $\\tau _{k}^{{\\bf 4}_{[1110]}} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & -1 \\\\0 & 0 & 1 & 0\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 4}_{[1110]}} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 0 & -1 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 4}_{[1110]}} =\\left(\\begin{array}{cccc}0 & -1 & 0 & 0 \\\\1 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 4}_{[1110]}}$ , $\\tau _{k}^{{\\bf 4}_{[2111]}} =\\left(\\begin{array}{cccc}0 & 1 & 0 & 0 \\\\-1 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+1}^{{\\bf 4}_{[2111]}} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 1\\end{array}\\right), \\,\\tau _{k+2}^{{\\bf 4}_{[2111]}} =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 1 \\\\0 & 0 & -1 & 0\\end{array}\\right),$ for $\\mathbb {H}^{{\\bf 4}_{[2111]}}$ .", "For quintet, the matrices are $\\tau _{k}^{{\\bf 5}_{[1111]}} = \\tau _{k+1}^{{\\bf 5}_{[1111]}} = \\tau _{k+2}^{{\\bf 5}_{[1111]}} = 1,$ for $\\mathbb {H}^{{\\bf 5}_{[1111]}}$ .", "Interestingly, we find again the non-Abelian matrices in several Hilbert subspaces.", "For example, the matrices $\\tau _{k}^{{\\bf 1}_{[2000]}}$ , $\\tau _{k+1}^{{\\bf 1}_{[2000]}}$ and $\\tau _{k+2}^{{\\bf 1}_{[2000]}}$ in the Hilbert subspace $\\mathbb {H}^{{\\bf 1}_{[2000]}}$ are non-commutative; $\\tau _{\\ell }^{{\\bf 1}_{[2000]}} \\tau _{\\ell +1}^{{\\bf 1}_{[2000]}} \\ne \\tau _{\\ell +1}^{{\\bf 1}_{[2000]}} \\tau _{\\ell }^{{\\bf 1}_{[2000]}}$ for $\\ell =k$ , $k+1$ .", "Therefore, the exchange of the $\\ell $ -th and $(\\ell +1)$ -th vortices ($\\ell =k$ , $k+1$ ) induces the non-Abelian representation of the braid group in $\\mathbb {H}^{{\\bf 1}_{[2000]}}$ .", "Similarly, the non-Abelian representation of the braid group is realized in the following Hilbert subspaces; $\\mathbb {H}^{{\\bf 1}_{[1100]}}$ , $\\mathbb {H}^{{\\bf 1}_{[2200]}}$ , $\\mathbb {H}^{{\\bf 1}_{[2110]}}$ , $\\mathbb {H}^{{\\bf 1}_{[2220]}}$ , $\\mathbb {H}^{{\\bf 1}_{[1111]}}$ and $\\mathbb {H}^{{\\bf 1}_{[2211]}}$ for singlet, $\\mathbb {H}^{{\\bf 2}_{[1000]}}$ , $\\mathbb {H}^{{\\bf 2}_{[2100]}}$ , $\\mathbb {H}^{{\\bf 2}_{[1110]}}$ , $\\mathbb {H}^{{\\bf 2}_{[2210]}}$ , $\\mathbb {H}^{{\\bf 2}_{[2111]}}$ and $\\mathbb {H}^{{\\bf 2}_{[2221]}}$ for doublet, $\\mathbb {H}^{{\\bf 3}_{[1100]}}$ , $\\mathbb {H}^{{\\bf 3}_{[2110]}}$ , $\\mathbb {H}^{{\\bf 3}_{[1111]}}$ and $\\mathbb {H}^{{\\bf 3}_{[2211]}}$ for triplet, $\\mathbb {H}^{{\\bf 4}_{[1110]}}$ and $\\mathbb {H}^{{\\bf 4}_{[2111]}}$ for quartet.", "As we have discussed in the text, the U(1) Dirac fermions are embedded in the U(2) Dirac fermions.", "When $\\hat{\\psi }_{\\ell }^{2}$ ($\\ell =k,k+1,k+2,k+3$ ) are set to zero, the Hilbert subspaces $\\mathbb {H}^{{\\bf 1}_{[0000]}}$ , $\\mathbb {H}^{{\\bf 2}_{[1000]}}$ , $\\mathbb {H}^{{\\bf 3}_{[1100]}}$ , $\\mathbb {H}^{{\\bf 4}_{[1110]}}$ and $\\mathbb {H}^{{\\bf 5}_{[1111]}}$ in U(2) Dirac vortices coincide with the Hilbert subspaces $\\mathbb {H}^{(4,0)}$ , $\\mathbb {H}^{(4,1)}$ , $\\mathbb {H}^{(4,2)}$ , $\\mathbb {H}^{(4,3)}$ and $\\mathbb {H}^{(4,4)}$ in U(1) Dirac vortices, respectively.", "The matrices between the two are equivalent; $\\tau _{\\ell }^{{\\bf 1}_{[0000]}} &=& \\tau _{\\ell }^{(4,0)}, \\\\\\tau _{\\ell }^{{\\bf 2}_{[1000]}} &=& \\tau _{\\ell }^{(4,1)}, \\\\\\tau _{\\ell }^{{\\bf 3}_{[1100]}} &=& \\tau _{\\ell }^{(4,2)}, \\\\\\tau _{\\ell }^{{\\bf 4}_{[1110]}} &=& \\tau _{\\ell }^{(4,3)}, \\\\\\tau _{\\ell }^{{\\bf 5}_{[1111]}} &=& \\tau _{\\ell }^{(4,4)},$ with $\\ell =k$ , $k+1$ , $k+2$ ." ], [ "Subspaces with $(\\tau _{k})^{2}=1$ in U(2) Dirac vortices", "We recall that, for both cases with U(1) and U(2) Dirac fermions, four-time exchange of vortices is equivalent to the identity; $(T_{k})^{4}=1$ .", "The same relation holds at the operator level: $(\\hat{\\tau }_{k}^{\\mathrm {s}})^{4}=1$ for U(1) Dirac vortices and $(\\hat{\\tau }_{k})^{4}=1$ for U(2) Dirac vortices.", "However, a matrix $\\tau _{k}$ representing $\\hat{\\tau }_{k}^{\\mathrm {s}}$ or $\\hat{\\tau }_{k}$ happens to satisfy a stronger relation, $(\\tau _{k})^{2}=1$ , in some Hilbert subspaces.", "There, two-time exchange of vortices is equivalent to identity.", "In this Appendix, we explain how such a relation can be satisfied, and check if the representation of the braid group is still non-Abelian.", "First, we consider the case of U(1) Dirac fermions.", "From Eq.", "(REF ), we find that $(\\hat{\\tau }_{k}^{\\mathrm {s}})^{2}$ is expressed in terms of the number operator of Dirac fermions in the $\\ell $ -th ($\\ell =k$ , $k+1$ ) vortices, $\\hat{\\psi }_{\\ell }^{\\mathrm {s}\\dag } \\hat{\\psi }_{\\ell }^{\\mathrm {s}}$ .", "If we define $N_{\\ell }^{\\mathrm {s}}=0$ , 1 as an expectation value of the number operator $\\hat{\\psi }_{\\ell }^{\\mathrm {s}\\dag }\\hat{\\psi }_{\\ell }^{\\mathrm {s}}$ , we find that the matrix $\\tau ^{\\mathrm {s}}_{k}$ representing the operator $\\hat{\\tau }^{\\mathrm {s}}_{k}$ yields $(\\tau _{k}^{\\mathrm {s}})^{2} =(1-2N_{k}^{\\mathrm {s}}) (1-2N_{k+1}^{\\mathrm {s}})\\, .$ The right-hand-side reduces to 1 only when $(N_{k}^{\\mathrm {s}}, N_{k+1}^{\\mathrm {s}})=(0, 0)$ or $(1,1)$ .", "Therefore, we conclude that the relation $(\\tau _{k}^{\\mathrm {s}})^{2}=1$ holds only when all the vortices are empty or fully occupied: $(N_{1}^{\\mathrm {s}}, \\cdots , N_{n}^{\\mathrm {s}})=(0, \\cdots , 0)$ or $(1, \\cdots , 1)$ .", "In both cases, $\\tau _{\\ell }^{\\mathrm {s}}=1$ for any $\\ell =1, \\cdots , n-1$ , and hence it gives just trivial representation of the braid group.", "In fact, as shown in Appendix , the matrices for the empty and fully-occupied states are $\\tau _{\\ell }^{(n,0)}=\\tau _{\\ell }^{(n,n)}=1$ with $\\ell = 1, \\cdots , n-1$ for $n=2$ , 3, 4.", "Second, let us consider the case of U(2) Dirac fermions.", "In this case, there are non-Abelian matrices satisfying $(\\tau _{\\ell })^{2}=1$ for any $\\ell =1,\\cdots ,n-1$ .", "From Eq.", "(REF ), we find again that $(\\hat{\\tau }_{k})^{2}$ is expressed in terms of the number operator of Dirac fermions in the $\\ell $ -th ($\\ell =k$ , $k+1$ ) vortices, $\\hat{\\psi }_{\\ell }^{a\\dag } \\hat{\\psi }_{\\ell }^{a}$ with $a=1$ , 2.", "Note that the indices $a=1$ , 2 of the pseudo-spin are introduced.", "Then, defining $N^{a}_{\\ell }=0$ , 1 as an expectation value of the number operator $\\hat{\\psi }_{\\ell }^{a\\dag } \\hat{\\psi }_{\\ell }^{a}$ , we find that the matrix $\\tau _{k}$ representing the operator $\\hat{\\tau }_{k}$ yields $(\\tau _{k})^{2} = (1-2N_{k}^{1}) (1-2N_{k}^{2}) (1-2N_{k+1}^{1}) (1-2N_{k+1}^{2}).$ The relation $(\\tau _{k})^{2}=1$ is fulfilled by the following combinations, $(N_{k}^{1}, N_{k}^{2}, N_{k+1}^{1}, N_{k+1}^{2}) &=& (0,0,0,0), \\quad (1,1,1,1), \\quad (1,1,0,0), \\quad (0,0,1,1), \\nonumber \\\\&& (1,0,1,0), \\quad (1,0,0,1), \\quad (0,1,1,0), \\quad (0,1,0,1).$ When we define $N_{\\ell }=N_{\\ell }^{1}+N_{\\ell }^{2}$ , the above combinations are further rewritten as $(N_{k}, N_{k+1}) &=& (0,0), \\quad (2,2), \\quad (2,0), \\quad (0,2), \\quad (1,1).$ Therefore, we conclude that the relation $(\\tau _{k})^{2}=1$ is satisfied when $N_{k}+N_{k+1}$ is an even number.", "This conclusion is consistent with the expectation from the transformation properties of the Dirac fermions under the operation $(T_k)^2$ .", "Under the two successive exchanges of $k$ -th an $(k+1)$ -th vortices, the Dirac fermion operators $\\hat{\\psi }_k^a$ and $\\hat{\\psi }_{k+1}^a$ are multiplied by $-1$ .", "If a state is composed of an even number of $k$ -th and $(k+1)$ -th fermions, the minus signs cancel and the state is unchanged under $(T_k)^2$ .", "Therefore, in order for that the condition $(\\tau _{\\ell })^{2}=1$ holds for any $\\ell = 1, \\cdots , n-1$ , a sum of the Dirac fermion number in the every neighboring vortices, $N_{\\ell }+N_{\\ell +1}$ , has to be an even number.", "We note that $(N_{1}, \\cdots , N_{n})=(0, \\cdots , 0)$ and $(2, \\cdots ,2)$ corresponding to the empty state and fully-occupied state, respectively, give just trivial representation of the braid group $\\tau _{\\ell }=1$ , like the case of U(1) Dirac fermions as discussed above.", "It is also the case for the state with the highest dimension in pseudo-spin representation.", "However, the other combinations of $(N_{1}, \\cdots , N_{n})$ induce non-Abelian matrices, namely non-Abelian representation of the braid group.", "Let us see examples in U(2) Dirac fermions with $n=3$ and 4. a) For $n=3$ , the Hilbert subspaces, in which $N_{\\ell }+N_{\\ell +1}$ is an even number for any $\\ell = 1$ , 2, are $\\mathbb {H}^{{\\bf 1}_{[000]}}, \\quad \\mathbb {H}^{{\\bf 1}_{[200]}}, \\quad \\mathbb {H}^{{\\bf 1}_{[220]}}, \\quad \\mathbb {H}^{{\\bf 1}_{[222]}}, \\quad \\mathbb {H}^{{\\bf 2}_{[111]}} \\quad \\mbox{and} \\quad \\mathbb {H}^{{\\bf 4}_{[111]}}.$ The matrices in each Hilbert subspace are $\\tau _{\\ell }^{{\\bf 1}_{[000]}}, \\quad \\tau _{\\ell }^{{\\bf 1}_{[200]}}, \\quad \\tau _{\\ell }^{{\\bf 1}_{[220]}}, \\quad \\tau _{\\ell }^{{\\bf 1}_{[222]}}, \\quad \\tau _{\\ell }^{{\\bf 2}_{[111]}} \\quad \\mbox{and} \\quad \\tau _{\\ell }^{{\\bf 4}_{[111]}}.$ Among them, the matrices $\\tau _{\\ell }^{{\\bf 1}_{[000]}}$ , $\\tau _{\\ell }^{{\\bf 1}_{[222]}}$ and $\\tau _{\\ell }^{{\\bf 4}_{[111]}}$ are trivial, because they correspond to the empty state, full-occupied state and the state with highest dimension in pseudo-spin representation, respectively.", "The other matrices $\\tau _{\\ell }^{{\\bf 1}_{[200]}}$ , $\\tau _{\\ell }^{{\\bf 1}_{[220]}}$ and $\\tau _{\\ell }^{{\\bf 2}_{[111]}}$ are non-Abelian matrices, and hence the Hilbert subspaces $\\mathbb {H}^{{\\bf 1}_{[200]}}$ , $\\mathbb {H}^{{\\bf 1}_{[220]}}$ and $\\mathbb {H}^{{\\bf 2}_{[111]}}$ lead to the non-Abelian representation of the braid group satisfying $(\\tau _{\\ell })^{2}=1$ for $\\ell =1$ , 2. b) For $n=4$ , from Appendix , the Hilbert subspaces, in which $N_{\\ell }+N_{\\ell +1}$ is an even number for any $\\ell = 1$ , 2, 3, are $\\mathbb {H}^{{\\bf 1}_{[0000]}}, \\quad \\mathbb {H}^{{\\bf 1}_{[2000]}}, \\quad \\mathbb {H}^{{\\bf 1}_{[2200]}}, \\quad \\mathbb {H}^{{\\bf 1}_{[2220]}}, \\quad \\mathbb {H}^{{\\bf 1}_{[1111]}}, \\quad \\mathbb {H}^{{\\bf 1}_{[2222]}}, \\quad \\mathbb {H}^{{\\bf 3}_{[1111]}}\\quad \\mbox{and} \\quad \\mathbb {H}^{{\\bf 5}_{[1111]}}.$ The matrices in each Hilbert subspace are $\\tau _{\\ell }^{{\\bf 1}_{[0000]}}, \\quad \\tau _{\\ell }^{{\\bf 1}_{[2000]}}, \\quad \\tau _{\\ell }^{{\\bf 1}_{[2200]}}, \\quad \\tau _{\\ell }^{{\\bf 1}_{[2220]}}, \\quad \\tau _{\\ell }^{{\\bf 1}_{[1111]}}, \\quad \\tau _{\\ell }^{{\\bf 1}_{[2222]}}, \\quad \\tau _{\\ell }^{{\\bf 3}_{[1111]}}\\quad \\mbox{and} \\quad \\tau _{\\ell }^{{\\bf 5}_{[1111]}}.$ Among them, the matrices $\\tau _{\\ell }^{{\\bf 1}_{[0000]}}$ , $\\tau _{\\ell }^{{\\bf 1}_{[2222]}}$ and $\\tau _{\\ell }^{{\\bf 5}_{[1111]}}$ are trivial, because they correspond to the empty state, full-occupied state and the state with highest dimension in pseudo-spin representation, respectively.", "The other matrices $\\tau _{\\ell }^{{\\bf 1}_{[2000]}}$ , $\\tau _{\\ell }^{{\\bf 1}_{[2200]}}$ , $\\tau _{\\ell }^{{\\bf 1}_{[2220]}}$ , $\\tau _{\\ell }^{{\\bf 1}_{[1111]}}$ and $\\tau _{\\ell }^{{\\bf 3}_{[1111]}}$ are non-Abelian matrices, and hence the Hilbert subspaces $\\mathbb {H}^{{\\bf 1}_{[2000]}}$ , $\\mathbb {H}^{{\\bf 1}_{[2200]}}$ , $\\mathbb {H}^{{\\bf 1}_{[2220]}}$ , $\\mathbb {H}^{{\\bf 1}_{[1111]}}$ and $\\mathbb {H}^{{\\bf 3}_{[1111]}}$ lead to the non-Abelian representation of the braid group satisfying $(\\tau _{\\ell })^{2}=1$ for $\\ell =1$ , 2, 3.", "We note that the condition $(\\tau _{k})^{2}=1$ with the braid relations (i) $\\tau _{k}\\tau _{\\ell }\\tau _{k}=\\tau _{\\ell }\\tau _{k}\\tau _{\\ell }$ for $|k-\\ell |=1$ and (ii) $\\tau _{k}\\tau _{\\ell }=\\tau _{\\ell }\\tau _{k}$ for $|k-\\ell |>1$ leads to the relations, (i') $(\\tau _{k}\\tau _{\\ell })^{3} =1$ for $|k-\\ell |=1$ and (ii') $(\\tau _{k}\\tau _{\\ell })^{2}=1$ for $|k-\\ell |>1$ for $k, \\ell =1, \\cdots , n-1$ .", "The relations (i') and (ii') imply that such matrices $\\tau _{\\ell }$ are regarded as generators of the symmetric group or the Coxeter group of the type $A_{n-1}$ .", "We recall that the Coxeter group was obtained in the SO(3) or more generally SO($N$ ) ($N$ odd numbers) symmetric Majorana vortices [8], [10].", "There, the matrices for exchanging vortices were tensor product of “the Ivanov matrices” found by Ivanov [6] and the generators of the Coxeter group of $A_{2m-1}$ type ($2m$ the number of vortices).", "In the present case of U(2) Dirac vortices, however, such tensor structure was not found." ] ]

[ [ "A strategy towards the extraction of the Sivers function with TMD\n evolution" ], [ "Abstract The QCD evolution of the unpolarized Transverse Momentum Dependent (TMD) distribution functions and of the Sivers functions have been discussed in recent papers.", "Following such results we reconsider previous extractions of the Sivers functions from semi-inclusive deep inelastic scattering data and propose a simple strategy which allows to take into account the Q^2 dependence of the TMDs in comparison with experimental findings.", "A clear evidence of the phenomenological success of the TMD evolution equations is given, mostly, by the newest COMPASS data off a transversely polarized proton target." ], [ " Introduction and formalism", "The exploration of the 3-dimensional structure of the nucleons, both in momentum and configuration space, is one of the major issues in hadron high energy physics, with dedicated experimental and theoretical efforts.", "In particular, several Semi-Inclusive Deep Inelastic Scattering (SIDIS) experiments are either running or being planned.", "From the measurements of azimuthal asymmetries, both with unpolarized and polarized nucleons, one obtains information on the Transverse Momentum Dependent Parton Distribution Functions (TMD PDFs) and on the Transverse Momentum Dependent Fragmentation Functions (TMD FFs).", "The TMD PDFs and the TMD FFs are often globally referred to simply as TMDs.", "The TMD PDFs convey information on the momentum distributions of partons inside protons and neutrons.", "The analysis of the experimental data is based on the so-called TMD factorization, which links measurable cross sections and spin asymmetries to a convolution of TMDs.", "In particular, the Sivers function, which describes the number density of unpolarized quarks inside a transversely polarized proton, has received much attention and has been extracted from SIDIS data by several groups, with consistent results [1], [2], [3], [4], [5], [6].", "However, all these phenomenological fits of the Sivers function (and other TMDs) have been performed so far using a simplified version of the TMD factorization scheme, in which the QCD scale dependence of the TMDs – which was unknown – is either neglected or limited to the collinear part of the unpolarized PDFs.", "While this might not be a serious numerical problem when considering only experimental data which cover limited ranges of low $Q^2$ values, it is not correct in principle, and taking into account the appropriate $Q^2$ evolution might be numerically relevant for predictions at higher $Q^2$ values, like future electron-ion or electron-nucleon colliders (EIC/ENC) and Drell-Yan experiments.", "Recently, the issue of the QCD evolution of unpolarized TMDs and of the Sivers function has been studied in a series of papers [7], [8], [9] and a complete TMD factorization framework is now available for a consistent treatment of SIDIS data and the extraction of TMDs.", "A first application of the new TMD evolution equations to some limited samples of the HERMES and COMPASS data [10] has indeed shown clear signs of the $Q^2$ TMD evolution.", "We follow here Refs.", "[8] and [9] adopting their formalism, which includes the explicit $Q^2$ dependence of the TMDs, and apply it to the extraction of the Sivers function from SIDIS data, exploiting the latest HERMES [11] and COMPASS [12] results.", "In the sequel of this Section we present the explicit formalism: in Subsection REF we describe the setup and structure of the TMD evolution equations, in Subsection REF we discuss the parameterizations used for the unknown input functions, while in Subsection REF we present analytical solutions of the TMD evolutions equations obtained under a specific approximation.", "In Section  we perform a best fit of the SIDIS Sivers asymmetries taking into account the different $Q^2$ values of each data point and the $Q^2$ dependence of the TMDs; we compare our results with a similar analysis performed without the TMD evolution.", "Differences between Sivers functions extracted from data with and without the TMD evolution are shown and commented.", "In all this we differ from Ref.", "[10], which explicitly shows the evolution of an existing fit of the Sivers SIDIS asymmetry [13] from the average value $\\langle Q^2\\rangle = 2.4$  GeV$^2$ for HERMES data [11] to the average value of $\\langle Q^2 \\rangle = 3.8$ GeV$^2$ for the most recent COMPASS data [12].", "Further comments and conclusions are given in Section .", "In Refs.", "[7] and [8], Collins, Aybat and Rogers have proposed a scheme to describe the $Q^2$ evolution of the TMD unpolarized distribution and fragmentation functions: within the framework of the Collins-Soper-Sterman (CSS) factorization formalism [14], [15], they can describe the non-perturbative, low transverse momentum region and, at the same time, consistently include the perturbative corrections affecting the region of larger energies and momentum transfers.", "However, this formalism cannot be directly applied to spin dependent distribution functions, like the Sivers function [16], for which the collinear limit does not exist.", "More recently, an extension of the unpolarized TMD-evolution formalism was presented in Ref.", "[9] to provide a framework in which also spin-correlated PDFs can be accounted for.", "For our purposes, we will use Eq.", "(44) of Ref.", "[9] which, compared to the unpolarized TMD evolution scheme, Eq.", "(26) of Ref.", "[8], requires the extra aid of a phenomenological input function embedding the missing information on the evolved function, that, in the case of the Sivers function, is both of perturbative and non-pertubative nature.", "Although the unpolarized PDF and FF TMD evolution equations are in principle known [8], in this paper we adopt the simplified functional form of the evolution equation, as proposed for the Sivers function in Ref.", "[9], for all TMD functions, for consistency.", "Thus, we strictly follow Ref.", "[9] and combine their Eqs.", "(44), (43) and (30), taking, as suggested [9], the renormalization scale $\\mu ^2$ and the regulating parameters $\\zeta _F$ and $\\zeta _D$ all equal to $Q^2$ .", "Then, the QCD evolution of the TMDs in the coordinate space can be written as $\\widetilde{F}(x, \\mbox{$b$}_T; Q) = \\widetilde{F}(x, \\mbox{$b$}_T; Q_0)\\exp \\left\\lbrace \\ln \\frac{Q}{Q_0} \\: \\widetilde{K}(b_T; Q_0)+ \\int _{Q_0}^Q \\frac{\\rm d \\mu }{\\mu } \\gamma _F \\left( \\mu , \\frac{Q^2}{\\mu ^2}\\right) \\right\\rbrace \\:, $ where $\\widetilde{F}$ can be either the unpolarized parton distribution, $\\widetilde{F}(x, \\mbox{$b$}_T; Q) = \\widetilde{f}_{q/p}(x, \\mbox{$b$}_T; Q)$ , the unpolarized fragmentation function $\\widetilde{F}(x, \\mbox{$b$}_T; Q) = \\widetilde{D}_{h/q}(z,\\mbox{$b$}_T; Q)$ , or the first derivative, with respect to the parton impact parameter $b_T$ , of the Sivers function, $\\widetilde{F}(x, \\mbox{$b$}_T; Q) =\\widetilde{f}_{1T}^{\\prime \\perp f}(x, \\mbox{$b$}_T; Q)$ .", "Notice that throughout the paper $b_T$ -dependent distribution and fragmentation functions will be denoted with a $\\sim $ on top.", "In the above equation the function $\\widetilde{K}$ is given in general by [9]: $\\widetilde{K}(b_T, \\mu ) = \\widetilde{K}(b_*, \\mu _b) + \\left[\\int _{\\mu }^{\\mu _b}\\frac{\\rm d \\mu ^{\\prime }}{\\mu ^{\\prime }} \\gamma _K(\\mu ^{\\prime }) \\right]- g_K(b_T) \\:, $ with, at ${\\cal O}(\\alpha _s)$ [14], [15], $\\widetilde{K}(b_*, \\mu _b) = - \\frac{\\alpha _s \\, C_F}{\\pi } \\left[\\ln (b_*^2 \\, \\mu _b^2) - \\ln 4 + 2 \\gamma _E \\right]$ $b_*(b_T) \\equiv \\frac{b_T}{\\sqrt{1 + b_T^2/b_{\\rm max}^2}}\\quad \\quad \\quad \\mu _b = \\frac{C_1}{b_*(b_T)} \\: \\cdot $ The first two terms in Eq.", "(REF ) are perturbative and depend on the scale $\\mu $ through the coupling $\\alpha _s(\\mu )$ , while the last term is non-perturbative, but scale independent.", "$C_1$ is a constant parameter which can be fixed to optimize the perturbative expansion, as explained in Ref. [15].", "Refs.", "[8] and [9] adopt the particular choice $C_1 = 2 e^{-\\gamma _E}$ which automatically implies $\\widetilde{K}(b_*, \\mu _b) = 0$ , considerably simplifying the $b_T$ dependence of the CSS kernel $\\widetilde{K}(b_T, \\mu )$ , Eq.", "(REF ).", "The anomalous dimensions $\\gamma _F$ and $\\gamma _K$ appearing respectively in Eqs.", "(REF ) and (REF ), are given, again at order ${\\cal O}(\\alpha _s)$ , by [15], [8] $\\gamma _F(\\mu ; \\frac{Q^2}{\\mu ^2}) = \\alpha _s(\\mu ) \\, \\frac{C_F}{\\pi }\\left( \\frac{3}{2} - \\ln \\frac{Q^2}{\\mu ^2} \\right)\\quad \\quad \\quad \\quad \\gamma _K(\\mu ) = \\alpha _s(\\mu ) \\, \\frac{2 \\, C_F}{\\pi } \\: \\cdot $ By making use of Eqs.", "(REF )-(REF ), the evolution of $\\widetilde{F}(x, \\mbox{$b$}_T; Q)$ in Eq.", "(REF ) can then be written as: $\\widetilde{F}(x, \\mbox{$b$}_T; Q) = \\widetilde{F}(x, \\mbox{$b$}_T; Q_0)\\:\\widetilde{R}(Q, Q_0, b_T)\\: \\exp \\left\\lbrace - g_K(b_T) \\ln \\frac{Q}{Q_0} \\right\\rbrace \\:, $ with $\\widetilde{R}(Q, Q_0, b_T)\\equiv \\exp \\left\\lbrace \\ln \\frac{Q}{Q_0} \\int _{Q_0}^{\\mu _b} \\frac{\\rm d \\mu ^{\\prime }}{\\mu ^{\\prime }} \\gamma _K(\\mu ^{\\prime }) +\\int _{Q_0}^Q \\frac{\\rm d \\mu }{\\mu }\\gamma _F \\left( \\mu , \\frac{Q^2}{\\mu ^2} \\right)\\right\\rbrace \\: \\cdot $ The $Q^2$ evolution is driven by the functions $g_K(b_T)$ and $\\widetilde{R}(Q,Q_0,b_T)$ .", "While the latter, Eq.", "(REF ), can be easily evaluated, numerically or even analytically, the former, is essentially unknown and will need to be taken from independent experimental inputs.", "The explicit expression of the TMDs in the momentum space, with the QCD $Q^2$ dependence, can be obtained by Fourier-transforming Eq.", "(REF ), obtaining [9]: $\\widehat{f}_{q/p}(x, k_\\perp ; Q) = \\frac{1}{2\\pi } \\int _0^\\infty \\!\\!\\!", "{\\rm d} b_T\\: b_T \\: J_0(k_\\perp b_T) \\: \\widetilde{f}_{q/p}(x, b_T; Q)$ $\\widehat{D}_{h/q}(z, p_\\perp ; Q) = \\frac{1}{2\\pi } \\int _0^\\infty \\!\\!\\!", "{\\rm d} b_T\\: b_T \\: J_0({\\rm k}_T b_T) \\: \\widetilde{D}_{h/q}(z, b_T; Q)$ $\\widehat{f}_{1T}^{\\perp f}(x, k_\\perp ; Q) = \\frac{-1}{2\\pi k_\\perp } \\int _0^\\infty \\!\\!\\!", "{\\rm d} b_T \\: b_T \\: J_1(k_\\perp b_T) \\:\\widetilde{f}_{1T}^{\\prime \\,\\perp q}(x, b_T; Q) \\:, $ where $J_0$ and $J_1$ are Bessel functions.", "In this paper we denote the distribution and fragmentation functions which depend on the transverse momenta (TMDs) with a “widehat\" on top.", "$\\widehat{f}_{q/p}$ is the unpolarized TMD distribution function for a parton of flavor $q$ inside a proton, and $\\widehat{D}_{h/q}$ is the unpolarized TMD fragmentation function for hadron $h$ inside a parton $q$ .", "$\\widehat{f}_{1T}^{\\perp q}$ is the Sivers distribution defined, for unpolarized partons inside a transversely polarized proton, as: $\\widehat{f}_{q/p^\\uparrow }(x, \\mbox{$k$}_\\perp , \\mbox{$S$}; Q) &=&\\widehat{f}_{q/p}(x, k_\\perp ; Q) - \\widehat{f}_{1T}^{\\perp q}(x, k_\\perp ; Q)\\frac{\\epsilon _{ij} \\, k_\\perp ^i \\,S^j}{M_p} \\\\&=& \\widehat{f}_{q/p}(x, k_\\perp ; Q)+ \\frac{1}{2} \\Delta ^N \\widehat{f}_{q/p^\\uparrow }(x, k_\\perp ; Q)\\frac{\\epsilon _{ij}\\, k_\\perp ^i \\, S^j}{k_\\perp } \\: \\cdot $ In our notation $\\mbox{$k$}_\\perp $ is the transverse momentum of the parton with respect to the parent nucleon direction and $\\mbox{$p$}_\\perp $ is the transverse momentum of the final hadron with respect to the parent parton direction.", "Notice that in Refs.", "[8] and [9] all transverse momenta are defined in a unique frame, the so-called hadron frame, in which the measured hadrons have zero transverse momentum.", "In this frame, the initial and the final parton transverse momenta are denoted, respectively, by ${\\bf k}_{1T}$ and ${\\bf k}_{2T}$ .", "They are related to our notation by: $\\mbox{$k$}_\\perp = {\\bf k}_{1T}$ and, at leading order in $p_\\perp $ , $\\mbox{$p$}_\\perp = - z \\, {\\bf k}_{2T}$ .", "This requires some attention when dealing with the fragmentation functions.", "Usually, the TMD FFs are defined in terms of the hadronic $p_\\perp $ , i.e.", "the transverse momentum of the final hadron $h$ with respect to the direction of the fragmenting parton $q$ , while, following Refs.", "[8] and [9], the Fourier transform (REF ) is performed from the impact parameter space of the fragmenting parton ($b_T$ ) into the corresponding partonic transverse momentum (k$_T = p_\\perp / z$ ) in the hadron frame.", "This will generate some extra $z^2$ factors, as explained in detail in Section REF ." ], [ " Parameterization of unknown functions", "Eqs.", "(REF )-(REF ) can be adopted as the appropriate functional forms, with the correct $Q^2$ dependence induced by Eqs.", "(REF )-(REF ), to be used in the extraction of phenomenological information on the unpolarized and Sivers TMDs.", "In order to do so, one should start with a parameterization of the unknown functions inside Eq.", "(REF ): $g_K(b_T)$ and $\\widetilde{F}(x, b_T; Q_0)$ .", "As already anticipated, $g_K(b_T)$ is a non-perturbative, but universal function, which in the literature is usually parameterized in a quadratic form.", "As in Refs.", "[9] and [10], we will adopt the results provided by a recent fit of Drell-Yan data [17], and assume $g_K(b_T) = \\frac{1}{2} \\, g_2 \\, b_T^2 \\quad \\quad {\\rm with} \\quad \\quad g_2 = 0.68 \\quad \\quad {\\rm corresponding~to} \\quad \\quad b_{\\rm max}=0.5 \\: {\\rm GeV}^{-1} \\:.$ We should now parameterize the function $\\widetilde{F}(x, b_T; Q_0)$ in configuration space.", "We wish to test the effect of the TMD evolution in the extraction of the Sivers functions from data; in particular we will compare the extraction based on TMD evolution with previous extractions which did not take such an evolution into account.", "Then, we parameterize the input function $\\widetilde{F}(x, b_T; Q_0)$ by requiring that its Fourier-transform, which gives the corresponding TMD function in the transverse momentum space, coincides with the previously adopted $k_\\perp $ -Gaussian form, with the $x$ dependence factorized out.", "That was also done in Refs.", "[8] and [9], assuming for the unpolarized TMD PDF $\\widetilde{f}_{q/p}(x, b_T; Q_0) = f_{q/p}(x,Q_0) \\exp \\left\\lbrace -\\alpha ^2\\,b_T^2 \\right\\rbrace \\:, $ where $f_{q/p}(x,Q_0)$ is the usual integrated PDF of parton $q$ inside proton $p$ , evaluated at $Q_0$ ; the value of $\\alpha ^2$ is fixed by requiring the desired behavior of the distribution function in the transverse momentum space at the initial scale $Q_0$ : taking $\\alpha ^2=\\langle k_\\perp ^2 \\rangle /4$ one recovers $\\widehat{f}_{q/p}(x,k_\\perp ; Q_0) = f_{q/p}(x,Q_0) \\, \\frac{1}{\\pi \\langle k_\\perp ^2\\rangle } \\,e^{-{k_\\perp ^2}/{\\langle k_\\perp ^2\\rangle }}\\,,$ in agreement with Refs.", "[5], [13], [18].", "Similar relations hold for the TMD FFs, with an additional $z^2$ factor due to the fact that the Fourier-transform (REF ) leads from the impact parameter space of the fragmenting parton in the hadron frame to the corresponding partonic transverse momentum k$_T$ , while the TMD FFs are functions of the transverse momentum $p_\\perp = z \\, {\\rm k}_T$ of the final hadron with respect to the fragmenting parton direction.", "This requires the initial parameterization $\\widetilde{D}_{h/q}(z, b_T; Q_0) = \\frac{1}{z^2} \\, D_{h/q}(z,Q_0) \\;\\exp \\left\\lbrace -\\beta ^2\\,b_T^2 \\right\\rbrace \\:, $ where $D_{h/q}(z,Q_0)$ is the usual integrated FF evaluated at the initial scale $Q_0$ , and $\\beta ^2 = \\langle p_\\perp ^2\\rangle /4z^2$ in order to recover the previously adopted behavior [5], [13], [18] of the fragmentation function in the $p_\\perp $ transverse momentum space at $Q_0$ : $\\widehat{D}_{h/q}(z,p _\\perp ; Q_0) = D_{h/q}(z,Q_0) \\, \\frac{1}{\\pi \\langle p_\\perp ^2\\rangle }\\, e^{-p_\\perp ^2/\\langle p_\\perp ^2\\rangle } \\:.$ Analogously, we parameterize the Sivers function at the initial scale $Q_0$ as $\\widetilde{f}_{1T}^{\\prime \\perp }(x, b_T; Q_0 ) = - 2 \\, \\gamma ^2 \\,f_{1T}^{\\perp }(x; Q_0 ) \\, b_T\\, e^{-\\gamma ^2 \\, b_T^2} \\:,$ which, when Fourier-transformed according to Eq.", "(REF ), yields: $\\widehat{f}_{1T}^{\\perp }(x, k_\\perp ; Q_0) = f_{1T}^{\\perp }(x; Q_0 ) \\, \\frac{1}{4 \\, \\pi \\, \\gamma ^2} \\, e^{-k_\\perp ^2 / 4 \\gamma ^2} \\:.", "$ Eq.", "(REF ) agrees with our previous parameterization of the Sivers function, at the initial scale $Q_0$  [5], [13], [18], taking: $4 \\, \\gamma ^2 \\equiv \\langle k_\\perp ^2 \\rangle _S = \\frac{M_1^2 \\, \\langle k_\\perp ^2 \\rangle }{M_1^2 +\\langle k_\\perp ^2 \\rangle }$ $f_{1T}^{\\perp }(x; Q_0) = - \\frac{M_p}{2 M_1} \\sqrt{2e} \\;\\Delta ^N \\!", "f_{q/p^\\uparrow }(x,Q_0) \\, \\frac{\\langle k_\\perp ^2 \\rangle _S}{\\langle k_\\perp ^2 \\rangle } \\:\\cdot $ $M_1$ is a mass parameter, $M_p$ the proton mass and $\\Delta ^N \\!f_{q/p^\\uparrow }(x,Q_0)$ is the $x$ -dependent term of the Sivers function, evaluated at the initial scale $Q_0$ and written as [5], [13], [18]: $\\Delta ^N \\!", "f_{q/p^\\uparrow }(x,Q_0) =2 \\, {\\cal N}_q(x) \\, f_{q/p} (x,Q_0)\\; , $ where ${\\cal N}_q(x)$ is a function of $x$ , properly parameterized (we will come back to details of the Sivers function parameterization in Section ).", "The final evolution equations of the unpolarized TMD PDFs and TMD FFs, in the configuration space, are obtained inserting Eqs.", "(REF ) and (REF ) into Eq.", "(REF ): $\\widetilde{f}_{q/p}(x, b_T; Q) = f_{q/p}(x,Q_0) \\;\\widetilde{R}(Q, Q_0, b_T)\\;\\exp \\left\\lbrace -b_T^2 \\left(\\alpha ^2\\, + \\frac{g_2}{2} \\ln \\frac{Q}{Q_0}\\right) \\right\\rbrace $ $\\widetilde{D}_{h/q}(z, b_T; Q) = \\frac{1}{z^2} D_{h/q}(z,Q_0) \\;\\widetilde{R}(Q, Q_0, b_T)\\;\\exp \\left\\lbrace - b_T^2 \\left(\\beta ^2\\, + \\frac{g_2}{2}\\ln \\frac{Q}{Q_0}\\right) \\right\\rbrace \\:,$ with $\\alpha ^2 = \\langle k_\\perp ^2 \\rangle /4$ , $\\beta ^2 = \\langle p_\\perp ^2 \\rangle /(4z^2)$ , $g_2$ given in Eq.", "(REF ) and $\\widetilde{R}(Q, Q_0, b_T)$ in Eq.", "(REF ).", "The evolution of the Sivers function is obtained through its first derivative, inserting Eq.", "() into Eq.", "(REF ): $\\widetilde{f}_{1T}^{\\prime \\perp }(x, b_T; Q) = -2 \\, \\gamma ^2 \\,f_{1T}^{\\perp }(x; Q_0) \\, \\widetilde{R}(Q,Q_0,b_T) \\, b_T\\,\\exp \\left\\lbrace -b_T^2 \\left( \\gamma ^2\\, + \\frac{g_2}{2} \\ln \\frac{Q}{Q_0}\\right) \\right\\rbrace \\,$ with $\\gamma ^2$ and $f_{1T}^{\\perp }(x; Q_0)$ given in Eqs.", "(REF )-(REF ).", "Eqs.", "(REF )-(REF ) show that the $Q^2$ evolution is controlled by the logarithmic $Q$ dependence of the $b_T$ Gaussian width, together with the factor $\\widetilde{R}(Q, Q_0, b_T)$ : for increasing values of $Q^2$ , they are responsible for the typical broadening effect already observed in Refs.", "[8] and [9].", "It is important to stress that although the structure of Eq.", "(REF ) is general and holds over the whole range of $b_T$ values, the input function $\\widetilde{F}(x,\\mbox{$b$}_T,Q_0)$ is only designed to work in the large-$b_T$ region, corresponding to low $k_\\perp $ values.", "Therefore, this formalism is perfectly suitable for phenomenological applications in the kinematical region we are interested in, but the parameterization of the input function should be revised in the case one wishes to apply it to a wider range of transverse momenta, like higher $Q^2$ processes where perturbative corrections become important." ], [ " Analytical solution of the TMD Evolution Equations", "The TMD evolution in Eqs.", "(REF )-(REF ) implies, apart from the explicit Gaussian dependence, a further non trivial dependence on the parton impact parameter $b_T$ through the evolution kernel $\\widetilde{R}(Q,Q_0,b_T)$ and the upper integration limit $\\mu _b$ , Eq.", "(REF ), which appears in Eq.", "(REF ); consequently, it needs to be evaluated numerically.", "However, the evolution equations can be solved analytically by making a simple approximation on this $b_T$ dependence.", "A close examination of Eq.", "(REF ) shows that $\\mu _b$ is a decreasing function of $b_T$ that very rapidly freezes to the constant value $C_1/b_{\\rm max}=\\mu _b(b_T\\rightarrow \\infty )$ : more precisely, the approximation $\\mu _b = const.$ holds for any $b_T \\mathrel {\\unknown.", "{\\hspace{1.0pt}\\sim }}$ >$$ 1$ GeV$ -1$.As very small values of $ bT$ correspond to very large values of $ k$,this approximation is safe in our framework, where the typical$ k$ are less than $ 1$ GeV.", "Neglecting the $ bT$-dependence of $ b$,the factor $ R(Q,Q0,bT)$ does not depend on $ bT$ anymore, seeEq.~(\\ref {RQQ0}), and can even be integrated analytically by using anexplicit representation of $ s(Q)$.", "In the sequel we will refer toit as $ R(Q,Q0)$, with $ R(Q,Q0) R(Q,Q0,bT)$.Fig.~\\ref {fig:R} shows the evolution factor $ R(Q,Q0,bT)$plotted as a function of $ bT$ at two fixed values of $ Q2$ (left panel),and $ R(Q,Q0)$ as a function of $ Q2$ (right panel).", "It is clear that$ R(Q,Q0)$ settles to a constant value for $ bT $\\sim $$>$ 1$ GeV$ -1$.In both cases, $ Q02=1$ GeV$ 2$.$ Thus, in this approximation, the TMD evolution equation (REF ) only depends on $b_T$ through the non-perturbative function $g_K(b_T)$ , which has been chosen to be a quadratic function of $b_T$ , Eq.", "(REF ), and through the $b_T$ dependence of the initial input function $\\widetilde{F}(x, \\mbox{$b$}_T; Q_0)$ which has been chosen to be Gaussian.", "It results in a $b_T$ -Gaussian form, with a width which depends logarithmically on $Q/Q_0$ , for the TMD evolution equation.", "For the unpolarized TMD PDFs one has $\\widetilde{f}_{q/p}(x, \\mbox{$b$}_T; Q) = f_{q/p}(x,Q_0) \\; R(Q, Q_0)\\;\\exp \\left\\lbrace -\\frac{b_T^2}{4} \\left( \\langle k_\\perp ^2 \\rangle + 2 \\, g_2\\ln \\frac{Q}{Q_0}\\right) \\right\\rbrace \\:.$ Figure: In the left panel, the evolution factor R(Q,Q 0 ,b T )R(Q,Q_0,b_T) is plotted as a function of b T b_T at twofixed values of Q 2 Q^2.", "In the right panel we show R(Q,Q 0 )≡R(Q,Q 0 ,b T →∞)R(Q,Q_0)\\equiv R(Q,Q_0,b_T\\rightarrow \\infty ) as a function of Q 2 Q^2.In both cases, Q 0 2 =1Q_0^2=1 GeV 2 ^2.Its Fourier-transform, Eqs.", "(REF ), delivers a Gaussian distribution in the transverse momentum space as well: $\\widehat{f}_{q/p}(x,k_\\perp ;Q)=f_{q/p}(x,Q_0)\\; R(Q,Q_0) \\;\\frac{ e^{-k_\\perp ^2/w^2}}{\\pi \\,w^2} \\:, \\quad \\quad $ where $f_{q/p}(x,Q_0)$ is the usual integrated PDF evaluated at the initial scale $Q_0$ and, most importantly, $w^2 \\equiv w^2(Q,Q_0)$ is the “evolving” Gaussian width, defined as: $w^2(Q,Q_0)=\\langle k_\\perp ^2 \\rangle + 2\\,g_2 \\ln \\frac{Q}{Q_0}\\: \\cdot $ It is worth noticing that the $Q^2$ evolution of the TMD PDFs is now determined by the overall factor $R(Q,Q_0)$ and, most crucially, by the $Q^2$ dependent Gaussian width $w(Q,Q_0)$ .", "The TMD FFs evolve in a similar way, Eq.", "(REF ), $\\widetilde{D}_{h/q}(z, b_T; Q) = \\frac{1}{z^2} D_{h/q}(z,Q_0) \\; R(Q, Q_0)\\;\\exp \\left\\lbrace - \\frac{b_T^2}{4 \\, z^2} \\left( \\langle p_\\perp ^2 \\rangle \\, + 2 \\, z^2 \\, g_2\\ln \\frac{Q}{Q_0}\\right) \\right\\rbrace \\:,$ leading to the TMD FF in momentum space, $\\widehat{D}_{h/q}(z,p_\\perp ;Q) = D_{h/q}(z,Q_0)\\; R(Q,Q_0) \\;\\frac{e^{-p_\\perp ^2/w^2_{\\!F}}}{\\pi w^2 _{\\!F}} \\:,$ with an evolving and $z$ -dependent Gaussian width $w_{\\!F}\\equiv w_{\\!F}(Q,Q_0)$ given by $w _{\\!F}^2\\equiv w _{\\!F}^2(Q,Q_0)=\\langle p_\\perp ^2\\rangle + 2 z^2 g_2 \\ln \\frac{Q}{Q_0}\\:\\cdot $ For the Sivers distribution function, by Fourier-transforming Eq.", "(REF ) (with $\\widetilde{R} \\rightarrow R$ ) as prescribed by Eq.", "(REF ), we obtain [see also Eqs.", "(REF ), (), (REF ) and (REF )]: $\\Delta ^N \\widehat{f}_{q/p^\\uparrow }(x,k_\\perp ;Q)=\\frac{k_\\perp }{M_1}\\,\\sqrt{2 e}\\,\\frac{\\langle k_\\perp ^2 \\rangle _S^2}{\\langle k_\\perp ^2 \\rangle }\\,\\Delta ^N f_{q/p^\\uparrow }(x,Q_0)\\,R(Q,Q_0)\\,\\frac{e^{-k_\\perp ^2/w_S^2}}{\\pi w_S^4} \\:,$ with $w^2_S(Q,Q_0)=\\langle k_\\perp ^2 \\rangle _S + 2 g_2 \\ln \\frac{Q}{Q_0}\\: \\cdot $ It is interesting to notice that the evolution factor $R(Q,Q_0)$ , controlling the TMD evolution according to Eqs.", "(REF ), (REF ) and (REF ) is the same for all functions (TMD PDFs, TMD FFs and Sivers ) and is flavor independent: consequently it will appear, squared, in both numerator and denominator of the Sivers azimuthal asymmetry and, approximately, cancel out.", "Therefore, we can safely conclude that most of the TMD evolution of azimuthal asymmetries is controlled by the logarithmic $Q$ dependence of the $k_\\perp $ Gaussian widths $w^2(Q,Q_0)$ , Eqs.", "(REF ), (REF ) and (REF ).", "We will come back to this in Section .", "Figure: The left panel shows the unpolarizedTMD PDF, f ^ u/p \\widehat{f}_{u/p}, evolved from the initial scale, Q 0 2 =1Q_0^2=1 GeV 2 ^2,to Q 2 =2.4Q^2=2.4 GeV 2 ^2, using TMD-evolution (red, solid line), DGLAP-evolution(blue, dashed line) and the analytical approximated TMD-evolution (greendot-dashed line).", "The right panel shows the same functions at the scaleQ 2 =20Q^2=20 GeV 2 ^2.", "Notice that, while there is hardly any differencebetween the DGLAP-evolved lines at Q 2 =2.4Q^2=2.4 and Q 2 =20Q^2=20 GeV 2 ^2, the TMDevolution induces a fast decrease in size of the TMD PDF functions at largeQ 2 Q^2 and a simultaneous widening of its Gaussian width.", "Here theanalytical approximated evolution gives results in good agreement withthe exact calculation even at large Q 2 Q^2.Figure: The left panel shows the ratio Sivers/PDF,Δ N f ^ u/p ↑ /2f ^ u/p \\Delta ^N \\widehat{f}_{u/p^{\\uparrow }}/ 2 \\widehat{f}_{u/p}, evolved from theinitial scale, Q 0 2 =1Q_0^2=1 GeV 2 ^2, to Q 2 =2.4Q^2=2.4 GeV 2 ^2, using TMD-evolution(red, solid line), DGLAP-evolution (blue, dashed line) and the analyticalapproximated TMD-evolution (green dot-dashed line).", "The right panel shows thesame functions at the scale Q 2 =20Q^2=20 GeV 2 ^2.", "Notice that, while there isalmost no difference between the DGLAP-evolved lines at Q 2 =2.4Q^2=2.4 and Q 2 =20Q^2=20GeV 2 ^2, the TMD evolution induces a fast decrease in size of the ratioSivers/PDF functions with growing Q 2 Q^2 and a simultaneous widening ofits Gaussian width.", "It is interesting to point out that the analyticalapproximation, for the Sivers function, visibly breaks down at large valuesof Q 2 Q^2.To illustrate the features of this new TMD evolution, we compare it with the results obtained evolving only the collinear part, $f_{q/p}(x,Q)$ , of the unpolarized TMD PDF according to the usual DGLAP equations and assuming the $k_\\perp $ dependent term of this function to be unaffected by evolution.", "In the left panel of Fig.", "REF we show the $k_\\perp $ behavior of the unpolarized TMD PDF $\\widehat{f}_{u/p}(x,k_\\perp ,Q^2)$ , at the fixed value $x=0.1$ , evaluated at the scale $Q^2=2.4$ GeV$^2$ (the average $Q^2$ value for the HERMES experiment).", "In the right panel we show the same function at a higher scale, $Q^2=20$ GeV$^2$ (which is the highest bin average $Q^2$ detected in the COMPASS experiment).", "In both cases the chosen initial scale is $Q_0^2=1$ GeV$^2$ .", "The red, solid line corresponds to the $k_\\perp $ distribution of the TMD PDF found by using the TMD-evolution of Eq.", "(REF ) while the blue, dashed line represents the result obtained by using DGLAP evolution equations.", "At the initial scale, $Q_0^2=1$ GeV$^2$ , solid and dashed curves coincide, by definition.", "However, while the DGLAP evolution is so slow that there is hardly any difference between the DGLAP-evolved lines at $Q^2=2.4$ GeV$^2$ and $Q^2=20$ GeV$^2$ , the TMD evolution induces a fast decrease of the maximum values of the TMD PDF function with growing $Q^2$ , and a simultaneous broadening of its Gaussian width, as observed in Refs.", "[8] and [9].", "It is interesting to notice that the approximated evolution of Eq.", "(REF ), corresponding to the green, dot-dashed line works really well, even for large $Q^2$ values.", "A similar study is performed in Fig.", "REF for the Sivers function.", "Here, by DGLAP evolution we mean that the Sivers function evolves like an unpolarized collinear PDF, only through the factor $f_{q/p}(x,Q)$ contained in its parameterization, Eq.", "(REF ).", "The parameters used for the plots are those given in Table REF , although any set of realistic parameters would lead to the same conclusions.", "The left panel shows the ratio between the Sivers function and the TMD PDF, $\\Delta ^N\\widehat{f}_{u/p^{\\uparrow }}(x,k_\\perp ;Q)/ (2 \\widehat{f}_{u/p}(x,k_\\perp ;Q))$ , evaluated at the scale $Q^2=2.4$ GeV$^2$ .", "Again, the red, solid line is obtained using the TMD-evolution of Eqs.", "(REF ) and (REF ), while the blue, dashed line is given by the DGLAP-evolution.", "The green dot-dashed line represents the results obtained using the approximated analytical TMD-evolution of Eqs.", "(REF ) and (REF ).", "The right panel shows the same functions at the scale $Q^2=20$ GeV$^2$ .", "Similarly to the case of TMD PDFs, while there is no difference between the DGLAP-evolved lines at $Q^2=2.4$ and $Q^2=20$ GeV$^2$ , the TMD evolution induces a fast decrease in the size of the TMD Sivers functions with growing $Q^2$ and a simultaneous widening of its Gaussian width.", "It is interesting to point out that the analytical TMD approximation, for the Sivers function visibly breaks down for large values of $k_\\perp $ ." ], [ " SIDIS data and TMD vs. non-TMD evolution", "Having established the phenomenological formalism necessary to implement the TMD evolution, as given in Refs.", "[7], [8], [9], we apply it to the Sivers function.", "This TMD distribution, $\\Delta ^N \\widehat{f}_ {q/p^\\uparrow }(x,k_\\perp ,Q) = (-2k_\\perp /M_p)\\widehat{f}_{1T}^\\perp $ , can be extracted from HERMES and COMPASS $\\ell \\, p \\rightarrow h \\, X$ SIDIS data on the azimuthal moment $A^{\\sin (\\phi _h-\\phi _S)}_{UT}$ , defined as $A^{\\sin (\\phi _h-\\phi _S)}_{UT} = 2 \\, \\frac{\\int d\\phi _S \\, d\\phi _h \\,[d\\sigma ^\\uparrow - d\\sigma ^\\downarrow ] \\, \\sin (\\phi _h-\\phi _S)}{\\int d\\phi _S \\, d\\phi _h \\,[d\\sigma ^\\uparrow + d\\sigma ^\\downarrow ]}\\, \\cdot $ This transverse single spin asymmetry (SSA) embeds the azimuthal modulation triggered by the correlation between the nucleon spin and the quark intrinsic transverse momentum.", "The “weighting” factor $\\sin (\\phi _h -\\phi _S)$ in Eq.", "(REF ) is appropriately chosen to single out, among the various azimuthal dependent terms appearing in $[d\\sigma ^\\uparrow - d\\sigma ^\\downarrow ]$ , only the contribution of the Sivers mechanism [18], [19].", "By properly taking into account all intrinsic motions this transverse single spin asymmetry can be written as [1] $A^{\\sin (\\phi _h-\\phi _S)}_{UT} = \\frac{\\displaystyle \\sum _q \\int {d\\phi _S \\, d\\phi _h \\, d^2 \\mbox{$k$}_\\perp }\\;\\Delta ^N \\widehat{f}_{q/p^\\uparrow } (x,k_\\perp ,Q) \\; \\sin (\\varphi -\\phi _S) \\;\\frac{d \\hat{\\sigma }^{\\ell q\\rightarrow \\ell q}}{dQ^2}\\; \\widehat{D}_q^h(z,p_\\perp ,Q) \\sin (\\phi _h -\\phi _S) }{\\displaystyle \\sum _q \\int {d\\phi _S \\,d\\phi _h \\, d^2 \\mbox{$k$}_\\perp }\\;\\widehat{f}_{q/p}(x,k _\\perp ,Q) \\; \\frac{d \\hat{\\sigma }^{\\ell q\\rightarrow \\ell q}}{dQ^2} \\; \\widehat{D}_q^h(z,p_\\perp ,Q) } \\: \\cdot $ With respect to the leptonic plane, $\\phi _S$ and $\\phi _h$ are the azimuthal angles identifying the transverse directions of the proton spin $\\mbox{$S$}$ and of the outgoing hadron $h$ respectively, while $\\varphi $ defines the direction of the incoming (and outgoing) quark transverse momentum, $\\mbox{$k$}_\\perp $ = $k_\\perp (\\cos \\varphi , \\sin \\varphi ,0)$ ; ${d \\hat{\\sigma }^{\\ell q\\rightarrow \\ell q}}/{dQ^2}$ is the unpolarized cross section for the elementary scattering $\\ell q\\rightarrow \\ell q$ .", "The aim of our paper is to analyze the available polarized SIDIS data from the HERMES and COMPASS collaborations in order to understand whether or not they show signs of the TMD evolution proposed in Ref.", "[9] and described in Section REF .", "Our general strategy is that of adopting the TMD evolution in the extraction of the Sivers functions, with the same parameterization and input functions as in Refs.", "[5], [13], and see if that can improve the quality of the fits.", "In doing so we will make use of the HERMES re-analysis of SIDIS experimental data on Sivers asymmetries for pion and kaon production and the newest SIDIS COMPASS data off a proton target, which cover a wider range of $Q^2$ values, thus giving a better opportunity to check the TMD evolution.", "In particular we perform three different data fits: a fit (TMD-fit) in which we adopt the TMD evolution equation discussed in the Section REF and REF , Eqs.", "(REF )-(REF ) and (REF )-(REF ); a second fit (TMD-analytical-fit) in which we apply the same TMD evolution, but using the analytical approximation discussed in Section REF , Eqs.", "(REF ), (REF ) and (REF ); a fit (DGLAP-fit) in which we follow our previous work, as done so far in Ref.", "[5], [13], using the DGLAP evolution equation only in the collinear part of the TMDs.", "As a result of the fit we will have explicit expressions of all the Sivers functions and their parameters.", "However, the goal of the paper is not that of obtaining a new extraction of the Sivers distributions, although in the sequel we will show, for comment and illustration purposes, the Sivers functions for $u$ and $d$ valence quarks, with the relative parameters.", "The procedure followed here aims at testing the effect of the TMD evolution, as compared with the simple DGLAP evolution so far adopted, in fitting the TMD SIDIS data.", "If it turns out, as it will, that this improves the quality of the fit, then a new extraction of the Sivers distributions, entirely guided by the TMD evolution, will be necessary.", "That will require a different approach from the very beginning, with different input functions and parameterizations.", "Here, we parameterize the Sivers function at the initial scale $Q_0=1$ GeV, as in Ref.", "[5], [13], in the following form: $\\Delta ^N \\widehat{f}_ {q/p^\\uparrow }(x,k_\\perp ,Q_0) = \\Delta ^N \\!", "f_ {q/p^\\uparrow }(x,Q_0)\\, h(k_\\perp )= 2 \\, {\\cal N}_q(x) \\, h(k_\\perp ) \\,\\widehat{f}_ {q/p} (x,k_\\perp ,Q_0)\\; , $ with $&&{\\cal N}_q(x) = N_q \\, x^{\\alpha _q}(1-x)^{\\beta _q} \\,\\frac{(\\alpha _q+\\beta _q)^{(\\alpha _q+\\beta _q)}}{\\alpha _q^{\\alpha _q} \\beta _q^{\\beta _q}}\\; , \\\\&&h(k_\\perp ) = \\sqrt{2e}\\,\\frac{k_\\perp }{M_{1}}\\,e^{-{k_\\perp ^2}/{M_{1}^2}}\\; ,$ where $\\widehat{f}_ {q/p} (x,k_\\perp ,Q_0)$ is defined in Eq.", "REF and $N_q$ , $\\alpha _q$ , $\\beta _q$ and $M_1$ (GeV) are (scale independent) free parameters to be determined by fitting the experimental data.", "Since $h(k_\\perp ) \\le 1$ for any $k_\\perp $ and $|{\\cal N}_q(x)| \\le 1$ for any $x$ (notice that we allow the constant parameter $N_q$ to vary only inside the range $[-1,1]$ ), the positivity bound for the Sivers function, $\\frac{|\\Delta ^N\\widehat{f}_ {q/p^\\uparrow }(x,k_\\perp )|}{2 \\widehat{f}_ {q/p} (x,k_\\perp )}\\le 1\\:,$ is automatically fulfilled.", "Similarly to PDFs, the FFs at the initial scale are parameterized with a Gaussian shape, Eq.", "(REF ).", "As in Refs.", "[20] and [5], the average values of $k_\\perp $ and $p_\\perp $ are fixed as $\\langle k_\\perp ^2\\rangle = 0.25 \\;{\\rm GeV}^2 \\quad \\quad \\quad \\langle p_\\perp ^2\\rangle = 0.20 \\;{\\rm GeV}^2 \\:.$ We take the unpolarized distributions $f_{q/p}(x,Q^2_0)$ from Ref.", "[21] and the unpolarized fragmentation functions $D_{h/q}(z,Q^2_0)$ from Ref.", "[22], with $Q^2_0=1.0$ GeV.", "As in Ref.", "[5], we adopt 11 free parameters: $&& N_{u_v} \\quad \\quad \\quad N_{d_v} \\quad \\quad \\quad N_s\\nonumber \\\\&& N_{\\bar{u}} \\quad \\quad \\quad \\;\\, N_{\\bar{d}} \\quad \\quad \\quad \\; N_{\\bar{s}}\\nonumber \\\\&& \\alpha _{u_v} \\quad \\quad \\quad \\,\\alpha _{d_v} \\quad \\quad \\quad \\,\\alpha _{sea} \\\\&& \\beta \\quad \\quad \\quad \\,\\,\\,\\;\\; M_1\\;({\\rm GeV}) \\:, \\nonumber $ where the subscript $v$ denotes valence contributions.", "In this choice we differ from Ref.", "[5], where valence and sea contributions were not separated.", "We perform best fits of 11 experimental data sets: HERMES [11] data for SIDIS production of pions ($\\pi ^{+}$ , $\\pi ^{-}$ , $\\pi ^{0}$ ) and kaons ($K^{+}$ and $K^{-}$ ), COMPASS data for SIDIS pion ($\\pi ^{+}$ , $\\pi ^{-}$ ) and kaon ($K^{+}$ and $K^{-}$ ) production from a $LiD$ (deuteron) target [23], and the preliminary COMPASS data for charged hadron production from an $NH_3$ (proton) target [12].", "The results of these 3 fits are presented in Table REF in terms of their $\\chi ^2$ s. Table: χ 2 \\chi ^2 contributions corresponding to the TMD-fit,the TMD-analytical-fitand the DGLAP-fit, for each experimental data set of HERMES andCOMPASS experiments.As it is clear from the first line of Table I, the best total $\\chi ^2_{tot}$ , which amounts to 256, is obtained by using the TMD evolution, followed by a slightly higher $\\chi ^2_{tot}$ of the analytical approximation, and a definitely larger $\\chi ^2_{tot} \\simeq 316$ corresponding to the DGLAP fit.", "To examine the origin of this difference between TMD and DGLAP evolution, we show the individual contributions to $\\chi ^2_{tot}$ of each experiment (HERMES, COMPASS on $NH_3$ and on $LiD$ targets), for all types of detected hadrons and for all variables observed ($x$ , $z$ and $P_T$ ).", "A global look at the numbers reported in Table I shows that the difference of about 60 $\\chi ^2$ -points between the TMD and the DGLAP fits is not equally distributed among all $\\chi ^2$ s per data point; rather, it is heavily concentrated in three particular cases, namely in the asymmetry for $\\pi ^+$ production at HERMES and for $h^+$ and $h^-$ production at COMPASS off a proton target, especially when this asymmetry is observed as a function of the $x$ -variable.", "It is important to stress that, as $x$ is directly proportional to $Q^2$ through the kinematical relation $Q^2=x\\,y\\,s$ , the $x$ behavior of the asymmetries is intimately connected to their $Q^2$ evolution.", "While the HERMES experimental bins cover a very modest range of $Q^2$ values, from $1.3$ GeV$^2$ to $6.2$ GeV$^2$ , COMPASS data raise to a maximum $Q^2$ of $20.5$ GeV$^2$ , enabling to test more severely the TMD $Q^2$ evolution in SIDIS.", "These aspects are illustrated in Fig.", "REF , where the SIDIS Sivers asymmetries $A^{\\sin (\\phi _h-\\phi _S)}_{UT}$ obtained in the three fits are shown in the same plot.", "It is evident that the DGLAP evolution seems to be unable to describe the correct $x$ trend, i.e.", "the right $Q^2$ behavior, while the TMD evolution (red solid line) follows much better the large $Q^2$ data points, corresponding to the last $x$ -bins measured by COMPASS.", "The approximate analytical TMD evolution (green dash-dotted line) works very well for low to moderate values of $Q^2$ while it starts to deviate from the exact behavior at large $Q^2$ values.", "Figure: The results obtained from our fit of the SIDIS A UT sin(φ h -φ S ) A_{UT}^{\\sin {(\\phi _h-\\phi _S)}} Sivers asymmetries applying TMD evolution (red, solid lines) are compared withthe analogous results found by using DGLAP evolution equations (blue, dashed lines).", "The green, dash-dotted lines correspond to the results obtained byusing the approximated analytical TMD evolution (see text for further details).The experimental data are from HERMES  (left panel) andCOMPASS  (right panel) Collaborations.Figure: The results obtained from the TMD-evolution fit (left panel) and from theDGLAP-evolution fit (right panel) of the SIDIS A UT sin(φ h -φ S ) A_{UT}^{\\sin {(\\phi _h-\\phi _S)}}Sivers asymmetries (red, solid lines) are compared with the HERMES experimentaldata  for charged and neutral pion production.The shaded area corresponds to the statistical uncertainty of the parameters,see Appendix A of Ref.", "for further details.Figure: The results obtained from the TMD-evolution fit (left panel) and from theDGLAP-evolution fit (right panel) of the SIDIS A UT sin(φ h -φ S ) A_{UT}^{\\sin {(\\phi _h-\\phi _S)}}Sivers asymmetries (red, solid lines) are compared with the COMPASS-pexperimental data  for charged hadron production.The shaded area corresponds to the statistical uncertainty of the parameters,see Appendix A of Ref.", "for further details.In Figs.", "REF we show, as an illustration of their qualities, our best fits (solid red lines) of the HERMES experimental data [11] on the Sivers asymmetries for pion production.", "Those on the left panels are obtained adopting the new TMD evolution, while those on the right use the simplified DGLAP evolution.", "Similar results are shown, for the recent COMPASS data off a proton target [12] for charged hadron production, in Fig.", "REF .", "The shaded area represents the statistical uncertainty of the fit parameters corresponding to a $\\Delta \\chi ^2=20$ (i.e.", "to $95.45\\%$ confidence level for 11 degrees of freedom, see Appendix A of Ref.", "[5] for further details).", "Notice that, in general, the error bands corresponding to the TMD-evolution fit are thinner than those corresponding to the DGLAP fit: this is caused by the fact that the TMD evolution implies a ratio Sivers/PDF which becomes smaller with growing $Q^2$ , as shown in Fig.", "REF , constraining the free parameters much more tightly than in the DGLAP-evolution fit, where the Sivers/PDF ratio remains roughly constant as $Q^2$ raises from low to large values.", "In Fig.", "REF we compare, for illustration purposes, the Sivers function – actually, its first moment, defined in Ref.", "[5] – at the initial scale $Q_0$ for $u$ and $d$ valence quarks, as obtained in our best fits with the TMD (left panel) and the DGLAP (right panel) evolution, Table REF .", "Notice that for this analysis we have chosen to separate valence from sea quark contributions, while in Ref.", "[5] the $u$ and $d$ flavors included all contributions.", "Figure: The first moment of the valence uu and dd Sivers functions, evaluated atQ=Q 0 Q=Q_0, obtained from our best fits of the A UT sin(φ h -φ S ) A_{UT}^{\\sin {(\\phi _h-\\phi _S)}}azimuthal moments as measured by HERMES  andCOMPASS , Collaborations.", "The extractionof the Sivers functions on the left side takes into account the TMD-evolution(left column of Table ), while for those on theright side it does not (right column of Table ).The shaded area corresponds to the statistical uncertainty of the parameters,see Appendix A of Ref.", "for further details.Table: Best values of the free parameters, Eq.", "(), forthe Sivers functions of uu and dd valence quarks, as obtained from ourTMD-fit, TMD-analytical-fit and DGLAP-fit, at Q 0 =1Q_0 = 1 GeV.", "The errorsreported in this table correspond to the maximum and minimum values of eachparameter in a restricted parameter space constrained by the conditionΔχ 2 =20\\Delta \\chi ^2=20, corresponding to 95.45%95.45\\% confidence level.They correspond to the shaded area in Fig.", ".This result deserves some comments.", "The comparison shows that the extracted $u$ and $d$ valence contributions, at the initial scale $Q_0 = 1$ GeV, are definitely larger for the TMD evolution fit.", "This reflects the TMD evolution property, according to which the Sivers functions are strongly suppressed with increasing $Q^2$ , which is not the case for the almost static collinear DGLAP evolution.", "Thus, in order to fit the same data at $Q^2$ bins ranging from 1.3 to 20.5 GeV$^2$ , the TMD evolving Sivers functions must start from higher values at $Q_0 = 1$ GeV.", "The Sivers distributions previously extracted, with the DGLAP evolution, in Refs.", "[5], [13] were given at $Q^2 = 2.4$  GeV$^2$ ; one should notice that if we TMD evolve the Sivers distributions on the left side of Fig.", "REF up to $Q^2 = 2.4$  GeV$^2$ we would obtain a result very close to that of Refs.", "[5], [13] (and to that of the right side of Fig.", "REF )." ], [ " Conclusions and further remarks", "We have addressed the issue of testing whether or not the recently proposed $Q^2$ evolution of the TMDs (TMD-evolution) can already be observed in the available SIDIS data on the Sivers asymmetry.", "It is a first crucial step towards the implementation, based on the TMD-evolution equations of Refs.", "[7], [8], [9], of a consistent QCD framework in which to study the TMDs and their full $Q^2$ dependence.", "That would put the study of TMDs – and the related reconstruction of the 3-dimensional parton momentum structure of the nucleons – on a firm basis, comparable to that used for the integrated PDFs.", "Previous extractions of the Sivers functions from SIDIS data included some simplified treatment of the $Q^2$ evolution, which essentially amounted to consider the evolution of the collinear and factorized part of the distribution and fragmentation functions (DGLAP-evolution).", "It induced modest effects, because of the slow $Q^2$ evolution and of the limited $Q^2$ range spanned by the available data.", "The situation has recently much progressed, for two reasons: the new TMD-evolution [8], [9] shows a strong variation with $Q^2$ of the functional form of the unpolarized and Sivers TMDs, as functions of the intrinsic momentum $k_\\perp $ ; in addition, some new COMPASS results give access to Sivers asymmetries at larger $Q^2$ values.", "It appears then possible to test the new TMD-evolution.", "In order to do so one has to implement the full machinery of the TMD-evolution equations in a viable phenomenological scheme.", "We have done so following Ref.", "[9] and the simplified version of the TMD-evolution given in Eqs.", "(REF )-(REF ).", "We have used them in our previous procedure adopted for the extraction of the Sivers functions [5], [13], [18], with the same input parameters; moreover, we have considered also the updated HERMES [11] and the new COMPASS [12] data.", "A definite statement resulting from our analysis is that the best fit of all SIDIS data on the Sivers asymmetry using TMD-evolution, when compared with the same analysis performed with the simplified DGLAP-evolution, exhibits a smaller value of the total $\\chi ^2$ , as shown in Table REF .", "Not only, but when analyzing the partial contributions to the total $\\chi ^2$ value of the single subsets of data, one realizes that such a smaller value mostly originates from the large $Q^2$ COMPASS data, which are greatly affected by the TMD evolution.", "We consider this as an indication in favor of the TMD evolution.", "A more comprehensive study of the TMD evolution and its phenomenological implications is now necessary.", "Both the general scheme and its application to physical processes need improvements.", "The recovery of the usual collinear DGLAP evolution equations, after integration of the TMD evolution results over the intrinsic momenta, has to be understood.", "Consider, as an example, the simple expression of the evolution of the unpolarized TMD PDF, as given in Eq.", "(REF ).", "Such an evolution describes how the TMD dependence on $k_\\perp $ changes with $Q^2$ , but does not induce any change in the $x$ dependence, which, at this order, remains fixed and factorized.", "The question whether or not one can recover the usual DGLAP evolution, which changes the $x$ dependence, for the integrated PDFs arises naturally at this point.", "A naive integration of Eq.", "(REF ) on $\\mbox{$k$}_\\perp $ , over the full integration range, would give $f_{q/p}(x,Q) =f_{q/p}(x,Q_0) \\, R(Q,Q_0)$ which is not the correct PDF evolution.", "However, the $\\mbox{$k$}_\\perp $ integration should have upper limits which depend on $x$ and $Q^2$ , and the full TMD evolution is more complicated than the simplified version used here, as explained at the beginning of Section REF .", "We have made a safe phenomenological use of the TMD evolution equations; it is true that they induce a strong change in the $k_\\perp $ dependence of the unpolarized and Sivers TMDs, leaving unchanged the $x$ dependent shape, thus neglecting the collinear DGLAP evolution, but this should not be a problem.", "Infact, as we have shown explicitly in Fig.", "REF (dashed curve), the collinear DGLAP evolution is negligible in the $Q^2$ region considered; Fig.", "REF is drawn for $x = 0.1$ , but a similar conclusion holds for all $x$ values involved in the SIDIS data used in the paper.", "Moreover, the extra factors $R(Q,Q_0)$ arising in the TMD evolution, cancel out, as already explained, in the expression of the Sivers asymmetries.", "A fresh analysis of TMD dependent data, both in polarized and unpolarized, SIDIS and Drell-Yan processes, has to be carefully performed including TMD evolution from the beginning in an unbiased way.", "Most importantly so, should predictions for future high energy experiments, like the planned EIC/ENC colliders, be considered or re-considered." ], [ "Acknowledgements", "We thank Umberto D'Alesio, Francesco Murgia and Alexei Prokudin for interesting and fruitful discussions.", "We acknowledge support of the European Community under the FP7 “Capacities - Research Infrastructures” program (HadronPhysics3, Grant Agreement 283286).", "We acknowledge partial support by MIUR under Cofinanziamento PRIN 2008." ] ]

[ [ "Global existence for the interaction of a Navier-Stokes fluid with a\n linearly elastic shell" ], [ "Abstract In my PhD thesis I show the existence of global-in-time weak solutions for a Navier-Stokes fluid interacting with a linearly elastic shell of Koiter type.", "This is achieved by the introduction of a new method for showing the compactness of bounded sequences of approximate weak solutions.", "This method might be of general interest in the study of fluid dynamical problems involving a free boundary.", "There is no damping term involved in the shell equations." ], [ "Erkl\"arung", "Ich erkl\"are hiermit, dass ich die vorliegende Arbeit ohne unzul\"assige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe.", "Die aus anderen Quellen direkt oder indirekt \"ubernommenen Daten und Konzepte sind unter Angabe der Quelle gekennzeichnet.", "Insbesondere habe ich hierf\"ur nicht die entgeltliche Hilfe von Vermittlungs- bzw.", "Beratungsdiensten (Promotionsberater oder anderer Personen) in Anspruch genommen.", "Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder \"ahnlicher Form einer anderen Pr\"ufungsbeh\"orde vorgelegt.", "Table: NO_CAPTIONZun\"achst m\"ochte ich meiner Frau Wiebke und meinen Eltern danken.", "Alle drei haben mich immer \"au\"serst wohlwollend unterst\"utzt.", "Zudem danke ich meinem Betreuer Prof. Dr. Michael Růžička f\"ur die fruchtbare Aufgabenstellung, zahlreiche hilfreiche Gespr\"ache und die freundschaftliche Zusammenarbeit.", "Ebenso danke ich Prof. Dr. Lars Diening und meinem Mitdoktoranden Dipl.-Math.", "Philipp N\"agele f\"ur regelm\"a\"sige Diskussionen, die ebenfalls sehr zum Gelingen dieser Arbeit beigetragen haben.", "[00]$B^{\\prime }$ Dualraum eines Banachraums $B$ [01]$B_0\\hookrightarrow B_1$ , $B_0\\hookrightarrow \\hookrightarrow B_1$ Stetige bzw.", "kompakte Einbettung von Banach-Räumen [02]$(B_0,B_1)_{\\theta ,p}$ , $[B_0,B_1]_{\\theta }$ Reeller bzw.", "komplexer Interpolationsraum [03]$L^p(\\Omega )$ , $W^{s,p}(\\Omega )$ Lebesgue- bzw.", "Sobolev-Slobodetskii-Raum.", "Im Falle $\\Omega \\subset \\mathbb {R}^3$ können diese Bezeichner auch für $L^p(\\Omega ,\\mathbb {R}^3)$ , $W^{s,p}(\\Omega ,\\mathbb {R}^3)$ stehen.", "[03]$W^{1,p}_{\\operatorname{div}}(\\Omega )$ , $W^{1,p}_{0,\\operatorname{div}}(\\Omega )$ Raum der divergenzfreien Elemente von $W^{1,p}(\\Omega ,\\mathbb {R}^3)$ (mit verschwindenden Randwerten), wobei $\\Omega \\subset \\mathbb {R}^3$ (mit Lipschitz-Rand) [04]$H^s(\\Omega )$$L^2$ -Skala $W^{s,2}(\\Omega )$ [05]$\\nabla $ Levi-Civita-Zusammenhang.", "Dieser kann je nach Kontext (euklidischer Raum oder Fläche) variieren.", "Ist ${\\bf u}$ ein Vektorfeld auf $\\mathbb {R}^3$ , so kann $\\nabla {\\bf u}$ mit der Funktionalmatrix identifiziert werden.", "[06]$\\Delta $ Dem Levi-Civita-Zusammenhang zugeordneter Laplace-Operator [07]$\\operatorname{div}$ , $\\nabla ^*$ Dem Levi-Civita-Zusammenhang zugeordnete Divergenz [08]$D{\\bf u}$ Symmetrischer Anteil der Funktionalmatrix eines Vektorfeldes ${\\bf u}$ auf $\\mathbb {R}^3$ , d.h. $D{\\bf u}:=1/2(\\nabla {\\bf u}+(\\nabla {\\bf u})^T)$ [09]$d\\Phi $ Differential einer Abbildung $\\Phi $ zwischen Mannigfaltigkeiten (Teilmengen des euklidischen Raums eingeschlossen).", "Dieses kann im euklidischen Fall mit der Funktionalmatrix identifiziert werden.", "[10]${\\bf u}\\cdot {\\bf v}$$\\mathbb {R}^3$ -Skalarpodukt [11]$A:B$ Matrixskalarpodukt [12]$\\partial M$ Rand einer Mannigfaltigkeit $M$ im Sinne von Mannigfaltigkeiten [13]$\\operatorname{int}M$ Inneres einer Mannigfaltigkeit $M$ im Sinne von Mannigfaltigkeiten, d.h. $\\operatorname{int}M:=M\\setminus \\partial M$ [14]$p^{\\prime }$ Definiert durch $\\frac{1}{p}+\\frac{1}{p^{\\prime }}=1$ , wobei $1\\le p\\le \\infty $ [15]$\\operatorname{supp}f$ Träger einer Funktion $f$ (per Definition abgeschlossen) [16]$C^k_0(U)$ Menge der $C^k$ -Funktionen $f$ mit $\\operatorname{supp}f\\subset U$ , wobei $U$ Teilmenge einer Mannigfaltigkeit (euklidischer Raum eingeschlossen) [17]$\\chi _A$ Charakteristische Funktion einer Menge $A$ [18]$c$ Generische Konstante [19]$\\kappa $ , $\\Lambda $ , $B_\\alpha $ , $\\Omega _\\eta $ Definition auf S.5 [20]$\\Psi _\\eta $ , $\\Phi _\\eta $ Definition auf S.6 [21]$\\tau (\\eta )$ Definition auf S.7 [22]$\\operatorname{tr_\\eta }$ Definition auf S.8 [23]$\\gamma (\\eta )$ Definition auf S.10 [24]$\\operatorname{tr^n_\\eta }$ Definition auf S.11 [25]$\\mathcal {F}_\\eta $ , $H(\\Omega _\\eta )$ Definition auf S.12 [26]$\\Omega _\\eta ^I$ , $L^p(I,L^r(\\Omega _{\\eta (t)}))$ Definition auf S.16 [27]$K$ Definition auf S.20 [28]$Y^I$ , $X_\\eta ^I$ , $T^I_\\eta $ Definition auf S.27 [29]$c_0$ Definition auf S.30 [30]$\\mathcal {R}_\\epsilon $ Definition auf S.40 [31]${\\bf u}_0^\\epsilon $ , $\\eta _1^\\epsilon $ Definition auf S.41 [32]$\\delta $ Definition auf S.76 [33]$\\mathcal {M}_\\eta $ Definition auf S.77 [3.8cm]" ], [ "Einleitung", "Die mathematische Analysis der Interaktion von Fluiden mit Festk\"orpern bei zeitlich variablen Phasengrenzen ist seit den 90er Jahren des letzten Jahrhunderts Gegenstand intensiver Forschung.", "Die ersten Arbeiten untersuchten die Bewegung starrer K\"orper in viskosen Fluiden; siehe zum Beispiel [18], [30].", "Die Einbeziehung elastischer K\"orper ist wesentlich schwieriger, bedingt durch potentielle Regularit\"atsinkompatibilit\"aten des zumeist hyperbolischen Festk\"orperanteils und des parabolischen Fluidanteils der Gleichungen.", "So verwendeten die ersten Existenzresultate in diese Richtung Regularisierungen des Festk\"orperanteils in Form von D\"ampfungstermen, siehe [48], [11], [9], oder einer endlichen Anzahl von Moden, siehe [19].", "Speziell in [11] wird die Interaktion eines Navier-Stokes-Fluids mit einer ged\"ampften elastischen Platte untersucht und die Langzeitexistenz schwacher L\"osungen gezeigt.", "Ein Durchbruch wurde in [15], [16] erzielt.", "Dort wird f\"ur die Interaktion eines Navier-Stokes-Fluids mit einem dreidimensionalen elastischen K\"orper, beschrieben durch lineare bzw.", "quasilineare hyperbolische Gleichungen ohne Regularisierung, die Kurzzeitexistenz f\"ur sehr regul\"are Daten gezeigt.", "Eine der zentralen Ideen dabei ist die Verwendung eines funktionalen Rahmens, der sich am hyperbolischen Anteil des Systems orientiert.", "Eine weitere wichtige Arbeit ist [32].", "Dort werden Absch\"atzungen gezeigt, die es erm\"oglichen auf die D\"ampfung in [11] zu verzichten.", "In [17] schlie\"slich wird die Interaktion eines Navier-Stokes-Fluids mit einer ann\"ahernd zweidimensionalen, im Ruhezustand gekr\"ummten elastischen Struktur, einer sogenannten Schale, untersucht.", "Die elastische Energie der Schale wird dabei durch die Koiter-Energie modelliert; siehe [35], [36], [13], [14] und die dortigen Referenzen.", "Im Vergleich mit [16] tritt dabei die zus\"atzliche Schwierigkeit auf, dass der (quasilineare) Gradient der Koiter-Energie in den Richtungen tangential zur ausgelenkten Schale elliptisch degeneriert ist.", "Allerdings konnte die Kurzzeitexistenz nur unter Vernachl\"assigung der Massentr\"agheit der Schale gezeigt werden.", "In diesem Fall wird die Auslenkung der Schale durch eine elliptische Gleichung mit zeitabh\"angiger rechter Seite beschrieben.", "In der vorliegenden Arbeit zeigen wir die Langzeitexistenz schwacher L\"osungen f\"ur die Interaktion eines Navier-Stokes-Fluids mit einer elastischen Koiter-Schale unter Ber\"ucksichtigung der Massentr\"agheit, wobei wir die Gleichungen der elastischen Struktur linearisieren und ihre Bewegung auf transversale Auslenkungen einschr\"anken.", "Die L\"osungen existieren, solange wir sicherstellen k\"onnen, dass sich verschiedene Teile der Schale nicht ber\"uhren.", "Die Gleichung der elastischen Struktur, die wir erhalten, ist eine Verallgemeinerung der instation\"aren, linearen Kirchhoff-Love-Plattengleichung f\"ur transversale Auslenkungen.", "Mithin ist unser Resultat eine direkte Verallgemeinerung von [11], [32].", "Unsere Konstruktion schwacher L\"osungen folgt in den grundlegenden Z\"ugen der in [11].", "Die zus\"atzliche Schwierigkeit unserer Aufgabe im Vergleich zu dieser Arbeit besteht nicht in der (wenig) komplizierteren Gleichung der elastischen Struktur, sondern vielmehr in der allgemeinen Geometrie der Schale.", "Diese erzwingt die Entwicklung neuer Techniken, die \"uber die vorliegende Arbeit hinaus von Interesse sein d\"urften.", "Als Nebenprodukt verk\"urzt unser Vorgehen den Beweis in [11], [32].Angesichts der Seitenzahl der vorliegenden Arbeit mag man sich \"uber diese Aussage wundern.", "W\"urde man sich jedoch auf den Fall einer Platte beschr\"anken, so w\"urden einige der folgenden Aussagen und Konstruktionen trivial oder gar hinf\"allig, und die Seitenzahl w\"urde sich erheblich reduzieren.", "Im scharfen Gegensatz zu diesen Arbeiten tritt in unserem Beweis nicht eine einzige l\"angere Absch\"atzung auf.", "Zudem ben\"otigen wir an keiner Stelle eine Regularisierung der Schalengleichung.", "Schlie\"slich ist unser Beweis so gestaltet, dass eine \"Ubertragung auf verallgemeinerte Newton'sche Fluide in greifbare N\"ahe r\"uckt.", "Eine der Schwierigkeiten dieser Arbeit besteht in der geringen Regularit\"at der Auslenkungen der Schale, die den mathematischen Rand des Fluids darstellt.", "In Kapitel 2 werden wir deshalb zun\"achst einige Vor\"uberlegungen bez\"uglich Gebieten mit geringer Randregularit\"at anstellen.", "In Kapitel 3 stellen wir die Koiter-Energie vor, und in Kapitel 4 folgt eine Spezifizierung der Problemstellung.", "Anschlie\"send geben wir in Kapitel 5 das zentrale Existenzresultat dieser Arbeit an und f\"uhren den zugeh\"origen Beweis.", "In Kapitel 6 skizzieren wir die ersten Schritte einer \"Ubertragung des Beweises auf verallgemeinerte Newton'sche Fluide, und Kapitel 7 gibt schlie\"slich einen Ausblick auf m\"ogliche zuk\"unftige Forschung." ], [ "Variable Gebiete", "Es sei $\\Omega \\subset \\mathbb {R}^3$ ein beschr\"anktes, nichtleeres Gebiet mit $C^4$ -Rand und \"au\"serer Einheitsnormale $\\nu \\in C^3(\\partial \\Omega ,\\mathbb {R}^3)$ .", "Offenbar ist $\\partial \\Omega $ eine geschlosseneD.h.", "kompakt und nicht berandet., nicht notwendig zusammenh\"angende Fl\"ache.", "Wir bezeichnen mit $dA$ das Fl\"achenma\"s von $\\partial \\Omega $ und f\"ur $\\alpha >0$ mit $S_{\\alpha }$ den offenen $\\alpha $ -Schlauch um $\\partial \\Omega $ .", "Es existiert ein maximales $\\kappa >0$ derart, dass die Abbildung $\\begin{aligned}\\Lambda : \\partial \\Omega \\times (-\\kappa ,\\kappa )\\rightarrow S_{\\kappa },\\ (q,s)\\mapsto q + s\\,\\nu (q)\\end{aligned}$ ein $C^3$ -Diffeomorphismus ist; siehe zum Beispiel Theorem 10.19 in [39].", "F\"ur die Inverse $\\Lambda ^{-1}$ schreiben wir auch $x\\mapsto (q(x),s(x))$ ; vgl.", "Abbildung REF .", "Man beachte, dass $\\kappa $ nicht notwendig klein ist.", "Ist $\\Omega $ der Ball vom Radius $R$ , so ist $\\kappa =R$ .", "$\\Lambda $ wird zum Rand seines Definitionsbereiches hin singul\"ar.", "Figure: NO_CAPTION Wir setzen $B_\\alpha :=\\Omega \\cup S_{\\alpha }$ f\"ur $0<\\alpha <\\kappa $ .", "Die Abbildung $\\Lambda (\\,\\cdot \\, ,\\alpha ): \\partial \\Omega \\rightarrow \\partial B_\\alpha $ ist ebenfalls ein $C^3$ -Diffeomorphismus, sodass $B_\\alpha $ ein beschr\"anktes Gebiet mit $C^3$ -Rand ist.", "F\"ur stetiges $\\eta :\\partial \\Omega \\rightarrow (-\\kappa ,\\kappa )$ setzen wir $\\begin{aligned}\\Omega _\\eta :=\\Omega \\setminus S_\\kappa \\ \\cup \\lbrace x\\in S_\\kappa \\ |\\ s(x)<\\eta (q(x))\\rbrace ;\\end{aligned}$ vgl.", "Abbildung REF .", "Offenbar ist $\\Omega _\\eta $ offen.", "Ist $\\eta \\in C^k(\\partial \\Omega )$ , $k\\in \\lbrace 1,2,3\\rbrace $ , so ist $\\partial \\Omega _\\eta $ eine $C^k$ -Fl\"ache, wie wir in K\"urze sehen werden.", "In diesem Falle seien $\\nu _\\eta $ die \"au\"sere Einheitsnormale und $dA_\\eta $ das Fl\"achenma\"s von $\\partial \\Omega _\\eta $ .", "Wir wollen nun einen Hom\"oomorphismus von $\\overline{\\Omega }$ auf $\\overline{\\Omega _\\eta }$ konstruieren, die sogenannte Hanzawa-Transformation.", "Sei dazu $\\beta \\in C^\\infty (\\mathbb {R})$ mit $\\beta =0$ in einer Umgebung von $-1$ und $\\beta =1$ in einer Umgebung von 0.", "F\"ur stetiges $\\eta :\\partial \\Omega \\rightarrow (-\\kappa ,\\kappa )$ definieren wir $\\Psi _\\eta : \\overline{\\Omega }\\rightarrow \\overline{\\Omega _\\eta }$ in $S_\\kappa \\cap \\overline{\\Omega }$ durch $\\begin{aligned}x&\\mapsto x +\\nu (q(x))\\,\\eta (q(x))\\,\\beta (s(x)/\\kappa )=q(x)+\\nu (q(x))\\Big (s(x)+\\eta (q(x))\\,\\beta (s(x)/\\kappa )\\Big ).\\end{aligned}$ In $\\Omega \\setminus S_\\kappa $ sei $\\Psi _\\eta $ die Identit\"at.", "Punkte $x\\in S_\\kappa \\cap \\overline{\\Omega }$ werden durch $\\Psi _\\eta $ im Wesentlichen um die L\"ange $\\eta (q(x))$ in Richtung $\\nu (q(x))$ verschoben.", "Allerdings m\"ussen wir die L\"ange, um die translatiert wird, zum \"`inneren\"' Rand von $S_\\kappa \\cap \\overline{\\Omega }$ hin geeignet gegen 0 gehen lassen, um eine Bijektion zu erhalten.", "Das erreichen wir durch die Multiplikation mit der Abschneidefunktion $\\beta $ , die noch n\"aher zu spezifizieren ist.", "Punkte $x\\in S_\\kappa \\cap \\overline{\\Omega }$ mit festem Fu\"spunkt $q(x)=q\\in \\partial \\Omega $ werden durch $\\beta _q: s\\mapsto s+\\eta (q)\\,\\beta (s/\\kappa ),\\ s=s(x)\\in [-\\kappa ,0]$ abgebildet.", "Insbesondere gilt $\\beta _q(-\\kappa )=-\\kappa $ und $\\beta _q(0)=\\eta (q)$ .", "Damit $\\beta _q$ eine (stetig differenzierbare) Inverse besitzt, muss $0<\\beta _q^{\\prime }(s)=1+\\eta (q)/\\kappa \\ \\beta ^{\\prime }(s/\\kappa )$ gelten.", "Wir fordern dazu $|\\beta ^{\\prime }(s)|<\\kappa /|\\eta (q)|$ f\"ur alle $s\\in [-1,0]$ und alle $q\\in \\partial \\Omega $ , was wegen $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\kappa $ m\"oglich ist.", "$\\Psi _\\eta $ ist dann bijektiv und die Inverse ist in $S_\\kappa \\cap \\overline{\\Omega }$ durch $\\Psi _\\eta ^{-1}:x\\mapsto q(x)+\\nu (q(x))\\,\\beta _{q(x)}^{-1}(s(x))$ gegeben.", "Offenbar h\"angt $\\beta _q^{-1}$ stetig und im Falle $\\eta \\in C^k(\\partial \\Omega )$ , $k\\in \\mathbb {N}$ , sogar $k$ -mal stetig differenzierbar von $q$ ab; bezeichnen wir mit $d$ das Differential bez\"uglich der Variablen $q$ und mit $^{\\prime }$ die Ableitung nach der skalaren Variable, so gilt wegen $ 0= ds=d(\\beta _q^{-1}(\\beta _q(s)))=(d\\beta _q^{-1})(\\beta _q(s)) +(\\beta _q^{-1})^{\\prime }(\\beta _q(s))\\,(d\\beta _q)(s)$ die Identit\"at $\\begin{aligned}(d\\beta _q^{-1})(\\beta _q(s))=-\\frac{\\beta (s/\\kappa )}{1+\\eta (q)/\\kappa \\ \\beta ^{\\prime }(s/\\kappa )}d\\eta (q).\\end{aligned}$ $\\Psi _\\eta $ ist somit ein Hom\"oomorphismus und im Falle $\\eta \\in C^k(\\partial \\Omega )$ , $k\\in \\lbrace 1,2,3\\rbrace $ , sogar ein $C^k$ -Diffeomorphismus.", "Auch der Hom\"oomorphismus $\\Phi _\\eta :=\\Psi _\\eta |_{\\partial \\Omega }:\\partial \\Omega \\rightarrow \\partial \\Omega _\\eta ,\\ q\\mapsto q+\\eta (q)\\,\\nu (q)$ mit der Inversen $x\\mapsto q(x)$ ist f\"ur $\\eta \\in C^k(\\partial \\Omega )$ , $k\\in \\lbrace 1,2,3\\rbrace $ , ein $C^k$ -Diffeomorphismus.", "Die Kettenregel und (REF ) zeigen, dass die Eintr\"age der Funktionalmatrizen von $\\Psi _\\eta $ , $\\Phi _\\eta $ und ihren Inversen von der Form $\\begin{aligned}b_0+\\sum _i b_i\\ (\\partial _i\\eta )\\circ q\\end{aligned}$ sind mit stetigen und beschr\"ankten Funktionen $b_0$ , $b_i$ , die f\"ur $\\tau (\\eta )\\rightarrow \\infty $ mit $\\begin{aligned}\\tau (\\eta ):=\\left\\lbrace \\begin{array}{cl} (1-\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}/\\kappa )^{-1} &\\text{, falls }\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\kappa \\\\\\infty & \\text{, sonst}\\end{array}\\right.\\end{aligned}$ teilweise singul\"ar werden, weil dann die Funktionen $\\beta _q$ singul\"ar werden und der Diffeomorphismus $\\Lambda $ in der N\"ahe seiner Singularit\"aten ausgewertet wird.", "Zudem sind die Tr\"ager der $b_i$ in $S_\\kappa $ enthalten, wobei der Abstand zum Rand von $S_\\kappa $ f\"ur $\\tau (\\eta )\\rightarrow \\infty $ gegen 0 geht.", "F\"ur $\\tau (\\eta )\\rightarrow \\infty $ wird somit die Abbildung $q$ in (REF ) in der N\"ahe ihrer Singularit\"aten ausgewertet.", "Neben m\"oglichen Irregularit\"aten der Auslenkung $\\eta $ k\"onnen die Abbildungen $\\Psi _\\eta $ und $\\Phi _\\eta $ also auch dadurch singul\"ar werden, dass die maximale Auslenkung an $\\kappa $ heranr\"uckt.", "Aus diesem Grund werden die Stetigkeitskonstanten der im Folgenden konstuierten linearen Abbildungen zwischen Funktionenr\"aumen stets von $\\tau (\\eta )$ abh\"angen.", "Die Abbildung $\\Psi _\\eta $ h\"angt von der Abschneidefunktion $\\beta $ ab, die wiederum in Abh\"angigkeit von $\\tau (\\eta )$ gew\"ahlt werden kann.", "Wann immer wir mit Folgen $(\\eta _n)$ von Auslenkungen mit $\\sup _n\\tau (\\eta _n)<\\infty $ zu tun haben werden, wollen und k\"onnen wir $\\beta $ unabh\"angig vom Folgenindex w\"ahlen.", "Dadurch stellen wir sicher, dass die Folge $(\\Psi _{\\eta _n})$ konvergiert, falls $(\\eta _n)$ konvergiert.", "Eine Bi-Lipschitz-Abbildung von Definitionsbereichen induziert Isomorphismen der jeweiligen $L^p$ - und $W^{1,p}$ -R\"aume.", "F\"ur $\\eta \\in H^2(\\partial \\Omega )$ ist wegen der Einbettung $H^2(\\partial \\Omega )\\hookrightarrow C^{0,\\theta }(\\partial \\Omega )$ für $\\theta <1$ die Abbildung $\\Psi _\\eta $ gerade nicht bi-Lipschitz-stetig, sodass ein kleiner Verlust unter dieser Transformation entsteht.", "Lemma 2.1 Es seien $1<p\\le \\infty $ und $\\eta \\in H^2(\\partial \\Omega )$ mit $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\kappa $ .", "Dann ist die lineare Abbildung $v\\mapsto v\\circ \\Psi _\\eta $ stetig von $L^p(\\Omega _\\eta )$ nach $L^r(\\Omega )$ und von $W^{1,p}(\\Omega _\\eta )$ nach $W^{1,r}(\\Omega )$ f\"ur alle $1\\le r<p$ .", "Eine analoge Aussage gilt mit $\\Psi _\\eta ^{-1}$ anstelle von $\\Psi _\\eta $ .", "Die Stetigkeitskonstanten h\"angen nur von $\\Omega $ , $p$ , $\\Vert \\eta \\Vert _{H^2(\\partial \\Omega )}$ , $\\tau (\\eta )$ und $r$ ab; sie bleiben beschr\"ankt, falls $\\Vert \\eta \\Vert _{H^2(\\partial \\Omega )}$ und $\\tau (\\eta )$ beschr\"ankt bleiben.", "Beweis: Wir k\"onnen ohne Einschr\"ankung $p<\\infty $ annehmen.", "Wir approximieren $\\eta $ durch Funktionen $(\\eta _n)\\subset C^2(\\partial \\Omega )$ in $H^2(\\partial \\Omega )$ , insbesondere gleichm\"a\"sig.", "Wegen (REF ) und der Einbettung $H^2(\\partial \\Omega )\\hookrightarrow W^{1,s}(\\partial \\Omega ) \\text{ f\"ur alle }1\\le s<\\infty $ sind die Eintr\"age der Funktionalmatrix von $\\Psi _{\\eta _n}^{-1}$ und somit auch die Funktionaldeterminante in $L^s(\\Omega _{\\eta _n})$ f\"ur jedes $s<\\infty $ in Abh\"angigkeit von $\\tau (\\eta _n)$ beschr\"ankt.", "Wir erhalten also f\"ur $v\\in C_0^\\infty (\\mathbb {R}^3)$ , $r<p$ und $1/r=1/p+1/s$ unter Verwendung des Transformationssatzes und der H\"older-Ungleichung $\\begin{aligned}\\Vert v\\circ \\Psi _{\\eta _n}\\Vert _{L^r(\\Omega )}=\\Vert v\\,(\\det d\\Psi _{\\eta _n}^{-1})^{1/r}\\Vert _{L^r(\\Omega _{\\eta _n})}&\\le \\Vert (\\det d\\Psi _{\\eta _n}^{-1})^{1/r}\\Vert _{L^s(\\Omega _{\\eta _n})}\\,\\Vert v\\Vert _{L^p(\\Omega _{\\eta _n})}.\\end{aligned}$ Aus der Konvergenz von $(\\eta _n)$ in $H^2(\\partial \\Omega )$ folgt $\\begin{aligned}\\Vert v\\circ \\Psi _{\\eta }\\Vert _{L^r(\\Omega )}&\\le \\Vert (\\det d\\Psi _{\\eta }^{-1})^{1/r}\\Vert _{L^s(\\Omega _{\\eta })}\\,\\Vert v\\Vert _{L^p(\\Omega _{\\eta })}\\le c(\\Omega ,p,\\Vert \\eta \\Vert _{H^2(\\partial \\Omega )},\\tau (\\eta ),r)\\,\\Vert v\\Vert _{L^p(\\Omega _{\\eta })}.\\end{aligned}$ Mit Hilfe der Dichtheit glatter Funktionen in $L^p(\\Omega _\\eta )$ folgern wir die Stetigkeit bez\"uglich der Lebesgue-R\"aume.", "Aufgrund der Kettenregel und (REF ) gilt $\\Vert \\nabla (v\\circ \\Psi _{\\eta _n})\\Vert _{L^r(\\Omega )}\\le c(\\Omega ,\\Vert \\eta _n\\Vert _{H^2(\\partial \\Omega )},\\tau (\\eta _n),r)\\,\\Vert (\\nabla v)\\circ \\Psi _{\\eta _n}\\Vert _{L^{(r+p)/2}(\\Omega )}.$ Zudem folgt aus der gleichm\"a\"sigen Konvergenz von $((\\nabla v)\\circ \\Psi _{\\eta _n})$ und der Konvergenz der Eintr\"age der Funktionalmatrix von $\\Psi _{\\eta _n}$ in $L^s(\\Omega )$ f\"ur $s<\\infty $ die Konvergenz von $(\\nabla (v\\circ \\Psi _{\\eta _n}))$ gegen $\\nabla (v\\circ \\Psi _{\\eta })$ in $L^r(\\Omega )$ .", "Wir erhalten somit die Stetigkeit bez\"uglich der Sobolev-R\"aume aus dem bereits Gezeigten sowie der Dichtheit glatter Funktionen in $W^{1,p}(\\Omega _\\eta )$ ; siehe Proposition REF .", "Der Beweis der analogen Aussage mit $\\Psi _\\eta ^{-1}$ anstelle von $\\Psi _\\eta $ geht genauso.", "$\\Box $ Bemerkung 2.2 Die Folge $(\\eta _n)$ konvergiere gegen ein $\\eta $ schwach in $H^2(\\partial \\Omega )$ und aufgrund der aus dem Satz von Arzela-Ascoli folgenden kompakten Einbettung $(\\theta <1)$ $H^2(\\partial \\Omega )\\hookrightarrow C^{0,\\theta }(\\partial \\Omega )\\hookrightarrow \\hookrightarrow C(\\partial \\Omega )$ insbesondere gleichm\"a\"sig.", "Zudem gelte $\\sup _n\\tau (\\eta _n)<\\infty $ .", "Setzen wir $v\\in L^p(\\Omega _\\eta )$ durch 0 auf $\\mathbb {R}^3$ fort, so konvergiert die Folge $(v\\circ \\Psi _{\\eta _n})$ gegen $v\\circ \\Psi _{\\eta }$ in $L^r(\\Omega )$ , $r<p$ .", "Das folgt mit Lemma REF und der Approximierbarkeit von $v$ durch $C_0^\\infty (\\mathbb {R}^3)$ -Funktionen $\\tilde{v}$ aus der Absch\"atzung (in der $L^r(\\Omega )$ -Norm) $\\begin{aligned}\\Vert v\\circ \\Psi _\\eta -v\\circ \\Psi _{\\eta _n}\\Vert \\le \\Vert \\tilde{v}\\circ \\Psi _\\eta -\\tilde{v}\\circ \\Psi _{\\eta _n}\\Vert + \\Vert (v-\\tilde{v})\\circ \\Psi _{\\eta }\\Vert +\\Vert (v-\\tilde{v})\\circ \\Psi _{\\eta _n}\\Vert .\\end{aligned}$ Ist $v\\in L^p(\\Omega )$ und setzen wir die Funktionen $v\\circ \\Psi ^{-1}_{\\eta _n}$ und $v\\circ \\Psi ^{-1}_{\\eta }$ durch 0 auf $\\mathbb {R}^3$ fort, so l\"asst sich auf vollkommen analoge Weise die Konvergenz der Folge $(v\\circ \\Psi ^{-1}_{\\eta _n})$ gegen $v\\circ \\Psi ^{-1}_{\\eta }$ in $L^r(\\mathbb {R}^3)$ , $r<p$ , einsehen.", "Wir konstruieren nun einen Spuroperator f\"ur Auslenkungen $\\eta \\in H^2(\\partial \\Omega )$ .", "Man beachte dabei, dass $\\partial \\Omega _\\eta $ wegen der Einbettung $H^2(\\partial \\Omega )\\hookrightarrow C^{0,\\theta }(\\partial \\Omega )$ für $\\theta <1$ gerade nicht Lipschitz-stetig ist.", "Die Einschr\"ankung $\\cdot \\,|_{\\partial \\Omega }$ ist fortan im Sinne des \"ublichen Spuroperators f\"ur regul\"are R\"ander zu verstehen.", "Korollar 2.3 Es seien $1<p\\le \\infty $ und $\\eta \\in H^2(\\partial \\Omega )$ mit $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\kappa $ .", "Dann ist die lineare Abbildung $\\operatorname{tr_\\eta }: v\\mapsto (v\\circ \\Psi _\\eta )|_{\\partial \\Omega }$ wohldefiniert und stetig von $W^{1,p}(\\Omega _\\eta )$ nach $W^{1-1/r,r}(\\partial \\Omega )$ f\"ur alle $1<r<p$ .", "Die Stetigkeitskonstante h\"angt nur von $\\Omega $ , $\\Vert \\eta \\Vert _{H^2(\\partial \\Omega )}$ , $\\tau (\\eta )$ und $r$ ab; sie bleibt beschr\"ankt, falls $\\Vert \\eta \\Vert _{H^2(\\partial \\Omega )}$ und $\\tau (\\eta )$ beschr\"ankt bleiben.", "Beweis: Die Behauptung folgt sofort aus Lemma REF und den Stetigkeitseigenschaften des \"ublichen Spuroperators; siehe Theorem REF .", "$\\Box $ Der Operator $\\operatorname{tr_\\eta }$ ist nichts anderes als die stetige Fortsetzung der \"`zur\"uckgeholten\"' Spur $v\\mapsto (q\\mapsto v(q+\\eta (q)\\,\\nu (q)), q\\in \\partial \\Omega ),$ die f\"ur glatte Funktionen $v$ wohldefiniert ist.", "Dabei bleibt die lokale Fl\"achenverzerrung der Abbildung $\\Phi _\\eta : q\\mapsto q+\\eta (q)\\,\\nu (q)$ au\"sen vor.", "Wir messen also die Spuren mit dem Fl\"achenma\"s von $\\partial \\Omega $ und nicht mit dem von $\\partial \\Omega _\\eta $ .", "Dieser Operator stellt genau die richtige Konstruktion f\"ur den Vergleich $\\operatorname{tr_\\eta }{\\bf u}=\\partial _t\\eta \\,\\nu $ der Geschwindigkeiten von Fluid und Schale am Rand dar.", "Man beachte, dass die Gleichungen der Elastizit\"at in Lagrange-Koordinaten formuliert werden.", "Aus Lemma REF und den Sobolev-Einbettungen f\"ur Gebiete mit regul\"arem Rand folgen Sobolev-Einbettungen f\"ur unsere speziellen Gebiete.", "Korollar 2.4 Es seien $1<p<3$ und $\\eta \\in H^2(\\partial \\Omega )$ mit $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\kappa $ .", "Dann gilt die kompakte Einbettung $W^{1,p}(\\Omega _\\eta )\\hookrightarrow \\hookrightarrow L^s(\\Omega _\\eta )$ f\"ur $1\\le s < p^*=3p/(3-p)$ .", "Die Stetigkeitskonstante h\"angt nur von $\\Omega $ , $\\Vert \\eta \\Vert _{H^2(\\partial \\Omega )}$ , $\\tau (\\eta )$ , $p$ und $s$ ab; sie bleibt beschr\"ankt, falls $\\Vert \\eta \\Vert _{H^2(\\partial \\Omega )}$ und $\\tau (\\eta )$ beschr\"ankt bleiben.", "Es seien im Folgenden stets $H$ die mittlere Kr\"ummung und $G$ die Gau\"s'sche K\"ummung von $\\partial \\Omega $ .", "Proposition 2.5 Es seien $1<p\\le \\infty $ und $\\eta \\in H^2(\\partial \\Omega )$ mit $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\kappa $ .", "Dann gilt f\"ur $\\varphi \\in W^{1,p}(\\Omega _\\eta )$ mit $\\operatorname{tr_\\eta }\\varphi =b\\,\\nu $ , $b$ eine skalare Funktion, und $\\psi \\in C^1(\\overline{\\Omega _\\eta })$ $\\int _{\\Omega _\\eta } \\varphi \\cdot \\nabla \\psi \\ dx=-\\int _{\\Omega _\\eta } \\operatorname{div}\\varphi \\ \\psi \\ dx+\\int _{\\partial \\Omega }b\\, (1-2H\\eta +G\\,\\eta ^2)\\, \\operatorname{tr_\\eta }\\psi \\ dA.$ Beweis: Wir k\"onnen ohne Einschr\"ankung $p<\\infty $ annehmen.", "Wir approximieren $\\varphi $ durch $(\\varphi _k)\\subset C^\\infty _0(\\mathbb {R}^3)$ in $W^{1,p}(\\Omega _\\eta )$ und $\\eta $ durch $(\\eta _n)\\subset C^2(\\partial \\Omega )$ in $H^2(\\partial \\Omega )$ .", "Dann erhalten wir mittels partieller Integration $\\int _{\\Omega _{\\eta _n}} \\varphi _k\\cdot \\nabla \\psi \\ dx=-\\int _{\\Omega _{\\eta _n}} \\operatorname{div}\\varphi _k\\ \\psi \\ dx + \\int _{\\partial \\Omega _{\\eta _n}} \\varphi _k\\cdot \\nu _{\\eta _n}\\,\\psi \\ dA_{\\eta _n}.$ Anwenden des Transformationssatzes, siehe (REF ), auf das Randintegral ergibt $\\int _{\\partial \\Omega }\\operatorname{tr_{\\eta _n}}\\varphi _k\\cdot (\\nu _{\\eta _n}\\circ \\Phi _{\\eta _n})\\ \\operatorname{tr_{\\eta _n}}\\psi \\ |\\det d\\Phi _{\\eta _n}|\\ dA.$ Mit Hilfe des Gram-Schmidt-Algorithmus lassen sich in einer Umgebung eines jeden Punktes von $\\partial \\Omega $ orthonormale, tangentiale $C^3$ -Vektorfelder ${\\bf e}_1,{\\bf e}_2$ konstruieren; siehe zum Beispiel [39].", "Ohne Einschr\"ankung gelte ${\\bf e}_1\\times {\\bf e}_2=\\nu $ .", "Wir setzen ${\\bf v}_i^n:=d\\Phi _{\\eta _n}{\\bf e}_i$ .", "Die Funktionen ${\\bf v}_i^n$ lassen sich als nichttangentiale lokale Vektorfelder auf $\\partial \\Omega $ auffassen.", "Die lokale Fl\"achenverzerrung von $\\Phi _{\\eta _n}$ ist identisch der Fl\"ache des durch die ${\\bf v}_i^n$ aufgespannten Parallelepipeds, also mit $|{\\bf v}_1^n\\times {\\bf v}_2^n|$ , w\"ahrend die Normale $\\nu _{\\eta _n}\\circ \\Phi _{\\eta _n}$ durch $({\\bf v}_1^n\\times {\\bf v}_2^n)/|{\\bf v}_1^n\\times {\\bf v}_2^n|$ gegeben ist.", "Die Felder ${\\bf v}_1^n\\times {\\bf v}_2^n$ sind offenbar unabh\"angig von der konkreten Wahl der ${\\bf e}_i$ , sodass wir durch die lokale Definition tats\"achlich globale, d.h. auf $\\partial \\Omega $ definierte, Vektorfelder ${\\bf v}^n$ erhalten.", "Das Randintegral l\"a\"st sich also schreiben als $\\int _{\\partial \\Omega }\\operatorname{tr_{\\eta _n}}\\varphi _k\\cdot {\\bf v}^n\\,\\operatorname{tr_{\\eta _n}}\\psi \\ dA.$ Ist $q\\in \\partial \\Omega $ und $c$ eine Kurve in $\\partial \\Omega $ mit $c(0)=q$ und $\\frac{d}{dt}\\big |_{t=0}\\ c(t) = {\\bf e}_i(q)$ , so ist $\\begin{aligned}{\\bf v}_i^n(q)&=\\frac{d}{dt}\\Big |_{t=0}\\Phi _{\\eta _n}(c(t))=\\frac{d}{dt}\\big |_{t=0}(c(t)+\\eta _n(c(t))\\,\\nu (c(t)))\\\\&={\\bf e}_i(q)+d\\eta _n{\\bf e}_i(q)\\, \\nu (q)+\\eta _n(q)\\,\\frac{d}{dt}\\Big |_{t=0}\\nu (c(t))\\\\&={\\bf e}_i(q)+d\\eta _n{\\bf e}_i(q)\\, \\nu (q)-\\eta _n(q)\\, h_i^j(q)\\, {\\bf e}_j(q),\\end{aligned}$ wobei $h^j_i$ die Komponenten der Weingartenabbildung bez\"uglich der Orthonormalbasis ${\\bf e}_1,{\\bf e}_2$ sind.", "Da $\\eta _n$ in $H^2(\\partial \\Omega )$ konvergiert, konvergiert ${\\bf v}_i^n$ in $L^r$ auf seinem Definitionsbereich f\"ur alle $r<\\infty $ .", "Somit konvergiert ${\\bf v}^n$ gegen ein ${\\bf v}$ in $L^r(\\partial \\Omega )$ f\"ur alle $r<\\infty $ .", "Lassen wir nun zun\"achst $n$ und anschlie\"send $k$ gegen unendlich gehen, so erhalten wir $\\int _{\\Omega _\\eta } \\varphi \\cdot \\nabla \\psi \\ dx=-\\int _{\\Omega _\\eta } \\operatorname{div}\\varphi \\ \\psi \\ dx + \\int _{\\partial \\Omega } b\\,\\nu \\cdot {\\bf v}\\,\\operatorname{tr_\\eta }\\psi \\ dA.$ Es bleibt, $\\nu \\cdot {\\bf v}$ zu bestimmen.", "Wir lesen von Gleichung (REF ) ab, dass $\\nu \\cdot ({\\bf v}_1^n\\times {\\bf v}_2^n)=1-(h_1^1+h^2_2)\\,\\eta _n+(h_1^1h_2^2-h_1^2h_2^1)\\,\\eta _n^2=1-2H\\eta _n+G\\,\\eta _n^2$ gilt, woraus wir durch Grenz\"ubergang die Behauptung folgern.$\\Box $ Das im Beweis konstruierte Feld ${\\bf v}\\in L^r(\\partial \\Omega )$ , $r<\\infty $ , wird auch im Folgenden noch Verwendung finden.", "Um die Abh\"angigkeit von $\\eta $ auszudr\"ucken, setzen wir ${\\bf v}={\\bf v}_\\eta $ .", "Da die Felder ${\\bf v}^n$ unabh\"angig von der Wahl der ${\\bf e}_i$ sind, gilt dies ebenso f\"ur ${\\bf v}_\\eta $ .", "Dies deckt sich mit der Koordinateninvarianz der Interpretation von ${\\bf v}_\\eta $ als \"`\"au\"sere Normale\"' an $\\partial \\Omega _\\eta $ , deren L\"ange identisch der \"`lokalen Fl\"achenverzerrung\"' von $\\Phi _\\eta $ ist.", "Bemerkung 2.6 Der Beweis von Proposition REF zeigt f\"ur $\\psi \\in L^1(\\partial \\Omega )$ und $\\eta $ hinreichend glatt die Identit\"at $\\int _{\\partial \\Omega _{\\eta }} \\big ((\\psi \\,\\nu )\\circ \\Phi _{\\eta }^{-1}\\big )\\cdot \\nu _\\eta \\ dA_\\eta =\\int _{\\partial \\Omega } \\psi \\,\\nu \\cdot {\\bf v}_\\eta \\ dA=\\int _{\\partial \\Omega } \\psi \\,(1-2H\\eta +G\\,\\eta ^2)\\ dA.$ Bemerkung 2.7 Die Gr\"o\"se $\\gamma (\\eta ):=1-2H\\eta +G\\,\\eta ^2$ ist positiv, solange $|\\eta |<\\kappa $ .", "Die Nullstellen dieses Polynoms liegen f\"ur $G\\ne 0$ n\"amlich bei $\\frac{H}{G}\\pm \\frac{\\sqrt{H^2-G}}{|G|}=\\frac{1}{2}\\bigg (\\frac{h_1+h_2}{h_1h_2}\\pm \\frac{|h_1-h_2|}{|h_1h_2|}\\bigg ),$ wobei $h_1$ , $h_2$ die Hauptkr\"ummungen sind.", "Die Betr\"age der Nullstellen sind somit identisch $\\frac{|h_1+h_2\\pm (h_1-h_2)|}{2|h_1h_2|}=|h_1|^{-1},|h_2|^{-1}.$ F\"ur $G\\ne 0$ ist die Behauptung somit bewiesen, wenn wir zeigen k\"onnen, dass $\\kappa \\le \\min (|h_1(q)|^{-1},|h_2(q)|^{-1})$ f\"ur alle $q\\in \\partial \\Omega $ .", "F\"ur $G=0$ ist der Beweis dann offensichtlich.", "Um die Absch\"atzung von $\\kappa $ einzusehen, betrachten wir den Diffeomorphismus $\\Lambda _\\alpha :=\\Lambda (\\,\\cdot \\, ,\\alpha ):\\partial \\Omega \\rightarrow \\Lambda _\\alpha (\\partial \\Omega )\\subset \\mathbb {R}^3$ mit $\\alpha \\in (-\\kappa ,\\kappa )$ .", "Die Absch\"atzung von $\\kappa $ folgt nun aus der Tatsache, dass das Differential von $\\Lambda _\\alpha $ an der Stelle $q\\in \\partial \\Omega $ singul\"ar wird f\"ur $|\\alpha |\\nearrow \\min (|h_1(q)|^{-1},|h_2(q)|^{-1})$ .", "Ist n\"amlich ${\\bf e}_1,{\\bf e}_2$ eine Eigenbasis der Weingarten-Abbildung in $q$ , so gilt $d\\Lambda _\\alpha \\,{\\bf e}_i=(1-\\alpha \\, h_i(q))\\,{\\bf e}_i$ .", "Wir betrachten nun den kanonisch normierten Raum $E^p(\\Omega _\\eta ):=\\lbrace \\varphi \\in L^p(\\Omega _\\eta )\\ |\\ \\operatorname{div}\\varphi \\in L^p(\\Omega _\\eta )\\rbrace $ mit $1\\le p\\le \\infty $ .", "F\"ur solche Vektorfelder l\"asst sich immerhin noch eine Spur in Normalenrichtung konstruieren.", "Proposition 2.8 Es seien $1<p<\\infty $ und $\\eta \\in H^2(\\partial \\Omega )$ mit $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\kappa $ .", "Dann existiert ein stetiger, linearer Operator $\\operatorname{tr^n_\\eta }: E^p(\\Omega _\\eta )\\rightarrow (W^{1,p^{\\prime }}(\\partial \\Omega ))^{\\prime }$ derart, dass f\"ur $\\varphi \\in E^p(\\Omega _\\eta )$ und $\\psi \\in C^1(\\overline{\\Omega _\\eta })$ gilt $\\int _{\\Omega _\\eta } \\varphi \\cdot \\nabla \\psi \\ dx=-\\int _{\\Omega _\\eta } \\operatorname{div}\\varphi \\ \\psi \\ dx + \\langle \\operatorname{tr^n_\\eta }\\varphi ,\\operatorname{tr_\\eta }\\psi \\rangle _{W^{1,p^{\\prime }}(\\partial \\Omega )}.$ Die Stetigkeitskonstante h\"angt nur von $\\Omega $ , $\\tau (\\eta )$ und $p$ ab; sie bleibt beschr\"ankt, falls $\\tau (\\eta )$ beschr\"ankt bleibt.", "Beweis: Es gen\"ugt $\\varphi \\in C^1(\\overline{\\Omega _\\eta })$ zu betrachten, da diese Funktionen dicht in $E^p(\\Omega _\\eta )$ liegen; siehe Proposition REF .", "Wir erhalten analog zum Vorgehen im Beweis von Proposition REF f\"ur $\\psi \\in C^1(\\overline{\\Omega }_\\eta )$ die Identit\"at $\\begin{aligned}\\int _{\\Omega _\\eta } \\varphi \\cdot \\nabla \\psi \\ dx+\\int _{\\Omega _\\eta } \\operatorname{div}\\varphi \\ \\psi \\ dx = \\int _{\\partial \\Omega } \\operatorname{tr_\\eta }\\varphi \\cdot {\\bf v}_\\eta \\,\\operatorname{tr_\\eta }\\psi \\ dA.\\end{aligned}$ Die linke Seite ist durch $2\\Vert \\varphi \\Vert _{E^p(\\Omega _\\eta )}\\Vert \\psi \\Vert _{W^{1,p^{\\prime }}(\\Omega _\\eta )}$ abgesch\"atzt.", "Offenbar gilt die Gleichung sogar f\"ur $\\psi \\in W^{1,p^{\\prime }}(\\Omega _\\eta )$ .", "Die Zuordnung $b\\mapsto b\\circ q$ definiert einen stetigen Fortsetzungsoperator von $W^{1,p^{\\prime }}(\\partial \\Omega )$ nach $W^{1,p^{\\prime }}(S_\\alpha \\cap \\Omega _\\eta )$ f\"ur ein festes $\\alpha $ mit $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\alpha <\\kappa $ .", "Der Transformationssatz, Gleichung (REF ), gibt uns n\"amlich $\\int _{S_\\alpha \\cap \\Omega _\\eta }|b\\circ q|^{p^{\\prime }}\\ dx=\\int _{\\partial \\Omega }|b|^{p^{\\prime }}\\int _{-\\alpha }^\\eta |\\det d\\Lambda |\\ dsdA,$ sodass die $L^{p^{\\prime }}$ -Norm von $b\\circ q$ in Abh\"angigkeit von $\\Omega $ und $\\alpha $ durch die $L^{p^{\\prime }}$ -Norm von $b$ abgesch\"atzt ist.", "Durch Approximation von $b$ durch hinreichend glatte Funktionen erhalten wir die Kettenregel $\\nabla (b\\circ q)=(\\nabla b)\\circ q\\ dq,$ sodass auch die $L^{p^{\\prime }}$ -Norm von $\\nabla (b\\circ q)$ in Abh\"angigkeit von $\\Omega $ , $\\alpha $ und $p^{\\prime }$ durch die $L^{p^{\\prime }}$ -Norm von $\\nabla b$ abgesch\"atzt ist.", "Ebenfalls durch Approximation von $b$ erhalten wir die Identit\"at $\\operatorname{tr_\\eta }(b\\circ q)=b$ .", "Durch Multiplikation der konstruierten Fortsetzungen mit der Abschneidefunktion $x\\mapsto \\beta (|s(x)|)$ , wobei $\\beta \\in C^\\infty (\\mathbb {R})$ , $\\beta =1$ in einer Umgebung von $[0,\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}]$ und $\\beta =0$ in einer Umgebung von $\\alpha $ , erhalten wir schlie\"slich einen stetigen, linearen Fortsetzungsoperator von $W^{1,p^{\\prime }}(\\partial \\Omega )$ nach $W^{1,p^{\\prime }}(\\Omega _\\eta )$ .", "Damit ist klar, dass die rechte Seite in obiger Identit\"at einen Spuroperator mit den behaupteten Eigenschaften definiert.", "$\\Box $ Proposition 2.9 Sei $\\eta \\in H^2(\\partial \\Omega )$ mit $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\kappa $ .", "Dann liegt der Teilraum der Funktionen mit in $\\Omega _\\eta $ enthaltenen Tr\"agern dicht im kanonisch normierten Raum $H(\\Omega _\\eta ):=\\lbrace \\varphi \\in L^2(\\Omega _\\eta )\\ |\\ \\operatorname{div}\\varphi =0,\\ \\operatorname{tr^n_\\eta }\\varphi =0 \\rbrace .$ Beweis: F\"ur $\\varphi \\in H(\\Omega _\\eta )$ gelte $\\begin{aligned}\\int _{\\Omega _\\eta } \\varphi \\cdot \\psi \\ dx =0\\end{aligned}$ f\"ur alle $\\psi \\in L^2(\\Omega _\\eta )$ mit $\\operatorname{div}\\psi =0$ und $\\operatorname{supp}\\psi \\subset \\Omega _\\eta $ .", "Gem\"a\"s Theorem REF existiert eine Funktion $p\\in L^2_{\\text{loc}}(\\Omega _\\eta )$ mit $\\varphi =\\nabla p$ .", "Aus Proposition REF folgt zudem $\\begin{aligned}\\int _{\\Omega _\\eta } \\varphi \\cdot \\nabla \\psi \\ dx =0\\end{aligned}$ f\"ur alle $\\psi \\in C^1(\\overline{\\Omega _\\eta })$ .", "Der Beweis von Proposition REF zeigt, dass $\\nabla p$ durch Funktionen $\\nabla \\psi $ , $\\psi \\in C^1(\\overline{\\Omega _\\eta })$ , in $L^2(\\Omega _\\eta )$ approximierbar ist, woraus wir $\\begin{aligned}\\int _{\\Omega _\\eta } |\\varphi |^2\\ dx = \\int _{\\Omega _\\eta } \\varphi \\cdot \\nabla p\\ dx = 0\\end{aligned}$ folgern.", "$\\Box $ Nun konstruieren wir einen Operator, der geeignete Randwerte $b\\,\\nu $ zu einem divergenzfreien Vektorfeld fortsetzt.", "Dazu skalieren wir das Feld $(b\\,\\nu )\\circ q$ derart, dass es divergenzfrei wird.", "Ein entprechender Ansatz f\"uhrt auf eine gew\"ohnliche Differentialgleichung und deren L\"osung auf Definition (REF ) unten.", "Proposition 2.10 Es seien $1<p<\\infty $ , $\\eta \\in H^2(\\partial \\Omega )$ mit $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\kappa $ und $\\alpha $ eine Zahl mit $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\alpha <\\kappa $ .", "Dann existiert ein stetiger, linearer Fortsetzungsoperator $\\mathcal {F}_\\eta : \\Big \\lbrace b\\in W^{1,p}(\\partial \\Omega )\\ \\big |\\ \\int _{\\partial \\Omega }b\\,\\gamma (\\eta )\\ dA=0\\Big \\rbrace \\rightarrow W^{1,p}_{\\operatorname{div}}(B_\\alpha );$ insbesondere gilt $\\operatorname{tr_\\eta }\\mathcal {F}_\\eta b=b\\,\\nu .$ Die Stetigkeitskonstante h\"angt nur von $\\Omega $ , $\\Vert \\eta \\Vert _{H^2(\\partial \\Omega )}$ , $\\alpha $ und $p$ ab; sie bleibt beschr\"ankt, falls $\\Vert \\eta \\Vert _{H^2(\\partial \\Omega )}$ und $\\tau (\\alpha )$ beschr\"ankt bleiben.", "Beweis: Sei $b\\in W^{1,p}(\\partial \\Omega )$ mit $\\begin{aligned}\\int _{\\partial \\Omega }b\\,\\gamma (\\eta )\\ dA=0.\\end{aligned}$ F\"ur $x\\in S_\\alpha $ setzen wir $\\begin{aligned}(\\mathcal {F}_\\eta b)(x):=\\exp \\Big (\\int _{\\eta (q(x))}^{s(x)}\\beta (q(x)+\\tau \\,\\nu (q(x)))\\ d\\tau \\Big )\\,(b\\,\\nu )(q(x)),\\end{aligned}$ wobei $\\beta :=-\\operatorname{div}(\\nu \\circ q)$ .", "Offenbar ist $\\beta $ eine $C^2$ -Funktion.", "Mit Hilfe der Kettenregel sehen wir, dass $\\mathcal {F}_\\eta b$ schwach differenzierbar ist und dass $\\begin{aligned}\\partial _i\\, \\mathcal {F}_\\eta b = \\Big [\\partial _i((b\\,\\nu )\\circ q) +(b\\,\\nu )\\circ q\\ \\Big (&\\int _{\\eta \\circ q}^s \\partial _i(\\beta (q+\\tau \\,\\nu \\circ q))\\ d\\tau +\\beta (q+s\\,\\nu \\circ q)\\ \\partial _i s\\\\&- \\beta (q+(\\eta \\,\\nu )\\circ q)\\ \\partial _i (\\eta \\circ q)\\Big )\\Big ]\\,e^{\\int _{\\eta \\circ q}^s\\beta (q+\\tau \\nu \\circ q) d\\tau }\\end{aligned}$ gilt.", "Da $\\partial _i((b\\,\\nu )\\circ q)\\in L^p(S_\\alpha )$ , $(b\\,\\nu )\\circ q\\in L^{r}(S_\\alpha )$ f\"ur ein $r>p$ , $\\partial _i (\\eta \\circ q)\\in L^r(S_\\alpha )$ f\"ur alle $r<\\infty $ und alle anderen Terme beschr\"ankt sind, ist $\\mathcal {F}_\\eta b$ in $W^{1,p}(S_\\alpha )$ abgesch\"atzt.", "Desweiteren giltF\"ur ein skalares Feld $f$ und ein Vektorfeld ${\\bf X}$ : $\\operatorname{div}(f\\,{\\bf X})=f\\operatorname{div}{\\bf X}+\\nabla f\\cdot {\\bf X}$ .", "$\\begin{aligned}\\operatorname{div}\\mathcal {F}_\\eta b&=\\operatorname{div}(\\nu \\circ q)\\, e^{\\int _{\\eta \\circ q}^s\\beta (q+\\tau \\nu \\circ q) d\\tau }\\,b\\circ q + (\\nu \\circ q)\\cdot \\nabla (e^{\\int _{\\eta \\circ q}^s\\beta (q+\\tau \\nu \\circ q) d\\tau }\\,b\\circ q)\\\\&= (-\\beta \\, e^{\\int _{\\eta \\circ q}^s\\beta (q+\\tau \\nu \\circ q) d\\tau } + \\partial _s e^{\\int _{\\eta \\circ q}^s\\beta (q+\\tau \\nu \\circ q) d\\tau })\\, b\\circ q=0.\\end{aligned}$ F\"ur das zweite Gleichheitszeichen haben wir die Definition von $\\beta $ und die Tatsache, dass f\"ur $x\\in S_\\alpha $ $dq\\, \\nu (q(x))=\\frac{d}{dt}\\Big |_{t=0}q(x+t\\,\\nu (q(x)))=\\frac{d}{dt}\\Big |_{t=0}q(x)=0$ und $ds\\, \\nu (q(x))=\\frac{d}{dt}\\Big |_{t=0}s(x+t\\,\\nu (q(x)))=\\frac{d}{dt}\\Big |_{t=0}s(x)+t=1$ gilt, verwendet.", "Approximieren wir $\\eta $ und $b$ durch $C^2$ -Funktionen $(\\eta _n)$ in $H^2(\\partial \\Omega )$ bzw.", "durch $C^1$ -Funktionen $(b_n)$ in $W^{1,p}(\\partial \\Omega )$ , so konvergieren die $C^1(\\overline{S_\\alpha })$ -Funktionen $\\varphi _n:=\\exp \\Big (\\int _{\\eta _n\\circ q}^{s}\\beta (q+\\tau \\,\\nu \\circ q)\\ d\\tau \\Big )\\, (b_n\\,\\nu )\\circ q$ gegen $\\mathcal {F}_\\eta b$ in $W^{1,p}(S_\\alpha )$ .", "Zudem konvergieren die Spuren $\\begin{aligned}\\operatorname{tr_\\eta }\\varphi _n=\\exp \\Big (\\int _{\\eta _n}^{\\eta }\\beta (q+\\tau \\,\\nu \\circ q)\\ d\\tau \\Big )\\,b_n\\,\\nu \\end{aligned}$ gegen $b\\,\\nu $ , woraus wir $\\operatorname{tr_\\eta }\\mathcal {F}_\\eta b=b\\,\\nu $ folgern.", "Es bleibt noch, $\\mathcal {F}_\\eta b$ auf $\\Omega \\setminus \\overline{S_\\alpha }$ fortzusetzen.", "Unter Verwendung von Proposition REF folgern wir, dass $\\begin{aligned}\\int _{\\partial (\\Omega \\setminus \\overline{S_\\alpha })}(\\mathcal {F}_\\eta b)\\cdot \\nu \\ dA=-\\int _{\\partial \\Omega }b\\, \\gamma (\\eta )\\ dA+\\int _{S_\\alpha \\cap \\Omega _\\eta } \\operatorname{div}\\mathcal {F}_\\eta b\\ dx=0,\\end{aligned}$ wobei $\\nu $ hier die innere Einheitsnormale an $\\partial (\\Omega \\setminus \\overline{S_\\alpha })$ bezeichnet.", "Da $\\partial (\\Omega \\setminus \\overline{S_\\alpha })$ ein $C^3$ -Rand ist, k\"onnen wir aufgrund dieser Identit\"at das Stokes-System in $\\Omega \\setminus \\overline{S_\\alpha }$ mit Randwerten $(\\mathcal {F}_\\eta b)|_{\\partial (\\Omega \\setminus \\overline{S_\\alpha })}$ l\"osen; siehe Theorem REF .", "Dies liefert uns die gew\"unschte Fortsetzung.$\\Box $ Bemerkung 2.11 Die Funktion $b\\in L^p(\\partial \\Omega )$ , $1<p<\\infty $ , erf\"ulle die Identit\"at (REF ).", "Dann k\"onnen wir eine Fortsetzung $\\mathcal {F}_\\eta b\\in L^p(B_\\alpha )$ mit verschwindender (distributioneller) Divergenz wie im Beweis der obigen Proposition konstruieren.", "Die lineare Abbildung $\\mathcal {F}_\\eta : \\Big \\lbrace b\\in L^{p}(\\partial \\Omega )\\ \\big |\\ \\int _{\\partial \\Omega }b\\, \\gamma (\\eta )\\ dA=0\\Big \\rbrace \\rightarrow \\lbrace \\varphi \\in L^p(B_\\alpha )\\ |\\ \\operatorname{div}\\varphi =0\\rbrace $ definiert wiederum einen stetigen, linearen Operator, dessen Stetigkeitskonstante wie in Proposition REF von den Daten abh\"angt.", "Bei der Konstruktion der Fortsetzung wenden wir den L\"osungsoperator des Stokes-Systems auf die formale Spur $\\varphi |_{\\partial (\\Omega \\setminus \\overline{S_\\alpha })}=\\exp \\Big (\\int _{\\eta \\circ q}^{-\\alpha }\\beta (q+\\tau \\,\\nu \\circ q)\\ d\\tau \\Big )\\, (b\\,\\nu )\\circ q$ an.", "Diese ist offenbar in $L^p(\\partial (\\Omega \\setminus \\overline{S_\\alpha }))$ abgesch\"atzt und erf\"ullt die Identit\"at $\\begin{aligned}\\int _{\\partial (\\Omega \\setminus \\overline{S_\\alpha })}\\varphi \\cdot \\nu \\ dA=0.\\end{aligned}$ Zum anschlie\"senden Nachweis der Divergenzfreiheit des resultierenden Vektorfelds auf $B_\\alpha $ verwenden wir die Tatsache, dass f\"ur $\\psi \\in C_0^\\infty (\\Omega )$ die Gleichung $\\begin{aligned}\\int _{S_\\alpha }\\varphi \\cdot \\nabla \\psi \\ dx=\\int _{\\partial (\\Omega \\setminus \\overline{S_\\alpha })}\\varphi \\cdot \\nu \\, \\psi \\ dA=-\\int _{\\Omega \\setminus \\overline{S_\\alpha }}\\varphi \\cdot \\nabla \\psi \\ dx\\end{aligned}$ gilt.", "Die Identit\"aten (REF ) und (REF ) lassen sich leicht durch Approximation von $b$ durch hinreichend glatte Funktionen einsehen.", "Ebenfalls durch Approximation von $b$ erhalten wir aus den Gleichungen (REF ) und (REF ) die Identit\"at $\\begin{aligned}\\operatorname{tr^n_\\eta }\\varphi =b\\, \\gamma (\\eta ),\\end{aligned}$ die die Bezeichnung Fortsetzungsoperator rechtfertigt.", "Proposition 2.12 Es seien $1<p<3$ und $\\eta \\in H^2(\\partial \\Omega )$ mit $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}<\\kappa $ .", "Dann ist die Fortsetzung durch 0 eine stetige, lineare Abbildung von $W^{1,p}(\\Omega _\\eta )$ nach $W^{1/4,p}(\\mathbb {R}^3)$ .", "Die Stetigkeitskonstante h\"angt nur von $\\Omega $ , $p$ , $\\Vert \\eta \\Vert _{H^2(\\partial \\Omega )}$ und $\\tau (\\eta )$ ab; sie bleibt beschr\"ankt, falls $\\Vert \\eta \\Vert _{H^2(\\partial \\Omega )}$ und $\\tau (\\eta )$ beschr\"ankt bleiben.", "Beweis: Wir zeigen zun\"achst, dass die Fortsetzung durch 0 eine stetige, lineare Abbildung von $W^{1,r}(\\Omega )$ nach $W^{s,r}(\\mathbb {R}^3)$ mit $1\\le r<3$ und $s<1/3$ ist.", "Dazu gen\"ugt es f\"ur $v\\in W^{1,r}(\\Omega )$ das Integral $\\begin{aligned}\\int _{\\mathbb {R}^3}\\int _{\\mathbb {R}^3} \\frac{|v(x)-v(y)|^r}{|x-y|^{3+sr}}\\ dydx =\\int _{\\Omega }\\int _{\\Omega } \\frac{|v(x)-v(y)|^r}{|x-y|^{3+sr}}\\ dydx +2\\int _{\\Omega }\\int _{\\mathbb {R}^3\\setminus \\Omega } \\frac{|v(x)|^r}{|x-y|^{3+sr}}\\ dydx\\\\=\\int _{\\Omega }\\int _{\\Omega } \\frac{|v(x)-v(y)|^r}{|x-y|^{3+sr}}\\ dydx+ 2\\int _{\\Omega } |v(x)|^r\\int _{\\mathbb {R}^3\\setminus \\Omega } \\frac{1}{|x-y|^{3+sr}}\\ dydx\\end{aligned}$ abzusch\"atzen.", "W\"ahrend der erste Term auf der rechten Seite durch $c\\,\\Vert v\\Vert _{W^{1,r}(\\Omega )}$ majorisiert ist, l\"asst sich das innere Integral des zweiten Terms durch $\\int _{|z|>d(x)}\\frac{1}{|z|^{3+sr}}\\ dz=\\frac{c(s,r)}{d(x)^{sr}}$ absch\"atzen, wobei $d(x)$ den Abstand von $x$ zum Rand $\\partial \\Omega $ bezeichnet.", "Eine Anwendung der H\"older-Ungleichung mit den Exponenten $r^*/r$ und $(r^*/r)^{\\prime }$ auf den zweiten Summanden zeigt somit, dass dieser durch $c(s,r)\\,\\Vert v\\Vert _{r^*}^r\\,\\Vert d(\\cdot )^{-s}\\Vert _{L^3(\\Omega )}^r$ dominiert ist.", "Die Identit\"at $\\int _{S_{\\kappa /2}\\cap \\Omega }|d(x)|^{-3s}\\ dx=\\int _{\\partial \\Omega }\\int _{-\\kappa /2}^0|\\det d\\Lambda |\\, \\alpha ^{-3s} \\ d\\alpha dA,$ eine Konsequenz des Transformationssatzes, zeigt zusammen mit der Ungleichung $3s<1$ , dass der Faktor $\\Vert d(\\cdot )^{-s}\\Vert _{L^3(\\Omega )}^r$ endlich ist.", "Damit ist die Behauptung bewiesen.", "Verketten der Abbildung aus Lemma REF mit obiger Fortsetzung ergibt nun eine stetige, lineare Abbildung $W^{1,p}(\\Omega _\\eta )\\rightarrow W^{1,r}(\\Omega )\\rightarrow W^{s,r}(\\mathbb {R}^3)$ f\"ur $r<p$ und $s<1/3$ .", "Sei nun $\\delta \\in C^4(\\partial \\Omega )$ mit $\\eta <\\delta <\\kappa $ in $\\partial \\Omega $ .", "Setzen wir die Abbildung $\\Psi _\\delta $ auf $\\overline{B_\\alpha }$ , $\\alpha >0$ hinreichend klein, fort, indem wir die Definition (REF ) $x\\in S_\\kappa \\cap \\overline{B_\\alpha }$ verwenden, so erhalten wir einen $C^3$ -Diffeomorphismus $\\widetilde{\\Psi }_\\delta : \\overline{B_\\alpha }\\rightarrow \\overline{\\Omega _{\\delta +\\alpha }}.$ Da die gebrochenen Sobolev-R\"aume Interpolationsr\"aume sind, ist die lineare Abbildung $v\\mapsto v\\circ \\widetilde{\\Psi }_\\delta ^{-1}$ stetig von $W^{s,r}(B_\\alpha )$ nach $W^{s,r}(\\Omega _{\\delta +\\alpha })$ .", "Aus demselben Grund ist die Einschr\"ankung von Funktionen stetig von $W^{s,r}(\\mathbb {R}^3)$ nach $W^{s,r}(B_\\alpha )$ .", "F\"ur geeignete $s<1/3$ und $r<p$ gilt zudem die Einbettung $W^{s,r}(\\Omega _{\\delta +\\alpha })\\hookrightarrow W^{1/4,p}(\\Omega _{\\delta +\\alpha }),$ siehe Theorem REF .", "Verkn\"upfen der obigen Abbildungen zeigt, dass die Fortsetzung durch 0 eine stetige, lineare Abbildung von $W^{1,p}(\\Omega _\\eta )$ nach $W^{1/4,p}(\\Omega _{\\delta +\\alpha })$ ist.", "Da $\\Omega _\\eta $ positiven Abstand von $\\mathbb {R}^3\\setminus \\Omega _{\\delta +\\alpha }$ hat, folgt die Behauptung der Proposition nun aus der Absch\"atzung $\\begin{aligned}|v|_{1/4,p;\\mathbb {R}^3}^p&=|v|_{1/4,p;\\Omega _{\\delta +\\alpha }}^p+\\int _{\\Omega _\\eta }\\int _{\\mathbb {R}^3\\setminus \\Omega _{\\delta +\\alpha }}\\frac{|v(x)|^p}{|x-y|^{3+p/4}}\\ dydx\\\\&\\le |v|_{1/4,p;\\Omega _{\\delta +\\alpha }}^p + c\\,\\Vert v\\Vert _{L^p(\\Omega _\\eta )}^p.\\end{aligned}$ $\\Box $ F\"ur Funktionen, die auf zeitlich variablen Gebieten definiert sind, sind die \"ublichen Bochner-R\"aume nicht die richtigen Objekte.", "Deshalb geben wir nun eine naheliegende, auf unsere Bed\"urfnisse zugeschnittene Konstruktion an.", "F\"ur $I:=(0,T)$ , $T>0$ , und $\\eta \\in C(\\bar{I}\\times \\partial \\Omega )$ mit $\\Vert \\eta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}<\\kappa $ setzen wir $\\Omega _\\eta ^I:=\\bigcup _{t\\in I}\\, \\lbrace t\\rbrace \\times \\Omega _{\\eta (t)}$ .", "$\\Omega _\\eta ^I$ ist ein Gebiet des $\\mathbb {R}^4$ .", "Wir definieren nun f\"ur $1\\le p,r\\le \\infty $ $\\begin{aligned}L^p(I,L^r(\\Omega _{\\eta (t)}))&:=\\lbrace v\\in L^1(\\Omega _\\eta ^I)\\ |\\ v(t,\\cdot )\\in L^r(\\Omega _{\\eta (t)})\\text{ f\"ur fast alle $t$ und}\\\\&\\hspace{139.41832pt}\\Vert v(t,\\cdot )\\Vert _{L^r(\\Omega _{\\eta (t)})}\\in L^p(I)\\rbrace ,\\\\L^p(I,W^{1,r}(\\Omega _{\\eta (t)}))&:=\\lbrace v\\in L^p(I,L^r(\\Omega _{\\eta (t)}))\\ |\\ \\nabla v\\in L^p(I,L^r(\\Omega _{\\eta (t)}))\\rbrace ,\\\\L^p(I,W_{\\operatorname{div}}^{1,r}(\\Omega _{\\eta (t)}))&:=\\lbrace {\\bf v}\\in L^p(I,W^{1,r}(\\Omega _{\\eta (t)}))\\ |\\ \\operatorname{div}{\\bf v}=0\\rbrace ,\\\\W^{1,p}(I,W^{1,r}(\\Omega _{\\eta (t)}))&:=\\lbrace {\\bf v}\\in L^p(I,W^{1,r}(\\Omega _{\\eta (t)}))\\ |\\ \\partial _t{\\bf v}\\in L^p(I,W^{1,r}(\\Omega _{\\eta (t)}))\\rbrace .\\end{aligned}$ Dabei wirken $\\nabla $ und $\\operatorname{div}$ nur auf die r\"aumlichen Koordinaten.", "Wir setzen zudem $\\Psi _\\eta : \\bar{I}\\times \\overline{\\Omega }\\rightarrow \\overline{\\Omega _\\eta ^I},\\ (t,x)\\mapsto (t,\\Psi _{\\eta (t)}(x))$ und $\\Phi _\\eta : \\bar{I}\\times \\partial \\Omega \\rightarrow \\bigcup _{t\\in \\bar{I}}\\, \\lbrace t\\rbrace \\times \\partial \\Omega _{\\eta (t)},\\ (t,x)\\mapsto (t,\\Phi _{\\eta (t)}(x)).$ Gilt zus\"atzlich $\\eta \\in L^\\infty (I,H^2(\\partial \\Omega ))$ , so erhalten wir \"`instation\"are\"' Versionen der obigen Aussagen, wenn wir diese zu (fast) jedem Zeitpunkt verwenden.", "Zum Beispiel folgt aus Korollar REF die Einbettung $L^2(I,H^{1}(\\Omega _{\\eta (t)}))\\hookrightarrow L^2(I,L^s(\\Omega _{\\eta (t)}))$ f\"ur $1\\le s<2^*$ .", "Man beachte, dass obige Konstruktion nicht ohne Weiteres einen Ersatz f\"ur Bochner-R\"aume von Funktionen mit Werten in einem Dualraum bietet.", "Solche R\"aume spielen in der Theorie schwacher L\"osungen von Evolutionsgleichungen eine wichtige Rolle, weil den Zeitableitungen solcher L\"osungen typischerweise nur als Elemente von R\"aumen dieser Art Sinn gegeben werden kann.", "Funktionale aus $(W^{1,r}_0)^{\\prime }$ lassen sich in der Form $f+\\partial _i f^i$ mit $f,f^i\\in L^{r^{\\prime }}$ schreiben; siehe zum Beispiel [2].", "Liegt eine derartige Charakterisierung des fraglichen Dualraums vor, so kann man versuchen, darauf aufbauend einen Ersatz nach obigem Schema zu entwickeln.", "Dennoch dr\"angt sich die Frage auf, ob nicht ein nat\"urlichere Konstruktion denkbar ist.", "Man k\"onnte auf die Idee kommen, R\"aume von Funktionen, die jeden Zeitpunkt in einen anderen Banach-Raum abbilden, zu betrachten.", "Objekte dieser Art treten in der Differentialgeometrie auf, und zwar in Form von Vektorb\"undeln.", "Ein Vektorb\"undel ordnet jedem Punkt einer Mannigfaltigkeit einen anderen Vektorraum zu.", "Um differenzierbaren Abbildungen von der Mannigfaltigkeit in das B\"undel Sinn zu geben, ben\"otigt letzteres nat\"urlich eine differenzierbare Struktur.", "In unserem Falle m\"usste die differenzierbare Strukturbzw.", "die lokalen Trivialisierungen des B\"undels, die \"ublicherweise die differenzierbare Struktur induzieren so gew\"ahlt werden, dass sie den Sinn der Zeitableitung, n\"amlich in einem inneren Punkt des sich bewegenden Gebiets die zeitliche \"Anderung der Funktion anzugeben, erfasst.", "Die Konstruktion einer derartigen Struktur ist bislang nicht gelungen.", "Eine verwandte M\"oglichkeit, die Problematik anzugehen, besteht auf den ersten Blick darin, das Gebiet auf einen Raumzeitzylinder zu transformieren und auf diesem die \"ublichen Bochner-R\"aume zu betrachten.", "Das scheitert in unserem Fall an der geringen Regularit\"at, die unser variabler Rand haben wird.", "Zudem verf\"alscht eine solche Transformation den Sinn der Zeitableitung.", "Prinzipiell scheint die Aussage zu gelten, dass durch eine Transformation auf einen Raumzeitzylinder nichts gewonnen wird, sondern dass im Gegenteil die Sache komplizierter wird.", "Die obige Konstruktion scheint somit das einzig sinnvolle Substitut f\"ur die \"ublichen Bochner-R\"aume zu sein.", "Die Frage nach einer M\"oglichkeit, den Verlust des \"ublichen Begriffs der schwachen Zeitableitung zu kompensieren, wird uns deshalb noch besch\"aftigen.", "F\"ur alle $1/2<\\theta <1$ gilt $\\begin{aligned}W^{1,\\infty }(I,L^2(\\partial \\Omega ))\\cap L^\\infty (I,H^2(\\partial \\Omega )) \\hookrightarrow C^{0,1-\\theta }(\\bar{I},H^{2\\theta }(\\partial \\Omega ))\\hookrightarrow C^{0,1-\\theta }(\\bar{I}, C^{0,2\\theta -1}(\\partial \\Omega )).\\end{aligned}$ W\"ahrend wir f\"ur die zweite Einbettung Theorem REF verwendet haben, folgt die erste aus der elementaren Absch\"atzung $\\begin{aligned}\\Vert u(t)-u(s)\\Vert _{(L^2(\\partial \\Omega ),H^2(\\partial \\Omega ))_{\\theta ,2}}&\\le \\Vert u(t)-u(s)\\Vert ^{\\theta }_{H^2(\\partial \\Omega )}\\,\\Vert u(t)-u(s)\\Vert _{L^2(\\partial \\Omega )}^{1-\\theta }\\\\ &\\le c\\Vert u\\Vert _{L^\\infty (I,H^2(\\partial \\Omega ))}^{\\theta }\\,\\Vert u\\Vert _{W^{1,\\infty }(I,L^2(\\partial \\Omega ))}^{1-\\theta }\\,|t-s|^{1-\\theta }.\\end{aligned}$ Proposition 2.13 Es seien $\\eta \\in W^{1,\\infty }(I,L^2(\\partial \\Omega ))\\cap L^\\infty (I,H^2(\\partial \\Omega ))$ mit $\\Vert \\eta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}<\\kappa $ und $\\alpha $ eine Zahl mit $\\Vert \\eta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}<\\alpha <\\kappa $ .", "Dann definiert die Anwendung des Fortsetzungsoperators aus Proposition REF zu (fast) allen Zeiten einen stetigen, linearen Fortsetzungsoperator $\\mathcal {F}_\\eta $ von $\\Big \\lbrace b\\in H^{1}(I,L^2(\\partial \\Omega ))\\cap L^2(I,H^2(\\partial \\Omega ))\\ |\\ \\int _{\\partial \\Omega }b(t,\\cdot )\\, \\gamma (\\eta (t,\\cdot ))\\ dA=0\\text{ f\"ur fast alle }t\\in I\\Big \\rbrace $ nach $\\lbrace \\varphi \\in H^{1}(I,L^2(B_\\alpha ))\\cap C(\\bar{I},H^1(B_\\alpha ))\\ |\\ \\operatorname{div}\\varphi =0\\rbrace .$ Die Stetigkeitskonstante h\"angt nur von $\\Omega $ , $\\Vert \\eta \\Vert _{ W^{1,\\infty }(I,L^2(\\partial \\Omega ))\\cap L^\\infty (I,H^2(\\Omega ))}$ und $\\alpha $ ab; sie bleibt beschr\"ankt, falls $\\Vert \\eta \\Vert _{ W^{1,\\infty }(I,L^2(\\partial \\Omega ))\\cap L^\\infty (I,H^2(\\Omega ))}$ und $\\tau (\\alpha )$ beschr\"ankt bleiben.", "Beweis: Die Stetigkeit nach $\\lbrace \\varphi \\in L^\\infty (I,H^1(B_\\alpha ))\\ |\\ \\operatorname{div}\\varphi =0\\rbrace $ folgt unter Beachtung der Einbettung $\\begin{aligned}H^1(I,L^2(\\partial \\Omega ))\\cap L^2(I,H^2_0(\\partial \\Omega ))\\hookrightarrow C(\\bar{I},H^1(\\partial \\Omega )),\\end{aligned}$ eine Konsequenz von Proposition REF ,Die Identit\"at $[L^2(\\partial \\Omega ),H^2(\\partial \\Omega )]_\\frac{1}{2}=H^1(\\partial \\Omega )$ folgt dabei unter Verwendung eines endlichen Atlas mit untergeordneter Zerlegung der Eins und der in [2] konstruierten Fortsetzungsoperatoren aus Theorem 6.4.5 in [8].", "sofort aus Proposition REF .", "Aus (REF ), (REF ) und Theorem REF erhalten wir zudem die Einbettungen $\\begin{aligned}H^1(I,L^2(\\partial \\Omega ))\\cap L^2(I,H^2_0(\\partial \\Omega ))\\hookrightarrow C(\\bar{I},L^4(\\partial \\Omega )),\\\\W^{1,\\infty }(I,L^2(\\partial \\Omega ))\\cap L^\\infty (I,H^2(\\partial \\Omega )) \\hookrightarrow C(\\bar{I},W^{1,4}(\\partial \\Omega )).\\end{aligned}$ Diese zeigen zusammen mit (REF ), (REF ) und den Stetigkeitseigenschaften des L\"osungsoperators des Stokes-Systems, dass $\\mathcal {F}_\\eta b$ in $C(\\bar{I},H^1(B_\\alpha ))$ liegt.", "Es bleibt also lediglich $\\mathcal {F}_\\eta b$ in $H^1(I,L^2(B_\\alpha ))$ abzusch\"atzen.", "In $I\\times S_\\alpha $ gilt $\\begin{aligned}\\partial _t\\, (\\mathcal {F}_\\eta b)(t,\\cdot ):= \\Big [(\\partial _tb)(t,q)\\ \\nu \\circ q - \\beta (q+\\eta (t,q)\\, \\nu \\circ q)\\ \\partial _t\\eta (t,q) \\ b(t,q)\\ \\nu \\circ q\\Big ]\\\\e^{\\int _{\\eta \\circ q}^s\\beta (q+\\tau \\nu \\circ q) d\\tau }.\\end{aligned}$ Wegen $\\partial _tb\\in L^2(I,L^2(\\partial \\Omega ))$ ist der erste Summand in $L^2(I,L^2(S_\\alpha ))$ abgesch\"atzt; die Absch\"atzung des zweiten Summanden in $L^2(I,L^2(S_\\alpha ))$ folgt aus $\\partial _t\\eta \\in L^\\infty (I,L^2(\\partial \\Omega ))$ und $b\\in L^2(I,L^\\infty (\\partial \\Omega ))$ .", "Die Zeitableitung der Spur von $\\mathcal {F}_\\eta b$ auf $I\\times \\partial (\\Omega \\setminus \\overline{S_\\alpha })$ ist identisch $\\Big [(\\partial _tb)(t,q)\\ \\nu \\circ q - \\beta (q+\\eta (t,q)\\,\\nu \\circ q)\\ \\partial _t\\eta (t,q) \\ b(t,q)\\ \\nu \\circ q\\Big ]\\,e^{\\int _{\\eta \\circ q}^{-\\alpha }\\beta (q+\\tau \\nu \\circ q) d\\tau }$ und somit in $L^2(I,L^2(\\partial (\\Omega \\setminus \\overline{S_\\alpha })))$ abgesch\"atzt.", "Aus den Stetigkeitseigenschaften des L\"osungsoperators des Stokes-Systems folgt nun die Behauptung.", "$\\Box $ Bemerkung 2.14 Die Argumente in obigem Beweis zeigen, dass die Anwendung des Fortsetzungsoperators aus Bemerkung REF zu (fast) allen Zeiten unter den Voraussetzungen von Proposition REF einen stetigen, linearen Fortsetzungsoperator $\\mathcal {F}_\\eta $ von $\\Big \\lbrace b\\in C(\\bar{I},L^2(\\partial \\Omega ))\\ |\\ \\int _{\\partial \\Omega }b(t,\\cdot )\\, \\gamma (\\eta (t,\\cdot ))\\ dA=0\\text{ f\"ur fast alle }t\\in I\\Big \\rbrace $ nach $\\lbrace \\varphi \\in C(\\bar{I},L^2(B_\\alpha ))\\ |\\ \\operatorname{div}\\varphi =0\\rbrace $ definiert.", "Die Stetigkeitskonstante h\"angt wie in Proposition REF von den Daten ab." ], [ "Koiter-Energie", "Es sei $M$ eine kompakte, orientierte und berandete $C^4$ -Fl\"ache im $\\mathbb {R}^3$ mit durch den umgebenden euklidischen Raum induzierter erster und zweiter Fundamentalform $g$ bzw.", "$h$ und Fl\"achenma\"s $dA$ .", "$M$ stelle die Mittelfl\"ache einer elastischen Schale der Dicke $2\\epsilon _0>0$ in ihrer Ruhelage dar, wobei $\\epsilon _0$ klein sei gegen die Kehrwerte der Hauptkr\"ummungen von $M$ .", "Die elastische Schale bestehe aus einem homogenen und isotropen Material, dessen lineares elastisches Verhalten durch die Lamé-Konstanten $\\lambda $ und $\\mu $ charakterisiert sei.", "Deformationen der Mittelfl\"ache und damit der Schale beschreiben wir durch ein (hinreichend glattes) Vektorfeld $\\eta :\\ M \\rightarrow \\mathbb {R}^3$ .", "Wir bezeichnen mit $g(\\eta )$ und $h(\\eta )$ die auf $M$ zur\"uckgeholte Metrik bzw.", "zweite Fundamentalform der gem\"a\"s $\\eta $ deformierten Fl\"ache.", "Die elastische Energie einer solchen Verschiebung kann durch die Koiter-Energie f\"ur eine nichtlinear elastische Schale $K_N(\\eta )=\\frac{1}{2}\\int _M \\epsilon _0\\,\\langle C,\\Sigma (\\eta )\\otimes \\Sigma (\\eta )\\rangle + \\frac{\\epsilon _0^3}{3}\\,\\langle C,\\Xi (\\eta )\\otimes \\Xi (\\eta )\\rangle \\ dA $ modelliert werden; siehe [35], [36], [13], [14] und die dortigen Referenzen.", "Dabei ist $\\begin{aligned}C_{\\alpha \\beta \\gamma \\delta }:=\\frac{4\\lambda \\mu }{\\lambda +2\\mu }\\,g_{\\alpha \\beta }\\,g_{\\gamma \\delta } +2\\mu \\,(g_{\\alpha \\gamma }\\,g_{\\beta \\delta } + g_{\\alpha \\delta }\\,g_{\\beta \\gamma })\\end{aligned}$ das Elastizit\"atstensorfeld der Schale, und $\\Sigma (\\eta ):=1/2\\,(g(\\eta )-g)$ sowie $\\Xi (\\eta ):=h(\\eta )-h$ bezeichnen die Differenzen der ersten bzw.", "zweiten Fundamentalformen.", "Diese Energie wird in [36] auf Basis der dreidimensionalen Elastizit\"at unter den zus\"atzlichen Annahmen kleiner Dehnungen und ebener Spannungszust\"ande parallel zur Mittelfl\"ache hergeleitet.", "Der mit $\\epsilon _0$ skalierende Anteil der Koiter-Energie, die L\"angungs- oder Membranenergie, erfasst ausschlie\"slich nichtisometrische Deformationen.", "Der mit $\\epsilon _0^3$ skalierende Anteil kann als Biegeenergie interpretiert werden.", "F\"ur die rigorose Rechtfertigung derartiger \"`zweidimensionaler Theorien\"' auf Basis der dreidimensionalen Elastizit\"at verweisen wir auf [13], [37], [26].", "Setzen wir $\\widetilde{C}:=\\frac{4\\lambda \\mu }{\\lambda +2\\mu }\\, g\\otimes g,$ so ist offenbar der Anteil $\\begin{aligned}\\frac{1}{2}\\int _M \\frac{\\epsilon _0^3}{3}\\,\\langle \\widetilde{C},h(\\eta )\\otimes h(\\eta )\\rangle \\ dA&=\\frac{\\epsilon _0^3}{3}\\frac{4\\lambda \\mu }{\\lambda +2\\mu }\\ \\frac{1}{2}\\int _M \\langle g,h(\\eta )\\rangle ^2\\ dA\\end{aligned}$ der Biegeenergie verwandt mit der Willmore-Energie $W(\\eta )=\\frac{1}{2}\\int _M H^2(\\eta )\\ dA_{\\eta }$ der deformierten Mittelfl\"ache.", "Hierbei bezeichnen $H(\\eta )=\\frac{1}{2}\\langle g(\\eta ),h(\\eta )\\rangle $ die auf $M$ zur\"uckgeholte mittlere Kr\"ummung der deformierten Mittelfl\"ache und $dA_{\\eta }$ das Fl\"achenma\"s bez\"uglich der Metrik $g(\\eta )$ .", "In der Koiter-Energie besteht \"uber das Hooke'sche Gesetz zwar ein linearer Zusammenhang zwischen Dehnungen und Spannungen, aber der Zusammenhang zwischen Deformationen und Dehnungen ist nichtlinear.", "Wir linearisieren nun die Abh\"angigkeit der Tensorfelder $\\Sigma (\\eta )$ und $\\Xi (\\eta )$ von $\\eta $ an der Stelle $\\eta =0$ und erhalten die Tensorfelder $\\sigma (\\eta )$ und $\\xi (\\eta )$ .", "Entsprechend definieren wir die Koiter-Energie (f\"ur eine linear elastische Schale) $\\begin{aligned}K(\\eta )=K(\\eta ,\\eta )=\\frac{1}{2}\\int _M \\epsilon _0\\,\\langle C,\\sigma (\\eta )\\otimes \\sigma (\\eta )\\rangle + \\frac{\\epsilon _0^3}{3}\\,\\langle C,\\xi (\\eta )\\otimes \\xi (\\eta )\\rangle \\ dA.\\end{aligned}$ $K$ ist eine quadratische Form in $\\eta $ , und es gilt $K\\ge 0$ ; siehe Theorem 4.4-1 in [14].", "Sie ist unter Umst\"anden sogar koerziv; siehe Theorem 4.4-2 in [14].", "Der Gradient dieser Energie ist ein elliptischer Operator vierter Ordnung, der in an $M$ tangentiale Richtung degeneriert ist.Diese Beobachtung spielt in [17] eine prominente Rolle.", "Da dort die Koiter-Energie f\"ur nichtlinear elastische Schalen betrachtet wird, h\"angen die Entartungsrichtungen von der L\"osung ab, sind also a-priori unbekannt.", "Speziell ist er von lediglich dritter Ordnung auf dem tangentialen Anteil.", "Der normale Anteil der Auslenkung ist deshalb im Allgemeinen regul\"arer als der tangentiale; vgl.", "Theorem 4.4-2 in [14].", "Motiviert durch das Vorgehen in [11] schr\"anken wir im Rahmen dieser Arbeit die Verschiebungen auf die Richtung der Einheitsnormale $\\nu $ von $M$ ein.", "Neben dem Elliptizit\"ats- und Regularit\"atsgewinn hat die Einschr\"ankung den Vorteil, dass wir ein einfaches Kriterium daf\"ur angeben k\"onnen, dass sich verschiedene Teile der Schale nicht ber\"uhren.", "Wir k\"onnen die Auslenkung $\\eta =:\\eta \\,\\nu $ durch eine skalare Funktion $\\eta $ beschreiben.", "Dadurch integrieren wir die vorliegende Zwangsbedingung in den Phasenraum, sodass kein Lagrange-Multiplikator in den Gleichungen auftreten wird.", "Wir setzen zudem $K(\\eta ):=K(\\eta \\,\\nu )$ .", "Die Tensorfelder $\\sigma (\\eta ):=\\sigma (\\eta \\,\\nu )$ und $\\xi (\\eta ):=\\xi (\\eta \\,\\nu )$ nehmen mit der Definition $k_{\\alpha \\beta }=h_\\alpha ^\\sigma \\, h_{\\sigma \\beta }$ die Form $\\begin{aligned}\\sigma (\\eta )=-h\\, \\eta ,\\ \\xi (\\eta )=\\nabla ^2 \\eta - k\\, \\eta \\end{aligned}$ an; siehe Theorem 4.2-1 und Theorem 4.2-2 in [14].", "Damit gilt f\"ur (hinreichend glatte) skalare Felder $\\eta ,\\zeta $ $\\begin{aligned}K(\\eta ,\\zeta )&=\\frac{1}{2}\\int _M \\epsilon _0\\,\\langle C,\\sigma (\\eta )\\otimes \\sigma (\\zeta )\\rangle + \\frac{\\epsilon _0^3}{3}\\,\\langle C,\\xi (\\eta )\\otimes \\xi (\\zeta )\\rangle \\ dA \\\\&= \\frac{1}{2}\\int _M \\epsilon _0 \\, \\eta \\,\\zeta \\, \\Big (\\frac{16\\lambda \\mu }{\\lambda +2\\mu } H^2 + 4\\mu |h|^2\\Big )\\\\&\\hspace{36.98866pt}+ \\frac{\\epsilon _0^3}{3} \\frac{4\\lambda \\mu }{\\lambda +2\\mu }(\\Delta \\eta \\,\\Delta \\zeta -|h|^2\\ (\\zeta \\, \\Delta \\eta + \\eta \\,\\Delta \\zeta ) -\\eta \\,\\zeta \\, |h|^4)\\\\&\\hspace{36.98866pt} + \\frac{\\epsilon _0^3}{3}\\,4\\mu \\,(\\langle \\nabla ^2\\eta ,\\nabla ^2\\zeta \\rangle - \\zeta \\,\\langle \\nabla ^2\\eta ,k\\rangle - \\eta \\,\\langle \\nabla ^2\\zeta ,k\\rangle - \\eta \\,\\zeta \\, |k|^2)\\ dA,\\end{aligned}$ wobei $H$ die mittlere Kr\"ummung von $M$ bezeichnet.", "F\"ur die zweite Gleichung haben wir beispielsweise $\\langle g\\otimes g,\\nabla ^2\\eta \\otimes \\nabla ^2\\zeta \\rangle =\\langle g,\\nabla ^2\\eta \\rangle \\,\\langle g,\\nabla ^2\\zeta \\rangle =\\Delta \\eta \\,\\Delta \\zeta $ und $g^{\\alpha \\bar{\\alpha }}\\,g^{\\gamma \\bar{\\gamma }}\\,g^{\\beta \\bar{\\beta }}\\,g^{\\delta \\bar{\\delta }}\\,g_{\\alpha \\gamma }\\,g_{\\beta \\delta }\\, \\nabla ^2_{\\bar{\\alpha }\\bar{\\beta }}\\eta \\,\\nabla ^2_{\\bar{\\gamma }\\bar{\\delta }}\\zeta =g^{\\alpha \\gamma }\\,g^{\\beta \\delta }\\,\\nabla ^2_{\\alpha \\beta }\\eta \\, \\nabla ^2_{\\gamma \\delta }\\eta =\\langle \\nabla ^2\\eta ,\\nabla ^2\\zeta \\rangle $ verwendet.", "Wenn wir annehmen, dass $\\zeta $ und $\\nabla \\zeta $ auf dem Rand vom $M$ verschwinden, erhalten wir durch partielle Integration, Gleichung (REF ), den $L^2$ -Gradienten der Energie $\\begin{aligned}dK(\\eta )\\,\\zeta &=2\\,K(\\eta ,\\zeta )\\\\&= \\int _M \\Big (\\epsilon _0\\,\\Big (\\frac{16\\lambda \\mu }{\\lambda +2\\mu } H^2 + 4\\mu \\, |h|^2\\Big )\\,\\eta \\\\&\\hspace{36.98866pt}+ \\frac{\\epsilon _0^3}{3}\\, \\frac{4\\lambda \\mu }{\\lambda +2\\mu }(\\Delta ^2 \\eta -|h|^2\\ \\Delta \\eta - \\Delta (|h|^2\\eta ) -\\eta \\, |h|^4)\\\\&\\hspace{36.98866pt} + \\frac{\\epsilon _0^3}{3}\\,4\\mu \\,( \\Delta ^2\\eta + \\nabla ^*[\\Delta ,\\nabla ]\\eta -\\langle \\nabla ^2\\eta ,k\\rangle -(\\nabla ^*)^2 (k\\,\\eta ) - \\eta \\,|k|^2)\\Big )\\, \\zeta \\ dA\\\\&=:(\\operatorname{grad}_{L^2}K(\\eta ),\\zeta )_{L^2}.\\end{aligned}$ Zum Beispiel erhalten wir durch jeweils zweimalige Verwendung von (REF )$\\operatorname{tr}_g$ bezeichnet hier die Spurbildung im Tensorsinne, nicht im Sinne von Randwerten.", "$\\begin{aligned}\\int _M \\Delta \\eta \\,\\Delta \\zeta \\ dA&=\\int _M \\Delta \\eta \\, \\operatorname{tr}_g\\nabla ^2\\zeta \\ dA=-\\int _M\\langle \\nabla \\Delta \\eta ,\\nabla \\zeta \\rangle \\ dA\\\\&=\\int _M\\operatorname{tr}_g\\nabla ^2\\Delta \\eta \\, \\zeta \\ dA=\\int _M\\Delta ^2\\eta \\, \\zeta \\ dA\\end{aligned}$ und $\\begin{aligned}\\int _M \\langle \\nabla ^2\\eta ,\\nabla ^2\\zeta \\rangle \\ dA=-\\int _M\\langle \\Delta \\nabla \\eta ,\\nabla \\zeta \\rangle \\ dA&=-\\int _M\\langle \\nabla \\Delta \\eta + [\\Delta ,\\nabla ]\\eta ,\\nabla \\zeta \\rangle \\ dA\\\\&=\\int _M(\\Delta ^2\\eta + \\nabla ^*[\\Delta ,\\nabla ]\\eta )\\, \\zeta \\ dA.\\end{aligned}$ Der Kommutator l\"asst sich in lokalen Koordinaten in der Form $([\\Delta ,\\nabla ]\\eta )_\\alpha = (2 H\\, h_{\\alpha }^\\beta - k_\\alpha ^\\beta )\\, \\partial _\\beta \\eta $ schreiben; siehe (REF ), (REF ).", "Insbesondere ist er ein Differentialoperator erster Ordnung, der im Falle $h=0$ verschwindet.", "Wir nehmen nun an, dass die Massenverteilung der Schale durch eine konstante Fl\"achenmassendichte $\\epsilon _0\\rho _S$ der Mittelfl\"ache $M$ beschrieben werden kann.", "Das Hamilton'sche Prinzip besagt, dass die Bewegung der Schale, hier gegeben durch ein zeitabh\"angiges skalares Feld $\\eta $ , ein station\"arer Punkt des Wirkungsintegrals $\\mathcal {A}(\\eta )=\\int _I \\epsilon _0\\rho _S\\int _M \\frac{(\\partial _t\\eta (t,\\cdot ))^2}{2}\\ dA-K(\\eta (t,\\cdot ))\\ dt$ ist, wobei $I:=(0,T)$ , $T>0$ .", "Der zeitliche Integrand ist die Differenz von kinetischer und potentieller Energie der Schale.", "Die Ableitung von $\\mathcal {A}$ an der Stelle $\\eta $ in Richtung eines skalaren Feldes $\\zeta $ , das zusammen mit $\\nabla \\zeta $ auf dem Rand von $I\\times M$ identisch 0 sei, ist durch $\\begin{aligned}d\\mathcal {A}(\\eta )\\,\\zeta =\\int _I\\epsilon _0\\rho _S\\int _M \\partial _t\\eta (t,\\cdot )\\, \\partial _t\\zeta (t,\\cdot )\\ dA-dK(\\eta (t,\\cdot ))\\,\\zeta (t,\\cdot ) \\ dt\\end{aligned}$ gegeben.", "Da die erste Variation verschwinden soll, erhalten wir mittels partieller Integration in der Zeit und (REF ) die Gleichung $\\begin{aligned}0&=\\epsilon _0\\rho _S\\,\\partial ^2_t\\eta +\\operatorname{grad}_{L^2}K(\\eta )=\\epsilon _0\\rho _S\\,\\partial ^2_t\\eta +\\epsilon _0^3\\,\\frac{8\\mu (\\lambda +\\mu )}{3(\\lambda +2\\mu )}\\,\\Delta ^2\\eta +B\\eta \\text{ in} I\\times M.\\end{aligned}$ Hierbei ist $B$ ein Operator zweiter Ordnung, der im Falle $h=0$ verschwindet.", "Es liegt also eine Verallgemeinerung der linearen Kirchhoff-Love-Plattengleichung f\"ur transversale Auslenkungen vor; vgl.", "zum Beispiel [12].", "Man sollte sich klar machen, dass diese Gleichung dispersiv, aber nicht hyperbolisch ist.", "Der Hauptteil faktorisiert in zwei Schr\"odinger-Operatoren $\\partial ^2_t +\\partial ^4_x=(i\\partial _t+\\partial ^2_x)(-i\\partial _t+\\partial ^2_x);$ im Gegensatz zum d'Alembert-Operator, der in zwei Transportoperatoren faktorisiert $\\partial ^2_t - \\partial ^2_x=(\\partial _t+\\partial _x)(\\partial _t-\\partial _x).$ Die Gleichung l\"asst mithin eine unendliche Ausbreitungsgeschwindigkeit zu.", "Der Einfachheit halber setzen wir fortan $\\epsilon _0\\rho _S=1$ ." ], [ "Problemstellung", "Wir \"ubernehmen die Bezeichnungen aus den vorangegangenen Abschnitten.", "Insbesondere sei $\\Omega \\subset \\mathbb {R}^3$ ein beschr\"anktes, nichtleeres Gebiet mit $C^4$ -Rand.", "Der Rand $\\partial \\Omega $ stelle die Mittelfl\"ache einer elastischen Schale in ihrer Ruhelage dar, deren Bewegung auf Auslenkungen l\"angs der \"au\"seren Einheitsnormale $\\nu $ von $\\partial \\Omega $ eingeschr\"ankt sei.", "$\\Gamma $ sei eine Vereinigung von Gebieten in $\\partial \\Omega $ mit $C^{1,1}$ -Rand, die nichttrivialen Schnitt mit allen Zusammenhangskomponenten von $\\partial \\Omega $ habe.", "Diesen Teil der Mittelfl\"ache setzen wir als fixiert voraus.", "Mit $M$ wollen wir den beweglichen Teil bezeichnen, d.h. $M:=\\partial \\Omega \\setminus \\Gamma $ .", "Die Mannigfaltigkeit $M$ ist kompakt und berandet.", "Desweiteren sei $I:=(0,T)$ , $T>0$ .", "Die zeitabh\"angige Auslenkung des Gebiets beschreiben wir durch eine Funktion $\\eta : \\bar{I}\\times M\\rightarrow (-\\kappa ,\\kappa )$ , die wir (wie immer) durch 0 auf $\\bar{I}\\times \\partial \\Omega $ fortsetzen.", "Desweiteren nehmen wir an, dass das Innere des variablen Gebiets $\\Omega _{\\eta }^I$ durch ein inkompressibles, viskoses Fluid bef\"ullt sei, dessen Geschwindigkeitsfeld ${\\bf u}$ und Druckfeld $\\pi $ durch die Navier-Stokes-Gleichungen beschrieben werden kann, d.h. es gelte $\\begin{aligned}\\rho _F\\big (\\partial _t {\\bf u}+ ({\\bf u}\\cdot \\nabla ){\\bf u}\\big ) &= \\operatorname{div}(2\\sigma D{\\bf u}- \\pi \\operatorname{id}) + {\\bf f}&&\\mbox{ in}\\Omega _{\\eta }^I, \\\\\\operatorname{div}{\\bf u}&= 0 &&\\mbox{ in } \\Omega _{\\eta }^I,\\\\{\\bf u}(\\,\\cdot \\,,\\,\\cdot \\, + \\eta \\,\\nu ) &= \\partial _t\\eta \\,\\nu &&\\mbox{ auf } I\\times M,\\\\{\\bf u}&= 0 &&\\mbox{ auf } I\\times \\Gamma .\\end{aligned}$ Die Bezeichnung $\\operatorname{id}$ steht dabei f\"ur die $3\\times 3$ -Einheitsmatrix, und ${\\bf f}$ ist eine gegebene \"au\"sere Kraftdichte.", "Zudem ist $({\\bf u}\\cdot \\nabla ){\\bf u}:=u^i\\,\\partial _i{\\bf u}.$ Die konstante Dichte $\\rho _F$ und die konstante Viskosit\"at $\\sigma $ setzen wir der Einfachheit halber fortan identisch 1.", "$2D{\\bf u}- \\pi \\operatorname{id}$ ist dann der Spannungstensor, und $2D{\\bf u}$ ist der viskose Spannungstensor.", "Aufgrund der Divergenzfreiheit von ${\\bf u}$ gilt $\\operatorname{div}2D{\\bf u}=\\Delta {\\bf u}$ .", "(REF )$_{3,4}$ ist die no-slip-Bedingung im Falle eines sich bewegenden Randes, d.h. die Geschwindigkeit des Fluids stimmt am Rand mit der Geschwindigkeit des Randes \"uberein.", "Die vom Fluid auf die Schale ausge\"ubte Kraft ist gegeben durch die Auswertung des Spannungstensors am Rand in Richtung der inneren Normale $-\\nu _{\\eta (t)}$ , also durch $\\begin{aligned}-2D{\\bf u}(t,\\cdot )\\,\\nu _{\\eta (t)} + \\pi (t,\\cdot )\\,\\nu _{\\eta (t)}.\\end{aligned}$ In die Gleichungen f\"ur die Auslenkung der Schale ($M:=\\partial \\Omega \\setminus \\Gamma $ ) $\\begin{aligned}\\partial ^2_{t} \\eta + \\operatorname{grad}_{L^2}K(\\eta ) &= g + {\\bf F}\\cdot \\nu &&\\text{ in } I\\times M, \\\\\\eta =0,\\ \\nabla \\eta &= 0 &&\\text{ auf } I\\times \\partial M\\end{aligned}$ geht diese Kraft zus\"atzlich zu einer gegebenen \"au\"seren Kraftdichte $g$ in der Form ${\\bf F}(t,\\cdot ) = \\big (-2D{\\bf u}(t,\\cdot )\\,\\nu _{\\eta (t)} +\\pi (t,\\cdot )\\,\\nu _{\\eta (t)}\\big )\\circ \\Phi _{\\eta (t)}\\ |\\det d\\Phi _{\\eta (t)}|$ ein.", "Da die Gleichungen der Elastizit\"at in Lagrangekoordinaten, d.h. im Referenzgebiet $M$ , formuliert werden, muss die auf $\\partial \\Omega _\\eta $ registrierte Fl\"achenkraftdichte (REF ) mit der Abbildung $\\Phi _{\\eta (t)}$ verkn\"upft und mit der lokalen Fl\"achenverzerrung skaliert werden.", "Schlie\"slich geben wir noch Anfangsdaten $\\begin{aligned}\\eta (0,\\cdot )=\\eta _0,\\ \\partial _t\\eta (0,\\cdot )=\\eta _1 \\text{ und } {\\bf u}(0,\\cdot )={\\bf u}_0\\end{aligned}$ vor.", "Man beachte, dass in dieses System nicht nur der Druckgradient eingeht, wie es bei einem festen Rand der Fall ist, sondern auch der Druck selber.", "So wie der Druckgradient als Lagrange-Multiplikator bez\"uglich der Zwangsbedingung $\\operatorname{div}{\\bf u}=0$ interpretiert werden kann, l\"asst sich sein Mittelwert als der mit der Zwangsbedingung $\\int _M\\partial _t\\eta \\, \\gamma (\\eta )\\ dA=0$ assoziierte Lagrange-Multiplikator verstehen.", "Diese Zwangsbedingung folgt mittels Lemma REF aus der Divergenzfreiheit von ${\\bf u}$ und der Kopplung (REF )$_3$ .", "Wir wollen nun rein formal, d.h. unter Vernachl\"assigung von Regularit\"atsfragen, Energieabsch\"atzungen f\"ur dieses parabolisch-dispersive System herleiten.", "Die Konstruktion schwacher L\"osungen beruht wesentlich auf diesen Absch\"atzungen.", "Nach dem Noether-Theorem ist der Energiesatz eng verkn\"upft mit der zeitlichen Translationsinvarianz des physikalischen Systems.", "Wir erhalten einen lokalen Energiesatz, wenn wir die Gleichung mit dem Vektorfeld multiplizieren, das sich ergibt, wenn wir den infinitesimalen Erzeuger der zeitlichen Translationen, also $\\partial _t$ , auf die Auslenkung anwenden.", "Wir multiplizieren also Gleichung (REF )$_1$ mit ${\\bf u}$ (und sp\"ater Gleichung (REF )$_1$ mit $\\partial _t\\eta $ ).", "Da wir an einem globalen Energiesatz interessiert sind, integrieren wir die resultierende Identit\"at \"uber $\\Omega _{\\eta (t)}$ und erhalten nach partieller Integration der Terme des SpannungstensorsDer \"Ubersichtlichkeit halber unterdr\"ucken wir hier und im Folgenden in der Notation die unabh\"angigen Variablen, d.h. wir setzen ${\\bf u}={\\bf u}(t,\\cdot )$ , etc.", "$\\begin{aligned}\\int _{\\Omega _{\\eta (t)}}\\partial _t{\\bf u}\\cdot {\\bf u}\\ dx + \\int _{\\Omega _{\\eta (t)}}({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot {\\bf u}\\ &dx = -\\int _{\\Omega _{\\eta (t)}}|\\nabla {\\bf u}|^2\\ dx+ \\int _{\\Omega _{\\eta (t)}}{\\bf f}\\cdot {\\bf u}\\ dx\\\\&+\\int _{\\partial \\Omega _{\\eta (t)}\\setminus \\Gamma }(2D{\\bf u}\\, \\nu _{\\eta (t)}- \\pi \\,\\nu _{\\eta (t)})\\cdot {\\bf u}\\ dA_{\\eta (t)}.\\end{aligned}$ Aufgrund der Divergenzfreiheit von ${\\bf u}$ verschwindet das r\"aumliche Integral des Druckterms.", "Bez\"uglich des r\"aumlichen Integrals des viskosen Spannungstensors haben wir die Korn'sche Gleichung $2\\int _{\\Omega _{\\eta (t)}}D{\\bf u}:D{\\bf u}\\ dx=2\\int _{\\Omega _{\\eta (t)}}D{\\bf u}:\\nabla {\\bf u}\\ dx=\\int _{\\Omega _{\\eta (t)}}\\nabla {\\bf u}:\\nabla {\\bf u}\\ dx$ verwendet.", "F\"ur das erste Gleichheitszeichen in dieser Identit\"at ist zu beachten, dass die Kontraktion eines symmetrischen Tensors mit einem antisymmetrischen Tensor verschwindet, sodass wir $\\nabla {\\bf u}$ durch seinen symmetrischen Anteil ersetzen k\"onnen.", "F\"ur das zweite Gleichheitszeichen sei auf Bemerkung REF im Anhang verwiesen.", "Die ersten beiden Integrale in (REF ) k\"onnen wir mit Hilfe des Reynolds'schen Transporttheorems, Proposition REF , zusammenfassen, wobei hier ${\\bf v}={\\bf u}$ gilt und wir $\\xi =\\frac{|{\\bf u}|^2}{2}$ w\"ahlen.", "Partielle Integration des zweiten Integrals ergibt n\"amlich aufgrund der Divergenzfreiheit von ${\\bf u}$ $\\begin{aligned}\\int _{\\Omega _{\\eta (t)}}({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot {\\bf u}\\ dx=-\\int _{\\Omega _{\\eta (t)}}({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot {\\bf u}\\ dx +\\int _{\\partial \\Omega _{\\eta (t)}}{\\bf u}\\cdot \\nu _{\\eta (t)}|{\\bf u}|^2\\ dA_{\\eta (t)}.\\end{aligned}$ Wir erhalten dann $\\begin{aligned}\\frac{1}{2}\\frac{d}{dt} \\int _{\\Omega _{\\eta (t)}}|{\\bf u}|^2\\ dx = &-\\int _{\\Omega _{\\eta (t)}}|\\nabla {\\bf u}|^2\\ dx+ \\int _{\\Omega _{\\eta (t)}}{\\bf f}\\cdot {\\bf u}\\ dx\\\\&+\\int _{\\partial \\Omega _{\\eta (t)}\\setminus \\Gamma }(2D{\\bf u}\\, \\nu _{\\eta (t)}- \\pi \\,\\nu _{\\eta (t)})\\cdot {\\bf u}\\ dA_{\\eta (t)}.\\end{aligned}$ Multiplikation der Gleichung (REF )$_1$ mit $\\partial _t\\eta $ , Integration \"uber $M$ und partielle Integration ergibt wegen $(\\operatorname{grad}_{L^2}K(\\eta ),\\partial _t\\eta )_{L^2}=2K(\\eta ,\\partial _t\\eta )$ $\\begin{aligned}\\frac{1}{2}\\frac{d}{dt} \\int _M|\\partial _t\\eta |^2\\ dA + \\frac{d}{dt}K(\\eta ) = \\int _Mg\\, \\partial _t\\eta \\ dA + \\int _M{\\bf F}\\cdot \\nu \\, \\partial _t\\eta \\ dA.\\end{aligned}$ Durch Addition von (REF ) und (REF ) erhalten wir unter Verwendung der Definition von ${\\bf F}$ , der Randbedingung (REF )$_3$ und des Transformationssatzes den Energiesatz $\\begin{aligned}\\frac{1}{2}\\frac{d}{dt} \\int _{\\Omega _{\\eta (t)}}|{\\bf u}|^2\\ dx + \\frac{1}{2}\\frac{d}{dt} &\\int _M|\\partial _t\\eta |^2\\ dA +\\frac{d}{dt}K(\\eta )\\\\&= - \\int _{\\Omega _{\\eta (t)}}|\\nabla {\\bf u}|^2\\ dx + \\int _{\\Omega _{\\eta (t)}}{\\bf f}\\cdot {\\bf u}\\ dx + \\int _Mg\\, \\partial _t\\eta \\ dA.\\end{aligned}$ Die zeitliche \"Anderung der Gesamtenergie des Systems, zusammengesetzt aus den kinetischen Energien von Fluid und Schale sowie der potentiellen Energie der Schale, ist identisch der negativen Energiedissipation durch das Fluid und der Leistung der \"au\"seren Kr\"afte.", "Wir definieren den Term $R$ durch $\\frac{d}{dt}K(\\eta )=2\\,K(\\eta ,\\partial _t\\eta )=:\\epsilon _0^3\\,\\frac{8\\mu (\\lambda +\\mu )}{6(\\lambda +2\\mu )}\\,\\frac{d}{dt}\\int _M|\\Delta \\eta |^2\\ dA + R;$ man beachte die Identit\"at (REF ).", "Offenbar gilt $|R|\\le c\\,(\\Vert \\eta \\Vert _{H^2(M)}^2 + \\Vert \\partial _t\\eta \\Vert _{L^2(M)}^2).$ Wir erhalten somit aus (REF ) $\\begin{aligned}\\frac{d}{dt} \\int _{\\Omega _{\\eta (t)}}|u|^2\\ dx &+ \\int _{\\Omega _{\\eta (t)}}|\\nabla {\\bf u}|^2\\ dx + \\frac{d}{dt} \\int _M|\\partial _t\\eta |^2\\ dA + \\frac{d}{dt}\\int _M|\\Delta \\eta |^2\\ dA \\\\ &\\le c\\,\\big (\\Vert {\\bf f}\\Vert _{L^2(\\Omega _{\\eta (t)})}^2 +\\Vert {\\bf u}\\Vert _{L^2(\\Omega _{\\eta (t)})}^2 + \\Vert g\\Vert _{L^2(M)}^2+ \\Vert \\partial _t\\eta \\Vert _{L^2(M)}^2 + \\Vert \\eta \\Vert _{H^2(M)}^2 \\big ).\\end{aligned}$ $\\Vert \\Delta \\,\\cdot \\,\\Vert _{L^2(M)}$ definiert eine \"aquivalente Norm auf $H^2_0(M)$ ; siehe Anhang A.2.", "Mit Hilfe des Gronwall'schen Lemmas, Proposition REF , erhalten wir also $\\begin{aligned}&\\Vert {\\bf u}(t,\\cdot )\\Vert _{L^2(\\Omega _{\\eta (t)})}^2 + \\int _0^t\\Vert \\nabla {\\bf u}(s,\\cdot )\\Vert ^2_{L^2(\\Omega _{\\eta (s)})}\\ ds +\\Vert \\partial _t\\eta (t,\\cdot )\\Vert _{L^2(M)}^2 + \\Vert \\eta (t,\\cdot )\\Vert _{H^2(M)}^2 \\\\&\\hspace{113.81102pt}\\le \\big (\\Vert {\\bf u}_0\\Vert _{L^2(\\Omega _{\\eta _0})}^2 + \\Vert \\eta _1\\Vert _{L^2(M)}^2 +\\Vert \\eta _0\\Vert _{H^2(M)}^2\\big )\\,e^{ct}\\\\&\\hspace{128.0374pt} + \\int _0^t\\big (\\Vert {\\bf f}(s,\\cdot )\\Vert _{L^2(\\Omega _{\\eta (s)})}^2 + \\Vert g(s,\\cdot )\\Vert _{L^2(M)}^2\\big )\\,e^{c(t-s)}\\ ds.\\end{aligned}$ Es gilt somit $\\begin{aligned}\\Vert \\eta \\Vert _{W^{1,\\infty }(I,L^2(M))\\cap L^\\infty (I,H_0^2(M))} +\\Vert {\\bf u}\\Vert _{L^\\infty (I,L^2(\\Omega _{\\eta (t)}))\\cap L^2(I,H^1(\\Omega _{\\eta (t)}))}\\le c(T,\\text{Daten}).\\end{aligned}$ Wir werden schwache L\"osungen in dieser Regularit\"atsklasse konstruieren.", "Die Einbettung (REF ) zeigt, dass der Rand unseres zeitlich variablen Gebietes Graph einer H\"older-stetigen, aber nicht Lipschitz-stetigen Funktion sein wird." ], [ "Existenz", "Wir setzen $\\begin{aligned}Y^I&:=W^{1,\\infty }(I,L^2(M))\\cap L^\\infty (I,H_0^2(M)),\\end{aligned}$ und f\"ur $\\eta \\in Y^I$ mit $\\Vert \\eta \\Vert _{L^\\infty (I\\times M)}<\\kappa $ $\\begin{aligned}X_\\eta ^I&:=L^\\infty (I,L^2(\\Omega _{\\eta (t)}))\\cap L^2(I,H^1_{\\operatorname{div}}(\\Omega _{\\eta (t)})).\\end{aligned}$ Wir wollen nun einen geeigneten Raum von Testfunktionen $b$ auf $M$ und $\\varphi $ auf $\\Omega _\\eta ^I$ definieren.", "Nat\"urlich m\"ussen die Funktionen $\\varphi $ divergenzfrei sein, und es soll $\\operatorname{tr_\\eta }\\varphi =b\\,\\nu $ f\"ur eine Testfunktion $b$ gelten.", "\"Ublicherweise w\"urde man $b$ und $\\varphi $ hinreichend oft stetig differenzierbar w\"ahlen.", "Aufgrund der geringen Regularit\"at von $\\eta $ ist aber nicht klar, ob \"uberhaupt ein ausreichender Satz solcher Testfunktionen existiert.", "Insbesondere ist die Fortsetzung in Proposition REF auch f\"ur $C^4$ -Randwerte $b$ im Allgemeinen nicht stetig differenzierbar.", "Wir bezeichnen deshalb mit $T_\\eta ^I$ den kanonisch normierten Raum aller Tupel $(b,\\varphi )\\in \\big (H^1(I,L^2(M))\\cap L^2(I,H^2_0(M))\\big )\\times \\big (H^1(\\Omega _{\\eta }^I)\\cap L^\\infty (I,L^4(\\Omega _{\\eta (t)}))\\big )$ mit $b(T,\\cdot )=0$ , $\\varphi (T,\\cdot )=0$Die Auswertung von $\\varphi $ bei $t=T$ ist sinnvoll, weil diese Funktion in $H^1((t,T)\\times Q)$ f\"ur jeden offenen Ball $Q\\subset \\subset \\Omega _{\\eta (T)}$ und hinreichend gro\"ses $t<T$ liegt., $\\operatorname{div}\\varphi =0$Der Divergenzoperator wirkt wie immer nur auf die r\"aumlichen Koordinaten.", "und $\\varphi -\\mathcal {F}_\\eta b\\in H_0$ .", "Dabei ist $H_0$ der Abschluss in $H^1(\\Omega _{\\eta }^I)\\cap L^\\infty (I,L^4(\\Omega _{\\eta (t)}))$ der bei $t=T$ verschwindenden, divergenzfreien Elemente dieses Raums mit in $\\Omega _{\\eta }^{\\bar{I}}$ enthaltenen Tr\"agern.", "Aus der letzten Forderung folgt offenbar $\\operatorname{tr_\\eta }\\varphi =\\operatorname{tr_\\eta }F_\\eta b=b\\,\\nu $ .", "Die Umkehrung gilt zumindest dann, wenn $\\eta $ hinreichend glatt ist und wir den Testfunktionen den kleineren Raum $H^1(I,H^2_0(M))\\times H^1(I,H^1(\\Omega _{\\eta (t)}))$ zugrunde legen.Dass dieser Raum kleiner ist, ist nicht offensichtlich.", "Wir zeigen aber in Bemerkung REF , dass f\"ur $(b,\\varphi )\\in T^I_\\eta $ die Fortsetzung von $\\varphi $ durch $(b\\,\\nu )\\circ q$ in $H^1(I,L^2(B_\\alpha ))$ (f\"ur geeignetes $\\alpha $ ) liegt.", "Dasselbe Argument zeigt, dass $\\varphi $ in $H^1(I,L^4(B_\\alpha ))$ liegt, wenn wir den Testfunktionen obigen Raum zugrunde legen.", "Unter Verwendung der Abbildung ${\\eta }$ aus Bemerkung REF sehen wir n\"amlich, dass es bei hinreichend glattem $\\eta $ gen\"ugt, die analoge Situation im Raumzeitzylinder $I\\times \\Omega $ zu betrachten.", "Dort k\"onnen wir standardm\"a\"sig unter Verwendung des L\"osungsoperators der Divergenzgleichung geeignete Approximationen konstruieren; siehe zum Beispiel III.4.1 in [28].", "F\"ur jedes $t\\in \\bar{I}$ liegen diese in $C_0^\\infty (\\Omega )$ und konvergieren in $H^1(\\Omega )$ .", "Da die dortige zeitunabh\"angige Konstruktion mit der Zeitableitung kommutiert, folgt die Konvergenz in $H^1(I,H^1(\\Omega _{\\eta (t)}))$ in trivialer Weise.Diese einfache Schlussfolgerung ist f\"ur den Raum $H^1(\\Omega _{\\eta }^I)\\cap L^\\infty (I,L^4(\\Omega _{\\eta (t)}))$ nicht m\"oglich.", "Gilt lediglich $\\eta \\in Y^I$ , so ist die Abbildung $\\eta $ nicht mehr anwendbar.", "Die Verwendung des L\"osungsoperators der Divergenzgleichung in $\\Omega _\\eta ^I$ ist dann aber auch problematisch, weil dessen \"ubliche Stetigkeitseigenschaften in Gebieten, deren Rand nicht Lipschitz-stetig ist, im Allgemeinen nicht gelten; vgl.", "zum Beispiel [1].", "M\"oglicherweise ist die Umkehrung in diesem Fall falsch; vgl.", "die Diskussion in Abschnitt III.4 in [28].", "Wie immer setzen wir auf $M$ definierte Funktionen stillschweigend durch 0 auf $\\partial \\Omega $ fort.", "Die Bedingung $\\operatorname{tr_\\eta }\\varphi = b\\,\\nu $ besagt dann insbesondere, dass $\\varphi $ auf $\\Gamma $ verschwindet.", "Wir haben durch ein lokales Argument gezeigt, dass die Auswertung von $\\varphi $ bei $t=T$ sinnvoll ist.", "Dieses Argument zeigt allgemeiner, dass die Auswertung bei $t\\in \\bar{I}$ in $L^2_{\\text{loc}}(\\Omega _{\\eta (t)})$ liegt.", "F\"ur die nachfolgende Definition ben\"otigen wir jedoch $\\varphi (0,\\cdot )\\in L^2(\\Omega _{\\eta _0})$ .", "Die G\"ultigkeit dieser Aussage sehen wir ein, indem wir $\\varphi $ durch $(b\\,\\nu )\\circ q$ auf $I\\times B_\\alpha $ , $\\Vert \\eta \\Vert _{L^\\infty (I\\times M)}<\\alpha <\\kappa $ , fortsetzen und zeigen, dass diese Fortsetzung in $H^1(I,L^2(B_\\alpha ))$ liegt.", "Insbesondere ist die Auswertung von $\\varphi $ bei fester Zeit $t\\in \\bar{I}$ sinnvoll, und es gilt $\\varphi (t,\\cdot )\\in L^2(\\Omega _{\\eta (t)})$ .", "Die Details finden sich in Bemerkung REF im Anhang.", "Felder ${\\bf f}\\in L_{\\text{loc}}^{2}([0,\\infty )\\times \\mathbb {R}^3)$ , $g\\in L_{\\text{loc}}^2([0,\\infty )\\times M)$ , $\\eta _0\\in H^2_0(M)$ mit $\\Vert \\eta _0\\Vert _{L^\\infty (M)}<\\kappa $ , $\\eta _1\\in L^2(M)$ und ${\\bf u}_0\\in L^2(\\Omega _{\\eta _0})$ mit $\\operatorname{div}{\\bf u}_0=0$ sowie $\\operatorname{tr^n_{\\eta _0}}{\\bf u}_0=\\eta _1\\,\\gamma (\\eta _0)$ nennen wir zul\"assige Daten.", "Definition 5.1 Ein Tupel $(\\eta ,{\\bf u})$ hei\"st schwache L\"osung von (REF ), (REF ) und (REF ) mit den zul\"assigen Daten $({\\bf f},g,{\\bf u}_0,\\eta _0,\\eta _1)$ auf dem Intervall $I$ , falls $\\eta \\in Y^I$ mit $\\Vert \\eta \\Vert _{L^\\infty (I\\times M)}<\\kappa $ und $\\eta (0,\\cdot )=\\eta _0$ , ${\\bf u}\\in X_\\eta ^I$ mit $\\operatorname{tr_\\eta }{\\bf u}=\\partial _t\\eta \\,\\nu $ und $\\begin{aligned}&- \\int _I\\int _{\\Omega _{\\eta (t)}}{\\bf u}\\cdot \\partial _t\\varphi \\ dxdt - \\int _I\\int _M(\\partial _t\\eta )^2\\, b\\, \\gamma (\\eta )\\ dAdt+\\int _I\\int _{\\Omega _{\\eta (t)}}({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot \\varphi \\ dxdt\\\\ &+ \\int _I\\int _{\\Omega _{\\eta (t)}}\\nabla {\\bf u}:\\nabla \\varphi \\ dxdt-\\int _I\\int _M\\partial _t\\eta \\, \\partial _tb\\ dAdt + 2\\int _I K(\\eta ,b)\\ dt \\\\&\\hspace{28.45274pt}=\\int _I\\int _{\\Omega _{\\eta (t)}}{\\bf f}\\cdot \\varphi \\ dxdt + \\int _I\\int _Mg\\, b\\ dAdt+\\int _{\\Omega _{\\eta _0}}{\\bf u}_0\\cdot \\varphi (0,\\cdot )\\ dx + \\int _M\\eta _1\\, b(0,\\cdot )\\ dA\\end{aligned}$ f\"ur alle Testfunktionen $(b,\\varphi )\\in T_\\eta ^I$ .", "Gleichung (REF ) ergibt sich formal durch Multiplikation von (REF ) mit einer Testfunktion $\\varphi $ , Integration \"uber Ort und Zeit, partielle Integration und Verwenden von (REF ).", "Genauer wird der Spannungstensor im Ort partiell integriert, Bemerkung REF verwendet, das auftretende Randintegral mittels (REF )$_1$ ersetzt und die Identit\"at $(\\operatorname{grad}_{L^2}K(\\eta ),b)_{L^2}=2K(\\eta ,b)$ ausgenutzt.", "Desweiteren werden die Terme mit den ersten Zeitableitungen von ${\\bf u}$ und den zweiten Zeitableitungen von $\\eta $ zeitlich partiell integriert.", "Bei dem ${\\bf u}$ -Term tritt dabei ein Randintegral auf, das sich mit Hilfe des Reynolds'schen Transporttheorems berechnen l\"asst.", "Mit ${\\bf v}={\\bf u}=(\\partial _t\\eta \\nu )\\circ \\Phi _{\\eta (t)}^{-1}$ und $\\xi ={\\bf u}\\cdot \\varphi $ erhalten wir $\\begin{aligned}\\frac{d}{dt}\\int _{\\Omega _{\\eta (t)}}{\\bf u}\\cdot \\varphi \\ dx &= \\int _{\\Omega _{\\eta (t)}}\\partial _t{\\bf u}\\cdot \\varphi \\ dx + \\int _{\\Omega _{\\eta (t)}}{\\bf u}\\cdot \\partial _t\\varphi \\ dx\\\\&\\hspace{14.22636pt}+\\int _{\\partial \\Omega _{\\eta (t)}}{\\bf u}\\cdot \\varphi \\ \\big ((\\partial _t\\eta \\,\\nu )\\circ \\Phi _{\\eta (t)}^{-1}\\big )\\cdot \\nu _{\\eta (t)}\\ dA_{\\eta (t)}.\\end{aligned}$ Gem\"a\"s Bemerkung REF ist das Randintegral identisch $\\int _M(\\partial _t\\eta )^2\\, b\\, \\gamma (\\eta )\\ dA,$ sodass sich (REF ) ergibt.", "Wir wollen uns noch davon \"uberzeugen, dass der dritte Term in (REF ) wohldefiniert ist.", "Es ist nicht schwer zu sehen, dass die Fortsetzung von $\\varphi $ durch $(b\\,\\nu )\\circ q$ in $H^1(I\\times B_\\alpha )$ , $\\Vert \\eta \\Vert _{L^\\infty (I\\times M)}<\\alpha <\\kappa $ , liegt, sodass aus Proposition REF folgt, dass $\\varphi \\in L^\\infty (I,L^3(\\Omega _{\\eta (t)}))$ gilt.", "W\"are ${\\bf u}\\in L^2(I,L^6(\\Omega _{\\eta (t)}))$ , so w\"urde dies gen\"ugen, damit $({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot \\varphi $ in $L^1(\\Omega _\\eta ^I)$ liegt.", "Aufgrund der geringen Randregularit\"at gilt aber lediglich ${\\bf u}\\in L^2(I,L^r(\\Omega _{\\eta (t)}))$ f\"ur alle $r<6$ ; siehe Korollar REF .", "Deshalb ist die zus\"atzliche Bedingung $\\varphi \\in L^\\infty (I,L^4(\\Omega _{\\eta (t)}))$ n\"otig.", "Bemerkung 5.2 F\"ur jede schwache L\"osung $(\\eta ,{\\bf u})$ , alle Testfunktionen $(b,\\varphi )\\in T^I_\\eta $ und fast alle $t\\in I$ gilt $\\begin{aligned}&- \\int _{0}^t\\int _{\\Omega _{\\eta (s)}}{\\bf u}\\cdot \\partial _t\\varphi \\ dxds - \\int _{0}^t\\int _M(\\partial _t\\eta )^2\\, b\\, \\gamma (\\eta )\\ dAds + \\int _{0}^t\\int _{\\Omega _{\\eta (s)}}({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot \\varphi \\ dxds\\\\& + \\int _{0}^t\\int _{\\Omega _{\\eta (s)}}\\nabla {\\bf u}:\\nabla \\varphi \\ dxds-\\int _{0}^t\\int _M\\partial _t\\eta \\, \\partial _tb\\ dAds + 2\\int _{0}^t K(\\eta ,b)\\ ds \\\\&\\hspace{42.67912pt}=\\int _{0}^t\\int _{\\Omega _{\\eta (s)}}{\\bf f}\\cdot \\varphi \\ dxds + \\int _{0}^t\\int _Mg\\, b\\ dAds +\\int _{\\Omega _{\\eta _0}}{\\bf u}_0\\cdot \\varphi (0,\\cdot )\\ dx\\\\&\\hspace{56.9055pt}+ \\int _M\\eta _1\\, b(0,\\cdot )\\ dA-\\int _{\\Omega _{\\eta (t)}}{\\bf u}(t,\\cdot )\\cdot \\varphi (t,\\cdot )\\ dx -\\int _M\\partial _t\\eta (t,\\cdot )\\, b(t,\\cdot )\\ dA.\\end{aligned}$ Tats\"achlich k\"onnen wir auf die Forderung $b(T,\\cdot )=0$ , $\\varphi (T,\\cdot )=0$ verzichten.", "Um die Identit\"at (REF ) einzusehen, verwenden wir n\"amlich in (REF ) die Testfunktion $(b\\,\\rho ^t_\\epsilon ,\\varphi \\,\\rho ^t_\\epsilon )$ , wobei $\\rho \\in C^\\infty (\\mathbb {R})$ , $\\rho (s)=1$ f\"ur $s\\le 0$ und $\\rho (s)=0$ f\"ur $s\\ge 1$ sowie $\\rho ^t_\\epsilon (s)=\\rho (\\epsilon ^{-1}(s-t))$ .", "Dann gilt $\\begin{aligned}-\\int _I\\int _M\\partial _t\\eta \\, &\\partial _s(b\\rho ^t_\\epsilon )\\ dAds\\\\&= -\\int _I\\rho ^t_\\epsilon \\int _M\\partial _t\\eta \\, \\partial _tb\\ dAds-\\int _I\\epsilon ^{-1}\\rho ^{\\prime }(\\epsilon ^{-1}(s-t))\\int _M\\partial _t\\eta \\, b\\ dAds\\\\&\\rightarrow -\\int _0^t\\int _M\\partial _t\\eta \\, \\partial _tb\\ dAds+\\int _M\\partial _t\\eta (t,\\cdot )\\, b(t,\\cdot )\\ dA\\end{aligned}$ f\"ur $\\epsilon \\rightarrow 0$ und fast alle $t\\in I$ .", "Die Konvergenz des ersten Terms folgt aus dem Satz \"uber die dominierte Konvergenz, w\"ahrend wir f\"ur die Konvergenz des zweiten Terms den Lebesgue'schen Differentiationssatz, der in Proposition REF enthalten ist, und die Identit\"at $\\int _\\mathbb {R}\\rho ^{\\prime }\\ ds=-1$ verwendet haben.", "Analog l\"asst sich die Konvergenz der anderen Terme in (REF ) einsehen, sodass wir (REF ) erhalten.", "Wir kommen nun zum zentralen Existenzresultat dieser Arbeit.", "Theorem 5.3 F\"ur beliebige zul\"assige Daten $({\\bf f},g,{\\bf u}_0,\\eta _0,\\eta _1)$ existieren ein $T^*\\in (0,\\infty ]$ und ein Tupel $(\\eta ,{\\bf u})$ derart, dass $(\\eta ,{\\bf u})$ f\"ur alle $T<T^*$ eine schwache L\"osung von (REF ), (REF ) und (REF ) auf dem Intervall $I=(0,T)$ ist und die Absch\"atzungGenau genommen verwenden wir auf $X_\\eta ^I$ die \"aquivalente Norm $\\Vert \\,\\cdot \\,\\Vert _{L^\\infty (I,L^2(\\Omega _{\\eta (t)}))} +\\Vert \\nabla \\,\\cdot \\,\\Vert _{L^2(\\Omega _{\\eta }^I)}$ .", "$\\begin{aligned}&\\Vert \\eta \\Vert _{Y^I}^2 + \\Vert {\\bf u}\\Vert _{X_\\eta ^I}^2\\le \\big (\\Vert {\\bf u}_0\\Vert _{L^2(\\Omega _{\\eta _0})}^2 +\\Vert \\eta _1\\Vert _{L^2(M)}^2 +\\Vert \\eta _0\\Vert _{H^2(M)}^2\\big )\\,e^{cT}\\\\&\\hspace{128.0374pt} + \\int _0^T\\big (\\Vert {\\bf f}(t,\\cdot )\\Vert _{L^2(\\Omega _{\\eta (t)})}^2 + \\Vert g(t,\\cdot )\\Vert _{L^2(M)}^2\\big )\\,e^{c(T-t)}\\ dt\\\\\\end{aligned}$ erf\"ullt.", "Entweder gilt $T^*=\\infty $ oder $\\lim _{t\\rightarrow T^*}\\Vert \\eta (t,\\cdot )\\Vert _{L^\\infty (M)}=\\kappa $ .", "Die rechte Seite von (REF ) als Funktion von $T$ , $\\Omega _\\eta ^I$ und den Daten wird im Folgenden wiederholt auftreten und deshalb kurz mit $c_0(T,\\Omega _\\eta ^I,{\\bf f},g,{\\bf u}_0,\\eta _0,\\eta _1)$ bezeichnet.", "Ein wichtiger Schritt im Beweis von Theorem REF ist der Nachweis der relativen Kompaktheit beschr\"ankter Folgen approximativer L\"osungen in $L^2$ .", "Im Falle der Navier-Stokes-Gleichungen auf einem Raumzeitzylinder folgt diese sofort aus dem Satz von Aubin-Lions, Proposition REF .", "In diesem Fall l\"asst sich n\"amlich direkt aus der Gleichung ablesen, dass die L\"osungen beschr\"ankte Zeitableitungen in $L^{4/3}(I,(H^{1}_{0,\\operatorname{div}})^{\\prime })$ besitzen.", "Auf den ersten Blick scheint eine \"ahnliche Aussage auch f\"ur unser gekoppeltes System zu gelten.", "Betrachten wir (REF ) mit $b(0,\\cdot )=0$ und $\\varphi (0,\\cdot )=0$ , so erhalten wir $\\begin{aligned}- \\int _I\\int _{\\Omega _{\\eta (t)}}&{\\bf u}\\cdot \\partial _t\\varphi \\ dxdt - \\int _I\\int _M(\\partial _t\\eta )^2\\, b\\, \\gamma (\\eta )\\ dAdt -\\int _I\\int _M\\partial _t\\eta \\, \\partial _tb\\ dAdt\\\\& = -\\int _I\\int _{\\Omega _{\\eta (t)}}({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot \\varphi \\ dxdt -\\int _I\\int _{\\Omega _{\\eta (t)}}\\nabla {\\bf u}:\\nabla \\varphi \\ dxdt- 2\\int _I K(\\eta ,b)\\ dt\\\\&\\hspace{14.22636pt} +\\int _I\\int _{\\Omega _{\\eta (t)}}{\\bf f}\\cdot \\varphi \\ dxdt + \\int _I\\int _Mg\\, b\\ dAdt.\\end{aligned}$ Es ist nun naheliegend, die rechte Seite als Definition der Zeitableitung des Tupels $(\\partial _t\\eta ,{\\bf u})$ zu interpretieren und zu folgern, dass diese durch die r\"aumlichen Regularit\"aten der L\"osung beschr\"ankt in einem geeigneten Dualraum liegt.", "Dieser Idee eine rigorose Bedeutung zu geben, ist allerdings schwierig.", "Erstens haben wir bislang kein Substitut f\"ur Bochner-R\"aume von Funktionen mit Werten in Dualr\"aumen konstruiert.", "Zweitens ist unklar, in welchem konkreten Sinne man \"uberhaupt von einer Zeitableitung des Tupels $(\\partial _t\\eta ,{\\bf u})$ sprechen k\"onnte.", "Vollends un\"ubersichtlich wird die Situation schlie\"slich durch den Umstand, dass die involvierten Funktionenr\"aume im Allgemeinen mit der L\"osung variieren und dadurch Folgen von (Dual-)R\"aumen auftreten.", "Angesichts dieser Schwierigkeiten wird in [11], [32] ein anderer Zugang gew\"ahlt; das System wird mit zeitlichen Differenzen(quotienten) getestet.", "Die Konstruktion der richtigen Testfunktionen ist allerdings subtil, da die L\"osung zum Zeitpunkt $t_0$ im Allgemeinen keine zul\"assige Testfunktion zum Zeitpunkt $t_1\\ne t_0$ ist.", "Auch die Analyse der resultierenden Identit\"at ist nicht einfach und ben\"otigt viele Seiten langwieriger Absch\"atzungen.", "Im ersten Anlauf [11] erforderte dieser Zugang die Einf\"uhrung eines D\"ampfungsterms.", "In unserem Fall einer allgemeinen Schalengeometrie ist die Konstruktion der richtigen Testfunktionen nochmals deutlich schwieriger und bislang daran gescheitert, dass, wie bereits angemerkt, die \"ublichen Stetigkeitseigenschaften des L\"osungsoperators der Divergenzgleichung in Gebieten, deren Rand nicht Lipschitz-stetig ist, nicht gelten.", "Wir wollen uns deshalb erneut dem nat\"urlicher erscheinenden Zugang \"uber die Zeitableitung zuwenden.", "Man k\"onnte auf die Idee kommen, die Kompaktheit der Fluidgeschwindigkeit mit Hilfe des Satzes von Aubin-Lions zun\"achst lokal, d.h. in kleinen Raumzeitzylindern, zu zeigen, und dann \"uber Interpolation und Spurbildung auf die globale Kompaktheit der Fluidgeschwindigkeit und die Kompaktheit der Schalengeschwindigkeit zu schlie\"sen. Durch die Inkompressibilit\"at und die resultierende unendliche Schallgeschwindigkeit ist unser System jedoch hochgradig nichtlokal, sodass lokale Argumente an dieser Stelle vermutlich nicht zum Erfolg f\"uhren k\"onnen.", "Wir m\"ussen das Argument \"uber die Zeitableitung also global durchf\"uhren.", "Das ist tats\"achlich m\"oglich, und der im Folgenden dargelegte Zugang ist au\"serdem direkter und k\"urzer als der in [11], [32].", "Wir wollen den Beweis des Satzes von Aubin-Lions auf unsere Situation \"uber\"-tragen.", "Aus diesem Grund skizzieren wir nun die Beweisvariante, die in [4] zu finden ist.", "Diese basiert auf dem Satz von Arzela-Ascoli und einem Interpolationsresultat.", "Die \"Ubertragung anderer Beweisvarianten, siehe zum Beispiel [46], ist zwar auch m\"oglich, jedoch scheint die Variante in [4] am besten geeignet zu sein.", "Proposition 5.4 Es seien $I\\subset \\mathbb {R}$ ein offenes, beschr\"anktes Intervall und $1\\le p\\le \\infty $ , $1<r\\le \\infty $ .", "F\"ur die Banach-R\"aume $B_0$ , $B$ und $B_1$ gelte $B_0\\hookrightarrow \\hookrightarrow B\\hookrightarrow B_1.$ Dann ist die Einbettung $W:=\\big \\lbrace v\\in L^p(I,B_0)\\ \\big |\\ v^{\\prime }\\in L^r(I,B_1)\\big \\rbrace \\hookrightarrow L^p(I,B)$ kompakt.", "Beweisskizze: Die Folge $(v_n)\\subset W$ sei beschr\"ankt.", "Es gen\"ugt zu zeigen, dass eine Teilfolge in $C(\\bar{I},B_1)$ konvergiert, denn aus dem Ehrling-Lemma erhalten wir f\"ur jedes $\\epsilon >0$ und $n,m\\in \\mathbb {N}$ die Absch\"atzung $\\Vert v_n-v_m\\Vert _{L^p(I,B)}\\le \\epsilon \\,\\Vert v_n-v_m\\Vert _{L^p(I,B_0)} + c(\\epsilon )\\,\\Vert v_n-v_m\\Vert _{L^p(I,B_1)}.$ Offenbar liegt $(v_n)$ beschr\"ankt in $C^{0,1-1/r}(\\bar{I},B_1)$ .", "Der Satz von Arzela-Ascoli liefert somit die Konvergenz einer Teilfolge in $C(\\bar{I},B_1)$ , wenn wir zeigen k\"onnen, dass f\"ur alle $t$ aus einer dichten Teilmenge von $I$ die Folge $(v_n(t))_n$ relativ kompakt in $B_1$ liegt.", "Das folgt aber aus den Einbettungen $\\begin{aligned}W\\hookrightarrow C(\\bar{I},(B_0,B_1)_{\\theta ,1/\\theta })\\ \\text{ und }\\ (B_0,B_1)_{\\theta ,1/\\theta }\\hookrightarrow \\hookrightarrow B_1\\end{aligned}$ f\"ur ein geeignetes $0<\\theta <1$ ; siehe Theorem 33 in [22] und Theorem 3.8.1 in [8].", "$\\Box $ Weder wollen wir versuchen, ein Substitut f\"ur den Raum $L^r(I,B_1)$ zu konstruieren, noch werden wir den Begriff der Zeitableitung des Tupels $(\\partial _t\\eta ,{\\bf u})$ konkretisieren.", "Stattdessen werden wir direkt mit der schwachen Formulierung, d.h. mit der getesteten formalen Zeitableitung, arbeiten und dabei stets zeigen, dass die jeweiligen Aussagen unabh\"angig von den Testfunktionen gelten, solange diese in geeigneten Normen beschr\"ankt bleiben.", "Die bereits angesprochene Problematik des Auftretens von Folgen formaler Dualr\"aume bekommen wir durch Verwenden zweier spezieller Familien von Testfunktionen in den Griff.", "Nat\"urlich m\"ussen wir auch den Beweis des Satzes von Arzela-Ascoli auf die vorliegende Situation \"ubertragen.", "Das Analogon der relativen Kompaktheit der Folgen $(v_n(t))_n$ in $B_1$ erhalten wir hingegen sehr einfach aus der Schranke f\"ur die kinetischen Energien.", "Die Aussage der folgenden Proposition wird im Beweis von Theorem REF nicht verwendet.", "Jedoch ist der Beweis der Proposition fast w\"ortlich auf die im Beweis von Theorem REF auftretenden Situationen \"ubertragbar.", "Proposition 5.5 Es seien $({\\bf f},g,{\\bf u}_0^n,\\eta _0^n,\\eta _1^n)$ eine Folge zul\"assiger Daten mit $\\begin{aligned}\\sup _n\\big (\\tau (\\eta _0^n)+\\Vert \\eta _0^n\\Vert _{H^2_0(M)}+\\Vert \\eta _1^n\\Vert _{L^2(M)}+\\Vert {\\bf u}^n_0\\Vert _{L^2(\\Omega _ {\\eta _0^n})}\\big )<\\infty \\end{aligned}$ und $(\\eta _n,{\\bf u}_n)$ eine Folge schwacher L\"osungen von (REF ), (REF ) und (REF ) mit den Daten $({\\bf f},g,{\\bf u}_0^n,\\eta _0^n,\\eta _1^n)$ auf dem Intervall $I=(0,T)$ mit F\"ur zeitabh\"angige Funktionen $\\eta $ ersetzen wir in der Definition von $\\tau (\\eta )$ die Norm $\\Vert \\eta \\Vert _{L^\\infty (\\partial \\Omega )}$ durch $\\Vert \\eta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}$ .", "$\\begin{aligned}\\sup _n\\big (\\tau (\\eta _n) + \\Vert \\eta _n\\Vert _{Y^I} +\\Vert {\\bf u}_n\\Vert _{X_{\\eta _n}^I}\\big )<\\infty .\\end{aligned}$ Dann liegt die Folge $(\\partial _t\\eta _n,{\\bf u}_n)$ relativ kompakt in $L^2(I\\times M)\\times L^2(I\\times \\mathbb {R}^3)$ .Wie immer setzen wir die Felder ${\\bf u}_n$ durch 0 auf $I\\times \\mathbb {R}^3$ fort.", "Beweis: Wegen (REF ) erhalten wir f\"ur eine TeilfolgeHier und im Rest der Arbeit werden wir bei der Auswahl von Teilfolgen stets stillschweigend bez\"uglich aller beteiligten Folgen zu derselben Teilfolge \"ubergehen und diese wieder mit $n$ indizieren.", "die Konvergenzen $\\begin{aligned}\\eta _n&\\rightarrow \\eta &&\\text{ schwach$^*$ in }L^\\infty (I,H^2_0(M))\\text{ und gleichm\"a\"sig},\\\\\\partial _t\\eta _n&\\rightarrow \\partial _t\\eta &&\\text{ schwach$^*$ in }L^\\infty (I,L^2(M)),\\\\{\\bf u}_n&\\rightarrow {\\bf u}&&\\text{ schwach$^*$ in } L^\\infty (I,L^2(\\mathbb {R}^3)),\\\\\\nabla {\\bf u}_n&\\rightarrow \\xi &&\\text{ schwach in } L^2(I\\times \\mathbb {R}^3),\\end{aligned}$ wobei wir die zun\"achst auf $\\Omega _{\\eta _n}^I$ definierten Felder $\\nabla {\\bf u}_n$ durch 0 auf $I\\times \\mathbb {R}^3$ fortsetzen.", "Wie man leicht \"uberpr\"uft ist der Grenzwert $\\xi $ nichts anderes als das Feld $\\nabla {\\bf u}$ , wenn wir dieses ebenfalls durch 0 fortsetzen.", "Die gleichm\"a\"sige Konvergenz der Folge $(\\eta _n)$ folgt aus der kompakten Einbettung $(1/2<\\theta <1)$ $Y^I\\hookrightarrow C^{0,1-\\theta }(\\bar{I}, C^{0,2\\theta -1}(\\partial \\Omega )) \\hookrightarrow \\hookrightarrow C(\\bar{I}\\times \\partial \\Omega ).$ K\"onnen wir die Konvergenz $\\begin{aligned}\\int _I\\int _{\\Omega _{\\eta _n(t)}}|{\\bf u}_n|^2\\ dxdt + \\int _I\\int _M|\\partial _t\\eta _n|^2\\ dAdt\\rightarrow \\int _I\\int _{\\Omega _{\\eta (t)}}|{\\bf u}|^2\\ dxdt + \\int _I\\int _M|\\partial _t\\eta |^2\\ dAdt\\end{aligned}$ zeigen, so folgt die Behauptung der Proposition mit Hilfe der Konvergenzen (REF ) und der trivialen Identit\"at $\\begin{aligned}&\\int _I\\int _{\\mathbb {R}^3} |{\\bf u}_n-{\\bf u}|^2\\ dxdt + \\int _I\\int _M|\\partial _t\\eta _n-\\partial _t\\eta |^2\\ dAdt \\\\& = \\int _I\\int _{\\Omega _{\\eta _n(t)}} |{\\bf u}_n|^2\\ dxdt + \\int _I\\int _M|\\partial _t\\eta _n|^2\\ dAdt +\\int _I\\int _{\\Omega _{\\eta (t)}}|{\\bf u}|^2\\ dxdt + \\int _I\\int _M|\\partial _t\\eta |^2\\ dAdt\\\\&\\hspace{14.22636pt} - 2\\int _I\\int _{\\mathbb {R}^3} {\\bf u}_n\\cdot {\\bf u}\\ dxdt -2\\int _I\\int _M\\partial _t\\eta _n\\ \\partial _t\\eta \\ dAdt.\\end{aligned}$ (REF ) wiederum ist eine Konsequenz der Konvergenzen $\\begin{aligned}\\int _I\\int _{\\Omega _{\\eta _n(t)}}{\\bf u}_n\\cdot \\mathcal {F}_{\\eta _n}\\partial _t\\eta _n\\ dxdt+\\int _I\\int _M|\\partial _t\\eta _n|^2\\ dAdt\\\\\\rightarrow \\int _I\\int _{\\Omega _{\\eta (t)}}{\\bf u}\\cdot \\mathcal {F}_{\\eta }\\partial _t\\eta \\ dxdt+ \\int _I\\int _M|\\partial _t\\eta |^2\\ dAdt\\end{aligned}$ und $\\begin{aligned}\\int _I\\int _{\\Omega _{\\eta _n(t)}} {\\bf u}_n\\cdot ({\\bf u}_n -\\mathcal {F}_{\\eta _n}\\partial _t\\eta _n)\\ dxdt \\rightarrow \\int _I \\int _{\\Omega _{\\eta (t)}} {\\bf u}\\cdot ({\\bf u}-\\mathcal {F}_{\\eta }\\partial _t\\eta )\\ dxdt.\\end{aligned}$ Die Zahl $\\alpha $ in der Definition der Fortsetzungsoperatoren $\\mathcal {F}$ , siehe Proposition REF , gen\"uge dabei der Ungleichung $\\sup _n\\Vert \\eta _n\\Vert _{L^\\infty (I\\times M)}<\\alpha <\\kappa $ .", "Die Beweise von (REF ) und (REF ) verlaufen sehr \"ahnlich.", "Wir beginnen mit (REF ).", "Eine beliebige Funktion $b\\in H^2_0(M)$ verletzt im Allgemeinen die Mittelwertbedingung (REF ) bez\"uglich $\\eta _n(t,\\cdot )$ und ist somit nicht divergenzfrei auf $\\Omega _{\\eta _n(t)}$ fortsetzbar.", "Aus diesem Grund ben\"otigen wir die Operatoren $\\mathcal {M}_{\\eta _n}$ aus Lemma REF .", "Aus diesem Lemma und Proposition REF folgt unter Beachtung von (REF ) die Absch\"atzung $\\begin{aligned}\\Vert \\mathcal {M}_{\\eta _n}b\\Vert _{H^1(I,L^2(M))\\cap L^2(I,H^2_0(M))}+\\Vert \\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b\\Vert _{H^1(I,L^2(B_\\alpha ))\\cap C(\\bar{I},H^1(B_\\alpha ))}\\le c\\,\\Vert b\\Vert _{H^2_0(M)}.\\end{aligned}$ Betrachten wir die Identit\"at (REF ) mit den L\"osungen $(\\eta _n,{\\bf u}_n)$ und den Testfunktionen $(\\mathcal {M}_{\\eta _n}b,\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)\\in T_{\\eta _n}^I$ , so folgern wir aus (REF ) und (REF ), dass die Integranden der auftretenden zeitlichen Integrale unabh\"angig von $b$ und $n$ in $L^{12/11}(I)$ beschr\"ankt sind, sofern $\\Vert b\\Vert _{H^2_0(M)}\\le 1$ gilt.Anstelle von \"`unabh\"angig von $b$ , sofern $\\Vert b\\Vert _{H^2_0(M)}\\le 1$ \"' schreiben wir fortan abk\"urzend \"`unabh\"angig von $\\Vert b\\Vert _{H^2_0(M)}\\le 1$ \"'.", "Nur f\"ur den Wirbelterm ist diese Behauptung nicht offensichtlich.", "F\"ur diesen folgt sie aus der Absch\"atzung $\\begin{aligned}\\big \\Vert \\int _{\\Omega _{\\eta _n(t)}}({\\bf u}_n\\cdot \\nabla ){\\bf u}_n&\\cdot \\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b\\ dx\\big \\Vert _{L^{12/11}(I)}\\\\&\\le \\Vert {\\bf u}_n\\Vert _{L^{12/5}(I,L^4(\\Omega _{\\eta _n(t)}))}\\,\\Vert \\nabla {\\bf u}_n\\Vert _{L^{2}(\\Omega _{\\eta _n}^I)}\\,\\Vert \\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b\\Vert _{L^{\\infty }(I,L^4(\\Omega _{\\eta _n(t)}))}\\end{aligned}$ unter Beachtung der Sobolev-Einbettung $H^1(\\Omega _{\\eta _n(t)})\\hookrightarrow L^5(\\Omega _{\\eta _n(t)}),$ die gleichm\"a\"sig in $n$ und $t$ gilt, siehe Korollar REF , und der Interpolationseinbettung ($\\theta =5/6$ ) $L^\\infty (I,L^2(\\Omega _{\\eta _n(t)}))\\cap L^2(I,L^5(\\Omega _{\\eta _n(t)}))\\hookrightarrow L^{12/5}(I,L^4(\\Omega _{\\eta _n(t)})).$ Letztere ist eine einfache Folgerung aus der H\"older'schen Ungleichung; vgl.", "Proposition REF .", "Die ersten acht Summanden der Identit\"at (REF ) sind somit unabh\"angig von $\\Vert b\\Vert _{H^2_0(M)}\\le 1$ und $n$ in $C^{0,1/12}(\\bar{I})$ beschr\"ankt.", "Dieselbe Aussage gilt mithin f\"ur die letzten beiden Summanden dieser Identit\"at, d.h. f\"ur die Funktionen $c_{b,n}(t):=\\int _{\\Omega _{\\eta _n(t)}}{\\bf u}_n(t,\\cdot )\\cdot (\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)(t,\\cdot )\\ dx +\\int _M\\partial _t\\eta _n(t,\\cdot )\\ (\\mathcal {M}_{\\eta _n}b)(t,\\cdot )\\ dA,$ da der neunte und zehnte Summand wegen (REF ) unabh\"angig von $\\Vert b\\Vert _{H^2_0(M)}\\le 1$ und $n$ beschr\"ankte reelle Zahlen sind.", "Aufgrund der Konvergenzen (REF ) und Lemma REF $(1.a)$ , $(2.a)$ konvergiert die Folge $(c_{b,n})_n$ f\"ur festes $b\\in H^2_0(M)$ im Distributionssinne gegen die Funktion $c_{b}(t):= \\int _{\\Omega _{\\eta (t)}}{\\bf u}(t,\\cdot )\\cdot (\\mathcal {F}_\\eta \\mathcal {M}_\\eta b)(t,\\cdot )\\ dx +\\int _M\\partial _t\\eta (t,\\cdot )\\, (\\mathcal {M}_\\eta b)(t,\\cdot )\\ dA.$ Eine Anwendung des Satzes von Arzela-Ascoli zeigt, dass $(c_{b,n})_n$ sogar gleichm\"a\"sig in $\\bar{I}$ gegen $c_b$ konvergiert.", "Wir wollen nun zeigen, dass diese gleichm\"a\"sige Konvergenz unabh\"angig von $\\Vert b\\Vert _{H^2_0(M)}\\le 1$ ist, d.h. dass die Funktionen $\\begin{aligned}&h_n(t):=&\\sup _{\\Vert b\\Vert _{H^2_0(M)}\\le 1} \\big (c_{b,n}(t)-c_{b}(t)\\big )\\end{aligned}$ gleichm\"a\"sig in $\\bar{I}$ gegen 0 konvergieren.", "Wegen (REF ) gilt f\"ur fast alle $t\\in I$ $\\sup _n \\big (\\Vert {\\bf u}_n(t)\\Vert _{L^2(\\mathbb {R}^3)}+\\Vert \\partial _t\\eta _n(t)\\Vert _{L^2(M)}\\big )<\\infty .$ Mit Hilfe eines Diagonalfolgenarguments schlie\"sen wir, dass eine abz\"ahlbare, dichte Teilmenge $I_0$ von $I$ und eine Teilfolge von $(\\eta _n,{\\bf u}_n)$ existieren derart, dass $({\\bf u}_n(t,\\cdot ))_n$ und $(\\partial _t\\eta _n(t,\\cdot ))_n$ f\"ur alle $t\\in I_0$ schwach in $L^2(\\mathbb {R}^3)$ bzw.", "$L^2(M)$ konvergieren.", "Wir zeigen nun, dass die Folge $(c_{b,n}(t))_n$ f\"ur festes, aber beliebiges $t\\in I_0$ unabh\"angig von $\\Vert b\\Vert _{H^2_0(M)}\\le 1$ konvergiert.", "Bezeichnet $\\eta ^*$ den schwachen Grenzwert von $(\\partial _t\\eta _n(t,\\cdot ))_n$ in $L^2(M)$ , so folgt diese Behauptung f\"ur den zweiten Summanden in $c_{b,n}(t)$ aus der Absch\"atzung $\\begin{aligned}&\\big |\\int _M\\partial _t\\eta _n(t,\\cdot )\\, (\\mathcal {M}_{\\eta _n} b)(t,\\cdot )-\\eta ^*\\,(\\mathcal {M}_\\eta b)(t,\\cdot )\\ dA\\big |\\\\&\\hspace{142.26378pt}\\le \\big |\\int _M\\big (\\partial _t\\eta _n(t,\\cdot )-\\eta ^*\\big )\\, (\\mathcal {M}_{\\eta _n} b)(t,\\cdot )\\ dA\\big |\\\\&\\hspace{156.49014pt}+ \\big |\\int _M\\eta ^*\\, \\big ((\\mathcal {M}_{\\eta _n}b)(t,\\cdot )-(\\mathcal {M}_{\\eta }b)(t,\\cdot )\\big )\\ dA\\big |\\\\&\\hspace{142.26378pt}\\le \\Vert \\partial _t\\eta _n(t,\\cdot )-\\eta ^*\\Vert _{(H^1(M))^{\\prime }}\\,\\Vert (\\mathcal {M}_{\\eta _n}b)(t,\\cdot )\\Vert _{H^1(M)}\\\\&\\hspace{156.49014pt}+\\Vert \\eta ^*\\Vert _{L^2(M)}\\Vert (\\mathcal {M}_{\\eta _n}b)(t,\\cdot )-(\\mathcal {M}_{\\eta }b)(t,\\cdot )\\Vert _{L^2(M)}\\end{aligned}$ unter Beachtung von (REF ), Einbettung (REF ), Lemma REF $(1.a)$ und der kompakten Einbettung $L^2(M)\\hookrightarrow \\hookrightarrow (H^1(M))^{\\prime }.$ Letztere folgt unter Verwendung des Satzes von Schauder \"uber die Kompaktheit adjungierter Operatoren aus den \"ublichen Sobolev-Einbettungen.", "Bezeichnet ${\\bf u}^*$ den schwachen Grenzwert von $({\\bf u}_n(t,\\cdot ))_n$ in $L^2(\\mathbb {R}^3)$ , so folgt die Behauptung f\"ur den zweiten Summanden in $c_{b,n}(t)$ vollkommen analog aus der Absch\"atzung $\\begin{aligned}&\\big |\\int _{\\Omega _{\\eta _n(t)}} {\\bf u}_n(t,\\cdot )\\cdot (\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)(t,\\cdot )\\ dx-\\int _{\\Omega _{\\eta (t)}} {\\bf u}^*\\cdot (\\mathcal {F}_{\\eta }\\mathcal {M}_{\\eta }b)(t,\\cdot )\\ dx\\big |\\\\&\\hspace{142.26378pt}\\le \\big |\\int _{B_\\alpha }\\big ({\\bf u}_n(t,\\cdot )-{\\bf u}^*\\big )\\cdot (\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)(t,\\cdot )\\ dx\\big |\\\\&\\hspace{156.49014pt}+\\big |\\int _{B_\\alpha }{\\bf u}^*\\cdot \\big ((\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)(t,\\cdot )-(\\mathcal {F}_{\\eta }\\mathcal {M}_{\\eta }b)(t,\\cdot )\\big )\\ dx\\big |\\\\&\\hspace{142.26378pt}\\le \\Vert {\\bf u}_n(t,\\cdot )-{\\bf u}^*\\Vert _{(H^1(B_\\alpha ))^{\\prime }}\\,\\Vert (\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)(t,\\cdot )\\Vert _{H^1(B_\\alpha )}\\\\&\\hspace{156.49014pt}+\\Vert {\\bf u}^*\\Vert _{L^2(B_\\alpha )}\\Vert (\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)(t,\\cdot )-(\\mathcal {F}_{\\eta }\\mathcal {M}_{\\eta }b)(t, \\cdot )\\Vert _{L^2(B_\\alpha )}\\end{aligned}$ unter Beachtung von (REF ), Lemma REF $(2.a)$ und der kompakten Einbettung $\\begin{aligned}L^2(B_\\alpha )\\hookrightarrow \\hookrightarrow (H^1(B_\\alpha ))^{\\prime }.\\end{aligned}$ Wir zeigen schlie\"slich, dass die von $\\Vert b\\Vert _{H^2_0(M)}\\le 1$ unabh\"angige Konvergenz der Folge $(c_{b,n}(t))_n$ nicht nur f\"ur $t\\in I_0$ , sondern sogar gleichm\"a\"sig f\"ur alle $t\\in \\bar{I}$ gilt.", "Aufgrund der gleichm\"a\"sigen Beschr\"anktheit von $(c_{b,n})$ in $C^{0,1/12}(\\bar{I})$ gilt f\"ur alle $t,t^{\\prime }\\in \\bar{I}$ und $n,m\\in \\mathbb {N}$ $\\begin{aligned}|c_{b,n}(t)-c_{b,m}(t)|&\\le |c_{b,n}(t)-c_{b,n}(t^{\\prime })| + |c_{b,n}(t^{\\prime })-c_{b,m}(t^{\\prime })| +|c_{b,m}(t^{\\prime })-c_{b,m}(t)|\\\\&\\le c\\,|t-t^{\\prime }|^{1/12} + |c_{b,n}(t^{\\prime })-c_{b,m}(t^{\\prime })|.\\end{aligned}$ Zu einem gegebenen $\\epsilon >0$ k\"onnen wir nun eine endliche Menge $I_0^\\epsilon \\subset I_0$ finden derart, dass zu jedem $t\\in \\bar{I}$ ein $t^{\\prime }\\in I_0^\\epsilon $ existiert mit $c\\,|t-t^{\\prime }|^{1/12}<\\epsilon /2.$ Zudem haben wir gerade gezeigt, dass $|c_{b,n}(t^{\\prime })-c_{b,m}(t^{\\prime })|<\\epsilon /2$ gilt, falls $t^{\\prime }\\in I_0^\\epsilon $ und $n,m\\ge N$ , wobei $N$ von $\\epsilon $ , aber nicht von $t^{\\prime }$ und $\\Vert b\\Vert _{H^2_0(M)}\\le 1$ abh\"angt.", "Somit konvergiert die Folge $(c_{b,n})_n$ gegen $c_b$ gleichm\"a\"sig in $\\bar{I}$ unabh\"angig von $\\Vert b\\Vert _{H^2_0(M)}\\le 1$ , d.h. die Folge $(h_n)$ konvergiert gleichm\"a\"sig in $\\bar{I}$ gegen 0.", "Mit der Definition $\\begin{aligned}&g_n(t):=\\sup _{\\Vert b\\Vert _{L^2(M)}\\le 1}\\big (c_{b,n}(t)-c_{b}(t)\\big )\\end{aligned}$ existiert wegen Lemma REF , einer Aussage vom Typ des Ehrling-Lemmas, f\"ur jedes $\\epsilon >0$ eine Konstante $c(\\epsilon )$ derart, dass $\\begin{aligned}\\int _I g_n(t)\\ dt \\le \\epsilon \\,c\\,\\big (\\Vert {\\bf u}_n\\Vert _{L^2(I,H^1(\\Omega _{\\eta _n(t)}))}+\\Vert {\\bf u}\\Vert _{L^2(I,H^1(\\Omega _{\\eta (t)}))}\\big ) + c(\\epsilon )\\int _I h_n(t)\\ dt\\end{aligned}$ gilt.", "Aufgrund der gleichm\"a\"sigen Konvergenz von $(h_n)$ gegen 0 folgt $\\begin{aligned}\\lim _n \\int _I g_n(t)\\ dt = 0.\\end{aligned}$ Durch Nulladdition erhalten wir die Identit\"at $\\begin{aligned}& \\int _{\\Omega _{\\eta _n(t)}}{\\bf u}_n(t,\\cdot )\\cdot (\\mathcal {F}_{\\eta _n}\\partial _t\\eta _n)(t,\\cdot )\\ dx+\\int _M|\\partial _t\\eta _n(t,\\cdot )|^2\\ dA\\\\& -\\int _{\\Omega _{\\eta (t)}}{\\bf u}(t,\\cdot )\\cdot (\\mathcal {F}_{\\eta }\\partial _t\\eta )(t,\\cdot )\\ dx-\\int _M|\\partial _t\\eta (t,\\cdot )|^2\\ dA\\\\&\\hspace{28.45274pt}= \\int _{\\Omega _{\\eta _n(t)}}{\\bf u}_n(t,\\cdot )\\cdot (\\mathcal {F}_{\\eta _n}\\partial _t\\eta _n)(t,\\cdot )\\ dx +\\int _M\\partial _t\\eta _n(t,\\cdot )\\, \\partial _t\\eta _n(t,\\cdot )\\ dA\\\\&\\hspace{42.67912pt}-\\int _{\\Omega _{\\eta (t)}}{\\bf u}(t,\\cdot )\\cdot (\\mathcal {F}_{\\eta }\\mathcal {M}_{\\eta }\\partial _t\\eta _n)(t,\\cdot )\\ dx -\\int _M\\partial _t\\eta (t,\\cdot )\\,(\\mathcal {M}_{\\eta }\\partial _t\\eta _n)(t,\\cdot )\\ dA\\\\&\\hspace{42.67912pt}+ \\int _{\\Omega _{\\eta (t)}}{\\bf u}(t,\\cdot )\\cdot \\big ((\\mathcal {F}_{\\eta }\\mathcal {M}_{\\eta }\\partial _t\\eta _n)(t,\\cdot )-\\mathcal {F}_{\\eta }\\partial _t\\eta )(t,\\cdot )\\big )\\ dx\\\\&\\hspace{42.67912pt} + \\int _M\\partial _t\\eta (t,\\cdot )\\,\\big ((\\mathcal {M}_\\eta \\partial _t\\eta _n)(t,\\cdot )-\\partial _t\\eta (t,\\cdot )\\big )\\ dA.\\end{aligned}$ Aufgrund der Gleichung $\\mathcal {M}_{\\eta _n}\\partial _t\\eta _n=\\partial _t\\eta _n$ sind die ersten beiden Zeilen der rechten Seite identisch $c_{b,n}(t)-c_{b}(t)$ mit $b=\\partial _t\\eta _n(t,\\cdot )$ .", "Wegen (REF ) ist ihr Betrag somit f\"ur fast alle $t$ durch $c\\,g_n(t)$ abgesch\"atzt.", "Integrieren wir (REF ) \"uber $I$ und verwenden (REF ) sowie die aus der schwachen Konvergenz von $(\\partial _t\\eta _n)$ in $L^2(I\\times M)$ resultierendenEin linearer Operator zwischen Banach-R\"aumen ist genau dann stetig bez\"uglich der Normtopologien, wenn er stetig ist bez\"uglich der schwachen Topologien.", "schwachen Konvergenzen von $(\\mathcal {M}_\\eta \\partial _t\\eta _n)$ und $(\\mathcal {F}_\\eta \\mathcal {M}_\\eta \\partial _t\\eta _n)$ in $L^2(I\\times M)$ bzw.", "$L^2(I\\times B_\\alpha )$ , so erhalten wir (REF ); man beachte dabei $\\mathcal {M}_{\\eta }\\partial _t\\eta =\\partial _t\\eta $ .", "Wir kommen nun zum Beweis von (REF ).", "Es sei $\\sigma >0$ hinreichend klein und $\\delta _\\sigma \\in C^4(\\bar{I}\\times \\partial \\Omega )$ mit $\\Vert \\delta _\\sigma -\\eta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}<\\sigma $ und $\\delta _\\sigma <\\eta $ in $\\bar{I}\\times \\partial \\Omega $ .", "F\"ur $\\varphi \\in H(\\Omega )$ setzen wir $c_{\\varphi ,n}^\\sigma (t):=\\int _{\\Omega _{\\eta _n(t)}}{\\bf u}_n(t,\\cdot )\\cdot {\\delta _\\sigma (t)}\\varphi \\ dx,\\quad c_{\\varphi }^\\sigma (t):=\\int _{\\Omega _{\\eta _n(t)}}{\\bf u}(t,\\cdot )\\cdot {\\delta _\\sigma (t)}\\varphi \\ dx;$ siehe Bemerkung REF .", "Wir k\"onnen nun wie gehabt vorgehen, um zu zeigen, dass die Funktionen $h_n^\\sigma (t):=\\sup _{\\Vert \\varphi \\Vert _{H^1_{0,\\operatorname{div}}(\\Omega )}\\le 1} \\big (c_{\\varphi ,n}^\\sigma (t)-c_{\\varphi }^\\sigma (t)\\big )$ gleichm\"a\"sig in $\\bar{I}$ gegen 0 konvergieren.", "Ist $\\varphi \\in H^1_{0,\\operatorname{div}}(\\Omega )$ und setzen wir das zun\"achst auf $\\Omega _{\\delta _\\sigma }^I$ definierte Feld ${\\delta _\\sigma }\\varphi $ durch 0 auf $I\\times B_\\alpha $ fort, so gilt $\\begin{aligned}\\Vert {\\delta _\\sigma }\\varphi \\Vert _{H^1(I,L^2(B_\\alpha ))\\cap C(\\bar{I},H^1(B_\\alpha ))}\\le c\\,\\Vert \\varphi \\Vert _{H^1_{0,\\operatorname{div}}(\\Omega )}.\\end{aligned}$ Das folgt unter Beachtung der verschwindenden Randwerte von ${\\delta _\\sigma }\\varphi $ aus den entsprechenden Absch\"atzungen auf $\\Omega _{\\delta _\\sigma }^I$ ; siehe Bemerkung REF .", "Die Identit\"at (REF ) mit den L\"osungen $(\\eta _n,{\\bf u}_n)$ und den Testfunktionen $(0,{\\delta _\\sigma }\\varphi )\\in T^I_{\\eta _n}$ ($n$ hinreichend gro\"s) zeigt unter Beachtung von (REF ) und (REF ), dass die Funktionen $c_{\\varphi ,n}^\\sigma $ unabh\"angig von $\\Vert \\varphi \\Vert _{H^1_{0,\\operatorname{div}}(\\Omega )}\\le 1$ und $n$ in $C^{0,1/12}(\\bar{I})$ beschr\"ankt sind.", "Aus den Konvergenzen (REF ) und Arzela-Ascoli erhalten wir wiederum die gleichm\"a\"sige Konvergenz in $\\bar{I}$ von $(c_{\\varphi ,n}^\\sigma )_n$ gegen $c_{\\varphi }^\\sigma $ .", "Die kompakte Einbettung (REF ) zeigt zudem, dass die Folge $(c_{\\varphi ,n}^\\sigma (t))_n$ f\"ur $t\\in I_0$ unabh\"angig von $\\Vert \\varphi \\Vert _{H^1_{0,\\operatorname{div}}(\\Omega )}\\le 1$ konvergiert.", "Wie zuvor schlie\"sen wir daraus, dass $(h_n^\\sigma )_n$ gleichm\"a\"sig gegen 0 konvergiert.", "Unter Verwendung von Lemma REF folgt $\\begin{aligned}\\lim _n\\int _I g^\\sigma _n(t)\\ dt=0,\\end{aligned}$ wobei $g_n^\\sigma (t):=\\sup _{\\Vert \\varphi \\Vert _{H(\\Omega )}\\le 1} \\big (c_{\\varphi ,n}^\\sigma (t)-c_{\\varphi }^\\sigma (t)\\big ).$ Die $L^2(\\Omega _{\\eta _n(t)})$ -Normen der Felder ${\\bf u}_n(t,\\cdot )-(\\mathcal {F}_{\\eta _n}\\partial _t\\eta _n)(t,\\cdot )\\in H(\\Omega _{\\eta _n(t)})$ sind wegen (REF ) f\"ur fast alle $t$ unabh\"angig von $t$ und $n$ beschr\"ankt.", "Gem\"a\"s Lemma REF existiert also f\"ur jedes $\\epsilon > 0$ ein $\\sigma >0$ derart, dass sich f\"ur fast alle $t$ und alle hinreichend gro\"sen $n$ Felder $\\psi _{t,n}\\in H(\\Omega _{\\eta _n(t)})$ mit unabh\"angig von $t$ und $n$ beschr\"ankten $L^2(\\Omega _{\\eta _n(t)})$ -Normen, $\\operatorname{supp}\\psi _{t,n}\\subset \\Omega _{\\delta _\\sigma (t)}$ und $\\begin{aligned}\\Vert {\\bf u}_n(t,\\cdot )-(\\mathcal {F}_{\\eta _n}\\partial _t\\eta _n)(t,\\cdot ) -\\psi _{t,n}\\Vert _{(H^{1/4}(\\mathbb {R}^3))^{\\prime }} <\\epsilon \\end{aligned}$ finden lassen; insbesondere gilt $\\psi _{t,n}\\in H(\\Omega _{\\delta _\\sigma (t)})$ mit unabh\"angig von $t$ und $n$ beschr\"ankter Norm.", "Durch Nulladdition erhalten wir die Identit\"at $\\begin{aligned}& \\int _{\\Omega _{\\eta _n(t)}} {\\bf u}_n(t,\\cdot )\\cdot \\big ({\\bf u}_n(t,\\cdot ) -(\\mathcal {F}_{\\eta _n}\\partial _t\\eta _n)(t,\\cdot )\\big )\\ dx -\\int _{\\Omega _{\\eta (t)}} {\\bf u}(t,\\cdot )\\cdot \\big ({\\bf u}(t,\\cdot ) - (\\mathcal {F}_{\\eta }\\partial _t\\eta )(t,\\cdot )\\big )\\ dx\\\\&= \\int _{\\Omega _{\\eta _n(t)}} {\\bf u}(t,\\cdot )\\cdot \\big ({\\bf u}_n(t,\\cdot ) -(\\mathcal {F}_{\\eta _n}\\partial _t\\eta _n)(t,\\cdot )\\big )\\, dx -\\int _{\\Omega _{\\eta (t)}} {\\bf u}(t,\\cdot )\\cdot \\big ({\\bf u}(t,\\cdot ) - (\\mathcal {F}_{\\eta }\\partial _t\\eta )(t,\\cdot )\\big )\\,dx\\\\&\\hspace{14.22636pt} + \\int _{\\Omega _{\\eta _n(t)}} {\\bf u}_n(t,\\cdot )\\cdot \\psi _{t,n}\\ dx -\\int _{\\Omega _{\\eta (t)}} {\\bf u}(t,\\cdot )\\cdot \\psi _{t,n}\\ dx\\\\&\\hspace{14.22636pt}+ \\int _{\\Omega _{\\eta _n(t)}} {\\bf u}_n(t,\\cdot )\\cdot \\big ({\\bf u}_n(t,\\cdot ) -(\\mathcal {F}_{\\eta _n}\\partial _t\\eta _n)(t,\\cdot )-\\psi _{t,n}\\big )\\ dx\\\\&\\hspace{14.22636pt} - \\int _{\\Omega _{\\eta (t)}} {\\bf u}(t,\\cdot )\\cdot \\big ({\\bf u}_n(t,\\cdot ) -(\\mathcal {F}_{\\eta _n}\\partial _t\\eta _n)(t,\\cdot )-\\psi _{t,n}\\big )\\ dx.\\end{aligned}$ Der Betrag der zweiten Zeile der rechten Seite ist durch $c\\,g_n^\\sigma (t)$ abgesch\"atzt, w\"ahrend die Betr\"age der letzten beiden Zeilen durch $\\epsilon \\, ( \\Vert {\\bf u}_n(t,\\cdot )\\Vert _{H^{1/4}(\\mathbb {R}^3)} +\\Vert {\\bf u}(t,\\cdot )\\Vert _{H^{1/4}(\\mathbb {R}^3)})$ dominiert sind; man beachte Proposition REF .", "Integrieren wir (REF ) \"uber $I$ und verwenden (REF ) sowie die aus (REF ) und Lemma REF $(2.c)$ folgende schwache Konvergenz der Folge $(\\chi _{\\Omega ^I_{\\eta _n}}({\\bf u}_n - \\mathcal {F}_{\\eta _n}\\partial _t\\eta _n))$ gegen $\\chi _{\\Omega ^I_{\\eta }}({\\bf u}-F_{\\eta }\\partial _t\\eta )$ in $L^2(I\\times \\mathbb {R}^3)$ , so erhalten wir (REF ).", "$\\Box $ Beim Versuch der Konstruktion einer L\"osung st\"o\"st man auf das Problem, dass der Definitionsbereich der L\"osung von der L\"osung selber abh\"angt.", "Das macht die Verwendung eines Galerkin-Ansatzes zun\"achst unm\"oglich, da die Ansatzfunktionen ebenso von der L\"osung abh\"angen w\"urden.", "Diesen Zirkel wollen wir durch ein Fixpunktargument aufbrechen.", "Da wir keine perturbative Aussage, sondern die Existenz zeitlich globaler L\"osungen zeigen m\"ochten, wird uns ein topologisches Resultat, eine Variante des Schauder'schen Fixpunktsatzes, weiterhelfen.", "Naheliegend mag nun folgendes Vorgehen sein.", "Wir geben eine Bewegung $\\delta $ des Randes vor und l\"osen die Fluidgleichungen auf dem resultierenden variablen Gebiet unter Beachtung der no-slip-Bedingung (REF )$_{3,4}$ ; vgl.", "zum Beispiel [27].", "Anschlie\"send l\"osen wir die Schalengleichung (REF ) mit der vom Fluid auf den Rand ausge\"ubten Kraft auf der rechten Seite und erhalten eine Randbewegung $\\eta $ .", "Schlie\"slich versuchen wir zu zeigen, dass die Abbildung $\\delta \\mapsto \\eta $ einen Fixpunkt besitzt.", "Der Haken an dieser Idee ist, dass wir globale, starke L\"osungen konstruieren m\"ussen, damit die Auswertung des Spannungstensors auf dem Rand sinnvoll ist.", "Das ist m\"oglich, wenn wir den Wirbelterm regularisieren.", "Da wir aber letztlich an schwachen L\"osungen interessiert sind, wollen wir uns diesen Mehraufwand sparen.", "Zudem ist fraglich, ob die auf diese Weise gewonnene Funktion $\\delta \\mapsto \\eta $ eine geeignete beschr\"ankte Menge in sich abbildet, weil die Bewegung des Randes nat\"urlich in die Energieabsch\"atzungen der L\"osungen eingeht.", "Es ist mithin geschickter, eine schwache Formulierung zu verwenden, die die Kopplung an die Wellengleichung bereits enth\"alt.", "Diese \"Uberlegung f\"uhrt uns auf die Aufgabe, zu geeignetem gegebenem $\\delta $ mit $\\delta (0,\\cdot )=\\eta _0$ Funktionen $\\eta \\in Y^I$ mit $\\Vert \\eta \\Vert _{L^\\infty (I\\times M)}<\\kappa $ und $\\eta (0,\\cdot )=\\eta _0$ sowie ${\\bf u}\\in X_\\delta ^I$ mit $\\operatorname{tr_{\\delta }}{\\bf u}=\\partial _t\\eta \\,\\nu $ zu finden, die der Gleichung $\\begin{aligned}&- \\int _I\\int _{\\Omega _{\\delta (t)}} {\\bf u}\\cdot \\partial _t\\varphi \\ dxdt -\\int _I\\int _M\\partial _t\\eta \\,\\partial _t\\delta \\, b\\, \\gamma (\\delta )\\ dAdt+\\int _I\\int _{\\Omega _{\\delta (t)}}({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot \\varphi \\ dxdt\\\\&+\\int _I\\int _{\\Omega _{\\delta (t)}} \\nabla {\\bf u}:\\nabla \\varphi \\ dxdt-\\int _I\\int _M\\partial _t\\eta \\, \\partial _tb\\ dAdt + 2\\int _I K(\\eta ,b)\\ dt \\\\&\\hspace{28.45274pt}=\\int _I\\int _{\\Omega _{\\delta (t)}} {\\bf f}\\cdot \\varphi \\ dxdt + \\int _I\\int _Mg\\, b\\ dAdt+\\int _{\\Omega _{\\eta _0}}{\\bf u}_0\\cdot \\varphi (0,\\cdot )\\ dx + \\int _M\\eta _1\\, b(0,\\cdot )\\ dA\\end{aligned}$ f\"ur alle Testfunktionen $(b,\\varphi )\\in T_\\delta ^I$ gen\"ugen.", "Um die Energieabsch\"atzungen zu bewahren, muss hier allerdings noch der Wirbelterm abge\"andert werden.", "Wenn wir n\"amlich obige Gleichung formal mit $(\\partial _t\\eta ,{\\bf u})$ testen, sehen wir (vgl.", "(REF )), dass das \"`erste ${\\bf u}$ \"' im Term $\\int _I\\int _{\\Omega _{\\delta (t)}}({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot {\\bf u}\\ dxdt$ am Rand mit der Geschwindigkeit des Randes übereinstimmen m\"usste.", "Wir k\"onnen dieses Problem beheben, indem wir in (REF ) den Wirbelterm durch partielle Integration umformen: $\\begin{aligned}\\int _I\\int _{\\Omega _{\\eta (t)}}({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot \\varphi \\ dxdt =\\frac{1}{2}\\int _I\\int _{\\Omega _{\\eta (t)}}({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot \\varphi \\ dxdt &-\\frac{1}{2}\\int _I\\int _{\\Omega _{\\eta (t)}}({\\bf u}\\cdot \\nabla )\\varphi \\cdot {\\bf u}\\ dxdt\\\\&+\\frac{1}{2}\\int _I\\int _M(\\partial _t\\eta )^2\\,b\\,\\gamma (\\eta )\\ dAdt.\\end{aligned}$ Wenn wir die gegebene Randbewegung $\\delta $ nach dieser Umformung einf\"uhren, erhalten wir anstelle von (REF ) die Gleichung $\\begin{aligned}&- \\int _I\\int _{\\Omega _{\\delta (t)}} {\\bf u}\\cdot \\partial _t\\varphi \\ dxdt -\\frac{1}{2}\\int _I\\int _M\\partial _t\\eta \\,\\partial _t\\delta \\, b\\,\\gamma (\\delta )\\ dAdt\\\\&+\\frac{1}{2}\\int _I\\int _{\\Omega _{\\delta (t)}}({\\bf u}\\cdot \\nabla ){\\bf u}\\cdot \\varphi \\ dxdt -\\frac{1}{2}\\int _I\\int _{\\Omega _{\\delta (t)}}({\\bf u}\\cdot \\nabla )\\varphi \\cdot {\\bf u}\\ dxdt\\\\&+\\int _I\\int _{\\Omega _{\\delta (t)}} \\nabla {\\bf u}:\\nabla \\varphi \\ dxdt -\\int _I\\int _M\\partial _t\\eta \\,\\partial _t b\\ dAdt +2\\int _I K(\\eta ,b)\\ dt \\\\&\\hspace{128.0374pt}=\\int _I\\int _{\\Omega _{\\delta (t)}} {\\bf f}\\cdot \\varphi \\ dxdt + \\int _I\\int _Mg\\ b\\ dAdt\\\\&\\hspace{142.26378pt} +\\int _{\\Omega _{\\eta _0}}{\\bf u}_0\\cdot \\varphi (0,\\cdot )\\ dx + \\int _M\\eta _1\\, b(0,\\cdot )\\ dA.\\end{aligned}$ Wenn wir diese formal mit $(\\partial _t\\eta ,{\\bf u})$ testen, heben sich die beiden \"`Wirbelterme\"' gegenseitig weg, w\"ahrend sich die ersten beiden Terme wie zuvor zu $\\int _I\\frac{1}{2}\\frac{d}{dt}\\int _{\\Omega _{\\delta (t)}}|{\\bf u}|^2\\ dxdt$ zusammenfassen lassen.", "Schlie\"slich m\"ussen wir noch eine Regularisierung von (REF ) vornehmen.", "Zun\"achst ben\"otigen wir eine Regularisierung der Randbewegung $\\delta $ , weil diese in einem Raum liegen muss, in den $Y^I$ kompakt einbettet, und deshalb nicht hinreichend regul\"ar sein kann.", "Zum Beispiel w\"aren die Ansatzfunktionen, die wir in K\"urze konstruieren werden, ohne eine Regularisierung von $\\delta $ unbrauchbar.", "Desweiteren ist eine Entsch\"arfung des Wirbelterms (durch eine Regularisierung des \"`ersten ${\\bf u}$ \"') unerl\"asslich.", "Bekanntlich ist die Eindeutigkeit schwacher L\"osungen der Navier-Stokes-Gleichungen ein offenes Problem, sodass wir bez\"uglich (REF ) erst recht keine Eindeutigkeit erwarten k\"onnen.", "Das ist ein Problem, weil wir dann nicht wissen, auf welche der L\"osungen die Funktion $\\delta \\mapsto \\eta $ , f\"ur die wir die Existenz eines Fixpunktes zeigen wollen, abbilden soll.", "Durch eine geeignete Entsch\"arfung des Wirbelterms wird dieses Problem gel\"ost.", "Andererseits ist die Eindeutigkeit von L\"osungen von (REF ) auch mit entsch\"arftem Wirbelterm nicht offensichtlich, und wir w\"urden gerne auf einen Beweis verzichten.H\"angt allerdings der Spannungstensor nichtlinear vom Scherratentensor $D{\\bf u}$ ab, so ist eine Untersuchung in diese Richtung unerl\"asslich.", "Stattdessen wollen wir eine Variante des Schauder'schen Fixpunktsatzes f\"ur mengenwertige Abbildungen, Theorem REF , verwenden.", "Dazu muss allerdings die Menge der L\"osungen zu gegebenen, festen Daten konvex sein.", "Diese Eigenschaft k\"onnen wir erzwingen, indem wir den Wirbelterm \"`linearisieren\"'.", "Genauer: Wir f\"uhren das Fixpunktargument nicht nur in $\\eta $ , sondern zus\"atzlich im ersten Argument des Wirbelterms durch, d.h. wir ersetzen $({\\bf u}\\cdot \\nabla ){\\bf u}$ durch $({\\bf v}\\cdot \\nabla ){\\bf u}$ , wobei ${\\bf v}$ ein gegebenes Feld aus einem Funktionenraum ist, in dem wir Kompaktheit der L\"osungen der Fluidgleichungen bekommen k\"onnen.", "Da also die Regularit\"at von ${\\bf v}$ gering sein wird, ist nach wie vor eine Regularisierung des ersten Arguments des Wirbelterms notwendig.", "Die resultierende schwache Formulierung entspricht im Wesentlichen der, die in [11] verwendet wird.", "Wir wollen nun entsprechende Regularisierungsoperatoren $\\mathcal {R}_\\epsilon $ , $\\epsilon >0$ , konstruieren.", "Wir w\"ahlen dazu einen Gl\"attungskern $\\omega \\in C_0^\\infty (\\mathbb {R}^3)$ mit $\\int _{\\mathbb {R}^3}\\omega \\ dx=1$ und $\\operatorname{supp}\\omega \\subset \\lbrace (t,x)\\in \\mathbb {R}^3\\ |\\ t>0\\rbrace $ und setzen $\\omega _\\epsilon :=\\epsilon ^{-3}\\omega (\\epsilon ^{-1}\\cdot )$ .", "Zudem sei $(\\varphi _k,U_k)_k$ ein endlicher Atlas von $\\partial \\Omega $ mit untergeordneter Zerlegung der Eins $(\\psi _k)_k$ ; siehe Anhang A.1.", "Funktionen $\\delta \\in C(\\bar{I}\\times \\partial \\Omega )$ mit $\\delta (0,\\cdot )=\\eta _0$ setzen wir durch $\\eta _0$ auf $(-\\infty ,T]\\times \\partial \\Omega $ fort, und wir definieren $\\begin{aligned}\\mathcal {R}_\\epsilon \\delta :=\\sum _k(\\omega _\\epsilon \\ast ((\\psi _k\\,\\delta )\\circ \\varphi ^{-1}_k))\\circ \\varphi _k +\\epsilon ^{1/2}.\\end{aligned}$ Der mit $k$ indizierte Summand sei dabei durch 0 ausserhalb $U_{k}$ fortgesetzt.", "Man beachte, dass $\\mathcal {R}_\\epsilon \\eta _0:=(\\mathcal {R}_\\epsilon \\delta )(0,\\cdot )$ aufgrund der speziellen Lokalisierung des Tr\"agers von $\\omega $ nur von $\\eta _0$ (und $\\epsilon $ ) abh\"angt.", "Wegen der H\"olderstetigkeit von $\\eta _0$ gilt zudem $\\mathcal {R}_\\epsilon \\eta _0\\ge \\eta _0$ f\"ur alle $0<\\epsilon \\le c(\\eta _0)$ mit einer nur von $\\eta _0$ abh\"angenden (kleinen) Konstante $c(\\eta _0)$ .", "Das folgt aus der Absch\"atzung $\\begin{aligned}|\\omega _\\epsilon \\ast ((\\psi _k\\,\\delta )\\circ \\varphi ^{-1}_k)&-(\\psi _k\\,\\delta )\\circ \\varphi ^{-1}_k|(0,x)\\\\&=|\\int _{\\mathbb {R}^4}\\omega _\\epsilon (-s,x-y)\\Big (((\\psi _k\\,\\eta _0)\\circ \\varphi ^{-1}_k)(y)-((\\psi _k\\,\\eta _0)\\circ \\varphi ^{-1}_k)(x)\\Big )\\ dyds|\\\\&\\le c\\int _{\\mathbb {R}^4}|\\omega _\\epsilon (-s,x-y)||x-y|^{3/4}\\ dyds=c\\ \\epsilon ^{3/4}.\\end{aligned}$ F\"ur die Ungleichung haben wir die $3/4$ -H\"olderstetigkeit von $\\eta _0$ und die Regularit\"at von $\\varphi _k$ , $\\psi _k$ verwendet.", "F\"ur hinreichend kleines $\\epsilon $ ist der Term $(\\omega _\\epsilon \\ast ((\\psi _k\\,\\delta )\\circ \\varphi ^{-1}_k))\\circ \\varphi _k$ in $C^4(\\bar{I}\\times \\partial \\Omega )$ , weil dann der Tr\"ager in $\\bar{I}\\times U_k$ enthalten ist.", "Aus Proposition REF folgt die Konvergenz von $(\\omega _\\epsilon \\ast ((\\psi _k\\,\\delta )\\circ \\varphi ^{-1}_k))$ gegen $(\\psi _k\\,\\delta )\\circ \\varphi ^{-1}_k$ in $L^\\infty (I\\times \\mathbb {R}^2)$ , woraus durch Aufsummieren sofort die Konvergenz von $(\\mathcal {R}_\\epsilon \\delta )$ gegen $\\delta $ in $L^\\infty (I\\times \\partial \\Omega )$ folgt.", "Falls $\\delta $ eine Zeitableitung $\\partial _t\\delta $ in $L^2(I\\times \\partial \\Omega )$ besitzt, folgt wegen $\\partial _t\\mathcal {R}_\\epsilon \\delta =\\mathcal {R}_\\epsilon \\partial _t\\delta $ die Konvergenz von $(\\partial _t\\mathcal {R}_\\epsilon \\delta )$ gegen $\\partial _t\\delta $ in $L^2(I\\times \\partial \\Omega )$ vollkommen analog aus Proposition REF .", "Es seien schlie\"slich $\\widetilde{\\omega }\\in C_0^\\infty (\\mathbb {R}^4)$ mit $\\int _{\\mathbb {R}^4}\\omega \\ dx=1$ und $\\widetilde{\\omega }_\\epsilon :=\\epsilon ^{-4}\\widetilde{\\omega }(\\epsilon ^{-1}\\cdot )$ .", "F\"ur Funktionen $v\\in L^2(I\\times \\mathbb {R}^3)$ setzen wir $\\mathcal {R}_\\epsilon v:=\\widetilde{\\omega }_\\epsilon \\ast v$ , wobei $v$ durch 0 auf $\\mathbb {R}^4$ fortgesetzt sei.", "Wegen Proposition REF konvergiert $(\\mathcal {R}_\\epsilon v)$ gegen $v$ in $L^2(I\\times \\mathbb {R}^3)$ .", "Der Definitionsbereich der Anfangsgeschwindigkeit ${\\bf u}_0$ des Fluids ist (im Allgemeinen) von $\\Omega _{\\mathcal {R}_\\epsilon \\eta _0}$ verschieden.", "Wir m\"ussen deshalb die Anfangswerte modifizieren.", "Die Tatsache, dass $(\\mathcal {R}_\\epsilon \\eta _0)$ die Funktion $\\eta _0$ von oben approximiert, erleichtert uns dabei die Arbeit.", "Wir setzen $\\eta _1$ gem\"a\"s Bemerkung REF zu einem divergenzfreien $L^2$ -Vektorfeld $\\psi $ auf $B_\\alpha $ fort, wobei $\\alpha $ eine Zahl mit $\\Vert \\eta _0\\Vert _{L^\\infty (M)}<\\alpha <\\kappa $ , und definieren ${\\bf u}_0^\\epsilon :=\\left\\lbrace \\begin{array}{cl} {\\bf u}_0 & \\text{in }\\Omega _{\\eta _0}, \\\\\\psi & \\text{in } \\Omega _{\\mathcal {R}_\\epsilon \\eta _0}\\setminus \\Omega _{\\eta _0}\\end{array}\\right..$ Offenbar gilt ${\\bf u}_0^\\epsilon \\in L^2(\\Omega _{\\mathcal {R}_\\epsilon \\eta _0})$ , und aus (REF ) und $\\operatorname{tr^n_{\\eta _0}}{\\bf u}_0=\\eta _1\\gamma (\\eta _0)$ folgern wir mit Hilfe von Proposition REF die Divergenzfreiheit von ${\\bf u}_0^\\epsilon $ .", "Mit der Definition $\\eta _1^\\epsilon := \\exp \\Big (\\int _{\\eta _0}^{\\mathcal {R}_\\epsilon \\eta _0} \\beta (\\cdot +\\tau \\nu )\\ d\\tau \\Big )\\ \\eta _1$ gilt zudem $\\operatorname{tr^n_{\\mathcal {R}_\\epsilon \\eta _0}}{\\bf u}_0^\\epsilon =\\eta _1^\\epsilon \\,\\gamma (\\mathcal {R}_\\epsilon \\eta _0)$ , was sich anhand der Definition von $\\psi $ leicht durch ein Approximationsargument wie im Beweis von Proposition REF einsehen l\"asst.", "Setzen wir ${\\bf u}_0$ und ${\\bf u}_0^\\epsilon $ durch 0 auf $\\mathbb {R}^3$ fort, so folgen aus der gleichm\"a\"sigen Konvergenz von $\\mathcal {R}_\\epsilon \\eta _0\\rightarrow \\eta _0$ in $\\partial \\Omega $ die Konvergenzen $\\begin{aligned}{\\bf u}_0^\\epsilon \\rightarrow {\\bf u}_0 &\\text{ in } L^2(\\mathbb {R}^3),\\\\\\eta _1^\\epsilon \\rightarrow \\eta _1&\\text{ in } L^2(M).\\end{aligned}$ Wir nehmen im Folgenden ein Intervall $I=(0,T)$ , $T>0$ , und beliebige, aber feste Funktionen ${\\bf v}\\in L^2(I\\times \\mathbb {R}^3)$ und $\\delta \\in C(\\bar{I}\\times \\partial \\Omega )$ mit $\\Vert \\delta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}<\\kappa $ und $\\delta (0,\\cdot )=\\eta _0$ als gegeben an.", "Unsere Vor\"uberlegungen f\"uhren uns auf die folgende Definition.", "Der \"Ubersichtlichkeit halber werden wir vorerst den Parameter $\\epsilon $ in der Notation unterdr\"ucken.", "Insbesondere stehen die Bezeichner ${\\bf u}_0$ , $\\eta _1$ vorerst f\"ur die modifizierten Anfangswerte ${\\bf u}_0^\\epsilon $ , $\\eta _1^\\epsilon $ .", "Definition 5.6 Ein Tupel $(\\eta ,{\\bf u})$ hei\"st schwache L\"osung des entkoppelten, regularisierten Systems zum Argument $(\\delta ,{\\bf v})$ auf dem Intervall $I$ , falls $\\eta \\in Y^I$ mit $\\eta (0,\\cdot )=\\eta _0$ , ${\\bf u}\\in X_{\\mathcal {R}\\delta }^I$ mit $\\operatorname{tr_{\\mathcal {R}\\delta }}{\\bf u}=\\partial _t\\eta \\,\\nu $ und $\\begin{aligned}&- \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf u}\\cdot \\partial _t\\varphi \\ dxdt - \\frac{1}{2}\\int _I\\int _M\\partial _t\\eta \\,\\partial _t\\mathcal {R}\\delta \\, b\\, \\gamma (\\mathcal {R}\\delta )\\ dAdt\\\\&+\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla ){\\bf u}\\cdot \\varphi \\ dxdt -\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla )\\varphi \\cdot {\\bf u}\\ dxdt\\\\& + \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\nabla {\\bf u}:\\nabla \\varphi \\ dxdt -\\int _I\\int _M\\partial _t\\eta \\, \\partial _tb\\ dAdt + 2\\int _I K(\\eta ,b)\\ dt\\\\&\\hspace{128.0374pt}=\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}\\cdot \\varphi \\ dxdt + \\int _I\\int _Mg\\, b\\ dAdt\\\\&\\hspace{142.26378pt} +\\int _{\\Omega _{\\mathcal {R}\\eta _0}}{\\bf u}_0\\cdot \\varphi (0,\\cdot )\\ dx +\\int _M\\eta _1\\, b(0,\\cdot )\\ dA\\end{aligned}$ f\"ur alle Testfunktionen $(b,\\varphi )\\in T_{\\mathcal {R}\\delta }^I$ .", "Die Existenz einer L\"osung des entkoppelten, regularisierten Systems wird in [11] im Falle einer einfachen Geometrie durch Transformation auf einen Raumzeitzylinder und einer anschlie\"senden Galerkin-Approximation gezeigt.", "Wie der Beweis der folgenden Proposition zeigt, ist die Transformation nicht sinnvoll, da sie f\"ur die Galerkin-Approximation irrelevant ist und die nat\"urliche Struktur der Gleichungen zerst\"ort.", "Die Existenz eines Diffeomorphismus des zeitlich variablen Gebiets auf einen Raumzeitzylinder ist aber auch hier n\"otig, und zwar zur Konstruktion der zeitlich variablen Ansatzfunktionen.", "Proposition 5.7 Es existiert eine schwache L\"osung $(\\eta ,{\\bf u})$ des entkoppelten, regularisierten Systems zum Argument $(\\delta ,{\\bf v})$ auf dem Intervall $I$ , die die Absch\"atzung $\\begin{aligned}&\\Vert \\eta \\Vert _{Y^I}^2 + \\Vert {\\bf u}\\Vert _{X_{\\mathcal {R}\\delta }^I}^2\\le c_0(T,\\Omega _{\\mathcal {R}\\delta }^I,{\\bf f},g,\\eta _0,\\eta _1)\\end{aligned}$ erf\"ullt.", "Insbesondere ist die linke Seite unabh\"angig vom Parameter $\\epsilon $ und vom Argument $(\\delta ,{\\bf v})$ beschr\"ankt.", "Vorarbeit.", "Wir wollen zun\"achst geeignete Ansatzfunktionen konstruieren.", "Dazu w\"ahlen wir eine Basis $(\\widehat{{\\bf X}}_k)_{k\\in \\mathbb {N}}$ von $H^1_{0,\\operatorname{div}}(\\Omega )$ und eine Basis $(\\widehat{Y}_k)_{k\\in \\mathbb {N}}$ des Raumes $\\Big \\lbrace Y\\in H^2_0(M)\\big |\\int _M Y\\ dA =0\\Big \\rbrace .$ Durch L\"osen des Stokes-Systems in $\\Omega $ mit Randwerten $\\widehat{Y}_k\\,\\nu $ (wie immer durch 0 auf $\\partial \\Omega $ fortgesetzt), siehe Theorem REF , erhalten wir divergenzfreie Fortsetzungen $\\widehat{{\\bf Y}}_k$ .", "F\"ur $t\\in \\bar{I}$ setzen wir ${\\bf X}_k(t,\\cdot ):={\\mathcal {R}\\delta (t)}\\widehat{{\\bf X}}_k,\\quad {\\bf Y}_k(t,\\cdot ):={\\mathcal {R}\\delta (t)}\\widehat{{\\bf Y}}_k;$ siehe Bemerkung REF .", "Die Felder ${\\bf X}_k(t,\\cdot )$ bilden offenbar eine Basis von $H^1_{0,\\operatorname{div}}(\\Omega _{\\mathcal {R}\\delta (t)})$ .", "Man beachte zudem, dass f\"ur $q\\in \\partial \\Omega $ das Differential $d\\Psi _{\\mathcal {R}\\delta (t)}(q)$ die Normale $\\nu (q)$ lediglich skaliert, weshalb die Definition $Y_k(t,\\cdot )\\,\\nu :=\\operatorname{tr_{\\mathcal {R}\\delta (t)}}{\\bf Y}_k(t,\\cdot )=d\\Psi _{\\mathcal {R}\\delta (t)}\\, (\\det d\\Psi _{\\mathcal {R}\\delta (t)})^{-1}\\,\\widehat{Y}_k\\,\\nu $ sinnvoll ist.", "Aus der Identit\"at $\\operatorname{tr^n_{\\mathcal {R}\\delta }}{\\bf Y}_k=Y_k\\,\\gamma (\\mathcal {R}\\delta )$ und Proposition REF schlie\"sen wir $\\int _M Y_k(t,\\cdot )\\,\\gamma (\\mathcal {R}\\delta (t,\\cdot ))\\ dA = 0.$ Aus der Basiseigenschaft der $\\widehat{Y}_k$ folgt, dass die Felder $Y_k(t,\\cdot )$ eine Basis des Raumes $\\begin{aligned}\\Big \\lbrace Y\\in H^2_0(M)\\ \\big |\\ \\int _MY\\, \\gamma (\\mathcal {R}\\delta (t,\\cdot ))\\ dA = 0\\Big \\rbrace \\end{aligned}$ bilden.", "Um die Notation zu vereinfachen w\"ahlen wir eine Aufz\"ahlungWir k\"onnen zum Beispiel ${\\bf W}_{2k}:={\\bf X}_k$ und ${\\bf W}_{2k-1}:={\\bf Y}_k$ w\"ahlen.", "$({\\bf W}_k)_{k\\in \\mathbb {N}}$ der Felder ${\\bf X}_k,{\\bf Y}_k$ und setzen $W_k\\,\\nu :=\\operatorname{tr_{\\mathcal {R}\\delta }}{\\bf W}_k$ .", "Wir wollen nun zeigen, dass $\\operatorname{span}\\lbrace (\\varphi \\, W_k,\\varphi \\,{\\bf W}_k)\\ |\\ \\varphi \\in C_0^1([0,T)),\\, k\\in \\mathbb {N}\\rbrace $ dicht liegt im Raum aller Tupel $(b,\\varphi )\\in \\big (H^1(I,L^2(M))\\cap L^2(I,H^2_0(M))\\big )\\times H^1(\\Omega _{\\mathcal {R}\\delta }^I)$ mit $b(T,\\cdot )=0$ , $\\varphi (T,\\cdot )=0$ , $\\operatorname{div}\\varphi =0$ und $\\operatorname{tr_{\\mathcal {R}\\delta }}\\varphi =b\\,\\nu $ .", "Offenbar bettet $T_{\\mathcal {R}\\delta }^I$ in diesen Raum ein.", "Aufgrund der durch ${\\mathcal {R}\\delta }$ induzierten Isomorphismen ist die Behauptung \"aquivalent zur Dichtheitheit von $\\operatorname{span}\\lbrace (\\varphi \\,\\widehat{W}_k,\\varphi \\,\\widehat{{\\bf W}}_k)\\ |\\ \\varphi \\in C_0^1([0,T)),\\, k\\in \\mathbb {N}\\rbrace $ im Raum $T$ aller Tupel $(b,\\varphi )\\in H^1(I,L^2(M))\\cap L^2(I,H^2_0(M))\\times H^1(I\\times \\Omega )$ mit $b(T,\\cdot )=0$ , $\\varphi (T,\\cdot )=0$ , $\\operatorname{div}\\varphi =0$ und $\\varphi |_{I\\times \\partial \\Omega } =b\\,\\nu $ .", "Sei also $(b,\\varphi )\\in T$ .", "Wir approximieren zun\"achst $b$ durch Funktionen $\\tilde{b}\\in C^\\infty _0([0,T),H^2_0(M))$ in $H^1(I,L^2(M))\\cap L^2(I,H^2_0(M))$ mit $\\int _M \\tilde{b}(t,\\cdot )\\ dA=0$ f\"ur $t\\in \\bar{I}$ .Wir k\"onnen $\\tilde{b}$ konstruieren, indem wir die vektorwertige Funktion $b$ durch 0 auf $[0,\\infty )$ fortsetzen und anschlie\"send $b(\\cdot +h)$ , $h>0$ , mit einem Gl\"attungskern falten.", "Die Mittelwertfreiheit von $\\tilde{b}(t,\\cdot )$ folgt dann aus der Mittelwertfreiheit von $b(t,\\cdot )$ .", "Nun k\"onnen wir $\\partial _t\\tilde{b}$ durch eine Folge $(\\varphi ^k_n\\,\\widehat{Y}_k)_n$ (Summation von 1 bis $n$ ), $\\varphi ^k_n\\in C_0^1([0,T))$ , in $L^2(I,H^2_0(M))$ approximieren.", "Ist n\"amlich $f\\in L^2(I,H^2_0(M))$ mit $\\int _M f(t,\\cdot )\\ dA=0$ f\"ur fast alle $t\\in I$ und $\\int _I\\varphi (t)\\, (\\widehat{Y}_k,f(t,\\cdot ))_{H^2(M)}\\ dt=0$ f\"ur jedes $\\varphi \\in C_0^1([0,T))$ und jedes $k\\in \\mathbb {N}$ , so verschwindet das Skalarprodukt im Integranden fast \"uberall, und aufgrund der Basiseigenschaft der Funktionen $\\widehat{Y}_k(t,\\cdot )$ ist $f$ somit die Nullfunktion.", "Wegen $\\begin{aligned}\\big \\Vert \\tilde{b}(t,\\cdot ) + \\int _t^T\\varphi ^k_n(s)\\, ds\\,\\widehat{Y}_k\\big \\Vert _{H^2(M)}&\\le \\int _0^T \\Vert \\partial _s\\tilde{b}(s,\\cdot ) - \\varphi ^k_n(s)\\,\\widehat{Y}_k\\Vert _{H^2(M)}\\ ds\\end{aligned}$ konvergiert die Folge $(-\\int _t^T\\varphi ^k_n(s) ds\\,\\widehat{Y}_k)_{n}$ gegen $\\tilde{b}$ in $C(\\bar{I}, H^2_0(M))$ , insgesamt also in $H^1(I,H^2_0(M))$ .", "Wir haben somit Linearkombinationen der Felder $\\widehat{Y}_k$ konstruiert, die $b$ in $H^1(I,L^2(M))\\cap L^2(I,H^2_0(M))$ approximieren.", "Aufgrund der Stetigkeitseigenschaften des L\"osungoperators des Stokes-Systems konvergieren die entsprechenden Linearkombinationen der Felder $\\widehat{{\\bf Y}}_k$ gegen ein ${\\bf Y}$ in $H^1(I\\times \\Omega )$ .", "Es gilt $({\\bf Y}-\\varphi )|_{I\\times \\partial \\Omega }=0$ , sodass lediglich zu zeigen bleibt, dass wir jedes $\\varphi $ mit $(0,\\varphi )\\in T$ durch eine Folge $(\\varphi ^k_n\\,\\widehat{{\\bf X}}_k)_{n}$ , $\\varphi ^k_n\\in C_0^1([0,T))$ , in $H^1(I\\times \\Omega )$ approximieren k\"onnen.", "Dazu k\"onnen wir aber genau wie bei der Approximation von $b$ vorgehen.", "Beweis: (von Proposition REF ) Wir verwenden die Galerkin-Methode, d.h. wir approximieren das vorliegende unendlichdimensionale dynamische System durch endlichdimensionale Systeme, also durch gew\"ohnliche Differentialgleichungen, indem wir die Gleichung auf endlichdimensionale Teilr\"aume der jeweiligen Funktionenr\"aume \"`projizieren\"'.", "Wir suchen dazu Funktionen $\\alpha _n^k:[0,T]\\rightarrow \\mathbb {R}$ , $k,n\\in \\mathbb {N}$ , derart, dass ${\\bf u}_n:=\\alpha _n^k\\,{\\bf W}_k$ und $\\eta _n(t,\\cdot ):=\\int _0^t\\alpha ^k_n\\, W_k\\ ds+\\eta _0$ (jeweils Summation von 1 bis $n$ ) die GleichungWie \"ublich unterdr\"ucken wir die unabh\"angigen Variablen in der Notation, d.h. ${\\bf u}={\\bf u}(t,\\cdot )$ , etc.", "$\\begin{aligned}&\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\partial _t{\\bf u}_n\\cdot {\\bf W}_j\\ dx +\\frac{1}{2} \\int _M\\partial _t\\eta _n\\, \\partial _t\\mathcal {R}\\delta \\, W_j\\, \\gamma (\\mathcal {R}\\delta )\\ dA\\\\&+\\frac{1}{2}\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla ){\\bf u}_n\\cdot {\\bf W}_j\\ dx -\\frac{1}{2}\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla ){\\bf W}_j\\cdot {\\bf u}_n\\ dx\\\\&+\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\nabla {\\bf u}_n:\\nabla {\\bf W}_j\\ dx +\\int _M\\partial ^2_t\\eta _n\\, W_j\\ dA + 2K(\\eta _n, W_j) \\\\&\\hspace{142.26378pt}=\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}_n\\cdot {\\bf W}_j\\ dx + \\int _Mg_n\\, W_j\\ dA\\end{aligned}$ f\"ur alle $1\\le j \\le n$ erf\"ullen.", "Dabei sind ${\\bf f}_n$ und $g_n$ Funktionen hoher Regularit\"at, die gegen ${\\bf f}$ und $g$ in $L^2_\\text{loc}([0,\\infty )\\times \\mathbb {R}^3)$ bzw.", "$L^2_\\text{loc}([0,\\infty )\\times M)$ konvergieren.", "Wir k\"onnen zum Beispiel ${\\bf f}_n:=\\mathcal {R}_{1/n}{\\bf f}$ und $g_n:=\\sum _\\alpha (\\omega _{1/n}\\ast ((\\psi _\\alpha g)\\circ \\varphi ^{-1}_\\alpha ))\\circ \\varphi _\\alpha $ setzen; vgl.", "Definition (REF ).", "Die Konvergenzen folgen dann direkt aus Proposition REF .", "Wir geben zudem Anfangswerte vor.", "Wir w\"ahlen die $\\alpha ^k_n(0)$ derart, dass $\\partial _t\\eta _n(0,\\cdot )\\rightarrow \\eta _1\\text{ in } L^2(M)$ und ${\\bf u}_n(0,\\cdot )\\rightarrow {\\bf u}_0\\text{ in }L^2(\\Omega _{\\mathcal {R}\\eta _0})$ gilt.", "Dazu w\"ahlen wir die Koeffizienten der ${\\bf Y}_k$ bei $t=0$ so, dass die erste der beiden Konvergenzen gilt.", "Das ist m\"oglich, weil die Felder $Y_k(0,\\cdot )$ eine Basis von (REF ) mit $t=0$ bilden und wir aus Proposition REF die Identit\"at $\\int _M \\eta _1\\, \\gamma (\\mathcal {R}\\eta _0)\\ dA=\\int _{\\Omega _{\\mathcal {R}\\eta _0}}\\operatorname{div}{\\bf u}_0\\ dx=0$ folgern k\"onnen.", "Theorem REF impliziert, dass der L\"osungsoperator des Stokes-Systems stetig von $\\Big \\lbrace Y\\,\\nu \\in L^2(M)\\ |\\ \\int _M Y\\,\\gamma (\\mathcal {R}\\eta _0)\\ dA=0\\Big \\rbrace $ nach $L^2(\\Omega _{\\mathcal {R}\\eta _0})$ abbildet.", "Somit konvergieren nicht nur die Linearkombinationen der $Y_k(0,\\cdot )$ gegen $\\eta _1$ in $L^2(M)$ , sondern auch die entsprechenden Linearkombinationen der ${\\bf Y}_k(0,\\cdot )$ gegen ein ${\\bf Y}$ in $L^2(\\Omega _{\\mathcal {R}\\eta _0})$ .", "Es gilt $\\operatorname{tr^n_{\\mathcal {R}\\eta _0}}({\\bf u}_0-{\\bf Y}) = 0$ .", "Da die ${\\bf X}_k(0,\\cdot )$ inbesondere eine Basis von $\\big \\lbrace {\\bf X}\\in L^2(\\Omega _{\\mathcal {R}\\eta _0})\\ |\\ \\operatorname{div}{\\bf X}=0,\\, \\operatorname{tr^n_{\\mathcal {R}\\eta _0}}{\\bf X}=0\\big \\rbrace $ bilden, k\"onnen wir nun deren Koeffizienten bei $t=0$ so w\"ahlen, dass die Folge $(\\alpha _n^k\\,{\\bf W}_k)_{n}$ gegen ${\\bf u}_0$ in $L^2(\\Omega _{\\mathcal {R}\\eta _0})$ konvergiert.", "Es liegt somit ein Anfangswertproblem f\"ur ein lineares System gew\"ohnlicher Integro-Differentialgleichungen der Form ($1\\le j\\le n$ , Summation von 1 bis $n$ ) $A_{jk}(t)\\,\\dot{\\alpha }^k(t) = B_{jk}\\,\\alpha ^k(t) + \\int _0^t C_{jk}(t,s)\\,\\alpha ^k(s)\\ ds +D_j(t),$ vor.", "Die Koeffizienten sind durch $\\begin{aligned}A_{jk}(t)&=\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf W}_k\\cdot {\\bf W}_j\\ dx + \\int _MW_k\\, W_j\\ dA,\\\\B_{jk}(t)&=\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\partial _t{\\bf W}_k\\cdot {\\bf W}_j\\ dx + \\frac{1}{2}\\int _MW_k\\, W_j\\ \\partial _t\\mathcal {R}\\delta \\, \\gamma (\\mathcal {R}\\delta )\\ dA\\\\&\\hspace{8.5359pt} +\\frac{1}{2} \\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla ){\\bf W}_k\\cdot {\\bf W}_j\\ dx - \\frac{1}{2} \\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla ){\\bf W}_j\\cdot {\\bf W}_k\\ dx \\\\&\\hspace{8.5359pt} + \\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\nabla {\\bf W}_k:\\nabla {\\bf W}_j\\ dx + \\int _M\\partial _t W_k\\, W_j\\ dA\\\\C_{jk}(t,s)&=2 K(W_k(s),W_j(t)),\\\\D_j(t)&=\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}_n\\cdot {\\bf W}_j\\ dx + \\int _Mg_n\\, W_j\\ dA\\end{aligned}$ gegeben und somit offenbar stetig.", "Die Matrix $A(t)$ ist symmetrisch, und aufgrund der linearen Unabh\"angigkeit unserer Ansatzfunktionen k\"onnen wir aus der Identit\"at ($\\beta ^j\\in \\mathbb {R}$ , $j=1,\\ldots ,n$ ) $\\beta ^j\\beta ^k A_{jk}(t)=\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}|\\beta ^j{\\bf W}_j|^2\\ dx+\\int _M|\\beta ^j W_j|^2\\ dA$ schlie\"sen, dass $A(t)$ positiv definit ist.", "Insbesondere ist diese Matrix invertierbar, womit Gleichung (REF ) von der in Anhang A.3 behandelten Form ist.", "Sie besitzt somit f\"ur jedes $n$ eine L\"osung auf dem Intervall $[0,T]$ .", "Um Energieabsch\"atzungen zu bekommen, testen wir die Gleichung mit $(\\partial _t\\eta _n,{\\bf u}_n)$ und erhalten $\\begin{aligned}&\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\partial _t{\\bf u}_n\\cdot {\\bf u}_n\\ dx +\\frac{1}{2} \\int _M(\\partial _t\\eta _n)^2\\, \\partial _t\\mathcal {R}\\delta \\, \\gamma (\\mathcal {R}\\delta )\\ dA\\\\&+\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}|\\nabla {\\bf u}_n|^2\\ dx+\\int _M\\partial ^2_t\\eta _n\\, \\partial _t\\eta _n\\ dA + 2K(\\eta _n,\\partial _t\\eta _n)\\\\&\\hspace{142.26378pt}=\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}_n\\cdot {\\bf u}_n\\ dx + \\int _Mg_n\\, \\partial _t\\eta _n\\ dA.\\end{aligned}$ Die ersten beiden Terme lassen sich wie gewohnt zusammenfassen, sodass sich $\\begin{aligned}\\frac{1}{2}\\frac{d}{dt}\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}|{\\bf u}_n|^2\\ dx + \\int _{\\Omega _{\\mathcal {R}\\delta (t)}}|\\nabla {\\bf u}_n|^2\\ dx+\\frac{1}{2}\\frac{d}{dt}&\\int _M|\\partial _t\\eta _n|^2\\ dA + \\frac{d}{dt} K(\\eta _n)\\ dA\\\\ &=\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}_n\\cdot {\\bf u}_n\\ dx + \\int _Mg_n\\, \\partial _t\\eta _n\\ dA\\end{aligned}$ ergibt.", "Nun k\"onnen wir wie am Ende des letzten Abschnitts vorgehen und erhalten $\\begin{aligned}\\Vert \\eta _n\\Vert _{Y^I}^2 +\\Vert {\\bf u}_n\\Vert _{X_{\\mathcal {R}\\delta }^I}^2\\le c_0(T,\\Omega _{\\mathcal {R}\\delta }^I,{\\bf f}_n,g_n,{\\bf u}_n(0,\\cdot ),\\eta _0,\\partial _t\\eta _n(0,\\cdot )).\\end{aligned}$ Aus dieser Absch\"atzung folgern wir f\"ur eine Teilfolge die KonvergenzenMan beachte, dass $\\begin{aligned}X_{\\mathcal {R}\\delta }^I&\\simeq L^\\infty (I,L^2(\\Omega ))\\cap L^2(I,H^1_{\\operatorname{div}}(\\Omega ))\\simeq (L^1(I,L^2(\\Omega ))+L^2(I,H^1_{\\operatorname{div}}(\\Omega )))^{\\prime }\\\\ &\\simeq (L^1(I,L^2(\\Omega _{\\mathcal {R}\\delta (t)}))+L^2(I,H^1_{\\operatorname{div}}(\\Omega _{\\mathcal {R}\\delta (t)})))^{\\prime }\\end{aligned}$ gilt, wobei die erste und die dritte Isomorphie durch die Abbildung ${\\mathcal {R}\\delta }$ induziert werden.", "$\\begin{aligned}\\eta _n&\\rightarrow \\eta \\hspace{8.5359pt}&&\\text{ schwach$^*$ in } L^\\infty (I,H^2_0(M)),\\\\\\partial _t\\eta _n&\\rightarrow \\partial _t\\eta &&\\text{ schwach$^*$ in } L^\\infty (I,L^2(M)),\\\\{\\bf u}_n&\\rightarrow {\\bf u}&&\\text{ schwach$^*$ in } X_{\\mathcal {R}\\delta }^I.\\end{aligned}$ Mit Hilfe der Unterhalbstetigkeit der Normen bez\"uglich der schwach*-Konvergenz erhalten wir (REF ).", "Desweiteren folgt aus $\\operatorname{tr_{\\mathcal {R}\\delta }}{\\bf u}_n=\\partial _t\\eta _n\\,\\nu $ und obigen Konvergenzen die Identit\"at $\\operatorname{tr_{\\mathcal {R}\\delta }}{\\bf u}=\\partial _t\\eta \\,\\nu $ .", "Wir multiplizieren nun Gleichung (REF ) mit $\\varphi (t)$ , wobei $\\varphi \\in C_0^1([0,T))$ , integrieren \"uber $I$ und anschlie\"send partiell in der Zeit und erhalten f\"ur $1\\le j\\le n$ $\\begin{aligned}&-\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf u}_n\\cdot \\partial _t(\\varphi \\,{\\bf W}_j)\\ dxdt -\\frac{1}{2}\\int _I\\int _M\\partial _t\\eta _n\\, \\partial _t\\mathcal {R}\\delta \\, \\varphi \\, W_j\\, \\gamma (\\mathcal {R}\\delta )\\ dAdt\\\\& +\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla ){\\bf u}_n\\cdot (\\varphi \\,{\\bf W}_j)\\ dxdt\\\\&-\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla )(\\varphi \\,{\\bf W}_j)\\cdot {\\bf u}_n\\ dxdt+\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\nabla {\\bf u}_n:\\nabla (\\varphi \\,{\\bf W}_j)\\ dxdt\\\\&+\\int _I\\int _M\\partial _t\\eta _n\\, \\partial _t (\\varphi \\, W_j)\\ dAdt + 2\\int _I K(\\eta _n, \\varphi \\, W_j)\\ dAdt \\\\&\\hspace{71.13188pt}=\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}_n\\cdot (\\varphi \\,{\\bf W}_j)\\ dxdt + \\int _I\\int _Mg_n\\, \\varphi \\, W_j\\ dAdt\\\\&\\hspace{85.35826pt} +\\int _{\\Omega _{\\mathcal {R}\\eta _0}}{\\bf u}_n(0)\\cdot (\\varphi (0){\\bf W}_j(0,\\cdot ))\\ dx +\\int _M\\partial _t\\eta _n(0)\\, \\varphi (0)W_j(0,\\cdot )\\ dA.\\end{aligned}$ Durch Grenz\"ubergang in $n$ in obiger Gleichung sehen wir, dass $\\eta $ und ${\\bf u}$ die Identit\"at (REF ) f\"ur alle Testfunktionen aus $\\operatorname{span}\\lbrace (\\varphi \\, W_j,\\varphi \\,{\\bf W}_j)\\ |\\ \\varphi \\in C_0^1([0,T)),\\, j\\in \\mathbb {N}\\rbrace $ erf\"ullen.", "Aufgrund der in der Vorarbeit gezeigten Dichtheit dieser Funktionen folgt die G\"ultigkeit der Identit\"at f\"ur alle Testfunktionen aus $T^I_{\\mathcal {R}\\delta }$ .", "Dabei ist zu beachten, dass f\"ur den \"`inneren Anteil\"' der Testfunktionen die Konvergenz in $H^1(\\Omega _{\\mathcal {R}\\delta }^I)$ wegen der Entsch\"arfung des Wirbelterms gen\"ugt.", "$\\Box $ Man beachte, dass die Eindeutigkeit schwacher L\"osungen des entkoppelten Systems nicht offensichtlich ist.", "Diese Eigenschaft zeigt man im Falle einer parabolischen Gleichung \"ublicherweise, indem man die Gleichung mit der L\"osung selber testet.", "Das ist m\"oglich, wenn wir wissen, dass schwache L\"osungen Zeitableitungen im Dualraum ihrer eigenen Regularit\"atsklasse besitzen.", "Das ist bei dem vorliegenden parabolisch-dispersiven System selbst formal nicht der Fall.", "Gleichung (REF ) zeigt n\"amlich, dass die formale Zeitableitung des Tupels $(\\partial _t\\eta ,{\\bf u})$ nicht im Dualraum der Regularit\"atsklasse von $(\\partial _t\\eta ,{\\bf u})$ , sondern in dem der Regularit\"atsklasse von $(\\eta ,{\\bf u})$ liegt.", "Andererseits zeigt man die Eindeutigkeit schwacher L\"osungen von Gleichungen vom Typ der Schalengleichung \"ublicherweise, indem man mit einem Term der Form $\\int \\eta (s,\\cdot )\\ ds$ testet, vgl.", "[25], was im Falle unseres gekoppelten Systems auch nicht m\"oglich ist.", "Dieses Problem kann durch eine weitere Regularisierung des Systems umgangen werden.", "Da wir jedoch dem Begriff der Zeitableitung bislang keine rigorose Bedeutung verliehen haben, ist die Eindeutigkeit auch dann nicht offensichtlich, sodass wir, wie bereits erw\"ahnt, gerne (vorerst) auf einen Beweis verzichten w\"urden.", "Aus der Linearit\"at von Gleichung (REF ) folgt, dass die L\"osungsmenge konvex ist.", "Dies erlaubt es uns, anstelle des Schauder'schen Fixpunktsatzes eine Variante dieses Satzes f\"ur mengenwertige Abbildungen, den Fixpunktsatz von Kakutani-Glicksberg-Fan, Theorem REF , zu verwenden.", "Wir wollen zun\"achst den nun naheliegenden L\"osungsbegriff fixieren.", "Definition 5.8 Ein Tupel $(\\eta ,{\\bf u})$ hei\"st schwache L\"osung des regularisierten Systems zum Parameter $\\epsilon $ auf dem Intervall $I$ , falls $\\eta \\in Y^I$ mit $\\Vert \\eta \\Vert _{L^\\infty (I\\times M)}<\\kappa $ und $\\eta (0,\\cdot )=\\eta _0$ , ${\\bf u}\\in X_{\\mathcal {R}_\\epsilon \\eta }^I$ mit $\\operatorname{tr_{\\mathcal {R}_\\epsilon \\eta }}{\\bf u}=\\partial _t\\eta \\,\\nu $ und $\\begin{aligned}&- \\int _I\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta (t)}}{\\bf u}\\cdot \\partial _t\\varphi \\ dxdt - \\frac{1}{2}\\int _I\\int _M\\partial _t\\eta \\, \\partial _t\\mathcal {R}_\\epsilon \\eta \\ b\\, \\gamma (\\mathcal {R}_\\epsilon \\eta )\\ dAdt\\\\&+\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta (t)}}(\\mathcal {R}_\\epsilon {\\bf u}\\cdot \\nabla ){\\bf u}\\cdot \\varphi \\ dxdt -\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta (t)}}(\\mathcal {R}_\\epsilon {\\bf u}\\cdot \\nabla )\\varphi \\cdot {\\bf u}\\ dxdt\\\\& + \\int _I\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta (t)}}\\nabla {\\bf u}:\\nabla \\varphi \\ dxdt-\\int _I\\int _M\\partial _t\\eta \\, \\partial _tb\\ dAdt + 2\\int _I K(\\eta ,b)\\ dt \\\\&\\hspace{113.81102pt}=\\int _I\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta (t)}}{\\bf f}\\cdot \\varphi \\ dxdt + \\int _I\\int _Mg\\, b\\ dAdt\\\\&\\hspace{128.0374pt} +\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta _0}}{\\bf u}_0^\\epsilon \\cdot \\varphi (0,\\cdot )\\ dx +\\int _M\\eta _1^\\epsilon \\, b(0,\\cdot )\\ dA\\end{aligned}$ f\"ur alle Testfunktionen $(b,\\varphi )\\in T_{\\mathcal {R}_\\epsilon \\eta }^I$ .", "Proposition 5.9 Es existieren ein $T>0$ und f\"ur jedes hinreichend kleine $\\epsilon >0$ eine schwache L\"osung $(\\eta _\\epsilon ,{\\bf u}_\\epsilon )$ des regularisierten Systems zum Parameter $\\epsilon $ auf dem Intervall $I=(0,T)$ , die die Absch\"atzung $\\begin{aligned}&\\Vert \\eta _\\epsilon \\Vert _{Y^I}^2 + \\Vert {\\bf u}_\\epsilon \\Vert _{X_{\\mathcal {R}_\\epsilon \\eta }^I}^2\\le c_0(T,\\Omega _{\\mathcal {R}_\\epsilon \\eta }^I,{\\bf f},g,{\\bf u}^\\epsilon _0,\\eta _0,\\eta ^\\epsilon _1)\\end{aligned}$ erf\"ullt.", "Die Zeit $T$ h\"angt nur von $\\tau (\\eta _0)$ und der durch (REF ) gegebenen Schranke f\"ur die $Y^I$ -Norm von $\\eta _\\epsilon $ ab.", "Beweis: Wir setzen $\\alpha :=(\\Vert \\eta _0\\Vert _{L^\\infty (M)}+\\kappa )/2$ und fixieren ein beliebiges, hinreichend kleines $\\epsilon >0$ , unterdr\"ucken diesen Parameter jedoch im Rahmen dieses Beweises in der Notation.", "Wir wollen Theorem REF anwenden.", "Dazu betrachten wir den Raum $Z:=C(\\bar{I}\\times \\partial \\Omega )\\times L^2(I\\times \\mathbb {R}^3)$ und die konvexe Menge $D:=\\lbrace (\\delta ,{\\bf v})\\in Z\\ |\\ \\delta (0,\\cdot )=\\eta _0,\\,\\Vert \\delta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}\\le \\alpha ,\\,\\Vert {\\bf v}\\Vert _{L^2(I\\times \\mathbb {R}^3)}\\le c_1\\rbrace $ mit einer hinreichend gro\"sen Konstante $c_1>0$ .", "Desweiteren betrachten wir die Abbildung $F:D\\subset Z\\rightarrow 2^Z,$ die jedem Tupel $(\\delta ,{\\bf v})$ die Menge der schwachen L\"osungen $(\\eta ,{\\bf u})$ des entkoppelten, regularisierten Systems zum Argument $(\\delta ,{\\bf v})$ zuordnet, die der Absch\"atzung $\\begin{aligned}\\Vert \\eta \\Vert _{Y^I} +\\Vert {\\bf u}\\Vert _{X_{\\mathcal {R}\\delta }^I} \\le c(\\delta )\\end{aligned}$ gen\"ugen, wobei $c(\\delta )$ die rechte Seite von $(\\ref {ab:ent})$ bezeichne.", "Wegen Proposition REF ist $F(\\delta ,{\\bf v})$ nichtleer.", "Es gelte $(\\eta ,{\\bf u})\\in F(\\delta ,{\\bf v})$ .", "Aufgrund von (REF ) ist dann $\\Vert {\\bf u}\\Vert _{L^2(I\\times \\mathbb {R}^3)}$ durch $c_1$ abgesch\"atzt, und die Norm von $\\eta $ in $Y^I\\hookrightarrow C^{0,1-\\theta }(\\bar{I}, C^{0,2\\theta -1}(\\partial \\Omega ))\\quad (1/2<\\theta <1)$ ist beschr\"ankt.", "Da zudem $\\eta (0,\\cdot )=\\eta _0$ gilt, kann das Zeitintervall $I=(0,T)$ unabh\"angig vom Regularisierungsparameter $\\epsilon $ so klein gew\"ahlt werden, dass $\\Vert \\eta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}\\le \\alpha $ .", "Also bildet $F$ die Menge $D$ in ihre Potenzmenge ab, $F(D)\\subset 2^D$ .", "Aufgrund der Linearit\"at der Gleichung ist $F(\\delta ,{\\bf v})$ konvex.", "Zudem ist diese Menge abgeschlossen in $Z$ .", "Falls n\"amlich eine Folge $(\\eta _n,{\\bf u}_n)\\subset F(\\delta ,{\\bf v})$ gegen ein $(\\eta ,{\\bf u})$ in $Z$ konvergiert, so gilt wegen $(\\ref {ab:schranke})$ f\"ur eine Teilfolge $\\begin{aligned}\\eta _n&\\rightarrow \\eta \\hspace{8.5359pt}&&\\text{ schwach$^*$ in } L^\\infty (I,H^2_0(M)),\\\\\\partial _t\\eta _n&\\rightarrow \\partial _t\\eta &&\\text{ schwach$^*$ in } L^\\infty (I,L^2(M)),\\\\{\\bf u}_n&\\rightarrow {\\bf u}&&\\text{ schwach$^*$ in } X_{\\mathcal {R}\\delta }^I.\\end{aligned}$ Wir k\"onnen also in Gleichung (REF ) zum Grenzwert \"ubergehen und erhalten $(\\eta ,{\\bf u})\\in F(\\delta ,{\\bf v})$ .", "Als n\"achstes wollen wir zeigen, dass $F(D)$ relativ kompakt ist in $Z$ .", "Sei dazu $(\\delta _n,{\\bf v}_n)$ eine Folge aus $D$ und $(\\eta _n,{\\bf u}_n)\\in F(\\delta _n,{\\bf v}_n)$ .", "Es ist zu zeigen, dass eine Teilfolge von $(\\eta _n,{\\bf u}_n)$ in $Z$ konvergiert.", "Die gleichm\"a\"sige Konvergenz einer Teilfolge von $(\\eta _n)$ ist offensichtlich.", "Da der Regularisierungsoperator $\\mathcal {R}:\\lbrace \\delta \\in C(\\bar{I}\\times \\partial \\Omega )|\\ \\delta (0,\\cdot )=\\eta _0 \\rbrace \\rightarrow C^3(\\bar{I}\\times \\partial \\Omega )\\hookrightarrow \\hookrightarrow C^2(\\bar{I}\\times \\partial \\Omega )$ Kompaktheit erzeugt, konvergiert eine Teilfolge von $(\\mathcal {R}\\delta _n)$ gegen ein $\\delta $ in $C^2(\\bar{I}\\times \\partial \\Omega )$ .", "Der Beweis der relativen Kompaktheit der Folge $({\\bf u}_n)$ in $L^2(I\\times \\mathbb {R}^3)$ l\"asst sich nun fast w\"ortlich von Proposition REF \"ubernehmen.", "Es ist lediglich die leicht abgewandelte Form (REF ) des Systems zu beachten und insbesondere teilweise die Folge $(\\eta _n)$ mit Grenzwert $\\eta $ durch die Folge $(\\mathcal {R}\\delta _n)$ mit Grenzwert $\\mathcal {R}\\delta $ zu ersetzen.", "Aufgrund der Regularisierungen k\"onnten nat\"urlich einige Argumente vereinfacht werden.", "Dieses Argument zeigt zus\"atzlich die relative Kompaktkeit der Folge $(\\partial _t\\eta _n)$ in $L^2(I\\times M)$ .", "Es bleibt zu zeigen, dass die Abbildung $F$ graphenabgeschlossen ist.Wir f\"uhren die Argumente im Beweis der Graphenabgeschlossenheit so, dass sie auch f\"ur R\"ander von asymptotisch geringer Regularit\"at g\"ultig sind.", "Das ist sinnvoll, weil diese Argumente im Beweis von Theorem REF wiederholt werden m\"ussen.", "Wir nehmen dazu eine Folge $(\\delta _n,{\\bf v}_n)\\subset D$ mit $(\\delta _n,{\\bf v}_n)\\rightarrow (\\delta ,{\\bf v})$ in $Z$ sowie eine Folge $(\\eta _n,{\\bf u}_n)\\in F(\\delta _n,{\\bf v}_n)$ mit $(\\eta _n,{\\bf u}_n)\\rightarrow (\\eta ,{\\bf u})$ in $Z$ als gegeben an.", "Es ist zu zeigen, dass $(\\eta ,{\\bf u})\\in F(\\delta ,{\\bf v})$ .", "Aus (REF ) und der soeben gezeigten relativen Kompaktheit folgern wir f\"ur eine Teilfolge die Konvergenzen $\\begin{aligned}\\eta _n&\\rightarrow \\eta &&\\text{gleichm\"a\"sig und schwach$^*$ in }L^\\infty (I,H^2_0(M)),\\\\\\partial _t\\eta _n&\\rightarrow \\partial _t\\eta &&\\text{ in}L^2(I\\times M)\\text{ und schwach$^*$ in }L^\\infty (I,L^2(M)),\\\\{\\bf u}_n&\\rightarrow {\\bf u}&&\\text{ in}L^2(I\\times \\mathbb {R}^3)\\text{ und schwach$^*$ in } L^\\infty (I,L^2(\\mathbb {R}^3)),\\\\\\nabla {\\bf u}_n&\\rightarrow \\nabla {\\bf u}&&\\text{ schwach in }L^2(I\\times \\mathbb {R}^3).\\end{aligned}$ Die zun\"achst auf $\\Omega _{\\mathcal {R}\\delta _n}^I$ bzw.", "$\\Omega _{\\mathcal {R}\\delta }^I$ definierten Felder $\\nabla {\\bf u}_n$ und $\\nabla {\\bf u}$ werden dabei durch 0 auf $I\\times \\mathbb {R}^3$ fortgesetzt.", "Aus der Unterhalbstetigkeit der Normen bez\"uglich schwacher bzw.", "schwach*-Konvergenz folgern wir die Absch\"atzung (REF ) f\"ur $\\eta $ und ${\\bf u}$ .", "Die Identit\"at $\\eta (0,\\cdot )=\\eta _0$ ist aufgrund der gleichm\"a\"sigen Konvergenz von $(\\eta _n)$ offensichtlich.", "Desweiteren gilt $\\begin{aligned}\\partial _t\\eta _n\\,\\nu =\\operatorname{tr_{\\mathcal {R}\\delta _n}}{\\bf u}_n = {\\bf w}_n|_{\\partial \\Omega },\\end{aligned}$ wobei ${\\bf w}_n:={\\bf u}_n\\circ \\Psi _{\\mathcal {R}\\delta _n}$ .", "Wegen Lemma REF konvergiert eine Teilfolge von $({\\bf w}_n)$ schwach gegen ein ${\\bf w}$ in $L^2(I,W^{1,1}(\\Omega ))$ , sodass die Identit\"at $\\partial _t\\eta \\,\\nu ={\\bf w}|_{\\partial \\Omega }$ folgt.", "Der erste Term auf der rechten Seite der Absch\"atzung $\\Vert {\\bf w}_n-{\\bf u}\\circ \\Psi _{\\mathcal {R}\\delta }\\Vert _{L^1(I\\times \\Omega )}\\le \\Vert ({\\bf u}_n-{\\bf u})\\circ \\Psi _{\\mathcal {R}\\delta _n}\\Vert _{L^1(I\\times \\Omega )}+\\Vert {\\bf u}\\circ \\Psi _{\\mathcal {R}\\delta _n}-{\\bf u}\\circ \\Psi _{\\mathcal {R}\\delta }\\Vert _{L^1(I\\times \\Omega )}$ wird f\"ur gro\"se $n$ wegen Lemma REF und (REF )$_3$ klein, w\"ahrend der zweite Term wegen Bemerkung REF klein wird.", "Es folgt ${\\bf w}={\\bf u}\\circ \\Psi _{\\mathcal {R}\\delta }$ , sodass wir insgesamt die Identit\"at $\\partial _t\\eta \\,\\nu =\\operatorname{tr_{\\mathcal {R}\\delta }}{\\bf u}$ gezeigt haben.", "Es bleibt lediglich zu zeigen, dass Gleichung (REF ) erf\"ullt ist.", "F\"ur alle $n$ und alle Testfunktionen $(b_n,\\varphi _n)\\in T_{\\mathcal {R}\\delta _n}^I$ gilt $\\begin{aligned}&- \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta _n(t)}}{\\bf u}_n\\cdot \\partial _t\\varphi _n\\ dxdt - \\frac{1}{2}\\int _I\\int _M\\partial _t\\eta _n\\, \\partial _t\\mathcal {R}\\delta _n\\, b_n\\, \\gamma (\\mathcal {R}\\delta _n)\\ dAdt\\\\&+\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta _n(t)}}(\\mathcal {R}{\\bf v}_n\\cdot \\nabla ){\\bf u}_n\\cdot \\varphi _n\\ dxdt -\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta _n(t)}}(\\mathcal {R}{\\bf v}_n\\cdot \\nabla )\\varphi _n\\cdot {\\bf u}_n\\ dxdt\\\\& + \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta _n(t)}}\\nabla {\\bf u}_n:\\nabla \\varphi _n\\ dxdt-\\int _I\\int _M\\partial _t\\eta _n\\, \\partial _tb_n\\ dAdt + 2\\int _IK(\\eta _n,b_n)\\ dt \\\\&\\hspace{113.81102pt}=\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta _n(t)}}{\\bf f}\\cdot \\varphi _n\\ dxdt + \\int _I\\int _Mg\\, b_n\\ dAdt\\\\&\\hspace{128.0374pt} +\\int _{\\Omega _{\\mathcal {R}\\eta _0}}\\mathcal {R}{\\bf u}_0\\cdot \\varphi _n(0,\\cdot )\\ dx +\\int _M\\mathcal {R}\\eta _1\\, b_n(0,\\cdot )\\ dA.\\end{aligned}$ Wir k\"onnen hier nicht ohne weiteres zum Grenzwert \"ubergehen, weil die Testfunktionen vom Index $n$ abh\"angen.", "F\"ur $(b,\\varphi )\\in T_{\\mathcal {R}\\delta }^I$ verwenden wir deshalb die speziellen Testfunktionen $(b_n,\\varphi _n):=(\\mathcal {M}_{\\mathcal {R}\\delta _n}b,\\mathcal {F}_{\\mathcal {R}\\delta _n}\\mathcal {M}_{\\mathcal {R}\\delta _n}b)\\in T_{\\mathcal {R}\\delta _n}^I$ wie sie auch im Beweis von Proposition REF zur Anwendung kamen.", "Die Zahl $\\alpha $ in Propositon REF w\"ahlen wir dabei identisch $(\\alpha +\\kappa )/2$ .", "Mit diesen Testfunktionen k\"onnen wir wegen der Konvergenzen (REF ) und wegen Lemma REF $(1.b)$ , $(2.b)$ in (REF ) zum Grenzwert \"ubergehen, sodass wir die G\"ultigkeit von (REF ) f\"ur $(b,\\varphi )=(b,\\mathcal {F}_{\\mathcal {R}\\delta }b)\\in T_{\\mathcal {R}\\delta }^I$ erhalten.", "Per Definition von $T_{\\mathcal {R}\\delta }^I$ bleibt lediglich zu zeigen, dass (REF ) f\"ur Testfunktionen $(0,\\varphi )\\in T_{\\mathcal {R}\\delta }^I$ mit $\\varphi (T,\\cdot )=0$ und $\\operatorname{supp}\\varphi \\subset \\Omega _{\\mathcal {R}\\delta }^{\\bar{I}}$ gilt.", "Aufgrund der gleichm\"a\"sigen Konvergenz von $(\\mathcal {R}\\delta _n)$ gilt f\"ur hinreichend gro\"se $n$ in diesem Fall aber $(0,\\varphi )\\in T_{\\mathcal {R}\\delta _n}^I$ , sodass wir wiederum in (REF ) zum Grenzwert \"ubergehen k\"onnen.", "Theorem REF garantiert uns nun die Existenz eines Fixpunktes von $F$ , d.h. es existiert ein Tupel $(\\eta ,{\\bf u})\\in D$ mit $(\\eta ,{\\bf u})\\in F(\\eta ,{\\bf u})$ .", "Dies zeigt die Behauptung der Proposition.$\\Box $ Wir k\"onnen nun unsere zentrale Behauptung beweisen, indem wir den Regularisierungsparameter $\\epsilon $ in Definition REF gegen 0 gehen lassen.", "Beweis: (von Theorem REF ) Wir haben gezeigt, dass ein $T>0$ und f\"ur jedes $\\epsilon =1/n$ , $n\\in \\mathbb {N}$ hinreichend gro\"s, eine schwache L\"osung $(\\eta _\\epsilon ,{\\bf u}_\\epsilon )$ des regularisierten Systems zum Parameter $\\epsilon $ auf dem Intervall $I=(0,T)$ existieren.", "Aus Absch\"atzung (REF ) und der kompakten Einbettung $Y^I\\hookrightarrow \\hookrightarrow C(\\bar{I}\\times \\partial \\Omega )$ erhalten wir f\"ur eine Teilfolge die Konvergenzen $\\begin{aligned}\\eta _\\epsilon ,\\,\\mathcal {R}_\\epsilon \\eta _\\epsilon &\\rightarrow \\eta &&\\text{ gleichm\"a\"sig undschwach$^*$ in }L^\\infty (I,H^2_0(M)),\\\\\\partial _t\\eta _\\epsilon ,\\, \\partial _t\\mathcal {R}_\\epsilon \\eta _\\epsilon &\\rightarrow \\partial _t\\eta &&\\text{ schwach$^*$in }L^\\infty (I,L^2(M)),\\\\{\\bf u}_\\epsilon &\\rightarrow {\\bf u}&&\\text{ schwach$^*$ in }L^\\infty (I,L^2(\\mathbb {R}^3)),\\\\\\nabla {\\bf u}_\\epsilon &\\rightarrow \\nabla {\\bf u}&&\\text{ schwach in }L^2(I\\times \\mathbb {R}^3).\\end{aligned}$ Die zun\"achst auf $\\Omega _{\\mathcal {R}_\\epsilon \\eta _\\epsilon }^I$ bzw.", "$\\Omega _{\\eta }^I$ definierten Felder $\\nabla {\\bf u}_\\epsilon $ und $\\nabla {\\bf u}$ werden dabei durch 0 auf $I\\times \\mathbb {R}^3$ fortgesetzt.", "Die Konvergenzen der Folge $(\\mathcal {R}_\\epsilon \\eta _\\epsilon )$ schlie\"sen wir unter Verwendung der Faltungsungleichung aus der Absch\"atzung $\\Vert \\mathcal {R}_\\epsilon \\eta _\\epsilon -\\eta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}\\le \\Vert \\mathcal {R}_\\epsilon (\\eta _\\epsilon -\\eta )\\Vert _{L^\\infty (I\\times \\partial \\Omega )}+\\Vert \\mathcal {R}_\\epsilon \\eta -\\eta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}.$ Wir k\"onnen nun den Beweis von Proposition REF fast w\"ortlich wiederholen, um die Konvergenzen $\\begin{aligned}\\partial _t\\eta _\\epsilon &\\rightarrow \\partial _t\\eta &&\\text{ in }L^2(I\\times M),\\\\{\\bf u}_\\epsilon &\\rightarrow {\\bf u}&&\\text{ in }L^2(I\\times \\mathbb {R}^3)\\end{aligned}$ zu zeigen.", "Desweiteren erhalten wir wie im Beweis von Proposition REF die Identit\"at $\\operatorname{tr_\\eta }{\\bf u}=\\partial _t\\eta \\,\\nu $ .", "Weiterhin zeigen wir unter Verwendung von (REF ), der Interpolationsungleichung $\\Vert {\\bf u}_\\epsilon -{\\bf u}\\Vert _{L^2(I,L^4(\\mathbb {R}^3))}\\le \\Vert {\\bf u}_\\epsilon -{\\bf u}\\Vert _{L^2(I\\times \\mathbb {R}^3)}^{1/6}\\,\\Vert {\\bf u}_\\epsilon -{\\bf u}\\Vert _{L^2(I,L^5(\\mathbb {R}^3))}^{5/6}$ und Korollar REF die Konvergenz $\\begin{aligned}{\\bf u}_\\epsilon \\rightarrow {\\bf u}\\text{ in }L^2(I,L^4(\\mathbb {R}^3)).\\end{aligned}$ Es folgt $\\begin{aligned}\\partial _t\\mathcal {R}_\\epsilon \\eta _\\epsilon &\\rightarrow \\partial _t\\eta &&\\text{ in }L^2(I\\times \\partial \\Omega ),\\\\\\mathcal {R}_\\epsilon {\\bf u}_\\epsilon &\\rightarrow {\\bf u}&&\\text{ in }L^2(I,L^4(\\mathbb {R}^3)).\\end{aligned}$ Aus der Unterhalbstetigkeit der Normen folgern wir die Absch\"atzung (REF ), und aufgrund der gleichm\"a\"sigen Konvergenz von $(\\eta _\\epsilon )$ ist die Identit\"at $\\eta (0,\\cdot )=\\eta _0$ offensichtlich.", "F\"ur alle $\\epsilon $ und alle $(b_\\epsilon ,\\varphi _\\epsilon )\\in T_{\\mathcal {R}_\\epsilon \\eta _\\epsilon }^I$ gilt $\\begin{aligned}&- \\int _I\\int _{\\Omega _{R_\\epsilon \\eta _\\epsilon (t)}} {\\bf u}_\\epsilon \\cdot \\partial _t\\varphi _\\epsilon \\ dxdt -\\frac{1}{2}\\int _I\\int _M\\partial _t\\eta _\\epsilon \\,\\partial _t\\mathcal {R}_\\epsilon \\eta _\\epsilon \\, b_\\epsilon \\, \\gamma (\\mathcal {R}_\\epsilon \\eta _\\epsilon )\\ dAdt\\\\&+\\frac{1}{2}\\int _I\\int _{\\Omega _{R_\\epsilon \\eta _\\epsilon (t)}}(\\mathcal {R}_\\epsilon {\\bf u}_\\epsilon \\cdot \\nabla ){\\bf u}_\\epsilon \\cdot \\varphi _\\epsilon \\ dxdt -\\frac{1}{2}\\int _I\\int _{\\Omega _{R_\\epsilon \\eta _\\epsilon (t)}}(\\mathcal {R}_\\epsilon {\\bf u}_\\epsilon \\cdot \\nabla )\\varphi _\\epsilon \\cdot {\\bf u}_\\epsilon \\ dxdt\\\\& + \\int _I\\int _{\\Omega _{R_\\epsilon \\eta _\\epsilon (t)}} \\nabla {\\bf u}_\\epsilon :\\nabla \\varphi _\\epsilon \\ dxdt-\\int _I\\int _M\\partial _t\\eta _\\epsilon \\, \\partial _tb_\\epsilon \\ dAdt + 2\\int _I K(\\eta _\\epsilon ,b_\\epsilon )\\ dt \\\\&\\hspace{142.26378pt}=\\int _I\\int _{\\Omega _{R_\\epsilon \\eta _\\epsilon (t)}} {\\bf f}\\cdot \\varphi _\\epsilon \\ dxdt +\\int _I\\int _Mg\\, b_\\epsilon \\ dAdt\\\\&\\hspace{156.49014pt} +\\int _{\\Omega _{\\eta _0^\\epsilon }}{\\bf u}_0^\\epsilon \\cdot \\varphi _\\epsilon (0,\\cdot )\\ dx +\\int _M\\eta _1^\\epsilon \\, b_\\epsilon (0,\\cdot )\\ dA.\\end{aligned}$ F\"ur $(b,\\varphi )\\in T_\\eta ^I$ verwenden wir wie im Beweis von Proposition REF die speziellen Testfunktionen $(b_\\epsilon ,\\varphi _\\epsilon ):=(\\mathcal {M}_{\\mathcal {R}_\\epsilon \\eta _\\epsilon }b,\\mathcal {F}_{\\mathcal {R}_\\epsilon \\eta _\\epsilon }\\mathcal {M}_{\\mathcal {R}_\\epsilon \\eta _\\epsilon }b)\\in T_{\\mathcal {R}_\\epsilon \\eta _\\epsilon }^I$ .", "Die Zahl $\\alpha $ in Propositon REF gen\"uge dabei der Ungleichung $\\sup _{\\epsilon }\\Vert \\eta _\\epsilon \\Vert _{L^\\infty (I\\times M)}<\\alpha <\\kappa $ .", "Da die Felder $(\\nabla \\,\\mathcal {F}_{\\mathcal {R}_\\epsilon \\eta _\\epsilon }\\mathcal {M}_{\\mathcal {R}_\\epsilon \\eta _\\epsilon }b)$ in $L^\\infty (I,L^{2}(B_\\alpha ))$ beschr\"ankt sind, gilt $\\begin{aligned}\\nabla \\,\\mathcal {F}_{\\mathcal {R}_\\epsilon \\eta _\\epsilon }\\mathcal {M}_{\\mathcal {R}_\\epsilon \\eta _\\epsilon }b\\rightarrow \\nabla \\,\\mathcal {F}_{\\eta }\\mathcal {M}_{\\eta }b\\quad \\text{ schwach$^*$ in}L^\\infty (I,L^2(B_\\alpha )).\\end{aligned}$ Unter Ausnutzung von (REF ), (REF ), (REF ), (REF ) und (REF ) sowie von Lemma REF $(1.b)$ , $(2.b)$ und (REF ) k\"onnen wir nun in (REF ) den Grenz\"ubergang vollziehen.", "(REF )$_1$ und (REF )$_2$ werden dabei f\"ur den zweiten Term ben\"otigt, w\"ahrend (REF ), (REF )$_3$ und (REF ) den Grenz\"ubergang im dritten Term und vierten Term erm\"oglichen.", "Wir k\"onnen also auf die G\"ultigkeit von (REF ) f\"ur $(b,\\varphi )=(b,\\mathcal {F}_{\\eta }b)\\in T_{\\eta }^I$ schlie\"sen. Der Grenz\"ubergang f\"ur Testfunktionen $(0,\\varphi )\\in T_\\eta ^I$ mit $\\varphi (T,\\cdot )=0$ und $\\operatorname{supp}\\varphi \\subset \\Omega ^{\\bar{I}}_{\\eta }$ ist aber jetzt klar.", "Das Existenzintervall der konstruierten L\"osung h\"angt lediglich von der Supremumsnorm von $\\eta $ zum Anfangszeitpunkt sowie der Schranke f\"ur die H\"older-Norm von $\\eta $ ab.", "Nach Konstruktion ist $\\Vert \\eta \\Vert _{L^\\infty (I\\times M)}<\\kappa $ .", "Die Gr\"o\"sen $\\Vert {\\bf u}(t)\\Vert _{L^2(\\Omega _{\\eta (t)})}$ , $\\Vert \\eta (t)\\Vert _{H^2(M)}$ und $\\Vert \\partial _t\\eta (t)\\Vert _{L^2(M)}$ sind gleichm\"a\"sig f\"ur fast alle $t\\in I$ beschr\"ankt.", "Wenn wir nun f\"ur fast alle $t$ L\"osungen zu diesen Anfangsdaten konstruieren, sind die H\"older-Normen der Auslenkungen des Randes wegen (REF ) unabh\"angig vom Anfangszeitpunkt $t$ nach oben und somit die Lebensdauer der L\"osungen nach unten beschr\"ankt.", "Liegt $t\\in I=(0,T)$ hinreichend nahe bei $T$ , erhalten wir also eine schwache L\"osung auf einem Intervall $(t,\\widetilde{T})$ , $\\widetilde{T}>T$ .", "Verketten dieser L\"osung mit $(\\eta ,{\\bf u})$ an der Stelle $t$ ergibt wegen Bemerkung REF eine L\"osung $(\\widetilde{\\eta },\\widetilde{{\\bf u}})$ auf dem Intervall $(0,\\widetilde{T})$ .", "Zudem erf\"ullt $(\\widetilde{\\eta },\\widetilde{{\\bf u}})$ die Absch\"atzung (REF ) auf dem Intervall $(0,\\widetilde{T})$ , denn f\"ur $t_0<t<\\widetilde{T}$ gilt $\\begin{aligned}&\\Vert \\widetilde{{\\bf u}}(t,\\cdot )\\Vert _{L^2(\\Omega _{\\eta (t)})}^2 +\\int _0^t\\Vert \\nabla \\widetilde{{\\bf u}}(s,\\cdot )\\Vert ^p_{L^p(\\Omega _{\\widetilde{\\eta }(s)})}\\ ds +\\Vert \\partial _t\\widetilde{\\eta }(t,\\cdot )\\Vert _{L^2(M)}^2 + \\Vert \\widetilde{\\eta }(t,\\cdot )\\Vert _{H^2(M)}^2 \\\\&\\hspace{28.45274pt}\\le \\big (\\Vert {\\bf u}(t_0)\\Vert _{L^2(\\Omega _{\\widetilde{\\eta }(t_0)})}^2 +\\Vert \\partial _t\\eta (t_0)\\Vert _{L^2(M)}^2 +\\Vert \\eta (t_0)\\Vert _{H^2(M)}^2\\big )e^{c(t-t_0)}\\\\&\\hspace{42.67912pt} + \\int _{t_0}^t\\big (\\Vert {\\bf f}(s,\\cdot )\\Vert _{L^2(\\Omega _{\\eta (s)})}^2 + \\Vert g(s,\\cdot )\\Vert _{L^2(M)}^2 \\big )e^{c(t-s)}\\ ds + \\int _0^{t_0}\\Vert \\nabla {\\bf u}(s,\\cdot )\\Vert ^p_{L^p(\\Omega _{\\widetilde{\\eta }(s)})}\\ ds\\\\&\\hspace{28.45274pt}\\le \\big (\\Vert {\\bf u}_0\\Vert _{L^2(\\Omega _{\\eta _0})}^2 + \\Vert \\eta _1\\Vert _{L^2(M)}^2 +\\Vert \\eta _0\\Vert _{H^2(M)}^2\\big )e^{ct}\\\\&\\hspace{142.26378pt} + \\int _0^t\\big (\\Vert {\\bf f}(s,\\cdot )\\Vert _{L^2(\\Omega _{\\eta (s)})}^2 + \\Vert g(s,\\cdot )\\Vert _{L^2(M)}^2\\big )e^{c(t-s)}\\ ds.\\end{aligned}$ Durch Wiederholen dieses Vorgangs erhalten wir eine maximale Zeit $T^*\\in (0,\\infty ]$ und ein Tupel $(\\eta ,{\\bf u})$ , das f\"ur jedes $T<T^*$ L\"osung auf dem Intervall $(0,T)$ ist und die Absch\"atzung (REF ) erf\"ullt.", "Ist $T^*$ endlich, so ist aufgrund dieser Absch\"atzung die H\"older-Norm von $\\eta $ in $[0,T^*]\\times M$ beschr\"ankt, und somit muss $\\Vert \\eta (t,\\cdot )\\Vert _{L^\\infty (M)}\\nearrow \\kappa $ f\"ur $t\\nearrow T^*$ gelten.", "$\\Box $" ], [ "Verallgemeinerte Newton'sche Fluide", "Wir wollen in diesem Abschnitt erste Schritte in Richtung einer interessanten und in hohem Ma\"se nichttrivialen Verallgemeinerung der bisherigen Resultate skizzieren, die eng mit dem bereits diskutierten Eindeutigkeitsproblem zusammenh\"angen.", "Bisher war der viskose Spannungstensor eine lineare Funktion des Scherratentensors $D{\\bf u}$ .", "F\"ur eine wichtige Klasse von Fluiden, die verallgemeinerten Newton'schen Fluide, die eine scherratenabh\"angige Viskosit\"at besitzen, gilt dieser lineare Zusammenhang nicht mehr.", "Ein wichtiges Beispiel daf\"ur ist Blut.", "Blut zeigt bei kleinen Scherraten eine hohe Viskosit\"at, was f\"ur die schnelle Gerinnung wichtig ist.", "Flie\"st es aber durch d\"unne Adern, so entstehen gr\"o\"sere Scherraten, und die Viskosit\"at wird kleiner, was den Durchfluss bei konstantem Druck erh\"oht.", "F\"ur eine detaillierte Diskussion verallgemeinerter Newton'scher Fluide verweisen wir auf [41] und die dort angef\"uhrten Referenzen.", "Typische Beispiele f\"ur die viskosen Spannungstensoren solcher Fluide sind $\\begin{aligned}S(D{\\bf u})&=\\mu _0(\\delta +|D{\\bf u}|)^{p-2}D{\\bf u},\\\\S(D{\\bf u})&=\\mu _0(\\delta ^2+|D{\\bf u}|^2)^{\\frac{p-2}{2}}D{\\bf u}\\end{aligned}$ mit $\\mu _0>0$ , $\\delta \\ge 0$ und $1<p<\\infty $ .", "Wir sind deshalb an Abbildungen $S$ mit $p$ -Struktur interessiert, d.h. f\"ur ein $1<p<\\infty $ und ein $\\delta \\ge 0$ gelte $S: M_{sym}\\rightarrow M_{sym}$ stetig, Wachstum: $S(D)\\le c_0(\\delta +|D|)^{p-2}|D|$ f\"ur alle $D\\in M_{sym}$ mit $c_0>0$ , Koerzivit\"at: $S(D):D\\ge c_1(\\delta +|D|)^{p-2}|D|^2$ f\"ur alle $D\\in M_{sym}$ mit $c_1>0$ , Strikte Monotonie: $(S(D)-S(E)):(D-E)>0$ f\"ur alle $D,E\\in M_{sym}, D\\ne E$ .", "Dabei bezeichnet $M_{sym}$ den Raum der reellen, symmetrischen $3\\times 3$ -Matrizen.", "Wir geben Daten genau wie im Falle eines linearen Spannungstensors vor, wobei wir nun ${\\bf f}\\in L^{p^{\\prime }}_\\text{loc}([0,\\infty )\\times \\mathbb {R}^3)$ annehmen.", "Anstelle von (REF )$_1$ stellen wir die Gleichung $\\partial _t {\\bf u}+ ({\\bf u}\\cdot \\nabla ){\\bf u}= \\operatorname{div}(S(D{\\bf u}) - \\pi \\operatorname{id}) + {\\bf f}\\ \\mbox{ in }\\ \\Omega _{\\eta }^I$ auf, w\"ahrend die Kraftdichte ${\\bf F}$ auf der rechten Seite von (REF )$_1$ die Form ${\\bf F}(t,\\cdot ) = \\big (-S(D{\\bf u}(t,\\cdot ))\\,\\nu _{\\eta (t)} +\\pi (t,\\cdot )\\,\\nu _{\\eta (t)}\\big )\\circ \\Phi _{\\eta (t)}\\, |\\det d\\Phi _{\\eta (t)}|$ annimmt.", "Den Rest der Gleichungen \"ubernehmen wir unver\"andert.", "Was die Analysis der Fluidgleichungen bei festem Rand (bzw.", "auf dem Torus) betrifft, verweisen wir auf [24] und die dortigen Referenzen; siehe insbesondere auch [10].", "Wir wollen uns mit einigen knappen Anmerkungen begn\"ugen.", "Abbildungen $S$ mit $p$ -Struktur definieren in geeigneten Sobolev-R\"aumen durch die Zuordnung $v\\mapsto \\operatorname{div}S(\\nabla v)$ monotone Operatoren.", "Diese Operatoren spielen als Erzeuger nichtlinearer Halbgruppen eine prominente Rolle; vgl.", "[50], [43], [6].", "Allerdings ist die abstrakte Theorie auf die Fluidgleichungen nicht direkt anwendbar, weil der Wirbelterm die Monotonie zerst\"ort.", "Dennoch l\"asst sich durch Kombination eines Kompaktheitsarguments und eines Arguments aus der Theorie monotoner Operatoren f\"ur hinreichend gro\"se $p$ die Existenz zeitlich globaler, schwacher L\"osungen zeigen.", "Die Einschr\"ankung an den Exponenten kommt dadurch zustande, dass das Argument eine L\"osung mit Zeitableitung im Dualraum ihrer eigenen Regularit\"atsklasse ben\"otigt und der Wirbelterm dabei restriktiv wirkt.", "F\"ur kleinere Exponenten sind deshalb andere Techniken vonn\"oten.", "Wir k\"onnen genau wie zuvor, unter Verwendung der Koerzivit\"at von $S$ , formale Energieabsch\"atzungen herleiten.", "Dabei erhalten wir anstelle von (REF ) die Ungleichung $\\begin{aligned}&\\Vert {\\bf u}(t,\\cdot )\\Vert _{L^2(\\Omega _{\\eta (t)})}^2 + \\int _0^t\\Vert D{\\bf u}(s,\\cdot )\\Vert ^p_{L^p(\\Omega _{\\eta (s)})}\\ ds +\\Vert \\partial _t\\eta (t,\\cdot )\\Vert _{L^2(M)}^2 + \\Vert \\eta (t,\\cdot )\\Vert _{H^2(M)}^2 \\\\&\\hspace{28.45274pt}\\le \\big (\\Vert {\\bf u}_0\\Vert _{L^2(\\Omega _{\\eta _0})}^2 + \\Vert \\eta _1\\Vert _{L^2(M)}^2 +\\Vert \\eta _0\\Vert _{H^2(M)}^2\\big )\\,e^{ct}\\\\&\\hspace{113.81102pt} + \\int _0^t\\big (\\Vert {\\bf f}(s,\\cdot )\\Vert _{L^{p^{\\prime }}(\\Omega _{\\eta (s)})}^{p^{\\prime }} + \\Vert g(s,\\cdot )\\Vert _{L^2(M)}^2\\big )\\,e^{c(t-s)}\\ ds.\\end{aligned}$ Üblicherweise wird der symmetrische Gradient in dieser Absch\"atzung mit Hilfe einer Korn-Ungleichung $\\Vert \\nabla {\\bf u}(s,\\cdot )\\Vert ^p_{L^p}\\le c\\,\\Vert D{\\bf u}(s,\\cdot )\\Vert ^p_{L^p}$ durch den vollen Gradienten ersetzt.", "Die Korn-Ungleichung ist allerdings in Gebieten, deren Rand nicht Lipschitz-stetig ist, im Allgemeinen falsch; vgl.", "[1].", "Deshalb sehen wir davon ab, begn\"ugen uns mit der Absch\"atzung $\\begin{aligned}\\Vert \\eta \\Vert _{W^{1,\\infty }(I,L^2(M))\\cap L^\\infty (I,H_0^2(M))} +\\Vert {\\bf u}\\Vert _{L^\\infty (I,L^2(\\Omega _{\\eta (t)}))} + \\Vert D{\\bf u}\\Vert _{L^p(\\Omega _{\\eta }^I)}\\le c(T,\\text{Daten})\\end{aligned}$ und setzen f\"ur $\\eta \\in Y^I$ mit $\\Vert \\eta \\Vert _{L^\\infty (I\\times M)}<\\kappa $ $\\begin{aligned}L^p(I,W^{1,p}_{\\operatorname{div},s}(\\Omega _{\\eta (t)}))&:=\\lbrace {\\bf v}\\ |\\ {\\bf v},D{\\bf v}\\in L^p(\\Omega _{\\eta }^I),\\, \\operatorname{div}{\\bf v}=0\\rbrace ,\\\\X^I_{\\eta ,p}&:= L^\\infty (I,L^2(\\Omega _{\\eta (t)}))\\cap L^p(I,W^{1,p}_{\\operatorname{div},s}(\\Omega _{\\eta (t)})).\\end{aligned}$ Den Raum $T^I_{\\eta ,p}$ der Testfunktionen definieren wir so wie $T^I_\\eta $ , wobei wir den Raum $L^2(I,H^2_0(M))$ durch $L^{\\max (2,p)}(I,H^2_0(M))$ und den Raum $H^1(\\Omega _{\\eta }^I)$ durch $H^1(I,L^2(\\Omega _{\\eta (t)}))\\cap L^{\\max (2,p)}(I,W^{1,\\max (2,p)}(\\Omega _{\\eta (t)}))$ ersetzen.", "Durch die Forderung $b\\in L^{p}(I,H^2_0(M))$ stellen wir sicher, dass die Fortsetzung $\\mathcal {F}_\\eta b$ in $L^{p}(I,W^{1,p}(\\Omega _{\\eta (t)}))$ liegt.", "Wir geben nun das Analogon von Definition REF an.", "Wir konstruieren genau wie zuvor modifizierte Anfangswerte $\\mathcal {R}_\\epsilon \\eta _0$ , $\\eta _1^\\epsilon $ und ${\\bf u}^\\epsilon $ und geben ein Zeitintervall $I=(0,T)$ , $T>0$ , und Funktionen ${\\bf v}\\in L^2(I\\times B)$ und $\\delta \\in C(\\bar{I}\\times \\partial \\Omega )$ mit $\\Vert \\delta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}<\\kappa $ und $\\delta (0,\\cdot )=\\eta _0$ vor.", "Zudem setzen wir $\\widetilde{Y}^I:=\\lbrace b\\in Y^I\\ |\\ b\\in H^1(I,H^2_0(M))\\rbrace .$ Wie zuvor unterdr\"ucken wir zun\"achst den Parameter $\\epsilon $ in der Notation.", "Definition 6.1 Ein Tupel $(\\eta ,{\\bf u})$ hei\"st schwache L\"osung des entkoppelten, regularisierten $p$ -Systems zum Argument $(\\delta ,{\\bf v})$ auf dem Intervall $I$ , falls $\\eta \\in \\widetilde{Y}^I$ mit $\\eta (0,\\cdot )=\\eta _0$ , ${\\bf u}\\in X_{\\mathcal {R}\\delta ,p}^I$ mit $\\operatorname{tr_{\\mathcal {R}\\delta }}{\\bf u}=\\partial _t\\eta \\,\\nu $ und $\\begin{aligned}&- \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf u}\\cdot \\partial _t\\varphi \\ dxdt - \\frac{1}{2}\\int _I\\int _M\\partial _t\\eta \\ \\partial _t\\mathcal {R}\\delta \\, b\\, \\gamma (\\mathcal {R}\\delta )\\ dAdt\\\\&+\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla )\\mathcal {R}{\\bf u}\\cdot \\varphi \\ dxdt -\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla )\\mathcal {R}\\varphi \\cdot {\\bf u}\\ dxdt\\\\& + \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}S(D{\\bf u}):D\\varphi \\ dxdt -\\int _I\\int _M\\partial _t\\eta \\, \\partial _tb\\ dAdt + 2\\int _IK(\\eta +\\partial _t\\eta ,b)\\ dt\\\\&\\hspace{142.26378pt}=\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}\\cdot \\varphi \\ dxdt + \\int _I\\int _Mg\\, b\\ dAdt\\\\&\\hspace{156.49014pt} +\\int _{\\Omega _{\\mathcal {R}\\eta _0}}{\\bf u}_0\\cdot \\varphi (0,\\cdot )\\ dx +\\int _M\\eta _1\\, b(0,\\cdot )\\ dA\\end{aligned}$ f\"ur alle Testfunktionen $(b,\\varphi )\\in T_{\\mathcal {R}\\delta ,p}^I$ .", "Aufgrund der Wachstumsbedingung an $S$ ist das Integral $\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}S(D{\\bf u}):D\\varphi \\ dxdt$ endlich.", "Eine wichtige Modifikation in dieser Definition gegen\"uber Definition REF ist der Zusatzterm $2\\int _I K(\\partial _t\\eta ,b)\\ dt,$ den man sich mit einem Faktor $\\epsilon $ versehen denke.", "Dieser entspricht einem zus\"atzlichen Ausdruck $\\operatorname{grad}_{L^2}K(\\partial _t\\eta )$ in der Schalengleichung.", "Ein \"ahnlicher Term wird in [11] zur D\"ampfung der Plattengleichung verwendet.", "Wie bereits angedeutet werden wir die Eindeutigkeit obiger L\"osungen ben\"otigen.", "Im Anschluss an den Beweis von Proposition REF wurde aber klar, dass ein Eindeutigkeitsbeweis aufgrund des gemischten Charakters der Gleichungen schwierig sein kann.", "Der Zusatzterm stellt nun gewisserma\"sen eine \"`Parabolisierung\"' der Schalengleichung dar.", "Insbesondere hat er die h\"ohere Regularit\"at $\\eta \\in \\widetilde{Y}^I$ zur Folge, wodurch, zusammen mit der zus\"atzlichen Regularisierung der Wirbelterme, nun zumindest formal die Zeitableitung von $(\\partial _t\\eta ,{\\bf u})$ im Dualraum von $L^2(I,H^2_0(M)))\\times L^p(I,W^{1,p}_{\\operatorname{div},s}(\\Omega _{\\mathcal {R}\\delta (t)}))$ , der Regularit\"atsklasse von $(\\partial _t\\eta ,{\\bf u})$ , liegt.", "Das Ziel ist, die Eindeutigkeit durch Aufstellen einer Energiegleichung, des Analogons von (REF ), zu beweisen.", "Diese Gleichung spielt auch beim Beweis der Existenz obiger L\"osungen eine wichtige Rolle.", "Augenscheinlich besitzen die schwachen L\"osungen unseres parabolisch-dispersiven Systems nicht gen\"ugend Regularit\"at, um der Energieidentit\"at Sinn zu verleihen.", "Das legt die Vermutung nahe, dass die Energiegleichung f\"ur derartige L\"osungen nicht gilt, was die N\"utzlichkeit der Parabolisierung des Systems unterstreicht.", "Sei nun $(\\eta ,{\\bf u})$ ein schwache L\"osung gem\"a\"s Definition REF , wobei allerdings der Term $S(D{\\bf u})$ durch eine beliebige Matrix $\\xi \\in L^{p^{\\prime }}(\\Omega _{\\mathcal {R}\\delta }^I)$ ersetzt sei.", "Wir m\"ussen die Energiegleichung ohne R\"uckgriff auf die Theorie der Bochner-R\"aume und das Konzept der distributionellen Zeitableitung im Dualraum beweisen.", "Wie im Beweis von Proposition REF wollen wir stattdessen direkt mit der Gleichung arbeiten.", "Wir setzen dazu $\\begin{aligned}V:=L^2(I,H^2_0(M))) \\times L^p(I,W^{1,p}_{\\operatorname{div},s}(\\Omega _{\\mathcal {R}\\delta (t)}))\\end{aligned}$ und stellen die folgende Behauptung auf.", "Behauptung 6.2 Es existieren hinreichend glatte Regularisierungen $(\\eta _k,{\\bf u}_k)_{k\\in \\mathbb {N}}$ mit $\\operatorname{tr_{\\mathcal {R}\\delta }}{\\bf u}_k=\\partial _t\\eta _k\\,\\nu $ und $\\begin{aligned}{\\bf u}_k&\\rightarrow {\\bf u}&&\\text{ in } L^2(\\Omega _{\\mathcal {R}\\delta }^I),\\\\(\\partial _t\\eta _k,{\\bf u}_k)&\\rightarrow (\\partial _t\\eta ,{\\bf u})&&\\text{ in } V,\\\\(\\partial _t^2\\eta _k,\\partial _t{\\bf u}_k)&\\rightarrow \\Sigma &&\\text{ in } V^{\\prime }.\\end{aligned}$ Dabei ist die \"`Zeitableitung\"' $(\\partial _t^2\\eta _k,\\partial _t{\\bf u}_k)\\in V^{\\prime }$ definiert durch $\\begin{aligned}\\langle (\\partial _t^2\\eta _k,\\partial _t{\\bf u}_k),(b,\\varphi )\\rangle _V&:=\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\partial _t{\\bf u}_k\\cdot \\varphi \\ dxdt +\\frac{1}{2}\\int _I\\int _M\\partial _t\\eta _k\\,\\partial _t\\mathcal {R}\\delta \\, b\\, \\gamma (\\mathcal {R}\\delta )\\ dAdt\\\\&\\hspace{14.22636pt} + \\int _I\\int _M\\partial _t^2\\eta _k\\, b\\ dAdt,\\\\\\end{aligned}$ w\"ahrend $\\Sigma \\in V^{\\prime }$ durch $\\begin{aligned}\\langle \\Sigma ,(b,\\varphi )\\rangle _V&:=-\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla )\\mathcal {R}{\\bf u}\\cdot \\varphi \\ dxdt +\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla )\\mathcal {R}\\varphi \\cdot {\\bf u}\\ dxdt\\\\&\\hspace{15.6491pt} - \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\xi :D\\varphi \\ dxdt - 2\\int _I K(\\eta +\\partial _t\\eta ,b)\\ dt\\\\&\\hspace{15.6491pt} + \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}\\cdot \\varphi \\ dxdt +\\int _I\\int _Mg\\, b\\ dAdt\\\\\\end{aligned}$ gegeben ist.", "Die Konstruktion derartiger Approximationen sollte zwar keine gr\"o\"seren Probleme bereiten, d\"urfte aber aufw\"andig und technisch sein.", "Wir wollen sie in dieser Arbeit nicht durchf\"uhren.", "Offenbar gilt f\"ur $s,t\\in \\bar{I}$ , $s<t$ $\\begin{aligned}\\langle (\\partial _t^2\\eta _k,\\partial _t{\\bf u}_k),(\\partial _t\\eta _k,{\\bf u}_k)\\chi _{(s,t)}\\rangle _V&=\\frac{1}{2} \\int _{\\Omega _{\\mathcal {R}\\delta (t)}} |{\\bf u}_k(t,\\cdot )|^2\\ dx +\\frac{1}{2} \\int _M|\\partial _t\\eta _k(t,\\cdot )|^2\\ dA\\\\&\\hspace{14.22636pt}-\\frac{1}{2} \\int _{\\Omega _{\\mathcal {R}\\delta (s)}} |{\\bf u}_k(s,\\cdot )|^2\\ dx +\\frac{1}{2} \\int _M|\\partial _t\\eta _k(s,\\cdot )|^2\\ dA.\\end{aligned}$ Wenn wir in dieser Gleichung $\\eta _k$ durch $\\eta _k-\\eta _l$ und ${\\bf u}_k$ durch ${\\bf u}_k-{\\bf u}_l$ ersetzen und die resultierende Identit\"at bez\"uglich $s$ integrieren, so folgt $\\begin{aligned}&\\int _{\\Omega _{\\mathcal {R}\\delta (t)}} |({\\bf u}_k-{\\bf u}_l)(t,\\cdot )|^2\\ dx + \\int _M|\\partial _t(\\eta _k-\\eta _l)(t,\\cdot )|^2\\ dA \\\\&\\hspace{14.22636pt}\\le c\\,\\big (\\Vert (\\partial _t^2(\\eta _k-\\eta _l),\\partial _t({\\bf u}_k-{\\bf u}_l))\\Vert _{V^{\\prime }}^2+\\Vert (\\partial _t(\\eta _k-\\eta _l),{\\bf u}_k-{\\bf u}_l)\\Vert _V^2 +\\Vert {\\bf u}_k-{\\bf u}_l\\Vert _{L^2(\\Omega _{\\mathcal {R}\\delta }^I)}^2\\big ).\\end{aligned}$ Setzen wir die Felder ${\\bf u}_k$ und ${\\bf u}$ durch 0 auf $I\\times \\mathbb {R}^3$ fort, so folgt aus dieser Absch\"atzung zusammen mit den Konvergenzen (REF ), dass die Folgen $({\\bf u}_k)$ und $(\\partial _t\\eta _k)$ gegen ${\\bf u}$ in $C(\\bar{I}, L^2(\\mathbb {R}^3))$ bzw.", "gegen $(\\partial _t\\eta )$ in $C(\\bar{I},L^2(M))$ konvergieren.", "Setzen wir in (REF ) $s=0$ , so konvergiert die linke Seite gegen $\\begin{aligned}\\langle \\Sigma ,(\\partial _t\\eta ,{\\bf u})\\chi _{(0,t)}\\rangle _V&=- \\int _0^t\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\xi :D{\\bf u}\\ dxds - \\int _0^t\\frac{d}{ds} K(\\eta ) + 2 K(\\partial _t\\eta )\\ ds \\\\&\\hspace{14.22636pt} + \\int _0^t\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}\\cdot {\\bf u}\\ dxds +\\int _0^t\\int _Mg\\, \\partial _t\\eta \\ dAds,\\\\\\end{aligned}$ w\"ahrend die rechte Seite gegen die Funktion $\\begin{aligned}& \\frac{1}{2} \\int _{\\Omega _{\\mathcal {R}\\delta (t)}} |{\\bf u}(t,\\cdot )|^2\\ dx +\\frac{1}{2} \\int _M|\\partial _t\\eta (t,\\cdot )|^2\\ dA\\\\&-\\frac{1}{2} \\int _{\\Omega _{\\mathcal {R}\\delta (0)}} |{\\bf u}(0,\\cdot )|^2\\ dx +\\frac{1}{2} \\int _M|\\partial _t\\eta (0,\\cdot )|^2\\ dA\\end{aligned}$ konvergiert.", "Mit der Definition $E_{\\eta ,{\\bf u}}(t):= \\frac{1}{2} \\int _{\\Omega _{\\mathcal {R}\\delta (t)}} |{\\bf u}(t,\\cdot )|^2\\ dx + \\frac{1}{2} \\int _M|\\partial _t\\eta (t,\\cdot )|^2\\ dA + K(\\eta (t,\\cdot ))$ gilt also f\"ur alle $t\\in \\bar{I}$ die Energiegleichung $\\begin{aligned}E_{\\eta ,{\\bf u}}(t)-E_{\\eta ,{\\bf u}}(0)&=- \\int _0^t\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\xi :D{\\bf u}\\ dxds - 2\\int _0^t K(\\partial _t\\eta )\\ ds \\\\&\\hspace{14.22636pt} + \\int _0^t\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}\\cdot {\\bf u}\\ dxds +\\int _0^t\\int _Mg\\, \\partial _t\\eta \\ dAds.\\end{aligned}$ Im Unterschied zur Energiebilanz (REF ) tritt hier durch die Parabolisierung des Systems ein zweiter dissipativer Term auf.", "Mithin ist der korrespondierende Ausdruck in der Schalengleichung als D\"ampfungsterm zu interpretieren.", "Wir wollen noch zeigen, dass $\\begin{aligned}E_{\\eta ,{\\bf u}}(0)=\\frac{1}{2} \\int _{\\Omega _{\\mathcal {R}\\eta _0}} |{\\bf u}_0|^2\\ dx + \\frac{1}{2} \\int _M|\\eta _1|^2\\ dA + K(\\eta _0)\\end{aligned}$ gilt.", "Wir wissen bereits, dass $\\eta \\in C(\\bar{I},H^2_0(M))$ und $\\eta (0,\\cdot )=\\eta _0$ gilt, sodass nur die ersten beiden Terme zu identifizieren sind.", "Wenn wir analog zum Beweis von (REF ) vorgehen, k\"onnen wir mit Hilfe der Stetigkeit von ${\\bf u}$ und $\\partial _t\\eta $ zeigen, dass Gleichung (REF ) mit ${\\bf u}(0,\\cdot )$ und $\\partial _t\\eta (0,\\cdot )$ anstelle von ${\\bf u}_0$ und $\\eta _1$ gilt, woraus wir (REF ) folgern.", "Um nun die Eindeutigkeit schwacher L\"osungen im Sinne von Definition REF einzusehen, bemerken wir, dass die Differenz zweier L\"osungen $(\\eta ,{\\bf u})$ und $(\\tilde{\\eta },\\tilde{{\\bf u}})$ Gleichung (REF ) mit verschwindenden Daten $({\\bf f},g,{\\bf u}_0,\\eta _0,\\eta _1)$ erf\"ullt, wenn wir den Term $S(D{\\bf u})$ durch $S(D{\\bf u})-S(D\\tilde{{\\bf u}})$ ersetzen.", "Es gilt somit f\"ur alle $t\\in \\bar{I}$ die Energiegleichung $\\begin{aligned}E_{\\eta -\\tilde{\\eta },{\\bf u}-\\tilde{{\\bf u}}}(t)&=- \\int _0^t\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}S(D{\\bf u})-S(D\\tilde{{\\bf u}}):(D{\\bf u}-D\\tilde{{\\bf u}})\\ dxds - 2\\int _0^t K(\\partial _t\\eta -\\partial _t\\tilde{\\eta })\\ ds\\end{aligned}$ Die linke Seite ist nichtnegativ, w\"ahrend die rechte Seite aufgrund der Monotonie von $S$ nichtpositiv ist.", "Mithin verschwinden beide Seiten und somit auch die Differenz der L\"osungen.", "Kommen wir nun zur Existenz schwacher L\"osungen.", "Der Wirbelterm spielt infolge der Regularisierungen praktisch keine Rolle.", "Aufgrund der komplexen Struktur des Systems ist der abstrakte Begriff des monotonen Operators hier dennoch unbrauchbar.", "Wir k\"onnen aber den klassischen Beweis ohne R\"uckgriff auf die abstrakte Theorie gewisserma\"sen elementar nachzeichnen.", "Somit wird die Energiegleichung auch hier eine wichtige Rolle spielen.", "Wir gehen zun\"achst genau wie im Falle des Navier-Stokes-Fluids vor, wobei die Felder $(\\widehat{{\\bf X}}_k)_{k\\in \\mathbb {N}}$ nun eine Basis von $W^{1,p}_{0,\\operatorname{div}}(\\Omega )$ bilden.", "Der Galerkin-Ansatz f\"uhrt uns auf nichtlineare Systeme gew\"ohnlicher Integro-Differentialgleichungen der in Anhang A.3 behandelten Form, zu denen wir lokale L\"osungen $(\\eta _n,{\\bf u}_n)$ auf Intervallen $I_n$ erhalten.", "Wir k\"onnen allerdings wie zuvor Energieabsch\"atzungen $\\begin{aligned}\\Vert \\eta _n\\Vert _{\\widetilde{Y}^{I_n}} +\\Vert {\\bf u}_n\\Vert _{X_{\\mathcal {R}\\delta ,p}^{I_n}}\\le c(T,\\text{Daten})\\end{aligned}$ herleiten, die zeigen, dass die L\"osungen auf dem ganzen Intervall $I$ existieren.", "Wegen der Wachstumsbedingung an $S$ schlie\"sen wir zudem, dass die Folge $(S(D{\\bf u}_n))$ in $L^{p^{\\prime }}(\\Omega _{\\mathcal {R}\\delta }^I)$ beschr\"ankt ist.", "Somit erhalten wir f\"ur eine Teilfolge die Konvergenzen $\\begin{aligned}\\eta _n&\\rightarrow \\eta \\hspace{8.5359pt}&&\\text{ schwach in } H^1(I,H^2_0(M)),\\\\\\partial _t\\eta _n&\\rightarrow \\partial _t\\eta &&\\text{ schwach$^*$ in } L^\\infty (I,L^2(M)),\\\\{\\bf u}_n&\\rightarrow {\\bf u}&&\\text{ schwach in } L^p(I,W^{1,p}_{\\operatorname{div}}(\\Omega _{\\mathcal {R}\\delta (t)}))\\\\& &&\\text{ undschwach$^*$ in } L^\\infty (I,L^2(\\Omega _{\\mathcal {R}\\delta (t)})),\\\\S(D{\\bf u}_n)&\\rightarrow \\xi &&\\text{ schwach in } L^{p^{\\prime }}(\\Omega _{\\mathcal {R}\\delta }^I).\\end{aligned}$ Bez\"uglich der dritten Konvergenz ist zu beachten, dass wegen des regul\"aren Randes die Korn'sche Ungleichung, Proposition REF , und somit $L^p(I,W^{1,p}_{\\operatorname{div},s}(\\Omega _{\\mathcal {R}\\delta (t)}^I))=L^p(I,W^{1,p}_{\\operatorname{div}}(\\Omega _{\\mathcal {R}\\delta (t)}^I))$ gilt.", "Wie zuvor folgt $\\operatorname{tr_{\\mathcal {R}\\delta }}{\\bf u}=\\partial _t\\eta \\,\\nu $ .", "Wenn wir in der Galerkin-Gleichung den Grenz\"ubergang vollziehen, sehen wir, dass das Tupel $(\\eta ,{\\bf u})$ die Identit\"at (REF ) mit $\\xi $ anstelle von $S(D{\\bf u})$ f\"ur alle Testfunktionen erf\"ullt.", "Es bleibt also lediglich, das Feld $\\xi $ zu identifizieren.", "Teilfolgen von $({\\bf u}_n(T,\\cdot ))$ und $(\\partial _t\\eta _n(T,\\cdot ))$ konvergieren schwach gegen Felder ${\\bf u}^*$ in $L^2(\\Omega _{\\mathcal {R}\\delta (T)})$ bzw.", "$\\eta ^*$ in $L^2(M)$ .", "Wenn wir die Funktionen ${\\bf W}_j$ und $W_j$ wie zuvor definieren, folgt daraus die Identit\"at $\\begin{aligned}&-\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf u}\\cdot \\partial _t(\\varphi \\,{\\bf W}_j)\\ dxdt -\\frac{1}{2}\\int _I\\int _M\\partial _t\\eta \\,\\partial _t\\mathcal {R}\\delta \\ \\varphi \\, W_j\\, \\gamma (\\mathcal {R}\\delta )\\ dAdt\\\\& +\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla )\\mathcal {R}{\\bf u}\\cdot (\\varphi \\,{\\bf W}_j)\\ dxdt\\\\&-\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(\\mathcal {R}{\\bf v}\\cdot \\nabla )\\mathcal {R}(\\varphi \\,{\\bf W}_j)\\cdot {\\bf u}\\ dxdt+\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\xi :D(\\varphi \\,{\\bf W}_j)\\ dxdt\\\\&+\\int _I\\int _M\\partial _t\\eta \\, \\partial _t (\\varphi \\, W_j)\\ dAdt + 2\\int _I\\int _MK(\\eta +\\partial _t\\eta , \\varphi \\, W_j)\\ dAdt \\\\&\\hspace{85.35826pt}=\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}\\cdot (\\varphi \\,{\\bf W}_j)\\ dxdt + \\int _I\\int _Mg\\, \\varphi \\, W_j\\ dAdt\\\\&\\hspace{99.58464pt} -\\int _{\\Omega _{\\mathcal {R}\\delta (T)}}{\\bf u}^*\\cdot (\\varphi (T)\\,{\\bf W}_j(T,\\cdot ))\\ dx -\\int _M\\eta ^*\\, \\varphi (T)\\,W_j(T,\\cdot )\\ dA\\end{aligned}$ f\"ur alle $j\\ge 1$ und $\\varphi \\in C_0^1((0,T])$ .", "Wir k\"onnen analog zum Beweis von (REF ) vorgehen und unter Verwendung der Stetigkeit von ${\\bf u}$ und $\\partial _t\\eta $ zeigen, dass diese Identit\"at auch mit ${\\bf u}(T,\\cdot )$ und $\\partial _t\\eta (T,\\cdot )$ anstelle von ${\\bf u}^*$ und $\\eta ^*$ gilt, woraus wir ${\\bf u}^*={\\bf u}(T,\\cdot )$ und $\\partial _t\\eta (T,\\cdot )=\\eta ^*$ folgern.", "Zudem konvergiert eine Teilfolge von $(\\eta _n(T,\\cdot ))$ schwach gegen $\\eta (T,\\cdot )$ in $H^2_0(M)$ .", "F\"ur die Galerkin-L\"osungen gilt der Energiesatz $\\begin{aligned}&\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}S(D{\\bf u}_n):D{\\bf u}_n\\ dxdt + 2\\int _I K(\\partial _t\\eta _n)\\ dt\\\\&\\hspace{28.45274pt}=-E_{\\eta _n,{\\bf u}_n}(T)+E_{\\eta _n,{\\bf u}_n}(0) + \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}_n\\cdot {\\bf u}_n\\ dxdt+\\int _I\\int _Mg_n\\,\\partial _t\\eta _n\\ dAdt.\\end{aligned}$ Wenn wir den $\\limsup $ dieser Gleichung nehmen und die schwache Unterhalbstetigkeit der Energie $E$ verwenden,Man beachte, dass jede stetige, nichtnegative quadratische Form, insbesondere also $K$ , schwach unterhalbstetig ist.", "Das folgt, wenn man den $\\liminf $ der Ungleichung $0\\le K(\\eta _n-\\eta ,\\eta _n-\\eta )=K(\\eta _n)-2K(\\eta ,\\eta _n)+K(\\eta )$ nimmt.", "so folgt $\\begin{aligned}&\\limsup _n \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}S(D{\\bf u}_n):D{\\bf u}_n\\ dxdt + 2\\int _I K(\\partial _t\\eta _n)\\ dt\\\\&\\hspace{28.45274pt}\\le -E_{\\eta ,{\\bf u}}(T)+E_{\\eta ,{\\bf u}}(0) + \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}{\\bf f}\\cdot {\\bf u}\\ dxdt +\\int _I\\int _Mg\\,\\partial _t\\eta \\ dAdt.\\end{aligned}$ Man beachte dabei, dass $\\eta _n(0,\\cdot )=\\eta _0$ f\"ur alle $n\\in \\mathbb {N}$ gilt.", "Aus der Energiegleichung f\"ur die schwache L\"osung $(\\eta ,{\\bf u})$ (mit $\\xi $ anstelle von $S(D{\\bf u})$ ) folgt $\\begin{aligned}&\\limsup _n \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}S(D{\\bf u}_n):D{\\bf u}_n\\ dxdt + 2\\int _I K(\\partial _t\\eta _n)\\ dt\\\\&\\hspace{113.81102pt}\\le \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\xi :D{\\bf u}\\ dxdt + 2\\int _I K(\\partial _t\\eta )\\ dt.\\end{aligned}$ Unter Ausnutzung dieser Absch\"atzung und der schwachen Konvergenzen schlie\"sen wir $\\begin{aligned}0&\\le \\limsup _n\\Big (\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}(S(D{\\bf u}_n)-S(D{\\bf u})):(D{\\bf u}_n-D{\\bf u})\\ dxdt\\\\&\\hspace{122.34692pt}+2\\int _I K(\\partial _t\\eta _n-\\partial _t\\eta ,\\partial _t\\eta _n-\\partial _t\\eta )\\ dt\\Big )\\\\&=\\limsup _n\\Big (\\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}S(D{\\bf u}_n):D{\\bf u}_n + S(D{\\bf u}):D{\\bf u}\\\\&\\hspace{135.15059pt} - S(D{\\bf u}_n):D{\\bf u}-S(D{\\bf u}):D{\\bf u}_n\\ dxdt\\\\&\\hspace{56.9055pt}+2\\int _I K(\\partial _t\\eta _n,\\partial _t\\eta _n) + K(\\partial _t\\eta ,\\partial _t\\eta )-2K(\\partial _t\\eta _n,\\partial _t\\eta )\\ dt\\Big )\\\\&\\le \\int _I\\int _{\\Omega _{\\mathcal {R}\\delta (t)}}\\xi :D{\\bf u}+ S(D{\\bf u}):D{\\bf u}- \\xi :D{\\bf u}- S(D{\\bf u}):D{\\bf u}\\ dxdt\\\\&\\hspace{62.59596pt}+2\\int _I K(\\partial _t\\eta ,\\partial _t\\eta ) + K(\\partial _t\\eta ,\\partial _t\\eta )-2K(\\partial _t\\eta ,\\partial _t\\eta )\\ dt\\\\&=0.\\end{aligned}$ Somit gilt f\"ur eine Teilfolge die Konvergenz $(S(D{\\bf u}_n)-S(D{\\bf u})):(D{\\bf u}_n-D{\\bf u})\\rightarrow 0$ fast \"uberall in $\\Omega _{\\mathcal {R}\\delta }^I$ .", "Mit Hilfe von Proposition REF schlie\"sen wir $D{\\bf u}_n\\rightarrow D{\\bf u}$ und somit $S(D{\\bf u}_n)\\rightarrow S(D{\\bf u})$ fast \"uberall in $\\Omega _{\\mathcal {R}\\delta }^I$ .", "Proposition REF liefert uns nun die Identifizierung $\\xi =S(D{\\bf u})$ .", "Unter der Annahme der Existenz obiger Approximationen ist damit die Existenz und Eindeutigkeit schwacher L\"osungen des entkoppelten, regularisierten p-Systems gezeigt.", "Auch bei der Durchf\"uhrung des Fixpunktarguments gehen wir zun\"achst genau wie im Falle des Navier-Stokes-Fluids vor.", "Wir ersetzen lediglich die Absch\"atzung (REF ) durch $\\begin{aligned}\\Vert \\eta \\Vert _{\\widetilde{Y}^I} +\\Vert {\\bf u}\\Vert _{X_{\\mathcal {R}\\delta ,p}^I} \\le c.\\end{aligned}$ F\"ur $(\\delta ,{\\bf v})\\in D$ ist die Menge $F(\\delta ,{\\bf v})$ aufgrund der Eindeutigkeit der schwachen L\"osungen einelementig, insbesondere konvex und abgeschlossen.", "Desweiteren k\"onnen wir den Beweis von Proposition REF fast w\"ortlich \"ubernehmen, um die Kompaktheit von $F$ zu zeigen.", "Zus\"atzlich zu (REF ) verwenden wir dabei die Absch\"atzung $\\begin{aligned}\\Vert \\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b\\Vert _{L^\\infty (I,W^{1,p}(B_\\alpha ))}\\le c\\,\\Vert b\\Vert _{H^2_0(M)},\\end{aligned}$ die aus Lemma REF und Proposition REF folgt.", "Zudem ersetzen wir in der Definition von $h_n^\\sigma $ den Raum $H^1_0(\\Omega )$ durch $W^{1,\\max (2,p)}_0(\\Omega )$ , sodass wir zus\"atzlich zu (REF ) \"uber die Absch\"atzung $\\Vert {\\delta _\\sigma }\\varphi \\Vert _{L^\\infty (I,W^{1,p}(B_\\alpha ))}\\le c\\,\\Vert \\varphi \\Vert _{W^{1,p}_0(\\Omega )}$ verf\"ugen.", "Bei der Anwendung von Lemma REF tritt die Einschr\"ankung $p>3/2$ auf.Im Prinzip l\"asst sich die Einschr\"ankung an dieser Stelle auf $p>6/5$ absenken, da die Kompaktheit von $(\\partial _t\\eta _n)$ in $L^2(I\\times M)$ an dieser Stelle noch nicht ben\"otigt wird.", "Die Einschr\"ankung kommt n\"amlich dadurch zustande, dass wir die r\"aumliche Regularit\"at der Folge $(\\partial _t\\eta _n)$ durch Spurbildung aus der r\"aumlichen Regularit\"at von $({\\bf u}_n)$ gewinnen und somit der Raum $W^{1-1/r,r}(M)$ f\"ur ein beliebiges $r<p$ kompakt nach $L^2(M)$ einbetten muss.", "Beim finalen Grenz\"ubergang wird die Einschr\"ankung (mit dieser Beweismethode) jedoch vermutlich nicht zu umgehen sein.", "Den Exponenten $p$ in Proposition REF w\"ahlen wir identisch $\\min (2,p)$ .", "Auch der Beweis der Graphenabgeschlossenheit von $F$ funktioniert fast genauso wie zuvor.", "Die Konvergenz der Folge $(\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)$ in $L^p(I,W^{1,p}(B_\\alpha ))$ erh\"alt man dabei wie im Beweis von Lemma REF .", "F\"ur die Identitfizierung des Grenzwerts der viskosen Spannungstensoren $(S(D{\\bf u}_n))\\subset L^{p^{\\prime }}(I\\times \\mathbb {R}^3)$ k\"onnen wir die Argumentation in obigem Existenzbeweis fast w\"ortlich \"ubernehmen.", "Im Wesentlichen ersetzen wir die dortigen Integrale \"uber $\\Omega _{\\mathcal {R}\\delta }^I$ durch Integrale \"uber $I\\times \\mathbb {R}^3$ , wobei wir die entsprechenden Felder durch 0 auf $I\\times \\mathbb {R}^3$ fortsetzen.", "Unter der Annahme der Existenz der Approximationen und $3/2<p<\\infty $ haben wir somit gezeigt, dass sich ein Intervall $I$ und zu jedem hinreichend kleinen Parameter $\\epsilon >0$ eine schwache L\"osungen des regularisierten p-Systems zum Parameter $\\epsilon $ auf dem Intervall $I$ gem\"a\"s der nachfolgenden Definition finden l\"asst.", "Definition 6.3 Ein Tupel $(\\eta ,{\\bf u})$ hei\"st schwache L\"osung des regularisierten $p$ -Systems zum Parameter $\\epsilon $ auf dem Intervall $I$ , falls $\\eta \\in \\widetilde{Y}^I$ mit $\\Vert \\eta \\Vert _{L^\\infty (I\\times M)}<\\kappa $ und $\\eta (0,\\cdot )=\\eta _0$ , ${\\bf u}\\in X_{\\mathcal {R}_\\epsilon \\eta ,p}^I$ mit $\\operatorname{tr_{\\mathcal {R}_\\epsilon \\eta }}{\\bf u}=\\partial _t\\eta \\,\\nu $ und $\\begin{aligned}&- \\int _I\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta (t)}}{\\bf u}\\cdot \\partial _t\\varphi \\ dxdt - \\frac{1}{2}\\int _I\\int _M\\partial _t\\eta \\, \\partial _t\\mathcal {R}_\\epsilon \\eta \\ b\\, \\gamma (\\mathcal {R}_\\epsilon \\eta )\\ dAdt\\\\&+\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta (t)}}(\\mathcal {R}_\\epsilon {\\bf u}\\cdot \\nabla )\\mathcal {R}_\\epsilon {\\bf u}\\cdot \\varphi \\ dxdt-\\frac{1}{2}\\int _I\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta (t)}}(\\mathcal {R}_\\epsilon {\\bf u}\\cdot \\nabla )\\mathcal {R}_\\epsilon \\varphi \\cdot {\\bf u}\\ dxdt\\\\& + \\int _I\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta (t)}}S(D{\\bf u}):D\\varphi \\ dxdt-\\int _I\\int _M\\partial _t\\eta \\, \\partial _tb\\ dAdt + 2\\int _IK(\\eta +\\epsilon \\,\\partial _t\\eta ,b)\\ dt\\\\&\\hspace{113.81102pt}=\\int _I\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta (t)}}{\\bf f}\\cdot \\varphi \\ dxdt + \\int _I\\int _Mg\\, b\\ dAdt\\\\&\\hspace{128.0374pt} +\\int _{\\Omega _{\\mathcal {R}_\\epsilon \\eta _0}}{\\bf u}_0^\\epsilon \\cdot \\varphi (0,\\cdot )\\ dx +\\int _M\\eta _1^\\epsilon \\, b(0,\\cdot )\\ dA\\end{aligned}$ f\"ur alle Testfunktionen $(b,\\varphi )\\in T_{\\mathcal {R}_\\epsilon \\eta ,p}^I$ .", "Um den Grenz\"ubergang $\\epsilon \\rightarrow 0$ vollziehen zu k\"onnen, werden nun allerdings andere Techniken ben\"otigt.", "F\"ur $p<11/5$ ist die Energieidentit\"at selbst f\"ur die Fluidgleichungen in einem Raumzeitzylinder ohne jegliche Kopplung vermutlich falsch.", "Dementsprechend sind f\"ur kleine $p$ schon in diesem einfacheren Fall andere Techniken vonn\"oten.", "Da mit $\\epsilon $ auch die Parabolisierung verschwindet, ist die Energieidentit\"at f\"ur das gekoppelte, nichtregularisierte System selbst f\"ur gro\"se $p$ vermutlich nicht richtig.", "Auch die asymptotisch geringe Randregularit\"at und das resultierende Versagen der Korn'schen Gleichung k\"onnten Probleme bereiten.", "Zur Behandlung der Fluidgleichungen f\"ur kleine $p$ in einem Raumzeitzylinder ohne Kopplung haben sich die Methoden der lokalen Druckfelder und der parabolischen $L^\\infty $ - bzw.", "$W^{1,\\infty }$ -Abschneidungen als besonders leistungsf\"ahig und flexibel erwiesen; siehe [24].", "Mit Hilfe dieser Techniken l\"asst sich die Existenz schwacher L\"osungen f\"ur $p>6/5$ zeigen.", "Das Ziel ist, auch hier diese Techniken erfolgreich einzubringen.", "Dies wird Gegenstand weiterer Forschung sein." ], [ "Ausblick", "Der n\"achste Schritt wird der vollst\"andige Existenzbeweis im Falle der verallgemeinerten Newton'schen Fluide sein.", "Hier ist noch einige Arbeit zu leisten.", "Zudem k\"onnte man unter Beibehaltung der Einschr\"ankung der Auslenkungen in Richtung der Normale an $\\partial \\Omega $ zur Koiter-Energie f\"ur nichtlinear elastische Schalen \"ubergehen.", "Trotz der Einschr\"ankung wird dabei allerdings ein Elliptizit\"atsverlust auftreten, da die Entartungsrichtungen im nichtlinearen Fall ja mit der L\"osung variieren und nicht l\"anger tangential an $\\partial \\Omega $ liegen werden.", "Interessant k\"onnte auch die Konstruktion starker L\"osungen f\"ur kurze Zeiten bei Newton'schen wie bei verallgemeinerten Newton'schen Fluiden sein.", "Insbesondere in letzterem Fall scheint sich die Technik der maximalen $L^p$ -Regularit\"at anzubieten, wie sie in [10] zur Konstruktion starker Kurzzeit-L\"osungen f\"ur verallgemeinerte Newton'sche Fluide in Raumzeitzylindern ohne zus\"atzliche Kopplung verwendet wird.", "Auch bei der Konstruktion von Kurzzeitl\"osungen sollte man die M\"oglichkeit, die Koiter-Energie f\"ur nichtlinear elastische Schalen zu verwenden, in Betracht ziehen.", "Eine weitere m\"ogliche Sto\"srichtung besteht darin, die Einschr\"ankung der Schalenauslenkung auf die Normalenrichtung aufzuheben.", "Hier tritt, wie bereits angemerkt, das Problem auf, dass der Gradient der Koiter-Energie in tangentiale Richtungen degeneriert ist.", "Zudem ist der Rand in diesem Fall im Allgemeinen kein Graph \"uber $\\partial \\Omega $ mehr.", "Anhang" ], [ "Differentialgeometrie", "Details und gegebenenfalls Beweise zu den folgenden Ausf\"uhrungen finden sich in [39], [38], [5].", "Wir nehmen im Folgenden an, dass alle auftretenden mathematischen Objekte so regul\"ar sind, dass die Definitionen sinnvoll und die durchgef\"uhrten Operationen zul\"assig sind.", "Sei $M$ eine kompakte Riemann'sche Mannigfaltigkeit (berandet oder nicht) endlicher Dimension mit Riemann'scher Metrik $g=\\langle \\cdot ,\\cdot \\rangle $ .", "Bez\"uglich beliebiger Koordinaten bezeichnen wir die Komponenten von $g$ mit $g_{\\alpha \\beta }$ , die Koordinatenvektorfelder mit $\\partial _\\alpha $ und die Koordinaten-1-Formen mit $dx^\\alpha $ .", "Das Skalarprodukt auf dem Tangentialb\"undel induziert ein Skalarprodukt $\\langle \\cdot ,\\cdot \\rangle $ auf s\"amtlichen Tensorb\"undeln; zum Beispiel f\"ur $(0,2)$ -Tensorfelder $T,S$ in Koordinaten $\\langle T,S \\rangle =g^{\\alpha \\gamma } g^{\\beta \\delta } T_{\\alpha \\beta } S_{\\gamma \\delta }$ .", "Dabei ist $(g^{\\alpha \\gamma })$ die inverse Matrix von $(g_{\\alpha \\gamma })$ .", "F\"ur beliebige Tensorfelder $T$ setzen wir $|T|^2:=\\langle T,T\\rangle $ .", "Desweiteren l\"asst sich ein Tensorprodukt definieren, das ein $(k,l)$ -Tensorfeld und ein $(r,s)$ -Tensorfeld auf ein $(k+r,l+s)$ -Tensorfeld abbildet; zum Beispiel f\"ur zwei $(1,1)$ -Tensorfelder $T,S$ in Koordinaten $(T\\otimes S)_{\\alpha \\gamma }^{\\beta \\delta }=T_\\alpha ^\\beta S_\\gamma ^\\delta $ .", "Die Riemann'sche Metrik induziert einen kanonischen Isomorphismus zwischen Tangential- und Kotangentialb\"undel und damit auch zwischen ko- und kontravarianten Tensorfeldern.", "Zum Beispiel l\"asst sich ein $(0,2)$ -Tensorfeld $T$ in ein $(1,1)$ -Tensorfeld umwandeln; in Koordinaten $T_\\alpha ^\\beta =g^{\\beta \\delta }T_{\\alpha \\delta }$ .", "Dieser Vorgang nennt sich Indexziehen.", "Desweiteren ist ein $(1,1)$ -Tensorfeld $T$ zu einem skalaren Feld kontrahierbar.", "Ein solches Feld l\"asst sich n\"amlich auch so interpretieren, dass punktweise Vektoren linear auf lineare Funktionale abgebildet werden, die Kovektoren aufnehmen.", "Ein lineares Funktional, das Kovektoren aufnimmt, ist (da jeder endlichdimensionale, normierte Raum reflexiv ist) aber ein Vektor.", "D.h. ein solches Tensorfeld definiert punktweise einen Endomorphismus des Tangentialraums.", "Von diesem Endomorphismenfeld nehmen wir punktweise die Spur, was ein skalares Feld ergibt; in Koordinaten $\\operatorname{tr}T=T^\\alpha _\\alpha $ .", "Allgemein l\"asst sich die Spur beliebiger Tensorfelder bez\"uglich eines ko- und eines kontravarianten Index nehmen, was den Rang des Tensorfelds um 2 vermindert; zum Beispiel f\"ur ein $(2,1)$ -Tensorfeld $T$ in Koordinaten $T^{\\alpha \\beta }_\\beta $ .", "Durch Verkn\"upfen von Indexziehen und Spurbildung l\"asst sich jedes Tensorfeld \"uber zwei verschiedene Indizes kontrahieren; zum Beispiel f\"ur ein $(0,2)$ -Tensorfeld $T$ in Koodinaten $\\operatorname{tr}_g T=g^{\\alpha \\beta }T_{\\alpha \\beta }$ .", "Es existiert ein kanonischer linearer Zusammenhang $\\nabla $ auf dem Tangentialb\"undel, der Levi-Civita-Zusammenhang, der dadurch charakterisiert ist, dass er symmetrisch und Riemann'sch ist.", "Sei $X$ ein Vektorfeld auf $M$ ; in Koordinaten $X=X^\\alpha \\partial _\\alpha $ .", "Dann bezeichnen wir mit $\\nabla X$ die totale kovariante Ableitung von $X$ bez\"uglich $\\nabla $ .", "Dies ist ein $(1,1)$ -Tensorfeld, d.h. es nimmt (punktweise) einen Vektor (die Richtung, in die abgeleitet wird) und einen Kovektor (der das Ergebnis der Ableitung aufnimmt) als Argument auf.", "In Koordinaten gilt $(\\nabla X)(\\partial _\\alpha ,dx^\\beta )=dx^\\beta (\\nabla _{\\partial _\\alpha } X)=\\partial _\\alpha X^\\beta + \\Gamma _{\\alpha \\delta }^\\beta X^\\delta $ , wobei die Christoffel-Symbole $\\Gamma _{\\alpha \\beta }^\\gamma $ durch $\\Gamma _{\\alpha \\beta }^\\gamma \\partial _\\gamma :=\\nabla _{\\partial _\\alpha }\\partial _\\beta $ definiert sind.", "F\"ur skalare Funktionen $f$ setzen wir zudem $\\nabla f:=df$ mit dem Differential $df$ von $f$ , ein Kovektorfeld; in Koordinaten $df(\\partial _\\alpha )=\\partial _\\alpha f$ .", "Durch Indexziehen erhalten wir aus $\\nabla f$ das Gradientenfeld $\\operatorname{grad}f$ ; in Koordinaten $(\\operatorname{grad}f)^\\alpha =g^{\\alpha \\beta }\\partial _\\beta f$ .", "Die kovariante Ableitung von Vektorfeldern induziert eine kovariante Ableitung beliebiger Tensorfelder; f\"ur ein Kovektorfeld $\\omega $ und Vektorfelder $X,Y$ zum Beispiel gilt $(\\nabla \\omega )(X,Y)=\\nabla _X(\\omega (Y))-\\omega (\\nabla _X Y)$ , in Koordinaten $(\\nabla \\omega )(\\partial _\\alpha ,\\partial _\\beta )=\\partial _\\alpha \\omega _\\beta - \\Gamma _{\\alpha \\beta }^\\delta \\omega _\\delta $ .", "Der $L^2$ -adjungierte Operator der totalen kovarianten Ableitung ist die Divergenz $\\nabla ^*=\\operatorname{tr}\\nabla $ .", "Dabei wird \"uber den Index der Ableitung und einen Index des Tensorfeldes, von dem wir die Divergenz nehmen wollen, kontrahiert; zum Beispiel f\"ur ein Vektorfeld $X$ in Koordinaten $\\begin{aligned}\\nabla ^* X=\\partial _\\alpha X^\\alpha + \\Gamma _{\\alpha \\delta }^\\alpha X^\\delta =(\\det (g_{\\beta \\gamma }))^{-\\frac{1}{2}}\\partial _\\alpha ((\\det (g_{\\beta \\gamma }))^{\\frac{1}{2}}X^\\alpha ).\\end{aligned}$ Zudem setzen wir $\\nabla ^2=\\nabla \\nabla $ .", "F\"ur ein skalares Feld $f$ zum Beispiel ist $\\nabla ^2 f$ die totale kovariante Ableitung des Differentials, ein symmetrisches $(0,2)$ -Tensorfeld, und $\\Delta f=\\nabla ^*\\nabla f=\\operatorname{tr}_g \\nabla ^2 f$ der Laplace(-Beltrami)-Operator auf $f$ angewendet.", "Dieser ist f\"ur beliebige Tensorfelder durch $\\Delta =\\nabla ^*\\nabla =\\operatorname{tr}_g \\nabla ^2$ definiert.", "Der Integralsatz von Stokes (f\"ur Differentialformen) beinhaltet den Spezialfall $\\begin{aligned}\\int _M \\operatorname{div}X\\ dV=0,\\end{aligned}$ falls das Vektorfeld $X$ auf dem (m\"oglicherweise leeren) Rand von $M$ verschwindet.", "Dabei ist $dV$ das durch die Metrik induzierte Ma\"s. Sind nun $T$ ein $(0,k)$ -Tensorfeld und $S$ ein $(0,k+1)$ -Tensorfeld, beide mit verschwindenden Randwerten, so gilt $\\begin{aligned}\\int _M \\langle \\nabla T,S\\rangle \\ dV = -\\int _M\\langle T,\\operatorname{tr}_g\\nabla S\\rangle \\ dV,\\end{aligned}$ wobei \"uber die ersten beiden Indizes von $\\nabla S$ kontrahiert wird.", "Diese Aussage folgt, wenn wir (REF ) auf das Vektorfeld $X:=(\\operatorname{tr}_g)^k\\ T\\otimes S$ anwenden, wobei wir den ersten Index von $T$ mit dem zweiten Index von $S$ kontrahieren, den zweiten mit dem dritten, etc.", "Dabei ist lediglich zu beachten, dass die Operatoren $\\nabla $ und $\\operatorname{tr}_g$ ebenso wie Kontraktionen bez\"uglich verschiedener Indizes kommutieren und dass $\\nabla X=(\\operatorname{tr}_g)^k\\ (\\nabla T)\\otimes S + (\\operatorname{tr}_g)^k\\ T\\otimes (\\nabla S)$ gilt.", "Wir definieren f\"ur Vektorfelder $X,Y,Z$ den Riemann'schen Kr\"ummungstensor $R$ durch $R(X,Y)Z:=\\nabla ^2_{X,Y}Z-\\nabla ^2_{Y,X}Z$ und den Ricci-Tensor $Rc$ durch Kontraktion von $R$ \"uber den ersten und den letzten Index; in Koordinaten $Rc_{\\alpha \\beta }=R_{\\gamma \\alpha \\beta }^{\\gamma }$ .", "F\"ur ein skalares Feld $f$ haben wir $[\\Delta ,\\nabla ] f=Rc(\\operatorname{grad}f,\\,\\cdot \\,),$ denn in lokalen Koordinaten gilt $\\begin{aligned}(\\Delta \\nabla f) (\\partial _\\gamma )&=g^{\\alpha \\beta }\\,(\\nabla ^3 f)(\\partial _\\alpha ,\\partial _\\beta ,\\partial _\\gamma )=g^{\\alpha \\beta }\\,(\\nabla ^3 f)(\\partial _\\alpha ,\\partial _\\gamma ,\\partial _\\beta )\\\\&=g^{\\alpha \\beta }\\,(\\nabla ^3 f)(\\partial _\\gamma ,\\partial _\\alpha ,\\partial _\\beta ) + g^{\\alpha \\beta }\\,\\langle R(\\partial _\\alpha ,\\partial _\\gamma )\\operatorname{grad}f,\\partial _\\beta \\rangle \\\\&=(\\nabla \\Delta f)(\\partial _\\gamma ) + Rc(\\operatorname{grad}f,\\partial _\\gamma ).\\end{aligned}$ F\"ur eine weitere Riemann'sche Mannigfaltigkeit $N$ und eine Abbildung $\\Phi :M\\rightarrow N$ bezeichnen wir mit $d\\Phi : TM \\rightarrow TN$ das Differential von $\\Phi $ .", "Dabei sind $TM$ und $TN$ die Tangentialb\"undel von $M$ bzw.", "$N$ .", "Ist $v$ ein Tangentialvektor an $M$ und $c:t\\mapsto c(t)$ ein Kurve in $M$ mit $\\frac{d}{dt}\\big |_{t=0}\\ c(t)=v$ , so gilt $d\\Phi \\, v=\\frac{d}{dt}\\big |_{t=0}\\ \\Phi \\circ c(t)$ .", "F\"ur Funktionen $f:N\\rightarrow \\mathbb {R}$ gilt der Transformationssatz $\\begin{aligned}\\int _N f\\ dA = \\int _M f\\circ \\Phi \\ |\\det d\\Phi |\\ dA.\\end{aligned}$ Dabei wird die Determinante von $d\\Phi $ im Punkt $q\\in M$ bez\"uglich zweier beliebiger Orthonormalbasen von $T_qM$ und $T_{\\Phi (q)}N$ gebildet.", "Dieser Kunstgriff ist notwendig, weil die Determinante von Homomorphismen zwischen verschiedenen Vektorr\"aumen keine koordinateninvariante Bedeutung besitzt.", "Die Determinanten bez\"uglich beliebiger Orthonormalbasen hingegen k\"onnen h\"ochstens um den Faktor $-1$ differieren.", "Ist $T$ ein Tensorfeld auf $N$ , so bezeichnen wir mit $\\Phi ^*T$ den Pullback von $T$ oder den zur\"uckgeholten Tensor.", "Ist $T$ ein skalares Feld, so ist dieser durch $\\Phi ^*T=T\\circ \\Phi $ definiert; f\"ur ein ein $(0,2)$ -Tensorfeld $T$ und Vektorfelder $X,Y$ auf $M$ gilt $(\\Phi ^*T)(X,Y)=T(d\\Phi \\, X, d\\Phi \\, Y)$ .", "Die kovariante Ableitung verh\"alt sich nat\"urlich unter Isometrien.", "Ist $\\Phi $ eine Isometrie, so bedeutet das speziell f\"ur die Divergenz eines Vektorfeldes $X$ auf $M$ die Identit\"at $\\nabla ^*X=(\\widetilde{\\nabla }^* Y)\\circ \\Phi $ , wobei $\\widetilde{\\nabla }$ der Levi-Civita-Zusammenhang von $N$ und $Y:=(d\\Phi \\, X)\\circ \\Phi ^{-1}$ der Pushforward von $X$ unter $\\Phi $ ist.", "Ist $(\\varphi _k,U_k)_{k}$ ein endlicher Atlas (insbesondere $U_k\\subset M$ offen), so existiert eine untergeordnete Zerlegung der Eins, d.h. es existieren differenzierbare Funktionen $\\psi _k: M\\rightarrow \\mathbb {R}$ mit $0\\le \\psi _k\\le 1$ , $\\operatorname{supp}\\psi _k\\subset U_k$ und $\\sum _k\\psi _k(q)=1$ f\"ur alle $q\\in M$ .", "Ist $M$ speziell eine kompakte, orientierte Fl\"ache in $\\mathbb {R}^3$ , so wird durch Einschr\"anken des euklidischen Skalarprodukts eine Riemann'sche Metrik auf $M$ definiert.", "Der Levi-Civita-Zusammenhang ist in diesem Fall durch Differenzieren von Vektorfeldern (l\"angs Kurven auf $M$ ) im $\\mathbb {R}^3$ und anschlie\"sende Orthogonalprojektion auf den Tangentialraum gegeben.", "Die Kr\"ummung der Fl\"ache wird durch die zweite Fundamentalform $h$ , ein symmetrisches $(0,2)$ -Tensorfeld, beschrieben.", "Diese ist f\"ur Vektorfelder $X,Y$ durch $h(X,Y):=\\frac{\\partial Y}{\\partial X}\\cdot \\nu $ definiert, wobei $\\frac{\\partial Y}{\\partial X}$ die Ableitung von $Y$ in Richtung $X$ im $\\mathbb {R}^3$ und $\\nu $ die Normale an $M$ bezeichnet.", "Durch Indexziehen erhalten wir ein Endomorphismenfeld, die Weingarten-Abbildung $W$ .", "Fassen wir die Normale $\\nu $ als Abbildung von $M$ in die 2-Sph\"are auf, so l\"asst sich das Differential $d\\nu $ als Endomorphismus des Tangentialb\"undels von $M$ interpretieren.", "Dann gilt $W=-d\\nu $ .", "Die Weingarten-Abbildung ist symmetrisch, und ihre Eigenwerte hei\"sen Hauptkr\"ummungen.", "Die Determinante der Weingarten-Abbildung ist die Gau\"s-Kr\"ummung $G$ ; in Koordinaten $G=\\det (h^\\alpha _\\beta )$ .", "Die gemittelte Spur ist die mittlere Kr\"ummung $H$ ; in Koordinaten $H=\\frac{1}{2}g^{\\alpha \\beta }h_{\\alpha \\beta }=\\frac{1}{2}h^\\alpha _\\alpha $ .", "Der Riemann'sche Kr\"ummungstensor l\"asst sich durch die zweite Fundamentalform ausdr\"ucken; in lokalen Koordinaten $R_{\\alpha \\beta \\gamma }^\\delta = h_{\\beta \\gamma }\\, h_\\alpha ^\\delta - h_{\\alpha \\gamma }\\,h_\\beta ^\\delta $ .", "Entsprechend hat der Ricci-Tensor in lokalen Koordinaten die Form $\\begin{aligned}Rc_{\\beta \\gamma }=h_{\\beta \\gamma }\\, h_\\alpha ^\\alpha - h_{\\alpha \\gamma }\\,h_\\beta ^\\alpha .\\end{aligned}$" ], [ "Sobolev-R\"aume", "Eine Einf\"uhrung in die Sobolev-R\"aume auf euklidischen Gebieten findet sich in [2]; siehe auch [8], [47], [44].", "F\"ur eine Einf\"uhrung in die Bochner-R\"aume sei auf [49], [43] verwiesen.", "F\"ur $d,k\\in \\mathbb {N}$ , $\\Omega \\subset \\mathbb {R}^d$ offen und $1\\le p\\le \\infty $ bezeichnen wir mit $W^{k,p}(\\Omega )$ den Sobolev-Raum der reellwertigen $L^p(\\Omega )$ -Funktionen, deren distributionelle Ableitungen bis zur Ordnung $k$ in $L^p(\\Omega )$ liegen.", "F\"ur $s\\in (k-1,k)$ und $1\\le p<\\infty $ setzen wir zudem $W^{s,p}(\\Omega ):=B^s_{pp}(\\Omega ):=(L^p(\\Omega ),W^{k,p}(\\Omega ))_{\\frac{s}{k},p}$ .", "Ist $\\Omega $ der $\\mathbb {R}^d$ oder ein beschr\"anktes Gebiet mit Lipschitz-Rand, so ist der Raum $W^{s,p}(\\Omega )$ isomorph zum (kanonisch normierten) Raum der $W^{k-1,p}(\\Omega )$ -Funktionen $f$ , f\"ur die die Gr\"o\"se $\\begin{aligned}|\\partial ^\\alpha f|_{\\sigma ,p;\\Omega }^p:=\\int _{\\Omega }\\int _{\\Omega }\\frac{|\\partial ^\\alpha f(x)-\\partial ^\\alpha f(y)|^p}{|x-y|^{d+\\sigma p}}\\ dxdy\\end{aligned}$ f\"ur alle $|\\alpha |=k-1$ endlich ist; siehe [2], [8], [47], [44].", "Theorem 8.1 Es sei $\\Omega \\subset \\mathbb {R}^d$ , $d\\in \\mathbb {N}$ , ein beschr\"anktes Gebiet mit Lipschitz-Rand.", "F\"ur $1\\le p<\\infty $ und $s>0$ mit $0<\\alpha :=s-d/p< 1$ gilt $W^{s,p}(\\Omega )\\hookrightarrow C^{0,\\alpha }(\\overline{\\Omega }).$ F\"ur $1\\le p, \\tilde{p}<\\infty $ und $0<\\tilde{s}<s$ mit $s-d/p>\\tilde{s}-d/\\tilde{p}$ gilt $W^{s,p}(\\Omega )\\hookrightarrow W^{\\tilde{s},\\tilde{p}}(\\Omega ).$ F\"ur $1\\le p<\\infty $ und $s>0$ haben wir die kompakte Einbettung $W^{s,p}(\\Omega )\\hookrightarrow \\hookrightarrow L^p(\\Omega ).$ Beweis: Die ersten beiden Behauptungen folgen mit Hilfe der in [2] konstruierten Fortsetzungsoperatoren aus Theorem 2.8.1 in [47].", "Die kompakte Einbettung ist wegen Theorem 3.8.1 in [8] eine Konsequenz der \"ublichen Sobolev-Einbettungen.", "$\\Box $ Ist $N$ eine geschlossene $C^k$ -Mannigfaltigkeit und $M\\subset N$ , so bestehe der Raum $W^{s,p}(M)$ , $0 \\le s \\le k$ , $1\\le p<\\infty $ oder $s=0$ , $p=\\infty $ aus den Funktionen(klassen) $f:M\\rightarrow \\mathbb {R}$ , f\"ur die in lokalen Koordinaten $(\\varphi ,U)$ von $N$ f\"ur alle $\\psi \\in C^k_0(U)$ gilt $(f\\,\\psi )\\circ \\varphi ^{-1}\\in W^{s,p}(\\varphi (U\\cap \\operatorname{int}M)).$ Wir setzen $L^p(M):=W^{0,p}(M)$ .", "Ist $(\\varphi _k,U_k)_{k}$ ein endlicher Atlas mit untergeordneter Zerlegung der Eins $(\\psi _k)$ , so ist die Norm von $W^{s,p}(M)$ durch $\\Vert f\\Vert _{W^{s,p}(M)}:=\\sum _k \\Vert (f\\,\\psi _k)\\circ \\varphi _k^{-1}\\Vert _{W^{s,p}(\\varphi _k(U_k\\cap \\operatorname{int}M))}$ gegeben.", "Ein $C^k$ -Diffeomorphismus, $k\\in \\mathbb {N}$ , zwischen Gebieten $\\Omega $ und $\\widetilde{\\Omega }$ des $\\mathbb {R}^d$ induziert Isomorphismen zwischen den R\"aumen $W^{s,p}(\\Omega )$ und $W^{s,p}(\\widetilde{\\Omega })$ , falls $0\\le s \\le k$ und $1\\le p<\\infty $ .", "Durch Wahl unterschiedlicher Atlanten erhalten wir also \"aquivalente Normen.", "Ist $s$ ganzzahlig und $N$ Riemann'sch mit Levi-Civita-Zusammenhang $\\nabla $ und induziertem Ma\"s $dV$ , so definiert auch die Gr\"o\"se $\\sum _{j=0}^s\\Big (\\int _{\\operatorname{int}M} |\\nabla ^j v|^p\\ dV\\Big )^{1/p}$ eine \"aquivalente Norm.", "Das ist eine simple Konsequenz der Kompaktheit von $N$ .", "Durch Verwenden eines endlichen Atlas von $N$ mit untergeordneter Zerlegung der Eins sieht man leicht, dass $W^{s,p}(M)=(L^p(M),W^{k,p}(M))_{\\frac{s}{k},p}$ gilt, wenn $p$ , $k$ und $s$ wie zu Beginn des Abschnitts gew\"ahlt sind.", "Desweiteren sei $W^{s,p}_0(M)$ der Abschluss von $C^k_0(\\operatorname{int}M)$ in $W^{s,p}(M)$ .", "Offenbar gilt $W^{s,p}_0(M)\\hookrightarrow W^{s,p}(N),$ wenn wir die Funktionen durch 0 auf $N$ fortsetzen.", "Eigenschaften wie Vollst\"andigkeit, Reflexivit\"at, Dichtheit regul\"arer Funktionen, etc.", "folgen ebenso wie Sobolev-Einbettungen sofort aus dem euklidischen Fall, wobei der Rand von $M$ gegebenenfalls hinreichend regul\"ar oder leer vorausgesetzt werden muss.", "Aus der $L^2$ -Theorie des Laplace-Operators auf Mannigfaltigkeiten, folgt, dass $\\Vert \\Delta \\, \\cdot \\,\\Vert _{L^2(M)}$ eine \"aquivalente Norm auf $H^2_0(M):=W^{2,2}_0(M)$ definiert, falls $M$ einen nichtleeren $C^{1,1}$ -Rand besitzt; siehe zum Beispiel [45].", "Theorem 8.2 Es seien $\\Omega \\subset \\mathbb {R}^d$ , $d\\in \\mathbb {N}$ , ein beschr\"anktes Gebiet mit Lipschitz-Rand und $1<p<\\infty $ .", "Dann besitzt die Abbildung $v\\mapsto v|_{\\partial \\Omega }$ , die f\"ur $v\\in C^1(\\overline{\\Omega })$ wohldefiniert ist, eine stetige Fortsetzung von $W^{1,p}(\\Omega )$ nach $W^{1-1/p,p}(\\partial \\Omega )$ .", "Ein Beweis findet sich in [34].", "Proposition 8.3 Es sei $\\omega \\in C^\\infty _0(\\mathbb {R}^d)$ , $d\\in \\mathbb {N}$ , mit $\\int _{\\mathbb {R}^d}\\omega \\ dx=1$ .", "F\"ur $\\epsilon >0$ setzen wir $\\omega _\\epsilon :=\\epsilon ^{-d}\\omega (\\epsilon ^{-1}\\cdot )$ .", "Falls $1\\le p <\\infty $ und $f\\in L^p(\\mathbb {R}^d)$ , so gilt $\\omega _\\epsilon \\ast f\\rightarrow f$ in $L^p(\\mathbb {R}^d)$ und fast \"uberall f\"ur $\\epsilon \\rightarrow 0$ .", "Ist $f$ stetig in Umgebung einer kompakten Menge $K\\subset \\mathbb {R}^d$ , so gilt $\\omega _\\epsilon \\ast f\\rightarrow f$ in $L^\\infty (K)$ f\"ur $\\epsilon \\rightarrow 0$ .", "F\"ur $d\\ge 2$ , $1\\le p,r <\\infty $ und $f\\in L^p(\\mathbb {R},L^r(\\mathbb {R}^{d-1})$ gilt $\\Vert \\omega _\\epsilon \\ast f\\Vert _{L^p(\\mathbb {R},L^r(\\mathbb {R}^{d-1})}\\le c\\,\\Vert f\\Vert _{L^p(\\mathbb {R},L^r(\\mathbb {R}^{d-1})}$ und $\\omega _\\epsilon \\ast f\\rightarrow f$ in $L^p(\\mathbb {R},L^r(\\mathbb {R}^{d-1})$ .", "Ein Beweis der Aussagen \"uber skalarwertige Funktionen findet sich in [31], Theorem 1.2.19 und Korollar 2.1.17.", "F\"ur den Beweis der Aussagen \"uber vektorwertige Funktionen verweisen wir auf [23].", "Proposition 8.4 Es seien $\\Omega \\subset \\mathbb {R}^d$ , $d\\in \\mathbb {N}$ , ein beschr\"anktes Gebiet mit $C^0$ -Rand und $1\\le p<\\infty $ .", "Dann ist $C^\\infty _0(\\mathbb {R}^d)$ dicht in $W^{1,p}(\\Omega )$ und in $E^p(\\Omega )$ .", "Beweisskizze: Es seien $B\\subset \\mathbb {R}^{d-1}$ ein offener Ball, $g:\\overline{B}\\rightarrow \\mathbb {R}$ eine stetige Funktion und $\\widetilde{\\Omega }:=\\lbrace (x^{\\prime },x_d)\\in \\mathbb {R}^d\\ |\\ x^{\\prime }\\in B,\\,g(x^{\\prime })< x_d\\rbrace $ .", "Zudem sei $v\\in W^{1,p}(\\widetilde{\\Omega })$ eine Funktion, deren Tr\"ager eine beschr\"ankte Teilmenge von $\\lbrace (x^{\\prime },x_d)\\in \\mathbb {R}^d\\ |\\ x^{\\prime }\\in B,\\, g(x^{\\prime })\\le x_d\\rbrace $ sei und die wir durch 0 auf $\\mathbb {R}^d$ fortsetzen.", "Wir setzen $v_t(x):=v(x^{\\prime },x_d+t)$ mit $t>0$ , sodass in $\\widetilde{\\Omega }$ f\"ur hinreichend kleine $\\epsilon >$ die Identit\"at $\\nabla (\\omega _\\epsilon \\ast v_t)=\\omega _\\epsilon \\ast \\nabla v_t$ , $\\omega _\\epsilon $ wie in Proposition REF , gilt.", "Aus Proposition REF folgt dann f\"ur $\\epsilon \\rightarrow 0$ die Konvergenz von $\\omega _\\epsilon \\ast v_t\\in C^\\infty _0(\\mathbb {R}^d)$ gegen $v_t$ in $W^{1,p}(\\widetilde{\\Omega })$ .", "Die Konvergenz von $v_t$ gegen $v$ in $W^{1,p}(\\widetilde{\\Omega })$ ist eine direkte Konsequenz der Stetigkeit der Translation in $L^p(\\mathbb {R}^d)$ .", "Die erste Behauptung folgt nun aus Obigem mittels Lokalisierung.", "Details finden sich in [34], [21].", "F\"ur die zweite Behauptung ersetzen wir einfach $\\nabla $ durch $\\operatorname{div}$ .", "$\\Box $ Theorem 8.5 (de Rham) Es seien $\\Omega \\subset \\mathbb {R}^d$ , $d\\in \\mathbb {N}$ , ein beschr\"anktes Gebiet mit Lipschitz-Rand und $f\\in (H^1_0(\\Omega ,\\mathbb {R}^d))^{\\prime }$ .", "Gilt $\\langle f,\\varphi \\rangle =0$ f\"ur alle $\\varphi \\in C^\\infty _0(\\Omega )$ mit $\\operatorname{div}\\varphi =0$ , so existiert genau eine Funktion $p\\in L^2(\\Omega )$ mit $\\int _\\Omega p\\ dx=0$ und $f=\\nabla p$ .", "F\"ur einen Beweis siehe [7].", "Proposition 8.6 (Korn'sche Ungleichung) Es seien $\\Omega \\subset \\mathbb {R}^d$ , $d\\in \\mathbb {N}$ , ein beschr\"anktes Gebiet mit Lipschitz-Rand und $1\\le p<\\infty $ .", "Dann definiert $\\Vert \\,\\cdot \\,\\Vert _{L^p(\\Omega )}+\\Vert D\\,\\cdot \\,\\Vert _{L^p(\\Omega )}$ eine \"aquivalente Norm auf $W^{1,p}(\\Omega ,\\mathbb {R}^d)$ .", "Diese Aussage wird in [42] bewiesen.", "Proposition 8.7 Es seien $I\\subset \\mathbb {R}$ ein offenes, beschr\"anktes Intervall und die komplexen Hilbert-R\"aume $H_0$ , $H_1$ ein Interpolationspaar.", "Dann gilt die Einbettung $\\Big \\lbrace v\\in L^2(I,H_0)\\ \\big |\\ \\frac{dv}{dt}\\in L^2(I,H_1)\\Big \\rbrace \\hookrightarrow C(\\bar{I},[H_0,H_1]_{\\frac{1}{2}}).$ F\"ur einen Beweis siehe [40].", "Proposition 8.8 Es seien $I\\subset \\mathbb {R}$ ein offenes Intervall und die Banach-R\"aume $B_0$ , $B_1$ ein Interpolationspaar.", "F\"ur $1\\le p,r,q \\le \\infty $ , $0<\\theta <1$ und $\\frac{1}{s}=\\frac{1-\\theta }{p}+\\frac{\\theta }{r}$ gilt die Einbettung $L^p(I,B_0)\\cap L^r(I,B_1)\\hookrightarrow L^s(I,(B_0,B_1)_{\\theta ,q}).$ Beweis: Unter Verwendung H\"older-Ungleichung mit Exponenten $\\tilde{p}=\\frac{p}{(1-\\theta )s}$ und $\\tilde{p}^{\\prime }=\\frac{r}{\\theta s}$ erhalten wir $\\begin{aligned}\\int _I\\Vert u\\Vert _{(B_0,B_1)_{\\theta ,q}}^s\\ dt \\le \\int _I\\Vert u\\Vert _{B_0}^{(1-\\theta )s}\\Vert u\\Vert _{B_1}^{\\theta s}\\ dt\\le \\Vert u\\Vert _{L^p(I,B_0)}^{(1-\\theta )s}\\Vert u\\Vert _{L^r(I,B_1)}^{\\theta s}.\\end{aligned}$ $\\Box $" ], [ "Gew\"ohnliche Integro-Differentialgleichungen", "Es seien $d\\in \\mathbb {N}$ , ${\\bf A}\\in C([0,\\infty )\\times \\mathbb {R}^d,\\mathbb {R}^d)$ und ${\\bf B}\\in C([0,\\infty )^2\\times \\mathbb {R}^d,\\mathbb {R}^d)$ .", "Wir suchen L\"osungen $\\alpha \\in C^1([0,T^*),\\mathbb {R}^d)$ der Gleichung $\\begin{aligned}\\dot{\\alpha }(t)={\\bf A}(t,\\alpha (t)) + \\int _0^t {\\bf B}(t,s,\\alpha (s))\\ ds\\end{aligned}$ f\"ur $t\\in [0,T^*)$ .", "Wir zeigen nun, dass zu jedem $\\alpha _0\\in \\mathbb {R}^d$ ein $T^*\\in (0,\\infty ]$ und eine L\"osung auf dem Intervall $[0,T^*)$ mit $\\alpha (0)=\\alpha _0$ existieren derart, dass aus $T^*<\\infty $ die Divergenz $\\lim _{t\\nearrow T^*}|\\alpha (t)|=\\infty $ folgt.", "Sind ${\\bf A}$ und ${\\bf B}$ affin-linear in $\\alpha $ , so ist $T^*=\\infty $ .", "Die Konstruktion der zeitlich lokalen L\"osung folgt de facto w\"ortlich dem Beweis des Existenzsatzes von Peano.", "Der Vollst\"andigkeit halber skizzieren wir sie dennoch.", "Wir k\"onnen zun\"achst ohne Einschr\"ankung annehmen, dass die rechte Seite von (REF ) unabh\"angig von $t$ und $\\alpha $ beschr\"ankt ist, da wir sonst die beiden Summanden mit Abschneidefunktionen multiplizieren k\"onnen, die in einer Umgebung von $(t=0$ , $\\alpha =\\alpha _0)$ identisch 1 sind.", "Die so gewonnene L\"osung ist f\"ur hinreichend kleine Zeiten eine L\"osung der urspr\"unglichen Gleichung.", "Wir definieren N\"aherungsl\"osungen $\\alpha _k$ , $k\\in \\mathbb {N}$ , indem wir $\\alpha _k(t):=\\alpha _0$ f\"ur $t<0$ und $\\alpha _k(t):=\\alpha _0+\\int _0^t{\\bf A}(s,\\alpha _k(s-1/k)) + \\int _0^s{\\bf B}(s,\\tau ,\\alpha _k(\\tau -1/k))\\ d\\tau \\ ds$ f\"ur $0\\le t\\le 1$ setzen.", "Aus der Beschr\"anktheit der rechten Seite von (REF ) folgt die Beschr\"anktheit von $(\\dot{\\alpha }_k)$ in $C([0,1],\\mathbb {R}^d)$ unabh\"angig von $k$ .", "Der Satz von Arzela-Ascoli liefert uns einen gleichm\"a\"sigen Grenzwert $\\alpha $ einer Teilfolge von $(\\alpha _k)$ .", "Aufgrund der Absch\"atzung $|\\alpha _k(t-1/k)-\\alpha _k(t)|\\le c/k$ konvergiert auch die entsprechende Teilfolge von $(\\alpha _k(\\cdot -1/k))$ gleichm\"a\"sig gegen $\\alpha $ .", "Nun k\"onnen wir in der Definition von $\\alpha _k$ den Grenz\"ubergang vollziehen und erhalten eine L\"osung von (REF ) f\"ur kleine Zeiten.", "Jede beschr\"ankte L\"osung $\\alpha $ auf einem Intervall $[0,T)$ , $T<\\infty $ , l\"asst sich auf ein Intervall $[0,\\widetilde{T})$ , $\\widetilde{T}>T$ , fortsetzen.", "Mit Hilfe von (REF ) folgt n\"amlich die Beschr\"ankheit von $\\dot{\\alpha }$ und mithin die gleichm\"a\"sige Stetigkeit von $\\alpha $ auf $[0,T)$ .", "Somit existiert der Grenzwert $\\alpha (T)=\\lim _{t\\nearrow T}\\alpha (t)$ , und wir k\"onnen zum Anfangswert $\\alpha (T)$ eine L\"osung konstruieren, die, mit der urspr\"unglichen L\"osung verkn\"upft, die Fortsetzung liefert.", "Die Behauptung $T^*=\\infty $ im affin-linearen Fall folgt mit Hilfe einer Variation des zum Beweis des Gronwall-Lemmas verwendeten Arguments.", "Wir wollen uns hier die Details sparen, weil wir die Langzeitexistenz der L\"osungen $\\alpha _n^k$ im Beweis von Proposition REF ebenso gut aus den dortigen Energieabsch\"atzungen folgern k\"onnen." ], [ "Weitere verwendete Fakten", "Proposition 8.9 (Gronwall'sches Lemma) Es seien $T>0$ , $\\alpha \\in C([0,T))$ und $c\\in \\mathbb {R}$ .", "Erf\"ullt $\\varphi \\in C^1([0,T))$ f\"ur alle $t\\in [0,T)$ die Ungleichung $\\varphi ^{\\prime }(t)\\le \\alpha (t)+c\\,\\varphi (t),$ so folgt f\"ur alle $t\\in [0,T)$ $\\varphi (t)\\le \\varphi (0)\\,e^{ct}+\\int _0^t\\alpha (s)\\,e^{c(t-s)}\\ ds.$ Proposition 8.10 (Reynolds'sches Transporttheorem) Es seien $\\Omega \\subset \\mathbb {R}^3$ ein beschr\"anktes Gebiet mit $C^1$ -Rand, $I\\subset \\mathbb {R}$ ein Intervall und $\\Psi \\in C^1(I\\times \\overline{\\Omega },\\mathbb {R}^3)$ derart, dass $\\Psi _t:=\\Psi (t,\\cdot ):\\overline{\\Omega }\\rightarrow \\Psi _t(\\overline{\\Omega })$ f\"ur alle $t\\in I$ ein Diffeomorphismus ist.", "Wir setzen $\\Omega _t:=\\Psi _t(\\Omega )$ und ${\\bf v}:=(\\partial _t\\Psi )\\circ \\Psi _t^{-1}$ .", "Dann gilt f\"ur $\\xi \\in C^1(\\bigcup _{t\\in I}\\lbrace t\\rbrace \\times \\overline{\\Omega }_t)$ und $t\\in I$ die Identit\"at $\\frac{d}{dt}\\int _{\\Omega _t} \\xi (t,x)\\ dx= \\int _{\\Omega _t} \\partial _t\\xi (t,x)\\ dx + \\int _{\\partial \\Omega _t} {\\bf v}\\cdot \\nu _t\\ \\xi (t,\\cdot )\\ dA_t.$ Dabei bezeichnen $dA_t$ das Fl\"achenma\"s und $\\nu _t$ die \"au\"sere Einheitsnormale von $\\partial \\Omega _t$ .", "Ein Beweis dieser Aussage findet sich in [7].", "Wir betrachten nun f\"ur ein beschr\"anktes Gebiet $\\Omega \\subset \\mathbb {R}^3$ das Stokes-System $\\begin{aligned}-\\Delta {\\bf u}+\\nabla \\pi &=0&&\\text{ in }\\Omega ,\\\\\\operatorname{div}{\\bf u}&=0&&\\text{ in }\\Omega ,\\\\{\\bf u}&={\\bf g}&&\\text{ auf }\\partial \\Omega .\\end{aligned}$ Theorem 8.11 Es seien $\\Omega \\subset \\mathbb {R}^3$ ein beschr\"anktes Gebiet mit $C^{\\max (2,k)}$ -Rand, $k\\in \\mathbb {N}$ , und $1<p<\\infty $ .", "Desweiteren sei ${\\bf g}\\in W^{k-1/p,p}(\\partial \\Omega )$ mit $\\int _{\\partial \\Omega }{\\bf g}\\cdot \\nu \\ dA=0,$ wobei $\\nu $ und $dA$ Einheitsnormale bzw.", "Fl\"achenma\"s von $\\partial \\Omega $ bezeichnen.", "Dann existiert genau eine (sehr schwache) L\"osung ${\\bf u}\\in W^{k,p}(\\Omega )$ des Stokes-Systems, d.h. es gilt $\\operatorname{div}{\\bf u}=0$ und $-\\int _\\Omega {\\bf u}\\cdot \\Delta \\varphi \\ dx+\\int _{\\partial \\Omega }{\\bf g}\\cdot \\partial _{\\nu }\\varphi \\ dA=0$ f\"ur alle $\\varphi \\in C^2(\\overline{\\Omega })$ mit $\\operatorname{div}\\varphi =0$ und $\\varphi =0$ auf $\\partial \\Omega $ .", "Die Zuordnung ${\\bf g}\\mapsto {\\bf u}$ definiert einen stetigen, linearen Operator von $W^{k-1/p,p}(\\partial \\Omega )$ nach $W^{k,p}(\\Omega )$ .", "Ist $\\partial \\Omega $ von der Klasse $C^{2,1}$ , so gilt die Aussage auch dann, wenn wir $W^{k-1/p,p}(\\partial \\Omega )$ durch $L^p(\\partial \\Omega )$ und $W^{k,p}(\\Omega )$ durch $L^p(\\Omega )$ ersetzen.", "Die erste Aussage wird in [28] bewiesen.", "Die zweite Aussage ist eine direkte Folgerung aus Theorem 3 in [33].", "Ein Beweis der folgenden Variante des Schauder'schen Fixpunktsatzes f\"ur mengenwertige Abbildungen ist in [29] nachzulesen.", "Die endlichdimensionale Version dieses Theorems, der Kakutani-Fixpunktsatz, wurde von John Nash bei der Beschreibung des Nash-Gleichgewichts verwendet.", "Theorem 8.12 (Kakutani-Glicksberg-Fan) Es seien $C$ eine konvexe Teilmenge eines normierten Raumes $Z$ und $F: C\\rightarrow 2^C$ eine oberhalbstetige mengenwertige Abbildung, d.h. f\"ur jede offene Teilmenge $W\\subset C$ sei die Menge $\\lbrace c\\in C\\ |\\ F(c)\\subset W\\rbrace \\subset C$ offen.", "Zudem sei $F(C)$ in einer kompakten Teilmenge von $C$ enthalten, und f\"ur alle $z\\in C$ sei die Menge $F(z)$ nichtleer, konvex und kompakt.", "Dann besitzt $F$ einen Fixpunkt, d.h. es existiert ein $c_0\\in C$ mit $c_0\\in F(c_0)$ .", "Es ist nicht schwierig zu sehen, dass die Forderung der Oberhalbstetigkeit \"aquivalent durch die Bedingung der Graphenabgeschlossenheit ersetzt werden kann.", "Letztere besagt, dass aus den Konvergenzen $c_n\\rightarrow c$ in $C$ und mit $F(c_n)\\ni z_n\\rightarrow z$ in $Z$ folgt, dass $z\\in F(c)$ .", "Wir bezeichnen mit $M_{sym}$ den Raum der reellen, symmetrischen $3\\times 3$ -Matrizen.", "Proposition 8.13 (Dal Maso-Murat) Sei $S:M_{sym}\\rightarrow M_{sym}$ stetig und strikt monoton, d.h. es gelte $(S(A)-S(B)):(A-B)>0$ f\"ur $A,B\\in M_{sym}$ , $A\\ne B$ .", "Desweiteren sei $(A_n)_{n\\in \\mathbb {N}}\\subset M_{sym}$ eine Folge mit $\\lim _{n}\\ (S(A_n)-S(A)):(A_n-A)=0$ f\"ur ein $A\\in M_{sym}$ .", "Dann gilt $\\lim _{n}A_n=A$ .", "Dies wird in [20] gezeigt.", "Proposition 8.14 (Vitali) Es seien $\\Omega \\subset \\mathbb {R}^d$ , $d\\in \\mathbb {N}$ , ein beschr\"anktes Gebiet und $(f_n)_{n\\in \\mathbb {N}}\\subset L^1(\\Omega )$ .", "Zudem konvergiere die Folge $(f_n)$ fast \"uberall in $\\Omega $ .", "Dann ist Konvergenz der Folge in $L^1(\\Omega )$ \"aquivalent zu der Aussage, dass zu jedem $\\epsilon >0$ ein $\\delta >0$ existiert derart, dass $\\sup _n\\int _K |f_n|\\ dx<\\epsilon $ f\"ur alle messbaren Mengen $K\\subset \\Omega $ mit $|K|<\\delta $ gilt.", "F\"ur einen Beweis sei auf [3] verwiesen." ], [ "Ausgelagertes", "Bemerkung 8.15 Im Kontext von Abschnitt 4 gilt formal, d.h. unter Vernachl\"assigung von Regularit\"atsfragen, $\\int _{\\Omega _{\\eta (t)}} (\\nabla {\\bf u})^T:\\nabla \\varphi \\ dx =0,$ falls $\\varphi $ ein Feld mit $\\operatorname{tr_\\eta }\\varphi =b\\,\\nu $ f\"ur eine skalare Funktion $b$ ist, insbesondere also falls $\\varphi ={\\bf u}$ .", "Partielle Integration liefert n\"amlich aufgrund der Divergenzfreiheit von ${\\bf u}$ $\\int _{\\Omega _{\\eta (t)}} (\\nabla {\\bf u})^T:\\nabla \\varphi \\ dx = \\int _{\\partial \\Omega _{\\eta (t)}}(\\varphi \\cdot \\nabla ){\\bf u}\\cdot \\nu _{\\eta (t)}\\ dA.$ Es gen\"ugt also zu zeigen, dass auf $\\partial \\Omega _{\\eta (t)}$ gilt $((\\nu \\circ q)\\cdot \\nabla ){\\bf u}\\cdot \\nu _{\\eta (t)}=0.", "$ Um diese Identit\"at zu beweisen, setzen wir ${\\bf e}_1:=\\nu \\circ q$ .", "Zudem w\"ahlen wir auf $\\partial \\Omega _{\\eta (t)}$ zwei linear unabh\"angige, tangentiale VektorfelderZur Erinnerung: Diese existieren lokal.", "und setzen diese konstant l\"angs ${\\bf e}_1$ fort.", "Die so konstruierten Felder nennen wir ${\\bf e}_2$ und ${\\bf e}_3$ .", "Per Konstruktion gilt $\\Gamma _{1i}^k\\,{\\bf e}_k:=\\nabla _{{\\bf e}_1}{\\bf e}_i=0$ und somit $\\Gamma _{1i}^k=0$ f\"ur alle $i,k$ .", "Schreiben wir ${\\bf u}=u^i\\,{\\bf e}_i$ , so gilt $((\\nu \\circ q)\\cdot \\nabla ){\\bf u}=\\nabla _{{\\bf e}_1}{\\bf u}=du^i\\,{\\bf e}_1\\ {\\bf e}_i+u^i\\,\\nabla _{{\\bf e}_1}{\\bf e}_i=du^i\\,{\\bf e}_1\\ {\\bf e}_i,$ also auf $\\partial \\Omega _{\\eta (t)}$ $\\begin{aligned}((\\nu \\circ q)\\cdot \\nabla ){\\bf u}\\cdot \\nu _{\\eta (t)}=du^1\\,{\\bf e}_1\\ {\\bf e}_1\\cdot \\nu _{\\eta (t)}.\\end{aligned}$ Andererseits gilt $0=\\operatorname{div}{\\bf u}=du^i\\,{\\bf e}_i+ u^k\\,\\Gamma _{ik}^i.$ Die Komponenten $u^2$ und $u^3$ und ihre tangentialen Ableitungen $du^2\\,{\\bf e}_2$ bzw.", "$du^3\\,{\\bf e}_3$ verschwinden auf $\\partial \\Omega _{\\eta (t)}$ .", "Wir folgern, dass auf $\\partial \\Omega _{\\eta (t)}$ $0=du^1\\,{\\bf e}_1+u^1\\,\\Gamma _{1i}^i=du^1\\,{\\bf e}_1$ gilt, was zusammen mit (REF ) die Behauptung zeigt.", "Bemerkung 8.16 F\"ur $\\delta \\in C^2(\\partial \\Omega )$ mit $\\Vert \\delta \\Vert _{L^\\infty (\\partial \\Omega )}<\\kappa $ und $\\varphi :\\Omega \\rightarrow \\mathbb {R}^3$ sei $\\delta \\varphi $ der Pushforward von $(\\det d\\Psi _\\delta )^{-1} \\varphi $ unter $\\Psi _\\delta $ , d.h. $\\delta \\varphi :=\\big (d\\Psi _\\delta \\, (\\det d\\Psi _\\delta )^{-1}\\varphi \\big )\\circ \\Psi ^{-1}_\\delta .$ Die Abbildung $\\delta $ mit der Inversen $\\delta ^{-1}\\varphi :=\\big (d\\Psi _\\delta ^{-1}\\, (\\det d\\Psi _\\delta )\\,\\varphi \\big )\\circ \\Psi _\\delta $ definiert offenbar Isomorphismen zwischen den Lebesgue- und Sobolev-R\"aumen auf $\\Omega $ bzw.", "$\\Omega _\\delta $ , solange die Differenzierbarkeitsordnung kleiner oder gleich 1 bleibt.", "Zudem erh\"alt die Abbildung Nullrandwerte.", "Der Diffeomorphismus $\\Psi _\\delta $ ist eine Isometrie von $\\overline{\\Omega }$ mit der Riemann'schen Metrik $h:=(d\\Psi _\\delta )^T d\\Psi _\\delta $ nach $\\overline{\\Omega _\\delta }$ mit der euklidischen Metrik.", "Aus der Nat\"urlichkeit des Levi-Civita-Zusammenhangs unter Isometrien sowie den Identit\"aten $\\sqrt{\\det h_{\\beta \\gamma }}=\\det d\\Psi _t$ und (REF ) folgt $(\\operatorname{div}\\delta \\varphi )\\circ \\Psi _\\delta =\\operatorname{div}_h((\\det d\\Psi _\\delta )^{-1}\\varphi )=(\\det d\\Psi _\\delta )^{-1}\\operatorname{div}\\varphi .$ $\\delta $ erh\"alt also auch die Divergenzfreiheit und definiert somit Isomorphismen zwischen den entsprechenden Funktionenr\"aumen auf $\\Omega $ bzw.", "$\\Omega _\\delta $ .", "Ist $\\delta \\in C^2(I\\times \\partial \\Omega )$ mit $\\Vert \\delta \\Vert _{L^\\infty (I\\times \\partial \\Omega )}<\\kappa $ , so definiert die Anwendung von ${\\delta (t)}$ zu jedem Zeitpunkt $t\\in I$ Isomorphismen zwischen geeigneten Funktionenr\"aumen auf $I\\times \\Omega $ bzw.", "$\\Omega _\\delta ^I$ , solange die Differenzierbarkeitsordnung wiederum kleiner oder gleich 1 bleibt.", "Lemma 8.17 Es sei $\\eta \\in Y^I$ mit $\\Vert \\eta \\Vert _{L^\\infty (I\\times M)}<\\kappa $ .", "Dann existiert ein linearer Operator $\\mathcal {M}_\\eta $ mit $\\begin{aligned}\\Vert \\mathcal {M}_\\eta b\\Vert _{L^r(I\\times M)}&\\le c\\, \\Vert b\\Vert _{L^r(I\\times M)},\\\\\\Vert \\mathcal {M}_\\eta b\\Vert _{C(\\bar{I},L^r(M)}&\\le c\\, \\Vert b\\Vert _{C(\\bar{I},L^r(M)},\\\\\\Vert \\mathcal {M}_\\eta b\\Vert _{L^r(I,H^2_0(M))}&\\le c\\, \\Vert b\\Vert _{L^r(I,H^2_0(M))},\\\\\\Vert \\mathcal {M}_\\eta b\\Vert _{H^1(I,L^2(M))}&\\le c\\, \\Vert b\\Vert _{H^1(I,L^2(M))}\\\\\\end{aligned}$ f\"ur $1\\le r\\le \\infty $ und $\\int _M (\\mathcal {M}_\\eta b)(t,\\cdot )\\,\\gamma (\\eta (t,\\cdot ))\\ dA = 0$ f\"ur fast alle $t\\in I$ .", "Die Konstante $c$ h\"angt nur von $\\Omega $ , $\\Vert \\eta \\Vert _{Y^I}$ und $\\tau (\\eta )$ ab; sie bleibt beschr\"ankt, wenn $\\Vert \\eta \\Vert _{Y^I}$ und $\\tau (\\eta )$ beschr\"ankt bleiben.", "Beweis: Wir fixieren eine beliebige Funktion $\\psi \\in C_0^\\infty (\\operatorname{int}M)$ mit $\\psi \\ge 0$ , $\\psi \\lnot \\equiv 0$ und setzen $\\begin{aligned}(M_\\eta b)(t,\\cdot ):= b- \\psi \\, \\frac{a(b(t,\\cdot ),\\eta (t,\\cdot ))}{a(\\psi ,\\eta (t,\\cdot ))}\\end{aligned}$ mit der Abk\"urzung $a(b(t,\\cdot ),\\eta (t,\\cdot )):=\\int _M b\\,\\gamma (\\eta (t,\\cdot ))\\ dA.$ Der Beweis der behaupteten Eigenschaften ist dann sehr einfach zu f\"uhren, wenn wir beachten, dass die Ungleichung $a(\\psi ,\\eta )\\ge c$ mit einer Konstante $c>0$ , die nur von $\\tau (\\eta )$ abh\"angt, gilt.", "Das folgt aber aus Bemerkung REF , da $\\gamma $ eine stetige Funktion von $\\eta $ ist.", "$\\Box $ Lemma 8.18 F\"ur die Folge $(\\eta _n)\\subset Y^I$ mit $\\sup _n\\Vert \\eta _n\\Vert _{L^\\infty (I\\times M)}<\\alpha <\\kappa $ gelten die Konvergenzen (REF )$_{(1,2)}$ .", "$(1.a)$ Ist $b\\in C(\\bar{I},L^2(M))$ , so konvergiert $(\\mathcal {M}_{\\eta _n}b)$ gegen $\\mathcal {M}_{\\eta }b$ in $C(\\bar{I},L^2(M))$ unabh\"angig von $\\Vert b\\Vert _{C(\\bar{I},L^2(M))}\\le 1$ .", "$(1.b)$ Konvergiert zus\"atzlich $(\\partial _t\\eta _n)$ in $L^2(I\\times M)$ und ist $b\\in H^1(I,L^2(M))\\cap L^2(I,H^2_0(M))$ , so konvergiert $(\\mathcal {M}_{\\eta _n}b)$ gegen $\\mathcal {M}_\\eta b$ in $H^1(I,L^2(M))\\cap L^2(I,H^2_0(M))$ .", "$(2.a)$ Ist $b\\in C(\\bar{I},L^2(M))$ , so konvergiert $(\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)$ gegen $\\mathcal {F}_{\\eta }\\mathcal {M}_{\\eta }b$ in $C(\\bar{I},L^2(B_\\alpha ))$ unabh\"angig von $\\Vert b\\Vert _{C(\\bar{I},L^2(M))}\\le 1$ .", "$(2.b)$ Unter den Voraussetzungen von $(1.b)$ konvergiert $(\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)$ gegen $\\mathcal {F}_{\\eta }\\mathcal {M}_\\eta b$ in $H^1(I\\times B_\\alpha )\\cap L^\\infty (I,L^4(B_\\alpha ))$ .", "$(2.c)$ Konvergiert $(b_n)$ gegen $b$ schwach in $L^2(I\\times M)$ , so konvergiert die Folge $(\\mathcal {F}_{\\eta _n}b_n)$ gegen $\\mathcal {F}_\\eta b$ schwach in $L^2(I\\times B_\\alpha )$ .", "Beweis: Behauptung $(1.a)$ folgt aus $\\begin{aligned}\\Vert a(b,\\eta _n)-a(b,\\eta )\\Vert _{L^\\infty (I)}&=\\big \\Vert \\int _M b\\,(\\gamma (\\eta _n)-\\gamma (\\eta ))\\ dA\\big \\Vert _{L^\\infty (I)}\\\\&\\le c\\,\\Vert b\\Vert _{C(\\bar{I},L^2(M))}\\,\\Vert \\gamma (\\eta _n)-\\gamma (\\eta )\\Vert _{L^\\infty (I\\times M)},\\end{aligned}$ der analogen Absch\"atzung mit $\\psi $ anstelle von $b$ und der Ungleichung $a(\\psi ,\\eta _n)\\ge c>0$ , wenn wir beachten, dass $(\\gamma (\\eta _n))$ gleichm\"a\"sig gegen $\\gamma (\\eta )$ konvergiert.", "Behauptung $(1.b)$ ist klar, wenn wir wissen, dass die Folgen $(a(b,\\eta _n))$ und $(a(\\psi ,\\eta _n))$ gegen $a(b,\\eta )$ bzw.", "$a(\\psi ,\\eta )$ in $H^1(I)$ konvergieren.", "Der Beweis dieser Konvergenzen ist aber sehr einfach zu f\"uhren.", "Kommen wir zu $(2.a)$ .", "Wir schlie\"sen aus (REF ) und $(1.a)$ , dass $(\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)$ in $C(\\bar{I},L^2(S_\\alpha ))$ unabh\"angig von $\\Vert b\\Vert _{C(\\bar{I},L^2(M))}\\le 1$ konvergiert.", "Ebenso konvergiert die formale Spur $\\begin{aligned}\\exp \\Big (\\int _{\\eta _n(t,q)}^{-\\alpha }\\beta (q+\\tau \\,\\nu \\circ q))\\ d\\tau \\Big )\\,(\\mathcal {M}_{\\eta _n}b)(t,q)\\,\\nu \\circ q\\end{aligned}$ von $\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b$ auf $I\\times \\partial (\\Omega \\setminus \\overline{S_\\alpha })$ in $C(\\bar{I},L^2(\\Omega \\setminus \\overline{S_\\alpha }))$ unabh\"angig von $\\Vert b\\Vert _{C(\\bar{I},L^2(M))}\\le 1$ .", "Aus den Stetigkeitseigenschaften des L\"osungsoperators des Stokes-Systems folgt die Behauptung; vgl.", "Bemerkung REF .", "Vollkommen analog erhalten wir $(2.c)$ .", "Wir zeigen nun $(2.b)$ .", "Aus (REF ), $(1.b)$ und der Einbettung $H^1(I,L^2(M))\\cap L^2(I,H^2_0(M))\\hookrightarrow C(\\bar{I},H^1(M))\\hookrightarrow L^\\infty (I,L^4(M))$ schlie\"sen wir die Konvergenz von $(\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)$ in $L^\\infty (I,L^4(S_\\alpha ))$ .", "Um die Konvergenz in $H^1(I\\times S_\\alpha )$ einzusehen, folgern wir zun\"achst aus Proposition REF mit $\\theta =2/3$ und Theorem REF die Einbettung $\\begin{aligned}L^\\infty (I,L^2(M))\\cap L^2(I,H^2_0(M)) \\hookrightarrow L^{3}(I,H^{4/3}(M))\\hookrightarrow L^3(I,L^\\infty (M)).\\end{aligned}$ Die Konvergenz in $L^2(I,H^1(S_\\alpha ))$ schlie\"sen wir nun aus (REF ), $(1.b)$ , (REF ) und der kompakten Einbettung $Y^I\\hookrightarrow \\hookrightarrow L^6(I,H^1_0(M)),$ die aus Proposition REF folgt.", "Die Konvergenz in $H^1(I,L^2(S_\\alpha ))$ ist eine Konsequenz von (REF ), $(1.b)$ , (REF ) und der Konvergenz von $(\\partial _t\\eta _n)$ in $L^6(I,L^2(M))$ .", "Letztere folgt aus der Interpolationsungleichung $\\Vert \\partial _t\\eta _n-\\partial _t\\eta \\Vert _{L^6(I,L^2(M))}\\le \\Vert \\partial _t\\eta _n-\\partial _t\\eta \\Vert _{L^\\infty (I,L^2(M))}^{2/3}\\, \\Vert \\partial _t\\eta _n-\\partial _t\\eta \\Vert _{L^2(I\\times M)}^{1/3}.$ Ebenso zeigen wir die Konvergenz der Spur in $H^1(I\\times \\partial (\\Omega \\setminus \\overline{S_\\alpha }))\\cap L^\\infty (I,L^4(\\partial (\\Omega \\setminus \\overline{S_\\alpha })))$ .", "Wie zuvor gen\"ugt es nun, die Stetigkeitseigenschaften des L\"osungsoperators des Stokes-Systems zu beachten, um die Behauptung zu erhalten.", "$\\Box $ Wir verwenden im Folgenden die zeitunabh\"angige Variante des Operators $\\mathcal {M}$ , die auf naheliegende Weise definiert ist.", "Lemma 8.19 F\"ur alle $N\\in \\mathbb {N}$ , $3/2<p\\le \\infty $ und $\\epsilon >0$ existiert eine Konstante $c$ derart, dass f\"ur alle $\\eta ,\\,\\tilde{\\eta }\\in H^2_0(M)$ mit $\\Vert \\eta \\Vert _{H^2_0(M)}+\\Vert \\tilde{\\eta }\\Vert _{H^2_0(M)}+\\tau (\\eta )+\\tau (\\tilde{\\eta })\\le N$ und alle ${\\bf v}\\in W^{1,p}(\\Omega _\\eta )$ , $\\tilde{{\\bf v}}\\in W^{1,p}(\\Omega _{\\tilde{\\eta }})$ die Absch\"atzung $\\begin{aligned}&\\sup _{\\Vert b\\Vert _{L^2(M)}\\le 1}\\bigg (\\int _{\\Omega _{\\eta }} {\\bf v}\\cdot \\mathcal {F}_{\\eta }\\mathcal {M}_{\\eta }b\\ dx - \\int _{\\Omega _{\\tilde{\\eta }}}\\tilde{{\\bf v}}\\cdot \\mathcal {F}_{\\tilde{\\eta }}\\mathcal {M}_{\\tilde{\\eta }}b\\ dx\\\\&\\hspace{56.9055pt} + \\int _M \\operatorname{tr_\\eta }{\\bf v}\\cdot \\nu \\ \\mathcal {M}_{\\eta }b - \\operatorname{tr_{\\tilde{\\eta }}}\\tilde{{\\bf v}}\\cdot \\nu \\ \\mathcal {M}_{\\tilde{\\eta }}b\\ dA\\bigg )\\\\&\\le \\epsilon \\, \\big (\\Vert {\\bf v}\\Vert _{W^{1,p}(\\Omega _{\\eta })} +\\Vert \\tilde{{\\bf v}}\\Vert _{W^{1,p}(\\Omega _{\\tilde{\\eta }})}\\big ) + c\\sup _{\\Vert b\\Vert _{H^2_0(M)}\\le 1}\\bigg (\\int _{\\Omega _{\\eta }} {\\bf v}\\cdot \\mathcal {F}_{\\eta }\\mathcal {M}_{\\eta }b\\ dx - \\int _{\\Omega _{\\tilde{\\eta }}}\\tilde{{\\bf v}}\\cdot \\mathcal {F}_{\\tilde{\\eta }}\\mathcal {M}_{\\tilde{\\eta }}b\\ dx\\\\&\\hspace{221.93158pt} + \\int _M \\operatorname{tr_\\eta }{\\bf v}\\cdot \\nu \\ \\mathcal {M}_{\\eta }b -\\operatorname{tr_{\\tilde{\\eta }}}\\tilde{{\\bf v}}\\cdot \\nu \\ \\mathcal {M}_{\\tilde{\\eta }}b\\ dA\\bigg )\\end{aligned}$ gilt.", "Ebenso existiert f\"ur alle $N\\in \\mathbb {N}$ , $6/5<p,r\\le \\infty $ und $\\epsilon >0$ eine Konstante $c$ derart, dass f\"ur alle $\\eta ,\\,\\tilde{\\eta }\\in H^2_0(M)$ und $\\delta \\in C^4(\\partial \\Omega )$ mit $\\Vert \\eta \\Vert _{H^2_0(M)}+\\Vert \\tilde{\\eta }\\Vert _{H^2_0(M)}+\\Vert \\delta \\Vert _{C^4(\\partial \\Omega )}+\\tau (\\eta )+\\tau (\\tilde{\\eta })+\\tau (\\delta )\\le N$ und alle ${\\bf v}\\in W^{1,p}(\\Omega _\\eta )$ , $\\tilde{{\\bf v}}\\in W^{1,p}(\\Omega _{\\tilde{\\eta }})$ die Absch\"atzung $\\begin{aligned}&\\sup _{\\Vert \\varphi \\Vert _{H(\\Omega )}\\le 1}\\bigg (\\int _{\\Omega _{\\eta }} {\\bf v}\\cdot \\delta \\varphi \\ dx - \\int _{\\Omega _{\\tilde{\\eta }}}\\tilde{{\\bf v}}\\cdot \\delta \\varphi \\ dx\\bigg )\\\\&\\le \\epsilon \\, \\big (\\Vert {\\bf v}\\Vert _{W^{1,p}(\\Omega _{\\eta })} +\\Vert \\tilde{{\\bf v}}\\Vert _{W^{1,p}(\\Omega _{\\tilde{\\eta }})}\\big ) +c\\sup _{\\Vert \\varphi \\Vert _{W^{1,r}_{0,\\operatorname{div}}(\\Omega )}\\le 1}\\bigg (\\int _{\\Omega _{\\eta }} {\\bf v}\\cdot \\delta \\varphi \\ dx - \\int _{\\Omega _{\\tilde{\\eta }}}\\tilde{{\\bf v}}\\cdot \\delta \\varphi \\ dx\\bigg )\\end{aligned}$ gilt.", "Beweis: Wir beweisen diese Aussagen vom Typ des Ehrling-Lemmas mittels des \"ublichen Widerspruchsarguments.", "Wir zeigen zun\"achst die erste Behauptung.", "W\"are diese falsch, so g\"abe es ein $3/2<p\\le \\infty $ , ein $\\epsilon >0$ , beschr\"ankte Folgen $(\\eta _n),\\,(\\tilde{\\eta }_n)\\subset H^2_0(M)$ mit $\\sup _n\\big (\\tau (\\eta _n)+\\tau (\\tilde{\\eta }_n)\\big )<\\infty $ sowie Folgen $({\\bf v}_n),\\,(\\tilde{{\\bf v}}_n)$ mit $\\Vert {\\bf v}_n\\Vert _{W^{1,p}(\\Omega _{\\eta _n})}+ \\Vert \\tilde{{\\bf v}}_n\\Vert _{W^{1,p}(\\Omega _{\\tilde{\\eta }_n})}=1$ und $\\begin{aligned}&\\sup _{\\Vert b\\Vert _{L^2(M)}\\le 1}\\bigg (\\int _{\\Omega _{\\eta _n}} {\\bf v}_n\\cdot \\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b\\ dx - \\int _{\\Omega _{\\tilde{\\eta }_n}}\\tilde{{\\bf v}}_n\\cdot \\mathcal {F}_{\\tilde{\\eta }_n}\\mathcal {M}_{\\tilde{\\eta }_n}b\\ dx\\\\&\\hspace{56.9055pt} + \\int _M \\operatorname{tr_{\\eta _n}}{\\bf v}_n\\cdot \\nu \\ \\mathcal {M}_{\\eta _n}b - \\operatorname{tr_{\\tilde{\\eta }_n}}\\tilde{{\\bf v}}_n\\cdot \\nu \\ \\mathcal {M}_{\\tilde{\\eta }_n}b\\ dA\\bigg )\\\\&\\hspace{28.45274pt}> \\epsilon + n\\sup _{\\Vert b\\Vert _{H^2_0(M)}\\le 1}\\bigg (\\int _{\\Omega _{\\eta _n}} {\\bf v}_n\\cdot \\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b\\ dx - \\int _{\\Omega _{\\tilde{\\eta }_n}}\\tilde{{\\bf v}}_n\\cdot \\mathcal {F}_{\\tilde{\\eta }_n}\\mathcal {M}_{\\tilde{\\eta }_n}b\\ dx\\\\&\\hspace{128.0374pt} + \\int _M \\operatorname{tr_{\\eta _n}}{\\bf v}_n\\cdot \\nu \\ \\mathcal {M}_{\\eta _n}b -\\operatorname{tr_{\\tilde{\\eta }_n}}\\tilde{{\\bf v}}_n\\cdot \\nu \\ \\mathcal {M}_{\\tilde{\\eta }_n}b\\ dA\\bigg ) .\\end{aligned}$ Wegen Korollar REF sind die Folgen $(\\operatorname{tr_{\\eta _n}}{\\bf v}_n)$ , $(\\operatorname{tr_{\\tilde{\\eta }_n}}\\tilde{{\\bf v}}_n)$ in $W^{1-1/r,r}(M)$ f\"ur ein $r>3/2$ und wegen Theorem REF insbesondere in $H^s(M)$ f\"ur ein $s>0$ beschr\"ankt.", "Es existieren also Teilfolgen mit $\\begin{aligned}\\operatorname{tr_{\\eta _n}}{\\bf v}_n\\cdot \\nu \\rightarrow d,\\ \\operatorname{tr_{\\tilde{\\eta }_n}}\\tilde{{\\bf v}}_n\\cdot \\nu \\rightarrow \\tilde{d}&\\text{\\quad in}L^2(M),\\\\\\eta _n\\rightarrow \\eta ,\\ \\tilde{\\eta }_n\\rightarrow \\tilde{\\eta }&\\text{\\quad schwach in }H^2_0(M) \\text{, insbesondere gleichm\"a\"sig}.\\end{aligned}$ Lemma REF und die \"ublichen Sobolev-Einbettungen zeigen, dass eine Teilfolge von $({\\bf w}_n:={\\bf v}_n\\circ \\Psi _{\\eta _n})$ gegen eine Funktion ${\\bf w}$ in $L^3(\\Omega )$ konvergiert.", "Setzen wir alle beteiligten Funktionen durch 0 auf $\\mathbb {R}^3$ fort, so folgt aus der Absch\"atzung $\\begin{aligned}\\Vert {\\bf v}_n-{\\bf w}\\circ \\Psi ^{-1}_{\\eta }\\Vert _{L^2(\\mathbb {R}^3)} \\le \\Vert ({\\bf w}_n-{\\bf w})\\circ \\Psi ^{-1}_{\\eta _n}\\Vert _{L^2(\\mathbb {R}^3)} + \\Vert {\\bf w}\\circ \\Psi ^{-1}_{\\eta _n} -{\\bf w}\\circ \\Psi ^{-1}_{\\eta }\\Vert _{L^2(\\mathbb {R}^3)},\\end{aligned}$ Lemma REF und Bemerkung REF , dass die Folge $({\\bf v}_n)$ gegen ${\\bf v}:={\\bf w}\\circ \\Psi ^{-1}_{\\tilde{\\eta }}$ in $L^2(\\mathbb {R}^3)$ konvergiert.", "Ebenso konvergiert die Folge $(\\tilde{{\\bf v}}_n)$ gegen ein $\\tilde{{\\bf v}}$ in $L^2(\\mathbb {R}^3)$ .", "Aus Lemma REF $(1.a)$ , $(2.a)$ schlie\"sen wir zudem, dass die Folgen $(\\mathcal {M}_{\\eta _n}b)$ , $(\\mathcal {M}_{\\tilde{\\eta }_n}b)$ in $L^2(M)$ und die Folgen $(\\mathcal {F}_{\\eta _n}\\mathcal {M}_{\\eta _n}b)$ , $(\\mathcal {F}_{\\tilde{\\eta }_n}\\mathcal {M}_{\\tilde{\\eta }_n}b)$ in $L^2(\\mathbb {R}^3)$ unabh\"angig von $\\Vert b\\Vert _{L^2(M)}\\le 1$ konvergieren.", "Somit strebt das Supremum auf der rechten Seite von (REF ) gegen $\\begin{aligned}\\sup _{\\Vert b\\Vert _{H^2_0(M)}\\le 1}\\bigg (\\int _{\\Omega _{\\eta }} {\\bf v}\\cdot \\mathcal {F}_{\\eta }\\mathcal {M}_{\\eta }b\\ dx - \\int _{\\Omega _{\\tilde{\\eta }}}\\tilde{{\\bf v}}\\cdot \\mathcal {F}_{\\tilde{\\eta }}\\mathcal {M}_{\\tilde{\\eta }}b\\ dx + \\int _M d\\ \\mathcal {M}_{\\eta }b - \\tilde{d}\\ \\mathcal {M}_{\\tilde{\\eta }}b\\ dA\\bigg ).\\end{aligned}$ Da die linke Seite von (REF ) beschr\"ankt ist, muss dieser Grenzwert jedoch verschwinden.", "Aufgrund der Dichtheit von $H^2_0(M)$ in $L^2(M)$ muss dann aber auch der Grenzwert $\\begin{aligned}\\sup _{\\Vert b\\Vert _{L^2(M)}\\le 1}\\bigg (\\int _{\\Omega _{\\eta }} {\\bf v}\\cdot \\mathcal {F}_{\\eta }\\mathcal {M}_{\\eta }b\\ dx - \\int _{\\Omega _{\\tilde{\\eta }}}\\tilde{{\\bf v}}\\cdot \\mathcal {F}_{\\tilde{\\eta }}\\mathcal {M}_{\\tilde{\\eta }}b\\ dx + \\int _M d\\ \\mathcal {M}_{\\eta }b - \\tilde{d}\\ \\mathcal {M}_{\\tilde{\\eta }}b\\ dA\\bigg )\\end{aligned}$ der linken Seite von (REF ) identisch 0 sein; im Widerspruch zu $\\epsilon >0$ .", "Der Beweis der zweiten Behauptung geht vollkommen analog.", "Wir wollen hier deshalb lediglich zeigen, dass f\"ur beschr\"ankte Folgen $(\\eta _n)\\subset H^2_0(M)$ , $(\\delta _n)\\subset C^4(\\partial \\Omega )$ und eine Folge $({\\bf v}_n)$ mit $\\sup _n\\big (\\Vert {\\bf v}_n\\Vert _{W^{1,p}(\\Omega _{\\eta _n})}+\\tau (\\eta _n)+\\tau (\\delta _n)\\big )<\\infty $ f\"ur eine Teilfolge die Konvergenz $\\begin{aligned}\\int _{\\Omega _{\\eta _n}} {\\bf v}_n\\cdot {\\delta _n}\\varphi \\ dx\\quad \\rightarrow \\quad \\int _{\\Omega _{\\eta }} {\\bf v}\\cdot {\\delta }\\varphi \\ dx\\end{aligned}$ f\"ur $n\\rightarrow \\infty $ unabh\"angig von $\\Vert \\varphi \\Vert _{H(\\Omega )}\\le 1$ gilt.", "Wie zuvor finden wir Teilfolgen mit $\\begin{aligned}{\\bf v}_n\\rightarrow {\\bf v}&\\text{\\quad in }L^2(\\mathbb {R}^3),\\\\\\eta _n\\rightarrow \\eta &\\text{\\quad schwach in }H^2_0(M) \\text{, insbesondere gleichm\"a\"sig},\\\\\\delta _n\\rightarrow \\delta &\\text{\\quad in } C^3(\\partial \\Omega ).\\end{aligned}$ Durch Nulladdition erhalten wir die Absch\"atzung $\\begin{aligned}\\big |\\int _{\\Omega _{\\eta _n}}{\\bf v}_n\\cdot {\\delta _n}\\varphi \\ dx-\\int _{\\Omega _{\\eta }}{\\bf v}\\cdot {\\delta }\\varphi \\ dx\\big |&\\le \\Vert {\\bf v}_n-{\\bf v}\\Vert _{L^2(\\mathbb {R}^3)}\\,\\Vert {\\delta _n}\\varphi \\Vert _{L^2(\\mathbb {R}^3)}\\\\&\\hspace{14.22636pt}+\\Vert {\\bf v}\\Vert _{W^{1,p}(\\Omega _\\eta )}\\,\\Vert {\\delta _n}\\varphi -\\delta \\varphi \\Vert _{(W^{1,p}(\\Omega _\\eta ))^{\\prime }}.\\end{aligned}$ Wir m\"ussen also lediglich zeigen, dass die Folge $({\\delta _n}\\varphi )$ gegen ${\\delta }\\varphi $ in $(W^{1,p}(\\Omega _\\eta ))^{\\prime }$ unabh\"angig von $\\Vert \\varphi \\Vert _{H(\\Omega )}\\le 1$ konvergiert.", "W\"are dies nicht der Fall, so g\"abe es aber ein $\\epsilon >0$ und eine Folge $(\\varphi _n)\\subset H(\\Omega )$ , die schwach gegen ein $\\varphi $ in $H(\\Omega )$ konvergiert, und f\"ur die $\\Vert {\\delta _n}\\varphi _n-{\\delta }\\varphi _n\\Vert _{(W^{1,p}(\\Omega _\\eta ))^{\\prime }}>\\epsilon $ gilt.", "Das steht aber im Wiederspruch zur Absch\"atzung $\\begin{aligned}\\Vert {\\delta _n}\\varphi _n-{\\delta }\\varphi _n\\Vert _{(W^{1,p}(\\Omega _\\eta ))^{\\prime }}\\le \\Vert {\\delta _n}(\\varphi _n-\\varphi )\\Vert _{(W^{1,p}(\\Omega _\\eta ))^{\\prime }} +\\Vert {\\delta _n}\\varphi -{\\delta }\\varphi _n\\Vert _{(W^{1,p}(\\Omega _\\eta ))^{\\prime }}.\\end{aligned}$ Die Folgen $({\\delta }\\varphi _n)$ und $({\\delta _n}\\varphi )$ konvergieren n\"amlich schwach bzw.", "stark in $L^2(\\mathbb {R}^3)$ gegen $\\delta \\varphi $ .", "Die starke Konvergenz von $({\\delta _n}\\varphi )$ l\"asst sich leicht durch Approximation von $\\varphi $ durch glatte Funktionen einsehen; vgl.", "Bemerkung REF .", "Zudem zeigt die Identit\"at $\\int _{\\mathbb {R}^3} {\\delta _n}(\\varphi _n-\\varphi )\\ h\\ dx = \\int _\\Omega d\\Psi _{\\delta _n}(\\varphi _n-\\varphi )\\ h\\circ \\Psi _{\\delta _n}\\ dx$ f\"ur $h\\in L^2(\\mathbb {R}^3)$ , dass $({\\delta _n}(\\varphi _n-\\varphi ))$ schwach in $L^2(\\mathbb {R}^3)$ gegen 0 konvergiert.", "Aus der kompakten Einbettung $L^2(\\Omega _\\eta )\\hookrightarrow \\hookrightarrow (W^{1,p}(\\Omega _\\eta ))^{\\prime },$ siehe Korollar REF , folgt nun, dass die rechte Seite von (REF ) f\"ur gro\"se $n$ klein wird.", "$\\Box $ Lemma 8.20 F\"ur alle $N\\in \\mathbb {N}$ , $s>0$ und $\\epsilon >0$ existiert eine kleine Zahl $\\sigma >0$ derart, dass f\"ur alle $\\eta \\in H^2_0(M)$ mit $\\Vert \\eta \\Vert _{H^2_0(M)}+\\tau (\\eta )\\le N$ und alle $\\varphi \\in H(\\Omega _\\eta )$ mit $\\Vert \\varphi \\Vert _{L^2(\\Omega _\\eta )}\\le 1$ ein $\\psi \\in H(\\Omega _\\eta )$ existiert mit $\\operatorname{supp}\\psi \\subset \\Omega _{\\eta -\\sigma }$ , $\\Vert \\psi \\Vert _{L^2(\\Omega _\\eta )}\\le 2$ und $\\Vert \\varphi -\\psi \\Vert _{(H^{s}(\\mathbb {R}^3))^{\\prime }}<\\epsilon $ .Bei der Konstruktion von $\\Omega _{\\eta -\\sigma }$ wird zuerst $\\eta $ durch 0 auf $\\partial \\Omega $ fortgesetzt und anschlie\"send $\\sigma $ subtrahiert.", "Zudem sei nochmals explizit darauf hingewiesen, dass wir die Felder $\\varphi ,\\psi $ wie immer durch 0 auf $\\mathbb {R}^3$ fortsetzen.", "Beweis: W\"are die Behauptung falsch, so g\"abe es $s,\\,\\epsilon >0$ , eine Nullfolge $(\\sigma _n)_{n\\in \\mathbb {N}}$ positiver Zahlen, eine beschr\"ankte Folge $(\\eta _n)\\subset H^2_0(M)$ mit $\\sup _n\\tau (\\eta _n)<\\infty $ und eine Folge $(\\varphi _n)\\subset H(\\Omega _{\\eta _n})$ mit $\\Vert \\varphi _n\\Vert _{L^2(\\Omega _{\\eta _n})}\\le 1$ , $\\Vert \\varphi _n-\\psi \\Vert _{(H^{s}(\\mathbb {R}^3))^{\\prime }}\\ge \\epsilon $ f\"ur alle $\\psi \\in H(\\Omega _{\\eta _n})$ mit $\\Vert \\psi \\Vert _{L^2(\\Omega _{\\eta _n})}\\le 2$ und $\\operatorname{supp}\\psi \\subset \\Omega _{\\eta _n-\\sigma _n}$ sowie $\\begin{aligned}\\eta _n&\\rightarrow \\eta &&\\text{ schwach in }H^2_0(M), \\text{insbesondere gleichm\"a\"sig},\\\\\\varphi _n&\\rightarrow \\varphi &&\\text{ schwach in }L^2(\\mathbb {R}^3).\\end{aligned}$ Aus der kompakten Einbettung $L^2(B)\\hookrightarrow \\hookrightarrow (H^{s}(B))^{\\prime },$ siehe Theorem REF , f\"ur einen geeigneten Ball $B\\subset \\mathbb {R}^3$ folgt die Konvergenz von $(\\varphi _n)$ in $(H^{s}(\\mathbb {R}^3))^{\\prime }$ unter Verwendung der in [2] konstruierten Fortsetzungsoperatoren.", "Ist $\\psi $ eine $C^1$ -Funktion in einer Umgebung von $\\Omega _\\eta $ , so folgt aus Proposition REF $\\begin{aligned}\\int _{\\Omega _\\eta } \\varphi \\cdot \\nabla \\psi \\ dx = \\lim _n \\int _{\\Omega _{\\eta _n}}\\varphi _n\\cdot \\nabla \\psi \\ dx =0,\\end{aligned}$ da jedes Folgenglied verschwindet.", "Somit gilt $\\operatorname{tr^n_\\eta }\\varphi = 0$ , sodass gem\"a\"s Proposition REF ein $\\psi \\in H(\\Omega _\\eta )$ mit $\\operatorname{supp}\\psi \\subset \\Omega _\\eta $ und $\\Vert \\varphi -\\psi \\Vert _{L^2(\\Omega _\\eta )}<\\epsilon /2$ existiert.", "Es folgt $\\begin{aligned}\\Vert \\varphi _n-\\psi \\Vert _{(H^{s}(\\mathbb {R}^3))^{\\prime }}\\le \\Vert \\varphi _n-\\varphi \\Vert _{(H^{s}(\\mathbb {R}^3))^{\\prime }} +\\Vert \\varphi -\\psi \\Vert _{L^2(\\Omega _{\\eta })}< \\epsilon ,\\end{aligned}$ falls $n$ hinreichend gro\"s ist.", "Das ist ein Widerspruch, sofern $\\Vert \\psi \\Vert _{L^2(\\Omega _{\\eta _n})}\\le 2$ gilt.", "Diese Absch\"atzung ist aber eine Konsequenz von $\\Vert \\varphi -\\psi \\Vert _{L^2(\\Omega _\\eta )}<\\epsilon /2$ , falls $\\epsilon $ hinreichend klein ist.", "Letzteres kann ohne Einschr\"ankung angenommen werden.", "$\\Box $ Bemerkung 8.21 Wir zeigen nun, dass f\"ur $\\eta \\in Y^I$ , $\\Vert \\eta \\Vert _{L^\\infty (I\\times M)}<\\alpha <\\kappa $ und $(b,\\varphi )\\in T^I_\\eta $ die Fortsetzung von $\\varphi $ durch $(b\\,\\nu )\\circ q$ auf $I\\times B_\\alpha $ in $H^1(I,L^2(B_\\alpha ))$ liegt.", "Wir approximieren dazu $\\varphi $ durch Funktionen $(\\varphi _k)\\subset C_0^\\infty (\\mathbb {R}^4)$ in $H^1(\\Omega _\\eta ^I)$ und $\\eta $ durch $(\\eta _n)\\subset C^4(\\bar{I}\\times \\partial \\Omega )$ derart, dass die Folgen $(\\eta _n)$ gegen $\\eta $ in $L^\\infty (I\\times \\partial \\Omega )$ und $(\\partial _t\\eta _n)$ gegen $\\partial _t\\eta $ in $L^2(I\\times \\partial \\Omega )$ konvergieren; vgl.", "Definition (REF ).", "Unter Verwendung des Reynolds'schen Transporttheorems mit ${\\bf v}=(\\partial _t\\eta _n\\nu )\\circ \\Phi _{\\eta _n(t)}^{-1}$ und $\\xi =\\varphi _k\\,\\psi $ , $\\psi \\in C_0^\\infty (I\\times \\mathbb {R}^3)$ , erhalten wir die Identit\"at $\\begin{aligned}0 = \\int _I\\frac{d}{dt}\\int _{\\Omega _{\\eta _n(t)}}\\varphi _k\\,\\psi \\ dxdt&=\\int _I\\int _{\\Omega _{\\eta _n(t)}} \\partial _t\\varphi _k\\, \\psi +\\varphi _k\\, \\partial _t\\psi \\ dxdt\\\\&\\hspace{14.22636pt}+ \\int _I\\int _{\\partial \\Omega } \\operatorname{tr_{\\eta _n}}(\\varphi _k\\,\\psi )\\, \\partial _t\\eta _n\\, \\gamma (\\eta _n)\\ dAdt.\\end{aligned}$ Lassen wir zun\"achst $n$ und anschlie\"send $k$ gegen unendlich gehen, so ergibt sich $\\begin{aligned}\\int _I\\int _{\\Omega _{\\eta (t)}} \\varphi \\, \\partial _t\\psi \\ dxdt=-\\int _I\\int _{\\Omega _{\\eta (t)}} \\partial _t\\varphi \\, \\psi \\ dxdt-\\int _I\\int _{\\partial \\Omega }b\\,\\nu \\,\\operatorname{tr_\\eta }\\psi \\ \\partial _t\\eta \\, \\gamma (\\eta )\\ dAdt.\\end{aligned}$ Analog zeigen wir $\\begin{aligned}\\int _I\\int _{B_\\alpha \\setminus \\overline{\\Omega _{\\eta (t)}}} (b\\,\\nu )\\circ q\\ \\partial _t\\psi \\ dxdt&=-\\int _I\\int _{B_\\alpha \\setminus \\overline{\\Omega _{\\eta (t)}}} (\\partial _tb\\ \\nu )\\circ q\\ \\psi \\ dxdt\\\\&\\hspace{14.22636pt}+\\int _I\\int _{\\partial \\Omega }b\\,\\nu \\, \\operatorname{tr_\\eta }\\psi \\ \\partial _t\\eta \\, \\gamma (\\eta )\\ dAdt.\\end{aligned}$ Die Addition der letzten beiden Gleichungen zeigt die Behauptung." ] ]

[ [ "On Cosmological Constants from alpha'-Corrections" ], [ "Abstract We examine to what extent perturbative alpha'-corrections can generate a small cosmological constant in warped string compactifications.", "Focusing on the heterotic string at lowest order in the string loop expansion, we show that, for a maximally symmetric spacetime, the alpha'-corrected 4D scalar potential has no effect on the cosmological constant.", "The only relevant terms are instead higher order products of 4D Riemann tensors, which, however, are found to vanish in the usual perturbative regime of the alpha'-expansion.", "The heterotic string therefore only allows for 4D Minkowski vacua to all orders in alpha', unless one also introduces string loop and/or nonperturbative corrections or allows for curvatures or field strengths that are large in string units.", "In particular, we find that perturbative alpha'-effects cannot induce weakly curved AdS_4 solutions." ], [ "Introduction", "At sufficiently low energies and for small string coupling, perturbative string theory is well approximated by an effective two-derivative supergravity Lagrangian supplemented by small corrections coming from a double expansion in the slope parameter $\\alpha ^\\prime $ and the string coupling $g_s$ .", "The terms of the $\\alpha ^\\prime $ -expansion are higher derivative corrections to the supergravity action that account for the extended nature of the strings.", "They are negligible if the curvature of the background manifold and derivatives of the fields are small in units of $\\alpha ^{\\prime }$ .", "The terms coming from the $g_s$ -expansion are loop corrections due to nontrivial topologies of the string world sheet, which are negligible in the semi-classical regime when the string coupling is small.", "From a phenomenological point of view, such sub-leading corrections can have important consequences, as they may allow for solutions with properties that are forbidden at the two-derivative supergravity level.", "A well-known example in type IIB string theory are the AdS$_4$ solutions at large internal volume [1], where $\\alpha ^\\prime $ -corrections [2] break the no-scale structure of the leading order Minkowski solutions found in [3] and contribute to a nonzero cosmological constant.", "In this example, however, the $\\alpha ^\\prime $ -corrections alone are not sufficient, and also non-perturbative quantum corrections from localized sources are needed in order to generate the AdS vacuum.", "For the heterotic string, an analogous scenario was investigated in [4], where the authors found that an interplay of the lowest order $\\alpha ^{\\prime }$ -correction [5] and non-perturbative effects could give rise to a similar large volume AdS vacuum in 4D, while the classical two-derivative supergravity action only admits Minkowski ground states.", "In view of these constructions, one might wonder whether there could also be situations where the perturbative $\\alpha ^{\\prime }$ -corrections alone already suffice to generate a small non-vanishing cosmological constant in a controlled compactification scheme.", "This question should be easiest to study for the heterotic string, where D-branes and orientifold planes are absent, and the leading $\\alpha ^{\\prime }$ -corrections are completely known and already appear at order $\\mathcal {O}(\\alpha ^{\\prime })$ .", "Looking at the heterotic effective action at string tree-level, however, one might quickly conclude that $\\alpha ^{\\prime }$ -corrections alone can never suffice to generate vacua other than Minkowski space.", "The apparent reason is that, in the absence of string loop or non-perturbative corrections, all terms in the heterotic effective action come from world sheets with spherical topology so that the action scales uniformly with the dilaton $\\phi $ : $S=\\int \\mathrm {d}^{10}x\\sqrt{-g}\\, \\mathrm {e}^{-2\\phi }\\lbrace \\ldots \\rbrace $ (cf.", "(REF )).", "As a consequence, the four-dimensional effective scalar potential likewise scales uniformly with the dilaton zero mode, and one would expect the 4D dilaton equation to be solved either if the potential vanishes on the solution or if there is a runaway to a free vacuum.", "It therefore seems obvious that heterotic string theory at string tree level can only lead to Minkowski solutions, and that a non-vanishing cosmological constant also requires string loop or non-perturbative quantum corrections.", "A related argument was employed by Dine and Seiberg in [6], [7] to suggest that realistic string vacua might be strongly coupled.A priori, all this also applies to the oriented closed string sector of the type II theories, but the inclusion of orientifold planes and D-branes leads to terms with a different dilaton scaling already at string tree-level.", "It is these different scalings that allow, e.g., for the classical AdS vacua of [8] and that have been exploited in attempts to construct “classical” de Sitter vacua in type II supergravity with (smeared) orientifold planes (see e.g.", "[9], [10], [11] for early discussions) that could evade the “no-go” theorems discussed in [12], [13], [14], [15], [16].", "It is the purpose of this paper to re-address this question and in particular the seemingly trivial counter-argument against non-Minkowski vacua sketched in the previous paragraph.", "The reason is that the higher curvature terms among the $\\alpha ^{\\prime }$ -corrections (e.g.", "the $\\alpha ^{\\prime }\\mathrm {tr}|R^+|^2$ -terms in the heterotic string) also lead to contributions to the four-dimensional Einstein equation and the equations of motion for the moduli that involve higher powers of external Riemann tensors and hence can not be interpreted as a part of the effective scalar potential.", "It is therefore a priori not clear whether the scaling argument sketched above is still valid or whether nontrivial effects might emerge from such higher order terms.", "That these effects exist follows from explicitly known AdS$_4$ -compactifications of the heterotic string when the effective action is truncated after the lowest order $\\alpha ^{\\prime }$ -corrections (see e.g.", "[17], [18]).", "In these solutions, the 4D cosmological constant turns out large, $\\Lambda \\sim \\frac{1}{\\alpha ^{\\prime }}$ , so that the effects of even higher $\\alpha ^{\\prime }$ -corrections are difficult to estimate offhand and would require more explicit calculations [17].", "In an interesting recent paper [19], on the other hand, it was investigated whether the $\\alpha ^{\\prime }$ -corrections of the heterotic string could also give rise to a small cosmological constant $\\Lambda \\sim \\alpha ^{\\prime } C$ , where $C$ is a 6D integral over fields such as the dilaton or the warp factor with four internal derivatives.", "Intriguingly, the authors found that de Sitter vacua of this type are excluded, but raised the possibility of warped AdS$_4$ compactifications as an $\\mathcal {O}(\\alpha ^{\\prime })$ -effect.", "Proving this requires only some of the 10D field equations, and it was left as an open problem to check whether really all field equations could be satisfied at the considered order in the $\\alpha ^{\\prime }$ -expansion.", "In this paper we investigate to what extent the usual scaling analysis of the 4D effective potential is invalidated by higher curvature terms in the $\\alpha ^{\\prime }$ -expansion and check whether this expansion can yield perturbatively small cosmological constants of order $\\mathcal {O}(\\alpha ^{\\prime })$ or higher.", "The main result of our analysis is that this is in general not possible at string tree-level.", "This follows from the four-dimensional Einstein equation and the dilaton equation, which can be combined to yield a constraint of the form $\\Lambda = \\sum _{m,n} c_{mn} \\alpha ^{\\prime m} \\Lambda ^n, \\qquad m,n>0,$ where $c_{mn}$ are numerical coefficients containing integrals over internal fields and their derivatives.", "Assuming a perturbative $\\alpha ^{\\prime }$ -expansion for $\\Lambda $ , one then obtains $\\Lambda = 0$ as the only solution to all orders in $\\alpha ^\\prime $ , as we will explain in more detail below.", "This paper is organized as follows.", "In Section REF we establish our notation and detail a simple argument (cf.", "[20]) showing that heterotic supergravity with the first order $\\alpha ^\\prime $ -corrections does not yield solutions with a nonzero cosmological constant to that order.", "In Appendix we investigate the proposed warped AdS solutions of [19] at order $\\mathcal {O}(\\alpha ^{\\prime })$ more directly, showing explicitly that the given $\\mathcal {O}(\\alpha ^{\\prime })$ -expression for the cosmological constant is really of higher order and thus could compete with neglected terms in the action.", "In Section REF we then show how the argument of Section REF can be extended to all orders in the $\\alpha ^\\prime $ -expansion and that it is completely independent of the details of the $\\alpha ^{\\prime }$ -corrected 4D scalar potential.", "We conclude with Section , where we discuss several ways to circumvent this “no-go theorem”, its relation to the Dine-Seiberg problem, and possible effects of violations of the effective potential description.", "Appendix collects some useful identities for the Riemann tensor, and Appendix contains the ten-dimensional version of the argument of Section REF ." ], [ "A “no-go theorem”", "In this section, we discuss a simple argument showing that tree-level heterotic string theory with its first order $\\alpha ^\\prime $ -corrections does not have 4D de Sitter or anti-de Sitter vacua with a perturbatively small cosmological constant at this order [20].", "We then show that the argument can in fact be extended to all orders in the $\\alpha ^\\prime $ -expansion.", "Our assumptions throughout the paper are as follows: We consider compactifications to four dimensions that respect maximal four-dimen-sional spacetime symmetry, i.e.", ": The 10D metric is a warped product of a maximally symmetric 4D spacetime (parameterized by coordinates $x^\\mu $ ; $\\mu ,\\nu ,\\ldots =0,\\ldots ,3$ ) and a 6D compact manifold (parameterized by $y^m$ ; $m,n,\\ldots =4,\\ldots ,9$ ), $\\mathrm {d}s^2 &=& \\mathrm {e}^{2A} \\mathrm {d}s_4^2 + \\mathrm {d}s_6^2,$ where $\\mathrm {e}^{2A}$ depends on the 6D coordinates only, and $\\mathrm {d}s_4^2$ describes an unwarped 4D Minkowski, de Sitter or anti-de Sitter spacetime.", "All 4D parts of tensor and spinor fields vanish (up to gauge choices) except for combinations that can be built from the 4D (unwarped) metric, its Riemann tensor or its volume form.", "This means, in particular, that there are no spacetime-filling fluxesWe express everything in terms of the Yang-Mills field strength $F$ and the NS 3-form $H$ , which have a too small rank to be spacetime-filling in 4D.", "The Hodge duals of purely 6D fluxes of these fields would of course generically have spacetime-filling components, but they do not appear explicitly in our formalism.", "and that all 4D covariant derivatives of all tensor fields, including the dilaton and the Riemann tensor, can be set to zero on the solution.Note that for maximally symmetric spaces, the Riemann tensor becomes an algebraic combination of metric tensors, and therefore its covariant derivative vanishes.", "Furthermore, the Lorentz-Chern-Simons 3-form does not contribute to the equations of motion in maximally symmetric backgrounds [21].", "String-loop and/or non-perturbative corrections to the action are disregarded.", "$\\alpha ^{\\prime }$ is a meaningful expansion parameter in the sense that all field variations are small over a string length and the $\\alpha ^{\\prime }$ -corrections can be organized in a perturbative expansion about the zero-slope limit.The $\\alpha ^\\prime $ -expansion differs from the derivative expansion in that some terms appear at higher orders than suggested by the number of their derivatives.", "An example is the term $\\mathrm {tr}|F|^2$ which, although a two derivative term, appears at $\\mathcal {O}(\\alpha ^{\\prime })$ .", "It should be noted though that our analysis does not depend on which of the two expansion schemes is used." ], [ "Heterotic supergravity with leading $\\alpha ^\\prime $ -corrections", "In string frame, the heterotic supergravity action with leading $\\alpha ^\\prime $ -corrections reads (for simplicity we set the 10D gravitational coupling $\\kappa ^2=\\frac{1}{2}$ ) $S = \\int \\mathrm {d}^{10} x \\sqrt{-g}\\, \\mathrm {e}^{-2\\phi } \\left\\lbrace { R + 4 (\\partial \\phi )^2 - \\frac{1}{2} |H|^2 - \\frac{\\alpha ^\\prime }{4} \\left[{\\mathrm {tr} |F|^2 - \\mathrm {tr} |R^+|^2 }\\right] } + \\mathcal {O }(\\alpha ^{\\prime 2})\\right\\rbrace $ with $|H|^2 = {6} H_{MNL}H^{MNL}$ , $\\mathrm {tr}|F|^2 = {2}\\mathrm {tr}F_{MN}F^{MN}$ and $\\mathrm {tr}|R^+|^2 = {2} R^+_{MNPQ}R^{+MNPQ}$ .", "Here, $\\phi $ denotes the dilaton, $F$ is the Yang-Mills field strength, and $R^+_{MNP}\\vphantom{}^Q$ is the Riemann tensor constructed from the torsionful connection $\\Gamma ^+\\vphantom{}^M_{NL} = \\Gamma ^{M}_{NL} - \\frac{1}{2} H_{NL}\\vphantom{}^{M}$ .", "$H$ is the $\\alpha ^{\\prime }$ -corrected 3-form field strength of the NS 2-form potential $B$ , $H = \\mathrm {d}B + {4}\\left({\\omega _{3 \\text{L}} - \\omega _{3 \\text{Y}}}\\right),$ where $\\omega _{3 \\text{L}}$ and $\\omega _{3 \\text{Y}}$ denote the Chern-Simons 3-forms formed from the spin connection and the Yang-Mills gauge field, respectively.", "For our argument, it is sufficient to look at the field equations of the dilaton and the external metric.", "This can be done either by using a 4D effective action approach or by working directly with the 10D field equations.", "We describe the 4D effective action approach here and sketch the analogous 10D argument in Appendix .", "For the 4D argument, we can restrict our attention to the zero mode, $\\tau $ , of the dilaton, which we define by separating off the higher Kaluza-Klein modes, $\\mathrm {e}^{-\\phi } = \\tau \\mathrm {e}^{-\\phi _\\textrm {KK}}.$ Here $\\phi _\\textrm {KK}$ denotes the sum of all remaining KK-modes, which we integrate out by simply setting them equal to their on-shell values.", "It does not matter for our argument whether $\\tau $ or one of the KK modes has the lowest mass (or whether they even combine with other degrees of freedom in the low energy EFT as suggested in [22]) as can be seen directly from the equivalent ten-dimensional analysis in Appendix .", "On-shell, all fields in 4D must be covariantly constant by maximal symmetry, so we can henceforth ignore any $x^\\mu $ -dependence of $\\tau $ and only need to keep track of $\\tau $ itself in the action, but not of its derivatives.", "The only other field whose dynamics we need to consider is the external metric $g_{\\mu \\nu }$ .", "Switching to four-dimensional Einstein frame, we define a new 4D metric $\\tilde{g}_{\\mu \\nu }$ by $\\tilde{g}_{\\mu \\nu } \\equiv \\mathcal {V} \\tau ^2 \\mathrm {e}^{-2A} g_{\\mu \\nu }.", "$ Here $\\mathcal {V} \\equiv \\int \\mathrm {d}^{6} y \\sqrt{g_6}\\, \\mathrm {e}^{-2\\phi _\\textrm {KK}+2A},$ which can again be treated as constant in 4D by maximal symmetry.", "Performing this rescaling, we then obtain an effective 4D action for $\\tilde{g}_{\\mu \\nu }$ and $\\tau $ of the form $S = \\int \\mathrm {d}^{4} x \\sqrt{-\\tilde{g}_4} \\left\\lbrace {\\tilde{R}_4 - V + W}\\right\\rbrace ,$ where we have split the action into the Einstein-Hilbert term and two extra contributions.", "$V$ contains all terms that are constructed from fields without external indices, whereas $W$ contains all terms that include fields with 4D spacetime indices.", "In the absence of $W$ , $V$ is just the usual effective potential.", "Using (REF ), these two terms are given by $V = - & \\int \\mathrm {d}^6 y \\sqrt{g_6}\\, \\mathrm {e}^{-2\\phi _\\textrm {KK}+4A} \\frac{1}{\\tau ^{2} \\mathcal {V}^2} \\\\ &{} \\times \\left[{ \\vphantom{|R^{\\textrm {(w)}}_{\\mu \\nu \\lambda }\\vphantom{}^\\rho |^2} R_6 - 20 (\\partial A)^2 - 8 \\nabla ^2 A + 4 (\\partial \\phi )^2 - \\frac{1}{2} |H|^2 - \\frac{\\alpha ^\\prime }{4} \\left({\\mathrm {tr} |F|^2 - |R^+_6|^2}\\right)}\\right] +\\mathcal {O}(\\alpha ^{\\prime 2})$ and $W = \\int \\mathrm {d}^6 y \\sqrt{g_6}\\, \\mathrm {e}^{-2\\phi _\\textrm {KK}} \\left[{\\frac{\\alpha ^\\prime \\tau ^2}{4} |\\tilde{R}_{\\mu \\nu \\lambda }\\vphantom{}^\\rho |^2 - \\frac{\\alpha ^\\prime }{2\\mathcal {V}} \\mathrm {e}^{2A} \\tilde{R}_4 (\\partial A)^2 }\\right] +\\mathcal {O}(\\alpha ^{\\prime 2}), $ where we have evaluated the curvature terms $R$ and $\\mathrm {tr} |R^+|^2$ for the tilded metric (REF ) and expressed them in terms of $\\tilde{R}_4$ and $|\\tilde{R}_{\\mu \\nu \\lambda }\\vphantom{}^\\rho |^2 = \\frac{1}{2} \\tilde{R}_{\\mu \\nu \\lambda \\rho }\\tilde{R}^{\\mu \\nu \\lambda \\rho }$ as well as a term $|R^+_6|^2$ containing various internal fields.", "Further details and the definition of $|R^+_6|^2$ can be found in Appendix .", "Using the scaling $V \\sim \\tau ^{-2}$ , one finds the 4D dilaton equation, $2 V + \\frac{\\alpha ^\\prime \\tau ^2}{2} |\\tilde{R}_{\\mu \\nu \\lambda }\\vphantom{}^\\rho |^2 \\int \\mathrm {d}^6 y \\sqrt{g_6}\\, \\mathrm {e}^{-2\\phi _\\textrm {KK}} = 0, $ and the trace of the four-dimensional Einstein equation, $\\tilde{R}_4 - 2 V - \\frac{\\alpha ^\\prime }{2\\mathcal {V}} \\tilde{R}_4 \\int \\mathrm {d}^6 y \\sqrt{g_6}\\, \\mathrm {e}^{-2\\phi _\\textrm {KK}+2A}\\, (\\partial A)^2 = 0, $ where we have neglected the variation with respect to the connection as it would give rise to covariant derivatives upon partial integration, which vanish due to maximal symmetry.", "Combining the two equations such that $V$ cancels out and substituting $\\tilde{R}_{\\mu \\nu \\lambda \\rho } = \\frac{2}{3} \\Lambda \\tilde{g}_{\\lambda [\\mu } \\tilde{g}_{\\nu ]\\rho }$ then yields an equation of the form $\\Lambda = \\alpha ^\\prime \\left({c_{11} \\Lambda + c_{12} \\Lambda ^2}\\right) + \\mathcal {O}(\\alpha ^{\\prime 2}), $ where $c_{11}$ and $c_{12}$ are given by $c_{11} = {2\\mathcal {V}}\\int \\mathrm {d}^6 y \\sqrt{g_6}\\, \\mathrm {e}^{-2\\phi _\\textrm {KK}+2A}\\, (\\partial A)^2,\\qquad c_{12} = - {3} \\int \\mathrm {d}^6 y \\sqrt{g_6}\\, \\mathrm {e}^{-2\\phi _\\textrm {KK}}.$ Given our assumption that we are in the regime of validity of the perturbative $\\alpha ^\\prime $ -expansion, (REF ) must be solved order by order with an ansatz of the form $\\Lambda = \\Lambda _0 + \\alpha ^\\prime \\Lambda _1 + \\mathcal {O}(\\alpha ^{\\prime 2})$ for the cosmological constant, where $\\Lambda _0$ denotes the solution of the leading order supergravity equations without $\\alpha ^{\\prime }$ -corrections, $\\alpha ^\\prime \\Lambda _1$ is a correction due to next-to-leading order terms in the $\\alpha ^\\prime $ -expansion, and so on.", "It is straightforward to see that plugging this ansatz into (REF ) yields $\\Lambda = \\mathcal {O}(\\alpha ^{\\prime 2}) $ as the only solution.", "Thus, perturbative heterotic string theory does not yield solutions with a nonzero cosmological constant up to corrections of order $\\mathcal {O}(\\alpha ^{\\prime 2})$ .", "Let us now compare this to the result of [19], where it was suggested that warped AdS vacua might be allowed in heterotic string theory as an $\\mathcal {O}(\\alpha ^\\prime )$ -effect, i.e., $\\Lambda = - \\alpha ^\\prime C + \\mathcal {O}(\\alpha ^{\\prime 2}), $ where $C$ is a non-negative constant which is built from a sum of squares of internal fields integrated over the internal manifold.", "At first sight, this seems to contradict the above argument that solutions with nonzero cosmological constant are not allowed at order $\\mathcal {O}(\\alpha ^{\\prime 1})$ .", "However, one can show directly by means of the supergravity equations of motion that the terms contained in $C$ actually vanish at the order considered here such that $C=\\mathcal {O}(\\alpha ^\\prime )$ .", "The right hand sides of (REF ) and (REF ) are therefore equal up to corrections of order $\\mathcal {O}(\\alpha ^{\\prime 2})$ .", "For convenience, we give the details in Appendix ." ], [ "General argument", "Let us now generalize the above argument to the heterotic string with $\\alpha ^\\prime $ -corrections of arbitrarily high order.", "The effective action for the massless fields then reads $S = \\int \\mathrm {d}^{10} x \\sqrt{-g}\\, \\mathrm {e}^{-2\\phi } \\left\\lbrace R + 4 (\\partial \\phi )^2 - \\frac{1}{2} |H|^2 + \\alpha ^\\prime \\text{-corrections}\\right\\rbrace , $ where all terms scale identically with respect to the dilaton if we neglect string loop or non-perturbative corrections as initially stated.", "Rescaling the metric as in (REF ), we obtain the action in four-dimensional Einstein frame $S = & \\int \\mathrm {d}^{4} x \\sqrt{-\\tilde{g}_4} \\left\\lbrace { \\tilde{R}_4 - V + W }\\right\\rbrace .$ As in the previous section, we have split the action into an Einstein-Hilbert term $\\tilde{R}_4$ , a term $V$ containing all terms that are constructed from fields without external spacetime indices, and a term $W$ containing everything else.", "In the absence of string loop or non-perturbative corrections, all terms in $V$ scale again as $V \\sim \\tau ^{-2}$ such that the dilaton equation yields $2 V + \\tau \\partial _\\tau W = 0.", "$ Taking the trace of the four-dimensional Einstein equation, we furthermore find $\\tilde{R}_4 - 2 V - W^\\prime = 0, \\qquad W^\\prime \\equiv \\frac{\\tilde{g}^{\\mu \\nu }}{\\sqrt{-\\tilde{g}_4}} \\frac{\\delta }{\\delta \\tilde{g}^{\\mu \\nu }} \\left({ \\int \\mathrm {d}^{4} x \\sqrt{-\\tilde{g}_4} \\, W}\\right), $ where, as indicated, $W^\\prime $ denotes all terms that are due to the variation of $W$ with respect to the external metric.", "Combining the two equations (REF ) and (REF ), we then find $\\tilde{R}_4 = - \\tau \\partial _\\tau W + W^\\prime .", "$ Although an explicit expression for the right hand side of this equation is only known for the first few orders in the $\\alpha ^\\prime $ -expansion, the general structure is rather simple: it is a sum of positive powers of the cosmological constant with coefficients built from integrals over internal fields and their derivatives.", "To see this, recall that our assumption of maximal 4D spacetime symmetry implies that only the metric, the epsilon tensor and the Riemann tensor are nontrivial, all with vanishing covariant derivative.", "Considering first the metric variations of $W$ that come from variations of connections (either within covariant derivatives or curvature tensors or Lorentz-Chern-Simons forms), one sees that these variations do not contribute to the right hand side of (REF ), as they would lead to terms with a total 4D covariant derivative, which vanish by assumption.", "The only contributions to $W^{\\prime }$ are therefore from variations of metric tensors that appear algebraically in $W$ or in the metric determinant.", "As there are no nontrivial contractions of just the epsilon tensor and/or the metric, all these terms must contain at least one Riemann tensor.Note that there is no constant term in $W$ : a constant has no external spacetime indices and hence would be part of $V$ , which however cancels out in (REF ).", "Similar remarks also apply to the dilaton variation of $W$ , so that the right hand side of (REF ) is a sum of terms that each involves at least one Riemann tensor.", "Because of $\\tilde{R}_{\\mu \\nu \\lambda \\rho } = \\frac{2}{3} \\Lambda \\tilde{g}_{\\lambda [\\mu } \\tilde{g}_{\\nu ]\\rho }$ , these then translate into positive powers of the cosmological constant, as claimed.", "Since at leading order the supergravity action does not contain any terms that depend on the Riemann tensor except for the Einstein-Hilbert term, the terms in $W$ and $W^\\prime $ are of order $\\mathcal {O}(\\alpha ^\\prime )$ or higher.", "We can therefore schematically rewrite (REF ) as $\\Lambda = \\sum _{m,n} c_{mn} \\alpha ^{\\prime m} \\Lambda ^n, \\qquad m,n > 0, $ with some numerical coefficients $c_{mn}$ that in general contain integrals over contractions of warp factor terms, internal field strengths and curvatures, and so on.", "Assuming again the validity of a perturbative $\\alpha ^{\\prime }$ -expansion, we need to solve (REF ) order by order with an ansatz of the form $\\Lambda = \\Lambda _0 + \\alpha ^\\prime \\Lambda _1 + \\alpha ^{\\prime 2} \\Lambda _2 + ... $ as in Section REF .", "This yields $\\Lambda = 0$ as the only solution to all orders in the perturbative $\\alpha ^\\prime $ -expansion.We might also try to solve (REF ) without expanding $\\Lambda $ as in (REF ).", "Assuming that $\\Lambda \\ne 0$ , we can then divide by $\\Lambda $ to get $1 \\le \\sum |c_{mn}\\alpha ^{\\prime m}\\Lambda ^{n-1}|$ .", "But this is again a contradiction to the assumption made in the beginning of Section .", "Hence, heterotic string theory yields Minkowski spacetime as the only maximally symmetric solution to all orders in the perturbative $\\alpha ^\\prime $ -expansion, unless one introduces loop and/or non-perturbative corrections.", "In particular, we don't find $\\alpha ^{\\prime }$ -generated $AdS_4$ vacua with perturbatively small curvatures to be possible." ], [ "Discussion", "Let us now discuss several implications of our findings.", "In particular, we will discuss possibilities to evade our above no-go argument, its relation to the Dine-Seiberg problem and the violation of the effective potential description due to higher order corrections to the supergravity action." ], [ "Evading the no-go theorem", "In Section REF , we have shown that heterotic string compactifications at string tree-level yield 4D Minkowski spacetime as the only maximally symmetric solution to all orders in a perturbative $\\alpha ^\\prime $ -expansion, unless one violates one of our initial assumptions.", "Let us now discuss these possible violations and how they evade our argument." ], [ "Loop and non-perturbative corrections/extended sources", "An obvious possibility to circumvent the argument of Section REF is the inclusion of terms that scale differently with respect to the dilaton than the tree-level terms considered here.", "Natural candidates are string loop or non-perturbative corrections e.g.", "from gaugino condensation [23], [24].", "With such terms turned on, the dilaton and Einstein equations read $- \\tau \\partial _\\tau V + \\tau \\partial _\\tau W = 0, \\qquad \\tilde{R}_4 - 2 V - W^\\prime = 0 $ and can in general not be combined such that $V$ cancels out.", "The right hand side of (REF ) may then contain terms which are independent of $\\Lambda $ , making solutions other than $\\Lambda = 0$ possible.", "It would be interesting to see whether including the first loop correction at order $\\mathcal {O}(\\alpha ^{\\prime 3}g_s)$ could allow for purely perturbative solutions with a non-zero cosmological constant for the heterotic string.", "A different dilaton scaling may also be introduced if one includes extended sources such as the various types of D-branes and orientifold planes in type II string theory.", "Being an open string tree-level action, the DBI action scales only with $\\mathrm {e}^{-\\phi }$ and so would in general also invalidate our argument.", "In fact, in type II string theory, a large number of compactification scenarios with a nonzero cosmological constant have been proposed using D-branes and orientifolds as well as non-perturbative quantum corrections starting with [25].", "Heterotic string theory, on the other hand, is much more limited in this respect, as it does not contain D-branes and O-planes but would require dealing with less common extended objects.", "Since spacetime-filling fluxes are in general not forbidden by maximal symmetry, they can be used to invalidate our argument around (REF ), where we explained that all terms in $W$ are contractions of Riemann tensors and must therefore contain factors of the cosmological constant.", "In heterotic string theory, there are no spacetime-filling fluxes if spacetime is assumed to be four-dimensional.", "Compactifying to three dimensions, however, allows for solutions with a nonzero cosmological constant, if spacetime components of $H$ are turned on (see e.g.", "[26]).", "In type II string theory, spacetime-filling RR-fluxes may also be present in compactifications to four dimensions and may lead to solutions with a nonzero cosmological constant.", "Another way to circumvent our no-go theorem is to leave the perturbative regime of the $\\alpha ^\\prime $ -expansion and consider solutions for which higher derivative terms are not small in units of $\\alpha ^\\prime $ .", "A truncation of the action at a finite order is then in general not guaranteed to be a good approximation to the full theory, because higher order terms are not automatically suppressed.This does of course not rule out that the truncated action could still capture the essential features of a solution or that the higher order terms happen to be small or even vanish in certain cases.", "This problem does of course not apply when supergravity is studied in its own right instead of being considered the low energy effective field theory of string theory.", "In any case, allowing curvature and derivatives of the fields to be large in units of $\\alpha ^\\prime $ , it is indeed possible to construct solutions with a nonzero cosmological constant that is large in units of $\\alpha ^{\\prime }$ .", "A good example are the heterotic AdS compactifications studied in [17], which are solutions to the heterotic supergravity action with linear $\\alpha ^\\prime $ -corrections that feature a curvature of order $\\mathcal {O}(\\frac{1}{\\alpha ^\\prime })$ .", "By construction, our argument does not make statements in this regime.", "Requiring spacetime to be maximally symmetric implies a very limited field content such that, in the absence of spacetime-filling fluxes, all terms showing up on the right hand side of (REF ) contain contractions of spacetime components of the Riemann tensor.", "All of these terms can then be rewritten as a power of $\\Lambda $ times some numerical factor, regardless of how the Riemann tensors are contracted.", "As explained earlier, this property ensures that the higher derivative curvature terms on the right hand side of (REF ) are much smaller than the Ricci scalar on the left hand side, leading to constraint (REF ) and the conclusion that only Minkowski solutions are possible.", "For spacetimes without maximal symmetry, however, this need not be the case.", "The presence of various (spacetime) tensor fields then leads to new terms in (REF ) which can be of the same order as the 4D Ricci scalar and thus generate a nonzero cosmological constant.", "Furthermore, it is not guaranteed anymore that higher derivative curvature terms in (REF ) are negligible, since whether they are much smaller than the 4D Ricci scalar or can compete with it depends on how they are contracted.", "This is due to the well-known fact that for general spaces the magnitude of individual components of the Riemann tensor and the Ricci scalar need not be the same, so that the Riemann tensor can have large components even when the Ricci scalar is very small.", "In heterotic string theory, the Ricci scalar may then compete, for example, with the $\\alpha ^\\prime |\\tilde{R}_{\\mu \\nu \\lambda }\\vphantom{}^\\rho |^2$ term and thus become nonzero.", "This is also the reason why it is not possible to extend our analysis to make a statement about the curvature of the internal space.", "An exception are compactifications on maximally symmetric spaces such as the six-sphere, which can be ruled out using an argument along the lines of Section REF , unless there exist six-form fluxes filling internal space.", "Since this only concerns a very restricted class of compactifications, our discussion does unfortunately not add much to the discussion of [27], where it is suggested that higher derivative corrections (or strong warping, see also [28]) could in principle support an everywhere negative internal Ricci scalar, which is difficult to realize otherwise." ], [ "The Dine-Seiberg problem", "In [6], [7], Dine and Seiberg used the dilaton behavior of the effective 4D scalar potential in the weak coupling limit to argue that, unless the effective potential is identically zero, there must in general either be a runaway to the free vacuum or a minimum at strong coupling.", "Using an analogous scaling analysis for the universal volume modulus, one may argue for similar difficulties regarding compactifications at large volume (cf.", "e.g.", "[29] for a recent discussion).", "Progress in moduli stabilization techniques have since then led to many interesting scenarios where an interplay of various scalar potential contributions suggest the existence of weakly coupled minima at controllably large volumes.", "Still many of the difficulties and complexities one encounters in these attempts can be traced back to the issues pointed out in [6], [7].", "The argument given in the present paper, although somewhat similar in its consequences, differs from the argument of [6], [7] in several ways.", "First of all we do not really use or discuss moduli stabilization.", "Nor do we trace the dependence of the scalar potential on the volume modulus.", "In fact, the detailed form of the scalar potential and its moduli dependence play no role for our argument (except that we exploit the overall dilaton scaling to eliminate the scalar potential completely from the equation of interest (eq.", "(REF ))).", "Instead, the only terms that matter for our argument are higher order products of 4D Riemann tensors, which did not play a role for the arguments in [6], [7].", "Moreover, it could have been the case that terms that appear to be of lower order in the $\\alpha ^{\\prime }$ -expansion compete with terms that are explicitly of higher order in $\\alpha ^{\\prime }$ without that the perturbative $\\alpha ^{\\prime }$ -expansion breaks down.", "An example for this are the $|H|^2$ and $|F|^2$ terms appearing in the heterotic supergravity action or gradient terms of the warp factor or the dilaton.", "As reviewed in Appendix , they are forced to be zero by the leading order equations of motion, if our initially stated assumptions hold.", "Including $\\alpha ^\\prime $ -corrections to the action, however, the equations of motion are modified such that the above terms can become nonzero and thus compete with higher order terms in the $\\alpha ^\\prime $ -expansion.", "This could have postponed the emergence of a nontrivial cosmological constant to a higher order than suggested by [19].", "Our argument from section REF , however, shows that this can not happen at any order in $\\alpha ^{\\prime }$ , regardless of the scalar potential." ], [ "Violation of effective scalar potential description", "The effective scalar potential description is a standard tool in effective field theory which is widely used in the moduli stabilization literature.", "For solutions yielding a maximally symmetric spacetime, the effective potential is usually expected to fulfill two assumptions: The equations of motion are satisfied at a point in moduli space which is an extremum of $V$ .", "The value of $V$ at this point is proportional to the cosmological constant.", "These assumptions are true if the effective action can be written in the form $S = & \\int \\mathrm {d}^{4} x \\sqrt{-\\tilde{g}_4} \\left\\lbrace { \\tilde{R}_4 - V }\\right\\rbrace ,$ where $\\tilde{R}_4$ is the only term in the Lagrangian that depends on the external metric, and $V$ is the only term that depends on the moduli.", "It is interesting to note that both assumptions are generically violated by higher order effects in the $\\alpha ^\\prime $ -expansion, unless the cosmological constant is zero.", "This follows from (REF ) which on-shell yields $\\partial _\\tau V \\ne 0, \\qquad V \\lnot \\sim \\Lambda .$ Hence, the equations of motion are in general satisfied at a point in moduli space which is not an extremum of $V$ .", "Moreover, $V$ is not proportional to the cosmological constant anymore.", "This effect is usually completely negligible when the cosmological constant is small.", "For inflation scenarios with a very high energy scale, these corrections might be more sizeable, but when they are, the validity of the perturbative $\\alpha ^{\\prime }$ -expansion would also be less obvious." ], [ "Acknowledgements", "This work was supported by the German Research Foundation (DFG) within the Cluster of Excellence \"QUEST\"." ], [ "Riemann tensor with warping", "Let us compute the components of the Riemann tensor for the warped spacetime $\\mathrm {d}s^2 = \\frac{\\mathrm {e}^{2A}}{\\mathcal {V}\\tau ^2}\\, \\tilde{g}_{\\mu \\nu } \\mathrm {d}x^\\mu \\mathrm {d}x^\\nu + \\mathrm {d}s_6^2.$ In order to express the full Riemann tensor $R_{MRN} \\vphantom{}^P$ in terms of the Riemann tensor $\\tilde{R}_{MRN} \\vphantom{}^P$ of the unwarped metric $\\tilde{g}_{MN}$ , we use the formula $R_{MRN} \\vphantom{}^P = - \\tilde{\\nabla }_M \\Gamma ^P_{RN} + \\tilde{\\nabla }_R \\Gamma ^P_{MN} + \\Gamma ^S_{NM} \\Gamma ^P_{RS} - \\Gamma ^S_{NR} \\Gamma ^P_{MS} + \\tilde{R}_{MRN} \\vphantom{}^P, $ where $M,N,\\ldots =0,\\ldots ,9$ denote 10D spacetime indices, $\\Gamma ^M_{NP} = \\tfrac{1}{2} g^{MR} ( \\tilde{\\nabla }_P g_{RN} + \\tilde{\\nabla }_N g_{RP} - \\tilde{\\nabla }_R g_{NP})$ and $\\tilde{\\nabla }_M$ is the covariant derivative associated with the unwarped metric $\\tilde{g}_{MN}$ (see e.g.", "[30]).", "This yields $& R_{\\mu \\nu \\lambda } \\vphantom{}^\\rho = - 2\\, \\frac{\\mathrm {e}^{2A}}{\\mathcal {V}\\tau ^2}\\, \\tilde{g}_{\\lambda [\\mu } \\delta _{\\nu ]}^\\rho (\\partial A)^2 + \\tilde{R}_{\\mu \\nu \\lambda } \\vphantom{}^\\rho , \\qquad R_{i j k} \\vphantom{}^l = \\tilde{R}_{i j k} \\vphantom{}^l, & \\\\& R_{\\mu j \\lambda } \\vphantom{}^l = - \\frac{\\mathrm {e}^{2A}}{\\mathcal {V}\\tau ^2}\\, \\tilde{g}_{\\mu \\lambda } \\nabla _j \\partial ^l A - \\frac{\\mathrm {e}^{2A}}{\\mathcal {V}\\tau ^2}\\, \\tilde{g}_{\\mu \\lambda } (\\partial _j A) (\\partial ^l A).", "&$ Assuming that $H$ has only internal components, it follows from (REF ) that introducing torsion with $\\Gamma ^M_{NL} \\rightarrow \\Gamma ^M_{NL} - \\tfrac{1}{2} H_{NL}\\vphantom{}^M$ modifies the internal components $R_{i j k} \\vphantom{}^l$ of the Riemann tensor and, in case of nontrivial warping, also some of the spacetime components $R^+_{\\mu \\nu \\lambda } \\vphantom{}^\\rho = R_{\\mu \\nu \\lambda } \\vphantom{}^\\rho , \\qquad R^+_{\\mu j \\lambda } \\vphantom{}^l = R_{\\mu j \\lambda } \\vphantom{}^l - \\tfrac{1}{2} \\Gamma ^m_{\\mu \\lambda } H_{jm}\\vphantom{}^l,$ where $\\Gamma ^m_{\\mu \\lambda } = - \\frac{\\mathrm {e}^{2A}}{\\mathcal {V}\\tau ^2}\\, \\tilde{g}_{\\mu \\lambda } \\nabla ^m A$ .", "We thus find $R & = \\mathcal {V}\\tau ^2 \\mathrm {e}^{-2A} \\tilde{R}_4 + \\tilde{R}_6 - 20 (\\partial A)^2 - 8 \\nabla ^2 A, \\\\\\mathrm {tr} |R^+|^2 & = \\mathcal {V}^2 \\tau ^4 \\mathrm {e}^{-4A} |\\tilde{R}_{\\mu \\nu \\lambda }\\vphantom{}^\\rho |^2 - 2 \\mathcal {V}\\tau ^2 \\mathrm {e}^{-2A} \\tilde{R}_4 (\\partial A)^2 + |R^+_6|^2, $ where $\\mathrm {tr} |R^+|^2 = \\frac{1}{2} R^+_{MNPQ} R^{+MNPQ}$ and $|\\tilde{R}_{\\mu \\nu \\lambda }\\vphantom{}^\\rho |^2 = \\frac{1}{2} \\tilde{R}_{\\mu \\nu \\lambda \\rho }\\tilde{R}^{\\mu \\nu \\lambda \\rho }$ .", "For convenience, we also introduced the shortcut notation $|R^+_6|^2 = 12 \\left[{(\\partial A)^2}\\right]^2 + 4 |R^+_{\\mu j\\lambda }\\vphantom{}^l|^2 + |R^+_{ikl}\\vphantom{}^m|^2$ to subsume all terms which in the tilded frame only depend on internal fields." ], [ "Leading order constraints on heterotic supergravity", "In [19], it was suggested that heterotic supergravity with leading $\\alpha ^\\prime $ -corrections could have solutions with a cosmological constant of the form $\\Lambda = - \\alpha ^\\prime C + \\mathcal {O}(\\alpha ^{\\prime 2}), $ where $C$ is a non-negative constant given by $C = \\frac{1}{2\\mathcal {V^\\prime }} \\int \\mathrm {d}^6 y \\sqrt{\\tilde{g}_6}\\, \\mathrm {e}^{6A-\\frac{\\phi }{2}} & \\left\\lbrace {\\vphantom{\\frac{1}{2}} 3 \\left[{(\\partial \\omega )^2}\\right]^2 + 2 |(\\partial _m \\omega )(\\partial _n \\omega ) - \\tilde{\\nabla }_m \\partial _n \\omega - \\tilde{g}_{mn} (\\partial \\omega )^2|^2}\\right.", "\\\\ & \\left.", "{+ \\frac{1}{2} \\mathrm {e}^{-4\\omega } |H_{mn}\\vphantom{}^l \\partial _l \\omega |^2}\\right\\rbrace $ with $\\mathcal {V^\\prime } = \\int \\mathrm {d}^6 y \\sqrt{\\tilde{g}_6}\\, \\mathrm {e}^{8A}$ and $\\omega = A + \\frac{\\phi }{4}$ .", "We will now show explicitly, using arguments similar to [31], [32], [33], that all terms in $C$ vanish up to higher order $\\alpha ^\\prime $ -corrections due to the leading order equations of motion.", "The result of [19] is therefore not in conflict with the argument given in Section REF .", "To omit confusion, we will stick to the metric conventions of [19] in this appendix, which differ from those used in the main text of our paper.", "The unwarped metric is then defined as $\\tilde{g}_{MN} = \\mathrm {e}^{-2A} g_{MN}$ , where $g_{MN}$ is the usual ten-dimensional Einstein frame metric.", "In the following, terms are always contracted with the Einstein frame metric, except for tilded objects and all terms in (REF ), which are contracted with the unwarped metric $\\tilde{g}_{MN}$ .", "The leading order dilaton equation in Einstein frame reads $\\nabla _M \\partial ^M \\phi + \\frac{1}{2} \\mathrm {e}^{-\\phi } |H|^2 = \\mathcal {O}(\\alpha ^\\prime ).$ Assuming that the dilaton only depends on the internal coordinates, we can write $\\nabla _M \\partial ^M \\phi = \\mathrm {e}^{-10A} \\tilde{\\nabla }_m \\mathrm {e}^{8A} \\tilde{g}^{mn} \\partial _n \\phi $ and integrate over internal space to find ${2}\\int \\mathrm {d}^6 y \\sqrt{\\tilde{g}_6}\\, \\mathrm {e}^{10A-\\phi } |H|^2 = \\mathcal {O}(\\alpha ^\\prime )$ and hence $\\mathrm {e}^{10A-\\phi } |H|^2 = \\mathcal {O}(\\alpha ^\\prime ).$ The traced internal and spacetime components of the leading order Einstein equation then read $-R_4 -2 R_6 + (\\partial \\phi )^2 = \\mathcal {O}(\\alpha ^\\prime ), \\qquad -3 R_4 - 2 R_6 + (\\partial \\phi )^2 = \\mathcal {O}(\\alpha ^\\prime ).$ Combining the two equations and rewriting $R_4$ in terms of the unwarped metric yields $R_4 = \\mathrm {e}^{-2A} \\tilde{R}_4 - \\frac{1}{2} \\mathrm {e}^{-10A} \\tilde{\\nabla }^2 \\mathrm {e}^{8A} = \\mathcal {O}(\\alpha ^\\prime ).", "$ We can now integrate over internal space to find $\\mathrm {e}^{8A} \\tilde{R}_4 = \\mathcal {O}(\\alpha ^\\prime )$ which with (REF ) implies that $\\tilde{\\nabla }^2 \\mathrm {e}^{8A} = \\mathcal {O}(\\alpha ^\\prime )$ .", "Hence the warp factor is a constant up to $\\alpha ^\\prime $ -corrections.", "The dilaton equation (REF ) then reduces to $\\mathrm {e}^{-2A} \\tilde{\\nabla }^2 \\phi = \\mathcal {O}(\\alpha ^\\prime )$ and therefore also $\\phi $ is a constant up to $\\alpha ^\\prime $ -corrections.", "We have thus shown that two-derivative terms involving the warp factor or the dilaton are at least of order $\\mathcal {O}(\\alpha ^\\prime )$ , which implies that the four-derivative terms appearing in (REF ) are of even higher order.", "It follows that $C=\\mathcal {O}(\\alpha ^\\prime )$ , and hence (REF ) yields $\\Lambda = \\mathcal {O}(\\alpha ^{\\prime 2}).$" ], [ "Ten-dimensional argument", "The result $\\Lambda = 0$ can also be derived directly from the ten-dimensional equations of motion.", "We write the ten-dimensional action (REF ) in the form $S = \\int \\mathrm {d}^{10}x \\sqrt{-g}\\, \\mathrm {e}^{-2\\phi } L,$ where $L$ includes all string theory $\\alpha ^{\\prime }$ -corrections to the ten-dimensional supergravity.", "We start by pulling out an overall warp factor $g_{MN} = e^{2\\omega } \\tilde{g}_{MN}$ .", "We will later on relate $\\omega $ to the warp factor $A$ used in the main text.", "Writing the action in terms of the tilded metric $\\tilde{g}$ , we get $S = \\int \\mathrm {d}^{10}x \\sqrt{-\\tilde{g}}\\, \\mathrm {e}^{8\\omega }\\mathrm {e}^{-2\\phi }\\tilde{L},$ notice the warp factor dependence in the action.", "The leading order terms of $\\tilde{L}$ are $\\tilde{L} = \\tilde{R} -18(\\tilde{\\nabla }^2\\omega + 4(\\partial \\omega )^2) + 4(\\partial \\phi )^2 - {2}e^{-4\\omega }|H|^2 + \\mathcal {O}(\\alpha ^{\\prime }).$ Assuming that $\\tilde{L}$ only depends on the derivatives of $\\phi $ , the dilaton equation is easily derived up to a total derivative, $0 = {\\sqrt{-\\tilde{g}}}{\\delta \\phi } = -2\\mathrm {e}^{8\\omega }\\mathrm {e}^{-2\\phi }\\tilde{L} + \\text{total derivative}.$ Using this, we can simplify the Einstein equation $0 = {\\sqrt{-\\tilde{g}}}{\\delta \\tilde{g}^{MN}} = \\mathrm {e}^{8\\omega }\\mathrm {e}^{-2\\phi }E_{MN} - {2}\\tilde{g}_{MN} \\mathrm {e}^{8\\omega }\\mathrm {e}^{-2\\phi } \\tilde{L} = \\mathrm {e}^{8\\omega }\\mathrm {e}^{-2\\phi } E_{MN} + \\tilde{g}_{MN}(\\text{total derivative}).$ The tensor $E_{MN}$ is simply the variation of the Lagrangian $\\tilde{L}$ .", "Now take the ten-dimensional manifold to be a direct product of a six-dimensional compact space and maximally symmetric spacetime.", "We also let $\\omega =\\phi /4 + A$ to switch to the unwarped Einstein frame.", "Since spacetime is assumed maximally symmetric, all external covariant derivatives vanish and the total derivative in (REF ) is a total derivative in internal space.", "We therefore look at the integrated traced Einstein equation, where the total derivatives drop out.", "To complete our analysis it is then enough for us to show that, when both indices lie in spacetime, $E_{\\mu \\nu }$ is a sum of terms that contain a positive power of the external curvature tensor.", "The only covariant quantities with external indices are the metric, the curvature tensor and the epsilon tensor.", "Keeping this in mind, there are only three possibilities that give a non-vanishing contribution to $E_{\\mu \\nu }$ : Terms where one or both of the free indices are that of a curvature tensor.", "These obviously carry a positive power of the Riemann tensor and we are done.", "Second are terms where the free indices are that of a metric, $E_{\\mu \\nu }\\sim g_{\\mu \\nu }B$ , coming from the ten-dimensional term $g_{MN}B$ , where $B$ is a ten-dimensional scalar.", "This could be problematic, since $B$ does not have to involve the Riemann tensor.", "Clearly, these terms cannot occur as a result of varying the determinant, we already got rid of those using the dilaton equation.", "However we could have such terms from varying curvature tensors or covariant derivatives or more generally the connection.", "But varying the connection always gives a total derivative because of the equation $\\delta \\Gamma _{MN}^R = {2}g^{RS}\\left(\\nabla _M \\delta g_{NS} + \\nabla _N \\delta g_{MS} - \\nabla _S \\delta g_{MN}\\right),$ and we see that $B$ must be a total derivative.", "Again this reduces to a total derivative in internal space and upon integration drops out.", "The final possibility are terms where both external indices come from epsilon symbols.", "Clearly, an epsilon symbol must have four spacetime indices, and these must contract with something, the only possibility is a curvature tensor.", "Other terms of the tensor $E_{MN}$ will be those, where the free indices are that of derivatives or fluxes etc.", "These all vanish in the maximally symmetric external spacetime.", "We have thus shown that all terms in the Einstein equation, traced with the external metric and integrated over internal space, contain a positive power of the Riemann tensor.", "This eventually leads to (REF ), and our result follows." ] ]

[ [ "Local and global structure of domain wall space-times" ], [ "Abstract We present a general proof on the equivalence of the comoving-coordinate approach, where the wall is fixed at a constant coordinate variable, and moving-wall approach, where the wall is moving in a background static space-time, in the domain wall space-times without reflection symmetry.", "We further provide a general procedure to construct the comoving coordinates in the domain wall space-times, where the two regions separated by an infinite thin wall have different cosmological constant $\\Lambda$ and Schwartzschild mass $M$.", "By solving Israel's junction conditions in the thin-wall limit, the gravitational fields of spherical, planar and hyperbolic domain wall space-times with M=0 in the two different comoving coordinate systems are obtained.", "We finally discuss the global structure of these domain wall space-times." ], [ "Introduction", "It is generally believed that phase transitions are occurred in the early Universe, so various types of topological defects can naturally form by Kibble mechanism [1] (see [2] for a review).", "Domain walls, a particular type of the topological defects, correspond to vacuumlike hypersurfaces interpolating between separate vacua.", "Beside the Kibble mechanism, domain walls can also form as the boundary of a true vacuum bubble created by quantum tunneling process of false vacuum decay, i.e.", "bubble nucleation [3], [4], and the dynamics of bubbles has been studied in the framework of general relativity [5], [6].", "In the study of gravitational effects of thin walls or dynamics of vacuum bubbles, it is useful to apply the thin-wall approximation, which is also considered in this paper.", "In the thin-wall approximation, the wall is regarded as an infinitely thin with $\\delta $ -function singularity in the energy-momentum tensor.", "The two regions $V^+$ and $V^-$ separated by the wall may have different physical parameters, e.g.", "cosmological constant {$\\Lambda _{+}, \\Lambda _{-}$ } and Schwartzschild mass {$M_{+}, M_{-}$ }.", "In this paper, we denote $V^{+}(V^{-})$ for exterior (interior) region to the wall and any quantity $Q$ with a subscript $+ (-)$ corresponds to the quantity at $V^{+}(V^{-})$ .", "Therefore gravitational effects of domain walls are described by Einstein's field equations off the wall together with Israel's junction conditions [7], [8].", "As far as we know, domain wall solutions have been studied based on two different approaches.", "The first approach, which we call moving-wall (MW) approach, starts from exact solutions of Einstein's field equations off the wall in the specific coordinates, e.g.", "the Schwartzschild coordinates, and then the wall's motion in the specific background metric is obtained by satisfying Israel junction conditions.", "The second approach, i.e.", "comoving-coordiantes (CC) approach, is to introduce the co-moving coordinates, where the wall is placed at a particular constant coordinate variable, say $z=z_0$ , and domain wall solutions are obtained by solving Einstein's field equations off the wall and Israel's Junction conditions.", "Since domain wall solutions obtained from these two approaches both satisfying Einstein's field equations and Israel's junction condition, one may expect that these two different approaches are equivalent up to a coordinate transformation.", "In the study of brane cosmologies, Mukohyama et al [9] has shown the coordinate transformation between two exact solutions of brane world, which are obtained by CC approach in Gaussian normal coordinates [10], [11] and MW approach [12], respectively.", "Moreover, Bowcock et al [13] demonstrated the equivalence of these two approaches in brane-cosmological models with $Z_2$ symmetryIn [13], these two different approaches are called the brane-based approach and the bulk-based approach., and then studied time evolution of 4-dimensional brane-universe in the MW approach without $Z_2$ symmetry.", "However, it is still not clear to us how to construct a comoving coordinates in domain wall space-times without $Z_2$ symmetry by given a trajectory of wall's motion in the MW approach.", "In this paper, we present a general proof on the equivalence of these two approaches in the 4-dimensional spherical, planar, and hyperbolical domain wall space-times without $Z_2$ symmetry, and then show how to construct comoving coordinates in domain wall space-times having different $\\Lambda $ and $M$ on each side of the wall.", "Although Einstein's field equations and also field equations of any classical field are general covariant according to general principle of relativity, finding a proper coordinate system to describe the dynamics of classical quantities and gravitational fields are still important, especially when one wants to compare the results to observations.", "For example, in the post-Newtonian approximation [15], [14], [16], equations of motion of many bodies are described in a particular coordinate system, where one can obtain Newtonian theory of gravity in the first-oder approximation of Einstein's field equations.", "Since the dynamics of our solar system are well described in Newtonian gravity, it indicates that this coordinate system is a proper choice to study our solar system.", "So higher-order effects of general relativity in this coordinate system can be calculated and compared to solar system observations.", "Furthermore, in the study of quantum fields in curved space-time, the coordinate choices become significant since there is no coordinate-invariant definition of the vacuum state, i.e.", "the vacuum state is coordinate-dependent [17].", "Besides showing the construction of comoving coordinates in domain wall space-times, our another motivation is to construct a proper comoving coordinate system in spherical, planar, and hyperbolic domain wall space-times, which may be useful to understand gravitational effects of domain walls on primordial quantum fluctuations during inflation.", "From the generalized Birkhoff theorem [18], [19], [20], the exact solutions of Einstein's field equation with cosmological constant $\\Lambda $ in spherically, planar, and hyperbolically symmetric space-times can be written as $g= -U(q) \\,\\textrm {d}T \\otimes \\textrm {d}T + \\frac{1}{U(q)} \\textrm {d}q \\otimes \\textrm {d}q + q^2 \\textrm {d}V_2, $ where $U(q)=k- 2M/q - (\\Lambda /3) q^2$ , $\\textrm {d}V_2 = (1-k x^2)^{-1} \\textrm {d}x \\otimes \\textrm {d}x +x^2 \\textrm {d}\\phi \\otimes \\textrm {d}\\phi $ , and $k$ denotes constant Gaussian curvature.", "It is clear that the metric (REF ) is static in the certain range of coordinates.", "In the MW approach, the metric $g_{+}\\,(g_{-})$ in $V^{+}\\, (V^{-})$ is described by $U_{+(-)} $ , where $U_{+(-)} = k- 2M_{+(-)} /q - (\\Lambda _{+(-)}/3) q^2$ .To concise our notation, we use subscript $\\pm $ on any quantity $Q$ to denote $Q_{+}(Q_{-})$ in this paper.", "Thus the Israel's junction conditions yield equations of motion of domain walls, which have been studied in [5], [6], and the wall's trajectories are described in terms of proper-time $\\tau $ .", "To verify the equivalence of MW and CC approaches, we consider the wall being placed at $r=r_0$ , and by using Einstein's field equations, metric continuity (with requiring coordinate time $\\eta $ on the wall being $\\tau $ ), and Israel's junction conditions, we derive that metric at $r=r_0$ , which is denoted by $\\hat{g}$ , satisfies the same equations as equations of motion of the wall obtained in the MW approach [6], [5].", "We further find that $\\hat{g}$ will uniquely determine the metric in $V^{+}$ and $V^{-}$ .", "It means that once the $\\hat{g}$ is known, we can then obtain $g_+$ and $g_-$ .", "Since there exists a degree of freedom on $\\hat{g}$ due to the choice of the time coordinate on the wall, we calculate $g_{+}$ and $g_{-}$ in the two different comoving coordinate systems by requiring $\\hat{g}_{00}=-1$ (case I) and $\\hat{g}_{00}=-\\alpha ^2/\\eta ^2$ (case II), where $\\hat{g}_{00}$ denotes the metric component $g_{00}$ on the wall and $\\alpha $ is a constant.", "So case I indicates that the coordinate time on the wall is the proper-time.", "Interestingly, we find that the metric solutions $g_{\\pm }$ in case I with $M{\\pm }=0$ are the same as the domain wall solutions obtained by Cvetič et al [21].", "In [21], they obtained domain wall solutions in the comoving coordiantes by using a metric ansatz: $g= A(z) (- \\textrm {d}t \\otimes \\textrm {d}t + \\textrm {d}z \\otimes \\textrm {d}z + S^2 (t)\\,\\textrm {d}V_2).$ However, we start from a general metric form in spherical, planar, and hyperbolic symmetric space-time and it turns out that the domain wall solutions in case I with $M{\\pm }=0$ agree with [21].", "In our previous work [22], we obtained a planar domain wall solution with reflection symmetry in de Sitter space-time, and $\\sigma _0=0$ yields the well known metric of steady-state Universe in the conformal time coordinate.", "We further study its gravitational effects on primordial quantum fluctuations during inflation, and found that its gravitational fields produce a primordial dipole effects in the power spectrum of primordial curvature perturbation [23].", "It is observed that this planar domain wall solution does satisfy $\\hat{g}_{00}=-\\alpha ^2/\\eta ^2$ and the coordinate time $\\eta $ on the wall corresponds to conformal time in de Sitter spacetime, we suggest that the choice of $\\hat{g}_{00}=-\\alpha ^2/\\eta ^2$ may provide a proper coordinates to investigate the gravitational effects of domain walls in the early Universe.", "Since our previous domain wall solution requires plane and reflection symmetry, it is quite limited to study of gravitational fields of realistic domain wall space-times.", "For example, false vacuum decay yields two space-time regions with different $\\Lambda $ separated by a spherical bubble.", "In the study of case II, we generalize our previous planar domain wall solution to spherical, planar, and hyperbolic domain wall space-times without reflection symmetry.", "The global structure of spherical, planar and hyperbolic domain wall space-times has been well studied in [21], [5] (see a review article [24] and the references in).", "Moreover, Ref.", "[24] has pointed out that the constant-$r$ sections of non- and ultraextreme domain wall space-times, which correspond to domain wall solutions in case I with $H^2\\ne 0$ , all represent (2+1)-dimensional de Sitter space-time ($\\textrm {d}\\textrm {S}_3$ ), whose topology is $\\textbf {R}$ (time) $\\times $ $\\textbf {S}^2$ (space).", "It turns out that non- and ultraextreme planar domain wall, which is locally plane-symmetric and geodesically incomplete, describes only a part of a spherical bubble [24].", "In Sec.", ", we present a coordinate transformation between the domain wall solutions in case I and solutions in case II, so topology of spherical, planar, and hyperbolic domain walls in case II is also $\\textbf {R}$ (time) $\\times $ $\\textbf {S}^2$ (space).", "Hence, one should expect that the planar domain wall solution in case II also represents a portion of a spherical domain wall space-time.", "The plan of this paper is as follows.", "In Sec.", ", we show the equivalence of CC and MW approaches and also the construction of comoving coordinates in domain wall space-time with different $M_{\\pm }$ and $\\Lambda _{\\pm }$ .", "Sec.", "discuss spherical, planar, and hyperbolic domain wall solutions in two different comoving coordinates, i.e.", "case I and case II, with $M_{\\pm }=0$ .", "Moreover, since the choice of $\\hat{g}_{00}=-\\alpha ^2/\\eta ^2$ is motivated to study gravitational effects of domain walls on quantum fluctuations during inflation, we will only discuss $\\Lambda _{\\pm }>0$ in case II.", "In Sec.", ", the global structure of domain wall space-times are discussed.", "Sec.", "gives a discussion and conclusion.", "In Appendix , we present some technical materials, which is useful for following our calculations in Sec.", "and Sec.", ".", "We use the units $\\hbar =c=1$ , and the metric signature is $(- + + +)$ .", "The Latin indices $a, b, \\cdots $ are referred to coordinate indices and the Greek indices $\\alpha ,\\beta ,\\gamma \\cdots $ referred to orthonormal frame indices.", "$g$ and $\\nabla $ denote metric tensor and Levi-Civita connection, respectively." ], [ "On the equivalence of comoving-coordinate and moving-wall approaches", "In the thin-wall approximation, the thickness $\\varepsilon $ of a thin wall is taken to be zero, so the infinitely thin wall becomes a 3-dimensional timelike, null or spacelike hypersurface $\\Sigma $ in 4-dimensional space-times, and its associated stress-energy tensor $T^a{_b}$ of the space-times has a $\\delta $ -function singularity on $\\Sigma $ .", "Here, we will assume $\\Sigma $ to be a 3-dimensional timelike hypersurface for our current interest.", "To describe the gravitational fields of domain walls, the metric off the walls satisfies vacuum Einstein's field equations with $\\Lambda $ $G_\\gamma = 2 \\Lambda * e_\\gamma , $ where $G_\\gamma = R_{\\alpha \\beta }\\wedge * (e^\\alpha \\wedge e^\\beta \\wedge e_\\gamma )$ are Einstein's 3-forms and $R_{\\alpha \\beta }$ are curvature 2-forms defined in terms of Levi-Civita connection $\\nabla $ [25].", "$e^\\alpha $ are orthonormal co-frames and $*$ denotes the Hodge map associated with $g$ .", "Moreover, by introducing the intrinsic metric $\\hat{h}$ of $\\Sigma $ In the following, we will put $\\hat{ }\\,$ on any quantity to restrict it on $\\Sigma $ .", "$\\hat{h}= \\hat{g} - \\tilde{n} \\otimes \\tilde{n},$ where $\\tilde{n}= g(n, -)$ is the metric dual of unit normal $n$ of $\\Sigma $ , and also the extrinsic curvature $\\pi _{ab}$ of $\\Sigma $ defined by $\\hat{\\pi }_{ab} = \\frac{1}{2}(\\mathcal {L}_n \\overline{h} )_{ab}|_\\Sigma , $ where $\\mathcal {L}_n$ denotes the Lie derivative along $n$ , and $\\overline{h}$ is any extension of $h$ to a neighborhood of $\\Sigma $ , the metric on the $\\Sigma $ should satisfy metric continuities, i.e.", "$g_{+}|_{\\Sigma }=g_{-}|_{\\Sigma }= \\hat{g}$ , and Israel's junction conditions $\\hat{\\pi }_{ab+} - \\hat{\\pi }_{ab-} = -\\frac{\\kappa \\sigma }{2} \\,\\hat{h}_{ab}, $ where $\\kappa = 8 \\pi G$ and $\\sigma =\\textrm {constant}$ is the surface tension of domain walls.", "In the spherical, planar, and hyperbolic symmetric space-time, the most general metric form can be written in double null-coordinates $(u, v)$ as $g= e^{2\\mu (u, v)} (- \\textrm {d}u \\otimes \\textrm {d}v) + B^2 (u, v) \\textrm {d}V_2, $ where $\\textrm {d}V_2 = (1-k x^2)^{-1} \\textrm {d}x \\otimes \\textrm {d}x +x^2 \\textrm {d}\\phi \\otimes \\textrm {d}\\phi $ , and constant Gaussian curvature $k= 1, 0, -1$ corresponds to 2-dimensional space-like spheres, planes, and hyperboloids, respectively.", "In [22], a general non-degenerate solution of Eq.", "(REF ) is obtained $g= 4 F(v) G(u) L(B) \\,\\textrm {d}u \\otimes \\textrm {d}v + B^2 \\textrm {d}V_2, $ with $B(u, v)$ satisfies $\\textrm {d}B= - L(B) \\,\\,( F(v) \\,\\textrm {d}v + G(u) \\,\\textrm {d}u), $ where $L(B)\\equiv k - \\frac{2 M}{B} - \\frac{\\Lambda }{3} B^2$ .", "It is clear that two arbitrary functions $F(v)$ and $G(u)$ are due to the freedom of choosing double-null coordinates.", "Eq.", "(REF ) can be integrated to get $\\mathcal {B}(B) = \\mathcal {F}(v) + \\mathcal {G}(u),$ where $\\mathcal {B}(B)\\equiv - \\int L^{-1}\\textrm {d}B$ , $\\mathcal {F}(v) \\equiv \\int F\\, \\textrm {d}v$ , and $\\mathcal {G}(u)\\equiv \\int G\\, \\textrm {d}u$ .", "$\\mathcal {B}(B)$ in some particular choices of parameters $k$ , $M$ , $\\Lambda $ are presented in Appendix .", "If the inverse function $\\mathcal {B}^{-1}$ of $\\mathcal {B}(B)$ exists, which may only true in certain range of $B$ , we then obtain $B(u, v)=\\mathcal {B}^{-1}(\\mathcal {F} + \\mathcal {G})$ .", "To obtain the domain wall solutions in comoving coordinates, it is convenient to introduce coordinate transformations $u=r+ \\eta $ and $v=\\frac{1}{r-\\eta }$ , and the wall is placed at $r=r_0$ .", "So metric $g_{+}$ and $g_{-}$ , which correspond to $r>r_0$ and $r<r_0$ regions, give $g_{\\pm }= A_{\\pm }( \\textrm {d}\\eta \\otimes \\textrm {d}\\eta - \\textrm {d}r\\otimes \\textrm {d}r) + B_{\\pm }^2 \\textrm {d}V_2,$ with $\\textrm {d}B_{\\pm } = - L_{\\pm }[(\\frac{F_{\\pm }}{(r-\\eta )^2} + G_{\\pm }) \\textrm {d}\\eta + (\\frac{- F_{\\pm }}{(r-\\eta )^2} + G_{\\pm }) \\textrm {d}r].", "\\nonumber $ where $A_{\\pm }\\equiv \\frac{4 F_{\\pm }\\, G_{\\pm } \\,L_{\\pm }}{(r-\\eta )^2}$ and $L_{\\pm } \\equiv (k-\\frac{2 M_\\pm }{B_{\\pm }}-\\frac{\\Lambda _{\\pm }}{3}B_{\\pm }^2)$ .", "The Israel's junction conditions give us two equations $&& \\zeta _1\\,\\frac{1}{\\sqrt{-\\hat{A}_+}}\\widehat{\\partial _r A_{+}}\\, - \\zeta _2\\,\\frac{1}{\\sqrt{-\\hat{A}_-}}\\widehat{\\partial _r A_{-}} =- \\kappa \\sigma \\hat{A},\\\\&& \\zeta _1 \\,\\frac{\\hat{B}_+\\,\\widehat{\\partial _r B_{+}}}{\\sqrt{-\\hat{A}_+}} \\, - \\zeta _2\\,\\frac{\\hat{B}_-\\,\\widehat{\\partial _r B_{-}}}{\\sqrt{-\\hat{A}_-}} = -\\frac{1}{2}\\kappa \\sigma \\hat{B}^2 ,$ where $\\hat{A}$ and $\\hat{B}$ denotes the metric components on $\\Sigma $ .", "Here, $\\lbrace \\zeta _1, \\zeta _2\\rbrace = \\pm 1$ due to the sign ambiguity of unit normal $n$ [21].", "In the following, we will only consider $\\zeta _1= \\zeta _2=1$ .", "Beside Israel's junction conditions, the metric continuities also give two equations $\\dot{\\hat{\\mathcal {F}}}_+\\,\\dot{\\hat{\\mathcal {G}}}_+\\, \\hat{L}_+=\\dot{\\hat{\\mathcal {F}}}_-\\,\\dot{\\hat{\\mathcal {G}}}_-\\hat{L}_-= \\hat{A}/4, \\\\\\mathcal {B}_+^{-1}(\\hat{\\mathcal {F}}_+ + \\hat{\\mathcal {G}}_+) = \\mathcal {B}_-^{-1}(\\hat{\\mathcal {F}}_- + \\hat{\\mathcal {G}}_-) = \\hat{B}, $ where $\\hat{\\dot{\\mathcal {F}}}_\\pm \\equiv \\widehat{\\partial _\\eta \\mathcal {F}_\\pm } = \\frac{\\hat{F}_\\pm }{(r_0-\\eta )^2}= \\dot{\\hat{\\mathcal {F}}}_\\pm , \\\\\\hat{\\dot{\\mathcal {G}}}_\\pm \\equiv \\widehat{\\partial _\\eta \\mathcal {G}_\\pm } = \\hat{G}_\\pm = \\dot{\\hat{\\mathcal {G}}}_\\pm .$ It can be showed that Eq.", "(REF ) is implied by Eq.", "().", "By differentiating Eq.", "() with respect to $\\eta $ and using Eqs ()-(), one can obtain Eq.", "(REF ).", "So we now have three independent equations ()-() for four unknown functions, $\\hat{\\mathcal {F}}_\\pm $ and $\\hat{\\mathcal {G}}_\\pm $ .", "Before we discuss these equations, it is useful to define $&&R_{\\pm }(u, v)= \\mathcal {F}_\\pm (v) + \\mathcal {G}_\\pm (u), \\\\&&T_{\\pm }(u, v) = \\mathcal {F}_\\pm (v) - \\mathcal {G}_\\pm (u), $ so Eqs.", "()-() become $\\hat{L}_+ \\dot{\\hat{T}}_+ - \\hat{L}_- \\dot{\\hat{T}}_- =-\\frac{\\kappa \\sigma }{2} \\hat{B} \\sqrt{-\\hat{A}}\\\\(\\dot{\\hat{R}}_+^{\\,2} - \\dot{\\hat{T}}_+^{\\,2})\\, \\hat{L}_+ = (\\dot{\\hat{R}}_-^{\\,2} - \\dot{\\hat{T}}_-^{\\,2})\\, \\hat{L}_-= \\hat{A},\\\\\\mathcal {B}_+^{-1}(\\hat{R}_+) = \\mathcal {B}_-^{-1}(\\hat{R}_-) = \\hat{B}.$ By using Eqs.", "()-(), one can express $\\dot{\\hat{T}}_\\pm $ in terms of $\\dot{\\hat{B}}$ and $\\hat{A}$ as $\\dot{\\hat{T}}_{\\pm } = h_{\\pm } \\sqrt{(-\\hat{A} \\hat{L}_\\pm + \\dot{\\hat{B}}^{\\,2})\\hat{L}_\\pm ^{-2}}, $ where $\\lbrace h_+, h_-\\rbrace =\\pm 1$ denotes the sign ambiguity coming from the quadratic in $\\dot{\\hat{T}}_\\pm ^{\\,2}$ .", "It is clear that Eq.", "(REF ) will be used to determine $\\hat{B}$ , so $\\hat{A}$ becomes a free function.", "The free choice of $\\hat{A}$ comes from the freedom of choosing time coordinate on $\\Sigma $ .", "The different choices of $\\hat{A}$ correspond to different time parametrization on $\\Sigma $ .", "In the following, we will consider two different time parametrization, which are $\\hat{A}=-1$ (case I) and $\\hat{A}=-\\alpha ^2 / \\eta ^2$ (case II).", "We first consider case I, where the coordinate time $\\eta $ on $\\Sigma $ corresponds to proper time.", "So Eq.", "(REF ) becomes $h_- \\sqrt{(\\hat{L}_- + \\dot{\\hat{B}}^{\\,2})}- h_+ \\sqrt{(\\hat{L}_+ + \\dot{\\hat{B}}^{\\,2})} =\\frac{\\kappa \\sigma }{2} \\hat{B}, $ which is a well-known equation of motion of domain walls in the MW approach [5], [6].", "The intrinsic metric $\\hat{g}$ obtained in CC approach are the same as in MW approach, so it is clear that these two approaches are equivalent.", "Eq.", "(REF ) can also be written as $\\dot{\\hat{B}}^2 = \\frac{(\\hat{L}_+ -\\hat{L}_-)^2}{\\kappa ^2\\sigma ^2 \\hat{B}^2} - \\frac{1}{2} (\\hat{L}_+ +\\hat{L}_-) + \\frac{\\kappa ^2\\sigma ^2}{16} \\hat{B}^2, $ and by substituting the definition of $\\hat{L}_\\pm $ into Eq.", "(REF ) gives $\\dot{\\hat{B}}^2 &=& H^2 \\hat{B}^2 - k - \\frac{\\Delta M}{\\hat{B}}\\Big (\\frac{4 (\\Lambda _- -\\Lambda _+)}{ 3 \\kappa ^2\\sigma ^2} -1+ \\frac{2 M_-}{\\Delta M}\\Big ) \\nonumber \\\\&&+ \\frac{4 (\\Delta M)^2}{\\kappa ^2\\sigma ^2 \\hat{B}^4}, $ where $\\Delta M\\equiv M_+ - M_-$ and $H^2 \\equiv \\frac{\\kappa ^2\\sigma ^2 }{16} + \\frac{(\\Lambda _- -\\Lambda _+)^2}{9 \\kappa ^2\\sigma ^2} + \\frac{(\\Lambda _- +\\Lambda _+) }{6}.", "$ Eq.", "(REF ) has been largely studied in the dynamics of bubbles and various exact solutions in some particular choices of parameters $M_\\pm $ and $\\Lambda _\\pm $ have been obtained [5], [6] (see also [26] and references in).", "In our previous work [22], a planar domain wall solution in de-Sitter space-time with refection symmetry is obtained, which satisfying the choice of $\\hat{A}=-\\alpha ^2 / \\eta ^2$ , and $\\sigma _0$ vanishing gives the well-known metric of steady-state Universe in conformal time.", "We further use this domain solution to study gravitational effects of planar domain walls on primordial quantum fluctuations [23].", "In order to generalize our previous planar domain wall solutions, we study spherical, planar, and hyperbolic domain wall space-times without reflection symmetry in case II.", "So Eq.", "(REF ) becomes $h_- \\sqrt{(\\frac{\\alpha ^2 \\hat{L}_-}{\\eta ^2} + \\dot{\\hat{B}}^{\\,2})}- h_+ \\sqrt{(\\frac{\\alpha ^2 \\hat{L}_+}{\\eta ^2}+ \\dot{\\hat{B}}^{\\,2})} =\\frac{\\kappa \\sigma }{2} \\frac{\\alpha \\hat{B}}{\\eta } $ and substituting the definition of $\\hat{L}_\\pm $ into Eq.", "(REF ) yields $\\dot{\\hat{B}}^2 &=& \\frac{\\alpha ^2}{\\eta ^2} \\Big [H^2 \\hat{B}^2 - k - \\frac{\\Delta M}{\\hat{B}}\\Big (\\frac{4 (\\Lambda _- -\\Lambda _+)}{ 3 \\kappa ^2\\sigma ^2} -1+ \\frac{2 M_-}{\\Delta M}\\Big ) \\nonumber \\\\&&+ \\frac{4 (\\Delta M)^2}{\\kappa ^2\\sigma ^2 \\hat{B}^4}\\Big ].", "$ Since solving Eqs.", "(REF ) or (REF ) only gives the metric $\\hat{g}$ on $\\Sigma $ , we should now discuss how to obtain the 4-dimensional metric $g_{\\pm }$ in the comoving coordinates for given an exact solution of $\\hat{B}$ .", "Suppose the two functions $\\hat{A}(\\eta )$ and $\\hat{B}(\\eta )$ are known, one can use Eqs.", "() and (REF ) to get $\\hat{R}_{\\pm }$ and $\\hat{T}_{\\pm }$ .", "From Eqs.", "(REF ) and (), we then obtain $\\hat{\\mathcal {F}}_\\pm $ and $\\hat{\\mathcal {G}}_\\pm $ .", "By noting that $\\mathcal {F}_{\\pm }(v)$ are functions of $\\frac{1}{r-\\eta }$ and $\\mathcal {G}_{\\pm }(u)$ are functions of $r + \\eta $ , we learn that $\\hat{\\mathcal {F}}_\\pm (\\eta , r_0)$ and $\\hat{\\mathcal {G}}_\\pm (\\eta , r_0)$ are sufficient to give $\\mathcal {F}_{\\pm }(v)$ and $\\mathcal {G}_{\\pm }(u)$ .", "It is easily to see that $A_\\pm (r, \\eta )$ and $B_\\pm (r, \\eta )$ can be derived from $\\mathcal {F}_{\\pm }(v)$ and $\\mathcal {G}_{\\pm }(u)$ , so we obtain domain wall solutions in the comoving coordinates.", "In Sec.", ", we derive $A_\\pm (r, \\eta )$ and $B_\\pm (r, \\eta )$ in the special cases of $M_\\pm =0$ ." ], [ "Domain wall solutions in comoving coordinates", "In this section we study domain wall solutions with $M_\\pm =0$ in the comoving coordinates by choosing $\\hat{g}_{00}=-1$ (case I) and $\\hat{g}_{00}=-\\alpha ^2/\\eta ^2$ (case II).", "The different choices of $\\hat{g}_{00}$ correspond to different boundary conditions of $A_\\pm (r, \\eta )$ and $B_\\pm (r, \\eta )$ .", "In case I, we find that $A_\\pm (r, \\eta )$ and $B_\\pm (r, \\eta )$ are the same as the domain wall solutions obtained in [21].", "In case II, we only concentrate on $\\Lambda _{\\pm }>0$ and generalize our previous planar domain wall solutions to spherical, planar and hyperbolic domain walls in de Sitter space-time without reflection symmetry.", "When $M_\\pm =0$ , Eq.", "(REF ) becomes $\\dot{\\hat{B}}^2 = \\hat{A} (k- H^2\\hat{B}^2), $ and its exact solutions in the choice of $\\hat{A}=-1$ have been studied in [5], [6], [26]." ], [ "Case I: ${\\hat{A}=-1}$ and {{formula:b2b21f9f-752f-469b-bbc0-a7c5057b4d36}}", "Since Eq.", "(REF ) has degenerate solutions in the case of $H=0$ , we shall discuss $H\\ne 0$ and $H=0$ separately.", "In [21], Cvetič et al classified the domain wall solutions into extreme walls ($q_0=0$ ), non- and ultraextreme walls ($q_0=\\beta ^2$ ) by the parameter $q_0$ .", "Actually, it can be showed that the parameter $H^2$ corresponds to $q_0$ by rewritting Eq.", "(REF ) to $\\kappa \\sigma = \\pm \\, 2\\sqrt{H^2 -\\frac{\\Lambda _+}{3}} \\mp 2 \\sqrt{H^2 -\\frac{\\Lambda _-}{3}}, $ which is the same as Eq.", "(2.34) in [21].", "The exact solutions of Eq.", "(REF ) in $k=\\lbrace -1, 0, 1\\rbrace $ yield $\\hat{B}=\\left\\lbrace \\begin{array}{ll}\\hat{B}^- =\\frac{1}{H}\\sinh H\\eta , \\\\\\hat{B}^0 =e^{H\\eta },\\\\\\hat{B}^+ =\\frac{1}{H}\\cosh H\\eta , \\end{array} \\right.$ where the superscripts $\\lbrace -, 0, +\\rbrace $ on $B$ and also any object in the following refer to $k=\\lbrace -1, 0, 1\\rbrace $ , respectively.", "Moreover, Eq.", "() gives $&&\\hat{R}^-_\\pm =\\left\\lbrace \\begin{array}{ll}{\\lambda ^{-1}_\\pm } \\tan ^{-1} (\\lambda _\\pm \\hat{B}^-), & \\Lambda _\\pm =3 \\lambda ^2_\\pm , \\\\\\frac{1}{H}\\sinh H\\eta , & \\Lambda _\\pm =0,\\\\{\\lambda ^{-1}_\\pm } \\coth ^{-1} ({\\lambda _\\pm } \\hat{B}^-), & \\Lambda _\\pm =-3 \\lambda ^2_\\pm ,\\end{array}\\right.", "\\\\&& \\hat{R}^0_\\pm = \\left\\lbrace \\begin{array}{ll}-\\lambda ^{-2}_\\pm (\\hat{B}^0)^{-1}, & \\Lambda _\\pm =3 \\lambda ^2_\\pm , \\\\\\,\\,\\,\\,\\lambda ^{-2}_\\pm (\\hat{B}^0)^{-1}, & \\Lambda _\\pm =-3 \\lambda ^2_\\pm ,\\end{array}\\right.", "\\\\&&\\hat{R}^+_\\pm =\\left\\lbrace \\begin{array}{ll}- {\\lambda ^{-1}_\\pm } \\coth ^{-1} ({\\lambda _\\pm } \\hat{B}^+), & \\Lambda _\\pm =3 \\lambda ^2_\\pm ,\\\\- \\frac{1}{H}\\cosh H\\eta , & \\Lambda _\\pm =0,\\\\- {\\lambda ^{-1}_\\pm } \\tan ^{-1} ({\\lambda _\\pm } \\hat{B}^+), &\\Lambda _\\pm =-3 \\lambda ^2_\\pm ,\\end{array}\\right.$ where the inverse hyperbolic function $\\coth ^{-1}(\\lambda _\\pm \\hat{B})$ in Eqs.", "(REF ) and () is obtained by considering $\\hat{B}>\\lambda ^{-1}_\\pm $ .", "In the case of $\\hat{B}<\\lambda ^{-1}_\\pm $ , we should obtain $\\tanh ^{-1}(\\lambda _\\pm \\hat{B})$ .In the situation of $\\hat{B}<\\lambda ^{-1}_\\pm $ , one should use $\\tanh ^{-1}(\\lambda _\\pm \\hat{B})$ and it turns out that the domain wall solutions $A_\\pm (r, \\eta )$ and $B_\\pm (r, \\eta )$ also yield Eqs.", "(REF )-().", "So these solutions are valid for both $\\hat{B}>\\lambda ^{-1}_\\pm $ and $\\hat{B}<\\lambda ^{-1}_\\pm $ By solving Eq.", "() yields $&&\\hat{T}^-_\\pm =\\left\\lbrace \\begin{array}{ll}h_\\pm {\\lambda ^{-1}_\\pm } \\tan ^{-1} (\\lambda _\\pm \\beta ^-), & \\Lambda _\\pm =3 \\lambda ^2_\\pm , \\\\h_\\pm \\frac{1}{H}\\cosh H\\eta , & \\Lambda _\\pm =0,\\\\-h_\\pm {\\lambda ^{-1}_\\pm } \\coth ^{-1} ({\\lambda _\\pm } \\beta ^-), & \\Lambda _\\pm =-3 \\lambda ^2_\\pm ,\\end{array}\\right.", "\\\\&& \\hat{T}^0_\\pm = \\left\\lbrace \\begin{array}{ll}- h_\\pm \\lambda ^{-2}_\\pm ({\\beta }^0)^{-1}, & \\Lambda _\\pm =3 \\lambda ^2_\\pm , \\\\h_\\pm \\lambda ^{-2}_\\pm ({\\beta }^0)^{-1}, & \\Lambda _\\pm =-3 \\lambda ^2_\\pm ,\\end{array}\\right.", "\\\\&&\\hat{T}^+_\\pm =\\left\\lbrace \\begin{array}{ll}- h_\\pm {\\lambda ^{-1}_\\pm } \\coth ^{-1} ({\\lambda _\\pm } \\beta ^+), & \\Lambda _\\pm =3 \\lambda ^2_\\pm ,\\\\h_\\pm \\frac{1}{H}\\sinh H\\eta , & \\Lambda _\\pm =0,\\\\h_\\pm {\\lambda ^{-1}_\\pm } \\tan ^{-1} ({\\lambda _\\pm } \\beta ^+), &\\Lambda _\\pm =-3 \\lambda ^2_\\pm ,\\end{array}\\right.$ where $\\left\\lbrace \\begin{array}{ll}\\beta ^- =\\sqrt{(H^2- \\Lambda _\\pm /3)^{-1}}\\cosh H\\eta , \\\\\\beta ^0 =H\\,\\sqrt{(H^2-\\Lambda _\\pm /3)^{-1}}\\,e^{H\\eta },\\\\\\beta ^+ =\\sqrt{(H^2-\\Lambda _\\pm /3)^{-1}}\\sinh H\\eta , \\end{array} \\right.$ with $H^2>\\lambda _\\pm ^2$ .In the case of $H^2=\\lambda ^2_\\pm $ , $\\hat{T}_\\pm $ becomes constants and the solutions of $A_\\pm (r, \\eta )$ and $B_\\pm (r, \\eta )$ correspond to $\\gamma _\\pm =0$ in Eqs.", "(REF )-().", "Substituting Eqs.", "(REF )-(REF ) into Eqs.", "(REF )-(), and noting that $\\mathcal {F}(v)$ and $\\mathcal {G}(u)$ are functions of $1/(r-\\eta )$ and $r + \\eta $ , respectively, we can obtain $\\mathcal {F}(v)$ and $\\mathcal {G}(u)$ .", "Since the calculations are straightforward and similar for $\\Lambda _\\pm >0$ and $\\Lambda _\\pm <0$ , we will only present the results for $\\Lambda _\\pm \\geqslant 0$ with $h_\\pm =1$ , which yields $&&\\mathcal {F}^-_\\pm =\\left\\lbrace \\begin{array}{ll}\\frac{1}{2 \\lambda _\\pm } \\tan ^{-1} \\lbrace \\frac{1}{\\sinh (\\gamma _\\pm + H (\\frac{1}{v}- r_0))}\\rbrace , &\\Lambda _\\pm >0, \\\\\\frac{1}{2 H} e^{-H(\\frac{1}{v} - r_0)}, & \\Lambda _\\pm =0,\\\\\\end{array}\\right.", "\\\\&& \\mathcal {F}^0_\\pm = \\left\\lbrace \\begin{array}{ll}- \\frac{1}{2 H \\lambda _\\pm } e^{-(H r_0-\\gamma _\\pm )}\\,e^{H/v}, & \\Lambda _\\pm >0, \\\\\\end{array}\\right.", "\\\\&&\\mathcal {F}^+_\\pm =\\left\\lbrace \\begin{array}{ll}\\frac{- \\coth ^{-1} \\lbrace \\cosh ( \\gamma _\\pm + H(\\frac{1}{v}- r_0))\\rbrace }{2\\lambda _\\pm }, & \\Lambda _\\pm >0, \\\\- \\frac{1}{2 H} e^{H(\\frac{1}{v} - r_0)}, & \\Lambda _\\pm =0,\\\\\\end{array}\\right.$ and $&&\\mathcal {G}^-_\\pm =\\left\\lbrace \\begin{array}{ll}- \\frac{1}{2 \\lambda _\\pm } \\tan ^{-1} \\lbrace \\frac{1}{\\sinh (\\gamma _\\pm + H (u - r_0))}\\rbrace , &\\Lambda _\\pm >0, \\\\- \\frac{1}{2 H} e^{-H( u - r_0)}, & \\Lambda _\\pm =0,\\\\\\end{array}\\right.", "\\\\&& \\mathcal {G}^0_\\pm = \\left\\lbrace \\begin{array}{ll}- \\frac{1}{2 H \\lambda _\\pm } e^{(H r_0-{\\gamma _\\pm })}\\,e^{-Hu}, & \\Lambda _\\pm >0, \\\\\\end{array}\\right.", "\\\\&&\\mathcal {G}^+_\\pm =\\left\\lbrace \\begin{array}{ll}- \\frac{\\coth ^{-1} \\lbrace \\cosh ( \\gamma _\\pm + H(u - r_0))\\rbrace }{2 \\lambda _\\pm }, & \\Lambda _\\pm >0, \\\\-\\frac{1}{2 H} e^{H( u- r_0)}, & \\Lambda _\\pm =0,\\\\\\end{array}\\right.$ where $\\gamma _\\pm =\\ln \\lbrace \\,\\lambda _\\pm ^{-1}H + \\sqrt{-1 +H^2/\\lambda _\\pm ^2}\\rbrace =\\cosh ^{-1}(\\sqrt{H^2/\\lambda _\\pm ^2}).\\nonumber $ The formula $\\tan ^{-1}x \\pm \\tan ^{-1}y = \\tan ^{-1}(\\frac{x \\pm y}{1\\mp x y})$ and $ \\coth ^{-1}x \\pm \\coth ^{-1} y = \\coth ^{-1}(\\frac{xy\\pm 1}{y\\pm x})$ have been used to calculate $\\mathcal {F}_\\pm (v)$ and $\\mathcal {G}_\\pm (u)$ .", "Since $\\mathcal {F}_\\pm (v)$ and $\\mathcal {G}_\\pm (u)$ are obtained, one can direct calculate $B_\\pm (r, \\eta )$ and $A_\\pm (r, \\eta )$ to get $&& B^-_\\pm =\\left\\lbrace \\begin{array}{ll}\\frac{\\sinh H\\eta }{\\lambda _\\pm \\cosh \\lbrace \\gamma _\\pm +H(r-r_0)\\rbrace }, &\\Lambda _\\pm >0, \\\\\\frac{1}{H} e^{-H(r - r_0)}\\sinh H\\eta , & \\Lambda _\\pm =0,\\\\\\end{array}\\right.", "\\\\&& B^0_\\pm = \\left\\lbrace \\begin{array}{ll}\\frac{H e^{H\\eta }}{\\lambda _\\pm \\cosh \\lbrace H(r-r_0)+\\gamma _\\pm \\rbrace }, & \\Lambda _\\pm >0, \\\\\\end{array}\\right.", "\\\\&&B^+_\\pm =\\left\\lbrace \\begin{array}{ll}\\frac{\\cosh H\\eta }{\\lambda _\\pm \\cosh \\lbrace \\gamma _\\pm + H(r-r_0)\\rbrace }, & \\Lambda _\\pm >0, \\\\\\frac{1}{H} e^{H(r - r_0)}\\cosh H\\eta , & \\Lambda _\\pm =0,\\\\\\end{array}\\right.$ and $&& A^-_\\pm =\\left\\lbrace \\begin{array}{ll}-\\frac{H^2}{\\lambda ^2_\\pm [\\cosh \\lbrace \\gamma _\\pm +H(r-r_0)\\rbrace ]^2}, &\\Lambda _\\pm >0, \\\\- e^{-2 H(r - r_0)}, & \\Lambda _\\pm =0,\\\\\\end{array}\\right.", "\\\\&& A^0_\\pm = \\left\\lbrace \\begin{array}{ll}-\\frac{H^2}{\\lambda ^2_\\pm [\\cosh \\lbrace H(r-r_0)+\\gamma _\\pm \\rbrace ]^2}, & \\Lambda _\\pm >0, \\\\\\end{array}\\right.", "\\\\&&A^+_\\pm =\\left\\lbrace \\begin{array}{ll}- \\frac{H^2}{\\lambda ^2_\\pm [\\cosh \\lbrace \\gamma _\\pm + H(r-r_0)\\rbrace ]^2}, & \\Lambda _\\pm >0, \\\\ - e^{2 H(r - r_0)}, & \\Lambda _\\pm =0,\\\\\\end{array}\\right.$ which agree with the results for $q_0=\\beta ^2$ in [21].We should point out that the $\\gamma _\\pm $ appeared in hyperbolic cosine are different from the integration constant $\\beta z^{\\prime \\prime }$ in [21] due to different process of normalization.", "It is not difficult to verify that the domain wall solutions for $\\Lambda _\\pm <0$ are also equivalent to Cvetič et al's results [21].", "When $3 H^2=\\Lambda _+= \\Lambda _- > 0$ , Eq.", "(REF ) yields $\\sigma =0$ , which means no domain wall in the space-times.", "So it corresponds to $\\gamma _+=\\gamma _-=0$ in the solutions (REF )-()." ], [ "H=0", "In the case of $H=0$ , which corresponds to extreme walls in [21], the exact solutions of Eq.", "(REF ) give $\\hat{B}=\\left\\lbrace \\begin{array}{ll}\\hat{B}^- =\\eta , \\\\\\hat{B}^0 =1,\\end{array} \\right.$ where no real solution exists in $k=1$ .", "Since the following calculations to obtain $\\mathcal {F}_\\pm $ and $\\mathcal {G}_\\pm $ are similar to the calculations in the previous subsection REF , we directly present the final results of $\\mathcal {F}_\\pm $ and $\\mathcal {G}_\\pm $ with $h_\\pm =1$ , which yield $&&\\mathcal {F}^-_\\pm =\\left\\lbrace \\begin{array}{ll}\\frac{1}{2 \\lambda _\\pm }\\ln \\lbrace r_0 + \\lambda _\\pm -\\frac{1}{v}\\rbrace , &\\Lambda _\\pm = - 3 \\lambda _\\pm ^2, \\\\ \\frac{1}{2}(b_\\pm + r_0 - \\frac{1}{ v} ), & \\Lambda _\\pm =0,\\\\\\end{array}\\right.", "\\\\&& \\mathcal {F}^0_\\pm = \\left\\lbrace \\begin{array}{ll}\\frac{1}{2}(\\lambda ^{-2}_{\\pm } + \\lambda ^{-1}_\\pm r_0 - \\frac{1}{\\lambda _\\pm v} ), & \\Lambda _\\pm = - 3 \\lambda _\\pm ^2, \\\\\\end{array}\\right.$ and $&&\\mathcal {G}^-_\\pm =\\left\\lbrace \\begin{array}{ll}- \\frac{1}{2 \\lambda _\\pm }\\ln \\lbrace u- r_0 - \\lambda _\\pm \\rbrace , &\\Lambda _\\pm = - 3 \\lambda _\\pm ^2, \\\\\\frac{1}{2}(-b_\\pm - r_0 + u ), & \\Lambda _\\pm =0,\\\\\\end{array}\\right.", "\\\\&& \\mathcal {G}^0_\\pm = \\left\\lbrace \\begin{array}{ll}\\frac{1}{2}(\\lambda ^{-2}_{\\pm } + \\lambda ^{-1}_\\pm r_0 - \\frac{u}{\\lambda _\\pm } ), & \\Lambda _\\pm = - 3 \\lambda _\\pm ^2, \\\\\\end{array}\\right.$ where $b_\\pm $ are constants and no real solution exists for $\\Lambda _\\pm >0$ .", "From Eqs.", "(REF )-(REF ), one can directly calculate $B_\\pm (r, \\eta )$ and $A_\\pm (r, \\eta )$ to obtain $&&B^-_\\pm =\\left\\lbrace \\begin{array}{ll}\\frac{- \\eta }{\\lambda _\\pm (r-r_0) -1}, &\\Lambda _\\pm = - 3 \\lambda _\\pm ^2, \\\\ \\eta , & \\Lambda _\\pm =0,\\\\\\end{array}\\right.", "\\\\&& B^0_\\pm = \\left\\lbrace \\begin{array}{ll}\\frac{-1}{\\lambda _\\pm (r-r_0) - 1}, & \\Lambda _\\pm = - 3 \\lambda _\\pm ^2, \\\\\\end{array}\\right.$ and $&&{A}^-_\\pm =\\left\\lbrace \\begin{array}{ll}- \\frac{1}{(\\lambda _\\pm (r-r_0) -1)^2}, &\\Lambda _\\pm = - 3 \\lambda _\\pm ^2, \\\\-1, & \\Lambda _\\pm =0,\\\\\\end{array}\\right.", "\\\\&& {A}^0_\\pm = \\left\\lbrace \\begin{array}{ll}- \\frac{1}{(\\lambda _\\pm (r-r_0) - 1)^2}.", "& \\Lambda _\\pm = - 3 \\lambda _\\pm ^2, \\\\\\end{array}\\right.$ which corresponds to extreme wall solutions $(q_0=0)$ in [21].", "So domain wall solutions in case I yield the same solutions as in [21].", "It indicates that one can transform domain wall solutions in static background metric to Cvetič et al's domain wall solutions [21] by coordinate transformations.", "In our previous work [22], we obtained a planar domain wall solution with reflection symmetry in de Sitter space-time, where $\\hat{A}=-\\alpha ^2/\\eta ^2 $ , and then study its gravitational effects on primordial quantum fluctuations during inflation [23].", "In case II, we generalize our previous planar domain wall solution to spherical, planar, and hyperbolic domain walls without reflection symmetry.", "In the choice of $\\hat{A}=-\\alpha ^2/\\eta ^2 $ , Eq.", "(REF ) becomes $\\dot{\\hat{B}}^2 = - \\frac{\\alpha ^2}{\\eta ^2} \\,(k- H^2\\hat{B}^2).", "$ and exact solutions of Eq.", "(REF ) yield $\\hat{B}=\\left\\lbrace \\begin{array}{ll}\\hat{B}^- =\\frac{\\eta }{2H} - \\frac{1}{2\\eta H}, \\\\ \\hat{B}^0 =-\\frac{1}{\\eta },\\\\\\hat{B}^+ =\\frac{\\eta }{2H} + \\frac{1}{2\\eta H}, \\end{array} \\right.$ where we have set $\\alpha ^2=1/H^2$ .", "One may notice that the coordinate time $\\eta $ is related to proper time $\\tau $ on $\\Sigma $ by $\\eta = -e^{-H \\tau }$ , where Eq.", "(REF ) becomes Eq.", "(REF ) (up to a sign choice).", "So $\\eta $ may be considered as conformal time in de Sitter space-time.", "In the following, we only discuss $\\Lambda _\\pm >0$ and $H^2 \\ne 0$ .", "From Eq.", "() we obtain $\\hat{R}=\\left\\lbrace \\begin{array}{ll}\\hat{R}_\\pm ^- = {\\lambda ^{-1}_\\pm } \\tan ^{-1} (\\lambda _\\pm \\hat{B}^-), \\\\\\hat{R}_\\pm ^0 = -\\lambda ^{-2}_\\pm (\\hat{B}^0)^{-1},\\\\\\hat{R}_\\pm ^+ = - {\\lambda ^{-1}_\\pm } \\coth ^{-1} ({\\lambda _\\pm } \\hat{B}^+), \\end{array} \\right.$ and then solving Eq.", "() gives $\\hat{T}=\\left\\lbrace \\begin{array}{ll}\\hat{T}_\\pm ^- = -{\\lambda ^{-1}_\\pm } \\tan ^{-1} \\lbrace \\lambda _\\pm \\mu _\\pm (\\frac{\\eta }{H}-\\hat{B}^-)\\rbrace , \\\\\\hat{T}_\\pm ^0 = h_\\pm \\lambda ^{-2}_\\pm (\\mu _\\pm \\hat{B}^0)^{-1},\\\\\\hat{T}_\\pm ^+ = {\\lambda ^{-1}_\\pm } \\coth ^{-1} \\lbrace \\lambda _\\pm \\mu _\\pm (\\frac{\\eta }{H}- \\hat{B}^+\\rbrace , \\end{array} \\right.$ where $\\mu _\\pm =\\sqrt{(1-\\lambda ^2_\\pm /H^2)^{-1}}$ and $h_\\pm =1$ are chosen in the case of spherical and hyperbolic walls.", "A tedious but straightforward calculation of $\\mathcal {F}(v)$ and $\\mathcal {G}(u)$ gives $\\mathcal {F}_\\pm =\\left\\lbrace \\begin{array}{ll}\\mathcal {F}^-_\\pm = \\frac{1}{2\\lambda _\\pm }\\tan ^{-1}\\lbrace \\frac{2}{ e^{\\gamma _\\pm } (\\frac{1}{v}-r_0)- e^{-\\gamma _\\pm } (\\frac{1}{v}-r_0)^{-1} }\\rbrace , \\\\\\mathcal {F}^0_\\pm = -\\frac{1}{2\\lambda _\\pm H} e^{\\lbrace h_\\pm \\gamma _\\pm \\rbrace } (\\frac{1}{v}-r_0),\\\\\\mathcal {F}^+_\\pm = \\frac{1}{2\\lambda _\\pm }\\coth ^{-1}\\lbrace \\frac{e^{-\\gamma _\\pm } (\\frac{1}{v}-r_0)^{-1}+ e^{\\gamma _\\pm } (\\frac{1}{v}-r_0)}{2}\\rbrace , \\end{array} \\right.$ and $\\mathcal {G}_\\pm =\\left\\lbrace \\begin{array}{ll}\\mathcal {G}^-_\\pm =- \\frac{1}{2\\lambda _\\pm }\\tan ^{-1}\\lbrace \\frac{2}{ e^{-\\gamma _\\pm } (u-r_0)- e^{\\gamma _\\pm }( \\frac{1}{u-r_0}) }\\rbrace , \\\\\\mathcal {G}^0_\\pm = \\frac{1}{2\\lambda _\\pm H}\\, e^{\\lbrace -h_\\pm \\gamma _\\pm \\rbrace } (u-r_0),\\\\\\mathcal {G}^+_\\pm =-\\frac{1}{2\\lambda _\\pm }\\coth ^{-1}\\lbrace \\frac{e^{\\gamma _\\pm } (\\frac{1}{u-r_0})+ e^{-\\gamma _\\pm } (u-r_0)}{2}\\rbrace .", "\\end{array} \\right.$ So one can then calculate $A(r, \\eta )$ and $B(r, \\eta )$ to get $B_\\pm =\\left\\lbrace \\begin{array}{ll}B^-_\\pm = \\frac{ \\eta ^2-(r-r_0)^2-1 }{2 \\lbrace H\\eta -\\sqrt{H^2-\\lambda _\\pm ^2}\\,\\,(r-r_0)\\rbrace }, \\\\B^0_\\pm = \\frac{ -H }{ H \\eta - h_\\pm \\sqrt{(H^2-\\lambda ^2_\\pm )}\\,\\,(r-r_0) },\\\\B^+_\\pm =\\frac{ \\eta ^2-(r-r_0)^2+1 }{2 \\lbrace H\\eta -\\sqrt{H^2-\\lambda _\\pm ^2}\\,\\,(r-r_0)\\rbrace }, \\end{array} \\right.$ and $A_\\pm =\\left\\lbrace \\begin{array}{ll}A^-_\\pm =- \\frac{ 1 }{ (H\\eta -\\sqrt{H^2-\\lambda _\\pm ^2}\\,\\,(r-r_0))^2 }, \\\\ A^0_\\pm =- \\frac{ 1 }{ ( H \\eta - h_\\pm \\sqrt{(H^2-\\lambda ^2_\\pm )}\\,\\,(r-r_0) )^2},\\\\A^+_\\pm =-\\frac{ 1 }{( H\\eta -\\sqrt{H^2-\\lambda _\\pm ^2}\\,\\,(r-r_0))^2 }.", "\\end{array} \\right.$ Eqs.", "(REF ) and (REF ) are the metric solutions of spherical, planar, hyperbolic domain walls in de Sitter space-times.", "In the case of $h_-=-h_+=1$ and $\\Lambda _-=\\Lambda _+$ , the solutions $B^0_\\pm (r, \\eta )$ and $A^0_\\pm (r, \\eta )$ do return to our previous planar domain wall solution [22].", "It is clear that planar domain wall space-times with different positive $\\Lambda $ on each side of the wall are conformally flat, and when $\\sigma =0$ , which corresponds to $H^2= \\lambda ^2_+=\\lambda ^2_-$ , $B^0_\\pm (r, \\eta )$ and $A^0_\\pm (r, \\eta )$ become the metric for describing inflationary Universe in the conformal time coordinate [17], [27].", "Since quantum fluctuations during inflation [28], [27], [29] and the well-known definition of the vacuum state, i.e.", "Bunch-Davies vacuum [30], are described in background metric $B^0_\\pm (r, \\eta )$ and $A^0_\\pm (r, \\eta )$ with $\\sigma =0$ , it suggests that this co-moving coordinates in case II is a proper coordinate choice to study gravitational effects of domain walls during inflation.", "In [23], we study quantum fluctuations of a scalar field in background planar domain wall metric, $B^0_\\pm (r, \\eta )$ and $A^0_\\pm (r, \\eta )$ , with reflection symmetry, and it yields that gravitational effects of planar domain wall will cause a primordial dipole in the power spectrum of primordial curvature perturbations.", "In the spherical domain wall solution, i.e.", "$B^+_\\pm (r, \\eta )$ and $A^+_\\pm (r, \\eta )$ , one may define a local neighborhood $\\mathcal {N}_p$ by restricting coordinates $\\lbrace \\eta , r, x, \\phi \\rbrace $ to the range $\\eta ^2-(r-r_0)^2\\ll 1$ and $x\\ll 1$ , so gravitational fields of spherical domain walls in $\\mathcal {N}_p$ can well approximate as the planar domain wall metric, i.e.", "$B^0_\\pm (r, \\eta )$ and $A^0_\\pm (r, \\eta )$ .", "The global structure of domain wall space-times in case I have been well studied in [5], [21], [24].", "In Sec.", ", we will first present coordinate transformations between the domain wall solutions Eqs.", "(REF )-() in case I and the solutions Eqs.", "(REF )-(REF ) in case II, and then study the global structure of domain wall space-times of case II." ], [ "Global structure of domain wall space-times", "The global structure of spherical, planar and hyperbolic domain walls has been largely discussed in [21], [5] (see the review article [24] and the references in).", "Ref.", "[24] has pointed out that the constant-$r$ sections of domain wall solutions in case I with $H^2\\ne 0$ , i.e.", "Eqs.", "(REF )-(), all represent (2+1)-dimensional de Sitter space-time ($\\textrm {d}\\textrm {S}_3$ ), whose topology is $\\textbf {R}$ (time) $\\times $ $\\textbf {S}^2$ (space).", "It turns out that planar domain wall solution, i.e.", "Eqs.", "() and (REF ), which is locally plane-symmetric and geodesically incomplete, describes only a part of a spherical bubble [24].", "Since the metric of spherical domain wall solution, Eqs.", "() and (), internal to the wall is geodesically complete, Ref.", "[21] study geodesic extension of spherically non- and ultraextreme domain wall space-time with non-positive $\\Lambda _\\pm $ in $(\\eta , r)$ directions.", "It is useful to study global and casual structure of space-times by using conforaml diagram [31], [32], which compactify space-time infinity into finite region.", "To study the global structure of domain wall solution in case II, we first present the coordinate transformation between the domain wall solutions in case I and case II.", "By performing the following coordinate transformations $\\begin{array}{ll}&\\eta = - \\cosh \\lbrace H(r^{\\prime }-r^{\\prime }_0)\\rbrace e^{-H t},\\\\ &r-r_0= \\sinh \\lbrace H(r^{\\prime }-r^{\\prime }_0)\\rbrace e^{-Ht},\\end{array}$ on domain wall metric in case II, we then obtain $d s^2_\\pm =$ $\\left\\lbrace \\begin{array}{ll}\\frac{H^2}{\\lambda ^2_\\pm [\\cosh \\lbrace \\gamma _\\pm +H(r^{\\prime }-r^{\\prime }_0)\\rbrace ]^2}(-\\textrm {d}t^2 + \\textrm {d}r^{\\prime 2}+ [\\frac{\\sinh H t}{H}]^2 \\textrm {d}H^2_2 ) \\\\\\frac{H^2}{\\lambda ^2_\\pm [\\cosh \\lbrace \\gamma _\\pm +H(r^{\\prime }-r^{\\prime }_0)\\rbrace ]^2}(-\\textrm {d}t^2 + \\textrm {d}r^{\\prime 2}+ e^{2 H t} \\textrm {d}X^2_2 ),\\\\\\frac{H^2}{\\lambda ^2_\\pm [\\cosh \\lbrace \\gamma _\\pm +H(r^{\\prime }-r^{\\prime }_0)\\rbrace ]^2}(-\\textrm {d}t^2 + \\textrm {d}r^{\\prime 2}+ [\\frac{\\cosh H t}{H}]^2 \\textrm {d}\\Omega ^2_2 ), \\end{array} \\right.$ which correspond to non- and ultraextreme domain wall solutions in case I.", "Here, $\\textrm {d}H^2_2$ , $\\textrm {d}X^2_2$ and $\\textrm {d}\\Omega ^2_2$ denote the line element of two-dimensional hyperboloid, plane, and sphere respectively.", "From the coordinate transformations (REF ), it is clear that both $\\lbrace \\eta , r, x, \\phi \\rbrace $ and $\\lbrace t, r^{\\prime }, x, \\phi \\rbrace $ are comoving coordinates with the wall sitting at $r=r_0$ and $r^{\\prime }=r^{\\prime }_0$ , respectively.", "Moreover, Eq.", "(REF ) restrict the $\\eta $ -coordinate range to $\\eta \\leqslant 0$ , so the future infinity ($t=\\infty $ ) and past infinity ($t=-\\infty $ ) in the conformal diagram of case I domain wall solutions ($H^2\\ne 0$ ) corresponds to $\\eta =0$ and $\\eta =-\\infty $ .", "The geodesic extensions of the comoving coordinate patch have been present in [21], [24].", "By transforming the domain wall solutions in case II to Eq.", "(REF ), we realize that the topology of spherical, planar, and hyperbolic domain walls in case II is also $\\textbf {R}$ (time) $\\times $ $\\textbf {S}^2$ (space)." ], [ "Discussion", "We have present a proof on the equivalence of the CC approach and MW approach, and shown how to construct a comoving coordinate by knowing the trajectories of domain walls in the MW approach.", "The spherical, planar and hyperbolic domain wall solutions with $M_\\pm =0$ are obtained in two different comoving coordinates, which are referred to case I and case II.", "The case I domain wall solutions yield the same solutions obtained in [21].", "Refs.", "[21], [24] studied the global and casual structure of case I domain wall solutions.", "As pointed out in [21], one can observe that constant-$r$ sections of spherical, planar and hyperbolic domain wall solutions with $H^2\\ne 0$ in case I represent (2+1)-dimensional de Sitter space-time $\\textrm {d}\\textrm {S}_3$ .", "Since it is not clear to us whether spherical, plane and hyperbolic domain walls in case II also represent $\\textrm {d}\\textrm {S}_3$ , we find the coordinate transformation between case I and case II domain wall solutions with $H^2\\ne 0$ .", "Hence, one can understand that the planar domain wall solution in case II represents a portion of a spherical domain wall space-time.", "From the coordinate transformation Eq.", "(REF ), we also learn the future infinity ($t=\\infty $ ) and past infinity ($t=-\\infty $ ) in the conformal diagram of case I domain wall solutions ($H^2\\ne 0$ ) corresponds to $\\eta =0$ and $\\eta =-\\infty $ in the case II domain wall solutions.", "In [22], we obtain a planar domain wall solution, which is conformally flat.", "When $\\sigma =0$ , the planar domain wall metric returns to the well-known metric of steady state Universe, which has been used to study quantum fluctuations during inflation, in conformal time coordinate.", "So we studied quantum fluctuations of a scalar field in the planar domain wall space-time and find the gravitational effects of the planar domain wall on power spectrum of primordial curvature perturbation [23].", "However, the planar domain wall solution requires plane and reflection symmetry, which is quite limited in the study of gravitational fields of realistic domain wall space-times.", "For example, false vacuum decay yields two space-time regions with different $\\Lambda $ separated by a spherical bubble.", "So the case II domain wall solutions, i.e.", "Eqs.", "(REF )-(REF ), generalized our previous planar domain wall solution [22] to spherical, planar and hyperbolic domain walls without refection symmetry.", "These solutions will be useful for further investigation on primordial quantum fluctuation of scalar fields during inflation.", "In this paper, we only discuss domain wall solutions in the case of $M_\\pm =0$ .", "There are various interesting physical problems, which may need to consider $M_\\pm \\ne 0$ .", "For example, in the study the evolution of remnants of the false vacuum surrounded by the true vacuum [5], one should consider $M_+\\ne 0$ .", "So finding domain wall solutions in the comoving coordinates with $M_\\pm \\ne 0$ may also be useful for studying some physical problems and their associated global structure of space-times.", "This will be considered as our future works.", "CHW is supported by the National Science Council of the Republic of China under the grants NSC 98-2811-M-032-005 and YHW is fully supported by the NCU Top University Project funded by the Ministry of Education, Taiwan ROC and Center for Mathematics and Theoretical Physics, National Central University.", "This work is also partially support by the National Center for Theoretical Sciences (NCTS), Hsinchu." ], [ "Integral form of $\\mathcal {B}(B)$", "Since the calculations of domain wall solutions in Sec.", "and Sec.", "involve integral function $\\mathcal {B}$ , which is $\\mathcal {B}(B_\\pm )= \\int \\frac{d B_\\pm }{-k + \\frac{2M_\\pm }{B_\\pm } + \\frac{\\Lambda _\\pm }{3} B_\\pm ^2}, $ we present the integration of Eq.", "(REF ) here.", "In the following, we omit the subscript $\\pm $ for simplification.", "To integrate Eq.", "(REF ) in the case of $k=1$ and $\\lbrace M, \\Lambda \\rbrace >0$ , we should first study the cubic equations $B^3 - \\frac{3}{\\Lambda } B + \\frac{6M}{\\Lambda }=0$ , and if determinant $D= -\\frac{1}{\\Lambda ^3} + \\frac{9 M^2}{\\Lambda ^2}<0$ , there are three distinct real roots (2 positive and 1 negative roots) $b_1 = 2 \\sqrt{\\frac{1}{\\Lambda }}\\cos {\\frac{\\phi }{3}}, \\hspace{14.22636pt}b_{2,3} = -2 \\sqrt{\\frac{1}{\\Lambda }}\\cos {\\frac{\\phi \\pm \\pi }{3}},$ where $\\phi = \\arccos \\lbrace -{3 M}\\sqrt{\\Lambda }\\rbrace $ .", "Integrating Eq.", "(REF ) yields $\\mathcal {B}(B)=\\left\\lbrace \\begin{array}{ll}\\frac{3\\ln \\lbrace (B- b_{++})^\\mu (B-b_+)^\\nu (B-b_-)^\\rho \\rbrace }{\\Lambda }, B> b_{++}, \\\\\\frac{3\\ln \\lbrace (b_{++}- B)^\\mu (B-b_+)^\\nu (B-b_-)^\\rho \\rbrace }{\\Lambda }, b_{++}>B>b_+,\\\\\\frac{3\\ln \\lbrace ( b_{++}- B)^\\mu (b_+- B)^\\nu (B-b_-)^\\rho \\rbrace }{\\Lambda }, b_{+}>B, \\end{array} \\right.$ with $\\mu $ , $\\nu $ and $\\rho $ satisfying $\\frac{\\mu }{(B-b_{++})} + \\frac{\\nu }{(B-b_{+})} + \\frac{\\rho }{(B-b_{-})} = \\frac{B}{B^3 - \\frac{3}{\\Lambda } B + \\frac{6M}{\\Lambda }},\\nonumber $ where $b_{++}$ denotes the larger positive root and $b_-$ denotes the negative root.", "From Eq.", "(REF ), we learn that one coordinate chart can only cover either $B>b_+$ or $B<b_+$ , so one need to introduce separate coordinate charts to cover the whole range of $B$ .", "In the case of $M=0$ and $k=1$ , the integration of Eq.", "(REF ) gives $\\mathcal {B}(B)=\\left\\lbrace \\begin{array}{ll}-\\lambda ^{-1}\\coth ^{-1}(\\lambda B), & B>\\lambda ^{-1},\\\\ -\\lambda ^{-1}\\tanh ^{-1}(\\lambda B), & B< \\lambda ^{-1},\\end{array} \\right.$ for $\\Lambda = 3\\lambda ^2$ , and $\\mathcal {B}(B)=-\\lambda ^{-1}\\tan ^{-1}(\\lambda B),$ for $\\Lambda = - 3\\lambda ^2$ .", "Similarly, in the case of $M=0$ and $k=-1$ , we obtain $\\mathcal {B}(B)=\\lambda ^{-1}\\tan ^{-1}(\\lambda B),$ for $\\Lambda = 3\\lambda ^2$ and $\\mathcal {B}(B)=\\left\\lbrace \\begin{array}{ll}\\lambda ^{-1}\\coth ^{-1}(\\lambda B), & B>\\lambda ^{-1},\\\\\\lambda ^{-1}\\tanh ^{-1}(\\lambda B), & B< \\lambda ^{-1},\\end{array} \\right.$ for $\\Lambda = -3\\lambda ^2$ .", "Finally, the case of $M=k=0$ yields $\\mathcal {B}(B)= -\\frac{3}{\\Lambda } B^{-1}.", "$ Eqs.", "(REF )-(REF ) will be used to calculate $A_\\pm (r, \\eta )$ and $B_\\pm (r, \\eta )$ in Sec.", ", and it is not difficult to verify that $A_\\pm $ and $B_\\pm $ yield the same results in the calculation of $B>\\lambda ^{-1}$ and $B<\\lambda ^{-1}$" ] ]

[ [ "The Complexity of Monotone Hybrid Logics over Linear Frames and the\n Natural Numbers" ], [ "Abstract Hybrid logic with binders is an expressive specification language.", "Its satisfiability problem is undecidable in general.", "If frames are restricted to N or general linear orders, then satisfiability is known to be decidable, but of non-elementary complexity.", "In this paper, we consider monotone hybrid logics (i.e., the Boolean connectives are conjunction and disjunction only) over N and general linear orders.", "We show that the satisfiability problem remains non-elementary over linear orders, but its complexity drops to PSPACE-completeness over N. We categorize the strict fragments arising from different combinations of modal and hybrid operators into NP-complete and tractable (i.e.", "complete for NC1or LOGSPACE).", "Interestingly, NP-completeness depends only on the fragment and not on the frame.", "For the cases above NP, satisfiability over linear orders is harder than over N, while below NP it is at most as hard.", "In addition we examine model-theoretic properties of the fragments in question." ], [ "Introduction", "Hybrid logic is an extension of modal logic with nominals, satisfaction operators and binders.", "The downarrow binder $$ , which is related to the freeze operator in temporal logic [11], provides high expressivity.", "The price paid is the undecidability of the satisfiability problem for the hybrid language with the downarrow binder $$ [4], [10], [1].", "In contrast, modal logic, and its extension with nominals and the satisfaction operator, is -complete [12], [1].", "In order to regain decidability, syntactic and semantic restrictions have been considered.", "It has been shown in [21] that the absence of certain combinations of universal operators ($\\Box $ , $\\wedge $ ) with $$ brings back decidability, and that the hybrid language with $$ is decidable over frames of bounded width.", "Furthermore, this language is decidable over transitive and complete frames [16], and over frames with an equivalence relation (ER frames) [15].", "Adding the at-operator $$ —which allows to jump to states named by nominals—leads to undecidability over transitive frames [16], but not over ER frames [15].", "Over linear frames and transitive trees, $$ on its own does not add expressivity, but combinations with $$ or the global modality—an additional $\\Diamond $ interpreted over the universal relation—do.", "These languages are decidable and of non-elementary complexity [9], [16]; if the number of state variables is bounded, then they are of elementary complexity [18], [23], [5].", "We aim for a more fine-grained distinction between fragments of different complexities by systematically restricting the set of Boolean connectives and combining this with restrictions to the modal/hybrid operators and to the underlying frames.", "In [14], we have focussed on four frame classes that allow cycles, and studied the complexity of satisfiability for fragments obtained by arbitrary combinations of Boolean connectives and four modal/hybrid operators.", "The main open question in [14] is the one for tight upper bounds for monotone fragments including the $\\Box $ -operator.", "Even though there are many logics for which the restriction to monotone Boolean connectives leads to a significant decrease in complexity, it is not straightforward, and therefore interesting to find out, where this happens for hybrid logics.", "In this study, we classify the computational complexity of satisfiability for monotone fragments of hybrid logic with arbitrary combinations of the operators $\\Diamond $ , $\\Box $ , $$ and $$ over linear orders and the natural numbers.", "Whereas the full logic is non-elementary and decidable [16] for both frame classes, we show that in the monotone case this high complexity is gained only over linear orders and drops to -completeness over the natural numbers.", "Informally speaking, the reason is that linearly ordered frames may consist of arbitrarily many dense parts that can be distinguished using the expressive power of all four operators.", "These dense parts and their distances are used to store information that cannot be stored in a frame without dense parts as, e.g., the natural numbers.", "For all other monotone fragments that contain the $\\Diamond $ -operator, we show -completeness independent on the frame class, for linear orders, all remaining fragments (i.e.", "the fragments without $\\Diamond $ ) can be shown to be 1-complete.", "The reason is, informally speaking, that all (sub-)formulas of the form $\\Box \\alpha $ are easily satisfied in a state without successor, which can essentially be used to reduce this problem to the satisfiability problem for monotone propositional formulae.", "This argument does not go through over the natural numbers, a total frame where every state has a successor.", "Over this frame class, we give a decision procedure that runs in logarithmic space for the fragment with all operators except $\\Diamond $ (and prove a matching lower bound), and in 1 for all other fragments.", "These results give rise to two interesting observations.", "First, the -completeness results are independent on the frame class.", "Second, for the fragment whose satisfiability problem is above , linear orders make the problem harder than the natural numbers, and for the richest fragment below , it is the opposite way round—the natural numbers make the problem harder than linear orders.", "Notice also that, in the case where Boolean operators are not restricted to monotone ones, all fragments are -hard.", "Our results are shown in Figure REF .", "Figure: Our complexity results for satisfiability overlinear frames ()() and the natural numbers ()()for hybrid logicwith monotone Boolean operators and different combinations of modal/hybrid operators" ], [ "Preliminaries", "Hybrid Logic.", "In the following, we introduce the notions and definitions of hybrid logic.", "The terminology is largely taken from [2].", "Let $$ be a countable set of atomic propositions, $$ be a countable set of nominals, $$ be a countable set of variables and $= \\cup \\cup $ .", "We adhere to the common practice of denoting atomic propositions by $p,q,\\ldots $ , nominals by $i,j,\\ldots $ , and variables by $x,y,\\ldots $ We define the language of hybrid (modal) logic $$ as the set of well-formed formulae of the form $\\varphi a \\mid \\top \\mid \\bot \\mid \\lnot \\varphi \\mid \\varphi \\wedge \\varphi \\mid \\varphi \\vee \\varphi \\mid \\Diamond \\varphi \\mid \\Box \\varphi \\mid x.\\varphi \\mid _t \\varphi $ where $a \\in $ , $x \\in $ and $t \\in \\cup $ .", "We define the usual Kripke semantics only to be able to refer to already existing results.", "We will then simplify the standard semantics for monotone formulae.", "Formulae of $$ are interpreted on (hybrid) Kripke structures $K=(W,R,\\eta )$ , consisting of a set of states $W$ , a transition relation $R\\colon W\\times W$ , and a labeling function $\\eta \\colon \\cup \\rightarrow \\wp (W)$ that maps $$ and $$ to subsets of $W$ with $|\\eta (i)| = 1$ for all $i \\in $ .", "The relational structure $(W,R)$ is the Kripke frame underlying $K$ .", "In order to evaluate $$ -formulae, an assignment $g\\colon \\rightarrow W$ is necessary.", "Given an assignment $g$ , a state variable $x$ and a state $w$ , an $x$ -variant $g^x_w$ of $g$ is defined by $g^x_w(x)=w$ and $g^x_w(x^{\\prime })=g(x^{\\prime })$ for all $x \\ne x^{\\prime }$ .", "For any $a \\in $ , let $[\\eta ,g](a)=\\lbrace g(a)\\rbrace $ if $a \\in $ and $[\\eta ,g](a)=\\eta (a)$ , otherwise.", "The satisfaction relation of hybrid formulae is defined as follows.", "K,g,w $~ ~if and only if~~ \\exists w^{\\prime } \\in W(wRw^{\\prime } \\mathbin {\\&} K,g,w^{\\prime } \\models \\varphi )$ px $K,g,w \\models a$   if and only if $w \\in [\\eta ,g](a)$ , $a \\in $ , px $K,g,w \\models \\top $ ,   and $K,g,w \\lnot \\models \\bot $ , px $K,g,w \\models \\lnot \\varphi $   if and only if $K,g,w \\lnot \\models \\varphi $ , px $K,g,w \\models \\varphi \\wedge \\psi $   if and only if $K,g,w \\models \\varphi $ and $K,g,w \\models \\psi $ , px $K,g,w \\models \\varphi \\vee \\psi $   if and only if $K,g,w \\models \\varphi $ or $K,g,w \\models \\psi $ , px $K,g,w \\models \\Diamond \\varphi $   if and only if $\\exists w^{\\prime } \\in W(wRw^{\\prime } \\mathbin {\\&} K,g,w^{\\prime } \\models \\varphi )$ , px $K,g,w \\models \\Box \\varphi $   if and only if $\\forall w^{\\prime } \\in W(wRw^{\\prime } \\Rightarrow K,g,w^{\\prime } \\models \\varphi )$ , px $K,g,w \\models _t \\varphi $   if and only if $K,g,[\\eta ,g](t) \\models \\varphi $ , px $K,g,w \\models x.", "\\varphi $   if and only if $K,g^x_w,w \\models \\varphi $ .", "A hybrid formula $\\varphi $ is said to be satisfiable if there exists a Kripke structure $K=(W,R,\\eta )$ , a $w \\in W$ and an assignment $g\\colon \\rightarrow W$ with $K,g,w \\models \\varphi $ .", "The at operator $_t$ shifts evaluation to the state named by $t\\in \\cup $ .", "The downarrow binder $x.$ binds the state variable $x$ to the current state.", "The symbols $_x$ , $x.$ are called hybrid operators whereas the symbols $\\Diamond $ and $\\Box $ are called modal operators.", "The scope of an occurrence of the binder $$ is defined as usual.", "For a state variable $x$ , an occurrence of $x$ or $_x$ in a formula $\\varphi $ is called bound if this occurrence is in the scope of some $$ in $\\varphi $ , free otherwise.", "$\\varphi $ is said to contain a free state variable if some $x$ or $_x$ occurs free in $\\varphi $ .", "Given two formulae $\\varphi ,\\alpha $ and a subformula $\\psi $ of $\\varphi $ , we use $\\varphi [\\psi /\\alpha ]$ to denote the result of replacing each occurrence of $\\psi $ in $\\varphi $ with $\\alpha $ .", "For considering fragments of hybrid logics, we define subsets of the language $$ as follows.", "Let $O$ be a set of hybrid and modal operators, i.e., a subset of $\\lbrace \\Diamond ,\\Box ,,\\rbrace $ .", "We define $(O)$ to denote the set of well-formed hybrid formulae using only the operators in $O$ , and $(O)$ to be the set of all formulae in $(O)$ that do not use $\\lnot $ .", "Properties of Frames.", "A frame $F$ is a pair $(W,R)$ , where $W$ is a set of states and $R\\subseteq W\\times W$ a transition relation.", "A frame $F = (W,R)$ is called transitive if $R$ is transitive (for all $u,v,w \\in W$ : $uRv \\wedge vRw uRw$ ), linear if $R$ is transitive, irreflexive and trichotomous ($\\forall u,v \\in W$ : $uRv$ or $u=v$ or $vRu$ ), In this paper we consider the class of all linear frames, denoted by $$ , and the singleton frame class $\\lbrace (,<)\\rbrace $ , denoted by .", "Obviously, $\\subseteq $ .", "Notational convenience.", "We can make some simplifying assumptions about syntax and semantics, of $(O)$ and $(O)$ , which do not restrict generality.", "(1) If $\\in O$ , then formulae do not contain any nominals.", "Those can be simulated by free state variables.", "(2) Free state variables are never bound later in the formula, and every state variable is bound at most once.", "The latter is no significant restriction because variables bound multiple times can be named apart, which is a well-established and computationally easy procedure.", "(3) Monotone formulae do not contain any atomic propositions.", "This restriction is correct because every monotone formula $\\varphi $ is satisfiable if and only if $\\varphi $ with all atomic propositions replaced by $\\top $ is satisfiable.", "This justifies the following restrictions.", "(4) For binder-free fragments, the domain of the labelling function $\\eta $ is restricted to nominals, and we re-define $\\eta \\colon \\rightarrow W$ .", "Furthermore, the absence of $$ makes assignments superfluous: we write $F,w \\models \\varphi $ instead of $F,g,w \\models \\varphi $ .", "(5) For binder fragments, the satisfaction relation $\\models $ is restricted to Kripke frames $F=(W,<)$ , where $<$ is a linear order, and assignments $g : \\rightarrow W$ , i.e., we write $F,g,w \\models \\varphi $ .", "(6) Over , we omit the single Kripke frame, i.e., we write $\\eta ,i \\models \\varphi $ with $\\eta : \\rightarrow $ and $i \\in $ for binder-free fragments, and $g,i \\models \\varphi $ with $g: \\rightarrow $ for binder fragments.", "Satisfiability Problems.", "The satisfiability problem for $(O)$ over the frame class ${F}$ is defined as follows: $[{F}](O)$ an $(O)$ -formula $\\varphi $ (without nominals, see above)Is there a Kripke structure $K$ based on a frame $(W,R) \\in {F}$ , an assignment $g\\colon \\rightarrow W$ and a $w\\in W$ such that $K,g,w\\models \\varphi $  ?", "The monotone satisfiability problem for $(O)$ over the frame class ${F}$ is defined as follows: $[{F}](O)$ an $(O)$ -formula $\\varphi $ without nominals and atomic propositionsIs there a Kripke frame $(W,R) \\in {F}$ , an assignment $g\\colon \\rightarrow W$ and a $w\\in W$ such that $F,g,w\\models \\varphi $  ?", "If ${F}$ is the class of all frames, we simply write $(O)$ or $(O)$ .", "Furthermore, we often omit the set parentheses when giving $O$ explicitly, e.g., $(\\Diamond ,\\Box ,,)$ .", "Complexity Theory.", "We assume familiarity with the standard notions of complexity theory as, e. g., defined in [17].", "In particular, we make use of the classes $Ł$ , $$ , $$ , $$ , and $$ .", "The complexity class $$ is the set of all languages $A$ that are decidable and for which there exists no $k\\in $ such that $A$ can be decided using an algorithm whose running time is bounded by $\\exp _k(n)$ , where $\\exp _k(n)$ is the $k$ -th iteration of the exponential function (e.g., $\\exp _3(n)=2^{2^{2^n}}$ ).", "Furthermore, we need two non-standard complexity classes whose definition relies on circuit complexity and formal languages, see for instance [22], [13].", "The class 1 is defined as the set of languages recognizable by a logtime-uniform family of Boolean circuits of logarithmic depth and polynomial size over $\\lbrace \\wedge ,\\vee ,\\lnot \\rbrace $ , where the fan-in of $\\wedge $ and $\\vee $ gates is fixed to 2.", "The class is defined as the set of languages reducible in logarithmic space to some deterministic context-free language.", "The following relations between the considered complexity classes are known.", "$1 \\subseteq \\subseteq \\subseteq \\subseteq \\subset $ .", "It is unknown whether contains or vice versa.", "A language $A$ is constant-depth reducible to $D$ , $AD$ , if there is a logtime-uniform 0-circuit family with oracle gates for $D$ that decides membership in $A$ .", "Unless otherwise stated, all reductions in this paper are -reductions.", "Known results.", "The following theorem summarizes results for hybrid languages with Boolean operators $\\wedge ,\\vee ,\\lnot $ that are known from the literature.", "Since $\\Box \\varphi \\equiv \\lnot \\Diamond \\lnot \\varphi $ , the $\\Box $ -operator is implicitly present in all fragments containing $\\Diamond $ and negation.", "[[1], [2], [3], [9], [16]]   $(\\Diamond ,,)$ and $(\\Diamond ,)$ are -complete.", "[1] $(\\Diamond ,\\Box )$ is -hard.", "[3] $[\\mathfrak {F}](\\Diamond ,,)$ , for $\\mathfrak {F} \\in \\lbrace ,\\rbrace $ , are in NONELEMENTARY.", "[9], [16] $[\\mathfrak {F}](\\Diamond ,)$ , $[\\mathfrak {F}](\\Diamond ,)$ and $[\\mathfrak {F}](\\Diamond )$ , with $\\mathfrak {F} \\in \\lbrace ,\\rbrace $ , are -complete.", "[2], [9] Our contribution.", "In this paper, we consider the monotone satisfiability problems $[\\mathfrak {F}](O)$ for $\\mathfrak {F} \\in \\lbrace ,\\rbrace $ and all $O \\subseteq \\lbrace \\Diamond ,\\Box ,,\\rbrace $ .", "The hard cases: Non-elementary and $$ results The hardest cases are those with the complete set of operators.", "In the non-monotone case, both satisfiability problems are non-elementary and decidable [16].", "We show that in the monotone case even this hardness is reached, but only on linear frames, i.e.", "$[](\\Diamond ,\\Box ,,)$ is non-elementary and decidable.", "In contrast, on the natural numbers the complexity decreases, i.e.", "we show that $[](\\Diamond ,\\Box ,,)$ is -complete.", "Our proofs use reductions to and from fragments of first-order logic on the natural numbers.", "Let $(<,P)$ be the set of all first-order formulae that use $<$ as the unique binary relation symbol, and $P$ as the unique unary relation symbol.I.e.", "$(<,P)$ is defined as set of all formulae $\\varphi $ as follows.", "$\\varphi \\top \\mid x < y \\mid P(x) \\mid \\lnot \\varphi \\mid \\varphi \\wedge \\varphi \\mid \\varphi \\vee \\varphi \\mid \\exists x \\, \\varphi \\mid \\forall x \\, \\varphi $ for variable symbols $x,y\\in $ .", "Let $[]_{}(<,P)$ denote the set of formulae from $(<,P)$ which are satisfied by a model that has as its universe, interprets $<$ as the less-than relation on $\\times $ , and has an arbitrary interpretation for the predicate symbol $P$ .", "It was shown by Stockmeyer [20] that $[]_{}(<,P)$ is non-elementary.", "Let $(<)$ be the fragment of $(<,P)$ in which the predicate symbol $P$ is not used.", "Accordingly, $[]_{}(<)$ denotes the set of formulae that are satisfiable over $$ and the natural interpretation of $<$ .", "It was shown by Ferrante and Rackoff [8] that $[]_{}(<)$ is in .", "Notice that in both fragments $x=y$ can be expressed as $\\lnot (x<y\\, \\vee \\, y<x)$ .", "Moreover, every $n\\in $ can be expressed by $x_n$ in the formula $\\exists x_0 \\cdots \\exists x_{n-1} [(\\bigwedge _{i=0,1,\\ldots ,n-1} x_i < x_{i+1}) \\wedge \\forall y (x_n<y \\vee \\bigvee _{i=0,1,\\ldots ,n} y=x_i)]$ .", "$[](\\Diamond ,\\Box ,,)$ is non-elementary and decidable.", "Decidability follows from Theorem REF .", "To establish non-elementary complexity, we give a reduction from $[]_{}(<,P)$ .", "We first show how to encode the intepretation of a predicate symbol, represented by a set $P\\subseteq $ , in a linear frame $F = (W,<)$ – without using atomic propositions and nominals as agreed in Section .", "Using free state variables, we can only distinguish linearly many states at any given time.", "We therefore use finite intervals (finite subchains of $(W,<)$ ) to encode whether $n\\in P$ .", "Such an interval—we call it a marker—has length 2 (resp.", "3) if for the corresponding $n$ holds $n\\notin P$ (resp.", "$n\\in P$ ).", "Accordingly, we call a marker of length 2 (resp.", "3) negative (resp. positive).", "These finite intervals are separated by dense intervals—those are intervals wherein every two states have an intermediate state, e.g., $[0,1]_{\\mathbb {Q}} = \\lbrace q \\in \\mathbb {Q} \\mid 0 \\leqslant q \\leqslant 1\\rbrace $ .", "For example, the set $P$ with $0,2\\notin P$ and $1\\in P$ is represented by the chain in Figure REF .", "Figure: An example with 0,2∉P0,2 \\notin P and 1∈P1 \\in P.In our fragment, it is possible to distinguish between dense and finite intervals.", "We now show how to achieve this.", "In order to encode the alternating sequence of finite and dense intervals that represents a subset $P\\subseteq $ , we use the free state variable $a$ to mark a state in a dense interval that is directly followed by the first marker.", "We furthermore use the following macros, where $x$ and $y$ are state variables that are already bound before the use of the macro, and $r,s,t,u$ are fresh state variables.", "The state named $y$ is a direct successor of the state named $x$ .", "It suffices to say that all successors of $x$ are equal to, or occur after, $y$ .", "${dirSuc}(x,y) := _x \\Box z.", "(_y z \\vee _y \\Diamond z)$ The state named $x$ has no direct predecessor.", "It suffices to say that, for all states $r$ equal to, or after, the left bound $a$ : if $r$ is before $x$ , then there is a state between $r$ and $x$ .", "We work around the implication by saying that one of the following three cases occurs: $r$ is after $x$ , or $r$ equals $x$ , or $r$ is before $x$ with a state in between.", "${noDirPred}(x) := _a\\Box r.(_x \\Diamond r \\vee _x r \\vee _r\\Diamond \\Diamond x)$ The state named $x$ has a direct predecessor.", "It suffices to say that there is a state $r$ after $a$ of which $x$ is a direct successor.", "${dirPred}(x) := _a\\Diamond r.{dirSuc}(r,x)$ The interval between states $x,y$ is dense.", "We say that, for all $r$ with $x < r$ : $r$ is after $y$ , or $r$ has no direct predecessor.", "${dense}(x,y) := _x\\Box r.(_y\\Diamond r \\vee {noDirPred}(r))$ The state $x$ is in a separator.", "This macro says that, for some successor $r$ of $x$ , the interval between $x$ and $r$ is dense.", "${sep}(x) := _x\\Diamond r.{dense}(x,r)$ The state $x$ is the begin of a negative marker.", "This macro says that $x$ has a direct successor that is the begin of a separator, and $x$ has no direct predecessor.", "The latter is necessary to avoid that, in the above example, the middle state of a positive marker is mistaken for the begin of a negative marker.", "${neg}(x) := _x\\Diamond r.({dirSuc}(x,r) \\wedge {sep}(r)) \\wedge {noDirPred}(x)$ The state $x$ is the begin of a positive marker.", "Similarly to the above macro, we express that $x$ has a direct-successor sequence $r,s$ with $s$ being the begin of a separator, and $x$ has no direct predecessor.", "${pos}(x) := _x\\Diamond r.({dirSuc}(x,r) \\wedge \\Diamond s.({dirSuc}(r,s) \\wedge {sep}(s))) \\wedge {noDirPred}(x)$ The state $x$ is in a separator whose end is a marker.", "This macro says that, for some successor $r$ of $x$ , the interval between $x$ and $r$ is dense and $r$ is the begin of a marker.", "${sepM}(x) := _x\\Diamond r.({dense}(x,r) \\wedge ({neg}(r) \\vee {pos}(r)))$ We now need the following two conjuncts to express that the part of the model starting at $a$ represents a sequence of infinitely many markers.", "$a$ is in a separator that ends with a marker.", "$\\psi _1 := {sepM}(a)$ Every marker has a direct successor marker.", "We say that every state $r$ after $a$ satisfies one of the following conditions.", "$r$ is in a separator—this also includes that $r$ is the end of a marker—that is followed by a marker.", "$r$ is the begin of a negative marker and its direct successor is the begin of a separator whose end is a marker.", "$r$ is the begin of a positive marker and its direct 2-step successor is the begin of a separator whose end is a marker.", "$r$ in the middle of a positive marker, i.e., $r$ has a direct predecessor which is the begin of a positive marker, and $r$ 's direct successor is in a separator whose end is a marker.", "$\\psi _2 :=\\, & _a \\Box r.\\Big ({sepM}(r) \\\\[2px]& \\vee \\Big ({neg}(r) \\wedge \\Diamond s.({dirSuc}(r,s) \\wedge {sepM}(s))\\Big ) \\\\[2px]& \\vee \\Big ({pos}(r) \\wedge \\Diamond s.({dirSuc}(r,s) \\wedge \\Diamond t.({dirSuc}(s,t) \\wedge {sepM}(t)))\\Big ) \\\\[2px]& \\vee \\Big ( (_a\\Diamond s.{dirSuc}(s,r) \\wedge {pos}(s)) \\wedge \\Diamond t.({dirSuc}(r,t) \\wedge {sepM}(t))\\Big )$ Finally, we encode formulae $\\varphi $ from $(<,P)$ .", "We assume w.l.o.g.", "that such formulae have the shape $\\varphi := Q_1x_1\\dots Q_nx_n.\\beta (x_1,\\dots ,x_n)$ , where $Q_i \\in \\lbrace \\exists ,\\forall \\rbrace $ and $\\beta $ is quantifier-free with atoms $P(x)$ and $x<y$ for variables $x,y$ , such that negations appear only directly before atoms.", "The transformation of $\\varphi $ reuses the $x_i$ as state variables and proceeds inductively as follows.", "$f(P(x_i)) &~:=~ {pos}(x_i) \\\\[4px]f(\\lnot P(x_i)) &~:=~ {neg}(x_i) \\\\[4px]f(x_i < x_j) &~:=~ _{x_i}\\Diamond x_j \\\\[4px]f(\\lnot (x_i < x_j)) &~:=~ _{x_i}x_j \\vee _{x_j}\\Diamond x_i \\\\[4px]f(\\alpha \\wedge \\beta ) &~:=~ f(\\alpha ) \\wedge f(\\beta ) \\\\[4px]f(\\alpha \\vee \\beta ) &~:=~ f(\\alpha ) \\vee f(\\beta ) \\\\[2px]f(\\exists x_i.\\alpha ) &~:=~ _a\\Diamond x_i.\\Big (({neg}(x_i) \\vee {pos}(x_i)) \\wedge f(\\alpha )\\Big ) \\\\f(\\forall x_i.\\alpha ) &~:=~ _a\\Box x_i.\\Big ({sep}(x_i) \\vee {dirPred}(x_i) \\vee f(\\alpha )\\Big ) \\\\$ The transformation of $\\varphi $ into $(\\Diamond ,\\Box ,,)$ is now achieved by the function $g$ defined as follows.", "$g(\\varphi ) := \\psi _1 \\wedge \\psi _2 \\wedge f(\\varphi )$ It is clear that the reduction function $g$ can be computed in polynomial time.", "The correctness of the reduction is expressed by the following claim.", "For every formula $\\varphi $ from $(<,P)$ holds: $\\varphi \\in []_{}(<,P)$ if and only if $g(\\varphi ) \\in [](\\Diamond ,\\Box ,,)$ .", "The proof of the claim should be clear.", "Since $[]_{}(<,P)$ is non-elementary [20], it follows that $[](\\Diamond ,\\Box ,,)$ is non-elementary, too.", "Finally, we note that our reduction uses a single free state variable $a$ , which could as well be bound to the first state of evaluation.", "The high complexity of $[](\\Diamond ,\\Box ,,)$ relies on the possibility that the linear frame alternatingly has dense and non-dense parts.", "If we have the natural numbers as frame for a hybrid language, we lose this possibility.", "As a consequence, the satisfiability problem for monotone hybrid logics over the natural numbers has a lower complexity than that over linear frames.", "$[](\\Diamond ,\\Box ,,)$ is -complete.", "Proof.", "Let $$ be the problem to decide whether a given quantified Boolean formula is valid.", "We show -hardness by a polynomial-time reduction from the $$ -complete $$ to $[](\\Diamond ,\\Box ,,)$ .", "Let $\\varphi $ be an instance of $$ and assume w.l.o.g.", "that negations occur only directly in front of atomic propositions.", "We define the transformation as $f\\colon \\varphi \\mapsto r. \\Diamond s. \\Diamond h(\\varphi )$ where $h$ is given as follows: let $\\psi ,\\chi $ be quantified Boolean formulae and let $x_k$ be a variable in $\\varphi $ , then $\\begin{array}{@{}l@{\\hspace*{28.45274pt}}l@{}}h(\\exists x_k \\psi ):=_r \\Diamond x_k.", "h(\\psi ), &h(\\forall x_k \\psi ):=_r \\Box x_k.", "h(\\psi ), \\\\[2px]h(\\psi \\wedge \\chi ):=h(\\psi ) \\wedge h(\\chi ), &h(\\psi \\vee \\chi ):=h(\\psi ) \\vee h(\\chi ), \\\\[2px]h(\\lnot x_k):=_s \\Diamond x_k, &h(x_k):=_s x_k.", "\\\\\\end{array}$ For example, the QBF $\\psi = \\forall x\\exists y(x \\wedge y) \\vee (\\lnot x \\wedge \\lnot y)$ is mapped to $f(\\varphi ) = r. \\Diamond s. \\Diamond _r\\Box x_0._r\\Diamond x_1.", "(_s x_0 \\wedge _s x_1) \\vee (_s \\Diamond x_0 \\wedge _s \\Diamond x_1)$ .", "Intuitively, this construction requires the existence of an initial state named $r$ , a successor state $s$ that represents the truth value $\\top $ , and one or more successor states of $s$ which together represent $\\bot $ .", "The quantifiers $\\exists ,\\forall $ are replaced by the modal operators $\\Diamond ,\\Box $ which range over $s$ and its successor states.", "Finally, positive literals are enforced to be true at $s$ , negative literals strictly after $s$ .", "For every model of $f(\\varphi )$ , it holds that $r$ is situated at the first state of the model and that state has a successor labelled by $s$ .", "By virtue of the function $h$ , positive literals have to be mapped to $s$ , whereas negative literals have to be mapped to some state other than $s$ .", "An easy induction on the structure of formulae shows that $\\varphi \\in $ iff $f(\\varphi ) \\in [](\\Diamond ,\\Box ,,)$ .", "We obtain -membership via a polynomial-time reduction from $[](\\Diamond ,\\Box ,,)$ to the satisfiability problem $[]_{}(<)$ for the fragment of first-order logic with the relation “$<$ ” interpreted over the natural numbers.", "Let the first order language contain all members of as variables and all members of as constants.", "Based on the standard translation from hybrid to first-order logic [21], we devise a reduction $H$ that maps hybrid formulae $\\varphi $ and variables or constants $z$ to first-order formulae.", "$\\begin{array}{@{}l@{\\hspace*{14.22636pt}}l@{}}H(p,z) := \\top \\text{ for $p\\in $}& H(v,z) := v=z ~~\\text{ for $v\\in \\cup $} \\\\[2px]H(\\alpha \\wedge \\beta ,z) := H(\\alpha ,z) \\wedge H(\\beta ,z)& H(\\alpha \\vee \\beta ,z) := H(\\alpha ,z) \\vee H(\\beta ,z) \\\\[2px]H(\\Diamond \\alpha ,z) := \\exists t (z<t \\wedge H(\\alpha ,t))& H(\\Box \\alpha ,z) := \\forall t (z<t \\rightarrow H(\\alpha ,t)) \\\\[2px]H(x.\\alpha ,z) := \\exists x (x=z \\wedge H(\\alpha ,z))& H(_x \\alpha ,z) := H(\\alpha ,x)\\end{array}$ In the $\\Diamond $ , $\\Box $ and $$ -cases we deviate from the usual definition of the standard translation because we do not insist on using only two variables in addition to —therefore it suffices to require that $t$ is a fresh variable—and we allow constants in the second argument.", "For a first-order formula $\\psi $ with variables in $$ and an assignment $g:\\rightarrow $ , let $\\psi [g]$ denote the first-order formula that is obtained from $\\psi $ by substituting every free occurrence of $x\\in $ by the first-order term that describes $g(x)$ .", "For every instance $\\varphi $ of $[](\\Diamond ,\\Box ,,)$ , every assignment $g:\\rightarrow $ and every $n\\in $ , it holds that:    $g,n \\models \\varphi $ if and only if $(,<)\\models H(\\varphi ,z)[g^{z}_n]$ , where $z$ is a new variable that does not occur in $\\varphi $ .", "We prove the claim inductively on the construction of $\\varphi $ .", "$\\varphi =v$ for $v\\in $ : Table: NO_CAPTIONJustifications for the equivalences: (1) is by the definition of $\\models $ for hybrid logic, (2) extends $g$ by the new variable $z$ , and (3) uses the definition of $\\models $ for first-order logic over $(,<)$ .", "$\\varphi =\\alpha \\wedge \\beta $ resp.", "$\\varphi =\\alpha \\vee \\beta $ : straightforward.", "$\\varphi =\\Diamond \\alpha $ : Table: NO_CAPTION(1) and (2) are by definition resp.", "by induction hypothesis.", "For (3), notice that the variable $t$ may appear free in $H(\\alpha ,t)$ but it does not appear free in $\\exists t (z<t \\wedge H(\\alpha , t))$ .", "The equivalence then follows by the semantics of the considered first-order logic.", "$\\varphi =\\Box \\alpha $ : Table: NO_CAPTION(1) and (2) are by definition resp.", "by induction hypothesis.", "The arguments for (3) are as in the case above.", "$\\varphi =x.", "\\alpha $ : Table: NO_CAPTION(1) and (2) are from the definition of $$ and from the induction hypothesis.", "Eventually, (3) follows from the semantics of FOL over $(,<)$ .", "$\\varphi =_x \\, \\alpha $ : Table: NO_CAPTION(1) and (2) are from the definition of $$ and from the induction hypothesis.", "Now, (3) follows from the semantics of FOL over $(,<)$ .", "Notice that $z$ does not appear free in $\\exists z (x=z \\wedge H(\\alpha ,z))$ .", "This proves Equivalence (4).", "This concludes the proof of the claim.", "Now, $\\varphi \\in [](\\Diamond ,\\Box ,,)$ if and only if $g,0\\models \\varphi \\vee \\Diamond \\varphi $ for some assignment $g$ .", "By the above claim, this is equivalent to $(,<)\\models H(\\varphi \\vee \\Diamond \\varphi ,z)[g^{z}_0]$ for some $g$ and a new variable $z$ , which can also be expressed as $(,<)\\models \\forall x (\\lnot (x<z) \\wedge H(\\varphi \\vee \\Diamond \\varphi ,z))$ .", "This shows that $[](\\Diamond ,\\Box ,,)$ is polynomial-time reducible to $[]_{}(<)$ , which was shown to be in in [8].", "Therefore, $[](\\Diamond ,\\Box ,,)$ is in .", "The easy cases: 1 and $Ł$ results In this section, we show that the fragments without the $\\Diamond $ -operator have an easy satisfiability problem.", "Our results can be structured into four groups.", "First, we consider fragments without modal operators.", "For these fragments we obtain 1-completeness.", "Simply said, without negation and $\\Diamond $ we cannot express that two nominals or state variables are not bound to the same state.", "Therefore, the model that binds all variables to the first state satisfies every satisfiable formula in this fragment.", "Let $F_0=(\\lbrace 0\\rbrace ,\\emptyset )$ and $g_0(y)=0$ for every $y \\in $ .", "Then $\\varphi \\in [](, )$ (resp.", "$\\varphi \\in [](, )$ ) if and only if $F_0,g_0,0\\models \\varphi $ .", "The implication direction from left to right follows from the monotonicity of the considered formulas.", "For the other direction, notice that $F_0\\in $ .", "For frame class , note that if $F_0,g_0,0\\models \\varphi $ and $\\varphi $ has no modal operators, then $g_0,0\\models \\varphi $ .", "Let $O \\subseteq \\lbrace , \\rbrace $ .", "Then $[](O)$ and $[](O)$ are 1-complete.", "1-hardness of $[{F}](\\emptyset )$ follows immediately from the 1-completeness of the Formula Value Problem for propositional formulae [6].", "It remains to show that $[](, )$ and $[](, )$ are in 1.", "In order to decide whether $\\varphi $ is in $[](, )$ , according to Lemma  it suffices to check whether the propositional formula obtained from $\\varphi $ deleting all occurrences of $x.$ and $_x$ , is satisfied by the assignment that sets all atoms to true.", "According to [6] this can be done in 1.", "Since $[](, )=[](, )$ by Lemma , we obtain the same for $[](, )$ .", "Second, we consider fragments with the $\\Box $ -operator over linear frames.", "We can show 1-completeness here, too.", "The main reason is that (sub-)formulas that begin with a $\\Box $ are satisfied in a state that has no successor.", "Therefore similar as above, every formula of this fragment that is satisfiable over linear frames is satisfied by a model with only one state.", "$[](\\Box , , )$ is 1-complete.", "1-hardness follows from Theorem .", "It remains to show that $[](\\Box , , )\\in 1$ .", "We show that essentially the $\\Box $ -operators can be ignored.", "$[](\\Box , , ) [](, )$ .", "For an instance $\\varphi $ of $[](\\Box , , )$ , let $\\varphi ^{\\prime \\prime }$ be the formula obtained from $\\varphi $ by replacing every subformula $\\Box \\psi $ of $\\varphi $ with the constant $\\top $ .", "Then $\\varphi ^{\\prime \\prime }$ is an instance of $[](, )$ .", "If $\\varphi \\in [](\\Box , , )$ , then $\\varphi ^{\\prime \\prime }\\in [](, )$ due to the monotonicity of $\\varphi $ .", "On the other hand, if $\\varphi ^{\\prime \\prime }\\in [](, )$ , then $K_0,g,0\\models \\varphi ^{\\prime \\prime }$ (Lemma ).", "Since $K_0,g,0\\models \\Box \\alpha $ for every $\\alpha $ , we obtain $K_0,g,0\\models \\varphi $ , hence $\\varphi \\in [](\\Box , , )$ .", "As such simple substitutions can be realized using an 0-circuit, the stated reduction is indeed a valid $$ -reduction from $[](\\Box ,,)$ to $[](,)$ .", "Since $[](, )\\in 1$ (Theorem ) and 1 is closed downwards under $$ , it follows from the Claim that $[](\\Box , , )\\in 1$ .", "It is clear that this argument does not apply to the natural numbers.", "Third, we show 1-completeness for the fragments with $\\Box $ and one of $$ and $$ over $$ .", "They receive separate treatment because, in , every state has a successor, and therefore $\\Box $ -subformulas cannot be satisfied as easily as above.", "It turns out that the complexity of the satisfiability problem increases only if both hybrid operators can be used.", "$[](\\Box ,)$ is 1-complete.", "Proof sketch.", "1-hardness follows from Theorem .", "For the upper bound, we distinguish occurrences of nominals that are either free, or that are bound by a $\\Box $ , or that are bound by an $$ .", "Simply said, a free occurrence of $i$ in $\\alpha $ is bound by $\\Box $ in $\\Box \\alpha $ and bound by $$ in $_x \\alpha $ (even if $x\\ne i$ ).", "Since the assignment $g$ is not relevant for the considered fragment, we write $K,w\\models \\alpha $ for short instead of $K,g,w\\models \\alpha $ .", "Let $\\alpha ^{\\prime }$ be the formula obtained from $\\alpha $ by replacing every occurrence of a nominal that is bound by $\\Box $ with $\\bot $ , and let $\\eta $ be a valuation.", "If $\\eta ,k\\models \\alpha $ , then $\\eta ,k\\models \\alpha ^{\\prime }$ .", "Moreover, it turns out that binding every nominal to the initial state suffices to obtain a satisfying model.", "$\\varphi \\in [](\\Box ,)$ if and only if $\\eta _0,0\\models \\varphi $ with $\\eta _0(x)=\\lbrace 0\\rbrace $ for every $x \\in $ .", "Both claims together yield that, in order to decide $\\varphi \\in [](\\Box ,)$ , it suffices to check whether $\\eta _0,0\\models \\varphi ^{\\prime }$ .", "No nominal in $\\varphi ^{\\prime }$ occurs bound by a $\\Box $ -operator.", "Therefore for every subformula $\\Box \\alpha $ of $\\varphi ^{\\prime }$ and for every $k$ holds: $\\eta _0,k\\models \\alpha $ if and only if $\\eta _0,0\\models \\alpha $ .", "All nominals that occur free or bound by an $$ evaluate to true in state 0 via $\\eta _0$ .", "Therefore, in order to decide $\\eta _0,0\\models \\varphi ^{\\prime }$ , it suffices to ignore all $\\Box $ and $$ -operators of $\\varphi ^{\\prime }$ and evaluate it as a propositional formula under assignment $\\eta _0$ that sets all atoms of $\\varphi ^{\\prime }$ to true.", "This can be done in 1 [6].", "The complete proof can be found in Appendix .", "Next, we consider $[](\\Box ,)$ .", "According to our remarks in Section about notational convenience, we assume that there are no nominals in $(\\Box ,)$ .", "$[](\\Box ,)$ is 1-complete.", "Proof sketch.", "Now, we distinguish occurrences of state variables as the occurrences in the proof sketch above.", "They are either free, or they are bound by a $\\Box $ , or they are bound by $$ .", "Note that this phrasing differs from the standard usage of the terms `free' and `bound' in the context of state variables.", "A free occurrence of $i$ in $\\alpha $ is bound by $\\Box $ in $\\Box \\alpha $ , as above.", "It is bound by $$ in $i .", "\\alpha $ only.", "Notice that $y$ occurs free in $x .", "y$ (for $x\\ne y$ ).", "Let $\\alpha ^{\\prime }$ be the formula obtained from $\\alpha $ by replacing every occurrence of a state variable that is bound by $\\Box $ with $\\bot $ , and let $g$ be an assignment.", "If $g,k\\models \\alpha $ , then $g,k\\models \\alpha ^{\\prime }$ .", "$\\varphi \\in [](\\Box ,)$ if and only if $g_0,0\\models \\varphi $ , for $g_0(x)=0$ for every $x \\in $ .", "Both claims together yield that, in order to decide $\\varphi \\in [](\\Box ,)$ , it suffices to check whether $g_0,0\\models \\varphi ^{\\prime }$ .", "No state variable in $\\varphi ^{\\prime }$ occurs bound by a $\\Box $ -operator.", "Therefore for every subformula $\\Box \\alpha $ of $\\varphi ^{\\prime }$ and for every $k$ holds: $g_0,k\\models \\alpha $ if and only if $g_0,0\\models \\alpha $ .", "All occurrences of state variables in $\\varphi ^{\\prime }$ that are bound by $$ evaluate to true, because no $\\Box $ occurs “between” the binding $i$ and the occurrence of $i$ , which means that the state where the variable is bound is the same as where the variable is used.", "All free occurrences of state variables evaluate to true in state 0 due to $g_0$ .", "Therefore, in order to decide $g_0,0\\models \\varphi ^{\\prime }$ , it suffices to ignore all $\\Box $ and $$ -operators of $\\varphi ^{\\prime }$ and evaluate it as a propositional formula under an assignment that sets all atoms to true.", "This can be done in 1 [6].", "The complete proof can be found in Appendix .", "The fourth part deals with the fragment with $\\Box $ and both $$ and $$ over the natural numbers.", "$[](\\Box , , )$ is $Ł$ -hard.", "This proof is very similar to the proof of Theorem 3.3. in [14].", "We give a reduction from the problem Order between Vertices () which is known to be $Ł$ -complete [7] and defined as follows.", "$$ A finite set of vertices $V$ , a successor-relation $S$ on $V$ , and two vertices $s,t\\in V$ .", "Is $s\\leqslant _S t$ , where $\\leqslant _S$ denotes the unique total order induced by $S$ on $V$ ?", "Notice that $(V,S)$ is a directed line-graph.", "Let $(V,S,s,t)$ be an instance of $$ .", "We construct an $(\\Box ,,)$ -formula $\\varphi $ that is satisfiable if and only if $s\\leqslant _S t$ .", "We use $V=\\lbrace v_0,v_1,\\ldots ,v_n\\rbrace $ as state variables.", "The formula $\\varphi $ consists of three parts.", "The first part binds all variables except $s$ to one state and the variable $s$ to a successor of this state.", "The second part of $\\varphi $ binds a state variable $v_l$ to the state labeled by $s$ iff $s\\leqslant _S v_l$ .", "Let $\\alpha $ denote the concatenation of all $_{v_{k}}v_{l}$ with $(v_k,v_l)\\in S$ and $v_l\\ne s$ , and $\\alpha ^n$ denotes the $n$ -fold concatenation of $\\alpha $ .", "Essentially, $\\alpha ^n$ uses the assignment to collect eventually all $v_i$ with $s\\leqslant _S v_i$ in the state labeled $s$ .", "The last part of $\\varphi $ checks whether $s$ and $t$ are bound to the same state after this procedure.", "That is, $\\varphi = v_0.v_1.v_2.", "\\cdots v_n.", "\\Box s. ~ \\alpha ^{n} ~ _s t.$ To prove the correctness of our reduction, we show that $\\varphi $ is satisfiable if and only if $s\\leqslant _S t$ .", "Assume $s\\leqslant _S t$ .", "For an arbitrary assignment $g$ , one can show inductively that $g,0\\models v_0.v_1.", "\\cdots v_n.", "\\Box s. ~ \\alpha ^i ~ _s r$ for $i=0,1,\\ldots ,n$ and for all $r$ that have distance $i$ from $s$ .", "Therefore it eventually holds that $g,0\\models \\varphi $ .", "For $s\\lnot \\leqslant _S t$ we show that $g,n\\lnot \\models \\varphi $ for any assignment $g$ and natural number $n$ .", "Let $g_0$ be the assignment obtained from $g$ after the bindings in the prefix $v_0.v_1.", "\\cdots v_n.", "\\Box s$ of $\\varphi $ , and let $g_i$ be the assignment obtained from $g_0$ after evaluating the prefix of $\\varphi $ up to and including $\\alpha ^i$ .", "It holds that $g_i(s)\\ne g_i(t)=0$ for all $i=0,1,\\ldots ,n$ .", "This leads to $g_{n},0\\lnot \\models _s t$ and therefore $g,0\\lnot \\models \\varphi $ .", "For the upper bound, we establish a characterisation of the satisfaction relation that assigns a unique assignment and state of evaluation to every subformula of a given formula $\\varphi $ .", "Using this new characterisation, we devise a decision procedure that runs in logarithmic space and consists of two steps: it replaces every occurrence of any state variable $x$ in $\\varphi $ with 1 if its state of evaluation agrees with that of its $x$ -superformula, and with 0 otherwise; it then removes all $\\Box $ -, $$ - and $$ -operators from the formula and tests whether the resulting Boolean formula is valid.", "$[](\\Box ,,)$ is in .", "The proof can be found in Appendix .", "The intermediate cases: $$ results After we have seen that all fragments without $\\Diamond $ have an easy satisfiability problem, we show that $\\Diamond $ together with the use of nominals makes the satisfiability problem $$ -hard.", "Recall that, owing to the presence of nominals, $(\\Diamond )$ is not just modal logic with the $\\Diamond $ -operator.", "The absence of $$ makes assignments superfluous: we write $K,w \\models \\varphi $ instead of $K,g,w \\models \\varphi $ .", "$[](\\Diamond )$ and $[](\\Diamond )$ both are $$ -hard.", "Proof.", "We reduce from $$ .", "Let $\\varphi =c_1 \\wedge \\ldots \\wedge c_n$ be an instance of $$ with clauses $c_1, \\dots , c_n$ (where $c_i=(l_1^i\\vee l_2^i\\vee l_3^i)$ for literals $l^i_j$ ) and variables $x_1, \\dots , x_m$ .", "We define the transformation as $f\\colon \\varphi \\mapsto \\Diamond (i_0 \\wedge \\Diamond i_1) ~\\wedge ~\\Bigg (\\bigwedge _{\\ell =1}^m \\Diamond (i_0 \\wedge x_{\\ell }) \\vee \\Diamond (i_1 \\wedge x_{\\ell })\\Bigg ) \\wedge h(\\varphi ),$ where $i_0,i_1$ and all $x_{\\ell }$ are nominals, and the function $h$ is defined as follows: let $l_k^j$ be a literal in clause $c_j$ , then $h(l_k^j)& := {\\left\\lbrace \\begin{array}{ll}(i_1 \\wedge x), \\text{ if } l_k^j=x \\\\(i_0 \\wedge x), \\text{ if } l_k^j=\\lnot x\\end{array}\\right.}", "\\\\[2px]h(c_j) & := \\Diamond ( h(l_1^j) \\vee h(l_2^j) \\vee h(l_3^j)),\\quad \\text{where } c_j = (l_1^j \\vee l_2^j \\vee l_3^j); \\\\[4px]h(c_1 \\wedge \\dots \\wedge c_n)& := h(c_1) \\wedge \\dots \\wedge h(c_n).$ Notice that $f$ turns variables in the $$ instance into nominals in the $[](\\Diamond )$ instance.", "The part $\\Diamond (i_0 \\wedge \\Diamond i_1)$ enforces the existence of two successors $w_1$ and $w_2$ of the state satisfying $f(\\varphi )$ .", "The part $\\bigwedge _{\\ell =1}^m \\Diamond (i_0 \\wedge x_{\\ell }) \\vee \\Diamond (i_1 \\wedge x_{\\ell })$ simulates the assignment of the variables in $\\varphi $ , enforcing that each $x_{\\ell }$ is true in either $w_1$ or $w_2$ .", "The part $h(\\varphi )$ then simulates the evaluation of $\\varphi $ on the assignment determined by the previous parts.", "With the following claim $$ -hardness of $[](\\Diamond )$ follows.", "$\\varphi \\in $ if and only if $h(\\varphi ) \\in [](\\Diamond )$ .", "We first show that $h(\\varphi ) \\in [](\\Diamond )$ implies $\\varphi \\in $ .", "If $K,w_0 \\models h(\\varphi )$ with $K=(W,<,\\eta )$ , then the following holds.", "Let $w_1=\\eta (i_0)$ , $w_2=\\eta (i_1)$ , and $\\lbrace w_0,w_1,w_2\\rbrace \\subseteq W$ with $w_0,w_1,w_2$ pairwise different; $w_0<w_1<w_2$ ; for all $x_j$ with $1 \\leqslant j \\leqslant m$ : $\\eta (x_j) \\subseteq \\lbrace w_1,w_2\\rbrace $ .", "We build a propositional logic assignment $\\beta = (\\beta _1 \\dots \\beta _m)$ that satisfies $\\varphi $ , where $\\beta _i \\in \\lbrace \\bot ,\\top \\rbrace $ is the truth value for $x_i$ , as follows.", "$\\beta _j=\\bot $ if $g(i_0) = g(x_j)$ , and $\\beta _j=\\top $ if $g(i_1) = g(x_j)$ .", "From the construction of $h(\\varphi )$ , it clearly follows that $\\beta $ satisfies $\\varphi $ .", "For the converse direction, suppose that $\\varphi $ is satisfied by the propositional logic assignment $\\beta = (\\beta _1 \\dots \\beta _m)$ .", "We construct a linear model $K:=(W,<,\\eta )$ containing a state $w$ such that $K,w \\models h(\\varphi )$ .", "$W &:= \\lbrace w,w_0,w_1\\rbrace \\\\< &:~~~ w<w_0<w_1 \\\\$ $\\eta (i_j) &:= w_j \\text{ for } j \\in \\lbrace 0,1\\rbrace \\\\\\eta (x_j) &:= {\\left\\lbrace \\begin{array}{ll}w_0, \\text{ if } \\beta _j=\\bot \\\\w_1, \\text{ if } \\beta _j=\\top \\end{array}\\right.", "}$ It follows from the construction of $K$ that $K,w \\models h(\\varphi )$ .", "The conjunct $h(\\varphi )$ is of the form $(h(l_1^1) \\vee h(l_1^2) \\vee h(l_1^3)) \\wedge \\dots \\wedge (h(l_n^1) \\vee h(l_n^2) \\vee h(l_n^3)).$ Hence, under $\\beta $ , at least one literal in every clause evaluates to true.", "The variable in this literal satisfies the same clause in $h(\\varphi )$ .", "Hence every clause in $h(\\varphi )$ is satisfied in $w$ in $K$ .", "Therefore, $K,w \\models h(\\varphi )$ .", "Using this claim, $$ -hardness of $[](\\Diamond )$ follows.", "It is straightforward to show that $$ reduces to $[](\\Diamond )$ using the same reduction.", "We will now establish -membership of the problems $[{F}](\\Diamond ,\\Box ,)$ , $[{F}](\\Diamond ,\\Box ,)$ , and $[{F}](\\Diamond ,,)$ for ${F} \\in \\lbrace ,\\rbrace $ .", "For the first two, this follows from the literature, see Theorem REF .", "For the third, we observe that all modal and hybrid operators in a formula $\\varphi $ from the fragment $(\\Diamond ,,)$ are translatable into FOL by the standard translation using no universal quantifiers.", "The existential quantifiers introduced by the binder can be skolemised away, which corresponds to removing all binding from $\\varphi $ and replacing each state variable with a fresh nominal.", "The correctness of this translation is proven in [21].", "Hence, $[{F}](\\Diamond ,,)$ polynomial-time reduces to $[{F}](\\Diamond ,)$ .", "$[](\\Diamond ,,)$ and $[](\\Diamond ,,)$ are in $$ .", "From the lower bounds in Lemma and the upper bounds in Theorem REF and Lemma , we obtain the following theorem.", "Let $\\lbrace \\Diamond \\rbrace \\subseteq O$ , and $O \\subsetneq \\lbrace \\Diamond , \\Box , , \\rbrace $ .", "Then $[](O)$ and $[](O)$ are $\\emph {-complete}$ .", "In addition to the NP-membership of the fragments captured by Theorem , we are interested in their model-theoretic properties.", "We show that these logics enjoy a kind of linear-size model property, precisely a quasi-quadratic size model property: over the natural numbers, every satisfiable formula has a model where two successive nominal states have at most linearly many intermediary states, and the states behind the last such state are indistinguishable.", "This property allows for an alternative worst-case decision procedure for satisfiability that consists of guessing a linear representation of a model of the described form and symbolically model-checking the input formula on that model.", "Over general linear frames, which may have dense intervals, we formulate the model property in a more general way and prove it using additional technical machinery to deal with density.", "However, the result then carries over to the rationals, where we are not aware of any upper complexity bound in the literature.", "In [19], Sistla and Clarke showed a variation of the linear-size model property for LTL(F), which corresponds to $(\\Diamond ,\\Box )$ over : whenever $\\varphi \\in (\\Diamond ,\\Box )$ is satisfiable over , then it is satisfiable in the initial state of a model over which has a linear-sized prefix init and a remainder final such that final is maximal with respect to the property that every type (set of all atomic propositions true in a state) occurs infinitely often, and final contains only linearly many types.", "Such a structure can be guessed in polynomial time, represented in polynomial space and model-checked in polynomial time.", "While it is straightforward to extend Sistla and Clarke's proof to cover nominals and the operator, it will not go through if density is allowed (frame class $$ ).", "We establish that $(\\Diamond ,\\Box ,)$ over $$ has a quadratic size model property, and we subsequently show how to extend the result to the other fragments from Theorem and how to restrict them to .", "$(\\Diamond ,\\Box ,)$ has the quasi-quadratic size model property with respect to $$ and $$ .", "The proof can be found in Appendix .", "As an immediate consequence, the model property in Theorem carries over to the subfragments $(\\Diamond ,\\Box )$ , $(\\Diamond ,)$ , $(\\Box ,)$ , $(\\Diamond )$ , $(\\Box )$ , $()$ , and $(\\emptyset )$ .", "Moreover, our arguments in the proofs of Theorems  and can be used to transfer it to $(\\Box ,,)$ .", "Together with the observations that $(\\Diamond ,,)$ is no more expressive than $(\\Diamond ,)$ (see the explanation before Lemma ), and $(\\Diamond ,\\Box ,)$ is no more expressive than $(\\Diamond ,\\Box )$ (because, without $$ , one cannot jump to named states), we obtain the following generalisation of Theorem .", "Let $O \\subsetneq \\lbrace \\Diamond ,\\Box ,,\\rbrace $ .", "Then $(O)$ has the quasi-quadratic size model property with respect to $$ and $$ .", "Conclusion We have completely classified the complexity of all fragments of hybrid logic with monotone Boolean operators obtained from arbitrary combinations of four modal and hybrid operators, over linear frames and the natural numbers.", "Except for the largest such fragment over linear frames, all fragments are of elementary complexity.", "We have classified their complexity into -complete, -complete and tractable and shown that the tractable cases are complete for either 1 or .", "Surprisingly, while the largest fragment is harder over linear frames than over , the largest $\\Diamond $ -free fragment is easier over linear frames than over .", "The question remains whether the -complete largest fragment over admits some quasi-polynomial size model property.", "Furthermore, this study can be extended in several possible ways: by allowing negation on atomic propositions, by considering frame classes that consist only of dense frames, such as , or by considering arbitrary sets of Boolean operators in the same spirit as in [14].", "For atomic negation, it follows quite easily that the largest fragment is of non-elementary complexity over , too, and that all fragments except $O = (\\Box ,,)$ are -complete.", "However, our proof of the quasi-quadratic model property does not immediately go through in the presence of atomic propositions.", "Over , we conjecture that all fragments, except possibly for the largest one, have the same complexity and model properties as over .", "Appendix Proof of Theorem Theorem $[](\\Box ,)$ is 1-complete.", "1-hardness follows from Theorem .", "For the upper bound, we distinguish occurrences of nominals that are either free, or that are bound by a $\\Box $ , or that are bound by an $$ .", "Simply said, a free occurrence of $i$ in $\\alpha $ is bound by $\\Box $ in $\\Box \\alpha $ and bound by $$ in $_x \\alpha $ (even if $x\\ne i$ ).", "Since the assignment $g$ is not relevant for the considered fragment, we write $K,w\\models \\alpha $ for short instead of $K,g,w\\models \\alpha $ .", "Let $\\alpha ^{\\prime }$ be the formula obtained from $\\alpha $ by replacing every occurrence of a nominal that is bound by $\\Box $ with $\\bot $ , and let $\\eta $ be a valuation.", "If $\\eta ,k\\models \\alpha $ , then $\\eta ,k\\models \\alpha ^{\\prime }$ .", "We use induction on the construction of $\\varphi $ .", "The base case for $\\varphi \\in \\cup $ is straightforward, as is the inductive step for $\\varphi =\\alpha \\vee \\beta $ and $\\varphi =\\alpha \\wedge \\beta $ , and even for $\\varphi =_x\\alpha $ .", "It remains to consider the case $\\varphi =\\Box \\alpha $ .", "If $\\eta ,k\\models \\Box \\alpha $ , then for all $k^{\\prime }>k$ : $\\eta ,k^{\\prime }\\models \\alpha $ (by semantics of $\\Box $ ) and by inductive hypothesis follows for all $k^{\\prime }>k$ : $\\eta ,k^{\\prime }\\models \\alpha ^{\\prime }$ .", "Assume that in $\\Box (\\alpha ^{\\prime })$ there occurs a nominal $i$ that is bound by the initial $\\Box $ -operator.", "Since for all $k>k^{\\prime }$ holds $\\eta ,k^{\\prime }\\models \\alpha ^{\\prime }$ , there is some $\\ell >\\max \\bigcup _{j\\in } \\eta (j)$ with $\\eta ,\\ell \\models \\alpha ^{\\prime }$ .", "Therefore $\\eta ,\\ell \\models \\alpha ^{\\prime }[i/\\bot ]$ , and by the monotonicity of $\\alpha ^{\\prime }$ and the properties of $\\eta $ it follows that for all $k^{\\prime }>k$ holds $\\eta ,k^{\\prime }\\models \\alpha ^{\\prime }[i/\\bot ]$ .", "In this way, all nominals bound by the initial $\\Box $ -operator can be replaced by $\\bot $ , and it follows that $\\eta ,k\\models (\\Box (\\alpha ^{\\prime }))^{\\prime }$ .", "Since $(\\Box (\\alpha ^{\\prime }))^{\\prime }=(\\Box \\alpha )^{\\prime }$ , the claim follows.", "$\\varphi \\in [](\\Box ,)$ if and only if $\\eta _0,0\\models \\varphi $ with $\\eta _0(x)=\\lbrace 0\\rbrace $ for every $x \\in $ .", "We use induction on the construction of $\\varphi $ .", "The base case for $\\varphi \\in \\cup $ is straightforward, as is the inductive step for $\\varphi =\\alpha \\vee \\beta $ and $\\varphi =\\alpha \\wedge \\beta $ , and even for $\\varphi =_x\\alpha $ .", "It remains to consider the case $\\varphi =\\Box \\alpha $ .", "If $\\eta _0,0\\models \\varphi $ , then $\\varphi \\in [](\\Box ,)$ .", "If $\\Box \\alpha \\in [](\\Box ,)$ , then there exists $k$ such that $\\eta ,k\\models (\\Box \\alpha )^{\\prime }$ (for some $\\eta $ , by the claim above).", "Let $\\alpha ^{\\ast }$ be the formula with $(\\Box \\alpha )^{\\prime }=\\Box (\\alpha ^{\\ast })$ .", "By the semantics of $\\Box $ we obtain that there exists $k$ such that for all $k^{\\prime }>k$ holds $\\eta ,k^{\\prime }\\models \\alpha ^{\\ast }$ .", "By inductive hypothesis follows $\\exists k \\forall k^{\\prime }>k: \\eta _0,0\\models \\alpha ^{\\ast }$ , what is equivalent to $\\eta _0,0\\models \\alpha ^{\\ast }$ .", "Notice that $\\alpha ^{\\ast }$ contains no nominal.", "By the monotonicity of $\\alpha $ , it follows that for all $k\\in $ holds $\\eta _0,k\\models \\alpha ^{\\ast }$ .", "When we re-replace the $\\bot $ 's by the replaced nominals, the satisfaction is kept because of the monotonicity of $\\alpha $ , and therefore for all $k\\in $ holds $\\eta _0,k\\models \\alpha $ .", "This implies $\\eta _0,0\\models \\Box \\alpha $ , which eventually yields $\\varphi \\in [](\\Box ,)$ .", "Both claims together yield that, in order to decide $\\varphi \\in [](\\Box ,)$ , it suffices to check whether $\\eta _0,0\\models \\varphi ^{\\prime }$ .", "No nominal in $\\varphi ^{\\prime }$ occurs bound by a $\\Box $ -operator.", "Therefore for every subformula $\\Box \\alpha $ of $\\varphi ^{\\prime }$ and for every $k$ holds: $\\eta _0,k\\models \\alpha $ if and only if $\\eta _0,0\\models \\alpha $ .", "All nominals that occur free or bound by an $$ evaluate to true in state 0 via $\\eta _0$ .", "Therefore, in order to decide $\\eta _0,0\\models \\varphi ^{\\prime }$ , it suffices to ignore all $\\Box $ and $$ -operators of $\\varphi ^{\\prime }$ and evaluate it as a propositional formula under assignment $\\eta _0$ that sets all atoms of $\\varphi ^{\\prime }$ to true.", "This can be done in 1 [6].", "Proof of Theorem Theorem $[](\\Box ,)$ is 1-complete.", "1-hardness follows from Theorem .", "For the upper bound, we distinguish occurrences of state variables as the occurrences in the proof sketch above.", "They are either free, or they are bound by a $\\Box $ , or they are bound by $$ .", "Note that this phrasing differs from the standard usage of the terms `free' and `bound' in the context of state variables.", "A free occurrence of $i$ in $\\alpha $ is bound by $\\Box $ in $\\Box \\alpha $ , as above.", "It is bound by $$ in $i .", "\\alpha $ only.", "Notice that $y$ occurs free in $x .", "y$ (for $x\\ne y$ ).", "Let $\\alpha ^{\\prime }$ be the formula obtained from $\\alpha $ by replacing every occurrence of a state variable that is bound by $\\Box $ with $\\bot $ , and let $g$ be an assignment.", "If $g,k\\models \\alpha $ , then $g,k\\models \\alpha ^{\\prime }$ .", "We use induction on the construction of $\\varphi $ .", "The base case for $\\varphi \\in $ is straightforward, as is the inductive step for $\\varphi =\\alpha \\vee \\beta $ , $\\varphi =\\alpha \\wedge \\beta $ , and for $\\varphi =x.\\alpha $ .", "It remains to consider the case $\\varphi =\\Box \\alpha $ .", "Let $g,k\\models \\Box \\alpha $ for $k \\in $ .", "Then for all $k^{\\prime }>k$ : $g,k^{\\prime }\\models \\alpha $ (by semantics of $\\Box $ ) and by inductive hypothesis follows for all $k^{\\prime }>k$ : $g,k^{\\prime }\\models \\alpha ^{\\prime }$ .", "Assume that in $\\Box (\\alpha ^{\\prime })$ there occurs a state variable $i$ that is bound by the initial $\\Box $ -operator.", "Since for all $k^{\\prime }>k$ holds $g,k^{\\prime }\\models \\alpha ^{\\prime }$ , there is some $\\ell >\\max \\bigcup _{x\\in }g(x)$ such that $g,\\ell \\models \\alpha ^{\\prime }$ .", "Therefore $g,\\ell \\models \\alpha ^{\\prime }[i/\\bot ]$ , and by the monotonicity of $\\alpha ^{\\prime }$ it follows that for all $k^{\\prime }>k$ holds $g,k^{\\prime }\\models \\alpha ^{\\prime }[i/\\bot ]$ .", "In this way, all state variables bound by the initial $\\Box $ -operator can be replaced by $\\bot $ , and it follows that $g,k\\models (\\Box (\\alpha ^{\\prime }))^{\\prime }$ , where $(\\Box \\alpha ^{\\prime })^{\\prime }=(\\Box \\alpha )^{\\prime }$ .", "$\\varphi \\in [](\\Box ,)$ if and only if $g_0,0\\models \\varphi $ , for $g_0(x)=0$ for every $x \\in $ .", "We use induction on the construction of $\\varphi $ .", "The base case for $\\varphi \\in $ is straightforward, as is the inductive step for $\\varphi =\\alpha \\vee \\beta $ , $\\varphi =\\alpha \\wedge \\beta $ , and for $\\varphi =x.\\alpha $ .", "It remains to consider the case $\\varphi =\\Box \\alpha $ .", "If $\\Box \\alpha \\in [](\\Box ,)$ , then there exists $k$ such that $g,k\\models (\\Box \\alpha )^{\\prime }$ (for some $\\eta $ and $g$ ).", "Let $\\alpha ^{\\ast }$ be the formula with $(\\Box \\alpha )^{\\prime }=\\Box \\alpha ^{\\ast }$ .", "By the semantics of $\\Box $ we obtain that there exists $k$ such that for all $k^{\\prime }>k$ holds $g_0,k^{\\prime }\\models \\alpha ^{\\ast }$ , and therefore $\\alpha ^{\\ast }\\in [](\\Box ,)$ .", "By inductive hypothesis follows $g_0,0\\models \\alpha ^{\\ast }$ .", "Notice that $\\alpha ^{\\ast }$ contains no free state variable.", "Therefore for all $k\\in $ holds $g_0,k\\models \\alpha ^{\\ast }$ .", "When we re-replace the $\\bot $ 's by the replaced state variables, the satisfaction is kept, and therefore for all $k\\in $ holds $g_0,k\\models \\alpha $ , which eventually implies $g_0,0\\models \\Box \\alpha $ , i.e.", "$g_0,0\\models \\varphi $ .", "Both claims together yield that in order to decide $\\varphi \\in [](\\Box ,)$ , it suffices to check whether $g_0,0\\models \\varphi ^{\\prime }$ .", "No state variable in $\\varphi ^{\\prime }$ occurs bound by a $\\Box $ -operator.", "Therefore for every subformula $\\Box \\alpha $ of $\\varphi ^{\\prime }$ and for every $k$ holds: $g_0,k\\models \\alpha $ if and only if $g_0,0\\models \\alpha $ .", "All occurrences of state variables in $\\varphi ^{\\prime }$ that are bound by $$ evaluate to true, because no $\\Box $ occurs “between” the binding $i$ and the occurrence of $i$ , which means that the state where the variable is bound is the same as where the variable is used.", "All free occurrences of state variables evaluate to true in state 0 due to $g_0$ .", "Therefore, in order to decide $g_0,0\\models \\varphi ^{\\prime }$ , it suffices to ignore all $\\Box $ and $$ -operators of $\\varphi ^{\\prime }$ and evaluate it as a propositional formula under an assignment that sets all atoms to true.", "This can be done in 1 [6].", "Proof of Theorem Theorem $[](\\Box ,,)$ is in .", "For this upper bound, we will establish a characterisation of the satisfaction relation that assigns a unique assignment and state of evaluation to every subformula of a given formula $\\varphi $ .", "Using this new characterisation, we will devise a decision procedure that runs in logarithmic space and consists of two steps: it replaces every occurrence of any state variable $x$ in $\\varphi $ with 1 if its state of evaluation agrees with that of its $x$ -superformula, and with 0 otherwise; it then removes all $\\Box $ -, $$ - and $$ -operators from the formula and tests whether the resulting Boolean formula is valid.", "In what follows, we want to restrict assignments to the finitely many free state variables occurring free in a given formula $\\varphi $ .", "For this purpose, we define the notion of a partial assignment $g : V \\rightarrow $ for $\\varphi $ where $V$ is a finite set of state variables with $_\\varphi \\subseteq V$ , i.e., $g$ is defined for all state variables free in $\\varphi $ .", "Here we include subscripts of the $$ -operator in the notion of a free state variable: for example, $x._x_yz$ has free state variables $y,z$ .", "The satisfaction relation $\\models $ for partial assignments is analogously defined to the definition in Section .", "For a partial assignment $g$ for $x.\\alpha $ and $i \\in $ , it holds that $g,i \\models x.\\alpha $ iff $g^x_i,i \\models \\alpha $ .", "Clearly, if $g$ is a partial assignment for $x.\\alpha $ , then $g^x_i$ is one for $\\alpha $ .", "The definition of the satisfaction relation implies that the satisfaction of $\\Box \\alpha $ at $g,i$ depends on the satisfaction of $\\Box \\alpha $ at infinitely many states (natural numbers) in $g$ .", "However, we will now show that the latter can be reduced to satisfaction in the smallest natural number to which $g$ does not bind any state variable.", "This will later imply that satisfiability of a given formula $\\varphi $ can be tested by evaluating its subformulas in their uniquely determined states $g,i$ of evaluation.", "Given a partial assignment $g : V \\rightarrow \\mathbb {N}$ , define $ n_g = \\max \\lbrace g(x) \\mid x \\in V\\rbrace + 1.", "$ For every $\\varphi \\in (\\Box ,,)$ , every partial assignment $g$ for $\\varphi $ and every $i \\in \\mathbb {N}$ , it holds that $g,i \\models \\Box \\varphi $ iff $g,n_g \\models \\varphi $ .", "We will prove this lemma later, using the following lemma.", "Let $\\varphi \\in (\\Box ,,)$ , let $i,j \\in \\mathbb {N}$ , and let $g,h$ be partial assignments for $\\varphi $ that satisfy the following two conditions: $g^{-1}(i) \\subseteq h^{-1}(j)$ .", "(All state variables free in $\\varphi $ and bound to $i$ by $g$ are bound to $j$ by $h$ .)", "For all $a,b \\in _\\varphi $ : if $g(a) = g(b)$ , then $h(a) = h(b)$ .", "(Whenever $g$ binds two state variables free in $\\varphi $ to one and the same state, so does $h$ .)", "Then $g,i \\models \\varphi $ implies $h,j \\models \\varphi $ .", "We proceed by induction on $\\varphi $ .", "In the base case $\\varphi \\in $ , we obtain the desired implication directly from (REF ).", "For the induction step, we distinguish between the possible cases for the outermost operator of $\\varphi $ .", "The Boolean cases are straightforward; the other cases are dealt with as follows.", "In case $\\varphi = \\Box \\psi $ , the following chain of (bi-)implications holds.", "$g,i \\models \\Box \\psi & ~\\Leftrightarrow ~ \\forall i^{\\prime } > i : g, i^{\\prime } \\models \\psi \\\\& ~\\Rightarrow ~ g,n_g \\models \\psi \\\\& ~\\Rightarrow ~ h,n_h \\models \\psi \\\\& ~\\Rightarrow ~ \\forall j^{\\prime } \\in \\mathbb {N} : h, j^{\\prime } \\models \\psi \\\\& ~\\Rightarrow ~ \\forall j^{\\prime } > j : h, j^{\\prime } \\models \\psi \\\\& ~\\Leftrightarrow ~ h,j \\models \\Box \\psi $ The first “$\\Rightarrow $ ” is immediate in case $i < n_g$ .", "Otherwise, if $i \\geqslant n_g$  , observe that $g^{-1}(\\mbox{i+1}) = \\emptyset = g^{-1}(n_g)$ .", "Hence we can apply the induction hypothesis (IH) to $\\psi ,\\mbox{i+1},n_g,g,h$ because $g$ is also a partial assignment for $\\psi $ , the assumption (REF ) of the IH is satisfied, and (REF ) follows from the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "The second “$\\Rightarrow $ ” is due to the IH applied to $\\psi ,n_g,n_h,g,h$ .", "Its assumption (REF ) is satisfied because $g^{-1}(n_g) = \\emptyset = h^{-1}(n_h)$ , and (REF ) follows from the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "The third “$\\Rightarrow $ ” is due to the IH applied to $\\psi ,n_h,j,h,h$ .", "Its assumption (REF ) is satisfied because $h^{-1}(n_h) = \\emptyset = h^{-1}(j)$ , and (REF ) is obvious because $h=h$ .", "In case $\\varphi = x.\\psi $ , the following chain of (bi-)implications holds.", "$g,i \\models x.\\psi & ~\\Leftrightarrow ~ g^x_i,i \\models \\psi \\\\& ~\\Rightarrow ~ h^x_j,j \\models \\psi \\\\& ~\\Leftrightarrow ~ h,j \\models x.\\psi $ The implication in the middle is obtained by observing that $g^x_i,h^x_i$ are partial assignments for $\\psi $ because $g,h$ are partial assignments for $\\varphi $ , and applying the IH to $\\psi ,i,j,g^x_i,h^x_j$ .", "Its assumption (REF ) is satisfied because of the following chain of equalities and inclusions, whose middle step follows from the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "${(g^x_i)}^{-1}(i) = g^{-1}(i) \\cup \\lbrace x\\rbrace \\subseteq h^{-1}(i) \\cup \\lbrace x\\rbrace = {(h^x_j)}^{-1}(j)$ Assumption (REF ) of the IH is satisfied for the following reason.", "Let $a,b \\in _\\psi $ with $g(a) = g(b)$ .", "In case $a=b=x$ , both $(g^x_i)(a) = (g^x_i)(b)$ and $(h^x_j)(a) = (h^x_j)(b)$ hold.", "In case $a=x$ and $b\\ne x$ , we have that $(g^x_i)(a) = (g^x_i)(b)$ implies $(g^x_i)(b) = i$ , which implies $g(b) = i$ because $b\\ne x$ .", "This implies $h(b) = j$ due to the assumption (REF ) for $\\varphi ,i,j,g,h$ and because $b \\in _\\varphi $ .", "Hence $(h^x_j)(a) = (h^x_j)(b)$ .", "The case $a\\ne x$ and $b=x$ is analogous to the previous one, and in case $a\\ne x$ and $b\\ne x$ , we have that $(g^x_i)(a) = (g^x_i)(b)$ implies $g(a) = g(b)$ , which implies $h(a) = h(b)$ due to the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "Hence $(h^x_j)(a) = (h^x_j)(b)$ .", "In case $\\varphi = _x.\\psi $ , the following chain of (bi-)implications holds.", "$g,i \\models _x\\psi & ~\\Leftrightarrow ~ g,g(x) \\models \\psi \\\\& ~\\Rightarrow ~ h,h(x) \\models \\psi \\\\& ~\\Leftrightarrow ~ h,j \\models _x\\psi $ The implication in the middle is obtained by observing that $g,h$ are also partial assignments for $\\psi $ , and applying the IH to $\\psi ,g(x),h(x),g,h$ .", "Its assumption (REF ) is satisfied: consider $y \\in g^{-1}(g(x))$ .", "Then $g(x) = g(y)$ , which implies $h(x) = h(y)$ due to the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "Hence $y \\in h^{-1}(h(x))$ .", "This establishes $g^{-1}(g(x)) \\subseteq h^{-1}(h(x))$ .", "The assumption (REF ) for the IH follows from the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "Before we can prove Lemma , we observe the following consequence of Lemma .", "For every $\\varphi \\in (\\Box ,,)$ , every partial assignment $g$ for $\\varphi $ and every $i \\in \\mathbb {N}$ with $g^{-1}(i) = \\emptyset $ , it holds that $g,i \\models \\varphi $ implies $g,j \\models \\varphi $ for all $j \\in \\mathbb {N}$ .", "It suffices to observe that the assumptions of Lemma are satisfied by $\\varphi ,i,j,g,g$ with $j \\in \\mathbb {N}$ arbitrary.", "(REF ) follows from $g^{-1}(i) = \\emptyset $ , and (REF ) holds trivially because $g=g$ .", "We can now proceed to prove Lemma   ($\\forall \\varphi ,g,i ~:~ g,i \\models \\Box \\varphi \\Leftrightarrow g,n_g \\models \\varphi $ ).", "[Proof of Lemma ] For the direction “$\\Rightarrow $ ”, assume that $g,i \\models \\Box \\varphi $ , i.e., for all $j > i$ , it holds that $g,j \\models \\varphi $ .", "In case $i < n_g$ , the consequence $g,n_g \\models \\varphi $ is immediate.", "Otherwise, in case $i \\geqslant n_g$ , we conclude $g,i+1 \\models \\varphi $ from $g,i \\models \\Box \\varphi $ .", "Since $g^{-1}(i+1) = \\emptyset $ in this case, we can use Corollary to conclude that $g,j \\models \\varphi $ for all $j \\in \\mathbb {N}$ , and in particular for $j = n_g$ .", "For the direction “$\\Leftarrow $ ”, assume that $g,n_g \\models \\varphi $ .", "Then Corollary implies that $g,j \\models \\varphi $ for all $j \\in \\mathbb {N}$ , and in particular for all $j > i$ .", "Hence $g,i \\models \\Box \\varphi $ .", "Using Lemma , we are now in a position to show that every satisfiable formula is satisfied by a canonical assignment $g_0^\\varphi $ in the state 0.", "We will furthermore use the characterisation of satisfaction for $\\Box $ -formulas in Lemma to establish that the question whether $g_0^\\varphi ,0 \\models \\varphi $ can be reduced to checking satisfaction of $\\varphi $ 's subformulas in uniquely determined states and assignments.", "Let $\\varphi \\in (\\Box ,,)$ .", "The canonical assignment $g_0^\\varphi $ for $\\varphi $ is the partial assignment for $\\varphi $ that maps all $x \\in _\\varphi $ to 0 and is undefined for all other state variables.", "Let $\\varphi \\in (\\Box ,,)$ .", "Then $\\varphi \\in [](\\Box ,,)$ iff $g_0^\\varphi , 0 \\models \\varphi $ .", "The “if” direction is obvious.", "The converse is a consequence of the following claim.", "For every $\\varphi \\in (\\Box ,,)$ , every partial assignment $g$ for $\\varphi $ and every $i \\in \\mathbb {N}$ : if $g,i \\models \\varphi $ , then $g_0^\\varphi , 0 \\models \\varphi $ .", "We proceed by induction on $\\varphi $ .", "The base case $\\varphi = x \\in $ is true because $g_0^x,0 \\models x$ holds.", "For the induction step, the Boolean cases are straightforward.", "The other cases are treated as follows.", "In case $\\varphi = \\Box \\psi $ , the following chain of implications holds.", "$g,i \\models \\Box \\psi & ~\\Rightarrow ~ g,n_g \\models \\psi \\\\& ~\\Rightarrow ~ g_0^\\psi ,1 \\models \\psi \\\\& ~\\Rightarrow ~ g_0^\\varphi ,1 \\models \\psi \\\\& ~\\Rightarrow ~ g_0^\\varphi ,0 \\models \\Box \\psi $ The first implication is due to Lemma , and the second uses Lemma for $\\psi ,g,g_0^\\psi ,n_g,1$ : remember that $g,g_0^\\psi $ are for $\\psi $ , and observe that the assumptions of Lemma are satisfied because $g^{-1}(n_g) = \\emptyset = {(g_0^\\psi )}^{-1}(1)$ and $g_0^\\psi (a) = 0 = g_0^\\psi (b)$ for all $a,b \\in _\\psi $ .", "The third implication holds because $g^\\psi _0 = g^\\varphi _0$ , and the fourth uses Lemma .", "In case $\\varphi = x.\\psi $ , the following chain of implications holds.", "$g,i \\models x.\\psi & ~\\Leftrightarrow ~ g^x_i,i \\models \\psi \\\\& ~\\Rightarrow ~ g^\\psi _0,0 \\models \\psi \\\\& ~\\Rightarrow ~ {(g^\\varphi _0)}^x_0,0 \\models \\psi \\\\& ~\\Leftrightarrow ~ g^\\varphi _0,0 \\models x.\\psi $ The first “$\\Rightarrow $ ” is due to the induction hypothesis, and the second uses $g^\\psi _0 = {(g^\\varphi _0)}^x_0$ .", "In case $\\varphi = _x\\psi $ , the following chain of implications holds.", "$g,i \\models _x\\psi & ~\\Leftrightarrow ~ g,g(x) \\models \\psi \\\\& ~\\Rightarrow ~ g^\\psi _0,0 \\models \\psi \\\\& ~\\Rightarrow ~ {(g^\\psi _0)}^x_0,0 \\models _x\\psi \\\\& ~\\Leftrightarrow ~ g^\\varphi _0,0 \\models _x\\psi $ The first “$\\Rightarrow $ ” is due to the induction hypothesis, and the second uses $g^\\psi _0 = {(g^\\varphi _0)}^x_0$ ; note that $g^\\psi _0 = g^\\varphi _0$ does not necessarily hold because $x$ might not be free in $\\psi $ .", "Using Theorem and Lemma , we can now assign a unique assignment and state of evaluation to every subformula of a given formula $\\varphi $ .", "This will lead us to characterize satisfiability of a given formula $\\varphi $ by validity of the Boolean formula obtained from $\\varphi $ by (a) replacing every free state variable $x$ with 0 or 1, depending on the compatibility between unique assignment and state of evaluation for $x$ , and (b) removing all non-Boolean operators.", "After establishing this criterion, we will show that the transformation can be achieved deterministically in logarithmic space.", "Fix a formula $\\varphi \\in (\\Box ,,)$ whose satisfiability is to be tested.", "We denote subformulas of $\\varphi $ as pairs $(\\psi ,p)$ , where $p \\in \\mathbb {N}$ denotes the position of $\\psi $ in (the string that represents) $\\varphi $ .", "This is necessary to distinguish between different occurrences of the same subformula in $\\varphi $ .", "The position of a subformula is always the position of its first character in the string representing $\\varphi $ .", "If the subformula is $(\\alpha \\wedge \\beta )$ or $(\\alpha \\vee \\beta )$ , then the position of the opening parenthesis is relevant.", "Consequently, $\\varphi $ has always position 0.", "For a position $p$ in $\\varphi $ , denote by $_1(p)$ and $_2(p)$ the position of the immediate subformulas of the subformula at position $p$ : if the subformula of $\\varphi $ at $p$ is $(\\alpha \\vee \\beta )$ or $(\\alpha \\wedge \\beta )$ , then $_1(p)$ and $_2(p)$ are the positions of $\\alpha $ and $\\beta $ , respectively; $\\Box \\alpha $ , $x.\\alpha $ or $_x\\alpha $ , then $_1(p)$ is the position of $\\alpha $ , and $_2(p)$ is undefined; is any other formula, then both $_1(p)$ and $_2(p)$ are undefined.", "We now define a unique state of evaluation $^\\varphi (\\psi ,p)$ for a subformula $\\psi $ of $\\varphi $ at position $p$ recursively on $p$ as follows.", "$^\\varphi (\\varphi ,0) = (g_0^\\varphi ,0)$ .", "For $\\circ \\in \\lbrace \\wedge ,\\vee \\rbrace $ , if $^\\varphi ((\\alpha \\circ \\beta ), p) = (g,i)$ , then $^\\varphi (\\alpha , _1(p)) = ^\\varphi (\\beta , _2(p)) = (g,i)$ .", "If $^\\varphi (\\Box \\alpha ,p) = (g,i)$ , then $^\\varphi (\\alpha ,_1(p)) = (g,n_g)$ .", "If $^\\varphi (x.\\alpha ,p) = (g,i)$ , then $^\\varphi (\\alpha ,_1(p)) = (g^x_i,i)$ .", "If $^\\varphi (_x\\alpha ,p) = (g,i)$ , then $^\\varphi (\\alpha ,_1(p)) = (g,g(x))$ .", "Observe that the first component in $^\\varphi (\\psi ,p)$ is always a partial assignment for $\\psi $ .", "Now consider a subformula $(x,p)$ of $\\varphi $ with $x \\in $ and $^\\varphi (x,p) = (g,i)$ .", "We define a function $^\\varphi $ mapping $x$ to $\\top $ if $g(x) = i$ (i.e., $x$ is satisfied at $^\\varphi (x,p)$ ), and to $\\bot $ otherwise.", "Using $^\\varphi $ , we now recursively define a function $^\\varphi $ mapping subformulas of $\\varphi $ to Boolean formulas with only monotone operators and without propositional variables: $^\\varphi (x,p) & = ^\\varphi (x,p),\\quad x \\in \\\\^\\varphi (c,p) & = c, \\quad c \\in \\lbrace \\top ,\\bot \\rbrace \\\\^\\varphi (\\alpha \\circ \\beta ,p) & = ^\\varphi (\\alpha , _1(p)) \\circ ^\\varphi (\\beta , _2(p)),\\quad \\circ \\in \\lbrace \\wedge ,\\vee \\rbrace \\\\^\\varphi (\\Delta \\alpha ,p) & = ^\\varphi (\\alpha , _1(p)),\\quad \\Delta \\in \\lbrace \\Box ,x,_x\\rbrace $ Furthermore, let $(\\varphi ) = ^\\varphi (\\varphi ,0)$ .", "Let $\\varphi \\in (\\Box ,,)$ .", "For all subformulas $(\\psi ,p)$ of $\\varphi $ , it holds that $^\\varphi (\\psi ,p) \\models \\psi $ iff $^\\varphi (\\psi ,p)$ is valid.", "We proceed by induction on $\\psi $ .", "Let $^\\varphi (\\psi ,p) = (g,i)$ .", "The base case $\\psi =x$ follows from the definition of $^\\varphi (x,p)$ and $^\\varphi (x,p)$ .", "For the inductive step, the cases $\\psi =\\top ,\\bot $ follow from the definition of $^\\varphi $ .", "The other cases are as follows.", "In case $\\psi = \\alpha \\vee \\beta $ , we observe the following chain of equivalent statements.", "$g,i\\models \\alpha \\vee \\beta & ~\\Leftrightarrow ~ g,i \\models \\alpha \\text{~or~} g,i \\models \\beta \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha ,_1(p)) \\models \\alpha \\text{~or~}^\\varphi (\\beta ,_1(p)) \\models \\beta \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha ,_1(p)) \\text{~is valid or~}^\\varphi (\\beta ,_1(p)) \\text{~is valid} \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha ,_1(p)) \\vee ^\\varphi (\\beta ,_1(p)) \\text{~is valid} \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha \\vee \\beta ,p) \\text{~is valid}$ The second equivalence is due to the definition of $^\\varphi $ , the third uses the induction hypothesis, and the fifth is due to the definition of $^\\varphi $ .", "The case $\\psi = \\alpha \\wedge \\beta $ is analogous.", "In case $\\psi = \\Box \\alpha $ , we observe the following chain of equivalent statements.", "$g,i\\models \\Box \\alpha & ~\\Leftrightarrow ~ g,n_g \\models \\alpha \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha ,_1(p)) \\models \\alpha \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha ,_1(p)) \\text{~is valid} \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\Box \\alpha ,p) \\text{~is valid}$ The first equivalence uses Lemma , the second is due to the definition of $^\\varphi $ , the third uses the induction hypothesis, and the fourth is due to the definition of $^\\varphi $ .", "The cases $\\psi = x.\\alpha $ and $\\psi = _x\\alpha $ are analogous to the previous one, but with the first equivalence via the definition of satisfaction.", "Let $\\varphi \\in (\\Box ,,)$ .", "Then $\\varphi \\in [](\\Box ,,)$ iff $(\\varphi )$ is valid.", "The following chain of equivalences holds.", "$\\varphi \\text{~is satisfiable}& ~\\Leftrightarrow ~ g^\\varphi _0,0 \\models \\varphi \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\varphi ,0) \\models \\varphi \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\varphi ,0) \\text{~is valid} \\\\& ~\\Leftrightarrow ~ (\\varphi ,0) \\text{~is valid}$ The first equivalence follows from Theorem , the second uses the definition of $^\\varphi $ , the third is due to Lemma , and the fourth uses the defintion of $$ .", "The function $$ is a reduction of $[](\\Box ,,)$ to the formula value problem for Boolean formulas with only monotone operators, which is in 1 [6].", "The correctness of this reduction is shown in Theorem .", "To establish that $[](\\Box ,,) \\in $ , it remains to show that $(\\varphi )$ can be computed in logarithmic space.", "The procedure BOOL, which will accomplish this task, will traverse its input formula $\\varphi $ from left to right, and send the character $c$ read at position $p$ to the output unchanged, unless one of the following two cases occurs.", "If $c$ belongs to a $\\Box $ -, $x.$ -, or $_x$ -operator, then $c$ is ignored.", "If $c$ is a free state variable $x$ , then $^\\varphi (x,p)$ is computed and sent to the output instead of $c$ .", "Given the definition of $$ , $^\\varphi $ and $^\\varphi $ , this is obviously a correct decision procedure provided that $^\\varphi (x,p)$ is computed by a correct subroutine REP, which we still have to describe.", "The procedure BOOL is given in Algorithm .", "Procedure BOOL $\\varphi \\in (\\Box ,,)$ output $(\\varphi )$ $p \\leftarrow 0$ $p < |\\varphi |$ an operator $\\Box $ , $x.$ or $_x$ starts at position $p$ $p \\leftarrow \\text{position immediately following that operator}$ a state variable $x$ starts at position $p$ output ${REP}(\\varphi , x, p)$ $p \\leftarrow \\text{position immediately following~} x$ output character at position $p$ $p \\leftarrow p+1$ To compute $^\\varphi (x,p)$ using the procedure REP, we make the following crucial observation about states of evaluation.", "The operators $\\Box $ and $_x$ are jumping operators: $^\\varphi (\\Box \\psi ,\\cdot )$ and $^\\varphi (\\psi ,\\cdot )$ may differ in their second component; the same holds for $^\\varphi (_x \\psi ,\\cdot )$ and $^\\varphi (\\psi ,\\cdot )$ .", "Such a difference does not occur between formulas starting with one of the other operators $x.$ , $\\wedge $ , $\\vee $ , and their direct subformulas.", "This observation can be used to compute $^\\varphi (x,p)$ because that value depends on the question whether there is a jumping operator between the position $q$ where $x$ is bound and the position $p$ of $x$ .", "Assume that this binder $x.$ leads the subformula $x.\\psi $ , and that $^\\varphi (x.\\psi ,q) = (g,i)$ and $^\\varphi (x,p) = (h,j)$ .", "We distinguish the following cases.", "Case 1.", "If there is no jumping operator between $(x,p)$ and $(x.\\psi ,q)$ , then it follows from the definition of $^\\varphi $ that $g(x) = i$ , $g(x) = h(x)$ , and $i=j$ – all three statements can be shown inductively on the positions in $\\varphi $ .", "They imply that $h(x) = j$ , hence $^\\varphi (x,p) = \\top $ .", "Case 2.", "Let $\\circ $ be the last jumping operator occurring between positions $q$ and $p$ .", "More precisely, let $r$ be the position between $q$ and $p$ such that the operator $\\circ $ at position $r$ is a jumping operator, that operator is in the scope of $(x.,q)$ and has $(x,p)$ in its scope, and there is no jumping operator in the scope of $(\\circ ,r)$ that has $(x,p)$ in its scope.", "Let $\\circ \\vartheta $ be the subformula at position $r$ .", "Case 2.1.", "If $\\circ = \\Box $ , then the definition of $^\\varphi $ implies that $^\\varphi (\\Box \\vartheta ,r) = (g, n_g)$ for some partial assignment $g$ .", "Since $x$ is not bound between $r$ and $p$ , and since no jumping operator occurs between $r$ and $p$ , we conclude from the definition of $^\\varphi $ that $h(x) \\ne n_g$ and $j = n_g$ .", "Hence $h(x) \\ne j$ , and $^\\varphi (x,p) = \\bot $ .", "Case 2.2.", "If $\\circ = _y$ , then let $(y.\\eta , s)$ be the subformula “above” $_y\\vartheta $ that binds $y$ , with $^\\varphi (y.\\eta ) = (g^{\\prime },i^{\\prime })$ and $^\\varphi (_y.\\vartheta ) = (h^{\\prime },j^{\\prime })$ .", "Then it holds that (a) $g(x) = h(x)$ , due to the definition of $^\\varphi $ and because $x$ is not bound between $q$ and $p$ , and (b) $j=h(y)=h^{\\prime }(y)=g^{\\prime }(y)$ , which follows from the definition of $^\\varphi $ for $_y$ -formulas and the fact that $y$ is not bound between $s$ and $p$ .", "Therefore we have that $^\\varphi (x,p) = \\top $ iff $g(x) = g^{\\prime }(x)$ .", "This new criterion compares states of evaluations of subformulas at smaller positions in $\\varphi $ , and it can be decided applying the same case distinction to those two subformulas.", "We therefore obtain a recursive procedure REP for deciding whether $^\\varphi = \\top $ .", "For every recursive call according to Case 2.2, a pair of subformulas at smaller positions in $\\varphi $ is compared.", "Therefore, the recursion has to terminate after at most $|\\varphi |$ steps.", "Since the result of a recursive call does not need to be processed any further, REP can be implemented using end-recursion, i.e., without a stack.", "Together with the fact that only a constant number of position counters are needed (and, consequently, determining the last jumping operator between two positions in $\\varphi $ can be implemented in logarithmic space), Algorithm REF runs in logarithmic space.", "The previous considerations imply its correctness.", "Figure: Procedure REP' Theorem $[](\\Box ,,)$ is in .", "Proof of Theorem Theorem $(\\Diamond ,\\Box ,)$ has the quasi-quadratic size model property with respect to $$ and $$ .", "We will develop a “quasi-quadratic size model property” for the logic $(\\Diamond ,\\Box ,)$ over $$ , and we will subsequently show how to extend the result to the other fragments from Theorem and how to restrict them to .", "In the appendix, we even sketch how to obtain an NP decision procedure for these fragments over $$ , $$ and the frame class $\\lbrace (,<)\\rbrace $ .", "Consider an arbitrary model $K = (W,<,\\eta )$ , and call all states in the range of $g$ nominal states.", "For every non-nominal state $w \\in W$ , let $\\delta (w)$ be the number of states between $w$ and the next nominal state $s$ .", "If the next nominal state is a direct successor, then $\\delta (w) = 0$ ; if there are infinitely many intermediary states—i.e., at least a part of the interval between $w$ and $s$ is dense—, then $\\delta (w) = \\infty $ .", "For every $m \\geqslant 0$ , we now define an equivalence relation $\\equiv _m$ on $W$ as follows.", "$w \\equiv _m w^{\\prime }$ if either $w=w^{\\prime }$ or both $w,w^{\\prime }$ are non-nominal states and $\\delta (w) > m$ and $\\delta (w^{\\prime }) > m$ .", "Figure REF gives an example for $m=3$ ; equivalence classes are denoted by dashed rectangles.", "The $i_j$ are nominal states, and of the 8 states between $i_2$ and $i_3$ , the rightmost three form separate equivalence classes, and the others form a single equivalence class.", "Figure: An example for m=3m=3The intuition behind this equivalence relation is that $w$ and $w^{\\prime }$ cannot be distinguished by formulas of modal depth $\\leqslant m$ .", "If $w \\equiv _m w^{\\prime }$ , we call $w$ and $w^{\\prime }$ $m$ -inseparable, and we denote the equivalence class of $w$ w.r.t.", "$\\equiv _m$ by $[w]_m$ .", "The definition of $\\equiv _m$ has the consequence $[w]_m \\subseteq [w]_{m-1}$ , for all $m > 0$ .", "It is possible to enforce dense parts in satisfying models, for instance via the following formula, which is satisfiable in a linear structure only if that structure ends with a state satisfying the nominal $j$ , and that state needs to be the end point of a dense interval.", "This formula is therefore not satisfiable over .", "$\\varphi _d = i \\wedge \\Diamond \\Diamond j \\wedge \\Box (j \\vee \\Diamond \\Diamond j)$ For this reason, an equivalence class can also consist of infinitely many states.", "In the case of a model satisfying $\\varphi _d$ , all points between $i$ and $j$ belong to the same equivalence class because all these points have an infinite distance to $j$ .", "The following lemma states that $m$ -inseparable states cannot be distinguished by formulas of modal depth $\\leqslant m$ .", "For every $m \\geqslant 0$ , every formula $\\varphi \\in (\\Diamond ,\\Box ,)$ with $(\\varphi ) \\leqslant m$ , every linear model $K = (W,<,\\eta )$ , and all $w,w^{\\prime } \\in W$ with $w \\equiv _m w^{\\prime }$ : $K,w \\models \\varphi \\quad \\Leftrightarrow \\quad K,w^{\\prime } \\models \\varphi .$ We proceed by induction on the structure of $\\varphi $ .", "The case for nominals is obvious because nominal states are $m$ -inseparable only from themselves.", "The Boolean cases are straightforward.", "$\\varphi = \\Diamond \\psi $ .", "For symmetry reasons, it suffices to show “$\\Rightarrow $ ”.", "Let $K,w \\models \\Diamond \\psi $ and $w \\equiv _m w^{\\prime }$ .", "Then there is some $v>w$ with $K,v \\models \\psi $ .", "We now distinguish several cases of how $w,w^{\\prime },v$ are located in relation to each other.", "$w^{\\prime }<w$ .", "Then $w<v$ implies $w^{\\prime }<v$ , and hence $K,w^{\\prime } \\models \\Diamond \\psi $ .", "$w\\leqslant w^{\\prime } < v$ .", "Then, still, $w^{\\prime }<v$ , and hence $K,w^{\\prime } \\models \\Diamond \\psi $ .", "$w<v\\leqslant w^{\\prime }$ .", "Since $w \\equiv _m w^{\\prime }$ , we have $w \\equiv _m v \\equiv _m w^{\\prime }$ .", "In case $|[w]_m| < \\infty $ , there are exactly $m$ states between $[w]_m$ and the next nominal state.", "Let $v^{\\prime }$ be the $<$ -least of them; then $w \\equiv _{m-1} v \\equiv _{m-1} w^{\\prime }\\equiv _{m-1} v^{\\prime }$ .", "Since $(\\psi ) = m-1$ , we get $K,v^{\\prime } \\models \\psi $ via the induction hypothesis.", "Hence, $K,w^{\\prime } \\models \\Diamond \\psi $ .", "In case $|[w]_m| = \\infty $ , we conclude that at least a subinterval of $[w]_m$ is dense, and therefore $w^{\\prime }$ has a successor $v^{\\prime }$ in $[w]_m \\subseteq [w]_{m-1}$ .", "We can continue the argument as in the previous case.", "$\\varphi = \\Box \\psi $ .", "As above, it suffices to show “$\\Rightarrow $ ”.", "Let $K,w \\models \\Box \\psi $ and $w \\equiv _m w^{\\prime }$ .", "Then, for all $v>w$ , we have that $K,v \\models \\psi $ .", "Again, we consider the two cases $|[w]_m| < \\infty $ and $|[w]_m| = \\infty $ , and fix the same $v^{\\prime }$ as above.", "Since $v^{\\prime }$ is $(m-1)$ -inseparable from $w$ and $w^{\\prime }$ , $\\psi $ is also satisfied by all states in $[w]_m$ .", "Therefore, $K,v \\models \\psi $ for all $v > w^{\\prime }$ , hence, $K,w^{\\prime } \\models \\Box \\psi $ .", "$\\varphi = _i\\psi $ .", "Then $K,w \\models _i\\psi ~\\Leftrightarrow ~ K,v \\models \\psi \\text{~for any~} v ~\\Leftrightarrow ~ K,w^{\\prime } \\models _i\\psi $ .", "We now use this inseparability result to reduce a satisfying model in size such that it can be represented in polynomial space.", "Fix a formula $\\varphi $ with $(\\varphi ) = m$ and a linear model $K$ with $K,w \\models \\varphi $ for some state $w$ .", "If it were not possible to enforce dense intervals, it would suffice to collapse every $m$ -equivalence class of $K$ to a single point, i.e., the quotient model of $K$ w.r.t.", "$\\equiv _m$ would satisfy $\\varphi $ at $[w]_m$ .", "This would serve our purpose over .", "In contrast, an infinite equivalence class (IEC)—which has to contain a dense subinterval—needs to remain dense for the next lemma to work.", "For a uniform representation, we replace any IEC with a copy of $(0,1)_$ , the open interval of all rationals between 0 and 1.", "Since a dense interval can be of higher cardinality than $(0,1)_$ —just consider , for example—, we cannot expect to map every point of an IEC M to a point in the associated copy of $(0,1)_$ .", "Instead, we use a surjective partial morphism $f: (M,<) \\rightarrow (0,1)_$ , i.e., a partial function that satisfies the equivalence $x=y \\Leftrightarrow f(x) = f(y)$ for all $x,y \\in M$ and whose range is all of .", "These conditions ensure that every $x \\in (f)$ has a successor $y \\in (f)$ with $f(x) < f(y)$ .", "Such a function always exists: since every IEC $[w]_m$ contains a dense subinterval, it also contains an isomorphic copy of $(0,1)_$ .", "The refined “quotient” model $K_m = (W_m, <_m, \\eta _m)$ is now constructed as follows.", "For every infinite $[w]_m$ , let ${[w]_m}$ be a fresh copy of $(0,1)_$ .", "We set $\\displaystyle W_k = \\biguplus _{|[w]_m| = \\infty } {[w]_m} \\quad \\uplus \\lbrace [w]_m : |[w]_m| < \\infty \\rbrace $ $[w]_m <_m [v]_m$ if $[w]_m$ and $[v]_m$ are finite and $w^{\\prime } < v^{\\prime }$ for some $w^{\\prime } \\in [w]_m$ and $v^{\\prime } \\in [v]_m$ $q <_m q^{\\prime }$ if $q,q^{\\prime } \\in {[w]_m}$ for some $w$ with $|[w]_m| = \\infty $ , and $q<q^{\\prime }$ on $(0,1)_$ $q <_m [v]_m$ if $q \\in {[w]_m}$ for some $w$ with $|[w]_m| = \\infty $ , $[v]_m$ is finite, and $w < v^{\\prime }$ for some $v^{\\prime } \\in [v]_m$ $[w]_m <_m q^{\\prime }$ if $q^{\\prime } \\in {[v]_m}$ for some $v$ with $|[v]_m| = \\infty $ , $[w]_m$ is finite, and $w^{\\prime } < v$ for some $w^{\\prime } \\in [w]_m$ $\\eta _m(i) = [\\eta (i)]_m$ We also define a model reduction function for $K$ to be a surjective partial function $f : K \\rightarrow K_m$ with the following conditions.", "If $|[w]_m| < \\infty $ , then $f(w^{\\prime }) = [w]_m$ for all $w^{\\prime } \\in [w]_m$ .", "If $|[w]_m| = \\infty $ , then $f(w^{\\prime }) = g(w^{\\prime })$ for all $w^{\\prime } \\in [w]_m$ , for some surjective partial morphism $g : [w]_m \\rightarrow {[w]_m}$ .", "For every $m \\geqslant 0$ , every formula $\\varphi \\in (\\Diamond ,\\Box ,)$ with $(\\varphi ) \\leqslant m$ , every linear model $K = (W,<,\\eta )$ , every model reduction function $f$ for $K$ and all $w \\in (f)$ : $K,w \\models \\varphi \\quad \\Leftrightarrow \\quad K_m,f(w) \\models \\varphi .$ We proceed by induction on $\\varphi $ .", "The atomic and Boolean cases are straightforward again.", "$\\varphi = \\Diamond \\psi $ .", "Let $K,w \\models \\varphi $ .", "Case 1: $|[w]_m| < \\infty $ .", "Let $w^{\\prime }$ be the $<$ -greatest member of $[w]_k$ .", "Due to Lemma , $K,w^{\\prime } \\models \\Diamond \\psi $ .", "Therefore there is some $v > w^{\\prime }$ with $K,v \\models \\psi $ and $v \\lnot \\equiv _m w$ .", "If $|[v]_m| < \\infty $ , then $v \\in (f)$ , and the induction hypothesis yields $K_m,f(v) \\models \\psi $ .", "Since $w<v$ with $v \\lnot \\equiv _m w$ , we obtain $f(w)<_m f(v)$ , hence $K_m,f(w) \\models \\psi $ .", "If $|[v]_m| = \\infty $ , we use $[v]_m \\subseteq [v]_{m-1}$ and conclude from Lemma that $K,v^{\\prime } \\models \\psi $ for all $v^{\\prime } \\in [v]_m$ .", "Take such a $v^{\\prime }$ with $v^{\\prime } \\in (f)$ and apply the induction hypothesis as in the case $|[v]_m| < \\infty $ .", "Case 2: $|[w]_m| = \\infty $ .", "Since $K,w \\models \\varphi $ , there is some $v > w$ with $K,v \\models \\psi $ .", "If $v \\lnot \\equiv _m w$ , then we argue as in Case 1.", "Otherwise, we use Lemma to conclude that $K,v^{\\prime } \\models \\psi $ for all $v^{\\prime } \\in [w]_m$ .", "Since the restriction of $f$ to $[w]_m$ is a surjective morphism and $(0,1)_$ is dense, there is some $v^{\\prime } > w$ with $v^{\\prime } \\in [w]_m$ , $v^{\\prime } \\in (f)$ and $f(w) <_m f(v^{\\prime })$ .", "From $K,v^{\\prime } \\models \\psi $ we conclude via the induction hypothesis that $K_m,f(v^{\\prime }) \\models \\psi $ , hence $K_m,f(w) \\models \\Diamond \\psi $ .", "$\\varphi = \\Box \\psi $ .", "Let $K,w \\models \\Box \\psi $ , i.e., $K,v \\models \\psi $ for all $v > w$ .", "Then $K,v \\models \\psi $ for all $v$ with $v \\in (f)$ and $f(v) >_m f(w)$ .", "Due to the induction hypothesis, $K_m,f(v) \\models \\psi $ for all $v$ with $v \\in (f)$ and $f(v) >_m f(w)$ .", "Since $f$ is surjective, we have $K_m,v^{\\prime } \\models \\psi $ for all $v^{\\prime } \\in W_m$ with $v^{\\prime } >_m f(w)$ .", "Hence $K_m,f(w) \\models \\Box \\psi $ .", "$\\varphi = _i\\psi $ .", "Let $K,w \\models _i\\psi $ , i.e., $K,\\eta (i) \\models \\psi $ .", "Then $K_m,\\eta _m(i) \\models \\psi $ due to the induction hypothesis and the definition of $K_m$ .", "Hence $K_m,f(w) \\models _i\\psi $ .", "At this point, it is important to notice that, if $K$ is a model over , then so is $K_m$ .", "Therefore, Lemma gives us a quasi-quadratic size model property for $(\\Diamond ,\\Box ,)$ over $$ as well as – and also over $\\lbrace (,<)\\rbrace $ , see appendix.", "We say that a model $K$ is of size quasi-quadratic in an integer $m$ if every interval between two consecutive nominal states in $K$ consists of at most $m$ states, possibly with one preceding isomorphic copy of $(0,1)_$ .", "We furthermore say that a fragment $(O)$ has the quasi-quadratic size model property with respect to a frame class F if, for every $\\varphi \\in [{F}](O)$ , there exists a model over a frame in F that is of size quasi-quadratic in $(\\varphi )$ and satisfies $\\varphi $ .", "Theorem $(\\Diamond ,\\Box ,)$ has the quasi-quadratic size model property with respect to $$ and $$ .", "Let $K=(W,<,\\eta )$ be a linear model and $w_0 \\in W$ with $K,w_0 \\models \\varphi $ .", "Consider $\\varphi ^{\\prime } = i \\wedge \\varphi $ for a fresh nominal $i$ .", "Let $m = (\\varphi ) = (\\varphi ^{\\prime })$ .", "Then $\\varphi ^{\\prime }$ is satisfiable in the $w_0$ of the model $K^{\\prime }$ obtained from $K$ by interpreting $i$ in $w_0$ .", "Now take an arbitrary model reduction function $f$ for $K^{\\prime }$ , which has to have $w_0$ in its domain, and apply Lemma to obtain $K_m,f(w_0) \\models \\varphi ^{\\prime }$ ." ], [ "The hard cases: Non-elementary and $$ results", "The hardest cases are those with the complete set of operators.", "In the non-monotone case, both satisfiability problems are non-elementary and decidable [16].", "We show that in the monotone case even this hardness is reached, but only on linear frames, i.e.", "$[](\\Diamond ,\\Box ,,)$ is non-elementary and decidable.", "In contrast, on the natural numbers the complexity decreases, i.e.", "we show that $[](\\Diamond ,\\Box ,,)$ is -complete.", "Our proofs use reductions to and from fragments of first-order logic on the natural numbers.", "Let $(<,P)$ be the set of all first-order formulae that use $<$ as the unique binary relation symbol, and $P$ as the unique unary relation symbol.I.e.", "$(<,P)$ is defined as set of all formulae $\\varphi $ as follows.", "$\\varphi \\top \\mid x < y \\mid P(x) \\mid \\lnot \\varphi \\mid \\varphi \\wedge \\varphi \\mid \\varphi \\vee \\varphi \\mid \\exists x \\, \\varphi \\mid \\forall x \\, \\varphi $ for variable symbols $x,y\\in $ .", "Let $[]_{}(<,P)$ denote the set of formulae from $(<,P)$ which are satisfied by a model that has as its universe, interprets $<$ as the less-than relation on $\\times $ , and has an arbitrary interpretation for the predicate symbol $P$ .", "It was shown by Stockmeyer [20] that $[]_{}(<,P)$ is non-elementary.", "Let $(<)$ be the fragment of $(<,P)$ in which the predicate symbol $P$ is not used.", "Accordingly, $[]_{}(<)$ denotes the set of formulae that are satisfiable over $$ and the natural interpretation of $<$ .", "It was shown by Ferrante and Rackoff [8] that $[]_{}(<)$ is in .", "Notice that in both fragments $x=y$ can be expressed as $\\lnot (x<y\\, \\vee \\, y<x)$ .", "Moreover, every $n\\in $ can be expressed by $x_n$ in the formula $\\exists x_0 \\cdots \\exists x_{n-1} [(\\bigwedge _{i=0,1,\\ldots ,n-1} x_i < x_{i+1}) \\wedge \\forall y (x_n<y \\vee \\bigvee _{i=0,1,\\ldots ,n} y=x_i)]$ .", "$[](\\Diamond ,\\Box ,,)$ is non-elementary and decidable.", "Decidability follows from Theorem REF .", "To establish non-elementary complexity, we give a reduction from $[]_{}(<,P)$ .", "We first show how to encode the intepretation of a predicate symbol, represented by a set $P\\subseteq $ , in a linear frame $F = (W,<)$ – without using atomic propositions and nominals as agreed in Section .", "Using free state variables, we can only distinguish linearly many states at any given time.", "We therefore use finite intervals (finite subchains of $(W,<)$ ) to encode whether $n\\in P$ .", "Such an interval—we call it a marker—has length 2 (resp.", "3) if for the corresponding $n$ holds $n\\notin P$ (resp.", "$n\\in P$ ).", "Accordingly, we call a marker of length 2 (resp.", "3) negative (resp. positive).", "These finite intervals are separated by dense intervals—those are intervals wherein every two states have an intermediate state, e.g., $[0,1]_{\\mathbb {Q}} = \\lbrace q \\in \\mathbb {Q} \\mid 0 \\leqslant q \\leqslant 1\\rbrace $ .", "For example, the set $P$ with $0,2\\notin P$ and $1\\in P$ is represented by the chain in Figure REF .", "Figure: An example with 0,2∉P0,2 \\notin P and 1∈P1 \\in P.In our fragment, it is possible to distinguish between dense and finite intervals.", "We now show how to achieve this.", "In order to encode the alternating sequence of finite and dense intervals that represents a subset $P\\subseteq $ , we use the free state variable $a$ to mark a state in a dense interval that is directly followed by the first marker.", "We furthermore use the following macros, where $x$ and $y$ are state variables that are already bound before the use of the macro, and $r,s,t,u$ are fresh state variables.", "The state named $y$ is a direct successor of the state named $x$ .", "It suffices to say that all successors of $x$ are equal to, or occur after, $y$ .", "${dirSuc}(x,y) := _x \\Box z.", "(_y z \\vee _y \\Diamond z)$ The state named $x$ has no direct predecessor.", "It suffices to say that, for all states $r$ equal to, or after, the left bound $a$ : if $r$ is before $x$ , then there is a state between $r$ and $x$ .", "We work around the implication by saying that one of the following three cases occurs: $r$ is after $x$ , or $r$ equals $x$ , or $r$ is before $x$ with a state in between.", "${noDirPred}(x) := _a\\Box r.(_x \\Diamond r \\vee _x r \\vee _r\\Diamond \\Diamond x)$ The state named $x$ has a direct predecessor.", "It suffices to say that there is a state $r$ after $a$ of which $x$ is a direct successor.", "${dirPred}(x) := _a\\Diamond r.{dirSuc}(r,x)$ The interval between states $x,y$ is dense.", "We say that, for all $r$ with $x < r$ : $r$ is after $y$ , or $r$ has no direct predecessor.", "${dense}(x,y) := _x\\Box r.(_y\\Diamond r \\vee {noDirPred}(r))$ The state $x$ is in a separator.", "This macro says that, for some successor $r$ of $x$ , the interval between $x$ and $r$ is dense.", "${sep}(x) := _x\\Diamond r.{dense}(x,r)$ The state $x$ is the begin of a negative marker.", "This macro says that $x$ has a direct successor that is the begin of a separator, and $x$ has no direct predecessor.", "The latter is necessary to avoid that, in the above example, the middle state of a positive marker is mistaken for the begin of a negative marker.", "${neg}(x) := _x\\Diamond r.({dirSuc}(x,r) \\wedge {sep}(r)) \\wedge {noDirPred}(x)$ The state $x$ is the begin of a positive marker.", "Similarly to the above macro, we express that $x$ has a direct-successor sequence $r,s$ with $s$ being the begin of a separator, and $x$ has no direct predecessor.", "${pos}(x) := _x\\Diamond r.({dirSuc}(x,r) \\wedge \\Diamond s.({dirSuc}(r,s) \\wedge {sep}(s))) \\wedge {noDirPred}(x)$ The state $x$ is in a separator whose end is a marker.", "This macro says that, for some successor $r$ of $x$ , the interval between $x$ and $r$ is dense and $r$ is the begin of a marker.", "${sepM}(x) := _x\\Diamond r.({dense}(x,r) \\wedge ({neg}(r) \\vee {pos}(r)))$ We now need the following two conjuncts to express that the part of the model starting at $a$ represents a sequence of infinitely many markers.", "$a$ is in a separator that ends with a marker.", "$\\psi _1 := {sepM}(a)$ Every marker has a direct successor marker.", "We say that every state $r$ after $a$ satisfies one of the following conditions.", "$r$ is in a separator—this also includes that $r$ is the end of a marker—that is followed by a marker.", "$r$ is the begin of a negative marker and its direct successor is the begin of a separator whose end is a marker.", "$r$ is the begin of a positive marker and its direct 2-step successor is the begin of a separator whose end is a marker.", "$r$ in the middle of a positive marker, i.e., $r$ has a direct predecessor which is the begin of a positive marker, and $r$ 's direct successor is in a separator whose end is a marker.", "$\\psi _2 :=\\, & _a \\Box r.\\Big ({sepM}(r) \\\\[2px]& \\vee \\Big ({neg}(r) \\wedge \\Diamond s.({dirSuc}(r,s) \\wedge {sepM}(s))\\Big ) \\\\[2px]& \\vee \\Big ({pos}(r) \\wedge \\Diamond s.({dirSuc}(r,s) \\wedge \\Diamond t.({dirSuc}(s,t) \\wedge {sepM}(t)))\\Big ) \\\\[2px]& \\vee \\Big ( (_a\\Diamond s.{dirSuc}(s,r) \\wedge {pos}(s)) \\wedge \\Diamond t.({dirSuc}(r,t) \\wedge {sepM}(t))\\Big )$ Finally, we encode formulae $\\varphi $ from $(<,P)$ .", "We assume w.l.o.g.", "that such formulae have the shape $\\varphi := Q_1x_1\\dots Q_nx_n.\\beta (x_1,\\dots ,x_n)$ , where $Q_i \\in \\lbrace \\exists ,\\forall \\rbrace $ and $\\beta $ is quantifier-free with atoms $P(x)$ and $x<y$ for variables $x,y$ , such that negations appear only directly before atoms.", "The transformation of $\\varphi $ reuses the $x_i$ as state variables and proceeds inductively as follows.", "$f(P(x_i)) &~:=~ {pos}(x_i) \\\\[4px]f(\\lnot P(x_i)) &~:=~ {neg}(x_i) \\\\[4px]f(x_i < x_j) &~:=~ _{x_i}\\Diamond x_j \\\\[4px]f(\\lnot (x_i < x_j)) &~:=~ _{x_i}x_j \\vee _{x_j}\\Diamond x_i \\\\[4px]f(\\alpha \\wedge \\beta ) &~:=~ f(\\alpha ) \\wedge f(\\beta ) \\\\[4px]f(\\alpha \\vee \\beta ) &~:=~ f(\\alpha ) \\vee f(\\beta ) \\\\[2px]f(\\exists x_i.\\alpha ) &~:=~ _a\\Diamond x_i.\\Big (({neg}(x_i) \\vee {pos}(x_i)) \\wedge f(\\alpha )\\Big ) \\\\f(\\forall x_i.\\alpha ) &~:=~ _a\\Box x_i.\\Big ({sep}(x_i) \\vee {dirPred}(x_i) \\vee f(\\alpha )\\Big ) \\\\$ The transformation of $\\varphi $ into $(\\Diamond ,\\Box ,,)$ is now achieved by the function $g$ defined as follows.", "$g(\\varphi ) := \\psi _1 \\wedge \\psi _2 \\wedge f(\\varphi )$ It is clear that the reduction function $g$ can be computed in polynomial time.", "The correctness of the reduction is expressed by the following claim.", "For every formula $\\varphi $ from $(<,P)$ holds: $\\varphi \\in []_{}(<,P)$ if and only if $g(\\varphi ) \\in [](\\Diamond ,\\Box ,,)$ .", "The proof of the claim should be clear.", "Since $[]_{}(<,P)$ is non-elementary [20], it follows that $[](\\Diamond ,\\Box ,,)$ is non-elementary, too.", "Finally, we note that our reduction uses a single free state variable $a$ , which could as well be bound to the first state of evaluation.", "The high complexity of $[](\\Diamond ,\\Box ,,)$ relies on the possibility that the linear frame alternatingly has dense and non-dense parts.", "If we have the natural numbers as frame for a hybrid language, we lose this possibility.", "As a consequence, the satisfiability problem for monotone hybrid logics over the natural numbers has a lower complexity than that over linear frames.", "$[](\\Diamond ,\\Box ,,)$ is -complete.", "Proof.", "Let $$ be the problem to decide whether a given quantified Boolean formula is valid.", "We show -hardness by a polynomial-time reduction from the $$ -complete $$ to $[](\\Diamond ,\\Box ,,)$ .", "Let $\\varphi $ be an instance of $$ and assume w.l.o.g.", "that negations occur only directly in front of atomic propositions.", "We define the transformation as $f\\colon \\varphi \\mapsto r. \\Diamond s. \\Diamond h(\\varphi )$ where $h$ is given as follows: let $\\psi ,\\chi $ be quantified Boolean formulae and let $x_k$ be a variable in $\\varphi $ , then $\\begin{array}{@{}l@{\\hspace*{28.45274pt}}l@{}}h(\\exists x_k \\psi ):=_r \\Diamond x_k.", "h(\\psi ), &h(\\forall x_k \\psi ):=_r \\Box x_k.", "h(\\psi ), \\\\[2px]h(\\psi \\wedge \\chi ):=h(\\psi ) \\wedge h(\\chi ), &h(\\psi \\vee \\chi ):=h(\\psi ) \\vee h(\\chi ), \\\\[2px]h(\\lnot x_k):=_s \\Diamond x_k, &h(x_k):=_s x_k.", "\\\\\\end{array}$ For example, the QBF $\\psi = \\forall x\\exists y(x \\wedge y) \\vee (\\lnot x \\wedge \\lnot y)$ is mapped to $f(\\varphi ) = r. \\Diamond s. \\Diamond _r\\Box x_0._r\\Diamond x_1.", "(_s x_0 \\wedge _s x_1) \\vee (_s \\Diamond x_0 \\wedge _s \\Diamond x_1)$ .", "Intuitively, this construction requires the existence of an initial state named $r$ , a successor state $s$ that represents the truth value $\\top $ , and one or more successor states of $s$ which together represent $\\bot $ .", "The quantifiers $\\exists ,\\forall $ are replaced by the modal operators $\\Diamond ,\\Box $ which range over $s$ and its successor states.", "Finally, positive literals are enforced to be true at $s$ , negative literals strictly after $s$ .", "For every model of $f(\\varphi )$ , it holds that $r$ is situated at the first state of the model and that state has a successor labelled by $s$ .", "By virtue of the function $h$ , positive literals have to be mapped to $s$ , whereas negative literals have to be mapped to some state other than $s$ .", "An easy induction on the structure of formulae shows that $\\varphi \\in $ iff $f(\\varphi ) \\in [](\\Diamond ,\\Box ,,)$ .", "We obtain -membership via a polynomial-time reduction from $[](\\Diamond ,\\Box ,,)$ to the satisfiability problem $[]_{}(<)$ for the fragment of first-order logic with the relation “$<$ ” interpreted over the natural numbers.", "Let the first order language contain all members of as variables and all members of as constants.", "Based on the standard translation from hybrid to first-order logic [21], we devise a reduction $H$ that maps hybrid formulae $\\varphi $ and variables or constants $z$ to first-order formulae.", "$\\begin{array}{@{}l@{\\hspace*{14.22636pt}}l@{}}H(p,z) := \\top \\text{ for $p\\in $}& H(v,z) := v=z ~~\\text{ for $v\\in \\cup $} \\\\[2px]H(\\alpha \\wedge \\beta ,z) := H(\\alpha ,z) \\wedge H(\\beta ,z)& H(\\alpha \\vee \\beta ,z) := H(\\alpha ,z) \\vee H(\\beta ,z) \\\\[2px]H(\\Diamond \\alpha ,z) := \\exists t (z<t \\wedge H(\\alpha ,t))& H(\\Box \\alpha ,z) := \\forall t (z<t \\rightarrow H(\\alpha ,t)) \\\\[2px]H(x.\\alpha ,z) := \\exists x (x=z \\wedge H(\\alpha ,z))& H(_x \\alpha ,z) := H(\\alpha ,x)\\end{array}$ In the $\\Diamond $ , $\\Box $ and $$ -cases we deviate from the usual definition of the standard translation because we do not insist on using only two variables in addition to —therefore it suffices to require that $t$ is a fresh variable—and we allow constants in the second argument.", "For a first-order formula $\\psi $ with variables in $$ and an assignment $g:\\rightarrow $ , let $\\psi [g]$ denote the first-order formula that is obtained from $\\psi $ by substituting every free occurrence of $x\\in $ by the first-order term that describes $g(x)$ .", "For every instance $\\varphi $ of $[](\\Diamond ,\\Box ,,)$ , every assignment $g:\\rightarrow $ and every $n\\in $ , it holds that:    $g,n \\models \\varphi $ if and only if $(,<)\\models H(\\varphi ,z)[g^{z}_n]$ , where $z$ is a new variable that does not occur in $\\varphi $ .", "We prove the claim inductively on the construction of $\\varphi $ .", "$\\varphi =v$ for $v\\in $ : Table: NO_CAPTIONJustifications for the equivalences: (1) is by the definition of $\\models $ for hybrid logic, (2) extends $g$ by the new variable $z$ , and (3) uses the definition of $\\models $ for first-order logic over $(,<)$ .", "$\\varphi =\\alpha \\wedge \\beta $ resp.", "$\\varphi =\\alpha \\vee \\beta $ : straightforward.", "$\\varphi =\\Diamond \\alpha $ : Table: NO_CAPTION(1) and (2) are by definition resp.", "by induction hypothesis.", "For (3), notice that the variable $t$ may appear free in $H(\\alpha ,t)$ but it does not appear free in $\\exists t (z<t \\wedge H(\\alpha , t))$ .", "The equivalence then follows by the semantics of the considered first-order logic.", "$\\varphi =\\Box \\alpha $ : Table: NO_CAPTION(1) and (2) are by definition resp.", "by induction hypothesis.", "The arguments for (3) are as in the case above.", "$\\varphi =x.", "\\alpha $ : Table: NO_CAPTION(1) and (2) are from the definition of $$ and from the induction hypothesis.", "Eventually, (3) follows from the semantics of FOL over $(,<)$ .", "$\\varphi =_x \\, \\alpha $ : Table: NO_CAPTION(1) and (2) are from the definition of $$ and from the induction hypothesis.", "Now, (3) follows from the semantics of FOL over $(,<)$ .", "Notice that $z$ does not appear free in $\\exists z (x=z \\wedge H(\\alpha ,z))$ .", "This proves Equivalence (4).", "This concludes the proof of the claim.", "Now, $\\varphi \\in [](\\Diamond ,\\Box ,,)$ if and only if $g,0\\models \\varphi \\vee \\Diamond \\varphi $ for some assignment $g$ .", "By the above claim, this is equivalent to $(,<)\\models H(\\varphi \\vee \\Diamond \\varphi ,z)[g^{z}_0]$ for some $g$ and a new variable $z$ , which can also be expressed as $(,<)\\models \\forall x (\\lnot (x<z) \\wedge H(\\varphi \\vee \\Diamond \\varphi ,z))$ .", "This shows that $[](\\Diamond ,\\Box ,,)$ is polynomial-time reducible to $[]_{}(<)$ , which was shown to be in in [8].", "Therefore, $[](\\Diamond ,\\Box ,,)$ is in ." ], [ "The easy cases: 1 and $Ł$ results", "In this section, we show that the fragments without the $\\Diamond $ -operator have an easy satisfiability problem.", "Our results can be structured into four groups.", "First, we consider fragments without modal operators.", "For these fragments we obtain 1-completeness.", "Simply said, without negation and $\\Diamond $ we cannot express that two nominals or state variables are not bound to the same state.", "Therefore, the model that binds all variables to the first state satisfies every satisfiable formula in this fragment.", "Let $F_0=(\\lbrace 0\\rbrace ,\\emptyset )$ and $g_0(y)=0$ for every $y \\in $ .", "Then $\\varphi \\in [](, )$ (resp.", "$\\varphi \\in [](, )$ ) if and only if $F_0,g_0,0\\models \\varphi $ .", "The implication direction from left to right follows from the monotonicity of the considered formulas.", "For the other direction, notice that $F_0\\in $ .", "For frame class , note that if $F_0,g_0,0\\models \\varphi $ and $\\varphi $ has no modal operators, then $g_0,0\\models \\varphi $ .", "Let $O \\subseteq \\lbrace , \\rbrace $ .", "Then $[](O)$ and $[](O)$ are 1-complete.", "1-hardness of $[{F}](\\emptyset )$ follows immediately from the 1-completeness of the Formula Value Problem for propositional formulae [6].", "It remains to show that $[](, )$ and $[](, )$ are in 1.", "In order to decide whether $\\varphi $ is in $[](, )$ , according to Lemma  it suffices to check whether the propositional formula obtained from $\\varphi $ deleting all occurrences of $x.$ and $_x$ , is satisfied by the assignment that sets all atoms to true.", "According to [6] this can be done in 1.", "Since $[](, )=[](, )$ by Lemma , we obtain the same for $[](, )$ .", "Second, we consider fragments with the $\\Box $ -operator over linear frames.", "We can show 1-completeness here, too.", "The main reason is that (sub-)formulas that begin with a $\\Box $ are satisfied in a state that has no successor.", "Therefore similar as above, every formula of this fragment that is satisfiable over linear frames is satisfied by a model with only one state.", "$[](\\Box , , )$ is 1-complete.", "1-hardness follows from Theorem .", "It remains to show that $[](\\Box , , )\\in 1$ .", "We show that essentially the $\\Box $ -operators can be ignored.", "$[](\\Box , , ) [](, )$ .", "For an instance $\\varphi $ of $[](\\Box , , )$ , let $\\varphi ^{\\prime \\prime }$ be the formula obtained from $\\varphi $ by replacing every subformula $\\Box \\psi $ of $\\varphi $ with the constant $\\top $ .", "Then $\\varphi ^{\\prime \\prime }$ is an instance of $[](, )$ .", "If $\\varphi \\in [](\\Box , , )$ , then $\\varphi ^{\\prime \\prime }\\in [](, )$ due to the monotonicity of $\\varphi $ .", "On the other hand, if $\\varphi ^{\\prime \\prime }\\in [](, )$ , then $K_0,g,0\\models \\varphi ^{\\prime \\prime }$ (Lemma ).", "Since $K_0,g,0\\models \\Box \\alpha $ for every $\\alpha $ , we obtain $K_0,g,0\\models \\varphi $ , hence $\\varphi \\in [](\\Box , , )$ .", "As such simple substitutions can be realized using an 0-circuit, the stated reduction is indeed a valid $$ -reduction from $[](\\Box ,,)$ to $[](,)$ .", "Since $[](, )\\in 1$ (Theorem ) and 1 is closed downwards under $$ , it follows from the Claim that $[](\\Box , , )\\in 1$ .", "It is clear that this argument does not apply to the natural numbers.", "Third, we show 1-completeness for the fragments with $\\Box $ and one of $$ and $$ over $$ .", "They receive separate treatment because, in , every state has a successor, and therefore $\\Box $ -subformulas cannot be satisfied as easily as above.", "It turns out that the complexity of the satisfiability problem increases only if both hybrid operators can be used.", "$[](\\Box ,)$ is 1-complete.", "Proof sketch.", "1-hardness follows from Theorem .", "For the upper bound, we distinguish occurrences of nominals that are either free, or that are bound by a $\\Box $ , or that are bound by an $$ .", "Simply said, a free occurrence of $i$ in $\\alpha $ is bound by $\\Box $ in $\\Box \\alpha $ and bound by $$ in $_x \\alpha $ (even if $x\\ne i$ ).", "Since the assignment $g$ is not relevant for the considered fragment, we write $K,w\\models \\alpha $ for short instead of $K,g,w\\models \\alpha $ .", "Let $\\alpha ^{\\prime }$ be the formula obtained from $\\alpha $ by replacing every occurrence of a nominal that is bound by $\\Box $ with $\\bot $ , and let $\\eta $ be a valuation.", "If $\\eta ,k\\models \\alpha $ , then $\\eta ,k\\models \\alpha ^{\\prime }$ .", "Moreover, it turns out that binding every nominal to the initial state suffices to obtain a satisfying model.", "$\\varphi \\in [](\\Box ,)$ if and only if $\\eta _0,0\\models \\varphi $ with $\\eta _0(x)=\\lbrace 0\\rbrace $ for every $x \\in $ .", "Both claims together yield that, in order to decide $\\varphi \\in [](\\Box ,)$ , it suffices to check whether $\\eta _0,0\\models \\varphi ^{\\prime }$ .", "No nominal in $\\varphi ^{\\prime }$ occurs bound by a $\\Box $ -operator.", "Therefore for every subformula $\\Box \\alpha $ of $\\varphi ^{\\prime }$ and for every $k$ holds: $\\eta _0,k\\models \\alpha $ if and only if $\\eta _0,0\\models \\alpha $ .", "All nominals that occur free or bound by an $$ evaluate to true in state 0 via $\\eta _0$ .", "Therefore, in order to decide $\\eta _0,0\\models \\varphi ^{\\prime }$ , it suffices to ignore all $\\Box $ and $$ -operators of $\\varphi ^{\\prime }$ and evaluate it as a propositional formula under assignment $\\eta _0$ that sets all atoms of $\\varphi ^{\\prime }$ to true.", "This can be done in 1 [6].", "The complete proof can be found in Appendix .", "Next, we consider $[](\\Box ,)$ .", "According to our remarks in Section about notational convenience, we assume that there are no nominals in $(\\Box ,)$ .", "$[](\\Box ,)$ is 1-complete.", "Proof sketch.", "Now, we distinguish occurrences of state variables as the occurrences in the proof sketch above.", "They are either free, or they are bound by a $\\Box $ , or they are bound by $$ .", "Note that this phrasing differs from the standard usage of the terms `free' and `bound' in the context of state variables.", "A free occurrence of $i$ in $\\alpha $ is bound by $\\Box $ in $\\Box \\alpha $ , as above.", "It is bound by $$ in $i .", "\\alpha $ only.", "Notice that $y$ occurs free in $x .", "y$ (for $x\\ne y$ ).", "Let $\\alpha ^{\\prime }$ be the formula obtained from $\\alpha $ by replacing every occurrence of a state variable that is bound by $\\Box $ with $\\bot $ , and let $g$ be an assignment.", "If $g,k\\models \\alpha $ , then $g,k\\models \\alpha ^{\\prime }$ .", "$\\varphi \\in [](\\Box ,)$ if and only if $g_0,0\\models \\varphi $ , for $g_0(x)=0$ for every $x \\in $ .", "Both claims together yield that, in order to decide $\\varphi \\in [](\\Box ,)$ , it suffices to check whether $g_0,0\\models \\varphi ^{\\prime }$ .", "No state variable in $\\varphi ^{\\prime }$ occurs bound by a $\\Box $ -operator.", "Therefore for every subformula $\\Box \\alpha $ of $\\varphi ^{\\prime }$ and for every $k$ holds: $g_0,k\\models \\alpha $ if and only if $g_0,0\\models \\alpha $ .", "All occurrences of state variables in $\\varphi ^{\\prime }$ that are bound by $$ evaluate to true, because no $\\Box $ occurs “between” the binding $i$ and the occurrence of $i$ , which means that the state where the variable is bound is the same as where the variable is used.", "All free occurrences of state variables evaluate to true in state 0 due to $g_0$ .", "Therefore, in order to decide $g_0,0\\models \\varphi ^{\\prime }$ , it suffices to ignore all $\\Box $ and $$ -operators of $\\varphi ^{\\prime }$ and evaluate it as a propositional formula under an assignment that sets all atoms to true.", "This can be done in 1 [6].", "The complete proof can be found in Appendix .", "The fourth part deals with the fragment with $\\Box $ and both $$ and $$ over the natural numbers.", "$[](\\Box , , )$ is $Ł$ -hard.", "This proof is very similar to the proof of Theorem 3.3. in [14].", "We give a reduction from the problem Order between Vertices () which is known to be $Ł$ -complete [7] and defined as follows.", "$$ A finite set of vertices $V$ , a successor-relation $S$ on $V$ , and two vertices $s,t\\in V$ .", "Is $s\\leqslant _S t$ , where $\\leqslant _S$ denotes the unique total order induced by $S$ on $V$ ?", "Notice that $(V,S)$ is a directed line-graph.", "Let $(V,S,s,t)$ be an instance of $$ .", "We construct an $(\\Box ,,)$ -formula $\\varphi $ that is satisfiable if and only if $s\\leqslant _S t$ .", "We use $V=\\lbrace v_0,v_1,\\ldots ,v_n\\rbrace $ as state variables.", "The formula $\\varphi $ consists of three parts.", "The first part binds all variables except $s$ to one state and the variable $s$ to a successor of this state.", "The second part of $\\varphi $ binds a state variable $v_l$ to the state labeled by $s$ iff $s\\leqslant _S v_l$ .", "Let $\\alpha $ denote the concatenation of all $_{v_{k}}v_{l}$ with $(v_k,v_l)\\in S$ and $v_l\\ne s$ , and $\\alpha ^n$ denotes the $n$ -fold concatenation of $\\alpha $ .", "Essentially, $\\alpha ^n$ uses the assignment to collect eventually all $v_i$ with $s\\leqslant _S v_i$ in the state labeled $s$ .", "The last part of $\\varphi $ checks whether $s$ and $t$ are bound to the same state after this procedure.", "That is, $\\varphi = v_0.v_1.v_2.", "\\cdots v_n.", "\\Box s. ~ \\alpha ^{n} ~ _s t.$ To prove the correctness of our reduction, we show that $\\varphi $ is satisfiable if and only if $s\\leqslant _S t$ .", "Assume $s\\leqslant _S t$ .", "For an arbitrary assignment $g$ , one can show inductively that $g,0\\models v_0.v_1.", "\\cdots v_n.", "\\Box s. ~ \\alpha ^i ~ _s r$ for $i=0,1,\\ldots ,n$ and for all $r$ that have distance $i$ from $s$ .", "Therefore it eventually holds that $g,0\\models \\varphi $ .", "For $s\\lnot \\leqslant _S t$ we show that $g,n\\lnot \\models \\varphi $ for any assignment $g$ and natural number $n$ .", "Let $g_0$ be the assignment obtained from $g$ after the bindings in the prefix $v_0.v_1.", "\\cdots v_n.", "\\Box s$ of $\\varphi $ , and let $g_i$ be the assignment obtained from $g_0$ after evaluating the prefix of $\\varphi $ up to and including $\\alpha ^i$ .", "It holds that $g_i(s)\\ne g_i(t)=0$ for all $i=0,1,\\ldots ,n$ .", "This leads to $g_{n},0\\lnot \\models _s t$ and therefore $g,0\\lnot \\models \\varphi $ .", "For the upper bound, we establish a characterisation of the satisfaction relation that assigns a unique assignment and state of evaluation to every subformula of a given formula $\\varphi $ .", "Using this new characterisation, we devise a decision procedure that runs in logarithmic space and consists of two steps: it replaces every occurrence of any state variable $x$ in $\\varphi $ with 1 if its state of evaluation agrees with that of its $x$ -superformula, and with 0 otherwise; it then removes all $\\Box $ -, $$ - and $$ -operators from the formula and tests whether the resulting Boolean formula is valid.", "$[](\\Box ,,)$ is in .", "The proof can be found in Appendix ." ], [ "The intermediate cases: $$ results", "After we have seen that all fragments without $\\Diamond $ have an easy satisfiability problem, we show that $\\Diamond $ together with the use of nominals makes the satisfiability problem $$ -hard.", "Recall that, owing to the presence of nominals, $(\\Diamond )$ is not just modal logic with the $\\Diamond $ -operator.", "The absence of $$ makes assignments superfluous: we write $K,w \\models \\varphi $ instead of $K,g,w \\models \\varphi $ .", "$[](\\Diamond )$ and $[](\\Diamond )$ both are $$ -hard.", "Proof.", "We reduce from $$ .", "Let $\\varphi =c_1 \\wedge \\ldots \\wedge c_n$ be an instance of $$ with clauses $c_1, \\dots , c_n$ (where $c_i=(l_1^i\\vee l_2^i\\vee l_3^i)$ for literals $l^i_j$ ) and variables $x_1, \\dots , x_m$ .", "We define the transformation as $f\\colon \\varphi \\mapsto \\Diamond (i_0 \\wedge \\Diamond i_1) ~\\wedge ~\\Bigg (\\bigwedge _{\\ell =1}^m \\Diamond (i_0 \\wedge x_{\\ell }) \\vee \\Diamond (i_1 \\wedge x_{\\ell })\\Bigg ) \\wedge h(\\varphi ),$ where $i_0,i_1$ and all $x_{\\ell }$ are nominals, and the function $h$ is defined as follows: let $l_k^j$ be a literal in clause $c_j$ , then $h(l_k^j)& := {\\left\\lbrace \\begin{array}{ll}(i_1 \\wedge x), \\text{ if } l_k^j=x \\\\(i_0 \\wedge x), \\text{ if } l_k^j=\\lnot x\\end{array}\\right.}", "\\\\[2px]h(c_j) & := \\Diamond ( h(l_1^j) \\vee h(l_2^j) \\vee h(l_3^j)),\\quad \\text{where } c_j = (l_1^j \\vee l_2^j \\vee l_3^j); \\\\[4px]h(c_1 \\wedge \\dots \\wedge c_n)& := h(c_1) \\wedge \\dots \\wedge h(c_n).$ Notice that $f$ turns variables in the $$ instance into nominals in the $[](\\Diamond )$ instance.", "The part $\\Diamond (i_0 \\wedge \\Diamond i_1)$ enforces the existence of two successors $w_1$ and $w_2$ of the state satisfying $f(\\varphi )$ .", "The part $\\bigwedge _{\\ell =1}^m \\Diamond (i_0 \\wedge x_{\\ell }) \\vee \\Diamond (i_1 \\wedge x_{\\ell })$ simulates the assignment of the variables in $\\varphi $ , enforcing that each $x_{\\ell }$ is true in either $w_1$ or $w_2$ .", "The part $h(\\varphi )$ then simulates the evaluation of $\\varphi $ on the assignment determined by the previous parts.", "With the following claim $$ -hardness of $[](\\Diamond )$ follows.", "$\\varphi \\in $ if and only if $h(\\varphi ) \\in [](\\Diamond )$ .", "We first show that $h(\\varphi ) \\in [](\\Diamond )$ implies $\\varphi \\in $ .", "If $K,w_0 \\models h(\\varphi )$ with $K=(W,<,\\eta )$ , then the following holds.", "Let $w_1=\\eta (i_0)$ , $w_2=\\eta (i_1)$ , and $\\lbrace w_0,w_1,w_2\\rbrace \\subseteq W$ with $w_0,w_1,w_2$ pairwise different; $w_0<w_1<w_2$ ; for all $x_j$ with $1 \\leqslant j \\leqslant m$ : $\\eta (x_j) \\subseteq \\lbrace w_1,w_2\\rbrace $ .", "We build a propositional logic assignment $\\beta = (\\beta _1 \\dots \\beta _m)$ that satisfies $\\varphi $ , where $\\beta _i \\in \\lbrace \\bot ,\\top \\rbrace $ is the truth value for $x_i$ , as follows.", "$\\beta _j=\\bot $ if $g(i_0) = g(x_j)$ , and $\\beta _j=\\top $ if $g(i_1) = g(x_j)$ .", "From the construction of $h(\\varphi )$ , it clearly follows that $\\beta $ satisfies $\\varphi $ .", "For the converse direction, suppose that $\\varphi $ is satisfied by the propositional logic assignment $\\beta = (\\beta _1 \\dots \\beta _m)$ .", "We construct a linear model $K:=(W,<,\\eta )$ containing a state $w$ such that $K,w \\models h(\\varphi )$ .", "$W &:= \\lbrace w,w_0,w_1\\rbrace \\\\< &:~~~ w<w_0<w_1 \\\\$ $\\eta (i_j) &:= w_j \\text{ for } j \\in \\lbrace 0,1\\rbrace \\\\\\eta (x_j) &:= {\\left\\lbrace \\begin{array}{ll}w_0, \\text{ if } \\beta _j=\\bot \\\\w_1, \\text{ if } \\beta _j=\\top \\end{array}\\right.", "}$ It follows from the construction of $K$ that $K,w \\models h(\\varphi )$ .", "The conjunct $h(\\varphi )$ is of the form $(h(l_1^1) \\vee h(l_1^2) \\vee h(l_1^3)) \\wedge \\dots \\wedge (h(l_n^1) \\vee h(l_n^2) \\vee h(l_n^3)).$ Hence, under $\\beta $ , at least one literal in every clause evaluates to true.", "The variable in this literal satisfies the same clause in $h(\\varphi )$ .", "Hence every clause in $h(\\varphi )$ is satisfied in $w$ in $K$ .", "Therefore, $K,w \\models h(\\varphi )$ .", "Using this claim, $$ -hardness of $[](\\Diamond )$ follows.", "It is straightforward to show that $$ reduces to $[](\\Diamond )$ using the same reduction.", "We will now establish -membership of the problems $[{F}](\\Diamond ,\\Box ,)$ , $[{F}](\\Diamond ,\\Box ,)$ , and $[{F}](\\Diamond ,,)$ for ${F} \\in \\lbrace ,\\rbrace $ .", "For the first two, this follows from the literature, see Theorem REF .", "For the third, we observe that all modal and hybrid operators in a formula $\\varphi $ from the fragment $(\\Diamond ,,)$ are translatable into FOL by the standard translation using no universal quantifiers.", "The existential quantifiers introduced by the binder can be skolemised away, which corresponds to removing all binding from $\\varphi $ and replacing each state variable with a fresh nominal.", "The correctness of this translation is proven in [21].", "Hence, $[{F}](\\Diamond ,,)$ polynomial-time reduces to $[{F}](\\Diamond ,)$ .", "$[](\\Diamond ,,)$ and $[](\\Diamond ,,)$ are in $$ .", "From the lower bounds in Lemma and the upper bounds in Theorem REF and Lemma , we obtain the following theorem.", "Let $\\lbrace \\Diamond \\rbrace \\subseteq O$ , and $O \\subsetneq \\lbrace \\Diamond , \\Box , , \\rbrace $ .", "Then $[](O)$ and $[](O)$ are $\\emph {-complete}$ .", "In addition to the NP-membership of the fragments captured by Theorem , we are interested in their model-theoretic properties.", "We show that these logics enjoy a kind of linear-size model property, precisely a quasi-quadratic size model property: over the natural numbers, every satisfiable formula has a model where two successive nominal states have at most linearly many intermediary states, and the states behind the last such state are indistinguishable.", "This property allows for an alternative worst-case decision procedure for satisfiability that consists of guessing a linear representation of a model of the described form and symbolically model-checking the input formula on that model.", "Over general linear frames, which may have dense intervals, we formulate the model property in a more general way and prove it using additional technical machinery to deal with density.", "However, the result then carries over to the rationals, where we are not aware of any upper complexity bound in the literature.", "In [19], Sistla and Clarke showed a variation of the linear-size model property for LTL(F), which corresponds to $(\\Diamond ,\\Box )$ over : whenever $\\varphi \\in (\\Diamond ,\\Box )$ is satisfiable over , then it is satisfiable in the initial state of a model over which has a linear-sized prefix init and a remainder final such that final is maximal with respect to the property that every type (set of all atomic propositions true in a state) occurs infinitely often, and final contains only linearly many types.", "Such a structure can be guessed in polynomial time, represented in polynomial space and model-checked in polynomial time.", "While it is straightforward to extend Sistla and Clarke's proof to cover nominals and the operator, it will not go through if density is allowed (frame class $$ ).", "We establish that $(\\Diamond ,\\Box ,)$ over $$ has a quadratic size model property, and we subsequently show how to extend the result to the other fragments from Theorem and how to restrict them to .", "$(\\Diamond ,\\Box ,)$ has the quasi-quadratic size model property with respect to $$ and $$ .", "The proof can be found in Appendix .", "As an immediate consequence, the model property in Theorem carries over to the subfragments $(\\Diamond ,\\Box )$ , $(\\Diamond ,)$ , $(\\Box ,)$ , $(\\Diamond )$ , $(\\Box )$ , $()$ , and $(\\emptyset )$ .", "Moreover, our arguments in the proofs of Theorems  and can be used to transfer it to $(\\Box ,,)$ .", "Together with the observations that $(\\Diamond ,,)$ is no more expressive than $(\\Diamond ,)$ (see the explanation before Lemma ), and $(\\Diamond ,\\Box ,)$ is no more expressive than $(\\Diamond ,\\Box )$ (because, without $$ , one cannot jump to named states), we obtain the following generalisation of Theorem .", "Let $O \\subsetneq \\lbrace \\Diamond ,\\Box ,,\\rbrace $ .", "Then $(O)$ has the quasi-quadratic size model property with respect to $$ and $$ ." ], [ "Conclusion", "We have completely classified the complexity of all fragments of hybrid logic with monotone Boolean operators obtained from arbitrary combinations of four modal and hybrid operators, over linear frames and the natural numbers.", "Except for the largest such fragment over linear frames, all fragments are of elementary complexity.", "We have classified their complexity into -complete, -complete and tractable and shown that the tractable cases are complete for either 1 or .", "Surprisingly, while the largest fragment is harder over linear frames than over , the largest $\\Diamond $ -free fragment is easier over linear frames than over .", "The question remains whether the -complete largest fragment over admits some quasi-polynomial size model property.", "Furthermore, this study can be extended in several possible ways: by allowing negation on atomic propositions, by considering frame classes that consist only of dense frames, such as , or by considering arbitrary sets of Boolean operators in the same spirit as in [14].", "For atomic negation, it follows quite easily that the largest fragment is of non-elementary complexity over , too, and that all fragments except $O = (\\Box ,,)$ are -complete.", "However, our proof of the quasi-quadratic model property does not immediately go through in the presence of atomic propositions.", "Over , we conjecture that all fragments, except possibly for the largest one, have the same complexity and model properties as over ." ], [ "Proof of Theorem ", "Theorem $[](\\Box ,)$ is 1-complete.", "1-hardness follows from Theorem .", "For the upper bound, we distinguish occurrences of nominals that are either free, or that are bound by a $\\Box $ , or that are bound by an $$ .", "Simply said, a free occurrence of $i$ in $\\alpha $ is bound by $\\Box $ in $\\Box \\alpha $ and bound by $$ in $_x \\alpha $ (even if $x\\ne i$ ).", "Since the assignment $g$ is not relevant for the considered fragment, we write $K,w\\models \\alpha $ for short instead of $K,g,w\\models \\alpha $ .", "Let $\\alpha ^{\\prime }$ be the formula obtained from $\\alpha $ by replacing every occurrence of a nominal that is bound by $\\Box $ with $\\bot $ , and let $\\eta $ be a valuation.", "If $\\eta ,k\\models \\alpha $ , then $\\eta ,k\\models \\alpha ^{\\prime }$ .", "We use induction on the construction of $\\varphi $ .", "The base case for $\\varphi \\in \\cup $ is straightforward, as is the inductive step for $\\varphi =\\alpha \\vee \\beta $ and $\\varphi =\\alpha \\wedge \\beta $ , and even for $\\varphi =_x\\alpha $ .", "It remains to consider the case $\\varphi =\\Box \\alpha $ .", "If $\\eta ,k\\models \\Box \\alpha $ , then for all $k^{\\prime }>k$ : $\\eta ,k^{\\prime }\\models \\alpha $ (by semantics of $\\Box $ ) and by inductive hypothesis follows for all $k^{\\prime }>k$ : $\\eta ,k^{\\prime }\\models \\alpha ^{\\prime }$ .", "Assume that in $\\Box (\\alpha ^{\\prime })$ there occurs a nominal $i$ that is bound by the initial $\\Box $ -operator.", "Since for all $k>k^{\\prime }$ holds $\\eta ,k^{\\prime }\\models \\alpha ^{\\prime }$ , there is some $\\ell >\\max \\bigcup _{j\\in } \\eta (j)$ with $\\eta ,\\ell \\models \\alpha ^{\\prime }$ .", "Therefore $\\eta ,\\ell \\models \\alpha ^{\\prime }[i/\\bot ]$ , and by the monotonicity of $\\alpha ^{\\prime }$ and the properties of $\\eta $ it follows that for all $k^{\\prime }>k$ holds $\\eta ,k^{\\prime }\\models \\alpha ^{\\prime }[i/\\bot ]$ .", "In this way, all nominals bound by the initial $\\Box $ -operator can be replaced by $\\bot $ , and it follows that $\\eta ,k\\models (\\Box (\\alpha ^{\\prime }))^{\\prime }$ .", "Since $(\\Box (\\alpha ^{\\prime }))^{\\prime }=(\\Box \\alpha )^{\\prime }$ , the claim follows.", "$\\varphi \\in [](\\Box ,)$ if and only if $\\eta _0,0\\models \\varphi $ with $\\eta _0(x)=\\lbrace 0\\rbrace $ for every $x \\in $ .", "We use induction on the construction of $\\varphi $ .", "The base case for $\\varphi \\in \\cup $ is straightforward, as is the inductive step for $\\varphi =\\alpha \\vee \\beta $ and $\\varphi =\\alpha \\wedge \\beta $ , and even for $\\varphi =_x\\alpha $ .", "It remains to consider the case $\\varphi =\\Box \\alpha $ .", "If $\\eta _0,0\\models \\varphi $ , then $\\varphi \\in [](\\Box ,)$ .", "If $\\Box \\alpha \\in [](\\Box ,)$ , then there exists $k$ such that $\\eta ,k\\models (\\Box \\alpha )^{\\prime }$ (for some $\\eta $ , by the claim above).", "Let $\\alpha ^{\\ast }$ be the formula with $(\\Box \\alpha )^{\\prime }=\\Box (\\alpha ^{\\ast })$ .", "By the semantics of $\\Box $ we obtain that there exists $k$ such that for all $k^{\\prime }>k$ holds $\\eta ,k^{\\prime }\\models \\alpha ^{\\ast }$ .", "By inductive hypothesis follows $\\exists k \\forall k^{\\prime }>k: \\eta _0,0\\models \\alpha ^{\\ast }$ , what is equivalent to $\\eta _0,0\\models \\alpha ^{\\ast }$ .", "Notice that $\\alpha ^{\\ast }$ contains no nominal.", "By the monotonicity of $\\alpha $ , it follows that for all $k\\in $ holds $\\eta _0,k\\models \\alpha ^{\\ast }$ .", "When we re-replace the $\\bot $ 's by the replaced nominals, the satisfaction is kept because of the monotonicity of $\\alpha $ , and therefore for all $k\\in $ holds $\\eta _0,k\\models \\alpha $ .", "This implies $\\eta _0,0\\models \\Box \\alpha $ , which eventually yields $\\varphi \\in [](\\Box ,)$ .", "Both claims together yield that, in order to decide $\\varphi \\in [](\\Box ,)$ , it suffices to check whether $\\eta _0,0\\models \\varphi ^{\\prime }$ .", "No nominal in $\\varphi ^{\\prime }$ occurs bound by a $\\Box $ -operator.", "Therefore for every subformula $\\Box \\alpha $ of $\\varphi ^{\\prime }$ and for every $k$ holds: $\\eta _0,k\\models \\alpha $ if and only if $\\eta _0,0\\models \\alpha $ .", "All nominals that occur free or bound by an $$ evaluate to true in state 0 via $\\eta _0$ .", "Therefore, in order to decide $\\eta _0,0\\models \\varphi ^{\\prime }$ , it suffices to ignore all $\\Box $ and $$ -operators of $\\varphi ^{\\prime }$ and evaluate it as a propositional formula under assignment $\\eta _0$ that sets all atoms of $\\varphi ^{\\prime }$ to true.", "This can be done in 1 [6]." ], [ "Proof of Theorem ", "Theorem $[](\\Box ,)$ is 1-complete.", "1-hardness follows from Theorem .", "For the upper bound, we distinguish occurrences of state variables as the occurrences in the proof sketch above.", "They are either free, or they are bound by a $\\Box $ , or they are bound by $$ .", "Note that this phrasing differs from the standard usage of the terms `free' and `bound' in the context of state variables.", "A free occurrence of $i$ in $\\alpha $ is bound by $\\Box $ in $\\Box \\alpha $ , as above.", "It is bound by $$ in $i .", "\\alpha $ only.", "Notice that $y$ occurs free in $x .", "y$ (for $x\\ne y$ ).", "Let $\\alpha ^{\\prime }$ be the formula obtained from $\\alpha $ by replacing every occurrence of a state variable that is bound by $\\Box $ with $\\bot $ , and let $g$ be an assignment.", "If $g,k\\models \\alpha $ , then $g,k\\models \\alpha ^{\\prime }$ .", "We use induction on the construction of $\\varphi $ .", "The base case for $\\varphi \\in $ is straightforward, as is the inductive step for $\\varphi =\\alpha \\vee \\beta $ , $\\varphi =\\alpha \\wedge \\beta $ , and for $\\varphi =x.\\alpha $ .", "It remains to consider the case $\\varphi =\\Box \\alpha $ .", "Let $g,k\\models \\Box \\alpha $ for $k \\in $ .", "Then for all $k^{\\prime }>k$ : $g,k^{\\prime }\\models \\alpha $ (by semantics of $\\Box $ ) and by inductive hypothesis follows for all $k^{\\prime }>k$ : $g,k^{\\prime }\\models \\alpha ^{\\prime }$ .", "Assume that in $\\Box (\\alpha ^{\\prime })$ there occurs a state variable $i$ that is bound by the initial $\\Box $ -operator.", "Since for all $k^{\\prime }>k$ holds $g,k^{\\prime }\\models \\alpha ^{\\prime }$ , there is some $\\ell >\\max \\bigcup _{x\\in }g(x)$ such that $g,\\ell \\models \\alpha ^{\\prime }$ .", "Therefore $g,\\ell \\models \\alpha ^{\\prime }[i/\\bot ]$ , and by the monotonicity of $\\alpha ^{\\prime }$ it follows that for all $k^{\\prime }>k$ holds $g,k^{\\prime }\\models \\alpha ^{\\prime }[i/\\bot ]$ .", "In this way, all state variables bound by the initial $\\Box $ -operator can be replaced by $\\bot $ , and it follows that $g,k\\models (\\Box (\\alpha ^{\\prime }))^{\\prime }$ , where $(\\Box \\alpha ^{\\prime })^{\\prime }=(\\Box \\alpha )^{\\prime }$ .", "$\\varphi \\in [](\\Box ,)$ if and only if $g_0,0\\models \\varphi $ , for $g_0(x)=0$ for every $x \\in $ .", "We use induction on the construction of $\\varphi $ .", "The base case for $\\varphi \\in $ is straightforward, as is the inductive step for $\\varphi =\\alpha \\vee \\beta $ , $\\varphi =\\alpha \\wedge \\beta $ , and for $\\varphi =x.\\alpha $ .", "It remains to consider the case $\\varphi =\\Box \\alpha $ .", "If $\\Box \\alpha \\in [](\\Box ,)$ , then there exists $k$ such that $g,k\\models (\\Box \\alpha )^{\\prime }$ (for some $\\eta $ and $g$ ).", "Let $\\alpha ^{\\ast }$ be the formula with $(\\Box \\alpha )^{\\prime }=\\Box \\alpha ^{\\ast }$ .", "By the semantics of $\\Box $ we obtain that there exists $k$ such that for all $k^{\\prime }>k$ holds $g_0,k^{\\prime }\\models \\alpha ^{\\ast }$ , and therefore $\\alpha ^{\\ast }\\in [](\\Box ,)$ .", "By inductive hypothesis follows $g_0,0\\models \\alpha ^{\\ast }$ .", "Notice that $\\alpha ^{\\ast }$ contains no free state variable.", "Therefore for all $k\\in $ holds $g_0,k\\models \\alpha ^{\\ast }$ .", "When we re-replace the $\\bot $ 's by the replaced state variables, the satisfaction is kept, and therefore for all $k\\in $ holds $g_0,k\\models \\alpha $ , which eventually implies $g_0,0\\models \\Box \\alpha $ , i.e.", "$g_0,0\\models \\varphi $ .", "Both claims together yield that in order to decide $\\varphi \\in [](\\Box ,)$ , it suffices to check whether $g_0,0\\models \\varphi ^{\\prime }$ .", "No state variable in $\\varphi ^{\\prime }$ occurs bound by a $\\Box $ -operator.", "Therefore for every subformula $\\Box \\alpha $ of $\\varphi ^{\\prime }$ and for every $k$ holds: $g_0,k\\models \\alpha $ if and only if $g_0,0\\models \\alpha $ .", "All occurrences of state variables in $\\varphi ^{\\prime }$ that are bound by $$ evaluate to true, because no $\\Box $ occurs “between” the binding $i$ and the occurrence of $i$ , which means that the state where the variable is bound is the same as where the variable is used.", "All free occurrences of state variables evaluate to true in state 0 due to $g_0$ .", "Therefore, in order to decide $g_0,0\\models \\varphi ^{\\prime }$ , it suffices to ignore all $\\Box $ and $$ -operators of $\\varphi ^{\\prime }$ and evaluate it as a propositional formula under an assignment that sets all atoms to true.", "This can be done in 1 [6]." ], [ "Proof of Theorem ", "Theorem $[](\\Box ,,)$ is in .", "For this upper bound, we will establish a characterisation of the satisfaction relation that assigns a unique assignment and state of evaluation to every subformula of a given formula $\\varphi $ .", "Using this new characterisation, we will devise a decision procedure that runs in logarithmic space and consists of two steps: it replaces every occurrence of any state variable $x$ in $\\varphi $ with 1 if its state of evaluation agrees with that of its $x$ -superformula, and with 0 otherwise; it then removes all $\\Box $ -, $$ - and $$ -operators from the formula and tests whether the resulting Boolean formula is valid.", "In what follows, we want to restrict assignments to the finitely many free state variables occurring free in a given formula $\\varphi $ .", "For this purpose, we define the notion of a partial assignment $g : V \\rightarrow $ for $\\varphi $ where $V$ is a finite set of state variables with $_\\varphi \\subseteq V$ , i.e., $g$ is defined for all state variables free in $\\varphi $ .", "Here we include subscripts of the $$ -operator in the notion of a free state variable: for example, $x._x_yz$ has free state variables $y,z$ .", "The satisfaction relation $\\models $ for partial assignments is analogously defined to the definition in Section .", "For a partial assignment $g$ for $x.\\alpha $ and $i \\in $ , it holds that $g,i \\models x.\\alpha $ iff $g^x_i,i \\models \\alpha $ .", "Clearly, if $g$ is a partial assignment for $x.\\alpha $ , then $g^x_i$ is one for $\\alpha $ .", "The definition of the satisfaction relation implies that the satisfaction of $\\Box \\alpha $ at $g,i$ depends on the satisfaction of $\\Box \\alpha $ at infinitely many states (natural numbers) in $g$ .", "However, we will now show that the latter can be reduced to satisfaction in the smallest natural number to which $g$ does not bind any state variable.", "This will later imply that satisfiability of a given formula $\\varphi $ can be tested by evaluating its subformulas in their uniquely determined states $g,i$ of evaluation.", "Given a partial assignment $g : V \\rightarrow \\mathbb {N}$ , define $ n_g = \\max \\lbrace g(x) \\mid x \\in V\\rbrace + 1.", "$ For every $\\varphi \\in (\\Box ,,)$ , every partial assignment $g$ for $\\varphi $ and every $i \\in \\mathbb {N}$ , it holds that $g,i \\models \\Box \\varphi $ iff $g,n_g \\models \\varphi $ .", "We will prove this lemma later, using the following lemma.", "Let $\\varphi \\in (\\Box ,,)$ , let $i,j \\in \\mathbb {N}$ , and let $g,h$ be partial assignments for $\\varphi $ that satisfy the following two conditions: $g^{-1}(i) \\subseteq h^{-1}(j)$ .", "(All state variables free in $\\varphi $ and bound to $i$ by $g$ are bound to $j$ by $h$ .)", "For all $a,b \\in _\\varphi $ : if $g(a) = g(b)$ , then $h(a) = h(b)$ .", "(Whenever $g$ binds two state variables free in $\\varphi $ to one and the same state, so does $h$ .)", "Then $g,i \\models \\varphi $ implies $h,j \\models \\varphi $ .", "We proceed by induction on $\\varphi $ .", "In the base case $\\varphi \\in $ , we obtain the desired implication directly from (REF ).", "For the induction step, we distinguish between the possible cases for the outermost operator of $\\varphi $ .", "The Boolean cases are straightforward; the other cases are dealt with as follows.", "In case $\\varphi = \\Box \\psi $ , the following chain of (bi-)implications holds.", "$g,i \\models \\Box \\psi & ~\\Leftrightarrow ~ \\forall i^{\\prime } > i : g, i^{\\prime } \\models \\psi \\\\& ~\\Rightarrow ~ g,n_g \\models \\psi \\\\& ~\\Rightarrow ~ h,n_h \\models \\psi \\\\& ~\\Rightarrow ~ \\forall j^{\\prime } \\in \\mathbb {N} : h, j^{\\prime } \\models \\psi \\\\& ~\\Rightarrow ~ \\forall j^{\\prime } > j : h, j^{\\prime } \\models \\psi \\\\& ~\\Leftrightarrow ~ h,j \\models \\Box \\psi $ The first “$\\Rightarrow $ ” is immediate in case $i < n_g$ .", "Otherwise, if $i \\geqslant n_g$  , observe that $g^{-1}(\\mbox{i+1}) = \\emptyset = g^{-1}(n_g)$ .", "Hence we can apply the induction hypothesis (IH) to $\\psi ,\\mbox{i+1},n_g,g,h$ because $g$ is also a partial assignment for $\\psi $ , the assumption (REF ) of the IH is satisfied, and (REF ) follows from the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "The second “$\\Rightarrow $ ” is due to the IH applied to $\\psi ,n_g,n_h,g,h$ .", "Its assumption (REF ) is satisfied because $g^{-1}(n_g) = \\emptyset = h^{-1}(n_h)$ , and (REF ) follows from the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "The third “$\\Rightarrow $ ” is due to the IH applied to $\\psi ,n_h,j,h,h$ .", "Its assumption (REF ) is satisfied because $h^{-1}(n_h) = \\emptyset = h^{-1}(j)$ , and (REF ) is obvious because $h=h$ .", "In case $\\varphi = x.\\psi $ , the following chain of (bi-)implications holds.", "$g,i \\models x.\\psi & ~\\Leftrightarrow ~ g^x_i,i \\models \\psi \\\\& ~\\Rightarrow ~ h^x_j,j \\models \\psi \\\\& ~\\Leftrightarrow ~ h,j \\models x.\\psi $ The implication in the middle is obtained by observing that $g^x_i,h^x_i$ are partial assignments for $\\psi $ because $g,h$ are partial assignments for $\\varphi $ , and applying the IH to $\\psi ,i,j,g^x_i,h^x_j$ .", "Its assumption (REF ) is satisfied because of the following chain of equalities and inclusions, whose middle step follows from the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "${(g^x_i)}^{-1}(i) = g^{-1}(i) \\cup \\lbrace x\\rbrace \\subseteq h^{-1}(i) \\cup \\lbrace x\\rbrace = {(h^x_j)}^{-1}(j)$ Assumption (REF ) of the IH is satisfied for the following reason.", "Let $a,b \\in _\\psi $ with $g(a) = g(b)$ .", "In case $a=b=x$ , both $(g^x_i)(a) = (g^x_i)(b)$ and $(h^x_j)(a) = (h^x_j)(b)$ hold.", "In case $a=x$ and $b\\ne x$ , we have that $(g^x_i)(a) = (g^x_i)(b)$ implies $(g^x_i)(b) = i$ , which implies $g(b) = i$ because $b\\ne x$ .", "This implies $h(b) = j$ due to the assumption (REF ) for $\\varphi ,i,j,g,h$ and because $b \\in _\\varphi $ .", "Hence $(h^x_j)(a) = (h^x_j)(b)$ .", "The case $a\\ne x$ and $b=x$ is analogous to the previous one, and in case $a\\ne x$ and $b\\ne x$ , we have that $(g^x_i)(a) = (g^x_i)(b)$ implies $g(a) = g(b)$ , which implies $h(a) = h(b)$ due to the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "Hence $(h^x_j)(a) = (h^x_j)(b)$ .", "In case $\\varphi = _x.\\psi $ , the following chain of (bi-)implications holds.", "$g,i \\models _x\\psi & ~\\Leftrightarrow ~ g,g(x) \\models \\psi \\\\& ~\\Rightarrow ~ h,h(x) \\models \\psi \\\\& ~\\Leftrightarrow ~ h,j \\models _x\\psi $ The implication in the middle is obtained by observing that $g,h$ are also partial assignments for $\\psi $ , and applying the IH to $\\psi ,g(x),h(x),g,h$ .", "Its assumption (REF ) is satisfied: consider $y \\in g^{-1}(g(x))$ .", "Then $g(x) = g(y)$ , which implies $h(x) = h(y)$ due to the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "Hence $y \\in h^{-1}(h(x))$ .", "This establishes $g^{-1}(g(x)) \\subseteq h^{-1}(h(x))$ .", "The assumption (REF ) for the IH follows from the assumption (REF ) for $\\varphi ,i,j,g,h$ .", "Before we can prove Lemma , we observe the following consequence of Lemma .", "For every $\\varphi \\in (\\Box ,,)$ , every partial assignment $g$ for $\\varphi $ and every $i \\in \\mathbb {N}$ with $g^{-1}(i) = \\emptyset $ , it holds that $g,i \\models \\varphi $ implies $g,j \\models \\varphi $ for all $j \\in \\mathbb {N}$ .", "It suffices to observe that the assumptions of Lemma are satisfied by $\\varphi ,i,j,g,g$ with $j \\in \\mathbb {N}$ arbitrary.", "(REF ) follows from $g^{-1}(i) = \\emptyset $ , and (REF ) holds trivially because $g=g$ .", "We can now proceed to prove Lemma   ($\\forall \\varphi ,g,i ~:~ g,i \\models \\Box \\varphi \\Leftrightarrow g,n_g \\models \\varphi $ ).", "[Proof of Lemma ] For the direction “$\\Rightarrow $ ”, assume that $g,i \\models \\Box \\varphi $ , i.e., for all $j > i$ , it holds that $g,j \\models \\varphi $ .", "In case $i < n_g$ , the consequence $g,n_g \\models \\varphi $ is immediate.", "Otherwise, in case $i \\geqslant n_g$ , we conclude $g,i+1 \\models \\varphi $ from $g,i \\models \\Box \\varphi $ .", "Since $g^{-1}(i+1) = \\emptyset $ in this case, we can use Corollary to conclude that $g,j \\models \\varphi $ for all $j \\in \\mathbb {N}$ , and in particular for $j = n_g$ .", "For the direction “$\\Leftarrow $ ”, assume that $g,n_g \\models \\varphi $ .", "Then Corollary implies that $g,j \\models \\varphi $ for all $j \\in \\mathbb {N}$ , and in particular for all $j > i$ .", "Hence $g,i \\models \\Box \\varphi $ .", "Using Lemma , we are now in a position to show that every satisfiable formula is satisfied by a canonical assignment $g_0^\\varphi $ in the state 0.", "We will furthermore use the characterisation of satisfaction for $\\Box $ -formulas in Lemma to establish that the question whether $g_0^\\varphi ,0 \\models \\varphi $ can be reduced to checking satisfaction of $\\varphi $ 's subformulas in uniquely determined states and assignments.", "Let $\\varphi \\in (\\Box ,,)$ .", "The canonical assignment $g_0^\\varphi $ for $\\varphi $ is the partial assignment for $\\varphi $ that maps all $x \\in _\\varphi $ to 0 and is undefined for all other state variables.", "Let $\\varphi \\in (\\Box ,,)$ .", "Then $\\varphi \\in [](\\Box ,,)$ iff $g_0^\\varphi , 0 \\models \\varphi $ .", "The “if” direction is obvious.", "The converse is a consequence of the following claim.", "For every $\\varphi \\in (\\Box ,,)$ , every partial assignment $g$ for $\\varphi $ and every $i \\in \\mathbb {N}$ : if $g,i \\models \\varphi $ , then $g_0^\\varphi , 0 \\models \\varphi $ .", "We proceed by induction on $\\varphi $ .", "The base case $\\varphi = x \\in $ is true because $g_0^x,0 \\models x$ holds.", "For the induction step, the Boolean cases are straightforward.", "The other cases are treated as follows.", "In case $\\varphi = \\Box \\psi $ , the following chain of implications holds.", "$g,i \\models \\Box \\psi & ~\\Rightarrow ~ g,n_g \\models \\psi \\\\& ~\\Rightarrow ~ g_0^\\psi ,1 \\models \\psi \\\\& ~\\Rightarrow ~ g_0^\\varphi ,1 \\models \\psi \\\\& ~\\Rightarrow ~ g_0^\\varphi ,0 \\models \\Box \\psi $ The first implication is due to Lemma , and the second uses Lemma for $\\psi ,g,g_0^\\psi ,n_g,1$ : remember that $g,g_0^\\psi $ are for $\\psi $ , and observe that the assumptions of Lemma are satisfied because $g^{-1}(n_g) = \\emptyset = {(g_0^\\psi )}^{-1}(1)$ and $g_0^\\psi (a) = 0 = g_0^\\psi (b)$ for all $a,b \\in _\\psi $ .", "The third implication holds because $g^\\psi _0 = g^\\varphi _0$ , and the fourth uses Lemma .", "In case $\\varphi = x.\\psi $ , the following chain of implications holds.", "$g,i \\models x.\\psi & ~\\Leftrightarrow ~ g^x_i,i \\models \\psi \\\\& ~\\Rightarrow ~ g^\\psi _0,0 \\models \\psi \\\\& ~\\Rightarrow ~ {(g^\\varphi _0)}^x_0,0 \\models \\psi \\\\& ~\\Leftrightarrow ~ g^\\varphi _0,0 \\models x.\\psi $ The first “$\\Rightarrow $ ” is due to the induction hypothesis, and the second uses $g^\\psi _0 = {(g^\\varphi _0)}^x_0$ .", "In case $\\varphi = _x\\psi $ , the following chain of implications holds.", "$g,i \\models _x\\psi & ~\\Leftrightarrow ~ g,g(x) \\models \\psi \\\\& ~\\Rightarrow ~ g^\\psi _0,0 \\models \\psi \\\\& ~\\Rightarrow ~ {(g^\\psi _0)}^x_0,0 \\models _x\\psi \\\\& ~\\Leftrightarrow ~ g^\\varphi _0,0 \\models _x\\psi $ The first “$\\Rightarrow $ ” is due to the induction hypothesis, and the second uses $g^\\psi _0 = {(g^\\varphi _0)}^x_0$ ; note that $g^\\psi _0 = g^\\varphi _0$ does not necessarily hold because $x$ might not be free in $\\psi $ .", "Using Theorem and Lemma , we can now assign a unique assignment and state of evaluation to every subformula of a given formula $\\varphi $ .", "This will lead us to characterize satisfiability of a given formula $\\varphi $ by validity of the Boolean formula obtained from $\\varphi $ by (a) replacing every free state variable $x$ with 0 or 1, depending on the compatibility between unique assignment and state of evaluation for $x$ , and (b) removing all non-Boolean operators.", "After establishing this criterion, we will show that the transformation can be achieved deterministically in logarithmic space.", "Fix a formula $\\varphi \\in (\\Box ,,)$ whose satisfiability is to be tested.", "We denote subformulas of $\\varphi $ as pairs $(\\psi ,p)$ , where $p \\in \\mathbb {N}$ denotes the position of $\\psi $ in (the string that represents) $\\varphi $ .", "This is necessary to distinguish between different occurrences of the same subformula in $\\varphi $ .", "The position of a subformula is always the position of its first character in the string representing $\\varphi $ .", "If the subformula is $(\\alpha \\wedge \\beta )$ or $(\\alpha \\vee \\beta )$ , then the position of the opening parenthesis is relevant.", "Consequently, $\\varphi $ has always position 0.", "For a position $p$ in $\\varphi $ , denote by $_1(p)$ and $_2(p)$ the position of the immediate subformulas of the subformula at position $p$ : if the subformula of $\\varphi $ at $p$ is $(\\alpha \\vee \\beta )$ or $(\\alpha \\wedge \\beta )$ , then $_1(p)$ and $_2(p)$ are the positions of $\\alpha $ and $\\beta $ , respectively; $\\Box \\alpha $ , $x.\\alpha $ or $_x\\alpha $ , then $_1(p)$ is the position of $\\alpha $ , and $_2(p)$ is undefined; is any other formula, then both $_1(p)$ and $_2(p)$ are undefined.", "We now define a unique state of evaluation $^\\varphi (\\psi ,p)$ for a subformula $\\psi $ of $\\varphi $ at position $p$ recursively on $p$ as follows.", "$^\\varphi (\\varphi ,0) = (g_0^\\varphi ,0)$ .", "For $\\circ \\in \\lbrace \\wedge ,\\vee \\rbrace $ , if $^\\varphi ((\\alpha \\circ \\beta ), p) = (g,i)$ , then $^\\varphi (\\alpha , _1(p)) = ^\\varphi (\\beta , _2(p)) = (g,i)$ .", "If $^\\varphi (\\Box \\alpha ,p) = (g,i)$ , then $^\\varphi (\\alpha ,_1(p)) = (g,n_g)$ .", "If $^\\varphi (x.\\alpha ,p) = (g,i)$ , then $^\\varphi (\\alpha ,_1(p)) = (g^x_i,i)$ .", "If $^\\varphi (_x\\alpha ,p) = (g,i)$ , then $^\\varphi (\\alpha ,_1(p)) = (g,g(x))$ .", "Observe that the first component in $^\\varphi (\\psi ,p)$ is always a partial assignment for $\\psi $ .", "Now consider a subformula $(x,p)$ of $\\varphi $ with $x \\in $ and $^\\varphi (x,p) = (g,i)$ .", "We define a function $^\\varphi $ mapping $x$ to $\\top $ if $g(x) = i$ (i.e., $x$ is satisfied at $^\\varphi (x,p)$ ), and to $\\bot $ otherwise.", "Using $^\\varphi $ , we now recursively define a function $^\\varphi $ mapping subformulas of $\\varphi $ to Boolean formulas with only monotone operators and without propositional variables: $^\\varphi (x,p) & = ^\\varphi (x,p),\\quad x \\in \\\\^\\varphi (c,p) & = c, \\quad c \\in \\lbrace \\top ,\\bot \\rbrace \\\\^\\varphi (\\alpha \\circ \\beta ,p) & = ^\\varphi (\\alpha , _1(p)) \\circ ^\\varphi (\\beta , _2(p)),\\quad \\circ \\in \\lbrace \\wedge ,\\vee \\rbrace \\\\^\\varphi (\\Delta \\alpha ,p) & = ^\\varphi (\\alpha , _1(p)),\\quad \\Delta \\in \\lbrace \\Box ,x,_x\\rbrace $ Furthermore, let $(\\varphi ) = ^\\varphi (\\varphi ,0)$ .", "Let $\\varphi \\in (\\Box ,,)$ .", "For all subformulas $(\\psi ,p)$ of $\\varphi $ , it holds that $^\\varphi (\\psi ,p) \\models \\psi $ iff $^\\varphi (\\psi ,p)$ is valid.", "We proceed by induction on $\\psi $ .", "Let $^\\varphi (\\psi ,p) = (g,i)$ .", "The base case $\\psi =x$ follows from the definition of $^\\varphi (x,p)$ and $^\\varphi (x,p)$ .", "For the inductive step, the cases $\\psi =\\top ,\\bot $ follow from the definition of $^\\varphi $ .", "The other cases are as follows.", "In case $\\psi = \\alpha \\vee \\beta $ , we observe the following chain of equivalent statements.", "$g,i\\models \\alpha \\vee \\beta & ~\\Leftrightarrow ~ g,i \\models \\alpha \\text{~or~} g,i \\models \\beta \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha ,_1(p)) \\models \\alpha \\text{~or~}^\\varphi (\\beta ,_1(p)) \\models \\beta \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha ,_1(p)) \\text{~is valid or~}^\\varphi (\\beta ,_1(p)) \\text{~is valid} \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha ,_1(p)) \\vee ^\\varphi (\\beta ,_1(p)) \\text{~is valid} \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha \\vee \\beta ,p) \\text{~is valid}$ The second equivalence is due to the definition of $^\\varphi $ , the third uses the induction hypothesis, and the fifth is due to the definition of $^\\varphi $ .", "The case $\\psi = \\alpha \\wedge \\beta $ is analogous.", "In case $\\psi = \\Box \\alpha $ , we observe the following chain of equivalent statements.", "$g,i\\models \\Box \\alpha & ~\\Leftrightarrow ~ g,n_g \\models \\alpha \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha ,_1(p)) \\models \\alpha \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\alpha ,_1(p)) \\text{~is valid} \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\Box \\alpha ,p) \\text{~is valid}$ The first equivalence uses Lemma , the second is due to the definition of $^\\varphi $ , the third uses the induction hypothesis, and the fourth is due to the definition of $^\\varphi $ .", "The cases $\\psi = x.\\alpha $ and $\\psi = _x\\alpha $ are analogous to the previous one, but with the first equivalence via the definition of satisfaction.", "Let $\\varphi \\in (\\Box ,,)$ .", "Then $\\varphi \\in [](\\Box ,,)$ iff $(\\varphi )$ is valid.", "The following chain of equivalences holds.", "$\\varphi \\text{~is satisfiable}& ~\\Leftrightarrow ~ g^\\varphi _0,0 \\models \\varphi \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\varphi ,0) \\models \\varphi \\\\& ~\\Leftrightarrow ~ ^\\varphi (\\varphi ,0) \\text{~is valid} \\\\& ~\\Leftrightarrow ~ (\\varphi ,0) \\text{~is valid}$ The first equivalence follows from Theorem , the second uses the definition of $^\\varphi $ , the third is due to Lemma , and the fourth uses the defintion of $$ .", "The function $$ is a reduction of $[](\\Box ,,)$ to the formula value problem for Boolean formulas with only monotone operators, which is in 1 [6].", "The correctness of this reduction is shown in Theorem .", "To establish that $[](\\Box ,,) \\in $ , it remains to show that $(\\varphi )$ can be computed in logarithmic space.", "The procedure BOOL, which will accomplish this task, will traverse its input formula $\\varphi $ from left to right, and send the character $c$ read at position $p$ to the output unchanged, unless one of the following two cases occurs.", "If $c$ belongs to a $\\Box $ -, $x.$ -, or $_x$ -operator, then $c$ is ignored.", "If $c$ is a free state variable $x$ , then $^\\varphi (x,p)$ is computed and sent to the output instead of $c$ .", "Given the definition of $$ , $^\\varphi $ and $^\\varphi $ , this is obviously a correct decision procedure provided that $^\\varphi (x,p)$ is computed by a correct subroutine REP, which we still have to describe.", "The procedure BOOL is given in Algorithm .", "Procedure BOOL $\\varphi \\in (\\Box ,,)$ output $(\\varphi )$ $p \\leftarrow 0$ $p < |\\varphi |$ an operator $\\Box $ , $x.$ or $_x$ starts at position $p$ $p \\leftarrow \\text{position immediately following that operator}$ a state variable $x$ starts at position $p$ output ${REP}(\\varphi , x, p)$ $p \\leftarrow \\text{position immediately following~} x$ output character at position $p$ $p \\leftarrow p+1$ To compute $^\\varphi (x,p)$ using the procedure REP, we make the following crucial observation about states of evaluation.", "The operators $\\Box $ and $_x$ are jumping operators: $^\\varphi (\\Box \\psi ,\\cdot )$ and $^\\varphi (\\psi ,\\cdot )$ may differ in their second component; the same holds for $^\\varphi (_x \\psi ,\\cdot )$ and $^\\varphi (\\psi ,\\cdot )$ .", "Such a difference does not occur between formulas starting with one of the other operators $x.$ , $\\wedge $ , $\\vee $ , and their direct subformulas.", "This observation can be used to compute $^\\varphi (x,p)$ because that value depends on the question whether there is a jumping operator between the position $q$ where $x$ is bound and the position $p$ of $x$ .", "Assume that this binder $x.$ leads the subformula $x.\\psi $ , and that $^\\varphi (x.\\psi ,q) = (g,i)$ and $^\\varphi (x,p) = (h,j)$ .", "We distinguish the following cases.", "Case 1.", "If there is no jumping operator between $(x,p)$ and $(x.\\psi ,q)$ , then it follows from the definition of $^\\varphi $ that $g(x) = i$ , $g(x) = h(x)$ , and $i=j$ – all three statements can be shown inductively on the positions in $\\varphi $ .", "They imply that $h(x) = j$ , hence $^\\varphi (x,p) = \\top $ .", "Case 2.", "Let $\\circ $ be the last jumping operator occurring between positions $q$ and $p$ .", "More precisely, let $r$ be the position between $q$ and $p$ such that the operator $\\circ $ at position $r$ is a jumping operator, that operator is in the scope of $(x.,q)$ and has $(x,p)$ in its scope, and there is no jumping operator in the scope of $(\\circ ,r)$ that has $(x,p)$ in its scope.", "Let $\\circ \\vartheta $ be the subformula at position $r$ .", "Case 2.1.", "If $\\circ = \\Box $ , then the definition of $^\\varphi $ implies that $^\\varphi (\\Box \\vartheta ,r) = (g, n_g)$ for some partial assignment $g$ .", "Since $x$ is not bound between $r$ and $p$ , and since no jumping operator occurs between $r$ and $p$ , we conclude from the definition of $^\\varphi $ that $h(x) \\ne n_g$ and $j = n_g$ .", "Hence $h(x) \\ne j$ , and $^\\varphi (x,p) = \\bot $ .", "Case 2.2.", "If $\\circ = _y$ , then let $(y.\\eta , s)$ be the subformula “above” $_y\\vartheta $ that binds $y$ , with $^\\varphi (y.\\eta ) = (g^{\\prime },i^{\\prime })$ and $^\\varphi (_y.\\vartheta ) = (h^{\\prime },j^{\\prime })$ .", "Then it holds that (a) $g(x) = h(x)$ , due to the definition of $^\\varphi $ and because $x$ is not bound between $q$ and $p$ , and (b) $j=h(y)=h^{\\prime }(y)=g^{\\prime }(y)$ , which follows from the definition of $^\\varphi $ for $_y$ -formulas and the fact that $y$ is not bound between $s$ and $p$ .", "Therefore we have that $^\\varphi (x,p) = \\top $ iff $g(x) = g^{\\prime }(x)$ .", "This new criterion compares states of evaluations of subformulas at smaller positions in $\\varphi $ , and it can be decided applying the same case distinction to those two subformulas.", "We therefore obtain a recursive procedure REP for deciding whether $^\\varphi = \\top $ .", "For every recursive call according to Case 2.2, a pair of subformulas at smaller positions in $\\varphi $ is compared.", "Therefore, the recursion has to terminate after at most $|\\varphi |$ steps.", "Since the result of a recursive call does not need to be processed any further, REP can be implemented using end-recursion, i.e., without a stack.", "Together with the fact that only a constant number of position counters are needed (and, consequently, determining the last jumping operator between two positions in $\\varphi $ can be implemented in logarithmic space), Algorithm REF runs in logarithmic space.", "The previous considerations imply its correctness.", "Figure: Procedure REP'Theorem $[](\\Box ,,)$ is in ." ], [ "Proof of Theorem ", "Theorem $(\\Diamond ,\\Box ,)$ has the quasi-quadratic size model property with respect to $$ and $$ .", "We will develop a “quasi-quadratic size model property” for the logic $(\\Diamond ,\\Box ,)$ over $$ , and we will subsequently show how to extend the result to the other fragments from Theorem and how to restrict them to .", "In the appendix, we even sketch how to obtain an NP decision procedure for these fragments over $$ , $$ and the frame class $\\lbrace (,<)\\rbrace $ .", "Consider an arbitrary model $K = (W,<,\\eta )$ , and call all states in the range of $g$ nominal states.", "For every non-nominal state $w \\in W$ , let $\\delta (w)$ be the number of states between $w$ and the next nominal state $s$ .", "If the next nominal state is a direct successor, then $\\delta (w) = 0$ ; if there are infinitely many intermediary states—i.e., at least a part of the interval between $w$ and $s$ is dense—, then $\\delta (w) = \\infty $ .", "For every $m \\geqslant 0$ , we now define an equivalence relation $\\equiv _m$ on $W$ as follows.", "$w \\equiv _m w^{\\prime }$ if either $w=w^{\\prime }$ or both $w,w^{\\prime }$ are non-nominal states and $\\delta (w) > m$ and $\\delta (w^{\\prime }) > m$ .", "Figure REF gives an example for $m=3$ ; equivalence classes are denoted by dashed rectangles.", "The $i_j$ are nominal states, and of the 8 states between $i_2$ and $i_3$ , the rightmost three form separate equivalence classes, and the others form a single equivalence class.", "Figure: An example for m=3m=3The intuition behind this equivalence relation is that $w$ and $w^{\\prime }$ cannot be distinguished by formulas of modal depth $\\leqslant m$ .", "If $w \\equiv _m w^{\\prime }$ , we call $w$ and $w^{\\prime }$ $m$ -inseparable, and we denote the equivalence class of $w$ w.r.t.", "$\\equiv _m$ by $[w]_m$ .", "The definition of $\\equiv _m$ has the consequence $[w]_m \\subseteq [w]_{m-1}$ , for all $m > 0$ .", "It is possible to enforce dense parts in satisfying models, for instance via the following formula, which is satisfiable in a linear structure only if that structure ends with a state satisfying the nominal $j$ , and that state needs to be the end point of a dense interval.", "This formula is therefore not satisfiable over .", "$\\varphi _d = i \\wedge \\Diamond \\Diamond j \\wedge \\Box (j \\vee \\Diamond \\Diamond j)$ For this reason, an equivalence class can also consist of infinitely many states.", "In the case of a model satisfying $\\varphi _d$ , all points between $i$ and $j$ belong to the same equivalence class because all these points have an infinite distance to $j$ .", "The following lemma states that $m$ -inseparable states cannot be distinguished by formulas of modal depth $\\leqslant m$ .", "For every $m \\geqslant 0$ , every formula $\\varphi \\in (\\Diamond ,\\Box ,)$ with $(\\varphi ) \\leqslant m$ , every linear model $K = (W,<,\\eta )$ , and all $w,w^{\\prime } \\in W$ with $w \\equiv _m w^{\\prime }$ : $K,w \\models \\varphi \\quad \\Leftrightarrow \\quad K,w^{\\prime } \\models \\varphi .$ We proceed by induction on the structure of $\\varphi $ .", "The case for nominals is obvious because nominal states are $m$ -inseparable only from themselves.", "The Boolean cases are straightforward.", "$\\varphi = \\Diamond \\psi $ .", "For symmetry reasons, it suffices to show “$\\Rightarrow $ ”.", "Let $K,w \\models \\Diamond \\psi $ and $w \\equiv _m w^{\\prime }$ .", "Then there is some $v>w$ with $K,v \\models \\psi $ .", "We now distinguish several cases of how $w,w^{\\prime },v$ are located in relation to each other.", "$w^{\\prime }<w$ .", "Then $w<v$ implies $w^{\\prime }<v$ , and hence $K,w^{\\prime } \\models \\Diamond \\psi $ .", "$w\\leqslant w^{\\prime } < v$ .", "Then, still, $w^{\\prime }<v$ , and hence $K,w^{\\prime } \\models \\Diamond \\psi $ .", "$w<v\\leqslant w^{\\prime }$ .", "Since $w \\equiv _m w^{\\prime }$ , we have $w \\equiv _m v \\equiv _m w^{\\prime }$ .", "In case $|[w]_m| < \\infty $ , there are exactly $m$ states between $[w]_m$ and the next nominal state.", "Let $v^{\\prime }$ be the $<$ -least of them; then $w \\equiv _{m-1} v \\equiv _{m-1} w^{\\prime }\\equiv _{m-1} v^{\\prime }$ .", "Since $(\\psi ) = m-1$ , we get $K,v^{\\prime } \\models \\psi $ via the induction hypothesis.", "Hence, $K,w^{\\prime } \\models \\Diamond \\psi $ .", "In case $|[w]_m| = \\infty $ , we conclude that at least a subinterval of $[w]_m$ is dense, and therefore $w^{\\prime }$ has a successor $v^{\\prime }$ in $[w]_m \\subseteq [w]_{m-1}$ .", "We can continue the argument as in the previous case.", "$\\varphi = \\Box \\psi $ .", "As above, it suffices to show “$\\Rightarrow $ ”.", "Let $K,w \\models \\Box \\psi $ and $w \\equiv _m w^{\\prime }$ .", "Then, for all $v>w$ , we have that $K,v \\models \\psi $ .", "Again, we consider the two cases $|[w]_m| < \\infty $ and $|[w]_m| = \\infty $ , and fix the same $v^{\\prime }$ as above.", "Since $v^{\\prime }$ is $(m-1)$ -inseparable from $w$ and $w^{\\prime }$ , $\\psi $ is also satisfied by all states in $[w]_m$ .", "Therefore, $K,v \\models \\psi $ for all $v > w^{\\prime }$ , hence, $K,w^{\\prime } \\models \\Box \\psi $ .", "$\\varphi = _i\\psi $ .", "Then $K,w \\models _i\\psi ~\\Leftrightarrow ~ K,v \\models \\psi \\text{~for any~} v ~\\Leftrightarrow ~ K,w^{\\prime } \\models _i\\psi $ .", "We now use this inseparability result to reduce a satisfying model in size such that it can be represented in polynomial space.", "Fix a formula $\\varphi $ with $(\\varphi ) = m$ and a linear model $K$ with $K,w \\models \\varphi $ for some state $w$ .", "If it were not possible to enforce dense intervals, it would suffice to collapse every $m$ -equivalence class of $K$ to a single point, i.e., the quotient model of $K$ w.r.t.", "$\\equiv _m$ would satisfy $\\varphi $ at $[w]_m$ .", "This would serve our purpose over .", "In contrast, an infinite equivalence class (IEC)—which has to contain a dense subinterval—needs to remain dense for the next lemma to work.", "For a uniform representation, we replace any IEC with a copy of $(0,1)_$ , the open interval of all rationals between 0 and 1.", "Since a dense interval can be of higher cardinality than $(0,1)_$ —just consider , for example—, we cannot expect to map every point of an IEC M to a point in the associated copy of $(0,1)_$ .", "Instead, we use a surjective partial morphism $f: (M,<) \\rightarrow (0,1)_$ , i.e., a partial function that satisfies the equivalence $x=y \\Leftrightarrow f(x) = f(y)$ for all $x,y \\in M$ and whose range is all of .", "These conditions ensure that every $x \\in (f)$ has a successor $y \\in (f)$ with $f(x) < f(y)$ .", "Such a function always exists: since every IEC $[w]_m$ contains a dense subinterval, it also contains an isomorphic copy of $(0,1)_$ .", "The refined “quotient” model $K_m = (W_m, <_m, \\eta _m)$ is now constructed as follows.", "For every infinite $[w]_m$ , let ${[w]_m}$ be a fresh copy of $(0,1)_$ .", "We set $\\displaystyle W_k = \\biguplus _{|[w]_m| = \\infty } {[w]_m} \\quad \\uplus \\lbrace [w]_m : |[w]_m| < \\infty \\rbrace $ $[w]_m <_m [v]_m$ if $[w]_m$ and $[v]_m$ are finite and $w^{\\prime } < v^{\\prime }$ for some $w^{\\prime } \\in [w]_m$ and $v^{\\prime } \\in [v]_m$ $q <_m q^{\\prime }$ if $q,q^{\\prime } \\in {[w]_m}$ for some $w$ with $|[w]_m| = \\infty $ , and $q<q^{\\prime }$ on $(0,1)_$ $q <_m [v]_m$ if $q \\in {[w]_m}$ for some $w$ with $|[w]_m| = \\infty $ , $[v]_m$ is finite, and $w < v^{\\prime }$ for some $v^{\\prime } \\in [v]_m$ $[w]_m <_m q^{\\prime }$ if $q^{\\prime } \\in {[v]_m}$ for some $v$ with $|[v]_m| = \\infty $ , $[w]_m$ is finite, and $w^{\\prime } < v$ for some $w^{\\prime } \\in [w]_m$ $\\eta _m(i) = [\\eta (i)]_m$ We also define a model reduction function for $K$ to be a surjective partial function $f : K \\rightarrow K_m$ with the following conditions.", "If $|[w]_m| < \\infty $ , then $f(w^{\\prime }) = [w]_m$ for all $w^{\\prime } \\in [w]_m$ .", "If $|[w]_m| = \\infty $ , then $f(w^{\\prime }) = g(w^{\\prime })$ for all $w^{\\prime } \\in [w]_m$ , for some surjective partial morphism $g : [w]_m \\rightarrow {[w]_m}$ .", "For every $m \\geqslant 0$ , every formula $\\varphi \\in (\\Diamond ,\\Box ,)$ with $(\\varphi ) \\leqslant m$ , every linear model $K = (W,<,\\eta )$ , every model reduction function $f$ for $K$ and all $w \\in (f)$ : $K,w \\models \\varphi \\quad \\Leftrightarrow \\quad K_m,f(w) \\models \\varphi .$ We proceed by induction on $\\varphi $ .", "The atomic and Boolean cases are straightforward again.", "$\\varphi = \\Diamond \\psi $ .", "Let $K,w \\models \\varphi $ .", "Case 1: $|[w]_m| < \\infty $ .", "Let $w^{\\prime }$ be the $<$ -greatest member of $[w]_k$ .", "Due to Lemma , $K,w^{\\prime } \\models \\Diamond \\psi $ .", "Therefore there is some $v > w^{\\prime }$ with $K,v \\models \\psi $ and $v \\lnot \\equiv _m w$ .", "If $|[v]_m| < \\infty $ , then $v \\in (f)$ , and the induction hypothesis yields $K_m,f(v) \\models \\psi $ .", "Since $w<v$ with $v \\lnot \\equiv _m w$ , we obtain $f(w)<_m f(v)$ , hence $K_m,f(w) \\models \\psi $ .", "If $|[v]_m| = \\infty $ , we use $[v]_m \\subseteq [v]_{m-1}$ and conclude from Lemma that $K,v^{\\prime } \\models \\psi $ for all $v^{\\prime } \\in [v]_m$ .", "Take such a $v^{\\prime }$ with $v^{\\prime } \\in (f)$ and apply the induction hypothesis as in the case $|[v]_m| < \\infty $ .", "Case 2: $|[w]_m| = \\infty $ .", "Since $K,w \\models \\varphi $ , there is some $v > w$ with $K,v \\models \\psi $ .", "If $v \\lnot \\equiv _m w$ , then we argue as in Case 1.", "Otherwise, we use Lemma to conclude that $K,v^{\\prime } \\models \\psi $ for all $v^{\\prime } \\in [w]_m$ .", "Since the restriction of $f$ to $[w]_m$ is a surjective morphism and $(0,1)_$ is dense, there is some $v^{\\prime } > w$ with $v^{\\prime } \\in [w]_m$ , $v^{\\prime } \\in (f)$ and $f(w) <_m f(v^{\\prime })$ .", "From $K,v^{\\prime } \\models \\psi $ we conclude via the induction hypothesis that $K_m,f(v^{\\prime }) \\models \\psi $ , hence $K_m,f(w) \\models \\Diamond \\psi $ .", "$\\varphi = \\Box \\psi $ .", "Let $K,w \\models \\Box \\psi $ , i.e., $K,v \\models \\psi $ for all $v > w$ .", "Then $K,v \\models \\psi $ for all $v$ with $v \\in (f)$ and $f(v) >_m f(w)$ .", "Due to the induction hypothesis, $K_m,f(v) \\models \\psi $ for all $v$ with $v \\in (f)$ and $f(v) >_m f(w)$ .", "Since $f$ is surjective, we have $K_m,v^{\\prime } \\models \\psi $ for all $v^{\\prime } \\in W_m$ with $v^{\\prime } >_m f(w)$ .", "Hence $K_m,f(w) \\models \\Box \\psi $ .", "$\\varphi = _i\\psi $ .", "Let $K,w \\models _i\\psi $ , i.e., $K,\\eta (i) \\models \\psi $ .", "Then $K_m,\\eta _m(i) \\models \\psi $ due to the induction hypothesis and the definition of $K_m$ .", "Hence $K_m,f(w) \\models _i\\psi $ .", "At this point, it is important to notice that, if $K$ is a model over , then so is $K_m$ .", "Therefore, Lemma gives us a quasi-quadratic size model property for $(\\Diamond ,\\Box ,)$ over $$ as well as – and also over $\\lbrace (,<)\\rbrace $ , see appendix.", "We say that a model $K$ is of size quasi-quadratic in an integer $m$ if every interval between two consecutive nominal states in $K$ consists of at most $m$ states, possibly with one preceding isomorphic copy of $(0,1)_$ .", "We furthermore say that a fragment $(O)$ has the quasi-quadratic size model property with respect to a frame class F if, for every $\\varphi \\in [{F}](O)$ , there exists a model over a frame in F that is of size quasi-quadratic in $(\\varphi )$ and satisfies $\\varphi $ .", "Theorem $(\\Diamond ,\\Box ,)$ has the quasi-quadratic size model property with respect to $$ and $$ .", "Let $K=(W,<,\\eta )$ be a linear model and $w_0 \\in W$ with $K,w_0 \\models \\varphi $ .", "Consider $\\varphi ^{\\prime } = i \\wedge \\varphi $ for a fresh nominal $i$ .", "Let $m = (\\varphi ) = (\\varphi ^{\\prime })$ .", "Then $\\varphi ^{\\prime }$ is satisfiable in the $w_0$ of the model $K^{\\prime }$ obtained from $K$ by interpreting $i$ in $w_0$ .", "Now take an arbitrary model reduction function $f$ for $K^{\\prime }$ , which has to have $w_0$ in its domain, and apply Lemma to obtain $K_m,f(w_0) \\models \\varphi ^{\\prime }$ ." ] ]

[ [ "Intrinsic Energy Dissipation in CVD-Grown Graphene Nanoresonators" ], [ "Abstract We utilize classical molecular dynamics to study the the quality (Q)-factors of monolayer CVD-grown graphene nanoresonators.", "In particular, we focus on the effects of intrinsic grain boundaries of different orientations, which result from the CVD growth process, on the Q-factors.", "For a range of misorientations orientation angles that are consistent with those seen experimentally in CVD-grown graphene, i.e.", "0$^{\\circ}$ to $\\sim20^{\\circ}$, we find that the Q-factors for graphene with intrinsic grain boundaries are 1-2 orders of magnitude smaller than that of pristine monolayer graphene.", "We find that the Q-factor degradation is strongly influenced by both the symmetry and structure of the 5-7 defect pairs that occur at the grain boundary.", "Because of this, we also demonstrate that find the Q-factors CVD-grown graphene can be significantly elevated, and approach that of pristine graphene, through application of modest (1%) tensile strain." ], [ "Introduction", "Since its recent discovery as the simplest two-dimensional crystal structure [1], graphene has been extensively studied not only for its unusual physical properties resulting from its two-dimensional structure [2], [3], [4], [5], but also for its potential as the basic building block of future applications, i.e.", "nanoelectromechanical systems (NEMS) [6], [7], [8], [9], [10].", "Graphene is viewed as an ideal material for NEMS-based sensing and detection applications due to its combination of extremely low mass and exceptional mechanical properties [11]; we note the recent review of Barton et al.", "[9] in this regard.", "However, one key issue limiting the applicability of graphene as a sensing component is its low quality (Q)-factor; the Q-factors of a 20-nm thick multilayer graphene sheet were found to range from 100 to 1800 as the temperature decreased from 300 K to 50 K [6].", "Similarly low Q-factors between 2 and 30 were also observed by Sanchez et al.", "[12] for multilayer graphene sheets, while higher Q-factors with values up to 4000 were reported using multilayer graphene oxide films [8].", "Theoretically, the Q-factors of graphene were recently studied using classical molecular dynamics (MD) simulations [13], [14], where spurious edge modes that are present in suspended graphene were proposed to have a key role in the low Q-factors that were observed experimentally.", "This hypothesis was recently validated by Barton et al.", "[15], who found Q-factors approaching 2000 for graphene resonators that were clamped on all slides, thus eliminating the spurious edge modes.", "These early experimental works on graphene nanoresonators utilized graphene flakes made via the scotch-tape method, which produces graphene sheets of varying thickness, though importantly, each graphene layer in the sheet is single crystalline.", "However, graphene research has been transformed by the recent development of the chemical-vapor-deposition (CVD) growth process to synthesize large area graphene sheets [16], [17], [18].", "CVD technology has the potential to revolutionize graphene-based sensing technology due to the resulting promise of wafer-scale graphene devices comprised of large arrays of single-layer graphene resonators [19].", "However, while large area graphene films are desirable for these and other graphene-based applications [20], the graphene films that result from CVD growth are polycrystalline, and thus are composed of many interconnected single crystalline graphene grains that intersect at grain boundaries having a range of misorientation angles [21], which generally are less than 20$^{\\circ }$  [22].", "While these grain boundaries are sometimes viewed favorably for tunable electronic devices [23], they are likely to have a deleterious effect for graphene nanoresonators because the misorientation and the resulting non-ideal bonding at the grain boundary causes an increase in phonon scattering, which creates another energy dissipation mechanism and a lower Q-factor.", "Some very recent experimental studies, such as those of van der Zande et al.", "[19] and Barton et al.", "[15] have studied CVD-grown graphene nanoresonators, and have found relatively high Q-factors on the order of about 2000.", "However, those works also improved the Q-factors by removing spurious edge modes [13], and therefore the intrinsic losses that occur in CVD-grown graphene due to the existence of the grain boundaries is unknown.", "Furthermore, previous theoretical studies [13], [14], [24] focused on the energy dissipation mechanisms in pure graphene without grain boundaries.", "Therefore, the objective of the present work is to quantify, via classical MD simulations, the intrinsic dissipation mechanisms introduced in CVD-grown graphene nanoresonators due to the presence of the grain boundaries." ], [ "Simulation Methodology", "The starting point of our simulations is to note that recent experimental studies have found that the misorientation angles at grain boundaries in CVD-grown graphene lie mostly between 0$^{\\circ }$ to 20$^{\\circ }$  [22].", "To systematically study the effects of different grain boundary orientations on the intrinsic loss mechanisms in graphene monolayers, we created graphene monolayers with a single grain boundary along the center line of the monolayer that runs along the armchair orientation with 6 different misorientation angles of 0$^{\\circ }$ (pristine graphene), 1.4$^{\\circ }$ , 5.31$^{\\circ }$ , 9.83$^{\\circ }$ , 12.83$^{\\circ }$ and 16.62$^{\\circ }$ .", "These configurations are shown in Fig.", "REF , where the diameter of all the graphene monolayers was 4 nm.", "The initial configuration including the single grain boundary was generated by rotating two semicircular graphene monolayers and then piecing them together.", "At that point, energy minimization using the conjugate gradient algorithm was employed to optimize the structure, where the carbon-carbon interactions were modeled using the AIREBO potential of [25], which is able to accurately simulate the forming and breaking of carbon bonds.", "A key point to note is that the creation of the grain boundary results in the formation of 5-7 defect ring pairs, which are colored in yellow in Fig.", "REF .", "We can observe that as the grain boundary misorientation angle increases, the density of the 5-7 ring pairs increases.", "For example, for the smallest misorientation angle of 1.4$^{\\circ }$ in Fig.", "REF (a), there exists only a single 5-7 unit ring pair near the center of the monolayer.", "In contrast, as the grain boundary misorientation angle increases, the density of the 5-7 ring pairs increases, as observed for the other cases in Fig.", "REF .", "Figure: (Color online) Schematic of the graphene monolayers containing grain boundaries with various misorientation angles of (a) 0 ∘ ^{\\circ };(b)1.4 ∘ ^{\\circ }; (c) 5.31 ∘ ^{\\circ }; (d) 9.83 ∘ ^{\\circ }; (e) 12.83 ∘ ^{\\circ }; (f) 16.62 ∘ ^{\\circ }.", "5-7 defect pairs are highlighted in yellow.The Q-factors were calculated and the intrinsic energy dissipation mechanisms studied using classical MD via the publicly available simulation code LAMMPS [26].", "After obtaining the equilibrium graphene monolayer structures with the various grain boundary misorientation angles as shown in Fig.", "REF , we then performed a thermal equilibration using a Nose-Hoover thermostat [27] for 500 ps using a time step of 1 fs, i.e.", "within an NVT ensemble.", "During the equilibration, the edges of the graphene sheet were constrained in plane while the rest of the sheet was left free to move.", "Previous theoretical [13] and experimental [15] works have demonstrated that spurious edge vibrational modes, which arise due to the undercoordinated nature of bonding at the edges of graphene, have a dominant role in reducing the Q-factors of suspended graphene nanoresonators; these edge modes would also be present for suspended CVD-grown nanoresonators, and thus should be eliminated to maximize the Q-factor.", "Because of this, after the thermal equilibration, the edges of the graphene monolayer were clamped at the equilibrium diameter that is established during the thermal equilibration to eliminate the possibility of spurious edge modes.", "During the thermal equilibration, out-of-plane buckling was observed due to the 5-7 defect pairs as observed in Fig.", "REF ; such buckles were previously observed in MD simulations by [28].", "We find that for the smallest grain boundary angle of 1.4$^{\\circ }$ at 3K, the height of the buckle is about 3.3Å.", "For larger grain boundary misorientation angles, there are more buckles along the grain boundary due to the larger number of 5-7 defect pairs, which interact and lead to a decrease in the buckling height.", "The impact of the buckles on the Q-factors will be elucidated later; in particular, we will demonstrate the utility of tensile mechanical strain in enhancing the Q-factor by flattening out the buckles.", "Figure: Out-of-plane buckling of 9.83 ∘ ^{\\circ } defective graphene sheet after thermal equilibration.", "Visualization performed using VMD .After the thermal equilibration, the graphene monolayer was actuated by assigning an initial sinusoidal velocity profile that ranged from zero at the clamped edges to a maximum at the center of the circular monolayer, and where the initial velocity was applied only in the vertical $z$ -direction to be perpendicular to the graphene sheets, and was chosen to be sufficiently small such that the resulting oscillation of the graphene monolayer would be purely harmonic, i.e.", "the resulting increase in total energy due to the applied sinusoidal velocity was only about 0.1%.", "While the buckling results in a non-planar graphene monolayer, for consistency, the direction of the applied initial velocities were the same for both pristine and defective graphene sheets.", "After the velocity profile was prescribed, the resulting free oscillation of the graphene monolayer was performed within an energy conserving (NVE) ensemble for 3000 ps.", "The Q-factors were calculated following the procedure described by [30] and [31].", "Specifically, as described by [30], the variation of the displacement of the center of mass of the graphene monolayer was tracked for the duration of the simulation after the initial velocity is applied.", "The decay in the root mean square displacement was then fit to the following exponential curve ($e^{-\\gamma \\omega t}$ ), which is then related to Q via Q$=0.5/\\zeta $ , where $\\zeta =\\gamma /\\omega $ is the damping ratio and $\\omega $ is the angular vibrational frequency.", "Specifically, because the vibrational motion is predominately in the $z$ -direction in our MD simulations, with little contribution from the motion in the $x$ and $y$ directions, we used the center of mass in the $z$ -direction only to fit the damping curve.", "This approach is utilized in the present work as it avoids the necessity of extracting the external energy as previously performed by [13], [14], [24].", "Figs.", "REF and REF show typical center of mass and natural frequency results for pristine and 5.31$^{\\circ }$ defective graphene sheets, respectively, that were used to obtain the Q-factors.", "We make three other relevant comments here.", "First, we chose to study graphene monolayers with a single grain boundary rather than study polycrystalline graphene with a distribution of grain boundary misorientation angles.", "By comparing the Q-factors of monolayer graphene with a single grain boundary and by varying the misorientation angle of the single grain boundary to pristine graphene, and by utilizing temperatures ranging from $\\sim $ 0K to 300K, we aim to quantify the effects of each grain boundary orientation on the intrinsic loss mechanisms, or Q-factor.", "Second, the models in this work represent extreme cases in that the defects run through the entire diameter of the graphene monolayer.", "Finally, we will also study the effects of tensile mechanical strain in enhancing the Q-factors of graphene.", "Previous studies on both graphene [13], [24] and other nanostructures [32] have demonstrated the effectiveness of strain in increasing the Q-factors.", "We will demonstrate the utility of strain in mitigating the effects of out of plane buckling due to the 5-7 unit ring defects along the grain boundaries, thus elevating the Q-factors.", "Figure: Center of mass and natural frequency for pristine graphene at 3K.Figure: Center of mass and natural frequency for defective graphene with a grain boundary misorientation angle of 5.31 ∘ ^{\\circ } at 3K." ], [ "Numerical Results and Discussion", "Before presenting the misorientation angle-dependent results for the Q-factors, we first note again that the 5-7 defects along the grain boundary induce out of plane buckling in the graphene monolayer as shown in Fig.", "REF .", "More specifically, we first show that both the orientation of the buckle (i.e.", "up or down), and its location along the grain boundary, significantly impacts not only the structural symmetry of the graphene sheet, but also the Q-factors; this is true for all grain boundary misorientation angles.", "Figure: (Color online) Schematic figures showing out-of-plane bucklings of 1.4 ∘ ^{\\circ } sheet with different defect pair positions after relaxation.To demonstrate this, we considered several possible spatial locations of the single 5-7 defect pair for the 1.4$^{\\circ }$ misorientation angle, as illustrated in Fig.", "REF .", "The first two cases, i.e.", "those depicted in Figs.", "REF (a) and (b), result in a significant asymmetry of the actuated oscillation.", "However, the buckle in the third case in Fig.", "REF (c) is near the edge of the graphene monolayer, and thus better preserves the circular oscillation symmetry, and results in a Q-factor of about 10,000, which is almost 1 order of magnitude larger than found for the cases in Fig.", "REF (a) and (b).", "However, to make the results of all the grain boundary misorientation cases to be comparable, we have placed the single 5-7 defect pair for the 1.4$^{\\circ }$ case in the center of the monolayer as shown in Fig.", "REF (a).", "Figure: (Color online) Variation of the Q-factor for monolayer graphene at 3K for different grain boundary misorientation angles.To illustrate the influence of the orientation of the buckling, for each misorientation angle (except for the 1.4$^{\\circ }$ sheet with one buckle which thus has only two possible buckling orientations, up or down), we tested five cases with different buckling patterns at 3K, where the different buckling patterns were generated by using different random velocity seeds during the thermal equilibration.", "From the results shown in Fig.", "REF , we can see that all cases have a Q-factor that is 1-2 orders of magnitude smaller than pristine, monolayer graphene (0$^{\\circ }$ ).", "Furthermore, the range between the highest and lowest Q-factor for each misorientation angle spans approximately one order of magnitude.", "For consistency, we utilize the configuration with the highest Q-factor for each misorientation angle, as indicated by the dashed line in Fig.", "REF , for the remainder of this study.", "The Q-factors of unstrained graphene, both pristine and with the different grain boundary orientations, are shown in Fig.", "REF as a function of temperature.", "We find that the Q-factors of pristine graphene follow the relationship $Q\\sim T^{-\\alpha }$ , where we find $\\alpha =1.28$ for the present case, which is similar to the result of [13] when the different approaches for calculating Q are accounted for.", "We can see that for all temperatures, pristine graphene has a higher Q-factor than CVD graphene, where the difference in Q-factor at low temperatures is between 1-2 orders of magnitude, as shown in Fig.", "REF .", "We also observe that the Q factors for all graphene sheets with grain boundaries obey a similar functional form ($Q\\sim T^{-\\alpha }$ ) with respect to temperature, while for both pristine and defected graphene sheets the Q factor drops rapidly by the time room temperature is reached.", "Figure: (Color online) Variation of the Q-factor for unstrained monolayer graphene as a function of temperature and grain boundary misorientation angle.More interestingly, we find that the relationship between the grain boundary misorientation angle and the Q-factor is non-monotonic.", "This is illustrated in Fig.", "REF , where the Q-factor is seen to first increase with increasing misorientation angle and then decrease when the angle becomes larger than about 10$^{\\circ }$ .", "We believe this is due to a competition between two effects.", "First, as the grain boundary misorientation angle increases, the density of 5-7 defect pairs along the grain boundary increases, which should result in a decrease in the Q-factor with increasing grain boundary misorientation angle.", "On the other hand, while there are more buckles in the graphene monolayer with increasing grain boundary misorientation angle, we find that due to the interaction of the increasing number of defect pairs with increasing grain boundary misorientation angle that the circular symmetry of the graphene monolayer is better preserved.", "Specifically, as seen in Fig.", "REF for the 1.4$^{\\circ }$ case, a single, non-symmetric 5-7 defect pair exists near the center of the graphene sheet, which we have already discussed results in a substantial reduction in Q-factor.", "However, increasing the misorientation angle results in more 5-7 defects on each side of the central 5-7 defect, which enhances the overall symmetry of the graphene sheet.", "Furthermore, the height of the buckles tends to decrease with increasing grain boundary misorientation angle, i.e.", "the buckling heights for the 1.4$^{\\circ }$ , 5.31$^{\\circ }$ , 9.83$^{\\circ }$ , 12.83$^{\\circ }$ and 16.62$^{\\circ }$ graphene sheets at 3K after relaxation are found to be 3.3Å, 2.5Å, 1.9Å, 1.5Åand 2.2Å, respectively.", "Thus, the increase in Q-factor due to the greater structural symmetry with increasing misorientation angle coupled with the corresponding reduction in buckling height counteracts the decrease in Q due to the increase in the defect density.", "To illustrate this concept, we show in Fig.", "REF the equilibrium buckled configuration for the 16.62$^{\\circ }$ case, where it can be seen that each of the four buckles forms from the combination of two smaller buckles.", "As a result, the buckles not only have larger buckling heights, but also break symmetry due to the fact that each buckle is composed of a smaller and larger sub-buckle.", "These factors couple to result in a smaller Q-factor for the 16.62$^{\\circ }$ case, and the overall non-monotonic trend seen in Fig.", "REF .", "Figure: (Color online) Out-of-plane buckling of 16.62 ∘ ^{\\circ } defective graphene sheet after thermal equilibration.Due to the deleterious effect of the grain boundaries on the Q-factor, we also examine how the Q-factors of CVD-grown graphene can be enhanced.", "As suggested in various works [19], [13], [24], [10], tensile mechanical strain is an effective approach to enhancing the Q-factors of nanostructures and NEMS.", "In our MD simulations, we imposed a modest, experimentally-accessible 1% tensile strain [33], [34] that was applied symmetrically (radially) outward from the center of the CVD-grown graphene sheets prior to the thermal equilibration and subsequent velocity-driven actuation.", "As shown by comparison of Figs.", "REF and REF , the Q-factors of strained graphene with grain boundaries can, in some cases, approach those of pristine graphene.", "In addition, as shown in Fig.", "REF , tensile strain increases the Q-factors of graphene with grain boundaries much more than pristine graphene, where the tensile strain-induced Q-factor enhancement for the graphene with grain boundaries can be larger than one order of magnitude.", "Fig.", "REF also shows that strained graphene follows the same $Q \\sim T^{-\\alpha }$ relationship as unstrained graphene, though with a slightly lower exponent $\\alpha $ of 1.11.", "Figure: (Color online) Variation in the Q-factor for monolayer graphene under 1% tensile strain as a function of temperature and grain boundary misorientation angle.We have found through our MD simulations that tensile strain increases the Q-factor by both increasing the natural frequency $\\omega $ , while simultaneously suppressing the damping $\\gamma $ in the expression $\\zeta =\\gamma /\\omega $ , where Q$=0.5/\\zeta $ .", "We found in analyzing the Q-factors that the tensile strain increased the natural frequency $\\omega $ about the same amount for both pristine graphene and graphene with grain boundaries.", "However, the damping factor $\\gamma $ showed a significantly greater reduction for graphene with grain boundaries as compared to pristine graphene.", "This is because the tensile strain reduces the out of plane buckling that results due to the 5-7 defect pairs, thus further enhancing the structural integrity and suppressing the damping of the oscillating graphene monolayer.", "The specific buckling heights at 3K due to 1% tensile strain for the grain boundary misorientation angles of 1.4$^{\\circ }$ , 5.31$^{\\circ }$ , 9.83$^{\\circ }$ , 12.83$^{\\circ }$ and 16.62$^{\\circ }$ : 2.5Å, 1.8Å, 1.4Å, 1.1Åand 1.7Å, respectively; these are clearly smaller than when no tensile strain is applied.", "Figure: (Color online) Variation in the Q-factor as a function of grain boundary misorientation angle at 3K due to 1% tensile strain, and the the ratio the Q-factors under tensile strain to the Q-factors of unstrained graphene.We also note that when tensile strain is applied the misorientation angle-dependent Q-factor follows a similar trend in Fig.", "REF as was previously observed for the unstrained graphene in Fig.", "REF .", "For strained graphene at 3K, we find that graphene with a misorientation angle of 12.83$^{\\circ }$ has the highest Q factor, while the graphene sheets with a 9.83$^{\\circ }$ misorientation angle has a similarly high Q-factor.", "Overall, this demonstrates the important fact that CVD-grown graphene can, under modest, experimentally accessible tensile strains, exhibit comparable performance for NEMS sensing applications as pristine graphene." ], [ "Conclusions", "We have utilized classical MD simulations to quantify the effects of grain boundaries in CVD-grown graphene on the Q-factors of graphene nanoresonators.", "Graphene with grain boundaries exhibit Q-factors that are 1-2 orders of magnitude smaller than pristine graphene.", "However, the Q-factors follow a non-monotonic dependence on the grain boundary misorientation angle due to the competing effects of increased 5-7 defect pair density on one hand, and the increased structural symmetry and reduction in out of plane buckling heights on the other hand.", "Furthermore, for practical applications, the Q-factors of CVD-grown graphene can be enhanced by about one order of magnitude through the application of 1% tensile strain, which results in Q-factors that approach those of pristine graphene." ], [ "Acknowledgements", "ZQ acknowledges support from a Boston University Dean's Catalyst Award.", "HSP acknowledges support of NSF grant CMMI-0856261." ] ]

[ [ "Forecast verification for extreme value distributions with an\n application to probabilistic peak wind prediction" ], [ "Abstract Predictions of the uncertainty associated with extreme events are a vital component of any prediction system for such events.", "Consequently, the prediction system ought to be probabilistic in nature, with the predictions taking the form of probability distributions.", "This paper concerns probabilistic prediction systems where the data is assumed to follow either a generalized extreme value distribution (GEV) or a generalized Pareto distribution (GPD).", "In this setting, the properties of proper scoring rules which facilitate the assessment of the prediction uncertainty are investigated and closed-from expressions for the continuous ranked probability score (CRPS) are provided.", "In an application to peak wind prediction, the predictive performance of a GEV model under maximum likelihood estimation, optimum score estimation with the CRPS, and a Bayesian framework are compared.", "The Bayesian inference yields the highest overall prediction skill and is shown to be a valuable tool for covariate selection, while the predictions obtained under optimum CRPS estimation are the sharpest and give the best performance for high thresholds and quantiles." ], [ "Introduction", "Extreme events in weather and climate such as high wind speeds, heavy precipitation or extremal temperatures are commonly associated with high impacts on both environment and society.", "However, the physical processes leading to the extremes are usually generated on small scales and their prediction is contaminated by large uncertainty.", "The need to determine uncertainties in the predictions of extreme events is stressed in the report from a recent workshop on extreme events in climate and weather [28].", "A prediction system for such events should therefore be probabilistic in nature, allowing for an assessment of the associated uncertainty [9], [19].", "The verification methods applied to these systems should thus necessarily be equipped to also handle the verification of uncertainty estimates.", "[37] argues that a general prediction system should strive to perform well on three types of goodness: there should be consistency between the forecaster's judgment and the forecast, there should be correspondence between the forecast and the observation, and the forecast should be informative for the user.", "On a similar note, [20] state that the goal of probabilistic forecasting should be to maximize the sharpness of the predictive distribution subject to calibration.", "Here, calibration refers to the statistical consistency between the predictive distribution and the observation, while sharpness refers to the concentration of the predictive distribution; the sharper the forecast, the higher information value will it provide.", "The prediction goal of [20] is thus equivalent to Murphy's second and third type of goodness.", "Verification methods that aim to attain these goals have been extensively studied in the literature, see e.g.", "[47] for an excellent overview.", "In this paper, we focus on the prediction of extreme events and our main objective is to assess the characteristics of proper scoring rules for extreme value distributions.", "The framework of proper scoring rules can also be used for the parameter estimation in that a scoring rule is optimized over the training data, see e.g.", "[23].", "Optimum score estimation returns unbiased parameter estimates [10] and for the ignorance score, this equals maximum likelihood estimation for independent observations.", "The continuous ranked probability score or CRPS [45], [30], [22] is of particular interest in our context, as it simultaneously assesses all of Murphy's types of goodness.", "A closed form expression of the CRPS has been calculated for a normal distribution [23], for a mixture of normals [27], for a truncated normal distribution [21], and for the three parameter two-piece normal distribution [44].", "We derive closed-form expressions for the CRPS for the generalized extreme value distribution (GEV) and the generalized Pareto distribution (GDP).", "In an application to peak wind prediction, we compare minimum CRPS estimation, maximum likelihood estimation, and a Bayesian approach under the GEV using predictive performance as the comparison criteria.", "Peak winds of short duration (few seconds) are a major cause of wind-related damage and crucial in wind hazard studies.", "Observations of peak winds are, however, sparse since only a small proportion of the weather stations provide peak wind speed observations.", "Using data from the observation station Valkenburg in the Netherlands, we investigate how observations of other weather variables, including mean wind speed, precipitation, and pressure, may be used as covariates to obtain a predictive distribution for the peak wind speed.", "Furthermore, we show how a Bayesian regression variable selection method can simplify the covariate selection procedure when the space of potential models is large.", "The remainder of the paper is organized as follows.", "In section , we discuss how the predictive performance of probabilistic forecasts for extreme events may be assessed and provide the closed form expressions for the CRPS under the GEV and the GPD.", "A detailed description of the derivation is given in the appendix.", "Our case study on probabilistic forecasting and forecast verification for peak wind speed is presented in section .", "The paper closes with a discussion in section ." ], [ "Assessing predictive performance", "[38] and [37] define a general framework for deterministic forecast verification based on two aspects of prediction quality: reliability and resolution.", "Reliability, or calibration, relates to the correspondence between the observation and the forecast.", "In our probabilistic setting, a forecast $F$ is perfectly calibrated if the conditional distribution of the observation $Y$ given the forecast is equal to $F$ .", "Resolution is a measure of information content which is closely related to the sharpness of the forecast.", "In our setting, sharpness is defined with respect to the predictive distribution, where a sharp predictive distribution reflects high information content or entropy.", "Note that sharpness has to be conditioned on $F$ being calibrated for it to be related to the resolution – otherwise it is merely an attribute of the forecast distribution.", "In the following, we discuss various methods to assess the calibration, or the reliability, and the resolution of a predictive distribution for data assumed to follow an extreme value distribution as in (REF ) or (REF ) below." ], [ "Calibration", "Let $F_1,\\ldots ,F_n$ denote the respective predictive distributions for the observations $y_1,\\ldots ,y_n$ .", "A standard method to assess the calibration of the forecasts is to apply the probability integral transform (PIT) [9].", "The PIT is given by the value of the predictive distribution at the observation, $F_i(y_i)$ .", "If the forecasts are calibrated, the sample $\\lbrace F_i(y_i)\\rbrace _{i=1}^n$ should follow the standard uniform distribution.", "This may e.g.", "be assessed graphically through a histogram, where a $\\cup $ -shaped histogram will indicate that the forecasts are underdispersive, while a $\\cap $ -shaped histogram will indicate that the forecasts are overdispersive, see e.g.", "[47].", "The PIT histogram is the continuous counterpart of the verification rank histogram or the Talagrand diagram [2], [29].", "The PIT values might also be displayed in terms of a PP-plot, where they are compared to $n$ equally spaced percentiles of the standard uniform distribution.", "In extreme value theory, a very popular assessment of the calibration of a non-stationary extreme value model is the so-called residual quantile plot [8].", "Residual quantile plots emphasize potential deviations in the upper tail of the distribution.", "To this end, the PIT values and the uniform percentiles are transformed to quantiles through some inverse distribution function $G^{-1}$ .", "In general, the inverse Gumbel is used for $G^{-1}$ , as it is the standard reference distribution for models of block maxima.", "For threshold models an exponential distribution is often used.", "The calibration might also be estimated over a restricted range of forecasts, for instance situations with high forecast probability of extreme events.", "A stratified PIT histogram or residual quantile plot would then be evaluated based on $\\lbrace F_i(y_i)\\rbrace _{i \\in I}$ for some $I \\subset \\lbrace 1,\\ldots ,n\\rbrace $ .", "An important aspect here is that the subset $I$ has to be chosen independently of the observations $y_1,\\ldots ,y_n$ , as the observed value is not known to the forecaster when a forecast is issued." ], [ "Scoring rules", "A variety of scores exist to assess the quality of a probabilistic forecast.", "An important characteristic of a qualified scoring rule is its propriety [35], [22], [7].", "A scoring rule is (strictly) proper if the expected score for an observation $Y$ is optimized if (and only if) the true distribution of $Y$ is issued as the forecast.", "Propriety will encourage honesty and prevent hedging which coincides with Murphy's first type of goodness [37].", "We will only consider proper scoring rules in the following.", "Furthermore, the scoring rules are negatively oriented such that a lower value means a better score.", "A widely used scoring rule is the logarithmic score proposed by [25] which in meteorological applications is known under the name ignorance score [41].", "It is defined as $S_{IGN}(f,y) = - \\log (f(y) dy) .$ The ignorance score applies to the predictive density function $f$ and is proportional to the log-likelihood of the data with respect to the predictive density.", "Note, that $dy$ is ignored when calculating the log-likelihood.", "However, it must be taken into account if transformed variables are compared.", "The associated expected score is the Shannon entropy, and the divergence function becomes the Kullback-Leibler divergence [22].", "It thus represents a score which is motivated by information theory.", "Both [42] and [27] have pointed out that, although the ignorance score is simple to calculate, it attributes a very strong penalty to events with low probability and is thus very sensitive to outliers.", "The continuous ranked probability score (CRPS) assesses the predictive skill of a forecast in terms of the entire predictive distribution $F$ [45], [30] and can thus assess both the calibration and the sharpness of the forecast simultaneously.", "It is defined as $S_{CRP}(F,y) = \\int _{-\\infty }^{\\infty } \\left[ F(t) - H(t-y)\\right]^2 dt$ and compares the forecast distribution $F$ and the empirical distribution of the observation $y$ .", "Here, $H(t-y)$ denotes the Heaviside step function using the half-maximum convention with $H(0)=0.5$ .", "Note that an observation error might easily be introduced in the CRPS, e.g.", "assuming Gaussian errors and using a Gaussian distribution instead of the Heaviside step function.", "The CRPS can be decomposed into a reliability and a resolution part [30], [22].", "That is, we can write $S_{CRP}(F,y) = \\mathbb {E}|X-y| - \\frac{1}{2} \\mathbb {E}|X-X^{\\prime }|,$ where $X$ and $X^{\\prime }$ are independent random variables with distribution $F$ .", "In principle, any (strictly) proper scoring rules may be decomposed into an uncertainty, a reliability and, a resolution part [22], [6].", "This representation of the CRPS is especially useful if a closed form expression of the CRPS is not available for $F$ .", "Let $\\mathbf {x}$ and $\\mathbf {x}^{\\prime }$ denote two independent samples of size $m$ from the predictive distribution $F$ .", "The representation in (REF ) can then easily be approximated by $S_{CRP}(F,y) \\approx \\sum _{i=1}^m |x_i - y| - \\frac{1}{2} \\sum _{i=1}^m |x_i - x^{\\prime }_i|.$ Even though the forecast is given by a full predictive distribution $F$ , it is often of interest to focus on the prediction of certain events, such as the probability of threshold exceedance.", "We can assess the forecasters ability to predict events over a given threshold $u$ with the Brier score, $S_B^u(F,y) = (p_u - \\mathbb {1}\\lbrace y \\ge u\\rbrace )^2,$ where $p_u = 1 - F(u)$ is the predicted probability of the realized value being greater or equal to the threshold $u$ [5].", "Note that the CRPS in (REF ) represents an integral of the Brier score over all possible thresholds.", "Similarly, we might want to focus on the predictive performance in the upper tail.", "This can be achieved by using the quantile score $S^{\\tau }_{Q}(F,y) = \\rho _\\tau (y-F^{-1}(\\tau )),$ where $y$ is the event that materializes, $\\tau $ is the quantile of interest, $\\rho _\\tau (u) = \\tau u$ if $u\\ge 0$ , and $\\rho _\\tau (u) = (\\tau -1) u$ otherwise [22], [16].", "A third alternative deviation of the CRPS is obtained using the quantile score, $S_{CRP}(F,y) = 2 \\int _0^1 \\rho _\\tau (y-F^{-1}(\\tau )) d\\tau ,$ see [33], [22], and [24].", "When comparing various forecasters, it might often be advantageous to compare skill scores rather than the scores themselves [35], [36].", "A skill score measures the relative gain of a forecast with respect to the reference forecast and it is defined as $SS(F,y) = \\frac{S(F,y)-S(F_{ref},y)}{S(F_{perfect},y)-S(F_{ref},y)} $ where $F^{ref}$ is a reference forecast used for all forecasters, and $F_{perfect}$ is the perfect forecast.", "In the case of CRPS and ignorance score the respective score of a perfect forecast is zero.", "A zero skill score represents no gain in predictive skill, while a perfect forecast would have a skill score of 100%.", "This approach is especially useful when comparing very high quantiles under the quantile score in (REF ) or very high thresholds under the Brier score in (REF ) as these will be based on relatively small amount of data." ], [ "Scoring rules for extreme value distributions", "Events are generally classified as being extreme by one of two criteria: the extremes are either given by a block maxima or they are defined as all values that exceed a given threshold.", "The asymptotic behavior of the first type can be modeled by the generalized extreme value distribution (GEV), $F_{GEV}(y) = \\left\\lbrace \\begin{array}{lc}\\exp \\big (-(1+\\xi \\frac{y-\\mu }{\\sigma })^{-1/\\xi }\\big ) ,& \\xi \\ne 0 \\\\[0.2cm]\\exp \\big (-\\exp (-\\frac{y-\\mu }{\\sigma })\\big ) ,& \\xi = 0 \\end{array} \\right.", ",$ where $1 + \\xi \\frac{y-\\mu }{\\sigma } > 0$ for $\\xi \\ne 0$ [8], [3].", "The three parameters of the GEV are location $\\mu $ , scale $\\sigma $ and shape $\\xi $ .", "For threshold excesses, the generalized Pareto distribution (GPD) is usually applied.", "It is given by $F_{GPD}(y) = \\left\\lbrace \\begin{array}{lc}1-(1+\\xi \\frac{y-u}{\\sigma _u})^{-1/\\xi },& \\xi \\ne 0 \\\\1-\\exp (-\\frac{y-u}{\\sigma _u}),& \\xi = 0 \\end{array} \\right.", ",$ where $u$ is a sufficiently large threshold, $\\sigma _u$ is the scale and $\\xi $ the shape parameter.", "For $\\xi \\ne 0$ , a closed form expression of the CRPS for the GEV is given by $S_{CRP}(F_{GEV_{\\xi \\ne 0}},y)= & \\Big [ \\mu - y - \\frac{\\sigma }{\\xi } \\Big ] \\Big [ 1-2F_{GEV_{\\xi \\ne 0}}(y)\\Big ] \\nonumber \\\\&- \\frac{\\sigma }{\\xi } \\Big [ 2^{\\xi } \\Gamma (1-\\xi ) - 2 \\Gamma _l\\big (1-\\xi ,-\\log F_{GEV_{\\xi \\ne 0}}(y) \\big ) \\Big ],$ where $\\Gamma $ denotes the gamma function and $\\Gamma _l$ the lower incomplete gamma function.", "For $\\xi = 0$ , this expression reads $S_{CRP}(F_{GEV_{\\xi = 0}},y) = \\mu - y + \\sigma [C-\\log 2] - 2 \\sigma Ei\\big (\\log F_{GEV_{\\xi = 0}}(y) \\big ),$ where $C \\approx 0.5772$ is the Euler-Mascheroni constant and $Ei(x) = \\int _{-\\infty }^x \\frac{e^t}{t} dt$ is the exponential integral [1].", "The derivations are given in Appendix A.", "In the case of the GPD, we get $S_{CRP}(F_{GPD_{\\xi \\ne 0}},y)= & \\Big [ u - y - \\frac{\\sigma _u}{\\xi }\\Big ] \\Big [ 1 - 2 F_{GPD_{\\xi \\ne 0}}(y) \\Big ] \\nonumber \\\\&- \\frac{2\\sigma _u}{\\xi (\\xi -1)} \\Big [ \\frac{1}{\\xi -2} + \\Big [1-F_{GPD_{\\xi \\ne 0}}(y)\\Big ]\\Big [1+\\xi \\frac{y-u}{\\sigma _u} \\Big ]\\Big ]$ for $\\xi \\ne 0$ and $S_{CRP}(F_{GPD_{\\xi = 0}},y) = y-u - \\sigma _u\\Big [2 F_{GPD_{\\xi = 0}}(y) -\\frac{1}{2}\\Big ]$ for $\\xi =0$ .", "The derivations for a GPD are given in Appendix B.", "Figure: The CRPS (solid lines), the ignorance score (dashed lines), and the quantile score for τ=0.5\\tau = 0.5 (dotted lines) as a function of the observation value for the GEV (top row) and the GPD (bottom row).", "The predictive distributions have zero location (μ=0\\mu = 0), standard scale (σ=1\\sigma = 1), and shape ξ=-0.5\\xi = -0.5 (first column), ξ=0\\xi = 0 (second column), or ξ=0.5\\xi = 0.5 (third column).", "The corresponding predictive densities are indicated in gray.Figure REF shows the CRPS, the ignorance score and the quantile score for $\\tau = 0.5$ for the GEV with shape parameter $\\xi =-0.5$ (Weibull type), $\\xi =0$ (Gumbel type), and $\\xi =0.5$ (Fréchet type), as well as for the GDP with the same parameter values.", "The ignorance score has a sharper minimum than the other scores and takes its minimum at the mode of the predictive distribution while the other two scores take their minimum at the median of the distribution.", "For the CRPS under the GEV, this can easily be shown by introducing the median, $F_{GEV}(0.5) = \\mu +\\frac{\\sigma }{\\xi }( (\\log 2)^{-\\xi }-1)$ for $\\xi \\ne 0$ and $F_{GEV}(0.5) = \\mu -\\sigma \\log ( \\log 2)$ for $\\xi =0$ , into the derivative of the score with respect to $y$ ." ], [ "Predicting peak wind speed", "We apply the verification measures discussed in the previous sections to predictions of daily peak wind speed at a location in the Netherlands over a period from January 1, 2001 to July 1, 2009.", "The goal is to derive a prediction model for daily peak wind speed using the observations of other meteorological variables as covariates with out-of-sample data used for the parameter estimation.", "In our application, the covariates are observed at the same time and location as the observation.", "We thus derive nowcasts rather than real forecasts in time for daily peak wind speed.", "In view of the sparse observational network for gust observations, such an analysis can be very useful, e.g.", "for forecast verification of wind gust warnings [15].", "Furthermore, the covariates might easily be replaced by the corresponding outputs from numerical weather prediction (NWP) models, thereby providing a probabilistic methodology for model output statistics, as NWP models only give diagnostic estimates for peak wind speed.", "For the modeling, we use generalized extreme value distributions under both a Bayesian forecasting framework and a frequentist framework where the parameters are estimated by minimizing a proper scoring rule.", "That is, we assume that peak wind speed observations follow a non-stationary GEV (REF ) where the parameters depend on covariates $\\mathbf {x}$ through functions of the form $\\mu (\\mathbf {x}) = \\mu _0 + \\mu _1 x_1 + \\mu _{2} x_{2} + \\ldots , \\quad h( \\sigma (\\mathbf {x})) = \\sigma _0 + \\sigma _1 x_1 + \\sigma _{2} x_{2} + \\ldots , \\quad \\xi = \\xi _0,$ where $h(\\cdot )$ is a link function in analogy to generalized linear models [14].", "The shape parameter is very sensitive to sampling uncertainty.", "Including covariates to the shape parameter largely increases the uncertainty not only of the shape parameters estimate itself, but also of the other parameters [15].", "The benefit in turn is generally small.", "For this reason, the shape parameter is kept independent of covariates.", "However, in a forecasting procedure, alike the other parameters, the shape parameter is estimated on a training data set and used to provide out-of-sample prediction.", "This out-of-sample prediction and verification is realized by a cross-validation procedure (see section REF ).", "As an alternative forecast, we apply a standard approach in meteorology and wind engineering, a generalized linear model [34] for the gust factor.", "The gust factor $G$ is defined as $G = \\frac{y_{\\mbox{\\sl fx}}}{x_{\\mbox{\\sl ff}}} - 1,$ where $x_{\\mbox{\\sl ff}}$ denotes the mean wind speed observation and $y_{\\mbox{\\sl fx}}$ the peak wind speed observation.", "The gust factor is generally assumed to follow a lognormal distribution, while the most simple approaches assume a constant gust factor.", "We base our approach on the work of [46] and [32] and use a GLM for lognormal residuals.", "This corresponds to a standard linear regression model for $\\log G$ with the expected value being modeled as $\\mathbb {E}[ \\log G ] = \\beta _0 + \\beta _1 x_1 + \\beta _2 x_{2} + \\ldots .$ The GLM is estimated using iteratively reweighted least squares as provided by the glm function in R [39]." ], [ "Data", "The Royal Netherlands Meteorological Institute freely provides daily weather data from numerous observation locations in the Netherlandshttp://www.knmi.nl/climatology/daily_data/download.html.", "The meteorological station data contain observations of temperature, sunshine, cloud cover and visibility, air pressure, precipitation, mean wind, and peak wind speed for the specified station.", "We have chosen the station number 210, Valkenburg, with hourly observations from January 1, 2001 to July 1, 2009 which gives us a total of 3104 daily observations.", "The daily observed peak wind $y_{\\mbox{\\sl fx}}$ is measured as a 24-hour (00UTC-00UTC) maximum over the observed hourly wind gusts.", "The gust observations are measured in 1 m/s which makes the data quasi discrete.", "In order to obtain a more continuous spectrum of values, we added a uniformly distributed noise to the gust observations.", "The hourly mean wind speed is given by the mean wind speed during the 10-minute period preceding the time of observation.", "We process these data to daily mean wind $x_{\\mbox{\\sl ff}}$ and daily wind variance $x_{\\mbox{\\sl ffVar}}$ by taking the average and the variance, respectively, over the 24 hourly observations as above.", "A similar procedure yields the mean rain rate $x_{\\mbox{\\sl rr}}$ and the maximum rain rate $x_{\\mbox{\\sl rMax}}$ over 24 hours.", "Finally, we also consider the mean 24-hour pressure $x_{\\mbox{\\sl P}}$ and the 24-hour pressure tendency $x_{\\mbox{\\sl dP}}$ as potential covariates.", "The pressure tendency $x_{\\mbox{\\sl dP}}$ is estimated by fitting a linear trend function (i.e., linear function in time) to the 24 hourly pressure observations.", "The slope estimate yields an estimate of $x_{\\mbox{\\sl dP}}$ .", "Note that the covariates are normalized before entering as a covariate.", "Normalization is not necessary but makes the inference procedures more stable.", "Figure: The goodness of fit of a seasonal GEV model applied to 24h peak wind speed observations at Valkenburg, the Netherlands from January 1, 2001 to July 1, 2009 measured by a QQ-plot (a) and a PP-plot (b).", "The dark gray line represents the seasonal GEV model estimates, the light gray lines indicate the 95% confidence interval derived by parametric bootstrapping, and the black dots denote the empirical estimates.Peak wind speed refers to the maximum wind speed in a given time period.", "In operational terms, this refers to the highest 3-second average value recorded in a particular time period (e.g.", "1 hour).", "Consequently, a block maxima approach using a GEV represents a natural approach to model the peak wind speed observations.", "Figure REF represents a goodness of fit for a GEV model fitted to the daily peak wind speed observations in our data set using maximum likelihood estimation.", "In order to account for seasonal variations, we included the annual cycle in terms of a annual sine and cosine function as covariate to the location and scale parameter.", "Both parameters exhibit significant annual changes.", "The relation of the shape parameter to the annual cycle is not significantly different from zero, neither any relation to a half-annual sine or cosine function.", "Although significant, the annual cycle accounts for only about 5% of the variance in the daily peak wind speed observations.", "The seasonally varying GEV exhibits a shape parameter of about $\\hat{\\xi }= -0.022$ with a standard error of $ 0.014$ and provides an overall acceptable fit even though it seems to slightly overestimate high quantiles, see Figure REF (a)." ], [ "Optimum score estimation", "In our frequentist approach, we apply both classic maximum likelihood estimation (GEV-MLE) and minimum CRPS estimation (GEV-CRPS) for the parameter estimation of the predictive distribution function.", "For the minimum CRPS estimation, we minimize the function $\\sum _i S_{CRP}(F_i,y_{\\mbox{\\sl fx},i})$ where the index runs over the training set, $S_{CRP}$ is given by either (REF ) or (REF ) depending on the value of the shape parameter $\\xi $ , and $F_i$ is the predictive distribution for a covariate vector $\\mathbf {x}_i$ .", "The predictive distribution for $Y$ is a GEV with $P(Y\\le y| \\mathbf {x}) = F_{GEV}(y;\\mu (\\mathbf {x}),\\sigma (\\mathbf {x}),\\xi ),$ where the parameters $\\mu (\\mathbf {x}), \\sigma (\\mathbf {x})$ , and $\\xi $ are given by (REF ).", "Here, we apply both the identity and the logarithmic link function for the scale parameter $\\sigma $ .", "The latter ensures a positive scale parameter and guarantees valid predictions.", "Numerical optimization was first performed using the quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) and the Nelder-Mead methods implemented in R [39] in parallel and the best estimates in terms of optimum score were used.", "However, as we included more covariates, the optimization became less stable.", "For that reason, all optimum score estimates are derived using a Simulated Annealing algorithm (SANN).", "The SANN method in R by default uses a variant of Simulated Annealing given in [4] and is useful for getting good estimates on rough surfaces.", "Although the SANN algorithm is slower than the BFGS and the Nelder-Mead, it provides more robust estimates.", "Figure: The in-sample ignorance score for the stationary GEV over the full data set.The plots show changes in the score when the location μ 0 \\mu _0, scale σ 0 \\sigma _0, and shape ξ 0 \\xi _0 are varied pairwise while the remaining parameter is fixed at the MLE.", "The black dots indicate the MLE and the white dots the minimum CRPS estimate.Figure: The in-sample CRPS for the stationary GEV over the full data set.The plots show changes in the score when the location μ 0 \\mu _0, scale σ 0 \\sigma _0, and shape ξ 0 \\xi _0 are varied pairwise while the remaining parameter is fixed at the minimum CRPS estimate.The white dots indicate the minimum CRPS estimates and the black dots the MLE.In order to compare the maximum likelihood and the minimum CRPS estimation, we investigate two-dimensional surfaces of the in-sample ignorance score and the CRPS over the entire data set.", "The link function for the scale parameter $\\sigma $ is the identity.", "Figure REF shows two-dimensional surfaces of the log-likelihood function, or the ignorance score, for pairwise combinations of $\\mu _0$ , $\\sigma _0$ , and $\\xi _0$ .", "In each plot, the other parameter is fixed at its respective maximum likelihood estimate (MLE).", "The log-likelihood function is displayed over a range that covers the parameter estimate plus-minus three standard errors as derived from the profile likelihood [8].", "Similar two-dimensional surfaces are displayed for the CRPS in Figure REF , where in each plot, the other parameter is fixed at its respective minimum CRPS estimates.", "The CRPS is displayed over the same parameter range as the log-likelihood function in Figure REF .", "The MLE and the minimum CRPS estimates differ only slightly, or by less than two standard errors for all parameters (cf.", "the black and white dots in Figure REF and REF ).", "The pairwise dependence of changes in the parameter values on the function surfaces is quite different for the two functions.", "It is particularly strong for the CRPS where the dependence is negatively oriented.", "Further, the CRPS function is less sharp than the log-likelihood function in the direction of $\\xi _0$ ." ], [ "Bayesian forecasting framework", "Let $l(\\mathbf {y}_{\\mbox{\\sl fx}}|\\mathbf {\\theta },\\mathbf {X})$ denote the likelihood function under the GEV model in (REF ), where $\\mathbf {y}_{\\mbox{\\sl fx}}$ are the observed peak winds over the training set, $\\mathbf {X}$ is the matrix of corresponding covariates, and $\\mathbf {\\theta }$ is a vector of the model parameters in (REF ) with a logarithmic link function for the scale parameter.", "Given a prior distribution $f(\\mathbf {\\theta })$ for $\\mathbf {\\theta }$ , the joint posterior distribution of the parameters given the data is $f(\\mathbf {\\theta }|\\mathbf {y}_{\\mbox{\\sl fx}},\\mathbf {X}) \\propto l(\\mathbf {y}_{\\mbox{\\sl fx}}|\\mathbf {\\theta },\\mathbf {X}) f(\\mathbf {\\theta }).$ The predictive density for a new observation $y^{\\prime }$ with a covariate vector $\\mathbf {x}^{\\prime }$ is given by $f(y^{\\prime }|\\mathbf {x}^{\\prime },\\mathbf {y}_{\\mbox{\\sl fx}},\\mathbf {X}) = \\int l(y^{\\prime }|\\mathbf {\\theta },\\mathbf {x}^{\\prime }) f(\\mathbf {\\theta }|\\mathbf {y}_{\\mbox{\\sl fx}},\\mathbf {X}) \\textup {d}\\mathbf {\\theta }.$ While this density is generally not a GEV density, it has the advantage that it reflects uncertainty both in the model and in the future observations, see e.g.", "[8].", "For the GEV model, the integral in (REF ) is usually intractable.", "However, the density may easily be approximated by a large sample from the predictive distribution, see [31].", "We use non-informative independent normal priors for the parameters and set $\\mu _j, \\sigma _j \\sim N(0,10^4)$ for all $j$ , and $\\xi _0 \\sim N(0,10^2)$ .", "For stationary GEV distributions, it is common to use the logarithmic link function for the scale parameter, see e.g.", "[43] and [18].", "We follow their practice and only use the logarithmic link function for this part of the analysis.", "Our priors correspond to setting $\\mu _0,\\log (\\sigma _0) \\sim N(0,10^4)$ and $\\xi _0 \\sim N(0,10^2)$ in the stationary case.", "[43] and [18] argue that, in the stationary case, a trivariate normal distribution would be preferred here, as a negative dependence between $\\sigma _0$ and $\\xi _0$ is a priori expected.", "As our inference is based on a relatively large data set, we refrain from this given the computational complexity that such a prior would induce in the non-stationary case.", "We then apply a Metropolis within Gibbs algorithm to obtain samples from the marginal posterior distributions using normal distributions with a small variance as proposal distributions for the updates.", "For each parameter estimation, we run 100.000 iterations of the Metropolis within Gibbs algorithm with a burn-in period of 25.000 iterations.", "Classical diagnostics such as traceplots and running mean plots (not shown) show good mixing and indicate that this is sufficient for convergence.", "In this case study, we consider six possible covariates and each of these can influence both the location and the scale parameter.", "We thus have a total of $2^{12}$ possible models in our model space.", "Besides considering specific models, we also perform a variable selection procedure over the entire model space.", "That is, we write each regression coefficient $\\mu _j$ for $j \\ge 1$ as $\\mu _j = z_j \\nu _j$ , where $z_j \\in \\lbrace 0,1\\rbrace $ and $\\nu _j \\in \\mathbb {R}$ such that the $z_j$ 's indicate which regression coefficients are non-zero.", "The regression equation for $\\mu $ becomes $\\mu (\\mathbf {x}) = \\mu _0 + z_1 \\nu _1 x_{\\mbox{\\sl ff}}+ z_2 \\nu _2 x_{\\mbox{\\sl ffVar}}+ z_3 \\nu _3 x_{\\mbox{\\sl rr}}+ z_4 \\nu _4 x_{\\mbox{\\sl rMax}}+ z_5 \\nu _5 x_{\\mbox{\\sl P}}+ z_6 \\nu _6 x_{\\mbox{\\sl dP}},$ and similarly for $\\log (\\sigma (\\mathbf {x}))$ .", "Each model indication parameter $z_j$ is a priori equal to either 0 or 1 with probability $1/2$ , while the regression parameters $\\nu _j$ have independent $N(0,10^4)$ priors as before.", "Again, the parameters are updated iteratively using a Metropolis within Gibbs updating scheme.", "Our algorithm is equivalent to a reversible jump MCMC algorithm [26] and the posterior inclusion probability of a covariate $x$ equals the average value of the corresponding indicator variable $z$ over the posterior sample.", "A related Bayesian inference framework is proposed in [13] where the authors suggest a birth-death MCMC procedure for covariate selection in generalized extreme value models and apply their framework to annual maximum precipitation data.", "However, the birth-death MCMC algorithm is quite complex to implement compared to the fairly simple regression variable selection method described in [31] which we have applied.", "For more details on the regression variable selection algorithm and Bayesian inference in general, see [31]." ], [ "Model selection", "In order to assess the predictive performance of our prediction approaches, we pursue a common approach in atmospheric science and use an out-of-sample verification.", "The prediction is performed in a cross-validation manner, in that we iteratively leave out one year of data, estimate the parameters of the statistical models based on the remaining data, and then perform an out-of-sample prediction.", "In this way, independent predictions for the complete time series are generated and used for verification and model selection.", "Table: Average posterior inclusion probabilities for the covariates in a Bayesian model selection framework over all years.", "The probabilities are given in percentages.Average posterior inclusion probabilities for the covariates under the Bayesian regression variable selection framework are given in Table REF .", "Three covariates, mean wind speed ($\\mbox{\\sl ff}$ ), wind variance ($\\mbox{\\sl ffVar}$ ), and maximum rain rate ($\\mbox{\\sl rMax}$ ) have very high inclusion probabilities for both the location parameter $\\mu $ and the scale parameter $\\sigma $ in (REF ).", "In addition, pressure ($\\mbox{\\sl P}$ ) and pressure tendency ($\\mbox{\\sl dP}$ ) also have high posterior inclusion probabilities for the location parameter $\\mu $ .", "Note that the inclusion probabilities are calculated after the burn-in period has been removed.", "Especially for the low inclusion probabilities, the chains show a high degree of mixing in the burn-in period.", "Although seasonality may be captured by the existing covariates, we have further investigated adding annual cycles to the model.", "The results indicate that a small improvement in predictive performance (i.e.", "of the order of 0.02) may be obtained by including an annual cycle in the location parameter.", "Consequently, only a small fraction of seasonality is left unexplained when no annual cycle is concidered explicitly.", "For clarity of exposition, we omit this factor in the following.", "Table: Average predictive performance of non-stationary peak wind speed predictions at Valkenburg from January 1, 2001 to July 1, 2009.", "The performance is measured in terms of the continuous ranked probability score (CRPS), which is given in meters per second, and the ignorance score (IGN).", "The covariates considered here are mean wind speed ff, wind variance ffVar, mean rain rate rr, maximum rain rate rMax, pressure P, and pressure tendency dP.", "VS stands for Bayesian variable selection approach.", "The link function h(σ)h(\\sigma ) used for the scale parameter is indicated in parenthesis and the optimal score in each column is indicated in bold.Table REF shows the average CRPS and ignorance score over all days in the test set for all the prediction methods considered in this study and different sets of covariates for each method.", "To calculate the CRPS in Table REF , we have applied (REF ) and (REF ) for the GEV models while for the Bayesian method, we have applied the approximation in (REF ).", "Since the logarithm of the gust factor $log(G)$ follows a normal distribution, the CRPS of the lognormal GLM is calculated using the closed-form expression in [23].", "To calculate the ignorance score for the Bayesian method, we obtain a non-parametric density estimate for the predictive density based on a large sample from the posterior predictive distribution.", "The ignorance score for the lognormal GLM reads $S_{IGN}(f_{\\cal N},\\log (G)) = - \\log (f_{\\cal N}(\\log (G))/(y-\\mbox{\\sl ff}) dy)$ , where $y$ is the respective gust value and $f_{\\cal N}$ is the predictive normal distribution of $\\log (G)$ .", "Note that the scores are calculated on cross-validated predictions (Sec.", "REF ), in order to provide an out-of-sample prediction and verification.", "Further, we estimate the sampling uncertainty of the CRPS and ignorance score via the bootstrap method [12].", "To that end, we recalculated the scores by resampling the prediction-observation pairs using the bootstrap method with replacement.", "The standard deviations of the CRPS and ignorance score within a 1000 member bootstrap sample vary between $0.01$ and $0.03$ for all the GEV methods.", "Including covariates in the prediction improves the predictive performance significantly compared to using a stationary model.", "The decrease in CRPS amounts to about 1.4 (1.7) when including ff (ff, ffVar, rMax, P, dP).", "With respect to a sampling uncertainty of about $0.03$ this decrease is highly significant.", "For the models compared here, all method show the highest predictive skill when only covariates with high posterior inclusion probability are included.", "Including P and dP in the scale parameter and rr in both the location and the scale parameter, or a subset of these, does not provide significant improvements.", "Although the lognormal distribution seems to provide a better fit to the data in the stationary case, the non-stationary GEV models clearly outperform the lognormal GLM except if only ff is used as covariate.", "The improvement of the CRPS amounts to about $0.1$ which is well above the sampling uncertainty of the CRPS.", "The differences between the various GEV approaches are more subtle and not very significant.", "For the frequentist methods, the identity link for the scale parameter yields slightly higher skill than the logarithmic link, especially in terms of the ignorance score for the best non-stationary model.", "For the more complex models, with three covariates or more, the optimum CRPS methods performs minimally better than the maximum likelihood methods in terms of the CRPS and the other way around for the ignorance score.", "The best scores are generally obtained within the Bayesian framework." ], [ "Predictive performance", "Based on the results in the previous section, we now focus on the non-stationary GEV with location parameter $\\mu (\\mathbf {x}) = \\mu _0 + \\mu _1 x_{\\mbox{\\sl ff}}+ \\mu _{2} x_{\\mbox{\\sl ffVar}}+ \\mu _3 x_{\\mbox{\\sl rMax}}+ \\mu _4 x_{\\mbox{\\sl P}}+ \\mu _5 x_{\\mbox{\\sl dP}}$ and scale parameter $h( \\sigma (\\mathbf {x})) = \\sigma _0 + \\sigma _1 x_{\\mbox{\\sl ff}}+ \\sigma _{2} x_{\\mbox{\\sl ffVar}}+ \\sigma _3 x_{\\mbox{\\sl rMax}},$ where $h$ is the identity link for the optimum score estimation and the logarithm for the Bayesian method.", "Although the differences are barely significant in table REF , there is evidence that the identity link model provides slightly better skill compared to the logarithm when using optimum score estimation.", "However, the following results also apply to the logarithm link model.", "For comparison, we also investigate the performance of the lognormal GLM method.", "Verification is then based on a variety of diagnoses.", "Figure: Residual quantile plots on Gumbel scale for models with the location parameter in () and the scale parameter in ().", "The prediction methods compared here are the GEV-MLE method with identity link (1st row), the GEV-CRPS method with identity link (2nd row), the Bayes method (3rd row), and the lognormal GLM (4th row).", "The first column shows the QQ-plots over all forecasts, the second column shows forecasts with x ff >q .75 x_{\\mbox{\\sl ff}}> q_{.75}, and the third column shows forecasts with x ff ≤q .75 x_{\\mbox{\\sl ff}}\\le q_{.75}.", "The vertical lines indicate (from left to right) the residual 0.5, 0.9, 0.95, and 0.99 quantiles.The calibration, or the reliability, of the predictions is assessed by using residual quantile plots based on the standard Gumbel distribution, see Figure REF .", "The first column of Figure REF shows the residual quantile plots (section REF ) for the complete set of observations.", "For the lowest 99% of the residuals, the bulk of the residuals lies within the 95% uncertainty band of the residual distribution for all three GEV methods.", "However, the models show a significant underestimation of the highest 1% of the residuals.", "The underestimation is even more prominent for the logarithmic GLM where significant underestimation is observed for the highest 5% of the residuals.", "An additional stratification is based on the covariate $x_{\\mbox{\\sl ff}}$ , where predictions with below normal mean wind speed ($x_{\\mbox{\\sl ff}}\\le q_{.75}$ ) and above normal mean wind speed ($x_{\\mbox{\\sl ff}}> q_{.75}$ ) are considered separately, where $q_{.75}$ is the $0.75$ quantile of the $x_{\\mbox{\\sl ff}}$ observations.", "In situations where the mean wind $x_{\\mbox{\\sl ff}}$ is above its 0.75 quantile, all models provide calibrated predictive distributions (Fig.", "REF , second column).", "The uncalibrated large residuals only occur during weak wind situations (Fig.", "REF , third column).", "Since strong gusts are very unlikely during these weather situations, this miscalibration is probably not highly relevant for the gust prediction.", "The three GEV methods in Figure REF thus all appear to be fairly well calibrated and we can compare the sharpness of the predictive distributions in concurrence with the forecasting principle of [20].", "The GEV-CRPS method yields the sharpest predictions with an average predictive standard deviation of $1.43 \\, (\\pm 0.44)$ , while this value is $1.48 \\, (\\pm 0.57)$ for the Bayesian method and $1.51 \\, (\\pm 0.48)$ for the GEV-MLE method.", "The values given in the parentheses are the standard deviations of the daily predictive standard deviations.", "The large day-to-day variation in the sharpness is to be expected as different levels of uncertainty are associated with different prediction scenarios.", "In general, the forecasts are more uncertain for higher expected peak wind speeds.", "An alternative approach to assess the sharpness is to calculate the average width of symmetric prediction intervals, see e.g.", "[23] and [40].", "Similarly, the average coverage of the prediction intervals may be used to assess calibration.", "Table: Average Brier score at three thresholds and quantile score at four quantiles of peak wind speed predictions at Valkenburg from January 1, 2001 to July 1, 2009.", "The GEV methods apply the location parameter in () and the scale parameter in ().", "The link function for the scale parameter is indicated in parenthesis.", "The best performance in each column is indicated in bold.Table: Average skill scores (in percentages) of peak wind speed predictions at Valkenburg from January 1, 2001 to July 1, 2009.", "The GEV methods apply the location parameter in () and the scale parameter in ().", "The link function for the scale parameter is indicated in parenthesis.", "The skill scores for the thresholds is based on the Brier score and the skill scores for the quantiles is based on the quantile score.", "The reference forecast is the stationary GEV-MLE method.", "The best performance in each column is indicated in bold.Table REF shows the Brier scores at three high thresholds and the quantile scores for four high quantiles for our four methods.", "Wind speeds of $14-18$ m/s are generally considered as near gale, wind speeds of $18-25$ m/s are defined as gale and strong gale, while storm values are 25m/s and above.", "The corresponding skill scores are shown in Table REF where the reference forecast is the stationary GEV-MLE method.", "The standard error for the Brier score is $(0.31,0.25,0.09)$ for the thresholds $(14,18,25)$ and the standard error for the quantile score is about $0.10$ for each quantile.", "These scores measure predictive performance at certain points of the predictive distribution or the quantile function, and may indicate local deficiencies.", "The lognormal GLM provides significantly less skill for all thresholds and probabilities, except for the $0.99$ quantile where the difference is no longer significant.", "All methods based on a non-stationary GEV distribution perform similarly, with the GEV-CRPS methods showing marginally better performance than the other two methods.", "Note that the scores in Table REF cannot be compared across columns as these are based on different subsets of the data.", "The apparent improvement in the scores for higher thresholds/quantiles is merely an effect of the decreasing data set on which the values are based.", "This is evident if the results of Table REF are compared to the results on Table REF .", "As the same reference forecast is used for all the columns in Table REF , we can compare the results across columns relative to the reference forecast.", "Here, we see that the performance of all the methods decreases somewhat with higher thresholds/quantiles compared to the stationary GEV-MLE reference method.", "Alternatively, weighted proper scoring rules may be applied to focus on areas of special interest in the predictive distribution.", "[11] consider weighted versions of the ignorance score while [24] discuss approaches to weight the CRPS.", "The weight function can here either be based on quantiles using the representation in (REF ) or it can be based on thresholds using the representation in (REF ).", "Note that the weight functions must be defined independent of the observed value to uphold propriety [24]." ], [ "Parameter estimation", "Verification in terms of scores and residual quantile plots provides average performance of the prediction method.", "Another aspect is the variance and covariances of the parameter estimates.", "Bayesian prediction provides estimates of the posterior distribution of the parameters, whereas optimum score estimation only gives approximate covariances based on the profile score function (i.e.", "profile likelihood [8] in maximum likelihood estimation).", "An indication of uncertainty in the parameter estimation is provided by the variability of the estimates over the different training data set used in cross-validation.", "In the cross-validation procedure we successively remove one year of data, which provides us with nine estimates of each parameter for each method.", "Figure: The estimated yearly shape parameters for the GEV methods using the location parameter in () and the scale parameter in ().", "The gray boxes indicate the 90%90\\% posterior intervals for for ξ\\xi under the Bayesian model, the black dots (•\\bullet ) indicate the parameter estimates under the GEV-CRPS(id) model, and the squares (□\\Box ) indicate the parameter estimates under the GEV-MLE(id) model.", "The vertical lines indicate the respective standard errors.", "For comparison, the shape parameter of the observations estimated via MLE, ξ ^=-0.013\\hat{\\xi } = -0.013, is indicated with a dotted line.Figure REF shows the parameter estimates for the yearly shape parameter $\\xi $ under the three GEV methods discussed in the previous section.", "The $90\\%$ posterior intervals obtained with the Bayesian method are fairly constant between years, while there is a large variation in the estimated values under both GEV-MLE and GEV-CRPS.", "Furthermore, the standard errors are significantly greater for the GEV-CRPS method.", "This is in accordance with Fig.", "REF where the minimum of the CRPS is less sensitive to changes in $\\xi $ and the discussion related to Fig.", "REF .", "The MLE and CRPS optimum score estimates of $\\xi $ are uncorrelated which suggests that the variability in the estimates is not due to interannual changes in the data.", "The majority of the estimates are significantly negative, indicating that the non-stationary GEV models are of Weibull type, in contrast to the goodness of fit analysis performed in Section REF , where $\\hat{\\xi }$ is only slightly negative.", "Figure: Optimum score estimates of yearly location parameters for the GEV methods using the location parameter in () and the scale parameter in ().", "The dots represent the maximum likelihood estimates (x-axis) and the optimum CRPS estimates (y-axis) for the different years.", "The gray lines indicate the respective standard errors.", "The gray bars on each axis indicate the 90%90\\% posterior intervals for the parameters under the Bayesian model.The parameter estimates for the location coefficients in (REF ) are shown in Figure REF .", "For clarity of the presentation and as the $90\\%$ posterior intervals are very similar for each run of the Bayesian method, we only show the $90\\%$ posterior interval of the joint posterior sample from all years.", "In general, there is no apparent correlation between the maximum likelihood estimates and the optimum CRPS estimates.", "Furthermore, the estimates for both methods are quite variable for different training sets and this variability cannot be explained solely by the uncertainty in the estimation which is of similar magnitude for both methods and across training sets.", "All three methods display the largest uncertainty in the estimate for the intercept coefficient.", "All methods show a strong positive dependence of the GEV location parameter on $\\mbox{\\sl ff}$ and $\\mbox{\\sl ffVar}$ , a weaker dependence on $\\mbox{\\sl rMax}$ , and a negative dependence on pressure $p$ and pressure tendency $\\mbox{\\sl dP}$ .", "The Bayesian posterior distributions for the scale coefficients in (REF ) cannot be directly compared to the corresponding estimates under the frequentist approaches as we have applied the identity link for the frequentist approaches while we use a logarithmic link for the Bayesian method.", "As for the location coefficients, no apparent correlation can be found between the MLE coefficient estimates and the minimum CRPS estimates and the variability between training sets is generally greater than expected based on the estimation uncertainty alone.", "Like the location parameter, the non-stationary GEV scale parameter positively depends on $\\mbox{\\sl ff}$ , $\\mbox{\\sl ffVar}$ , and $\\mbox{\\sl rMax}$ .", "Hence, predictive uncertainty increases for large expected peak wind speed." ], [ "Discussion", "As e.g.", "[9] and [19] have argued before us, predictions for uncertain events ought to include an estimate of the associated uncertainty.", "This is easily obtained within a probabilistic framework where the predictions take the form of probability distributions.", "Proper scoring rules such as the Brier score or the CRPS are widely used to assess the predictive skill in probabilistic weather forecasting, see e.g.", "[47].", "In this paper, we discuss how proper scoring rules may be used for out-of-sample model selection and verification for extreme value distributions.", "We present a closed-form expressions of the CRPS for this class of distributions.", "With these expressions at hand, the CRPS may be used for optimum score estimation as an alternative to maximum likelihood estimation or Bayesian inference.", "In a case study, we compare various approaches to derive non-stationary distributions for daily peak wind speed observations.", "The non-stationarity is built in by including covariates, i.e.", "meteorological parameters that are observed together with the gust measurement.", "Our competing methodologies comprise a lognormal GLM for the wind gust factor and non-stationary GEV models for the daily wind gusts.", "The non-stationary GEV approaches provide significantly higher skill than the corresponding lognormal GLM.", "This confirms findings in [15] that extreme value theory provides an appropriate and theoretically consistent statistical model for wind gusts.", "Due to the close relationship between wind speed and peak wind, mean wind speed is the main covariate in our model.", "However, the predictive performance is significantly improved by also including information on additional weather variables.", "The most informative covariates besides mean wind speed are the variability of the mean wind speed, the maximum rain rate, and the pressure tendency throughout the day.", "With six potential covariates for both the location and the scale parameter, our model space includes a total of $2^{12}$ models.", "This large space can easily be searched using a Bayesian variable selection procedure which returns different covariate sets for the location and for the scale.", "Overall, the robustness and the predictive performance of the Bayesian framework is very good though it should be noted that the Bayesian inference is considerably slower than the frequentist methods.", "[23] compare minimum CRPS estimation and maximum likelihood estimation in a prediction framework and show that the former returns forecasts with slightly better overall predictive performance and significantly improved calibration.", "Their results confirm well with our findings in that the optimum CRPS estimation outperforms the maximum likelihood approach in nearly all aspects of our verification even though the differences are sometimes small.", "A drawback of the minimum CRPS estimation is its weak discriminant power with respect to the GEV shape parameter.", "In our case study, we have performed an extensive analysis of the properties of the various scoring rules and associated inference methods for the GEV distribution.", "The GEV is the self-evident distribution in the case of block maxima, i.e.", "in the case of peak wind speed.", "In the case of threshold excesses, a peak-over-threshold (POT) or Poisson point process approach would be more appropriate.", "We assume that similar results will hold for the POT approach using a GPD distribution and the closed-form expression given in this paper.", "Since the Poisson point process is generally represented in terms of GEV parameters, optimum score estimation using the CRPS should be possible as well." ], [ "Acknowledgments", "Special thanks go to Michael Scheuerer for valuable discussions and comments on early versions of the manuscript.", "The authors also thank Tilmann Gneiting, Andreas Hense, Alex Lenkoski, and Michael Weniger for helpful discussions, and the associate editor and an anonymous reviewer for their useful comments.", "We acknowledge the support of the Volkswagen Foundation through the project “Mesoscale Weather Extremes - Theory, Spatial Modeling and Prediction (WEX-MOP)”.", "tocsectionAppendix A: Derivation of the CRPS for a GEV For the deviation of the CRPS under the GEV, we apply the representation in (REF ), where the CRPS is presented as an integral over the quantile score in (REF ).", "We can write the quantile score as $S_Q^\\tau (F,y) = \\tau \\big [ y - F^{-1}(\\tau )\\big ] - \\mathbb {1}\\lbrace \\tau \\ge F(y)\\rbrace \\big [ y - F^{-1}(\\tau )\\big ]$ from which it is easily seen that $S_{CRP}(F,y) = y \\big [ 2F(y) -1 \\big ] -2 \\int _{0}^1 \\tau F^{-1}(\\tau ) d \\tau + 2 \\int _{F(y)}^1 F^{-1}(\\tau ) d \\tau .$ The quantile function of the GEV is given by $F_{GEV}^{-1}(\\tau ) = \\left\\lbrace \\begin{array}{lc}\\mu - \\frac{\\sigma }{\\xi } \\big [ 1- \\left( -\\log \\tau \\right)^{-\\xi } \\big ] ,& \\xi \\ne 0 \\\\[0.2cm]\\mu - \\sigma \\log \\left( -\\log \\tau \\right),& \\xi = 0 \\end{array}\\right.", ".$ For a non-zero shape parameter $\\xi \\ne 0$ and with $F = F_{GEV_{\\xi \\ne 0}}$ , we obtain $2 \\int _{0}^1 \\tau F^{-1}(\\tau ) d \\tau = \\mu - \\frac{\\sigma }{\\xi } + 2 \\int _0^1 \\tau \\big ( - \\log \\tau \\big )^{-\\xi } d \\tau $ and $2 \\int _{F(y)}^1 F^{-1}(\\tau ) d \\tau = 2 \\Big [ \\mu - \\frac{\\sigma }{\\xi } \\Big ] \\big [ 1 - F(y) \\big ] + 2 \\frac{\\sigma }{\\xi } \\int _{F(y)}^1 \\big ( - \\log \\tau \\big )^{-\\xi } d \\tau .$ To solve the two remaining integrals in (REF ) and (REF ), we use that $\\int \\tau (-\\log \\tau )^{-\\xi } d\\tau = 2^{\\xi -1} \\Gamma _u(1-\\xi ,-2\\log \\tau )$ and $\\int (-\\log \\tau )^{-\\xi } d\\tau = \\Gamma _u(1-\\xi ,-\\log \\tau ),$ where $\\Gamma _u(a, \\tau )= \\int _{\\tau }^\\infty t^{a-1} e^{-t} dt$ is the upper incomplete gamma function.", "By combining (REF ), (REF ), (REF ), and the integral equations above, we get $S_{CRP}(F_{GEV_{\\xi \\ne 0}},y)= \\Big [ y - \\mu + \\frac{\\sigma }{\\xi } \\Big ] \\big [ 2F_{GEV_{\\xi \\ne 0}}(y)-1 \\big ]- \\frac{\\sigma }{\\xi } \\Big [ 2^{\\xi } \\Gamma (1-\\xi )- 2\\Gamma _l\\big (1-\\xi , -\\log F_{GEV_{\\xi \\ne 0}}(y) \\big ) \\Big ],$ where $\\Gamma _l(a, \\tau ) = \\Gamma (a)-\\Gamma _u(a,\\tau )$ is the lower incomplete gamma function.", "For a Gumbel type GEV with $\\xi =0$ , similar calculations show that $S_{CRP}(F_{GEV_{\\xi = 0}},y)= & [y - \\mu ] \\big [2F_{GEV_{\\xi = 0}}(y) - 1 \\big ] + 2\\sigma \\int _0^1 \\tau \\log (-\\log \\tau ) d \\tau \\\\& - 2\\sigma \\int _{F_{GEV_{\\xi = 0}}}^1 \\log (-\\log \\tau ) d \\tau .", "\\nonumber $ The first integral is solved as $\\int \\tau \\log (-\\log \\tau ) d\\tau = \\frac{1}{2}\\Big [\\tau ^2 \\log (-\\log \\tau ) - Ei (2\\log \\tau ) \\Big ]$ where $Ei(x)= \\int _{-\\infty }^x \\frac{e^t}{t} dt$ is the exponential integral also given by $Ei(x ) = C + \\log (|x|) + \\sum _{k=1}^\\infty \\frac{x^k}{k!", "\\, k}$ with $C \\approx 0.5772$ being the Euler-Mascheroni constant.", "Further, we use $\\int \\log (-\\log \\tau ) d \\tau = \\tau \\log (-\\log \\tau ) - Ei(\\log \\tau ) ,$ to solve the second integral in (REF ).", "This leads to a closed-form expression for the CRPS under a Gumbel-type GEV with $S_{CRP}(& F_{GEV_{\\xi = 0}},y) \\\\& = [y - \\mu ] \\big [2F_{GEV_{\\xi = 0}}(y) - 1 \\big ] \\\\& \\quad + \\sigma \\lim _{\\nu \\rightarrow 1} \\Big [ \\nu ^2 \\log (-\\log \\nu ) - Ei(2 \\log \\nu ) \\Big ] - \\lim _{\\eta \\rightarrow 0}\\Big [ \\eta ^2 \\log (-\\log \\eta ) - Ei(2 \\log \\eta ) \\Big ] \\\\& \\quad - 2\\sigma \\lim _{\\nu \\rightarrow 1} \\Big [ \\nu \\log ( -\\log \\nu ) - Ei(\\log \\nu ) \\Big ] - 2 \\sigma \\Big [ F_{GEV_{\\xi =0}}(y) \\frac{y - \\mu }{\\sigma } + Ei\\big (\\log F_{GEV_{\\xi =0}}(y) \\big ) \\Big ] \\\\& = - (y - \\mu ) - 2 \\sigma Ei\\big (\\log F_{GEV_{\\xi =0}}(y) \\big ) \\\\& \\quad + \\sigma \\lim _{\\nu \\rightarrow 1} \\Big [ (\\nu -1)^2 \\log (-\\log \\nu ) + C - \\log 2 + \\sum _{k=1}^{\\infty } \\frac{(2 \\log \\nu )^k + 2 (\\log \\nu )^k}{k!", "k} \\Big ]\\\\&= -\\ (y - \\mu ) - 2 \\sigma Ei \\big (\\log F_{GEV_{\\xi = 0}}(y)\\big ) + \\sigma [C-\\log 2],$ where we use that $\\lim _{\\eta \\rightarrow - \\infty }Ei(\\eta ) = 0$ .", "The series expansion to compute $Ei ( \\log F_{GEV_{\\xi = 0}}(y))$ converges rapidly for $F_{GEV_{\\xi = 0}}(y)$ much greater than 0.", "However, it may fail for very small values of $F_{GEV_{\\xi = 0}}(y)$ .", "We use the GNU Scientific Library [17] to compute $Ei(\\cdot )$ .", "tocsectionAppendix B: Derivation of the CRPS for a GPD The derivation of the closed-form expression of the CRPS for a GPD is very similar to the derivations for a GEV in Appendix A with significantly simpler calculations.", "The GPD quantile function is given by $F_{GPD}^{-1}(p) = \\left\\lbrace \\begin{array}{lc}u- \\frac{\\sigma _u}{\\xi } + \\frac{\\sigma _u}{\\xi }(1-p)^{-\\xi } ,& \\xi \\ne 0 \\\\u- \\sigma _u \\log (1-p) ,& \\xi = 0 \\end{array} \\right.", ".$ By applying the representation of the CRPS given in (REF ), we get $S_{CRP}(F_{GPD_{\\xi \\ne 0}},y)&= \\Big [ y - u+\\frac{\\sigma _u}{\\xi } \\Big ] \\big [ 2F_{GPD_{\\xi \\ne 0}}(y) - 1 \\big ] \\\\& \\quad - \\frac{2 \\sigma _u}{\\xi } \\Big [ \\int _{F_{GPD_{\\xi \\ne 0}}(y)}^1 (1-\\tau )^{-\\xi } d \\tau - \\int _0^1 \\tau (1-\\tau )^{-\\xi } d \\tau \\Big ] \\\\&= \\Big [ y - u+\\frac{\\sigma _u}{\\xi } \\Big ] \\big [ 2F_{GPD_{\\xi \\ne 0}}(y) - 1 \\big ] \\\\& \\quad - 2 \\frac{\\sigma _u}{\\xi (\\xi -1)} \\Big [ \\frac{1}{\\xi -2} + \\Big [ 1-F_{GPD_{\\xi \\ne 0}}(y) \\Big ] \\Big [ 1+\\xi \\frac{y-u}{\\sigma _u} \\Big ] \\Big ].$ Here, the second integral can be solved using integration by parts.", "For $\\xi = 0$ , we obtain the following expression $S_{CRP}(F_{GPD_{\\xi = 0}},y)&= (y-u)\\big [ 2F_{GPD_{\\xi = 0}}(y) - 1 \\big ] \\\\& \\quad + 2 \\sigma _u \\Big [ \\int _0^1 \\tau \\log (1-\\tau ) d \\tau - \\int _{F_{GPD_{\\xi = 0}}(y)}^1 \\log (1-\\tau ) d \\tau \\Big ] \\\\&= y-u - \\sigma _u\\Big [ 2F_{GPD_{\\xi = 0}}(y) -\\frac{1}{2} \\Big ],$ where we use the integral equation $\\int \\log (1-x) dx = x \\big [ \\log (1-x) - 1 \\big ] - \\log (1-x)$ and integration by parts." ] ]

[ [ "On the nature of the fourth generation neutrino and its implications" ], [ "Abstract We consider the neutrino sector of a Standard Model with four generations.", "While the three light neutrinos can obtain their masses from a variety of mechanisms with or without new neutral fermions, fourth-generation neutrinos need at least one new relatively light right-handed neutrino.", "If lepton number is not conserved this neutrino must have a Majorana mass term whose size depends on the underlying mechanism for lepton number violation.", "Majorana masses for the fourth generation neutrinos induce relative large two-loop contributions to the light neutrino masses which could be even larger than the cosmological bounds.", "This sets strong limits on the mass parameters and mixings of the fourth generation neutrinos." ], [ "Introduction", "In the framework of the Standard Model (SM), fermions are grouped into three families, each containing a doublet of quarks and a doublet of leptons.", "The number of families is not a constructive parameter of the theory, and it could well be four or more; for this reason, the enlargement of the SM with new generations has been commonly considered [1], and it has proven to help in dealing with several problems, such as the lack of CP violation in the SM to explain the baryon asymmetry of the universe [2] or the structure of the leptonic mass matrices [3].", "The currently available SM observables, however, constrain quite tightly the properties of such new families [4], and the global electroweak fits seem to disfavour a scenario with more than five generations [5], [6]; maybe the most striking result against the existence of additional families is the LEP measurement of the number of neutrinos at the $Z$ peak, which forbids more than three light neutrinos [4], but even this can be dodged if the neutrinos of the new generations are too heavy to be produced in $Z$ decays.", "All in all, the existence of new generations is not actually excluded, and it seems worth being considered [1], even more now that the LHC is working and exploring the relevant mass range.", "On the other hand, right-handed neutrinos constitute a common new physics proposal, usually linked to the generation of neutrino masses.", "This is particularly interesting nowadays, ever since we gathered compelling evidence that neutrinos do have masses, that they lie well below the other fermions' ones, and that their mixing patterns differ extraordinarily from those of the quark sector (for a review on the matter of neutrino masses see, for example, [7]).", "The most straightforward way to construct a mass term for the neutrinos within the SM is just to rely on the Higgs mechanism, and so to write the corresponding Yukawa couplings; for that aim, one needs some fermionic fields which carry no SM charge: right-handed neutrinos.", "However, we do not know whether neutrinos are Dirac or Majorana.", "If they are Dirac, the smallness of the neutrino mass scale remains unexplained, for it would be just a product of the smallness of the corresponding Yukawa couplings.", "In order to provide such an explanation, many models and mechanisms have been proposed: in the so-called see-saw models, the lightness of the neutrino mass scale is a consequence of the heaviness of another scale.", "For instance, this scale is the lepton-number-violating (LNV) Majorana mass of the extra right-handed neutrinos in type I see-saw [8], [9], [10], [11].", "On the other hand, radiative models propose that neutrino masses are originated via suppressed, high-order processes [12], [13], [14], [15].", "Although some of these proposals do not require right-handed neutrinos, for the sake of generality it is a good idea to consider their possible involvement in the generation of neutrino masses.", "In this work we aim to discuss the naturality of the various scenarios arising when new generations and right-handed neutrinos are brought together.", "Several previous works have considered such association, either explicitly, in order to provide a mechanism for mass generation, or implicitly, when assuming Dirac neutrinos in their analyses [16], [17], [18], [19], [20], [21], [22], [23], [24], [25].", "We argue that unless a symmetry is invoked which separates the new family from the first three, the coexistence of both Dirac and Majorana neutrinos is not stable under radiative corrections and doesn't seem natural [26], [25].", "Furthermore, the presence of a fourth family plus a right-handed Majorana neutrino triggers the generation of Majorana masses for the light species through a well-known mechanism [27], [28], [29], [16], [17], [23], [25]; the upper bounds on the light neutrino masses can thus be translated into bounds on the mixings with the new, heavy generations.", "This paper is structured as follows.", "In section we start by reviewing the different mechanisms which can provide light neutrino masses.", "In section we discuss the naturalness of those mechanisms to generate the fourth-family neutrino mass and conclude that at least one right-handed neutrino is needed.", "Assuming that light neutrinos are Majorana, we use naturalness arguments to provide a lower bound on the Majorana mass of the right-handed neutrino.", "In section we consider a minimal four generation SM with only one relatively light right-handed neutrino and Majorana masses for light neutrinos parametrized by the Weinberg operator [30], [31].", "We describe the radiative, two-loop contribution of the heavy fourth-family neutrinos to the light neutrino mass matrix.", "In section we discuss the phenomenological consequences of this minimal four-generation scenario with heavy Majorana neutrinos (lepton flavour violation, universality bounds, light neutrino masses, neutrinoless double beta decay,...), and we conclude in section .", "Appendix A is devoted to describe an explicit example in which a finite Majorana mass for the fourth-generation right-handed neutrino is radiatively generated." ], [ "Light neutrino masses", "The huge hierarchy between neutrino masses and those of all other fermions has triggered the appearance of many different mechanisms to explain the lightness of neutrinos.", "Here we briefly review some of these mechanisms, with special emphasis on the frameworks that are able to explain neutrino masses including a fourth generation, which will be discussed in the next section." ], [ "Dirac masses", "If there are right-handed neutrinos and a conserved global symmetry (for instance $B-L$ ) prevents them from having a Majorana mass, neutrinos are Dirac particles, as all other fermions in the SM.", "However, in this scenario there is no explanation for the smallness of neutrino masses, having to impose by hand extremely tiny Yukawa couplings, approximately 6 (11) orders of magnitude smaller than the electron (top) one.", "Therefore, although in principle it is possible, a Dirac nature does not seem the most natural option for neutrinos (but see, for example, [3], for a proposal in this direction which avoids tiny Yukawas)." ], [ "Seesaw", "Seesaw models are minimal extensions of the SM which can naturally lead to tiny (Majorana) neutrino masses, keeping the SM gauge symmetry, $SU(3)_\\mathrm {C} \\otimes SU(2)_\\mathrm {L}\\otimes U(1)_\\mathrm {Y}$ and renormalizability, but giving up the (accidental) lepton number conservation of the SM.", "Let's explain briefly the different types to fix notation." ], [ "Type I: fermionic singlets", "In type I see-saw [8], [32], [9], [10], [11], $n$ SM fermionic singlets with zero hypercharge are added to the SM; these have the quantum numbers of right-handed neutrinos, and can be denoted by $\\nu _{\\mathrm {R} i}$ .", "Note that to explain neutrino data, which requires al least two massive neutrinos, a minimum of two extra singlets are needed.", "Having no charges under the SM, Majorana masses for right-handed neutrinos are allowed by gauge invariance, so the new terms in the Lagrangian are: $\\mathcal {L}_{\\nu _\\mathrm {R}} = i \\, \\overline{\\nu _\\mathrm {R}} \\gamma ^{\\mu }\\partial _{\\mu } \\nu _\\mathrm {R} - \\left( \\dfrac{1}{2}\\overline{\\nu _\\mathrm {R}^\\mathrm {c}} M \\nu _\\mathrm {R} +\\overline{\\ell } \\, \\tilde{\\phi } \\, Y \\,\\nu _\\mathrm {R} +\\mathrm {H. c.} \\right)\\, ,$ where $\\ell $ and $\\phi $ are respectively the lepton and Higgs SM doublets, $\\tilde{\\phi } = i \\, \\tau _{2} \\, \\phi ^{*}$ with $\\tau _2$ , the second Pauli matrix, acting on the $SU (2)_\\mathrm {L}$ indices, $M$ is a $n \\times n$ symmetric matrix, $Y$ is a general $3 \\times n$ matrix and we have omitted flavour indices for simplicity.", "After spontaneous symmetry breaking (SSB), $\\langle \\phi \\rangle = v_\\phi $ with $v_\\phi =$ 174 GeV, the neutrino mass terms are given by $\\mathcal {L}_{\\nu \\: \\mathrm {mass}} = -\\dfrac{1}{2} \\,\\begin{pmatrix}\\overline{\\nu _\\mathrm {L}} & \\overline{\\nu _\\mathrm {R}^\\mathrm {c}}\\end{pmatrix} \\,\\begin{pmatrix}0 & m_\\mathrm {D} \\\\m_\\mathrm {D}^\\mathrm {T} & M\\end{pmatrix} \\,\\begin{pmatrix}\\nu _\\mathrm {L}^\\mathrm {c} \\\\\\nu _\\mathrm {R}\\end{pmatrix} + \\mathrm {H.c.} \\ ,$ where $m_D = Y v_{\\phi }$ .", "The mass scale for right-handed neutrinos is in principle free, however if $M \\gg m_\\mathrm {D}$ , upon block-diagonalization one obtains $n$ heavy leptons which are mainly SM singlets, with masses $\\sim M$ , and the well-known see-saw formula for the effective light neutrino Majorana mass matrix, $m_{\\nu } \\simeq -m_\\mathrm {D} \\, M^{-1} \\, m_\\mathrm {D}^\\mathrm {T} \\, ,$ which naturally explains the smallness of light neutrino masses as a consequence of the presence of heavy SM singlet leptons." ], [ "Type II: scalar triplet", "The type II see-saw [33], [34], [35], [36], [37] only adds to the SM field content one scalar triplet with hypercharge $Y=1$ (we adopt the convention that $Q = Y + T_3$ ) and assigns to it lepton number $L=-2$ .", "In the doublet representation of $SU(2)_\\mathrm {L}$ the triplet can be written as a $2 \\times 2$ matrix, whose components are $\\chi = \\begin{pmatrix}\\chi ^{+}/\\sqrt{2} & \\chi ^{++} \\\\\\chi _{0} & - \\chi ^{+}/\\sqrt{2}\\end{pmatrix} \\, .$ Gauge invariance allows a Yukawa coupling of the scalar triplet to two lepton doublets, $\\mathcal {L}_{\\chi } = \\left( (Y^\\dagger _\\chi )_{\\alpha \\beta } \\, \\overline{\\tilde{\\ell }}_\\alpha \\chi \\ell _\\beta + \\mathrm {H. c.} \\right) - V(\\phi ,\\chi ) \\, ,$ where $Y_\\chi $ is a symmetric matrix in flavour space, and $\\tilde{\\ell }= i\\tau _2 \\, \\ell ^\\mathrm {c}$ .", "The scalar potential has, among others, the following terms: $V (\\phi , \\chi ) = m_\\chi ^2 \\, \\mathrm {Tr}[ \\chi \\chi ^\\dagger ] -\\left( \\mu \\,\\tilde{\\phi }^{\\dagger } \\chi ^{\\dagger } \\phi + \\mathrm {H.c.} \\right) + \\ldots $ The $\\mu $ coupling violates lepton number explicitly, and it induces a vacuum expectation value (VEV) for the triplet via the VEV of the doublet, even if $m_\\chi > 0$ .", "In the limit $m_\\chi \\gg v_\\phi $ this VEV can be approximated by: $\\left< \\chi \\right> \\equiv v_\\chi \\simeq \\cfrac{\\mu v_\\phi ^{2}}{m_{\\chi }^{2}} \\, ;$ then, the Yukawa couplings in equation (REF ) lead to a Majorana mass matrix for the left-handed neutrinos $m_\\nu = 2 Y_\\chi v_\\chi = 2 Y_\\chi \\cfrac{\\mu v_\\phi ^{2}}{m_{\\chi }^{2}} \\, .$ Neutrino masses are thus proportional to both $Y_\\chi $ and $\\mu $ .", "Such dependence can be understood from the Lagrangian, since the breaking of lepton number $L$ results from the simultaneous presence of the Yukawa and $\\mu $ couplings.", "As long as $m_{\\chi }^{2}$ is positive and large, $v_\\chi $ will be small, in agreement with the constraints from the $\\rho $ parameter, $v_\\chi \\lesssim 6 \\; \\mathrm {GeV}$ [38].This bound is calculated after the inclusion of the one-loop corrections to the $\\rho $ parameter, and is slightly looser than other previously obtained from electroweak global fits (see, for example, [39]).", "Note also that the authors of [38] use a different normalisation for the VEV, and hence the difference between their value and the one we present here.", "Moreover, the parameter $\\mu $ , which has dimensions of mass, can be naturally small, because in its absence lepton number is recovered, increasing the symmetry of the model." ], [ "Type III: fermionic triplets", "In the type III see-saw model [40], [41], the SM is extended by fermion $SU(2)_{\\mathrm {L}}$ triplets $\\Sigma _{\\alpha }$ with zero hypercharge.", "As in type I, at least two fermion triplets are needed to have two non-vanishing light neutrino masses.", "We choose the spinors $\\Sigma _\\alpha $ to be right-handed under Lorentz transformations and write them in $SU(2)$ Cartesian components $\\vec{\\Sigma }_{\\alpha }=(\\Sigma _\\alpha ^1,\\Sigma _\\alpha ^2,\\Sigma _\\alpha ^3)$ .", "The Cartesian components can be written in terms of charge eigenstates as usual $\\Sigma ^+_\\alpha =\\frac{1}{\\sqrt{2}} (\\Sigma ^1_\\alpha -i\\Sigma ^2_\\alpha )\\ ,\\qquad \\Sigma ^0_\\alpha =\\Sigma ^3_\\alpha \\ , \\qquad \\Sigma ^-_\\alpha =\\frac{1}{\\sqrt{2}} (\\Sigma ^1_\\alpha +i\\Sigma ^2_\\alpha )\\ ,$ and the charged components can be further combined into negatively charged Dirac fermions $E_\\alpha =\\Sigma _\\alpha ^-+\\Sigma _\\alpha ^{+\\mathrm {c}}$ .", "Using standard four-component notation the new terms in the Lagrangian are given by $\\mathcal {L}_{\\Sigma } = i \\,\\overline{\\vec{\\Sigma }_\\alpha } \\gamma ^{\\mu }D_{\\mu } \\cdot \\vec{\\Sigma }_\\alpha -\\left( \\cfrac{1}{2} \\, M_{\\alpha \\beta }\\overline{\\vec{\\Sigma }^{\\mathrm {c}}_\\alpha }\\cdot \\vec{\\Sigma }_\\beta +Y_{\\alpha \\beta } \\, \\overline{\\ell _\\alpha } \\, \\left(\\vec{\\tau }\\cdot \\vec{\\Sigma }_\\beta \\right)\\tilde{\\phi }+ \\mathrm {H.c.} \\right) \\, ,$ where $Y$ is the Yukawa coupling of the fermion triplets to the SM lepton doublets and the Higgs, and $M$ their Majorana mass matrix, which can be chosen to be diagonal and real in flavour space.", "After SSB the neutrino mass matrix can be written as $\\mathcal {L}_{\\nu \\: \\mathrm {mass}} = - \\cfrac{1}{2} \\,\\begin{pmatrix}\\overline{\\nu _\\mathrm {L}} & \\overline{\\Sigma ^{0\\mathrm {c}}}\\end{pmatrix} \\,\\begin{pmatrix}0 & m_{\\mathrm {D}} \\\\m_{\\mathrm {D}}^{\\mathrm {T}} & M\\end{pmatrix} \\,\\begin{pmatrix}\\nu _{\\mathrm {L}}^{\\mathrm {c}} \\\\\\Sigma ^{0}\\end{pmatrix}+ \\mathrm {H. c.} \\, ,$ which is the same as in the type I see-saw just replacing the singlet right-handed neutrinos by the neutral component of the triplets, $\\Sigma ^0_{\\alpha }$ , and therefore leads to a light neutrino Majorana mass matrix $m_{\\nu } \\simeq -m_{\\mathrm {D}}\\, M^{-1} \\, m_{\\mathrm {D}}^{\\mathrm {T}}\\, .$ However, since the triplet has also charged components with the same Majorana mass, in this case there are stringent lower bounds on the new mass scale, $M \\gtrsim 100 \\; \\mathrm {GeV}$ .", "Here we briefly summarize non minimal mechanisms which also lead to Majorana light neutrino masses.", "Most of these models do not include right-handed neutrinos and are designed to obtain tiny Majorana masses for the left-handed SM neutrinos, so we can anticipate that they will not be appropriate for the fourth generation.", "a) Radiative mechanisms.", "Small Majorana neutrino masses may also be induced by radiative corrections [12], [13], [14], [15].", "Typically, on top of loop factors of at least $1/(4\\pi )^2$ , there are additional suppressions due to couplings or ratios of masses, leading to the observed light neutrino masses with a new physics scale not far above the electroweak one.", "b) Supersymmetry.", "There is an intrinsically supersymmetric way of breaking lepton number by breaking the so-called R parity [42], [43], [44], [45], [46], [47], [48], [49], [50], [51] (for a review see [52]).", "In this scenario, the SM doublet neutrinos mix with the neutralinos, i.e., the supersymmetric (fermionic) partners of the neutral gauge and Higgs bosons.", "As a consequence, Majorana masses for neutrinos (generated at tree level and at one loop) are naturally small because they are proportional to the small R-parity-breaking parameters." ], [ "Weinberg operator", "As we have mentioned, it does not seem very natural that neutrinos are Dirac particles; assuming that they are Majorana, we will be often interested in abstracting from the actual mechanism of mass generation.", "In such case, if the light degrees of freedom are those of the SM we can parametrise the Majorana masses in terms of the well-known dimension 5 Weinberg operator In supersymmetric models, $\\tilde{\\phi }= H_u$ , since there are two Higgs doublets.", "[30], [31]: $\\mathcal {L}_{5}= \\cfrac{1}{2} \\, \\frac{c_{\\alpha \\beta }}{\\Lambda _W}(\\overline{\\ell _\\alpha }\\tilde{\\phi }) \\,(\\phi ^{\\dagger }\\tilde{\\ell }_\\beta )+ \\mathrm {H. c.}\\, ,$ where $\\Lambda _W \\gg v_\\phi $ is the scale of new physics and $c_{\\alpha \\beta }$ are model-dependent coefficients with flavour structure, which in some models can carry additional suppression due to loop factors (as is the case in radiative mechanisms) and/or ratios of mass parameters (for instance in type II see-saw $c \\propto \\mu /m_{\\chi }$ ).", "In those cases we will assume that $\\Lambda _W$ is directly related to the masses of the new particles and absorb all suppression factors in $c_{\\alpha \\beta }$ .", "Upon electroweak symmetry breaking, the Weinberg operator leads to a Majorana mass matrix for the light neutrinos of the form $m_{\\nu } = c \\, \\frac{v_\\phi ^{2}}{\\Lambda _W}\\, .$ Notice that if $c_{\\alpha \\beta }$ is suppressed, the scale $\\Lambda _W$ does not need to be extremely large in order to fit light neutrino masses and, thus, the Weinberg operator can parametrize a variety of Majorana neutrino mass models, including those with masses generated radiatively." ], [ "Fourth-generation neutrino masses", "If there exists a fourth generation, the fourth-generation neutrinos must be massive (with masses $\\gtrsim m_{Z}/2$ in order to avoid the strong limits for the number of active neutrinos found at LEP).", "In principle all mass mechanisms available for the light neutrinos are also available to the fourth-generation neutrinos, however, the fact that they must be quite massive changes completely the discussion of the naturalness of the different mechanisms.", "Let us discuss them: a) Dirac masses.", "Since the fourth generation must be at the electroweak scale, this mechanism of mass generation is quite natural for the fourth-generation neutrinos as long as lepton number is conserved.", "b) Fermionic singlets with Majorana mass.", "If lepton number is not conserved there is no reason to forbid a Majorana mass term for right-handed neutrinos (see the discussion below).", "However if $m_{R}\\gg m_{D}$ the see-saw formula applies and the spectrum contains a relatively light, almost-active neutrino with mass $m_{4}\\sim m_{D}^{2}/m_{R}$ , which must be heavier than $m_{Z}/2$ .", "Therefore $m_{R}<m_{D}^{2}/m_{Z}$ and the mass of the right-handed neutrino cannot be much larger than the electroweak scale.", "On the other hand, if $m_{R} \\ll m_{D}$ there are two almost degenerate neutrinos and we are in the pseudo-Dirac limit, which does not pose any problem.", "c) Scalar triplet.", "In principle, as in the case of light neutrinos, scalar triplets could also be used to obtain Majorana masses for the fourth-generation neutrinos.", "However, the strong limits on the triplet's VEV coming from the $\\rho $ parameter $v_\\chi \\lesssim 6 \\; \\mathrm {GeV}$ will yield fourth-generation neutrino masses too small.", "This limit could be relaxed a bit if radiative corrections to the $\\rho $ parameter coming from triplet masses are large and such that cancel in part the deviations induced by the triplet's VEV, but this will require quite a high degree of fine tuning among rather different quantities.", "Therefore, this mechanism alone is not a natural mechanism for the fourth-generation neutrino masses.", "d) Fermionic triplets.", "This is similar to b), but together with the right-handed neutrinos there come new charged fermions degenerate with the neutral component.", "Since production limits tell us that the charged fermions must be heavier than about $100 \\; \\mathrm {GeV}$ , in this case the pseudo-Dirac limit is not possible.", "Moreover these new fermions cannot be extremely heavy, because otherwise the active neutrino will be too light.", "We conclude that this mechanism is viable but much more constrained than b).", "e) Radiative mechanisms and SUSY with broken R parity.", "Neutrino masses in these models are strongly suppressed with respect to the electroweak scale by either loop factors, couplings and/or ratios of masses.", "Therefore they are not viable for the fourth generation.", "f) Weinberg operator.", "In principle the Weinberg operator could also be used to give Majorana masses to the fourth-generation neutrinos.", "However, it will provide masses $\\mathcal {O}( {v_\\phi ^2}{\\Lambda _W} )$ which should be $\\gtrsim m_Z/2$ , so the scale of new physics $\\Lambda _W$ can not be much larger than the electroweak scale $v_\\phi $ and the effective theory does not make sense.", "Therefore the Weinberg operator does not provide a useful parametrization of the fourth-generation neutrino mass.", "We can therefore conclude that only a), b) (which includes a) in some limit) and possibly d) are good mechanisms for the fourth-generation neutrino masses.", "It seems then that to describe correctly the fourth-generation neutrino one needs at least one right-handed neutrino (either SM singlet or triplet) which has standard Dirac couplings to the doublets.", "If this RH neutrino is a SM triplet, we have seen that its Majorana mass is in the range 100 GeV $\\lesssim m_R \\lesssim $ few TeV.", "However, if the right-handed neutrino is a SM singlet it could have a very small or even vanishing Majorana mass term.", "Is it natural to have Dirac neutrinos for the fourth generation?", "The answer is simple: yes, provided there is a symmetry that protects them from acquiring a Majorana mass term.", "This is not the situation if the light neutrinos are Majorana, as most of the SM extensions that we considered in section , and they can mix freely with the heavy fourth family.", "We argue that in such a case a Majorana mass term for the fourth right-handed neutrino should be allowed just on symmetry grounds, and in fact, based on naturality arguments, a lower bound for this Majorana mass can be given.", "Figure: NO_CAPTIONThe two-loop process that provides a Majorana mass for the fourth-generation right-handed neutrino in the framework of type I see-saw.", "The indices $i$ and $j$ represent any of the four families; the index $k$ , however, represents only the right-handed neutrinos associated to the generation of masses for the light families, $k = 1, 2, 3$ .", "Let us consider first the case in which the three light neutrinos obtain their masses via a type-I see-saw containing heavy right-handed neutrinos with masses $m_{\\mathrm {R}k}$ (with $k=1,2,3$ ) of the order of $10^{12}$ –$10^{15}$  GeV.", "Since lepton number is not conserved it is natural to consider a Majorana mass term for the fourth right-handed neutrino, $\\nu _{\\mathrm {R} 4}$ .", "However, in order to satisfy the LEP bounds on the number of light active neutrinos, $m_{\\mathrm {R}4}$ should be, at most, of the order of a few TeV and, therefore, much smaller than $m_{\\mathrm {R}k}$ .", "Thus, one might think that perhaps it is more natural to set directly $m_{\\mathrm {R}4}=0$ and consider only Dirac neutrinos for the fourth generation.", "The question that arises then is whether this choice is stable or not under radiative corrections and what is the natural size one might expect for $m_{\\mathrm {R}4}$ , since setting $m_{\\mathrm {R}4}=0$ does not increase the symmetries of the Lagrangian.", "The answer can be obtained from the diagram in figure  which gives a logarithmically divergent contribution to $m_{\\mathrm {R}4}$ induced by the presence of the three heavy Majorana neutrino masses, $m_{\\mathrm {R}k}$ [26], [25].", "Thus, above the $m_{\\mathrm {R}k}$ scale, $m_{\\mathrm {R}4}$ and $m_{\\mathrm {R}k}$ mix under renormalization and do not run independently.", "Therefore, even if one finds a model in which $m_{\\mathrm {R}4}=0$ at some scale $\\Lambda _C > m_{\\mathrm {R}k}$ , $m_{\\mathrm {R}4}$ will be generated by running from $\\Lambda _C$ to $m_{\\mathrm {R}k}$ .", "This running can easily be estimated from the diagram in figure  and, barring accidental cancellations, one should require $m_{\\mathrm {R} 4} \\gtrsim \\frac{1}{(4\\pi )^{4}} \\sum _{i j k }Y_{i4} Y_{i k}^* m_{\\mathrm {R} k} Y_{j k}^* Y_{j4}\\ln (\\Lambda _C/m_{\\mathrm {R} k})\\gtrsim \\frac{1}{(4\\pi )^{4}} \\sum _{i j k }Y_{i4} Y_{i k}^* m_{\\mathrm {R} k} Y_{j k}^* Y_{j4}\\, ,$ where $i,j=1,2,3,4$ , $k=1,2,3$ and in the last step we have taken $\\ln (\\Lambda _C/m_{\\mathrm {R} k}) \\gtrsim 1$ .", "Of course, given a particular renormalizable model yielding $m_{\\mathrm {R}4}=0$ at tree level (see appendix  for an explicit example) one should be able to compute the full two-loop mass $m_{\\mathrm {R}4}$ , which will be finite and will contain the logarithmic contributions we have just discussed.", "Eq.", "(REF ) sets the lower bound that we had announced.", "Let us now estimate its value; bearing in mind that in type I see-saw the light neutrino masses are given byNotice that by integrating out the three heavy right-handed neutrinos we obtain a Majorana neutrino mass matrix for the four active neutrinos which is of the order of the light neutrino masses.", "$(m_{\\nu })_{ij} \\sim \\sum _k Y_{ik} Y_{jk} v_{\\phi }^{2}/m_{\\mathrm {R} k}$ .", "Then, by taking all $m_{\\mathrm {R}k}$ of the same order we can rewrite the bound as $ m_{\\mathrm {R} 4}\\gtrsim \\sum _{ij}\\frac{Y_{i4} (m^*_\\nu )_{ij} Y_{j4}}{(4\\pi )^{4}}\\frac{m^2_{\\mathrm {R} k}}{v_{\\phi }^{2}}\\, .$ To give a conservative estimate we consider only the contribution of the first three generations because we expect their Yukawa couplings to the fourth right-handed neutrino to be somewhat suppressed due to universality and LFV constraints [16], [17], [53], [23] (say, $Y_{k 4} \\sim 10^{-2}$ ).", "Once we fix the neutrino masses and the Yukawa couplings between the fourth-generation neutrino and the first three, $m_{\\mathrm {R} 4}$ grows quadratically with $m_{\\mathrm {R} k}$ .", "For $m_\\nu =0.01 \\; \\mathrm {eV}$ and $Y_{k 4} = 0.01$ we obtain that $m_{\\mathrm {R} 4}$ is of order keV, GeV, PeV for $m_{\\mathrm {R} k} = 10^9, 10^{12}, 10^{15} \\; \\mathrm {GeV}$ , respectively.", "The contribution of the fourth active neutrino is not necessarily suppressed by the Yuwawa couplings and, in principle, by using it, even more restrictive bounds on $m_{\\mathrm {R} 4}$ could be set.", "However, as $(m_\\nu )_{44}$ is model-dependentIn this case, $(m_\\nu )_{44}$ is the see-saw mass induced by only the three heavy right-handed neutrinos, thus, it could even be zero if the Yukawas between the fourth lepton doublet and the three right-handed neutrinos vanish for some reason., we keep the most conservative bound.", "Figure: NO_CAPTIONThe process that provides a Majorana mass for the right-handed neutrino associated to the fourth generation in the framework of type II see-saw.", "Let us consider now the case in which the three light neutrinos obtain their masses through the type II see-saw mechanism (see section REF ), i.e., through their coupling to a scalar triplet, $\\chi $ , which develops a VEV.", "As discussed in section , this triplet cannot be the only source Majorana masses for the fourth-generation neutrinos and, at least, one right-handed neutrino is needed.", "We will assume then that there is a right-handed neutrino which has Yukawa couplings to the four SM doublets.", "In this scenario one can easily see that the right-handed neutrino will acquire, at two loops (as seen in figure ), a Majorana mass.", "This just reflects the fact that the right-handed neutrino mass $m_{\\mathrm {R}4}$ and the trilinear coupling of the triplet, $\\mu $ , mix under renormalization.", "Applying the same arguments used in the case of see-saw type I for light neutrino masses and the estimate of the diagram in figure we can write $m_{\\mathrm {R}4} \\gtrsim \\frac{\\mu }{(4\\pi )^{4}}\\sum _{ij} Y_{i4} (Y^*_\\chi )_{ij} Y_{j4}\\, ,$ where $Y_\\chi $ are the Yukawa couplings of the triplet to the lepton doublets and, as before, we have taken $\\ln (\\Lambda _C/m_{\\chi }) \\gtrsim 1$ .", "As in the type I see-saw case the result can also be expressed in terms of the light neutrino masses $(m_{\\nu })_{ij} \\sim (Y_\\chi )_{ij} \\mu v_{\\phi }^{2}/m_{\\chi }^{2}$ ; thus $ m_{\\mathrm {R}4}\\gtrsim \\sum _{ij}\\frac{Y_{i4} (m^*_\\nu )_{ij} Y_{j4}}{(4\\pi )^{4}}\\frac{m^2_\\chi }{v^2_{\\phi }} \\, ,$ which shows a similar structure to that obtained for type I see-saw, eq.", "(REF ).", "The same result is obtained for type III see-saw, whose couplings are analogous to those of type I.", "The similarity of the two results suggests that bounds of this type are quite general and should appear in all kinds of four-generation models with light Majorana neutrinos.", "In fact, as discussed in section REF , light Majorana neutrino masses can be parametrized in many models by means of the Weinberg operator, eq.", "(REF ), which yields neutrino masses given by eq.", "(REF ).", "Then, one could draw a two-loop diagram analogous to the diagrams in figures and but with the propagators of heavy particles pinched and substituted by one insertion of the Weinberg operator.", "This diagram is quadratically divergent and, therefore, its contribution to $m_{\\mathrm {R}4}$ can not be reliably computed in the effective field theory because it depends on the details of the matching with the full theory from which the effective one originates (in fact it vanishes in dimensional regularization or in any other regularization scheme allowing symmetric integration), but one can use naive dimensional analysis to estimate contributions of order $m_{\\mathrm {R}4} \\sim \\frac{\\Lambda _W}{(4\\pi )^{4}}\\sum _{ij} Y_{i4} c^*_{ij} Y_{j4} \\sim \\frac{ Y_{i4} (m^*_\\nu )_{ij} Y_{j4}}{(4\\pi )^{4}}\\frac{\\Lambda ^2_W}{v^2_\\phi }\\, ,$ which is precisely the result obtained in the see-saw models discussed above if one identifies $\\Lambda _W \\sim m_{\\mathrm {R}k},m_\\chi $ .", "However, it is important to remark that in the low energy effective theory $m_{\\mathrm {R}4}$ is a free parameter, and eq.", "(REF ) is only a naive dimensional analysis estimate of what one would expect in a more complete theory." ], [ "Light neutrino masses induced by new generations", "After the discussion above, to describe correctly the neutrino sector of models with four generations we need just one relatively light right-handed neutrino, $\\nu _{\\mathrm {R}}$ , to give Dirac mass terms to the fourth-generation neutrinos, while Majorana masses for light neutrinos can be parametrized by the Weinberg operator.", "We will work in this minimal four-generation scenario, thus, the relevant part of the Lagrangian for our discussion is $\\mathcal {L}_{Y} = -\\bar{\\ell }Y_{e} e_{\\mathrm {R}} \\phi -\\bar{\\ell } y \\nu _{\\mathrm {R}} \\tilde{\\phi } -\\cfrac{1}{2} \\, \\overline{\\nu _{\\mathrm {R}}^{\\mathrm {c}}} m_{\\mathrm {R}}\\nu _{\\mathrm {R}} +\\frac{1}{2v_{\\text{$\\phi $}}^{2}}(\\overline{\\ell }\\tilde{\\phi })m_{\\mathrm {L}}(\\phi ^{\\dagger }\\tilde{\\ell )}+ \\mathrm {H.c.}\\ ,$ where $\\ell $ and $e_{\\mathrm {R}}$ contain the four generation components while $\\nu _{\\mathrm {R}}$ is the only right-handed neutrino.", "Thus $Y_{e}$ is a completely general $4\\times 4$ complex matrix, $y$ is a 4 component column vector, $m_{\\mathrm {R}}$ is just a number and $m_{\\mathrm {L}}$ is a general complex symmetric $4\\times 4$ matrix.", "The Dirac limit is recovered when $m_{\\mathrm {R}}=0$ and $m_{\\mathrm {L}}=0$ .", "Since light neutrino masses are very small, we will assume $m_{\\mathrm {L}}\\ll v_{\\phi }$ while $m_{\\mathrm {R}}$ , as we immediately see, cannot be very large to ensure there are only three light active neutrinos.", "Moreover, as shown in the previous section, we do not expect it to be zero if $m_{\\mathrm {L}}$ is not zero.", "Above we have taken for $m_{\\mathrm {L}}$ a general complex symmetric $4\\times 4$ matrix in spite of the fact that to describe the light neutrino sector we just need a $3\\times 3$ matrix.", "This is because in most of the neutrino mass models one also obtains contributions to the fourth-generation Weinberg operator.", "For instance, we give below the values of $m_{\\mathrm {L}}$ one obtains for the different types of seesaw.", "If the three light neutrino masses are generated by the seesaw mechanism type I or type III, we need three of the right-handed neutrinos much heavier than the fourth.", "We can always choose a basis in which the Majorana mass matrix of right-handed neutrinos is diagonal and integrate out the three heavy right-handed neutrinos.", "The result can be writen in terms of the Weinberg operator in (REF ) with $(m_{\\mathrm {L}})_{\\alpha \\beta } = -\\sum _{k=1,2,3}\\frac{\\left(Y_{\\nu }\\right)_{\\alpha k} \\left(Y_{\\nu }\\right)_{\\beta k}}{m_{\\mathrm {R} k}} v_{\\phi }^{2} \\, ,$ where $m_{\\mathrm {R} k}$ are the eigenvalues of the diagonal Majorana mass matrix of the three heavy right-handed neutrinos, while $\\left(Y_{\\nu }\\right)_{\\alpha k}$ are the Yukawa couplings of the three heavy right-handed neutrinos with the four lepton doublets.", "Then, the $4\\times 4$ mass matrix (REF ) is projective and has at most rank 3.", "If the three light neutrino masses are generated by the VEV of a triplet (type II see-saw), we will have $(m_{\\mathrm {L}})_{\\alpha \\beta } = 2(Y_\\chi )_{\\alpha \\beta } v_\\chi \\, ,$ being $(Y_\\chi )_{\\alpha \\beta }$ the Yukawa couplings of the 4 lepton doublets to the triplet and $v_\\chi \\sim \\mu v_{\\phi }^{2}/m_{\\chi }^{2}$ its VEV.", "In this case, $m_{\\mathrm {L}}$ is a completely general $4\\times 4$ symmetric complex matrix.", "After SSB the neutrino mass matrix (in the basis $(\\nu ^\\mathrm {c}_{\\mathrm {L} \\alpha },\\nu _{\\mathrm {R}})$ ) is $M=\\left(\\begin{array}{cc}m_{\\mathrm {L}} & y v_{\\phi }\\\\y^{\\mathrm {T}} v_{\\phi } & m_{\\mathrm {R}}\\end{array}\\right)\\, .$ To diagonalize this mass matrix we perform first a $4 \\times 4$ rotation in order to separate heavy from light degrees of freedom, so we change from the flavour basis ($\\nu _{e},\\nu _{\\mu },\\nu _{\\tau },\\nu _{E}$ ) to a new basis $\\nu _{\\text{1}}^{\\prime },\\nu _{\\text{2}}^{\\prime },\\nu _{\\text{3}}^{\\prime },\\nu _{\\text{4}}^{\\prime }$ in which the first three states are light (with masses given by $m_{\\mathrm {L}}$ ) and only $\\nu _{4}^{\\prime }$ mixes with $\\nu _{\\mathrm {R}}$ .", "Then, we have $\\nu _{\\alpha }=\\sum _{i}V_{\\alpha i}\\nu _{i}^{\\prime }$ ($i=1,\\text{$\\cdots $},4$ , $\\alpha =e,\\mu ,\\tau ,E$ ), where $V$ is a orthogonal matrix, and we define $ N_{\\alpha } \\equiv V_{\\alpha 4} = \\frac{y_\\alpha }{\\sqrt{\\sum _{\\beta }y_{\\beta }^{2}}}\\, .$ Now, we are free to choose $\\nu _{\\text{1}}^{\\prime },\\nu _{\\text{2}}^{\\prime },\\nu _{\\text{3}}^{\\prime }$ in any combination of $\\nu _{e},\\nu _{\\mu },\\nu _{\\tau },\\nu _{E}$ as long as they are orthogonal to $\\nu _{4}^{\\prime }$ , i.e., $\\sum _{\\alpha }V_{\\alpha k} N_{\\alpha } = 0$ for $k \\text{=1,2,3}$ .", "The orthogonality of $V$ almost fixes all its elements in terms of $N_{\\alpha }$ , but still leaves us some freedom to set three of them to zero.", "Following [16], [17] we choose $V_{\\tau 1} = V_{E1} = V_{E2}=0$ for convenience.", "The transpose of the matrix $V$ is: ${V^\\mathrm {T}=\\left(\\begin{array}{cccc}\\cfrac{N_{\\mu }}{\\sqrt{N_{e}^{2}+N_{\\mu }^{2}}} & \\cfrac{-N_{e}}{\\sqrt{N_{e}^{2}+N_{\\mu }^{2}}} & 0 & 0\\\\\\cfrac{N_{e}N_{\\tau }}{\\sqrt{(N_{e}^{2}+N_{\\mu }^{2})(1-N_{E}^{2})}} &\\cfrac{N_{\\mu }N_{\\tau }}{\\sqrt{(N_{e}^{2}+N_{\\mu }^{2})(1-N_{E}^{2})}} &\\cfrac{-N_{e}^{2}-N_{\\mu }^{2}}{\\sqrt{(N_{e}^{2}+N_{\\mu }^{2})(1-N_{E}^{2})}} & 0\\\\\\cfrac{N_{e}N_{E}}{\\sqrt{(1-N_{E}^{2})}} & \\cfrac{N_{\\mu }N_{E}}{\\sqrt{(1-N_{E}^{2})}} &\\cfrac{N_{\\tau }N_{E}}{\\sqrt{(1-N_{E}^{2})}} & - {\\sqrt{(1-N_{E}^{2})}}\\\\N_{e} & N_{\\mu } & N_{\\tau } & N_{E}\\, .\\end{array}\\right)}$ After this rotation the neutrino mass matrix is $\\tilde{M}=\\left(\\begin{array}{cc}\\tilde{m}_{\\mathrm {L}} & \\begin{array}{cc}\\omega _{1} & 0\\\\\\omega _{2} & 0\\\\\\omega _{3} & 0\\end{array}\\\\\\begin{array}{ccc}\\omega _{1} & \\omega _{2} & \\omega _{3}\\\\0 & 0 & 0\\end{array} & \\begin{array}{cc}\\omega _{4} & m_{\\mathrm {D}}\\\\m_{\\mathrm {D}} & m_{\\mathrm {R}}\\end{array}\\end{array}\\right)\\, ,$ where $(\\tilde{m}_{\\mathrm {L}})_{k k^\\prime }=(V m_{\\mathrm {L}}V^{\\mathrm {T}})_{k k^\\prime }$ is a $3\\times 3$ matrix with $k,k^\\prime =1,2,3$ , $\\text{$\\omega $}_{k}=(V m_{\\mathrm {L}} V^{\\mathrm {T}})_{4k}$ , $\\omega _{4}=(Vm_{\\mathrm {L}}V^{\\mathrm {T}})_{44}$ and $m_{\\mathrm {D}}=v_\\phi \\sqrt{\\sum _{\\alpha }y_{\\alpha }^{2}}$ .", "Since $\\tilde{m}_{\\mathrm {L}},\\omega _k,\\omega _4 \\ll m_{\\mathrm {R}},m_{\\mathrm {D}}$ , the matrix $\\tilde{M}$ can be block-diagonalized using the see-saw formula.", "Then, the mass matrix of the light neutrinos (at tree level) will be $m_{\\nu }^{(0)}=\\tilde{m}_{\\mathrm {L}} -\\frac{m_{\\mathrm {R}}}{m_{\\mathrm {R}} \\omega _{4} -m_{\\mathrm {D}}^{2}} \\vec{\\omega } \\cdot \\vec{\\omega }^{\\mathrm {T}}\\, ,$ while the heavy sector will be obtained after diagonalizing the $2\\times 2$ matrix $M_H =\\left(\\begin{array}{cc}\\omega _{4} & m_{\\mathrm {D}}\\\\m_{\\mathrm {D}} & m_{\\mathrm {R}}\\end{array}\\right)\\ .$ Neglecting $\\omega _4$ , this diagonalization leads to two Majorana neutrinos $\\nu _{4}&=&i\\cos \\theta (-\\nu _{4}^{\\prime }+\\nu _{4}^{\\prime \\mathrm {c}})+i \\sin \\theta (\\nu _{\\mathrm {R}}-\\nu _{\\mathrm {R}}^{\\mathrm {c}})\\\\\\nu _{\\bar{4}}&=&-\\sin \\theta (\\nu _{4}^{\\prime }+\\nu _{4}^{\\prime \\mathrm {c}})+\\cos \\theta (\\nu _{\\mathrm {R}}+\\nu _{\\mathrm {R}}^{\\mathrm {c}})$ with masses $m_{4,\\bar{4}}=\\cfrac{1}{2}\\left(\\sqrt{m_{\\mathrm {R}}{}^{2}+ 4m_{\\mathrm {D}}^{2}}\\mp m_{\\mathrm {R}}\\right) \\ ,$ and mixing angle $\\tan ^{2}\\theta =m_{4}/m_{\\bar{4}}$ .", "The imaginary unit factor $i$ and the relative signs in $\\nu _{4}$ are necessary to keep the mass terms positive and preserve the canonical Majorana condition $\\nu _{4}=\\nu _{4}^{\\mathrm {c}}$ .", "If $m_{\\mathrm {R}}\\ll m_{\\mathrm {D}}$ , we have $m_{4}\\approx m_{\\bar{4}}$ , $\\tan \\theta \\approx 1$ , and we say we are in the pseudo-Dirac limit while when $m_{\\mathrm {R}}\\gg m_{\\mathrm {D}}$ , $m_{4}\\approx m_{\\mathrm {D}}^{2}/m_{\\mathrm {R}}$ and $m_{\\bar{4}}\\approx m_{\\mathrm {R}}$ , $\\tan \\theta \\approx m_{\\mathrm {D}}/m_{\\mathrm {R}}$ and we say we are in the see-saw limit.", "Eq.", "(REF ) can be used as long as $m_{\\mathrm {R}}\\omega _{4}-m_{\\mathrm {D}}^{2}$ is different from zero.", "However, we expect $m_{\\mathrm {R}}$ to be below few TeV and $\\omega _{4}$ below 1 eV.", "Therefore $m_{\\mathrm {R}}\\omega _{4}\\ll m_{\\mathrm {D}}^{2}$ unless $m_{\\mathrm {D}}$ is very small but, in that case, the fourth-generation neutrinos will be too light.", "Thus, the correction to the $3\\times 3$ neutrino mass matrix is projective (only one eigenvalue different from zero) and it is naturally order $m_{\\mathrm {L}}^{2}$ and, therefore, negligible.", "Summarizing, there are two heavy neutrinos 4 and $\\bar{4}$ (with a small pollution from $m_{\\mathrm {L}}$ which can be neglected) and a tree-level mass matrix for the light neutrinos $m^{(0)}_\\nu \\simeq \\tilde{m}_{\\mathrm {L}}$ .", "Therefore, neglecting the small $\\omega _i$ 's in eq.", "(REF ), the 5$\\times $ 5 unitary matrix which relates the flavour with the mass eigenstate basis can be written as $U = U_H \\cdot U_L$ , being $U_H$ the rotation in the heavy sector which diagonalizes the mass matrix $M_H$ in eq.", "(REF ) and $U_L$ given by $U_L=\\left(\\begin{array}{cc}V & \\begin{array}{c}0\\\\0\\\\0\\\\0\\end{array}\\\\\\begin{array}{cccc}0 & 0 & 0 &0\\end{array}& \\begin{array}{c}1\\end{array}\\end{array}\\right)\\left(\\begin{array}{cc}W & \\begin{array}{cc}0 & 0\\\\0 & 0\\\\0 & 0\\end{array}\\\\\\begin{array}{ccc}0 & 0 & 0 \\\\0 & 0 & 0\\end{array} & \\begin{array}{cc}1 & 0\\\\0 & 1\\end{array}\\end{array}\\right)\\, ,$ where $V$ rotates from the $\\nu _i^{\\prime }$ basis to the flavour basis (see eq.", "(REF )) and $W$ is the matrix which diagonalizes $\\tilde{m}_{\\mathrm {L}}$ .", "Within this approximation, the mixing among the light and the heavy sector, which we wish to constrain, depends on $(U_L)_{\\alpha 4} = V_{\\alpha 4} = N_\\alpha $ .", "Having fixed the tree-level neutrino mass spectrum and given the huge hierarchies present we should consider the stability of the results against radiative corrections.", "One can check that there are no rank-changing one-loop corrections to the neutrino mass matrices.", "This result can be easily understood in the $\\nu ^{\\prime }_i$ basis that we defined before, since the light neutrinos $(\\nu _{\\text{1}}^{\\prime },\\nu _{\\text{2}}^{\\prime },\\nu _{\\text{3}}^{\\prime })$ are decoupled from the heavy sector, $\\nu _{4}, \\nu _{\\bar{4}}$ , so there are not one-loop diagrams involving the fourth-generation neutrinos with light ones as external legs.", "Figure: NO_CAPTIONTwo-loop diagram generating light neutrino masses in the presence of a Majorana fourth generation.", "However it has been shown [27], [16], [17], [23] that two-loop corrections induced by the fourth-generation fermions can generate neutrino masses for the light neutrinos even if they were not present at tree level, see figure .", "In the $\\nu ^{\\prime }_i$ basis the result reads (see [23] for details) $(m_{\\nu })_{ij}^{(2)}=-\\frac{g^{4}}{m_{W}^{4}} m_{\\mathrm {R}}m_{\\mathrm {D}}^{2}\\sum _{\\text{$\\alpha $}}V_{\\alpha i}V_{\\alpha 4}m_{\\alpha }^{2}\\sum _{\\beta }V_{\\beta j}V_{\\beta 4}m_{\\beta }^{2}I_{\\alpha \\beta }\\, ,$ where the sums run over the charged leptons $\\alpha ,\\beta =e,\\mu ,\\tau ,E$ while $i,j=1,2,3$ , and $I_{\\alpha \\beta }$ is a loop integral which was discussed in [23].", "When $m_{\\mathrm {R}}=0,$ $(m_{\\nu })_{ij}^{(2)}=0$ , as it should, because in that case lepton number is conserved.", "Also when $m_{\\mathrm {D}}=0$ we obtain $(m_{\\nu })_{ij}^{(2)}=0$ , since then the right-handed neutrino decouples completely and lepton number is again conserved.", "To see more clearly the structure of this mass matrix we can approximate $m_{e}=m_{\\mu }=m_{\\tau }=0$ ; then, since we have chosen $V_{\\tau 1}=V_{E1}=V_{E2}=0$ , the only non-vanishing element in $(m_{\\nu })_{ij}^{(2)}$ is $(m_{\\nu })_{33}^{(2)}$ and it is proportional to $V_{E3}^{2}N_{E}^{2}m_{E}^{4}I_{EE}$ .", "Therefore, the largest contribution to $(m_\\nu )^{(2)}$ is given by: $(m_\\nu )_{33}^{(2)} &=& - \\frac{g^4}{m_W^4} N_E^2 (N_e^2 +N_\\mu ^2 + N_\\tau ^2)m_{\\mathrm {R}} m_{\\mathrm {D}}^2 m_E^4 I_{EE}\\nonumber \\\\& \\approx & \\frac{g^4}{2 (4\\pi )^4}(N_e^2 +N_\\mu ^2 + N_\\tau ^2)m_{\\mathrm {R}} \\, \\frac{m_{\\mathrm {D}}^2 m_E^2}{m_W^4}\\ln \\frac{m_E}{m_{\\bar{4}}} \\ ,$ where in the last line we have used the approximated expression of the loop integral $I_{EE}$ in the case $m_E \\gg m_{4,\\bar{4}} \\gg m_W$ for definiteness, but other mass relations lead to analogous conclusions.", "Keeping all the charged lepton masses one can easily show that the eigenvalues of the light neutrino mass matrix are proportional to $m_{\\mu }^{4},\\, m_{\\tau }^{4},\\, m_{E}^{4}$ which gives a huge hierarchy between neutrino masses.", "Therefore, as discussed in [23], [26], these radiative corrections cannot explain by themselves the observed spectrum of masses and mixings, although they lead to a strong constraint for this kind of SM extensions which we will analyze in the next section." ], [ "Direct searches", "Figure: NO_CAPTIONThe shaded region shows the allowed values for the Majorana and Dirac masses of the heavy neutrinos given the LEP bound $m_N > 33.5 \\; \\mathrm {GeV}$ on stable neutrinos with both Dirac and Majorana masses.", "We also display a dashed line in the 62.1 GeV limit for unstable neutrinos.", "Two more lines are drawn for completeness, giving an idea of the combination of parameters that produces two possible, allowed masses for the lightest heavy neutrino.", "Let us now discuss the constraints that several phenomenological tests impose on the parameters of this minimal four-generation (4G) model.", "Direct searches for the new heavy leptons can be used to set limits on the Yukawa couplings and the Majorana mass of the $\\nu _\\mathrm {R}$ .", "In the case of the heavy charged lepton, searches at LEP [4] yield $m_E > 100.8 \\;\\mathrm {GeV}$ (assuming it decays rapidly to $\\nu W$ ; a slightly poorer bound is obtained if the lepton is long-lived and can be tracked inside the detectors), which can be immediately translated into a bound on the corresponding Yukawa.", "For the heavy neutrinos, we can have different bounds depending on their stability and the Dirac or Majorana character of their masses [4]: stable neutrinos, for example (understood here as `stable enough to get out of the detectors after production'), are only constrained by the requirement that they don't show up in the invisible decays of the $Z$ boson.", "Unstable (visible) neutrinos get tighter bounds due to the non-observation of their decay products.", "As we are not making any a priori assumption about the neutrino mass structure, we will select here the most conservative from this set of bounds; that corresponds to a stable neutrino with both Dirac and Majorana mass terms, for which we demand $m_N > 33.5 \\; \\mathrm {GeV}$ [54].", "The weakest bound for an unstable neutrino, which applies if it has again both Dirac and Majorana mass terms, will also be of use; we need in that case $m_N > 62.1 \\; \\mathrm {GeV}$ [20].", "As these bounds apply to the physical masses of the neutrinos, which as seen in eqs.", "(REF –REF ) are nonlinear combinations of the Dirac and Majorana components, we display in figure REF the translation of the 33.5 and 62.1 GeV bounds into the $m_\\mathrm {D} - m_\\mathrm {R}$ plane, together with several other lines to give an idea of the relations between physical masses and Lagrangian parameters.", "As explained in section , the neutrino Yukawas $y_\\alpha $ encode the mixings between the flavour-eigenstate neutrinos $\\nu _\\alpha $ and the mass eigenstates $\\nu _{4, \\bar{4}}$ .", "Thus, we can use mixing-mediated LFV processes to constrain the values of the light neutrino Yukawas $y_e$ , $y_\\mu $ , $y_\\tau $ .", "It is important to note, however, that the situation is not the same for `stable' and unstable neutrinos; so-called stable neutral leptons are constrained to decay outside the detectors, which implies that the mean free path must go beyond $\\mathcal {O} (\\mathrm {m})$ .", "The lightest of our heavy neutrinos can only decay through mixing (the main channel being $\\nu _4 \\rightarrow \\ell _\\alpha W$ , $\\alpha =e,\\mu ,\\tau $ , with a possibly virtual $W$ depending on the mass of the $\\nu _4$ ), so this statement is actually a constraint on the Yukawas, implying $y_\\alpha \\sim N_\\alpha \\lesssim 10^{-6}$ .", "This constraint is much stronger than any other phenomenolgical bound, and so it ends the discussion for stable neutrinos, which must have very small mixings that won't be observable in low-energy experiments in the near future (see below).", "For the rest of this section we will consider the case of unstable neutrinos, which present a richer variety of constraints." ], [ "Lepton flavour violation", "Let us now discuss the bounds on violation of lepton family number that can shed light on the relevant mixings of our model; the most stringent limits are derived from the non-observation of radiative decays of the form $\\ell _\\alpha \\rightarrow \\ell _\\beta \\gamma $  Bounds obtained from present data on $\\mu $ –$e$ conversion in nuclei [4] are of the same order.", "However, there are plans to improve the sensitivity in $\\mu $ –$e$ conversion in 4 and even 6 orders of magnitude [55], therefore we expect from this process much stronger bounds in the future..", "In our model, the ratios for such processes are given by $ B (\\ell _\\alpha \\rightarrow \\ell _\\beta \\gamma ) \\equiv \\frac{\\Gamma (\\ell _\\alpha \\rightarrow \\ell _\\beta \\gamma )}{\\Gamma (\\ell _\\alpha \\rightarrow \\ell _\\beta \\nu \\bar{\\nu })}= \\frac{3 \\alpha }{2 \\pi } \\, \\left| \\sum _{a=4,\\bar{4}} U_{\\beta a}U^\\ast _{\\alpha a} \\, H \\left( {m_a^2}{m_W^2}\\right) \\right|^2\\, ,$ where $H(x)$ is a loop function that can be found in [53], and the sum proceeds over all the heavy neutrinosNote this expression contains the contributions from the light neutrinos; by using unitarity of the mixing matrix, they are included in the definition of $H(x)$ .", "(one in the Dirac case, two if they are Majorana).", "The weakest bounds are obtained if only one neutrino with light mass runs inside the loop; this corresponds either to the Dirac limit with a low mass or to a hard see-saw limit, with the heavy neutrino almost decoupled due to its small mixing.", "We will assume this scenario in our calculations in order to produce conservative bounds.", "Table REF summarises the experimental limits and the constraints that can be extracted from these processes.", "Table: NO_CAPTION Summary of the constraints derived from low-energy radiative decays." ], [ "Universality tests", "A second class of constraints upon family mixing arises from the tests of universality in weak interactions.", "For our purposes, these are either direct comparison of decay rates of one particle into two different weak-mediated channels, or comparison of the decay rates of two different particles into the same channel Data from neutrino oscillations can also be used to constrain the elements of the leptonic mixing matrix [57], however, they lead to weaker bounds than the ones obtained here..", "If the weak couplings are to be the same for all families these rates should differ only in known kinematic factors or calculable higher-order corrections.", "The relevant ratios are: $R_{\\pi \\rightarrow e / \\pi \\rightarrow \\mu } &\\equiv \\frac{\\Gamma (\\pi \\rightarrow e \\nu ) }{\\Gamma (\\pi \\rightarrow \\mu \\nu ) }\\, , & R_{\\tau \\rightarrow \\mu / \\tau \\rightarrow e}&\\equiv \\frac{\\Gamma (\\tau \\rightarrow \\mu \\nu \\bar{\\nu })}{\\Gamma (\\tau \\rightarrow e \\nu \\bar{\\nu }) }\\, , \\\\R_{\\tau \\rightarrow e / \\mu \\rightarrow e} &\\equiv \\frac{\\Gamma (\\tau \\rightarrow e \\nu \\bar{\\nu })}{\\Gamma (\\mu \\rightarrow e \\nu \\bar{\\nu })}\\, , & R_{\\tau \\rightarrow \\mu / \\mu \\rightarrow e}&\\equiv \\frac{\\Gamma (\\tau \\rightarrow \\mu \\nu \\bar{\\nu })}{\\Gamma (\\mu \\rightarrow e \\nu \\bar{\\nu })} \\, ,$ and their theoretical values in a 3G SM can be consulted, for example, in [58].", "Comparison of the experimental values and the 3G predictions yields values very close to 1, as can be seen in table REF ; in our 4G model, family mixing induces deviations from this behaviour that must be kept under control.", "Essentially, these deviations result from the fact that the flavour-eigenstate neutrinos $\\nu _e, \\nu _\\mu , \\nu _\\tau $ have a small component of the heavy neutrinos $\\nu _4, \\nu _{\\bar{4}}$ , which cannot be produced in the decays of pions, taus or muons; the corresponding mixings $N_e, N_\\mu , N_\\tau $ are then forced to be small.", "In table REF we also show the constraints that this processes impose on the mixing parameters.", "Table: NO_CAPTION Summary of the constraints derived from universality tests in weak decays.", "The ratios marked as “SM” represent the theoretical predictions of a 3G Standard Model.", "Figure: NO_CAPTIONThese two graphs present the allowed regions for the mixing parameters in our model at 90% confidence level, according to several LFV tests.", "The left plot displays the constraints in the $N_e - N_\\mu $ plane, which are much more stringent and suffice to bound both $N_e$ and $N_\\mu $ .", "The right plot displays the $N_e - N_\\tau $ plane; the $N_\\mu -N_\\tau $ plane offers slighly poorer constraints and is not displayed.", "In figure REF we collect all the relevant LFV constraints from tables REF and REF .", "As can be read from the graphs, the final bounds we can set on the mixings of the light families are $N_e &< 0.08 \\nonumber \\\\N_\\mu &< 0.03 \\\\N_\\tau &< 0.3 \\nonumber $ Figure: NO_CAPTIONSummary of the constraints on the mixings of the model, as defined in equation (REF ).", "The three horizontal lines present the upper bounds in equation (REF ), derived from universality tests and limits on LFV processes.", "The two shaded areas display the allowed region derived from the fact that the two-loop diagram does not disturb the correct structure for the light neutrino masses, assumed to arise from any other mechanism.", "This last bound applies to any of the mixings." ], [ "Light neutrino masses", "Finally, there is a further constraint that can be set upon the mixings of the model: as explained in section , the two-loop mechanism which gives small Majorana masses for the light neutrinos cannot explain by itself the observed pattern of masses in this simple model; it, nevertheless, still has the potential to generate too large masses, which would exclude the model.", "Of course, one could always invoke cancellations between these two-loop masses and other contributions (for example, the Weinberg operator), but we think that this wouldn't be a natural situation and choose not to consider it.", "If we bar such cancellations we need to impose that the two-loop masses don't go above some limit, and thus a bound can be set upon the parameters that participate in the two-loop mechanism, essentially the mixings and the Majorana mass, as seen in equation (REF ).", "Figure REF shows the allowed regions for this constraint; the curves are constructed using the lowest possible values of the fourth-generation Dirac masses, in order to provide conservative limits (this implies using a different value of $m_{\\mathrm {D}}$ for each $m_{\\mathrm {R}}$ , as we must also impose that $m_4$ is above 62.1 GeV).", "We show two possible limits: $m_\\nu ^{(2)} < 0.05 \\; \\mathrm {eV}$ ensures that the largest two-loop mass is below the atmospheric mass scale; this, of course, does not guarantee that it doesn't distort the neutrino spectrum, which may contain smaller masses, so this bound can be contemplated as rather conservative (particular models may need to impose a stronger bound to be phenomenologically viable).", "An even more conservative constraint is obtained if we impose $m_\\nu ^{(2)} < 0.3 \\; \\mathrm {eV}$ , meaning that the largest of the two-loop masses is not above the bound imposed by cosmology to each mass of the degenerate spectrum, $\\sum _k m_k \\lesssim 1$ eV [59], [60].", "Two-loop masses as large as 0.3 eV will in most cases spoil the structure of neutrino masses, but there may be pathological cases in which such situation is allowed (for example, if the Weinberg operator generates a massless neutrino and two massive ones near the 0.3 eV limit; then the two-loop diagram might provide the third mass to fit the mass splittings).", "Even with these conservative assumptions, the two-loop bound proves to be much stronger than those derived from universality and LFV for most of the parameter space.", "It is, therefore, a limit to be kept in mind when considering 4G models with Majorana neutrinos." ], [ "Neutrinoless double beta decay ($0\\nu 2\\beta $ )", "In our framework, the contributions to the amplitude of neutrinoless double beta decay ($0\\nu 2\\beta $ ) can be written as: $A = A_L + A_{\\rm {md}} + A_4 \\ ,$ where $A_L$ stands for the light neutrino contribution (i.e., neutrino masses $m_k \\ll p_{\\rm {eff}} \\sim 100\\,$ MeV), given by $A_L \\propto \\sum _k^{\\rm {light}} m_k U_{e k}^2 M^{0\\nu 2\\beta }(m_k)\\simeq m_{ee} M^{0\\nu 2\\beta }(0) \\ ,$ with $M^{0\\nu 2\\beta }(0)\\propto 1/p_{\\rm {eff}} ^2$ the nuclear matrix element.", "The cosmology upper bound on the sum of neutrino masses, $\\sum _k m_k \\lesssim 1$ eV [59], [60], combined with neutrino oscillation data, leads to an upper limit on each neutrino mass $m_k \\lesssim 0.3$ eV and on the element of the neutrino mass matrix relevant to $0\\nu 2\\beta $ decay, $m_{ee} \\lesssim 0.3$ eV.", "$A_{\\rm {md}}$ represents the additional, model dependent contribution due to the unknown physics which generates the three light neutrino masses parametrized by the Weinberg operator.", "We assume that this last term is negligible compared to $A_L$ , as it is the case if the underlying mechanism for neutrino masses is any of the standard three see-saw types [61].", "We focus then on the contribution from the fourth-generation neutrinos ($\\nu _4,\\nu _{\\bar{4}}$ ), given by $A_4 &\\propto & N_{e}^2 \\left(m_4 \\cos ^2 \\theta M^{0\\nu 2\\beta }(m_4)- m_{\\bar{4}} \\sin ^2 \\theta M^{0\\nu 2\\beta }(m_{\\bar{4}}) \\right)\\nonumber \\\\& \\propto & N_{e}^2\\left( \\frac{\\cos ^2 \\theta }{m_4} - \\frac{\\sin ^2 \\theta }{m_{\\bar{4}}} \\right)= N_{e}^2 \\frac{m_{\\bar{4}}^2 - m_4^2}{m_4 m_{\\bar{4}} (m_4 + m_{\\bar{4}})}= N_{e}^2 \\frac{m_{\\mathrm {R}}}{m_{\\mathrm {D}}^2}\\, ,$ where we have used that $M^{0\\nu 2\\beta }(m_a) \\propto 1/m_a^2$ for $a=4,\\bar{4}$ , $\\tan ^2\\theta = m_4/m_{\\bar{4}}$ , $m_4 m_{\\bar{4}} = m_{\\mathrm {D}}^2$ and $m_{\\bar{4}} - m_4 = m_{\\mathrm {R}}$ .", "From eqs.", "(REF ) and (REF ) we see that the fourth-generation neutrino contribution to the $0\\nu 2\\beta $ amplitude can be dominant provided $N_e^2 m_{\\mathrm {R}}/m_{\\mathrm {D}}^2> m_{ee} / (100 \\, {\\rm MeV})^{2}$ .", "Notice, in fact, that the value of $m_{ee}$ could be zero if normal hierarchy is realised and the neutrino phases have the appropriate values; in this extreme case the only contribution would be that of the fourth generation, which would dominate $0 \\nu 2 \\beta $ .", "Now we can exploit the dependence on $N_e^2 m_{\\mathrm {R}}$ of both $A_4$ and $(m_\\nu )_{33}^{(2)}$ in eq.", "(REF ) to constrain the fourth-generation neutrino contribution to the $0\\nu 2\\beta $ decay amplitude Note that $(m_\\nu )_{33}^{(2)}$ receives contributions from $N_e$ , $N_\\mu $ and $N_\\tau $ , while $0 \\nu 2 \\beta $ only involves $N_e$ .", ", namely $A_4 \\le \\left( \\frac{4\\pi m_W}{g m_{\\mathrm {D}}} \\right)^4\\frac{2 (m_\\nu )_{33}^{(2)}}{m_E^2 \\ln \\frac{m_E}{m_{\\bar{4}}}}\\lesssim 190 (m_\\nu )_{33}^{(2)} \\left( \\frac{50 \\, {\\rm GeV}}{m_{\\mathrm {D}}} \\right)^4 \\, {\\rm GeV}^{-1} \\, ,$ where we have taken into account the LEP limit, $m_E \\gtrsim 100$ GeV and set $\\ln (m_E/m_{\\bar{4}}) \\simeq 1$ .", "From this equation, it is clear that the largest fourth-generation contributions to the amplitude $A_4$ correspond to a small Dirac neutrino mass, $m_{\\mathrm {D}}$ .", "Imposing that the two-loop mass matrix element $(m_\\nu )_{33}^{(2)}$ is below the cosmology upper bound, 0.3 eV, we obtain $A_4 < 6 \\times 10^{-8} (50 \\, {\\rm GeV}/m_{\\mathrm {D}})^4\\, {\\rm GeV}^{-1}$ , while if we require that the two-loop contribution is at most the atmospheric mass scale, 0.05 eV, we find $A_4 < 10^{-8} (50 \\, {\\rm GeV}/m_{\\mathrm {D}})^4 \\, {\\rm GeV}^{-1}$ .", "On the other hand, the non-observation of $0 \\nu 2 \\beta $ implies that $A_4 < 10^{-8} \\, {\\rm GeV}^{-1}$ [62], while future sensitivity is expected to improve this limit one order of magnitude.", "Bringing these two results together, we see that, once the constraint from light neutrino masses is taken into account, the contribution of the fourth-generation neutrinos to the $0 \\nu 2 \\beta $ decay amplitude can reach observable values only in the small region of parameter space $m_{\\mathrm {D}} \\lesssim 100 \\; \\mathrm {GeV}$ (see figure REF ), even though it is the dominant one for a larger set of allowed masses and mixings." ], [ "Four generations and the Higgs boson", "It is well known that due to the presence of a new generation there is an enhancement of the Higgs-gluon-gluon vertex, which arises from a triangle diagram with all quarks running in the loop.", "This vertex is enhanced approximately by a factor 3 in the presence of a heavy fourth generation, therefore the Higgs production cross section through gluon fusion at the LHC is enhanced by a factor of 9.", "However, Higgs decay channels are also strongly modified, in particular the Higgs to gluon decays are equally enhanced, while the $\\gamma \\gamma $ channel is reduced because of a cancellation between the quark and $W$ contributions.", "Moreover some of these channels, $\\gamma \\gamma $ for instance, suffer from important electroweak radiative corrections [63].", "With the first LHC data, ATLAS and CMS ruled out at $95\\%$ C.L the range $120-600$ GeV for a SM4 Higgs boson, assuming very large masses for the fourth-generation particles [64].", "However, different authors have noticed that if fourth-generation neutrinos are light enough ($m_{W}/2\\lesssim m_{\\nu _{4}}\\lesssim m_{W}$ ), the decay mode of the Higgs into fourth-generation neutrinos can be dominant for $m_H \\lesssim 2 m_W$ [65], [66], [67], [68].", "Moreover, if the lightest fourth-generation neutrino is long-lived this decay channel is invisible and the excluded range for the SM4 Higgs boson is reduced to $160-500$ GeV [67], [68].", "In general, if the fourth-generation neutrinos have both Dirac ($m_{\\mathrm {D}}$ ) and Majorana ($m_{\\mathrm {R}}$ ) masses, the Higgs can decay to more channels: $\\nu _{4}\\nu _{4}$ , $\\nu _{\\bar{4}}\\nu _{\\bar{4}}$ and $\\nu _{4}\\nu _{\\bar{4}}$ .", "Recently, ATLAS and CMS have analysed new data, including more Higgs decay channels [69], [70], and they have a preliminary low-mass ($\\sim 125$ GeV) hint of the Higgs boson in several channelsAlso Fermilab CDF and D0 have presented some preliminary results pointing to some excess around this mass which can be assigned mainly to $H\\rightarrow b\\bar{b}$ decays in $HW$ and $HZ$ associated production [71]..", "In particular there is an excess in the $\\gamma \\gamma $ channel with respect to the SM3 prediction.", "For such a light Higgs, the expected ratio of number of events into $\\gamma \\gamma $ for SM4 over SM3 is about $1.5-2.5$  at leading order [72], [73].", "However, a global fit to all relevant observables (Higgs searches and electroweak precision data), assuming Dirac neutrinos and a Higgs mass of 125 GeV, shows that data are better described by the SM3 [74].", "On the other hand, as commented above, within the SM4 the cancellations in the $\\gamma \\gamma $ channel at leading order render next-to-leading order radiative corrections important.", "These corrections tend to decrease even further the two-photon production rate $\\sigma (gg\\rightarrow H)\\times BR(H\\rightarrow \\gamma \\gamma )|_{SM4}$ .", "Therefore, were the 125 GeV Higgs hint confirmed, by combining the $\\gamma \\gamma $ , $ZZ^*$ , $WW^*$ and the $f\\bar{f}$ channels a perturbative SM4 with just one SM Higgs doublet would be excluded, even in the case $m_{\\nu _{4}}<m_{W}$ [75], [76].", "Otherwise, in principle it seems possible that if $m_{\\nu _{4}}\\lesssim m_{W}$ and $\\nu _4$ is long-lived, some portion of the low Higgs mass parameter space, previously allowed to be between $114-160$ GeV, is still allowed by the new data.", "Moreover, if one does not trust the convergence of perturbation theory in the $\\gamma \\gamma $ channel and drops it from the global analysis, including Higgs searches, $R_b$ and oblique parameters, the SM4 with Dirac neutrinos is strongly constrained but still viable [77].", "Considering neutrino Majorana masses will presumably open up even more the allowed parameter space of the model.", "The previous bounds from LEP on the masses of unstable (in collider sense) fourth-generation neutrinos were $m_{\\nu _{4}}>62.1$ GeV.", "Using CDF inclusive like-sign dilepton analysis, $\\nu _4$ masses below $m_W$ can be excluded for Higgs masses up to $2 m_W$ [67], therefore in this case the ATLAS and CMS analysis for the Higgs boson still apply, and at least the range $120-600$ GeV for a SM4 Higgs boson is excluded.", "To know definitely whether the SM4 Higgs boson is excluded or not, we will have to wait for new data and a combined analysis of the different channels, $\\gamma \\gamma $ , $ZZ^*$ , $WW^*$ and $f\\bar{f}$ , including correctly all radiative corrections.", "However, even if the SM-like four-generation Higgs is excluded, many possibilities may arise in extensions of a four-generation scenario, for instance, with an extra Higgs doublet (see [78], [79] where the observed signatures of LHC are explained in the framework of 4G two-Higgs-doublet models).", "We have addressed the question of the generation and nature of neutrino masses in the context of the SM with four families of quarks and leptons.", "The three light neutrinos can obtain their masses from a variety of mechanisms with or without new neutral fermions, but the huge hierarchy among such masses and those of the remaining fermions is more naturally explained assuming that they have Majorana nature.", "On the other hand, current bounds on fourth-generation neutrino masses imply that, although in principle the same mechanisms are also available, most of them are not natural or provide too small fourth-generation neutrino masses; therefore, we have argued that at least one right-handed neutrino is needed.", "This would suggest that, contrary to the light neutrinos, fourth-generation ones are naturally Dirac.", "However, we have shown that if lepton number is not conserved in the light neutrino sector, the right-handed neutrino must have a Majorana mass term whose size depends on the underlying mechanism for LNV, unless Yukawa couplings of the light leptons to the right-handed neutrino are forbidden.", "We have estimated the natural size of such Majorana mass term within two frameworks for the light neutrino masses, namely see-saw type I and type II.", "We have seen that, even if we set it to zero by hand in the Lagrangian at tree level, it is generated at two-loops , and although it depends on the Yukawa couplings and the LNV scale responsible for light neutrino masses, it can be up to the TeV scale.", "We have developed a model where this Majorana mass is forbidden at tree level by a global symmetry, and it is generated radiatively and finite once this symmetry is broken spontaneously (see appendix ).", "We have then considered a minimal four-generation scenario, with neutrino Majorana masses parametrized by the Weinberg operator and one right-handed neutrino $\\nu _{\\mathrm {R}}$ , which has Yukawa couplings to the four lepton doublets and non-zero Majorana mass.", "We have analyzed the phenomenological constraints on the parameter space of such a model, derived from direct searches for four-generation leptons, universality tests, charged lepton flavour-violating processes and neutrinoless double beta decay.", "We have pointed out that the Majorana mass for the fourth-generation neutrino induces relatively large two-loop contributions to the light neutrino masses, which can easily exceed the atmospheric scale and the cosmological bounds.", "Indeed, this sets the strongest limits on the masses and mixings of fourth-generation neutrinos, collected in figure REF .", "To summarize, in the context of a SM with four generations, we have shown that if light neutrinos are Majorana particles, it is natural that also the fourth-generation neutrino has the Majorana character.", "We did so by calculating the fourth-neutrino Majorana masses induced by the three light neutrino ones.", "This has important implications for the neutrino and Higgs sectors of these models, which are being actively tested at the LHC." ], [ "Acknowledgments", "We thank Enrique Fernández-Martínez and Jacobo López-Pavón for fruitful discussions on the matter of neutrinoless double beta decay.", "This work has been partially supported by the Spanish MICINN under grants FPA-2007-60323, FPA-2008-03373, FPA2011-23897, FPA2011-29678-C02-01, Consolider-Ingenio PAU (CSD2007-00060) and CPAN (CSD2007- 00042) and by Generalitat Valenciana grants PROMETEO/2009/116 and PROMETEO/2009/128.", "A.A. and J.H.-G. are supported by the MICINN under the FPU program." ], [ "A model for calculable right-handed neutrino masses", "In this appendix we present a model which gives a realistic pattern of neutrino masses in the context of the SM with four-generations and in which the right-handed neutrino mass of the fourth generation is generated radiatively and finite.", "This is an illustration of the general (model-independent) mechanism discussed in section which allowed us to estimate the size of Majorana neutrino masses for the fourth-generation right-handed neutrinos if the three light active neutrinos are Majorana particles.", "Let us consider the SM with four generations and four right-handed neutrinos $\\nu _{\\mathrm {R} i}$ ($i=1,\\cdots ,4$ ).", "To implement the ordinary see-saw, we need three of them very heavy while one of them should be much lighter in order to avoid a too light fourth-generation active neutrino.", "Then, it is natural to require that one of the fourth right-handed neutrino is massless at tree level and let its mass be generated by radiative corrections.", "For that purpose we add three extra chiral singlets $s_{\\mathrm {L} a}$ $(a=1,\\cdots ,3$ ).", "In order to break lepton number we will also include a complex scalar singlet $\\sigma $ We assign lepton number in the following way $\\ell _{j} \\rightarrow e^{i\\alpha } \\ell _{j} \\;, \\quad e_{\\mathrm {R} j} \\rightarrow e^{i\\alpha } e_{\\mathrm {R} j} \\;,\\quad \\nu _{\\mathrm {R} j} \\rightarrow e^{i \\alpha } \\nu _{\\mathrm {R} j} \\;,\\quad \\sigma \\rightarrow e^{i \\alpha } \\sigma \\; ;$ the $s_{\\mathrm {L} a}$ do not carry lepton number.", "With these assignments and the requirement that lepton number is conserved we have the following Yukawa Lagrangian $\\mathcal {L}_{Y} = -\\overline{\\ell } \\, Y_{e} e_{\\mathrm {R}} \\phi -\\overline{\\ell } \\, Y_{\\nu } \\nu _{\\mathrm {R}} \\tilde{\\phi } -\\sigma \\, \\overline{\\nu _{\\mathrm {R}}} \\, y^* \\, s_{\\mathrm {L}} -\\frac{1}{2} \\, \\overline{s_{\\mathrm {L}}^{\\mathrm {c}}} M^*s_{\\mathrm {L}} + \\mathrm {H.c.} \\; ,$ where $Y_{e}$ and $Y_{\\nu }$ are the ordinary four-generation Yukawa couplings, $y_{ia}$ , along this appendix, is a general $4 \\times 3$ matrix while $M$ is a symmetric $3\\times 3$ matrix, which without loss of generality can be taken diagonal and positive.", "We choose the scalar potential in such a way that lepton number is conserved and subsequently spontaneously broken by the VEV of $\\sigma $ , $v_\\sigma =\\langle \\sigma \\rangle $ .", "Thus, the model will contain a singlet Majoron.", "Alternatively, we could also choose to softly break lepton number in the potential to avoid the Majoron without changing the point we would like to illustrate.", "Before spontaneous symmetry breaking only $s_{\\mathrm {L} a}$ are massive.", "We will take $M$ very large (around GUT scale).", "After $\\sigma $ gets a VEV (which is somewhat free, but we can take it just a bit below $M$ ), we will have a mass matrix for the combined system $\\nu _{\\mathrm {R}}-s_{\\mathrm {L}}$ of see-saw type.", "Therefore, if $y \\, v_{\\sigma } \\ll M$ the four right-handed neutrinos will get a $4 \\times 4$ Majorana mass matrix $M_{\\mathrm {R}}^{(0)} \\simeq v_{\\sigma }^{2} \\, y M^{-1} y^{\\mathrm {T}} \\; ;$ this is basically the see-saw formula but applied to the right-handed neutrinos and changing the VEV of the Higgs doublet for that of the singlet $\\sigma $ .", "This matrix has rank 3 and, therefore, only three of the right-handed neutrinos will obtain a tree-level mass.", "The other neutrino will remain massless at tree level.", "However, at two loops, due to the mechanism described in section , also the fourth right-handed neutrino will acquire a Majorana mass.", "We depict the diagram giving rise to this mass in figure ; the diagram is obviously finite by power counting and the generated mass matrix can be estimated as $M_{\\mathrm {R}}^{(2)} \\sim \\frac{v_{\\sigma }^{2}}{(4\\pi )^{4}} \\,(Y^\\dagger _{\\nu } Y_{\\nu })^{\\mathrm {T}}\\, y M^{-1} y^{\\mathrm {T}} \\,Y^\\dagger _{\\nu } Y_{\\nu } \\ln \\left(\\frac{M}{y v_\\sigma }\\right)\\; .$ Figure: NO_CAPTIONThe process which generates masses for the right-handed neutrinos at two loops.", "Since $Y_{e}$ does not enter in these calculations we can choose a basis in which $Y_{e}$ is arbitrary but $Y_{\\nu }$ is diagonal and real.", "If we take the logarithm order 1, $\\ln \\left(\\frac{M}{y v_\\sigma }\\right)\\sim 1$ , we see that $M_{\\mathrm {R}}^{(2)}$ is also projective but in a different direction, given by $y^{\\prime }= (Y^\\dagger _{\\nu } Y_{\\nu })^{\\mathrm {T}} y$ ; then we can write the full right-handed neutrino mass matrix as $M_{\\mathrm {R}} \\sim v_{\\sigma }^{2} \\, \\left( y M^{-1} y^{\\mathrm {T}}+ \\frac{1}{(4\\pi )^{4}} \\, y^{\\prime } M^{-1}{y^{\\prime }}^{\\mathrm {T}} \\right) \\: ,$ which, in general, has rank 4 and gives a Majorana mass to the fourth right-handed neutrino.", "To see how it works, let us discuss a simplified example, with the following structure for the $s_{\\mathrm {L}}$ Yukawas: $y = \\begin{pmatrix}y_{1} & 0 & 0 \\\\0 & y_{2} & 0 \\\\0 & 0 & y_{3} \\\\0 & 0 & y_{4}\\end{pmatrix} \\: .$ Let us also choose $M$ diagonal and with elements $M_i$ ; then, at tree level we obtain an almost diagonal mass matrix, $M_{\\mathrm {R}}^{(0)} = v_\\sigma ^2 \\, \\begin{pmatrix}\\frac{y_{1}^{2}}{M_{1}} & 0 & 0 & 0 \\\\0 & \\frac{y_{2}^{2}}{M_{2}} & 0 & 0 \\\\0 & 0 & \\frac{y_{3}^{2}}{M_{3}} & \\frac{y_{3}y_{4}}{M_{3}} \\\\0 & 0 & \\frac{y_{3}y_{4}}{M_{3}} & \\frac{y_{4}^{2}}{M_{3}}\\end{pmatrix} \\: ,$ which has a zero eigenvalue.", "At two loops we will have $M_{\\mathrm {R}}^{(2)} = \\frac{v_{\\sigma }^{2}}{(4\\pi )^{4}} \\, \\begin{pmatrix}\\frac{y_{1}^{\\prime 2}}{M_{1}} & 0 & 0 & 0 \\\\0 & \\frac{y_{2}^{\\prime 2}}{M_{2}} & 0 & 0 \\\\0 & 0 & \\frac{y_{3}^{\\prime 2}}{M_{3}} &\\frac{y_{3}^{\\prime }y_{4}^{\\prime }}{M_{3}} \\\\0 & 0 & \\frac{y_{3}^{\\prime }y_{4}^{\\prime }}{M_{3}} &\\frac{y_{4}^{\\prime 2}}{M_{3}}\\end{pmatrix} \\: ,$ with $y_{i}^{\\prime } = y_{i} {(Y_{\\nu })_i}^{2}$ , and $(Y_{\\nu })_i$ the diagonal elements of $Y_{\\nu }$ .", "$M_{\\mathrm {R}}^{(2)}$ has also rank 3.", "However, the sum of $M_{\\mathrm {R}}^{(0)}$ and $M_{\\mathrm {R}}^{(2)}$ has rank 4, and the fourth $\\nu _{\\mathrm {R}}$ acquires a mass.", "We can estimate it by considering $M_{\\mathrm {R}}^{(2)}$ a small perturbation to $M_{\\mathrm {R}}^{(0)}$ and find that $m_{\\mathrm {R} 4} \\sim \\frac{v_{\\sigma }^{2}}{(4\\pi )^{4} M_{3}} \\,\\frac{y_{4}^{2} \\, y_{3}^{2}}{y_{3}^{2}+y_{4}^{2}} \\,\\left( {(Y_{\\nu })^2_4} - {(Y_{\\nu })^2_3} \\right)^{2} \\: ,$ while the mass of the third right-handed neutrino is of order (the other two are also order $y^2 v^2_\\sigma /M$ as can be seen from the mass matrix) $ m_{\\mathrm {R} 3} \\sim (y_{3}^{2}+y_{4}^{2}) \\, \\frac{v_{\\sigma }^{2}}{M_{3}} \\: .$ Therefore, if we rewrite the fourth-generation right-handed neutrino mass $m_{\\mathrm {R} 4}$ in terms of $m_{\\mathrm {R} 3}$ we have $ m_{\\mathrm {R} 4} \\sim \\frac{m_{\\mathrm {R} 3}}{(4\\pi )^{4}}\\frac{y_{4}^{2} \\, y_{3}^{2}}{(y_{3}^{2}+y_{4}^{2})^2} \\,\\left( {(Y_{\\nu })^2_4} - {(Y_{\\nu })^2_3} \\right)^{2} \\: ,$ which is roughly the structure that one would expect from the effective theory obtained by integrating the new fermions $s_{\\mathrm {L} a}$ , i.e, $m_{\\mathrm {R} 4}$ obtains a contribution proportional to the heavy right-handed Majorana masses $m_{\\mathrm {R} 3}$ suppressed by a two-loop factor and Yukawa couplings.", "After all, the diagram in figure  reduces to the diagram in figure  when the fermion lines of $s_{\\mathrm {L} a}$ are contracted to a point.", "The result also shows that, as expected, the exact coefficient depends on the details of the model.", "These expressions could be generalized to a more general structure of Yukawa couplings, leading to similar, although more complicated expressions.", "As for other features of this model, we will just mention that as lepton number is broken spontaneously, a Majoron will appear.", "Since the Majoron is a singlet and $v_{\\sigma }$ is large their couplings to standard model particles are suppressed and, therefore, this Majoron should not create any problem.", "On the other hand, it could have some advantages in cosmological contexts; if lepton number is also broken softly (for instance with a mass term $\\sigma ^{2}$ ) the Majoron will become a massive pseudo-Majoron, which could constitute a good dark matter candidate.", "In any case, this simple example illustrates how the general mechanism discussed in section works in a complete renormalizable model; if $m_{\\mathrm {R} 4}$ is zero at tree level and light neutrinos are Majorana (therefore lepton number is not conserved), in general $m_{\\mathrm {R} 4}$ will be generated at two-loops with the behaviour discussed in section ." ] ]

[ [ "Search for neutrino emission in gamma-ray flaring blazars with the\n ANTARES telescope" ], [ "Abstract The ANTARES telescope observes a full hemisphere of the sky all the time with a duty cycle close to 100%.", "This makes it well suited for an extensive observation of neutrinos produced in astrophysical transient sources.", "In the surrounding medium of blazars, i.e.", "active galactic nuclei with their jets pointing almost directly towards the observer, neutrinos may be produced together with gamma-rays by hadronic interactions, so a strong correlation between neutrinos and gamma-rays emissions is expected.", "The time variability information of the studied source can be obtained by the gamma-ray light curves measured by the LAT instrument on-board the Fermi satellite.", "If the expected neutrino flux observation is reduced to a narrow window around the assumed neutrino production period, the point-source sensitivity can be drastically improved.", "The ANTARES data collected in 2008 has been analysed looking for neutrinos detected in the high state period of ten bright and variable Fermi sources assuming that the neutrino emission follows the gamma-ray light curves.", "First results show a sensitivity improvement by a factor 2-3 with respect to a standard time-integrated point source search.", "The analysis has been done with an unbinned method based on the minimization of a likelihood ratio applied to data corresponding to a live time of 60 days.", "The width of the flaring periods ranges from 1 to 20 days.", "Despite the fact that the most significant studied source is compatible with background fluctuations, recently detected flares promise interesting future analyse." ], [ "Introduction", "Neutrino astronomy is an incipient field of observation of the universe.", "Since neutrinos do not interact electromagnetically, but weakly only, they are not absorbed and point directly to their original sources, playing the role of unique messengers.", "However, due to their weak interaction, neutrino telescopes are very low event rate experiments requiring high fluxes in order to detect a clear signal of a neutrino astrophysical source.", "Up to now no claim of an astronomy source of neutrinos further than the Sun has been made, with the exception of the well known SN1987A.", "Assuming a direct correlation between the neutrino emission and the light emission of a source in an hadronic acceleration scenario [1], the standard analysis can be improved by a factor 2–3 in the sensitivity if we limit this search to the periods of maximum probability of neutrino emission by the source.", "The time variability visible in the gamma-ray light curves measured by the Fermi-LAT [2] can be used for this purpose.", "The ANTARES detector is a neutrino telescope placed at the bottom of the Mediterranean Sea (42$^\\circ $ 48 N, 6$^\\circ $ 10 E), at a depth of 2475 m, connected by a submarine cable of 42 km to the shore in Toulon (France).", "This cable connects through a junction box 12 lines which are separated by 60–70 m and vertically suspended by a buoy.", "Each line has 25 floors spaced by 14.5 m, except for line 12 which has only 20 floors.", "A floor consists of a triplet of optical modules (OMs) each one housing a photomultiplier (PMT) facing 45$^\\circ $ downwards.", "The full detector is a tri-dimensional array of 885 PMTs [3] [4] which was completed in 2008 when the last line was connected.", "Neutrinos are detected via the Cherenkov light induced by relativistic muons produced in the detector surroundings by CC interactions of muon neutrinos with nuclei in water.", "The signals from the Cherenkov photons detected by the PMTs are digitized (`hits') [5] and sent to the shore station for reconstruction and physics analysis.", "The analyzed data correspond to the period from September 6th to December 31st, 2008 (54720–54831 modified Julian day).", "Some periods have been excluded from the selection in order to avoid conditions with high optical backgrounds due to the bioluminescence activity, leaving 60.8 days of life-time.", "While most of the atmospheric muon background is suppressed by selecting only up-going events, a fraction of mis-reconstructed events still remains which have to be rejected in a different way.", "In the track-reconstruction process an algorithm based on the maximization of a likelihood is used.", "The likelihood function is built from the difference between the expected and the measured arrival times of the hits from the Cherenkov photons emitted along the muon track.", "Its maximization considers the Cherenkov photons that scatter in the water and the additional photons that are generated by secondary particles (e. g. electromagnetic showers created along the muon trajectory).", "A good measure of the track-fit quality is the value of the log-likelihood per degree of freedom ($\\lambda $ ), providing a useful tool to reject the mis-reconstructed muons by applying a cut in that $\\lambda $ value.", "This cut leaves the atmospheric neutrinos as the dominant background.", "An additional cut requiring the error estimated for the reconstructed muon track direction to be less than 1$^\\circ $ is also applied in the selection of events.", "The angular resolution has been estimated by Monte Carlo simulations.", "The cumulative distribution of the angular difference between the reconstructed muon direction and the neutrino direction, with an assumed spectrum proportional to ${E_{\\nu }}^{-2}$ , where $E_{\\nu }$ is the neutrino energy, is shown in Fig.", "REF .", "For the studied period, the median resolution is estimated to be 0.4 $\\pm $  0.1 degrees." ], [ "Time-dependent search algorithm", "An unbinned method based on a likelihood ratio maximization has been used to perform this time-dependent point-source analysis.", "The data is parameterized as a two components mixture of signal and background.", "The goal is to determine, at a given point in the sky and at a given time, the relative contribution of each component and to calculate the probability to have a signal above a given background model.", "The likelihood ratio $\\lambda $ is the ratio of the probability density for the hypothesis of background and signal ($H_{\\mathrm {sig}+\\mathrm {bkg}}$ ) over the probability density of only background ($H_{\\mathrm {bkg}}$ ): $\\begin{split}\\lambda &= \\sum _{i=1}^{N} \\log \\frac{P(x_{i}|H_{\\mathrm {sig}+\\mathrm {bkg}})}{P(x_{i}|H_{\\mathrm {bkg}})}\\end{split}$ $\\begin{split}&= \\sum _{i=1}^{N} \\log \\frac{\\frac{n_{\\mathrm {sig}}}{N}P_{\\mathrm {sig}}(\\alpha _{i},\\delta _{i},t_{i})+(1-\\frac{n_{\\mathrm {sig}}}{N})P_{\\mathrm {bkg}}(\\alpha _{i},\\delta _{i},t_{i})}{P_{\\mathrm {bkg}}(\\alpha _{i},\\delta _{i},t_{i})}\\nonumber \\end{split}$ where $n_{\\mathrm {sig}}$ and $N$ are respectively the unknown number of signal events and the total number of events in the considered data sample.", "$P_{\\mathrm {sig}}(\\alpha _{i},\\delta _{i},t_{i})$ and $P_{\\mathrm {bkg}}(\\alpha _{i},\\delta _{i},t_{i})$ are the probability density function (PDF) for signal and background respectively, and $\\delta _{i}$ is the declination of the studied source.", "For a given event $i$ , $t_{i}$ and $\\alpha _{i}$ represent the time of the event and the angular difference between the coordinate of this event and the studied source.", "$P_{\\mathrm {sig}}(\\alpha _{i},\\delta _{i},t_{i})$ and $P_{\\mathrm {bkg}}(\\alpha _{i},\\delta _{i},t_{i})$ are described by two components: $\\begin{split}P_{\\mathrm {sig}}(\\alpha _{i},\\delta _{i},t_{i}) = P_{\\mathrm {sig}}^{\\mathrm {dir}}(\\alpha _{i},\\delta _{i}) \\times P_{\\mathrm {sig}}^{\\mathrm {time}}(t_{i})\\end{split}$ $\\begin{split}P_{\\mathrm {bkg}}(\\alpha _{i},\\delta _{i},t_{i}) = P_{\\mathrm {bkg}}^{\\mathrm {dir}}(\\alpha _{i},\\delta _{i}) \\times P_{\\mathrm {bkg}}^{\\mathrm {time}}(t_{i})\\nonumber \\end{split}$ where $P_{\\mathrm {sig}}^{\\mathrm {dir}}$ is the probability to have a signal event from the studied source, computed via Monte Carlo, $P_{\\mathrm {sig}}^{\\mathrm {time}}$ is the probability of a neutrino event derived from the gamma-ray light curve correlation, and $P_{\\mathrm {bkg}}^{\\mathrm {dir}}$ and $P_{\\mathrm {bkg}}^{\\mathrm {time}}$ are the corresponding background PDFs.", "The shape of the time PDF for the signal event, $P_{\\mathrm {sig}}^{\\mathrm {time}}$ , is extracted directly from the gamma-ray light curve assuming that it is proportional to the gamma-ray flux.", "For the signal event directional PDF, $P_{\\mathrm {sig}}^{\\mathrm {dir}}$ , the one-dimensional point-spread function is used, which is the probability density of reconstructing an event at an angular distance $\\alpha _i$ from the true source position.", "Then, the directional, $P_{\\mathrm {bkg}}^{\\mathrm {dir}}$ , and time PDF for the background, $P_{\\mathrm {bkg}}^{\\mathrm {time}}$ , are derived from the data using respectively the observed declination distribution of selected events in the sample and the observed time distribution of all the reconstructed muons.", "The latter distribution (not normalized to 1) is shown in Fig.", "REF , where periods without data (due to detector maintenance, etc) or with very poor quality data (high bioluminescence activity or bad calibration) are shown as empty bins.", "Figure: Cumulative distribution of the angle between the true Monte Carlo neutrino direction and the reconstructed muon direction for E -2 E^{-2} upgoing neutrino events selected for this analysis.Figure: Time distribution of the reconstructed events.", "Black: distribution for all reconstructed events.", "Red filled: distribution of selected upgoing events (λ>-5.4\\lambda > -5.4 and β<1 ∘ \\beta < 1^\\circ ).", "If 0, there are no data available (i. e. detector in maintenance) or the data have a very poor quality (high bioluminescence or bad calibration).The $\\lambda _{\\mathrm {data}}$ value obtained from the data is then compared to the distribution of $\\lambda $ given by the null hypothesis ($n_{\\mathrm {sig}}=0$ ).", "The comparison of $\\lambda _{\\mathrm {data}}$ with the background only $\\lambda $ distribution is used to reject the null hypothesis, where the confidence level corresponds to the fraction of the scrambled trials above $\\lambda _{\\mathrm {data}}$ (p-value).", "The discovery potential is then defined as the average number of signal events required to achieve a p-value lower than the equivalent to $5\\sigma $ in 50% of trials.", "The average number of events required for a $5\\sigma $ discovery (50% C.L.)", "produced in one source located at a declination of -40$^\\circ $ as a function of the total width of the flare period is shown in Fig.", "REF .", "A comparision with the numbers obtained with a standard time-integrated point source search, shows an improvement of the discovery potential by about a factor 2–3.", "Figure: Average number of events required for a 5σ\\sigma discovery (50% C.L.)", "of a source located at a declination of -40 ∘ ^\\circ as a function of the width of the flare period (solid line), a simple Heavyside function.", "This is compared to the number required for a time-integrated search (dashed line).", "The average typical length of the studied source flares goes from 1 up to 100 days." ], [ "Search for neutrino emission from gamma-ray flares", "The sources for which this time-dependent analysis has been applied have been selected from the Fermi blazar sources reported in the first year Fermi LAT catalogue [6] and in the LBAS catalogue (LAT Bright AGN sample [7]).", "The selection has been done based on their variability, brightness and visibility: sources visible to ANTARES with a significant time-variability in the studied period and a flux in the high state greater than $20\\times 10^{-8}$  photons cm$^{-2}$ s$^{-1}$ above 300 MeV in the averaged 1 day-binned light curve were selected.", "The final selection list of sources includes four BLLacs and six flat spectrum radio quasars (FSRQ), listed in Table REF .", "Table REF lists their fluxes, visibility and live times.", "Table: List of the bright, variable Fermi blazars selected for this analysis.", "F 300 F_{300} is the gamma-ray flux above 300 MeV in 10 -8 10^{-8} photons cm 2 ^{2}s -1 ^{-1}.", "Live Time is the effective time of observation of the source in days.", "N(5σ)N(5\\sigma ) are the number of neutrino events needed to be detected in ANTARES with 5σ5\\sigma , while NN is the number of events observed in the source's direction and flare time window.", "Fluence is the upper limit (90% C. L.) on the neutrino fluence in GeV cm -2 ^{-2}, calculated according to the classical (frequentist) method for upper limits .The light curves used for $P_{\\mathrm {sig}}^{\\mathrm {time}}$ in (REF ) are those published on the Fermi web page for monitored sources [8] (one-day-binned time evolution of the average gamma-ray flux above a threshold of 100 MeV).", "Figure REF shows the high state periods (blue histogram) obtained for the Fermi LAT gamma-ray light curve of 3C454 for almost two years of data.", "These states are defined using a simple and robust method based on three steps: First, the baseline and its error is determined with an iterative linear and gaussian fits.", "After each fit, the points where the flux is above the fitted base line plus one sigma are suppressed.", "This is done 3 times.", "Then all points (green dots in Fig.", "REF ) where the measured flux minus its error is above the baseline plus two times the baseline sigma and at the same time the measured flux value is above the baseline plus three times the baseline sigma, are used as priors from which the flares are defined.", "Then, for each selected prior, the adjacent points for which the emission is compatible with a flare are added, i. e. the adjacent points which have their flux minus their error above the baseline plus one sigma.", "Finally, an additional delay of 0.5 days is added before and after the flare in order to take into account the 1-day binning of the light curve.", "Hence, a flare has a minimum width of 2 days.", "Assuming that the neutrino emission follows that in gamma-rays, the signal time PDF is simply this de-noised light curve after normalization.", "Figure: Gamma-ray light curve (red dots) of the blazar 3C454.3 measured by the LAT instrument on board the Fermi satellite above 100 MeV for almost two years of data.", "Blue histogram: high state periods.", "Green line and dots: baseline and significant dots above this baseline used for the determination of the flare periods.The most significant source is 3C279, which has a pre-trial p-value of 1.03%.", "The unbinned method finds one high-energy neutrino event located 0.56$^\\circ $ from the source location during a large flare in November 2008 (Fig.", "REF ).", "This event has been reconstructed with 89 hits spread on 10 lines with a track fit quality $\\lambda =-4.4$ and an error estimate of $\\beta =0.3^\\circ $ .", "The post-trial probability is computed taking into account the ten searches.", "The final probability of 10% is compatible with a background fluctuation.", "Other source results are sumarized in Table REF .", "Figure: Gamma-ray light curve (black dots) of the blazar 3C279 measured by the LAT instrument on board the Fermi satellite above 100 MeV.", "Blue histogram: high state periods.", "Green dashed line: fit of a baseline.", "Red histogram: time of the ANTARES neutrino event in coincidence with 3C279." ], [ "Conclusion", "The first time-dependent search for cosmic neutrinos in ANTARES has been presented.", "It has used data taken with the full 12-line ANTARES detector during the last four months of 2008.", "Time-dependent searches are significantly more sensitive than a standard point-source search for a variable source.", "This search has been applied to ten bright and variable Fermi LAT blazars.", "The most significant observation of a flare is 3C279 with a p-value of about 10% after trials for which one neutrino event has been detected in time and space coincident with the gamma-ray emission.", "Limits on the neutrino fluence have been obtained for the ten selected sources.", "The most recent measurements of Fermi in 2009-11 show very large flares yielding a more promising search for neutrinos [10], and ongoing analyses are being performed together with improvements in the denoising of the light curves." ], [ "Acknowledgments", "We gratefully acknowledge the financial support of the Spanish Ministerio de Ciencia e Innovación (MICINN), grants FPA2009-13983-C02-01, ACI2009-1020 and Consolider MultiDarkCSD2009-00064 and of the Generalitat Valenciana, Prometeo/2009/026." ] ]

[ [ "Transition map and shadowing lemma for normally hyperbolic invariant\n manifolds" ], [ "Abstract For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries: the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma.", "We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in Hamiltonian systems.", "This consists in the following steps: (i) computation of the scattering map and of the transition map for a flow, (ii) reduction to the scattering map and to the transition map, respectively, for the return map to some surface of section, (iii) construction of sequences of windows within the surface of section, with the successive pairs of windows correctly aligned, alternately, under the transition map, and under some power of the inner map, (iv) detection of trajectories which follow closely those windows.", "We illustrate this strategy with two models: the large gap problem for nearly integrable Hamiltonian systems, and the the spatial circular restricted three-body problem." ], [ "Introduction", "Consider a normally hyperbolic invariant manifold for a flow or a map, and assume that the stable and unstable manifolds of the normally hyperbolic invariant manifold have a transverse intersection along a homoclinic manifold.", "One can distinguish an inner dynamics, associated to the restriction of the flow or of the map to the normally hyperbolic invariant manifold, and an outer dynamics, associated to the homoclinic orbits.", "There exist pseudo-orbits obtained by alternately following the inner dynamics and the outer dynamics for some finite periods of time.", "An important question on the dynamics is whether there exist true orbits with similar behavior.", "In this paper, we develop a toolkit of instruments and techniques to detect true orbits near a normally hyperbolic invariant manifold, that alternatively follow the inner dynamics and the outer dynamics, for all time.", "Some of the tools discussed below have already been used in other works.", "The aim of this paper is to provide a general recipe on how to make a systematic use of these tools in general situations.", "The first tool is the scattering map, which is defined on the normally hyperbolic invariant manifold, and assigns to the foot of an unstable fiber passing through a point in the homoclinic manifold, the foot of the corresponding stable fiber that passes through the same point in the homoclinic manifold.", "This tool, sometimes referred as the homoclinic map, has been used in [10], [12], [7], and subsequently refined and analyzed in [5].", "The scattering map can be defined both in the flow case and in the map case.", "In Section we recall some background on normally hyperbolic invariant manifolds and Lambda Lemma.", "In Section we describe the relationship between the scattering map for a flow and the scattering map for the return map to a surface of section.", "We note that the scattering map is defined in terms of the geometric structure, however it is not dynamically defined – there is no actual orbit that is given by the scattering map.", "The second tool that we discuss is the transition map, that actually follows the homoclinic orbits for a prescribed time.", "The transition map can be computed in terms of the scattering map.", "Again, we will have a transition map for the flow and one for the return map, and we will describe the relationships between them.", "The transition map is presented in Section .", "The third tool is the topological method of correctly aligned windows (see [22]), which is used to detect orbits with prescribed itineraries in a dynamical system.", "A window is a homeomorphic copy of a multi-dimensional rectangle, with a choice of an exit direction and of an entry direction.", "A window is correctly aligned with another window if the image of the first window crosses the second window all the way through and across its exit set.", "This method is reviewed briefly in Section .", "The fourth tool is a shadowing lemma type of result for a normally hyperbolic invariant manifold, presented in Section .", "The assumption is that a bi-infinite sequence of windows lying in the normally hyperbolic invariant manifold is given, with the consecutive pairs of windows being correctly aligned, alternately, under the transition map (outer map), and under some power of the inner map.", "The role of the windows is to approximate the location of orbits.", "Then there exists a true orbit that follows closely these windows, in the prescribed order.", "To apply this lemma for a normally hyperbolic invariant manifold for a map, one needs to reduce the dynamics from the continuous case to the discrete case by considering the return map to a surface of section, and construct the sequence of correctly aligned windows for the return map.", "For this situation, the relationships between the scattering map for the flow and the scattering map for the return map, and between the transition map for the flow and the transition map for the return map, explored in Section and Section , are useful.", "A remarkable feature of these tools is that they can be used for both analytic arguments and rigorous numerical verifications.", "The scattering map and the transition map can be computed explicitly in concrete systems.", "The main advantage is that they can be used to reduce the dimensionality of the problem: from the phase space of a flow to a normally hyperbolic invariant manifold for the flow, and further to the normally hyperbolic invariant manifold for the return map to a surface of section.", "The shadowing lemma also plays a key role in reducing the dimensionality of the problem: it requires the verification of topological conditions in the normally hyperbolic invariant manifold for the return map to conclude the existence of trajectories in the phase space of the flow.", "In numerical applications, reducing the number of dimensions of the objects computed is very crucial.", "The potency of these tools in numerical application is illustrated in [4].", "The main motivation for developing these tools resides with the instability problem for Hamiltonian systems.", "In the Appendix we describe two models where the above techniques can be applied to show the existence of unstable orbits.", "The first model is the large gap problem for nearly integrable Hamiltonian systems.", "The second model is the spatial circular restricted three-body problem.", "In conclusion, we provide a practical recipe for finding trajectories with prescribed itineraries for a normally hyperbolic invariant manifold with the property that its stable and unstable manifolds have a transverse intersection along a homoclinic manifold: Compute the scattering map associated to the homoclinic manifold.", "For some prescribed forward and backwards integration times, compute the corresponding transition map.", "If necessary, reduce the dynamics from a flow to the return map via some surface of section.", "Determine the normally hyperbolic invariant manifold relative to the surface of section, and compute the inner map – the restriction of the return map relative to the normally hyperbolic invariant manifold.", "Compute the scattering map and the transition map for the return map.", "Construct windows within the normally hyperbolic invariant manifold relative to the surface of section, with the the property that the consecutive pairs of windows are correctly aligned, alternately, under the transition map and under some power of the inner map.", "Apply the shadowing lemma stated in Theorem REF to conclude that there exist orbits that follow closely these windows." ], [ "Preliminaries", "In this section we review the concepts of normal hyperbolicity for flows and maps, normally hyperbolic invariant manifold for the return map to a surface of section, and we state a version of the Lambda Lemma that will be used in the subsequent sections." ], [ "Normally hyperbolic invariant manifolds", "In this section we recall the concept of a normally hyperbolic invariant manifolds for a map and for a flow, following [11], [17].", "Let $M$ be a $C^r$ -smooth, $m$ -dimensional manifold (without boundary), with $r\\ge 1$ , and $\\Phi :M\\times \\mathbb {R}\\rightarrow M$ a $C^r$ -smooth flow on $M$ .", "Definition 2.1 A submanifold (possibly with boundary) $\\Lambda $ of $M$ is said to be a normally hyperbolic invariant manifold for $\\Phi $ if $\\Lambda $ is invariant under $\\Phi $ , there exists a splitting of the tangent bundle of $TM$ into sub-bundles $TM=E^u\\oplus E^s\\oplus T\\Lambda ,$ that are invariant under $d\\Phi ^t$ for all $t\\in \\mathbb {R}$ , and there exist a constant $C>0$ and rates $0<\\beta <\\alpha $ , such that for all $x\\in \\Lambda $ we have $\\begin{split}v\\in E^s_x \\Leftrightarrow \\Vert D\\Phi ^t(x)(v)\\Vert \\le Ce^{-\\alpha t}\\Vert v\\Vert \\textrm { for all } t\\ge 0,\\\\v\\in E^u_x \\Leftrightarrow \\Vert D\\Phi ^t(x)(v)\\Vert \\le Ce^{\\alpha t}\\Vert v\\Vert \\textrm { for all } t\\le 0,\\\\v\\in T_x\\Lambda \\Leftrightarrow \\Vert D\\Phi ^t(x)(v)\\Vert \\le Ce^{\\beta |t|}\\Vert v\\Vert \\textrm { for all } t\\in \\mathbb {R}.\\end{split}$ It follows that there exist stable and unstable manifolds of $\\Lambda $ , as well as stable and unstable manifolds of each point $x\\in \\Lambda $ , which are defined by $\\begin{split}W^s(\\Lambda )=&\\lbrace y\\in M\\,|\\, d(\\Phi ^t(y),\\Lambda )\\le C_ye^{-\\alpha t} \\textrm { for all } t\\ge 0\\rbrace ,\\\\W^u(\\Lambda )=&\\lbrace y\\in M\\,|\\, d(\\Phi ^t(y),\\Lambda )\\le C_ye^{\\alpha t} \\textrm { for all } t\\le 0\\rbrace ,\\\\W^s(x)=&\\lbrace y\\in M\\,|\\, d(\\Phi ^t(y),\\Phi ^t(y)(x))\\le C_{x,y}e^{-\\alpha t}\\textrm { for all } t\\ge 0\\rbrace ,\\\\W^u(x)=&\\lbrace y\\in M\\,|\\, d(\\Phi ^t(y),\\Phi ^t(y)(x))\\le C_{x,y}e^{\\alpha t} \\textrm { for all } t\\le 0\\rbrace ,\\end{split}$ for some constants $C_y,C_{x,y}>0$ .", "The stable and unstable manifolds of $\\Lambda $ are foliated by stable and unstable manifolds of points, respectively, i.e., $W^s(\\Lambda )=\\bigcup _{x\\in \\Lambda }W^s(x)$ and $W^u(\\Lambda )=\\bigcup _{x\\in \\Lambda }W^u(x)$ .", "In the sequel we will assume that $\\Lambda $ is a compact and connected manifold.", "With no other assumptions, $E^s_x$ and $E^u_x$ depend continuously (but non-smoothly) on $x\\in M$ ; thus the dimensions of $E^s_x$ and $E^u_x$ are independent of $x$ .", "Below we only consider the case when the dimensions of the stable and unstable bundles are equal.", "We denote $n=\\dim (E^s_x)=\\dim (E^u_x)$ , $l=\\dim (T_x\\Lambda )$ , where $2n+l=m$ .", "The smoothness of the invariant objects defined by the normally hyperbolic structure depends on the rates $\\alpha $ and $\\beta $ .", "Let $\\ell $ be a positive integer satisfying $1\\le \\ell <\\min \\lbrace r,\\alpha /\\beta \\rbrace $ .", "The manifold $\\Lambda $ is $C^\\ell $ -smooth.", "The stable and unstable manifolds $W^s(\\Lambda )$ and $W^u(\\Lambda )$ are $C^{\\ell -1}$ -smooth.", "The splittings $E^s_x$ and $E^u_x$ depend $C^{\\ell -1}$ -smoothly on $x$ .", "The stable and unstable fibers $W^s(x)$ and $W^u(x)$ are $C^r$ -smooth.", "The stable and unstable fibers $W^s(x)$ and $W^u(x)$ depend $C^{\\ell -1-j}$ -smoothly on $x$ when $W^s(x),W^u(x)$ are endowed with the $C^j$ -topology.", "In the sequel we will assume that the rates are such that there exists an integer $\\ell \\ge 2$ as above, and that all the manifolds and maps considered below are at least $C^k$ -smooth, with $2\\le k\\le \\ell $ .", "The notion of normal hyperbolicity for maps is very similar.", "Let $F:M\\rightarrow M$ be a $C^r$ -smooth map on $M$ .", "Definition 2.2 A submanifold $\\Lambda $ of $M$ is said to be a normally hyperbolic invariant manifold for $F$ if $\\Lambda $ is invariant under $F$ , there exists a splitting of the tangent bundle of $TM$ into sub-bundles $TM=E^u\\oplus E^s\\oplus T\\Lambda ,$ that are invariant under $dF$ , and there exist a constant $C>0$ and rates $0<\\lambda <\\mu ^{-1}<1$ , such that for all $x\\in \\Lambda $ we have $\\begin{split}v\\in E^s_x \\Leftrightarrow \\Vert DF^k_x(v)\\Vert \\le C\\lambda ^k\\Vert v\\Vert \\textrm { for all } k\\ge 0,\\\\v\\in E^u_x \\Leftrightarrow \\Vert DF^k_x(v)\\Vert \\le C\\lambda ^{-k}\\Vert v\\Vert \\textrm { for all } k\\le 0,\\\\v\\in T_x\\Lambda \\Leftrightarrow \\Vert DF^k_x(v)\\Vert \\le C\\mu ^{|k|}\\Vert v\\Vert \\textrm { for all } k\\in \\mathbb {Z}.\\end{split}$ There exist stable and unstable manifolds of $\\Lambda $ , as well as the stable and unstable manifolds of each point $x\\in \\Lambda $ , that are defined similarly as in the flow case, and they carry analogous properties.", "The smoothness properties of these invariant objects are analogous of those for a flow, if we set $1\\le \\ell <\\min \\lbrace r, (\\log \\lambda ^{-1})(\\log \\mu )^{-1}\\rbrace $ ." ], [ "Normal hyperbolicity relative to the return map", "Let $\\Phi :M\\times \\mathbb {R}\\rightarrow M$ be a $C^r$ -smooth flow defined on an $m$ -dimensional manifold $M$ .", "Denote by $X$ the vector field associated to $\\Phi $ , where $X(x)=\\frac{\\partial }{\\partial t}\\Phi (x,t)_{\\mid t=0}$ .", "As before, assume that $\\Lambda \\subseteq M$ is an $l$ -dimensional normally hyperbolic invariant manifold for $\\Phi $ .", "The dimensions of $T\\Lambda $ , $E^u$ and $E^s$ are $l,n,n$ , respectively, with $l+2n=m$ .", "Let $\\Sigma $ be an $(m-1)$ -dimensional local surface of section, i.e., $\\Sigma $ is a $C^1$ submanifold of $M$ such that $X(x)\\notin T_x\\Sigma $ for all $x\\in \\Sigma $ .", "Let $\\Lambda _\\Sigma =\\Lambda \\cap \\Sigma $ .", "Then $\\Lambda _\\Sigma $ is a $(l-1)$ -dimensional submanifold in $\\Sigma $ , assuming that the intersection is non-empty.", "Assume that each forward and backward orbit through a point in $\\Lambda _\\Sigma $ intersects again $\\Lambda _\\Sigma $ .", "Since $X(x)\\notin T_x\\Sigma $ for all $x\\in \\Sigma $ , then the intersection of the forward and backward orbits with $\\Sigma $ are transverse.", "Also, $X(x)\\notin T_x\\Lambda _\\Sigma $ for all $x\\in \\Lambda _\\Sigma $ .", "Additionally, assume that the function $\\tau :\\Lambda _\\Sigma \\rightarrow (0,\\infty ), \\textrm { given by } \\tau (x)=\\inf \\lbrace t>0\\,|\\, \\Phi (x,\\tau (x))\\in \\Lambda _\\Sigma \\rbrace ,$ is a continuous function.", "Following [11], we will refer to $\\Lambda _\\Sigma $ with these properties as a thin surface of section.", "By the Implicit Function Theorem, $\\tau $ can be extended to a $C^1$ -smooth function in a neighborhood $U_\\Sigma $ of $\\Lambda _\\Sigma $ in $\\Sigma $ such that $\\Phi (x,\\tau (x))\\in \\Sigma $ for all $x\\in U_\\Sigma $ .", "The Poincaré first return map to $\\Sigma $ is the map $F:U_\\Sigma \\rightarrow \\Sigma $ given by $F(x)=\\Phi ^{\\tau (x)}(x)$ .", "Let $\\Lambda _\\Sigma ^X\\subseteq \\Lambda $ be the union of the orbits of the flow through points in $\\Lambda _\\Sigma $ .", "Since $\\Lambda _\\Sigma ^X$ is a $C^1$ -submanifold of $\\Lambda $ , and is invariant under $\\Phi $ , then is a normally hyperbolic invariant manifold for the flow $\\Phi $ .", "The theorem below implies that the manifold $\\Lambda _\\Sigma $ is normally hyperbolic for the return map $F$ .", "Theorem 2.3 (Fenichel, [11]) Let $\\Lambda _\\Sigma $ be a thin surface of section for the vector field $X$ on $M$ .", "Then $\\Lambda _\\Sigma $ is normally hyperbolic with respect to $F$ if and only if $\\Lambda _\\Sigma ^X$ is a normally hyperbolic invariant manifold with respect to $\\Phi $ .", "The invariant sub-bundles $T\\Lambda $ , $E^u$ , $E^s$ associated to the normal hyperbolic structure on $\\Lambda $ correspond to sub-bundles $T\\Lambda _\\Sigma $ , $E^u_\\Sigma $ , $E^s_\\Sigma $ in the following way.", "Let $\\pi :TM=\\textrm {span}(X)\\oplus T\\Sigma \\rightarrow T\\Sigma $ be the projection onto $T\\Sigma $ .", "Then $T\\Lambda _\\Sigma =\\pi (T_\\Lambda )$ , $E^u_\\Sigma =\\pi (E^u)$ , and $E^s_\\Sigma =\\pi (E^s)$ .", "Note that the surface of section described above is only a local surface of section.", "It is in general very difficult, or even impossible, to obtain a global surface of section for a flow.", "However, one can obtain global surfaces of section for Hamiltonian flows on 3-dimensional strictly convex energy surfaces [18]." ], [ "Lambda Lemma", "We describe a Lambda Lemma type of result for normally hyperbolic invariant manifolds that appears in J.-P. Marco [19].", "By a normal form in a neighborhood $V$ of $\\Lambda $ in $M$ we mean a $C^k$ -smooth coordinate system $(c,s,u)$ on $V$ such that $V$ is diffeomorphic through $(c,s,u)$ with a product $\\Lambda \\times \\mathbb {R}^n\\times \\mathbb {R}^n$ , where $\\Lambda =\\lbrace (c,s,u)\\,|\\, c\\in \\Lambda ,\\, u=s=0\\rbrace $ , and $W^u(x)=\\lbrace (c,s,u)\\,|\\,c=c(x), s=0\\rbrace $ , $W^s(x)=\\lbrace (c,s,u)\\,|\\, c=c(x), u=0\\rbrace $ for each $x\\in \\Lambda $ of coordinates $(c(x),0,0)$ .", "Theorem 2.4 (Lambda Lemma) Suppose that $\\Lambda $ is a normally hyperbolic invariant manifold for $F$ and $(c,s,u)$ is a normal form in a neighborhood of $\\Lambda $ .", "Consider a submanifold $\\Delta $ of $M$ of dimension $n$ which intersects the stable manifold $W^s(\\Lambda )$ transversely at some point $z= (c, s, 0)$ .", "Set $F^N(z)=z_N=(c_N, s_N, 0)$ for $N\\in \\mathbb {N}$ .", "Then there exists $\\delta >0$ and $N_0>0$ such that for each $N\\ge N_0$ the connected component $\\Delta _N$ of $F^N(\\Delta )$ in the $\\delta $ -neighborhood $V(\\delta )=\\Lambda \\times B^s_{\\delta }(0)\\times B^u_{\\delta }(0)$ of $\\Lambda $ in $M$ admits a graph parametrization of the form $\\Delta _N:=\\lbrace (C_N(u), S_N(u),u) \\,|\\, u\\in B^u_{\\delta }(0)\\rbrace $ such that $\\Vert C_N-c_N\\Vert _{C^1(B^u_{\\delta }(0))}\\rightarrow 0, \\, \\textrm { and } \\Vert S_N\\Vert _{C^1(B^u_{\\delta }(0))}\\rightarrow 0 \\textrm { as } N\\rightarrow \\infty .", "$ In this section we review the scattering map associated to a normally hyperbolic invariant manifold for a flow or for a map, and discuss the relationship between the scattering map for a flow and the scattering map for the corresponding return map to some surface of section." ], [ "Scattering map for continuous and discrete dynamical systems", "Consider a flow $\\Phi : M\\times \\mathbb {R}\\rightarrow M$ defined on a manifold $M$ that possesses a normally hyperbolic invariant manifold $\\Lambda \\subseteq M$ .", "As the stable and unstable manifolds of $\\Lambda $ are foliated by stable and unstable manifolds of points, respectively, for each $x\\in W^u(\\Lambda )$ there exists a unique $x_-\\in \\Lambda $ such that $x\\in W^u(x_-)$ , and for each $x\\in W^s(\\Lambda )$ there exists a unique $x_+\\in \\Lambda $ such that $x\\in W^s(x_+)$ .", "We define the wave maps $\\Omega _{+}:W^s(\\Lambda )\\rightarrow \\Lambda $ by $\\Omega _{+}(x)=x_{+}$ , and $\\Omega _{-}:W^u(\\Lambda )\\rightarrow \\Lambda $ by $\\Omega _{-}(x)=x_{-}$ .", "The maps $\\Omega _+$ and $\\Omega _-$ are $C^{\\ell }$ -smooth.", "We now describe the scattering map, following [7].", "Assume that $W^u(\\Lambda )$ has a transverse intersection with $W^s(\\Lambda )$ along a $l$ -dimensional homoclinic manifold $\\Gamma $ .", "The manifold $\\Gamma $ consists of a $(l-1)$ -dimensional family of trajectories asymptotic to $\\Lambda $ in both forward and backwards time.", "The transverse intersection of the hyperbolic invariant manifolds along $\\Gamma $ means that $\\Gamma \\subseteq W^u(\\Lambda ) \\cap W^s(\\Lambda )$ and, for each $x\\in \\Gamma $ , we have $ \\begin{split}T_xM=T_xW^u(\\Lambda )+T_xW^s(\\Lambda ),\\\\T_x\\Gamma =T_xW^u(\\Lambda )\\cap T_xW^s(\\Lambda ).\\end{split} $ Let us assume the additional condition that for each $x\\in \\Gamma $ we have $ \\begin{split}T_xW^s(\\Lambda )=T_xW^s(x_+)\\oplus T_x(\\Gamma ),\\\\T_xW^u(\\Lambda )=T_xW^u(x_-)\\oplus T_x(\\Gamma ),\\end{split}$ where $x_-,x_+$ are the uniquely defined points in $\\Lambda $ corresponding to $x$ .", "The restrictions $\\Omega _+^\\Gamma ,\\Omega _-^\\Gamma $ of $\\Omega _+,\\Omega _-$ , respectively, to $\\Gamma $ are local $C^{\\ell -1}$ - diffeomorphisms.", "By restricting $\\Gamma $ even further, if necessary, we can ensure that $\\Omega _+^\\Gamma ,\\Omega _-^\\Gamma $ are $C^{\\ell -1}$ -diffeomorphisms.", "A homoclinic manifold $\\Gamma $ for which the corresponding restrictions of the wave maps are $C^{\\ell -1}$ -diffeomorphisms will be referred as a homoclinic channel.", "Definition 3.1 Given a homoclinic channel $\\Gamma $ , the scattering map associated to $\\Gamma $ is the $C^{\\ell -1}$ -diffeomorphism $S^\\Gamma =\\Omega ^\\Gamma _+\\circ (\\Omega ^\\Gamma _-)^{-1}$ defined on the open subset $U_-:=\\Omega ^\\Gamma _-(\\Gamma )$ in $\\Lambda $ to the open subset $U_+:=\\Omega ^\\Gamma _+(\\Gamma )$ in $\\Lambda $ .", "See Figure REF .", "In the sequel we will regard $S^\\Gamma $ as a partially defined map, so the image of a set $A$ by $S^\\Gamma $ is $S^\\Gamma (A\\cap U_-)$ .", "Figure: Scattering map.If we flow $\\Gamma $ backwards and forward in time we obtain the manifolds $\\Phi ^{-t_u}(\\Gamma )$ and $\\Phi ^{t_s}(\\Gamma )$ that are also homoclinic channels, where $t_u,t_s>0$ .", "The associated wave maps are $\\Omega _+^{\\Phi ^{-t_u}(\\Gamma )},\\Omega _-^{\\Phi ^{-t_u}(\\Gamma )}$ , and $\\Omega _+^{\\Phi ^{t_s}(\\Gamma )},\\Omega _-^{\\Phi ^{t_s}(\\Gamma )}$ , respectively.", "The scattering map can be expressed with respect to these wave maps as $S^\\Gamma =\\Phi ^{-t_s}\\circ (\\Omega ^{\\Phi ^{t_s}(\\Gamma )}_+)\\circ \\Phi ^{t_s+t_u}\\circ (\\Omega ^{\\Phi ^{-t_u}(\\Gamma )}_-)^{-1}\\circ \\Phi ^{-t_u}.$ We recall below some important properties of the scattering map.", "Proposition 1 Assume that $\\dim M=2n+l$ is even (i.e., $l$ is even) and $M$ is endowed with a symplectic (respectively exact symplectic) form $\\omega $ and that $\\omega _{\\mid \\Lambda }$ is also symplectic.", "Assume that $\\Phi ^t$ is symplectic (respectively exact symplectic).", "Then, the scattering map $S^\\Gamma $ is symplectic (respectively exact symplectic).", "Proposition 2 Assume that $T_1$ and $T_2$ are two invariant submanifolds of complementary dimensions in $\\Lambda $ .", "Then $W^u(T_1)$ has a transverse intersection with $W^s(T_2)$ in $M$ if and only if $S(T_1)$ has a transverse intersection with $T_2$ in $\\Lambda $ .", "In the case of a discrete dynamical system consisting of a diffeomorphism $F:M\\rightarrow M$ defined on a manifold $M$ , the scattering map is defined in a similar way.", "We assume that $F$ has a normally hyperbolic invariant manifold $\\Lambda \\subseteq M$ .", "The wave maps are defined by $\\Omega _{+}:W^s(\\Lambda )\\rightarrow \\Lambda $ with $\\Omega _{+}(x)=x_{+}$ , and $\\Omega _{-}:W^u(\\Lambda )\\rightarrow \\Lambda $ with $\\Omega _{-}(x)=x_{-}$ .", "Assume that $W^u(\\Lambda )$ and $W^s(\\Lambda )$ have a differentiably transverse intersection along a homoclinic $l$ -dimensional $C^{\\ell -1}$ -smooth manifold $\\Gamma $ .", "We also assume the transverse foliation condition (REF ).", "A homoclinic manifold $\\Gamma $ for which the corresponding restrictions of the wave maps are $C^{\\ell -1}$ -diffeomorphisms is referred as a homoclinic channel.", "Definition 3.2 Given a homoclinic channel $\\Gamma $ , the scattering map associated to $\\Gamma $ is the $C^{\\ell -1}$ -diffeomorphism $S^\\Gamma =\\Omega ^\\Gamma _+\\circ (\\Omega ^\\Gamma _-)^{-1}$ defined on the open subset $U_-:=\\Omega ^\\Gamma _-(\\Gamma )$ in $\\Lambda $ to the open subset $U_+:=\\Omega ^\\Gamma _+(\\Gamma )$ in $\\Lambda $ .", "Note that for $M,N>0$ , the manifolds $F^{-M}(\\Gamma )$ and $F^N(\\Gamma )$ are also homoclinic channels.", "The associated wave maps are $\\Omega _-^{F^{-M}(\\Gamma )},\\Omega _+^{F^{-M}(\\Gamma )}$ , and $\\Omega _-^{F^{N}(\\Gamma )},\\Omega _+^{F^{N}(\\Gamma )}$ .", "The scattering map can be expressed with respect to these wave map as $S^\\Gamma =F^{-N}\\circ (\\Omega ^{F^N(\\Gamma )}_+)\\circ F^{M+N}\\circ (\\Omega ^{F^{-M}(\\Gamma )}_-)^{-1}\\circ F^{-M}.$ The scattering map for the discrete case satisfies symplectic and transversality properties similar to those in Proposition REF and Proposition REF for the continuous case." ], [ "Scattering map for the return map", "Let $\\Phi :M\\times \\mathbb {R}\\rightarrow M$ be a $C^r$ -smooth flow defined on an $m$ -dimensional manifold $M$ , and $X$ be the vector field associated to $\\Phi $ .", "Let $\\Lambda \\subseteq M$ be an $l$ -dimensional normally hyperbolic invariant manifold for $\\Phi $ .", "Assume that $\\Sigma $ is a local surface of section and $\\Lambda _\\Sigma =\\Lambda \\cap \\Sigma $ satisfies the conditions in Subsection REF .", "Consider $\\Gamma $ a homoclinic channel for $\\Phi $ .", "First, we assume that $\\Gamma $ has a non-empty intersection with $\\Sigma $ .", "Note that $\\Gamma $ is a $(l-1)$ -parameter family of orbits; we further assume that each trajectory intersects $\\Sigma $ transversally.", "Since $\\Gamma $ is a homoclinic channel, each orbit intersects $\\Sigma $ exactly once.", "Let $\\Gamma _\\Sigma =\\Gamma \\cap \\Sigma $ .", "It is easy to see that $\\Gamma _\\Sigma $ is a homoclinic channel for $F$ .", "Thus, we have a scattering map $S^\\Gamma $ for $\\Gamma $ associated to the flow $\\Phi $ , and we also have a scattering map $S^{\\Gamma _\\Sigma }$ for $\\Gamma _\\Sigma $ associated to the map $F$ .", "We want to understand the relationship between $S^\\Gamma $ and $S^{\\Gamma _\\Sigma }$ .", "Associated to the homoclinic channels $\\Gamma $ and $\\Gamma _\\Sigma $ there exist wave maps $\\Omega _\\pm ^{\\Gamma }: \\Gamma \\rightarrow \\Lambda $ and $\\Omega _\\pm ^{\\Gamma _\\Sigma }: \\Gamma _\\Sigma \\rightarrow \\Lambda _\\Sigma $ , respectively.", "These maps are diffeomorphisms.", "Let $x\\in \\Gamma _\\Sigma $ , and let $x_-=\\Omega _-^{\\Gamma }(x)$ , $x_+=\\Omega _+^{\\Gamma }(x)$ , and $\\hat{x}_-=\\Omega _-^{\\Gamma _\\Sigma }(x)$ , $\\hat{x}_+=\\Omega _+^{\\Gamma _\\Sigma }(x)$ .", "We have $S^\\Gamma (x_-)=x_+$ and $S^{\\Gamma _\\Sigma }(\\hat{x}_-)=\\hat{x}_+$ .", "We want to relate $x_-$ with $\\hat{x}_-$ , and $x_+$ with $\\hat{x}_+$ .", "These points are all in $\\Lambda $ .", "It is clear that $\\hat{x}_-=\\Omega ^{\\Gamma _\\Sigma }_-\\circ (\\Omega ^{\\Gamma }_-)^{-1}(x_-)$ , and $\\hat{x}_+=\\Omega ^{\\Gamma _\\Sigma }_+\\circ (\\Omega ^{\\Gamma }_+)^{-1}(x_+)$ .", "Denote by $P^{\\Gamma }_-:\\Omega ^{\\Gamma }_-(\\Gamma )\\rightarrow \\Omega ^{\\Gamma _\\Sigma }_-$ the map given by $P^{\\Gamma }_-=\\Omega ^{\\Gamma _\\Sigma }_-\\circ (\\Omega ^{\\Gamma }_-)^{-1}$ , and denote by $P^{\\Gamma }_+:\\Omega ^{\\Gamma }_+(\\Gamma )\\rightarrow \\Omega ^{\\Gamma _\\Sigma }_+$ the map given by $P^{\\Gamma }_+=\\Omega ^{\\Gamma _\\Sigma }_+\\circ (\\Omega ^{\\Gamma }_+)^{-1}$ .", "We want to express these maps in terms of the dynamics restricted to $\\Lambda $ .", "Let $V$ be a flow box at $\\hat{x}_-$ (for definition see [21]).", "This means each trajectory through a point $y\\in V$ intersects $\\Sigma $ exactly once.", "Then there exists a differentiable function $\\hat{\\tau }:V\\rightarrow \\mathbb {R}$ defined by $\\tau (z)=0$ if $z\\in \\Sigma $ and $\\Phi ^{\\hat{\\tau }(y)} (y)\\in \\Sigma $ for each $y\\in V$ .", "The function $\\hat{\\tau }$ can be extended in a unique way on each trajectory passing though $V$ .", "Due to the relationship between the invariant bundles for the flow and the invariant bundles for the map described in Subsection REF , the fiber $E^u_{\\Sigma }(\\hat{x}_-)$ is the projection onto $T\\Sigma $ of the image of the fiber $E^u(x_-)$ under $D\\Phi ^{\\hat{\\tau }(x_-)}_{x_-}$ .", "This means that $\\Phi ^{\\hat{\\tau }(x_-)}(x_-)=\\hat{x}_-$ .", "In other words, $\\hat{x}_-$ is at the intersection of the trajectory through $x_-$ with $\\Sigma $ .", "Thus, the projection $P^{\\Gamma }_-$ that takes $x_-$ to $\\hat{x}_-$ is given by $P^{\\Gamma }_-(x_-)=\\Phi ^{\\hat{\\tau }(x_-)}(x_-)$ .", "This projection map is invertible.", "If $\\hat{y}_-$ is a point in $\\Omega ^{\\Gamma _\\Sigma }_-$ , there exists a unique point $y_-\\in \\Omega ^{\\Gamma }_-(\\Gamma )$ such that $\\Phi ^{\\hat{\\tau }(y_-)}(y_-)=\\hat{y}_-$ .", "If there exist two such points, $y_-$ and $y^{\\prime }_-$ , to them they correspond two points $y$ , $y^{\\prime }$ in $\\Gamma _\\Sigma $ such that $y\\in W^u_{F}(y_-)$ and $y^{\\prime }\\in W^u_{F}(y^{\\prime }_-)$ .", "The points $y,y^{\\prime }$ should belong to the same unstable fiber $W^u_{F}(\\hat{y}_-)$ .", "Then it means that $y$ , $y^{\\prime }$ are on the same trajectory.", "As they are also in $\\Gamma $ and $\\Gamma $ is a homoclinic channel, than $y=y^{\\prime }$ and $y_-=y^{\\prime }_-$ .", "In summary, the projection map $P^{\\Gamma }_-:\\Omega ^{\\Gamma }_-(\\Gamma )\\rightarrow \\Omega ^{\\Gamma _\\Sigma }_-$ is given by $P^{\\Gamma }_-(x_-)=\\Phi ^{\\tau (x_-)}(x_-)$ .", "Similarly, the projection map $P^{\\Gamma }_+:\\Omega ^{\\Gamma }_+(\\Gamma )\\rightarrow \\Omega ^{\\Gamma _\\Sigma }_+$ is given by $P^{\\Gamma }_+(x_+)=\\Phi ^{\\tau (x_+)}(x_+)$ .", "See Figure REF .", "Figure: Scattering map for the return map.Now we can formulate the relationship between the scattering map $S^\\Gamma $ associated to the flow $\\Phi $ , and the scattering map $S^{\\Gamma _\\Sigma }$ associated to the map $F$ .", "Proposition 3 Assume that $\\Gamma $ is a homoclinic channel for the flow $\\Phi $ , and $\\Gamma _\\Sigma =\\Gamma \\cap \\Sigma $ is the corresponding homoclinic channel for the map $F$ .", "Let $S^\\Gamma $ be the scattering map corresponding to $\\Gamma $ , and let $S^{\\Gamma _\\Sigma }$ be the scattering map corresponding to $\\Gamma _\\Sigma $ .", "Then: $S^{\\Gamma _\\Sigma }=P_+^\\Gamma \\circ S^{\\Gamma }\\circ (P_-^\\Gamma )^{-1}.$ We have that $S^{\\Gamma }(x_-)=x_+$ , $S^{\\Gamma _\\Sigma }(\\hat{x}_-)=\\hat{x}_+$ , $P^{\\Gamma }_-(x_-)=\\hat{x}_-$ , and $P^{\\Gamma }_-(x_+)=\\hat{x}_+$ .", "Thus $S^{\\Gamma _\\Sigma }(\\hat{x}_-)=P^{\\Gamma }_+(x_+)=P^{\\Gamma }_+\\circ S^{\\Gamma }(x_-)=P^{\\Gamma }_+\\circ S^{\\Gamma }\\circ (P^{\\Gamma }_-)^{-1}(\\hat{x}_-)$ ." ], [ "Transition map", "The scattering map for a flow $\\Phi $ is geometrically defined: $S^\\Gamma (x_-)=x_+$ means that $W^u(x_-)$ intersects $W^s(x_+)$ at a unique point $x\\in \\Gamma $ , with $W^u(x_-)$ and $W^s(x_+)$ being $n$ -dimensional manifolds.", "However, there is no trajectory of the system that goes from near $x_-$ to near $x_+$ .", "Instead, the trajectory of $x$ approaches asymptotically the backwards orbit of $x_-$ in negative time, and approaches asymptotically the forward orbit of $x_+$ in positive time.", "For applications we need a dynamical version of the scattering map.", "That is, we need a map that takes some backwards image of $x_-$ into some forward image of $x_+$ .", "We will call this map a transition map.", "The transition map depends on the amounts of times we want to flow in the past and in the future.", "The transition map carries the same geometric information as the scattering map.", "Since in perturbation problems the scattering map can be computed explicitly, the transition map is also computable.", "The notion of transition map below is similar to the transition map defined in [3], however, their version is not related to the scattering map." ], [ "Transition map for continuous and discrete dynamical systems", "Consider a flow $\\Phi : M\\times \\mathbb {R}\\rightarrow M$ defined on a manifold $M$ that possesses a normally hyperbolic invariant manifold $\\Lambda \\subseteq M$ .", "Assume that $W^u(\\Lambda )$ and $W^s(\\Lambda )$ have a transverse intersection, and that there exists a homoclinic channel $\\Gamma $ .", "Given $t_u,t_s>0$ , the time-map $\\Phi ^{t_s+t_u}$ is a diffeomorphism from $\\Phi ^{-t_u}(\\Gamma )$ to $\\Phi ^{t_s}(\\Gamma )$ .", "Using (REF ) we can express the restriction of $\\Phi ^{t_s+t_u}$ to $\\Phi ^{-t_u}(\\Gamma )$ in terms of the scattering map as $\\Phi ^{t_s+t_u}_{\\mid \\Phi ^{-t_u}(\\Gamma )}:(\\Omega ^{\\Phi ^{-t_u}(\\Gamma )}_-)^{-1}(\\Phi ^{-t_u}(U_-))\\rightarrow (\\Omega ^{\\Phi ^{t_s}(\\Gamma )}_+)^{-1}(\\Phi ^{t_s}(U_+)), $ given by $\\Phi ^{t_s+t_u}_{\\mid \\Phi ^{-t_u}(\\Gamma )}=(\\Omega ^{\\Phi ^{t_s}(\\Gamma )}_+)^{-1}\\circ \\Phi ^{t_s}\\circ S^\\Gamma \\circ \\Phi ^{t_u}\\circ (\\Omega ^{\\Phi ^{-t_u}(\\Gamma )}_-),$ where $S^\\Gamma :U_-\\rightarrow U_+$ is the scattering map associated to the homoclinic channel $\\Gamma $ .", "We use this to define the transition map as an an approximation of $\\Phi ^{t_s+t_u}_{\\mid \\Phi ^{-t_u}(\\Gamma )}$ provided that $t_u,t_s$ are sufficiently large.", "Definition 4.1 Let $\\Gamma $ be a homoclinic channel for $\\Phi $ .", "Let $t_u, t_s>0$ fixed.", "The transition map $S_{t_u,t_s}^{\\Gamma }$ is a diffeomorphism $S_{t_u,t_s}^{\\Gamma }:\\Phi ^{-t_u}(U_-)\\rightarrow \\Phi ^{t_s}(U_+) $ given by $S_{t_u,t_s}^{\\Gamma }= \\Phi ^{t_s}\\circ S^\\Gamma \\circ \\Phi ^{t_u} ,$ where $S^\\Gamma :U_-\\rightarrow U_+$ is the scattering map associated to the homoclinic channel $\\Gamma $ .", "Alternatively, we can express the transition map as $S_{t_u,t_s}^{\\Gamma }=\\Omega ^{\\Phi ^{t_s(\\Gamma )}}_+\\circ \\Phi ^{t_u+t_s}\\circ (\\Omega ^{\\Phi ^{-t_u(\\Gamma )}}_-)^{-1}$ The symplectic property and the transversality property of the scattering map lend themselves to similar properties of the transition map.", "In the case of a dynamical system given by a map $F:M\\rightarrow M$ , the transition map can be defined in a similar manner to the flow case, and enjoys similar properties.", "As before, we assume that $\\Lambda \\subseteq M$ is a normally hyperbolic invariant manifold for $F$ .", "Definition 4.2 Let $\\Gamma $ be a homoclinic channel for $F$ .", "Let $N_u, N_s>0$ fixed.", "The transition map $S_{N_u,N_s}^{\\Gamma }$ is a diffeomorphism $S_{N_u,N_s}^{\\Gamma }:F^{-N_u}(U_-)\\rightarrow F^{N_s}(U_+) $ given by $S_{N_u,N_s}^{\\Gamma }= F^{N_s}\\circ S^\\Gamma \\circ F^{N_u} ,$ where $S^\\Gamma :U_-\\rightarrow U_+$ is the scattering map associated to the homoclinic channel $\\Gamma $ ." ], [ "Transition map for the return map", "We will consider the reduction of the transition map to a local surface of section.", "Let $\\Sigma $ be a local surface of section and $\\Lambda _\\Sigma =\\Lambda \\cap \\Sigma $ .", "By Theorem REF , $\\Lambda _\\Sigma $ is normally hyperbolic with respect to the first return map to $\\Sigma $ .", "Assume that $\\Gamma $ intersects $\\Sigma $ as in Subsection REF , and let $\\Gamma _\\Sigma =\\Gamma \\cap \\Sigma $ .", "Let $x$ be a point in $\\Gamma _\\Sigma $ .", "Then $\\Phi ^{-t_u}(x)$ lies on $W^u(\\Phi ^{-t_u}(x_-))$ , approaches asymptotically $\\Lambda $ as $t_u\\rightarrow \\infty $ , and intersects $\\Sigma $ infinitely many times.", "Similarly, $\\Phi ^{t_s}(x)$ lies on $W^u(\\Phi ^{t_s}(x_+))$ , approaches asymptotically $\\Lambda $ as $t_s\\rightarrow \\infty $ , and intersects $\\Sigma $ infinitely many times.", "We want to choose and fix some times $t_u,t_s$ , depending on $x\\in \\Gamma $ , such that $\\Phi ^{-t_u}(x), \\Phi ^{t_s}(x)$ are both in $\\Sigma $ , and moreover, $\\Phi ^{-t_u}(x), \\Phi ^{t_s}(x)$ are sufficiently close to $\\Phi ^{-t_u}(x_-), \\Phi ^{t_s}(x_+)$ , respectively.", "Let $\\upsilon >0$ be a small positive number.", "We define $t_u=t_u(x)$ to be the smallest time such that $\\Phi ^{-t_u(x)}(x) \\in \\Sigma $ , and the distance between $\\Phi ^{-t_u}(x)$ and $\\Phi ^{-t_u}(x_-)$ , measured along the unstable fiber $W^u(\\Phi ^{-t_u}(x_-))$ , is less than $\\upsilon $ .", "Let $N_u>0$ be such that $\\Phi ^{-t_u}(x)=F^{-N_u}(x)$ .", "Similarly, we define $t_s=t_s(x)$ to be the smallest time such that $\\Phi ^{t_s}(x) \\in \\Sigma $ , and the distance between $\\Phi ^{t_s}(x)$ and $\\Phi ^{t_s}(x_+)$ , measured along the stable fiber $W^s(\\Phi ^{t_s}(x_+))$ , is less than $\\upsilon $ .", "Let $N_s>0$ be such that $\\Phi ^{t_s}(x)=F^{N_S}(x)$ .", "At this point, we have a transition map $S^{\\Gamma }_{t_u,t_s}$ associated to the flow $\\Phi $ and to the homoclinic channel $\\Gamma $ for the flow, and a transition map $S^{\\Gamma _\\Sigma }_{N_u,N_s}$ associated to the map $F$ and to the homoclinic channel $\\Gamma _\\Sigma $ for the map.", "We have that $\\Phi ^{-t_u}(\\Gamma )$ and $\\Phi ^{t_s}(\\Gamma )$ are both homoclinic channels for the flow $\\Phi $ , and $F^{-N_u}(\\Gamma )$ and $F^{N_s}(\\Gamma )$ are both homoclinic channels for the map $F$ .", "Let us consider the projection mappings $P_-^{F^{-N_u}}(\\Gamma ), P_+^{F^{-N_u}}(\\Gamma ) $ associated to the homoclinic channel $F^{-N_u}(\\Gamma )$ , and the projection mappings $P_-^{F^{N_s}}(\\Gamma ), P_+^{F^{N_s}}(\\Gamma ) $ associated to the homoclinic channel $F^{N_s}(\\Gamma )$ .", "These projections mappings are defined as in Subsection REF .", "The relationship between the transition map for the flow $\\Phi $ and the transition map for the return map $F$ is given by the following: Proposition 4 Assume that $\\Gamma $ is a homoclinic channel for the flow $\\Phi $ , and $\\Gamma _\\Sigma =\\Gamma \\cap \\Sigma $ is the corresponding homoclinic channel for the map $F$ .", "Let $t_u$ , $t_s$ , $N_u$ , $N_s>0$ be fixed.", "Let $S^\\Gamma _{N_u,N_s}$ be the transition map corresponding to $\\Gamma $ for the flow $\\Phi $ , and let $S_{N_u,N_s}^{\\Gamma _\\Sigma }$ be the transition map corresponding to $\\Gamma _\\Sigma $ for the return map $F$ .", "Then $S^{\\Gamma _\\Sigma }_{N_u,N_s}=P^{F^{N_s}(\\Gamma )}_+\\circ S^{\\Gamma }_{t_u,t_s}\\circ (P^{F^{-N_u(\\Gamma )}}_-)^{-1}.$ We have that $S^{\\Gamma _\\Sigma }(\\hat{x}_-)=\\hat{x}_+$ .", "Note that $\\hat{x}_-=F^{N_u}\\circ P^{F^{-N_u(\\Gamma )}}_- \\circ \\Phi ^{-t_u}(x_-)$ and $\\hat{x}_+=F^{-N_s}\\circ P^{F^{N_s(\\Gamma )}}_+ \\circ \\Phi ^{t_s}(x_+)$ .", "Thus $S^{\\Gamma _\\Sigma }(\\hat{x}_-)&=&\\hat{x}_+\\\\&=&F^{-N_s}\\circ P^{F^{N_s(\\Gamma )}}_+ \\circ \\Phi ^{t_s}(x_+) \\\\&=&F^{-N_s}\\circ P^{F^{N_s(\\Gamma )}}_+ \\circ \\Phi ^{t_s}\\circ S(x_-)\\\\&=&F^{-N_s}\\circ P^{F^{N_s(\\Gamma )}}_+ \\circ \\Phi ^{t_s}\\circ S \\circ \\Phi ^{t_u}\\circ (P_-^{F^{-N_u}(\\Gamma )})^{-1}\\circ F^{-N_u} (\\hat{x}_-).$ Hence $F^{N_s}\\circ S^{\\Gamma _\\Sigma }\\circ F^{N_u}= P^{F^{N_s(\\Gamma )}}_+ \\circ \\Phi ^{t_s}\\circ S^\\Gamma \\circ \\Phi ^{t_u}\\circ (P_-^{F^{-N_u}(\\Gamma )})^{-1}.$ The conclusion of the proposition now follows from the definition of the transition map in the flow case and the definition of the transition map in the map case." ], [ "Topological method of correctly aligned windows", "We review briefly the topological method of correctly aligned windows.", "We follow [22].", "See also [14], [13].", "Definition 5.1 An $(m_1,m_2)$ -window in an $m$ -dimensional manifold $M$ , where $m_1+m_2=m$ , is a compact subset $R$ of $M$ together with a $C^0$ -parametrization given by a homeomorphism $\\chi $ from some open neighborhood of $[0,1]^{m_1}\\times [0,1]^{m_2}$ in $\\mathbb {R}^{m_1}\\times \\mathbb {R}^{m_2}$ to an open subset of $M$ , with $R=\\chi ([0,1]^{m_1}\\times [0,1]^{m_2})$ , and with a choice of an `exit set' $R^{\\rm exit} =\\chi \\left(\\partial [0,1]^{m_1}\\times [0,1]^{m_2} \\right)$ and of an `entry set' $R^{\\rm entry}=\\chi \\left([0,1]^{m_1}\\times \\partial [0,1]^{m_2}\\right).$ We adopt the following notation: $R_\\chi =\\chi ^{-1}(R)$ , $(R^{\\rm exit})_\\chi =\\chi ^{-1}(R^{\\rm exit} )$ , and $(R^{\\rm entry})_\\chi =\\chi ^{-1}(R^{\\rm entry})$ .", "(Note that $R_\\chi =[0,1]^{m_1}\\times [0,1]^{m_2}$ , $(R^{\\rm exit} )_\\chi =\\partial [0,1]^{m_1}\\times [0,1]^{m_2}$ , and $(R^{\\rm entry})_\\chi =[0,1]^{m_1}\\times \\partial [0,1]^{m_2}$ .)", "When the local parametrization $\\chi $ is evident from context, we suppress the subscript $\\chi $ from the notation.", "Definition 5.2 Let $R_1$ and $R_2$ be $(m_1,m_2)$ -windows, and let $\\chi _1$ and $\\chi _2$ be the corresponding local parametrizations.", "Let $F$ be a continuous map on $M$ with $F(\\textrm {im}(\\chi _1))\\subseteq \\textrm {im}(\\chi _2)$ .", "We say that $R_1$ is correctly aligned with $R_2$ under $F$ if the following conditions are satisfied: (i) There exists a continuous homotopy $h:[0,1]\\times (R_1){\\chi _1} \\rightarrow {\\mathbb {R}}^{m_1} \\times {\\mathbb {R}}^{m_2}$ , such that the following conditions hold true $h_0&=&F_\\chi , \\\\h([0,1],(R^{\\rm exit}_1)_{\\chi _1}) \\cap (R_2)_{\\chi _2} &=& \\emptyset , \\\\h([0,1],(R_1)_{\\chi _1}) \\cap (R_2^{\\rm entry})_{\\chi _2} &=& \\emptyset ,$ where $F_\\chi = \\chi _2^{-1}\\circ F\\circ \\chi _1$ , and (ii) the map $A_{y_0}:\\mathbb {R}^{m_1}\\rightarrow \\mathbb {R}^{m_1}$ defined by $A_{y_0}(x)=\\pi _{m_1}\\left(h_{1}(x, y_0)\\right)$ satisfies $A_{y_0}\\left( \\partial [0,1]^{m_1}\\right)\\subseteq \\mathbb {R}^{m_1}\\setminus [0,1]^{m_1},\\\\\\deg ({A_{y_0}},0)\\ne 0,$ where $\\pi _{m_1}: \\mathbb {R}^{m_1}\\times \\mathbb {R}^{m_2}\\rightarrow \\mathbb {R}^{m_1}$ is the projection onto the first component, and $\\deg $ is the Brouwer degree of the map $A_{y_0}$ at 0.", "The following result allows the detection of orbits with prescribed itineraries.", "Theorem 5.3 Let $R_i$ be a collection of $(m_1,m_2)$ -windows in $M$ , where $i\\in \\mathbb {Z}$ or $i\\in \\lbrace 0,\\ldots , d-1\\rbrace $ , with $d>0$ (in the latter case, for convenience, we let $R_{i}=R_{(i\\,{\\rm mod}\\, d)}$ for all $i\\in \\mathbb {Z}$ ).", "Let $F_i$ be a collection of continuous maps on $M$ .", "If $R_i$ is correctly aligned with $R_{i+1}$ , for all $i$ , then there exists a point $p\\in R_0$ such that $(F_{i}\\circ \\ldots \\circ F_{0})(p)\\in R_{i+1},$ Moreover, if $R_{i+k}=R_{i}$ for some $k>0$ and all $i$ , then the point $p$ can be chosen periodic in the sense $(F_{k-1}\\circ \\ldots \\circ F_{0})(p)=p.$ Often, the maps $F_i$ represent different powers of the return map associated to a certain surface of section.", "The orbit of the point $p$ found above is not necessarily unique.", "The correct alignment of windows is robust, in the sense that if two windows are correctly aligned under a map, then they remain correctly aligned under a sufficiently small perturbation of the map.", "Proposition 5 Assume $R_1,R_2$ are $(m_1,m_2)$ -windows in $M$ .", "Let $G$ be a continuous maps on $M$ .", "Assume that $R_1$ is correctly aligned with $R_2$ under $G$ .", "Then there exists $\\epsilon >0$ , depending on the windows $R_1,R_2$ and $G$ , such that, for every continuous map $F$ on $M$ with $\\Vert F(x) - G(x)\\Vert < \\epsilon $ for all $x \\in R_1$ , we have that $R_1$ is correctly aligned with $R_2$ under $F$ .", "Also, the correct alignment satisfies a natural product property.", "Given two windows and a map, if each window can be written as a product of window components, and if the components of the first window are correctly aligned with the corresponding components of the second window under the appropriate components of the map, then the first window is correctly aligned with the second window under the given map.", "For example, if we consider a pair of windows in a neighborhood of a normally invariant normally hyperbolic invariant manifold, if the center components of the windows are correctly aligned and the hyperbolic components of the windows are also correctly aligned, then the windows are correctly aligned.", "Although the product property is quite intuitive, its rigorous statement is rather technical, so we will omit it here.", "The details can be found in [13]." ], [ "A shadowing lemma for normally hyperbolic invariant manifolds", "In this section we present a shadowing lemma-type of result.", "It is assumed the existence of a sequence of windows in the normally hyperbolic invariant manifold $\\Lambda $ , with the windows of the same dimension $l$ as $\\Lambda $ .", "It is assumed that the sequence of windows is made up of pairs of windows $D_i^-,D^+_{i+1}$ that are correctly aligned under the transition map $S^\\Gamma _{N^-_i, N^+_{i+1}}$ , alternately with pairs of windows $D^+_{i+1},D^-_{i+1}$ that are correctly aligned under some power $F^{N_i^0}$ of the restriction of $F$ to $\\Lambda $ .", "Here, the superscript $\\pm $ for the windows $D^\\pm _{i}$ suggest that $D^\\pm _{i}$ is typically obtained by taking some positive (negative) iteration of some other window that lies in the codomain (domain) of the scattering map.", "It is required that the numbers $N^0_i,N^-_i,N^+_i$ can be chosen arbitrarily large, but uniformly bounded relative to $j$ .", "The conclusion is that there exists a true orbit in the full space dynamics that follows these windows arbitrarily closely.", "The resulting orbit is not necessarily unique.", "The result below provides a method to reduce the problem of the existence of orbits in the full dimensional phase space to a lower dimensional problem of the existence of pseudo-orbits in the normally hyperbolic invariant manifold.", "Theorem 6.1 Assume that there exists a bi-infinite sequence $\\lbrace D^+_i,D^-_i\\rbrace _{i\\in \\mathbb {Z}}$ of $l$ -dimensional windows contained in a compact subset of $\\Lambda $ such that, for any integers $n^0_1,n^-_1,n^+_1>0$ , there exist integers $n^0_2>n^0_1$ , $n^-_2>n^-_1$ , $n^+_2>n^+_1$ and sequences of integers $\\lbrace N^0_i,N^-_i,N^+_i,\\rbrace _{i\\in \\mathbb {Z}}$ with $n^0_1<N^0_i<n^0_2$ , $n^-_1<N^-_i<n^-_2$ , $n^+_1<N^+_i<n^+_2$ such that the following properties hold for all $i\\in \\mathbb {Z}$ : (i) $F^{-N^+_{i}}(D^+_{i})\\subseteq U_+$ and $F^{N^-_i}(D^-_{i})\\subseteq U_-$ .", "(ii) $D^-_{i}$ is correctly aligned with $D^+_{i+1}$ under the transition map $S^\\Gamma _{N^-_i,N^+_{i+1}}=F^{N^+_{i+1}}\\circ S\\circ F^{N^-_{i}}$ .", "(iii) $D^+_{i}$ is correctly aligned with $D^-_{i}$ under the iterate $F^{N^0_{i}}$ of $F_{\\mid \\Lambda }$ .", "Then, for every $\\varepsilon >0$ , there exist an orbit $\\lbrace F^{N}(z)\\rbrace _{N\\in \\mathbb {Z}}$ of $F$ for some $z\\in M$ , and an increasing sequence of integers $\\lbrace N_i\\rbrace _{i\\in \\mathbb {Z}}$ with $N_{i+1}=N_i+N^+_{i+1}+N^0_{i+1}+N^-_{i+1}$ such that, for all $i$ : $ \\begin{split}d(F^{N_i}(z),\\Gamma )<\\varepsilon ,\\\\d(F^{N_{i}-N^-_{i}}(z), D^-_{i})<\\varepsilon ,\\\\d(F^{N_i+N^+_{i+1}}(z), D^+_{i+1})<\\varepsilon .\\end{split}$ The idea of this proof is to `thicken' the windows $D^+_{i},D^-_{i}$ in $\\Lambda $ to full-dimensional windows $R^-_i,R^+_i$ in $M$ , so that the successive windows in the sequence $\\lbrace R^-_i,R^+_i\\rbrace _i$ are correctly aligned under some appropriate iterations of the map $F$ .", "The argument is done in several steps.", "In the first three steps, we only specify the relative sizes of the windows involved in each step.", "In the fourth step, we explain how to make the choices of the sizes of the windows uniform.", "Step 1.", "At this step, we take a pair of $l$ -dimensional windows $D^-_{i}$ and $D^+_{i+1}$ as in the statement of the theorem and, through applying iteration and the wave maps we construct two $(l+2n)$ -dimensional windows $\\bar{R}^-_{i}$ and $\\bar{R}^+_{i+1}$ near $\\Gamma $ such that $\\bar{R}^-_{i}$ is correctly aligned with $\\bar{R}^+_{i+1}$ under the identity map.", "Note that conditions (i) and (ii) imply that $\\hat{D}^-_{i}:=F^{N^-_{i}}(D^-_{i})\\subseteq U_-\\subseteq \\Lambda $ is correctly aligned with $\\hat{D}^+_{i+1}:=F^{-N^+_{i+1}}(D^+_{i+1})\\subseteq U_+\\subseteq \\Lambda $ under the scattering map $S$ .", "Let $\\bar{D}^-_{i}=(\\Omega _-^\\Gamma )^{-1}(\\hat{D}^-_{i})$ and $\\bar{D}^+_{i+1}=(\\Omega _+^\\Gamma )^{-1}(\\hat{D}^+_{i+1})$ be the copies of $\\hat{D}^-_{i}$ and $\\hat{D}^+_{i+1}$ , respectively, in the homoclinic channel $\\Gamma $ .", "By making some arbitrarily small changes in the sizes of their exit and entry directions, we can alter the windows $\\hat{D}^-_{i}$ and $\\hat{D}^+_{i+1}$ such that $\\hat{D}^-_{i}$ is correctly aligned with $\\bar{D}^-_{i}$ under $(\\Omega ^-_\\Gamma )^{-1}$ , $\\bar{D}^-_{i}$ is correctly aligned with $\\bar{D}^+_{i+1}$ under the identity mapping, and $\\bar{D}^+_{i+1}$ is correctly aligned with $\\hat{D}^+_{i+1}$ under $\\Omega ^+_\\Gamma $ .", "We `thicken' the $l$ -dimensional windows $\\bar{D}^-_{i}$ and $\\bar{D}^+_{i+1}$ in $\\Gamma $ , which are correctly aligned under the identity mapping, to $(l+2n)$ -dimensional windows that are correctly aligned under the identity map.", "We now explain the `thickening' procedure.", "First, we describe how to thicken $\\bar{D}^-_i$ to a full dimensional window $\\bar{R}^-_i$ .", "We choose some $0<\\bar{\\delta }^-_{i}<\\varepsilon $ and $0<\\bar{\\eta }^-_{i}<\\varepsilon $ .", "At each point $x\\in \\bar{D}^-_i$ we choose an $n$ -dimensional closed ball $\\bar{B}^-_{\\bar{\\delta }^-_{i}}(x)$ of radius $\\bar{\\delta }^-_{i}$ centered at $x$ and contained in $W^u({x_-})$ , where $x_-=\\Omega _{-}^\\Gamma (x)$ .", "We take the union $\\bar{\\Delta }^-_{i}:=\\bigcup _{x\\in \\bar{D}^-_{i}}\\bar{B}^{u}_{\\bar{\\delta }^-_{i}}(x)$ .", "Note that $\\bar{\\Delta }^-_{i}$ is contained in $W^u(\\Lambda )$ and is homeomorphic to an $(l+n)$ -dimensional rectangle.", "We define the exit set and the entry set of this rectangle as follows: $\\begin{split}(\\bar{\\Delta }^-_{i})^{\\rm exit}:=\\bigcup _{x\\in (\\bar{D}^-_{i})^{\\rm exit}}\\bar{B}^{u}_{\\bar{\\delta }^-_{i}}(x) \\cup \\bigcup _{x\\in \\bar{D}^-_{i}}\\partial \\bar{B}^{u}_{\\bar{\\delta }^-_{i}}(x),\\\\(\\bar{\\Delta }^-_{i})^{\\rm entry}:=\\bigcup _{x\\in (\\bar{D}^-_{i})^{\\rm entry}}\\bar{B}^{u}_{\\bar{\\delta }^-_{i}}(x).\\end{split}$ We consider the normal bundle $N^+$ to $W^u(\\Lambda )$ .", "At each point $y\\in \\bar{\\Delta }^-_{i}$ , we choose an $n$ -dimensional closed ball $\\bar{B}^+_{\\bar{\\eta }^-_{i}}(y)$ centered at $y$ and contained in the image of $N^+_y\\subseteq T_yM$ under the exponential map $\\exp _y:N^+_y\\rightarrow M$ .", "We let $\\bar{R}^-_{i}:=\\bigcup _{y\\in \\bar{\\Delta }^-_{i}}\\bar{B}^{s}_{\\bar{\\eta }^-_{i}}(y)$ .", "By the Tubular Neighborhood Theorem (see, for example [2]), we have that for $\\bar{\\eta }^-_{i}>0$ sufficiently small, the set $\\bar{R}^-_{i}$ is a homeomorphic copy of an $(l+2n)$ -rectangle.", "We now define the exit set and the entry set of $\\bar{R}^-_{i}$ as follows: $\\begin{split} (\\bar{R}^-_{i})^{\\rm exit}:=\\bigcup _{y\\in (\\bar{\\Delta }^-_{i})^{\\rm exit}}\\bar{B}^{s}_{\\bar{\\eta }^-_{i}}(y),\\\\(\\bar{R}^-_{i})^{\\rm entry}:=\\bigcup _{y\\in (\\bar{\\Delta }^-_{i})^{\\rm entry}}\\bar{B}^{s}_{\\bar{\\eta }^-_{i}}(y) \\cup \\bigcup _{y\\in (\\bar{\\Delta }^-_{i})}\\partial \\bar{B}^{s}_{\\bar{\\eta }^-_{i}}(y).\\end{split}$ Second, we describe in a similar fashion how to thicken $\\bar{D}^+_{i+1}$ to a full dimensional window $\\bar{R}^+_{i+1}$ .", "We choose $0<\\bar{\\delta }^+_{i+1}<\\varepsilon $ and $0<\\bar{\\eta }^+_{i+1}<\\varepsilon $ .", "We consider the $(l+n)$ -dimensional rectangle $\\bar{\\Delta }^+_{i+1}:=\\bigcup _{x\\in \\bar{D}^+_{i+1}}\\bar{B}^{s}_{\\bar{\\eta }^+_{i+1}}(x)\\subseteq W^s(\\Lambda )$ , where $\\bar{B}^+_{\\bar{\\eta }^+_{i+1}}(x)$ is the $n$ -dimensional closed ball of radius $\\bar{\\eta }^+_{i+1}$ centered at $x$ and contained in $W^s(x_+)$ , with $x_+=\\Omega _{+}^\\Gamma (x)$ .", "The exit set and entry set of this window are defined as follows: $\\begin{split}(\\bar{\\Delta }^+_{i+1})^{\\rm exit}:=\\bigcup _{x\\in (\\bar{D}^+_{i+1})^{\\rm exit}}\\bar{B}^{s}_{\\bar{\\eta }^+_{i+1}}(x),\\\\(\\bar{\\Delta }^+_{i+1})^{\\rm entry}:=\\bigcup _{x\\in (\\bar{D}^+_{i+1})^{\\rm entry}}\\bar{B}^{s}_{\\bar{\\eta }^+_{i+1}}(x)\\cup \\bigcup _{x\\in (\\bar{D}^+_{i+1})}\\partial \\bar{B}^{s}_{\\bar{\\eta }^+_{i+1}}(x).\\end{split}$ We let $\\bar{R}^+_{i+1}:=\\bigcup _{y\\in \\bar{\\Delta }^+_{i+1}}\\bar{B}^{u}_{\\bar{\\delta }^+_{i+1}}(y)$ , where $\\bar{B}^-_{\\bar{\\delta }^+_{i+1}}(y)$ is the $n$ -dimensional closed ball centered at $y$ and contained in the image of $N^-_y\\subseteq T_yM$ under the exponential map $\\exp _y:N^-_y\\rightarrow M$ , and $N^-$ is the normal bundle to $W^s(\\Lambda )$ .", "The Tubular Neighborhood Theorem implies that for $\\bar{\\delta }^+_{i+1}>0$ sufficiently small the set $\\bar{R}^+_{i+1}$ is a homeomorphic copy of a $(l+2n)$ -rectangle.", "The exit set and the entry set of $\\bar{R}^+_{i+1}$ are defined by: $\\begin{split} (\\bar{R}^+_{i+1})^{\\rm exit}:=\\bigcup _{y\\in (\\bar{\\Delta }^+_{i+1})^{\\rm exit}}\\bar{B}^{u}_{\\bar{\\delta }^+_{i+1}}(y)\\cup \\bigcup _{y\\in (\\bar{\\Delta }^+_{i+1}}\\partial \\bar{B}^{u}_{\\bar{\\delta }^+_{i+1}}(y),\\\\(R^+_{i+1})^{\\rm entry}:=\\bigcup _{y\\in (\\bar{\\Delta }^+_{i+1})^{\\rm entry}}\\bar{B}^{u}_{\\bar{\\delta }^+_{i+1}}(y).\\end{split}$ This completes the description of the thickening of the $l$ -dimensional window $\\bar{D}^-_{i}$ into an $(l+2n)$ -dimensional window $\\bar{R}^-_{i}$ , and of the thickening of the $l$ -dimensional window $\\bar{D}^+_{i+1}$ into an $(l+2n)$ -dimensional window $\\bar{R}^+_{i+1}$ .", "Note that, by construction, $\\bar{R}^-_{i}$ and $\\bar{R}^+_{i+1}$ are both contained in an $\\varepsilon $ -neighborhood of $\\Gamma $ .", "Now we want to make $\\bar{R}^-_{i}$ correctly aligned with $\\bar{R}^+_{i+1}$ under the identity map.", "This is achieved by choosing $\\bar{\\delta }^+_{i+1}$ sufficiently small relative to $\\bar{\\delta }^-_{i}$ , and by choosing $\\bar{\\eta }^-_{i}$ sufficiently small relative to $\\bar{\\eta }^+_{i+1}$ .", "Thus, we have $\\bar{\\delta }^-_{i}>\\bar{\\delta }^+_{i+1}$ and $\\bar{\\eta }^-_{i}<\\bar{\\eta }^+_{i+1}$ (we stress that these inequalities alone may not suffice for the correct alignment).", "Choosing $\\bar{\\delta }^+_{i+1}$ and $\\bar{\\eta }^-_{i}$ small enough agrees with the constraints imposed by the Tubular Neighborhood Theorem.", "Step 2.", "At this step, we expand the given $l$ -dimensional window $D^-_i$ to an $(l+2n)$ -dimensional window $R_i^-$ such that $R_i^-$ is correctly aligned with $\\bar{R}_i^-$ under some positive iterate, and we also expand the given $l$ -dimensional window $D^+_{i+1}$ to an $(l+2n)$ -dimensional window $R_{i+1}^+$ such that $\\bar{R}_{i+1}^+$ is correctly aligned with $\\bar{R}_{i+1}^+$ under some positive iterate.", "We take a negative iterate $F^{-M}(\\bar{R}^-_{i})$ of $\\bar{R}^-_{i}$ , where $M>0$ .", "We have that $F^{-M}(\\Gamma )$ is $\\varepsilon $ -close to $\\Lambda $ on a neighborhood in the $C^1$ -topology, for all $M$ sufficiently large.", "The vectors tangent to the fibers $W^u(x_-)$ in $\\bar{R}^-_i$ are contracted, and the vectors transverse to $W^u(\\Lambda )$ along $\\bar{R}^-_i\\cap W^u(\\Lambda )$ are expanded by the derivative of $F^{-M}$ .", "We choose and fix $M=N^-_i$ sufficiently large.", "We obtain that, in particular, $F^{-N^-_i}(\\bar{R}^-_{i})$ is $\\varepsilon $ -close to $D^-_i=F^{-N^-_{i}}(\\hat{D}^-_{i})$ .", "We now construct a window $R^-_{i}$ about $D^-_{i}$ that is correctly aligned with the window $F^{-N^-_{i}}(\\bar{R}^-_{i})$ under the identity.", "Note that each closed ball $\\bar{B}^{u}_{\\delta ^-_{i}}(x)$ , which is a part of $\\bar{\\Delta }^- _{i}$ , gets exponentially contracted as it is mapped into $W^u(F^{-{N^-_i}}(x_-))$ by $F^{-{N^-}_{i}}$ .", "By the Lambda Lemma (Proposition REF ), each closed ball $\\bar{B}^s_{\\eta ^-_{i}}(y)$ with $y\\in \\bar{\\Delta }^-_{i}$ , which is a part of $\\bar{R}^-_{i}$ , $C^1$ -approaches a subset of $W^s(F^{-{M}}(y_-))$ under $F^{-{M}}$ , as $M\\rightarrow \\infty $ .", "For $N^-_i$ sufficiently large, we may assume that $F^{-{N^-_i}}(\\bar{B}^s_{\\eta ^-_{i}}(y))$ is $\\varepsilon $ -close to a subset of $W^s(F^{-{N^-_i}}(y^-))$ in the $C^1$ -topology, for all $y\\in \\bar{\\Delta }^- _{i}$ .", "As $\\hat{D}^-_{i}$ is correctly aligned with $\\bar{D}^-_{i}$ under $(\\Omega _-^\\Gamma )^{-1}$ , we have that $D^-_{i}=F^{-N^-_{i}}(\\hat{D}^-_{i})$ is correctly aligned with $F^{-N^-_{i}}(\\bar{D}^-_{i})$ under $(\\Omega _-^{F^{-N^-_{i}}(\\Gamma )})^{-1}$ .", "In other words, $D^-_{i}$ is correctly aligned under the identity mapping with the projection of $F^{-N^-_{i}}(\\bar{D}^-_{i})$ onto $\\Lambda $ along the unstable fibres.", "Let us consider $0<\\delta ^-_{i}<\\varepsilon $ and $0<\\eta ^-_{i}<\\varepsilon $ .", "To define the window $R^-_{i}$ we use a local linearization of the normally hyperbolic invariant manifold.", "For $\\Lambda $ normally hyperbolic, let $N\\Lambda =(E^u\\oplus E^s)_{\\mid \\Lambda }=\\bigcup _{p\\in \\Lambda }\\lbrace p\\rbrace \\times E^u_p\\times E^s_p$ be the normal bundle to $\\Lambda $ , and $NF=TF_{\\mid N\\Lambda }$ , where $TF(p,v^u,v^s)=(F(p),DF_p(v^u),DF_p(v^s))\\textrm { for all } p\\in \\Lambda ,v^u\\in E^u, v^s\\in E^s.$ By Theorem 1 in [20], there exists a homeomorphisms $h$ from an open neighborhood of the zero section of $N\\Lambda $ to a neighborhood of $\\Lambda $ in $M$ such that $F\\circ h=h\\circ NF$ .", "Since $D^-_i$ is contractible the bundles are trivial on $D^-_i$ and we can identify $(E^u\\oplus E^s)_{D^-_i}$ with $D^-_i \\times E^u_{x } \\times E^s_{x }$ .", "At each point $x\\in D^-_i$ we define a rectangle $H^-_i(x)$ of the type $h(\\lbrace x\\rbrace \\times \\bar{B}^u_{\\delta ^-_{i}}(0)\\times \\bar{B}^s_{\\eta ^-_{i}}(0))$ , where $\\bar{B}^u_{\\delta ^-_{i}}(0)$ is the closed ball centered at 0 of radius $\\delta ^-_{i}$ in the unstable space $E^u_x$ , and $\\bar{B}^s_{\\eta ^-_{i}}$ is the closed ball centered at 0 of radius $\\eta ^-_{i}$ in the stable space $E^s_x$ .", "We set the exit and entry sets of $H^-_{i}(x)$ as $(H^-_{i}(x))^{\\rm exit}=h(\\lbrace x\\rbrace \\times \\partial \\bar{B}^u_{\\delta ^-_{i}}(0)\\times \\bar{B}^s_{\\eta ^-_{i}}(0))$ and $(H^-_{i}(x))^{\\rm entry}=h(\\lbrace x\\rbrace \\times \\bar{B}^u_{\\delta ^-_{i}}(0)\\times \\partial \\bar{B}^s_{\\eta ^-_{i}}(0))$ .", "Then we define the window $R^-_{i}$ as follows: $\\begin{split}R^-_{i}=\\bigcup _{x\\in D^-_{i}}H^-_{i}(x),\\\\(R^-_{i})^{\\rm exit}=\\bigcup _{x\\in (D^-_{i})^{\\rm exit}}H^-_{i}(x) \\cup \\bigcup _{x\\in D^-_{i}}(H^-_{i}(x))^{\\rm exit},\\\\(R^-_{i})^{\\rm entry}=\\bigcup _{x\\in (D^-_{i})^{\\rm entry}}H^-_{i}(x) \\cup \\bigcup _{x\\in D^-_{i}}(H^-_{i}(x))^{\\rm entry}.\\end{split}$ In order to ensure the correct alignment of $R^-_{i}$ with $F^{-N^-_i}(\\bar{R}^-_{i})$ under the identity map, it is sufficient to choose $\\delta ^-_i,\\eta ^-_i$ such that $\\bigcup _{x\\in D^-_{i}}h(\\lbrace x\\rbrace \\times \\bar{B}^u_{\\delta ^-_{i}}(0)\\times \\lbrace 0\\rbrace )$ is correctly aligned with $F^{-N^-_{i}}(\\bar{\\Delta }^-_{i})$ under the identity map (the exit sets of both windows being in the unstable directions), and that each closed ball $F^{-N^-_{i}}(\\bar{B}^s_{\\eta ^-_{i}})$ intersects $R_{i}$ in a closed ball that is contained in the interior of $F^{-N^-_{i}}(\\bar{B}^s_{\\eta ^-_{i}})$ .", "The existence of suitable $\\delta ^-_i,\\eta ^-_i$ follows from the exponential contraction of $\\bar{\\Delta }^-_{i}$ under negative iteration, and from the Lambda Lemma applied to $\\bar{B}^s_{\\eta ^-_{i}}(y)$ under negative iteration.", "In a similar fashion, we construct a window $R^+_{i+1}$ contained in an $\\varepsilon $ -neighborhood of $\\Lambda $ such that $\\bar{R}^+_{i+1}$ is correctly aligned with $R^+_{i+1}$ under $F^{N^+_{i+1}}$ .", "The window $R^+_{i+1}$ , and its entry and exit sets, are defined by: $\\begin{split}R^+_{i+1}=\\bigcup _{x\\in D^+_{i+1}}H^+_{i+1}(x),\\\\(R^+_{i+1})^{\\rm exit}=\\bigcup _{x\\in (D^+_{i+1})^{\\rm exit}}H^+_{i+1}(x) \\cup \\bigcup _{x\\in D^+_{i+1}}(H^+_{i+1}(x))^{\\rm exit},\\\\(R^+_{i+1})^{\\rm entry}=\\bigcup _{x\\in (D^+_{i+1})^{\\rm entry}}H^+_{i+1}(x) \\cup \\bigcup _{x\\in D^+_{i+1}}(H^+_{i+1}(x))^{\\rm entry},\\end{split}$ where $H^+_{i+1}(x)=h(\\lbrace x\\rbrace \\times \\bar{B}^u_{\\delta ^+_{i+1}}(0)\\times \\bar{B}^s_{\\eta ^+_{i+1}}(0))$ , $(H^+_{i+1}(x))^{\\rm exit}$ , and $(H^+_{i+1}(x))^{\\rm entry}$ are defined as before for some appropriate choices of radii $\\delta ^+_{i+1},\\eta ^+_{i+1}>0$ .", "Step 3.", "At this step, we take the $(l+2n)$ -dimensional window $R_{i+1}^+$ and $\\bar{R}_{i+1}^-$ as constructed in the previous step, and we make $R_{i+1}^+$ correctly aligned with $\\bar{R}_{i+1}^-$ under some positive iterate.", "Suppose that we have constructed the window $R^+_{i+1}$ about the $l$ -dimensional rectangle $D^+_{i+1}\\subseteq \\Lambda $ and the window $R^-_{i+1}$ about the $l$ -dimensional rectangle $R^-_{i+1}\\subseteq \\Lambda $ .", "Under positive iterations, the rectangle $\\bar{B}^u_{\\delta ^+_{i+1}}(0)\\times \\bar{B}^s_{\\eta ^+_{i+1}}(0)\\subseteq E^u\\oplus E^s$ gets exponentially expanded in the unstable direction and exponentially contracted in the stable direction by $DF$ .", "Thus $\\bar{B}^u_{\\delta ^+_{i+1}}(0)\\times \\bar{B}^s_{\\eta ^+_{i+1}}(0)$ is correctly aligned with $\\bar{B}^u_{\\delta ^-_{i+1}}(0)\\times \\bar{B}^s_{\\eta ^-_{i+1}}(0)$ under the power $DF^{N^0_{i+1}}$ of $DF$ , provided $N^0_{i+1}$ is sufficiently large.", "This implies that $F^{N^0_{i+1}}(h(\\lbrace x\\rbrace \\times \\bar{B}^u_{\\delta ^+_{i+1}}(0)\\times \\bar{B}^s_{\\delta ^+_{i+1}}(0)))$ is correctly aligned with $h({F^{N^0_{i+1}}(x)}\\times \\bar{B}^u_{\\delta ^-_{i+1}}(0)\\times \\bar{B}^s_{\\delta ^-_{i+1}}(0))$ under the identity map (both rectangles are contained in $h({F^{N^0_{i+1}}(x)}\\times E^u\\times E^s)$ ).", "Since $D^+_{i+1}$ is correctly aligned with $D^-_{i+1}$ under $F^{N^0_i}$ , the product property of correctly aligned windows implies that $R^+_{i+1}$ is correctly aligned with $R^-_{i+1}$ under $F^{N^0_{i+1}}$ , provided that $N^0_{i+1}$ is sufficiently large.", "Step 4.", "At this step we will use the previous steps to construct a bi-infinite sequences of windows $\\lbrace R^\\pm _{i},\\bar{R}^\\pm _{i}\\rbrace _{i\\in \\mathbb {Z}}$ such that, for each $i$ , the windows $\\lbrace R^\\pm _{i}\\rbrace $ are obtained by thickening the rectangles $\\lbrace D^\\pm _i\\rbrace \\subseteq \\Lambda $ , the windows $\\lbrace \\bar{R}^\\pm _{i+1}\\rbrace $ are obtained by thickening some rectangles $\\lbrace \\bar{D}^\\pm _i\\rbrace \\subseteq \\Gamma $ , and, moreover, $R^-_{i}$ is correctly aligned with $\\bar{R}^-_{i}$ under $F^{N^-_{i}}$ , $\\bar{R}^-_{i}$ is correctly aligned with $\\bar{R}^+_{i+1}$ under the identity map, $\\bar{R}^+_{i+1}$ is correctly aligned with $R^+_{i+1}$ under $F^{N^+_{i+1}}$ , and $ R^+_{i+1}$ is correctly aligned with $R^-_{i+1}$ under $F^{N^0_i}$ .", "We can assume without loss of generality that $\\Lambda $ and $\\Gamma $ are compact.", "We fix an $\\varepsilon $ -neighborhood $V$ of $\\Lambda $ .", "Using the compactness of $\\Lambda $ and $\\Gamma $ and the uniform boundedness of the iterates $N^-_i,N^+_i,N^0_i$ , we now show how to choose the sizes of the stable and unstable components of the windows $\\lbrace R^\\pm _{i},\\bar{R}^\\pm _{i}\\rbrace _{i\\in \\mathbb {Z}}$ constructed in the previous steps in a uniformly bounded manner.", "For each point $x$ in $\\Lambda $ we consider a $(2n)$ -dimensional window $h(\\lbrace x\\rbrace \\times \\bar{B}^u_\\delta \\times \\bar{B}^s_\\eta )$ , for some $0<\\delta ,\\eta <\\varepsilon $ , where $h$ is the local conjugacy between $F$ and $DF$ near $\\Lambda $ .", "Then $F^{N^0_i}(h(\\lbrace x\\rbrace \\times \\bar{B}^u_\\delta \\times \\bar{B}^s_\\eta )$ is correctly aligned with $h(\\lbrace F^{N^0_i}(x)\\rbrace \\times \\bar{B}^u_\\delta (0)\\times \\bar{B}^s_\\eta (0))$ , for all $n^0_1\\le N^0_i\\le n^0_2$ , provided that $n^0_1$ is chosen sufficiently large.", "For each $i$ , we thicken $D^+_{i}$ and $D^-_{i}$ into full dimensional windows $R^+_{i}$ and of $R^-_{i}$ respectively, as described in Step 2, where for the sizes of the components of these windows we choose $\\delta ^\\pm _i=\\delta $ and $\\eta ^\\pm _i=\\eta $ for all $i$ .", "Since $D^+_{i}$ is correctly aligned with $D^-_{i}$ under $F^{N^0_{i}}$ , then, as in Step 3, it follows that $R^+_{i}$ is correctly aligned with $R^-_{i}$ under $F^{N^0_{i}}$ .", "We also define the set $\\Upsilon ^0=\\bigcup _{x\\in \\Lambda } h(\\lbrace x\\rbrace \\times \\bar{B}^s_\\eta (0)\\times \\bar{B}^s_\\delta (0)).$ This set cannot be realized as a window since it does not have exit/entry directions associated to the $\\Lambda $ components.", "However, for each $x\\in \\Lambda $ , the set $h(\\lbrace x\\rbrace \\times \\bar{B}^s_\\eta (0)\\times \\bar{B}^u_\\delta (0))$ is a well defined window, with the exit given by the hyperbolic unstable directions.", "Note that $\\Upsilon ^0(x)\\subseteq h(\\lbrace x\\rbrace \\times \\bar{W}^u(x)\\times \\bar{W}^s(x))$ for each $x\\in \\Gamma $ .", "We let $\\bar{\\Delta }^-=\\bigcup _{x\\in \\Gamma }\\bar{B}^u_{\\bar{\\delta }^-}(x)$ , with $\\bar{B}^u_{\\bar{\\delta }^-}(x)$ being the closed ball centered at $x$ of radius $\\bar{\\delta }^-$ in $W^u(x_-)$ .", "For each point $y\\in \\bar{\\Delta }^-$ we consider the closed ball $\\bar{B}^s_{\\bar{\\eta }^-}(y)$ centered at $y$ of radius $\\bar{\\eta }^-$ in the image under $\\exp _y$ of the normal subspace $N_y$ to $W^u(\\Lambda )$ at $y$ .", "Similarly, we let $\\bar{\\Delta }^+=\\bigcup _{x\\in \\Gamma }\\bar{B}^s_{\\bar{\\eta }^+}(x)$ , where $\\bar{B}^u_{\\bar{\\eta }^+}(x)\\subseteq W^s(x_+)$ , and for each $y\\in \\bar{\\Delta }^+$ we consider the closed ball $\\bar{B}^u_{\\bar{\\delta }^+}(y)$ in the image under $\\exp _y$ of the normal subspace $N_y$ to $W^s(\\Lambda )$ at $y$ .", "We define the sets $\\Upsilon ^-=\\bigcup _{y\\in \\bar{\\Delta }^-} \\bar{B}^s_{\\bar{\\eta }^-}(y),\\,\\Upsilon ^+=\\bigcup _{y\\in \\bar{\\Delta }^+} \\bar{B}^u_{\\bar{\\delta }^+}(y).$ These sets cannot be realized as windows as there are no well defined exit/entry directions associated to their $\\Gamma $ components.", "However, for each $x\\in \\Gamma $ , the set $\\Upsilon ^-(x)= \\bigcup _{y\\in B^u_{\\bar{\\delta }^-}(x)} \\bar{B}^s_{\\bar{\\eta }^-}(y)$ is a well defined $(2n)$ -dimensional window, with the exit given by the hyperbolic unstable directions.", "Note that $\\Upsilon ^-(x)\\subseteq \\bigcup _{y\\in W^u(x_-)}\\exp _y(N_y)$ .", "The intersection of $\\bigcup _{y\\in W^u(x_-)}\\exp _y(N_y)$ with $\\Upsilon ^+$ defines a window $\\Upsilon ^+(x)$ with the exit given by the hyperbolic unstable directions.", "Due to the compactness of $\\Gamma $ , there exist $\\delta ^\\pm ,\\eta ^\\pm $ such that $\\Upsilon ^-(x)$ is correctly aligned with $\\Upsilon ^+(x)$ for all $x\\in \\Gamma $ .", "We choose and fix such $\\delta ^\\pm ,\\eta ^\\pm $ .", "We define the windows $\\bar{R}^\\pm _i$ at Step 1 with the choices of $\\delta ^\\pm _i=\\delta ^\\pm $ , and $\\eta ^\\pm _i=\\eta ^\\pm $ , for all $i$ .", "It follows that $\\bar{R}^-_i$ is correctly aligned with $\\bar{R}^+_i$ under the identity map for all $i$ .", "Due to the compactness of $\\Gamma $ and the uniform expansion and contraction of the hyperbolic directions, there exist $n^-_1,n^+_1$ such that, for all $N^-_i>n^-_1$ , $N^+_i>n^+_2$ , we have $F^{-N^-_i}(\\Gamma )\\subseteq V$ and $F^{N^+_i}(\\Gamma )\\subseteq V$ for all $i\\in \\mathbb {Z}$ , where $V$ is the neighborhood of $\\Lambda $ where the local linearization is defined.", "For any such $n^-_1,n^+_1$ , the assumptions of Lemma REF provide us with some $n^-_2>n^-_i$ , $n^+>n^-_i$ .", "Moreover, we choose $n^-_1,n^+_1$ such that for all $N^-_i>n^-_1$ , $N^+_i>n^+_2$ we have (i) $\\Upsilon ^0(F^{-N^-_i}(x_-))$ is correctly aligned with $F^{-N^-_i}(\\Upsilon ^-)\\cap h(\\lbrace \\tilde{F}^{-N^-_i}(x_-)\\rbrace \\times \\bar{W}^u(F^{-N^-_i}(x_-)) \\times \\bar{W}^s(F^{-N^-_i}(x_-)))]$ under the identity map, (ii) $F^{N^+_i}(\\Upsilon ^+(x))$ is correctly aligned with $\\Upsilon ^0\\cap h(\\lbrace \\tilde{F}^{N^+_i}(x_+)\\rbrace \\times \\bar{W}^u(F^{N^+_i}(x_+)) \\times \\bar{W}^s(F^{N^+_i}(x_+)))]$ under the identity map.", "From these choices, it follows that the windows $R^-_i, R^+_i$ constructed in Step 2 satisfy that $R^-_i$ is correctly aligned with $\\bar{R}^-_i$ under $F^{N^-_i}$ , and $\\bar{R}^+_i$ is correctly aligned with $R^+_i$ under $F^{N^+_i}$ .", "This concludes the construction of windows $\\lbrace R^\\pm _{i},\\bar{R}^\\pm _{i}\\rbrace _{i\\in \\mathbb {Z}}$ of uniform sizes, such that $R^-_{i}$ is correctly aligned with $\\bar{R}^-_{i}$ under $F^{N^-_{i}}$ , $\\bar{R}^-_{i}$ is correctly aligned with $\\bar{R}^+_{i+1}$ under the identity map, $\\bar{R}^+_{i+1}$ is correctly aligned with $R^+_{i+1}$ under $F^{N^+_{i+1}}$ , and $ R^+_{i+1}$ is correctly aligned with $R^-_{i+1}$ under $F^{N^0_i}$ .", "The windows $R^\\pm _i$ are contained in $\\varepsilon $ -neighborhoods of the given rectangles $D^\\pm _i$ , respectively, and the windows $R^\\pm _i$ are contained in $\\varepsilon $ -neighborhoods of some rectangles $\\bar{D}^\\pm _i$ , respectively.", "By Theorem REF , there exits an orbit $F^{N}(z)$ that visits the windows $\\lbrace R^\\pm _{i},\\bar{R}^\\pm _{i}\\rbrace _{i\\in \\mathbb {Z}}$ in the prescribed order.", "More precisely, if $F^{N_i}(z)$ is the corresponding point in $\\bar{R}^-_i\\cap R^+_{i+1}$ , then $F^{N_i+N^+_{i+1}}(z)$ is in $R^+_{i+1}$ , $F^{N_i+N^+_{i+1}+N^0_{i+1}}(z)$ is in $ R^-_{i+1}$ , and$F^{N_i+N^+_{i+1}+N^0_{i+1}+N^-_{i+1}}(z)$ is in $\\bar{R}^-_{i+1}\\cap \\bar{R}^+_{i+2}$ , for all $i$ .", "This means that $N_{i+1}=N_i+N^+_{i+1}+N^0_{i+1}+N^-_{i+1}$ for all $i$ .", "The existence of the shadowing orbit concludes the proof." ], [ "Acknowledgement", "Part of this work has been done while M.G.", "was visiting the Centre de Recerca Matemàtica, Barcelona, Spain, for whose hospitality he is very grateful.", "In this section we discuss the application of the techniques discussed in this paper to show the existence of unstable orbits in dynamical systems.", "We describe two models for Hamiltonian instability and we explain how the transition map and the shadowing lemma are utilized.", "The first model is related to the Arnold diffusion problem for Hamiltonian systems [1].", "This problem conjectures that generic Hamiltonian systems that are close to integrable possess trajectories the move `wildly' and `arbitrarily far'.", "The model, from [6], describes a rotator and a pendulum with a small, periodic coupling.", "Consider the time-dependent Hamiltonian, $H_\\mu (p, q, I,\\phi ,t)&=&h_0(I)+P_{\\pm }(p,q)+\\mu h(p, q,I,\\phi , t; \\mu ),$ where $(p, q, I,\\phi , t)\\in (\\mathbb {R}\\times \\mathbb {T}^1)^2\\times \\mathbb {T}^1$ , and assume: (i) $V$ , $h_0$ and $h$ are uniformly $C^r$ with $r$ sufficiently large; (ii) $P_{\\pm }(p,q)=\\pm (\\frac{1}{2}p^2+V(q))$ where $V$ is periodic in $q$ of period $2\\pi $ and has a unique non-degenerate global maximum at $(0,0)$ ; thus $P_{\\pm }(p,q)$ has a homoclinic orbit $(p^0 (\\sigma ),q^0 (\\sigma ))$ to $(0,0)$ , with $\\sigma \\in \\mathbb {R}$ ; (iii) $h_0$ satisfies a uniform twist condition $\\partial ^2h_0/\\partial I^2>\\theta $ , for some $\\theta >0$ and all $I$ in some interval $(I^-, I^+)$ , with $I^-<I^+$ independent of $\\mu $ ; (iv) $h$ is a trigonometric polynomial in $(\\phi ,t)$ , periodic of period $2\\pi $ in both $\\phi ,t$ , (v) The Melnikov potential associated to $(p^0 (\\sigma ),q^0 (\\sigma ))$ , given by $\\mathcal {L}(I,\\phi ,t)=&\\displaystyle -\\int _{-\\infty }^{\\infty }&\\left[h(p^0(\\sigma ),q^0(\\sigma ),I,\\phi +\\omega (I)\\sigma ,t+\\sigma ;0)\\right.\\\\& &\\left.-h(0,0,I,\\phi +\\omega (I)\\sigma ,t+\\sigma ;0)\\right]d\\sigma ;$ where $\\omega (I)=(\\partial h_0/\\partial I)(I)$ , satisfies the following non-degeneracy conditions: (v.a) For each $I\\in (I^-,I^+)$ , and each $(\\phi , t)$ in some open set in $\\mathbb {T}^1\\times \\mathbb {T}^1$ , the map $\\tau \\in \\mathbb {R} \\rightarrow \\mathcal {L}(I,\\phi -\\omega (I)\\tau ,t-\\tau )\\in \\mathbb {R}$ has a non-degenerate critical point $\\tau ^*$ , which can be parameterized as $\\tau ^*=\\tau ^*(I,\\phi ,t);$ (v.b) For each $(I,\\phi , t)$ as above, the function $(I,\\phi ,t)\\rightarrow \\frac{\\partial {\\mathcal {L}}}{\\partial \\phi }(I,\\phi -\\omega (I)\\tau ^*,t-\\tau ^*)$ is non-constant, negative for $P_{-}$ , and positive for $P_{+}$ ; (v.c) The perturbation term $h$ satisfies some additional non-degeneracy conditions described in hypothesis (H5) of Theorem 7 in [6].", "The objective is to show that there exists $\\mu _0>0$ such that, for each $0<\\mu <\\mu _0$ , the Hamiltonian has a trajectory $x(t)$ such that $I(x(0))<I^-$ and $I(x(T))>I^+$ for some $T>0$ .", "In the sequel, we show that the arguments in [6], [13], [8], [15] can be combined with the techniques developed in this paper to prove the existence of diffusing trajectories as above.", "(1) We can make the Hamiltonian $H_\\mu $ into an autonomous Hamiltonian by adding an extra variable $A$ symplectically conjugate with the time $t$ .", "The variable $A$ has no dynamical role.", "We then restrict to an energy level, which can be parametrized by the variables $(p, q,I,\\phi , t)$ with $A$ an implicit function of these variables.", "When $\\mu =0$ the set $\\tilde{\\Lambda }_0=\\lbrace (p, q,I,\\phi , t),\\, p=q=0, I\\in [I^-,I^+]\\rbrace $ is a 3-dimensional normally hyperbolic invariant manifold with boundary for the flow of $H_0$ .", "It is foliated by 2-dimensional invariant tori $\\tilde{\\mathcal {T}}_{I^{\\prime }}=\\lbrace (p, q,I,\\phi , t),\\, p=q=0, I=I^{\\prime }\\rbrace $ with $I^{\\prime }\\in [I^-,I^+]$ .", "The stable and unstable manifolds of $\\tilde{\\Lambda }_0$ coincide, i.e.", "$W^u(\\tilde{\\Lambda }_0)=W^s(\\tilde{\\Lambda }_0)$ .", "Also $W^u(\\tilde{\\mathcal {T}}_{I^{\\prime }})=W^s(\\tilde{\\mathcal {T}}_{I^{\\prime }})$ for all $I^{\\prime }\\in [I^-,I^+]$ .", "This follows from (ii).", "(2) There exists $\\mu _0>0$ small such that for each $0<\\mu <\\mu _0$ the manifold $\\tilde{\\Lambda }_0$ has a continuation $\\tilde{\\Lambda }_\\mu $ that is a normally hyperbolic invariant manifold for the flow of $H_\\mu $ .", "Inside $\\Lambda _\\mu $ there are finitely many resonant regions, due to (iv).", "Outside the resonant regions, the conditions (i), (iii) allow one to apply the KAM theorem, and obtain KAM tori that are at a distance of order $O(\\mu ^{3/2})$ from one another.", "The resonant regions yield gaps of size $O(\\mu ^{j/2})$ between the KAM tori, where $j$ is the order of the resonance.", "Only the resonances of order 1 and 2 are of interest, as they produce gaps of size $O(\\mu )$ and $O(\\mu ^{1/2})$ respectively.", "Inside each resonant region, there exist primary KAM tori and secondary KAM tori (homotopically trivial relative to $\\Lambda _\\varepsilon $ ).", "They can be chosen to be $O(\\mu ^{3/2})$ from one another.", "(3) The conditions (v) on the Melnikov potential imply that if $\\mu _0$ is chosen small enough then $W^u(\\tilde{\\Lambda }_\\mu )$ intersects $W^s(\\tilde{\\Lambda }_\\mu )$ transversally at an angle $O(\\mu )$ .", "Choose and fix a homoclinic channel $\\tilde{\\Gamma }_\\mu $ as in (REF ) and (REF ), and let $S^{\\tilde{\\Gamma }_\\mu }$ denote the corresponding scattering map for the flow of $H_\\mu $ , as in Definition REF .", "It turns out that the scattering map has the property that there exists a constant $C>0$ such that for each $I^{\\prime }\\in [I^-,I^+]$ there exists $I^{\\prime \\prime }\\in [I^-,I^+]$ with $|I^{\\prime \\prime }-I^{\\prime }|>C\\mu $ and $x^{\\prime },x^{\\prime \\prime }\\in \\tilde{\\Lambda }_\\mu $ with $I(x^{\\prime })=I^{\\prime }, I(x^{\\prime \\prime })=I^{\\prime \\prime }$ , such that $S^{\\tilde{\\Gamma }_\\mu }(x^{\\prime })=x^{\\prime \\prime }$ .", "In the above, one can always choose $I^{\\prime \\prime }>I^{\\prime }$ or $I^{\\prime \\prime }>I^{\\prime }$ , as one wishes.", "That is, there are always points whose $I$ -coordinates in increased or decreased by the scattering map by $O(\\mu )$ .", "(4) Fix a Poincaré surface of section $\\Sigma =\\lbrace (p, q,I,\\phi , t),\\, t=0\\, (\\textrm {mod } 2\\pi ) \\rbrace $ , and let $F_\\mu $ be the first return map to $\\Sigma $ .", "By Theorem REF $\\Lambda _0=\\tilde{\\Lambda }_0\\cap \\Sigma $ is a normally hyperbolic invariant manifold with boundary for $F_\\mu $ .", "The map $F_\\mu $ restricted to $\\Lambda _\\mu $ satisfies a uniform twist condition, due to (iii).", "Each 2-dimensional torus $\\tilde{\\mathcal {T}}_{I^{\\prime }}$ in $\\tilde{\\Lambda }_\\mu $ intersects $\\Sigma $ in a 1-dimensional $\\mathcal {T}_{I^{\\prime }}$ in $ \\Lambda _\\mu $ .", "From Subsection REF it follows that $\\Gamma _\\mu =\\tilde{\\Gamma }_\\mu \\cap \\Sigma $ is a homoclinic channel.", "The scattering map $S^{\\Gamma _\\mu }:U^-_\\varepsilon \\rightarrow U^+_\\varepsilon $ associated to $\\Gamma ^\\mu $ for $F_\\mu $ is related to $S^{\\tilde{\\Gamma }_\\mu }$ by the relation given in Proposition REF .", "(5) Choose a sequence $\\lbrace \\mathcal {T}_{I_j}\\rbrace _{j\\in \\mathbb {Z}}$ of KAM tori (primary or secondary) in $\\Lambda _\\mu $ with the following properties: (a) Each leave $\\mathcal {T}_{I_j}$ is within $O(\\mu )$ , relative to the $I$ -variable, from the next leave $\\mathcal {T}_{I_{j+1}}$ , (b) $S^{\\Gamma _\\mu }$ takes each $\\mathcal {T}_{I_j}$ transversally across $\\mathcal {T}_{I_{j+1}}$ , for all $n$ .", "From Proposition REF , $W^u(\\mathcal {T}_{I_j})$ intersects transversally $W^s(\\mathcal {T}_{I_{j+1}})$ at some point $z_j\\in \\Gamma _\\mu $ .", "Thus, the set $\\lbrace \\mathcal {T}_{I_j}\\rbrace _{j\\in \\mathbb {Z}}$ , together with the transverse homoclinic connections among consecutive tori, forms a transition chain.", "(6) Now we explain how Theorem REF is applied to this problem.", "We have to show that the assumptions of the theorem are met for some sequence of 2-dimensional windows $\\lbrace D_j^-,D_j^+\\rbrace $ in $\\Lambda _\\varepsilon $ .", "Following the arguments in [13], for each $j$ we can construct a pair of windows $\\hat{D}_j^-\\subseteq U_\\varepsilon ^-$ , $\\hat{D}_{j+1}^+\\subseteq U_\\varepsilon ^+$ , with $\\hat{D}_j^-$ in some neighborhood of $\\mathcal {T}_{I_j}$ and $\\hat{D}_{j+1}^+$ in some neighborhood of $\\mathcal {T}_{I_{j+1}}$ such that $\\hat{D}_j^-$ is correctly aligned with $\\hat{D}_{j+1}^+$ under $S^{\\Gamma _\\mu }$ .", "The exit and entry directions of each window are 1-dimensional.", "The window $\\hat{D}_j^-$ is chosen to have its entry set components on some pair of tori $\\mathcal {T}^{\\prime }_{I_j}$ , $\\mathcal {T}^{\\prime \\prime }_{I_j}$ neighboring $\\mathcal {T}_{I_j}$ on both its sides in $\\Lambda _\\varepsilon $ , and the window $\\hat{D}_{j+1}^+$ is chosen to have its exit components set on some pair of tori $\\mathcal {T}^{\\prime }_{I_{j+1}}$ , $\\mathcal {T}^{\\prime \\prime }_{I_{j+1}}$ neighboring $\\mathcal {T}_{I_{j+1}}$ on both its sides in $\\Lambda _\\varepsilon $ .", "Choose some positive integers $n_1^0,n_1^-,n^+_1$ .", "Choose an open neighborhood $\\mathcal {N}(\\Lambda _\\varepsilon )$ of $\\Lambda _\\varepsilon $ in $\\Sigma $ on which the local linearization of the normally hyperbolic invariant manifold from Theorem 1 in [20] applies.", "Then for each $j$ there exists $N_j^->n_1^-$ such that $F^{-N^-_j}(z_{j})$ is contained in $\\mathcal {N}(\\Lambda _\\varepsilon )$ , and there exists $N_{j+1}^+>n_1^+$ such that $F^{N^+_{j+1}}(z_j)$ is contained in $\\mathcal {N}(\\Lambda _\\varepsilon )$ .", "Due to the compactness of $\\Gamma _\\mu ,\\Lambda _\\mu $ , one can choose the numbers $N_j^-, N_{j+1}^+$ uniformly bounded above by some $n_2^-,n^+_2$ , respectively, for all $j$ .", "Now let $D_j^-:=F^{-N^-_j}(\\hat{D}_j^-)$ and $D_{j+1}^+:=F^{N^+_{j+1}}(\\hat{D}_{j+1}^+)$ .", "By construction, these sets satisfy condition (i) of Theorem REF .", "By Definition REF , we have that $D_j^-$ is correctly aligned with $D_{j+1}^+$ under the transition map $S^{\\Gamma _\\mu }_{N_j^-,N_{j+1}^+}$ .", "This ensures condition (ii) of Theorem REF .", "Now using the twist condition, for each $j$ there exists a large enough $N^0_j$ such that $D_{j}^+$ is correctly aligned with $D_{j}^-$ .", "For this we use the fact that the exit set components of $D_{j}^+$ and the entry set components of $D_{j}^-$ lie on $\\mathcal {T}^{\\prime }_{I_j}$ , $\\mathcal {T}^{\\prime \\prime }_{I_j}$ , as in [15].", "This way we fulfill condition (iii) of Theorem REF .", "We can always choose $N_j^0$ as large as we want, and in particular $N_j^0>n^0_1$ .", "The number of iterates $N^0_j$ to achieve this correct alignment depends only on the twist condition on $\\Lambda _\\varepsilon $ , on the sizes of the windows, and on the location of the windows $D_{j}^+, D_{j}^-$ about $\\mathcal {T}_{I_j}$ .", "Since the sizes of the windows are uniformly bounded, we can choose the number $N^0_j$ so that they are all bounded above by some $n^0_2$ , i.e., $N_j^0<n^0_2$ for all $j$ .", "At this point, given positive integers $n_1^0,n_1^-,n^+_1$ , we have obtained positive integers $n_2^0>n_1^0$ , $n_2^->n_1^-$ , $n^+_2>n^+_1$ , and sequences $N^0_j,N^-_j,N^+_j$ as specified by Theorem REF .", "Then, for every $\\varepsilon >0$ , there exists an orbit $\\lbrace F^N(z)\\rbrace _{n\\in \\mathbb {Z}}$ that visits the $\\varepsilon $ -neighborhoods of the windows $\\lbrace D_j^-,D_j^+\\rbrace $ in the prescribed order.", "In particular, we obtain diffusing orbits and symbolic dynamics.", "We should note that the usage of the KAM theorem to construct correctly aligned windows in the above argument is not necessary, as it is done here only to simplify the exposition.", "Instead, one can only use the averaging method as in [13] and construct the windows about the almost-invariant level sets of the averaged energy.", "This alternative based on the averaging method lowers the regularity requirements from condition (i) above.", "As an alternative to the diffusion mechanism described above, we can cross the large gaps corresponding to the resonances of order 1 and 2, by following Birkhoff connecting orbits, as in [15], or Mather connecting orbits, as in [16].", "This mechanism allows one to get rid of the assumption (v.c) above.", "The second model is the spatial circular restricted three-body problem, in the case of the Sun-Earth system.", "We follow [4].", "This problem considers the spatial motion of an infinitesimal mass under the gravitational influence of Sun and Earth, of masses $m_1, m_2$ , respectively, that are assumed to move on circular orbits about their center of mass.", "Let $\\mu =m_2/(m_1+m_2)$ .", "We can translate the equations of motion of the infinitesimal body relative to a co-rotating frame that moves together with $m_1, m_2$ , and then describe the dynamics by a Hamiltonian system of Hamiltonian function $H(x,y,z,p_x,p_y,p_z)=\\frac{1}{2}(p_x^2+p_y^2+p_z^2)+y p_x-xp_y-\\frac{1-\\mu }{r_1}-\\frac{\\mu }{r_2},$ where $r_1^2=(x-\\mu )^2+y^2+z^2$ and $r_2^2=(x-\\mu +1)^2+y^2+z^2$ .", "The system has five equilibrium points; one of them, denoted $L_1$ , lies between the two primaries.", "Its stability is of saddle $\\times $ center $\\times $ center type.", "We focus our attention on the dynamics near $L_1$ .", "At a given energy level, near $L_1$ we can distinguish a family of quasi-periodic orbits that are uniquely defined by their out-of-plane amplitude of the motion.", "The objective is to show that there exits trajectories that move from near one such a quasi-periodic orbit to near another, in the prescribed order, and so to change the out-of-plane amplitude of the motion near $L_1$ with zero cost.", "The mechanism to achieve such trajectories involves starting about a quasi-periodic orbit about $L_1$ of some out-of-plane amplitude, moving the infinitesimal mass around one of the primaries, returning to $L_1$ , and moving about some other quasi-periodic orbit about $L_1$ of a different out-of-plane amplitude, and so on.", "Since this problem is not close to integrable, the methods from perturbation theory do not apply.", "The approach discussed below is semi-numerical.", "(1) Fix an energy level $h$ slightly above the energy level of $L_1$ , so that the dynamical channel between the $m_1$ -region and the $m_2$ -region is open.", "Restrict the study of the dynamics to the 5-dimensional energy manifold $M_h=\\lbrace H=h\\rbrace $ .", "(2) Compute numerically the 3-dimensional normally hyperbolic invariant manifold $\\tilde{\\Lambda }\\subseteq M_h$ by writing the Hamiltonian in Birkhoff normal form in a neighborhood of $L_1$ .", "By taking a high-order truncation of the normal form we obtain a system of action-angle coordinates $(I_1,I_2,\\phi _1,\\phi _2)$ on $\\tilde{\\Lambda }$ , where we can choose $I_1$ as the out-of-plane amplitude of a quasi-periodic orbit in $\\tilde{\\Lambda }$ , and $I_2$ implicitly defined by the restriction to the energy level.", "As the truncated normal form is integrable, the numerically computed $\\tilde{\\Lambda }\\subseteq M_h$ is filled with 2-dimensional invariant tori $\\tilde{\\mathcal {T}}_{I_1}$ parametrized by the out-of-plane amplitude $I_1$ .", "Dynamically, these invariant tori are only almost invariant.", "(3) The Birkhoff normal form is also used to compute the stable and unstable manifolds $W^s(\\tilde{\\Lambda })$ , $W^u(\\tilde{\\Lambda })$ of the normally hyperbolic invariant manifold $\\tilde{\\Lambda }$ .", "One shows numerically that $W^s(\\tilde{\\Lambda })$ , $W^u(\\tilde{\\Lambda })$ intersect transversally along some homoclinic channel $\\tilde{\\Gamma }$ .", "In particular, one can compute the scattering map $S^{\\tilde{\\Gamma }}$ for the flow as in [9].", "We choose some $\\varepsilon >0$ sufficiently small, and choose the $t_u,t_s$ to be the first times when $\\phi ^{-t_u}(\\tilde{\\Gamma })$ , $\\phi ^{-t_s}(\\tilde{\\Gamma })$ , respectively, land in the $\\varepsilon $ -neighborhood of $\\tilde{\\Lambda }$ .", "We then compute the corresponding transition map $S^{\\tilde{\\Gamma }}_{t^u,t^s}$ as in Definition REF .", "(4) We reduce the dimension of the problem by choosing a suitable Poincaré section, e.g.", "$\\Sigma =\\lbrace \\phi _2=0\\rbrace $ .", "Let $F$ denote the first return map to $\\Sigma $ .", "We have that $\\Lambda =\\Sigma \\cap \\tilde{\\Lambda }$ is normally hyperbolic for $F$ , and $\\Gamma =\\Sigma \\cap \\tilde{\\Gamma }$ is a homoclinic channel.", "The intersections of the 2-dimensional tori $\\tilde{\\mathcal {T}}_{I_1}$ with the Poincaré section yields 1-dimensional tori $\\mathcal {T}_{I_1}$ in $\\Lambda $ , each torus corresponding to a fixed value of $I_1$ .", "The inner dynamics induced on the Poincaré section is a monotone twist map.", "Using the relationship between the transition map for a flow and the transition map for the return map from Proposition REF , we compute the transition map $S^{\\Gamma }_{N^u,N^s}$ for $F$ .", "The image of each 1-dimensional torus $\\mathcal {T}_{I^{\\prime }_1}$ under $S^{\\Gamma }_{N^u,N^s}$ is a curve along which the value of $I_1$ variable sometimes goes above and sometimes below the original value $I^{\\prime }_1$ .", "See Figure REF .", "Figure: The effect of the transition map on action level sets.", "(5) To design trajectories that visit some finite collection of $I_1$ -level sets in the prescribed order, we first use the transition map $S^{\\Gamma }_{N^u,N^s}$ to move between level sets.", "We obtain pairs of points $x_j,x_{j+1}$ with $I(x_j)=(I_1)_j$ , $I(x^{\\prime })=(I_1)_{j+1}$ , and $S^{\\Gamma }_{N^u,N^s}(x_{j})=x_{j+1}$ , moving from a level set $(I_1)_{j}$ to a level set $(I_1)_{j+1}$ .", "Using continuation, we construct a window $D^-_j$ about the level set $(I_1)_{j}$ , and a window $D^+_{j+1}$ about the level set $(I_1)_{j+1}$ , such that $D^-_j$ is correctly aligned with $D^+_{j+1}$ under $S^{\\Gamma }_{N^u,N^s}$ .", "Similarly, we construct a window $D^-_{j+1}$ about the level set $(I_1)_{j+1}$ , and a window $D^+_{j+2}$ about a level set $(I_1)_{j+2}$ , such that $D^-_{j+1}$ is correctly aligned with $D^+_{j+2}$ under $S^{\\Gamma }_{N^u,N^s}$ .", "To align $D^+_{j+1}$ with $D^-_{j+1}$ we take some large number of iterates $N^0_j$ such that $D^+_{j+1}$ is correctly aligned with $D^-_{j+1}$ under $F^{N^0_j}$ .", "We obtain the situation described in Theorem REF .", "In particular, we obtain diffusing orbits for which the action variable $I_1$ increases as much as possible.", "We should note that, since the correct alignment of windows is robust, the fact that the tori $\\mathcal {T}_{I_1}$ are only almost-invariant does not matter.", "This makes the method suitable for a computer assisted proof.", "In both examples, the main advantage of using the transition map and the Theorem REF is that, via these tools, one only needs to construct windows of lower dimension (the dimension of the normally hyperbolic invariant manifold) that are correctly aligned either under the transition map, or under some power of the inner map.", "Both these maps are defined on the lower dimensional normally hyperbolic invariant manifold.", "The result from Theorem REF is the existence of a certain trajectory that lives in the full dimensional phase space.", "Received xxxx 20xx; revised xxxx 20xx." ] ]

[ [ "Seeing the light : experimental signatures of emergent electromagnetism\n in a quantum spin ice" ], [ "Abstract The \"spin ice\" state found in the rare earth pyrochlore magnets Ho2Ti2O7 and Dy2Ti2O7 offers a beautiful realisation of classical magnetostatics, complete with magnetic monopole excitations.", "It has been suggested that in \"quantum spin ice\" materials, quantum-mechanical tunnelling between different ice configurations could convert the magnetostatics of spin ice into a quantum spin liquid which realises a fully dynamical, lattice-analogue of quantum electromagnetism.", "Here we explore how such a state might manifest itself in experiment, within the minimal microscopic model of a such a quantum spin ice.", "We develop a lattice field theory for this model, and use this to make explicit predictions for the dynamical structure factor which would be observed in neutron scattering experiments on a quantum spin ice.", "We find that \"pinch points\", seen in quasi-elastic scattering, which are the signal feature of a classical spin ice, fade away as a quantum ice is cooled to its zero-temperature ground state.", "We also make explicit predictions for the ghostly, linearly dispersing magnetic excitations which are the \"photons\" of this emergent electromagnetism.", "The predictions of this field theory are shown to be in quantitative agreement with Quantum Monte Carlo simulations at zero temperature." ], [ "Introduction", "The idea that a strongly interacting quantum magnet might support a spin liquid phase which remains disordered even at zero-temperature has fascinated — and frustrated — physicists ever since the seminal “resonating valence bond” (RVB) paper of Anderson in 1973 [1].", "Such a phase, it was argued, need not support the spin waves found in conventional magnets, but could instead exhibit “spinons” with fractional quantum numbers.", "Forty years later, the search for quantum spin liquids goes on, but with strong grounds for encouragement : a growing number of quantum magnets have been identified which do not order down to the lowest temperatures measured, many of which have low-temperature properties which hint at spinons [2], [3].", "At the same time, the “spin ice” materials Ho$_2$ Ti$_2$ O$_7$ and Dy$_2$ Ti$_2$ O$_7$ have emerged as text-book examples of classical (i.e.", "entropy-driven) spin liquids [4], [5], [6] These highly-frustrated magnetic insulators show algebraic correlations of spins over macroscopic distances [7], [8], [9], [10] and support magnetic monopole excitations which provide classical analogues to the spinons envisaged by Anderson [11], [12], [13], [14], [15], [16], [17].", "Recently, the idea of a “quantum spin ice” has also attracted considerable interest.", "The family of rare earth pyrochlores to which Ho$_2$ Ti$_2$ O$_7$ and Dy$_2$ Ti$_2$ O$_7$ belong includes other systems in which quantum effects play a much more important role [6].", "Perhaps the most widely studied system of this type is Tb$_2$ Ti$_2$ O$_7$ .", "Like the classical spin ices, the magnetism of Tb$_2$ Ti$_2$ O$_7$ is controlled by the competition between strong Ising anisotropy, and dipolar interactions which are ferromagnetic on nearest-neighbour bonds, so it is expected to be an “ice”.", "However, in Tb$_2$ Ti$_2$ O$_7$ , anisotropic exchange interactions also play an important role, and endow the spins with dynamics [18], [19], [20], [21].", "A diffuse, liquid-like structure is observed in neutron scattering for a wide range of temperatures, with no conventional magnetic order observed down to 50mK, despite the fact that the typical scale of interactions between spins is closer to 11K [23], [22].", "Muon spin rotation experiments, meanwhile, suggest that spins continue to fluctuate down to the lowest temperatures [24], and the most recent quasi-elastic neutron scattering experiments find evidence of power-law spin correlations at 50mK [25].", "Taken together, these facts make Tb$_2$ Ti$_2$ O$_7$ a prime example of a three-dimensional, quantum spin liquid.", "The magnetism of Yb$_2$ Ti$_2$ O$_7$ has also proved very interesting, with neutron scattering finding no evidence of order at temperatures above 210mK, and evidence for frustrated, anisotropic exchange interactions favouring significant dynamics within an “ice-like” manifold of states  [26], [27], [28], [29], [30].", "Comparable studies of Pr$_2$ Sn$_2$ O$_7$ suggest that it does not order down to 500mK, but with spins continuing to fluctuate [31], [32], [33].", "There is also reason to believe that other Pr metal oxides, including Pr$_2$ Zr$_2$ O$_7$ , may prove a worthwhile hunting ground for quantum spin liquids [32], [33], [34], [35].", "And, while the dynamics of the “classical” spin ices Ho$_2$ Ti$_2$ O$_7$ and Dy$_2$ Ti$_2$ O$_7$ become very slow at low temperatures, neither system has ever been observed to order, despite the fact that the dipolar interactions present in these systems are expected to favour an ordered state [36], [37].", "All of this begs the question of how the classical spin liquid found in spin ice might evolve into a quantum spin liquid as quantum effects become more important ?", "Figure: (Color online).Spin correlations in a spin ice, as measured by quasi-elastic neutron scattering :(a) Correlations within the classical spin ice configurations,showing the characteristic “pinch point” singularities.", "(b) Correlations in a quantum ice at T=0T=0, showing the suppression of pinch pointsby quantum fluctuations.", "(c) Correlations in a quantum ice at an intermediate temperature T=ca 0 -1 T=ca_0^{-1},showing how pinch points are progressively restored by the thermal excitation ofmagnetic photons.In all cases, results are shown for equal-time structure factors in the(h,h,0)(h,h,0) plane, for a polarised neutron scattering experiment in the spin-flip channelconsidered by Fennell et al .Temperature is measured in units where cc is the speed of light associatedwith magnetic “photon” excitations, a 0 a_0 the lattice constant, and ℏ=k B =1\\hbar = k_B =1.In fact spin ice is just one example of a much broader class of systems which obey the “ice rules”.", "First introduced by Bernal and Fowler in 1933 to describe the correlations of protons in water ice [38], the ice rules have since found application in models of frustrated charge order [39], [40], proton bonded ferroelectrics [41] and dense polymer melts [42].", "All of these systems possess a local “two-in, two-out” constraint, which can most conveniently be written in terms of a zero-divergence condition on a notional magnetic field $\\nabla \\cdot {\\bf B} = 0 \\, .$ In the case of spin ice, ${\\bf B}$ has the physical meaning of the local magnetisation of the system, and we can associate a field ${\\bf B}_i$ with each spin on the lattice.", "For this reason, spin ice offers a beautiful realisation of classical magnetostatics, with local violations of the ice rules entering as point magnetic charges (magnetic monopoles [11], [12], [13], [14], [15], [16], [17]) and spin correlations which exhibit “pinch point” singularities in k-space $\\langle S_{\\mu } (-\\mathbf {k}) S_{\\nu } (\\mathbf {k}) \\rangle _{\\sf classical}\\propto \\left( \\delta _{\\mu \\nu } - \\frac{k_{\\mu } k_{\\nu }}{k^2} \\right) \\, ,$ [Fig.", "REF (a)] corresponding to algebraic (dipolar) correlations in real space [7], [8], [9], [10], [41].", "Since the ice rules can be satisfied by an exponentially large number of proton (spin, charge, polymer...) configurations [43], they explain the residual entropy observed in both water ice [44] and spin ice [45] at low temperatures.", "Given this enormous reservoir of entropy, both spin ice and water ice are natural places to look for a quantum liquid ground state.", "The key ingredient needed to convert a classical ice into a quantum liquid is tunnelling between different ice configurations [Fig.", "REF ].", "This opens the door to a “quantum ice” : a unique, quantum mechanical ground state, formed through the coherent superposition of an exponentially large number of classical ice configurations.", "Such a state could have a vanishing entropy at zero temperature, and so satisfy the third law of thermodynamics, without sacrificing the algebraic correlations and fractional excitations (magnetic monopoles) associated with the degeneracy of the ice states.", "If realised in a spin ice, it would provide a concrete, three–dimensional example of the long-sought quantum spin liquid.", "Precisely this scenario was proposed by Moessner and Sondhi in the context of three-dimensional quantum dimer models [46], by Hermele, Balents and Fisher in a quantum, ice-type model derived from an easy-axis antiferromagnet on a pyrochlore lattice [47], and by Castro-Neto, Pujol and Fradkin in the context of a simplified model of water ice [48].", "All of these models included tunnelling between ice (or dimer) configurations of the type illustrated in Fig.", "REF .", "In a spin ice, the dominant tunnelling process involves flipping loops of spins which point nose-to-tail on an hexagonal plaquette, and the resulting dynamics are described symbollically by $\\mathcal {H}_{\\sf tunnelling} =-g \\sum _{}\\big [|\\!", "\\circlearrowright \\rangle \\langle \\circlearrowleft \\!", "| +|\\!", "\\circlearrowleft \\rangle \\langle \\circlearrowright \\!", "|\\big ]$ where $g$ is the strength of the tunnelling matrix element, and $\\mathcal {H}_{\\sf tunnelling}$ acts on the space of all possible ice (or dimer) configurations.", "Both Moessner and Sondhi [46] and Hermele et al.", "[47] also introduced an additional control parameter $\\mu $ to the Hamiltonian $\\mathcal {H}_{\\mu } &=& \\mathcal {H}_{\\sf tunneling} + \\delta \\mathcal {H}_{\\mu } \\, ,$ where $\\delta \\mathcal {H}_{\\mu } &=&\\mu \\sum _{}\\big [|\\!", "\\circlearrowleft \\rangle \\langle \\circlearrowleft \\!", "| +|\\!", "\\circlearrowright \\rangle \\langle \\circlearrowright \\!", "|\\big ] \\, .$ This makes it possible to fine-tune the model to an exactly soluble Rokhsar-Kivelson (RK) point $g=\\mu $ , where the ground state wave is an equally-weighted sum of all possible ice (dimer) configurations [49].", "The authors then argued, by continuity, that a quantum liquid phase would occur for a finite range of parameters $\\mu \\lesssim 1$ bordering on the RK point [46], [47].", "Figure: (Color online).An illustration of the simplest tunnelling process between differentspin-ice configurations.The ice rules dictate that each tetrahedron within the lattice has twospins which point “in”, and two which point “out”.Where these spins form a closed loop on a hexagonal plaquette— here shaded red — the sense of each spin within the loop can be reversedto give a new configuration which also obeys the ice rules.The most striking feature of this quantum liquid is “light”.", "Attempts to construct models with “artificial light” — gapless photon excitations of an effective, low-energy $U(1)$ gauge field — have a long history [50].", "In recent years, it has been realised that large families of lattice models could, in principle, be described by such theories.", "These include abstract models of “quantum order” [51], [52], Bose-Hubbard models bordering on superfluidity [53], systems of screened dipoles [54], and suitably adapted sigma models [56].", "Reviews of these ideas can be found in papers by Montrunich and Senthil [55] and Wen and Levin [57].", "Figure: (Color online).Ghostly magnetic “photon” excitation as it might appear in an inelastic neutron scatteringexperiment on a quantum spin ice realising a quantum ice ground state.The photon dispersion ω(𝐤)\\omega (\\mathbf {k}) is taken from lattice gauge theorydeveloped in Section  of this paper, convolutedwith a Gaussian representing the finite energy resolution of the instrument.The intensity of scattering vanishes as I∝ω(𝐤)I \\propto \\omega (\\mathbf {k})at low energies.The way in which “light” arises in three-dimensional quantum ice and quantum dimer models is particularly simple.", "The ice-rules constraint Eq.", "(REF ) is most conveniently resolved as $\\mathbf {\\mathcal {B}}({\\bf r}) = \\nabla \\times \\mathbf {\\mathcal {A}}({\\bf r}) \\, .$ The new feature which enters where there is tunnelling between ice configurations is the fluctuation in time of the gauge field $\\mathbf {\\mathcal {A}}({\\bf r})$ .", "In conventional electromagnetism, this gives rise to an electric field $\\mathbf {\\mathcal {E}}({\\bf r}) = -\\frac{\\partial \\mathbf {\\mathcal {A}}({\\bf r}) }{\\partial t} \\, .$ Then, following the heuristic arguments of Moessner and Sondhi [46] — or the more microscopic derivation of Hermele et al.", "[47] — it is reasonable to suppose that a quantum liquid found bordering the RK point ($\\mathcal {H}_{\\mu }$  [Eq.", "(REF )] with $\\mu \\lesssim 1$ ), would be governed by the Maxwell action $\\mathcal {S}_{\\sf Maxwell}= \\frac{1}{8\\pi }\\int dt d^3 \\mathbf {r}\\bigg [ \\mathbf {\\mathcal {E}}({\\bf r})^2 - c^2 \\mathbf {\\mathcal {B}}({\\bf r})^2 \\bigg ]$ Any state described by $\\mathcal {S}_{\\sf Maxwell}$ [Eq.", "(REF )] automatically supports linearly-dispersing transverse excitations of the gauge field $\\mathbf {\\mathcal {A}}({\\bf r})$ — “photons”, with a speed of “light” $c$ .", "On the lattice, such a magnetic photon would have a dispersion $\\omega (\\bf k)$ of the form illustrated in Fig.", "REF .", "Moreover, the fact that the spins now fluctuate in time, as well as space, introduces an additional power of k in energy-integrated (i.e.", "equal time) spin correlations [47], [48], $\\langle S_{\\mu } (-\\mathbf {k}) S_{\\nu } (\\mathbf {k}) \\rangle _{\\sf quantum}\\propto k\\left( \\delta _{\\mu \\nu } - \\frac{k_{\\mu } k_{\\nu }}{k^2} \\right) ,$ which serves to eliminate the pinch points seen in quasi-elastic neutron scattering experiments [Fig.", "REF (b)] [58] .", "More formally, this theory is a compact, frustrated $U(1)$ gauge theory on a diamond lattice, and we will refer to the liquid state it describes as the quantum $U(1)$ liquid below.", "The degree of fine-tuning in these arguments, and the need to introduce additional parameter $\\mu $ [Eq.", "(REF )], might seem to render them of purely academic interest.", "However the idea of a quantum $U(1)$ liquid found strong support in finite-temperature quantum Monte Carlo simulations of an ice-type model of frustrated charge order on the pyrochlore lattice [59].", "Subsequently, it has proved possible to determine the ground state phase diagrams of both the quantum dimer model of Moessner and Sondhi [46], and the quantum ice model of Hermele et al.", "[47], from zero-temperature quantum Monte Carlo simulations [60], [61], [58].", "Both models contains extended regions of a quantum liquid phase, connecting to the RK point.", "In both cases, this quantum liquid has low energy excitations which are described by a lattice analogue of quantum electromagnetism [60], [61], [58].", "Significantly, in the case of the quantum ice model, this quantum liquid phase encompasses the “physical” point of the model $\\mu =0$ , and so does not require any fine-tuning [Fig.", "REF ] [58].", "The theoretical possibility of a three-dimensional spin-liquid state with excitations described by a lattice analogue of quantum electromagnetism is now well-established.", "What remains is to connect these ideas with experiments.", "The purpose of this paper is therefore to set out predictions for the correlations which would be measured in neutron scattering experiments, if such a state were realised in a spin-ice material.", "For concreteness, we work with the minimal lattice model introduced by Hermele et al.", "[47], transcribed to coordinates appropriate for a spin ice.", "More realistic generalisations of this model will be considered elsewhere [62].", "Figure: (Color online).Zero-temperature phase diagram of the model of tunnelling between ice statesℋ μ {\\mathcal {H}}_\\mu [Eq.", "], as determined by quantum Monte Carlosimulation in shannon12.The “quantum ice” point, μ=0\\mu =0, lies deep withina quantum liquid phase with low-energy excitations described bya lattice analogue of quantum electromagnetismThis extends from a “squiggle” ordered phase, found for μ<-0.5g\\mu < -0.5 g,to the exactly-soluble RK point μ=g\\mu = g.(Here gg is the strength of tunnelling between ice states).In Section  of the paper, we develop the mathematical formalism needed to describe the spin correlations and low-energy spin excitations in a spin ice with a quantum $U(1)$ -liquid ground state.", "Using this theory, we make predictions for the photon dispersion $\\omega (\\mathbf {k})$ and dynamical structure factor $S^{\\alpha \\beta }(\\mathbf {k}, \\omega )$ , which would be measured in neutron scattering experiments.", "In Section , we make explicit comparison of the predictions of this theory with zero-temperature Quantum Monte Carlo simulations of the minimal, microscopic model of a quantum spin ice with tunnelling between different ice configurations, ${\\mathcal {H}}_\\mu $ [Eq.", "(REF )].", "We find essentially perfect, quantitative agreement between simulation results and the field theory solved on a finite-size lattice, for a range of parameters $0 \\le \\mu \\le g$ which interpolate from the minimal model of a quantum spin ice ($\\mu = 0$ ), to the classical correlations of the RK point ($\\mu = g$ ).", "This analysis reinforces the conclusions reached in [shannon12] about the existence of a quantum $U(1)$ -liquid in this model, and puts the field-theory description on a quantitative footing.", "In Section  we make predictions for neutron scattering experiments carried out at finite temperature.", "In particular we analyse the way in which the characteristic “pinch point” structure in quasi-elastic scattering is lost as the system is cooled towards its zero-temperature ground state.", "We conclude that the loss of the pinch points coincides with the progressive loss of the Pauling entropy as the system cools into a unique, quantum coherent, liquid ground state.", "Thus the signature features of the ice problem: pinch points and the Pauling entropy die together at low temperatures.", "We also give a brief discussion of the uniform magnetic susceptibility and heat capacity, in the low temperature quantum regime.", "Finally, in Section  we conclude with a discussion of some of the remaining issues relating to experiment.", "As far as possible, each section of the paper is written so as to be self-contained.", "Readers uninterested in the mathematical development of the theory are therefore invited pass directly to Section  and Section , referring to Section  as required." ], [ "From quantum ice to quantum electromagnetism", "At first sight, an assembly of magnetic ions on a lattice does not look like a promising place to search for a gauge theory which perfectly mimics quantum electromagnetism.", "However in the simplest microscopic model for quantum mechanical tunnelling between spin configurations obeying the “two in, two out” ice rule, this is exactly what happens [47], [59], [58].", "In what follows we retrace the steps which lead from a spin ice system to a theory of electromagnetism on a lattice.", "In Section REF we review the relevant microscopic models.", "In Section REF , we show how a lattice gauge theory resembling electromagnetism arises in these problems, recasting the earlier field-theoretical arguments of Hermele et al.", "[47] in terms appropriate for a spin ice.", "In Section REF we explicitly construct the magnetic “photon” excitations of this lattice gauge theory.", "In Section REF we use the mapping between spins and photons to calculate the correlations between spins in a quantum spin liquid described by this lattice gauge theory.", "Throughout this analysis we set $\\hbar =k_B = 1$ , restoring dimensional factors only where we quote a result for the speed of light." ], [ "Spins on a pyrochlore lattice", "The materials which we will seek to describe have magnetic ions which a) have a crystal-field ground state which is a doublet, and b) occupy the sites of the pyrochlore lattice shown in Fig.", "REF .", "In the case of the spin ices Ho$_2$ Ti$_2$ O$_7$ and Dy$_2$ Ti$_2$ O$_7$ , this doublet has Ising character (rare-earth moments point into, or out of, the tetrahedra which make up the lattice), and the dominant interactions between these Ising spins are dipolar [63].", "However, since these dipolar interactions are effectively self-screened, the correlations present in spin-ice are extremely well described by models with only nearest-neighbour interactions between spins [4], [64], [36], [37], [65].", "This approximation gains further justification in “quantum spin ice” materials such as Yb$_2$ Ti$_2$ O$_7$ , where magnetic moments are smaller than for Ho$_2$ Ti$_2$ O$_7$ and Dy$_2$ Ti$_2$ O$_7$ , and exchange interactions play a much larger role.", "As a starting point, we can therefore consider the Hamiltonian for a (pseudo) spin-1/2 degree of freedom on a pyrochlore lattice, with the most general nearest-neighbour exchange interactions allowed by symmetry [66] $\\mathcal {H}_{\\sf S=1/2}& = & \\sum _{\\langle ij\\rangle } \\Big \\lbrace J_{zz} \\mathsf {S}_i^z \\mathsf {S}_j^z - J_{\\pm }(\\mathsf {S}_i^+ \\mathsf {S}_j^- + \\mathsf {S}_i^- \\mathsf {S}_j^+)\\nonumber \\\\ &&+ J_{\\pm \\pm } \\left[\\gamma _{ij} \\mathsf {S}_i^+ \\mathsf {S}_j^+ + \\gamma _{ij}^*\\mathsf {S}_i^-\\mathsf {S}_j^-\\right]\\nonumber \\\\ &&+ J_{z\\pm }\\left[ \\mathsf {S}_i^z (\\zeta _{ij} \\mathsf {S}_j^+ + \\zeta ^*_{ij} \\mathsf {S}_j^-) +{i\\leftrightarrow j}\\right]\\Big \\rbrace $ Here we have followed the notation of Ross et al.", "[27], in which the $\\mathsf {S}_i^z$ is aligned with the local trigonal axes of the pyrochlore lattice on each site $i$ , and $\\gamma _{ij}$ and $\\zeta _{ij} $ are $4 \\times 4$ complex unimodular matrices encoding the rotations between these local coordinate frames.", "In the “quantum spin ice” Yb$_2$ Ti$_2$ O$_7$ , where the ground state doublet of Yb has XY-character [67], [68], [69], Eq.", "(REF ) gives a good account of diffuse structure observed in neutron scattering experiments provided that the exchange interactions $J_{\\pm }$ , $J_{z\\pm }$ and $J_{\\pm \\pm }$ are taken into account [26], [29].", "It also gives an excellent description of spin wave excitations about the saturated state of Yb$_2$ Ti$_2$ O$_7$ in applied magnetic field, with parameters $J_{zz} = 0.17 \\pm 0.04$  meV, $J_{\\pm } = 0.05 \\pm 0.01$  meV, $J_{z\\pm } =- 0.14 \\pm 0.01$  meV and $J_{\\pm \\pm } = 0.05 \\pm 0.01$  meV obtained from fits to data [27].", "The phase diagram associated with $\\mathcal {H}_{\\sf S=1/2}$ [Eq.", "(REF )] is explored in [ross11-PRX,savary12,lee-arXiv].", "We can further simplify the problem by setting $J_{\\pm \\pm }=0$ , and focusing on the limit $J_{zz} >> J_{\\pm }, J_{z\\pm } > 0$ .", "In this limit, the role of $J_{zz}$ is to enforce the “ice rules” constraint, while $J_{\\pm }$ generates dynamics, and $J_{z\\pm }$ lifts the degeneracy of ice-rule obeying states.", "Performing degenerate perturbation theory in the basis of (spin) ice configurations, and dropping terms which lead only to a constant energy shift, leads to the effective Hamiltonian [27] $\\mathcal {H}_{\\sf eff} = \\mathcal {H}_{\\sf tunnelling} + \\mathcal {H}_{\\sf J_3}$ with $\\mathcal {H}_{\\sf tunnelling}= - g \\sum _{}\\left[\\mathsf {S}^{+}_{1} \\mathsf {S}^{-}_{2}\\mathsf {S}^{+}_{3}\\mathsf {S}^{-}_{4}\\mathsf {S}^{+}_{5} \\mathsf {S}^{-}_{6}+ h.c. \\right]$ where $\\sum _{} $ runs over all hexagonal plaquettes in the pyorchlore lattice [cf.", "Fig REF ] with $g = \\frac{12 J_{\\pm }^3}{J_{zz}^2} \\, ,$ and $\\mathcal {H}_{\\sf J_3}= - J_3 \\sum _{\\langle ij \\rangle _3} \\mathsf {S}_i^z \\mathsf {S}_j^z$ where $\\sum _{\\langle ij \\rangle _3}$ runs over third-neighbours bonds (parallel to the nearest-neighbour bonds), with $J_3 = \\frac{3 J_{z\\pm }^2}{J_{zz}} > 0$ We note that, by construction, the Hamiltonian Eq.", "(REF ) acts only on spin configurations satisfying the ice rules.", "This implies that, in performing the degenerate perturbation theory, virtual excitations of magnetic monopoles have been projected out of the problem.", "This approximation will have little effect on the conclusions drawn in this paper, and could in principle be relaxed.", "It is also important to note that these spin ice configurations may possess a non-zero net magnetisation ${\\bf M}$ .", "The tunnelling term $\\mathcal {H}_{\\sf tunnelling}$ [Eq.", "(REF )] generates dynamics by performing a cyclic exchange of spins on a hexagonal plaquette [cf.", "Fig.", "REF ].", "This tunnelling process can be written symbolically as acting on a closed loop of spins [cf.", "Eq.", "(REF )].", "Under these dynamics, the total magnetisation ${\\bf M}$ is a conserved quantity.", "We make the final simplification of neglecting $\\mathcal {H}_{\\sf J_3}$ [Eq.", "(REF )] and focusing exclusively on the spin-liquid favoured by the tunnelling term $\\mathcal {H}_{\\sf tunnelling}$ — Eq.", "(REF ) or, symbolically, Eq.", "(REF ).", "The neglected term $\\mathcal {H}_{\\sf J_3}$ favours the six ice states with the maximum possible magnetisation per site ${\\bf m} = (\\pm 1/\\sqrt{3},0,0) \\times S$ , etc., where $S$ is the moment of the magnetic ion.", "We have confirmed through zero-temperature quantum Monte Carlo simulation of $\\mathcal {H}_{\\sf eff}$ [Eq.", "(REF )], that the system remains in quantum $U(1)$ liquid ground state up to a value of $J_3 \\approx 0.27\\, g$ , at which point it undergoes first-order transition into this ordered, ferromagnetic state.", "These results will be discussed elsewhere [62].", "We note that a gauge mean-field theory for $\\mathcal {H}_{\\sf S=1/2} $ [Eq.", "(REF )] predicts an intermediate “Coulombic ferromagnet” phase, in which the quantum $U(1)$ liquid spontaneously acquires a finite magnetisation for any finite $J_{z\\pm }$  [27], [70].", "This does not appear to be a ground state of the effective model $\\mathcal {H}_{\\sf eff}$ [Eq.", "(REF )].", "Following Hermele et al.", "[47], it is useful to augment the minimal model $\\mathcal {H}_{\\sf tunnelling}$ with an additional, artificial, interaction term $\\delta {\\mathcal {H}}_\\mu $ [Eq.", "(REF )].", "This renders the model exactly soluble for $\\mu = g$ .", "Thus the most general microscopic model we consider in this paper can be written symbolically as $\\mathcal {H}_{\\mu }&=&-g \\sum _{}\\big [|\\!", "\\circlearrowright \\rangle \\langle \\circlearrowleft \\!", "| +|\\!", "\\circlearrowleft \\rangle \\langle \\circlearrowright \\!", "|\\big ]\\nonumber \\\\ && \\quad +\\mu \\sum _{}\\big [|\\!", "\\circlearrowleft \\rangle \\langle \\circlearrowleft \\!", "| +|\\!", "\\circlearrowright \\rangle \\langle \\circlearrowright \\!", "|\\big ].$ where $\\mathcal {H}_{\\mu }$ acts on the space of all possible (spin) ice configurations.", "This Hamiltonian is known to support a quantum $U(1)$ liquid ground state for $-0.5g < \\mu \\le g$  [58].", "It is important to note that this effective description of tunnelling between ice configurations might equally have been derived for the model of hardcore bosons on the pyrochlore lattice considered by Banerjee et al.", "[59] $\\mathcal {H}_{\\sf charge-ice}&=& - t \\sum _{\\langle ij \\rangle }\\left( b_i^{\\dagger } b^{\\phantom{\\dagger }}_j+ b_j^{\\dagger } b^{\\phantom{\\dagger }}_i \\right)\\nonumber \\\\&& \\quad + V \\sum _{\\langle ij \\rangle } \\left( n_i - \\frac{1}{2} \\right) \\left( n_j - \\frac{1}{2} \\right)$ with $V \\gg t$ .", "At $1/2$ -filling [$\\langle n \\rangle \\equiv 1/2$ ], $V$ selects charge configurations with exactly two bosons in each tetrahedron of the lattice, and $\\mathcal {H}_{\\sf charge-ice}$ is exactly equivalent to the pseudospin-$1/2$ model Eq.", "(REF ), in the case where $J_{z\\pm } = J_{\\pm \\pm } \\equiv 0$.", "The leading tunnelling matrix element between different (charge) ice configurations is then $g = \\frac{12 t^3}{V^2}$ We will return to this model below in the context of predictions for experiment and simulations performed at finite temperature [59].", "The manifold of ice configurations on the pyrochlore lattice is equivalent to the set of possible close-packed loop coverings of the diamond lattice [71].", "Exactly parallel arguments, leading to a formally identical Hamiltonian, can also be constructed for the closely related quantum dimer model on the diamond lattice [46], [72].", "This model also exhibits a quantum $U(1)$ liquid ground states for a smaller — but none the less finite — range of parameters $0.75 g < \\mu \\le g$  [sikora09,sikora11]." ], [ "Electromagnetism on a diamond lattice", "The mappings described in Section REF permit us to reduce complicated interactions between magnetic ions to a problem of tunnelling between spin configurations obeying the “ice rules” [cf.", "Fig.", "REF ].", "If we think of these spins as field lines of a fictitious magnetic field ${\\bf B}$ , these rules can conveniently be written as $\\nabla \\cdot {\\bf B} = 0$ This naturally suggests an analogy with magnetostatics, with magnetic field lines constrained to lie on the bonds of a diamond lattice [cf.", "Fig.", "REF (a)].", "And in the presence of tunnelling between ice configurations, this analogy can be extended to a fully dynamical quantum electromagnetism.", "Here we review the mapping from an ice with tunnelling, to a compact, $U(1)$ lattice gauge theory, before moving on to an analysis of its “photon” excitations [Section REF ] and spin correlations [Section REF ] .", "In so doing we follow closely the arguments of Hermele et al [47], but recast the discussion in terms of the magnetic fields ${\\bf B}$ usually associated with the spins of a spin ice.", "We begin by transcribing the spin variables of ${\\mathcal {H}}_{\\sf tunnelling}$ [Eq.", "(REF )] in terms of a quantum rotor variable $\\theta _i$ , and its conjugate number operator $n_i$ ${\\mathsf {S}}^z_i &=& \\left( n_i-\\frac{1}{2} \\right) \\\\{\\mathsf {S}}^{+}_i &=& \\sqrt{n_i} \\exp {[i \\theta _i]} \\sqrt{1-n_i} \\\\{\\mathsf {S}}^{-}_i &=& \\sqrt{1-n_i} \\exp {[-i \\theta _i]} \\sqrt{n_i}$ where $[\\theta _i, n_j]=i \\delta _{ij} .$ The number operator $n_i$ could equally be associated with the density of (hard-core) bosons in a charge ice, and in order to remain in the physical subspace where $n_i = 0$ or 1, we add the term $\\mathcal {H}_{\\sf U}=\\frac{U}{2} \\sum _i (n_i - 1/2)^2$ to the Hamiltonian, subsequently taking the limit $U \\rightarrow \\infty $ .", "With this restriction in place, the Hamiltonian becomes $\\mathcal {H}_{\\sf rotor}&=& \\frac{U}{2} \\sum _i (n_i - 1/2)^2 \\nonumber \\\\&-& 2g \\sum _{}\\cos {\\left( \\theta _1 - \\theta _2 + \\theta _3- \\theta _4 + \\theta _5 - \\theta _6 \\right)} \\nonumber \\\\$ It is from this rotor form of the Hamiltonian that we will make the passage to a $U(1)$ gauge theory on the diamond lattice.", "Figure: (Color online).The different fields used in Section  to construct lattice gauge theoryof the spin liquid state, and the different lattices on which they are defined.", "(a) The “magnetic” field ℬ 𝐫𝐫 ' \\mathcal {B}_{\\mathbf {r}\\mathbf {r}^{\\prime }} [Eq.", "()], and its conjugatefield 𝒢 𝐫𝐫 ' \\mathcal {G}_{\\mathbf {r}\\mathbf {r}^{\\prime }} [Eq.", "()] are defined on the linksof the diamond lattice, shown here in red.Each link of this diamond lattice corresponds to a site of the original pyrochlore lattice,and ℬ 𝐫𝐫 ' \\mathcal {B}_{\\mathbf {r}\\mathbf {r}^{\\prime }} encodes the orientation of the spin on this site.", "(b) The compact U(1)U(1) gauge field 𝒜 𝐬𝐬 ' \\mathcal {A}_{\\mathbf {s}\\mathbf {s}^{\\prime }} and the conjugate“electric” field ℰ 𝐬𝐬 ' \\mathcal {E}_{\\mathbf {s}\\mathbf {s}^{\\prime }} are defined on the links of a second,dual, diamond lattice, shown here in blue.The midpoints of these bonds also form a second, dual, pyrochlore lattice, corresponding tothe centres of hexagonal plaquettes in the original pyrochlore lattice.", "(c) An illustration of taking the lattice curl on the hexagonal plaquettesof the diamond lattice.The resulting vector lives on the links of the dual, diamond lattice.The site $i$ of the pyrochlore lattice can be thought of as the midpoint of the bond $\\mathbf {r} \\rightarrow \\mathbf {r}^{\\prime }$ of a diamond lattice [cf.", "Fig.", "REF ].", "Since this diamond lattice is bipartite, it is possible to define directed variables on these bonds $\\mathcal {B}_{\\mathbf {r} \\mathbf {r}^{\\prime }}= - \\mathcal {B}_{\\mathbf {r}^{\\prime } \\mathbf {r}} \\quad \\mathcal {G}_{\\mathbf {r} \\mathbf {r}^{\\prime }}= - \\mathcal {G}_{\\mathbf {r}^{\\prime } \\mathbf {r}}$ through the mapping $\\mathcal {B}_{\\mathbf {r} \\mathbf {r}^{\\prime }} &=& \\pm \\left( \\hat{n}_i-\\frac{1}{2} \\right) \\\\\\mathcal {G}_{\\mathbf {r} \\mathbf {r}^{\\prime }} &=& \\pm \\theta _i$ where the sign is taken to be positive if $\\mathbf {r}$ belongs to the $A$ -sublattice, and negative if $\\mathbf {r}$ belongs to the $B$ -sublattice.", "Taking this convention into account, we are left with a pair of canonically conjugate variables $\\left[\\mathcal {G}_{\\mathbf {r} \\mathbf {r}^{\\prime }}, \\mathcal {B}_{\\mathbf {r}^{\\prime \\prime } \\mathbf {r}^{\\prime \\prime \\prime }}\\right]= i \\left(\\delta _{\\mathbf {r} \\mathbf {r}^{\\prime \\prime }} \\delta _{\\mathbf {r}^{\\prime } \\mathbf {r}^{\\prime \\prime \\prime }}-\\delta _{\\mathbf {r} \\mathbf {r}^{\\prime \\prime \\prime }} \\delta _{\\mathbf {r}^{\\prime } \\mathbf {r}^{\\prime \\prime }} \\right).$ The field, $\\mathcal {B}_{\\mathbf {r} \\mathbf {r}^{\\prime }}$ will take on the role of a magnetic field in our lattice field theory.", "However in order to recreate “electromagnetism” we need also to discover an analogue to the electric field.", "The missing field, $\\mathcal {E}_{\\mathbf {s} \\mathbf {s}^{\\prime }}$ , inhabits the bonds $\\mathbf {s} \\rightarrow \\mathbf {s}^{\\prime }$ of a second diamond lattice, interpenetrating the first [Fig.", "REF (b)].", "It is defined through a lattice curl $\\mathcal {E}_{\\mathbf {s} \\mathbf {s}^{\\prime }}= (\\nabla _{\\scriptsize } \\times \\mathcal {G})_{\\mathbf {s} \\mathbf {s}^{\\prime }}= \\sum _{\\circlearrowleft } \\mathcal {G}_{\\mathbf {r} \\mathbf {r}^{\\prime }}$ where the sum $\\sum _{\\circlearrowleft }$ is taken with anticlockwise sense around the hexagonal plaquette of pyrochlore lattice sites encircling the bond $\\mathbf {s} \\rightarrow \\mathbf {s}^{\\prime }$ .", "It follows that $\\mathcal {E}_{\\mathbf {s} \\mathbf {s}^{\\prime }}$ is also a directed variable $\\mathcal {E}_{\\mathbf {s} \\mathbf {s}^{\\prime }}=-\\mathcal {E}_{\\mathbf {s}^{\\prime } \\mathbf {s}}$ We are now in a position to transcribe $\\mathcal {H}_{\\sf rotor}$ completely in terms of “electromagnetic” fields $\\mathcal {H}_{\\sf rotor}&=& \\frac{U}{2} \\sum _{\\langle \\mathbf {r} \\mathbf {r}^{\\prime } \\rangle }\\mathcal {B}_{\\mathbf {r} \\mathbf {r}^{\\prime }}^2- 2g \\sum _{\\langle \\mathbf {s} \\mathbf {s}^{\\prime } \\rangle }\\cos {(\\mathcal {E}_{\\mathbf {s} \\mathbf {s}^{\\prime }})}$ where the sum $\\sum _{\\langle \\mathbf {r} \\mathbf {r}^{\\prime } \\rangle }$ runs over all bonds of the original diamond lattice, while the sum $\\sum _{\\langle \\mathbf {s} \\mathbf {s}^{\\prime } \\rangle } $ runs over all bonds of the second, dual diamond lattice.", "The fact that the Hamiltonian is invariant under the transformation $\\mathcal {E}_{\\mathbf {s} \\mathbf {s}^{\\prime }} \\rightarrow \\mathcal {E}_{\\mathbf {s} \\mathbf {s}^{\\prime }} + 2\\pi $ makes it evident that this theory is compact.", "It is also important to note that each of the components of the total magnetic field $(\\mathcal {B}_x, \\mathcal {B}_y,\\mathcal {B}_z)&=& \\sum _{\\langle \\mathbf {r} \\mathbf {r}^{\\prime } \\rangle }\\mathcal {B}_{\\mathbf {r} \\mathbf {r}^{\\prime }}\\hat{\\mathbf {e}}_{\\mathbf {r} \\mathbf {r}^{\\prime }}$ where $\\hat{\\mathbf {e}}_{\\mathbf {r} \\mathbf {r}^{\\prime }}$ is a unit vector directed from $\\mathbf {r}$ to $\\mathbf {r}^{\\prime }$ , is a conserved quantity under the dynamics of $\\mathcal {H}_{\\sf rotor}$  [Eq.", "(REF )].", "More generally, reversing the sign of $\\mathcal {B}_{\\mathbf {r} \\mathbf {r}^{\\prime }}$ on a closed loop of spins will tunnel one ice configuration to another, without changing the total magnetisation of the system.", "In deriving Eq.", "(REF ), we have assumed that the ice rules hold, i.e.", "$(\\nabla \\cdot \\mathcal {B})_\\mathbf {r}&=& \\sum _{\\langle \\mathbf {r}^{\\prime } \\rangle } \\mathcal {B}_\\mathbf {r r^{\\prime }} = 0$ where the sum $\\sum _{\\langle \\mathbf {r}^{\\prime } \\rangle }$ runs over all sites neighbouring $\\mathbf {r}$ .", "This condition is automatically satisfied if we write $\\mathcal {B}_{\\mathbf {r} \\mathbf {r^{\\prime }}}$ as the lattice curl of a gauge field $\\mathcal {A}_{\\mathbf {s} \\mathbf {s^{\\prime }}}$ .", "However we must also respect the requirement that the field $\\mathcal {B}_{\\mathbf {r} \\mathbf {r}^{\\prime }}$ take on half-integer values [cf.", "Eq.", "(REF )].", "This can be accomplished by introducing a static background field $\\mathcal {B}_{\\mathbf {r} \\mathbf {r^{\\prime }}}^0$ , taken from any spin configuration which satisfies the ice rules, and writing $\\left( \\mathcal {B}_{\\mathbf {r} \\mathbf {r^{\\prime }}} - \\mathcal {B}_{\\mathbf {r} \\mathbf {r^{\\prime }}}^0 \\right)&=& \\left( \\nabla _{\\scriptsize } \\times \\mathcal {A} \\right)_{\\mathbf {r} \\mathbf {r}^{\\prime }}$ to give $\\mathcal {H}_{\\sf rotor}&=& \\frac{U}{2} \\sum _{\\langle \\mathbf {r} \\mathbf {r}^{\\prime } \\rangle }\\left[ \\left(\\nabla _{\\scriptsize } \\times \\mathcal {A}\\right)_{\\mathbf {r} \\mathbf {r}^{\\prime }}+ \\mathcal {B}_{\\mathbf {r} \\mathbf {r^{\\prime }}}^0 \\right]^2 \\nonumber \\\\&& \\quad - 2g \\sum _{\\langle \\mathbf {s} \\mathbf {s}^{\\prime } \\rangle }\\cos {\\left( \\mathcal {E}_{\\mathbf {s} \\mathbf {s}^{\\prime }} \\right)}$ The fields $\\mathcal {E}_{\\mathbf {s} \\mathbf {s}^{\\prime }}$ and $\\mathcal {A}_{\\mathbf {s} \\mathbf {s}^{\\prime }}$ are canonically conjugate $\\left[ \\mathcal {E}_{\\mathbf {s} \\mathbf {s}^{\\prime }}, \\mathcal {A}_{\\mathbf {s}^{\\prime \\prime } \\mathbf {s}^{\\prime \\prime \\prime }} \\right]= i \\left( \\delta _{\\mathbf {s} \\mathbf {s}^{\\prime \\prime }} \\delta _{\\mathbf {s}^{\\prime } \\mathbf {s}^{\\prime \\prime \\prime }}- \\delta _{\\mathbf {s} \\mathbf {s}^{\\prime \\prime \\prime }} \\delta _{\\mathbf {s}^{\\prime } \\mathbf {s}^{\\prime \\prime }} \\right)$ Moreover, the theory has a local gauge symmetry since one can make the transformation $\\mathcal {A}_{\\mathbf {s} \\mathbf {s}^{\\prime }}\\rightarrow \\mathcal {A}_{\\mathbf {s} \\mathbf {s}^{\\prime }}+ \\lambda _{\\mathbf {s}} - \\lambda _{\\mathbf {s}^{\\prime }}$ on any bond without changing the value of $(\\nabla _{\\scriptsize } \\times \\mathcal {A})_{\\mathbf {r} \\mathbf {r}^{\\prime }}$ — each value of $\\lambda _{\\mathbf {s}}$ occurs twice, with opposite signs.", "The situation now bears more than a passing resemblance to quantum electromagnetism.", "At this point a subtlety enters the problem.", "In passing from Eq.", "(REF ) to Eq.", "(REF ), we have performed a series of changes of variable without making any new approximations.", "However it still remains to take the limit $U \\rightarrow \\infty $ .", "If the “magnetic” field $\\mathcal {B}_{\\mathbf {r}\\mathbf {r}^{\\prime }}$ were an integer variable, it could be eliminated from the problem by setting $\\mathcal {B}_{\\mathbf {r}\\mathbf {r}^{\\prime }} = 0$ on all bonds.", "This would be energetically favourable at large $U/g$ , and would imply a phase transition from a spin liquid phase at small $U/g$ , into a phase in which spinon excitations (magnetic monopoles) were confined at large $U/g$ [cf.", "guth80].", "However the fact that $\\mathcal {B}_{\\mathbf {r}\\mathbf {r}^{\\prime }}$ takes on half-integer values “frustrates” the lattice theory, and makes it possible for a spin liquid phase to survive in the limit $U \\rightarrow \\infty $ .", "Keeping this in mind, we now follow Hermele et al.", "[47] in assuming that an average over fast fluctuations of the gauge field a) softens the restriction that $\\mathcal {B}_{\\mathbf {r}\\mathbf {r}^{\\prime }}$ take on half-integer values and, b) restricts $\\mathcal {E}_{\\mathbf {s}\\mathbf {s}^{\\prime }}$ to small values.", "Provided that both of these assumptions hold true, we can drop the reference field $\\mathcal {B}_{\\mathbf {r}\\mathbf {r}^{\\prime }}^0$ and expand the cosine in Eq.", "(REF ), to obtain $\\mathcal {H}_{\\sf U(1)}= \\frac{\\mathcal {U}}{2} \\sum _{\\langle \\mathbf {r} \\mathbf {r}^{\\prime } \\rangle }\\left[ \\left( \\nabla _{\\scriptsize } \\times \\mathcal {A} \\right)_{\\mathbf {r} \\mathbf {r}^{\\prime }} \\right]^2+ \\frac{\\mathcal {K}}{2} \\sum _{\\langle \\mathbf {s}\\mathbf {s}^{\\prime } \\rangle } \\mathcal {E}_{\\mathbf {s} \\mathbf {s}^{\\prime }}^2$ where both the normalisation of the field $\\mathcal {B}_{\\mathbf {r}\\mathbf {r}^{\\prime }}$ , and the parameters of $\\mathcal {H}_{U(1)}$ may be renormalized from their bare values $|\\mathcal {B}_{\\mathbf {r}\\mathbf {r}^{\\prime }}| \\sim 1/2$ , $\\mathcal {U} \\sim U$ , $\\mathcal {K} \\sim g$ .", "This, finally, is the Hamiltonian for non-compact quantum electromagnetism on a diamond lattice.", "At first sight, the final step of this derivation might seem to involve an uncomfortably large leap of faith [10].", "However this will be justified a posteriori in Section REF by the excellent, quantitative, agreement of the predictions of $\\mathcal {H}_{U(1)}$ [Eq.", "(REF )] with quantum Monte Carlo simulation of the microscopic model ${\\mathcal {H}}_\\mu $ [Eq.", "(REF )].", "In order to extend this comparison to finite values of the control parameter $\\mu $ , we will augment $\\mathcal {H}_{U(1)}$ with a term $\\delta \\mathcal {H}_{\\sf U(1)}= \\frac{\\mathcal {W}}{2} \\sum _{\\langle \\mathbf {s} \\mathbf {s}^{\\prime } \\rangle }\\left[ \\left( \\nabla _{\\scriptsize } \\times \\left( \\nabla _{\\scriptsize }\\times \\mathcal {A} \\right) \\right)_{\\mathbf {s} \\mathbf {s}^{\\prime }} \\right]^2$ which mimics the effect of the “RK” potential [Eq.", "(REF )].", "Since this term is permitted by the gauge symmetry, in principle it might also be generated dynamically by an average over fast fluctuations of $\\mathcal {A}_{\\mathbf {s}\\mathbf {s}^{\\prime }}$ ." ], [ "Constructing the photon", "The lattice gauge theory described in Section REF supports three types of excitation : magnetic charges (point sources of $\\mathcal {B}$ ) and electric charges (point sources of $\\mathcal {E}$ ), together with photons (transverse excitations of $\\mathcal {A}$ ) which mediate Coulomb interactions between these emergent charges [moessner03,hermele04].", "The magnetic charges are the magnetic monopoles of the classical theory [12], now quantised and endowed with dynamics [40], [75], [74].", "They correspond to the “spinon” excitations of the spin liquid.", "Since they involve spin configurations lying outside the ice manifold, they have an energy gap $2 \\Delta _{\\mathcal {B}} \\sim J^{zz}$ [cf.", "Eq (REF )].", "The electric charges are gapped, topological excitations which can be constructed as a wave packet of ice configurations with suitably chosen phases [46], [47].", "These also have an energy gap $\\Delta _{\\mathcal {E}} \\sim {\\mathcal {K}} \\sim g = 12 J_{\\pm }^3/J_{zz}^2$ [cf.", "Eq (REF )].", "However the energy of the photons vanishes linearly at small wave vector $\\omega (\\mathbf {k} \\rightarrow 0) = c |\\mathbf {k}|.$ and being gapless, the photons will control the low energy and low temperature properties of the system.", "We therefore concentrate on exploring the consequences of the photons in this paper, leaving other excitations for future work.", "In what follows, we will explicitly construct a photon basis for the lattice gauge theory developed in Section REF , with a view to calculating the spin-spin correlation functions of the original model of a quantum spin ice.", "We take as a starting point $\\mathcal {H}^\\prime _{\\sf U(1)}&=& \\frac{\\mathcal {U}}{2} \\sum _{\\mathbf {r} \\in A, n}\\left[ \\left(\\nabla _{\\scriptsize } \\times \\mathcal {A}\\right)_{(\\mathbf {r}, n)} \\right]^2\\nonumber \\\\&&+ \\frac{1}{2\\mathcal {K}} \\sum _{\\mathbf {s} \\in A^{\\prime }, m}\\left[ \\frac{\\partial \\mathcal {A}_{(\\mathbf {s}, m)}}{\\partial t} \\right]^2\\nonumber \\\\&& + \\frac{\\mathcal {W}}{2} \\sum _{\\mathbf {s} \\in A^{\\prime }, m}\\left[ \\left( \\nabla _{\\scriptsize } \\times \\nabla _{\\scriptsize } \\times \\mathcal {A} \\right)_{(\\mathbf {s}, m)} \\right]^2$ where we have used the fact that, in the absence of electric charges $\\mathcal {E}_{(\\mathbf {s},m)}&=& - \\frac{1}{\\mathcal {K}} \\frac{\\partial \\mathcal {A}_{(\\mathbf {s}, m)}}{\\partial t}$ To avoid double counting of bonds, the sums over diamond lattice sites $\\lbrace \\mathbf {r}\\rbrace $ and $\\lbrace \\mathbf {s}\\rbrace $ are restricted to a single sublattice, with bonds labelled $(\\mathbf {r}, n) = (\\mathbf {r}, \\mathbf {r}+\\mathbf {e}_n)\\quad , \\quad (\\mathbf {s}, m)=(\\mathbf {s}, \\mathbf {s}+\\mathbf {e}_m)$ where $\\mathbf {e}_0 &=& \\frac{a_0}{4} \\left( 1, 1, 1 \\right) \\nonumber \\\\\\mathbf {e}_1 &= & \\frac{a_0}{4} \\left( 1, -1, -1 \\right) \\nonumber \\\\\\mathbf {e}_2 &= & \\frac{a_0}{4} \\left( -1, 1, -1 \\right) \\nonumber \\\\\\mathbf {e}_3 &= & \\frac{a_0}{4} \\left(-1, -1, 1 \\right)$ and $a_0$ is the linear dimension of the cubic unit cell of the lattice, shown in Fig.", "REF .", "We proceed to quantise $\\mathcal {A}_{\\mathbf {s} m}$ by analogy with conventional electromagnetism, introducing a Bose operator $\\big [ a^{\\phantom{\\dagger }}_{\\lambda }, a^{\\dagger }_{\\lambda ^{\\prime }} \\big ]= \\delta _{\\lambda \\lambda ^{\\prime }}$ where the four sites of the tetrahedron in the primitive unit cell of the pyrochlore lattice translate into four bands $\\lambda =1 \\ldots 4$ .", "We write $\\mathcal {A}_{(\\mathbf {s}, m)}&=&\\sqrt{\\frac{2}{N}} \\sum _{\\mathbf {k}} \\sum _{\\lambda =1}^{4}\\sqrt{\\frac{\\mathcal {K}}{\\omega _{\\lambda }(\\mathbf {k})}} \\nonumber \\\\&& \\times \\left( \\exp \\left[-i \\mathbf {k} \\cdot (\\mathbf {s}+\\mathbf {e}_m/2)\\right]\\eta _{m \\lambda }(\\mathbf {k}) a^{\\phantom{\\dagger }}_{\\lambda }(\\mathbf {k}) \\right.", "\\nonumber \\\\&& \\quad + \\left.", "\\exp \\left[i \\mathbf {k} \\cdot (\\mathbf {s}+\\mathbf {e}_m/2)\\right]\\eta ^{\\ast }_{\\lambda m}(\\mathbf {k}) a^{\\dagger }_{\\lambda }(\\mathbf {k}) \\right) \\nonumber \\\\$ where the sum $\\sum _{\\lambda =1}^{4}$ runs over all four branches of photons and ${\\eta }({\\mathbf {k}})$ is a unitary, $4\\times 4$ matrix whose columns, $\\eta _{\\lambda }({\\mathbf {k}})$ , play the same role as the polarisation vector in conventional electromagnetism.", "By obvious extension $\\mathcal {E}_{(\\mathbf {s}, m)}&=& i \\sqrt{\\frac{2}{N}} \\sum _{\\mathbf {k}} \\sum _{\\lambda =1}^{4}\\sqrt{\\frac{\\omega _{\\lambda }(\\mathbf {k})}{\\mathcal {K}}} \\nonumber \\\\&& \\times \\left( \\exp \\left[-i \\mathbf {k} \\cdot (\\mathbf {s}+\\mathbf {e}_m/2)\\right]\\eta _{m \\lambda }(\\mathbf {k}) a^{\\phantom{\\dagger }}_{\\lambda }(\\mathbf {k}) \\right.", "\\nonumber \\\\&& \\quad - \\left.", "\\exp \\left[i \\mathbf {k} \\cdot (\\mathbf {s}+\\mathbf {e}_m/2)\\right]\\eta ^{\\ast }_{\\lambda m}(\\mathbf {k}) a^{\\dagger }_{\\lambda }(\\mathbf {k}) \\right) \\nonumber \\\\$ The Hamiltionian (Eq.", "REF ) is already quadratic in $a^{\\phantom{\\dagger }}_\\lambda $ .", "What remains is to eliminate all terms which do not conserve photon number, by constructing a suitable matrix $\\eta ^{\\ast }_{\\lambda m}(\\mathbf {k})$ .", "To do this, we need to evaluate the Fourier transform of the lattice curl $(\\nabla _{\\scriptsize } \\times A)_{(\\mathbf {r}, n)}$.", "This operator is defined on a six-bond plaquette, composed of pairs of bonds which enter with opposite signs in the directed sum around the plaquette [Fig.", "REF , Fig.", "REF ].", "These bonds have midpoints located at $\\mathbf {r} - \\mathbf {e}_n/2 \\pm \\mathbf {h}_{nm}$ where $\\mathbf {h}_{nm}\\equiv \\frac{a_0}{\\sqrt{8}} \\frac{\\hat{\\mathbf {e}}_n\\times \\hat{\\mathbf {e}}_m}{|\\hat{\\mathbf {e}}_n\\times \\hat{\\mathbf {e}}_m |}$ Hence $&& \\left( \\nabla _{\\scriptsize } \\times \\mathcal {A} \\right)_{(\\mathbf {r}, n)}=\\sqrt{\\frac{2}{N}} \\sum _{\\mathbf {k}} \\sum _{\\lambda =1}^{4}\\sqrt{\\frac{\\mathcal {K}}{\\omega _{\\lambda }(\\mathbf {k})}}\\nonumber \\\\&& \\qquad \\times \\bigg \\lbrace \\exp [-i \\mathbf {k} \\cdot (\\mathbf {r}-\\mathbf {e}_n/2)] a^{\\phantom{\\dagger }}_{\\lambda }(\\mathbf {k})\\nonumber \\\\&& \\qquad \\times \\sum _{m} (-2 i \\sin (\\mathbf {k} \\cdot \\mathbf {h}_{nm})) \\eta _{m \\lambda }(\\mathbf {k})\\nonumber \\\\&& \\qquad + \\exp \\left[ i \\mathbf {k} \\cdot (\\mathbf {r}-\\mathbf {e}_n/2) \\right] a^{\\dagger }_{\\lambda }(\\mathbf {k})\\nonumber \\\\&& \\qquad \\times \\sum _{m} \\left( 2 i \\sin (\\mathbf {k} \\cdot \\mathbf {h}_{nm}) \\right)\\eta ^{\\ast }_{\\lambda m}(\\mathbf {k}) \\bigg \\rbrace $ where, by inspection, $\\mathbf {h}_{nn} \\equiv 0$ .", "We can rewrite the sum $\\sum _{m}$ in Eq.", "(REF ) in a more convenient form by introducing an Hermitian, anti-symmetric matrix $&&{Z}(\\mathbf {k})= -2i \\times \\nonumber \\\\&&\\begin{pmatrix}0 & \\sin (\\mathbf {k} \\cdot \\mathbf {h}_{01}) & \\sin (\\mathbf {k} \\cdot \\mathbf {h}_{02}) & \\sin (\\mathbf {k} \\cdot \\mathbf {h}_{03}) \\\\-\\sin (\\mathbf {k} \\cdot \\mathbf {h}_{01}) & 0 & \\sin (\\mathbf {k} \\cdot \\mathbf {h}_{12}) & \\sin (\\mathbf {k} \\cdot \\mathbf {h}_{13}) \\\\-\\sin (\\mathbf {k} \\cdot \\mathbf {h}_{02}) & -\\sin (\\mathbf {k} \\cdot \\mathbf {h}_{12}) & 0 & \\sin (\\mathbf {k} \\cdot \\mathbf {h}_{23}) \\\\-\\sin (\\mathbf {k} \\cdot \\mathbf {h}_{03}) & -\\sin (\\mathbf {k} \\cdot \\mathbf {h}_{13}) & -\\sin (\\mathbf {k} \\cdot \\mathbf {h}_{23}) & 0\\end{pmatrix}\\nonumber \\\\$ acting on the four component vectors $\\eta _{\\lambda }(\\mathbf {k})$ .", "Since ${Z}({\\mathbf {k}})$ is Hermitian, we are free to construct the matrix ${\\eta }({\\mathbf {k}})$ from the eigenvectors of ${Z}({\\mathbf {k}})$ , such that ${Z}(\\mathbf {k})\\cdot \\begin{pmatrix} \\eta _{\\lambda 0} \\\\ \\eta _{\\lambda 1} \\\\ \\eta _{\\lambda 2} \\\\ \\eta _{\\lambda 3} \\end{pmatrix}&=& \\zeta _{\\lambda }(\\mathbf {k})\\begin{pmatrix} \\eta _{\\lambda 0} \\\\ \\eta _{\\lambda 1} \\\\ \\eta _{\\lambda 2} \\\\ \\eta _{\\lambda 3} \\end{pmatrix}$ A specific choice of ${\\eta }({\\mathbf {k}})$ corresponds to a choice of gauge, since using Eq.", "(REF ), the divergence of $\\mathcal {A}_{{\\bf s} {\\bf s^{\\prime }}}$ is now fixed.", "The choice here, which is made for maximum convenience in constructing the photon dispersion, is the radiation (or Coulomb) gauge $\\nabla \\cdot \\mathcal {A}=0.$ It follows from Eqs.", "(REF ) and (REF ) that $&&(\\nabla _{\\scriptsize } \\times \\mathcal {A})_{(\\mathbf {r}, n)}= \\sqrt{\\frac{2}{N}} \\sum _{\\mathbf {k}} \\sum _{\\lambda =1}^{4}\\sqrt{\\frac{\\mathcal {K}}{\\omega _{\\lambda }(\\mathbf {k})}}\\nonumber \\\\&& \\times \\bigg (\\exp [ -i \\mathbf {k} \\cdot (\\mathbf {r}-\\mathbf {e}_n/2) ]a_{\\lambda }(\\mathbf {k}) \\zeta _{\\lambda }(\\mathbf {k})\\eta _{n \\lambda }(\\mathbf {k})\\nonumber \\\\&& \\quad + \\exp [i \\mathbf {k} \\cdot (\\mathbf {r}-\\mathbf {e}_n/2)]a^{\\dagger }_{\\lambda }(\\mathbf {k})\\zeta _{\\lambda }(\\mathbf {k})\\eta _{\\lambda n}^{\\ast }(\\mathbf {k})\\bigg ).$ Squaring and summing over $\\mathbf {r}$ and $n$ , we arrive at $&&\\sum _{(\\mathbf {r}, n)} (\\nabla _{\\scriptsize } \\times \\mathcal {A})_{(\\mathbf {r}, n)}^2= \\frac{1}{2} \\sum _{\\mathbf {k}} \\sum _{\\lambda =1}^{4} \\sum _{\\lambda ^{\\prime }=1}^{4}\\nonumber \\\\&& \\times \\sqrt{\\frac{\\mathcal {K}}{\\omega _{\\lambda }(\\mathbf {k})}}\\sqrt{\\frac{\\mathcal {K}}{\\omega _{\\lambda ^{\\prime }}(\\mathbf {k})}} \\zeta _{\\lambda }(\\mathbf {k}) \\zeta _{\\lambda ^{\\prime }}(\\mathbf {k}^{\\prime })\\nonumber \\\\&& \\times \\bigg \\lbrace a_{\\lambda }(\\mathbf {k}) a_{\\lambda ^{\\prime }}(-\\mathbf {k}) \\zeta _{\\lambda }(\\mathbf {k}) \\zeta _{\\lambda ^{\\prime }}(-\\mathbf {k})\\sum _n \\eta _{n \\lambda }(\\mathbf {k}) \\eta _{n\\lambda ^{\\prime }}(-\\mathbf {k})\\nonumber \\\\&& + a^{\\dagger }_{\\lambda }(\\mathbf {k}) a^{\\dagger }_{\\lambda ^{\\prime }}(\\mathbf {k})\\zeta _{\\lambda }(\\mathbf {k}) \\zeta _{\\lambda ^{\\prime }}(-\\mathbf {k})\\sum _n \\eta _{\\lambda n}^{\\ast }(\\mathbf {k})\\eta _{\\lambda ^{\\prime } n}^{\\ast }(-\\mathbf {k})\\nonumber \\\\&& + a_{\\lambda }(\\mathbf {k}) a^{\\dagger }_{\\lambda ^{\\prime }}(\\mathbf {k})\\zeta _{\\lambda }(\\mathbf {k}) \\zeta _{\\lambda ^{\\prime }}(\\mathbf {k})\\sum _n \\eta _{n \\lambda }(\\mathbf {k})\\eta _{\\lambda ^{\\prime } n}^{\\ast }(\\mathbf {k})\\nonumber \\\\&& + a^{\\dagger }_{\\lambda }(\\mathbf {k}) a_{\\lambda ^{\\prime }}(\\mathbf {k})\\zeta _{\\lambda }(\\mathbf {k}) \\zeta _{\\lambda ^{\\prime }}(\\mathbf {k})\\sum _n \\eta _{\\lambda n}^{\\ast }(\\mathbf {k})\\eta _{n \\lambda ^{\\prime }}(\\mathbf {k}) \\bigg \\rbrace .$ This rather dense expression can be simplified using the unitarity of ${\\eta }(\\mathbf {k})$ $\\sum _n \\eta _{\\lambda n}^{\\ast }(\\mathbf {k}) \\eta _{n \\lambda ^{\\prime }}(\\mathbf {k})=\\delta _{\\lambda \\lambda ^{\\prime }}.$ and the fact that ${Z}(-\\mathbf {k})={Z}(\\mathbf {k})^{\\ast }$ from which it follows that $\\eta _{\\lambda }(-\\mathbf {k}) &=& \\eta _{\\lambda }^{\\ast }(\\mathbf {k}) \\\\\\zeta _{\\lambda }(\\mathbf {k}) &=& \\zeta _{\\lambda }(-\\mathbf {k}).$ Whence, $&& \\sum _{(\\mathbf {r}, n)} (\\nabla _{\\scriptsize } \\times \\mathcal {A})_{(\\mathbf {r}, n)}^2= \\frac{\\mathcal {K}}{2} \\sum _{\\mathbf {k}}\\sum _{\\lambda =1}^{4} \\frac{\\zeta _{\\lambda }(\\mathbf {k})^2}{\\omega _{\\lambda }(\\mathbf {k})}\\nonumber \\\\&& \\quad \\times \\bigg \\lbrace a_{\\lambda }(\\mathbf {k}) a_{\\lambda }(-\\mathbf {k})+ a_{\\lambda }^{\\dagger }(\\mathbf {k})a_{\\lambda }^{\\dagger }(-\\mathbf {k})\\nonumber \\\\&& \\qquad + a_{\\lambda }^{\\dagger }(\\mathbf {k}) a_{\\lambda }(\\mathbf {k})+ a_{\\lambda }(\\mathbf {k})a_{\\lambda }^{\\dagger }(\\mathbf {k})\\bigg \\rbrace .$ Applying the same procedure again to Eq.", "(REF ), we find $&&\\sum _{(\\mathbf {s}, m)} (\\nabla _{\\scriptsize } \\times \\nabla _{\\scriptsize } \\times \\mathcal {A})_{(\\mathbf {s}, m)}^2= \\frac{\\mathcal {K}}{2} \\sum _{\\mathbf {k}} \\sum _{\\lambda =1}^{4}\\frac{\\zeta _{\\lambda }(\\mathbf {k})^4}{\\omega _{\\lambda }(\\mathbf {k})}\\nonumber \\\\&& \\times \\bigg \\lbrace a_{\\lambda }(\\mathbf {k}) a_{\\lambda }(-\\mathbf {k})+ a_{\\lambda }^{\\dagger }(\\mathbf {k})a_{\\lambda }^{\\dagger }(-\\mathbf {k})\\nonumber \\\\&& \\qquad + a_{\\lambda }^{\\dagger }(\\mathbf {k}) a_{\\lambda }(\\mathbf {k})+ a_{\\lambda }(\\mathbf {k})a_{\\lambda }^{\\dagger }(\\mathbf {k}) \\bigg \\rbrace .$ The remaining, electric field, term in $\\mathcal {H}_0$ [Eq.", "(REF )] yields $&\\sum _{(\\mathbf {s}, m)} &\\left(\\frac{\\partial \\mathcal {A}_{(\\mathbf {s}, m)}}{\\partial t}\\right)^2= \\frac{1}{2} \\sum _{\\mathbf {k}} \\sum _{\\lambda =1}^{4}\\omega _{\\lambda }(\\mathbf {k})\\nonumber \\\\&& \\times \\bigg \\lbrace -a_{\\lambda }(\\mathbf {k}) a_{\\lambda }(-\\mathbf {k})- a_{\\lambda }^{\\dagger }(\\mathbf {k})a_{\\lambda }^{\\dagger }(-\\mathbf {k}) \\nonumber \\\\&& \\qquad + a_{\\lambda }^{\\dagger }(\\mathbf {k}) a_{\\lambda }(\\mathbf {k})+ a_{\\lambda }(\\mathbf {k})a_{\\lambda }^{\\dagger }(\\mathbf {k}) \\bigg \\rbrace .$ Inserting all of this into the Hamiltonian Eq.", "(REF ) gives $&\\mathcal {H}^\\prime _{\\sf U(1)}&= \\sum _{\\mathbf {k}} \\sum _{\\lambda =1}^{4}\\bigg [\\bigg (\\frac{\\mathcal {U} \\mathcal {K} \\zeta _{\\lambda }(\\mathbf {k})^2}{4 \\omega _{\\lambda }(\\mathbf {k})}+ \\frac{\\mathcal {W} \\mathcal {K} \\zeta _{\\lambda }(\\mathbf {k})^4}{4 \\omega _{\\lambda }(\\mathbf {k})}+ \\frac{\\omega _{\\lambda }(\\mathbf {k})}{4}\\bigg )\\nonumber \\\\&& \\qquad \\times \\left(a_{\\lambda }(\\mathbf {k}) a_{\\lambda }^{\\dagger }(\\mathbf {k})+ a_{\\lambda }^{\\dagger }(\\mathbf {k}) a_{\\lambda }(\\mathbf {k})\\right)\\nonumber \\\\&& \\qquad + \\left(\\frac{\\mathcal {U} \\mathcal {K} \\zeta _{\\lambda }(\\mathbf {k})^2}{4 \\omega _{\\lambda }(\\mathbf {k})}+ \\frac{\\mathcal {W} \\mathcal {K} \\zeta _{\\lambda }(\\mathbf {k})^4}{4 \\omega _{\\lambda }(\\mathbf {k})}- \\frac{\\omega _{\\lambda }(\\mathbf {k})}{4}\\right)\\nonumber \\\\&& \\qquad \\times \\left(a_{\\lambda }(\\mathbf {k}) a_{\\lambda }(-\\mathbf {k})+ a_{\\lambda }^{\\dagger }(\\mathbf {k}) a_{\\lambda }^{\\dagger }(-\\mathbf {k})\\right)\\bigg ]$ To diagonalize the Hamiltonian we require $\\frac{\\mathcal {U} \\mathcal {K} \\zeta _{\\lambda }(\\mathbf {k})^2}{4 \\omega _{\\lambda }(\\mathbf {k})}+ \\frac{\\mathcal {W} \\mathcal {K} \\zeta _{\\lambda }(\\mathbf {k})^4}{4 \\omega _{\\lambda }(\\mathbf {k})}= \\frac{\\omega _{\\lambda }(\\mathbf {k})}{4}.$ which implies $\\mathcal {H}^\\prime _{\\sf U(1)}= \\sum _{\\mathbf {k}} \\sum _{\\lambda =1}^{4} \\omega _{\\lambda }(\\mathbf {k})\\left( a^{\\dagger }_{\\lambda } (\\mathbf {k}) a_{\\lambda } (\\mathbf {k})+\\frac{1}{2} \\right)$ with dispersion relation fixed by Eq.", "(REF ) $\\omega _{\\lambda }(\\mathbf {k})= \\mathcal {K}\\sqrt{\\frac{\\mathcal {U}}{\\mathcal {K}}\\zeta _{\\lambda }(\\mathbf {k})^2+ \\frac{\\mathcal {W}}{\\mathcal {K}}\\zeta _{\\lambda }(\\mathbf {k})^4}.$ All that now remains is to determine the eigenvalues of the matrix $Z(\\mathbf {k})$ , $\\zeta _{\\lambda }(\\mathbf {k})$ .", "We find $\\zeta _{1}(\\mathbf {k}) &=& +\\sqrt{2} \\sqrt{\\sum _{mn} \\sin {(\\mathbf {k} \\cdot \\mathbf {h}_{mn})}^2} \\\\\\zeta _{2}(\\mathbf {k}) &=& -\\sqrt{2} \\sqrt{\\sum _{mn} \\sin {(\\mathbf {k} \\cdot \\mathbf {h}_{mn})}^2} \\\\\\zeta _{3}(\\mathbf {k}) &=& 0 \\\\\\zeta _{4}(\\mathbf {k}) &=& 0.$ It follows that the four bands of excitations $\\zeta _{\\lambda }(\\mathbf {k})$ correspond to two, degenerate, physical photon modes, and two unphysical, zero energy modes.", "The unphysical modes arise because of the gauge redundancy in $\\mathcal {A}$ and make no contribution to either the Hamiltonian or to any gauge invariant correlation functions.", "Keeping only the physical photon modes from Eq.", "(REF ), we finally arrive at $\\mathcal {H}^\\prime _{\\sf U(1)}= \\sum _{\\mathbf {k}} \\sum _{\\lambda =1}^{2}\\omega (\\mathbf {k})\\left(a^{\\dagger }_{\\lambda } (\\mathbf {k}) a^{\\phantom{\\dagger }}_{\\lambda } (\\mathbf {k})+ \\frac{1}{2}\\right)$ where $\\lambda $ now has the interpretation of the polarisation of the photon.", "The photon dispersion $\\omega (\\mathbf {k})$ is independent of polarisation and can be written $&&\\omega (\\mathbf {k}) = \\mathcal {K}\\sqrt{\\frac{\\mathcal {U}}{\\mathcal {K}}\\zeta (\\mathbf {k})+ \\frac{\\mathcal {W}}{\\mathcal {K}}\\zeta (\\mathbf {k})^2}\\nonumber \\\\$ where $\\zeta (\\mathbf {k})=\\zeta _{1}(\\mathbf {k})= -\\zeta _{2}(\\mathbf {k})= \\sqrt{2} \\sqrt{\\sum _{mn} \\sin {(\\mathbf {k} \\cdot \\mathbf {h}_{mn})}^2}$ with $\\mathbf {h}_{mn}$ defined by Eq.", "(REF ).", "For all $\\ \\mathcal {U}/\\mathcal {K} > 0$ the photon dispersion is linear in the long-wavelength limit $\\omega (\\mathbf {k}\\approx \\mathbf {0})\\approx \\sqrt{\\mathcal {U}\\mathcal {K}} \\ a_0 |\\mathbf {k}|$ This means that there is a well-defined speed of light $c = \\sqrt{\\mathcal {U}\\mathcal {K}}\\ a_0\\ \\hbar ^{-1}$ where we have restored the dimensional factor of $\\hbar $ .", "However in the limiting case $\\mathcal {U}/\\mathcal {K} \\rightarrow 0$ , $c \\rightarrow 0$ , and the dispersion of the photon becomes quadratic in the long-wavelength limit $\\omega (\\mathbf {k})\\approx \\sqrt{\\mathcal {W}\\mathcal {K}} a_0^2 |\\mathbf {k}|^2.$ Precisely this limit is realised at the RK point $\\mu =g$ of the quantum ice model ${\\mathcal {H}}_\\mu $ [Eq.", "(REF )], and defines the boundary of the quantum $U(1)$ liquid phase [47], [58].", "The photon dispersion relations in the two extreme cases $\\mathcal {U}/\\mathcal {K}=0$ and $\\mathcal {W}/\\mathcal {K}=0$ are plotted in Fig.", "REF and Fig.", "REF .", "Figure: (Color online).Dispersion ω(𝐤)\\omega (\\mathbf {k}) of magnetic photon excitations, calculated for the latticefield theory Eq.", "() in the “quantum ice” limit 𝒲/𝒦→0\\mathcal {W}/\\mathcal {K} \\rightarrow 0.The dispersion is plotted in the (h,h,l)(h,h,l) plane, following Eq.", "().The dispersion is linear in |𝐤||\\mathbf {k}| in the long-wavelength limit, with aspeed of light c=𝒰𝒦a 0 c=\\sqrt{\\mathcal {U}\\mathcal {K}}a_0." ], [ "From photons to structure factors", "Spin correlations in real materials can be measured directly by neutron scattering.", "Here we convert the analysis of photons in Section REF into concrete predictions for the dynamical structure factors measured in such an experiment.", "We also consider the structure factors which might be measured in, e.g., X-ray scattering experiments on a charge ice of the type considered by Banerjee et al. [59].", "Specifically, we will consider $S_{\\sf spin}^{\\alpha \\beta } (\\mathbf {k}, \\omega )&=& \\int dt e^{-i \\omega t}\\langle {\\mathsf {S}}^\\alpha (-\\mathbf {k}, t) {\\mathsf {S}}^\\beta (\\mathbf {k}, 0) \\rangle $ and $S_{\\sf charge} (\\mathbf {k}, \\omega )&=& \\int dt e^{-i \\omega t}\\langle n(-\\mathbf {k}, t) n(\\mathbf {k}, 0) \\rangle $ Possessing the full photon wave function [Eq.", "(REF )] permits us to calculate these dynamical structure factors on a lattice, passing directly from the correlations of $\\mathcal {A}_{(\\mathbf {s},m)}$ to those of ${\\mathsf {S}}_{(\\mathbf {r},n)}$ or $n_{(\\mathbf {r},n)}$ .", "We first consider the charge ice and, following banerjee08, introduce an additional (dimensionless) scale factor $\\kappa \\lesssim 1$ to take account of any renormalization of the field $\\mathcal {B}$ when an average is taken over fast fluctuations of $\\mathcal {A}_{(\\mathbf {s},m)}$ [cf.", "Eq.", "(REF ) to Eq.", "(REF )] $S_{\\sf charge} (\\mathbf {k}, t) &=& \\kappa ^2\\sum _{mn}\\langle \\tilde{\\mathcal {B}}_n(-\\mathbf {k}, t) \\tilde{\\mathcal {B}}_m(\\mathbf {k}, 0)\\rangle $ where $\\tilde{\\mathcal {B}}_n(\\mathbf {k},t)= \\frac{1}{\\sqrt{N}} \\sum _{\\mathbf {r}}\\exp [-i \\mathbf {k} \\cdot (\\mathbf {r} +\\mathbf {e}_n/2)]\\mathcal {B}_n(\\mathbf {r},t)$ with $\\mathcal {B}_n(\\mathbf {r}) \\equiv \\mathcal {B}(\\mathbf {r}-\\mathbf {e}_n/2)= \\left( \\nabla _{\\scriptsize } \\times \\mathcal {A} \\right)_{(\\mathbf {r}, n)}$ The time evolution of $\\mathcal {A}_{(\\mathbf {s}, m)}$ follows directly from $\\mathcal {H}^\\prime _{\\sf U(1)}$ [Eq.", "(REF )] $\\tilde{\\mathcal {B}}_n(\\mathbf {k}, t)&=& \\frac{\\sqrt{2}}{4} \\sum _{\\lambda =1}^{4}\\sqrt{\\frac{\\mathcal {K}}{\\omega _{\\lambda }(\\mathbf {k})}}\\zeta _{\\lambda }(\\mathbf {k})\\nonumber \\\\&& \\times \\left(\\eta _{n \\lambda }( -\\mathbf {k})a_{\\lambda }(-\\mathbf {k})e^{-i \\omega _{\\lambda }(\\mathbf {k}) t}\\right.", "\\nonumber \\\\&& \\left.", "\\qquad +\\eta _{\\lambda n}^{\\ast }(\\mathbf {k})a_{\\lambda }^{\\dagger }(\\mathbf {k})e^{i \\omega _{\\lambda }(\\mathbf {k}) t}\\right).$ such that $S_{\\sf charge} (\\mathbf {k}, \\omega )&=& \\frac{\\kappa ^2}{8} \\sum _{mn}\\sum _{\\lambda =1}^{4} \\frac{\\mathcal {K}}{\\omega _{\\lambda }(\\mathbf {k})}\\zeta _{\\lambda }(\\mathbf {k})^2 \\eta _{m \\lambda }(\\mathbf {k}) \\eta _{\\lambda n}^{\\ast }\\nonumber \\\\&& \\quad \\times \\langle a^{\\phantom{\\dagger }}_{\\lambda }(\\mathbf {k}) a_{\\lambda }^{\\dagger }(\\mathbf {k})+ a_{\\lambda }^{\\dagger }(\\mathbf {k}) a^{\\phantom{\\dagger }}_{\\lambda }(\\mathbf {k})\\rangle \\nonumber \\\\&& \\quad \\times \\delta \\left(\\omega - \\omega _{\\lambda }(\\mathbf {k}) \\right)$ where we have dropped all terms which fail to preserve photon number or polarisation.", "Figure: (Color online).Dispersion ω(𝐤)\\omega (\\mathbf {k}) of magnetic photon excitations, calculated for the latticefield theory Eq.", "() in the limit 𝒰/𝒦→0\\mathcal {U}/\\mathcal {K} \\rightarrow 0.The dispersion is plotted in the (h,h,l)(h,h,l) plane, following Eq.", "().The dispersion is quadratic in 𝐤\\mathbf {k} the long wavelength limit.This situation is realised in the microscopic “quantum ice” modelℋ μ {\\mathcal {H}}_\\mu [Eq.", "()], at the RK point, μ=g\\mu =g.The unphysical photon polarisations $\\lambda =3$ ,4 do not contribute to Eq.", "(REF ), since $\\zeta _\\lambda (\\mathbf {k})^2/\\omega _\\lambda (\\mathbf {k})|_{\\lambda =3,4} \\equiv 0$ For the physical polarisations $\\lambda =1$ ,2 $\\langle a_{\\lambda }^{\\dagger }(\\mathbf {k}) a^{\\phantom{\\dagger }}_{\\lambda }(\\mathbf {k})\\rangle = \\frac{1}{e^{\\frac{\\omega (\\mathbf {k})}{T}}-1} \\equiv n_B (\\mathbf {k})$ since the photons are bosons.", "Noting that $\\sum _{\\lambda =1}^{4} \\zeta _{\\lambda }(\\mathbf {k})^2\\eta _{m \\lambda }(\\mathbf {k}) \\eta _{\\lambda n}^{\\ast } = \\sum _{mn} ({Z}(\\mathbf {k})^2)_{mn}$ we arrive at a result for the dynamical structure factor of a quantum charge ice $S_{\\sf charge} (\\mathbf {k},\\omega )&& = \\frac{\\kappa ^2}{2} \\frac{\\mathcal {K}}{\\omega (\\mathbf {k})}\\sum _{mn} \\sum _l\\sin {( \\mathbf {k} \\cdot \\mathbf {h}_{ml} )}\\sin {( \\mathbf {k} \\cdot \\mathbf {h}_{nl} )}\\nonumber \\\\&& \\quad \\times \\bigg ( \\delta \\left(\\omega - \\omega _{\\lambda }(\\mathbf {k}) \\right) (1+n_B(\\mathbf {k}))\\nonumber \\\\&& \\quad + \\ \\delta \\left(\\omega + \\omega _{\\lambda }(\\mathbf {k}) \\right) n_B(\\mathbf {k}) \\bigg )$ where the vectors $\\mathbf {h}_{nm}$ are defined by Eq.", "(REF ).", "In comparing with quantum Monte Carlo simulation, we will also make extensive use of the zero-temperature, equal-time (i.e.", "energy-integrated) structure factor $S_{\\sf charge} (\\mathbf {k},t=0)_{T=0} = \\int d\\omega \\ S_{\\sf charge} (\\mathbf {k},\\omega )_{T=0}$ This can be written as a function of just two, dimensionless, ratios of parameters, $\\overline{\\mathcal {U}}$ and $\\overline{\\mathcal {W}}$ $S_{\\sf charge} (\\mathbf {k},t=0)_{T=0}= \\frac{\\overline{S}_{\\sf 0}(\\mathbf {k})}{\\sqrt{\\overline{\\mathcal {U}}\\zeta (\\mathbf {k})^2+\\overline{\\mathcal {W}}\\zeta (\\mathbf {k})^4}}$ where $\\zeta (\\mathbf {k})$ is defined by Eq.", "(REF ), $\\overline{S}_{\\sf 0}(\\mathbf {k})= \\sum _{mn}\\sum _l \\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{ml})}\\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{nl})} \\, ,$ and the dimensionless ratios of parameters are given by $&&\\overline{\\mathcal {U}} = \\frac{\\mathcal {U}}{\\mathcal {K} \\kappa ^4}\\quad , \\quad \\overline{\\mathcal {W}} =\\frac{\\mathcal {W}}{\\mathcal {K} \\kappa ^4} \\, .$ It is this form of the result, evaluated at the discrete set of wave vectors $\\lbrace \\mathbf {k} \\rbrace $ appropriate for a finite-size cluster with given boundary conditions, which we will fit to simulation results in Section REF .", "Calculating the dynamical structure factor $S_{\\sf spin}^{\\alpha \\beta } (\\mathbf {k}, \\omega )$ for a spin ice means generalising Eq.", "(REF ) to take account of neutron polarisation, and the local easy axes of spins in a spin ice [8].", "However the underlying field-theoretical description of the problem ${\\mathcal {H}}^\\prime _{\\sf U(1)}$ [Eq.", "(REF )] is unchanged, and the two results differ only in the way in which the contraction of fields on different sublattices $\\langle \\mathcal {B}_n(-\\mathbf {k}) \\mathcal {B}_m(\\mathbf {k}) \\rangle $ contribute to correlation functions.", "In a charge ice we simply sum over $m$ , $n$ as in Eq.", "(REF ).", "In a spin ice we must account for the easy axes which lie along the vectors $\\hat{\\mathbf {e}}_n$ [Eq.", "(REF ] and then calculate the projection of the spin along the axis of interest [8].", "Thus, the equal time structure factor is $&& S^{\\alpha \\beta }_{\\sf spin}(\\mathbf {k}, t=0) = \\nonumber \\\\&&\\quad \\kappa ^2 \\sum _{mn}\\bigg (\\hat{\\mathbf {e}}_m \\cdot \\hat{{\\bf \\alpha }} \\bigg )\\bigg (\\hat{\\mathbf {e}}_n \\cdot \\hat{{\\bf \\beta }}\\bigg ) \\langle \\mathcal {B}_n(-\\mathbf {k}) \\mathcal {B}_m(\\mathbf {k}) \\rangle $ where $\\hat{{\\bf \\alpha }}$ and $\\hat{{\\bf \\beta }}$ are unit vectors in the $\\alpha $ and $\\beta $ directions.", "Following the same procedure as described above for the charge ice we come to the general result for the dynamical structure factor in a spin ice $&& S^{\\alpha \\beta }_{\\sf spin}(\\mathbf {k}, \\omega ) =\\frac{\\kappa ^2}{2}\\frac{\\mathcal {K}}{\\omega (\\mathbf {k})}\\sum _{mn} \\sum _l \\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{ml})}\\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{nl})}\\nonumber \\\\&& \\quad \\times \\bigg (\\hat{\\mathbf {e}}_m \\cdot \\hat{{\\bf \\alpha }} \\bigg )\\bigg (\\hat{\\mathbf {e}}_n \\cdot \\hat{{\\bf \\beta }}\\bigg )\\bigg ( \\delta \\left(\\omega - \\omega _{\\lambda }(\\mathbf {k}) \\right) (1+n_B(\\mathbf {k}))\\nonumber \\\\&& \\quad + \\ \\delta \\left(\\omega + \\omega _{\\lambda }(\\mathbf {k}) \\right) n_B(\\mathbf {k}) \\bigg )$ For concreteness, where we come to plot results, we will follow the conventions of Fennell et al.", "[9], who used neutrons with polarisation parallel to $\\bf {n}_{\\nu }=(1, -1, 0) \\nonumber $ to measure the energy-integrated structure factor $S_{\\sf spin}^{\\alpha \\beta } (\\mathbf {k}, t=0)$, for transfered momentum $\\mathbf {k}$ in the $(h,h,l)$ plane.", "We also follow the conventions of Ref.", "fennell09 in choosing a coordinate system in which ${\\bf x} \\parallel \\mathbf {k} \\ , \\ {\\bf y} \\parallel {\\bf n}_\\nu \\times \\mathbf {k} \\ , \\ {\\bf z} \\parallel {\\bf n}_{\\nu },$ and consider the “spin-flip” channel $S_{\\sf spin}^{yy}(\\mathbf {k}, \\omega )$ .", "In this convention, the non spin-flip channel measures $S_{\\sf spin}^{zz}(\\mathbf {k}, \\omega )$ .", "It follows from Eq.", "(REF ) that the dynamical structure factor in the spin-flip channel is given by $&& S_{\\sf spin}^{yy} (\\mathbf {k}, \\omega )=\\frac{\\kappa ^2}{2}\\frac{\\mathcal {K}}{\\omega (\\mathbf {k})}\\sum _{mn} \\sum _l \\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{ml})}\\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{nl})}\\nonumber \\\\&& \\quad \\times \\left(\\frac{\\hat{\\mathbf {e}}_m \\cdot (\\mathbf {n}_\\nu \\times \\mathbf {k})}{|(\\mathbf {n}_\\nu \\times \\mathbf {k})|}\\right)\\left(\\frac{\\hat{\\mathbf {e}}_n \\cdot (\\mathbf {n}_\\nu \\times \\mathbf {k})}{|(\\mathbf {n}_\\nu \\times \\mathbf {k})|}\\right)\\nonumber \\\\&& \\quad \\times \\big ( \\delta \\left(\\omega - \\omega _{\\lambda }(\\mathbf {k}) \\right) (1+n_B(\\mathbf {k}))\\nonumber + \\delta \\left(\\omega + \\omega _{\\lambda }(\\mathbf {k}) \\right) n_B(\\mathbf {k}) \\big )$ and the corresponding zero-temperature, energy-integrated (i.e.", "equal-time) structure factor is $&& S_{\\sf spin}^{yy} (\\mathbf {k}, t=0)_{T=0}=\\frac{\\kappa ^2}{2}\\frac{\\mathcal {K}}{\\omega (\\mathbf {k})}\\nonumber \\\\&& \\quad \\times \\sum _{mn} \\sum _l \\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{ml})}\\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{nl})}\\nonumber \\\\&& \\quad \\times \\left(\\frac{\\hat{\\mathbf {e}}_m \\cdot (\\mathbf {n}_\\nu \\times \\mathbf {k})}{|(\\mathbf {n}_\\nu \\times \\mathbf {k})|}\\right)\\left(\\frac{\\hat{\\mathbf {e}}_n \\cdot (\\mathbf {n}_\\nu \\times \\mathbf {k})}{|(\\mathbf {n}_\\nu \\times \\mathbf {k})|}\\right)$ Once again, we will make extensive use of this result when comparing with quantum Monte Carlo simulation.", "At finite temperature we obtain, for the energy integrated structure factor in SF channel $&& S_{\\sf spin}^{yy} (\\mathbf {k}, t=0)=\\frac{\\kappa ^2}{2}\\frac{\\mathcal {K}}{\\omega (\\mathbf {k})} \\coth {\\left( \\frac{\\omega (\\mathbf {k})}{2T} \\right)}\\nonumber \\\\&& \\quad \\times \\sum _{mn} \\sum _l \\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{ml})}\\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{nl})}\\nonumber \\\\&& \\quad \\times \\left(\\frac{\\hat{\\mathbf {e}}_m \\cdot (\\mathbf {n}_\\nu \\times \\mathbf {k})}{|(\\mathbf {n}_\\nu \\times \\mathbf {k})|}\\right)\\left(\\frac{\\hat{\\mathbf {e}}_n \\cdot (\\mathbf {n}_\\nu \\times \\mathbf {k})}{|(\\mathbf {n}_\\nu \\times \\mathbf {k})|}\\right).$ Where neutron scattering is performed with unpolarized neutrons, experiments measure an average over different components of the dynamical structure factor $I ({\\bf k}, \\omega )\\propto \\sum _{\\alpha \\beta }\\left( \\delta _{\\alpha \\beta }-\\frac{k_{\\alpha } k_{\\beta }}{k^2} \\right)S^{\\alpha \\beta }_{\\sf spin} (\\mathbf {k}, \\omega )$ This is the result plotted where we illustrate photon dispersions in Fig.", "REF and Fig.", "REF .", "The corresponding quasi-elastic (energy integrated) form of Eq.", "(REF ) is given by $I ({\\bf k})\\propto \\sum _{\\alpha \\beta }\\left( \\delta _{\\alpha \\beta }-\\frac{k_{\\alpha } k_{\\beta }}{k^2} \\right)S^{\\alpha \\beta }_{\\sf spin} (\\mathbf {k}, t=0) .$ It is important to note the predictions for quasi-elastic neutron scattering $S_{\\sf spin}^{yy} (\\mathbf {k}, \\omega )$  [Eq.", "(REF )] and $I ({\\bf k}, \\omega )$  [Eq.", "(REF )], and their energy-integrated counterparts $S_{\\sf spin}^{yy} (\\mathbf {k}, t=0)$  [Eq.", "(REF )] and $I ({\\bf k})$  [Eq.", "(REF )] only include contributions from the low-energy photon excitations of a quantum spin ice.", "In a real quantum spin ice material, their would also be contributions at higher energy from gapped “electric charges”, and magnetic monopole excitations.", "These are not treated in the present theory." ], [ " “Electromagnetism” in a Quantum Spin Ice at $T=0$", "The arguments presented in Section REF explain how a spin liquid state with correlations described by an effective electromagnetism can arise in a quantum spin ice, but stop short of offering proof that this happens in any real material or microscopic model.", "In what follows, we validate our use of Gaussian electromagnetism $\\mathcal {H}_{\\sf U(1)}$ [Eq.", "(REF )] as a description of the quantum ice model $\\mathcal {H}_{\\mu }$ [Eq.", "(REF )], by making explicit comparison with the results of zero-temperature quantum Monte Carlo simulation.", "However before considering results on a lattice, it is useful to ask how correlations in quantum spin ice might differ from those in a classical spin ice, within a simple continuum field theory.", "This is considered in Section REF .", "We then turn to simulation of the lattice model $\\mathcal {H}_{\\mu }$ [Eq.", "(REF )] in Section REF , demonstrating that the lattice field theory $\\mathcal {H}_{\\sf U(1)}$ [Eq.", "(REF )] provides an excellent quantitative description of the results for $S_{\\sf spin}^{\\alpha \\beta } (\\mathbf {k}, t=0)$ .", "In Section REF we use the same lattice field theory to make predictions for the magnetic photon excitations which could be observed in inelastic neutron scattering experiments.", "Finally in Section REF we use the finite-size scaling of ground state energies in simulation to put an absolute scale on the speed of light $c$ associated with these magnetic photons.", "Throughout this analysis we set $\\hbar =1$ , restoring dimensional factors of $\\hbar $ only where we quote results for the speed of light.", "Figure: (Color online).Comparison between the predictions of the lattice field theoryℋ 𝖴(1) ' {\\mathcal {H}}^\\prime _{\\sf U(1)} [Eq.", "()]and quantum Monte Carlo simulation of the microscopic modelℋ μ {\\mathcal {H}}_\\mu  [Eq.", "()],for a quantum charge ice at T=0T=0.First column :equal-time structure factor S 𝖼𝗁𝖺𝗋𝗀𝖾 (𝐤,t=0)S_{\\sf charge} ({\\mathbf {k}}, t=0) calculated using Green's functionMonte Carlo (GFMC) simulation of a 2000-site cubic cluster, for a range of μ\\mu ranging fromμ/g=1\\mu /g = 1 (RK point) to μ/g=0\\mu /g = 0 (quantum ice).Second column :best fit of the finite-size (FS) prediction of the lattice field theory to simulation,following Eq.", "().There is excellent, quantitative, agreement between theory and simulation forall values of μ/g\\mu /g.Third column :prediction of lattice field theory in the thermodynamic limit, for parameters obtainedfrom fits to simulation." ], [ "Structure factors within continuum theory", "The long-wavelength properties of a quantum $U(1)$ liquid are well-described by a continuum field theory of the form considered in Ref.", "moessner03 $\\mathcal {S}_{\\sf eff}= \\frac{1}{8\\pi }\\int dt d^3 \\mathbf {r}&\\bigg [& \\mathbf {\\mathcal {E}}({\\bf r})^2 - c^2 \\mathbf {\\mathcal {B}}({\\bf r})^2\\nonumber \\\\&& \\quad - \\rho _{\\sf c}\\ \\bigg ( \\nabla \\times \\mathbf {\\mathcal {B}}({\\bf r}) \\bigg )^2 \\bigg ]$ This therefore provides a convenient starting point for discussing the evolution of spin correlations in quantum spin ice.", "We emphasise that such a theory can be derived as a continuum limit of ${\\mathcal {H}}^\\prime _{\\sf U(1)}$ [Eq.", "(REF )] [47].", "And where we go on to make comparison with quantum Monte Carlo simulation in Section REF , we will use the appropriate results on a lattice, i.e.", "Eq.", "(REF ) and Eq.", "(REF ).", "For $\\rho _{\\sf c} = 0$ , $\\mathcal {S}_{\\sf eff}$ reduces to the familiar Maxwell action of quantum electromagnetism.", "Crucially, this action supports photon excitations with dispersion $\\omega (\\mathbf {k}) = c |\\mathbf {k}|$ .", "The additional term $\\rho _{\\sf c}\\ ( \\nabla \\times \\mathbf {\\mathcal {B}}({\\bf r}) )^2$ is invariant under gauge transformations $ \\mathbf {\\mathcal {A}}({\\bf r}) \\rightarrow \\mathbf {\\mathcal {A}}({\\bf r}) + \\nabla \\phi ({\\bf r})$ , and is an irrelevant perturbation in the RG sense [47].", "However it introduces a new length scale into the problem $\\lambda _{\\sf c} = 2 \\pi \\frac{\\sqrt{\\rho _{\\sf c}}}{c} \\, ,$ which controls the curvature of the photon dispersion $\\omega (\\mathbf {k})=c |\\mathbf {k}| \\sqrt{1 +\\left( \\frac{\\lambda _c}{2 \\pi } \\right)^2 |\\mathbf {k}|^2} \\, ,$ and has an important impact on how correlations evolve as a function of distance.", "The role of $\\lambda _c$ can most easily be understood in the limit $c \\rightarrow 0$ , where correlations of $\\mathbf {\\mathcal {B}}({\\bf r})$ are controlled entirely by $\\rho _{\\sf c}$ .", "Precisely this limit is realised in the microscopic model $\\mathcal {H}_{\\mu }$ [Eq.", "(REF )] at the exactly soluble “RK” point $\\mu = g$.", "At the RK point, all ice configurations are degenerate, and the photons have dispersion $\\omega (\\mathbf {k}) = \\sqrt{\\rho _{\\sf c}} |\\mathbf {k}|^2$  [moessner03,hermele04].", "Correlations of the magnetic field $C^{\\mathbf {\\mathcal {B}}}_{\\mu \\nu }(\\mathbf {k})= \\langle \\mathbf {\\mathcal {B}}_{\\mu } (-\\mathbf {k})\\mathbf {\\mathcal {B}}_{\\nu } (\\mathbf {k})\\rangle $ can be calculated from Eq.", "(REF ), and for $c=0$ these behave as $C^{\\mathbf {\\mathcal {B}}}_{\\mu \\nu }(\\mathbf {k})\\approx \\frac{8 \\pi ^4}{\\sqrt{\\rho _{\\sf c}}}\\left(\\delta _{\\mu \\nu } - \\frac{k_{\\mu } k_{\\nu }}{k^2}\\right)$ exhibiting the pinch-point singularities characteristic of the “Coulombic”, classical $U(1)$ liquid phase [7], [8], [41].", "On Fourier transform, Eq.", "(REF ) corresponds to dipolar correlations in a three-dimensional space $C^{\\mathbf {\\mathcal {B}}}_{\\mu \\nu }(\\mathbf {r})\\propto \\frac{3 r_\\mu r_\\nu /r^2 - \\delta _{\\mu \\nu }}{r^3}$ The quantum $U(1)$ liquid phase, with its linearly dispersing photons, is stabilised by the emergence of finite value of the speed of light $c$ for $\\mu < g$  [moessner03,hermele04,shannon12].", "In this case, we find $C^{\\mathbf {\\mathcal {B}}}_{\\mu \\nu }(\\mathbf {k})= \\frac{8 \\pi ^4 k}{c \\sqrt{ 1+\\left( \\frac{\\lambda _{\\sf c} k}{2 \\pi } \\right)^2}}\\left( \\delta _{\\mu \\nu } - \\frac{k_{\\mu } k_{\\nu }}{k^2} \\right).$ [cf.", "hermele04,castro-neto06].", "For wavelengths $\\lambda \\ll \\lambda _{\\sf c}$ , Eq.", "(REF ) reduces to Eq.", "(REF ), and the system exhibits “classical” dipolar correlations of the form Eq.", "(REF ).", "However for long wavelengths $\\lambda \\gg \\lambda _c$ the additional factor of $k$ in the numerator of Eq.", "(REF ) “hollows out” the pinch point singularities.", "In this limit, $r \\gg \\lambda _{\\sf c}$ , Eq.", "(REF ) corresponds to dipolar correlations in a four-dimensional space $C^{\\mathbf {\\mathcal {B}}}_{\\mu \\nu }(\\mathbf {r})\\propto \\frac{2 r_\\mu r_\\nu /r^2 - \\delta _{\\mu \\nu }}{r^4} \\, ,$ the additional dimension arising because of fluctuations in time [47], [48].", "We therefore associate $\\lambda _{\\sf c}$ with the length-scale over which the system crosses over from “classical” ice correlations, decaying as $1/r^3$ , to “quantum” ice correlations decaying as $1/r^4$ .", "The length-scale $\\lambda _{\\sf c}$ will also play an important role where we compare the predictions of field theory with simulation of microscopic model ${\\mathcal {H}}_\\mu $  [Eq.", "(REF )] as a function of $\\mu $ .", "We can gain some insight into the $\\mu $ dependence of $\\lambda _{\\sf c}$ , from degenerate perturbation theory about the RK point [47], [60], [61], [58].", "We find that $c^2 \\sim (g-\\mu )$ , while $\\rho _c \\approx const.$ , and it follows from Eq.", "(REF ) that $\\lambda _c$ diverges as $\\lambda _c \\sim 1/\\sqrt{g-\\mu }$ .", "Exactly at the RK point, where $g=\\mu $ , $\\lambda _c$ is infinite and correlations have the classical form Eq.", "(REF ) at all length scales, as expected.", "However, as we move away from the RK point into the quantum liquid phase for $\\mu /g < 1$ , there will be a progressive evolution of correlations from classical (pinch points) at short distances to quantum (no pinch points) at long distances.", "This expectation is born out by quantum Monte Carlo simulations, described below.", "For the purposes of these simulations, $\\lambda _{\\sf c}$ also sets the minimum size of cluster which is needed to capture quantum effects at a given $\\mu $ .", "At $\\mu =0$ we find that $\\lambda _{\\sf c} \\approx 0.8 a_0$, and hence a cluster of linear dimension $L=5 a_0 \\ (N=2000)$ is comfortably big enough to observe the quantum spin liquid phase [58]." ], [ "Comparison with Quantum Monte Carlo simulation", "We now turn to zero-temperature quantum Monte Carlo simulation of the microscopic model ${\\mathcal {H}}_\\mu $ [Eq.", "(REF )].", "We have previously argued that this model supports a quantum $U(1)$ liquid ground state for a range of parameters $-0.5g < \\mu < g$ — cf.", "Fig.", "REF and shannon12.", "In this earlier work, evidence for the ground state phase diagram was taken from the finite-size scaling of energy spectra.", "Our main tool here will be the equal time structure factor $S (\\mathbf {k},t=0)$ , calculated from simulation, and from the lattice field theory ${\\mathcal {H}}^\\prime _{\\sf U(1)}$ [Eq.", "(REF )].", "These two independent calculations are found to be in excellent, quantitative agreement, confirming the conclusions of shannon12.", "Making a direct comparison between the field theory and simulation also serves to put the field theory on a quantitative footing, providing information about the evolution of the parameters of the field theory as a function of the microscopic parameter $\\mu $ .", "Simulations were performed using a Green's Function Monte Carlo (GFMC) technique based on the statistical sampling of ice configurations.", "This sampling is weighted using a variational estimate of the ground state wave function, which is optimised in a separate variational Monte Carlo (VMC) calculation.", "In this sense, GMFC can be thought of a systematic method of improving upon variational calculations.", "There is no sign problem associated with ${\\mathcal {H}}_\\mu $ , since all of its off-diagonal matrix elements are equal to 0 or $-g$ , with $g > 0$ .", "Where simulations converge, the results obtained are numerically exact.", "Our implementation of VMC and GFMC calculations for quantum ice [58] exactly parallels our earlier work on the quantum dimer model on a diamond lattice [60], [61], with correlation functions calculated using techniques described in Ref.", "calandra98.", "We refer the interested reader to these papers for further details of the method.", "In the left-hand column of Fig.", "REF , we present GFMC simulation results for the equal-time correlations in a quantum charge ice $S_{\\sf charge} ({\\mathbf {k}}, t=0)_{T=0} = \\langle n ({\\mathbf {k}}) n (-{\\mathbf {k}}) \\rangle _{T=0}$ Simulations were performed for a 2000-site cubic cluster possessing the full symmetry of the lattice, for parameters $\\mu /g = 1$ , $0.75$ , $0.5$ , $0.25$ , 0.", "The classical, dipolar correlations at the RK point $\\mu /g = 1$ are clearly visible as sharp “bow-tie” motifs in $S_{\\sf charge} ({\\mathbf {k}}, t=0)$, centred on pinch-points at ${\\mathbf {k}} = (1,1,1)$ , etc.", "As expected, these pinch points are progressively eliminated as $\\mu /g \\rightarrow 0$ , and quantum effects come to dominate the long lengthscale physics of the problem.", "This erosion of the pinch points is accompanied by a gradual redistribution of spectral weight, with high intensity regions evolving from a triangular into an oval shape.", "In the central column of Fig.", "REF , we present the best fit to simulation results obtained from the lattice field theory.", "Fits were made using the result $S_{\\sf charge} ({\\mathbf {k}}, t=0)_{T=0}$  [Eq.", "(REF )], evaluated for the same 2000-site cluster, as a function of the two dimensionless parameters $\\overline{\\mathcal {U}}$ and $\\overline{\\mathcal {W}}$ [Eq.", "(REF )].", "The two results are indistinguishable by eye, and differ maximally by a few percent, for values of ${\\mathbf {k}}$ close to the Brillouin zone boundary.", "The quality of these fits implies that they can be used to accurately parameterize the lattice field theory ${\\mathcal {H}}^\\prime _{\\sf U(1)}$ [Eq.", "(REF )], and the values of $\\overline{\\mathcal {U}}$ and $\\overline{\\mathcal {W}}$ obtained are shown in Fig.", "REF .", "We note that the values obtained at the RK point, $\\overline{\\mathcal {U}} =0$ and $\\overline{\\mathcal {W}} = 1$ , are uniquely determined by the known form of correlations within the classical ice states [8].", "A separate evaluation of the speed of light $c \\propto \\sqrt{\\mathcal {U}K}$ from finite size scaling of the ground state energy is given in Section REF below.", "In Fig.", "REF we show equivalent results for the equal-time structure factor of a spin ice $S_{\\sf spin}^{yy} (\\mathbf {k}, t=0)_{T=0}= \\langle {\\mathsf {S}}^y (\\mathbf {k}) {\\mathsf {S}}^y (\\mathbf {-k}) \\rangle _{T=0}$ in the spin-flip channel considered by Fennell et al. [9].", "Superficially, these results look very different to those presented in Fig.", "REF .", "This is because the local easy axis is different for each of the four sublattices, leading to a staggering of correlations not present in the charge ice problem.", "However the information content of the two structure factors is exactly the same.", "At the RK point $\\mu /g=1$ , correlations are classical, and $S_{\\sf spin}^{yy} (\\mathbf {k}, t=0)$ exhibits a characteristic “snow flake” motif in the $(h,h,l)$ plane, also seen in neutron scattering experiments on Ho$_2$ Ti$_2$ O$_7$ [fennell09].", "Pinch point singularities are clearly visible at the reciprocal lattice vectors ${\\mathbf {k}} = (1,1,1)$ , etc.", "Once again, these pinch points are progressively eroded as the system is tuned away from the RK point into the quantum spin-liquid regime for $\\mu /g < 1$ .", "Probably the most striking change, however, occurs at $\\mathbf {k} = (0,0,0)$ .", "Here, for a classical spin ice $S_{\\sf spin}^{yy} (\\mathbf {k}\\rightarrow \\mathbf {0}, t=0)_{T=0} \\rightarrow const.$ However, in a quantum spin ice, $S_{\\sf spin}^{yy} (\\mathbf {k}=\\mathbf {0}, t=0)_{T=0} \\equiv 0 \\, ,$ and spectral weight is progressively excavated from the region of reciprocal space around $\\mathbf {k} = (0,0,0)$ for $\\mu /g < 1$ .", "This has important consequences for the evolution of correlations at finite temperature, discussed in Section REF and for the uniform magnetic susceptibility, discussed in Section REF .", "We wish to emphasise that the results shown in Fig.", "REF are not the outcome of separate simulations of a quantum spin ice.", "They are taken from the same simulations of the quantum ice model ${\\mathcal {H}}_\\mu $ [Eq.", "(REF )], recast in the coordinates appropriate for a spin ice.", "It follows that the parameters obtained from fits to field theory at finite size are exactly the same as those for a charge ice, given in Fig.", "REF ." ], [ "Seeing the light : photons and inelastic neutron scattering", "Inelastic neutron scattering provides a direct method of measuring the dynamical structure factor $S_{\\sf spin}^{\\alpha \\beta }(\\mathbf {k}, \\omega )$ , and so of resolving photon excitations in a quantum spin ice.", "These photons disperse linearly out of those reciprocal lattice vectors where pinch points are observed in quasi-elastic scattering experiments.", "However, since these experiments measure the energy integral of the dynamical structure factor, the suppression of pinch points in a quantum spin ice at $T=0$ has important implications for the observation of its photon excitations.", "Specifically, for non-interacting photons, the suppression of energy-integrated structure factor must imply the suppression of the weight in the photon peak itself.", "This is illustrated in Fig.", "REF .", "To see how this works, we consider the result for the dynamical structure factor in a quantum spin ice $S_{\\sf spin}^{\\alpha \\beta }(\\mathbf {k}, \\omega )$  [Eq.", "(REF )], in the (physically relevant) limit where $\\mathcal {W}=0$ .", "In this case weight in the photon peak is determined by the ratio $\\frac{\\overline{S}^{\\alpha \\beta }_0(\\mathbf {k})}{\\omega (\\mathbf {k})}$ where, $\\overline{S}^{\\alpha \\beta }_0(\\mathbf {k}) &=&\\sum _{mn} \\sum _l \\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{ml})}\\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{nl})}\\nonumber \\\\&& \\quad \\times \\bigg (\\hat{\\mathbf {e}}_m \\cdot \\hat{{\\bf \\alpha }} \\bigg )\\bigg (\\hat{\\mathbf {e}}_n \\cdot \\hat{{\\bf \\beta }}\\bigg )$ and $\\omega _{\\lambda }(\\mathbf {k})=\\sqrt{\\mathcal {U} \\mathcal {K}} \\zeta _{\\lambda }(\\mathbf {k}).$ We can use the spectral representation of ${Z}(\\mathbf {k})$ [Eq.", "(REF )] to write $&& \\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{ml})}\\sin {(\\mathbf {k}\\cdot \\mathbf {h}_{nl})}\\nonumber \\\\&& \\qquad = \\frac{1}{4} \\sum _{\\lambda =1}^{4}\\frac{\\omega _{\\lambda }(\\mathbf {k})^2}{\\mathcal {KU}}\\ \\eta _{m \\lambda } (\\mathbf {k}) \\eta _{\\lambda n}^{\\ast }(\\mathbf {k})$ Since the only contributions to the RHS of Eq.", "(REF ) come from the two dispersing modes $\\lambda =1$ , 2, [cf. Eq.", "(REF )], Eq.", "(REF ) simplifies to $\\overline{S}^{\\alpha \\beta }_0(\\mathbf {k})&=& \\frac{1}{4} \\frac{\\omega (\\mathbf {k})^2 }{\\mathcal {KU}}\\sum _{\\lambda =1}^{2} \\sum _{mn}\\ \\eta _{m \\lambda } (\\mathbf {k}) \\eta _{\\lambda n}^{\\ast }(\\mathbf {k})\\nonumber \\\\&& \\quad \\times \\bigg (\\hat{\\mathbf {e}}_m \\cdot \\hat{{\\bf \\alpha }}\\bigg )\\bigg (\\hat{\\mathbf {e}}_n \\cdot \\hat{{\\bf \\beta }}\\bigg )$ Expanding in the first Brillouin zone, for $\\mathbf {k} \\approx 0$ , we find $\\sum _{mn} \\eta _{m \\lambda } (\\mathbf {k}) \\eta _{\\lambda n}^{\\ast }(\\mathbf {k})\\bigg (\\hat{\\mathbf {e}}_m \\cdot \\hat{{\\bf \\alpha }}\\bigg )\\bigg (\\hat{\\mathbf {e}}_n \\cdot \\hat{{\\bf \\beta }}\\bigg )\\approx \\frac{1}{3}$ for $\\alpha = \\beta = y,z$ and zero otherwise.", "It follows that $&& S^{yy}_{\\sf spin}(\\mathbf {k} \\approx \\mathbf {0}, \\omega \\approx 0) \\nonumber \\\\&& \\quad = S^{zz}_{\\sf spin}(\\mathbf {k} \\approx \\mathbf {0}, \\omega \\approx 0) \\nonumber \\\\&& \\qquad \\propto \\omega (\\mathbf {k}) \\ \\delta (\\omega -\\omega (\\mathbf {k})).$ Therefore at low energies, in the first Brillouin zone, inelastic neutron scattering experiments will resolve the magnetic photon excitation as a ghostly, linearly dispersing peak, with intensity vanishing as $I \\propto \\omega (\\mathbf {k})$ , as noted in [savary12].", "However at higher energies and in other Brillouin zones, the momentum dependence of $\\eta _{m \\lambda } (\\mathbf {k}) \\eta _{\\lambda n}^{\\ast }(\\mathbf {k})$ in Eq.", "(REF ) will lead to a significant variation in the intensity of the signal at fixed $\\omega $ .", "This behaviour is illustrated in Fig.", "REF , where we have plotted the intensity of scattering $I(\\mathbf {k},\\omega )$ [Eq.", "(REF )] for an experiment performed using unpolarised neutrons.", "The corresponding quasi elastic scattering, and the path within the $[h,h,l]$ plane, are shown in Fig.", "(REF ).", "The phenomenology of this photon excitation stands in stark contrast to conventional antiferromagnets, whose linearly-dispersing spin-wave excitations have the greatest intensity approaching the zero-energy magnetic Bragg peak associated with magnetic order.", "The difference between these two problems stems from the fact that the photon is a quantised excitation of $\\mathcal {A}$ , while neutron scattering measures correlations of $\\mathcal {B}$ .", "The lattice curl needed to relate one to the other introduces additional factors of $\\zeta _{\\lambda }(\\mathbf {k})$ in $S_{\\sf spin}^{\\alpha \\beta }(\\mathbf {k}, \\omega )$  [Eq.", "(REF )], which leads to the suppression of spectral weight at low energies.", "A better point of comparison, in fact, is the scattering of neutrons by real photons in models of a hot early universe[77].", "In both cases, photons are associated with a periodically fluctuating magnetic field, transverse to the direction of their propagation.", "And in both cases neutrons scatter inelastically from these locally fluctuating magnetic fields.", "In a spin ice, this scattering can occur in both the spin-flip (SF) channel, in which case there is a transfer of angular momentum to the sample, and in the non spin-flip (NSF) channel (cf.", "Fig.", "REF , Fig.", "REF and Fig.", "REF ).", "It is also interesting to note that the same phenomenology of linearly-dispersing excitations, with a vanishing spectral weight at long wavelengths, is encountered in quantum spin nematics [78].", "In this case, low-energy spin fluctuations are controlled by a time derivative of the underlying nematic order parameter [79], and so vanish for $\\omega \\rightarrow 0$ ." ], [ "Estimating the speed of light", "The signal feature of the quantum $U(1)$ liquid is its photon excitations.", "One important consequence of these, so far as the simulation of finite-size systems is concerned, is a characteristic finite-size correction to the ground state energy per site $E_0/N$ , coming from the zero-point energy of the photons $\\frac{\\delta E_0(L)}{N} = \\frac{1}{N}\\left[ E_0(L) - E_0(\\infty ) \\right] = x_1 L^{-4} + \\ldots $ where $L \\sim N^\\frac{1}{3}$ is the linear dimension of the cluster, and the coefficient $x_1$ is proportional to the speed of light $c$ [cf.", "sikora11].", "This means that it is possible to extract the speed of light from the finite-size scaling of the ground state energy found in simulations of ${\\mathcal {H}}_\\mu $  [Eq.", "(REF )], shown in Fig.", "REF .", "Approaching this problem from the lattice field theory ${\\mathcal {H}}^\\prime _{\\sf U(1)}$  [Eq.", "(REF )], we know that $c = \\sqrt{\\mathcal {U} \\mathcal {K}} a_0= \\kappa ^2 \\mathcal {K} \\sqrt{\\overline{\\mathcal {U}}} a_0.$ where the dimensionless parameter $\\overline{\\mathcal {U}} = \\frac{\\mathcal {U}}{\\mathcal {K} \\kappa ^4}$ can be determined separately from fits to structure factors (cf Fig.", "REF ).", "We also have enough information from the fits to the structure factor to evaluate the sum $\\frac{1}{N}\\sum _{\\mathbf {k}} \\frac{\\omega (\\mathbf {k})}{\\kappa ^2 \\mathcal {K}}= \\frac{1}{\\kappa ^2 \\mathcal {K}}\\left(\\frac{E_0}{N} + const \\right)$ where $\\frac{E_0}{N}$ is the ground state energy per site found from Monte Carlo simulations.", "For $\\overline{\\mathcal {U}}=0$ the LHS of Eq.", "(REF ) does not depend on $L$ .", "This is consistent with simulations of the microscopic model at $\\mu =g$ .", "For $\\overline{\\mathcal {U}}>0$ we expect a scaling law $\\sim \\frac{1}{L^4}$ for large $L$ .", "We write $\\epsilon (L)\\equiv \\frac{1}{N}\\sum _{\\mathbf {k}} \\frac{\\omega (\\mathbf {k})}{\\kappa ^2 \\mathcal {K}}=\\epsilon (\\infty )-x_2 L^{-4}$ and it follows that $\\frac{x_1}{x_2}=\\kappa ^2 \\mathcal {K}$ with $c=\\frac{x_2}{x_1} \\sqrt{\\overline{\\mathcal {U}}} a_0.$ where the coefficients $x_1$ and $x_2$ can be found from the finite-size scaling of the ground state energy in simulation [Fig.", "REF ], and through the numerical evaluation of the $\\sum _{\\mathbf {k}}$ in Eq.", "(REF ) for a finite-size system.", "Figure: (Color online).Finite-size scaling of the finite-size correction to the ground state energy per siteδE 0 /N\\delta E_0/N found in quantum Monte Carlo simulations of the quantum ice modelℋ μ {\\mathcal {H}}_\\mu [Eq.", "()].Results are shown for cubic clusters of N=432N=432, 1024 and 2000 sites, forparameters μ/g=0\\mu /g=0, 0.250.25, 0.50.5, 0.750.75, as function of the linear dimensionof the system L=(a 0 /2)(N/2) 1 3 L = (a_0/2) (N/2)^\\frac{1}{3}.The fact that δE 0 /N∼1/L 4 \\delta E_0/N \\sim 1/L^4 implies the existence of a linearly dispersingexcitation — the “photon” of the underlying lattice gauge theory.We find that, for $0 \\le \\mu \\le 1$ , the evolution of the speed of light as a function of $\\mu $ is well-described by $c^2 = \\alpha \\delta \\overline{\\mu } + \\beta \\delta \\overline{\\mu }^2 + {\\mathcal {O}}(\\delta \\overline{\\mu }^3)$ where $\\delta \\overline{\\mu } &=& 1 - \\mu /g \\\\\\alpha &=& 0.22\\ g^2\\ a_0^2 \\\\\\beta &=& 0.13\\ g^2\\ a_0^2$ In particular, for $\\mu =0$ , the physical point of our model, we find $c=(0.6 \\pm 0.1) \\ g \\ a_0 \\ \\hbar ^{-1}$ where we have restored the dimensional factor of $\\hbar $ .", "We have also calculated an upper bound on $c$ from a single mode approximation, in the spirit of Ref.", "moessner03.", "We find $c \\le (0.6 \\pm 0.1) \\ g\\ a_0 \\ \\hbar ^{-1}$ Within errors, the two numbers are indistinguishable.", "It is interesting to use this result to make an order of magnitude estimate of the speed of light in a quantum spin ice material.", "Considering Yb$_2$ Ti$_2$ O$_7$ , as (presently) the best-characterised material, and inserting the exchange parameters obtained by Ross et al.", "[27] into the expression for the tunnelling matrix element [Eq.", "(REF )], we obtain $g_{\\text{\\tiny Yb$_2$Ti$_2$O$_7$}} \\approx 0.05 \\ \\text{meV}.$ From Eq.", "(REF ), and the known size of the unit cell $a_0 = 10.026$ Å [thompson11], we find a speed of light $c \\sim 0.3 \\ \\text{meV Å} \\sim 50 \\ \\text{ms$^{-1}$}$ which implies a photon bandwidth $\\Delta \\omega \\sim 0.1 \\ \\text{meV} \\, ,$ within the range accessible to modern inelastic neutron scattering experiments [80].", "The accuracy of this estimate is limited by the approximations made in setting up the minimal model of a quantum spin ice ${\\mathcal {H}}_{\\sf tunnelling}$ [Eq.", "(REF )], and so it should only be regarded as a “ballpark” figure.", "It should also be remembered that Yb$_2$ Ti$_2$ O$_7$ is believed to order ferromagnetically at the lowest temperatures [29], [70].", "However as long as a given system remains an “ice”, the inclusion of further tunnelling processes beyond ${\\mathcal {H}}_{\\sf tunnelling}$ should only increase the speed of light.", "In Section REF we have demonstrated that the field theory ${\\mathcal {H}}^\\prime _{\\sf U(1)}$  [Eq.", "(REF )] — quantum electromagnetism on a pyrochlore lattice — gives an excellent account of the results of zero-temperature quantum Monte Carlo simulations of the minimal microscopic model of a quantum spin ice, ${\\mathcal {H}}_\\mu $  [Eq.", "(REF )].", "These results confirm the conjecture that this model could support a spin-liquid phase, down to $T=0$ .", "Encouraged by this, we now use the same field theory to explore how correlations in this spin liquid state develop at finite temperature.", "In Section REF we assess how the thermal excitation of magnetic photons changes the temperature dependence of the energy-integrated structure factors measured in quasi-elastic scattering.", "We find that pinch-points eliminated by quantum fluctuations at zero temperature, are progressively restored as the temperature of the spin liquid is raised.", "In Section REF we compare the results of the lattice field theory with published results for quantum Monte Carlo simulations of quantum charge ice at finite temperature [59].", "We find that both the form and the temperature dependence of the correlations are well described by the lattice field theory.", "Finally, in Section REF we conclude with a few remarks about the finite temperature behaviour of the heat capacity and uniform magnetic susceptibility in a quantum spin ice.", "Throughout this analysis we set $\\hbar =k_B = 1$ , restoring dimensional factors of $\\hbar $ and $k_B$ only where we quote results for the coefficient of the heat capacity associated with photons." ], [ "Temperature dependence of structure factors", "The qualitative changes in correlations between spins at finite temperature can most easily be understood within the continuum field theory ${\\mathcal {S}}_{\\sf eff}$ [Eq.", "(REF )].", "The thermal excitation of photons enhances correlations of the magnetic field ${\\mathcal {B}}$ at small $|\\mathbf {k}|$ $C^{\\mathcal {B}}_{\\mu \\nu }(\\mathbf {k})&=& \\frac{8 \\pi ^4 k}{c \\sqrt{ 1+\\left( \\frac{\\lambda _c k}{2 \\pi } \\right)^2}}\\left( \\delta _{\\mu \\nu } - \\frac{k_{\\mu } k_{\\nu }}{k^2} \\right) \\nonumber \\\\&& \\times \\coth {\\left( \\frac{c k \\sqrt{ 1+\\left( \\frac{\\lambda _c k}{2 \\pi } \\right)^2}}{2T} \\right)}.$ and introduces a thermal de Broglie wavelength for the photons.", "$\\lambda _{\\sf T}=\\frac{\\pi c}{T}$ Over sufficiently long distances, this enhancement of correlations exactly cancels their suppression by quantum fluctuations.", "Assuming that $\\lambda _{\\sf c} \\ll \\lambda _{\\sf T}$ , and expanding Eq.", "(REF ) for small wave number, we find $C^{\\mathcal {B}}_{\\mu \\nu }(|\\mathbf {k}| \\ll 2\\pi /\\lambda _{\\sf T})= \\frac{16 \\pi ^4 T}{c^2} \\left( \\delta _{\\mu \\nu } - \\frac{k_{\\mu } k_{\\nu }}{k^2} \\right)+ \\ldots $ This implies that, for these small wave vectors, the pinch point is restored, but with a prefactor that depends linearly on temperature.", "This result has very simple interpretation.", "At finite temperature photons are only coherent quantum excitations over a length scale $\\lambda _{\\sf T}$ .", "Therefore, while correlations in a quantum spin ice may decay as $1/r^4$ over distances $\\lambda _{\\sf c} \\ll r \\ll \\lambda _{\\sf T}$ , at long distances for $r \\gg \\lambda _{\\sf T}$ the classical $1/r^3$ decay of the spin correlations is restored.", "Figure: (Color online)Angle-integrated scattering intensity I(k≈0,T)I (k \\approx 0, T) [Eq.", "()]calculated from the lattice field theory ℋ 𝖴(1) ' {\\mathcal {H}}^\\prime _{\\sf U(1)} [Eq.", "()],for comparison with neutron scattering experiments on a powder sample ofa quantum spin ice.Results are plotted for temperatures ranging from T=0T=0 to T=1.0ca 0 -1 T= 1.0 c a_0^{-1},where cc is the speed of light, and a 0 a_0 the linear dimension of the cubic unit cell, withtemperature measured in units such that ℏ=k B =1\\hbar = k_B = 1.The progressive elimination of pinch points as the sample is cooledmanifests itself as a steady loss of scattering for |𝐤|→0|\\mathbf {k}| \\rightarrow 0.All of these arguments generalise to the lattice field theory ${\\mathcal {H}}^\\prime _{\\sf U(1)}$ [Eq.", "(REF )], and to expressions for the equal-time structure factor at finite temperatures derived from $S_{\\sf spin}^{\\alpha \\beta } (\\mathbf {k}, \\omega )$ [Eq.", "(REF )].", "Thus we anticipate that they will apply equally to a quantum spin ice at finite temperatures.", "This suggests a simple diagnostic for a quantum spin ice in quasi-elastic neutron scattering experiments — as the sample is cooled, and photons become coherent over longer length scales, the pinch points observed at reciprocal lattice vectors are progressively “bleached out”.", "This slow, cold, death of pinch points is illustrated in Fig.", "REF .", "Since there is also a characteristic loss of spectral weight in $S_{\\sf spin}^{\\alpha \\beta } (\\mathbf {k}, t=0)$ for $\\mathbf {k} \\approx \\mathbf {0}$ , exactly the same process could be seen in the angle integrated structure factor measured in neutron scattering experiments on powder samples.", "In this case, the intensity of scattering is given by $I (k, T)\\propto \\sum _{\\alpha \\beta } \\int d\\mathbf {\\Omega }\\left( \\delta _{\\alpha \\beta }-\\frac{k_{\\alpha } k_{\\beta }}{k^2} \\right)\\ S_{\\sf spin}^{\\alpha \\beta } (\\mathbf {k}, t=0)\\nonumber \\\\$ For classical spin ice, or a quantum spin ice at sufficiently high temperature, $ I (k \\approx 0, T) \\approx const.", "$ However, as a quantum spin ice is cooled to zero temperature, the growing coherence of photons will manifest itself as a progressive loss of spectral weight at small $k$ , $ I (k = 0, T) \\sim T $ until, for $T=0$ , spectral weight at $k=0$ is eliminated entirely $ I (k \\approx 0, T=0) \\propto k $ This progression is illustrated in Fig.", "REF ." ], [ "Comparison with quantum Monte Carlo simulation", "It is also interesting to compare the predictions of the lattice field theory ${\\mathcal {H}}^\\prime _{\\sf U(1)}$ [Eq.", "(REF )], with the results of finite-temperature quantum Monte Carlo simulations of a quantum charge ice described by ${\\mathcal {H}}_{\\sf t-V}$ [Eq.", "(REF )], as published by Banerjee et al. [59].", "Banerjee et al.", "performed their simulations for hard-core bosons on a pyrochlore lattice at half filling, with hopping integral $t=1$ , and nearest-neighbour repulsion $V=19.4$ , at temperatures $T = 1.05 g$ and $T = 1.57 g$ , where $g = 12 t^3/V^2$ is the size of the leading tunnelling matrix element between different charge ice configurations.", "In Fig.", "REF we plot simulation results for $S_{\\sf charge} (\\mathbf {k}, t=0)$ at these temperatures, calculated within a single sublattice of pyrochlore lattice sites, together with the best fit to Eq.", "(REF ), projected onto a single sublattice.", "We assume that the parameters of the field theory depend relatively weakly on temperature, and attribute the temperature dependence of correlations entirely to the thermal excitations of photons.", "Under these assumptions, the lattice field theory gives a good account of both the form and the temperature dependence of $S_{\\sf charge} (\\mathbf {k}, t=0)$ , within the error bars on points taken from simulation.", "These fits suggest a speed of light $c=(1.8 \\pm 0.2) \\ g \\ a_0 \\ \\hbar ^{-1}$ which is $\\sim 3$ times larger than that found in Section REF from finite size scaling of the ground state energy of ${\\mathcal {H}}_\\mu $ [Eq.", "(REF )].", "This discrepancy can probably be attributed to the fact that the simulations of Banerjee et al.", "were performed close to the melting point of the charge ice [59], where both interactions between photons, and tunnelling processes involving more than six lattice sites, are likely to play an important role.", "Since all of these processes will contribute to the rate at which the gauge field fluctuates in time, they can be expected to increase the speed of light.", "Figure: (Color online).Comparison of the predictions of the lattice field theory ℋ 𝖴(1) ' {\\mathcal {H}}^\\prime _{\\sf U(1)}[Eq.", "()] with the results of finite-temperature quantum Monte Carlo simulationsof a quantum charge ice described by ℋ 𝗍-𝖵 {\\mathcal {H}}_{\\sf t-V} [Eq.", "()].Results are shown for the equal time, on-sublattice structure factorS 00 (𝐤)=〈n 0 (-𝐤)n 0 (𝐤)〉S_{00}(\\mathbf {k})=\\langle n_0 (\\mathbf {-k}) n_0(\\mathbf {k}) \\rangle Simulations are taken from Banerjee et al.", ",and were performed at temperatures T=1.05gT = 1.05 g and T=1.57gT = 1.57 g,where g=12t 3 /V 2 g = 12 t^3/V^2 is the size of the leading tunnelling matrixelement between different charge ice configurations.The temperature dependence of the spin correlations makes it possibleto estimate the speed of light c≈1.8ga 0 ℏ -1 c \\approx 1.8 \\ g \\ a_0 \\ \\hbar ^{-1}." ], [ "Heat capacity and magnetic susceptibility at low temperatures", "Neutron scattering experiments have the potential to give decisive information about emergent electromagnetism in a quantum spin ice.", "However these experiments are expensive and difficult to perform, and depend critically on the size and quality of available samples.", "We therefore conclude with a few brief remarks on potential signatures of a quantum $U(1)$ liquid in thermodynamic quantities.", "The results given will hold in the low-temperature regime where the physics of a quantum spin-ice can be described as a gas of photons.", "At higher temperatures the thermal excitation of the gapped spinons (monopoles) and electric charges also play an important role.", "We have seen in Section REF how quantum fluctuations lead to an equal-time structure factor which, in the limit $k \\rightarrow 0$, vanishes at low temperatures as $\\lim _{k \\rightarrow 0} S(k, T) \\propto T$ This in turn implies a bulk magnetic susceptibility $\\chi (T)$ which is independent of temperature at low temperatures $\\chi ^{-1}(T \\ll g) = \\frac{3\\mathcal {U}}{\\kappa ^2}.$ where $\\mathcal {U}$ is the coefficient of $\\mathcal {B}^2$ in the effective Hamiltonian $\\mathcal {H}_{\\sf U(1)}$ [Eq.", "(REF )], and $\\kappa \\approx 1$ is the dimensionless scale factor introduced in Eq.", "(REF ).", "This result provides another means of parameterizing the lattice field theory.", "It is also a potentially useful diagnostic for experiment, since, a classical spin ice which remains in thermodynamic equilibrium at low temperatures, will exhibit an effective Curie law [81], [11] $\\chi ^{-1}(T) \\sim T$ This result follow directly from the fact that there are more spin ice configurations with vanishing magnetisation $\\mathbf {M} = 0$ than with any finite magnetisation $\\mathbf {M} \\ne 0$ , and so, in the absence of any other considerations, a state with $\\mathbf {M} = 0$ is selected by an entropic term $\\delta {\\mathcal {F}} = T \\delta {\\mathcal {S}}\\ \\sim T\\ \\mathbf {M}^2$ in the free energy [8].", "Nonetheless, any comparison with a classical spin ice should be approached with some caution, as these systems need not remain in equilibrium at low temperatures [82], [83], and the character of the spin fluctuations which control $\\chi (T)$ changes as a function of temperature [84], [16].", "As noted elsewhere [47], [70], the fact that photons are linearly dispersing excitations implies that they must make a $T^3$ contribution to the heat capacity at low temperatures.", "While this contribution has exactly the same temperature dependence as that from acoustic phonons, the large amount of entropy available in ice states, and low speed of light [cf.", "Section REF ], mean that the heat capacity at low temperatures will be dominated by photons.", "The photon contribution to the heat capacity per mole is $C_{\\sf photon}[mole]= B T^3$ with the coefficient B given by $B= \\left( \\frac{\\pi ^2}{30}\\right) R \\left( \\frac{k_B a_0}{\\hbar c}\\right)^3.$ From the characterisation of Yb$_2$ Ti$_2$ O$_7$ by Ross et al.", "[27], and the analysis of the speed of light in Section REF , we estimate $B \\approx 65\\ \\text{J}\\ \\text{mol}^{-1}\\ \\text{K}^{-4}$ which is several orders of magnitude larger than the expected phonon contribution.", "This should be compared with the value $1\\ \\text{J}\\ \\text{mol}^{-1}\\ \\text{K}^{-4}$ obtained in Ref. savary12.", "We note that, since the photons are magnetic excitations, measurements of the heat capacity in an applied magnetic field may also prove instructive." ], [ "Discussion and Conclusions", "In this paper we have developed a detailed theory for the simplest microscopic model which could describe quantum tunnelling between different spin ice configurations [Eq.", "(REF )].", "The striking claim that this type of model could support a spin liquid phase which perfectly mimics quantum electromagnetism [47] has been verified by quantum Monte Carlo simulations [59], [58].", "Here we have explored how such a quantum spin liquid might manifest itself in experiment, parameterizing an “electromagnetic” lattice gauge theory from quantum Monte Carlo simulations at zero temperature, and using this to calculate the dynamical structure factor $S^{\\alpha \\beta }(\\mathbf {k},\\omega )$ [Eq.", "(REF )] which would be measured in neutron scattering experiments at finite temperature.", "We find that a key signature of the emergent electromagnetism is the suppression of pinch points singularities in the energy-integrated structure factor $S^{\\alpha \\beta }(\\mathbf {k}, t=0)$ as the system is cooled to its zero-temperature ground state [Fig.", "REF ].", "This will coincide with the appearance of a gapless, linearly dispersing, mode — the photon of the lattice gauge theory —- in inelastic neutron scattering [Fig.", "REF ].", "In sharp contrast with a conventional antiferromagnet, the dispersing feature associated with this photon vanishes as $\\omega \\rightarrow 0$ .", "These photons will also strongly influence the low-temperature thermodynamic properties of the system, giving rise to a temperature-independent contribution to the magnetic susceptibility [Eq.", "(REF )] and an anomalously large $T^3$ contribution to specific heat [Eq.", "(REF )] [70].", "Neither the idea of “artificial light” [50], [53], [51], [52], [54], [55], nor the observation that there could be quantum tunnelling between different spin ice configurations [85], is new.", "However the possibility that one might lead to another is both new and exciting, and adds to the general frisson surrounding pyrochlore magnets.", "Without attempting to review all of this fast-developing field — but with the possibility of observing photons in mind — it is interesting to ask whether any of the materials currently studied “fit the bill”.", "The most widely studied example of a three-dimensional spin liquid is the insulating pyrochlore oxide Tb$_2$ Ti$_2$ O$_7$  [6].", "Tb$_2$ Ti$_2$ O$_7$ does not order down to 50 mK [gardner03], despite having a Curie-Weiss temperature $\\theta _{\\sf CW} \\sim 14$  K [gingras00], and a strong tendency to order under magnetic field or pressure [86], [87].", "In a series of papers, Gingras and coauthors have argued that Tb$_2$ Ti$_2$ O$_7$ is a “quantum spin ice”, in which spins fluctuate strongly about the crystallographic $[111]$ axes.", "These claims were made on the basis of a characteristic checkerboard structure observed in diffuse neutron scattering experiments at high temperatures [88], [19], and a subsequent microscopic analysis of crystal field levels [20], [89], and find support in the recent observation of partial magnetisation plateau for magnetic field applied along a $[111]$ axis [90], [21].", "Within this framework, the field at which the plateau is observed implies that the energy scale relevant for “quantum spin ice” behaviour in Tb$_2$ Ti$_2$ O$_7$ is $J_{\\sf eff} \\approx 0.2 K$ [90].", "Unfortunately, the interpretation of experiment at these low temperatures is muddied by questions of sample quality, with inconsistent results for spin-freezing obtained by different authors [24], [92], [91], [23], [93].", "Published thermodynamic data at low temperatures is also less than conclusive, showing hints of a saturation of $\\chi (T)$ at low temperatures, but strong sample dependence [92], [23], [93], [94], [95].", "And the picture is further complicated by strong fluctuations of the lattice [96], [97], with alternative theories of Tb$_2$ Ti$_2$ O$_7$ building on lattice effects [98], [99], [100] At present there is no published neutron scattering data for Tb$_2$ Ti$_2$ O$_7$ with the combination of k-resolution, energy resolution and low temperature needed to compare with the predictions in Section REF and Section REF of this paper.", "However recent evidence of “pinch-point” structure in quasi-elastic neutron scattering on single crystals of Tb$_2$ Ti$_2$ O$_7$  [25], taken together with inelastic neutron scattering experiments on powder samples [101], suggest that the comparison might be interesting.", "The latter find evidence of a quasi-elastic feature evolving into two bands of excitations at temperatures $T < 0.4$ K [101].", "If — and it remains a big IF — the behaviour of Tb$_2$ Ti$_2$ O$_7$ is connected with the physics of the quantum ice described in this paper, it would be tempting to identify these bands with the excitations of electromagnetism on a lattice — gapless photons, together with gapped “electric” and “magnetic” charges (spinons).", "But more, and more delicate, experiments will be needed to determine whether this is indeed the case.", "And ultimately, Tb$_2$ Ti$_2$ O$_7$ will remain a fascinating system to study.", "regardless of whether or not it is a quantum spin ice.", "Recently, there has also been intense experimental and theoretical interest in the closely-related Yb pyrochlore, Yb$_2$ Ti$_2$ O$_7$ .", "Originally identified in the pioneering survey of Blöte et al.", "[102] as a ferromagnet with $T_c=0.21$ K and $\\theta _{\\sf CW}=0.4$ K, Yb$_2$ Ti$_2$ O$_7$ differs from the classical spin ice materials Ho$_2$ Ti$_2$ O$_7$ and Dy$_2$ Ti$_2$ O$_7$ in that the lowest lying crystal field state is a Kramers doublet with easy-plane anisotropy [67], [68], [32].", "An XY ferromagnet on a pyrochlore lattice — modern estimates suggest $\\theta _{\\sf CW} \\approx 0.65$ K for Yb$_2$ Ti$_2$ O$_7$ [yasui03,hodges01] — would naturally be expected to order ferromagnetically at low temperatures.", "However Yb$_2$ Ti$_2$ O$_7$ exhibits a far more complicated phenomenology.", "Neutron scattering experiments at temperatures below 10K find diffuse liquid-like structure which offers evidence of anisotopic exchange interactions [26], [29].", "At temperature of order 1K, rod like structure emerges, reminiscent of a dimensional crossover [103], [104], [28], [26], [105].", "Some authors have found evidence of a first order transition into a ferromagnetically ordered state at $T_c=0.24$ K [106], [29], although this has been contested, and may not occur in all samples [67], [108], [107], [103], [104].", "That Yb$_2$ Ti$_2$ O$_7$ orders ferromagnetically in applied magnetic field is, however, uncontroversial.", "And this has made it possible for Ross et al.", "[27] to accurately characterise an exchange Hamiltonian for Yb$_2$ Ti$_2$ O$_7$ [Eq.", "(REF )] from fits to spin wave excitations in the polarised state.", "The parameters obtained confirm that the dominant interactions in Yb$_2$ Ti$_2$ O$_7$ favour “ice” states, but that these are complimented by terms which will drive significant fluctuations at low temperatures.", "Reassuringly, this description of Yb$_2$ Ti$_2$ O$_7$ is also in quantitative agreement with measurements of thermodynamic properties over a wide range of temperatures [30].", "This makes Yb$_2$ Ti$_2$ O$_7$ the best-characterised “quantum spin ice”, and as such, it is a natural place to look for emergent electromagnetism.", "However neutron scattering data with sufficient resolution to compare with the predictions of this paper are not, as yet, available.", "Tb$_2$ Ti$_2$ O$_7$ and Yb$_2$ Ti$_2$ O$_7$ are by no means the only pyrochlore systems with spin-liquid properties [6], and some of these other systems, notably Pr$_2$ Sn$_2$ O$_7$  [109], [31], [32] and Pr$_2$ Zr$_2$ O$_7$  [32], [33], [34] are also worth investigating as potential realisations of a quantum ice.", "It might also be interesting to revisit two-dimensional ice-type materials, such as the proton bonded ferroelectric copper formate tetrahydrate [41].", "While two-dimensional quantum ice models are known to order at low temperatures [110], [111], [112], [113], [114], [115], they are described by the same class of lattice gauge theory, and possess the same spinon excitations as their three-dimensional counterparts [40], [111], [115].", "These excitations will be confined in the ordered state, but might be visible at finite energy, and above the ordering temperature.", "Although the theoretical possibility of emergent electromagnetism in quantum ice [47], [48], [59], [58] and quantum dimer [46], [72], [60], [61] models is now well-established, many theoretical questions remain open.", "In this paper we have considered only the simplest microscopic model of a quantum spin ice [Eq.", "(REF )], and fully characterised only its photon excitations.", "The study of more realistic models, and of other excitations, is still in its infancy [70], [74], [30].", "We have also made no attempt to resolve the question of how the quantum ice state which we find at low temperatures, becomes a classical ice at high temperatures.", "All of these issues remain for future study.", "But we believe that the best motivation for studying them is experiment, and hope that the results in this paper will encourage further experiments on spin liquid materials which might realise artificial light." ], [ "Acknowledgements", "The authors are pleased to acknowledge helpful conversations with Steven Bramwell, John Chalker, Peter Fulde, Bruce Gaulin, Michel Gingras, Paul McClarty, Roderich Moessner, Karlo Penc, Frank Pollmann, Lucile Savary and Alan Tennant.", "We are particularly grateful to Tom Fennell and Radu Coldea for critical readings of the manuscript.", "This work was supported by EPSRC Grants EP/C539974/1 and EP/G031460/1.", "OS and NS gratefully acknowledge the hospitality of the guest program of MPI-PKS Dresden." ] ]

[ [ "Generalized Error Exponents For Small Sample Universal Hypothesis\n Testing" ], [ "Abstract The small sample universal hypothesis testing problem is investigated in this paper, in which the number of samples $n$ is smaller than the number of possible outcomes $m$.", "The goal of this work is to find an appropriate criterion to analyze statistical tests in this setting.", "A suitable model for analysis is the high-dimensional model in which both $n$ and $m$ increase to infinity, and $n=o(m)$.", "A new performance criterion based on large deviations analysis is proposed and it generalizes the classical error exponent applicable for large sample problems (in which $m=O(n)$).", "This generalized error exponent criterion provides insights that are not available from asymptotic consistency or central limit theorem analysis.", "The following results are established for the uniform null distribution: (i) The best achievable probability of error $P_e$ decays as $P_e=\\exp\\{-(n^2/m) J (1+o(1))\\}$ for some $J>0$.", "(ii) A class of tests based on separable statistics, including the coincidence-based test, attains the optimal generalized error exponents.", "(iii) Pearson's chi-square test has a zero generalized error exponent and thus its probability of error is asymptotically larger than the optimal test." ], [ "Introduction", "As an example of the application of the results, consider the following hypothesis testing problem.", "An i.i.d.", "sequence ${\\mbox{\\protect $Y$}}_1^n=\\lbrace Y_1, \\ldots , Y_n\\rbrace $ with $Y_i \\in [0,1]$ is observed.", "There are two hypotheses: Under the null hypothesis $H0$ , the probability measure induced by $Y_i$ is denoted by $P$ .", "Under the alternative hypothesis $H1$ , it is only known that the probability measure $Q$ induced by $Y_i$ satisfies $Q \\in \\mathcal {Q}$ .", "For simplicity of exposition, we assume in this section that $P$ is absolutely continuous with respect to the Lebesgue measure on $[0,1]$ , and the density is positive almost everywhere; $Q$ is absolutely continuous with respect to $P$ .", "The goal is to design a test $\\mathbf {\\phi }: [0,1]^n \\rightarrow \\lbrace 0,1\\rbrace $ with small probabilities of false alarm and missed detection: $P_F: ={\\sf P}_P\\lbrace \\phi _n({\\mbox{\\protect $Y$}}_1^n)=1\\rbrace , P_M: =\\sup _{Q \\in \\mathcal {Q}}{\\sf P}_Q\\lbrace \\phi _n({\\mbox{\\protect $Y$}}_1^n)=0\\rbrace .$ We consider a universal hypothesis testing problem, also called goodness of fit.", "It has the following form of $\\mathcal {Q}$ : $\\mathcal {Q}=\\lbrace Q: d(Q, P) \\ge \\varepsilon \\rbrace $ where $d$ is a distance function that could change with $n$ , and $\\varepsilon >0$ .", "As discussed in [3], if the distance function is the total variation distance or any distance function dominating the total variation distance, then there is no test that is asymptotically consistent: i.e.", "$P_F \\rightarrow 0$ and $P_M \\rightarrow 0$ as $n \\rightarrow \\infty $ .", "On the other hand, there is a consistent test if the distance function is the total variation distance defined on a finite partition of $[0,1]$ : Let $\\mathbf {A}=\\lbrace \\mathcal {A}_1, \\ldots , \\mathcal {A}_m\\rbrace $ be a partition of $[0,1]$ .", "The total variation distance defined on this partition is given by $d_\\mathbf {A}(Q,P)=\\sup _{\\mathcal {A} \\subset \\mathbf {A}}\\lbrace |Q(A)-P(A)|\\rbrace .$ As the number of observations $n$ increases, it is desirable for a test to not only have a decreasing probability of error, but also be effective against an increasingly larger alternative set $\\mathcal {Q}$ .", "Therefore, we consider a sequence of distance functions defined with increasingly finer partitions.", "We restrict ourselves to partitions in which the cells of the partition have equal probabilities under $P$ : $ P(A_j)=1/m \\textrm { for } 1\\le j \\le m.$ One reason to consider uniform cells, as argued in [4], is that the total-variation distance based on this partition gives the best possible distinguishability with respect the Kolmogorov-Smirnov distance: Consider the maximum Kolmogorov-Smirnov distance between the null distribution and any alternative distribution that has zero partition-based total variation distance to the null distribution.", "Then among any partitions with the same number of cells, the maximum Kolmogorov-Smirnov distance is minimized by the partition with uniform cells.", "The dependence between $n$ and $m$ plays a significant role on test analysis and synthesis: the small sample case in which $n/m\\rightarrow 0$ has a different nature than the large sample case in which $n/m \\rightarrow \\infty $ .", "In the large sample case, the number of samples per cell increases to infinity, and thus eventually the underlying probability that $Y_i$ falls in each cell of $\\mathbf {A}$ can be estimated.", "This does not hold for the small sample case, in which $m$ increases faster than $n$ .", "The goal of this paper is to find an appropriate analysis criterion for the small sample problem.", "In this section, we review related results with emphasis on the type of analysis used and the asymptotic settings considered.", "Many of the results reviewed apply to cases more general than (REF ).", "Examples of partitioned-based tests for small sample problems include Pearson's chi-square test, Generalized Likelihood Ratio Test (GLRT) and the coincidence-based test proposed in [5].", "Existing results differ in the asymptotic setting considered, which can be roughly classified into three cases: 1) $m$ is fixed; 2) $m$ is increasing and $m=O(n)$ ; 3) $n=o(m)$ and $m=o(n^2)$ .", "There is no need to consider the case $n=O(\\sqrt{m})$ because the converse result (lower-bounds on probability of error) established in [5] indicates that no asymptotically consistent test exists if $n=O(\\sqrt{m})$ .", "There are three predominant types of analysis: 1.", "Asymptotic consistency / sample complexity analysis: This type of analysis characterizes how fast $m$ can increase with $n$ , while still ensuring that $\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty } P_F<\\delta , \\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty } P_M < \\delta $ for any small $\\delta \\ge 0$ .", "Finer results on $P_F$ and $P_M$ are obtained in Central Limit Theorem (CLT) and large deviations analysis.", "2.", "CLT analysis: CLTs are applied to obtain asymptotic approximations of the distributions of the test statistic under both hypotheses.", "It is usually assumed that $\\varepsilon \\rightarrow 0$ as a function of $n$ , i.e., the set of alternative distributions becomes closer to the null distribution as $n$ increases.", "This ensures that the decision boundary of the test is close to both the null distribution and the alternative distributions, so that the probabilities of false alarm and missed detection can be analyzed using the CLT.", "Under this choice of $\\varepsilon $ , $P_F$ and $P_M$ usually converge to nonzero values.", "The results characterize how the limits of $P_F$ and $P_M$ differ for different tests.", "3. large deviations analysis: The normalized limits (or asymptotic expansions) of $\\log (P_F(\\phi ))$ and $\\log (P_M(\\phi ))$ are studied.", "The distance $\\varepsilon >0$ is held to be a constant in large deviations analysis.", "The proper normalization of $\\log (P_F(\\phi ))$ and $\\log (P_M(\\phi ))$ must first be identified, and then the normalized limits are calculated.", "Consider the case where $m$ is fixed.", "Pearson's chi-square statistic and GLRT statistic are asymptotically distributed as a chi-square distribution whose degree of freedom is $m-1$ .", "These results and their extensions can be found in [6], [7], [8], [9], [10], [11].", "The performance of Pearson's chi-square test and GLRT is analyzed in [12] using the large deviations analysis.", "The following error exponent criterion is used to evaluate a test $\\mathbf {\\phi }$ : $\\begin{aligned}I_F(\\mathbf {\\phi }) & : =-\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty } \\frac{1}{n}\\log (P_F(\\phi _n)),\\\\I_M(\\mathbf {\\phi }) &: =-\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty } \\frac{1}{n}\\log (P_M(\\phi _n)).\\end{aligned}$ The GLRT is shown to have optimal error exponents while Pearson's chi-square test does not.", "Our use of the term error exponent follows [13].", "Next consider the case $m=O(n)$ .", "Pearson's chi-square test and GLRT are both asymptotically consistent (For example, see [14]).", "Pearson's chi-square statistic and the GLRT statistic both have asymptotically normal distributions.", "The first work in this line is [15].", "Extensions and applications of this result can be found in [16], [17], [18], [19], [20], [21].", "A lower-bound on the best achievable probability of error in CLT analysis is given in [14]: Under the condition $0 \\!<\\!", "\\mathop {\\rm lim{\\,}inf}_{n \\rightarrow \\infty } \\!\\frac{\\varepsilon }{\\sqrt{m}} \\!\\le \\!", "\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty } \\!\\frac{\\varepsilon }{\\sqrt{m}} \\!< \\!\\infty $ , Pearson's chi-square test is asymptotically optimal.", "That is, for any test whose limit of $P_F$ is no larger than that of Pearson's chi-square test, the limit of its $P_M$ is asymptotically no smaller than that of Pearson's chi-square test.", "This result applies to the range of $m$ satisfying $m=o(n^2)$ .", "An achievability result (A lower-bound on the error exponent) and a complementing converse result (An upper-bound on the error exponent) in the large deviations analysis have been obtained in [3]: There exists a test for which $P_F$ and $P_M$ both decay exponentially fast with respect to $n$ , i.e., $I_F$ and $I_M$ defined in (REF ) are both nonzero, if and only if $m=O(n)$ .", "Other large deviations and moderate-deviations analyses of GLRT and Pearson's chi-square test can be found in [22], [23], [24], [25], [26], [27] Finally consider the small sample case where $n=o(m)$ and $m=o(n^2)$ .", "Pearson's chi-square test is known to be asymptotically consistent [14].", "Two others tests shown to be asymptotically consistent is the test based on counting pairwise-collisions [28] and the coincidence-based test [5].", "An approach to extend tests designed for uniform cells (REF ) to non-uniform cells has been proposed in [29].", "Results on the asymptotic distribution of Pearson's chi-square statistic and the GLRT statistic have been obtained in [30], [31].", "To the best of our knowledge, the proper normalization for the large deviations analysis has not been identified before in the small sample case.Combining the upper-bounds on probability of error given in [5], [29] with the Chernoff inequality gives a loose upper-bound on the asymptotic probability error and does not yield the proper normalization.", "We note that the classical error exponent analysis is not suitable." ], [ "Our contributions", "The new large deviations framework proposed here is motivated by and analogous to the classical error exponent (REF ) in the large sample case.", "While the classical error exponent is defined with the normalization $n$ , our main results imply that for the small sample problem the following generalized error exponent is best for asymptotic analysis, defined with respect to the normalization $r(m,n)=n^2/m$ : $\\begin{aligned}J_F(\\mathbf {\\phi }) &: =-\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty } \\frac{1}{r(m,n)}\\log (P_F(\\phi _n)),\\\\\\quad J_M(\\mathbf {\\phi })& : =-\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty } \\frac{1}{r(m,n)}\\log (P_M(\\phi _n)).\\end{aligned}$ The generalized error exponents give the following approximation to the probabilities of false alarm and missed detection: $P_F \\unknown.", "e^{-r(n,m) J_F}, \\quad P_M \\unknown.", "e^{-r(n,m) J_m}.$ The generalized error exponent provides new insights that are not available from asymptotic consistency, or CLT analysis.", "More precisely, the following results are established: 1.", "The best achievable probability of error $P_e\\!=\\!\\max \\lbrace P_F,P_M\\rbrace $ , decays as $-\\log (P_e)=r(n,m)J(1+o(1))$ , where $r(n,m)=n^2/m$ .", "This is applicable not only for the case where the set of alternative distributions is defined by the total variation distance in (REF ), but also for a broad collection of distance functions.", "2.", "A class of tests based on the separable statistics, including the coincidence-based test $\\phi ^{{*}}$ , is shown to achieve the optimal pair of generalized error exponents $J_F$ and $J_M$ : $J_M(\\phi ^{{*}})=\\max \\lbrace J_M(\\phi ): J_F(\\phi ) \\ge J_F(\\phi ^{{*}})\\rbrace .$ The exact formulae for these generalized error exponents are obtained.", "3.", "The performance of Pearson's chi-square test is asymptotically worse than the optimal test." ], [ "Organization of the paper", "The paper is organized as follows: The universal hypothesis testing problems and tests are presented in Section .", "The main achievability and converse results on generalized error exponents are described in Section .", "Extensions of the coincidence-based test are given in Section .", "Performance characterization of Pearson's chi-square test is given in Section .", "In Section , it is shown that the generalized error exponent criterion is also applicable when the set of alternative distributions is defined using many other distance functions.", "The paper is concluded in Section ." ], [ "Models and Preliminaries", "Here we introduce a more general model based on a sequence of universal hypothesis testing problems, each with a finite number of outcomes (a finite alphabet).", "Consider an i.i.d.", "sequence of observations ${\\mbox{\\protect $Z$}}_1^n: =\\lbrace Z_1, \\ldots , Z_n\\rbrace $ where $Z_i \\in [m]: =\\lbrace 1,2,\\ldots , m\\rbrace $ .", "Let ${{\\cal P}_m}$ denote the collection of probability mass functions (p.m.f.s) on $[m]$ .", "We have two hypotheses: Under the null hypothesis $H0$ , the p.m.f of $Z_i$ is given by $p$ , the uniform distribution on $[m]$ : $ p_j=1/m \\textrm { for } j \\in [m].$ Under the alternative hypothesis $H1$ , the p.m.f.", "of $Z_i$ belongs to a set ${\\mathcal {Q}_n}$ given by ${\\mathcal {Q}_n}: =\\lbrace q\\in {{\\cal P}_m}: d(q, p) \\ge \\varepsilon \\rbrace $ where $d$ is taken to be the total variation distance $d_{TV}$ defined for any pair of p.m.f.s on $[m]$ : $d_{TV}(q, p)=\\sup _{B \\subseteq [m]}\\lbrace |q(B)-p(B)|\\rbrace .$ A test $\\mathbf {\\phi }=\\lbrace \\phi _n\\rbrace _{n \\ge 1}$ is given by a sequence of binary-valued functions $\\phi _n: [m]^n \\rightarrow \\lbrace 0,1\\rbrace $ .", "The test decides in favor of $H0$ if $\\phi _n({\\mbox{\\protect $Z$}}_1^n)=0$ .", "The test is required to be powerful against the set ${\\mathcal {Q}_n}$ of alternative p.m.f.s, and thus its performance is evaluated using the probabilities of false alarm $P_F(\\phi _n)$ and worst-case probability of missed detection $P_M(\\phi _n)$ : $\\begin{aligned}&P_F(\\phi _n): ={\\sf P}_p\\lbrace \\phi _n({\\mbox{\\protect $Z$}}_1^n)=1\\rbrace , \\\\&P_{M,q}(\\phi _n): ={\\sf P}_q\\lbrace \\phi _n({\\mbox{\\protect $Z$}}_1^n)=0\\rbrace ,\\\\&P_M(\\phi _n): =\\sup _{q\\in {\\mathcal {Q}_n}}{\\sf P}_q\\lbrace \\phi _n({\\mbox{\\protect $Z$}}_1^n)=0\\rbrace .\\end{aligned}$ An important class of tests is based on the separable statistics (see [30]).", "A separable statistic is a test statistic of the form $S_n=\\sum _{j=1}^m f_j(n\\Gamma ^n_j),$ where $\\Gamma ^n_j : =\\frac{1}{n}\\sum _{i=1}^n \\mathbb {I}\\lbrace Z_i=j\\rbrace $ is the empirical distribution.", "General theorems on asymptotic distributions and asymptotic moments of separable statistics are available in [30].", "Large deviations analysis for the case $m=O(n)$ is given in [25], [26].", "We are not aware of previous general large deviations results for the small sample case where $n=o(m)$ .", "In this paper, we examine two tests based on separable statistics: Pearson's chi-square test [32] and the coincidence-based test introduced in [5].", "After normalization, the test statistic of Pearson's chi-square test is given by $S_n^{\\sf P}= \\frac{n}{m}\\sum _{j=1}^m \\frac{(n\\Gamma ^n_j-np_j)^2}{np_j}.$ The test is given by $\\phi ^{{\\sf P}}_n({\\mbox{\\protect $Z$}}_1^n)=\\mathbb {I}\\lbrace S_n^{\\sf P}\\ge \\tau _n\\rbrace $ .", "The test statistic of the coincidence-based test is given by, $S_n^{*}=-\\sum _{j=1}^m \\mathbb {I}\\lbrace n\\Gamma ^n_j=1\\rbrace .$ This test statistic $S_n^{*}$ counts the number of symbols in $[m]$ that appear in the sequence exactly once.", "The coincidence-based test is given by $\\phi ^{{*}}_n({\\mbox{\\protect $Z$}}_1)=\\mathbb {I}\\lbrace S_n^{*}\\ge {\\sf E}_p[S_n^{*}]+\\tau _n\\rbrace $ .", "The coincidence-based test is applicable only when the null distribution is uniform.", "An important difference between $S_n^{*}$ and $S_n^{\\sf P}$ is that $f_j$ is bounded in $S_n^{*}$ , while this is not true in $S_n^{\\sf P}$ .", "In Section , we show that this difference has a significant impact on tests' performance," ], [ "Applications to continuously-valued observations", "Tests designed for finite-valued observations can be applied to solve a universal hypothesis testing problem with continuously-valued observations by first partitioning observation space.", "Consider a measurable space $(\\mathbf {Y}, \\mathcal {B})$ , and let ${\\mbox{\\protect $Y$}}_1^n=\\lbrace Y_1, \\ldots , Y_n\\rbrace $ be an i.i.d.", "sequence of observations with $Y_i \\in \\mathbf {Y}$ .", "We have two hypotheses: $H0: Y_i \\sim P, \\quad H1: Y_i \\sim Q \\in \\mathcal {Q}$ To apply a test designed for the finite-valued observations, we start with a partition of $\\mathbf {Y}$ : $\\mathbf {A}=\\lbrace A_1, \\ldots , A_{m}\\rbrace $ where $\\cup _{1 \\le j \\le m} A_j=\\mathbf {Y}$ .", "The observation $Y_i$ is mapped to a finite-valued observation via $\\mathcal {T}: \\mathbf {Y} \\rightarrow [m]$ : $Z_i : =\\mathcal {T}(Y_i)=j$ if $Y_i \\in A_j$ .", "Then a test defined for finite-valued observations can be applied towards $\\lbrace Z_i\\rbrace $ .", "Assume that the partition is chosen so that the marginal of ${Z_i}$ is uniform, $ P(A_j)=\\frac{1}{m}.$ Then tests designed for a uniform null distribution are applicable, such as the coincidence-based test.", "This partition-based approach gives tests that are optimal for the model introduced in Section .", "More precisely, suppose that the set of alternative distributions is defined as $\\mathcal {Q}=\\lbrace Q: d_{\\mathbf {A}}(Q,P) \\ge \\varepsilon \\rbrace $ where $d_\\mathbf {A}$ is defined in (REF ).", "Then in terms of the probability of false alarm and worst-case probability of missed detection, without loss of optimality we can restrict our attention to tests whose test statistics take constant value on each cell $A_j$ of the partition.", "This is exactly the collection of partition-based tests we have described.", "The model introduced in Section  assumes that the alternative distribution $Q$ is absolutely continuously with respect to $P$ .", "The partition-based tests are still applicable when $Q$ is not absolutely continuous with respect to $P$ , provided that the tests for finite-valued observations are designed for a more general model where we allow $p$ not to have full support: Instead of (REF ), let the null distribution $p$ be $p_j=1/k \\textrm { for } 1\\le j \\le k, p_j=0 \\textrm { for } k<j\\le m.$ The generalized error exponent analysis still applies except the normalization should be $n^2/k$ instead of $n^2/m$ ." ], [ "Generalized Error Exponents", "In this section, we describe the main results on the proper normalization for large deviations analysis for the small sample universal hypothesis testing problem.", "The following assumption is imposed throughout: Assumption 1 $n=o(m)$ and $m=o(n^2)$ .", "To show that the proper normalization to be used in the definition of generalized error exponent is $n^2/m$ , we need to establish: There is a test for which both generalized error exponents are non-zero, and therefore this normalization is not too large.", "For any test, at least one of the generalized error exponents is finite, and therefore this normalization is not too small.", "These are established in Theorem REF and Theorem REF .", "These two theorems characterize the achievable region of $(J_F, J_M)$ .", "This is depicted in Fig.", "REF .", "The boundary of the achievable region is given by the following formulae: For $\\tau \\in [0,\\underline{\\kappa }(\\varepsilon )-1]$ , $\\begin{aligned}&J^*_F(\\tau ) : =\\sup _{\\theta \\ge 0} \\lbrace \\theta \\tau -{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\bigl (e^{2\\theta }-(1+2\\theta )\\bigr )\\rbrace ,\\\\&J^*_M(\\tau ) : =\\sup _{\\theta \\ge 0} \\lbrace \\theta (\\underline{\\kappa }(\\varepsilon ) -1 - \\tau )-{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\bigl (e^{-2\\theta } -(1-2\\theta )\\bigr )\\underline{\\kappa }(\\varepsilon )\\rbrace ,\\end{aligned}$ where $\\underline{\\kappa }: \\mathbb {R}_{+} \\rightarrow \\mathbb {R}_{+}$ is the $C^{1}$ function, $\\underline{\\kappa }(\\varepsilon )=\\left\\lbrace \\begin{array}{c c}1+4\\varepsilon ^2, & \\varepsilon < 0.5,\\\\1+{\\varepsilon }/{(1-\\varepsilon )},& \\varepsilon \\ge 0.5.\\end{array}\\right.$ Theorem 1 (Achievability) The coincidence-based test $\\phi ^{{*}}$ achieves the generalized error exponents given in (REF ), i.e., for any $\\tau \\in [0, \\underline{\\kappa }(\\varepsilon )-1]$ , if the sequence of thresholds $\\lbrace \\tau _n\\rbrace $ is chosen so that, $\\tau =\\lim _{n \\rightarrow \\infty } m\\tau _n/n^2,$ then the coincidence-based test has the generalized error exponents: $J_F(\\phi ^{{*}})=J^*_F(\\tau ), \\quad J_F(\\phi ^{{*}})=J^*_M(\\tau ).$ Theorem 2 (Converse) Consider any $\\tau \\in [0,\\kappa (\\varepsilon )-1]$ .", "For any test $\\phi $ satisfying $J_F(\\phi )\\ge J_F^*(\\tau ),\\nonumber $ the following upper-bound on the generalized error exponent of missed detection holds: $J_M(\\phi )\\le J_M^*(\\tau ).\\nonumber $ Figure: Achivable region when ε=0.35\\varepsilon =0.35 and ε=0.45\\varepsilon =0.45 given by the lower-bound in Theorem  and upper-bound in Theorem .", "The lower and upper bound meet over the entire region.We now the approximation in (REF ) given by the generalized error exponent analysis to the actual empirical performance of the coincidence-based test $\\phi ^{{*}}$ .", "The results are shown in Fig.", "REF for $\\varepsilon =0.35$ and Fig.", "REF for $\\varepsilon =0.45$ .", "We choose the threshold $\\tau $ based on (REF ) so that $J_F$ and $J_M$ are the same.", "The generalized error exponents are estimates of the slope of $\\log (P_F)$ and $\\log (P_M)$ with respect to $r(n,m)$ .", "It can be observed that the slope from the theoretical approximation by generalized error exponents approximately matches the slope of the simulated value.", "The remaining difference between the theoretical and the empirical slope in Fig.", "REF is mainly due to two reasons: First, the threshold chosen is based on the first order approximation.", "It can be observed from the figure that the slope for $P_M$ is slightly smaller than the predicted slope while the one for $P_F$ is larger.", "A slightly larger threshold might yield slopes that are closer to the predicted.", "Second, the generalized error exponent is only the first term in the asymptotic expansion of $\\log (P_F)$ and $\\log (P_M)$ .", "Higher order terms might capture the remaining difference.", "Figure: Performance of φ * \\phi ^{{*}} with ε=0.35\\varepsilon =0.35.Figure: Performance of φ * \\phi ^{{*}} with ε=0.45\\varepsilon =0.45." ], [ "Rate function and worst-case distributions", "Similar to the large deviations for the large sample case, we can define a rate function for the small sample case.", "Consider the coincidence-based test $\\phi ^{{*}}$ .", "Consider the following restricted set of alternative distributions: $\\mathcal {P}^b_m=\\lbrace q\\in {{\\cal P}_m}: \\max _j q_j \\le \\gamma /m\\rbrace ,$ where $\\gamma $ is a large positive constant satisfying $\\gamma \\ge \\max \\lbrace 2/(1-\\varepsilon ), 4\\varepsilon \\rbrace $ .", "This restricted set of distributions has bounded likelihood ratios with respect to the uniform distribution $p$ .", "The rate function for this test is associated with a sequence of distributions ${\\mbox{\\protect $q$}}=\\lbrace q^{(1)},q^{(2)}, q^{(3)}, \\ldots \\rbrace $ with $q^{(n)} \\in \\mathcal {P}^b_m$ as follows: $J_{\\mbox{\\protect $q$}}(\\phi ^{{*}}, \\tau )\\!=\\!-\\!\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty }\\frac{m}{n^2} \\log ({\\sf P}_{q^{(n)}}\\lbrace S_n^{*}\\le {\\sf E}_{p}[S_n^{*}]\\!+\\!\\frac{n^2}{m}\\tau \\rbrace ).\\nonumber $ We show that $J$ is a function of the following quantity: $\\kappa ({\\mbox{\\protect $q$}}): =\\mathop {\\rm lim{\\,}inf}_{n}\\sum _{j}\\frac{(q^{(n)}_j)^2}{p_j}.$ Theorem 3 ${J}_{\\mbox{\\protect $q$}}(\\phi ^{{*}},\\tau ) = \\sup _{\\theta \\ge 0} \\lbrace \\theta (-1-\\tau )-{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(e^{-2\\theta }-1)\\kappa ({\\mbox{\\protect $q$}})\\rbrace .$ Its proof is given in Appendix .", "The rate function can be applied to identify the sequence of worst-case alternative distributions, for which the probability of missed detection is asymptotically the largest.", "Note that $J_{\\mbox{\\protect $q$}}(\\phi ^{{*}},\\tau )$ is monotonically increasing in $\\kappa ({\\mbox{\\protect $q$}})$ .", "Therefore, the smaller the quantity $\\kappa ({\\mbox{\\protect $q$}})$ , the larger the probability of missed detection associated with ${\\mbox{\\protect $q$}}$ .", "The sequence of distributions achieving the minimum $\\kappa ({\\mbox{\\protect $q$}})$ is given in the following lemma: Lemma 1 When $p$ is the uniform distribution, we have $\\inf _{q\\in {\\mathcal {Q}_n}} \\bigl (\\sum _{j=1}^m \\frac{q_j^2}{p_j}\\bigr )=(1+\\underline{\\kappa }(\\varepsilon ))(1+o(1)).$ The infimum is achieved by the following bi-uniform distribution: 1.", "When $\\varepsilon < 0.5$ , $q_{j}^*=\\left\\lbrace \\begin{array}{c c}{1}/{m}+{\\varepsilon }/{\\lfloor m/2 \\rfloor }, & j \\le \\lfloor m/2 \\rfloor ,\\\\ {1}/{m}-{\\varepsilon }/{\\lceil m/2 \\rceil }, & j> \\lfloor m/2 \\rfloor .\\end{array}\\right.$ 2.", "When $\\varepsilon \\ge 0.5$ , $q_{j}^{*}=\\left\\lbrace \\begin{array} {c c} {1}/{\\lfloor m(1-\\varepsilon )\\rfloor }, & j \\le \\lfloor m(1-\\varepsilon )\\rfloor ,\\\\ 0, & j > \\lfloor m(1-\\varepsilon )\\rfloor .\\end{array}\\right.$ Thus, the worst case distributions are identified as bi-uniform distributions whose p.m.f.s only take two possible values.", "[Proof of Lemma REF ] The main task is to show that any optimizer $q^*$ is a bi-uniform distribution.", "The formulae (REF ) and (REF ) follow from solving the optimization in (REF ) restricted to bi-uniform distributions.", "Let $\\mathcal {J_+}=\\lbrace j: q^*_j \\ge p_j\\rbrace $ , $\\mathcal {J_-}=\\lbrace j: q^*_j <p_j\\rbrace $ .", "The following quadratic programming problem has a unique optimal solution $x^*=q^*$ : $\\begin{array}{c c}\\min & \\sum _{j \\in \\mathcal {J_+}} x_j^2,\\\\{\\sf {s.t.}}", "& \\sum _{j \\in \\mathcal {J_+}} x_j=\\sum _{j \\in \\mathcal {J_+}}q^*_j,\\\\& x_j=q^*_j \\textrm { for j \\in \\mathcal {J_-}},\\\\& x_j \\ge p_j \\textrm { for j \\in \\mathcal {J_+}}.\\end{array}$ By Jensen's inequality, $x^*$ must satisfy $x^*_j=x^*_{j^{\\prime }} \\textrm { for all $ j, j' J+$}$ .", "Thus, $q^*$ also satisfies $q^*_j=q^*_{j^{\\prime }} \\textrm { for all $ j, j' J+$}$ .", "The same conclusion holds for $j \\in \\mathcal {J_-}$ .", "Consequently, $q^*$ must be a bi-uniform distribution." ], [ "Sketch of the proofs for Theorem ", "The large deviations characterization of $P_F$ for the coincidence-based test follows from the following asymptotic approximation of the logarithmic moment generating function of its test statistic: $\\begin{aligned}&\\log \\bigl ({\\sf E}_{p}[\\exp \\lbrace \\theta (n-S_n^{*})\\rbrace ]\\bigr )={\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}\\bigl (m\\sum _{j=1}^mp_j^2\\bigr )(e^{-2\\theta }-1)+O(\\frac{n^3}{m^2})+O(1).\\nonumber \\end{aligned}$ A characterization of $P_M$ is obtained in similar way except we need to work with the set of alternative distributions.", "We show that the probability of missed detection is dominated by that associated with the worst-case distributions given in Lemma REF .", "The details are given in Appendix .", "The main idea to prove the converse result is the following: A sequence of events $\\lbrace B_{n, \\tau ,\\delta }\\rbrace $ is constructed so that (i) the probability of these events can be lower-bounded based on the condition on $P_F$ ; (ii) the probability of missed detection conditioned on these events is lower-bounded.", "The key to the proof is the following inequality: $\\begin{aligned}&P_M(\\phi _n)\\ge \\sup _{q\\in {\\mathcal {Q}_n}}{\\sf P}_{q}\\bigl (\\lbrace \\phi _n=0\\rbrace \\cap B_{n, \\tau ,\\delta }\\bigr )\\ge \\sup _{q\\in {\\mathcal {Q}_n}}\\frac{q^n}{p^n}(\\lbrace \\phi _n=0\\rbrace \\cap B_{n, \\tau ,\\delta }){\\sf P}_p(\\lbrace \\phi _n=0\\rbrace \\cap B_{n, \\tau ,\\delta }).\\nonumber \\end{aligned}$ A lower-bound on the second term follows from the construction of the events and the assumption on the probability of false alarm.", "To lower-bound the first term, we construct a collection of distributions over which the largest likelihood ratio is always lower-bounded on the event $B_{n,\\tau ,\\delta }$ .", "These distributions are obtained by taking the worst-case distribution $q^*$ given in (REF ) and permuting the symbols in $[m]$ .", "Let $U_m$ denote the collection of all subsets of $[m]$ whose cardinality is $\\lfloor m/2 \\rfloor $ .", "For each set $\\mathcal {U} \\in U_m$ , define the distribution $q_{\\mathcal {U}}$ as $q_{\\mathcal {U},j}=\\left\\lbrace \\begin{array}{l l}{1}/{m}+{\\varepsilon }/{\\lfloor m/2 \\rfloor }, & j \\in \\mathcal {U};\\\\ {1}/{m}-{\\varepsilon }/{\\lceil m/2 \\rceil }, & j \\in [m]\\setminus \\mathcal {U}.\\end{array}\\right.$ Then a lower-bound is established for $\\sup _{\\mathcal {U} \\in U_m}\\frac{q_{\\mathcal {U}}^n}{p^n}(\\lbrace \\phi _n=0\\rbrace \\cap B_{n, \\tau ,\\delta }).$ The details are given in Appendix ." ], [ "Extensions of the Coincidence-Based Test", "This section collects together extensions of Section  in terms of tests and models.", "We first propose a collection of tests that extend the coincidence-based test, and provide the freedom for fine-tuning the performance for finite samples.", "We then propose an extension of the coincidence-based test for non-uniform $p$ ." ], [ "Extensions considering symbols appearing more than once", "The coincidence-based test uses only the number of symbols that appear in the sequence exactly once.", "We now add terms to the test statistic that also depend on the number of symbols appearing more than once to create a broader collection of tests.", "Conditions will be established under which these tests have optimal generalized error exponents.", "Consider the class of test statistics of the following form: $S_n^{*+}=S_n^{*}+ \\sum _{l=2}^{\\bar{l}} v_l \\mathbb {I}\\lbrace n\\Gamma _j^n=l\\rbrace .$ The test is given by $\\phi ^{{*+}}({\\mbox{\\protect $Z$}}_1)=\\mathbb {I}\\lbrace S_n^{*+}-{\\sf E}_{p}[S_n^{*+}] \\ge \\tau _n\\rbrace .$ Theorem 4 If $\\bar{l}<\\infty $ , $v_2=0$ , and $v_l \\ge 0$ for all $3 \\le l \\le \\bar{l}$ , then the test $\\phi ^{{*+}}$ achieves the optimal generalized error exponents given in (REF ).", "Its proof is given in Appendix .", "The additional terms for $l\\ge 3$ in the separable statistic give us ways to fine-tune the test for a better finite-sample performance.", "One interesting question is to obtain finer asymptotic approximations of $\\log (P_F)$ and $\\log (P_M)$ that provide guidance on how to select the weights $\\lbrace v_l\\rbrace $ .", "For the case with $v_2 \\ne 0$ , we have the following conjecture: Conjecture 1 If $S_n^{*+}$ satisfies $\\bar{l}<\\infty $ , $v_2>-2$ , and $v_l \\ge 0$ for all $3 \\le l \\le \\bar{l}$ , then the test is optimal in terms of the generalized error exponent." ], [ "Extensions to non-uniform $p$", "The coincidence-based test can be extended to the case where $p$ is not necessarily uniform but the likelihood ratio between $p$ and the uniform distribution remains bounded.", "Assumption 2 There exists a constant $\\eta >0$ such that $\\max _j mp_j \\le \\eta $ holds for all $n$ .", "The following separable statistic is considered, $S_n^{\\sf W}=\\sum _{j=1}^m f_j(n\\Gamma ^n_j)$ with $f_j(n\\Gamma ^n_j)=\\left\\lbrace \\begin{array}{c c}{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2 p_j^2, & n\\Gamma ^n_j=0,\\\\-np_j, & n\\Gamma ^n_j=1,\\\\1, & n\\Gamma ^n_j=2,\\\\0, & \\textrm {others}.\\end{array}\\right.$ The weighted coincidence-based test is given by $\\phi ^{{\\sf W}}_n=\\mathbb {I}\\lbrace S_n^{\\sf W}\\ge \\tau _n\\rbrace $ .", "The choice of coefficients given in (REF ) ensures ${\\sf E}_\\nu [S_n^{\\sf W}]$ approximates the $\\ell _2$ -distance between $\\nu $ and $p$ : Lemma 2 For $\\nu \\in \\mathcal {P}^b_m$ , the expectation of $S_n^{\\sf W}$ is given by: ${\\sf E}_{\\nu }[S_n^{\\sf W}]={\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m} [ m \\sum _{j=1}^m (\\nu _j-p_j)^2]+O(\\frac{n^3}{m^2}).$ The proposed test has nonzero generalized error exponents: Theorem 5 Suppose Assumption REF and Assumption REF hold.", "For $\\tau \\in (0, 2\\varepsilon ^2)$ where $\\tau $ is defined in (REF ), the test $\\phi ^{{\\sf W}}$ has nonzero generalized error exponents: $J_F(\\phi ^{{\\sf W}}) >0, \\quad J_M(\\phi ^{{\\sf W}}) >0.$ Its proof is given in Appendix ." ], [ "Pearson's Chi-Square Test", "In this section, we investigate the performance of Pearson's chi-square test given in (REF ).", "We find that this test has a zero generalized error exponent, and therefore its probability of error is asymptotically larger than that of the coincidence-based test.", "Pearson's chi-square test is asymptotically consistent in the small sample case: Proposition 1 (Asymptotic consistency) Under Assumption REF , there exists a sequence of thresholds $\\lbrace \\tau _n\\rbrace $ , with which the Pearson's chi-square test is asymptotically consistent: $\\lim _{n \\rightarrow \\infty }P_F(\\phi ^{{\\sf P}}_n)=0, \\quad \\lim _{n \\rightarrow \\infty }P_M(\\phi ^{{\\sf P}}_n)=0.$ We give a proof that highlights the relationship between Pearson's chi-square test and the coincidence-based test.", "[Proof of Proposition REF ] Let $\\tau _n=n+{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}(\\underline{\\kappa }(\\varepsilon )-1)$ .", "Applying approximations of moments of separable statistic given in Lemma REF and Lemma REF , we obtain $ \\begin{aligned}{\\sf E}_{p}[S_n^{\\sf P}]\\!&=\\!n\\!+\\!O(\\frac{n^3}{m^2}), \\\\\\hbox{\\sf Var}\\,_{p}[S_n^{\\sf P}]\\!&=\\!2\\frac{n^2}{m}(m\\sum _{j=1}^mp_j^2)(1+o(1)).\\!\\end{aligned}$ Applying Chebyshev's inequality gives $\\lim _{n \\rightarrow \\infty }P_F(\\phi ^{{\\sf P}}_n)=0$ .", "We bound $P_M(\\phi ^{{\\sf P}}_n)$ by coupling Pearson's chi-square statistic $S_n^{\\sf P}$ with the coincidence-based test statistic $S_n^{*}$ : $\\begin{aligned}S_n^{\\sf P}&=\\sum _{j=1}^m (n\\Gamma ^n_j-np_j)^2=\\sum _{j=1}^m (n\\Gamma ^n_j)^2-\\frac{n^2}{m}\\ge 2\\sum _{j=1}^n \\mathbb {I}\\lbrace n\\Gamma ^n_j\\ge 2\\rbrace n\\Gamma ^n_j+ \\sum _{j=1}^m\\mathbb {I}\\lbrace n\\Gamma ^n_j=1\\rbrace -\\frac{n^2}{m}=2n+S_n^{*}-\\frac{n^2}{m},\\end{aligned}$ where the inequality follows from $(n\\Gamma ^n_j)^2\\ge 2(n\\Gamma ^n_j)$ when $n\\Gamma ^n_j > 1$ .", "Consequently, $\\lbrace S_n^{\\sf P}\\le \\tau _n\\rbrace \\subseteq \\lbrace S_n^{*}\\le \\tau _n-2n+\\frac{n^2}{m}\\rbrace .$ The asymptotic approximation on the expectation of $S_n^{*}$ obtained from Lemma REF gives $\\tau _n-2n+\\frac{n^2}{m}={\\sf E}_{p}[S_n^{*}]+{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}(\\underline{\\kappa }(\\varepsilon )-1)+O(\\frac{n^3}{m^2}).$ It follows from Theorem REF that the coincidence-based test is asymptotically consistent.", "Thus $\\lim _{n \\rightarrow \\infty }\\sup _{ q\\in {\\mathcal {Q}_n}}{\\sf P}_q\\lbrace S_n^{*}\\le \\tau _n-2n+\\frac{n^2}{m}\\rbrace =0.$ Applying (REF ), we obtain $\\lim _{n \\rightarrow \\infty }\\sup _{ q\\in {\\mathcal {Q}_n}}{\\sf P}_q\\lbrace S_n^{\\sf P}\\le \\tau _n\\rbrace =0.$ However, the probability of false alarm of Pearson's chi-square test is asymptotically larger than that the coincidence-based test: We show that its generalized error exponent of false alarm is zero: Theorem 6 Suppose Assumption REF hold.", "Assume in addition that $m=o(n^2/\\log (n)^2)$ .", "If the sequence of thresholds is chosen so that $\\lim _{n \\rightarrow \\infty } P_M(\\phi ^{{\\sf P}}_n) =0,$ then the generalized error exponent of false alarm is zero, i.e., $J_F(\\phi ^{{\\sf P}}) =0.$ We conjecture that the conclusion holds without the assumption $m=o(n^2/\\log (n)^2)$ .", "We now compare Pearson's chi-square test and the coincidence-based test.", "Note that Pearson's chi-square test statistic can be written as $\\begin{aligned}S_n^{\\sf P}=&-\\frac{n^2}{m}+\\sum _{j=1}^m \\mathbb {I}\\lbrace n\\Gamma _j^n=1\\rbrace +\\sum _{j=1}^m 4\\mathbb {I}\\lbrace n\\Gamma _j^n=2\\rbrace +\\sum _{l=3}^\\infty \\sum _{j=1}^m l^2\\mathbb {I}\\lbrace n\\Gamma _j^n=l\\rbrace .\\end{aligned}$ The main difference between these two tests are how the coefficients of $\\mathbb {I}\\lbrace n\\Gamma _j^n=l\\rbrace $ for $l\\ge 2$ are chosen: Remove all the terms corresponding to $l \\ge 3$ and consider the following separable statistic: $S_n^{\\sf P0}=-\\frac{n^2}{m}+\\sum _{j=1}^m \\mathbb {I}\\lbrace n\\Gamma _j^n=1\\rbrace +\\sum _{j=1}^m 4\\mathbb {I}\\lbrace n\\Gamma _j^n=2\\rbrace .$ Then we have the following relationship between these three test statistics: ${\\Omega }^{\\sf P} : =\\lbrace S_n^{\\sf P}\\le \\check{\\tau }_n\\rbrace \\subset {\\Omega }^{*}: =\\lbrace S_n^{*}\\le \\tau _n\\rbrace \\subset {\\Omega }^{\\sf P0}: =\\lbrace S_n^{\\sf P0}\\le \\check{\\tau }_n\\rbrace $ where the thresholds $\\tau _n$ and $\\check{\\tau }_n$ satisfy $\\check{\\tau }_n=\\tau _n+2n-\\frac{n^2}{m}$ .", "This is depicted in Fig.", "REF .", "Note that the region which Pearson's chi-square test decides in favor of $H1$ is larger than the coincidence-based test, and the probability that the empirical distribution fall into this region is asymptotically larger than $\\exp \\lbrace -\\alpha n^2/m\\rbrace $ for any $\\alpha >0$ .", "This is made precise in the proof of Theorem REF .", "On the other hand, we can show that the test associated with $\\phi ^{{\\sf P0}}$ has $J_M=0$ by considering a sequence of alternative distributions whose likelihood ratios with respect to $p$ increase to infinity.", "In sum, we have $J_F(\\phi ^{{\\sf P}})=0, J_M(\\phi ^{{\\sf P}})>0$ ; $J_F(\\phi ^{{*}})>0, J_M(\\phi ^{{*}})>0$ ; $J_F(\\phi ^{{\\sf P0}})>0, J_M(\\phi ^{{\\sf P0}})=0$ .", "Figure: Decision regions in the space of p.m.f.", "for Pearson's chi-square test, the coincidence-based test and the test given in ().", "[Proof of Theorem REF ] The requirement $P_M(\\phi ^{{\\sf P}}_n) \\rightarrow 0$ imposes an upper-bound on the threshold $\\tau _n$ for $\\phi ^{{\\sf P}}$ : Lemma 3 In order for (REF ) to hold, for large enough $n$ , we must have $\\tau _n\\le \\bar{\\tau }_n : ={\\sf E}_{p}[S_n^{\\sf P}]+\\frac{n^2}{m}\\underline{\\kappa }(\\varepsilon )+2\\frac{n}{\\sqrt{m}}.\\nonumber $ Consider the event that the first symbol appears many times: $A_n: =\\lbrace n\\Gamma ^n_1=\\lfloor \\frac{n\\sqrt{2\\underline{\\kappa }(\\varepsilon )}}{\\sqrt{m}}\\rfloor \\rbrace .$ In the event $A_n$ , the first term $f_1(n\\Gamma ^n_1)$ in the summation in the definition of $S_n^{\\sf P}$ given in (REF ) is approximately $2\\frac{n^2}{m}\\underline{\\kappa }(\\varepsilon )$ .", "This drives the value of $S_n^{\\sf P}$ above the threshold $\\tau _n$ .", "Thus the probability of false alarm conditioned on this event converges to one, as summarized in Lemma REF .", "On the other hand, the probability of $A_n$ does not decay exponentially fast with respect to $n^2/m$ , as summarized in Lemma REF .", "Lemma 4 ${\\sf P}_p\\lbrace S_n^{\\sf P}\\ge \\bar{\\tau }_n|A_n\\rbrace =1-o(1).$ Lemma 5 $-\\lim _{n \\rightarrow \\infty }\\frac{m}{n^2}\\log ({\\sf P}_p\\lbrace A_n\\rbrace )=0.\\nonumber $ Combining Lemma REF , Lemma REF and Lemma REF together, we conclude $J_F(\\phi ^{{\\sf P}})\\le -\\mathop {\\rm lim{\\,}inf}_{n \\rightarrow \\infty }\\frac{m}{n^2}\\log \\bigl ({\\sf P}_p\\lbrace S_n^{\\sf P}\\ge \\bar{\\tau }_n|A_n\\rbrace {\\sf P}_p\\lbrace A_n\\rbrace \\bigr )=0.$ The proofs of these three lemmas are given in Appendix ." ], [ "Alternative Distributions Based on $f$ -Divergence", "The set of alternative distributions studied in previous sections is defined using the total variation distance.", "The generalized error exponent analysis with the same normalization $r(n,m)=n^2/m$ also applies to other distance functions, as we show in Proposition REF and Proposition REF .", "In this section, the set of alternative distributions ${\\mathcal {Q}_n}$ defined in (REF ) is based on a general distance function $d$ rather than $d=d_{\\sf TV}$ .", "Examples include the KL divergence $d_{\\sf KL}(q,p)=\\sum _{j} q_j\\log (q_j/p_j),$ and its generalization known as $f$ -divergence, $d_f(q,p)=\\sum _j p_j f(q_j/p_j),$ where $f$ is a convex function with $f(1)=0$ .", "Conditions under which the generalized error exponent analysis applies are contained in the following: Proposition 2 Suppose the distance function $d$ satisfies $d(q,p) \\ge \\alpha d_{TV}(q,p)$ for some $\\alpha >0$ .", "$\\mathop {\\rm lim{\\,}inf}_{k}\\inf \\lbrace \\sum _j\\frac{q_j^2}{p_j}: d(q,p) \\ge \\varepsilon , q\\in {{\\cal P}_m}\\rbrace >0.$ Then $n^2/m$ is the appropriate normalization for the large deviations analysis for small $\\varepsilon >0$ : There exists a test $\\mathbf {\\phi }$ such that $J_F(\\mathbf {\\phi })>0, J_M(\\mathbf {\\phi })>0.$ There is a constant $\\bar{J}$ satisfying $0<\\bar{J}<\\infty $ such that for any test $\\mathbf {\\phi }$ , we have $\\min \\lbrace J_F(\\mathbf {\\phi }),J_M(\\mathbf {\\phi })\\rbrace \\le \\bar{J}.$ For the set of alternative distributions defined in (REF ) with the $f$ -divergence $d=d_f$ , the generalized error exponent can be applied subject to conditions on $f$ : Proposition 3 Suppose $f$ satisfies the following conditions: For some $0<x<1$ , ${\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(f(1-x)+f(1+x))>f(1).$ There is a constant $\\alpha >0$ such that for all $x$ , $f(x) \\le \\alpha (x-1)^2.$ Then $n^2/m$ is the appropriate normalization for the large deviations analysis for small $\\varepsilon >0$ : There exists a test $\\mathbf {\\phi }$ such that $J_F(\\mathbf {\\phi })>0, J_M(\\mathbf {\\phi })>0.$ There is a constant $\\bar{J}$ satisfying $0<\\bar{J}<\\infty $ such that for any test $\\mathbf {\\phi }$ , we have $\\min \\lbrace J_F(\\mathbf {\\phi }),J_M(\\mathbf {\\phi })\\rbrace \\le \\bar{J}.$ [Proof of Proposition REF ] The converse result in Theorem REF is proved by showing that the worst-case probability of missed detection over the set of distributions given in (REF ) is lower-bounded regardless of the test used.", "The first condition in Proposition REF guarantees that these distributions are still in the set ${\\mathcal {Q}_n}$ of alternative distributions.", "For the achievability result, the critical step is to show that the rate function is positive for any alternative distribution whose likelihood ratio with respect to $p$ is bounded.", "The second condition in Proposition REF guarantees that $\\kappa $ defined in (REF ) is positive, which in turn ensures a positive rate function.", "[Proof of Proposition REF ] The proof is similar to that of Proposition REF .", "The first condition of Proposition REF ensures that the collection of bi-uniform distributions given in (REF ) used in the proof of the converse result is in the set ${\\mathcal {Q}_n}$ of alternative distributions: For $q_{\\mathcal {U}}$ defined in (REF ) with $\\varepsilon $ replaced by $\\varepsilon ^{\\prime }$ , for even $m$ , for small enough $\\varepsilon $ , we have $d_f(q_{\\mathcal {U}},p) = {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}f(1+2\\varepsilon ^{\\prime })+{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}f(1-2\\varepsilon ^{\\prime }) \\ge \\varepsilon $ The second condition implies that $\\alpha \\sum _j\\frac{q_j^2}{p_j} \\ge d_f(q,p) \\ge \\varepsilon .$ Thus, the rate function is positive for any alternative distribution whose likelihood ratio with respect to $p$ is bounded." ], [ "Conclusions and Discussions", "We have shown that the classical error exponent criterion, which appears in the large deviation analysis for universal hypothesis testing problems with large number of samples, can be extended to the small sample case, provided the normalization is modified to account for both the sample size $n$ and the alphabet size $m$ .", "We offer a few discussions on the results and point out directions for future research: 1.", "The analysis in this paper is of asymptotic nature.", "The generalized error exponent gives the leading term in the asymptotic expansion of the probability of error.", "Finer approximations are valuable especially for characterizing the finite sample performance when $n/m$ is not very small.", "For example, finer approximations can reveal the difference among the class of tests described in Section REF that has the same generalized error exponents.", "2.", "The size of alphabet $m$ , is used in this and previous work to capture the “complexity\" of the hypothesis testing problem in the case where the null distribution is uniform.", "It remains to see how this can be generalized to other cases, where the null distribution is far from uniform or has a countably infinite support.", "A possible generalization of the size of alphabet is the Rényi entropy of $p$ , which is equal to $\\log (m)$ when $p$ is uniform.", "3.", "It is desirable to establish general large deviation characterizations of separable statistics for small sample problems, similar to those established for $n \\asymp m$ in [25], [26].", "Such results could provide more insights on how the coefficients of a separable statistic affect the test's performance.", "For example, how the performance of a test with test statistic $\\sum _{j=1}^m (n\\Gamma _j^n-np_j)^\\rho $ varies with $\\rho $ ?", "4.", "We have focused on the simple goodness-of-fit problem in this paper, in which $p$ is fully specified.", "A natural extension is the composite goodness-of-fit problem in which $p$ is not fully specified but assumed to be in a known set.", "A similar generalized error exponent concept should exist for the composite case.", "5.", "There are many other problems for which the approach presented in this paper is relevant.", "Examples include the classification problem [33], [34], [35], the problem of testing whether two distributions are close [36], [37], and probability estimation over a large or unknown alphabet [38], [39], [40].", "In the recent work [41] it is shown how to adapt the methods presented to the classification problem.", "The generalized error exponent analysis is applied to characterize the different ways in which the number of training samples and the number of test samples affect the performance of classification algorithms.", "6.", "Topological structure often contains critical information that is easily ignored in the approaches focused on in this work.", "In particular, in this paper we have not considered any notion of distance between points in the alphabet.", "Other approaches such as the support vector machine, or more recent work such as [42] are based primarily on topology.", "It will be desirable to create a coherent bridge between the approach developed here and topological approaches to hypothesis testing.", "It is likely that current information-theoretic tools can help to create these bridges, such as concepts from lossy source-coding.", "We are also considering extensions of the work described here to the feature selection problem of [43], [44] in which $m$ is interpreted as the number of features rather than the alphabet size." ], [ "Organization of the Appendix", "Approximations to the moments of the separable statistics are given in Appendix , and the results are used in the rest of the proofs.", "The proofs of Theorem REF and Theorem REF are given in Appendix .", "Similar arguments are used in the proofs of Theorem REF and Theorem REF given in Appendix .", "The proof of Theorem REF given in Appendix  can be read almost independently of Appendix  and .", "The lemmas supporting the proof of Theorem REF are given in Appendix , and can be read independently of Appendix , and ." ], [ "Moments of Separable Statistics", "This section provides a survey of results on asymptotic approximations to moments of separable statistics.", "The results hold for the distributions in the set $\\mathcal {P}^b_m$ defined in (REF ).", "Lemma 6 (Expectation of a separable statistic) Consider the separable statistic $\\sum _{j=1}^m f_j(n\\Gamma ^n_j)$ .", "Suppose we have $\\max _j |f_j(x)|\\le a_0e^{a_0x}$ for some $a_0>0$ .", "Then its expectation for $\\nu \\in \\mathcal {P}^b_m$ is given by: $\\begin{aligned}&{\\sf E}_\\nu [\\sum _{j=1}^m f_j(n\\Gamma ^n_j)]=\\sum _j f_j(0)+n\\sum _{j=1}^m \\nu _j(f_j(1)-f_j(0))+{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}(m\\sum _{j=1}^m \\nu _j^2)\\bigl (f_j(0)-2f_j(1)+f_j(2)\\bigr )+O(\\frac{n^3}{m^2}).\\end{aligned}$ We have $\\nu _j^3{n \\atopwithdelims ()3}|f_j(3)|=O(\\frac{n^3}{m^3})$ and $\\begin{aligned}&\\sum _{x=4}^\\infty \\nu _j^x{n \\atopwithdelims ()x}|f_j(x)|\\le a_0\\sum _{x=4}^\\infty (\\frac{e^{a_0} \\gamma n}{m})^x \\le \\frac{a_0}{|\\log (e^{a_0}\\gamma n/m)|}(\\frac{e^{a_0} \\gamma n}{m})^3=O(\\frac{n^3}{m^3}).\\end{aligned}$ Consequently, $\\begin{aligned}&{\\sf E}_\\nu [\\sum _{j=1}^m f_j(n\\Gamma ^n_j)]\\\\=&\\sum _{j=1}^m [f_j(0)(1\\!-\\!\\nu _j)^n \\!+\\!", "f_j(1)n\\nu _j(1\\!-\\!\\nu _j)^{n-1} \\!\\\\&+\\!", "f_j(2){\\!n\\!", "\\atopwithdelims ()\\!2\\!", "}\\nu _j^2(1\\!-\\!\\nu _j)^{n-2}\\!+\\!O(\\frac{n^3\\!}{m^3\\!})]", "\\\\=&\\sum _j \\!f_j(0)\\!+\\!n\\!\\sum _{j=1}^m \\!\\nu _j(f_j(1)\\!-\\!f_j(0))\\!\\\\&+\\!\\frac{n^2}{2}\\!\\!\\sum _{j=1}^m \\!\\nu _j^2(f_j(0)\\!-\\!2f_j(1)\\!+\\!f_j(2))\\!+\\!O(\\!\\frac{n^3\\!}{m^2\\!", "}).\\end{aligned}\\nonumber $ Lemma REF leads to Lemma REF , as well as the following asymptotic approximation of the expectation of $S_n^{*}$ : Lemma 7 For any $\\nu \\in \\mathcal {P}^b_m$ : $\\begin{aligned}{\\sf E}_{\\nu }[S_n^{*}]&=-n+\\frac{n^2}{m}\\bigl (m\\sum _{j=1}^m\\nu _j^2\\bigr )+O(\\frac{n^3}{m^2}).\\end{aligned}\\nonumber $ This will be used in the proof of Theorem REF .", "Lemma 8 (Variance of a separable statistic) Consider a symmetric separable statistic $\\sum _{j=1}^m f(n\\Gamma ^n_j)$ .", "Suppose $|f(x)|\\le a_0e^{a_0x}$ for some $a_0>0$ .", "Moreover, suppose $f(0)=0$ and $f(2) \\ne 2 f(1)$ .", "Then its variance for $\\nu \\in \\mathcal {P}^b_m$ is given by $\\hbox{\\sf Var}\\,_\\nu [\\sum _{j=1}^m f(n\\Gamma ^n_j)]={\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m} (f(2)-2f(1))^2 (m\\sum _{j=1}^m \\nu _j^2)(1+o(1)).$ Lemma REF is the combination of Equation 2.11 and Equation 2.20 in [30]." ], [ "Proofs of Theorem ", "The proof of Theorem REF and Theorem REF is based on the Chernoff bound and the Gärtner-Ellis Theorem.", "The key step is to obtain an asymptotic approximation to the logarithmic moment generating function of the test statistic.", "To simplify the presentation, instead of $S_n^{*}$ we work with the following statistic: $\\tilde{S}_n^{*}=\\sum _{j=1}^m \\mathbb {I}\\lbrace n\\Gamma ^n_j=1\\rbrace -n.$ Its logarithmic moment generating is denoted by $\\Lambda _{\\nu , \\tilde{S}_n^{*}}(\\theta ): =\\log \\bigl ({\\sf E}_{\\nu }[\\exp \\lbrace \\theta \\tilde{S}_n^{*}\\rbrace ]\\bigr ).$ Asymptotic approximations or bounds to $\\Lambda _{\\nu , \\tilde{S}_n^{*}}(\\theta )$ are presented in the next two sections." ], [ "Approximation to the logarithmic moment generating function for distributions in $\\mathcal {P}^b_m$", "Bounds and approximations for $\\Lambda _{\\nu , \\tilde{S}_n^{*}}$ are first obtained for the restricted set of distributions $\\mathcal {P}^b_m$ defined in (REF ).", "Proposition 4 For any $\\nu \\in \\mathcal {P}^b_m$ , the logarithmic moment generating function for the statistic $\\tilde{S}_n^{*}$ has the following asymptotic expansion $\\Lambda _{\\nu , \\tilde{S}_n^{*}}(\\theta )={\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}\\bigl (m\\sum _{j=1}^m\\nu _j^2\\bigr )(e^{-2\\theta }-1)+O(\\frac{n^3}{m^2})+O(1).$ The approximation errors $O(\\frac{n^3}{m^2})$ and $O(1)$ are uniform over the set $\\mathcal {P}^b_m$ .", "The proof uses the Poissonization technique, and the procedure is applicable for many separable statistics including $S_n^{*}$ : Let $\\lbrace X_j\\rbrace $ be a sequence of independent Poisson random variables with parameter $\\lambda \\nu _j$ for some $\\lambda >0$ .", "Then for any integers $u_1, \\ldots , u_m$ satisfying $\\sum _{j=1}^m u_j=n$ , we have ${\\sf P}\\lbrace n\\Gamma ^n_j=u_j, \\textrm {for all j} \\rbrace ={\\sf P}\\lbrace X_j=u_j, \\textrm {for all j} | \\sum _{j=1}^m X_j=n\\rbrace .\\nonumber $ Therefore, the moment generating function of a separable statistic $\\sum _{j=1}^m f_j(n\\Gamma ^n_j)$ admits the following representation: ${\\sf E}_{\\nu }[\\exp \\lbrace \\theta \\sum _{j=1}^m f_j(n\\Gamma ^n_j)\\rbrace ]={\\sf E}[\\exp \\lbrace \\theta \\sum _{j=1}^m f_j(X_j)\\rbrace | \\sum _{j=1}^m X_j=n].\\nonumber $ It is related to the moment generating function $A_\\lambda (\\theta )$ for $\\sum _{j=1}^m f_j(X_j)$ as follows: $\\begin{aligned}&A_\\lambda (\\theta )\\!", ": =\\!", "{\\sf E}[\\exp \\lbrace \\theta \\sum _{j=1}^m f_j(X_j)\\rbrace ]\\\\&=\\sum _{n=0}^\\infty \\frac{\\lambda ^n}{n!", "}e^{-\\lambda }{\\sf E}[\\exp \\lbrace \\theta \\sum _{j=1}^m f_j(X_j)\\rbrace | \\sum _{j=1}^m X_j=n]\\\\&=\\sum _{n=0}^\\infty \\frac{\\lambda ^n}{n!", "}e^{-\\lambda }{\\sf E}_\\nu [\\exp \\lbrace \\theta \\sum _{j=1}^m f_j(n\\Gamma ^n_j)\\rbrace ].\\end{aligned}$ It follows from the independence of the variables $\\lbrace X_j\\rbrace $ that the moment generating function $A_\\lambda (\\theta )$ has the following formula: $A_\\lambda (\\theta )=\\prod _{j=1}^m (\\sum _{k=0}^\\infty \\frac{(\\lambda \\nu _j)^k}{k!}", "e^{-\\lambda \\nu _j} e^{\\theta f_j(k)}).$ Since $A_\\lambda (\\theta )$ is analytic in $\\lambda $ , the moment generating function of $\\sum _{j=1}^m f_j(n\\Gamma ^n_j)$ can be obtained via Cauchy's theorem: ${\\sf E}_{\\nu }[\\exp \\lbrace \\theta \\sum _{j=1}^m f_j(n\\Gamma ^n_j)\\rbrace ]=\\frac{n!", "}{2\\pi } \\oint e^{\\lambda } A_\\lambda (\\theta ) \\frac{d\\lambda }{\\lambda ^{n+1}},$ where the integration is carried out along any closed contour around $\\lambda =0$ in the complex plane.", "These arguments lead to the following lemma: Lemma 9 The moment generating function of the separable statistic $\\sum _{j=1}^m f_j(n\\Gamma ^n_j)$ is given by $\\begin{aligned}&{\\sf E}_{\\nu }[\\exp \\lbrace \\theta \\sum _{j=1}^m f_j(n\\Gamma ^n_j)\\rbrace ]\\\\&=\\frac{n!", "}{2\\pi } \\oint e^{\\lambda } \\prod _{j=1}^m \\big (\\sum _{k=0}^\\infty \\frac{(\\lambda \\nu _j)^k}{k!", "}e^{-\\lambda \\nu _j} e^{\\theta f_j(k)} \\big ) \\frac{d\\lambda }{\\lambda ^{n+1}}.\\nonumber \\end{aligned}$ [Proof of Proposition REF ] Applying Lemma REF with $f_j(1)=1, f_j(k)=0 \\textrm { for $ k1$}$ , we obtain ${\\sf E}_{\\nu }[\\exp \\lbrace \\theta (\\tilde{S}_n^{*})\\rbrace ]=e^{-\\theta n}\\frac{n!", "}{2pi} \\oint g(\\lambda )d\\lambda $ where $g(\\lambda )=e^{\\lambda } \\prod _{j=1}^m (1-(\\lambda \\nu _j)e^{-\\lambda \\nu _j}+(\\lambda \\nu _j)e^{-\\lambda \\nu _j} e^{\\theta })\\frac{1}{\\lambda ^{n+1}}.\\nonumber $ The rest of the proof is an application of the saddle point method [45].", "It consists of two steps: The first step is to pick a particular contour around $\\lambda =0$ to carry out the integration.", "It is desirable to have a contour along which $g(\\lambda )$ behaves violently: $g(\\lambda )$ is large on a small interval on the contour and significantly smaller at the rest, so that the value of integral can be approximated by integrating over this small interval.", "Such a contour can be found, by identifying a saddle point of $g(\\lambda )$ at which the derivative of $g(\\lambda )$ vanishes, and then pick a contour that goes through the saddle point.", "The second step is to apply the Laplace method to estimate the integral along the contour.", "We now apply the first step of the saddle point method: identifying the saddle point and defining the contour for integration.", "Note that the derivative of $g$ is given by $\\frac{d}{d \\lambda } g(\\lambda )=g(\\lambda ) [\\sum _{j=1}^m \\frac{\\nu _j(e^{\\theta }-1+e^{\\lambda \\nu _j})}{\\lambda \\nu _j(e^{\\theta }-1)+e^{\\lambda \\nu _j} }-\\frac{n+1}{\\lambda }].$ To simplify the derivation, we select a point that is close to a saddle point, defined as the solution to $\\sum _{j=1}^m \\frac{\\lambda \\nu _j(e^{\\theta }-1+e^{\\lambda \\nu _j})}{\\lambda \\nu _j(e^{\\theta }-1)+e^{\\lambda \\nu _j} }=n.$ If $\\lambda $ on the left-hand side was taken to be a saddle point, then the right-hand side would be $n+1$ instead of $n$ , and we will see this error is negligible for our purposes.", "Equation (REF ) has one unique real-valued nonnegative solution, which we denote by $\\lambda _0$ .", "To see this, note that when restricting $\\lambda $ to $[0, \\infty )$ , the left-hand-side is a continuous and strictly increasing function of $\\lambda $ .", "Moreover, its value is 0 when $\\lambda =0$ , increases to $\\infty $ when $\\lambda $ increases to $\\infty $ .", "We now obtain an asymptotic expansion of $\\lambda _0$ .", "We first show that $\\lambda _0=O(n)$ .", "When $\\theta \\ge 0$ , using the fact that $0 \\le xe^{-x}\\le e^{-1}$ and $0 \\le e^{-x} \\le 1$ for $x \\ge 0$ , we obtain $\\frac{1}{1+e^{-1}(e^{\\theta }-1)}\\le \\frac{e^{\\theta }-1+e^{\\lambda \\nu _j}}{\\lambda \\nu _j(e^{\\theta }-1)+e^{\\lambda \\nu _j}}\\le e^{\\theta }.$ Substituting this into (REF ) leads to $ne^{-\\theta } \\le \\lambda _0\\le n(1+e^{-1}(e^{\\theta }-1)).$ When $\\theta <0$ , we obtain $e^{\\theta } \\le \\frac{e^{\\theta }-1+e^{\\lambda \\nu _j}}{\\lambda \\nu _j(e^{\\theta }-1)+e^{\\lambda \\nu _j}} \\le \\frac{1}{1+e^{-1}(e^{\\theta }-1)}.$ Substituting this into (REF ) leads to $n(1+e^{-1}(e^{\\theta }-1))\\le \\lambda _0\\le ne^{-\\theta } .$ It follows from the bounds (REF ), (REF ) and $\\nu \\in \\mathcal {P}^b_m$ that $\\lambda _0\\nu _j=o(1)$ .", "Thus the demominator of (REF ) satisfies $\\lambda _0\\nu _j(e^{\\theta }-1)+e^{\\lambda _0\\nu _j} =1+o(1).$ Substituting this into (REF ) leads to $\\sum _{j=1}^m \\lambda _0\\nu _j(e^{\\theta }-1+e^{\\lambda _0\\nu _j})=n(1+o(1)).$ Consequently, $\\lambda _0=n e^{-\\theta }(1+o(1)).\\nonumber $ To obtain a refined approximation, let $w: =\\lambda _0e^\\theta /n-1$ .", "Consequently, $\\lambda _0=n e^{-\\theta } (1+w).$ An approximation for $w$ will be obtained: Since $\\lambda _0\\nu _j=O(\\frac{n}{m})$ , we have that the numerator and denominator in the summand of (REF ) satisfy $\\begin{aligned}\\lambda _0\\nu _j(e^{\\theta }-1+e^{\\lambda _0\\nu _j}) &=\\lambda _0\\nu _j(e^\\theta + \\lambda _0\\nu _j +O(\\frac{n^2}{m^2})),\\\\\\lambda _0\\nu _j(e^{\\theta }-1)+e^{\\lambda _0\\nu _j} &=1+ \\lambda _0\\nu _j e^\\theta +O(\\frac{n^2}{m^2}).\\end{aligned}$ Thus, $\\begin{aligned}&\\sum _{j=1}^m \\frac{\\lambda _0\\nu _j(e^{\\theta }-1+e^{\\lambda _0\\nu _j})}{\\lambda _0\\nu _j(e^{\\theta }-1)+e^{\\lambda _0\\nu _j} }\\\\=&\\sum _j [\\lambda _0\\nu _j e^\\theta +\\lambda _0^2 \\nu _j^2 (1-e^{2\\theta })+O(\\frac{n^3}{m^3})].\\end{aligned}$ Substituting this and (REF ) into (REF ) leads to $w+ n \\sum _j \\nu _j^2 (1+w)^2 (e^{-2\\theta }-1)=O(\\frac{n^2}{m^3}),$ which gives $w=n\\sum _j \\nu _j^2 (1-e^{-2\\theta }) (1+O(\\frac{n}{m}))=O(\\frac{n}{m}).$ The integration in (REF ) is now carried out along the closed contour given by $\\lambda =\\lambda _0e^{i\\psi }=ne^{-\\theta }(1+w) e^{i \\psi }$ : $\\begin{aligned}{\\sf E}_\\nu [\\exp \\lbrace \\theta (\\tilde{S}_n^{*})\\rbrace ]=&e^{-\\theta n}\\frac{n!", "}{2\\pi } \\int _{-\\pi }^\\pi g(\\lambda _0e^{i \\psi }) \\lambda _0e^{i\\psi }d\\psi \\\\=&\\frac{n!", "}{2\\pi }\\lambda _0^{-n} e^{-\\theta n}\\operatorname{Re}\\bigl [\\int _{-\\pi }^\\pi h(\\psi ) d\\psi \\bigr ].\\end{aligned}$ where $h(\\psi ): =e^{-i n \\psi }\\prod _{j=1}^m \\bigl (\\lambda _0\\nu _j(e^{\\theta }-1)e^{i\\psi }+e^{\\lambda _0\\nu _je^{i\\psi } }\\bigr ).$ We now apply the second step of the saddle point method: estimating the integral by the Laplace method.", "We begin with a rough estimate of $h(\\psi )$ .", "It follows from $\\lambda _0=n^{-\\theta }(1+o(1))$ that $\\begin{aligned}h(\\psi )&=e^{-i n \\psi }\\prod _{j=1}^m \\bigl (\\lambda _0\\nu _j(e^{\\theta }-1)e^{i\\psi }\\!+\\!1\\!+\\!\\lambda _0\\nu _je^{i\\psi }\\!+\\!O(\\frac{n^2}{m^2})\\bigr )\\\\&=e^{-i n \\psi }\\prod _{j=1}^m \\bigl (1+\\lambda _0\\nu _je^{\\theta }e^{i\\psi }+\\!O(\\frac{n^2}{m^2})\\bigr )\\\\&=e^{-i n \\psi }\\exp \\lbrace \\sum _{j=1}^m \\bigl (\\lambda _0\\nu _je^{\\theta }e^{i\\psi }+O(\\frac{n^2}{m^2})\\bigr )\\rbrace \\\\&=e^{-i n \\psi }e^n \\exp \\lbrace -n (1-e^{i\\psi })+O(\\frac{n^2}{m})\\rbrace .\\end{aligned}$ Therefore, for any $\\psi \\ne 0$ , $|h(\\psi )|$ is exponentially smaller than the value of $h(\\psi )$ at $\\psi =0$ .", "This suggests that the integral in (REF ) can be approximated by integrating over a small interval around $\\psi =0$ .", "Split the integral in (REF ) into three parts: $\\begin{aligned}I_1&=\\operatorname{Re}[\\int _{-\\pi /3}^{\\pi /3} h(\\psi ) d\\psi ],\\\\ I_2&=\\operatorname{Re}[\\int _{-\\pi }^{-\\pi /3}\\!", "h(\\psi ) d\\psi ],\\\\ I_3&=\\operatorname{Re}[\\int _{\\pi /3}^{\\pi }\\!", "h(\\psi ) d\\psi ].\\end{aligned}$ We first estimate $I_1$ .", "Denote $H(\\psi )=\\log (h(\\psi ))$ .", "Simple calculus gives $\\begin{aligned}H(\\psi )&=-in\\psi +\\sum _{j=1}^m \\log (\\lambda _0\\nu _j(e^\\theta -1)e^{i\\psi }+\\exp \\lbrace \\lambda _0\\nu _j e^{i\\psi }\\rbrace ),\\\\H^{\\prime }(\\psi )&=-in+i\\sum _{j=1}^m \\frac{\\lambda _0\\nu _j(e^\\theta -1)e^{i\\psi }+\\lambda _0\\nu _je^{i\\psi }\\exp \\lbrace \\lambda _0\\nu _j e^{i\\psi }\\rbrace }{\\lambda _0\\nu _j(e^\\theta -1)e^{i\\psi }+\\exp \\lbrace \\lambda _0\\nu _j e^{i\\psi }\\rbrace },\\\\H^{\\prime \\prime }(\\psi )&=-\\sum _{j=1}^m\\exp \\lbrace \\lambda _0\\nu _j e^{i\\psi }\\rbrace \\frac{1}{(\\lambda _0\\nu _j(e^\\theta -1)e^{i\\psi }+\\exp \\lbrace \\lambda _0\\nu _j e^{i\\psi }\\rbrace )^2}\\\\&\\quad \\times \\bigl (\\lambda _0\\nu _j(e^\\theta -1)e^{i\\psi }(1-\\lambda _0\\nu _je^{i\\psi }+\\lambda _0^2\\nu _j^2e^{2i\\psi }) +\\lambda _0\\nu _je^{i\\psi }\\exp \\lbrace \\lambda _0\\nu _j e^{i\\psi }\\rbrace \\bigr ).\\end{aligned}$ It is clear that $\\operatorname{Im}(H(0))=0$ .", "It follows from (REF ) that $H^{\\prime }(0)=0$ .", "Estimates of $\\operatorname{Re}(H(0))$ and $H^{\\prime \\prime }(\\psi )$ are obtained from substituting (REF ) and (REF ) into the expression of $H(\\psi )$ and $H^{\\prime \\prime }(\\psi )$ and applying asymptotic analysis.", "In sum, $\\begin{aligned}&\\operatorname{Im}(H(0))\\!=\\!0, \\\\&\\operatorname{Re}(H(0))\\!=\\!n(1+w)\\!-\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2(\\sum _{j=1}^m\\nu _j^2) (1-e^{-2\\theta })\\!+\\!O(\\frac{n^3}{m^2}),\\\\&H^{\\prime }(0)=0, H^{\\prime \\prime }(\\psi )\\!=\\!-ne^{i\\psi }+O(\\frac{n^2}{m}).\\end{aligned}$ To obtain an upper-bound on $I_1$ , note that for large enough $n$ and for any $\\psi \\in [-\\pi /3, \\pi /3]$ , we have $\\operatorname{Re}(H^{\\prime \\prime }(\\psi ))\\le -0.4n$ .", "It then follows from the mean value theorem that $\\operatorname{Re}(H(\\psi )) \\le H(0)-0.2n\\psi ^2.$ Consequently, for large enough $n$ and $m$ , $\\begin{aligned}I_1&\\le e^{H(0)}\\int _{-\\pi /3}^{-\\pi /3} e^{-0.2 n\\psi ^2} d\\psi \\le e^{H(0)} \\int _{-\\infty }^{\\infty } e^{-0.2 n\\psi ^2} d\\psi = e^{H(0)}\\frac{\\sqrt{\\pi }}{\\sqrt{0.2n}}.\\end{aligned}$ To obtain a lower-bound on $I_1$ , we begin with an bound on $\\operatorname{Im}(H^{\\prime \\prime }(\\psi ))$ : Since $\\operatorname{Im}(H^{\\prime \\prime }(\\psi ))=-n\\sin (\\psi )+O(\\frac{n^2}{m})$ , applying $|\\sin (\\psi )| \\!\\le \\!|\\psi |$ , we have that for large enough $n$ , for any $\\psi \\!\\in \\!", "[-\\pi /3, \\pi /3]$ , $|\\operatorname{Im}(H^{\\prime \\prime }(\\psi ))|\\le 1.1n|\\psi |$ .", "It also follows from (REF ) that $\\operatorname{Re}(H^{\\prime \\prime }(\\psi ))\\ge -1.1n$ .", "Applying the mean value theorem, we conclude that there exists some $c>0$ such that for $\\psi \\in [-\\pi /3, \\pi /3]$ , $\\begin{aligned}\\operatorname{Re}(H(\\psi )) &\\ge H(0)-1.1n\\psi ^2, \\\\|\\operatorname{Im}(H(\\psi ))|&\\le 1.1n|\\psi |^3+c\\frac{n^2}{m}\\psi ^2.\\end{aligned}$ Use the short-hand notation $t_n=0.1 \\min \\lbrace n^{\\!-\\!1/3},\\!", "\\sqrt{m}/(\\sqrt{c}n)\\rbrace $ .", "For $\\psi \\in [-t_n,t_n]$ , we have $\\cos (\\operatorname{Im}(H(\\psi )))\\ge 0.5$ , and thus $\\operatorname{Re}(e^{H(\\psi )}) \\ge 0.5e^{\\operatorname{Re}(H(\\psi ))}$ .", "The integration for $I_1$ is further split into three parts: $\\begin{aligned}I_1=&\\operatorname{Re}[\\int _{-\\pi /3}^{-t_n}e^{H(\\psi )}d\\psi ]+\\operatorname{Re}[\\int _{t_n}^{\\pi /3}e^{H(\\psi )}d\\psi ]+\\operatorname{Re}[\\int _{-t_n}^{t_n}e^{H(\\psi )}d\\psi ].\\end{aligned}$ The absolute value of the first term is upper-bounded as follows: $\\begin{aligned}|\\int _{-\\pi /3}^{-t_n}e^{H(\\psi )}d\\psi |&\\le e^{H(0)} \\int _{-\\infty }^{-t_n}e^{-0.2n\\psi ^2}d\\psi = t_n e^{H(0)} \\int _{-\\infty }^{-1}e^{-0.2nt_n^2\\bar{\\psi }^2}d \\bar{\\psi }\\\\&\\le t_n e^{H(0)} \\int _{-\\infty }^{-1}e^{-0.2nt_n^2|\\bar{\\psi }|}d \\bar{\\psi }=e^{H(0)}O(\\frac{1}{nt_n})=e^{H(0)}o(\\frac{1}{\\sqrt{n}}).\\end{aligned}$ The second term is bounded in a similar way.", "The third term is lower-bounded as follows: $\\begin{aligned}\\operatorname{Re}[\\int _{-t_n}^{t_n}e^{H(\\psi )}d\\psi ]\\ge & \\int _{-t_n}^{t_n}0.5 e^{\\operatorname{Re}(H(\\psi ))} d\\psi \\ge 0.5 e^{H(0)}\\int _{-t_n}^{t_n} e^{-1.1n \\psi ^2}d\\psi \\\\\\ge & 0.5 e^{H(0)}[\\int _{-\\infty }^{\\infty } e^{-1.1n \\psi ^2}d\\psi -2\\int _{-\\infty }^{-t_n} e^{-1.1n \\psi ^2}d\\psi ]\\\\\\ge & 0.5 e^{H(0)} (\\frac{\\sqrt{\\pi }}{\\sqrt{1.1n}}+O(\\frac{1}{nt_n}))= 0.5 e^{H(0)} \\frac{\\sqrt{\\pi }}{\\sqrt{1.1n}}(1+o(1)).\\!\\end{aligned}\\nonumber $ where the last inequality follows from an argument similar to (REF ).", "Combining these bounds together, we obtain $\\begin{aligned}I_1 &\\ge \\operatorname{Re}[\\int _{-t_n}^{t_n}e^{H(\\psi )}d\\psi ] -|\\operatorname{Re}[\\int _{-\\pi /3}^{-t_n}e^{H(\\psi )}d\\psi ]|-|\\operatorname{Re}[\\int _{t_n}^{\\pi /3}e^{H(\\psi )}d\\psi ]| \\\\&\\ge e^{H(0)} \\frac{0.5\\sqrt{\\pi }}{\\sqrt{1.1n}}(1+o(1)).\\end{aligned}\\nonumber $ Combing this and (REF ) leads to, $I_1 = e^{H(0)} \\frac{1}{\\sqrt{n}}e^{O(1)}=e^{n(1+o(1))} \\frac{1}{\\sqrt{n}}e^{O(1)}.$ where the last equality follows from the estimate of $H(0)$ given in (REF ) and (REF ).", "We now estimate $I_2$ and $I_3$ .", "For $\\psi \\in [-\\pi , -\\pi /3] \\cup [\\pi /3, \\pi ]$ , we obtain from (REF ) that $|h(\\psi )|\\le \\exp \\lbrace 0.5n+O(\\frac{n^2}{m})\\rbrace $ , which implies $\\operatorname{Re}[I_2]+\\operatorname{Re}[I_3] =O(e^{0.6n})$ .", "This shows that $I_2$ and $I_3$ are much smaller than $I_1$ .", "Thus, the integral in (REF ) can be approximated by the estimate of $I_1$ : Substituting (REF ) and (REF ) into (REF ), we obtain $\\begin{aligned}&{\\sf E}_{\\nu }[\\exp \\lbrace \\theta (\\tilde{S}_n^{*})\\rbrace ]\\\\&=\\frac{n!", "}{2\\pi }\\lambda _0^{-n} e^{-\\theta n}I_1(1+o(1)) \\\\&=\\frac{n!", "}{2\\pi }\\lambda _0^{-n} e^{-\\theta n}e^{H(0)} \\frac{1}{\\sqrt{n}}e^{O(1)}(1+o(1))\\\\&=\\frac{n!", "}{n^n\\sqrt{2\\pi n} }\\bigl (1+n\\sum _j \\nu _j^2(1-e^{-2\\theta })+O(\\frac{n^2}{m^2})\\bigr )^{-n}\\\\&\\quad \\times \\exp \\lbrace {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2(\\sum _{j=1}^m\\nu _j^2) (1-e^{-2\\theta })+O(\\frac{n^3}{m^2})\\rbrace e^{O(1)})\\\\&=\\frac{n!e^{n}}{n^n \\sqrt{2\\pi n}}\\exp \\lbrace -{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2(\\sum _{j=1}^m\\nu _j^2)(1-e^{-2\\theta })+O(\\frac{n^3}{m^2})\\rbrace e^{O(1)}.\\end{aligned}\\nonumber $ Stirling formula gives $\\frac{n!e^{n}}{n^n 2\\pi \\sqrt{n}}=1+O(\\frac{1}{n})$ .", "The claim of the proposition is obtained on taking logarithm on both sides." ], [ "Approximation to the logarithmic moment generating function for distributions not in $\\mathcal {P}^b_m$", "We also need to consider distributions in ${\\mathcal {Q}_n}\\setminus \\mathcal {P}^b_m$ .", "For any $q\\in {\\mathcal {Q}_n}\\setminus \\mathcal {P}^b_m$ , the set of indices $\\mathcal {S}_0: =\\lbrace j \\in [m]: q_j \\ge \\gamma m^{-1}\\rbrace $ is non-empty.", "Now fix a small constant $\\eta >0$ , and consider each index $j$ in $\\mathcal {S}_0$ in two separate cases, according to whether $n q_j \\ge \\eta $ .", "Denote $\\mathcal {W}_\\eta (q)=\\lbrace j: nq_j \\ge \\eta \\rbrace , \\quad \\beta (q) =\\sum _{j \\in \\mathcal {W}_\\eta (q)} q_j.\\nonumber $ Proposition REF below addresses the case where $\\beta (q)$ is large.", "It implies that the probability of missed detection associated with such a distribution is much smaller than that associated with the worst-case distributions: The probability decays exponentially fast with respect to $n$ , which is larger than $n^2/m$ .", "Proposition REF considers the alternate case, and shows that if $\\beta (q)$ is not large, then a bound similar to that in Proposition REF holds.", "Proposition 5 For all sufficiently small $\\eta >0$ , any $\\theta \\in (0,0.5]$ , and any $\\underline{\\beta }>0$ , there exists $n_0$ such that for any $n>n_0$ , and any $\\nu $ satisfying $\\beta (\\nu ) \\ge \\underline{\\beta }$ , the following holds, $\\Lambda _{\\nu , \\tilde{S}_n^{*}}(\\theta )\\le - \\beta (\\nu ) \\alpha (\\theta ) n,$ where $\\alpha (\\theta ) >0 $ .", "Proposition 6 For any $\\delta >0$ , $\\theta \\in (0,0.5]$ , $\\overline{\\eta }>0$ , there exist $\\eta \\in (0,\\overline{\\eta })$ , $\\overline{\\beta }>0$ , and $n_0$ such that for any $n>n_0$ , and any $\\nu $ satisfying $\\beta (\\nu ) \\le \\overline{\\beta }$ , the following holds, $\\Lambda _{\\nu , \\tilde{S}_n^{*}}(\\theta )\\le {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}(m\\sum _{j \\notin \\mathcal {W}_\\eta (\\nu )}q_j^2)(e^{-2\\theta }-1)(1-\\delta ).$ The proofs of Proposition REF and Proposition REF use steps similar to those leading to the upper-bound in Proposition REF .", "However, the approximation given by (REF ) and (REF ) is no longer valid, so a different approximation is required.", "The conclusions on the existence and uniqueness of the solution $\\lambda _0$ and the bounds in (REF ) are still valid, and our proof begins from there.", "To simplify the presentation, we use the following notation similar to the small “$o$ \" notation: We write $x=o^\\eta (1)$ whenever there exists a function $s(\\eta )$ that does not depend on $\\theta $ , $n$ , and $\\nu $ , such that $|x|\\le s(\\eta )$ and $\\lim _{\\eta \\rightarrow 0}s(\\eta )=0$ .", "Consider any $\\eta $ and $\\nu $ .", "Write $\\mathcal {W}_\\eta =\\mathcal {W}_\\eta (\\nu )$ .", "For any $j \\notin \\mathcal {W}_\\eta $ , we obtain the expansion of the summand in (REF ) via the mean value theorem: $\\frac{\\lambda _0\\nu _j(e^{\\theta }-1+e^{\\lambda _0\\nu _j})}{\\lambda _0\\nu _j(e^{\\theta }-1)+e^{\\lambda _0\\nu _j} }=\\lambda _0\\nu _j e^{\\theta }+\\lambda _0^2 \\nu _j^2(1-e^{2\\theta })(1+o^\\eta (1)).$ For any $j \\in \\mathcal {W}_\\eta $ , the following equality holds: $\\frac{\\lambda _0\\nu _j(e^{\\theta }-1+e^{\\lambda _0\\nu _j})}{\\lambda _0\\nu _j(e^{\\theta }-1)+e^{\\lambda _0\\nu _j} }=D_j \\lambda _0\\nu _j e^{\\theta },$ where $D_j : =\\frac{e^{-\\theta }+e^{-\\lambda _0\\nu _j}(1-e^{-\\theta })}{1+\\lambda _0\\nu _je^{-\\lambda _0\\nu _j} (e^{\\theta }-1)} \\ge e^{-2\\theta }.$ Substituting these estimates into (REF ) leads to $\\lambda _0(1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1))e^{\\theta }+\\lambda _0^2 \\sum _{j \\notin \\mathcal {W}_\\eta } \\nu _j^2 (1-e^{2\\theta })(1+o^\\eta (1))=n.$ Applying $\\lambda _0\\sum _{j \\notin \\mathcal {W}_\\eta } \\nu _j^2 \\le \\eta \\sum _{j \\notin \\mathcal {W}_\\eta } \\nu _j \\le \\eta $ gives, $\\lambda _0=\\frac{ne^{-\\theta }}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}(1+o^\\eta (1)).$ Introducing a variable $w$ as before, $\\lambda _0=\\frac{ne^{-\\theta }}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}(1+w).$ On substituting (REF ) into (REF ), we obtain $\\begin{aligned}w=\\frac{n\\bigl (\\sum _{j \\notin \\mathcal {W}_\\eta } \\nu _j^2 (1-e^{-2\\theta })\\bigr )}{(1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1))^2}(1+o^\\eta (1))=o^\\eta (1).\\end{aligned}$ In the proofs of both propositions, we integrate (REF ) along the closed contour corresponding to $\\lambda =\\lambda _0e^{i\\psi }$ from $\\psi =-\\pi $ to $\\psi =\\pi $ , and use the same definition of $h(\\psi )$ given in (REF ) and $H(\\psi )=\\log (h(\\psi ))$ .", "The integral is given in (REF ) and our task is to estimate it.", "We now give the details.", "[Proof of Proposition REF ] We first show that any $\\psi $ , $\\operatorname{Re}(H(\\psi )) \\le H(0)=\\sum _j [\\lambda _0\\nu _j +\\log \\bigl (1+\\lambda _0\\nu _j e^{-\\lambda _0\\nu _j }( e^\\theta -1)\\bigr )].", "$ Thus to bound the integral in (REF ), we only need to bound $H(0)$ .", "For $\\psi \\!\\in \\!", "[-{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\pi ,\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\pi ]$ , the summand in the expression of $\\operatorname{Re}(H(\\psi ))$ given in (REF ) is bounded as follows: $\\begin{aligned}&\\operatorname{Re}[\\log \\bigl (\\lambda _0\\nu _j(e^\\theta -1)e^{i\\psi }+e^{\\lambda _0\\nu _j e^{i\\psi }}\\bigr )]\\\\&=\\operatorname{Re}[\\log (e^{\\lambda _0\\nu _j e^{i\\psi }})+\\log \\bigl (1+\\lambda _0\\nu _j(e^\\theta -1)e^{i\\psi }e^{-\\lambda _0\\nu _j e^{i\\psi }}\\bigr )] \\\\&\\le \\lambda _0\\nu _j \\cos \\psi +\\log \\bigl (1+\\lambda _0\\nu _j e^{-\\lambda _0\\nu _j \\cos \\psi }( e^\\theta -1)\\bigr ).\\end{aligned}$ The right-hand side is a convex function of $\\cos \\psi $ for $ \\psi \\in [-{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\pi , {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\pi ]$ .", "Thus, it achieves its maximum value at $\\cos \\psi =1$ or $\\cos \\psi =0$ .", "Note that its value at $\\cos \\psi =1$ is exactly equal to the summand in $H(0)$ .", "Moreover, we can show that its value at $\\cos \\psi =1$ is no smaller than its value at $\\cos \\psi =0$ : $\\begin{aligned}&\\lambda _0\\nu _j+\\log \\bigl (1+\\lambda _0\\nu _j(e^\\theta -1)e^{-\\lambda _0\\nu _j}\\bigr )-\\log \\bigl (1+\\lambda _0\\nu _j(e^\\theta -1)\\bigr )\\\\&=\\lambda _0\\nu _j+\\log \\bigl (\\frac{1+\\lambda _0\\nu _j(e^\\theta -1)e^{-\\lambda _0\\nu _j}}{1+\\lambda _0\\nu _j(e^\\theta -1)}\\bigr ) \\le \\lambda _0\\nu _j+\\log (e^{-\\lambda _0\\nu _j})=0,\\end{aligned}\\nonumber $ where the inequality follows from $\\theta \\ge 0$ .", "This leads to (REF ) for $\\psi \\in [-{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\pi , {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\pi ]$ .", "For $\\psi \\in [-\\pi , -{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\pi ] \\cup [{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\pi ,\\pi ]$ , we have $|e^{\\lambda _0\\nu _j e^{i\\psi }}| \\le 1$ .", "Consequently, $|\\lambda _0\\nu _j(e^\\theta -1)e^{i\\psi }+e^{\\lambda _0\\nu _j e^{i\\psi }}|\\le 1+\\lambda _0\\nu _j (e^\\theta -1),$ which leads to $\\operatorname{Re}[\\log \\bigl (\\lambda _0\\nu _j(e^\\theta -1)e^{i\\psi }+e^{\\lambda _0\\nu _j e^{i\\psi }}\\bigr )] \\le \\log \\bigl (1+\\lambda _0\\nu _j (e^\\theta -1)\\bigr ).$ The right-hand side of (REF ) is equal to the value of the right-hand side of (REF ) at $\\cos \\psi =0$ , which has been shown in the previous paragraph to be smaller than $H(0)$ .", "This leads to (REF ) for $\\psi \\in [-\\pi , -{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\pi ] \\cup [{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\pi ,\\pi ]$ .", "We now approximate the right-hand side of (REF ): For $ j \\notin \\mathcal {W}_\\eta $ , we have $\\begin{aligned}&\\lambda _0\\nu _j +\\log \\bigl (1+\\lambda _0\\nu _j e^{-\\lambda _0\\nu _j }( e^\\theta -1)\\bigr )=\\lambda _0\\nu _j^{\\theta }+{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\lambda _0^2 \\nu _j^2 (1-e^{2 \\theta })(1+o^\\eta (1)).\\end{aligned}$ For $j \\in \\mathcal {W}_\\eta $ , we have the inequality $\\begin{aligned}&\\lambda _0\\nu _j +\\log \\bigl (1+\\lambda _0\\nu _j e^{-\\lambda _0\\nu _j }( e^\\theta -1)\\bigr ) \\le \\lambda _0\\nu _je^\\theta +\\lambda _0\\nu _j (1-e^{-\\lambda _0\\nu _j })(1- e^\\theta ).\\end{aligned}$ Substituting these two estimates, and (REF ), (REF ) into (REF ) leads to $&&{\\sf E}_{\\nu }[\\exp \\lbrace \\theta (\\tilde{S}_n^{*})\\rbrace ] \\le \\frac{n!", "}{2\\pi }\\lambda _0^{-n} e^{-\\theta n} \\exp \\lbrace H(0)\\rbrace \\\\&&\\le n!\\lambda _0^{-n} e^{-\\theta n} \\exp \\lbrace \\sum _{j \\notin \\mathcal {W}_\\eta } [\\lambda _0\\nu _j e^{\\theta }+{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\lambda _0^2 \\nu _j^2 (1-e^{2 \\theta })(1+o^\\eta (1))]\\rbrace \\nonumber \\\\&&\\quad \\times \\exp \\lbrace \\sum _{j \\in \\mathcal {W}_\\eta }[\\lambda _0\\nu _j e^\\theta +\\lambda _0\\nu _j (1-e^{-\\lambda _0\\nu _j })(1- e^\\theta )]\\rbrace \\nonumber \\\\&&= \\frac{n!", "e^n}{n^n } \\bigl (1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)\\bigr )^{n} (1+w)^{-n} \\nonumber \\\\&&\\quad \\times \\exp \\lbrace -\\frac{ {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2 \\sum _{j \\notin \\mathcal {W}_\\eta }\\nu _j^2 (1-e^{-2\\theta })(1+o^\\eta (1))}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}\\rbrace \\nonumber \\\\&&\\quad \\times \\exp \\lbrace n [\\frac{(1+w)+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j(1-e^{-\\lambda _0\\nu _j })(e^{-\\theta }-1)}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}]\\rbrace \\nonumber \\\\&&\\le \\frac{n!", "e^n}{n^n }\\exp \\lbrace -n\\log (1+w)+\\frac{nw}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}\\rbrace \\nonumber \\\\&&\\quad \\times \\exp \\lbrace -\\frac{ {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2 \\sum _{j \\notin \\mathcal {W}_\\eta }\\nu _j^2 (1-e^{-2\\theta })(1+o^\\eta (1))}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}]\\rbrace \\nonumber \\\\&&\\quad \\times \\exp \\lbrace n [\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1) -1+\\frac{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j(1-e^{-\\lambda _0\\nu _j })(e^{-\\theta }-1)}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}]\\rbrace .$ We now bound each exponential term on the right-hand side of ().", "Applying (REF ) and the lower-bound on $D_j$ in (REF ) gives the following bound on the second term: $-\\frac{ {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2 \\sum _{j \\notin \\mathcal {W}_\\eta }\\nu _j^2 (1-e^{-2\\theta })}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}\\le -{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}e^{-2\\theta }nw(1+o^\\eta (1)).$ The first exponential term satisfies $-n\\log (1+w)+\\frac{nw}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}= -n w o^\\eta (1),$ which follows from (REF ) and $w=o^\\eta (1)$ .", "Combining (REF ) and (REF ) implies that for small enough $\\eta $ , the sum of the first and second term is negative.", "The exponent in the last term on the right-hand side of () is bounded as follows: $\\begin{aligned}&\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1) \\!-\\!1\\!+\\!\\frac{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j(1-e^{-\\lambda _0\\nu _j })(e^{-\\theta }-1)}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}\\\\=&\\frac{\\bigl (\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)\\bigr )^2+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j(1-e^{-\\lambda _0\\nu _j })(e^{-\\theta }-1)}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}\\\\\\le &\\frac{(\\sum _{j \\in \\mathcal {W}_\\eta }\\nu _j)\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)^2+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j(1-e^{-\\lambda _0\\nu _j })(e^{-\\theta }-1)}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)} \\\\\\le &\\frac{\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j [(D_j-1)^2+(1-e^{-\\lambda _0\\nu _j })(e^{-\\theta }-1)]}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}\\end{aligned}$ where the first inequality follows from Jensen's inequality and the second follows from $\\sum _{j \\in \\mathcal {W}_\\eta }\\nu _j \\le 1$ .", "We first bound the summand in the numerator on the right-hand side of (REF ).", "Consider any $j \\in \\mathcal {W}_\\eta $ .", "Let $x: =\\lambda _0\\nu _j$ .", "Applying the formula of $D_j$ in (REF ) gives $\\begin{aligned}&(D_j-1)^2+(1-e^{-x})(e^{-\\theta }-1)=\\frac{e^{-x}+e^{-\\theta }-e^{-x}e^{-\\theta }}{\\bigl (1+xe^{-x}(e^\\theta -1)\\bigr )^2}[(1-e^{-x})(e^{-\\theta }-1)+\\bigl (xe^{-x}(e^\\theta -1)\\bigr )^2].\\end{aligned}$ Let $t(x)=(1-e^{-x})(e^{-\\theta }-1)+\\bigl (xe^{-x}(e^\\theta -1)\\bigr )^2$ .", "Note that $j \\in \\mathcal {W}_\\eta $ implies $n \\nu _j \\ge \\eta $ , which combined with (REF ) implies $x = \\lambda _0\\nu _j \\ge \\eta e^{-\\theta }$ .", "Since for $\\theta \\in (0,0.5]$ , $t(x)$ is strictly decreasing on $[0,\\infty )$ , we obtain $t(x) \\le t(\\eta {e^{-\\theta }})<0$ .", "Substituting this into (REF ) and using the elementary fact that $\\frac{e^{-x}+e^{-\\theta }-e^{-x}e^{-\\theta }}{\\bigl (1+xe^{-x}(e^\\theta -1)\\bigr )^2}\\!\\le \\!", "e^{-3\\theta \\!", "},$ we obtain $(D_j-1)^{2\\!}+(1-e^{-x\\!", "})(e^{-\\theta \\!", "}-1) \\!\\le \\!", "-e^{-3\\theta }t(\\eta e^{-\\eta \\!", "}).$ The denominator of on the right-hand side of (REF ) is positive and upper-bounded by 1 because $D_j \\le 1$ .", "Combining the bounds on the numerator and denominator gives a bound on the exponent in the last term on the right-hand side of () $\\begin{aligned}&\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1) -1+\\frac{(1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j(1-e^{-\\lambda _0\\nu _j })(e^{-\\theta }-1)}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}\\le - \\beta (\\nu ) \\alpha (\\theta ) \\le 0,\\end{aligned}$ where $\\alpha (\\theta )=\\frac{1}{3} e^{-3\\theta }[(1-e^{-\\eta e^{-\\theta }})(e^{-\\theta }-1)+ \\bigl (\\eta e^{-\\theta }e^{-\\eta e^{-\\theta }}(e^\\theta -1)\\bigr )^2].$ Combining (REF ), (REF ) and (REF ) and using the fact that the right-hand sides of (REF ) (REF ) are negative, we obtain: ${\\sf E}_{\\nu }[\\exp \\lbrace \\theta (\\tilde{S}_n^{*})\\rbrace ] \\le \\frac{n!", "e^n}{\\sqrt{2\\pi n}n^n } \\sqrt{2\\pi n} \\exp \\lbrace - n {\\beta (\\nu )} \\alpha (\\theta ) \\rbrace .$ Taking the logarithm on both side and applying Stirling's formula leads to $\\Lambda _{\\nu ,S_n^{*}}(\\theta ) \\le - n {\\beta (\\nu )} \\alpha (\\theta ) + {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\log (2\\pi n)+O(\\frac{1}{n}).$ Since $\\beta (\\nu ) \\ge \\underline{\\beta }$ , the second term ${\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\log (2\\pi n)$ becomes negligible comparing to the first term for large $n$ .", "This leads to the claim of the proposition.", "[Proof of Proposition REF ] We pick $\\overline{\\beta }$ so that $\\overline{\\beta }=o^\\eta (1)$ .", "It then follows that $\\sum _{j \\in \\mathcal {W}_\\eta } \\nu _j (D_j-1)=o^\\eta (1)$ Substituting this into (REF ) and (REF ) gives $\\lambda _0=ne^{-\\theta }(1+o^\\eta (1)), \\quad w=n(\\sum _{j \\notin \\mathcal {W}_\\eta } \\nu _j^2) (1-e^{-2\\theta })(1+o^\\eta (1)).$ The rest of the proof is similar to the proof of Proposition REF .", "Applying (REF ) to $j \\in \\mathcal {W}_\\eta $ , we obtain $\\begin{aligned}|h(\\psi )\\le & |e^{-i n \\psi }\\prod _{j\\notin \\mathcal {W}_\\eta } \\bigl (\\lambda _0\\nu _j(e^{\\theta }-1)e^{i\\psi }+e^{\\lambda _0\\nu _je^{i\\psi } }\\bigr )| \\prod _{j \\in \\mathcal {W}_\\eta }\\exp \\lbrace \\lambda _0\\nu _j +\\log \\bigl (1+\\lambda _0\\nu _j e^{-\\lambda _0\\nu _j }( e^\\theta -1)\\bigr )\\rbrace \\\\\\le &|e^{-i n \\psi }|\\exp \\lbrace \\bigl (\\sum _{j \\notin \\mathcal {W}_\\eta } \\lambda _0\\nu _je^{\\theta }\\cos \\psi (1+o^\\eta (1))\\bigr )+ \\sum _{j \\in \\mathcal {W}_\\eta } \\lambda _0\\nu _j e^\\theta \\rbrace \\\\=&e^n \\exp \\lbrace -n (1-\\cos \\psi + o^\\eta (1))\\rbrace .\\end{aligned}$ It is clear from (REF ) that the integrand is large at the interval around 0.", "Thus, we again split the integral in (REF ) into three parts $I_1$ , $I_2$ and $I_3$ as in (REF ).", "We will show later that $I_2$ and $I_3$ are much smaller than $I_1$ .", "We first upper-bound $I_1$ .", "Similar to (REF ), we have $\\operatorname{Im}(H(0))=0, \\operatorname{Re}(H^{\\prime }(0))=0, \\operatorname{Im}(H^{\\prime }(0))=0.$ We now estimate $H^{\\prime \\prime }(\\psi )$ , whose exactly formula is given in (REF ).", "Consider $j \\in \\mathcal {W}_\\eta $ .", "For $\\psi \\in [-\\pi /3, \\pi /3]$ , we have the following inequality: $|1+\\lambda _0\\nu _j(e^\\theta -1)e^{i\\psi }\\exp \\lbrace -\\lambda _0\\nu _j e^{i\\psi }\\rbrace |\\ge 1,$ $\\begin{aligned}|&\\lambda _0\\nu _j(e^\\theta -1)e^{i\\psi }(1-\\lambda _0\\nu _je^{i\\psi }+\\lambda _0^2\\nu _j^2e^{2i\\psi })\\exp \\lbrace -\\lambda _0\\nu _j e^{i\\psi }\\rbrace +\\lambda _0\\nu _je^{i\\psi }|\\!\\le \\!100 \\lambda _0\\nu _j e^\\theta .\\end{aligned}$ Substituting these into (REF ), we obtain $|H^{\\prime \\prime }(\\psi )| \\le 100\\overline{\\beta }n(1+o^\\eta (1))=no^\\eta (1)$ .", "Substituting this and the estimate (REF ) into the expression of $H^{\\prime \\prime }(\\psi )$ leads to $H^{\\prime \\prime }(\\psi )=-n (e^{i\\psi }+o^\\eta (1)).$ Note that the assumption of the proposition allows us to take very small $\\eta $ .", "We choose it small enough so that the term $o^\\eta (1)$ in the above equation is smaller than $0.05$ .", "Then for large enough $n$ , for any $\\psi \\in [-\\pi /3, \\pi /3]$ , we have $\\operatorname{Re}(H^{\\prime \\prime }(\\psi ))\\le -0.4n$ .", "It follows from the mean value theorem that $\\operatorname{Re}(H(\\psi )) \\le H(0)-0.2n\\psi ^2.$ Consequently, for large enough $n$ and $m$ , we have $\\begin{aligned}&I_1\\le e^{H(0)}\\int _{-\\pi /3}^{-\\pi /3} e^{-0.4 \\psi ^2} d\\psi \\le e^{H(0)} \\int _{-\\infty }^{\\infty } e^{-0.4 \\psi ^2} d\\psi = e^{H(0)}\\frac{\\sqrt{\\pi }}{\\sqrt{0.4n}}.\\end{aligned}$ We now bound the tails $I_2$ and $I_3$ .", "For $\\psi \\in [-\\pi , -\\pi /3] \\cup [\\pi /3, \\pi ]$ , we obtain from (REF ) that $|h(\\psi )|\\le \\exp \\lbrace 0.5n(1+o^\\eta (1))\\rbrace $ .", "Thus, for small enough $\\eta $ , we have $\\operatorname{Re}[I_2]+\\operatorname{Re}[I_3] =O(e^{0.6n}).$ Substituting the estimate for $I_1$ , $I_2$ and $I_3$ into (REF ) gives ${\\sf E}_{\\nu }[\\exp \\lbrace \\theta (\\tilde{S}_n^{*})\\rbrace ]\\le \\frac{n!", "}{\\sqrt{1.6n\\pi }} \\lambda _0^{-n} e^{-\\theta n}e^{H(0)}(1+o(1)).\\nonumber $ Note that the right-hand side is almost the same as the right-hand side of (REF ) except for the multiplication term $\\frac{1}{\\sqrt{1.6np}}(1+o(1))$ .", "Thus, we can bound it using the right-hand side of () after taking into account this additional multiplication term.", "We obtain $\\begin{aligned}&{\\sf E}_{\\nu }[\\exp \\lbrace \\theta (\\tilde{S}_n^{*})\\rbrace ] \\le \\frac{n!", "e^n}{n^n \\sqrt{1.6n\\pi }} \\exp \\lbrace -\\frac{ {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2 \\!\\sum _{j \\notin \\mathcal {W}_\\eta }\\!\\nu _j^2 (1\\!-\\!e^{-2\\theta })(1\\!+\\!o^\\eta (1))}{1+\\sum _{j\\in \\mathcal {W}_\\eta }\\nu _j (D_j-1)}\\rbrace (1\\!+\\!o^\\eta (1)).\\end{aligned}$ Substituting (REF ) and Stirling's formula into the right-hand side of the above inequality leads to $\\begin{aligned}&{\\sf E}_{\\nu }[\\exp \\lbrace \\theta (S_n^{*}\\!-\\!n)\\rbrace ]\\le \\frac{1}{\\sqrt{0.8}} \\exp \\lbrace -{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2(\\sum _{j \\notin \\mathcal {W}_\\eta }\\!\\!\\nu _j^2)(1\\!-\\!e^{-2\\theta })(1\\!+\\!o^\\eta (1))\\rbrace (1\\!+\\!o(1)).\\!\\end{aligned}$ Taking logarithm on both sides gives the claim of this proposition." ], [ "Proof of Theorem ", "[Proof of Theorem REF ] Let $\\Lambda _{{\\mbox{\\protect $q$}}}(\\theta )$ be the limit of the logarithmic moment generating function of $\\Lambda _{q^{(n)}, \\tilde{S}_n^{*}}$ : $\\Lambda _{{\\mbox{\\protect $q$}}}(\\theta ): =\\lim _{n \\rightarrow \\infty } \\frac{m}{n^2}\\Lambda _{q^*, \\tilde{S}_n^{*}}(\\theta ).\\nonumber $ It follows from Proposition REF that the limit exists and is given by the following $C^{1}$ function: $\\Lambda _{{\\mbox{\\protect $q$}}}(\\theta )= {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(e^{-2\\theta }-1)\\kappa ({\\mbox{\\protect $q$}}).$ Denote its Fenchel-Legendre transformation $\\Lambda _{{\\mbox{\\protect $q$}}}^*(t) : =\\sup _{\\theta } [\\theta t-\\Lambda _{{\\mbox{\\protect $q$}}}(\\theta )].$ It follows from the Gärtner-Ellis Theorem [46] that $\\begin{aligned}&-\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty }\\frac{m}{n^2} \\log ({\\sf P}_{q^{(n)}}\\lbrace S_n^{*}\\le {\\sf E}_{p}[S_n^{*}]\\!+\\!\\frac{n^2}{m}\\tau \\rbrace )\\\\=&-\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty }\\frac{m}{n^2} \\log ({\\sf P}_{q^{(n)}}\\lbrace \\tilde{S}_n^{*}\\ge -{\\sf E}_{p}[S_n^{*}]-n-\\frac{n^2}{m}\\tau \\rbrace )\\\\=&\\inf _{t \\ge -\\tau -1}\\Lambda _1^*(t)=\\Lambda _1^*(-\\tau -1)\\\\=&\\sup _{\\theta \\ge 0} \\lbrace \\theta (-1-\\tau )-{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(e^{-2\\theta }-1)\\kappa ({\\mbox{\\protect $q$}})\\rbrace .\\end{aligned}$ where $-\\tau -1$ is the normalized limit of $-{\\sf E}_{p}[S_n^{*}]-n-\\frac{n^2}{m}\\tau $ by Lemma REF .", "[Proof of Theorem REF ] The proof for the result on the generalized error exponent of false alarm $J_F(\\phi ^{{*}})$ is very similar to that of Theorem REF .", "Let $\\Lambda _{0}(\\theta )$ be the limit of the logarithmic moment generating function of $\\Lambda _{p, \\tilde{S}_n^{*}}$ : $\\Lambda _{0}(\\theta ): =\\lim _{n \\rightarrow \\infty } \\frac{m}{n^2}\\Lambda _{p, \\tilde{S}_n^{*}}(\\theta ).\\nonumber $ It follows from Proposition REF that the limit exists and is given by the following $C^{1}$ function: $\\Lambda _{0}(\\theta )= {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(e^{-2\\theta }-1).$ Let $\\Lambda _0^*(t)=\\sup _{\\theta }[\\theta t-\\Lambda _0(\\theta )]$ .", "It follows from the Gärtner-Ellis Theorem that $\\begin{aligned}&-\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty } \\frac{m}{n^2} \\log ( {\\sf P}_{p}(\\phi ^{{*}}_n=1))\\\\=&-\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty }\\frac{m}{n^2} \\log ({\\sf P}_{p}\\lbrace \\tilde{S}_n^{*}\\le -{\\sf E}_{p}[S_n^{*}]-n-\\frac{n^2}{m}\\tau \\rbrace )\\\\=&\\inf _{t \\le -\\tau -1}\\Lambda _0^*(t)=\\Lambda _0^*(-\\tau -1)\\\\=&\\sup _\\theta \\lbrace \\theta (-\\tau -1)-{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(e^{-2\\theta }-1)\\rbrace =J_F^*(\\tau ).\\end{aligned}$ For the result on the generalized error exponent of missed detection $J_M(\\phi ^{{*}})$ , we prove an upper-bound and a lower-bound.", "For the upper-bound, consider the sequence of distributions given in (REF ) and (REF ) and let ${\\mbox{\\protect $q$}}^*$ denote this sequence.", "The rate function associated with ${\\mbox{\\protect $q$}}^*$ satisfies $ J_{{\\mbox{\\protect $q$}}^*}(\\phi ^{{*}}, \\tau )=J^*_M(\\tau ).$ On the other hand, since each element of ${\\mbox{\\protect $q$}}^*$ is in the set of alternative distributions, it follows from the definition of $J_M(\\phi ^{{*}})$ and $J_{{\\mbox{\\protect $q$}}^*}(\\phi ^{{*}}, \\tau )$ that $J_M(\\phi ^{{*}}) \\le J_{{\\mbox{\\protect $q$}}^*}(\\phi ^{{*}}, \\tau )$ To obtain the lower-bound on $J_M(\\phi ^{{*}})$ , we apply Proposition REF and Proposition REF .", "We only need to prove it for the case $\\tau \\in [0,\\underline{\\kappa }(\\varepsilon ))$ .", "The case $\\tau =\\underline{\\kappa }(\\varepsilon )$ will then follow from a continuity argument.", "Take $\\theta _0$ to be the maximizer in the optimization problem defining $J_M^*(\\tau ) $ (see (REF )).", "It is not difficult to see that $\\theta _0 > 0$ .", "It follows from Lemma REF that $m\\sum _{j \\notin \\mathcal {W}_\\eta }q_j^2 \\ge (1+\\underline{\\kappa }(\\frac{\\varepsilon -\\beta (q)}{1-\\beta (q)}))(1-\\beta (q))(1+o(1)).$ Thus, for any $\\delta >0$ , we can choose $\\eta , \\beta _0$ small enough so that for any $q\\in {\\mathcal {Q}_n}$ satisfying $\\beta (q)\\le \\beta _0$ , we have $m\\sum _{j \\notin \\mathcal {W}_\\eta }q_j^2 \\ge (1+\\underline{\\kappa }(\\varepsilon ))(1-\\delta )$ .", "It then follows from Proposition REF that for large enough $n$ , $\\Lambda _{q,\\tilde{S}_n^{*}}(\\theta _0)\\le {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}(1+\\underline{\\kappa }(\\varepsilon ))(e^{-2\\theta _0}-1)(1-\\delta )^2+O(1).$ For $q$ satisfying $\\beta (q) \\ge \\beta _0$ , it follows from Proposition REF that for large enough $n$ , $\\Lambda _{q,\\tilde{S}_n^{*}}(\\theta _0)\\le -\\beta _0 \\alpha (\\theta _0) n. $ We can pick $n$ large enough so that the right-hand side of (REF ) is smaller than the right-hand side of (REF ).", "Applying the Chernoff bound leads to $\\begin{aligned}&\\log (\\sup _{q\\in {\\mathcal {Q}_n}}{\\sf P}_q(\\phi ^{{*}}_n=0))\\\\\\le & -\\theta _0 ({\\sf E}_{p}[\\tilde{S}_n^{*}]-\\tau _n)+\\sup _{q\\in {\\mathcal {Q}_n}}\\Lambda _{q,\\tilde{S}_n^{*}}(\\theta _0)\\\\\\le & \\theta _0(\\tau _n\\!", "-\\!", "{\\sf E}_{p}[\\tilde{S}_n^{*}]) \\!+\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}(1\\!+\\!\\underline{\\kappa }(\\varepsilon ))(e^{-2\\theta _0}-1)(1\\!-\\!\\delta )^2\\!+\\!O(1).\\end{aligned}\\nonumber $ Thus, $J_M(\\phi ^{{*}}) \\ge \\theta _0(-1-\\tau )-{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}( e^{-2\\theta _0}-1)(1+\\underline{\\kappa }(\\varepsilon ))(1-\\delta )^2.$ This holds for any $\\delta >0$ .", "Consequently, $J_M(\\phi ^{{*}}) \\ge J_M^*(\\tau )$ ." ], [ "Proof of Theorem ", "The performance of $\\phi ^{{*+}}$ is analyzed by connecting it to the performance of $\\phi ^{{*}}$ .", "We first show that its probability of missed detection is no larger than that of $\\phi ^{{*}}$ .", "We then apply a result similar to Proposition REF to analyze its probability of false alarm.", "Consider the statistic $\\tilde{S}_n^{*+}=-S_n^{*+}-n.$ Define $\\Lambda _{\\nu , \\tilde{S}_n^{*+}}(\\theta ): =\\log \\bigl ({\\sf E}_{\\nu }[\\exp \\lbrace \\theta (\\tilde{S}_n^{*+})\\rbrace ]\\bigr ).$ Proposition 7 For any $\\nu \\in \\mathcal {P}^b_m$ , the logarithmic moment generating function for the statistic $\\tilde{S}_n^{*+}$ has the following asymptotic expansion $\\begin{aligned}\\Lambda _{\\nu , \\tilde{S}_n^{*+}}(\\theta )=&\\frac{n^2}{m}\\bigl (m\\sum _{j=1}^m\\nu _j^2\\bigr )\\lbrace -\\theta +{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}[e^{-2\\theta }-(1-2\\theta )]\\rbrace +O(\\frac{n^3}{m^2})+O(1).\\end{aligned}$ [Proof of Proposition REF ] The proof follows exactly the same step as that of Proposition REF except some of the approximations are different.", "We now only describe the key steps and highlight the difference: First, the estimate of the saddle point is the same as (REF ) and (REF ).", "Second, different from (REF ), we have the following expression of the moment generating function: $\\begin{aligned}{\\sf E}_\\nu [\\exp \\lbrace \\theta (\\tilde{S}_n^{*+})\\rbrace ]=\\frac{n!", "}{2\\pi }\\lambda _0^{-n} e^{-\\theta n}\\operatorname{Re}\\bigl [\\int _{-\\pi }^\\pi h(\\psi ) d\\psi \\bigr ]\\nonumber .\\end{aligned}$ where instead of (REF ), $\\begin{aligned}&h(\\psi ): =e^{-i n \\psi }\\prod _{j=1}^m \\bigl (\\lambda _0\\nu _j(e^{\\theta }-1)e^{i\\psi }+e^{\\lambda _0\\nu _je^{i\\psi } }+ \\sum _{l=2}^{\\bar{l}} \\frac{(\\lambda _0\\nu _j)^l}{l!", "}(e^{\\theta v_l}-1)\\bigr ).\\end{aligned}$ It follows from $\\lambda _0=n^{-\\theta }(1+o(1))$ that the last term is negligible when $v_2=0$ and $\\bar{l}< \\infty $ .", "$\\sum _{l=2}^{\\bar{l}} \\frac{(\\lambda _0\\nu _j)^l}{l!", "}(e^{\\theta v_l}-1)=O(\\frac{n^3}{m^3})$ The asymptotic approximation of $h(\\psi )$ is the same as that in (REF ): $\\begin{aligned}h(\\psi )&=e^{-i n \\psi }\\prod _{j=1}^m \\bigl (\\lambda _0\\nu _j(e^{\\theta }-1)e^{i\\psi }\\!+\\!1\\!+\\!\\lambda _0\\nu _je^{i\\psi }\\!+\\!O(\\frac{n^2}{m^2})\\bigr ).\\nonumber \\end{aligned}$ Finally, the approximations of $H(0), H^{\\prime }(0), H^{\\prime \\prime }(\\psi )$ are the same as in (REF ).", "Therefore, $\\Lambda _{\\nu , \\tilde{S}_n^{*+}}$ has the same asymptotic approximation as that of $\\Lambda _{\\nu , \\tilde{S}_n^{*}}$ up to an approximation error of $O(\\frac{n^3}{m^2})$ .", "[Proof of Theorem REF ] Since $v_l \\ge 0$ for $l \\ge 2$ , we have $S_n^{*+}\\ge S_n^{*}.$ Thus, for the same sequence of thresholds $\\tilde{\\tau }_n$ , we have ${\\sf P}_{q}\\lbrace S_n^{*+}\\le \\tilde{\\tau }_n\\rbrace \\le {\\sf P}_{q}\\lbrace S_n^{*}\\le \\tilde{\\tau }_n\\rbrace $ On the other hand, since $\\Lambda _{\\nu , \\tilde{S}_n^{*+}}$ has the same asymptotic approximation as that of $\\Lambda _{\\nu , \\tilde{S}_n^{*}}$ up to an approximation error of $O(\\frac{n^3}{m^2})$ , we have $\\begin{aligned}& \\log {\\sf P}_p\\lbrace S_n^{*+}\\ge -n+\\tilde{\\tau }_n\\rbrace \\\\=&\\log {\\sf P}_p\\lbrace \\tilde{S}_n^{*+}\\le -\\tilde{\\tau }_n\\rbrace \\\\\\le & \\theta (-\\tilde{\\tau }_n)+\\Lambda _{p,\\tilde{S}_n^{*+}}(-\\theta )\\\\=&-\\theta \\tilde{\\tau }_n+\\frac{n^2}{m}\\bigl (\\theta +{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}[e^{2\\theta }-(1+2\\theta )]\\bigr )+O(\\frac{n^3}{m^2})+O(1).\\end{aligned}\\nonumber $ which is the same bound as that for $\\log {\\sf P}_p\\lbrace S_n^{*}\\ge -n+\\tilde{\\tau }_n\\rbrace $ ." ], [ "Proof of Theorem ", "The proof of Theorem REF follows exactly the same steps as those in the proof of Theorem REF .", "We use Proposition REF , Proposition REF and Proposition REF in place of Proposition REF , Proposition REF and Proposition REF .", "Denote $\\Lambda _{\\nu , S_n^{\\sf W}}(\\theta ): =\\log \\bigl ({\\sf E}_{\\nu }[\\exp \\lbrace \\theta S_n^{\\sf W}\\rbrace ]\\bigr ).$ Proposition 8 For any $\\nu \\in \\mathcal {P}^b_m$ , the logarithmic moment generating function for the statistic $S_n^{\\sf W}$ has the following asymptotic expansion $\\begin{aligned}\\Lambda _{\\nu ,S_n^{\\sf W}}(\\theta )=&{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2(\\sum _{j=1}^m(p_j-\\nu _j)^2) \\theta +{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2(\\sum _{j=1}^m\\nu _j^2) [e^{\\theta }-(1+\\theta )]+O(\\frac{n^3}{m^2})+O(1).\\nonumber \\end{aligned}$ Proposition 9 For all sufficiently small $\\eta >0$ , any $\\theta \\in [-1,0)$ and any $\\underline{\\beta }>0$ .", "There exists $n_0$ such that for any $n>n_0$ , and any $\\nu $ satisfying $\\beta (\\nu ) \\le \\underline{\\beta }$ , the following holds, $\\Lambda _{\\nu ,S_n^{\\sf W}}(\\theta )\\le - \\beta (q) \\alpha ^{\\prime }(\\theta ) n$ where $\\alpha ^{\\prime }(\\theta )>0$ for $\\theta \\in [-1,0)$ .", "Proposition 10 For any $\\delta >0$ , $\\theta \\in [-1,0)$ , $\\overline{\\eta }>0$ , there exists $\\eta \\in (0, \\overline{\\eta })$ , $\\overline{\\beta }>0$ , and $n_0$ such that for any $n>n_0$ , and any $\\nu $ satisfying $\\beta (q) \\le \\overline{\\beta }$ , the following holds, $\\begin{aligned}&\\Lambda _{\\nu ,S_n^{\\sf W}}(\\theta \\le \\frac{n^2}{m}[(m\\!\\!\\sum _{j \\notin \\mathcal {W}_\\eta (\\nu )}\\!", "(p_j-\\nu _j)^2)\\theta +{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(m\\!\\!\\sum _{j\\notin \\mathcal {W}_\\eta (\\nu )}\\!\\!\\nu _j^2)(e^{\\theta }-(1+\\theta ))](1-\\delta ).\\end{aligned}$ We only outline the proof for Proposition REF .", "[Proof of Proposition REF ] The steps are the same as thos in the proof of Proposition REF .", "Again, we describe the main steps and highlight the difference.", "First, the estimate of the saddle point is different than that in (REF ) and (REF ).", "We have $\\begin{aligned}\\lambda _0&=n(1+w), \\\\w&=n(\\sum _j \\nu _j p_j \\theta - \\sum _j \\nu _j^2 (e^\\theta -1)) (1+O(\\frac{n}{m})).\\end{aligned}\\nonumber $ Second, different from (REF ), we have the following expression of the moment generating function: ${\\sf E}_\\nu ^n[\\exp \\lbrace \\theta S_n^{\\sf W}\\rbrace ]=\\frac{n!", "}{2\\pi }\\lambda _0^{-n}\\operatorname{Re}[\\int _{-\\pi }^\\pi h(\\psi )d\\psi ]$ where $\\begin{aligned}h(\\psi )&= e^{\\!-i n \\psi }\\!\\prod _{j=1}^m \\!", "[\\exp \\lbrace \\!\\lambda _0\\nu _je^{i\\psi }\\!\\rbrace \\!+\\!(e^{\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2p_j^2 \\theta \\!", "}\\!-\\!\\!1)+\\!\\lambda _0e^{i\\psi } \\nu _j(e^{\\!-np_j \\theta \\!}\\!\\!-\\!\\!1)\\!+\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\lambda _0^2 e^{2i\\psi \\!}", "\\nu _j^2 (e^{\\theta }\\!\\!-\\!\\!1) ]\\\\&=e^{-i n \\psi }\\exp \\lbrace n e^{i\\psi }+O(\\frac{n^2}{m})\\rbrace ,\\end{aligned}\\nonumber $ Finally, the approximation of $\\operatorname{Re}(H(0))$ is different from that in (REF ) $\\begin{aligned}\\operatorname{Re}(H(0))\\!=&n (1\\!+\\!w)\\!+\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2(\\sum _{j=1}^m(p_j\\!-\\!\\nu _j)^2) \\theta \\!+ \\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2(\\sum _{j=1}^m\\nu _j^2) (e^\\theta \\!-\\!1\\!-\\!\\theta )\\!+\\!O(\\frac{n^3}{m^2}).\\!\\end{aligned}$ The rest of the steps are the same as those in Proposition REF .", "[Proof of Theorem REF ] We first prove the lower-bound on $J_F(\\phi ^{{\\sf W}})$ .", "Substituting the asymptotic approximation of $\\Lambda _{p,S_n^{\\sf W}}(\\theta )$ given in Proposition REF into the Chernoff bound, we obtain that for $\\theta \\ge 0$ , $\\begin{aligned}&\\log {\\sf P}_p(\\phi ^{{\\sf W}}_n=1)\\\\\\le & -\\theta \\tau _n+\\Lambda _{p,S_n^{\\sf W}}(\\theta )\\nonumber \\\\=&-\\theta \\tau _n+n^2(\\sum _{j=1}^mp_j^2){\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}[e^{\\theta }-(1+\\theta )]+O(\\frac{n^3}{m^2})+O(1).\\end{aligned}\\nonumber $ Since $m\\sum _{j=1}^m p_j^2\\le \\gamma ^2$ , which is a consequence of Assumption REF , we have $J_F(\\phi ^{{\\sf W}}) \\ge \\sup _{\\theta \\ge 0} \\lbrace {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\tau \\theta - {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\gamma ^2 [e^{\\theta }-(1+\\theta )]\\rbrace >0.$ Lower-bounding $J_M(\\phi ^{{\\sf W}})$ requires us to obtain a uniform bound on the probability ${\\sf P}_q(\\phi _n=0)$ over $q\\in {\\mathcal {Q}_n}$ .", "We apply Proposition REF and Proposition REF .", "Using an argument similar to the proof in Theorem REF , we conclude that for any $\\delta >0$ , and $\\theta \\in (0,1]$ , for large enough $n$ , $\\begin{aligned}&\\log {\\sf P}_q(\\phi ^{{\\sf W}}_n\\!=\\!0)\\\\\\!\\le & \\theta \\tau _n+\\Lambda _{q,S_n^{\\sf W}}(-\\theta )\\\\\\!=&\\theta \\tau _n\\!\\!-\\!\\!\\frac{n^2\\!", "}{m} [{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\theta m\\!\\sum _{j=1}^m\\!", "(q_j\\!\\!-\\!p_j)^2\\!-\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(m\\!\\sum _{j=1}^m\\!q_j^2)\\bigl (e^{\\!-\\theta \\!}\\!\\!-\\!", "(1\\!\\!-\\!\\!\\theta )\\bigr )](1\\!-\\!\\delta ).\\end{aligned}\\nonumber $ We need to upper-bound the right-hand side uniformly over all $q\\in {\\mathcal {Q}_n}$ .", "Using the inequalities $q_j^2 \\le 2p_j^2+2(p_j-q_j)^2$ and $e^{-\\theta }-(1-\\theta ) \\le {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\theta ^2$ for $\\theta >0$ , we obtain $\\begin{aligned}&\\frac{m}{n^2}\\log {\\sf P}_q(\\phi ^{{\\sf W}}_n=0) \\nonumber \\\\\\le & \\theta \\frac{m\\tau _n\\!}{n^2}\\!-\\!", "[{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\theta m\\sum _{j=1}^m(q_j\\!-\\!p_j)^2-\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\theta ^2 \\bigl (m\\!\\sum _{j=1}^m p_j^2\\!+\\!m\\!\\sum _{j=1}^m(q_j\\!-\\!p_j)^2\\bigr )](1\\!-\\!\\delta )\\!+\\!O(1)\\nonumber \\\\=&{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\theta [-(m\\sum _{j=1}^m(q_j-p_j)^2)(1-\\theta )+\\theta (m\\sum _{j=1}^m p_j^2)](1-\\delta ) +\\theta \\frac{m\\tau _n}{n^2}+O(1).\\end{aligned}\\nonumber $ Applying $m\\sum _{j=1}^m (q_j-p_j)^2 \\ge 4\\varepsilon ^2$ and $m\\sum _{j=1}^m p_j^2\\le \\gamma ^2$ leads to, $\\frac{m}{n^2}\\log [P_M(\\phi ^{{\\sf W}}_n)] \\le {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\theta [-4\\varepsilon ^2(1-\\theta )+\\theta \\gamma ^2](1-\\delta )+\\frac{m\\tau _n}{n^2}+O(1).$ Taking $\\theta =(4\\varepsilon ^2(1-\\delta )-2\\tau )/[(8\\varepsilon ^2+2\\gamma ^2)(1-\\delta )]$ , and taking the limit on both sides gives $J_M(\\phi ^{{\\sf W}}) \\ge \\frac{1}{4} 4\\varepsilon ^2\\frac{4\\varepsilon ^2(1-\\delta )-2\\tau }{(8\\varepsilon ^2+2\\gamma ^2)(1-\\delta )}.$ Since this holds for all $\\delta >0$ , and $2\\tau < 4\\varepsilon ^2$ , we conclude that $J_M(\\phi ^{{\\sf W}}) \\ge \\frac{1}{4} 4\\varepsilon ^2\\frac{2\\varepsilon ^2-\\tau }{(8\\varepsilon ^2+2\\gamma ^2)({\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}+\\tau /(4\\varepsilon ^2))}>0.$" ], [ "Proof of Theorem ", "We first give an outline of the proof: Consider any $\\tau \\in [0, \\underline{\\kappa }(\\varepsilon )]$ .", "Given $\\delta >0$ , a sequence of events $\\lbrace B_{n, \\tau ,\\delta }\\rbrace $ is constructed so that the following is satisfied: (i) The probability of the event is close to the probability of false alarm: $\\mathop {\\rm lim{\\,}sup}_{n \\rightarrow \\infty }-\\frac{m}{n^2}\\log ({\\sf P}_p(B_{n, \\tau ,\\delta }))\\le J_F^*(\\tau )-\\delta .$ (ii) For any ${\\mbox{\\protect $z$}}_1^n$ satisfying $\\lbrace {\\mbox{\\protect $Z$}}_1^n={\\mbox{\\protect $z$}}_1^n\\rbrace \\subseteq B_{n, \\tau ,\\delta }$ , the following uniform bound on the likelihood ratio holds: $\\sup _{q\\in {\\mathcal {Q}_n}}\\frac{q^n}{p^n}({\\mbox{\\protect $z$}}_1^n) \\ge \\exp \\lbrace -\\frac{n^2}{m}(J_M^*(\\tau )-J_F^*(\\tau )+\\delta )\\rbrace .$ The lower-bound on $P_M$ is then obtained from the following inequality: $\\begin{aligned}P_M(\\phi _n)\\ge & \\sup _{q\\in {\\mathcal {Q}_n}}{\\sf P}_{q}\\bigl (\\lbrace \\phi _n=0\\rbrace \\cap B_{n, \\tau ,\\delta }\\bigr )\\\\\\ge & \\sup _{q\\in {\\mathcal {Q}_n}}\\frac{q^n}{p^n}(\\lbrace \\phi _n=0\\rbrace \\cap B_{n, \\tau ,\\delta }){\\sf P}_p(\\lbrace \\phi _n=0\\rbrace \\cap B_{n, \\tau ,\\delta })\\\\\\ge & \\sup _{q\\in {\\mathcal {Q}_n}}\\frac{q^n}{p^n}(\\lbrace \\phi _n=0\\rbrace \\cap B_{n, \\tau ,\\delta })\\bigl ({\\sf P}_p(B_{n, \\tau ,\\delta })-{\\sf P}_p(\\lbrace \\phi _n=1\\rbrace )\\bigr ) .\\end{aligned}$ The first term on the right-hand side is lower-bounded in (REF ).", "The second term can be shown to have the same large deviations limit as that of ${\\sf P}_p(B_{n ,\\tau ,\\delta })$ : ${\\sf P}_p(\\lbrace \\phi _n=0\\rbrace \\cap B_{n, \\tau ,\\delta }) \\ge {\\sf P}_p(B_{n, \\tau ,\\delta })-{\\sf P}_p(\\lbrace \\phi _n=1\\rbrace )$ The inequality in (REF ) ensures that ${\\sf P}_p(\\lbrace \\phi _n=1\\rbrace )$ is negligible comparing to ${\\sf P}_p(B_{n, \\tau ,\\delta })$ .", "The technique of using uniform lower-bounds on likelihood ratio (LR) to prove lower-bounds of probability of missed detection has been applied in [5], [3]: In this prior work, a uniform bound on LR is obtained over all possible ${\\mbox{\\protect $z$}}_1^n$ .", "To prove the tight hardness result as in Theorem REF , we require the bound on LR to hold uniformly for the sequences in the event $B_n$ , instead of all sequences.", "This gives us the freedom to optimize $B_n$ to obtain the tightest bound.", "The technique to prove (REF ) has been previously used in providing hardness results for composite and hypothesis testing problems [5], [3], [35].", "First, construct a collection of distributions so that for each distribution $q$ , the likelihood ratio $q/p$ has a simple expression.", "Second, show that for all observations ${\\mbox{\\protect $z$}}_1^n : =\\lbrace z_1, \\ldots , z_n\\rbrace $ in the event $B_n$ , the average of ${\\sf P}_{q}\\lbrace {\\mbox{\\protect $Z$}}_1^n={\\mbox{\\protect $z$}}_1^n\\rbrace /{\\sf P}_{p}\\lbrace {\\mbox{\\protect $Z$}}_1^n={\\mbox{\\protect $z$}}_1^n\\rbrace $ over the collection of distributions can be lower-bounded, which in turn lower-bounds the left-hand side of (REF ).", "The proof for $\\varepsilon < 0.5$ and $\\varepsilon \\ge 0.5$ uses different constructions of distributions.", "We now carry out these two steps: Construction of the event $B_{n, \\tau , \\delta }$ and lower-bounding the likelihood ratio." ], [ "Construction of $B_{n, \\tau , \\delta }$", "Define the event $\\begin{aligned}B_{n, \\tau ,\\delta }\\!=&\\!\\!\\bigl \\lbrace \\sum _{j=1}^m \\mathbb {I}\\lbrace n\\Gamma ^n_j \\!\\!=\\!\\!1\\rbrace \\!\\ge \\!n-(1\\!+\\tau \\!+\\delta )\\frac{n^{2\\!", "}}{m}, \\quad \\sum _{j=1}^m \\mathbb {I}\\lbrace n\\Gamma ^n_j\\!\\!", "=\\!", "\\!2\\rbrace \\!\\ge \\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(1\\!+\\tau \\!-\\delta ) \\frac{n^{2\\!", "}}{m}\\bigr \\rbrace .\\!\\end{aligned}$ The probability of the event $B_{n,\\tau ,\\delta }$ has the following asymptotic approximation: Lemma 10 For $\\tau =0$ and any $\\delta >0$ , $\\lim _{n \\rightarrow \\infty } {\\sf P}_{p}(B_{n, \\tau ,\\delta })=1.$ For any $\\tau , \\delta $ satisfying $\\tau >\\delta >0$ , $\\lim _{n \\rightarrow \\infty }-\\frac{m}{n^2}\\log {\\sf P}_p(B_{n, \\tau ,\\delta }) = J_F^*(\\tau -\\delta ).$ [Proof of Lemma REF ] First consider the case where $\\tau =0$ .", "Applying Theorem REF with $\\tau $ replaced by $\\delta $ gives ${\\sf P}_p\\big \\lbrace \\sum _{j=1}^m \\mathbb {I}\\lbrace n\\Gamma ^n_j =1\\rbrace \\le n-(1+\\delta )\\frac{n^2}{m}\\big \\rbrace =1-o(1).$ The following asymptotic approximations on the expectation and variance of the statistic $\\sum _{j=1}^m \\mathbb {I}\\lbrace n\\Gamma ^n_j = 2\\rbrace $ follows from Lemma REF and Lemma REF : $\\begin{aligned}{\\sf E}_{p}[\\sum _{j=1}^m\\mathbb {I}\\lbrace n\\Gamma ^n_j = 2\\rbrace ]\\!", "&=\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}(1+o(1)), \\\\\\hbox{\\sf Var}\\,_{p}[\\sum _{j=1}^m\\mathbb {I}\\lbrace n\\Gamma ^n_j = 2\\rbrace ]\\!", "&=\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}(1+o(1)).\\!\\end{aligned}\\nonumber $ Applying Chebyshev's inequality leads to ${\\sf P}_{p}\\bigl \\lbrace \\sum _{j=1}^m\\mathbb {I}\\lbrace n\\Gamma ^n_j = 2\\rbrace \\le {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}(1-\\delta )\\bigr \\rbrace =O(\\frac{m}{n^2}).$ The claim of this lemma for $\\tau =0$ follows from combining this inequality with (REF ).", "Next consider the case where $\\tau >0$ .", "We first obtain a large deviations characterization of $S^{(2)}: =\\sum _{j=1}^m \\mathbb {I}\\lbrace n\\Gamma ^n_j = 2\\rbrace $ by deriving an approximation to the logarithmic moment generating function.", "The steps are the same as those in the proof of Proposition REF .", "Again, we describe the main steps and highlight the difference.", "First, the estimate of the saddle point is different than that in (REF ) and (REF ).", "We have $\\begin{aligned}\\lambda _0&=n(1+w), \\\\w&=-n\\sum _j \\nu _j^2 (e^\\theta -1) (1+O(\\frac{n}{m})).\\end{aligned}\\nonumber $ Second, different from (REF ), we have the following expression of the moment generating function: ${\\sf E}_\\nu ^n[\\exp \\lbrace \\theta S^{(2)}\\rbrace ]=\\frac{n!", "}{2\\pi }\\lambda _0^{-n}\\operatorname{Re}[\\int _{-\\pi }^\\pi h(\\psi )d\\psi ]$ where $\\begin{aligned}h(\\psi )&\\!=\\!", "e^{\\!-i n \\psi }\\!\\prod _{j=1}^m \\!", "[\\exp \\lbrace \\lambda _0\\nu _je^{i\\psi }\\rbrace \\!", "+\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\lambda _0^2 e^{2i\\psi \\!}", "\\nu _j^2 (e^{\\theta }\\!\\!-\\!\\!1) ]=e^{-i n \\psi }\\exp \\lbrace n e^{i\\psi }+O(\\frac{n^2}{m})\\rbrace \\end{aligned}\\nonumber $ Finally, the approximation of $\\operatorname{Re}(H(0))$ is different from that in (REF ) $\\begin{aligned}\\operatorname{Re}(H(0))=&n (1+w)+ {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}n^2(\\sum _{j=1}^m\\nu _j^2) (e^\\theta -1)+O(\\frac{n^3}{m^2}).\\end{aligned}$ The rest of the steps are the same as those in Proposition REF .", "We obtain $\\Lambda _{\\nu , S^{(2)}}(\\theta )={\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}\\bigl (m\\sum _{j=1}^m\\nu _j^2\\bigr )(e^{-2\\theta }-1)+O(\\frac{n^3}{m^2})+O(1).\\\\$ Applying the same steps as those for the characterization of $J_F(\\phi ^{{*}})$ in Theorem REF , we have $\\begin{aligned}&\\lim _{n \\rightarrow \\infty }-\\frac{m}{n^2}\\log {\\sf P}_p\\big \\lbrace \\sum _{j=1}^m \\mathbb {I}\\lbrace n\\Gamma ^n_j = 2\\rbrace \\ge {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(1+\\tau -\\delta ) \\frac{n^2}{m}\\big \\rbrace = J_F^*(\\tau -\\delta ).\\nonumber \\end{aligned}$ Applying Theorem REF with $\\tau $ replaced by $\\tau +\\delta $ , we obtain $\\begin{aligned}&\\lim _{n \\rightarrow \\infty }-\\frac{m}{n^2}\\log {\\sf P}_p\\big \\lbrace \\sum _{j=1}^m \\mathbb {I}\\lbrace n\\Gamma ^n_j =1\\rbrace \\le n-(1+\\tau +\\delta )\\frac{n^2}{m}\\big \\rbrace = J_F^*(\\tau +\\delta ).\\end{aligned}$ Note that $J_F^*(\\tau +\\delta ) > J_F^*(\\tau -\\delta )$ .", "Thus the probability that the first constraint in the definition of $B_{n,\\tau ,\\delta }$ is violated is negligible comparing to the probability that the second constraint is satisfied.", "This shows that the probability of $B_{n,\\tau ,\\delta }$ can be approximated by the probability that the second constraint in the definition of $B_{n,\\tau ,\\delta }$ is satisfied.", "This leads to the claim of the lemma." ], [ "A lower-bound on the likelihood ratio for $\\varepsilon \\ge 0.5$", "When $\\varepsilon \\ge 0.5$ , we use the following construction of distributions: Let $U_m$ denote the collection of all subsets of $[m]$ whose cardinality is $\\lfloor m(1-\\varepsilon ) \\rfloor $ .", "For each $\\mathcal {U} \\in U_m$ , define the distribution $q_{\\mathcal {U},j}=\\left\\lbrace \\begin{array} {l l} \\frac{1}{\\lfloor m(1-\\varepsilon )\\rfloor }, & j \\in \\mathcal {U};\\\\ 0, & j \\in [m]\\setminus \\mathcal {U}.\\end{array}\\right.\\nonumber $ Consider the mixture $\\bar{q}^n\\!=\\!\\frac{1}{|U_m|}\\sum _{\\mathcal {U} \\in U_m}q_{\\mathcal {U}}^n$ .", "The following bound on $\\bar{q}^n/p^n$ holds: Lemma 11 Suppose $\\varepsilon \\ge 0.5$ .", "For any sequence ${\\mbox{\\protect $z$}}_1^n= \\lbrace z_1, \\ldots , z_n\\rbrace $ satisfying $\\lbrace {\\mbox{\\protect $Z$}}_1^n={\\mbox{\\protect $z$}}_1^n\\rbrace \\subseteq B_{n, \\tau ,\\delta }$ , the following holds: $\\log \\bigl (\\frac{\\bar{q}^n}{p^n}({\\mbox{\\protect $z$}}_1^n)\\bigr )\\ge -{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}[\\underline{\\kappa }(\\varepsilon )-\\log (1+\\underline{\\kappa }(\\varepsilon ))(1+\\tau -\\delta )]+O(\\frac{n^3}{m^2}).$ [Proof of Lemma REF ] Let $\\mathcal {S}: =\\lbrace j: \\textrm { $ j$ appears in $$z$1n$}\\rbrace $ .", "Let $s=|\\mathcal {S}|$ .", "It follows from $\\lbrace {\\mbox{\\protect $Z$}}_1^n={\\mbox{\\protect $z$}}_1^n\\rbrace \\subseteq B_{n,\\tau ,\\delta }$ that $n-{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m} (1+\\tau +3\\delta ) \\le s \\le n-{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m} (1+\\tau -\\delta ).$ The likelihood ratio $\\frac{q_{\\mathcal {U}}^n}{p^n}$ has the expression: $\\frac{q_{\\mathcal {U}}^n}{p^n}({\\mbox{\\protect $z$}}_1^n)=(\\frac{m}{\\lfloor m(1-\\varepsilon )\\rfloor })^n \\mathbb {I}_{\\mathcal {S} \\subseteq \\mathcal {U}}$ .", "Thus, $\\frac{\\bar{q}^n}{p^n}({\\mbox{\\protect $z$}}_1^n)=(\\frac{m}{\\lfloor m(1-\\varepsilon )\\rfloor })^n(\\frac{1}{|U_m|}\\sum _{\\mathcal {U} \\in U_m}\\mathbb {I}_{\\mathcal {S} \\subseteq \\mathcal {U}} ),$ where $\\frac{1}{|U_m|}\\sum _{\\mathcal {U} \\in U_m}\\mathbb {I}_{\\mathcal {S} \\subseteq \\mathcal {U}} =\\frac{{m-s \\atopwithdelims ()\\lfloor m(1-\\varepsilon ) \\rfloor -s}}{{m \\atopwithdelims ()\\lfloor m(1-\\varepsilon ) \\rfloor }}.\\nonumber $ Stirling's formula gives $\\begin{aligned}&\\frac{{m-s \\atopwithdelims ()\\lfloor m(1-\\varepsilon ) \\rfloor -s}}{{m \\atopwithdelims ()\\lfloor m(1-\\varepsilon ) \\rfloor }}=(\\frac{\\lfloor m(1-\\varepsilon )\\rfloor }{m})^s\\exp \\lbrace -{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{s^2}{m}\\frac{\\varepsilon }{1-\\varepsilon }+O(\\frac{k^3}{m^2})\\rbrace (1+O(\\frac{1}{m})).\\nonumber \\end{aligned}$ Substituting this into (REF ) leads to $\\frac{\\bar{q}^n}{p^n}({\\mbox{\\protect $z$}}_1^n)= (1-\\varepsilon )^s\\exp \\lbrace -{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{s^2}{m}\\frac{\\varepsilon }{1-\\varepsilon }+O(\\frac{n^3}{m^2})\\rbrace (1+O(\\frac{n}{m})).$ The claim of this lemma follows from applying the inequality (REF ) and the fact that $\\underline{\\kappa }(\\varepsilon )=\\frac{\\varepsilon }{1-\\varepsilon }$ when $\\varepsilon \\ge 0.5$ ." ], [ "A lower-bound on the likelihood ratio for $\\varepsilon < 0.5$", "When $\\varepsilon <0.5$ , we use the following construction of distributions: Let $U_m$ denote the collection of all subsets of $[m]$ whose cardinality is $\\lfloor m/2 \\rfloor $ .", "For each set $\\mathcal {U} \\in U_m$ , define the distribution $q_{\\mathcal {U}}$ as $q_{\\mathcal {U},j}=\\left\\lbrace \\begin{array}{l l}\\frac{1}{m}+\\frac{\\varepsilon }{\\lfloor m/2 \\rfloor }, & j \\in \\mathcal {U};\\\\ \\frac{1}{m}-\\frac{\\varepsilon }{\\lceil m/2 \\rceil }, & j \\in [m]\\setminus \\mathcal {U}.\\end{array}\\right.\\nonumber $ This collection of distributions can be obtained by taking the worst-case distribution $q^*$ given in (REF ), and permuting the symbols in the alphabet $[m]$ .", "Let $q^n_{\\mathcal {U}}$ be the $n$ -order product of $q_{\\mathcal {U}}$ .", "Define the following mixture distribution, $\\bar{q}^n=\\frac{1}{|U_m|}\\sum _{\\mathcal {U} \\in U_m}q^n_\\mathcal {U}.$ The LR $\\bar{q}^n/p^n$ can be lower-bounded on $B_{n, \\tau ,\\delta }$ : Lemma 12 Suppose $\\varepsilon <0.5$ .", "The following holds for any sequence ${\\mbox{\\protect $z$}}_1^n$ satisfying $\\lbrace {\\mbox{\\protect $Z$}}_1^n={\\mbox{\\protect $z$}}_1^n\\rbrace \\subseteq B_{n, \\tau ,\\delta }$ : $\\begin{aligned}\\log \\bigl (\\frac{\\bar{q}^n}{p^n}(\\!", "{\\mbox{\\protect $z$}}_1^n\\!", ")\\bigr )\\!\\ge &\\!-\\!\\frac{n^2}{2m}[\\underline{\\kappa }(\\varepsilon )\\!-\\!\\log (1\\!+\\!\\underline{\\kappa }(\\varepsilon ))(1\\!+\\!\\tau \\!-\\!\\delta )](1\\!+\\!o(1))-\\!", "\\frac{n^2}{m}2\\delta \\log (1-2\\varepsilon ).\\end{aligned}$ [Proof of Lemma REF ] For simplicity of exposition we restrict to the case where $m$ is even.", "Define $\\begin{aligned}\\mathcal {S}_1\\!&: =\\!\\lbrace j\\!", ": \\!\\textrm { j appears in {\\mbox{\\protect $z$}}_1^n exactly \\emph {once}}\\rbrace ,\\\\ \\mathcal {S}_2\\!&: =\\!\\lbrace j\\!", ": \\!\\textrm { j appears in {\\mbox{\\protect $z$}}_1^n exactly \\emph {twice}}\\rbrace .\\end{aligned}$ Let $s_1=|\\mathcal {S}_1|$ , $s_2=|\\mathcal {S}_2|$ .", "It follows from $\\lbrace {\\mbox{\\protect $Z$}}_1^n={\\mbox{\\protect $z$}}_1^n\\rbrace \\subseteq {B}_{n, \\tau , \\delta }$ that $n \\ge s_1 \\ge n-\\frac{n^2}{m}(1+\\tau +\\delta ), \\quad s_2 \\ge {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n^2}{m}(1+\\tau -\\delta ).$ Consider any set $\\mathcal {U} \\in U_m$ .", "Let $k_{\\mathcal {U},1}=|\\mathcal {U} \\cap \\mathcal {S}_1|$ , and $k_{\\mathcal {U},2}=|\\mathcal {U} \\cap \\mathcal {S}_2|$ .", "Then $\\frac{q^n_{\\mathcal {U}}}{p^n}({\\mbox{\\protect $z$}}_1^n) \\ge (1-2\\varepsilon )^n (\\frac{1+2\\varepsilon }{1-2\\varepsilon })^{k_{\\mathcal {U},1}+2k_{\\mathcal {U},2}}\\nonumber .$ Consequently, $\\frac{\\bar{q}^n}{p^n}({\\mbox{\\protect $z$}}_1^n)\\ge G(s_1,s_2)$ where $\\begin{aligned}G(s_1,s_2): =& \\frac{1}{|U_m|}(1-2\\varepsilon )^n \\sum _{k_1=1}^{s_1}\\sum _{k_2=1}^{s_2} \\bigl ( (\\frac{1+2\\varepsilon }{1-2\\varepsilon })^{k}(\\frac{1+2\\varepsilon }{1-2\\varepsilon })^{2k_{2}}|\\lbrace \\mathcal {U} \\in U_m:k_{\\mathcal {U},1}=k_1,k_{\\mathcal {U},2}=k_2\\rbrace |\\bigr )\\\\=& \\frac{1}{{m \\atopwithdelims ()m/2}}(1-2\\varepsilon )^n \\sum _{k_1=1}^{s_1}\\sum _{k_2=1}^{s_2}\\bigl ((\\frac{1+2\\varepsilon }{1-2\\varepsilon })^{k}(\\frac{1+2\\varepsilon }{1-2\\varepsilon })^{2k_{2}} {s_1 \\atopwithdelims ()k_1}{s_2 \\atopwithdelims ()k_2}{m-(s_1+s_2) \\atopwithdelims ()m/2-(k_1+k_2)}\\bigr ).\\end{aligned}$ The summand on the right-hand side of (REF ) takes its maximum value approximately when $k_1=\\bar{k}_1 : =\\lceil \\frac{1+2\\varepsilon }{2}s_1\\rceil , \\quad k_2=\\bar{k}_2 : =\\lceil {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(1+\\frac{4\\varepsilon }{1+4\\varepsilon ^2})\\rceil .$ We apply the Laplace method to approximate the summation: Denote $\\begin{aligned}y(\\Delta _1, \\Delta _2)&=(\\frac{1+2\\varepsilon }{1-2\\varepsilon })^{\\bar{k}_1+\\Delta _1+2(\\bar{k}_2+\\Delta _{2})}{s_1 \\atopwithdelims ()\\bar{k}_1+\\Delta _1}{s_2 \\atopwithdelims ()\\bar{k}_2+\\Delta _2}{m-(s_1+s_2) \\atopwithdelims ()m/2-(\\bar{k}_1+\\Delta _1+\\bar{k}_2+\\Delta _2)}/{m \\atopwithdelims ()m/2}.\\end{aligned}$ Stirling's formula gives $\\begin{aligned}&\\!{m\\!-\\!(s_1\\!+\\!s_2)\\!", "\\atopwithdelims ()\\!\\frac{m}{2}\\!-\\!", "(\\bar{k}_1\\!+\\!\\Delta _1\\!+\\!\\bar{k}_{2\\!", "}\\!+\\!\\Delta _{2\\!})\\!}\\!/\\!{\\!m\\!-\\!(s_1\\!+\\!s_2)\\!", "\\atopwithdelims ()\\!\\frac{m}{2}\\!-\\!(\\bar{k}_1\\!+\\!\\bar{k}_2)\\!", "}\\!=\\!\\exp \\lbrace 1\\!+\\!O(\\!\\frac{(\\Delta _1\\!\\!+\\!\\!\\Delta _2)(\\bar{k}_1\\!+\\!\\bar{k}_2)}{m}\\!", ")\\!+o(1)\\!\\rbrace .\\end{aligned}$ Let $\\begin{aligned}y_1(\\Delta _1)&=(\\frac{1+2\\varepsilon }{1-2\\varepsilon })^{\\Delta _1}{s_1 \\atopwithdelims ()\\bar{k}_1+\\Delta _1}/{s_1 \\atopwithdelims ()\\bar{k}_1},\\\\ y_2(\\Delta _2)&=(\\frac{1+2\\varepsilon }{1-2\\varepsilon })^{2\\Delta _{2}}{s_2 \\atopwithdelims ()\\bar{k}_2+\\Delta _2}/{s_2 \\atopwithdelims ()\\bar{k}_2}\\end{aligned}$ Note that $y(\\bar{k}_1,\\bar{k}_2)$ is the largest summand.", "Keeping only the $\\lceil \\sqrt{s_1}\\rceil \\lceil \\sqrt{s_2}\\rceil $ number of terms in the summation in (REF ) whose index $(k_1,k_2)$ is close to $(\\bar{k}_1, \\bar{k}_2)$ , and applying (REF ), we obtain $\\begin{aligned}\\frac{\\bar{q}^n}{p^n}({\\mbox{\\protect $z$}}_1^n)&\\ge \\sum _{\\Delta _1=-\\lceil \\sqrt{s_1} \\rceil }^{\\lceil \\sqrt{s_1} \\rceil }\\sum _{\\Delta _2=-\\lceil \\sqrt{s_2} \\rceil }^{\\lceil \\sqrt{s_2} \\rceil } y(\\Delta _1, \\Delta _2)\\\\&=\\bigl (\\!\\!\\sum _{\\Delta _1=-\\lceil \\sqrt{s_1} \\rceil }^{\\lceil \\sqrt{s_1} \\rceil }\\!\\!\\!y_1(\\Delta _1)\\bigr )\\bigl (\\!\\!\\sum _{\\Delta _2=-\\lceil \\sqrt{s_2} \\rceil }^{\\lceil \\sqrt{s_2} \\rceil }\\!\\!\\!y_2(\\Delta _2)\\bigr ) y(0,0)\\exp \\lbrace 1\\!+\\!O(\\frac{n^{\\frac{3}{2}}}{m})\\rbrace .\\end{aligned}$ We first approximate $\\sum _{\\Delta _1=-\\lceil \\sqrt{s_1} \\rceil }^{\\lceil \\sqrt{s_1} \\rceil }y_1(\\Delta _1)$ .", "Note that for $\\Delta _1>0$ , $\\log (y_1(\\Delta _1))=\\Delta _1 \\log (\\frac{1+2\\varepsilon }{1-2\\varepsilon })+\\sum _{t=1}^{\\Delta _1} \\log (\\frac{s-\\bar{k}_1-t}{\\bar{k}_1+t}).$ Approximating the above summation by integrals leads to $\\log (y_1(\\Delta _1))=- {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(\\frac{1}{s_1-\\bar{k}_1}+\\frac{1}{\\bar{k}_1})\\Delta _1^2(1+o(1))+O(1).$ Approximating the summation over $\\Delta _1$ using integrals, and applying the above approximation of $y_1(\\Delta _1)$ leads to $\\begin{aligned}&\\sum _{\\Delta _1=-\\lceil \\!\\sqrt{s_1} \\rceil }^{\\lceil \\sqrt{s_1} \\rceil }\\!\\!\\!\\!\\!\\!y_1(\\Delta _1)\\!=\\!e^{O(1)}\\!\\!\\!\\int _{-\\infty }^{\\infty }\\!\\!e^{\\!- \\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}(\\frac{1}{s_1-\\bar{k}_1}+\\frac{1}{\\bar{k}_1})\\Delta _1^2}d \\Delta _1\\!=\\!e^{O(1)}\\!\\sqrt{\\!\\frac{(s_1\\!-\\!\\bar{k}_1)\\bar{k}_1}{s_1}}\\!=\\!e^{O(1)\\!", "}\\!\\sqrt{s_1},\\end{aligned}$ where the last equality follows from (REF ).", "A similar approximation for the summation over $y_2$ holds: $\\sum _{\\Delta _2=-\\lceil \\sqrt{s_2} \\rceil }^{\\lceil \\sqrt{s_2} \\rceil }y_2(\\Delta _2)=e^{O(1)}\\sqrt{s_2}.$ Substituting these into (REF ) gives $\\begin{aligned}&G(s_1,s_2)\\!=\\!e^{O(1)\\!+\\!O(\\frac{n^{3/2}}{m})}\\sqrt{s_1s_2} (1\\!-\\!2\\varepsilon )^n (\\frac{1+2\\varepsilon }{1-2\\varepsilon })^{\\bar{k}_1}(\\frac{1+2\\varepsilon }{1-2\\varepsilon })^{2\\bar{k}_{2}} {s_1 \\atopwithdelims ()\\bar{k}_1}{s_2 \\atopwithdelims ()\\bar{k}_2}{m-(s_1+s_2) \\atopwithdelims ()m/2-(\\bar{k}_1+\\bar{k}_2)}/{m \\atopwithdelims ()m/2}.\\end{aligned}$ Stirling's formula gives the following asymptotic approximations the combinatorial terms in (REF ): $\\begin{aligned}&{{s_1} \\atopwithdelims ()\\bar{k}_1}\\!=\\!\\frac{(1+2\\varepsilon )^{-\\bar{k}_1} (1-2\\varepsilon )^{\\bar{k}_1-{s_1}}2^{s_1}}{\\sqrt{2\\pi \\bar{k}_1({s_1}-\\bar{k}_1)/{s_1}}}(1+o(1)),\\\\&{{s_2} \\atopwithdelims ()\\bar{k}_2}\\!=\\!\\frac{(1+2\\varepsilon )^{-2\\bar{k}_2} (1-2\\varepsilon )^{2(\\bar{k}_2-{s_2})}(1+4\\varepsilon ^2)^{s_2}2^{s_2}}{\\sqrt{2\\pi \\bar{k}_2({s_2}-\\bar{k}_2)/{s_2}}}(1+o(1)),\\\\&{m\\!-\\!", "(s_1\\!+\\!s_2) \\atopwithdelims ()m/2\\!-\\!", "(\\bar{k}_1\\!+\\!\\bar{k}_2)}\\!=\\!2^{m-s_1-s_2}\\exp \\lbrace -\\frac{s_1^2(2\\varepsilon )^2}{2m}(1+o(1))\\rbrace \\frac{\\sqrt{2}}{\\sqrt{\\pi m}}(1+o(1)),\\\\&{m \\atopwithdelims ()m/2}\\!=\\!\\frac{2^m}{\\sqrt{2\\pi m}}(1+o(1)).\\nonumber \\end{aligned}$ Substituting these approximations and the value of $\\bar{k}_1$ and $\\bar{k}_2$ into (REF ) leads to $\\begin{aligned}G(s_1,s_2)=& (1\\!-\\!2\\varepsilon )^{n\\!-\\!s_1\\!-\\!2s_2} \\exp \\lbrace \\!-\\frac{s_1^2(\\!2\\varepsilon \\!)^2\\!", "}{2m}(1\\!+\\!o(1))\\!+\\!s_2\\!\\log (\\!1+4\\varepsilon ^2\\!", ")\\rbrace \\!\\exp \\lbrace O(1)\\!+\\!O(\\!\\frac{n^{3/2\\!}}{m}\\!", ")\\rbrace .\\nonumber \\end{aligned}$ Combining this with (REF ), (REF ) gives the claim of the lemma." ], [ "Proof of Theorem ", "Consider first the case $\\tau >0$ .", "Consider any $\\delta \\in (0, \\tau )$ , and any test $\\phi $ such that $J_F(\\phi ) \\ge J_F^*(\\tau )$ .", "Applying (REF ) and Lemma REF , we obtain $\\lim _{n \\rightarrow \\infty }-\\frac{m}{n^2}\\log {\\sf P}_p(\\lbrace \\phi _n=0\\rbrace \\cap B_{n, \\tau ,\\delta }) = J_F^*(\\tau -\\delta ).$ When $\\varepsilon \\ge 0.5$ , we apply (REF ), (REF ), and Lemma REF to obtain $\\begin{aligned}J_M(\\phi ) &\\le {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}[\\underline{\\kappa }(\\varepsilon )-\\log (1+\\underline{\\kappa }(\\varepsilon ))(1+\\tau -\\delta )]+J_F^*(\\tau -\\delta )\\\\&=J_M^*(\\tau -\\delta )+r_2(\\delta ).\\vspace{-2.84544pt}\\end{aligned}$ where $r_2$ again vanishes as $\\delta \\rightarrow 0$ , $\\begin{aligned}r_2(\\delta )=&{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}[-\\delta \\log (1+\\kappa (\\varepsilon ))+(1+\\tau )\\log (1-\\frac{\\delta }{1+\\tau })-\\delta \\log (1+\\tau -\\delta )+\\delta ].\\end{aligned}$ We have used the following explicit expressions of $J_F^*$ and $J_M^*$ : $\\begin{aligned}J_F^*(\\tau )&={\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}[-\\tau +(1+\\tau )\\log (1+\\tau )],\\!\\!\\!", "\\\\J_M^*(\\tau )&={\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}[\\underline{\\kappa }(\\varepsilon )-\\tau +(1+\\tau )\\log (\\frac{1+\\tau }{1+\\underline{\\kappa }(\\varepsilon )})].\\end{aligned}$ Since (REF ) holds for any $\\delta \\!>\\!0$ and $J_M^*(\\tau )$ is continuous, we conclude $J_M(\\phi ) \\!\\le \\!", "J_M^*(\\tau )$ .", "When $\\varepsilon < 0.5$ , we apply (REF ), (REF ), and Lemma REF to obtain $\\begin{aligned}J_M(\\phi )\\le & {\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}[\\underline{\\kappa }(\\varepsilon )\\!-\\!\\log (1\\!+\\!\\underline{\\kappa }(\\varepsilon ))(1\\!+\\!\\tau \\!-\\!\\delta )+4\\delta \\log (1-2\\varepsilon )+J_F^*(\\tau -\\delta )\\\\=& J_M^*(\\tau -\\delta )+r_1(\\delta ).\\end{aligned}\\vspace{-2.84544pt}$ where $\\begin{aligned}r_1(\\delta )\\!=&\\!", "{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}[-\\delta \\!\\log (1+\\kappa (\\varepsilon )\\!", ")+(\\!1+\\tau \\!", ")\\!\\log (1-\\frac{\\delta }{\\!1\\!+\\!\\tau \\!", "})-\\delta \\!", "\\log (1+\\tau -\\delta )+\\delta +4\\delta \\!\\log (1-2\\varepsilon )].\\vspace{-2.84544pt}\\end{aligned}$ Since the inequality (REF ) holds for any $\\delta >0$ , $J_M^*(\\tau )$ is continuous in $\\tau $ , and $r_1(\\delta ) \\rightarrow 0$ as $\\delta \\rightarrow 0$ , we conclude that $J_M(\\phi ) \\le J_M^*(\\tau )$ .", "The proof for the case where $\\tau =0$ is exactly the same as that for the case $\\tau >0$ , except (REF ) is used in place of (REF ).", "We omit the details." ], [ "Proof of Lemma ", "[Proof of Lemma REF ] Applying Lemma REF to the distribution $q^* \\in {\\mathcal {Q}_n}$ given in (REF ) and (REF ) gives ${\\sf E}_{q^*}[S_n^{\\sf P}]={\\sf E}_{p}[S_n^{\\sf P}]+\\frac{n^2}{m}\\kappa (\\varepsilon )(1+o(1))$ .", "It follows from Chebyshev's inequality that for $\\tau _n > {\\sf E}_{p}[S_n^{\\sf P}]+\\frac{n^2}{m}\\underline{\\kappa }(\\varepsilon )$ , ${\\sf P}_q^*\\lbrace \\phi ^{{\\sf P}}_n({\\mbox{\\protect $Z$}}_1^n)=1\\rbrace \\le \\frac{\\hbox{\\sf Var}\\,_{q^*}[S_n^{\\sf P}]}{(\\tau _n-{\\sf E}_{p}[S_n^{\\sf P}]-\\frac{n^2}{m}\\underline{\\kappa }(\\varepsilon ))^2}.$ Thus, in order for $\\lim _{n \\rightarrow \\infty }{\\sf P}_q^*\\lbrace \\phi ^{{\\sf P}}_n({\\mbox{\\protect $Z$}}_1^n)=1\\rbrace =1$ to hold, we must have $\\begin{aligned}&(\\tau _n-{\\sf E}_{p}[S_n^{\\sf P}]-\\frac{n^2}{m}\\underline{\\kappa }(\\varepsilon ))^2 \\le \\hbox{\\sf Var}\\,_{q^*}[S_n^{\\sf P}](1+o(1))=2\\frac{n^2}{m}(1+\\underline{\\kappa }(\\varepsilon ))(1+o(1)).\\end{aligned}$ where the last equality follows from Lemma REF .", "This leads to the claim of Lemma REF .", "[Proof of Lemma REF ] Consider the statistic $\\bar{S_n^{\\sf P}}=S_n^{\\sf P}-\\frac{n}{m}\\frac{(n\\Gamma ^n_1-np_1)^2}{np_1}=S_n^{\\sf P}-2\\frac{n^2}{m}\\underline{\\kappa }(\\varepsilon )+O(\\frac{n}{\\sqrt{m}}).$ The conditional distribution of $\\bar{S_n^{\\sf P}}$ in the event $A$ under $p$ is the same as the distribution of $S^{\\sf P}_{n^{\\prime }}$ under $p^{\\prime }$ , where the number of samples is $n^{\\prime }=n-\\lfloor \\frac{n\\sqrt{2\\underline{\\kappa }(\\varepsilon )}}{\\sqrt{m}}\\rfloor $ and $p^{\\prime }$ is the uniform distribution over $[m-1]$ .", "It then follows from Lemma REF and Lemma REF that $\\begin{aligned}{\\sf E}_{p}[\\bar{S_n^{\\sf P}}|A]&={\\sf E}_{p^{\\prime }}[S^{\\sf P}_{n^{\\prime }}]=n-\\lfloor \\frac{n\\sqrt{2\\underline{\\kappa }(\\varepsilon )}}{\\sqrt{m}}\\rfloor +O(\\frac{n^2}{m}),\\\\\\hbox{\\sf Var}\\,_{p}[\\bar{S_n^{\\sf P}}|A]&=\\hbox{\\sf Var}\\,_{p^{\\prime }}[S^{\\sf P}_{n^{\\prime }}]=2\\frac{n^2}{m}(1+o(1)).\\end{aligned}$ It follows from Chebyshev's inequality, Lemma REF and Lemma REF that for large enough $n$ , $\\begin{aligned}&{\\sf P}_p\\lbrace S_n^{\\sf P}\\le {\\sf E}_{p}[S_n^{\\sf P}]+\\frac{n^2}{m}\\underline{\\kappa }(\\varepsilon )+2\\frac{n}{\\sqrt{m}}|A_n\\rbrace \\\\&={\\sf P}_p\\lbrace \\bar{S_n^{\\sf P}} \\!+\\!", "2\\frac{n^2}{m}\\underline{\\kappa }(\\varepsilon ) \\le n\\!+\\!\\frac{n^2}{m}\\underline{\\kappa }(\\varepsilon )\\!+\\!2\\frac{n}{\\sqrt{m}}\\!+\\!O(\\frac{n}{\\sqrt{m}})|A_n\\rbrace \\\\&={\\sf P}_p\\lbrace \\bar{S_n^{\\sf P}} \\le {\\sf E}_{p}[\\bar{S_n^{\\sf P}}|A]-\\frac{n^2}{m}\\underline{\\kappa }(\\varepsilon )+O(\\frac{n}{\\sqrt{m}})|A_n\\rbrace \\\\&\\le \\frac{2\\frac{n^2}{m}(1+O(\\frac{n}{\\sqrt{m}}))}{\\bigl (\\frac{n^2}{m}\\underline{\\kappa }(\\varepsilon )+O(\\frac{n}{\\sqrt{m}})\\bigr )^2}=O(\\frac{m}{n^2}).\\end{aligned}\\nonumber $ [Proof of Lemma REF ] A simple combinatorial argument gives ${\\sf P}_p\\lbrace A_n\\rbrace ={n \\atopwithdelims ()\\lfloor {n\\frac{\\sqrt{2\\underline{\\kappa }(\\varepsilon )}}{\\sqrt{m}}}\\rfloor } p_1^{\\lfloor {n\\frac{\\sqrt{2\\underline{\\kappa }(\\varepsilon )}}{\\sqrt{m}}}\\rfloor }(1-p_1)^{n-\\lfloor {n\\frac{\\sqrt{2\\underline{\\kappa }(\\varepsilon )}}{\\sqrt{m}}}\\rfloor }.\\nonumber $ Applying Stirling's formula and substituting $p_1=\\frac{1}{m}$ leads to ${\\sf P}_p\\lbrace A_n\\rbrace =\\exp \\lbrace -{\\mathchoice{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}1{1}{2}}{\\genfrac{}{}{}3{1}{2}}{\\genfrac{}{}{}3{1}{2}}}\\frac{n\\sqrt{2\\underline{\\kappa }(\\varepsilon )}}{\\sqrt{m}} \\log (m)(1+o(1))\\rbrace (1+o(1)).$ Since $m=o(\\frac{n^2}{\\log (n)^2})$ and $m=o(n^2)$ , we have $\\frac{n\\sqrt{2\\underline{\\kappa }(\\varepsilon )}}{\\sqrt{m}} \\log (m)=\\frac{n\\sqrt{2\\underline{\\kappa }(\\varepsilon )}}{\\sqrt{m}}o(2\\log (n))=o(\\frac{n^2}{m}).$ Substituting this into (REF ) leads to the claim of this lemma." ] ]

[ [ "Unbounded normal operators in octonion Hilbert spaces and their spectra" ], [ "Abstract Affiliated and normal operators in octonion Hilbert spaces are studied.", "Theorems about their properties and of related algebras are demonstrated.", "Spectra of unbounded normal operators are investigated." ], [ "Introduction", "Unbounded normal operators over the complex field have found many-sided applications in functional analysis, differential and partial differential equations and their applications in the sciences [4], [11], [12], [14], [31].", "On the other hand, hypercomplex analysis is fast developing, particularly in relation with problems of theoretical and mathematical physics and of partial differential equations [2], [7], [9].", "The octonion algebra is the largest division real algebra in which the complex field has non-central embeddings [3], [1], [13].", "It is intensively used especially in recent years not only in mathematics, but also in applications [5], [10], [8], [15], [16].", "In previous works analysis over quaternion and octonions was developed and spectral theory of bounded normal operators and unbounded self-adjoint operators was described [18], [19], [20], [21], [22].", "Some results on their applications in partial differential equations were obtained [23], [24], [25], [26], [27].", "This work continuous previous articles and uses their results.", "The present paper is devoted to unbounded normal operators and affiliated operators in octonion Hilbert spaces, that was not yet studied before.", "Frequently in practical problems, for example, related with partial differential operators, spectral theory of unbounded normal operators is necessary.", "This article contains the spectral theory of unbounded affiliated and normal operators.", "Notations and definitions of papers [18], [19], [20], [21], [22] are used below.", "The main results of this paper are obtained for the first time." ], [ "Affiliated and normal operators", "1.", "Definitions.", "Let $X$ be a Hilbert space over the Cayley-Dickson algebra ${\\cal A}_v$ , $~2\\le v$ , and let $\\sf A$ be a von Neumann algebra contained in $L_q(X)$ .", "We say that an operator $Q\\in L_q(X)$ quasi-commutes with $\\sf A$ if the algebra $alg_{{\\cal A}_v} (Q,B)$ over ${\\cal A}_v$ generated by $Q$ and $B$ is quasi-commutative for each $B\\in \\sf A$ .", "A closed $\\bf R$ homogeneous ${\\cal A}_v$ additive operator $T$ with a dense ${\\cal A}_v$ vector domain ${\\cal D}(T)\\subset X$ is said to be affiliated with $\\sf A$ , when $(1)$ $U^*TUx=Tx$ for every $x\\in {\\cal D}(T)$ and each unitary operator $U\\in L_q(X)$ quasi-commuting with $\\sf A$ .", "The fact that $T$ is affiliated with $\\sf A$ is denoted by $T\\eta \\sf A$ .", "An ${\\cal A}_v$ vector subspace $\\bigcup _n \\mbox{ }_nF{\\cal D}(T)$ is called a core of an operator $T$ , if $\\mbox{}_nF$ is an increasing sequence of ${\\cal A}_v$ graded projections and $\\bigcup _n \\mbox{ }_nF{\\cal D}(T)$ is dense in ${\\cal D}(T)$ .", "2.", "Note.", "Definition 1 implies that $U{\\cal D}(T)={\\cal D }(T)$ .", "If $V$ is a dense ${\\cal A}_v$ vector subspace in ${\\cal D}(T)$ and $T|_V\\eta \\sf A$ , then $T\\eta \\sf A$ .", "Indeed, $\\lim _nUy^n=Uy$ for each unitary operator $U\\in L_q(X)$ and every sequence $y^n \\in V$ converging to a vector $y\\in {\\cal D}(T)$ and with $\\lim _n Ty^n=Ty$ .", "Therefore, the limit $\\lim _n TUy^n= \\lim _n U Ty^n= UTy$ exists.", "The operator $T$ is closed, hence $Uy\\in {\\cal D}(T)$ and $UTy= TUy$ .", "Thus ${\\cal D}(T)\\subset U^* {\\cal D}(T)$ .", "The proof above for $U^*$ instead of $U$ gives the inclusion ${\\cal D}(T)\\subset U {\\cal D}(T)$ , consequently, $U{\\cal D}(T)={\\cal D}(T)$ and hence ${\\cal D}(U^*TU)={\\cal D}(T)$ and $TUy=UTy$ for every $y\\in {\\cal D}(T)$ .", "For an $\\bf R$ homogeneous ${\\cal A}_v$ additive (i.e.", "quasi-linear) operator $A$ in $X$ with an ${\\cal A}_v$ vector domain ${\\cal D}(A)$ the notation can be used: $(1)$ $Ax=\\sum _j A^{i_j}x_j$ for each $(2)$ $x=\\sum _j x_ji_j\\in {\\cal D}(A)\\subset X$    with $x_j\\in X_j$ and $(3)$ $A^{i_j}x_j := A(x_ji_j)$ for each $j=0,1,2,...$ .", "That is $A^{i_j}(i_j^*\\pi ^j) = A\\pi ^j$ , where $\\pi ^j: X\\rightarrow X_ji_j$ is an $\\bf R$ linear projection with $\\pi ^j(x)=x_ji_j$ so that $\\sum _j \\pi ^j =I$ .", "It can be lightly seen, that Definition 1 is natural.", "Indeed, if $\\sf A$ is a quasi-commutative algebra over the Cayley-Dickson algebra ${\\cal A}_v$ with $2\\le v$ , then an algebra ${\\sf A}_0i_0\\oplus {\\sf A}_ki_k$ is commutative for $k\\ge 1$ over the complex field ${\\bf C}_{i_k} := {\\bf R}\\oplus {\\bf R}i_k$ , since there is the decomposition ${\\sf A}= {\\sf A}_0i_0\\oplus {\\sf A}_1i_1\\oplus ... {\\sf A}_m i_m \\oplus ...$ with pairwise isomorphic real algebras ${\\sf A}_0, {\\sf A}_1,..., {\\sf A}_m,...$ .", "Let ${\\cal B}(Y,{\\cal A}_v)$ denote the algebra of all bounded Borel functions from a topological space $Y$ into the Cayley-Dickson algebra ${\\cal A}_v$ , let also ${\\cal B}_u(Y,{\\cal A}_v)$ denote the algebra of all Borel functions from $Y$ into ${\\cal A}_v$ with point-wise addition and multiplication of functions and multiplication of functions $f$ on the left and on the right on Cayley-Dickson numbers $a, b \\in {\\cal A}_v$ , $~2\\le v$ .", "3.", "Lemma.", "If $T$ is a closed symmetrical operator in a Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ , $~2\\le v$ , then ranges ${\\cal R}(T\\pm MI)$ of $(T\\pm MI)$ are closed for each $M\\in {\\cal S}_v$ , when $v\\le 3$ or $M\\in \\lbrace i_1,i_2,... \\rbrace $ .", "If $T$ is closed and $0\\le ~ <Tz;z>$ for each $z\\in {\\cal D}(T)$ , then $T+I$ has a closed range.", "Proof.", "Let $\\mbox{}_nx$ be a sequence in ${\\cal D}(T)$ so that $ \\lbrace (T\\pm MI)\\mbox{ }_nx : ~n \\rbrace $ tends to a vector $y\\in X$ .", "But $<Tz;z>$ is real for each $z\\in {\\cal D}(T)$ , hence $\\Vert z \\Vert ^2\\le (<Tz;z>^2 + <z;z>^2)^{1/2} = |<(T\\pm MI)z;z>|$ $\\le \\Vert (T\\pm MI)z \\Vert \\Vert z \\Vert ,$ consequently, $ \\Vert \\mbox{}_nx - \\mbox{}_mx \\Vert \\le \\Vert (T\\pm MI)(\\mbox{}_nx-\\mbox{ }_mx) \\Vert $ and $\\mbox{}_nx$ converges to some vector $x\\in X$ .", "On the other hand, the sequence $\\lbrace T\\mbox{}_nx : ~ n \\rbrace $ converges to $\\mp Mx+y$ and the operator $T$ is closed, consequently, $x\\in {\\cal D}(T)$ and $Tx=\\mp Mx+y$ .", "Indeed, $M^*(Mx) = M^* \\sum _jx_j(Mi_j)=\\sum _jx_ji_j=x,$ since $ \\Vert Mx \\Vert = \\Vert x \\Vert $ and $M(x_ji_j) = x_j(Mi_j)$ and $M^*(Mi_j)=i_j$ for each $x_j\\in X_j$ and $M\\in {\\cal S}_v$ for $2\\le v \\le 3$ or $M\\in \\lbrace i_1, i_2,... \\rbrace $ .", "Thus $(T\\pm MI)x=y$ , hence the operators $(T\\pm MI)$ have closed ranges for such $M$ .", "If an operator $T$ is closed and $0\\le ~ <Tz;z>$ for each $z\\in {\\cal D}(T)$ , then $ \\Vert z \\Vert ^2 \\le ~ <z;z> + <Tz;z> \\le \\Vert (T+I)z\\Vert \\Vert z \\Vert $ and analogously to the proof above we get that $(T+I)$ has a closed range.", "4.", "Proposition.", "If $T$ is a closed symmetrical operator on a Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ , $~2 \\le v$ , then the following statements are equivalent: $(1)$ an operator $T$ is self-adjoint; $(2)$ operators $(T^*\\pm MI)$ have $ \\lbrace 0 \\rbrace $ as null space for each $M\\in {\\cal S}_v$ , when $v\\le 3$ or $M\\in \\lbrace i_1,i_2,... \\rbrace $ ; $(3)$ operators $(T\\pm MI)$ have $X$ as range for every $M\\in {\\cal S}_v$ , when $v\\le 3$ or $M\\in \\lbrace i_1,i_2,... \\rbrace $ ; $(4)$ operators $(T\\pm MI)$ have ranges dense in $X$ for all $M\\in {\\cal S}_v$ with $v\\le 3$ or $M\\in \\lbrace i_1,i_2,... \\rbrace $ .", "Proof.", "$(1)\\Rightarrow (2)$ .", "We have $<Tx;x> = <x;Tx> \\in \\bf R$ for each $x\\in {\\cal D}(T)$ , when $T^*=T$ , hence $<(T^*\\pm MI)x;x> = <(T\\pm MI)x;x> = <Tx;x> \\pm M \\Vert x \\Vert ^2$ , consequently, $<(T^*\\pm MI)x;x> =0$ only when $x=0$ , since $ \\Vert Mx\\Vert = \\Vert x \\Vert $ for each $M\\in {\\cal S}_v$ with $v\\le 3$ or $M\\in \\lbrace i_1, i_2,... \\rbrace $ .", "Thus each operator $(T^*\\pm MI)$ has $ \\lbrace 0 \\rbrace $ as null space.", "$(2)\\Rightarrow (3)$ .", "Ranges ${\\cal R}(T\\pm MI)$ are closed due to Lemma 3.", "Therefore, it is sufficient to show that these ranges are dense in a Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ .", "But $<Tx;y> = \\mp M <x;y>$ , when $<(T\\pm MI)x;y> =0$ for all $x\\in {\\cal D}(T)$ , consequently, $y\\in {\\cal D}(T^*)$ and $T^*y=\\pm My$ .", "Therefore, $y=0$ , since the operators $T^*\\pm MI$ have $ \\lbrace 0 \\rbrace $ as null space.", "Thus the operators $(T\\pm MI)$ have dense ranges for each $M\\in {\\cal S}_v$ when $v\\le 3$ or $M\\in \\lbrace i_1, i_2, ... \\rbrace $ .", "$(3)\\Leftrightarrow (4)$ .", "This follows from the preceding demonstrations.", "$(3)\\Rightarrow (1)$ .", "When $T$ is a closed and symmetrical operator, one has that $T\\subseteq T^*$ and a graph $\\Gamma (T)$ is a closed subspace of the closed $\\bf R$ -linear space $\\Gamma (T^*)$ .", "The equality $<y;x> + <T^*y;Tx> =0$ is valid for each $x\\in {\\cal D}(T)$ , when $(y,T^*y)\\in \\Gamma (T^*)$ is orthogonal to $\\Gamma (T)$ .", "Operators $(T\\pm MI)$ have range $X$ , consequently, there exists a vector $x\\in {\\cal D}(T)$ so that $Tx\\in {\\cal D}(T)$ and $y= (T+MI)(T-MI) x = (T^2+I)x$ , since $T\\subseteq T^*$ and $T(MI)\\subseteq (MI)T^*$ .", "For such vector $x$ one gets $<y;y> = <y;(T^2+I)x> = <y;x> + <T^*y;Tx>=0$ , hence $<y;T^*y> = (0,0)$ and $\\Gamma (T) = \\Gamma (T^*)$ and hence $T^*=T$ .", "That is, the operator $T$ is self-adjoint.", "5.", "Note.", "If an operator $T$ in a Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ , $~2\\le v$ , is self-adjoint, the fact that operators $(T\\pm MI)$ have dense everywhere defined bounded inverses with bound not exceeding one follows from Proposition 4 and the inequality at the beginning of the demonstration of Lemma 3 for $M\\in {\\cal S}_v$ with $v\\le 3$ or $M\\in \\lbrace i_1, i_2,... \\rbrace $ .", "For an operator $T$ let $alg_{{\\cal A}_v}(I,T)=:\\sf Q$ be a family of operators generated by $I$ and $T$ over the Cayley-Dickson algebra ${\\cal A}_v$ .", "Consider this family of operators on a common domain ${\\cal D}^{\\infty }(T) := \\cap _{n=1}^{\\infty } {\\cal D}(T^n)$ .", "Then the family $\\sf Q$ on ${\\cal D}^{\\infty }(T)$ can be considered as an ${\\cal A}_v$ vector space.", "Take the decomposition ${\\sf Q}={\\sf Q}_0i_0\\oplus {\\sf Q}_1i_1\\oplus ... \\oplus {\\sf Q}_mi_m\\oplus ...$ of this ${\\cal A}_v$ vector space with pairwise isomorphic real vector spaces ${\\sf Q}_0, {\\sf Q}_1,...,{\\sf Q}_m,... $ and for each operator $B\\in \\sf Q$ put $(1)\\quad B=\\sum _j \\mbox{}^jB\\mbox{ with }\\mbox{}^jB={\\hat{\\pi }}^j(B)\\in {\\sf Q}_ji_j$ for each $j$ , where ${\\hat{\\pi }}^j: {\\sf Q}\\rightarrow {\\sf Q}_ji_j$ is the natural $\\bf R$ linear projection, real linear spaces ${\\sf Q}_mi_m$ and $i_m{\\sf Q}_m$ are considered as isomorphic.", "Evidently, there is the inclusion of domains of these $\\bf R$ linear operators ${\\cal D}(\\mbox{}^kB)\\supset {\\cal D}(B)$ for each $k$ , particularly, ${\\cal D}(\\mbox{}^kT)\\supset {\\cal D}(T)$ .", "6.", "Lemma.", "Let $\\lbrace \\mbox{}_nE: ~ n \\rbrace $ be an increasing sequence of ${\\cal A}_v$ graded projections on a Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ and let also $G$ be an $\\bf R$ homogeneous ${\\cal A}_r$ additive operator with dense domain $\\bigcup _n \\mbox{ }_nE(X)=: \\cal E$ such that $G\\mbox{}_nE$ is a bounded self-adjoint operator on $X$ , where $2\\le v$ .", "Then $G$ is pre-closed and its closure $T$ is self-adjoint.", "Moreover, if an operator $T$ is closed with core $\\cal E$ and $T\\mbox{ }_nE$ is a bounded self-adjoint operator for each $n\\in \\bf N$ , then $T$ is self-adjoint.", "Proof.", "For each $x, y \\in \\cal E$ there exists a natural number $m$ so that $<Gx;y> = <G\\mbox{ }_mEx;y> = <x;G\\mbox{}_mEy>=<x;Gy>$ , hence $y\\in {\\cal D}(G^*)$ and $G^*$ is densely defined so that $G$ is pre-closed.", "Consider now the closure $T$ of $G$ .", "For each $x, y \\in {\\cal D}(T)$ there exist sequences $\\mbox{}_nx$ and $\\mbox{}_ny$ in $\\cal E$ for which $\\lim _n \\mbox{}_nx=x$ , $ ~ \\lim _n \\mbox{ }_ny=y$ , $~ \\lim _n T \\mbox{ }_nx=Tx$ and $~ \\lim _n T \\mbox{ }_ny=Ty$ .", "Therefore, the equalities $<T\\mbox{ }_nx; \\mbox{ }_ny> = <T\\mbox{ }_mE\\mbox{ }_nx;\\mbox{ }_ny> = <\\mbox{ }_nx; T\\mbox{ }_mE\\mbox{ }_ny> = <\\mbox{}_nx; T \\mbox{ }_ny>$ are satisfied, consequently, $\\lim _n<T\\mbox{ }_nx; \\mbox{ }_ny> = <Tx;y>$ and $\\lim _n <\\mbox{ }_nx; T\\mbox{ }_ny> = <x;Ty>$ .", "Thus one gets $<Tx;y> = <x;Ty>$ and that the operator $T$ is symmetric.", "Mention that $(T\\pm MI)\\mbox{ }_nE(X) = \\mbox{}_nE(X)$ for each $M\\in {\\cal S}_v$ with $v\\le 3$ or $M\\in \\lbrace i_1, i_2,... \\rbrace $ , since $T\\mbox{ }_nE$ is bounded and self-adjoint, for which the operator $T\\mbox{ }_nE \\pm M\\mbox{ }_nE$ has a bounded inverse on $\\mbox{}_nE(X)$ .", "This implies that $T\\pm MI$ has a dense range and it coincides with $X$ , consequently, $T$ is self-adjoint due to Lemma 3 and Proposition 4.", "7.", "Theorem.", "Let $\\mu $ be a $\\sigma $ -finite measure $\\mu : {\\cal F} \\rightarrow [0,\\infty ]$ on a $\\sigma $ -algebra $\\cal F$ of subsets of a set $S$ and let $L^2(S,{\\cal F},\\mu ,{\\cal A}_v)$ be a Hilbert space completion of the set of all step $\\mu $ measurable functions $f: S\\rightarrow {\\cal A}_v$ with the ${\\cal A}_v$ valued scalar product $<f;g> =\\int f(x){\\tilde{g}}(x)\\mu (dx)$ for each $f, g \\in L^2(S,{\\cal F},\\mu ,{\\cal A}_v)$ , where $2\\le v$ .", "Suppose that $\\sf A$ is its left multiplication algebra $M_gf=gf$ .", "Then an $\\bf R$ -linear ${\\cal A}_v$ -additive operator $T$ is affiliated with $\\sf A$ if and only if a measurable finite $\\mu $ almost everywhere on $S$ function $g: S\\rightarrow {\\cal A}_v$ exists so that $T=M_g$ and $g_k(t)f_j(t)=(-1)^{\\kappa (j,k)} f_j(t)g_k(t)$ for $\\mu $ almost all $t\\in S$ and each $f\\in L^2(S,{\\cal F},\\mu ,{\\cal A}_v)$ for each $j, k=0,1,...$ , where $\\kappa (j,k)=0$ for $j=0$ or $k=0$ or $j=k$ , while $\\kappa (j,k)=1$ for each $j\\ne k\\ge 1$ .", "Moreover, an operator $T\\eta \\sf A$ is self-adjoint if and only if $g$ is real-valued $\\mu $ almost everywhere on $S$ .", "Proof.", "If $g: S\\rightarrow {\\cal A}_v$ is a $\\mu $ measurable $\\mu $ essentially bounded on $S$ function, then $M_g$ is a bounded $\\bf R$ -linear ${\\cal A}_v$ -additive operator on $L^2(S,{\\cal F},\\mu ,{\\cal A}_v)$ , since $\\Vert M_gf \\Vert _2 \\le \\Vert g \\Vert _{\\infty } \\Vert f\\Vert _2$ , where $\\Vert g \\Vert _{\\infty } = ess ~\\sup _{x\\in S} |g(x)|$ and $ \\Vert f \\Vert _2 := \\sqrt{<f;f>}$ .", "That is $M_g\\in \\sf A$ .", "Each operator $G\\in \\sf A$ is an $({\\cal A}_v)_{{\\bf C}_{\\bf i}}$ combination of unitary operators in $\\sf A$ (see Theorem II.2.20 [28]) and $UT\\subseteq TU$ for each unitary operator $U\\in \\sf A$ , since $T\\eta \\sf A$ and ${\\sf A}\\subseteq {\\sf A}^{\\star }$ , where ${\\bf C}_{\\bf i} = {\\bf R}\\oplus {\\bf R}{\\bf i}$ .", "If $F$ is an ${\\cal A}_v$ graded projection operator corresponding to the characteristic function $\\chi _P$ of a $\\mu $ -measurable subset $P$ in $S$ , this implies the inclusion $FT\\subseteq TF$ , consequently, $Ff\\in {\\cal D}(T)$ for each $f\\in {\\cal D}(T)$ .", "If $f\\in {\\cal D}(T)$ and $\\mbox{}_nF$ corresponds to the multiplication by the characteristic function $\\chi _K$ of the set $K = \\lbrace x\\in S: ~ |f(x)|\\le n \\rbrace $ , then $\\mbox{}_nF$ is an ascending sequence of projections in the algebra $\\sf A$ converging to the unit operator $I$ relative to the strong operator topology, since $f$ is finite almost everywhere, $ \\mu \\lbrace x: ~ | f(x)| =\\infty \\rbrace =0$ .", "Therefore, $\\mbox{}_nFf\\in \\cal E$ for each $n$ , where $\\cal E$ denotes the set of all $\\mu $ essentially bounded functions $f\\in {\\cal D}(T)$ .", "Moreover, the limits exist: $\\lim _n \\mbox{ }_nFf=f\\mbox{ and } \\lim _n T\\mbox{ }_nFf= \\lim _n\\mbox{ }_nFTf = Tf.$ Thus ${\\cal E} = \\bigcup _n \\mbox{ }_nF{\\cal D}(T)$ , where $\\bigcup _n \\mbox{ }_nF{\\cal D}(T)$ is dense in ${\\cal D}(T)$ .", "Thus $\\cal E$ is a core of $T$ .", "Each step function $u: S\\rightarrow {\\cal A}_v$ on $(S,{\\cal F})$ has the form $u(s) = \\sum _{l=1}^m c_l \\chi _{B_l},$ where $B_l\\in \\cal F$ and $c_l\\in {\\cal A}_v$ for each $l=1,...,m$ , $~m\\in \\bf N$ , where $\\chi _B$ denotes the characteristic function of a subset $B$ in $S$ so that $\\chi _B(s)=1$ for each $s\\in B$ and $\\chi _B(s)=0$ for all $s$ outside $B$ .", "A subset $N$ in $S$ is called $\\mu $ null if there exists $H\\in \\cal F$ so that $N\\subset H$ and $\\mu (H)=0$ .", "If consider an algebra ${\\cal F}_{\\mu }$ of subsets in $S$ which is the completion of $\\cal F$ by $\\mu $ null subsets, then each step function in $L^2(S,{\\cal F},\\mu ,{\\cal A}_v)$ may have $B_l\\in {\\cal F}_{\\mu }$ for each $l$ .", "For each functions $f, g \\in \\cal E$ there are the equalities $(1)$ $((f_ji_j)\\mbox{ }^kT)g_k= (M_{f_ji_j}\\mbox{ }^kT)g_k=(-1)^{\\kappa (j,k)} (\\mbox{}^kT M_{f_ji_j})g_k = \\pm \\mbox{}^lT f_jg_k$ $ =(-1)^{\\kappa (j,k)} M_{g_ki_k} \\mbox{ }^jTf_j =(-1)^{\\kappa (j,k)}g_ki_k [\\mbox{ }^jTf_j]$ , where $i_ji_k=\\pm i_l$ , $ ~ \\kappa (j,k)=0$ for $j=0$ or $k=0$ or $j=k$ , while $\\kappa (j,k)=1$ for each $j\\ne k\\ge 1$ , $f=\\sum _jf_ji_j$ with real-valued functions $f_j$ for each $j$ .", "Let $S_k\\in \\cal F$ be a sequence of pairwise disjoint subsets with $0<\\mu (S_k)<\\infty $ for each $k$ and with union $\\bigcup _k S_k=S$ .", "For the characteristic function $\\chi _{S_k}$ a sequence $ \\lbrace f^{k,j}: ~j\\in {\\bf N} \\rbrace \\subset \\cal E$ of real-valued functions exists converging to $\\chi _{S_k}$ in $L^2(S,{\\cal F},\\mu ,{\\cal A}_v)$ .", "The set $S_k^0 := \\lbrace t\\in S_k: ~ f^{k,j}(s)=0 ~ \\forall j \\rbrace $ has $\\mu $ measure zero, since $0=\\lim _j M_gf^{k,j} =M_g\\chi _{S_k}=g,\\mbox{ where }g=\\chi _{S_k^0}.$ Put $h(s) =[(Tf^{k,j})(s)][f^{k,j}(s)]^{-1}$ for $s\\in S_k\\setminus S_k^0$ , where $j$ is the least natural number so that $f^{k,j}(s)\\ne 0$ .", "Thus $h$ is a measurable function defined $\\mu $ almost everywhere on $S$ .", "In accordance with Formula $(1)$ the equality $(2)\\quad f^{k,j}(s)[Tf]_l(s)=\\sum _{p,q; ~ i_qi_p=i_l} \\lbrace (-1)^{\\kappa (q,p)} f_p(s)i_p[Tf^{k,j}]_q(s)i_q+f_q(s)i_q[Tf^{k,j}]_p(s)i_p \\rbrace $ is accomplished for all $k, j\\in \\bf N$ and $l =0, 1, 2,...$ except for a set of measure zero, consequently, $[Tf](s) = h(s)f(s)=M_hf(s)$ almost everywhere on $S$ so that $\\mbox{}^kTf=[Tf]_ki_k$ for each $k$ .", "The operator $M_h$ is closed and affiliated with $\\sf A$ as follows from the demonstration above.", "On the other hand, $M_h$ is an extension of $T|_{\\cal E}$ , consequently, $T\\subseteq M_h$ .", "The family of all functions $z\\in L^2(S,{\\cal F},\\mu ,{\\cal A}_v)$ vanishing on $\\lbrace s\\in S: ~ |h(s)|>m \\rbrace $ for some $m=m(z)\\in \\bf N$ forms a core for $M_h$ .", "For such a function $z$ take a sequence $f^k$ of functions in $\\cal E$ tending to $z$ in $L^2(S,{\\cal F},\\mu ,{\\cal A}_v)$ .", "Evidently $f^k$ can be replaced by $yf^k$ , where $y$ is the characteristic function of the set $ \\lbrace s\\in S: ~ z(s)\\ne 0 \\rbrace $ of all points at which $z$ does not vanish.", "Thus we can choose a sequence $f^k$ vanishing for each $k$ when $z$ does.", "Therefore, $(3)$ $\\lim _k Tf^k = \\lim _k M_hM_yf^k = M_hM_yz=M_hz$ , since the operator $M_hM_y$ is bounded.", "For a closed operator $T$ this implies that $z\\in {\\cal D}(T)$ and $Tz=M_hz$ , consequently, $T=M_h$ .", "If an operator $T$ is self-adjoint, then $M_{yh}$ is a bounded self-adjoint operator, hence the function $yh$ is real-valued $\\mu $ almost everywhere on $S$ .", "For a bounded multiplication operator $M_g$ the ${\\cal A}_v$ graded projections $\\mbox{}_tE$ corresponding to multiplication by characteristic function of the set $\\lbrace s\\in S: ~g(s)\\le t \\rbrace $ forms a spectral ${\\cal A}_v$ graded spectral resolution $\\lbrace \\mbox{}_tE: ~t \\in {\\bf R} \\rbrace $ of the identity for the operator $T$ (see also Theorem 2.28 [20]).", "8.", "Definition.", "Suppose that $V$ is an extremely disconnected compact Hausdorff topological space and $V\\setminus W$ is an open dense subset in $V$ .", "If a function $f: V\\setminus W \\rightarrow {\\cal A}_v$ is continuous and $\\lim _{x\\rightarrow y} |f(x)|=\\infty $ for each $y\\in W$ , where $x\\in V\\setminus W$ , $~ 1\\le v$ , then $f$ is a called a normal function on $V$ .", "If a normal function on $V$ is real-valued it will be called a self-adjoint function on $V$ .", "The families of all normal and self-adjoint functions on $V$ we denote by ${\\cal N}(V,{\\cal A}_v)$ and ${\\cal Q}(V)$ respectively, let also $W_+ :=W_+(f) := \\lbrace y\\in W: ~ \\lim _{x\\rightarrow y} f(x)=\\infty \\rbrace ,$ $W_-:=W_-(f) := \\lbrace y\\in W: ~ \\lim _{x\\rightarrow y} f(x)= - \\infty \\rbrace $ for a self-adjoint function $f$ on $V$ .", "9.", "Lemma.", "Let $f$ and $g$ be normal functions on $V$ (see Definition 8) defined on $V\\setminus W_f$ and $V\\setminus W_g$ respectively so that $f(x)=g(x)$ for each $x$ in a dense subset $U$ in $V\\setminus (W_f\\cup W_g)$ .", "Then $W_f=W_g$ and $f=g$ .", "Proof.", "The subset $V\\setminus (W_f\\cup W_g)$ is dense in $V$ , consequently, $U$ is dense in $V$ .", "If $y\\in W_g$ , then for each $N>0$ there exists a neighborhood $E$ of $y$ in $V$ so that $|g(x)|>N$ for each $x\\in E\\cap (V\\setminus W_g)$ , hence $|f(x)|=|g(x)|>N$ for each $x\\in E\\cap (V\\setminus (W_g\\cup W_f))$ , consequently, $W_g\\subset W_f$ and symmetrically $W_f\\subset W_g$ .", "Thus $W_f=W_g$ and $f-g$ is defined and continuous on $V\\setminus W_f$ and is zero on $U$ , hence $f=g$ .", "10.", "Lemma.", "Suppose that $T$ is a self-adjoint operator acting on a Hilbert space $X$ over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ , $~2\\le v\\le 3$ , so that $T$ is affiliated with some quasi-commutative von Neumann algebra ${\\sf A}$ over ${\\cal A}_v$ , where $\\sf A$ is isomorphic to $C(\\Lambda ,{\\cal A}_v)$ with an extremely disconnected compact Hausdorff topological space $\\Lambda $ .", "Then there is a unique self-adjoint function $h$ on $\\Lambda $ such that $h\\hat{\\cdot }e\\in C(\\Lambda ,{\\cal A}_v)$ and a function $h\\hat{\\cdot }e$ represents $TE$ , when $E$ is an ${\\cal A}_v$ graded projection for which $TE\\in L_q(X)$ is a bounded operator on $X$ , a function $e\\in C(\\Lambda ,{\\cal A}_v)$ corresponds to $E$ , $~ h\\hat{\\cdot }e(x)=h(x)$ for $e(x)=1$ , while $~h\\hat{\\cdot }e(x)=0$ otherwise.", "There exists an ${\\cal A}_v$ graded resolution of the identity $ \\lbrace \\mbox{}_bE: ~ b \\rbrace $ so that $\\bigcup _{n=1}^{\\infty } \\mbox{}_nF(X)$ is a core for $T$ , where $\\mbox{}_nF := \\mbox{}_nE-\\mbox{ }_{-n}E$ and $(1)$ $Tx=\\int _{-n}^n d\\mbox{}_bE.b x$ for every $x\\in \\mbox{ }_nF(X)$ and each $n$ in the sense of norm convergence of Riemann sums.", "Proof.", "Take $Y=X\\oplus {\\bf i}X$ , where $\\bf i$ is a generator commuting with $i_j$ for each $j$ such that ${\\bf i}^2=-1$ .", "Take an extension $T$ onto ${\\cal D}(T)\\oplus {\\cal D}(T){\\bf i}$ so that $T(x+y{\\bf i})=Tx+(Ty){\\bf i}$ for each $x, y \\in {\\cal D}(T)$ .", "Then ${\\cal R}(T+{\\bf i}I)=Y$ and ${\\cal R}(T-{\\bf i}I)=Y$ , where ${\\cal R}(T\\pm {\\bf i}I)=(T\\pm {\\bf i}I)Y$ , $~ker (T\\pm {\\bf i}I)= \\lbrace 0 \\rbrace $ and inverses $B_{\\pm } := (T\\pm {\\bf i}I)^{-1}$ are everywhere defined on $Y$ and of norm not exceeding one in accordance with §II.2.74 and Proposition II.2.75 [28].", "Then the equalities $<B_+(T+{\\bf i}I)x; (T-{\\bf i}I)y> = <x;(T-{\\bf i}I)y> = <(T+{\\bf i}I)x;y> = <(T+{\\bf i}I)x,B_-(T-{\\bf i}I)y>$ are satisfied for each $x$ and $y\\in {\\cal D}(T)$ , since the operator $T$ is self-adjoint, hence $B_- = B_+^*$ .", "Then an arbitrary vector $z\\in X$ has the form $z=B_- B_+ x$ for the corresponding vector $x\\in {\\cal D}(T)$ with $Tx\\in {\\cal D}(T)$ , since ${\\cal R}(B_{\\pm }) = Y$ .", "In the latter case we get $B_- B_+ x = (TT- {\\bf i}IT+T{\\bf i}I + I)x = B_+ B_- x$ , since to $\\mbox{}_nF$ a real-valued function $\\mbox{}_nf\\in C(\\Lambda ,{\\bf R})$ corresponds for each $n$ .", "On the other hand, $B_-=B_+^*$ , consequently, the operator $B_+$ is normal.", "Consider a quasi-commutative von Neumann algebra $\\breve{\\sf A}$ over the complexified algebra $({\\cal A}_v)_{\\bf C_i}$ containing $I$ , $B_+$ and $B_- \\in \\sf A$ .", "If $U$ is a unitary operator in ${\\sf A}^{\\star }$ (see §II.2.71 [28]), then $Ux=UB_+(T+{\\bf i}I)x=(B_+) U (T+{\\bf i}I)x\\mbox{ hence}$ $(T+{\\bf i}I)Ux= U(T+{\\bf i}I)x,$ consequently, $T$ is affiliated with $\\breve{\\sf A}$ by Definition 1, since $({\\bf i}I)Ux= U({\\bf i}I)x$ for a unitary operator $U$ as follows from the definition of a unitary operator and the ${\\cal A}_v$ valued scalar product on $X$ extended to the $({\\cal A}_v)_{\\bf C_i}$ valued scalar product on $Y$ : $<a+b{\\bf i};c+q{\\bf i}> = (<a;c> + <b;q>) +(<b;c> - <a;q>){\\bf i}$ for each vectors $a, b, c, q\\in X$ .", "The algebra $\\breve{\\sf A}$ has the decomposition $\\breve{\\sf A}= {\\sf A}^0\\oplus {\\sf A}^1{\\bf i}$ , where ${\\sf A}^0$ and ${\\sf A}^1$ are quasi-commutative isomorphic algebras over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ with $2\\le v \\le 3$ .", "In view of Theorem 2.24 [20] $\\breve{\\sf A}$ is isomorphic with $C(\\Lambda ,({\\cal A}_v)_{\\bf C_i})$ for some extremely disconnected compact Hausdorff topological space $\\Lambda $ , since the generator $\\bf i$ has the real matrix representation ${\\bf i} ={{0 ~~ 1}{-1 ~ 0}}$ and the real field is the center of the algebra ${\\cal A}_v$ with $2\\le v\\le 3$ .", "Let $f_+$ and $f_-$ be functions corresponding to $B_+$ and $B_-$ respectively.", "Put $h_{\\pm } =1/f_{\\pm }$ at those points where $f_{\\pm }$ does not vanish.", "Therefore, these functions $h_{\\pm }$ are continuous on their domains of definitions as well as $h=\\frac{1}{2}(h_+ + h_-)$ .", "The function $h$ corresponds to $T$ .", "It will be demonstrated below that $h$ is real-valued and then the ${\\cal A}_v$ graded spectral resolution of $T$ will be constructed.", "At first it is easy to mention that $f_+^*=f_-$ , since $B_+^*=B_-$ , consequently, $h$ is real-valued.", "The functions $f_+$ and $f_-$ are continuous and conjugated, hence the set $W :=f_+^{-1}(0)=f_-^{-1}(0)$ is closed.", "If the set $W $ contains some non-void open subset $J$ , then $cl (J)\\subset W$ so that $cl (J)$ is clopen (i.e.", "closed and open simultaneously) in $\\Lambda $ , consequently, $W$ is nowhere dense in $\\Lambda $ .", "The projection $P$ corresponding to this subset $cl (J)$ would have the zero product $B_+P=0$ contradicting the fact that $ker (B_+)= \\lbrace 0 \\rbrace $ .", "Thus each point $x\\in W$ is a limit point of points $y\\in \\Lambda \\setminus W$ , but $h_+$ and $h_-$ are defined on the latter set.", "Therefore, the set $\\Lambda \\setminus W$ is dense in $\\Lambda $ and the functions $h_+$ and $h_-$ are defined on $\\Lambda \\setminus W$ .", "Then $TB_+B_- y = (T+{\\bf i}I-{\\bf i}I) B_+B_-y = B_-y - ({\\bf i}I)B_+B_-y$ for each $y\\in \\Lambda $ , hence $TB_+B_- = B_- - ({\\bf i}I)B_+B_-$ and analogously $TB_-B_+ = B_+ + ({\\bf i}I)B_-B_+$ and inevitably $(2)$ $2({\\bf i}I)B_+B_- = B_- - B_+$ and $(3)$ $TB_+B_- = \\frac{1}{2} (B_+ + B_-)$ .", "From Formula $(2)$ and the definitions of functions $f_{\\pm }$ and $h$ one gets $(h(y)+{\\bf i})^{-1} = f_+(y)$ and $(h(y)-{\\bf i})^{-1} =f_-(y)$ for each $y\\in \\Lambda \\setminus W$ , since $(h(y)+{\\bf i})f_+(y) = \\frac{1}{2} +\\frac{1}{2f_-}f_+ +{\\bf i}f_+ =\\frac{1}{2}+\\frac{w-{\\bf i}}{2}\\frac{1}{w+{\\bf i}} +{\\bf i}\\frac{1}{w+{\\bf i}}= 1 -\\frac{2{\\bf i}}{2}\\frac{1}{w+{\\bf i}}+{\\bf i}\\frac{1}{w+{\\bf i}}=1$ , where $w\\pm {\\bf i}=\\frac{1}{f_{\\pm }}$ corresponds to $(T\\pm {\\bf i}I)$ .", "But $f_+(y)$ tends to zero when $y\\in \\Lambda \\setminus W$ tends to $x\\in W$ , consequently, $\\lim _{y\\rightarrow x} |h(y)|=\\infty $ .", "This means that $h$ is a self-adjoint function on $\\Lambda $ .", "We put $U_b := \\lbrace x\\in \\Lambda \\setminus W: ~h(x)>b \\rbrace $ and $V_b=U_b\\cup W_+(h)$ for $b\\in {\\bf R}$ (see Definition 8).", "The function $h$ is continuous on the open subset $\\Lambda \\setminus W$ in $\\Lambda $ , hence the subset $U_b$ is open in $\\Lambda $ .", "For a point $x\\in W_+$ there exists an open set $Q$ containing $x$ such that $h(y)>\\max (b,0)$ for each $y\\in Q\\cap (\\Lambda \\setminus W)$ , since $\\lim _{y\\rightarrow x} |h(y)|=\\infty $ , where $W_+=W_+(h)$ (see Definition 8).", "Therefore, this implies the inclusion $Q\\cap (\\Lambda \\setminus W)\\subset U_b\\subset V_b$ .", "Suppose that there would exist a point $z\\in Q\\cap W_-$ , where $W_- =W_-(h)$ , then a point $y\\in Q$ with $h(y)<0$ would exist contradicting the choice of $Q$ .", "This means that $Q\\cap W\\subset W_+$ and $Q\\subset V_b$ , consequently, $V_b$ is open in $\\Lambda $ .", "We consider next the subset $\\Lambda _b := \\Lambda \\setminus cl(V_b)$ .", "In accordance with §2.24 [20] the set $\\Lambda _b$ contains every clopen subset $K_b := \\lbrace y\\in \\Lambda : ~ h(y)\\le b \\rbrace $ , where $h(y)= - \\infty $ for $y\\in W_-$ so that $W_-\\subset K_b$ for each $~b\\in \\bf R$ .", "To demonstrate this suppose that $y\\in V_b$ , then $y\\in \\Lambda \\setminus K_b$ and $cl (V_b)\\subset \\Lambda \\setminus K_b$ and $K_b\\subset \\Lambda \\setminus cl (V_b)=\\Lambda _b$ , since $\\Lambda \\setminus K_b$ is closed.", "Moreover, we have $h(y)\\le b$ for each $y\\in \\Lambda _b\\cap (\\Lambda \\setminus W)$ , since $y\\notin U_b$ , while the set $~\\Lambda _b$ is clopen in $\\Lambda $ .", "Therefore, $y\\in W\\setminus W_+$ and $h(y)\\le b$ for each $y\\in \\Lambda _b\\cap W$ and $W_-=W\\setminus W_+$ .", "Thus $\\Lambda _b$ is the largest clopen subset in $\\Lambda $ so that $h(y)\\le b$ and $\\Lambda _b\\cap W=W_-$ .", "Denote by $e_b$ the characteristic function of the subset $\\Lambda _b$ and $\\mbox{}_bE$ be an ${\\cal A}_v$ graded projection operator in $\\sf A$ corresponding to $e_b$ , where $b\\in \\bf R$ (see also §2.24 [20]).", "The subset $W$ is nowhere dense in $\\Lambda $ , consequently, $\\vee _b e_b=1$ and $\\wedge _b e_b =0$ such that $\\vee _b \\mbox{ }_bE =1$ and $\\wedge _b \\mbox{ }_bE =0$ .", "Thus $\\lbrace \\mbox{}_bE: ~ b \\rbrace $ is the ${\\cal A}_v$ graded resolution of the identity so that $\\mbox{}_bE_s=(sI)\\mbox{ }_bE=\\mbox{}_bE(sI)$ corresponds to $e_bs=se_b$ for each marked Cayley-Dickson number $s\\in {\\cal A}_v$ .", "This resolution of the identity is unbounded when $h\\notin C(\\Lambda ,{\\cal A}_v)$ .", "Put $F = \\mbox{}_bE - \\mbox{ }_aE$ for $a<b\\in \\bf R$ , hence $e_b - e_a=:u$ is the characteristic function of $\\Lambda _b\\setminus \\Lambda _a$ corresponding to $F$ .", "Therefore, the inclusion $\\Lambda _b\\setminus \\Lambda _a\\subset \\Lambda \\setminus W$ follows and $f_+(y)f_-(y)\\ne 0$ when $u(y)=1$ , since $\\Lambda _b\\cap W=\\Lambda _a\\cap W=W_-$ .", "Then $(4)\\quad h(y)=\\frac{f_++f_-}{2f_+f_-}(y)$ for each $y\\in \\Lambda \\setminus W$ .", "The function $f_+f_-$ is continuous and vanishes nowhere on the clopen subset $\\Lambda _b\\setminus \\Lambda _a$ , consequently, a positive continuous function $\\psi $ on $\\Lambda $ exists so that $\\psi f_+f_-=u$ and $\\psi u=\\psi $ .", "Consider an element $\\Psi $ of the algebra $\\sf A$ corresponding to $\\psi $ , hence $(5)$ $\\Psi B_+B_-=F$ .", "On the other hand, from the construction above it follows that $a\\le h(y)\\le b$ for each $y\\in \\Lambda _b\\setminus \\Lambda _a$ and from Formula $(4)$ one gets $af_+f_-u\\le \\frac{(f_+ + f_-)u}{2} \\le b f_+f_-u$ and $a\\psi f_+f_-u=au\\le \\frac{(f_+ +f_-)\\psi u}{2}= \\frac{(f_+ +f_-)\\psi }{2} \\le b\\psi f_+f_-u=bu$ , since the real field is the center of the Cayley-Dickson algebra ${\\cal A}_v$ .", "Thus $(6)$ $aF\\le \\frac{(B_++B_-)\\Psi }{2}\\le bF$ .", "Therefore, Formulas $(3,5,6)$ imply that $(7)$ $aF\\le TF\\le bF$ .", "Therefore, the operator $TF$ is bounded and the element $h\\hat{\\cdot }u\\in C(\\Lambda ,{\\cal A}_v)$ corresponds to it due to $(3-5)$ .", "If $E$ is an ${\\cal A}_v$ graded projection belonging to $\\sf A$ so that $TE\\in L_q(X)$ and $U$ is a unitary operator in ${\\sf A}^{\\star }$ such that $U^{-1}TEU=U^{-1}TUE=TE$ one has $TE\\in \\sf A$ (see §II.2.71 [28]).", "Let each function $\\mbox{}_ng\\in C(\\Lambda ,{\\cal A}_v)$ be corresponding to the ${\\cal A}_v$ graded projection $\\mbox{}_nF$ and $\\Lambda _n :=\\mbox{}_ng^{-1}(1)$ .", "Then $\\bigcup _n \\Lambda _n$ is dense in $\\Lambda $ , since $\\vee _n \\mbox{ }_nF=I$ .", "If a function $e\\in C(\\Lambda ,{\\cal A}_v)$ corresponds to $E$ , then $(h\\hat{\\cdot }\\mbox{ }_ng)e$ corresponds to the operator $T\\mbox{ }_nFE$ , where $\\mbox{}_nFE= E\\mbox{ }_nF$ .", "Suppose that $e(y)=1$ for some $y\\in \\Lambda $ .", "For each (open) neighborhood $H$ of $y$ there exist $n\\in \\bf N$ and $x\\in \\Lambda _n$ such that $x\\in H$ , hence $((h\\hat{\\cdot }\\mbox{ }_ng)e)(x)=h(x)$ and $|h(x)| \\le \\Vert TE\\mbox{ }_nF \\Vert \\le \\Vert TE \\Vert $ .", "Thus $y\\notin W$ and $|h(y)|\\le \\Vert TE \\Vert $ and hence $h\\hat{\\cdot }e \\in C(\\Lambda ,{\\cal A}_v)$ .", "Then one gets also $(h_j\\hat{\\cdot }\\mbox{ }_ng_k)e_l= h_j\\hat{\\cdot }(\\mbox{ }_ng_ke_l)=(h_j\\hat{\\cdot }e_l)\\mbox{ }_ng_k$ for each $j, k, l$ .", "If $s\\in C(\\Lambda ,{\\cal A}_v)$ represents $TE$ , then $s\\mbox{ }_ng$ represents $TE\\mbox{ }_nF$ , that is $s\\mbox{ }_ng= (h\\hat{\\cdot }e)\\mbox{ }_ng$ for every $n$ and $s=h\\hat{\\cdot }e$ .", "On the other hand, $(2\\mbox{ }_nF -I)T (2\\mbox{ }_nF -I)=T$ , since $T$ and $\\mbox{}_nF$ are $\\bf R$ -linear operators, also $(\\mbox{}_nF-I)(X)= (I-\\mbox{}_nF)(X)$ and $X=\\mbox{}_nF(X)\\oplus (\\mbox{}_nF-I)(X)$ , hence $\\mbox{}_nFT\\subset T\\mbox{ }_nF$ .", "Moreover, $\\lim _n \\mbox{ }_nFx=x$ and $\\lim _n T\\mbox{ }_nFx =\\lim _n \\mbox{ }_nFTx=T x$ hence $\\bigcup _n \\mbox{ }_nF(X)$ is a core for $T$ .", "In view of Theorem I.3.9 [28] we get that $ \\lbrace E\\mbox{}_bE|_{E(X)} : ~ b\\in {\\bf R} \\rbrace $ is an ${\\cal A}_v$ graded resolution of the identity on $E(X)$ , since $(h\\hat{\\cdot }e)ee_b\\le b e e_b$ and $b(e-ee_b) \\le (h\\hat{\\cdot }e)(e-ee_b)$ .", "Applying Theorem I.3.6 [28] for $\\mbox{}_nF$ in place of $E$ for $T\\mbox{ }_nF|_{\\mbox{}_nF(X)}$ and $ \\lbrace \\mbox{}_nF \\mbox{}_bE|_{\\mbox{}_nF(X)}: ~ b \\rbrace $ leads to the formula $Tx =\\int _{-n}^n d\\mbox{}_bE.b x$ for each $x\\in \\mbox{}_nF(X)$ and every $n\\in \\bf N$ .", "11.", "Remark.", "Lemma 10 means that $\\lim _n \\int _{-n}^n d\\mbox{}_bE.b x = \\lim _n\\int _{-n}^n d\\mbox{}_bE.b \\mbox{ }_nFx =\\lim _nT\\mbox{ }_nFx=Tx=\\int _{-\\infty }^{\\infty }d\\mbox{}_bE.b x$ for each $x\\in {\\cal D}(T)$ interpreting the latter integral as improper.", "Under conditions imposed in Lemma 10 one says that the function $h\\in {\\cal Q}(\\Lambda )$ represents an affiliated operator $T\\eta \\sf A$ .", "Mention that an ${\\cal A}_v$ graded projection operator $\\mbox{}_bE$ is $\\bf R$ -homogeneous and ${\\cal A}_v$ -additive, hence ${\\bf R}$ -linear.", "Therefore, in the particular case of a real-valued Borel function $h(b)$ one has $d\\mbox{}_bE.h(b)x=h(b)d\\mbox{}_bE.1x$ or simplifying notation $h(b)d\\mbox{}_bEx$ .", "12.", "Lemma.", "Suppose that $\\sf A$ is a quasi-commutative von Neumann algebra over ${\\cal A}_v$ , where $~2\\le v\\le 3$ , so that $\\sf A$ is isomorphic to $C(\\Lambda ,{\\cal A}_v)$ for some extremely disconnected compact Hausdorff topological space $\\Lambda $ .", "Then each function $h\\in {\\cal Q}(\\Lambda )$ represents some self-adjoint operator $T$ affiliated with $\\sf A$ .", "Proof.", "From §10 it follows that a self-adjoint function $h$ determines an ${\\cal A}_v$ graded resolution of the identity $ \\lbrace \\mbox{}_bE: ~ b\\in {\\bf R} \\rbrace $ in $\\sf A$ .", "Moreover, $h\\hat{\\cdot }g_n\\in C(\\Lambda ,{\\cal A}_v)$ , where $g_n=e_n - e_{-n}$ , $~e_n\\in C(\\Lambda ,{\\cal A}_v)$ represents $\\mbox{}_nE$ .", "Consider an operator $\\mbox{}_nT$ corresponding to $h\\hat{\\cdot }g_n$ .", "Certainly one has $(h\\hat{\\cdot }g_m)g_n=h\\hat{\\cdot }g_n$ for each $n\\le m$ , consequently, $\\mbox{}_mT\\mbox{}_nF=\\mbox{}_nT$ , where $\\mbox{}_nF$ corresponds to $g_n$ .", "Put $Gx=\\mbox{}_nTx$ for every vector $x\\in \\mbox{ }_nF(X)$ and $n\\in \\bf N$ .", "Therefore, $G$ is an $\\bf R$ linear ${\\cal A}_v$ additive operator on $\\bigcup _{n=1}^{\\infty }\\mbox{}_nF(X)=:\\cal K$ .", "Therefore, the operator $G$ is pre-closed and its closure $T$ is a self-adjoint operator with core $\\cal K$ as Lemma 6 asserts.", "For a unitary operator $U$ in ${\\sf A}^{\\star }$ and $x\\in \\mbox{ }_nF(X)$ we get $Ux\\in \\mbox{ }_nF(X)$ , hence $TUx = \\mbox{ }_nTUx= U\\mbox{ }_nTx $ .", "Therefore, $T\\eta \\sf A$ due to Definition 1 and Remark 2.", "If $u\\in {\\cal Q}(\\Lambda )$ represents $T$ , then $u\\hat{\\cdot }g_ n$ represents $T\\mbox{ }_nF$ in accordance with Lemma 10, consequently, $h\\hat{\\cdot }g_n = u\\hat{\\cdot }g_n$ for each $n$ .", "In view of Lemma 9 $h=u$ , since $h$ and $u$ are consistent on a dense subsets in $\\Lambda $ .", "Thus the function $h$ represents the self-adjoint operator $T$ .", "13.", "Lemma.", "Suppose that $\\lbrace \\mbox{}_bE: ~ b \\rbrace $ is an ${\\cal A}_v$ graded resolution of the identity on a Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ , $~2\\le v\\le 3$ , also $\\sf A$ is a quasi-commutative von Neumann algebra over ${\\cal A}_v$ so that $\\sf A$ contains $\\lbrace \\mbox{}_bE: ~ b \\rbrace $ , where $2\\le v \\le 3$ .", "Then there exists a self-adjoint operator $T$ affiliated with $\\sf A$ so that $(1)\\quad Tx = \\int _{-n}^n d\\mbox{}_bE.b x$ for each $x\\in \\mbox{}_nF(X)$ and every $n\\in \\bf N$ , where $\\mbox{}_nF= \\mbox{}_nE-\\mbox{}_{-n}E$ , and $\\lbrace \\mbox{}_bE: ~ b \\rbrace $ is the ${\\cal A}_v$ graded resolution of the identity for $T$ given by Lemma 10.", "Proof.", "Take a function $e_b\\in C(\\Lambda ,{\\cal A}_v)$ corresponding to $\\mbox{}_bE$ and a subset $\\Lambda _b=e_b^{-1}(1)$ clopen in $\\Lambda $ .", "Consider the subsets $W_+ := \\Lambda \\setminus \\bigcup _b \\Lambda _b$ and $W_- := \\bigcap _b \\Lambda _b$ .", "From their definition it follows that $W_+$ and $W_-$ are closed in $\\Lambda $ .", "These two subsets $W_{\\pm }$ are nowhere dense in $\\Lambda $ , since $\\wedge _b \\mbox{ }_bE=0$ and $\\vee _b \\mbox{ }_bE=I$ , hence $W :=W_+\\cup W_-$ is nowhere dense in $\\Lambda $ .", "For a point $x\\in \\Lambda \\setminus W$ we put $h(x) := \\inf \\lbrace b: ~ x\\in \\Lambda _b \\rbrace $ .", "Given a positive number $\\epsilon >0$ and $y\\in \\Lambda \\setminus W$ so that $h(y)=b$ one gets $|h(z)-h(y)|\\le \\epsilon $ for each $z\\in \\Lambda _{b+\\epsilon }\\setminus \\Lambda _{b-\\epsilon }$ .", "This means that the function $h$ is continuous on $\\Lambda \\setminus W$ .", "Then $\\lim _{y\\rightarrow x} h(y)=\\pm \\infty $ for $x\\in W_{\\pm }$ , where $y\\in \\Lambda \\setminus W$ .", "Thus the function $h$ is self-adjoint $h\\in {\\cal Q}(\\Lambda )$ and by Lemma 12 corresponds to a self-adjoint operator $T$ affiliated with $\\sf A$ .", "Certainly we get that $\\lbrace \\mbox{}_bE: ~ b \\rbrace $ is the ${\\cal A}_v$ graded resolution of the identity for the operator $T$ and Formula $(1)$ is valid, since $\\Lambda _b$ is the largest clopen subset in $\\Lambda $ on which the function $h$ takes values not exceeding $b$ .", "If $\\Psi $ is another such clopen subset and $e$ is its characteristic function, $E\\in \\sf A$ corresponds to $e$ , then $\\Psi \\subset \\Lambda _c$ for each $c\\ge b$ .", "This leads to the conclusion that $E\\le \\wedge _{c>b} \\mbox{ }_cE$ and $\\Psi \\subset \\Lambda _b$ .", "14.", "Lemma.", "Suppose that $T$ is a closed operator on a Hilbert space $X$ over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ with $2\\le v \\le 3$ and $\\lbrace \\mbox{}_bE: ~ b \\rbrace $ is an ${\\cal A}_v$ graded resolution of the identity on $X$ , where ${\\cal E} := \\bigcup _n \\mbox{ }_nF(X)$ is a core for $T$ while $\\mbox{}_nF=\\mbox{}_nE - \\mbox{ }_{-n}E$ and $(1)\\quad Tx = \\int _{-n}^n d\\mbox{}_bE.b x$ for each $x\\in \\mbox{}_nE(X)$ and all $n$ , then $T$ is self-adjoint and $\\mbox{}_bE$ is the ${\\cal A}_v$ graded resolution of the identity for $T$ .", "Proof.", "Formula $(1)$ implies that $T\\mbox{ }_nF$ is bounded and everywhere defined and is the strong operator limit of finite real-linear combinations of $ \\lbrace \\mbox{}_bE: ~ b \\rbrace $ .", "Thus $\\mbox{}_bE (T\\mbox{ }_nF) =(T\\mbox{ }_nF) \\mbox{ }_bE$ and $T\\mbox{}_nF$ is self-adjoint, hence $T$ is self-adjoint by Lemma 6.", "For each vector $x\\in {\\cal D}(T)$ there exists a subsequence $ \\lbrace n_p:~ p \\rbrace $ of natural numbers and a sequence $ \\lbrace \\mbox{}_p x : ~ p \\rbrace $ of vectors such that $\\lim _p \\mbox{}_px = \\lim _p\\mbox{}_{n_p}F \\mbox{ }_px=x$ and $\\lim _p T\\mbox{ }_px = Tx$ , since $\\cal E$ is a core for the operator $T$ .", "Therefore, $\\mbox{}_nFT\\subseteq T\\mbox{ }_nF$ for each $n$ , since $\\mbox{}_nFTx = \\lim _p \\mbox{}_nF (T\\mbox{ }_{n_p}F) \\mbox{ }_px =\\lim _p (T\\mbox{ }_{n_p}F)\\mbox{}_nF\\mbox{ }_px = \\lim _pT\\mbox{}_nF\\mbox{ }_px = T\\mbox{}_nFx$ .", "On the other hand, the limits exist: $\\lim _n ~ (T\\mbox{ }_nF)\\mbox{}_bEx = \\lim _n \\mbox{ }_bE(T\\mbox{ }_nF)x = \\mbox{ }_bETx$ and $\\lim _n \\mbox{ }_nF\\mbox{ }_bEx =\\mbox{ }_bEx$ .", "The operator $T$ is closed, hence $\\mbox{}_bEx \\in {\\cal D}(T)$ and $T \\mbox{ }_bEx =\\mbox{ }_bE Tx$ .", "This leads to the conclusion that $\\mbox{}_bET\\subseteq T\\mbox{ }_bE$ and $(2\\mbox{ }_bE-I)T(2\\mbox{}_bE-I)=T$ , since these operators are $\\bf R$ linear and $X=\\mbox{}_bE(X)\\oplus (\\mbox{}_bE-I)(X)$ .", "Therefore, $\\mbox{}_bE(B_{\\pm }) = (B_{\\pm }) \\mbox{}_bE$ for each $b\\in \\bf R$ .", "Take the quasi-commutative von Neumann algebra ${\\sf G}={\\sf G}(T) = cl [~ alg_{{\\cal A}_v} \\lbrace B_-, B_+; ~\\mbox{}_bE: ~ b \\in {\\bf R} \\rbrace \\oplus alg_{{\\cal A}_v} \\lbrace B_-, B_+;~ \\mbox{}_bE: ~ b \\in {\\bf R} \\rbrace {\\bf i} ]$ .", "In view of Lemma 10 the operator $T$ is affiliated with $\\sf G$ .", "But Lemma 13 means that there exists a self-adjoint operator $H$ affiliated with $\\sf G$ so that $Hx = \\int _{-n}^n d\\mbox{}_bE.b x $ for each $x\\in \\mbox{}_nF(X)$ and every natural number $n$ .", "Therefore, $H=T$ and $ \\lbrace \\mbox{}_bE: ~ b \\rbrace $ is the ${\\cal A}_v$ graded resolution of the identity for $T$ , since $H|_{\\cal E} = T|_{\\cal E}$ and $\\cal E$ is a core for $T$ and $H$ simultaneously.", "15.", "Note.", "The quasi-commutative von Neumann algebra ${\\sf G}={\\sf G}(T)$ generated by $B_-$ and $B_+$ in §14 with which the self-adjoint operator $T$ is affiliated will be called the von Neumann algebra generated by $T$ .", "16.", "Theorem.", "Let $\\sf A$ be a quasi-commutative von Neumann algebra acting on a Hilbert space $X$ over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ , $~ 2\\le v\\le 3$ , let also ${\\cal Q}({\\sf A})$ be a family of all self-adjoint operators affiliated with $\\sf A$ .", "Suppose that $\\sf A$ is isomorphic to $C(\\Lambda ,{\\cal A}_v)$ for an extremely disconnected compact Hausdorff topological space $\\Lambda $ and ${\\cal Q}(\\Lambda )$ is the family of all self-adjoint functions on $\\Lambda $ .", "Then $(a)$ there exists a bijective mapping $\\phi $ from ${\\cal Q}({\\sf A})$ onto ${\\cal Q}(\\Lambda )$ which is an extension of the isomorphism of $\\sf A$ with $C(\\Lambda ,{\\cal A}_v)$ for which $\\phi (T)\\hat{\\cdot }e$ corresponds to $TE$ for each projection $E$ in $\\sf A$ with $TE\\in \\sf A$ , where $e\\in C(\\Lambda ,{\\cal A}_v)$ corresponds to $E$ and $((\\phi (T)\\hat{\\cdot }e)(y)= (\\phi (T))(y)$ for $e(y)=1$ while $(\\phi (T)\\hat{\\cdot }e)(y)=0$ for $e(y)=0$ ; $(b)$ an ${\\cal A}_v$ graded resolution $\\lbrace \\mbox{}_bE: ~ b \\rbrace $ of the identity exists in the quasi-commutative von Neumann subalgebra $\\sf G$ generated by an operator $T$ in ${\\cal Q}({\\sf A})$ so that $(1)\\quad Tx = \\int _{-n}^n d\\mbox{}_bE.bx$ for each $x\\in \\mbox{}_nF(X)$ and every natural number $n$ , where $\\mbox{}_nF =\\mbox{}_nE- \\mbox{ }_{-n}E$ and $\\bigcup _n \\mbox{ }_nF(X)=:\\cal E$ is a core for $T$ ; $(c)$ if $\\lbrace \\mbox{}_bQ: ~ b\\in {\\bf R} \\rbrace $ is an ${\\cal A}_v$ graded resolution of the identity on $X$ so that $(2)\\quad Tx = \\int _{-n}^n d\\mbox{}_bQ.bx$ for each $x\\in \\mbox{}_nP(X)$ and every natural number $n$ , where $\\mbox{}_nP= \\mbox{}_nQ- \\mbox{ }_{-n}Q$ and $\\bigcup _n \\mbox{ }_nP(X)$ is a core for $T$ , then $\\mbox{}_bE=\\mbox{}_bQ$ for all $b$ ; $(d)$ if $\\lbrace \\mbox{}_bE : ~b \\in {\\bf R} \\rbrace $ is an ${\\cal A}_v$ graded resolution of the identity in $\\sf A$ , then there exists an operator $T\\in {\\cal Q}({\\sf A})$ for which Formula $(1)$ is valid; $(e)$ if a function $e_b \\in C(\\Lambda ,{\\cal A}_v)$ corresponds to $\\mbox{}_bE$ and $\\Lambda _b=e_b^{-1}(1)$ , then $\\Lambda _b$ is the largest clopen subset of $\\Lambda $ on which $\\phi (T)$ takes values not exceeding $b$ in the extended sense.", "The latter theorem is the reformulation of the results obtained above in this section.", "17.", "Definition.", "A closed densely defined operator $T$ in a Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ will be called normal when two self-adjoint operators $T^*T$ and $TT^*$ are equal, where $2\\le v$ .", "18.", "Remark.", "For an unbounded $\\bf R$ homogeneous ${\\cal A}_v$ additive operators the following properties are satisfied: $(1)$ if $A\\subseteq B$ and $C\\subseteq D$ , then $A+C \\subseteq B+D$ ; $(2)$ if $A\\subseteq B$ , then $CA\\subseteq CB$ and $AC\\subseteq BC$ ; $(3)$ $(A+B)C=AC+BC$ and $CA+CB\\subseteq C(A+B)$ .", "The latter inclusion does not generally reduce to the equality.", "For example, an operator $C$ may be densely defined, but not everywhere, one can take $A=I$ and $B=-I$ .", "This gives $C(A+B)=0$ on $X$ , but $CA+CB$ is zero only on ${\\cal D}(C)$ .", "These rules imply that if $CA\\subseteq AC$ for each $C$ in some family $\\cal Y$ , then $TA\\subseteq AT$ for each sum of products of operators from $\\cal Y$ .", "Apart from algebras of bounded operators a family $\\cal Y$ may be not extendable to an algebra, since a distributive law generally may be invalid, as it was seen above.", "Another property is the following: $(4)$ if $\\lbrace \\mbox{}_bT: ~ b\\in \\Psi \\rbrace $ is a net of bounded operators in $L_q(X)$ so that $\\lim _b \\mbox{ }_bT=T$ in the strong operator topology and $\\mbox{}_bTA\\subset B\\mbox{ }_bT$ for each $b$ in a directed set $\\Psi $ , where $B$ is a closed operator, then $TA\\subseteq BT$ .", "Indeed, if $x\\in {\\cal D}(A)$ , then $\\mbox{}_bTx\\in {\\cal D}(B)$ and $\\lim _b B\\mbox{ }_bTx=\\lim _b \\mbox{ }_bTAx=TAx$ and hence $\\lim _b \\mbox{ }_bTx=Tx$ .", "Then $Tx\\in {\\cal D}(B)$ and $BTx=TAx$ , since the operator $B$ is closed, from which property $(4)$ follows.", "We sum up these properties as the lemma.", "19.", "Lemma.", "Suppose $A$ is a closed operator acting in a Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ and $CA\\subseteq AC$ for every operator $C$ in a self-adjoint subfamily $\\cal Y$ of $L_q(X)$ , where $2\\le v$ .", "Then $TA\\subseteq AT$ for each operator $T$ in the von Neumann algebra over ${\\cal A}_v$ generated by $\\cal Y$ .", "20.", "Definition.", "When $A$ is a closed $\\bf R$ homogeneous ${\\cal A}_v$ additive operator in a Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ , $~2\\le v$ , and $E$ is an ${\\cal A}_v$ graded projection on $X$ so that $EA\\subseteq AE$ and $AE$ is a bounded operator on $X$ , we say that $E$ is a bounding ${\\cal A}_v$ graded projection for $A$ .", "An increasing sequence $\\lbrace \\mbox{}_nE: ~ n\\in {\\bf N} \\rbrace $ such that each $\\mbox{}_nE$ is a bounding ${\\cal A}_v$ graded projection for $A$ and $\\vee _n \\mbox{ }_nE=I$ will be called a bounding ${\\cal A}_v$ graded sequence for $A$ .", "21.", "Lemma.", "Let $E$ be a bounding ${\\cal A}_v$ graded projection for a closed densely defined operator $T$ in a Hilbert space over the Cayley-Dickson algebra ${\\cal A}_v$ , $~2\\le v$ .", "Then $E$ is bounding for $T^*$ , $T^*T$ and $TT^*$ and $(1)$ $(TE)^* = T^*E^*.$ Moreover, if $\\lbrace \\mbox{}_nE: ~ n \\rbrace $ is a bounding sequence for $T$ , then $\\bigcup _n \\mbox{ }_nE(X)$ is a core for $T$ and $T^*$ and $T^*T$ and $TT^*$ .", "Proof.", "The conditions of this lemma mean that $ET\\subseteq TE$ and $TE$ is bounded, hence the operator $ET$ is pre-closed and densely defined and bounded.", "Therefore, the operator $ET$ has the closure $ TE$ , also $(2)$ $(TE)^* = (ET)^*$ is closed and hence the operators $(TE)^*$ and $(ET)^*$ are closed by Theorem I.3.34 [28].", "For each vectors $x\\in E(X)$ and $y\\in {\\cal D}(T)$ the equality $<Ty;x> = <y;(E^*T)^*x>$ is satisfied so that $x\\in {\\cal D}(T^*)$ and $T^*x=(ET)^*x$ , consequently, $T^*E=(E^*T)^*E$ .", "At the same time we have $(I-E){\\overline{E^*T}}=0$ and hence $(E^*T)^* = {\\overline{(E^*T)}}^*=(E^*T)^*E=T^*E$ .", "Therefore, $E^*T^*\\subseteq (TE)^*$ and $E$ is bounding for $T^*$ in accordance with Equality $(2)$ .", "Analogously $E$ is bounding for $T^*T$ and similarly for $TT^*$ , since $E(T^*T)\\subseteq (T^*T)E$ .", "This implies that $\\lbrace \\mbox{}_nE: ~ n \\rbrace $ is a bounding sequence for $T^*$ , $T^*T$ and $TT^*$ if it is such for $T$ .", "Then $\\lim _n \\mbox{ }_nEx=x$ , $~\\mbox{}_nEx\\in {\\cal D}(T)$ and $\\lim _nT\\mbox{ }_nEx= \\lim _n \\mbox{}_nETx=Tx$ for every $x\\in {\\cal D}(T)$ .", "Thus $\\bigcup _{n=1}^{\\infty } \\mbox{}_nE({\\cal D}(T))$ is a core for $T$ , consequently, $\\bigcup _{n=1}^{\\infty } \\mbox{ }_nE(X)$ is a core for $T$ and $T^*$ , $~T^*T$ and $TT^*$ as well, since $\\mbox{}_nE(X)\\subseteq {\\cal D}(T)$ for each natural number $n$ .", "22.", "Remark.", "Mention that in accordance with Theorem 2.28 [20] an ${\\cal A}_v$ graded projection operator can be chosen self-adjoint $E^*=E$ for a quasi-commutative von Neumann algebra over ${\\cal A}_v$ , since $E$ corresponds to a characteristic function $e$ which is real-valued.", "23.", "Theorem.", "Suppose that $\\sf A$ is a quasi-commutative von Neumann algebra over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ , $~2\\le v\\le 3$ , acting on a Hilbert space $X$ over ${\\cal A}_v$ , also $T$ and $B\\eta \\sf A$ .", "Then $(1)$ each finite set of operators affiliated with $\\sf A$ has a common bounding sequence in $\\sf A$ ; $(2)$ an operator $B+T$ is densely defined and pre-closed and its closure is $B{\\hat{+}}T\\eta \\sf A$ ; $(3)$ $BT$ is densely defined and pre-closed with the closure $B{\\hat{\\cdot }}T\\eta \\sf A$ affiliated with $\\sf A$ ; $(4)$ $\\mbox{}^jB{\\hat{\\cdot }}\\mbox{ }^kT = (-1)^{\\kappa (j,k)}\\mbox{ }^kT{\\hat{\\cdot }}\\mbox{ }^j B$ for each $j, k$ , also $B^* B = B^* {\\hat{\\cdot }} B = B B^*$ ; $(5)$ $((bI)B{\\hat{+}} T)^* = B^*(b^*I){\\hat{+}} T^* $ for each quaternion or octonion number $b\\in {\\cal A}_v$ ; $(6)$ $(B{\\hat{\\cdot }}T)^*=T^*{\\hat{\\cdot }}B^*$ ; $(7)$ if $B\\subseteq T$ , then $B=T$ ; if $B$ is symmetric, then $B^*=B$ ; $(8)$ the family ${\\cal N}({\\sf A})$ of all operators affiliated with $\\sf A$ forms a quasi-commutative $*$ -algebra with unit $I$ under the operations of addition $\\hat{+}$ and multiplication $\\hat{\\cdot }$ given by $(2,3)$ .", "Proof.", "Take an arbitrary unitary operator $U$ in the super-commutant ${\\sf A}^{\\star }$ of ${\\sf A}$ (see §II.2.71 [28]).", "From $TU=UT$ it follows that $T^*U=UT^*$ , hence $T^*\\eta \\sf A$ .", "Then $(T^*T)U= U(T^*T)$ , since $U^*U=UU^*=I$ , consequently, $(T^*T)\\eta {\\sf A}$ .", "For an ${\\cal A}_v$ graded projection $E$ in $\\sf A$ an operator $(2E-I)$ is unitary such that $T(2E-I)=(2E-I)T$ , i.e.", "$T(2E-I)x=(2E-I)Tx$ for each $x\\in {\\cal D}(T)\\subset X$ , hence $ET\\subseteq TE$ .", "In view of Theorem I.3.34 [28] the operator $T^*T$ is self-adjoint.", "Take an ${\\cal A}_v$ graded resolution $\\lbrace \\mbox{}_bE: ~ b \\rbrace $ of the identity for $T^*T$ and put $\\mbox{}_nF = \\mbox{}_nE- \\mbox{ }_{-n}E$ for each natural number $n$ .", "From Theorem 16 the inclusion $\\mbox{}_bE\\in \\sf A$ follows.", "The operator $T^*T\\mbox{ }_nF$ is bounded and everywhere defined, consequently, the operator $T\\mbox{ }_nF$ is everywhere defined and closed, since $T$ is closed and $\\mbox{}_nF$ is bounded.", "In accordance with the closed graph theorem 1.8.6 [12] for $\\bf R$ linear operators one gets that $T\\mbox{ }_nF$ is bounded.", "It can lightly be seen also from the estimate $ \\Vert T\\mbox{ }_nFx\\Vert ^2 = <\\mbox{}_nFx; T^*T\\mbox{ }_nFx>\\le \\Vert T^*T\\mbox{ }_nF \\Vert \\Vert x \\Vert ^2$ .", "A sequence $ \\lbrace \\mbox{}_nF: ~ n \\rbrace $ of projections is increasing with least upper bound $I$ for which $\\mbox{}_nFT\\subseteq T\\mbox{}_nF$ .", "Therefore, the limits exist: $\\lim _n \\mbox{ }_nFx=x$ and $\\lim _n T\\mbox{}_nFx= \\lim _n \\mbox{}_nFT x = Tx$ for each vector $x$ in the domain ${\\cal D}(T)$ , consequently, $\\bigcup _{n=1}^{\\infty } \\mbox{}_nF(X)$ is a core for $T$ and $ \\mbox{}_nF$ is a bounding ${\\cal A}_v$ graded sequence in $\\sf A$ for the operator $T$ .", "Let now $\\lbrace \\mbox{}_nE: ~ n \\rbrace $ be a bounding sequence in $\\sf A$ for $ \\lbrace \\mbox{}_pT: ~ p=1,...,m-1 \\rbrace \\subset \\sf A$ and let $\\mbox{}_nF$ be a bounding sequence in $\\sf A$ for $\\mbox{}_mT\\in \\sf A$ .", "Then $ \\lbrace \\mbox{}_nE\\mbox{ }_nF: ~ n \\rbrace $ is a bounding sequence in $\\sf A$ for $\\mbox{}_1T,...,\\mbox{}_mT$ , particularly, $\\bigcup _{n=1}^{\\infty } \\mbox{}_nE\\mbox{ }_nF(X)$ is a common core for $\\mbox{}_1T,...,\\mbox{}_mT$ .", "This implies that two operators $T+B$ and $T^*+B^*$ are densely defined, but $T^*+B^*\\subseteq (T+B)^*$ , consequently, $(T+B)^*$ is densely defined and $T+B$ is pre-closed (see also Theorem I.3.34 [28]).", "As soon as $ \\lbrace \\mbox{}_nE: ~n \\rbrace $ is a bounding sequence in $\\sf A$ for $T$ , $B$ , $T^*$ and $B^*$ , the inclusions are satisfied $\\mbox{}_nET\\subseteq T\\mbox{}_nE$ and $\\mbox{}_nEB\\subseteq B\\mbox{}_nE$ and hence $\\mbox{}_nE(TB)\\subseteq (TB)\\mbox{}_nE$ and $T\\mbox{}_nEB\\mbox{}_nE\\subseteq TB\\mbox{}_nE\\mbox{ }_nE$ .", "The operators $T\\mbox{ }_nE$ and $B\\mbox{ }_nE$ are bounded and defined everywhere, consequently, $(9)$ $T\\mbox{}_nEB\\mbox{ }_nE=TB\\mbox{}_nE$ (see also Theorem 2.28 [20]).", "This means that $ \\lbrace \\mbox{}_nE: ~ n \\rbrace $ is a bounding sequence for $TB$ and analogously for $BT$ and $B^*T^*$ .", "That is, the operator $B^*T^*$ is densely defined.", "Since $B^*T^*\\subseteq (TB)^*$ , one gets that the operator $(TB)^*$ is densely defined and $TB$ is pre-closed.", "From Formula $(9)$ we infer that $\\mbox{}^kT\\mbox{ }^jB\\mbox{ }_nE = (-1)^{\\kappa (j,k)}\\mbox{ }^jB\\mbox{ }^kT \\mbox{ }_nE$ for each $j, k$ , since $\\sf A$ is quasi-commutative.", "Therefore, the operators $T{\\hat{\\cdot }}B$ and $B{\\hat{\\cdot }}T$ agree on their common core $\\bigcup _{n=1}^{\\infty } \\mbox{ }_nE(X)$ and inevitably $(10)\\quad \\mbox{}^jB{\\hat{\\cdot }}\\mbox{ }^kT =(-1)^{\\kappa (j,k)} \\mbox{ }^kT{\\hat{\\cdot }} \\mbox{}^jB.$ The operators $T^*T$ and $TT^*$ are self-adjoint, consequently, $T^*T=T^*{\\hat{\\cdot }}T$ and $TT^*=T{\\hat{\\cdot }}T^*$ and $T^*{\\hat{\\cdot }}T= T{\\hat{\\cdot }}T^*$ .", "Then $U^*x$ and $Ux\\in {\\cal D}(T+B)$ for each $x\\in {\\cal D}(T)\\cap {\\cal D}(B)={\\cal D}(T+B)$ , consequently, $U{\\cal D}(T+B)={\\cal D}(T+B)$ and $(T+B)U=U (T+B)$ .", "Thus $(T{\\hat{+}} B)U = U (T{\\hat{+}} B)$ , as well as $(T{\\hat{+}}B)\\eta \\sf A$ .", "For each vector $y\\in {\\cal D}(TB)$ the inclusions follow $y\\in {\\cal D}(B)$ and $By\\in {\\cal D}(T)$ .", "Therefore, $Uy\\in {\\cal D}(B)$ and $BUy = UB y\\in {\\cal D}(T)$ , consequently, $Uy\\in {\\cal D}(TB)$ .", "Then we get $U{\\cal D}(TB)={\\cal D}(TB)$ , since $U^*y\\in {\\cal D}(TB)$ .", "But $(TB)Uy= U(TB)y$ , hence $(T{\\hat{\\cdot }}B)Uy= U(T{\\hat{\\cdot }}B)y$ , and inevitably $(T{\\hat{\\cdot }}B)\\eta \\sf A$ .", "Having a bounding ${\\cal A}_v$ graded sequence $ \\lbrace \\mbox{}_nE: ~ n\\rbrace $ for $T$ and $T^*$ one gets $\\mbox{}_nET^*\\subseteq T^*\\mbox{}_nE$ and $\\mbox{}_nE^*T^*\\subseteq (T\\mbox{ }_nE)^*$ , consequently, $T^*\\mbox{ }_nE$ and $(T\\mbox{ }_nE)^*$ are bounded everywhere defined extensions of operators $ \\mbox{}_nET^*$ and $\\mbox{}_nE^*T^*$ respectively.", "In view of Lemma 21 the equality $(T\\mbox{}_nE)^* = \\mbox{}_nE^*T^*$ follows.", "We now consider a bounding ${\\cal A}_v$ graded sequence $ \\lbrace \\mbox{}_nE: ~ n \\rbrace $ for $T$ , $~T^*$ , $~B$ , $~B^*$ , $ ~ ((bI)T{\\hat{+}}B)$ , $ ~ ((bI)T{\\hat{+}}B)^*$ , $ ~ (T{\\hat{\\cdot }}B)$ , $ ~ (T{\\hat{\\cdot }}B)^*$ and $T^*{\\hat{\\cdot }}B^*$ .", "From the preceding demonstration we get the equalities $ ~ (T^* (b^*I) {\\hat{+}}B^*)\\mbox{ }_nE =(T^* (b^*I)) \\mbox{ }_nE {\\hat{+}} B^*\\mbox{ }_nE = [((bI)T{\\hat{+}}B)\\mbox{ }_nE^*]^*= ((bI)T{\\hat{+}}B)^*\\mbox{ }_nE$ and $(T{\\hat{\\cdot }}B)^*\\mbox{ }_nE = [(T{\\hat{\\cdot }}B)\\mbox{ }_nE^*]^*=(B^* {\\hat{\\cdot }}T^*)\\mbox{ }_nE$ due to $(9,10)$ .", "The operators $((bI)T{\\hat{+}}B)^*$ and $(T^*(b^*I){\\hat{+}}B^*)$ agree on their common core $\\bigcup _{n=1}^{\\infty } \\mbox{ }_nE(X)$ , consequently, $((bI)T{\\hat{+}}B)^* = (T^*(b^*I){\\hat{+}}B^*)$ .", "Analogously we infer that $(T{\\hat{\\cdot }}B)^* = B^*{\\hat{\\cdot }}T^*$ .", "Then $T\\mbox{ }_nE\\subseteq B\\mbox{ }_nE$ and hence $T\\mbox{ }_nE = B\\mbox{ }_nE$ as soon as $T\\subseteq B$ and $ \\lbrace \\mbox{}_nE: ~ n \\rbrace $ is a bounding ${\\cal A}_v$ graded sequence in $\\sf A$ for $T$ and $B$ .", "Therefore, the operators $T$ and $B$ are consistent on their common core $\\bigcup _{n=1}^{\\infty } \\mbox{}_nE(X)$ and hence $T=B$ .", "If $T$ is symmetric, then $T\\subseteq T^*$ and from the preceding conclusion one obtains $T=T^*$ .", "For any three operators $T, B, C \\in {\\cal N}({\\sf A})$ we take a common bounding ${\\cal A}_v$ graded sequence $ \\lbrace \\mbox{}_nE: ~ n \\rbrace $ and get $(T{\\hat{\\cdot }}B){\\hat{\\cdot }}C =T{\\hat{\\cdot }}(B{\\hat{\\cdot }}C)$ , since $[(T{\\hat{\\cdot }}B){\\hat{\\cdot }}C]\\mbox{ }_nE=[T{\\hat{\\cdot }}(B{\\hat{\\cdot }}C)]\\mbox{ }_nE$ for all $n$ (see also [30]).", "From this Statement $(8)$ of the theorem follows.", "24.", "Lemma.", "Suppose that $ \\lbrace \\mbox{}_nF: ~ n \\rbrace $ is a bounding ${\\cal A}_v$ graded sequence for the closed operator $T$ on a Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ , $~2\\le v$ , and $T\\mbox{ }_nF$ is normal for each natural number $n$ .", "Then $T$ is normal.", "Proof.", "In view of Lemma 21 we have the equalities $(T\\mbox{ }_nF)^* = \\mbox{}_nF^*T^*$ and $\\mbox{}_nFT^*= T^* \\mbox{}_nF$ and $T^*T\\mbox{ }_nF =T^*\\mbox{}_nFT\\mbox{ }_nF$ and $(T^* T)\\mbox{ }_nF = (TT^*)\\mbox{ }_nF$ .", "Therefore, the self-adjoint operators $T^*T$ and $TT^*$ agree on their common core $\\bigcup _{n=1}^{\\infty } \\mbox{ }_nF(X)$ , consequently, $T^*T = TT^*$ , i.e.", "the operator $T$ is normal.", "25.", "Remark.", "The condition $T^*T=TT^*$ is equivalent to $\\sum _{j,k} [(T^*)^{i_k}T^{i_j} - T^{i_j}(T^*)^{i_k}]=0$ on ${\\cal D}(T^*T)={\\cal D}(TT^*)$ , since $\\sum _j \\pi ^j=I$ (see §2).", "If $BT\\subseteq TB$ in a Hilbert space $X$ over the Cayley-Dickson algebra, then ${\\bf B}{\\bf T}\\subseteq {\\bf T}{\\bf B}$ for ${\\bf B}={\\bf i} B={{0 ~~ B}{-B ~ 0}}$ and ${\\bf T}={\\bf i}T={{0 ~~ T}{-T ~ 0}}$ defined on ${\\cal D}({\\bf B})={\\cal D}(B)\\oplus {\\cal D}(B){\\bf i}$ and ${\\cal D}({\\bf T}) ={\\cal D}(T)\\oplus {\\cal D}(T){\\bf i}$ with ${\\bf i} ={{0 ~~1}{-1 ~ 0}}$ (see also §18).", "26.", "Lemma.", "Let $BT\\subseteq TB$ and ${\\cal D}(T)\\subseteq {\\cal D}(B)$ , let also $T$ be a self-adjoint operator and let $B$ be a closed operator in a Hilbert space $X$ over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ , $~2\\le v\\le 3$ .", "Then $\\mbox{}_bEB\\subseteq B\\mbox{ }_bE$ for each $\\mbox{}_bE$ in the spectral ${\\cal A}_v$ graded resolution $ \\lbrace \\mbox{}_bE: ~ b \\rbrace $ of $T$ .", "Proof.", "There is the decomposition $alg_{{\\cal A}_v}(I,B,T,{\\bf i}I, {\\bf i}T, {\\bf i}T) = alg_{{\\cal A}_v} (I,B,T)\\oplus alg_{{\\cal A}_v} (I,B,T){\\bf i}$ and this family is defined in the Hilbert space $X\\oplus X{\\bf i}$ .", "From Formula 18$(3)$ we have the inclusion $(BT+B{\\bf i}I)\\subseteq B(T+{\\bf i}I)$ .", "Now we consider these operators with domains in $X\\oplus X{\\bf i}$ denoted by ${\\cal D}(T)$ , ${\\cal D}(B)$ , etc.", "Take an arbitrary vector $x\\in {\\cal D}(B(T+{\\bf i}I))$ , then $x\\in {\\cal D}(T)$ and $Tx+{\\bf i}x\\in {\\cal D}(B)$ .", "But from the suppositions of this lemma we have the inclusion ${\\cal D}(T)\\subseteq {\\cal D}(B)$ , consequently, $Tx\\in {\\cal D}(B)$ and hence $x\\in {\\cal D}(BT+B({\\bf i}I))$ and $B(T+{\\bf i}I)x=BTx+B({\\bf i}x)$ .", "Thus $B(T+{\\bf i}I)\\subseteq BT+B({\\bf i}I)$ , consequently, $B(T+{\\bf i}I)=BT+B({\\bf i}I)$ .", "Denote by $Q_-$ and $Q_+$ the bounded everywhere defined inverses to $(T-{\\bf i}I)$ and $(T+{\\bf i}I)$ respectively.", "In view of 18$(1-3)$ and the preceding proof we infer $Q_{\\pm }B=Q_{\\pm }B(T\\pm {\\bf i}I)Q_{\\pm }=Q_{\\pm }(BT\\pm B({\\bf i}I))Q_{\\pm }\\subseteq Q_{\\pm }(TB\\pm B({\\bf i}I))Q_{\\pm }=Q_{\\pm }(T\\pm {\\bf i}I)BQ_{\\pm }$ and hence $(Q_{\\pm })B\\subseteq B(Q_{\\pm })$ .", "From §10 one has $Q_+ = (Q_-)^*$ and applying Lemma 19 one gets $QB\\subseteq BQ$ for each element $Q$ in the von Neumann algebra over ${\\cal A}_v$ generated by $Q_+$ and $Q_-$ .", "To an ${\\cal A}_v$ graded projection operator $\\mbox{}_bE$ on $X$ an operator $\\mbox{}_bE\\oplus \\mbox{}_bE$ on $X\\oplus X{\\bf i}$ corresponds.", "Particularly, this means the inclusion $\\mbox{}_bEB\\subseteq B\\mbox{}_bE$ in $X$ for each $b$ .", "27.", "Theorem.", "Suppose that $T$ is an operator on a Hilbert space $X$ over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ , $~2\\le v\\le 3$ .", "An operator $T$ is normal if and only if it is affiliated with a quasi-commutative von Neumann algebra $\\sf A$ over ${\\cal A}_v$ .", "Moreover, there exists the smallest such algebra $\\mbox{}_0{\\sf A}$ .", "Proof.", "In view of Theorem 23 if an operator is affiliated with a quasi-commutative von Neumann algebra $\\sf A$ over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ with $2\\le v \\le 3$ , then it is normal.", "Suppose that an operator $T$ is normal.", "Then Lemma 26 is applicable, since $TT^*T=T^*TT$ and ${\\cal D}(T^*T)\\subseteq {\\cal D}(T)$ .", "For an ${\\cal A}_v$ graded spectral resolution $\\lbrace \\mbox{}_bE: ~ b\\in {\\bf R} \\rbrace $ this implies that $\\mbox{}_bET\\subseteq T\\mbox{ }_bE$ for each $b$ , hence $\\mbox{}_nFT\\subseteq T\\mbox{ }_nF$ for each natural number $n$ , where $\\mbox{}_nF = \\mbox{}_nE - \\mbox{ }_{-n}E$ .", "Then we also have $T^*T^*T = T^*TT^*$ and ${\\cal D}(T^*T)={\\cal D}(TT^*)\\subseteq {\\cal D}(T^*)$ , consequently, $\\mbox{}_nFT^*\\subseteq T^*\\mbox{ }_nF$ .", "From §23 it follows that the operators $T\\mbox{ }_nF$ and $\\mbox{}_nFT$ are bounded, since such the operator $T^*T\\mbox{ }_nF = TT^*\\mbox{ }_nF$ is.", "But then $\\mbox{}_nF^*T^*\\subseteq (T\\mbox{}_nF)^*$ and hence both operators $(T\\mbox{ }_nF)^*$ and $T^*\\mbox{ }_nF$ are bounded extensions of the densely defined operator $\\mbox{ }_nFT^*$ when choosing $\\mbox{}_bE$ and hence $\\mbox{}_nF$ self-adjoint for each $b$ and $n$ .", "Thus one gets the equalities $(T\\mbox{ }_nF)^* = T^*\\mbox{}_nF$ and $(T^*\\mbox{ }_nF)^* = T\\mbox{ }_nF$ .", "On the other hand, there are the inclusions $T\\mbox{ }_nFT\\mbox{ }_mF \\subseteq TT\\mbox{ }_nF$ and $T\\mbox{ }_mFT\\mbox{ }_nF\\subseteq TT\\mbox{ }_nF$ for each $n\\le m$ .", "The operators $T\\mbox{ }_nFT\\mbox{ }_mF$ and $T\\mbox{ }_mFT\\mbox{ }_nF$ are everywhere defined, consequently, $T\\mbox{ }_nFT\\mbox{ }_mF = TT\\mbox{ }_nF= T\\mbox{ }_mFT\\mbox{}_nF.$ There are the equalities $T^* \\mbox{ }_mF T \\mbox{ }_nF=T^*T\\mbox{ }_nF=TT^*\\mbox{ }_nF=T\\mbox{ }_nFT^*\\mbox{ }_mF$ , hence $\\mbox{}_0{\\sf A} := cl [ alg_{{\\cal A}_v} \\lbrace \\mbox{}_nF, ~ T\\mbox{}_nF, ~ T^*\\mbox{ }_nF: ~ n\\in {\\bf N} \\rbrace ]$ is a quasi-commutative von Neumann algebra over the algebra ${\\cal A}_v$ , since there is the inclusion $alg_{{\\cal A}_v} \\lbrace \\mbox{}_nF, ~ T\\mbox{ }_nF, ~T^*\\mbox{ }_nF: ~ n\\in {\\bf N} \\rbrace \\subset L_q(X)$ .", "Moreover, one has that $\\bigcup _{n=1}^{\\infty } \\mbox{ }_nF(X)=:\\cal Y$ is a core for $T$ , since $\\vee _{n=1}^{\\infty } \\mbox{}_nF=I$ and $\\mbox{}_nFT\\subseteq T\\mbox{ }_nF$ .", "If $U$ is a unitary operator in $(\\mbox{}_0{\\sf A})^{\\star }$ and $x\\in \\cal Y$ , then the equalities $TUx=TU\\mbox{ }_nFx=T\\mbox{ }_nFUx = UT\\mbox{ }_nFx =UTx$ are fulfilled for some natural number $n$ .", "In accordance with Remark 2 $T\\eta \\mbox{ }_0{\\sf A}$ and $T^*\\eta \\mbox{ }_0{\\sf A}$ also.", "If $T\\eta {\\cal R}$ , then $T^*\\eta {\\cal R}$ and $T^*T\\eta {\\cal R}$ are affiliations as well.", "From Note 15 it follows that the self-adjoint operator $T^*T$ generates a quasi-commutative von Neumann algebra $\\sf A$ contained in $\\cal R$ , consequently, $\\mbox{}_nF\\in \\cal R$ and hence $T\\mbox{ }_nF, ~ T^*\\mbox{ }_nF\\in \\cal R$ .", "This implies the inclusion $\\mbox{}_0{\\sf A}\\subset \\cal R$ .", "28.", "Definition.", "The algebra $\\mbox{}_0\\sf A$ from the preceding section will be called the von Neumann algebra over the Cayley-Dickson algebra ${\\cal A}_v$ with $2\\le v$ generated by the normal operator $T$ .", "29.", "Theorem.", "Suppose that $\\sf A$ is a quasi-commutative von Neumann algebra over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ , $~2\\le v\\le 3$ , also $\\phi $ is an isomorphism of $\\sf A$ onto $C(\\Lambda ,{\\cal A}_v)$ , where $\\Lambda $ is a compact Hausdorff topological space, $T\\eta \\sf A$ .", "Then there exists a unique normal function $\\phi (T)$ on $\\Lambda $ so that $\\phi (TE)=\\phi (T){\\hat{\\cdot }}\\phi (E)$ , when $E$ is an ${\\cal A}_v$ graded bounding projection in $\\sf A$ for $T$ , where $(\\phi (T){\\hat{\\cdot }} \\phi (E))(z)=\\phi (T)(z) \\phi (E)(z)$ , if $\\phi (T)(z)$ is defined and zero in the contrary case, $z\\in \\Lambda $ .", "If ${\\cal N}(\\Lambda ,{\\cal A}_v)$ is the family of all ${\\cal A}_v$ valued normal function on $\\Lambda $ and $f, g \\in {\\cal N}(\\Lambda ,{\\cal A}_v)$ , then there are unique normal functions ${\\tilde{f}}$ , $~sf$ , $~fs$ , $~f{\\hat{+}}g$ and $f{\\hat{\\cdot }}g$ so that ${\\tilde{f}}(z)=\\widetilde{f(z)}$ , $~(sf)(z)=sf(z)$ , $~(fs)(z)=f(z)s$ , $~(f{\\hat{+}}g) (z) = f(z)+g(z)$ and $(f{\\hat{\\cdot }}g)(z) = f(z){\\hat{\\cdot }}g(z)$ , when $f$ and $g$ are defined at $z\\in \\Lambda $ , $~s\\in {\\cal A}_v$ .", "Endowed with the operations $f\\mapsto {\\tilde{f}}$ and $(s,f)\\mapsto sf$ and $(f,s)\\mapsto fs$ and $(f,g)\\mapsto f{\\hat{+}}g$ and $(f,g)\\mapsto f{\\hat{\\cdot }}g$ , the family ${\\cal N}(\\Lambda ,{\\cal A}_v)$ is a quasi-commutative algebra with unit 1 and involution $f\\mapsto \\tilde{f}$ , it is associative over the quaternion skew field ${\\bf H}={\\cal A}_2$ and alternative over the octonion algebra ${\\bf O}={\\cal A}_3$ .", "The natural extension of $\\phi $ is a $*$ -isomorphism of ${\\cal N}({\\sf A})$ onto ${\\cal N}(\\Lambda ,{\\cal A}_v)$ .", "Proof.", "In view of Theorem 23 an operator $T$ affiliated with $\\sf A$ has an ${\\cal A}_v$ graded bounding sequence $ \\lbrace \\mbox{}_nE: ~ n \\rbrace $ in $\\sf A$ .", "If $\\phi $ has the properties described above, if also $f$ and $g$ are normal functions defined at a point $z$ and corresponding to $\\phi (T)$ , then $f(z)\\phi (\\mbox{}_nE)(z) = \\phi (T\\mbox{ }_nE)(z) =g(z) \\phi (\\mbox{}_nE)(z)$ for every natural number $n$ .", "Thus if $\\phi (\\mbox{}_nE)(z)=1$ for some natural number $n$ , then certainly $f(z)=g(z)$ .", "Applying Lemma 9 we obtain, that $f=g$ , since the sequence $\\mbox{}_nE$ is monotone increasing to $I$ , while $f$ and $g$ agree on a dense subset of $\\Lambda $ .", "Thus an isomorphism $\\phi $ is unique.", "On the other hand, Theorem 16 provides $\\phi (T)$ with the required properties for each $T\\in {\\cal Q}({\\sf A})$ , where ${\\cal Q}({\\sf A})$ denotes a family of all self-adjoint operators affiliated with $\\sf A$ .", "This means that $\\phi $ is defined on ${\\cal Q}({\\sf A})$ .", "For any pair of operators $T$ , $B$ in ${\\cal Q}({\\sf A})$ it is possible to choose a bounding sequence $\\lbrace \\mbox{}_nE: ~ n \\rbrace $ for both $T$ and $B$ by Theorem 23.", "Therefore, the operators $T\\mbox{ }_nE$ , $ ~ B\\mbox{ }_nE$ , $~(T+B)\\mbox{ }_nE$ and $TB\\mbox{ }_nE$ belong to the quasi-commutative von Neumann algebra $\\sf A$ over the Cayley-Dickson algebra ${\\cal A}_v$ so that $(T{\\hat{+}}B)\\mbox{ }_nE= T\\mbox{ }_nE + B\\mbox{ }_nE$ and $(T{\\hat{\\cdot }}B)\\mbox{ }_nE = TB\\mbox{ }_nE.$ Therefore, we deduce the equalities $\\phi (T{\\hat{+}}B)(z)\\phi (\\mbox{}_nE)(z) =\\phi ((T{\\hat{+}}B)\\mbox{}_nE)(z) = \\phi (T)(z) \\phi (\\mbox{}_nE)(z)+ \\phi (B)(z) \\phi (\\mbox{}_nE)(z)$ and $\\phi (T{\\hat{\\cdot }}B)(z)\\phi (\\mbox{}_nE)(z) =\\phi ((T{\\hat{\\cdot }}B)\\mbox{}_nE)(z) = \\phi (T)(z)\\phi (B)(z) \\phi (\\mbox{}_nE)(z)$ , when $\\phi (T)$ , $~\\phi (B)$ , $~ \\phi (T{\\hat{+}}B)$ and $\\phi (T{\\hat{\\cdot }} B)$ are defined at $z$ .", "The pairs $[\\phi (T{\\hat{+}}B); \\phi (T)+\\phi (B)]$ and $[\\phi (T{\\hat{\\cdot }} B); \\phi (T)\\phi (B)]$ agree on dense subsets of $\\Lambda $ .", "Therefore, images $\\phi (T{\\hat{+}}B)$ and $\\phi (T{\\hat{\\cdot }}B)$ are finite, when $\\phi (T)$ and $\\phi (B)$ are defined, hence $\\phi (T{\\hat{+}}B)$ and $\\phi (T{\\hat{\\cdot }}B)$ are normal extensions of $\\phi (T)+\\phi (B)$ and $\\phi (T)\\phi (B)$ correspondingly.", "Then $\\phi (T){\\hat{+}} \\phi (B)$ and $\\phi (T){\\hat{\\cdot }} \\phi (B)$ are defined as normal extensions of $\\phi (T)+\\phi (B)$ and $\\phi (T)\\phi (B)$ respectively.", "Each function $w\\in {\\cal Q}(\\Lambda )$ corresponds to some operator $T\\in {\\cal Q}({\\sf A})$ and by Theorem 16 the operations ${\\hat{+}}$ and $\\hat{\\cdot }$ are applicable to all functions in ${\\cal Q}(\\Lambda )$ .", "An iterated application of Lemma 9 shows that ${\\cal Q}(\\Lambda )$ endowed with these operations is a commutative algebra over the real field $\\bf R$ with unit 1, since each function in ${\\cal Q}(\\Lambda )$ is real valued.", "Moreover, $\\phi $ is an isomorphism of ${\\cal Q}({\\sf A})$ onto ${\\cal Q}(\\Lambda )$ .", "If $v$ is finite, for each $j=0,...,2^v-1$ the $\\bf R$ -linear projection operator $\\hat{\\pi }^j := \\hat{\\pi }^j_v: {\\sf A}\\rightarrow {\\sf A}_ji_j$ is expressible as a sum of products with generators and real constants due to Formulas $(1,2)$ below so that $\\hat{\\pi }^j(A)=i_jA_j=A_ji_j$ : $(1)$ $\\hat{\\pi }^j(A) = (- i_j (Ai_j) - (2^v-2)^{-1} \\lbrace - A+\\sum _{k=1}^{2^v-1}i_k(Ai_k^*) \\rbrace )/2$ for each $j=1,2,...,2^v-1$ , $(2)\\quad \\hat{\\pi }^0(A) = (A+(2^v-2)^{-1} \\lbrace -A + \\sum _{k=1}^{2^v-1}i_k(Ai_k^*) \\rbrace )/2,$ where $2\\le v\\in \\bf N$ , $(3)$ $ A=\\sum _j A_ji_j$ , $ ~ A_j\\in {\\sf A}_j$ for each $j$ , $~A\\in \\sf A$ .", "If $\\sf A$ is embedded into $L_q(X)$ for a Hilbert space $X$ over the algebra ${\\cal A}_v$ , we can consider projections $\\pi ^j :{\\sf A}\\rightarrow {\\sf A}_ji_j$ and get decomposition $(3)$ so that $A_j\\in \\sf A$ for each $j$ .", "The function $\\sum _j \\phi (A_j)i_j$ is defined on $\\Lambda \\setminus \\bigcup _j W_j$ when $\\phi (A_j)$ is defined on $X\\setminus W_j$ .", "But $\\Lambda \\setminus \\bigcup _j W_j$ is everywhere dense in $\\Lambda $ , since $|z|=\\sqrt{\\sum _j z_j^2}$ is the norm on ${\\cal A}_v$ .", "If $p\\in W_j$ , then $\\lim _{x\\in \\Lambda \\setminus \\bigcup _j W_j, ~ x\\rightarrow p } |\\phi (A)(x)| =\\sqrt{\\sum _j \\phi (A_j)^2(x)}=\\infty .$ That is $\\sum _j \\phi (A_j)i_j$ is an element ${\\hat{\\sum }}_j \\phi (A_j) i_j= \\phi (A)\\in {\\cal N}(\\Lambda ,{\\cal A}_v)$ .", "In accordance with Theorem 6.2.26 [6] a topological space $\\Lambda $ is extremely disconnected if and only if for each pair of nonintersecting open subsets $U$ and $V$ in $\\Lambda $ their closures always $cl (U)\\cap cl (V)=\\emptyset $ do not intersect.", "By Theorem 8.3.10 [6] if $(S,{\\cal U})$ is a uniform space and $(Y,{\\cal V})$ is a complete uniform space, then each uniformly continuous mapping $f: (P,{\\cal U}_P)\\rightarrow (Y,{\\cal V})$ , where $P$ is an everywhere dense subset in $S$ relative to a topology induced by a uniformity $\\cal U$ , has a uniformly continuous extension $f:(S,{\\cal U})\\rightarrow (Y,{\\cal V})$ .", "A set $\\xi ^{-1}(0)$ is closed in $\\Lambda $ , when $\\xi \\in {\\cal N}(\\Lambda ,{\\cal A}_v)$ .", "Indeed, a function $\\xi $ is continuous on $\\Lambda \\setminus W_{\\xi }$ , consequently, $\\xi ^{-1}(0)$ is closed in $\\Lambda \\setminus W_{\\xi }$ .", "At the same time the limit $\\lim _{x\\rightarrow p, ~ x\\in \\Lambda \\setminus W_{\\xi }}|\\xi (x)| = \\infty $ is infinite for each $p\\in W_{\\xi }$ , hence $p\\notin cl [\\xi ^{-1}(0)]$ .", "This fact implies that the interior $K:= Int [\\xi ^{-1}(0)]$ is clopen in $\\Lambda $ .", "Thus the function $g$ defined such that $g(x)=1$ on $K$ , while $g(x) := \\xi (x)/|\\xi (x)|$ on $\\Lambda \\setminus (\\xi ^{-1}(0)\\cup W_{\\xi })$ , is continuous on $[\\Lambda \\setminus (\\xi ^{-1}(0)\\cup W_{\\xi })]\\cup K$ .", "This function $g$ has a continuous extension $q\\in C(\\Lambda ,{\\cal A}_v)$ by Theorems 6.2.26 and 8.3.10 [6], since $\\Lambda \\setminus (\\lbrace \\xi ^{-1}(0)\\setminus K \\rbrace \\cup W_{\\xi })$ is a dense open subset in $\\Lambda $ .", "Indeed, Cantor's cube $D^{\\sf m}$ is universal for all zero-dimensional spaces of topological weight ${\\sf m}\\ge \\aleph _0$ , where $D$ is a discrete two-element space (see Theorem 6.2.16 [6]).", "If ${\\sf m}<\\aleph _0$ , then $\\Lambda $ is discrete and this case is trivial.", "Therefore, if ${\\sf m}\\ge \\aleph _0$ , the compact topological space $\\Lambda $ has an embedding into $D^{\\sf m}$ as a closed subset.", "Then a uniformity $\\cal U$ compatible with its topology can be chosen non-archimedean.", "That is, each pseudo-metric $\\rho $ in a family $\\bf P$ of all pseudo-metrics inducing a uniformity $\\cal U$ on $\\Lambda $ satisfies the inequality $\\rho (x,y) \\le \\max (\\rho (x,z); ~ \\rho (z,y))$ for each $x, y, z\\in \\Lambda $ .", "The function $g(x)$ has values in the unit sphere in ${\\cal A}_v$ , which is closed in the Cayley-Dickson algebra ${\\cal A}_v$ and hence is complete.", "If $z\\in (\\xi ^{-1}(0)\\cup W_{\\xi })\\setminus K$ and $\\lbrace x_{\\alpha }: \\alpha \\in \\Sigma \\rbrace \\subset [\\Lambda \\setminus (\\xi ^{-1}(0)\\cup W_{\\xi })]\\cup K$ is a Cauchy net converging to $z$ , then $\\lbrace x_{\\alpha }: \\alpha \\in \\Sigma \\rbrace $ can be chosen such that the limit $\\lim _{\\alpha } g(x_{\\alpha })$ exists, where $\\Sigma $ is a directed set.", "For any other Cauchy net $ \\lbrace y_{\\alpha } : \\alpha \\in \\Sigma \\rbrace \\subset [\\Lambda \\setminus (\\xi ^{-1}(0)\\cup W_{\\xi })]\\cup K$ converging to $z$ the limit of the net $ \\lbrace g(y_{\\alpha }) : \\alpha \\in \\Sigma \\rbrace $ exists and will be the same for $ \\lbrace g(x_{\\alpha }) : \\alpha \\in \\Sigma \\rbrace $ , since each pseudo-metric $\\rho \\in \\bf P$ is non-archimedean and the function $g$ is continuous on $[\\Lambda \\setminus (\\xi ^{-1}(0)\\cup W_{\\xi })]\\cup K$ (see also Propositions 1.6.6 and 1.6.7 and Theorem 8.3.20 [6]).", "For a point $x\\in \\Lambda \\setminus W_{\\xi }$ the equality $q(x)|\\xi (x)|=\\xi (x)$ is fulfilled.", "Since $|\\xi |\\in {\\cal Q}(\\Lambda )$ , one gets also normal extensions $q_j{\\hat{\\cdot }} |\\xi |$ for $\\xi _j$ defined on $\\Lambda \\setminus W_{\\xi }$ .", "Thus the functions $\\xi , \\xi _j$ are in ${\\cal N}(\\Lambda ,{\\cal A}_v)$ for each $j$ .", "Choosing $A_j\\in {\\cal Q}({\\sf A})$ as $\\phi (A_j)=\\xi _j$ leads to the equalities $\\phi ({\\hat{\\sum }}_j A_ji_j) = {\\hat{\\sum }}_j \\xi _ji_j=\\xi $ .", "Thus this function $\\phi $ maps ${\\cal N}({\\sf A})$ onto ${\\cal N}(\\Lambda ,{\\cal A}_v)$ .", "As soon as functions $f, g \\in {\\cal N}(\\Lambda ,{\\cal A}_v)$ are defined on $\\Lambda \\setminus W_f$ and $\\Lambda \\setminus W_g$ respectively, then components $f_j$ and $g_j$ have normal extensions for each $j$ as follows from the proof above.", "Therefore, their sum $f+g$ and product $fg$ defined on $\\Lambda \\setminus (W_f\\cup W_g)$ have the normal extensions ${\\hat{\\sum }}_j (f_j {\\hat{+}} g_j) i_j =f{\\hat{+}}g$ and ${\\hat{\\sum }}_{j,k} (f_j {\\hat{\\cdot }} g_k) i_ji_k= f{\\hat{\\cdot }}g$ .", "The set ${\\cal N}(\\Lambda ,{\\cal A}_v)$ supplied with these operations is the algebra over the Cayley-Dickson algebra ${\\cal A}_v$ with $2\\le v$ .", "Moreover, the algebra ${\\cal N}(\\Lambda ,{\\cal A}_v)$ is quasi-commutative by its construction and with unit 1 and adjoint operation $f\\mapsto {\\tilde{f}}$ .", "This algebra ${\\cal N}(\\Lambda ,{\\cal A}_v)$ is associative over the quaternion skew field ${\\bf H}={\\cal A}_2$ and alternative over the octonion algebra ${\\bf O}={\\cal A}_3$ , since the multiplication of functions is defined point-wise, while the quaternion skew field ${\\bf H}$ is associative and the octonion algebra ${\\bf O}$ is alternative.", "Therefore, the mapping $\\phi $ has an extension up to a $*$ -isomorphism of ${\\cal N}({\\sf A})$ onto ${\\cal N}(\\Lambda ,{\\cal A}_v)$ .", "For $T\\eta \\sf A$ and a bounding ${\\cal A}_v$ graded projection $E$ in $\\sf A$ for $T$ we get $\\phi (TE) = \\phi (T ~ {\\hat{\\cdot }} ~ E) = \\phi (T) ~ {\\hat{\\cdot }} ~\\phi (E)$ .", "30.", "Definition.", "The spectrum $sp (T)$ of a closed densely defined operator $T$ on a Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ with $2\\le v$ is the set of all those Cayley-Dickson numbers $z\\in {\\cal A}_v$ for which an operator $(T-zI)$ is not a bijective $\\bf R$ linear ${\\cal A}_v$ additive mapping of ${\\cal D}(T)$ onto $X$ , where ${\\cal D}(T)$ as usually denotes a ${\\cal A}_v$ vector domain of definition of $T$ .", "31.", "Remark.", "For an operator $T$ from Definition 30 if $z\\notin sp (T)$ , then $(T-zI)$ is bijective from ${\\cal D}(T)$ onto $X$ and has an $\\bf R$ linear ${\\cal A}_v$ additive inverse $B=(T-zI)^{-1}: X\\rightarrow {\\cal D}(T)$ .", "The graph of $B$ is closed, since it is such for $(T - zI)$ .", "In accordance with the closed graph theorem 1.8.6 [12] this inverse operator $B$ is bounded.", "Thus we get $z\\notin sp (T)\\Leftrightarrow (T-zI)$ has a bounded inverse from $X$ onto ${\\cal D}(T)$ .", "If $T\\eta \\sf A$ for some quasi-commutative von Neumann subalgebra in $L_q(X)$ and $z\\notin sp(T)$ , then $B\\in \\sf A$ .", "From the boundedness of $B$ and closedness of $(T-zI)$ it follows that $I=(T-zI)B= (T-zI)~ {\\hat{\\cdot }} ~B = B ~ {\\hat{\\cdot }} ~(T-zI)$ .", "Therefore, $z\\notin sp (T)$ is equivalent to: $(T-zI)$ has an inverse $B$ in the algebra ${\\cal N}(\\Lambda ,{\\cal A}_v)$ and $B\\in \\sf A$ .", "32.", "Proposition.", "Let $T$ be a normal operator affiliated with a quasi-commutative von Neumann algebra $\\sf A$ over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ with $2\\le v\\le 3$ .", "Then $sp (T)$ coincides with the range of $\\phi (T)$ , where $\\phi $ denotes the isomorphism of ${\\cal N}({\\sf A})$ onto ${\\cal N}(\\Lambda ,{\\cal A}_v)$ extending the isomorphism of $\\sf A$ with $C(\\Lambda ,{\\cal A}_v)$ (see §29).", "Proof.", "In accordance with Definition 30 $z\\notin sp (T)$ if and only if there is a bounded $B$ inverse to $(T-zI)$ .", "For a unitary operator $U$ in ${\\sf A}^{\\star }$ one has the equality $U^*(T-zI)U=(T-zI)$ , since $(T-zI)\\eta \\sf A$ .", "Therefore, $U^*BU=B$ for each such unitary operator $U$ and $B\\in \\sf A$ .", "But an operator $B\\in \\sf A$ exists so that $(T-zI) \\mbox{ }^{ \\hat{.}}", "\\mbox{ }B=I$ .", "Thus the equality $\\phi (T-zI) \\mbox{ }^{ \\hat{.}}", "\\mbox{ }\\phi (B) = 1$ is satisfied if and only if $z\\notin sp (T)$ .", "This means that there exists such $\\phi (B)\\in C(\\Lambda ,{\\cal A}_v)$ if and only if a Cayley-Dickson number $z$ does not belong to the range of $\\phi (T)$ , hence $sp (T)$ is the range of $\\phi (T)$ .", "33.", "Definition.", "A self-adjoint operator $T$ is called positive when $<Tx;x> ~ \\ge 0$ for each vector $x\\in {\\cal D}(T)$ in its domain.", "34.", "Note.", "A function $f-z1$ for $f\\in {\\cal N}(\\Lambda ,{\\cal A}_v)$ and $z\\in {\\cal A}_v$ has not an inverse in ${\\cal N}(\\Lambda ,{\\cal A}_v)$ if and only if $f-z1$ vanishes on some non-void clopen subset in $\\Lambda $ , where $2\\le v \\le 3$ .", "Considering this in ${\\cal N}({\\sf A})$ we get a non-zero ${\\cal A}_v$ graded projection operator $F\\in \\sf A$ so that $(T-zI) ~{\\hat{\\cdot }} ~ F =0$ .", "The latter is equivalent to the existence of a non-zero vector $x$ on which $(T-zI)x=0$ .", "In this situation one says that $z$ is in the point spectrum of $T$ .", "Thus the spectrum of $T$ relative to ${\\cal N}({\\sf A})$ is its point spectrum.", "35.", "Proposition.", "Let $T$ be a self-adjoint operator in a Hilbert space $X$ over either the quaternion skew field or the octonion algebra.", "Then $T$ is positive if and only if $z\\ge 0$ for each $z\\in sp (T)$ .", "Proof.", "In view of Lemma 10 it is sufficient to consider the variant, when $T$ is affiliated with a quasi-commutative von Neumann algebra $\\sf A$ .", "There exists an isomorphism $\\phi $ of ${\\cal N}({\\sf A})$ onto ${\\cal N}(\\Lambda ,{\\cal A}_v)$ extending an isomorphism of $\\sf A$ onto $C(\\Lambda ,{\\cal A}_v)$ (see Theorem 29), where $\\Lambda $ is an extremely disconnected compact Hausdorff topological space.", "Whenever $\\phi (T)$ is defined on $\\Lambda \\setminus W_{\\phi (T)}$ and $\\phi (T)(y)<0$ for some point $y\\in \\Lambda \\setminus W_{\\phi (T)}$ , there exists a non-void clopen subset $P$ containing $y$ and a negative constant $b<0$ so that $P\\subset \\Lambda \\setminus W_{\\phi (T)}$ and $\\phi (T)(x)<b<0$ for each $x\\in P$ .", "If $F$ is an ${\\cal A}_v$ graded projection in $\\sf A$ corresponding to the characteristic function $\\chi _P$ of $P$ , then $F$ is a non-zero ${\\cal A}_v$ graded bounding projection for the operator $T$ so that $TF\\le b F$ .", "For each unit vector $z\\in {\\cal R}(F)$ in the range of $F$ , the inequality $<Tz;z> ~ \\le b< 0$ would be fulfilled, hence $T$ would be not positive.", "This implies the inequality $sp (T)\\ge 0$ if $T\\ge 0$ .", "As $\\phi (T)$ has range consisting of non-negative real numbers, its positive square root $g$ is a normal self-adjoint mapping on $\\Lambda $ .", "For any $\\phi (B)=g\\in {\\cal Q}(\\Lambda )$ the equality $B^2=B ~ {\\hat{\\cdot }} ~ B =T$ is valid (see Theorem 23).", "Therefore, for each unit vector $z\\in {\\cal D}(T)$ , one gets $z\\in {\\cal D}(B)$ and $<Tz;z> = <Bz;Bz>\\ge 0$ , hence $T\\ge 0$ whenever $sp (T)\\ge 0$ .", "36.", "Note.", "The set of positive elements in ${\\cal N}({\\sf A})$ forms a positive cone.", "Therefore, ${\\cal Q}({\\sf A})$ is a partially ordered real vector space relative to the partial ordering induced by this cone.", "But the unit operator $I$ is not an order unit for ${\\cal Q}({\\sf A})$ in the considered case." ], [ "Normal operators and homomorphisms", "37.", "Definition.", "A subset $P$ of a topological space $W$ is called nowhere dense in $W$ if its closure has empty interior.", "A subset $B$ in $W$ is called meager or of the first category if it is a countable union $B=\\bigcup _{j=1}^{\\infty } P_j$ of nowhere dense subsets $P_j$ in $W$ .", "There is said that the mapping ${\\sf B}(\\Phi ,{\\bf R})\\ni f\\mapsto f(T)\\in L_q(X)$ with the monotone sequential convergence property is $\\sigma $ -normal.", "38.", "Lemma.", "Suppose that $\\Lambda $ is an extremely disconnected compact Hausdorff topological space.", "Then each Borel subset of $\\Lambda $ differs from a unique clopen subset by a meager set.", "Each bounded Borel function $g$ from $\\Lambda $ into the Cayley-Dickson algebra ${\\cal A}_v$ with $card (v)\\le \\aleph _0$ differs from a unique continuous function $f$ on a meager set.", "There exists a conjugation-preserving $\\sigma $ -normal homomorphism $\\theta $ from the algebra ${\\cal B}(\\Lambda ,{\\cal A}_v)$ of bounded Borel functions onto $C(\\Lambda ,{\\cal A}_v)$ .", "Its kernel $ker (\\theta )$ consists of all bounded Borel functions vanishing outside a meager set.", "Proof.", "As $card (v)\\le \\aleph _0$ the Cayley-Dickson algebra is separable and of countable topological weight as the topological space relative to its norm topology, since $card (\\bigcup _{n\\in \\bf N} \\aleph _0^n) = \\aleph _0$ .", "Consider the family $\\cal F$ of all subsets contained in $\\Lambda $ which differ from a clopen subset by a meager set.", "Take an arbitrary $V\\in \\cal F$ and a clopen subset $Q$ so that $(V\\setminus Q)\\cup (Q\\setminus V)=: P$ is meager, i.e.", "$\\Lambda \\setminus V$ and $\\Lambda \\setminus Q$ differ by this same meager set.", "Since $\\Lambda \\setminus Q$ is clopen, one gets $(\\Lambda \\setminus V)\\in \\cal F$ .", "In addition each open set $U$ in $\\Lambda $ belongs to $\\cal F$ , since $cl (U)$ is clopen and $cl (U)\\setminus U$ is nowhere dense in $\\Lambda $ .", "For a sequence $\\lbrace V_j, Q_j, P_j: ~ j= 0, 1, 2,... \\rbrace $ of such sets the inclusion is valid: $ [(\\bigcup _{j=1}^{\\infty } V_j)\\setminus (\\bigcup _{j=1}^{\\infty }Q_j) ] \\cup [(\\bigcup _{j=1}^{\\infty } Q_j)\\setminus (\\bigcup _{j=1}^{\\infty } V_j) ] \\subseteq \\bigcup _{j=1}^{\\infty }P_j .$ The set $\\bigcup _{j=1}^{\\infty } P_j$ is meager and the set $\\bigcup _{j=1}^{\\infty } Q_j$ is open, hence $(\\bigcup _{j=1}^{\\infty } V_j)\\in \\cal F$ .", "Thus the family $\\cal F$ contains the $\\sigma $ -algebra generated by open subsets, consequently, ${\\cal F}$ contains the Borel $\\sigma $ -algebra ${\\cal B}(\\Lambda )$ of all Borel subsets in $\\Lambda $ .", "Theorem 3.9.3 [6] tells that the union $\\bigcup _{j=1}^{\\infty } K_j$ of a sequence of nowhere dense subsets $K_j$ in a $\\check{C}$ hech-complete topological space $W$ is a co-dense subset, i.e.", "its complement set $W\\setminus \\bigcup _{j=1}^{\\infty } K_j$ is everywhere dense in $W$ .", "On the other hand, each topological space metrizable by a complete metric is $\\check{C}$ hech-complete, also each locally compact Hausdorff space is $\\check{C}$ hech-complete [6].", "This implies that the complement of a meager set is dense in $\\Lambda $ .", "Therefore, two continuous functions agree on the complement of a meager set only if they are equal.", "This means that there exists at most one continuous function agreeing with a given bounded Borel function on the complement of a meager set.", "Let now $V$ be a Borel subset of $\\Lambda $ , let also $g=\\chi _V$ be the characteristic function of $V$ .", "Take a clopen subset $Q$ in $\\Lambda $ so that $(Q \\setminus V)\\cup (V\\setminus Q) =: P$ is a meager subset in $\\Lambda $ .", "The characteristic function $f=\\chi _Q$ of $Q$ is continuous and $(g-f)$ is zero on $\\Lambda \\setminus P$ .", "Therefore, there exists at most one clopen subset in $\\Lambda $ differing from $V$ by a meager set.", "Therefore, a step function being a finite ${\\cal A}_v$ vector combination of characteristic functions of disjoint Borel subsets in $\\Lambda $ differs from a continuous function on the complement of a meager set.", "The set of step functions is dense in the ${\\cal A}_v$ vector space ${\\cal B}(\\Lambda ,{\\cal A}_v)$ relative to the supremum-norm.", "This means that if $g$ is a Borel function $g: \\Lambda \\rightarrow {\\cal A}_v$ , $~ \\Vert g\\Vert := \\sup _{x\\in \\Lambda } |g(x)|<\\infty $ , then there exists a sequence of Borel step functions $\\mbox{}_ng$ so that $\\lim _{n\\rightarrow \\infty } \\Vert \\mbox{}_ng - g \\Vert =0$ .", "Let $\\mbox{}_nf$ be a sequence of continuous functions agreeing with $\\mbox{}_ng$ on the complement of a meager set $P_n$ .", "Therefore, the inequality $\\Vert \\mbox{}_nf -\\mbox{ }_mf \\Vert \\le \\Vert \\mbox{}_ng - \\mbox{ }_mg \\Vert $ if fulfilled, since $\\mbox{}_nf - \\mbox{ }_mf$ and $\\mbox{}_ng - \\mbox{ }_mg$ agree on the complement of a meager set $P_n\\cup P_m$ .", "The set $P_n\\cup P_m$ is meager and $| \\mbox{}_nf(x) - \\mbox{ }_mf (x) | \\le \\Vert \\mbox{}_ng - \\mbox{ }_mg \\Vert $ for each $x\\in \\Lambda \\setminus (P_n\\cup P_m)$ .", "Therefore, $ \\lbrace \\mbox{}_nf : n \\rbrace $ is a Cauchy sequence converging in supremum norm to a continuous function $f\\in C(\\Lambda ,{\\cal A}_v)$ .", "Therefore, $f$ and $g$ agree on $\\Lambda \\setminus (\\bigcup _{n=1}^{\\infty } P_n)$ , where a set $(\\bigcup _{n=1}^{\\infty } P_n)$ is meager.", "If functions $g^1$ and $g^2\\in {\\cal B}(\\Lambda ,{\\cal A}_v)$ differ from $f^1$ and $f^2\\in C(\\Lambda ,{\\cal A}_v)$ on meager sets $P^1$ and $P^2$ , then ${\\tilde{g}}^1$ , $~ bg^1+g^2$ , $~g^1b+g^2$ and $g^1g^2$ differ from ${\\tilde{f}}^1$ , $~bf^1+f^2$ , $~f^1b+f^2$ and $f^1f^2$ on a subset of $P^1\\cup P^2$ .", "Thus the mapping $\\theta : {\\cal B}(\\Lambda ,{\\cal A}_v)\\ni g\\mapsto f\\in C(\\Lambda ,{\\cal A}_v)$ is a conjugate preserving surjective homomorphism so that $\\theta (g)=0$ if and only if $g$ vanishes on the complement of a meager set.", "Particularly, if $ \\lbrace \\mbox{}_ng: ~ n \\rbrace $ is a monotone increasing sequence in ${\\cal B}(\\Lambda ,{\\bf R})\\hookrightarrow {\\cal B}(\\Lambda ,{\\cal A}_v)$ of bounded Borel real-valued functions tending point-wise to the bounded Borel function $g$ , each continuous real-valued function $\\mbox{}_nf \\in C(\\Lambda ,{\\cal A}_v)$ differs from $\\mbox{}_ng$ on the meager set $P_n$ , then $\\mbox{}_nf(x)\\le \\mbox{ }_{n+1}f(x)$ for each $x\\in \\Lambda \\setminus (P_n\\cup P_{n+1})$ in a dense set so that $\\mbox{}_nf \\le \\mbox{ }_{n+1}f$ .", "Therefore, $\\mbox{}_nf\\le f$ for each natural number $n$ , while $f$ differs from $g$ on the meager set $P$ .", "Thus the sequences $\\lbrace \\mbox{}_ng(x): n \\rbrace $ and $\\lbrace \\mbox{}_nf(x): n \\rbrace $ tend to $f(x)$ for each $x\\in \\Lambda \\setminus (P\\cup (\\bigcup _{n=1}^{\\infty }P_n))$ .", "This means that the function $f$ is the least upper bound in $C(\\Lambda ,{\\bf R})$ of the sequence $ \\lbrace \\mbox{}_nf: n \\rbrace $ and the homomorphism $\\theta $ is $\\sigma $ -normal.", "39.", "Corollary.", "Suppose that $U$ is an open dense subset in an extremely disconnected compact Hausdorff topological space $\\Lambda $ and $f$ is a continuous bounded function $f: U \\rightarrow {\\cal A}_v$ , $~card (v)\\le \\aleph _0$ .", "Then there exists a unique continuous function $\\xi : \\Lambda \\rightarrow {\\cal A}_v$ extending $f$ from $U$ onto $\\Lambda $ .", "Proof.", "Put $g(x)=f(x)$ for each $x\\in U$ , while $g(x)=0$ for each $x\\in \\Lambda \\setminus U$ , hence $g$ is a bounded Borel function on $\\Lambda $ .", "From Lemma 38 we know that a unique continuous function $\\xi : \\Lambda \\rightarrow {\\cal A}_v$ exists so that $\\xi $ and $g$ differ on the complement of a meager set.", "If $\\xi (x)\\ne f(x)$ for some $x\\in U$ , then by continuity of $f-\\xi $ on $U$ we get that $f(x)\\ne \\xi (x)$ on a non-void clopen subset $W$ of $U$ .", "But $W$ is not meager.", "This contradicts the choice of $\\xi $ .", "Thus $\\xi $ is the continuous extension of $f$ from $U$ on $\\Lambda $ .", "40.", "Remark.", "If $\\Lambda $ is metrizable, the condition $card (v)\\le \\aleph _0$ can be dropped in Lemma 38 and Corollary 39 in accordance with §31 in chapter 2 volume 1 [17].", "We denote by ${\\cal B}_u(\\Lambda ,{\\cal A}_v)$ the family of all Borel functions $f: \\Lambda \\rightarrow {\\cal A}_v$ .", "41.", "Lemma.", "Suppose that $\\Lambda $ is an extremely disconnected compact Hausdorff space.", "Let either $\\Lambda $ be metrizable or $card (v)\\le \\aleph _0$ .", "Then a conjugation-preserving surjective homomorphism $\\psi : {\\cal B}_u(\\Lambda ,{\\cal A}_v) \\rightarrow {\\cal N}(\\Lambda ,{\\cal A}_v)$ exists so that its kernel $ker (\\psi ) $ consists of all those functions in ${\\cal B}_u(\\Lambda ,{\\cal A}_v)$ vanishing on the complement of a meager set.", "Proof.", "As $f$ and $g$ are normal functions defined on $\\Lambda \\setminus W_f$ and $\\Lambda \\setminus W_g$ respectively and $f(x)=g(x)$ for each $x\\in \\Lambda \\setminus (W_f\\cup W_g\\cup P)$ , where $P$ is a meager subset of $\\Lambda $ , then $W_f=W_g$ and $f=g$ in accordance with Lemma 9, since the set $(W_f\\cup W_g\\cup P)$ is meager in $\\Lambda $ , while $\\Lambda \\setminus (W_f\\cup W_g\\cup P)$ is dense in $\\Lambda $ .", "This implies that at most one normal function can be agreeing with any function on the complement of a meager set.", "For each Borel function $g: \\Lambda \\rightarrow {\\cal A}_v$ and each ball $B({\\cal A}_v,y,q) := \\lbrace z\\in {\\cal A}_v: ~ |z-y|\\le q \\rbrace $ with center $y$ and radius $q>0$ , its inverse image $g^{-1}(B({\\cal A}_v,y,q))$ is a Borel subset $V_q$ of $\\Lambda $ .", "In view of Lemma 38 a clopen subset $Q_n$ exists so that $(V_n\\setminus Q_n)\\cup (Q_n\\setminus V_n)$ is meager in $\\Lambda $ .", "Take the Borel function $\\mbox{}_ng$ such that $\\mbox{}_ng$ is equal to $g$ on $V_n$ and is zero on $\\Lambda \\setminus V_n$ , consequently, it satisfies the inequality $ \\Vert \\mbox{}_ng \\Vert \\le n$ .", "Applying Lemma 38 one gets a continuous function $\\mbox{}_nf: \\Lambda \\rightarrow {\\cal A}_v$ so that that agrees with $\\mbox{}_ng$ on the complement of a meager subset $P_n$ of $\\Lambda $ .", "The function $\\mbox{}_nf$ vanishes on the subset $\\Lambda \\setminus (V_n\\cup P_n)$ , since $\\mbox{}_ng$ vanishes on $\\Lambda \\setminus V_n$ .", "Certainly the set $\\Lambda \\setminus (V_n\\cup P_n)$ contains the subset $(\\Lambda \\setminus Q_n)\\setminus (P_n\\cup (V_n\\setminus Q_n)\\cup (Q_n\\setminus V_n))$ .", "The set $(P_n\\cup (V_n\\setminus Q_n)\\cup (Q_n\\setminus V_n))$ is meager and $\\mbox{}_nf$ is continuous on $\\Lambda $ , hence this mapping $\\mbox{}_nf$ vanishes on $\\Lambda \\setminus V_n$ .", "The inclusion $V_n\\subseteq V_{n+1}$ for each natural number $n$ implies the inequality $e_n(x)\\le e_{n+1}(x)$ for each point $x$ in $\\Lambda $ outside a meager set, where $e_n := \\chi _{Q_n}$ is the characteristic function of $Q_n$ .", "But continuity gives $e_n\\le e_{n+1}$ and hence $Q_n\\subseteq Q_{n+1}$ for every $n\\in \\bf N$ .", "The functions $\\mbox{}_{n+1}g$ and $\\mbox{}_ng$ are consistent on $V_n$ , consequently, continuous functions $\\mbox{}_{n+1}f$ and $\\mbox{}_nf$ agree on $Q_n$ .", "Then the inequality $n\\le |\\mbox{}_mf(x)|$ for each $x\\in Q_m\\setminus Q_n$ and $n<m$ is fulfilled, since $n\\le |\\mbox{}_mg(y)|$ when $y\\in V_m\\setminus V_n$ .", "The equality $\\bigcup _{n=1}^{\\infty } V_n=\\Lambda $ leads to the inclusion $W_f=\\Lambda \\setminus (\\bigcup _{n=1}^{\\infty } Q_n)\\subseteq \\bigcup _{n=1}^{\\infty } (V_n\\triangle Q_n)$ , where $A\\triangle B := (A\\setminus B)\\cup (B\\setminus A)$ for two sets.", "Therefore, the set $W_f$ is closed and meager, consequently, $W_f$ is nowhere dense in $\\Lambda $ .", "If $f(x)=\\mbox{ }_nf(x)$ for each $x\\in Q_n$ and $n\\in \\bf N$ , then the mapping $f$ is continuous on $\\Lambda \\setminus W_f$ .", "For a point $x\\in W_f$ there exists a natural number $n$ so that $x\\in \\Lambda \\setminus Q_n$ .", "When $y\\in \\Lambda \\setminus (W_f\\cup Q_n)$ the inequality $n\\le |f(y)|$ is valid.", "Thus the function $f$ is normal.", "By the construction above two functions $f$ and $g$ agree on the complement of $W_f\\cup (\\bigcup _{n=1}^{\\infty } P_n)$ , where the latter set is meager.", "This induces a conjugation-preserving surjective homomorphism $\\psi :{\\cal B}(\\Lambda ,{\\cal A}_v)\\ni g \\mapsto f \\in {\\cal N}(\\Lambda ,{\\cal A}_v)$ with kernel consisting of those Borel functions vanishing on the complement of a meager set in $\\Lambda $ .", "42.", "Proposition.", "Let $\\sf A$ be a quasi-commutative von Neumann algebra over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ with $2\\le v\\le 3$ , let also $\\mbox{}_nT$ be a sequence of operators in ${\\cal Q}({\\sf A})$ with upper bound $\\mbox{}_0T$ in ${\\cal Q}({\\sf A})$ .", "Then $ \\lbrace \\mbox{}_nT: ~ n=1,2,3,... \\rbrace $ has a least upper bound $T$ in ${\\cal Q}({\\sf A})$ .", "Proof.", "An algebra $\\sf A$ is $*$ -isomorphic with $C(\\Lambda ,{\\cal A}_v)$ , where $\\Lambda $ is a compact extremely disconnected Hausdorff space (see Theorem I.2.52 [28]).", "Take normal functions $\\mbox{}_nf$ in ${\\cal N}(\\Lambda ,{\\cal A}_v)$ representing $\\mbox{}_nT$ .", "Then $B {\\hat{+}}\\mbox{ }_1T$ is the least upper bound of $ \\lbrace \\mbox{}_nT: ~ n \\rbrace $ , when $B$ is the leat upper bound of $\\lbrace \\mbox{}_nT {\\hat{+}} -\\mbox{}_1T: ~ n \\rbrace $ .", "This means that without loss of generality we can consider $\\mbox{}_nT\\ge 0$ for each $n\\in \\bf N$ .", "If a normal function $\\mbox{}_nf$ is defined on $\\Lambda \\setminus W_n$ , then $W_n=(W_n)_+$ , hence $W_n\\subseteq W_{n+1}$ for each $n$ .", "If $x\\notin W := \\bigcup _{n=0}^{\\infty } W_n$ , then $\\mbox{}_nf(x)$ is defined for each natural number $n$ and the sequence $\\lbrace \\mbox{}_nf:~ n \\rbrace $ has an upper bound $\\mbox{}_0f(x)$ so that $\\mbox{}_0f\\le h$ for each $h\\in \\psi ^{-1}(\\phi (\\mbox{}_0T))$ (see Proposition 32 and Lemma 41).", "Thus $ \\lbrace \\mbox{}_nf: ~ n \\rbrace $ converges to some $g(x)$ .", "Put $g$ to be zero on $W$ , then $g$ is a Borel function on $\\Lambda $ .", "Applying Lemma 41 one gets a normal function $f\\in {\\cal N}(\\Lambda ,{\\cal A}_v)$ agreeing with $g$ on $\\Lambda \\setminus P$ , where $P$ is a meager subset in $\\Lambda $ .", "Thus the sequence $ \\lbrace \\mbox{}_nf(x): ~ n \\rbrace $ converges to $f(x)$ on the dense subset $\\Lambda \\setminus (P\\cup W)$ .", "This means that $f$ is the least upper bound for $ \\lbrace \\mbox{}_nf: ~ n \\rbrace $ .", "Then an element $T$ in ${\\cal Q}({\\sf A})$ represented by $f\\in {\\cal Q}(\\Lambda )$ is the least upper bound of $ \\lbrace \\mbox{}_nT: ~ n=1,2,... \\rbrace $ .", "43.", "Notes.", "Using Proposition 42 we can extend our definition of a $\\sigma $ -normal homomorphism to ${\\cal N}({\\sf A})$ , $~ {\\cal N}(\\Lambda ,{\\cal A}_v)$ , $~ {\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ and $~ {\\cal B}_u(\\Lambda ,{\\cal A}_v)$ .", "In view of Lemma 41 this homomorphism from ${\\cal B}_u(\\Lambda ,{\\cal A}_v)$ onto ${\\cal N}(\\Lambda ,{\\cal A}_v)$ is $\\sigma $ -normal.", "Consider an increasing sequence $ \\lbrace \\mbox{}_ng: ~ n \\rbrace $ of Borel functions on $\\Lambda $ tending point-wise to the Borel function $\\mbox{}_0g$ and take the $\\sigma $ -normal functions $\\mbox{}_nf$ corresponding to $\\mbox{}_ng$ .", "Then the sequence $\\lbrace \\mbox{}_nf: ~ n\\rbrace $ has the least upper bound $\\mbox{}_0f$ in ${\\cal N}(\\Lambda ,{\\cal A}_v)$ .", "Indeed, the functions $\\mbox{}_ng$ and $\\mbox{}_nf$ agree on the complement of a meager set $P_n$ .", "Therefore, the limit exists $\\lim _n \\mbox{ }_nf(x) =\\mbox{}_0f(x)$ for each point $x\\in \\Lambda \\setminus (\\bigcup _{n=0}^{\\infty }P_n)$ and the subset $\\Lambda \\setminus (\\bigcup _{n=0}^{\\infty }P_n)$ is dense in $\\Lambda $ .", "If a function $h$ is an upper bound for $ \\lbrace \\mbox{}_nf:~ n=1,2,3,...\\rbrace $ , then $\\mbox{}_nf(x)\\le h(x)$ for each natural number $n=1,2,3,...$ and points $x\\in \\Lambda \\setminus (\\bigcup _{n=0}^{\\infty }P_n)$ in the complement of the meager set $(\\bigcup _{n=0}^{\\infty }P_n)$ .", "Thus $\\mbox{}_0f(x)\\le h(x)$ for all $x$ in the complement of this meager set, hence the mapping $h{\\hat{+}} -\\mbox{ }_0f$ has non-negative values on a dense set.", "Thus $\\mbox{}_0f\\le h$ and $\\mbox{}_0f$ is the least upper bound in ${\\cal N}(\\Lambda ,{\\cal A}_v)$ of the sequence $ \\lbrace \\mbox{}_nf: ~n=1,2,3,... \\rbrace $ .", "Using Lemma 41 one can define $g(T)$ for an arbitrary Borel function $g$ on $sp (T)$ for any normal operator $T$ in a Hilbert space either over the quaternion skew field or the octonion algebra ${\\cal A}_v$ with $2\\le v\\le 3$ .", "In accordance with Theorem 27 operators $T$ , $T^*$ and $I$ generate a quasi-commutative von Neumann algebra $\\sf A$ over the algebra ${\\cal A}_v$ with $2\\le v\\le 3$ such that $T$ is affiliated with $\\sf A$ .", "It is known from Theorems 2.24 and 2.28 [20] that $\\sf A$ is isomorphic with $C(\\Lambda ,{\\cal A}_v)$ for some extremely disconnected compact Hausdorff topological space $\\Lambda $ .", "In accordance with Theorem 29 there exists an isomorphism $\\phi $ of ${\\cal N}({\\sf A})$ onto ${\\cal N}(\\Lambda ,{\\cal A}_v)$ .", "As $\\phi (T)$ is defined on $\\Lambda \\setminus W$ , the function $q$ defined to be zero on $W$ and $g\\circ \\phi (T)$ on $\\Lambda \\setminus W$ is Borel, $q\\in {\\cal B}_u(\\Lambda ,{\\cal A}_v)$ .", "But Lemma 41 says that a function $h$ exists so that $h\\in {\\cal N}(\\Lambda ,{\\cal A}_v)$ and $h$ agrees with $q$ on the complement of a meager set in $\\Lambda $ .", "We now define $g(T)$ as $\\phi ^{-1}(h)$ .", "It may happen that $sp(T)$ is a subset of a Borel set $V$ (in ${\\cal A}_r$ , for example) and $g$ is a Borel function on $V$ , then $g(T)$ will denote $g|_{sp(T)}(T)$ , where $g|_B$ denotes the restriction of $g$ to a subset $B\\subset V$ .", "As usually ${\\cal B}(Y,{\\cal A}_v)$ denotes the algebra of all bounded Borel functions from a topological space $Y$ into the Cayley-Dickson algebra ${\\cal A}_v$ , while ${\\cal B}_u(Y,{\\cal A}_v)$ denotes the algebra of all Borel functions from $Y$ into ${\\cal A}_v$ .", "44.", "Theorem.", "Let $\\sf A$ be a quasi-commutative von Neumann algebra over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ , $~ 2\\le v\\le 3$ , so that $\\sf A$ is generated by a normal operator $T$ acting on a Hilbert space $X$ over the algebra ${\\cal A}_v$ .", "Then the mapping $\\theta : g\\mapsto g(T)$ of the algebra ${\\cal B}_u(sp (T),{\\cal A}_v)$ into ${\\cal N}({\\sf A})$ is a $\\sigma $ -normal homomorphism such that $\\theta (1)=I$ and $\\theta (id)=T$ .", "Moreover, the mapping $V\\mapsto E(V)$ of Borel subsets in ${\\cal A}_v$ into $\\sf A$ is an ${\\cal A}_v$ graded projection-valued measure on a Hilbert space $X$ , where $E(V)=\\chi _V(T)$ , $~ \\chi _V$ denotes the characteristic function of $V\\in {\\cal B}({\\cal A}_v)$ .", "Suppose that $h: {\\cal A}_v \\rightarrow {\\cal A}_v$ is a Borel bounded function, ${\\cal B}({\\cal A}_v,{\\cal A}_v)$ denotes the algebra over ${\\cal A}_v$ of all such Borel bounded functions, then $(1)\\quad \\Vert h(T) \\Vert \\le \\sup _{y\\in {\\cal A}_v} |h(y)| =: \\Vert h \\Vert \\mbox{ and}$ $(2)\\quad <h(T)x;x> = \\int _{{\\cal A}_v} d\\mu _x(y).h(y)$ for each vector $x\\in X$ and every Borel bounded function $h\\in {\\cal B}({\\cal A}_v,{\\cal A}_v)$ , where $\\mu _x(V).h(y) := <E(V).h(y)x;x>$ .", "For $f\\in {\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ a vector $x$ belongs to a domain ${\\cal D}(f(T))$ if and only if $(3)\\quad \\int _{{\\cal A}_v} d\\mu _x(y).|f(y)|^2 =: \\Vert f(T)x \\Vert ^2< \\infty $ and Formula $(2)$ is valid with $f$ in place of $h$ .", "Let $\\phi $ be an extension on ${\\cal N}({\\sf A})$ of the isomorphism $\\psi $ of $\\sf A$ with $C(\\Lambda ,{\\cal A}_v)$ , let also $\\nu _x$ be a regular Borel measure on $\\Lambda $ so that $(4)\\quad <Bx;x> = \\int _{\\Lambda } d \\nu _x(t).", "(\\phi (B))(t)$ for each $B\\in \\sf A$ , then $x\\in {\\cal D}(Q)$ with $Q\\in {\\cal N}({\\sf A})$ if and only if $(5)\\quad \\int _{\\Lambda } d \\nu _x(t).|(\\phi (Q))(t)|^2 =: \\Vert Qx \\Vert ^2<\\infty .$ If additionally $T$ is a self-adjoint operator, its spectral resolution is $ \\lbrace \\mbox{}_bE: ~ b\\in {\\bf R} \\rbrace $ , where $\\mbox{}_bE = I - E((b,\\infty ))$ , and $x\\in {\\cal D}(f(T))$ if and only if $(6)\\quad \\int _{-\\infty }^{\\infty } d<\\mbox{}_bE.|f(b)|^2x;x> <\\infty $ and for such vector $x\\in {\\cal D}(f(T))$ in a domain of $f(T)$ the equality $(7)\\quad <f(T)x;x> = \\int _{-\\infty }^{\\infty } d<\\mbox{}_bE.f(b)x ;x> $ is valid.", "Proof.", "In accordance with Remark 43 there is a $\\sigma $ -normal homomorphism $\\omega $ from ${\\cal B}_u (sp (T))$ into ${\\cal B}_u(\\Lambda ,{\\cal A}_v)$ .", "On the other hand, the mapping assigning $h\\in {\\cal N}(\\Lambda ,{\\cal A}_v)$ to $\\omega (g)$ is a $\\sigma $ -normal homomorphism, where $g\\in {\\cal B}_u (sp (T))$ .", "Therefore, the mapping $g\\mapsto g(T)$ is a $\\sigma $ -normal homomorphism from ${\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ into ${\\cal N}({\\sf A})$ .", "When $g=1$ is the constant unit function, the function $h$ is also constant unit on $\\Lambda $ , consequently, $g(T)=I$ .", "When $g=id$ , then $h=\\phi (T)$ , so that $id (T)=T$ , where $id(x)=x$ for each $x\\in {\\cal A}_v$ .", "As $V$ is a Borel subset of ${\\cal A}_v$ and $g=\\chi _V$ is its characteristic function, the identities are satisfied: $g(T)^* ={\\tilde{g}}(T) = g(T)$ and $g(T) = g^2(T) = g(T)^2$ .", "This means that $g(T)$ is an ${\\cal A}_v$ graded projection $E(V)$ in $\\sf A$ .", "Particularly, $\\chi _{ \\emptyset }=0$ and $E(\\emptyset )=0$ , $\\chi _{{\\cal A}_v}\\equiv 1$ , $\\chi _{{\\cal A}_v}(T)=I$ and $E({\\cal A}_r)=I$ .", "Recall that ${\\cal A}_v$ graded projection valued measures were defined in §§I.2.73 and I.2.58.", "We use in this section the simplified notation $E$ instead of ${\\hat{\\bf E}}$ .", "For any countable family $\\lbrace V_j: ~ j \\rbrace $ of disjoint Borel subsets in ${\\cal A}_ v$ and their characteristic functions $\\mbox{}_jg := \\chi _{V_j}$ , their sums $\\mbox{}_nh := \\mbox{}_1g+...+\\mbox{}_ng$ form an increasing sequence tending point-wise to the characteristic function $h := \\chi _V$ , where $V=\\bigcup _{j=1}^{\\infty } V_j$ .", "Then $ \\lbrace \\sum _{j=1}^n E(V_j): ~ n \\rbrace $ has the least upper bound $E(V)$ so that $E(V) = \\sum _{j=1}^{\\infty } E(V_j)$ , since $g\\mapsto g(T)$ is a $\\sigma $ -normal homomorphism.", "Using our notation we get: $<h(T)x;x> = <E(V)x;x> = \\mu _x(V) = \\int _{{\\cal A}_v} d \\mu _x(t).h(t)$ hence Equation $(2)$ is valid for Borel step functions $h$ .", "For any bounded Borel function $h$ on ${\\cal A}_v$ with values in ${\\cal A}_v$ we have the inequality $ \\Vert \\omega (h) \\Vert \\le \\Vert h\\Vert = \\sup _{x\\in {\\cal A}_r} |h(x)|$ and the function $f\\in {\\cal N}(\\Lambda ,{\\cal A}_v)$ corresponding to $\\omega (h)$ belongs to $C(\\Lambda ,{\\cal A}_v)$ by Lemma 38.", "We put $f=\\theta (h)$ .", "Then we infer, that $ \\Vert h(T) \\Vert = \\Vert \\phi ^{-1} (f) \\Vert = \\Vert f \\Vert \\le \\Vert h \\Vert ,$ since $ \\Vert f \\Vert \\le \\Vert \\omega (h) \\Vert \\le \\Vert h \\Vert $ .", "Each bounded Borel function in ${\\cal B}({\\cal A}_v,{\\cal A}_v)$ is a norm limit of Borel step functions, consequently, Equality $(2)$ is valid for each $h\\in {\\cal B}({\\cal A}_v,{\\cal A}_v)$ .", "We consider now a self-adjoint operator $T$ and the characteristic function $g=\\chi _{(b,\\infty )}$ of $(b,\\infty )$ .", "Then $\\theta (g)$ is the characteristic function of $\\phi (T)^{-1}((b,\\infty ))$ which is an open subset contained in $\\Lambda \\setminus W$ and hence in $\\Lambda $ .", "Moreover, the function in $C(\\Lambda ,{\\cal A}_v)$ corresponding to $\\theta (g)$ is the characteristic function $1- e_b$ of $cl [\\phi (T)^{-1}((b,\\infty ))]$ .", "This means that $E((b,\\infty )) = g(T)\\in \\sf A$ corresponds to $1-e_b$ and from Theorem 16 one gets $\\mbox{}_bE = I- E((b,\\infty ))$ .", "This implies that $<(\\mbox{}_cE-\\mbox{}_bE)x;x> = \\mu _x((b,c])$ for any $b\\le c$ and $\\int _{-\\infty }^{\\infty } d<\\mbox{}_bE.|f(b)|^2 x;x> = \\int d \\mu _x(b).|f(b)|^2$ for each Borel function $f\\in {\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ , consequently, the last assertions of this theorem reduce to Formulas $(2,3)$ .", "Denote by $q_n := \\chi _{V_n}$ the characteristic function of the subset $V_n :=|f|^{-1}([0,n])$ , where $f\\in {\\cal B}_u({\\cal A}_r,{\\cal A}_v)$ is a Borel function, we put $\\mbox{}_nF := q_n (T)$ .", "Therefore, $\\mbox{}_nf(T)=f(T)\\mbox{ }_nF$ due to the first part of the proof, where $\\mbox{}_nf = fq_n$ , since the functions $f$ and $q_n$ commute, $fq_n=q_n ~f$ .", "This implies that for a vector $x\\in X$ we get the equalities: $(6)\\quad \\Vert f(T) \\mbox{ }_nF x \\Vert ^2 = < |f_n|^2(T)x;x> = \\int _{V_n}d\\mu _x(t).|f(t)|^2 \\mbox{ and}$ $(7)\\quad \\Vert f(t)\\mbox{ }_n Fx - f(t)\\mbox{ }_m F x \\Vert ^2 =\\int _{V_n\\setminus V_m} d \\mu _x(t).|f(t)|^2 .$ The sequence $ \\lbrace \\mbox{}_nF: ~ n \\rbrace $ is increasing with least upper bound $I$ , since $ \\lbrace q_n : n \\rbrace $ is the increasing net of non-negative Borel functions tending point-wise to 1.", "Then we infer that $\\mbox{}_nFf(T)x = f(T) \\mbox{ }_nF x$ for each vector $x$ in a domain ${\\cal D}(f(T))$ , since $\\mbox{}_nF f(T) \\subseteq f(T)\\mbox{ }_nF$ .", "This implies that the limit exists $\\lim _n f(T) \\mbox{ }_nF x = f(T)x$ and Formula $(3)$ follows from $(6)$ .", "Vise versa, if the integral $\\int _{{\\cal A}_v} d\\mu _x(t).|f(t)|^2 $ converges, then one gets a Cauchy sequence $ \\lbrace f(T) \\mbox{ }_nF x: ~ n \\rbrace $ due to Formula $(7)$ converging to some vector in the Hilbert space $X$ over the Cayley-Dickson algebra ${\\cal A}_v$ .", "But $x\\in {\\cal D}(f(T))$ , since $\\lim _n \\mbox{}_nFx=x$ and the function $f(T)$ of the operator $T$ is closed.", "Quite analogous demonstration leads to Formula $(5)$ .", "We have a non-negative measure $\\mu _x(V).1 = <E(V).1x;x> = <E^2(V).1x;x> =<E(V).1x;E^*(V).1x> ~ \\ge 0$ for each Borel subset $V$ .", "As $x$ is a vector in the domain ${\\cal D}(f(T))$ , the function $f$ belongs to the Hilbert space $L^2({\\cal A}_v, \\mu _x, {\\cal A}_v)$ , while $L^2({\\cal A}_v, \\mu _x, {\\cal A}_v)\\subset L^1({\\cal A}_v, \\mu _x,{\\cal A}_v)$ , since $\\mu _x.1$ is a finite non-negative measure, where $L^p({\\cal A}_v, \\mu _x, {\\cal A}_v)$ with $1\\le p<\\infty $ denotes an ${\\cal A}_v$ vector space which is the norm completion of the family of all step Borel functions $u$ from ${\\cal A}_v$ into ${\\cal A}_v$ , where the norm is prescribed by the formula: $ \\Vert u \\Vert := \\@root p \\of {\\int _{{\\cal A}_v} d\\mu _x(t).|u(t)|^p} .$ Finally one deduces that $<f(T)x;x> = \\lim _n <f(T)\\mbox{ }_nFx;x> =\\lim _n \\int _{{\\cal A}_v}q_n(t) d\\mu _x(t).f(t) = \\int _{{\\cal A}_v} d\\mu _x(t).f(t) .$ 45.", "Remark.", "The ${\\cal A}_v$ graded projection $E(V)$ of the preceding theorem will also be referred as the spectral ${\\cal A}_v$ graded projection for $T$ corresponding to the Borel subset $V$ of ${\\cal A}_v$ , where $2\\le v \\le 3$ .", "46.", "Theorem.", "Suppose that $T$ is a normal operator affiliated with a von Neumann algebra acting on a Hilbert space over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ with $2\\le v \\le 3$ and $\\psi $ is a $\\sigma $ -normal homomorphism of the algebra ${\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ of Borel functions into ${\\cal N}({\\sf A})$ so that $\\psi (1)=I$ and $\\psi (id) =T$ , where $id : {\\cal A}_v\\rightarrow {\\cal A}_v$ denotes the identity mapping $id (t)=t$ on ${\\cal A}_v$ .", "Then $\\psi (f) =f^r(T)$ for each $f\\in {\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ , where $f^r=f|_{sp (T)}$ denotes the restriction of $f$ to $sp (T)$ .", "Proof.", "This homomorphism $\\psi $ is adjoint preserving, since $\\psi $ is $\\sigma $ -normal.", "Positive elements of ${\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ have positive roots, hence $\\psi $ is order preserving.", "Moreover, $\\psi : {\\cal B}({\\cal A}_v,{\\cal A}_v)\\rightarrow \\sf A$ and does not increase norm, since $\\psi (1)=I$ .", "At first we consider the case when $T$ is bounded.", "Put $\\mbox{}_0g := \\chi _{{\\cal A}_v\\setminus B}$ , where $B=B({\\cal A}_v,0,2 \\Vert T \\Vert )$ is the closed ball in ${\\cal A}_v$ with center 0 and radius $2 \\Vert T \\Vert $ .", "Then $0\\le (2 \\Vert T \\Vert )^n \\mbox{}_0g\\le |id|^n$ for each natural number $n=1,2,3,...$ .", "Therefore, $\\psi (|id |^n) = T^n$ such that $0\\le (2 \\Vert T \\Vert )^n \\psi (\\mbox{}_0g)\\le | T | ^n$ , hence $ \\Vert \\psi (\\mbox{}_0g) \\Vert \\le 2^{-n}$ for each natural number $n$ , consequently, $\\psi (\\mbox{}_0g)=0$ .", "As $\\mbox{}_1g$ is the characteristic function of the ball $B$ , we get $\\psi (\\mbox{}_1g) = I$ , such that $\\psi (\\mbox{}_1gh) = \\psi (h)$ for each Borel function $h\\in {\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ .", "Consider the restriction $\\mbox{}_0h = h|_B$ of $h$ to $B$ and put $\\psi ^0(\\mbox{}_0h) =\\psi (h)$ .", "Therefore, this mapping $\\psi ^0$ is a $\\sigma $ -normal homomorphism of ${\\cal B}_u(B,{\\cal A}_v)$ into ${\\cal N}({\\sf A})$ with the properties: $\\psi ^0(\\chi _B) =I$ and $\\psi ^0 (\\mbox{}_0id)= T$ and $\\psi ^0: {\\cal B}(B,{\\cal A}_v)\\rightarrow \\sf A$ .", "At the same time it is known from the exposition presented above that $C(B,{\\cal A}_v)$ is a $C^*$ -algebra over the algebra ${\\cal A}_v$ with unit being the constant function $\\chi _B$ .", "Therefore, by Theorem 1.3.17 [28] one gets that $\\psi ^0 (f) = \\psi ^0 (f(\\mbox{}_0id )) = f(T)$ for each continuous function $f\\in C(B,{\\cal A}_v)$ .", "It is known from §I.3.17 that the characteristic function $\\mbox{}_1h$ of the open subset $B\\setminus sp (T)$ of $B$ is the point-wise limit of an increasing sequence $ \\lbrace \\mbox{}_nf: ~n \\rbrace $ of positive functions $\\mbox{}_nf$ on $B$ , consequently, $\\psi ^0(\\mbox{}_1h)$ is the least upper bound in $\\sf A$ of the sequence $\\lbrace \\psi ^0(\\mbox{}_nf): ~ n \\rbrace $ , since the homomorphism $\\psi ^0$ is $\\sigma $ -normal.", "Each function $\\mbox{}_nf$ is continuous and vanishes on $sp (T)$ , consequently, $\\psi ^0(\\mbox{}_nf) = \\mbox{}_nf(T)$ and $\\mbox{}_nf (T)=0$ and hence $\\psi ^0(\\mbox{}_1h)=0$ .", "For the characteristic function $\\chi _{sp(T)}$ of $sp (T)\\subset B$ one obtains the equalities: $\\psi ^0(\\chi _{sp (T)})=I$ and $\\psi ^0(\\chi _{sp (T)}\\mbox{ }_0h) = \\psi ^0(\\mbox{}_0h)$ for each $\\mbox{}_0h\\in {\\cal B}_u(B,{\\cal A}_v)$ .", "If put $\\psi ^1(q|_{sp (T)}) = \\psi ^0(q)$ for each $q\\in {\\cal B}_u(B,{\\cal A}_v)$ , then $\\psi ^1 : {\\cal B}_u(sp (T),{\\cal A}_v)\\rightarrow {\\cal N}({\\sf A})$ is a $\\sigma $ -normal homomorphism so that $\\psi ^1(1)=I$ and $\\psi ^1(id)=T$ and $\\psi ^1({\\cal B}(sp(T),{\\cal A}_v))\\subset \\sf A$ .", "In view of Theorem I.3.21 [28] applied to to the restriction $\\psi ^1|_{{\\cal B}(sp (T),{\\cal A}_v)}$ the equality $\\psi ^1(f)=f(T)$ is valid for each Borel bounded function $f\\in {\\cal B}(sp (T),{\\cal A}_v)$ .", "On the other hand, each positive function $g$ is the point-wise limit of an increasing sequence of positive functions in ${\\cal B}(sp (T),{\\bf R})\\subset {\\cal B}(sp (T),{\\cal A}_v)$ .", "Therefore, $\\psi ^1(g)=g(T)$ for each positive Borel function $g\\in {\\cal B}_u(sp (T),{\\bf R})$ , since the homomorphism $\\psi ^1: {\\cal B}_u(sp (T),{\\cal A}_v)\\rightarrow {\\cal N}({\\sf A})$ is $\\sigma $ -normal.", "Using the decomposition $h=\\sum _j h_ji_j$ of each Borel function $h\\in {\\cal B}_u(sp (T),{\\cal A}_v)$ with real-valued Borel functions $h_j$ and $h_j=h_j^+ - h_j^-$ with non-negative Borel functions $h_j^+$ and $h_j^-$ we infer that $\\psi ^1 (h)=h(T)$ .", "This implies that if $q\\in {\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ and $\\mbox{}_0q = q|_B$ and $q^r =q|_{sp (T)}$ , then $\\psi (q) = \\psi ^0(\\mbox{}_0q) = \\psi ^1(q^r)=q^r(T).$ We take now an arbitrary normal operator $T\\in {\\cal N}({\\sf A})$ and a bounding ${\\cal A}_v$ graded projection $E$ for $T$ in $\\sf A$ .", "The mapping $\\phi $ posing $(Y{\\hat{\\cdot }} E)|_{E(X)}\\in {\\cal N}({\\sf A}E)$ acting on $E(X)$ to $Y\\in {\\cal N}({\\sf A})$ is a $\\sigma $ -normal homomorphism of ${\\cal N}({\\sf A})$ into ${\\cal N}({\\sf A}E)$ .", "Taking the composition of $\\phi $ with $\\psi $ yields a $\\sigma $ -normal homomorphism $\\psi ^2: {\\cal B}_u({\\cal A}_v,{\\cal A}_v)\\rightarrow {\\cal N}({\\sf A}E)$ mapping 1 onto $E|_{E(X)}$ and $id $ onto $T|_{E(X)}$ .", "But the composition of $\\phi $ with the mapping $f\\mapsto f^r(T)$ of ${\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ into ${\\cal N}({\\sf A}E)$ is another homomorphism.", "The restriction $T|_{E(X)}$ is bounded, consequently, from the first part of this proof we infer that $(\\psi (f){\\hat{\\cdot }} E)|_{E(X)} = \\psi ^2 (T|_{E(X)}) =(f^r(T){\\hat{\\cdot }} E)|_{E(X)} .$ Theorem 23 states that there exists a common bounding ${\\cal A}_v$ graded sequence $\\lbrace \\mbox{}_nE: ~ n \\rbrace $ for $T$ and $\\psi (f)$ and $f^r(T)$ , where $f$ is a given element of ${\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ .", "Then we deduce that $(\\psi (f){\\hat{\\cdot }} \\mbox{ }_nE)|_{\\mbox{}_nE(X)} = (\\psi (f)\\mbox{ }_nE)|_{\\mbox{}_nE(X)} = (f^r(T){\\hat{\\cdot }} \\mbox{}_nE)|_{\\mbox{}_nE(X)} = (f^r(T)\\mbox{ }_nE)|_{\\mbox{}_nE(X)}$ so that $\\psi (f)\\mbox{ }_nE= f^r(T) \\mbox{ }_nE$ for each $n$ .", "Therefore, $\\psi (f) = f^r(T)$ , since $\\bigcup _{n=1}^{\\infty }\\mbox{ }_nE(X)$ is a core for both $\\psi (f)$ and $f^r(T)$ .", "47.", "Note.", "The procedure of §43 assigning $g(T)\\in \\sf A$ can be applied to ${\\cal N}(\\Omega ,{\\cal A}_v)$ , where $\\sf Y$ is another quasi-commutative von Neumann algebra over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ with $2\\le v \\le 3$ so that $T$ is affiliated with $\\sf Y$ and ${\\sf Y}\\cong C(\\Omega ,{\\cal A}_v)$ , $~\\Omega $ is an extremely disconnected compact Hausdorff topological space.", "The operator in ${\\cal N}({\\sf A})$ formed in this way is $g(T)\\in {\\cal N}({\\sf Y})$ by Theorem 46.", "48.", "Corollary.", "Let $T$ be a normal operator satisfying conditions of Theorem 46 and let $f$ and $g$ be in ${\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ , where $2\\le v\\le 3$ .", "Then $(1)\\quad (f\\circ g)(T) = f(g(T)).$ Proof.", "Consider the von Neumann quasi-commutative von Neumann algebra over the algebra ${\\cal A}_v$ with $2\\le v \\le 3$ generated by $T$ , $T^*$ and $I$ , then $g(T)\\in {\\cal N}({\\sf A})$ and $f\\mapsto f\\circ g$ is a $\\sigma $ -normal homomorphism $\\phi $ of ${\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ into ${\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ .", "Taking the composition of $\\phi $ with the $\\sigma $ -normal homomorphism $h\\mapsto h(T)$ of ${\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ into ${\\cal N}({\\sf A})$ leads to a $\\sigma $ -normal homomorphism $\\xi : f\\mapsto (f\\circ g)(T)$ of ${\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ into ${\\cal N}({\\sf A})$ with $\\xi (1)=I$ and $\\xi (id) = g(T)$ .", "Then Formula $(1)$ follows from Theorem 46.", "49.", "Proposition.", "Suppose that $\\psi $ is a $\\sigma $ -normal homomorphism of ${\\cal N}({\\sf A})$ into ${\\cal N}({\\sf B})$ so that $\\psi (I)=I$ , where $\\sf A$ and $\\sf B$ are von Neumann quasi-commutative algebras over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ , where $2\\le v\\le 3$ .", "Then $\\psi (f(T)) = f(\\psi (T))$ for each $T\\in {\\cal N}({\\sf A})$ and each $f\\in {\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ .", "Proof.", "Each quasi-commutative algebra $\\sf A$ over the algebra ${\\cal A}_v$ with $2\\le v \\le 3$ has the decomposition ${\\sf A}=\\bigoplus _j {\\sf A}_j i_j$ , where ${\\sf A}_j$ and ${\\sf A}_k$ are real isomorphic commutative algebras for each $j,k=0,1,2,...,2^v-1$ .", "On the other hand, the minimal subalgebra $alg_{\\bf R} (i_0,i_j,i_k)$ for each $1\\le j\\ne k$ is isomorphic with the quaternion skew field.", "Therefore, knowing restrictions $\\psi |_{{\\sf A}_0\\oplus {\\sf A}_ji_j\\oplus {\\sf A}_ki_k\\oplus {\\sf A}_li_l}$ for each $1\\le j<k$ will induce $\\psi $ on $\\sf A$ , where $i_l=i_ji_k$ .", "Indeed, homomorphisms $\\psi _{j,k} := \\psi |_{{\\sf A}_0\\oplus {\\sf A}_ji_j\\oplus {\\sf A}_ki_k\\oplus {\\sf A}_li_l}$ for different $1\\le j<k$ and $1\\le j^{\\prime }<k^{\\prime }$ are in bijective correspondence: $\\psi _{j^{\\prime },k^{\\prime }}\\circ \\theta ^{j,k}_{j^{\\prime },k^{\\prime }} = \\omega ^{j,k}_{j^{\\prime },k^{\\prime }}\\circ \\psi _{j,k}$ , where $\\theta ^{j,k}_{j^{\\prime },k^{\\prime }}: {\\sf A}_0\\oplus {\\sf A}_ji_j\\oplus {\\sf A}_ki_k\\oplus {\\sf A}_li_l \\rightarrow {\\sf A}_0\\oplus {\\sf A}_{j^{\\prime }}i_{j^{\\prime }}\\oplus {\\sf A}_{k^{\\prime }}i_{k^{\\prime }}\\oplus {\\sf A}_{l^{\\prime }}i_{l^{\\prime }}$ and $\\omega ^{j,k}_{j^{\\prime },k^{\\prime }}: {\\sf B}_0\\oplus {\\sf B}_ji_j\\oplus {\\sf B}_ki_k\\oplus {\\sf B}_li_l \\rightarrow {\\sf B}_0\\oplus {\\sf B}_{j^{\\prime }}i_{j^{\\prime }}\\oplus {\\sf B}_{k^{\\prime }}i_{k^{\\prime }}\\oplus {\\sf B}_{l^{\\prime }}i_{l^{\\prime }}$ denote isomorphisms of von Neumann algebras over isomorphic quaternion skew fields.", "There exists the mapping $f\\mapsto \\psi (f(T))$ of ${\\cal B}_u({\\cal A}_r,{\\cal A}_v)$ into ${\\cal N}({\\sf B})$ which is a $\\sigma $ -normal homomorphism so that $1\\mapsto I$ and $id \\mapsto \\psi (T)$ .", "Applying Theorem 46 we get this assertion.", "50.", "Corollary.", "Let $\\sf A$ be a quasi-commutative von Neumann algebra on a Hilbert space $X$ over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ , where $2\\le v\\le 3$ .", "Let also $E$ be an ${\\cal A}_v$ graded projection in $\\sf A$ and $T\\eta \\sf A$ and $f\\in {\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ .", "Then the identity $f((T{\\hat{\\cdot }}E)|_{E(X)}) = (f(T){\\hat{\\cdot }} E)|_{E(X)}$ is fulfilled.", "Proof.", "Consider the mapping $B\\mapsto (B{\\hat{\\cdot }}E)|_{E(X)}$ which is a $\\sigma $ -normal homomorphism $\\psi $ of ${\\cal N}({\\sf A})$ onto ${\\cal N}({\\sf A}E|_{E(X)})$ so that $\\psi (I)=E|_{E(X)}$ .", "But $E|_{E(X)}$ is the identity operator on $E(X)$ .", "Then Proposition 49 implies that $f((T{\\hat{\\cdot }}E)|_{E(X)}) =f(\\psi (T)) = \\psi (f(T))= (f(T){\\hat{\\cdot }} E)|_{E(X)}$ .", "51.", "Note.", "Consider a situation when $T$ and $B$ are normal operators, where $T$ and $B$ may be unbounded operators whose spectra are contained in the domain of a Borel function $g\\in {\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ and $g$ has an inverse Borel function $f$ , where $2\\le v \\le 3$ .", "Then $g(T) = g(B)$ if and only if $T=B$ .", "To demonstrate this mention that if $g(T)=g(B)$ , then $T=(f\\circ g)(T) = f(g(T)) = f(g(B)) = (f\\circ g)(B)=B$ due to Corollary 48.", "Observe particularly, that if $T^2=B^2$ with positive operators $T$ and $B$ , then $T=B$ .", "This means that a positive operator has a unique positive square root.", "Let $y$ be a non-zero vector in $X$ so that $Ty= b y$ for some $b\\in {\\cal A}_v$ for a normal operator $T$ , let also $f\\in {\\cal B}_u({\\cal A}_v,{\\cal A}_v)$ be a Borel function whose domain contains $sp (T)$ .", "Suppose in addition that $T$ is strongly right ${\\cal A}_v$ linear, that is by our definition $T(xu)=(Tx)u$ for each $x\\in X$ and $u\\in {\\cal A}_v$ .", "Take an ${\\cal A}_v$ graded projection $E$ with range $ \\lbrace x: ~ Tx=bx \\rbrace $ , which is closed since $T$ is a closed operator.", "Therefore, $TE=bE$ on $X$ , since $ET\\subseteq TE$ , and hence $f(T)y = [(f(T){\\hat{\\cdot }}E)|_{E(X)}] y = f((T{\\hat{\\cdot }} E)|_{E(X)})y = f(bE|_{E(X)})y =f(bI) y$ in accordance with Corollary 50.", "52.", "Example.", "Let $X$ be a separable Hilbert space over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ with $2\\le v\\le 3$ and an orthonormal basis $ \\lbrace e_n: ~ n\\in {\\bf N} \\rbrace $ , let also $\\sf A$ be the algebra of bounded diagonal operators $Te_n=t_ne_n$ , where $t_n \\in {\\cal A}_v$ for each natural number $n\\in {\\bf N} = \\lbrace 1, 2, 3,... \\rbrace $ .", "Then ${\\sf A}$ is isomorphic with $C(\\Lambda ,{\\cal A}_v)$ , where $\\Lambda = \\beta {\\bf N}$ is the Stone-$\\check{C}$ hech compactification of the discrete space $\\bf N$ due to Theorems 3.6.1, 6.2.27 and Corollaries 3.6.4 and 6.2.29 [6], since ${\\bf N}\\subset \\Lambda $ and $\\Lambda $ is extremely disconnected.", "Take points $s_n$ corresponding to the pure states $T\\mapsto <Te_n;e_n>$ of this algebra $\\sf A$ , where $n\\in {\\bf N}$ .", "Then the set $ \\lbrace s_n: ~ n\\in {\\bf N} \\rbrace $ is dense in $\\Lambda $ , since from $<Te_n;e_n>=0$ for every $n$ it follows that $T=0$ and each continuous function $f:\\Lambda \\rightarrow {\\cal A}_v$ vanishing on $ \\lbrace s_n: ~ n \\rbrace $ is zero.", "Consider the characteristic function $\\chi _{s_n}$ of the singleton $s_n$ .", "The projection corresponding to ${\\cal A}_ve_n$ lies in $\\sf A$ and a continuous function corresponds to this projection $\\chi _{s_n}\\in C(\\Lambda ,{\\cal A}_v)$ .", "Thus $s_n$ is an open subset of $\\Lambda $ .", "Therefore, $ \\lbrace s_n: ~ n \\rbrace $ is an open dense subset in $\\Lambda $ and its complement $Z=\\Lambda \\setminus \\lbrace s_n: ~ n \\rbrace $ is a closed nowhere dense subset in $\\Lambda $ .", "One can define the function $h(s_n)=t_n\\in {\\cal A}_v$ with $\\lim _n |t_n|=\\infty $ , hence this function $h$ is normal and defined on $\\Lambda \\setminus Z$ .", "As $t_n=n\\xi _n$ so that $\\xi _n\\in {\\cal A}_r$ with $|\\xi _n|=1$ for each $n$ we get a normal function $f$ corresponding to an operator $Q$ affiliated with $\\sf A$ and $Qe_n=n\\xi _n e_n$ .", "Choosing $t_n=(n^{1/4}-n)\\xi _n$ we obtain a normal function $g$ corresponding to an operator $B$ affiliated with $\\sf A$ so that $Be_n=(n^{1/4}-n)\\xi _ne_n$ .", "Take the vector $x= \\sum _{n=1}^{\\infty } n^{-1} z_n e_n$ with $z_n\\in \\lbrace \\pm 1, ~ \\pm \\xi _n, ~ \\pm \\xi _n^* \\rbrace $ for each $n$ , then $x\\in X$ and $\\nu _x( \\lbrace s_n \\rbrace )=n^{-2}$ .", "Thus one gets $\\int _{\\Lambda } d\\nu _x(b).|f(b)|^2 =\\infty \\mbox{ and}$ $\\int _{\\Lambda } d\\nu _x(b).|(f+g)(b)|^2 = \\sum _{n=1}^{\\infty }n^{-3/2} <\\infty .$ The function $f+g$ is normal when defined on $\\Lambda \\setminus Z$ , since $|(f+g)(s_n)|=n^{1/4}\\rightarrow \\infty $ and $f{\\hat{+}}g$ corresponds to $Q{\\hat{+}}B$ .", "In view of Theorem 44 this vector $x$ does not belong to ${\\cal D}(Q)$ , but $x\\in {\\cal D}(Q{\\hat{+}} B)$ .", "Thus $Q+B\\ne Q{\\hat{+}}B$ .", "Take now $f$ , $Q$ and $x$ as above and put $h(s_n)=n^{-3/4}\\xi _n$ .", "The operator $C$ corresponding to $h$ is bounded and $hf$ is normal when defined on $\\Lambda \\setminus Z$ .", "Thus the functions $hf$ corresponds to the product $C{\\hat{\\cdot }}Q$ and $\\int _{\\Lambda } d\\nu _x(b).|(hf)(b)|^2 =\\sum _{n=1}^{\\infty } n^{-3/2}<\\infty ,$ hence $x\\in {\\cal D}(C{\\hat{\\cdot }}Q)$ .", "Contrary $x\\notin {\\cal D}(CQ)$ , since $x\\notin {\\cal D}(Q)$ .", "Thus $CQ\\ne C{\\hat{\\cdot }}Q$ , consequently, the operator $CQ$ is not closed.", "In accordance with Section 23 this product in the reverse order $QC$ is automatically closed.", "53.", "Note.", "Let $T$ and $Q$ be two positive operators affiliated with a quasi-commutative von Neumann algebra $\\sf A$ , which acts on a Hilbert space $X$ over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ , where $2\\le v\\le 3$ , so that $\\sf A$ is isomorphic with $C(\\Lambda ,{\\cal A}_v)$ .", "Positive normal functions $f$ and $g$ correspond to these operators $T$ and $Q$ such that $f$ and $g$ are defined on $\\Lambda \\setminus W_f$ and $\\Lambda \\setminus W_g$ respectively.", "Therefore, their sum $f+g$ is defined on $\\Lambda \\setminus (W_f\\cup W_g)$ and is normal and corresponds to $T {\\hat{+}} Q$ , hence $0\\le \\int _{\\Lambda } d\\nu _x(t).|f(t)|^2 \\le \\int _{\\Lambda }d\\nu _x(t).|f(t)+g(t)|^2 <\\infty $ for each vector $x\\in {\\cal D}(T{\\hat{+}}Q)$ .", "This implies that $x\\in {\\cal D}(T)$ and symmetrically $x\\in {\\cal D}(Q)$ , consequently, $x\\in {\\cal D}(T+Q)$ and $T+Q = T{\\hat{+}}Q$ .", "If drop the condition that $T$ and $Q$ are positive, but suppose additionally that $W_f\\cap W_g=\\emptyset $ , then two disjoint open subsets $U_f$ and $U_g$ in $\\Lambda $ exist containing $W_f$ and $W_g$ correspondingly.", "Therefore, $cl (U_f)\\subset \\Lambda \\setminus U_g$ and the clopen set $cl (U_f)$ contains $W_f$ , since a Hausdorff topological space $\\Lambda $ is extremely disconnected.", "Thus $f$ is bounded on $X\\setminus cl (U_f)$ and $g$ is bounded on $cl (U_f)$ .", "Take the ${\\cal A}_v$ graded projection in $\\sf A$ corresponding to the characteristic function $\\chi _{cl (U_f)}$ of $cl (U_f)$ .", "Then two operators $QE$ and $T(I-E)$ belong to $L_q(X)$ .", "But the operator $TE$ is closed, since $T$ is closed and $E$ is bounded, consequently, $TE=T{\\hat{\\cdot }}E$ .", "We deduce that $(T{\\hat{+}} Q) E = TE {\\hat{+}} QE = TE + QE = (T+Q)E \\mbox{ and}$ $(T{\\hat{+}} Q)(I-E) = (T+Q)(I-E).$ Take a vector $x\\in {\\cal D}(T{\\hat{+}}Q)$ , then $Ex$ and $(I-E)x$ are in the domain ${\\cal D}(T{\\hat{+}}Q)$ , hence $Ex$ and $(I-E)x$ are in ${\\cal D}(T+Q)$ , consequently, $x\\in {\\cal D}(T+Q)$ and inevitably we get that $T{\\hat{+}}Q = T+Q$ .", "54.", "Proposition.", "Suppose that $\\sf A$ is a quasi-commutative von Neumann algebra over either the quaternion skew field or the octonion algebra ${\\cal A}_v$ with $2\\le v \\le 3$ and $T\\eta \\sf A$ .", "Let $(1)\\quad P(z)=\\sum _{k, s, ~ n_1+...+n_k\\le n} \\lbrace a_{s,n_1}z^{n_1}...a_{s,n_k} z^{n_k} \\rbrace _{q(2k)}$ be a polynomial on ${\\cal A}_v$ with ${\\cal A}_v$ coefficients $a_{s,m}$ , where $k,s \\in \\bf N$ , $0\\le n_l\\in \\bf Z$ for each $l$ , $~n$ is a marked natural number, $z^0 := 1$ , $~z\\in {\\cal A}_v$ , $q(m)$ is a vector indicating on an order of the multiplication of terms in the curled brackets, $a_{1,n_1}...a_{1,n_k}\\ne 0$ for $n_1+...+n_k=n$ and constants $c>0$ and $R>0$ exist so that $(2)\\quad c|z|^n\\le |\\sum _{k, s, ~ n_1+...+n_k= n} \\lbrace a_{s,n_1}z^{n_1}...a_{s,n_k}z^{n_k} \\rbrace _{q(2k)}|$ for each $|z|>R$ .", "Then the operator $\\sum _{k, s, ~n_1+...+n_k\\le n} \\lbrace a_{s,n_1}T^{n_1}...a_{s,n_k} T^{n_k} \\rbrace _{q(2k)}$ is closed and equal to $P(T)$ .", "Proof.", "We have that $\\sf A$ is isomorphic with $C(\\Lambda ,{\\cal A}_v)$ , where $\\Lambda $ is a Hausdorff extremely disconnected compact topological space (see Theorem I.2.52 [28]).", "An operator $T$ corresponds to some normal function $f$ defined on $\\Lambda \\setminus W_f$ so that a set $W_f$ is nowhere dense in $\\Lambda $ .", "Therefore, the composite function $P(f(z))$ is defined on $\\Lambda \\setminus W_f$ and normal.", "Thus $P(f(z))$ corresponds to the polynomial $P(T)$ of the operator $T$ .", "On the other hand, a vector $x$ is in ${\\cal D}(P(T))$ if and only if the integral $(3)\\quad \\int _{\\Lambda } d \\nu _x(t).|P(f(t))|^2 <\\infty $ converges.", "Consider the sets $\\Lambda _m := cl \\lbrace t: ~ |f(t)|< m \\rbrace $ .", "Since $\\lim _{|z|\\rightarrow \\infty } |P(z)|=\\infty $ and due to Condition $(2)$ there exists a positive number $m>0$ such that $(4)\\quad \\frac{1}{2}c|f(t)|^n\\le \\frac{1}{2} |\\sum _{k, s, ~n_1+...+n_k= n} \\lbrace a_{s,n_1}f(t)^{n_1}...a_{s,n_k} f(t)^{n_k} \\rbrace _{q(2k)}| \\le |P(f(t))|$ $\\forall $ $t\\in \\Lambda \\setminus (W_f\\cup \\Lambda _m)$ , consequently, $(5)\\quad \\int _{\\Lambda \\setminus (W_f\\cup \\Lambda _m)} d\\nu _x(t).|f(t)|^{2n}<\\infty .$ A measure $\\nu _x$ is non-negative and finite on $\\Lambda $ , the function $f$ is bounded on $\\Lambda _m$ , consequently, $f\\in L^{2n}(\\Lambda ,\\nu _x,{\\cal A}_v)$ , hence $f\\in L^k(\\Lambda ,\\nu _x,{\\cal A}_v)$ for each $1\\le k \\le 2n$ .", "The power $f^n$ of $f$ is defined on $\\Lambda \\setminus W_f$ and is normal, hence $f^n$ represents $\\overline{T^n}$ .", "In view of Theorem 44 the inclusion $x\\in {\\cal D}(\\overline{T^k})$ is fulfilled for each $k=1,...,n$ , particularly, $x\\in {\\cal D}(\\overline{T})={\\cal D}(T)$ .", "We have $\\overline{T}=T$ .", "Suppose that $\\overline{T^k}=T^k$ for $k=1,...,j-1$ .", "As $y\\in {\\cal D}(\\overline{T^j})$ , then $y\\in {\\cal D}(T)$ .", "Take a bounding ${\\cal A}_v$ graded sequence $\\mbox{}_lE$ in $\\sf A$ for $\\overline{T^{j-1}}$ and $T$ .", "Therefore, we deduce that $\\overline{T^j}\\mbox{ }_mE = (\\overline{T^{j-1}} {\\hat{\\cdot }}T)\\mbox{ }_mE = \\overline{T^{j-1}} \\mbox{ }_mE T\\mbox{ }_mE$ , consequently, $\\mbox{}_mE\\overline{T^j}y = \\overline{T^j}\\mbox{}_mEy=\\overline{T^{j-1}} \\mbox{ }_mE T\\mbox{}_mEy=\\overline{T^{j-1}} \\mbox{ }_mE Ty$ .", "On the other hand, the limits exist $\\lim _m \\mbox{ }_mE\\overline{T^j}y =\\overline{T^j}y $ and $\\lim _m \\mbox{ }_mET^jy =T^jy $ .", "The operator $\\overline{T^{j-1}}$ is closed, consequently, $Ty\\in {\\cal D}(\\overline{T^{j-1}})$ and $\\overline{T^{j-1}}Ty=\\overline{T^j}y$ , hence $\\overline{T^j}\\subseteq \\overline{T^{j-1}}T$ .", "By our inductive assumption $\\overline{T^{j-1}} = T^{j-1}$ .", "Therefore, $\\overline{T^j}\\subseteq T^j$ and hence $\\overline{T^j}= T^j$ .", "Thus by induction we get $\\overline{T^k}= T^k$ for each $k=1,...,n$ , hence $x\\in {\\cal D}(T^k)$ for every $k=1,...,n$ , consequently, $x\\in {\\cal D}(P(T))$ and inevitably $P(T) = \\sum _{k, s, ~n_1+...+n_k\\le n} \\lbrace a_{s,n_1}T^{n_1}...a_{s,n_k} T^{n_k} \\rbrace _{q(2k)}.$ 55.", "Remark.", "For $v\\ge 4$ the Cayley-Dickson algebra ${\\cal A}_v$ has divisors of zero.", "Therefore, we have considered mostly the quaternion and octonion cases $2\\le v \\le 3$ .", "It would be interesting to study further spectral operators over the Cayley-Dickson algebras ${\\cal A}_v$ with $v\\ge 4$ , but it is impossible to do this in one article or book if look for comparison on the operator theory over the complex field." ] ]

[ [ "Etude d'un mod\\`ele de dynamique des populations" ], [ "Abstract We study an infinite dimensional dynamical system that was proposed by J.C. Yoccoz and N.G.", "Yoccoz for modeling the population dynamics of some small rodents.", "We show an attractor exist in a large domain of the parameter space.", "Thanks to simulations, we describe the possible dynamics of the system, in particular some Henon-type strange attractors, but also a new kind of attractor.", "We study the complexity of the new attractor and the reasons causing it, from both geometrical and dynamical points of view.", "Thus, we show a chaotic behaviour can be obtained in a population dynamics model with a strong density-dependence and some delay, but without seasonal variations over years." ], [ "Introduction", "La dynamique des populations est au cœur de l'interface entre systèmes dynamiques et biologie.", "Ainsi, l'un des modèles biologiques les plus simples et les plus importants — le modèle logistique — correspond à la dynamique des polynômes quadratiques, dont l'étude mathématique est des plus intéressantes.", "Ce seul cas montre combien il est aisé d'obtenir un comportement complexe sans introduire beaucoup de complexité dans le modèle.", "Mais le modèle logistique est bien trop peu réaliste pour que sa complexité dynamique puisse être directement interprétée dans le cadre d'une population biologique réellement observée.", "D'un point de vue mathématique, les systèmes de dimension 1 présentent un nombre limité de dynamiques possibles.", "Il est donc intéressant de considérer des modèles de dimension supérieure tels que l'application de Hénon (qui est une petite perturbation de la dynamique d'un polynôme quadratique, en dimension 2), et qui sont encore mal compris du point de vue théorique.", "Le pas suivant dans cette démarche consiste en l'étude de systèmes dynamiques de dimension infinie, qui seront de « petites perturbations » des modèles précédents, i.e.", "le modèle logistique.", "En effet, si l'on veut intégrer le phénomène biologique de maturation des jeunes individus, il est nécessaire de considérer la fonction d'évolution de l'effectif en temps continu, et non seulement sa valeur à un instant donné, ce qui donne un système dynamique de dimension infinie (ou de grande dimension, si l'on discrétise ce système).", "Un autre phénomène intéressant à considérer est l'influence des rythmes saisonniers sur un tel système, lorsqu'il se combine avec cet effet de retard induit par le temps de maturation des jeunes.", "Le modèle que nous considérons combine ces deux effets avec une forme de densité-dépendance un peu différente de celle du modèle logistique.", "Nous commencerons par définir le modèle étudié, tel qu'il a été énoncé dans [8], puis sous une forme légèrement modifiée, en motivant celle-ci aussi bien par des raisons biologiques que des raisons de simplicité pratique.", "Nous verrons ensuite ce que l'on peut montrer simplement par une étude théorique a priori, point de départ d'une étude future plus approfondie (mais surtout bien plus difficile).", "La dernière et plus importante partie de notre étude sera consacrée à l'analyse des résultats de simulations numériques, en vue de comprendre l'influence des paramètres sur la dynamique du système et d'analyser plus finement un des attracteurs étranges que nous avons pu observer.", "Ce travail est bien sûr loin d'être complet, et se veut surtout être une introduction (et une motivation) pour de futurs travaux, aussi bien mathématiques que biologiques.", "Je tiens à remercier particulièrement Jean-Christophe Yoccoz pour le temps qu'il m'a consacré, ses nombreux conseils et la clarté des explications théoriques qu'il m'a données.", "Je remercie également Gilles Yoccoz pour ses conseils bibliographiques pour la partie biologique de ce mémoire." ], [ "Modèle initial continu", "Le modèle suivant, défini dans [8], décrit l'évolution temporelle d'une population de campagnols.", "$ N(t) = \\int _{A_0}^{A_1} {S(a) m_{\\rho }(t-a) N(t-a) m(N(t-a)) da}$ $t$ est le temps (en années), $N$ la population active (i.e.", "d'âge supérieur à $A_0$ ), $A_0$ l'âge de maturation, $A_1$ l'âge maximal, $S$ le taux de survie, $m_{\\rho }$ le paramètre de saison (décrit une probabilité de reproduction en fonction de la saison), $m(N)$ le taux de fécondité individuel annuel pour une population active de taille $N$ .", "On a choisi des formes simples pour les fonctions $S$ , $m_{\\rho }$ et $m$  : $S(a) = 1-\\frac{a}{A_1}$ $m_{\\rho } (t) ={\\left\\lbrace \\begin{array}{ll}0 \\text{ si } 0 \\le t \\le \\rho \\text{ mod.", "1} \\\\1 \\text{ si } \\rho \\le t \\le 1\\end{array}\\right.", "}$ $m(N) = {\\left\\lbrace \\begin{array}{ll}m_0 \\text{ si } N \\le 1 \\\\m_0 N^{-\\gamma } \\text{ sinon}\\end{array}\\right.", "}$ Cependant, pour éviter des artéfacts dûs à la non-régularité du système, il nous a semblé préférable de remplacer $m$ et $m_{\\rho }$ par des fonctions un peu plus régulières." ], [ "Lissage de la fécondité", "La fonction $m$ définie par (REF ) est continue mais pas $C^1$ .", "Il y a une forte rupture de pente à la valeur critique $N=1$ .", "On pourrait aisément « recoller » les deux parties de la courbe pour obtenir une fonction $C^{\\infty }$ , mais cela ne serait pas très pratique pour les simulations numériques.", "Nous utilisons ainsi une parabole intermédiaire qui rend $m$ $C^1$ .", "$m(N) = {\\left\\lbrace \\begin{array}{ll}m_0 \\text{ si } N \\le N_{1,\\gamma } \\\\m_0 \\times \\left( A_{\\gamma } + B_{\\gamma } N + C_{\\gamma } N^2\\right) \\text{si } N_{1,\\gamma } < N \\le N_{2,\\gamma } \\\\m_0 N^{-\\gamma } \\text{ si } N_{2,\\gamma } < N\\end{array}\\right.", "}$ Pour garder un modèle réaliste et suffisamment proche du modèle initial, il faut conserver la décroissance de la fécondité $N \\mapsto m(N)$ , et s'assurer que la parabole rejoint les valeurs extrêmes $m_0$ et $m_0 N^{-\\gamma }$ suffisamment près de $N=1$ .", "Pour définir complètement les paramètres $N_{1,\\gamma }$ , $A_{\\gamma }$ , $B_{\\gamma }$ , $C_{\\gamma }$ et $N_{2,\\gamma }$ , on impose également $m(N_{2,\\gamma }) = m_0 /2$ .", "Cette dernière contrainte permet de limiter la zone intermédiaire, ce qui simplifiera notamment les calculs explicites de la section .", "On a ainsi les conditions suivantes : $\\left\\lbrace \\begin{aligned}A_{\\gamma } + B_{\\gamma } N_{1,\\gamma } + C_{\\gamma } N_{1,\\gamma }^2 &= 1 \\\\B_{\\gamma } + 2 C_{\\gamma } N_{1,\\gamma } &= 0 \\\\N_{2,\\gamma }^{-\\gamma } &= \\frac{1}{2} \\\\A_{\\gamma } + B_{\\gamma } N_{2,\\gamma } + C_{\\gamma } N_{2,\\gamma }^2 &= \\frac{1}{2}\\\\B_{\\gamma } + 2 C_{\\gamma } N_{2,\\gamma } &= - \\gamma N_{2,\\gamma }^{-\\gamma -1} = - \\frac{\\gamma }{2 N_{2,\\gamma }}\\end{aligned} \\right.", "$ Figure: Comparaison des fonctions defécondité.De plus, on veut $C_{\\gamma } \\le 0$ pour garantir la décroissance de $m$ .", "On choisit donc : $\\left\\lbrace \\begin{aligned}N_{2,\\gamma } &= 2^{1/\\gamma } \\\\C_{\\gamma } &= \\frac{- \\gamma ^2}{8 \\times 4^{1/\\gamma }} \\\\A_{\\gamma } &= \\frac{1}{2} \\left( 1+\\gamma - \\frac{\\gamma ^2}{4} \\right) \\\\B_{\\gamma } &= 2^{-1/\\gamma } \\times \\left( \\frac{\\gamma ^2}{4} - \\frac{\\gamma }{2} \\right) \\\\N_{1,\\gamma } &= 2^{1/\\gamma } \\times \\left( 1 - \\frac{2}{\\gamma }\\right)\\end{aligned} \\right.", "$ La figure REF représente les deux fonctions $m$ — continue et $C^1$ — pour $\\gamma = 825$ ." ], [ "Lissage du facteur saisonnier", "La fonction $m_{\\rho }$ définie par (REF ) n'est pas continue, il est légitime de vouloir considérer un facteur saisonnier un peu plus régulier.", "On a choisi, arbitrairement, de le rendre $C^1$ en effectuant le passage de 0 à 1 à l'aide d'un cosinus.", "Pour cela, on ajoute un paramètre $\\epsilon $ qui est la durée du printemps et celle de l'automne.", "La durée de l'été est désormais $1 - \\rho - \\epsilon $ et non plus $1- \\rho $ .", "Prendre $\\epsilon = 0$ ramène bien sûr au cas précédent.", "Figure: Comparaison des fonctionsm ρ,ϵ m_{\\rho ,\\epsilon }.$m_{\\rho ,\\epsilon } (t\\text{ mod.", "}1) ={\\left\\lbrace \\begin{array}{ll}\\frac{1}{2}(1+ \\cos (\\pi \\times (\\frac{t}{\\epsilon }+\\frac{1}{2}))) &\\text{ si } 0 \\le t < \\epsilon /2 \\\\0 &\\text{ si } \\epsilon /2 \\le t < \\rho - \\epsilon /2 \\\\\\frac{1}{2} \\left( 1+ \\cos \\left(\\pi \\times (\\frac{t-\\rho }{\\epsilon }-\\frac{1}{2}) \\right) \\right) &\\text{ si } \\rho - \\epsilon /2 \\le t < \\rho + \\epsilon /2 \\\\1 &\\text{ si } \\rho + \\epsilon /2 \\le t < 1 - \\epsilon /2 \\\\\\frac{1}{2}(1+ \\cos (\\pi \\times (\\frac{t-1}{\\epsilon }+\\frac{1}{2})))&\\text{ si } 1-\\epsilon /2 \\le t < 1\\end{array}\\right.", "}$ Cette définition n'étant valable que lorsque $\\epsilon \\le \\rho \\le 1-\\epsilon $ , on posera $\\epsilon = \\min (\\rho ,1-\\rho )$ lorsque ce n'est pas le cas a priori.", "La figure REF représente $m_{\\rho ,\\epsilon }$ pour $\\rho = 041$ et deux valeurs de $\\epsilon $ .", "Remarquons enfin que l'on aurait également pu rendre $m_{\\rho ,\\epsilon }$ $C^{\\infty }$ dès que $\\epsilon >0$ en utilisant autre chose qu'un cosinus.", "Le choix que nous avons fait tient compte de la simplicité des calculs numériques futurs.", "Pour les détails concernant les simulations numériques, voir le paragraphe REF en annexe." ], [ "Espèces concernées", "Le modèle que nous venons de décrire a été élaboré en relation avec l'étude de la dynamique de certaines populations de petits rongeurs.", "Ceux-ci se caractérisent en effet par un fort investissement dans la reproduction (portées importantes et maturité sexuelle très rapide) et de grandes variations annuelles de la taille de la population.", "Plusieurs espèces de campagnols ont ainsi été étudiées, notamment le campagnol roussâtre Clethrionomys glaerolus (figure REF  ; [14], [2], [15]) et Microtus townsendii [5].", "Figure: Campagnol roussâtre (bankvole, Clethrionomys glareolus).", "Ce petit rongeur, de lafamille des Microtinés, vit dans les forêts tempérées etse nourrit principalement de graines.C'est plus particulièrement une population de Microtus epiroticussibling vole en anglais., introduite accidentellement il y a une cinquantaine d'années dans l'archipel arctique de Svalbard (en Finlande, dans le Spitzberg), qui est visée par ce modèle.", "Ces campagnols possèdent en effet une fécondité est extrêmement élevée pour des mammifères.", "De plus, cette espèce étant menacée d'extinction, l'étude de sa démographie permettrait également de mieux la protéger.", "La figure REF montre ainsi de grandes fluctuations de population, et des effectifs minimaux très faibles, de l'ordre de quelques individus.", "Figure: Microtusepiroticus à Svalbard : nombre d'individus capturés entre1991 et 2002.", "(Source : NINA and University of Tromsø.", "2003© Statistics Norway)." ], [ "Mécanismes envisagés", "Plusieurs causes possibles à ces phénomènes ont été étudiées.", "Il a été montré expérimentalement qu'une augmentation de la quantité de nourriture disponible augmente la densité mais n'a pas d'influence sur la densité-dépendance (C. glaerolus, Finlande [15]).", "Plus précisément, la nourriture disponible est liée aux variations inter-annuelles, en jouant sur la survie de l'année suivante ; on observe des variations saisonnières de taille comparable aux variations inter-annuelles et qui se caractérisent par une survie plus faible au printemps (C. glaerolus, Belgique [2]).", "Le facteur climatique semble lui aussi jouer un grand rôle : une comparaison entre M. epiroticus à Svalbard et Chionomys nivalissnow vole en anglais.", "dans les Alpes françaises indique une relation entre un environnement stable et un faible turnoveri.e.", "forte survie et faible fécondité [9].", "En effet, la population vivant dans l'Arctique, où les conditions hivernales sont très variables, a une reproduction extrêmement rapide, tandis que la population alpine, dont l'environnement est stable, se reproduit peu et a une forte survie.", "On a ainsi essayé d'inclure une stochasticité environnementale et démographique dans les modèles afin d'expliquer les fluctuations de population observées (C. glaerolus, Alpes [14]).", "Un phénomène important pourrait être relié à ces facteurs environnementaux : la plasticité de l'âge à maturité.", "Il a ainsi été montré que les femelles nées en début de saison de reproduction ont avantage à se reproduire rapidement, malgré le coût élevé d'une reproduction précoce (M. townsendii, Vancouver [5]) : les femelles naissant plus tôt peuvent se reproduire avant le fin de la saison de reproduction, augmentant ainsi la contribution de leur mère à la population totale." ], [ "Paramètres du modèle", "Les paramètres de croissance et de reproduction des deux populationsM.", "epiroticus à Svalbard et M. arvalis en Finlande.", "ont été évalués dans [10].", "À partir de ces conclusions, nous pouvons justifier le modèle et le choix de paramètre « typiques »." ], [ "Densité-dépendance", "La forme particulière de densité-dépendance se justifie car le facteur limitant est ici le nombre très restreint de sites de reproduction.", "Ainsi, seule la population (femelleComme souvent en dynamique des populations, seule la population femelle est considérée dans la mesure où elle est le facteur limitant de la reproduction.", "Il n'est intéressant de considérer les mâles que si la femelle a peu de chances de rencontrer un mâle (par exemple si la densité de population est faible), ou bien si le sex-ratio est loin de $1:1$ .)", "mature doit entrer en ligne de compte, et en cas de surpopulation, les quelques sites disponibles perdent beaucoup en qualité.", "Ceci est confirmé par les conclusions de [15], qui indiquent dans le cas de C. glaerolus une densité-dépendance plus forte en été, à cause de la maturation des femelles.", "Nous choisissons de ne pas faire dépendre $\\gamma $ de la saison puisque la variation observée est liée à la prise en compte de la densité totale, et non de la densité de femelles matures comme nous le faisons ici." ], [ "Saison de reproduction", "La période de reproduction correspond à la saison de croissance des plantes, c'est-à-dire du printemps à l'automne.", "Nous considérons avec ce modèle un climat parfaitement prévisible, identique d'une année sur l'autre.", "Il s'agit de savoir si l'on peut observer un comportement chaotique dans des conditions parfaitement stables.", "Sa durée varie donc selon les lieux.", "À Svalbard comme dans les Alpes, celle-ci dure de 3 à 4 mois (Juin à Septembre) [9], i.e.", "$\\rho \\approx 07$ .", "Dans la plupart des environnements tempérés, celle-ci est beaucoup plus longue.", "Ainsi, en Belgique, C. glaerolus se reproduit en général entre la deuxième semaine d'Avril et la fin du mois d'Octobre [2], mais peut varier de telle sorte que l'on a $0 35 < \\rho < 0 45$ ." ], [ "Fécondité", "Un élément important du modèle est la forte fécondité des campagnols.", "Pour ajuster les paramètres du modèle, nous avons besoin d'évaluer le nombre de jeunes femelles par femelle mature et par an, en l'absence de densité-dépendance.", "Pour M. epiroticus [10], la période de gestation est de 20 jours, et la taille des portées varie de 45 à 7, celle-ci augmentant pour une même femelle au fur et à mesure de ses reproductions.", "Le sex ratio est très proche de $1:1$ .", "Dans la mesure où une femelle peut se reproduire pendant la période d'allaitement, on en déduit une valeur maximale $m_0 \\approx 18 \\times 6 \\times 0 5 = 54$ .", "Dans le cas de M. townsendii [5], on observe de 5 à 6 portées par saison de reproduction (celle-ci durant de Mars à Novembre, soit environ 8 mois : $\\rho \\approx 03$ ), chacune comportant 5 à 8 individus.", "On a donc dans ce cas $m_0 /3 \\approx 5 \\times 65 \\times 0 5$ soit $m_0 \\approx 24$ .", "Il n'est pas étonnant de constater que cette valeur est bien inférieure à celle de M. epiroticus, qui représente un cas extrême parmi les mammifères.", "On peut donc prendre comme valeur $m_0 = 50$ , mais une fécondité légèrement inférieure serait sans doute plus réaliste." ], [ "Âge de première reproduction", "L'âge de maturité des femelles est supérieur à 17 jours, auquel il faut rajouter la durée de gestation, soit 20 jours supplémentaires [10].", "L'âge de première reproduction de M. epiroticus est donc au minimum 37 jours, i.e.", "$A_0 \\ge 0 10$ .", "En moyenne, on observe plutôt une première portée à un âge d'environ 50 jours, i.e.", "$A_0 \\approx 014$ .", "Une telle précocité ne se retrouve pas chez les autres Microtinés, à l'exception de M. arvalis en Finlande.", "On utilisera donc 01 comme valeur minimale, tandis que $A_0= 0 20$ (correspondant à 75 jours) est plus classique." ], [ "Survie", "Pour M. epiroticus à Svalbard, la survie hivernale est de l'ordre de 01 (et très variable selon les années), et la survie estivale 085 [9].", "Le taux de mortalité par année est donc de 0046 en hiver et 052 en été.", "L'âge maximal $A_1$ est toujours inférieur à deux ans.", "Chez C. glaerolus, en Belgique, les variations saisonnières de la survie ont été étudiées plus en détails [2].", "Le taux de survie est plus élevé en hiver (095 par semaine, soit 007 par an) qu'en été (090 par semaine, soit 0004 par an).", "Elle est également légèrement différente chez les femelles matures et immatures.", "Le taux de mortalité est supposé dans le modèle constant avec l'âge, et indépendant des saisons, ce qui est loin d'être le cas en général.", "L'âge maximal de 2 ans est également une légère sur-estimation de ce qu'il est en réalité.", "Pour plus de réalisme, la fonction de survie est sans doute l'un des premiers éléments du modèle à modifier." ], [ "Problèmes posés", "L'étude de ce modèle n'a pas pour but de faire des prévisions précises concernant l'avenir de la population de Microtus epiroticus à Svalbard.", "Nous nous efforcerons de considérer des paramètres réalistes pour de petits rongeurs, pas nécessairement M. epiroticus.", "Il s'agit surtout d'effectuer un travail théorique dans un cadre assez simple, afin de savoir si l'on peut observer une grande variabilité d'effectifs (voire une dynamique chaotique) dans un modèle complètement déterministe, dans un environnement régulier.", "Et si oui, quelles sont les facteurs biologiques déterminants (fécondité, âge de maturité, durée de l'hiver) ?", "Un autre objectif est de bien comprendre ce modèle très simple avant de le complexifier en introduisant d'autres mécanismes pouvant jouer un rôle dans la dynamique de cette population, parmi ceux que nous avons évoqués précédemment." ], [ "Étude théorique", "Les résultats de cette partie proviennent principalement de [8], où les fonctions $m_{\\rho }$ et $m$ considérées étaient données par (REF ) et (REF ).", "Nous avons considéré ici un cadre un peu plus général, valable pour les fonctions $m_{\\rho }$ et $m$ « lissées »." ], [ "Définition du système dynamique", "Pour $t_0 \\in /$ , notons $Y_{t_0}$ l'ensemble des fonctions continues $N$ sur $[-A_1 ; 0]$ à valeurs positives et vérifiant la condition $ N(0) = \\int _{A_0}^{A_1} S(a) N(-a)m(N(-a))m_{\\rho }(t_0 - a) da.$ Le système dynamique défini par (REF ) est donné par le semi-groupe $(T_s)_{s\\ge 0}$  : $ T^s (t,N) = ( t+s (1) , N^s_t ) $ $N^s_t(-a) ={\\left\\lbrace \\begin{array}{ll}N(s-a) \\text{ si } 0 \\le s \\le a \\le A_1& \\\\\\int _{A_0}^{A_1} S(b) N(s-a-b) m(N(s-a-b)) m_{\\rho }(t+s-a-b) db\\text{ sinon}&\\end{array}\\right.", "}$ Ceci est bien défini pour $0 \\le s\\le A_0$ , on l'étend à $s \\ge 0$ par la propriété de semi-groupe.", "L'espace des phases est alors $ Y^{\\sharp } = \\lbrace (t,N) \\, t \\in / ,\\, N \\in Y_t\\rbrace .$ Dans la suite, on écrira parfois $N^s$ au lieu de $N^s_t$ lorsque cela ne crée pas de confusion.", "On s'intéresse à l'application $T^1 : Y_0 \\rightarrow Y_0$ donnant l'évolution de la population d'une année sur l'autreLe choix de $t_0=0$ est arbitraire, on pourrait aussi bien considérer $T^1 : Y_{t_0} \\rightarrow Y_{t_0}$ , qui donnerait le même type de dynamique.. À $N \\in Y_{t_0}$ donnée, on associe ainsi une unique fonction continue $\\overline{N}$ définie sur $[-A_1 ; +\\infty [$ prolongeant $N$ et compatible avec $T$ (c'est-à-dire la solution de l'équation (REF )).", "On munit $Y^{\\sharp }$ de sa topologie naturelle, induite par la topologie produit sur $/ \\times \\mathcal {C}([-A_1 ; 0])$ , l'espace $\\mathcal {C}([-A_1 ; 0])$ des fonctions continues de $[-A_1 ; 0]$ dans $$ étant muni de la topologie de la convergence uniforme.", "Il découle alors de la continuité uniforme de $N \\rightarrow N \\times m(N)$ la propriété suivante : Pour tout $s\\ge 0$ , $T^s$ est un opérateur continu $Y^{\\sharp } \\rightarrow Y^{\\sharp }$ .", "De plus, $N \\rightarrow N \\times m(N)$ étant $K_f$ -Lipschitzienne (la constante $K_f$ peut être calculée explicitement en fonction des paramètres du modèle), l'application $T^s : Y_t \\rightarrow Y_{t+s}$ est $K$ -Lipschitzienne, avec $K = \\max (1,(A_1-A_0)\\times K_f)$ .", "Ceci découle directement de la définition de $N^s_t$ .", "La constante $K$ dépend uniquement des paramètres du modèle, et pas de $t\\in / $ ." ], [ "Existence d'un attracteur", "Nous allons montrer que pour des valeurs raisonnables des paramètres, un tel système dynamique possède un attracteur, ce qui nécessite plusieurs lemmes techniques.", "Nous n'utiliserons pas les formes explicites des fonctions $m_{\\rho }$ et $m$ (pour rester généraux, dans la mesure où celles-ci pourraient être modifiées ultérieurement), mais uniquement les hypothèses suivantes : $m_0 \\ge m(N) &\\ge \\frac{m_0}{2} &\\text{ si } N \\le 1 \\\\m_0 N^{- \\gamma } \\ge m(N) &\\ge \\left( \\frac{1}{2} \\wedge N^{-\\gamma } \\right) m_0 &\\text{ si } N \\ge 1 \\\\ 1 \\ge m_{\\rho }(t) &\\ge 0 \\, &\\forall t $ $m_{\\rho }(t)=1 \\text{ sur un intervalle de longueur }1-\\rho -\\epsilon .$ On voit aisément que les fonctions définies par (REF ) et (REF ) (resp.", "(REF ) et (REF )) vérifient ces hypothèses.", "Le symbole $\\wedge $ est employé ici et dans la suite à la place de $\\min $ , de même que $\\vee $ signifie $\\max $ .", "Posons $ c_0 :&= \\int _{A_0+\\rho +\\epsilon }^{A_0+1} S(a)da \\\\ &= (1-\\rho -\\epsilon ) \\left( 1 - \\frac{ 1 + \\rho + \\epsilon +2A_0}{2A_1} \\right).$ Nous nous plaçons désormais dans le cas où les paramètre vérifient les conditions suivantes : $\\gamma \\ge 1 \\\\A_1 \\ge (2A_0) \\vee (A_0+1)\\\\c_0 \\times m_0 >2 \\\\\\rho + \\epsilon < 1$ Ces conditions sont très raisonnables, et toujours vérifiées au cours des simulations que nous avons faites.", "En effet, si on impose $\\rho +\\epsilon \\le \\frac{6}{10}$ , $A_1 \\ge 2$ , $A_0 \\le 1/2$ , $\\gamma \\ge 1$ , alors $c_0 \\ge \\frac{14}{100}$ et donc $m_0\\ge 15$ suffit pour satisfaire (REF ).", "Il n'y a donc pas à s'inquiéter du manque de finesse de cette majoration.", "Soit $t_0 \\in / $ , $N \\in Y_{t_0}$ .", "On a alors, pour tout $0\\le s \\le A_0$  : $ N(s) \\le N_{\\max } := m_0 \\frac{A_1}{2}\\left( 1-\\frac{A_0}{A_1} \\right)^2$ On a toujours $N m(N) \\le m_0$ , d'après (REF ), (), et car $\\gamma \\ge 1$ .", "Comme de plus $m_{\\rho }\\le 1$ (), on a $ N(s) \\le m_0 \\int _{A_0}^{A_1}S(a)da = N_{\\max }.", "$ Soit $N \\in Y_{t_0}$ telle que $N \\le N_{\\max }$ .", "On a $i(N)=\\min _{[-A_1 ; 0]} N >0$ .", "Si $i(N) \\le N_{\\max }^{1-\\gamma }$ , alors $N(s) \\ge \\frac{c_0m_0}{2} i(N)$ pour $0 \\le s \\le A_0$ .", "Si $i(N) \\ge N_{\\max }^{1-\\gamma }$ , alors $N(s) \\ge \\frac{c_0 m_0}{2}N_{\\max }^{1-\\gamma }$ pour $0 \\le s \\le A_0$ .", "Commençons par montrer que $N m(N) \\ge \\frac{m_0}{2} \\times \\left( i(N) \\wedge N_{\\max }^{1-\\gamma } \\right)$ sur $[-A_1 ; 0]$  : si $N \\le 1$ , $\\begin{split} N m(N) &\\ge N \\times \\frac{m_0}{2} \\\\ &\\ge i(N) \\times \\frac{m_0}{2},\\end{split}$ et si $N \\ge 1$ , $\\begin{split} N m(N) &\\ge \\frac{m_0}{2} \\times N^{1-\\gamma } \\\\ &\\ge \\frac{m_0}{2} \\times N_{\\max }^{1-\\gamma }.\\end{split}$ Par conséquent, $ N(s) \\ge \\frac{m_0}{2} \\times (i(N) \\wedge N_{\\max }^{1-\\gamma }) \\times \\int _{A_0}^{A_1} S(a)m_{\\rho }(t_0+s-a) da.", "$ En fonction de la valeur de $t_0+s$ , on peut trouver un sous-intervalle de $[A_0 ; A_0+1] \\subset [A_0 ;A_1]$ , de longueur $1-\\rho -\\epsilon $ sur lequel $m_{\\rho }(t_0+s-\\cdot )$ vaut 1.", "La dernière intégrale est donc minorée par la même intégrale restreinte à ce sous-intervalle, qui est plus grande que $c_0$ car $S$ est décroissante.", "Ceci achève la preuve du lemme.", "Si $\\gamma \\ge 1$ , $\\frac{c_0 m_0}{2} >1$ , $N \\in Y_{t_0}$ , alors pour $s$ assez grand (dépendant de $N$ ), on a : $ \\frac{c_0m_0}{2} N_{\\max }^{1-\\gamma } \\le N^s(a) \\le N_{max} ,\\, \\forall a\\in [-A_1 ; 0].", "$ Remarquons que l'on peut remplacer la condition (REF ) par $c_0 m_0 >1$ dans le cas où $m$ est définie par (REF ) (cf.", "[8]).", "La constante 2 a été choisie arbitrairement dans l'opération de lissage de $m$ , celle-ci pourrait être prise plus proche de 1 sans difficulté supplémentaire, mais toujours strictement supérieure à 1.", "Soit $N \\in Y_{t_0}$ .", "Posons $L=m_0 \\left( 3-\\frac{A_0}{A_1}\\right)$ .", "Alors, si $0 \\le s_0 \\le s_1 \\le A_0$ , on a ${\\overline{N}(s_1) - \\overline{N}(s_0)} \\le L {s_1 -s_0}.", "$ Remarquons tout d'abord que $m_{\\rho } \\le 1$ , $0 \\le S \\le 1$ et $ {S(s_1-u) - S(s_0-u)} \\le A_1^{-1} {s_1-s_0} .$ Plaçons-nous dans le cas où $s_1-A_1 \\le s_0-A_0$ (c'est vrai car on a supposé $A_1 \\ge 2 \\times A_0$ ), et écrivons la définition de $\\overline{N}(s_i)$ en fonction de $N$ .", "$\\begin{split}{\\overline{N}(s_1) - \\overline{N}(s_0)} &= \\Bigl | - \\int _{s_0-A_1}^{s_1 -A_1} S(s_0-u) N(u)m(N(u))m_{\\rho }(t_0+u)du \\\\ &+\\int _{s_1 -A_1}^{s_0 -A_0} [S(s_1-u)-S(s_0-u)] N(u)m(N(u))m_{\\rho }(t_0+u)du \\\\ &+ \\int _{s_0 -A_0}^{s_1 -A_0} S(s_1-u)N(u)m(N(u))m_{\\rho }(t_0+u)du \\Bigr |\\end{split} \\\\&\\le m_0 {s_1-s_0} + m_0\\left(1-\\frac{A_0}{A_1}\\right){s_0-s_1} + m_0 {s_1-s_0}\\\\&\\le L {s_0-s_1}$ Nous pouvons maintenant définir $\\begin{split}\\mathcal {K}_{t_0} = & \\Bigl \\lbrace N \\in Y_{t_0}; \\, \\forall s \\in [-A_1 ; 0] , \\, c_0 m_0 N_{\\max }^{1-\\gamma } \\le N(s) \\le N_{\\max } , \\\\&\\forall s_0 , s_1 \\in [-A_1 ; 0], \\, {N(s_0) - N(s_1) } \\le L { s_0 - s_1} \\Bigr \\rbrace \\end{split}$ qui est une partie compacte de $Y_{t_0}$ pour la topologie de la convergence uniforme, d'après le théorème d'Ascoli.", "Les lemmes que nous venons de démontrer peuvent se formuler de la façon suivante : Soit $N \\in Y_0$ , $(T^s)_{s \\ge 0}$ le semi-groupe défini par l'équation (REF ).", "On se place dans les conditions précédemment énoncées pour les différents paramètres du modèle.", "Si $N \\in \\mathcal {K}_0$ , alors $N^s \\in \\mathcal {K}_s$ pour tout $s \\ge 0$ .", "En particulier $T^1(\\mathcal {K}_0) \\subset \\mathcal {K}_0$ .", "En général, il existe $s_0 \\ge 0$ (dépendant de $N$ ) tel que $N^s \\in \\mathcal {K}_s$ pour tout $s\\ge s_0$ .", "$N^s \\in Y_s$ par définition, $N^s \\le N_{max}$ d'après le lemme REF , $N^s$ reste $L$ -lipschitzienne d'après le lemme REF .", "La partie 2 du lemme REF donne la minoration, en utilisant que $N \\in Y_0$ .", "On utilise le corollaire REF pour montrer l'existence de $s_0$ , le reste de la preuve étant identique.", "L'attracteur du système dynamique $(Y^{\\sharp },(T^s)_{s\\ge 0})$ est défini par : $\\Lambda = \\left\\lbrace (t,N) \\, t \\in / , \\, N \\in \\Lambda _t \\right\\rbrace \\\\\\text{avec } \\Lambda _t = \\bigcap _{n \\ge 0} T^n (\\mathcal {K}_t)$ La propriété suivante justifie l'appellation d'attracteur pour $\\Lambda $ .", "$\\Lambda $ est une partie compacte de $Y^{\\sharp }$ .", "Pour tout $s \\ge 0$ , $T^s(\\Lambda ) = \\Lambda $ .", "Pour tout voisinage $U$ de $\\Lambda $ , et toute condition initiale $(0,N)$ , $N \\in Y_0$ , il existe $s_0$ (dépendant de $N$ et $U$ ) tel que $T^s(0,N) \\in U$ pour tout $s \\ge s_0$ .", "D'après la continuité de $T^1$ et la compacité de $\\mathcal {K}_t$ , $\\Lambda _t$ est compact pour tout $t \\ge 0$ .", "De plus, $T^{\\epsilon } \\xrightarrow[\\epsilon \\rightarrow 0]{} Id$ uniformément sur $Y^{\\sharp }$ (c'est une conséquence du lemme REF , car on a alors ${T^{\\epsilon }(N)-N}_{\\infty } \\le L \\epsilon $ ).", "Par conséquent, $\\Lambda $ est compact.", "Par construction, $T^s(\\lbrace t\\rbrace \\times \\Lambda _t) = \\lbrace t+s\\rbrace \\times \\Lambda _{t+s}$ pour tous $s,t \\in /$ , d'où $T^s(\\Lambda ) =\\Lambda $ pour tout $s \\ge 0$ .", "On peut supposer $N \\in \\mathcal {K}_0$ d'après la proposition REF .", "La suite $T^n(0,N)$ est alors contenue dans $\\mathcal {K} = \\lbrace (t,N), t \\in / , N \\in \\mathcal {K}_t \\rbrace $ qui est compact.", "Tout point d'accumulation de cette suite est nécessairement dans $\\Lambda $ , et donc $T^n(0,N) \\in U$ pour $n$ entier assez grand.", "De même, en considérant la suite $\\left( T^{\\alpha n}(0,N)\\right)_{n \\in }$ avec $\\alpha >0$ réel quelconque, on montre que $T^{\\alpha n}(0,N) \\in U$ pour $n \\ge n(\\alpha ,U)$ entier.", "Munissons $Y^{\\sharp }$ de la distance $d((s,N),(t,\\widetilde{N})) = {s-t} +{N-\\widetilde{N}}_{\\infty }, $ qui engendre bien la topologie de $Y^{\\sharp }$ précédemment définie.", "Puisque les éléments de $\\mathcal {K}_0$ sont $L$ -lipschitziens, on a pour tous $s,t \\ge 0$ , $ d(T^s(0,N),T^t(0,N)) &={s-t} + {N^{s} - N^{t}}_{\\infty } \\\\ &\\le {s-t} +{\\overline{N}(s+\\cdot ) - \\overline{N}(t+\\cdot )}_{\\infty } \\\\&\\le {s-t}(1+L).", "$ De plus, comme $\\Lambda $ est compact, il existe $\\epsilon >0$ tel que $\\Lambda \\subset \\Lambda ^{(\\epsilon )} \\subset U$ où l'on a noté $\\Lambda ^{(\\epsilon )}$ l'épaississement de $\\epsilon $ de $\\Lambda $ (c'est-à-dire l'ensemble des points situés à distance $< \\epsilon $ d'un point de $\\Lambda $ )on recouvre $\\Lambda $ par des boules contenues dans $U$ , un nombre fini suffit par compacité, $\\epsilon $ est alors le $\\min $ des rayons de ces boules.. Prenons $\\alpha = \\frac{\\epsilon }{2(1+L)}$ , alors pour tout $t\\ge n(\\alpha ,\\Lambda ^{(\\epsilon /2)})$ réel, $T^t(0,N) \\in \\Lambda ^{(\\epsilon )} \\subset U$ ce qui achève la preuve.", "Il est alors aisé de faire le lien avec les définitions données REF et REF données en annexe, sous la forme du corollaire suivant.", "Le compact $\\Lambda $ défini par l'équation (REF ) est un attracteur pour le système dynamique $\\left( \\left(T^s\\right)_{s\\ge 0} , Y^{\\sharp }\\right)$ .", "Son bassin d'attraction est $Y^{\\sharp }$ .", "La seule difficulté supplémentaire par rapport à la proposition REF est qu'il faut montrer l'existence d'un voisinage de $\\Lambda $ revenant tout entier dans lui-même en un temps fini $N$ .", "Pour l'instant, nous savons seulement que toute condition initiale arrive en temps fini dans un voisinage donné, mais ce temps peut être arbitrairement grand en fonction de la condition initiale dès que l'on est hors de $\\Lambda $ .", "D'après le $2.$ de la proposition REF , il revient au même de considérer le système dynamique discret $(T^1,Y^{\\sharp })$ .", "Nous nous placerons désormais dans ce cas.", "Soit $N_1 \\ge A_1$ un entier et $\\alpha >0$ tel que $(1-\\alpha )\\frac{c_0 m_0}{2} > 1$ .", "On définit alors l'ensemble $ V = \\left(T^{N_1}\\right) ^{-1} \\left( \\left\\lbrace (t,N) t \\in /,\\, N \\in Y_t,\\, N>N_{\\max }^{1-\\gamma }\\times \\frac{c_0 m_0}{2} \\times (1-\\alpha ) \\right\\rbrace \\right)$ Par continuité de $T^{N_1}$ , c'est un ouvert.", "Par définition de $\\mathcal {K}_t$ , il contient $\\lbrace (t,N) t\\in /,\\, N \\in \\mathcal {K}_t \\rbrace $ , et donc $\\Lambda $ .", "Les lemmes REF , REF et REF montrent que $T^{2N_1}(V) \\subset \\lbrace (t,N)N \\in \\mathcal {K}_t \\rbrace $ , d'où $T^{2N_1}(V) \\subset V$ .", "On a également $ \\bigcap _{n\\ge 0} T^n(V) \\subset \\bigcap _{n\\ge 0} T^n(\\lbrace (t,N) N \\in \\mathcal {K}_t \\rbrace ) = \\Lambda $ Comme de plus $\\Lambda \\subset V$ et $T^n(\\Lambda )=\\Lambda $ , on a $\\Lambda = \\bigcap _{n \\ge 0} T^n(V)$ .", "Le $3.$ de la proposition REF montre que si $x=(t,N) \\in Y^{\\sharp }$ , il existe un temps $t_0$ à partir duquel $T^s(x) \\in \\Lambda $ , et donc nécessairement $\\omega (x)\\subset \\Lambda $ ." ], [ "Modèle non-saisonnier", "Le cas $m_{\\rho } \\equiv 1$ ($\\rho = 0$ ) peut être traité plus en détails, au moins pour de petites valeurs de $\\gamma $ .", "Il existe une valeur d'équilibre (i.e.", "une solution constante en temps continu) $ N_{eq} = m^{-1}\\left(\\frac{2A_1}{(A_1-A_0)^2}\\right) $ pourvu que la quantité $ \\frac{2A_1}{(A_1-A_0)^2} = \\left(\\int _{A_0}^{A_1}S(a) da \\right)^{-1}$ soit plus petite que $m_0$ .", "Si elle est plus petite que $\\frac{m_0}{2}$ , lorsque $m$ est définie par (REF ), on peut réécrire $ N_{eq} = \\left[ m_0\\frac{(A_1-A_0)^2}{2A_1}\\right]^{1/\\gamma }.", "$ C'est toujours le cas pour des valeurs raisonnables des paramètrese.g.", "$\\gamma \\ge 1$ , $A_1 \\ge 2$ , $A_1 \\ge 2 A_0$ , $m_0 \\ge 8$ .. Pour déterminer la stabilité de cet équilibre, définissons $ F(\\lambda ) &= \\int _{A_0}^{A_1} S(a) e^{-a \\lambda }da \\\\ &= \\left( \\frac{1}{\\lambda } \\left( 1-\\frac{A_0}{A_1} \\right) - \\frac{1}{\\lambda ^2 A_1} \\right) e^{-A_0\\lambda } + \\frac{1}{\\lambda ^2 A_1} e^{- A_1 \\lambda }, $ les valeurs propres de la différentielleOn considère le système sous la forme $\\frac{dN}{dt}=f(N)$ , au voisinage de $N_{eq}$ (on peut expliciter $f$ , au voisinage de $N_{eq}$ , en considérant $T^{\\epsilon }$ quand $\\epsilon $ tend vers 0).", "Les solutions de la forme $N_{eq}+h$ vérifient $\\frac{dh}{dt} =Df_{N_{eq}}h$ .", "Si $\\lambda $ est valeur propre de la différentielle à l'equilibre, un vecteur propre associé $h$ est nécessairement sous la forme $h(t)=h_0 e^{\\lambda t}$ .", "En introduisant cette relation dans (REF ), on peut ainsi déterminer l'ensemble des valeurs propres.", "à l'équilibre sont les solutions de $ F(\\lambda ) = \\left[ \\frac{2A_1}{(A_1-A_0)^2} (1-\\gamma )\\right]^{-1} c_{\\gamma }.$ On peut alors définir des valeurs $\\gamma _0(A_0) < \\gamma _1(A_0)< \\cdots < \\gamma _k(A_0) < \\cdots $ telles que : si $\\gamma < \\gamma _0(A_0) 1 +\\frac{(A_1-A_0)^2}{2A_1}{F(-i u_0)}^{-1}$ , l'équilibre est stable.", "si $\\gamma _0(A_0) < \\gamma < \\gamma _1(A_0) 1 + \\frac{(A_1-A_0)^2}{2A_1}{F(-i u_2)}^{-1}$ , il y a exactement deux valeurs propres (complexes conjuguées) instables.", "... si $\\gamma _{k-1}(A_0) < \\gamma <\\gamma _k(A_0) 1 + \\frac{(A_1-A_0)^2}{2A_1}{F(-iu_{2k})}^{-1}$ , il y a exactement $2k$ valeurs propres instables$k$ paires de valeurs propres complexes conjuguées.. [Éléments de preuve :] Les valeurs propres de la différentielle sont stables si et seulement si leur partie rélle est négative, puisque le vecteur propre associé est de la forme $t \\rightarrow \\exp (\\lambda t)$ .", "Comme $c_{\\gamma }$ est un réel (négatif si $\\gamma > 1$ ), ce sont des solutions de l'équation $\\Im (F(\\lambda ))=0$ .", "Les valeurs des bifurcations correspondant à $\\lambda $ imaginaire pur, on s'intéresse à l'équation $\\Im (F(-iu))=0$ , avec $u>0$ .", "On montre alors que l'ensemble de ces solutions peut s'écrire $u_0 < u_1 < u_2 < \\cdots $ avec $ \\left\\lbrace \\begin{aligned} \\Re (F(-i u_{2k})) &<\\Re (F(-i u_{2k+2})) &< \\cdots &< 0 \\\\ \\Re (F(-i u_{2k+1})) &>\\Re (F(-i u_{2k+3})) &> \\cdots &> 0 \\end{aligned} \\right.$ En effet, $ \\Im (F(-iu)) &= \\frac{-1}{u^2 A_1}\\sin (A_1 u) + \\frac{1}{u} (1-\\frac{A_0}{A_1}) \\cos (A_0 u) +\\frac{1}{u^2 A_1} \\sin (A_0 u) \\\\ \\Re (F(-iu)) &= \\frac{-1}{u^2 A_1}\\cos (A_1 u) + \\frac{1}{u} (1-\\frac{A_0}{A_1}) \\sin (A_0 u) +\\frac{1}{u^2 A_1} \\cos (A_0 u) $ et donc l'ensemble des solutions n'a pas de point d'accumulation (en 0, on le vérifie par un développement limité de $\\Im (F(-iu))$  ; ailleurs, cela découle clairement de l'analycité de la fonction qui n'est pas identiquement nulle), ce qui permet d'énumérer les solutions.", "Il faut vérifier par un calcul direct que les inégalités annoncées sont vraies pour les premières valeurs de $k$ .", "Pour $k$ grand, le terme en $1/u$ est dominant, et donc $\\Im (F(-iu))$ s'annule presque en même temps que $\\cos (A_0 u)$ , et en ce point $\\Re (F(-iu))$ se comporte comme $\\frac{1}{u} (1 - \\frac{A_0}{A_1})\\sin (A_0 u)$ .", "On en déduit l'alternance des signes et la décroissance des valeurs absolues.", "Nous ne considérons que les $u_k$ tels que $\\Re (F(-iu_k))<0$ car $c_{\\gamma }<0$ .", "À chaque $u_{2k}$ , on associe alors un $\\gamma _k$ tel que $c_{\\gamma _k} = F(-i u_{2k})$ , i.e.", "$\\gamma _k(A_0, A_1) 1 + \\frac{(A_1-A_0)^2}{2A_1}{F(-iu_{2k})}^{-1}.", "$ La décroissance des valeurs absolues des $F(-iu_{2k})$ montre que les $\\gamma _k$ sont ordonnés par ordre croissant.", "Nous venons de montrer que lorsque $\\gamma $ varie, une paire de valeurs propres traverse l'axe imaginaire en chaque $\\gamma _k$ , et seulement en ces points-là.", "Lorsque $\\gamma $ tend vers 1 par valeurs supérieures, $c_{\\gamma }$ décroît vers $-\\infty $ , et donc les valeurs propres $\\lambda $ doivent rendre $F(\\lambda )$ de plus en plus grand en module et négatif.", "Or, le module de $F(\\lambda )$ est borné sur le demi-plan $\\Re (\\lambda )>0$ (d'après l'équation (REF )), donc pour $\\gamma $ assez proche de 1, toutes les valeurs propres ont une partie réelle négative.", "C'est donc le cas pour tout $\\gamma <\\gamma _0$ .", "En réalité, $\\gamma _0, \\gamma _1, \\ldots $ dépendent peu de $A_0$ , et leurs valeurs typiques sont $\\gamma _0 \\approx 62$ et $\\gamma _1 \\approx 30$ .", "Lorsque $\\gamma $ grandit et traverse $\\gamma _0$ ($A_0$ étant fixé), on s'attend à voir une bifurcation de Hopf (voir annexe REF ) : une orbite périodique attractive est créée au voisinage de l'équilibre pour $\\gamma = \\gamma _0$ , et attire toutes les solutions proches de l'équilibre (sauf l'équilibre lui-même) pour $\\gamma > \\gamma _0$ proche de $\\gamma _0$ ." ], [ "Simulations numériques", "La mise en œuvre de simulations du modèle (REF ) a demandé un travail préliminaire de discrétisation et de mise au point des paramètres de simulation qui est détaillé en annexe .", "Les précisions concernant le traitement des données sont données en annexe , dans l'ordre de présentation des résultats.", "Elles sont cependant indispensables pour une réelle compréhension de ceux-ci, car les nombreuses approximations qui ont été nécessaires ont souvent une réelle influence sur les résultats obtenus." ], [ "Explorations de l'espace des paramètres", "Pour commencer, on fait varier un paramètre en gardant les autres fixes, et l'on observe la façon dont la dynamique stationnaire évolue.", "Trois paramètres semblent déterminants : $A_0$ (qui introduit un effet de retard correspondant au temps de maturation), $\\rho $ (qui mesure l'importance du facteur saisonnier) et $\\gamma $ (qui traduit l'influence de la densité sur la fécondité).", "Pour les autres paramètres, on a fixé les valeurs suivantes : $A_1 = 2$ .", "$m_0 = 50$ et la fécondité est prise $C^1$ .", "$\\epsilon _{ete} = 01$ (ou 0 dans le premier cas, car le facteur saisonnier n'avait pas encore été régularisé).", "On pourra ainsi représenter chaque simulation par le triplet $(A_0; \\rho ; \\gamma )$ qui lui correspond." ], [ "$A_0=0.18$ , {{formula:2a1bc5a7-a105-4c24-a07f-83d416cdc250}} , {{formula:98e7f9b4-8410-4377-95f1-24ce55d83004}} variable", "Pour la première exploration, nous sommes partis des valeurs (018 ; 041 ; 825) et nous avons fait varier $\\gamma $ .", "Contrairement aux simulations effectuées ultérieurement, le facteur saisonnier n'est pas régularisé (i.e.", "$\\epsilon =0$ ).", "Les résultats sont représentés sur un diagramme de bifurcation, figure REF  : pour chaque valeur de $\\gamma $ sont représentées les valeurs de $N(t)$ aux temps entiers (i.e.", "à la fin de l'été), en se limitant à $t$ assez grand (on a fixé arbitrairement $19001 \\le t \\le 20000$ ).", "Les points bleus correspondent à une même condition initiale (obtenue aléatoirement), que nous notons (I) (voir figure REF ).", "La partie rouge correspond à d'autres simulations, détaillées ci-après.", "Figure: Diagramme de bifurcation(018;041;γ) 2≤γ≤16 (018; 041; \\gamma )_{2 \\le \\gamma \\le 16}.", "Pourcertaines valeurs de γ\\gamma , plusieurs attracteurs coexistent.On les a obtenus en utilisant la condition (I) (en bleu) et enprolongeant aussi loin que possible (en rouge) certaines branchesinterrompues dans le diagramme bleu.", "Noter que les saisons sontdiscontinues (ϵ ete =0\\epsilon _{ete} = 0)." ], [ "Orbites périodiques attractives", "Pour $2 \\le \\gamma \\le 72$ , le comportement observé est assez classique.", "On a d'abord un équilibre attractif, puis une orbite de période 2, et ainsi de suite avec des doublements de périodes successifs (de plus en plus rapprochés) au fur-et-à mesure que $\\gamma $ se rapproche de la valeur limite $\\gamma _{\\infty ,1} \\approx 736$ .", "Il s'agit de bifurcations par doublement de période (voir la section REF ) qui ont lieu pour $\\gamma _0 <\\gamma _1 < \\cdots < \\gamma _n < \\cdots < \\gamma _{\\infty ,1}$ , et qui se traduisent par des doublements de période successifs.", "Pour $\\gamma \\ge \\gamma _{\\infty ,1}$ , le diagramme de bifurcations permet de distinguer essentiellement deux comportements.", "D'une part, il y a toujours des orbites périodiques attractives sur certains intervalles de valeurs de $\\gamma $ , tout comme il y a des « fenêtres de périodicité » dans le cas des polynômes quadratiques (voir section REF ).", "C'est en particulier le cas pour $900 \\le \\gamma \\le 939$ , $967 \\le \\gamma \\le 986$ et $1223 \\le \\gamma \\le 16$ .", "On peut situer plus précisément ces fenêtres à l'aide d'un calcul de dimensions fractales." ], [ "Dimensions fractales", "On peut aisément calculer une valeur approchée de la dimension fractale des ensembles limites correspondant aux différentes valeurs de $\\gamma $ .", "Le graphique obtenu est représenté figure REF .", "Figure: Dimension fractale desattracteurs (018;041;γ) 2≤γ≤16 (018; 041; \\gamma )_{2 \\le \\gamma \\le 16}.", "On s'est limité à la condition initiale (I).Les points où la dimension fractale est nulle correspond aux orbites périodiques attractives, ce qui nous permet de les détecter bien plus facilement qu'en observant le diagramme de bifurcation.", "Dans le domaine intermédiaire ($736 \\le \\gamma \\le 1223$ ), on constate qu'il y a alternativement des attracteurs de dimension environ égale à 1 et des fenêtres de périodicité.", "On peut raisonnablement penser qu'il existe un ouvert dense dans l'espace des paramètres pour lequel il existe une orbite périodique attractivequi peut éventuellement coexister avec un autre attracteur.. C'est ouvert n'est en revanche certainement pas de mesure totale.", "La précision de ces calculs étant limitée par le faible nombre de points considérés pour chaque ensemble, il n'est pas aisé de déterminer s'il y a ou non réellement des attracteurs de dimension non-entière.", "Si oui, elle n'est pas très grande, certainement inférieure à 15, et probablement supérieure à 1, dans la mesure où le calcul effectué sous-estime légèrement la dimension de l'attracteur (en particulier, le petit nombre de points utilisés peut créer de nombreux « trous » correspondant à la mesure physique sur l'attracteur, et non à un trou réel dans sa géométrie).", "Il y a un autre argument théorique en faveur d'une dimension fractale supérieure à 1 lorsqu'elle n'est pas nulle.", "En effet, s'il y a une orbite périodique hyperbolique instable, sa variété instable est contenue dans l'attracteur, qui doit donc avoir une dimension au moins égale à 1.", "Il est donc difficilement concevable que dans un grand domaine de l'espace des paramètres on puisse avoir un attracteur de dimension fractale comprise strictement entre 0 et 1." ], [ "Attracteurs de type Hénon", "En-dehors des fenêtres de périodicité apparaît un comportement stationnaire non-périodique, le long d'un attracteur qui semble constitué de deux morceaux de courbes.", "Un exemple est représenté avec la figure REF , où l'on a tracé les points ($N(t)$ , $N(t+1)$ , $N(t+2)$ ) pour chaque valeur entière de $t$ ($10002 \\le t \\le 19999$ ).", "On s'intéresse alors à la dynamique de l'applicationCette application n'est pas parfaitement bien définie, le système étudié étant de dimension infinie, alors que la visualisation considérée est une projection de celui-ci en dimension 3.", "$f$  : ($N(t)$ , $N(t+1)$ , $N(t+2)$ ) $\\mapsto $ ($N(t+1)$ , $N(t+2)$ , $N(t+3)$ ).", "Par abus de notation, on écrira $T^1$ au lieu de $f$ , sans perdre de vue que nous ne pouvons pas visualiser directement $T^1$ .", "En utilisant deux couleurs suivant la parité de $t$ , on constate que chacune des deux parties de l'attracteur est envoyée sur l'autre.", "En revanche, il ne semble pas possible (pour cette valeur de $\\gamma $ ) de séparer de la même façon l'attracteur en un plus grand nombre de composantes.", "Il semble donc que l'application $f^2$ restreinte à chacune des deux composantes de l'attracteur soit topologiquement mélangeante (voir définition REF ).", "Figure: (018;041;861)(018; 041; 861).", "Condition initiale (I).", "Les deux composantes sontinvariantes par T 2 T^2, qui y semble topologiquement mélangeante.Dimension fractale estimée : d f ≈106d_f \\approx 106.Un zoom sur l'attracteur fait apparaître des structures semblables à celles de l'attracteur de Hénon (voir annexe REF ).", "Il est assez probable que la dynamique soit du même type, mais cela n'à pas été étudié précisément.", "La dimension fractale n'est pas clairement différente de 1, mais elle est sûrement sous-estimée à cause du petit nombre de points que nous avons calculé.", "Si elle s'avérait être clairement différente de 1, cela renforcerait l'hypothèse d'une dynamique de type Hénon.", "Il n'y a pas toujours deux composantes.", "On le voit sur le diagramme de bifurcations notamment au voisinage des valeurs « limites » de $\\gamma $ (i.e.", "juste après une zone où il y a une orbite périodique attractive).", "Ainsi, pour $\\gamma =862$ , on peut compter 10 composantes distinctes, et $f^{10}$ semble topologiquement mélangeante sur chacune d'entre elles (figure REF ).", "Dans les deux cas (figures REF et REF ), il semble donc qu'on ait une décomposition spectrale (voir théorème REF ) de l'attracteur $\\Lambda $ en un nombre fini de composantes (respectivement 2 et 10).", "Figure: (018;041;989)(018; 041; 989).", "Condition initiale (I).", "Chacune des composantes estinvariante par T 10 T^{10}, qui y semble topologiquementmélangeante.", "La numérotation des composantes correspond auxvaleurs de t10t 10 .", "Comme à lafigure , il y a deux groupes suivant laparité de tt.", "Dimension fractale estimée : d f ≈092d_f \\approx 092.Lorsque $\\gamma $ évoluevoir aussi l'animation film_gamma_100_18_200_410_00_50__1.avi (diagramme de bifurcation en quatre dimensions)., on observe que l'attracteur grandit petit-à-petit à l'intérieur d'un même objetÀ de légères déformations près, notamment un changement de taille., que l'on visualise à peu près en entier avec la figure REF .", "Tant que $\\gamma \\le 996$ , on remarque même que les orbites périodiques attractives sont contenues dans le même objet.", "Lorsque l'attracteur est continu mais en morceaux distincts, envoyés périodiquement l'un dans l'autre, chaque morceau grandit et ceux-ci fusionnent petit-à petit.", "Lorsque deux composantes fusionnent, la stabilité de chacune vis-à-vis de $f$ (composée le bon nombre de fois) semble instantanément perdue, et il y a alors mélange topologique à l'intérieur de chaque composante.", "Cette fusion des composantes connexes s'apparente à la cascade inverse qui suit la cascade harmonique directe, comme c'est le cas pour la famille quadratique réelle (annexe REF )." ], [ "Discontinuités du diagramme", "Les discontinuités observées à $\\gamma = 370$ et $\\gamma = 1223$ sont plutôt surprenantes.", "L'hypothèse la plus plausible serait qu'il existe à ces valeurs de $\\gamma $ deux attracteurs distincts, et la condition initiale (I) passe brusquement du bassin de l'un au bassin de l'autre.", "Pour tester cette hypothèse, nous avons choisi comme nouvelles conditions initiales les états stationnaires obtenus de part et d'autre de ces discontinuités, et nous avons fait varier $\\gamma $ pour déterminer s'il y a effectivement coexistence de deux attracteurs pour certaines valeurs de $\\gamma $ .", "En repartant de l'équilibre obtenu avec $\\gamma = 370$ , nous avons pu continuer la branche du diagramme jusqu'à $\\gamma = 420$ .", "Ensuite, on retrouve l'orbite de période 2 déjà trouvée.", "Le même procédé nous permet de continuer jusqu'à $\\gamma = 423$ , mais nous n'avons plus pu retrouver l'équilibre ensuite en utilisant le même procédé.", "Dans l'autre sens, on prolonge le domaine où se trouve une orbite de période 2 attractive jusqu'à $\\gamma =360$ .", "On peut ainsi tracer un nouveau diagramme de bifurcations autour de ces valeurs, avec cette fois les deux états stationnaires (figure REF ).", "Figure: Diagramme de bifurcation(018;041;γ) 350≤γ≤430 (018; 041; \\gamma )_{350 \\le \\gamma \\le 430}.", "Deux attracteurs coexistent pour certaines valeurs deγ\\gamma  : un point fixe (au centre) et une orbite de période 2(en haut et en bas).", "Avec la condition (I), on passe de l'un àl'autre pour 370<γ<375370 < \\gamma < 375.La seconde discontinuité a lieu autour de $\\gamma =1223$  : on observe l'attracteur pour $\\gamma = 1222$ et une orbite attractive de période 4 pour $\\gamma = 1223$ .", "En repartant de l'état final obtenu pour $\\gamma = 1222$ , on observe pour les valeurs supérieures de $\\gamma $ (au moins jusqu'à 16) un comportement similaire à ce qu'on constatait pour $\\gamma \\le 1222$ , c'est-à-dire le même attracteur, avec parfois des orbites périodiques attractives (mais dans un domaine de valeurs de $N$ différent de l'orbite de période 4).", "Inversement, l'orbite 4-périodique attractive persiste jusqu'à $\\gamma \\approx 10078$ .", "Les attracteurs ainsi détectés sont représentés en rouge sur la figure REF .", "Dans ces deux cas, plusieurs attracteurs coexistent, mais il y a toujours des discontinuités dans le diagramme.", "Pour certaines valeurs de $\\gamma $ (envion 360 et 10078), des orbites périodiques deviennent attractives.", "A l'inverse, pour $\\gamma \\approx 423$ , l'équilibre devient instable.", "Il pourrait donc s'agir d'une bifurcation du type de $f_{(+1),\\tau }$ (voir annexe REF , figure REF ).", "Il est également possible que le diagramme soit en réalité continu, mais que les bassins d'attraction des orbites périodiques attractives soient trop réduits pour que l'on puisse les atteindre par des simulations, avec la méthode que nous avons employée ici.", "Un petit travail théorique serait nécessaire pour éclairer ce point." ], [ "$A_0=018$ , {{formula:869dbd62-67cd-491e-956a-81eab984eefa}} , {{formula:cb5d9479-e60b-4300-9aba-e49396dff50b}} variable", "Suite à une rapide exploration en faisant varier $\\rho $ , et au vu de l'intérêt des valeurs $(018; 030; 825)$ (voir figure REF ), nous avons effectué une deuxième exploration à $\\gamma $ variable, autour de ces nouvelles valeurs.", "Le diagramme de bifurcation ainsi obtenu est représenté figure REF .", "Il est très semblable au diagramme REF , mise à part l'absence de deux composantes bien distinctes pour la plupart des valeurs de $\\gamma $ , et le faible nombre de fenêtres de périodicité.", "Figure: Diagramme de bifurcation(018;030;γ) 2≤γ≤15 (018; 030; \\gamma )_{2 \\le \\gamma \\le 15}.", "Lessaisons sont continues (ϵ ete =01\\epsilon _{ete} = 01).", "Conditioninitiale (I)." ], [ "$A_0=018$ , {{formula:a31c4262-9ef0-479e-9446-9432ff66ff02}} variable, {{formula:4b9f04e9-6e15-471d-9cdd-5be981306dce}}", "Le facteur saisonnier semble jouer un rôle déterminant dans la dynamique du système (le modèle non-saisonnier est particulièrement simple, alors que pour des valeurs de $\\rho $ plus proches de la réalité, on observe des comportements bien plus complexes, e.g.", "figure REF ).", "Le diagramme de bifurcation obtenu en faisant varier $\\rho $ est représenté figure REF .", "Une autre condition initiale, notée (II), a été utilisée pour ces simulations (voir figure REF ).", "On observe comme précédemment une discontinuité dans le diagramme, pour $\\rho $ proche de 01, mais nous n'avons pas essayé de prolonger les deux branches interrompues.", "Figure: Diagramme de bifurcation(018;ρ;825) 0≤ρ≤05 (018; \\rho ; 825)_{0 \\le \\rho \\le 05}.", "Lessaisons sont continues (ϵ ete =01\\epsilon _{ete} = 01).", "Conditioninitiale (II).Figure: Un comportement complexe :(018;030;825)(018; 030; 825).", "Les saisons sont continues(ϵ ete =01\\epsilon _{ete} = 01).", "Condition initiale (II).", "Il sembleque T 1 T^1 soit mélangeante sur cet attracteur.", "Dimensionfractale estimée : d f ≈119d_f \\approx 119." ], [ "Bifurcation de Hopf", "On constate sur ce diagramme un phénomène que nous n'avions pas trouvé sur les diagrammes précédents : une bifurcation de Hopf pour $\\rho \\approx 0152$ .", "On passe en effet d'une orbite attractive de période 2 à deux « cycles » attractifs stables, qui persistent jusquà $\\rho \\approx 0165$ (la figure REF en représente un exemple).", "Cet attracteur est de dimension fractale 1 et chaque lacet est parfaitement connexe.", "Il n'est en revanche pas totalement certain que $T^2$ soit bien topologiquement mélangeante sur chacun, bien que l'on n'ait vu aucune périodicité évidente.", "La dynamique de $f$ sur ces cycles n'est pas forcément simple (i.e.", "topologiquement conjuguée à une rotation), et le cycle ne coïncide peut-être pas exactement avec l'attracteur.", "Répondre à ces questions demanderait une étude plus poussée." ], [ "$A_0$ variable, {{formula:24f71017-9ac1-49ef-8795-dfbc11670f6a}} , {{formula:474c4a64-3768-4fce-8c34-03dacbae31ba}}", "Pour cette dernière exploration préliminaire, une difficulté supplémentaire a été de trouver une façon de faire varier la valeur de $A_0$ plus finement que le pas de discrétisation.", "Celle-ci a conduit à utiliser la méthode évoquée à la fin du paragraphe REF .", "Les résultats obtenus sont représentés dans le diagramme de bifurcations de la figure REF .", "Figure: Diagramme de bifurcation (A 0 ;030;825) 0≤A 0 ≤04 (A_0;030; 825)_{0 \\le A_0 \\le 04}.", "Conditioninitiale (II)." ], [ "Décomposition spectrale", "L'utilisation de 2 couleurs de visualisation montre que la 2-périodicité de la figure REF ne se retrouve pas à la figure REF .", "Il y ainsi initialement deux composantes connexes bien distinctes, l'une étant l'image de l'autre par l'application $T^1$ .", "Sur chaque composante, $T^2$ semble mélangeante.", "On a ainsi une décomposition spectrale avec 2 composantes (voir théorème REF ).", "Lorsque celles-ci fusionnent, on perd cette 2-périodicité et $T^1$ devient topologiquement mélangeante.", "On retrouve le même comportement que précédemment avec le diagramme $(018; 041;\\gamma )$ .", "Figure: Un comportement complexe :(015;030;825)(015; 030; 825).", "Condition initiale (II).", "Lesdeux composantes sont invariantes par T 2 T^2, qui sembletopologiquement mélangeante sur chacune." ], [ "Étude du cas $(015; 030; 825)$", "Essayons de comprendre la dynamique de la figure REF .", "Nous avons vu qu'il y a deux composantes connexes distinctes, il suffit donc de considérer l'une des deux pour comprendre la dynamique de $T^1$ .", "Elle est représentée figure REF .", "Figure: (015;030;825)(015; 030;825) Condition initiale (II).", "On n'a conservé qu'unecomposante connexe." ], [ "Visualisation en dimension 3", "La représentation que nous avons choisie (arbitrairement) est-elle correcte ?", "Cette question est fondamentale : nous projetons en effet un objet de dimension infinieaprès discrétisation, on se limite à une dimension finie très grande, ici 201. dans un espace de dimension 3." ], [ "Injectivité de la projection", "Pour tenter d'y répondre, nous pouvons évaluer la qualité de la « projection » $\\pi $  : $^{201}$ $\\rightarrow $ $^3$ , $x_{201}(t)=(N(t+k/100))_{k=0\\ldots 200}$ $\\mapsto $ ($N(t)$ , $N(t+1)$ , $N(t+2)$ )$=x_3(t)$ où $N(t)$ désigne la population mature à l'instant $t$ , en différents points de l'attracteur.", "Nous voulons nous assurer que des points proches dans $^3$ sont également proches dans $^{201}$ , c'est-à-dire majorer $\\sup _{t \\ne t^{\\prime } \\in }\\frac{{x_{201}(t)-x_{201}(t^{\\prime })}_{^{201}}}{{x_{3}(t)-x_{3}(t^{\\prime })}_{^3}}$ pour différents choix de normes ($L^1$ , $L^2$ ou $L^{\\infty }$ ).", "Le résultat, représenté à la figure REF , montre que cette quantité est raisonnablement bornée.", "Avec les normes $L^1$ ou $L^2$ , le résultat semble un petit peu meilleur, mais reste du même ordre de grandeur.", "Une zone de l'attracteur semble en revanche être un peu moins bien représentée par cette projection, il s'agit du point 24 (et plus généralement des points 21 à 30).", "En se reportant à la figure REF où sont localisés ces points (voir section REF ), on constate qu'il s'agit de la zone de pli.", "Une vue rapprochée sur cette zone de l'attracteur montre en effet des filaments entrelacés, et certains rapprochements de filaments semblent dûs à la projection.", "Figure: Injectivité de laprojection : norme L ∞ L^{\\infty }, δ t =0\\delta _t = 0." ], [ "Choix de l'origine des temps", "Nous l'avons arbitrairement fixée à la fin de l'été, mais ce choix est-il judicieux ?", "Nous avons donc fait les mêmes calculs que précédemment en décalant l'origine des temps.", "Il semble que l'instant choisi initialement n'est pas mauvais.", "La « meilleure » origine semble se situer autour de $\\delta _t=04$ , mais la différence avec $\\delta _t = 0$ n'est pas flagrante (figure REF ).", "On peut expliquer ces résultats en observant l'évolution en temps continu de $N(t)$ .", "En effet, la population mature atteint tous les deux ans — un peu après le milieu de l'été — un maximum élevé, suivi d'une chute brutale d'effectif.", "La valeur $\\delta _t =-04$ correspond à l'instant du pic de population, qui est suivi d'une simple diminution linéaire de $N(t)$ (dûe à la mortalité naturelle, en l'absence de naissances), si bien que les instants qui suivent sont encore des origines des temps de bonne qualité.", "Figure: Injectivité de laprojection : norme L ∞ L^{\\infty }, δ t \\delta _t variable.Il est également intéressant, en vue de comprendre la dynamique en temps continu du système, de visualiser l'évolution de l'attracteur tridimensionnell'animation film_delta.avi permet une bonne compréhension de la façon dont l'attracteur se déforme, pour passer d'une composante à l'autre quand $\\delta _t$ varie de $-1$ à 0 ou de 0 à 1. lorsque l'on fait varier l'origine des temps $\\delta _t$ dans l'intervalle $[-1;1]$ , la valeur 0 correspondant à la fin de l'été.", "La figure REF en donne un exemple, pour $\\delta _t = -04$ .", "Figure: Visualisation de l'attracteur(015;030;825)(015; 030; 825) avec une origine des tempsδ=-04\\delta = -04." ], [ "Échelle logarithmique", "Une autre piste possible est de visualiser la projection tridimensionnelle de l'attracteur suivant une échelle logarithmique, c'est-à-dire de considérer les points $(\\log N(t), \\log N(t+1), \\log N(t+2))$ pour $t$ entier grand.", "Figure: Visualisation de l'attracteur (015;030;825)(015; 030; 825) avec une échelle logarithmique.Figure: Qualité de la projection enéchelle logarithmique.Au vu de la figure REF , l'apport d'un tel changement d'échelle n'est pas évident.", "L'attracteur est légèrement déformé, mais garde le même aspect, et certaines zones semblent toujours aussi « emmêlées ».", "Une évaluation quantitative de la qualité de cette nouvelle projection, comme effectué précédemment, confirme l'aspect visuel : il n'y a pas de gain significatif." ], [ "Géométrie de l'attracteur", "La figure REF représente de façon simplifiée la géométrie de l'attracteur de la figure REF , en distinguant neuf régions principales.", "Celles-ci sont nommées en fonction de leur forme et de leur position dans l'attracteur, identifié au corps d'un animal dont la tête serait située à droite (chevelure-cou-pli-pointe) et la queue à gauche.", "Figure: Représentation grossière de lagéométrie de l'attracteur (015;030;825)(015; 030; 825).Quatre phénomènes principaux semblent pouvoir expliquer la complexité de l'attracteur que nous observons.", "Les deux premiers étaient déjà présents dans le solénoïde : un très fort pincement et un étirement.", "Le troisième est également présent dans l'attracteur de Hénon : un pli (il y en a peut-être plusieurs ici).", "Le quatrième semble nouveau, et ressemble à un ou plusieurs « embranchements ».", "Avec la numérotation introduite dans la section précédente, on peut en situer trois : 67–80, 63–55, 58–74.", "Cela ne signifie pas pour autant que ces embranchements sont distincts.", "Localement, l'attracteur ressemble au produit d'une droite et d'un ensemble de Cantor (figure REF ), sauf en certains points où l'on observe des « pointes » (figure REF ).", "Par ces aspects, il ressemble beaucoup à l'attracteur de Hénonvoir annexe REF ..", "Figure: Zoom sur un filament, au voisinage dupoint 23.Figure: Zooms successifs sur une pointe, auvoisinage du point 54." ], [ "Dynamique sur l'attracteur", "Considérons l'application $T^2$ .", "Comment agit-elle sur les points de l'attracteur ?", "Pour essayer de le comprendre, 80 points ont été choisisCe choix a été fait arbitrairement, en essayant de répartir ces points uniformément suivant la mesure de Hausdorff sur l'attracteur, et non la mesure physique.", "sur l'attracteur, numérotés de 1 à 80, comme représenté sur la figure REF (des vues plus rapprochées sont en annexe ).", "La position des images directes et réciproques de ces 80 points est indiquée dans le tableau REF .", "Figure: Position des 80 points choisis surl'attracteur.Table: Dynamique sous T 2 T^2 des 80points sur l'attracteur.", "On a noté 1 3+ 1^{3+} pour T 6 (1)T^6(1) ;54–55 pour un point situé entre 54 et 55, plus proche de 54 ;76–76 pour un point proche de 76 mais différent de 76.On peut penser qu'il y a un point fixe dans la région du point 15, une orbite de période 2 dans la région 42–31 et une orbite de période 3 dans le « triangle » 66–74–77.", "Nous étudierons le point fixe plus en détails dans une des sections suivantes.", "Partant de ces trois orbites remarquables, on peut essayer de comprendre schématiquement la dynamique interne à l'attracteur." ], [ "Sortie de la « tête »", "L'équilibre (entre pli et pointe) est répulsif avec expansion négative (la valeur propre dominante est négative), tandis que l'orbite de période 2 (dans le cou, entre les points 42 et 31) est répulsive positive.", "Dans cette région de la tête et de la pointe arrière, on peut schématiser la dynamique de la façon suivante : Pointe arrière $\\rightarrow $ creux-cou-pli-pointe (sans inversion haut-bas).", "Chevelure $\\rightarrow $ cou-pli-pointe (sans inversion).", "Bas du cou (sous l'orbite de période 2 = 42–31) $\\rightarrow $ plat-creux-bas du cou (sans inversion).", "Haut du cou (au-dessus de la période 2) $\\rightarrow $ haut du cou-pli (sans inversion).", "Pli $\\rightarrow $ bas de la pointe (avec inversion).", "Haut de la pointe $\\rightarrow $ bas du pli (avec inversion).", "Ainsi, si l'on part d'assez haut, on arrive en bas de la pointe puis en bas de la chevelure (69) et enfin dans le creux (54).", "Si l'on part plus bas, on arrive directement dans le creux, sans être passé par la pointe." ], [ "Sortie du « corps »", "Dans le « corps » (queue, triangle, plat, creux), l'orbite de période 3 (aux milieux des sommets du triangle) joue un rôle majeur.", "Elle est répulsive positive, et agit sur le triangle comme une rotation d'angle $2\\pi /3$ (75–72 $\\rightarrow $ 76–79 $\\rightarrow $ 67–59) combinée avec un peu d'expansion.", "On sort ainsi du triangle soit par le bas de la pointe arrière, soit par le plat ou le creux, soit par la queue.", "La dynamique se schématise alors ainsi : Bas de la pointe arrière $\\rightarrow $ plat-creux.", "Creux $\\rightarrow $ droite de la queue (1–7)-côté du triangle (75–73) (avec inversion droite-gauche).", "Plat $\\rightarrow $ gauche de la queue (sans inversion haut-bas).", "Queue $\\rightarrow $ pointe arrière.", "On sort ainsi du corps, pour y revenir rapidement (si l'on arrive trop bas dans la pointe arrière, soit près du triangle, soit à la pointe 49), ou (le plus souvent) après un passage dans le pli et éventuellement la pointe.", "Ce bref aperçu de la dynamique nous permet de comprendre comment s'instaure le mélange et le chaos de la dynamique sur l'attracteur." ], [ "Visualisation en temps continu", "Il est intéressant, du point de vue mathématique comme du point de vue biologique, de mettre en relation les différentes régions de l'attracteur (en dimension 3) avec la dynamique en temps continu dans ces régions.", "La forme de $N(t)$ pour $t\\in [t_0 - 5 ; t_0 +5]$ peut ainsi être mise en correspondance avec les points de l'attracteur et leurs régions de provenance et de destination par $T^2$ .", "On a représenté ces données pour les 80 points de la « carte » en annexe ." ], [ "Aspect général", "La dynamique en temps continu garde en général un aspect identique dans toutes les zones de l'attracteur : un pic chaque année pour $t\\approx 06 1$ (croissance très rapide suivie d'une décroissance linéaire moins brusque), élevé les années impaires (entre 25 et 65), plus faibles voire inexistant les années paires (entre 1 et 2).", "Juste avant les maxima se trouvent des minima locaux plus ou moins bas : ceux-ci sont toujours légèrement supérieurs à 1 (mais inférieurs à 15) avant un maximum faible, mais pouvant atteindre 05 avant un maximum élevé.", "Il y a ainsi une pseudo-périodicité de 2 ans, avec une très forte variation d'amplitude (les maxima étant à 65, les minima entre 05 et 1).", "On retrouve en partie l'aspect général de la figure REF (alternance de maxima et de minima, avec un facteur allant jusqu'à 8 entre les deux).", "Ce comportement s'explique par l'alternance entre une explosion de la population dûe à la très forte fécondité, qui est ainsi suivie d'une chute linéaire de la population mature (dûe à la mortalité naturelle, en l'absence quasi-totale de naissances)." ], [ "Différences entre régions", "Les régions de l'attracteur se différencient par l'amplitude des pics (il y a un facteur 2 entre les amplitudes possibles des maxima élevés) et des creux (inférieurs ou supérieurs à 1), ainsi que par l'amplitude relative des maxima secondaires (inexistants ou bien valant jusqu'à 2).", "Les pics faibles se trouvent au niveau de la queue (lorsqu'il se produit à $t=0$ ) ou du plat (lorsqu'il se produit à $t=2$ ).", "Les pics très élevés ($N\\ge 6$ ) se trouvent dans le pli, la pointe, le haut de la pointe arrière et le cou, avec des maxima secondaires quasi-inexistants et des minima inférieurs à 1.", "Le deuxième pic est légèrement inférieur au premier dans la pointe, mais le rapport s'équilibre quand on se rapproche du pli.", "On peut encore affiner cette analyse en s'aidant des figures placées en annexe , qui permettent de faire la différence entre filaments dans une région à l'aide de l'évolution en temps continu dans le passé ou dans le futur proche." ], [ "Dimension fractale de l'attracteur", "Pour l'évaluer, nous avons calculé pour différentes valeurs de $r$ le nombre de boîtes de côté $r$ (et dont les points ont des coordonnées qui sont des multiples entiers de $r$ ) contenant des points de l'attracteur (avec 100000 points).", "En notant $N(r)$ ce nombre de points, on a alors tracé $\\log _{10}(N(r))$ en fonction de $\\log _{10}(r)$ .", "Tant que $N(r)$ est assez petit devant 100000 et assez grand devant 1, les points obtenus sont presque alignés, et la pente (en valeur absolue) de la droite de régression est une bonne estimation de la dimension fractale de l'attracteur.", "Le résultat est représenté à la figure REF .", "On obtient donc une dimension de l'ordre de 133.", "Cette valeur correspond bien à l'impression visuelle que l'on a : localement, l'attracteur semble être le produit d'une droite et d'un ensemble de Cantor de dimension proche de $1/3$ , soit une dimension fractale d'environ $4/3$ (voir figure REF ).", "Il faut bien sûr prendre ce résultats avec beaucoup de précautions, dans la mesure où cette régression est faite dans la zone qui semble — visuellement — présenter une « bonne pente » (voir annexe REF ).", "Figure: Évaluation de ladimension fractale de l'attracteur.Une des conséquences de cette évaluation est la justification a posteriori de la possibilité d'utiliser trois dimensions seulement pour visualiser l'attracteur.", "En effet, le théorème de Whitney REFvoir annexe REF .", "affirme qu'un objet de dimension $d$ peut être visualisé avec $N$ dimensions pourvu que $N>2d$ .", "L'attracteur ayant une dimension strictement comprise entre 1 et 15, l'utilisation de 3 dimensions pour le visualiser semble raisonnablemais bien sûr, comme nous avons calculé la dimension fractale de la projection tridimensionnelle de l'attracteur, nous n'avons pas la dimension fractale de l'attracteur lui-même mais une légère sous-estimation de celle-ci.." ], [ "Sensibilité aux conditions initiales", "Pour l'évaluer, il est intéressant de regarder la dynamique future et passée d'une petite boule centrée sur un point de l'attracteur.", "Les figures REF et REF permettent de juger du résultat au voisinage de l'équilibre.", "En se référant à l'exemple du solénoïdevoir annexe REF ., on peut tenter d'interpréter la forme des courbes obtenues.", "Tout d'abord, il y a clairement une très forte sensibilité aux conditions initiales, en tout point de l'attracteur, aussi bien dans le passé que dans le futur.", "Dans le cas de la figure REF , on constate ainsi des écarts de l'ordre de 4 en moins de 15 ans, aussi bien dans le passé que dans le futur, alors que les courbes étaient initialement séparées de moins de 004.", "Les dynamiques futures des différents points se répartissent de façon à peu près homogène, au moins au cours des 10 premières années.", "Ceux-ci se séparent en effet selon leur répartition initiale dans la direction instable, puisque la direction stable est contractée dans le futur.", "L'homogénéité dans le futur traduit une répartition à peu près uniforme des points d'une orbite dans la direction instable.", "Cette propriété est à rapprocher du cas du solénoïde, où la mesure physique (qui donne la répartition des points d'une orbite sur l'attracteur) possède une densité par rapport à la mesure de Lebesgue dans la direction instable.", "La dynamique passée semble bien différente, les différentes courbes se séparant « par paquets », et non plus de façon homogène.", "Ainsi, en à peine 3 ans, on observe déjà une différence de 1 pour l'une des courbes, tandis que la plupart des points ont une orbite toujours très proche de l'équilibre.", "Ceci nous donne des informations sur la mesure physique dans la direction stable, puisque la direction instable est contractée dans le passé.", "Ainsi, comme dans le cas du solénoïde, il semble que la mesure physique possède une densité par rapport à la mesure de Hausdorff sur un ensemble de Cantor de dimension fractale 03.", "Les résultats observés ici sont cependant moins clairs que dans le cas du solénoïde, pour ${t} \\ge 10$ .", "Ceci est sans doute dû au passage des orbites dans un pli, phénomène qui ne se produit pas dans le cas du solénoïde.", "On observe le même type de résultat en de nombreux autres points de l'attracteur, d'autant plus nettement que l'on n'est pas au voisinage d'une « pointe ».", "Figure: Dynamique passée etfuture de points au voisinage de l'équilibre.", "Les points ontété choisis pour leur proximité de l'équilibre dansl'intervalle de temps [0;2][0;2].Figure: Position des points auvoisinage de l'équilibre : l'ensemble des points de la bouleconsidérée est représenté en vert." ], [ "Point fixe, variété instable", "Un point fixe (instable, bien sûr) a été repéré sur l'attracteur.", "La figure REF donne sa localisation approximative en dimension 3, et la fonction continue correspondante est représentée figure REF .", "Figure: Position de l'équilibre,et de sa « préimage ».Figure: L'équilibre : tempscontinu." ], [ "Différentielle de $T^2$ à l'équilibre", "Celle-ci nous fournit beaucoup de renseignements sur le système dynamique.", "On peut calculer aisément ses valeurs propres.", "Une seule est de module strictement supérieur à 1 et correspond à l'expansion dans la direction instable : $\\lambda _1 \\approx -229$ .", "Le vecteur propre associé est tracé figure REF .", "La seconde plus grande valeur propre (la précision de ce calcul est faible) est $\\lambda _2 \\approx 0043$ , et le vecteur propre associé est représenté figure REF .", "Les modules des valeurs propres suivantes décroissent ensuite rapidement, comme le montre la figure REF .", "Figure: Différentielle àl'équilibre : premier vecteur propre.Figure: Différentielle àl'équilibre : second vecteur propre.Figure: Différentielleà l'équilibre : décroissance des valeurs propres.Le point fixe est donc hyperbolique, puisqu'il n'a aucune valeur propre de module proche de 1.", "De plus, on constate que l'expansion est assez raisonnable (elle est sans doute un peu plus forte dans certaines zones, un peu moins dans d'autres, mais reste de cet ordre de grandeur), tandis que la contraction est beaucoup plus forte.", "Cela nous donne un argument supplémentaire pour penser que 3 dimensions suffisent à représenter l'attracteur : les valeurs propres suivantes ayant un module encore plus petit, l'attracteur n'est vraiment étendu que dans 2 ou 3 dimensions, les autres étant peu importantes." ], [ "Variété instable", "On peut déterminer la variété instable en regardant les images par $f=T^2$ d'un segment situé dans la direction instable au voisinage de l'origine (figure REF ).", "Figure: Variété instable.Il est intéressant, d'un point de vue dynamique, de visualiser comment celle-ci se déploie à l'intérieur de l'attracteur, à la fois d'un point de vue dynamique et d'un point de vue géométrique.", "En effet, la variété instable est une courbe continue, ce qui nous donne une idée plus précise de la géométrie de l'attracteur que lorsque nous ne disposons que d'un nuage de points.", "Ceci est fait avec une animationconsulter les fichiers var_u_0.1_2_1_18.avi et var_u_0.1_2_18.avi.", "dont la figure REF donne des extraits.", "Figure: Déploiement de lavariété instable à l'équilibre f n (W u (x eq ))f^n(W^u(x_{eq})).On peut décrire $f^n(W^u(x_{eq}))$ pour les valeurs successives de $n$ de la façon suivante : $1 \\le n \\le 11$  : une petite portion rectiligne autour de l'équilibre.", "$n=12$  : un filament plié, dans la direction de la pointe intermédiaire située dans le cou (43) (figure REF a).", "$n=13$  : le filament est un peu étendu et atteint le bas de la pointe.", "$n=14$  : le bas de la chevelure est atteint, ainsi que le cou.", "$n=15$  : le creux (avec la pointe 54) est atteint.", "Les filaments sont dédoublés (figure REF b).", "$n=16$  : un côté du triangle (71–75) et la droite de la queue sont atteints.", "Les filaments se dédoublent ailleurs (figure REF c).", "$n=17$  : un autre côté du triangle (76–80) et la pointe arrière sont atteints (figure REF d).", "$n=18$  : le dernier côté du triangle (59–66) est atteint.", "C'est la dernière région importante à être touchée.", "Notons tout de même que le filament 58–74 (c'est l'un des embranchements) n'est pas encore atteint." ], [ "Formation du pli", "Une des caractéristiques du système est l'existence d'un pliIl y en a peut-être plusieurs, mais il n'est pas évident de distinguer si deux régions pliées sont ou non indépendantes.", "Cela demanderait une étude plus approfondie.. Sous l'action de $T$ , la région quasi-rectiligne de la figure REF a se courbe progressivement pour arriver dans la région « pliée » de la figure REF b.", "Figure: Localisation du pli et de sa préimage." ], [ "Approche géométrique", "On peut visualiser la formation de ce pli en calculant la courbure au niveau du pli (sur un même filament de l'attracteur) à différents instants.", "Le résultatCette figure est extraite d'une animation donnant plus d'informations sur la formation géométrique du pli.", "est reproduit figure REF .", "Hormis quelques irrégularités, il se forme clairement un pli pour $15 < t < 1 6$ , et celui-ci s'accentue fortement pour former un pli très marqué à $t=4$ .", "L'évolution du maximum de courbure est reportée figure REF a. Cette étude permet également de localiser très précisément l'endroit plié, en notant à chaque instant la position du maximum de courbure sur le segment.", "On constate avec la figure REF b que celui-ci est situé au point 690 du segment initialement choisi.", "Figure: Formation du pli : courbure.Figure: Formation du pli : maximum de courbure." ], [ "Étude des discontinuités de la courbure", "Les différentes figures montrant la formation du pli présentent cependant quelques irrégularités.", "Ainsi, on peut se demander à quoi sont dûes les discontinuités de la courbure figure REF .", "Au vu de l'imprécision de la méthode de localisation du pli, il est quasiment certain que la ligne brisée considérée saute d'un filament à l'autre.", "Il est également possible que le manque de régularité des fonctions du modèle (qui sont $C^1$ et non $C^2$ ) engendre des ruptures de courbure au niveau de l'attracteur lui-même.", "Pour tester cette seconde hypothèse, nous avons tenté d'utiliser la variété instable globale de l'équilibre, qui devrait nous assurer que l'on considère un seul filament.", "Les résultats obtenus sont représentés en annexe , avec les figures REF à REF .", "Il y a toujours des discontinuités dans la courbure le long du filament.", "On peut sans aucun doute attribuer le pic de courbure aux environs de $j=500$ à un saut d'un filament à un autre (figure REF ).", "Il semble très difficile d'extraire avec suffisamment de précision un seul filament de l'attracteur dans la zone de pli.", "Cette tâche est peut-être simplement rendue impossible par l'aspect fractal de celui-ci.", "En revanche, la similitude des figures REF et REF confirme nos observations sur le processus de pliage.", "La figure REF b est d'ailleurs plus simple à interpréter, puisque l'on a uniquement le pic artificiel à $j\\approx 500$ et le pic réel à $j \\approx 875$ .", "La figure REF présente également des irrégularités facilement interprétables.", "En effet, la courbure ne croît pas toujours avec le temps, et le maximum de courbure se déplace par moments, même après la formation du pli à $t=16$ .", "Ces irrégularités peuvent s'expliquer par le non-uniformité du pincement et de l'étirement dans la dynamique.", "Ainsi, le pincement fait augmenter fortement la courbure, et l'étirement entraîne au contraire sa diminution.", "Sur deux ans, cela se traduit par une forte augmentation de la courbure, mais en temps continu, il est des périodes où la tendance s'inverse.", "Un argument en faveur de cette hypothèse est que ces périodes coïncident avec les instants des pics de population, c'est-à-dire les instants d'expansion maximale (voir figures REF et REF )." ], [ "Approche dynamique", "On peut aborder la question du pli d'un point de vue dynamique et non plus purement géométrique en observant l'évolution de la population en temps continu (figure REF ).", "Les points $x_j$ du filament non-plié dans $^3$ correspondent à l'intervalle $[-2;0]$ .", "Figure: Évolution en temps continu auniveau du pli.Figure: Vecteurs tangents au filament.Il est également intéressant de considérer la différentielle de $T^2$ le long du pli pour mieux comprendre ce qui se passe dans cette région de l'attracteur.", "Les vecteurs tangents (figure REF ) nous indiqueront alors comment $T^2$ plie le filament lui-même." ], [ "Filament non encore plié", "À $t=0$ (i.e.", "sur le filament non-plié $x_1 , \\ldots x_{1000}$ ), la plus grande valeur propre de la différentielleOn a calculé ici les valeurs et vecteurs propres de la différentielle $D$ de $T^2$ , et non ceux de ${D}D$ , qui permettent d'évaluer exactement les directions contractées ou dilatées, dans la mesure où ${x}{{D}D(x)}={D(x)}^2$ .", "Ces deux résultats ont cependant de bonnes chances d'être très semblables.", "de $T^2$ est négative et comprise entre $-5 2$ et $-3 5$ (figure REF ).", "Elle atteint un minimum en $x_{675}$ , c'est-à-dire à proximité du futur maximum de courbure $x_{690}$ .", "Figure: Différentielle de T 2 T^2sur le filament non-plié : première valeur propre.La seconde valeur propre est très éloignée de 1, et reste comprise entre 004 et 007.", "Le premier vecteur propre correspond donc à la direction instable.", "Il est représenté en différents points du filament sur la figure REF .", "On peut le décomposer en trois parties : (1) $0 \\le t \\le 1 4$  ; (2) $1 4\\le t \\le 1 6$  ; (3) $1 6 \\le t \\le 2$ .", "Il faut tenir compte de la normalisation du vecteur propre pour interpréter correctement son évolution le long du pli.", "La première partie varie très peu le long du filament, ce qui confirme l'hypothèse de formation du pli à $t \\approx 1 4$ que l'on a formulée d'après la figure REF a.", "La troisième partie est rectiligne et remonte en même temps que le pic de la deuxième partie.", "Elle traverse 0 en $x_{686}$ , à l'instant précis où le pic atteint son maximum.", "Le pliage semble ainsi correspondre à une forte expansion localisée en $t=1 5$ , accompagnée d'une absence d'expansion (ou plutôt une contraction puisque les autres vecteurs propres ont de très petites valeurs propres) sur l'intervalle $[1 6 ; 2]$ .", "Ces deux phénomènes sont difficiles à distinguer dans la mesure où la normalisation $L^1$ du vecteur propre entraîne une compensation entre la hauteur du pic et celle de la partie rectiligne.", "On reconnaît tout de même une forme de pliage en nous limitant à la partie (3) du vecteur propre, l'expansion ayant lieu dans des sens opposés de part et d'autre du pli.", "Figure: Différentielle de T 2 T^2sur le filament non-plié : premier vecteur propre.On remarque également que la direction instable à $t=0$ (figure REF ) est semblable au vecteur tangent à $t=2$ (figure REF b).", "Cela signifie que le filament est suffisamment transverse à la variété instable en $x_{690}$ .", "Les quelques irrégularités observées sur les courbes des figure REF et REF semblent confirmer l'hypothèse avancée lors de l'étude de la courbure du pli : la ligne brisée considérée doit « sauter » d'un filament à l'autre, deux filaments côte à côte n'ayant pas précisément la même courbure." ], [ "Filament plié", "À $t=2$ (i.e.", "sur le filament plié $T^2(x_1) , \\ldots T^2(x_{1000})$ ), les deux plus grandes valeurs propres de la différentielle de $T^2$ sont représentées figure REF .", "Elle est légèrement supérieure à 2 au voisinage de $T^2(x_{690})$ , mais il n'y pas d'expansion tout au long du filament.", "Celle-ci est en effet de module inférieur à 1 entre $T^2(x_1)$ et $T^2(x_{444})$ où elle traverse le cercle unité, puis croît lorsqu'on se déplace le long du filament plié vers $x_{1000}$ où elle dépasse 4.", "Il n'y a donc pas uniforme hyperbolicité sur l'attracteur.", "Figure: Différentielle de T 2 T^2 surle filament plié : premières valeurs propres.Dans le domaine où les valeurs propres sont toutes de module strictement inférieur à 1, la courbe $\\lambda _{max}(j)$ n'est pas continue.", "En traçant aussi la seconde valeur propre, on constate qu'il s'agit d'un échange dans l'ordre des deux premiers vecteurs propres.", "Tant que la première valeur propre n'est pas suffisamment grande devant la seconde, le premier vecteur propre ne suffit pas à décrire la différentielle de $T^2$ .", "Il est cependant intéressant de le considérer autour de $T^2(x_{690})$ (figure REF ).", "Il a alors le même aspect qu'à la figure REF , avec un pic autour de $t=1 6$ .", "Figure: Différentielle de T 2 T^2 sur lefilament plié : premier vecteur propre.Compte-tenu du vecteur tangent au temps $t=4$ (figure REF a), le filament plié est faiblement transverse à la variété stable en $T^2(x_{690})$ .", "Après le pliage, il semble donc que la principale action de $T^2$ soit de renforcer le pli, par une forte contraction, l'expansion étant très faible dans la direction tangente au filament en $T^2(x_{690})$ .", "Il y a par ailleurs assez peu de différences entre les différentes directions instables représentées figure REF .", "Le pli étant déjà formé, il y a expansion dans la direction tangente au filament loin de $T^2(x_{690})$ .", "En effet, le vecteur tangent en $T^2(x_{900})$ (figure REF b) coïncide avec la direction instable au temps $t=2$ .", "C'est également le cas du vecteur tangent en $T^2(x_{400})$ , qui est précisément opposé au vecteur tangent en $T^2(x_{900})$ .", "Figure: Vecteurs tangents aufilament après pliage (t=4t=4)." ], [ "Visualisation animée de l'attracteur", "Une dernière façon de comprendre la formation du pli (et la dynamique générale de l'attracteur) est d'utiliser une animationIl s'agit du fichier film_delta.avi.", "représentant la projection dans $^3$ de l'attracteur avec origine des temps $\\delta _t$ pour des valeurs successives de $\\delta _t$ (voir section REF ).", "On constate pour $-06 \\le \\delta \\le -0 35$ qu'une portion jusque là quasiment rectiligne se plie littéralement au cours d'un rapide déplacement dans $^3$ de la zone « inférieure » (deux coordonnées proches de 1, la troisième grande) vers la zone « supérieure » (une coordonnée proche de 1, les deux autres grandes).", "Le pli est inexistant pour $\\delta _t= - 0 6$ (figure REF a), quasiment formé pour $\\delta _t = -05$ mais pas encore placé en $x_{690}$ (figure REF b) et l'est totalement à $\\delta _t =-0 35$ (i.e.", "$t=165$ avec les conventions de cette section : voir figure REF c).", "Figure: L'attracteur et le pli pourdifférentes valeurs de δ t \\delta _t.La suite de la déformation, pour $\\delta _t > -04$ , ne fait qu'accentuer ce pli en étirant l'attracteur dans une direction et en le contractant dans les autres (au voisinage du pli).", "Cette contraction se ressent plus particulièrement autour de $\\delta _t= 0 5$ (i.e.", "$t \\approx 2 5$  : figure REF d)." ], [ "Non-hyperbolicité de\nl'attracteur", "La figure REF représente le spectre de la différentielle de $T^2$ en chacun des 80 points de la carte (en mettant la structure $L^2$ canonique sur chaque espace tangent).", "Cela nous permet de caractériser l'action de $T^2$ sur les différentes zones de l'attracteur.", "Figure: Spectre de la différentielle deT 2 T^2 en les 80 points de la carte.La figure REF permet de situer dans $^3$ les zones d'expansion (dans la ou les direction(s) instable(s)) et de « non-hyperbolicité »non-hyperbolicité avec la structure $L^2$ sur chaque espace tangent.", "Il est possible qu'une autre structure rende l'attracteur hyperbolique..", "Figure: Spectre de la différentielle selon les zones de l'attracteur.L'expansion est assez forte dans l'ensemble, à l'exception de trois régions : le pli (25, 28, 22, 27), le creux (61, 62) et le triangle (78 ; 65 ; 75, 72, 73).", "Ces trois zones semblent ainsi jouer un rôle particulier dans la dynamique.", "Nous avons déjà commencé l'étude du pli, il faudra également s'intéresser aux deux autres zones.", "Le « triangle » possède sans aucun doute une orbite de période 3 pour $T^2$ .", "Il faudrait évaluer plus précisément l'hyperbolicité de cette orbite, qui joue un grand rôle dans la dynamique globale.", "Le « creux » semble quant à lui être une autre zone de pli dans l'attracteur.", "Il faudrait déterminer si c'est effectivement le cas, et si ce pli est ou non distinct du pli que nous avons déjà mis au jour.", "Enfin, deux points présentent à la fois une forte expansion et un défaut d'hyperbolicité (c'est-à-dire une seconde valeur propre proche de 1) : 42 et 70.", "Leurs images par $T^2$ étant proche du point 62, il est possible que leur particularité soit simplement liée au « creux »." ], [ "Stabilité de la structure dans l'espace des\nparamètres", "Après avoir détaillé certains éléments de la structure d'un seul attracteur, celui que nous avons observé pour $(A_0,\\rho ,\\gamma ) = (015; 030; 825)$ , on peut s'interroger sur la persistance de cette structure quand on fait varier légèrement les paramètres.", "Nous pouvons d'ores-et-déjà esquisser une réponse, visuelle, à l'aide d'une animationVoir les fichiers film_A0_100__200_300_10_50_08250_1_2coul_0.1_0.3.avi pour une vue d'ensemble, et film_zoomA0_100__200_300_10_50_08250_1.avi pour un zoom sur la région pliée (zone 3).", "représentant l'attracteur $(A_0; 030; 825)$ dans $^3$ quand $A_0$ varie (voir aussi la section REF , consacrée à cette exploration, notamment le diagramme de bifurcation REF ).", "Pour $0135 \\le A_0 \\le 0 160$ , l'attracteur grandit continûment en deux parties, à partir d'une forme très simple (uniquement la région de l'équilibre et du pli, i.e.", "zones 2 et 3).", "Pour $A_0 \\approx 0 139$ , les autres régions apparaissent brusquement, dans une configuration assez similaire à $A_0 = 0 15$ .", "La croissance de l'attracteur se fait par l'allongement de filaments s'achevant pour une pointe, un peu de la même façon que lorsqu'on a observé comment la variété instable s'étend à l'intérieur de l'attracteur (voir section REF ).", "L'essentiel de la structure est conservé, même si la complexité de la dynamique va apparemment en s'accroissant.", "En $A_0 = 0 16$ , les deux composantes connexes se rejoignent pour n'en former qu'une, qui continue à grandir de la même façon.", "Mises à part quelques interruptions (une ou deux fenêtres de périodicité), l'attracteur reste intact (tout en se complexifiant au fur-et-à-mesure de sa croissance) jusqu'à $A_0 \\approx 0 19$ .", "Il disparaît alors par brusques paliers pour laisser place à une grande fenêtre de périodicité, puis un cycle se refermant sur un point fixe.", "On ne le retrouve plus trace ensuite de cet attracteur.", "Un zoom sur la région du pli (zone 3) nous a permis de déterminer si la pointe où se situent les points 29 et 30 continue à grandir en se pliant, parallèlement aux filaments des points 21 à 28.", "Contrairement à ce que l'on aurait pu penser, ce n'est pas le cas : la double pointe 29–30 reste d'un seul côté du pli sans le franchir.", "Il n'y a donc pas de complexification de la structure de cette manière-là." ], [ "Perspectives", "Il reste beaucoup de travail à faire pour comprendre ce modèle.", "À la lumière des simulations dont nous venons d'analyser les résultats, nous pouvons cependant déjà esquisser quelques pistes afin de poursuivre ce travail." ], [ "Simulations numériques", "Tout d'abord, plusieurs aspects restent à étudier numériquement afin de comprendre au mieux le comportement du modèle.", "Certaines questions étaient déjà évoqués dans [8], d'autres sont venues à la lueur des nouvelles simulations numériques." ], [ "Explorations", " Lors de l'augmentation de $A_0$ dans le diagramme $(A_0,0.30, 8.25)$ , les deux filaments se rejoignent-ils préciséments ou bien se rapprochent-ils suffisamments pour induire le mélange ?", "Dans le cas d'une exploration particulière, chercher le plus grand nombre possible d'attracteurs, notamment en « suivant » automatiquement chacun jusqu'à la perte de stabilité, et en utilisant plusieurs conditions initiales.", "Ce travail pourrait déjà être fait dans le cas des explorations déjà faites, notamment pour éclaircir la question des discontinuités du diagramme.", "Exploration plus exhaustive de l'espace des paramètres.", "Tester d'autres transitions de l'équilibre stable vers la dynamique chaotique.", "Déterminer des régions de l'espace des paramètres (en 2 ou 3 dimensions) où se produisent les divers comportements observés (orbite périodique attractive, cycle attractif, attracteur de type Hénon et autres attracteurs étranges), et les bifurcations qui se produisent à l'interface entre ces différentes zones.", "Pour chaque comportement observé, on pourrait chercher à caractériser un peu plus précisément la dynamique dans un cas particulier au moins (notamment pour les cycles et les attracteurs de type Hénon).", "Déterminer par un calcul numérique la nature des bifurcations observées sur le diagramme, en calculant les valeurs propres de la différentielle (soit en linéarisant l'équation, soit expérimentalement).", "Ceci serait particulièrement utile au niveau des discontinuités apparentes du diagramme." ], [ "Étude de l'attracteur étrange", " Géométrie de l'attracteur : Décrire géométriquement l'attracteur avec plus de précision, et notamment comment se séparent les différents filaments (en se limitant aux plus gros d'entre eux).", "Ceci devrait permettre d'élaborer un modèle simple pouvant générer un tel attracteur.", "Il faudrait en particulier caractériser les « embranchements ».", "Y a-t-il des zones plus « denses » que d'autres (au sens de la mesure de Lebesgue et non de la mesure physique) ?", "Dynamique sur l'attracteur : il reste beaucoup de travail pour comprendre précisément l'intéraction entre cette dynamique en trois dimensions et la dynamique en temps continu.", "Dimension fractale : est-elle la même dans toutes les régions de l'attracteur ?", "Sensibilité aux conditions initiales : calculer les exposants de Lyapunov.", "Équilibre et orbites périodiques : S'assurer de la correction de l'estimation de la seconde plus grande valeur propre de la différentielle à l'équilibre, éventuellement en linéarisant directement l'équation.", "Localiser quelques orbites périodiques de faible période (en particulier l'orbite de période 3 pour $T^2$ , située dans la zone 8 et l'orbite de période 2 dans les zones 4-5) et y effectuer la même étude que pour l'équilibre, notamment en évaluant les valeurs propres de la différentielle de $T^m$ et en traçant la variété instable.", "Y a-t-il d'autres équilibres, associés ou non à ces orbites, situés en-dehors de l'attracteur ?", "Le tracé que nous avons fait représente l'ensemble $\\omega $ -limite : où sont situés les points périodiques ?", "sont-ils denses ?", "Hyperbolicité : utiliser les orbites périodiques présentes sur l'attracteur pour la tester avec plus de généralité.", "Pli(s) : Peut-on éviter les discontinuités de la courbure en augmentant encore la précision de localisation du pli ?", "Y a-t-il un lien entre pics de courbure et défaut d'injectivité de la projection ?", "Localiser les autres plis s'il y en a (notamment en étudiant le « creux » et sa préimage, qui se sont distingués dans l'étude de l'hyperbolicité).", "Mesure physique : Évaluer la mesure physique sur l'attracteur : quelles zones sont plus chargées, quelles zones le sont moins ?", "Essayer de « suivre » précisément la structure mise en évidence sur cet attracteur lorsque l'on fait varier légèrement les paramètres (l'équilibre, le pli, les valeurs propres de la différentielle dans ces régions, etc.).", "Jusqu'où peut-on la suivre ?" ], [ "D'autres attracteurs étranges ?", "Le cas $(018; 030; 825)$ (figure REF ) semble plus complexe mais peut-être encore plus intéressant que l'attracteur que nous avons étudié.", "Parmi les explorations déjà effectuées (ou à venir), on pourrait chercher à approfondir l'étude de la dynamique de certains objets remarquables, à la lumière du travail déjà effectué." ], [ "Conjectures", "Au vu des résultats numériques, on peut énoncer quelques conjectures, en vue de rendre rigoureuses les observations qualitatives que nous venons de faire, et dont nous sommes à peu près sûrs.", "La dynamique est chaotique pour certaines valeurs des paramètres.", "L'attracteur est étrange.", "La dynamique chaotique est persistante, au voisinage de cet attracteur étrange.", "La dimension fractale de l'attracteur étrange est comprise strictement entre 1 et 15.", "L'attracteur est « quasi-hyperbolique » (cf.", "propriétés d'un système hyperbolique et de l'attracteur de Hénon en annexe REF ).", "On peut le décomposer en un nombre fini de parties dans lesquelles $T^1$ est transitive (« décomposition spectrale »).", "Il y a une cascade sous-harmonique lorsqu'on se déplace dans l'espace des paramètres à partir de $\\gamma $ petit, $\\rho $ proche de 0 ou 1, $A_0$ petit, pour se diriger vers des valeurs plus grandes de $\\gamma $ ou de $\\rho $ , ou bien vers des valeurs intermédiaires de $\\rho $ ." ], [ "Questions biologiques", "Il y a de nombreuses manières de complexifier le modèle pour le rendre plus réaliste, par exemple en ne supposant plus que la survie est indépendante de l'âge.", "On peut également essayer d'introduire des effets maternels." ], [ "Conclusion", "Revenons tout d'abord au problème purement biologique que nous nous sommes posé initialement.", "Il s'est avéré qu'un modèle simple combinant maturation et des saisons régulières peut engendrer des comportements chaotiques extrêmement complexes, pour des valeurs assez raisonnables des paramètres.", "La principale condition, et qui se trouve remplie pour ce qui concerne Microtus epiroticus, est une très forte fécondité.", "Il est clair que ce modèle pourrait difficilement être prédictif, tant il a été simplifié sans se soucier finement du cycle de vie des populations qui nous intéressent.", "En revanche, nous pouvons d'ores-et-déjà tirer des conclusions qualitatives, la plus importante étant que l'imprévisibilité des effectifs futurs à long terme peut avoir lieu dans un environnement stable, où tous les hivers sont strictement identiques.", "De plus, de nombreuses difficultés techniques soulevées lors de la mise en œuvre de simulations sur ce premier modèle seront très utiles pour des calculs numériques sur des modèles plus complexes dérivés ou non de celui-ci.", "D'un point de vue mathématique, nous avons eu un aperçu de la richesse des comportements que peut générer un modèle somme toute assez élémentaire.", "En nous attardant sur l'un de ces « attracteurs étranges », nous avons pu évaluer la complexité d'un seul de ces comportements, en mettant en évidence des phénomènes très mal compris, voire jamais abordés encore.", "Nous avons finalement posé beaucoup plus de questions que nous n'avons apporté de réponses.", "De nombreuses simulations restent ainsi encore à faire, soit pour confirmer une hypothèse expliquant les résultats obtenus, soit pour nous aider à en formuler au sujet des questions pour lesquelles nous n'arrivons même pas à esquisser une réponse.", "Le travail théorique restant à faire est lui aussi immense.", "Avec quelques outils élémentaires, nous avons pu définir l'attracteur global du système, mais nous n'avons aucune information à son sujet.", "Les expérimentations numériques nous ont permis de formuler quelques conjectures, mais ont surtout pour objectif de nous indiquer par quelles voies il serait possible de s'attaquer à la résolution de celles-ci.", "Au vu des quelques figures que nous avons pu tracer, il semble assez probable que les « attracteurs étranges » — s'il s'avèrent en être effectivement — que nous avons observés ont certainement un grand intérêt dans l'étude théorique des systèmes dynamiques non-uniformémement hyperboliques." ], [ "Discrétisation du modèle", "On veut passer du modèle continu décrit par l'équation (REF ) à un modèle discret, qui pourra être simulé numériquement.", "Fixons un entier $p>0$ , ce sera le nombre de classes d'âges considérées par année.", "On note $n_i$ le nombre de naissances qui ont lieu dans l'intervalle de temps $\\left[\\frac{i-1}{p};\\frac{i}{p}\\right[$ , $N_i$ l'effectif mature moyen au cours de ce même intervalle, $e_i$ le facteur saisonnier égal à la moyenneEn pratique, on a pris la moyenne des valeurs aux extrémités de l'intervalle.", "dans cet intervalle de $m_{\\rho ,\\epsilon }$ (définie par (REF ), $i$ étant considéré mod.", "$p$ ).", "Soit $s_i$ la proportion des individus matures et vivants parmi ceux qui sont nés $i$ pas de temps auparavant, et $m$ la fonction de fécondité définie par les relations (REF ).", "On a alors les relations suivantes : $\\left\\lbrace \\begin{aligned}n_i &= \\frac{m(N_i)\\times N_i \\times e_i}{p} \\\\N_i &= \\sum _{k=1}^{2p} s_k \\times n_{i-k}\\end{aligned} \\right.", "$ La condition initiale $\\left(n_i\\right)_{1 \\le i \\le 2p}$ étant donnée, ceci permet de calculer les $n_i$ pour tout $i > 2p$ .", "Remarquons que le calcul de $s_i$ n'est pas toujours évident.", "Pour les calculs, dans le cas où $A_0$ est un multiple entier de $1/p$ ($A_0 = i_0 / p$ ), on a pris $s_i = \\left( 1-\\frac{i}{p\\times A_1} \\right) _{i \\ge i_0}$ .", "Lorsque $A_0$ n'est pas adapté au pas de discrétisation, on a pris pour $s_i$ la moyenne des $s_{j,q}$ pour $(i-1)\\times k +1 \\le j \\le i\\times k$ , où $s_{j,q}$ désigne le coefficient de survie calculé avec un pas $q=p \\times k$ (le plus souvent, $k=100$ ).", "C'est le cas du diagramme REF .", "Pour des raisons pratiques de calcul, on a souvent utilisé le vecteur des naissances $n_i$ au lieu des effectifs matures $N_i$ , car cela évite de calculer deux fois $n_i$ au cours de la simulation.", "Il est bien sûr très simple de passer des naissances aux effectifs matures, mais en perdant les deux premières années.", "Ce choix explique l'apparition des naissances dans les résultats, alors que celles-ci ne sont pas explicitement utilisées dans le modèle." ], [ "Choix des conditions initiales", "Pour choisir une condition initiale « au hasard », on a choisi de déterminer un vecteur de naissances aléatoire.", "Les naissances successives sont tirées suivant des lois uniformes indépendantes.", "Deux méthodes ont été employées.", "Pour la condition initiale (I), on a imposé qu'il n'y ait pas de naissances en hiver (avec $\\rho =041$ ), et que l'effectif mature soit égal à 20 à l'instant $t=0$ (avec $A_0 = 0 18$ , $A_1 = 2$ ).", "La figure REF représente les naissances ($-2 \\ge t < 0$ ) et la première génération de populations matures ($0 \\le t \\le 2$ ) qui en découle (avec $A_0 = 0 18$ , $A_1 = 2$ , $\\gamma = 8 25$ ).", "Figure: Condition initiale (I).Pour la condition initiale (II), qui a été utilisée le plus souvent, on n'a pas tenu compte de l'hiver (le choix d'une valeur particulière de $\\rho $ n'étant ps justifié), et on a imposé un effectif mature égal à 1 à l'instant $t=0$ (cette valeur étant plus raisonnable au vu de la dynamique stationnaire du système).", "La figure REF représente dans les mêmes conditions que précédemment les naissances et la première génération correspondante.", "Figure: Condition initiale (II).On pourrait bien sûr concevoir d'autres méthodes de choix aléatoire d'une condition initiale, évitant mieux les biais possibles, mais ce n'est pas très important pour l'usage que nous en avons eu.", "Cela ne serait utile que dans le cadre d'une étude de la « taille » de bassins d'attractions de plusieurs attracteurs." ], [ "Choix du pas de discrétisation", "Le paramètre $p$ (nombre de classes d'âge par année, appelé $an$ pour plus de clarté) est décisif pour les simulations.", "La complexité de l'algorithme est en effet proportionnelle au carré de ce pas.", "Le choix $an=100$ est le résultat d'un compromis entre rapidité du calcul et précision, après quelques simulations test.", "Cette valeur est de plus raisonnable pour ce modèle : au vu de l'ordre de grandeur des paramètres $A_0$ et $\\rho $ , il ne semble pas utile d'être plus précis.", "De plus, la situation biologique de départ étant discrète, il est inutile de considérer une échelle de temps inférieure à trois jours." ], [ "Lissage des fonctions", "Le choix des fonctions $m(N)$ et $m_{\\rho }(t)$ étant assez arbitraire et peu réaliste, le lissage des fonctions a surtout été fait à titre préventif, pour que d'éventuelles discontinuités ou irrégularités dans la dynamique ne soient pas dûes au manque de régularité des fonctions utilisées dans le modèle.", "Il a été motivé par l'observation d'angles sur certains attracteurs qui semblaient lisses par ailleurs, au cours de simulations préliminaires.", "Il semble en réalité que ces changements n'ont pas modifié qualitativement les propriétés globales du système dans l'espace des paramètres.", "En revanche, il est certain que pour des valeurs fixées des paramètres, la plus infime modification des fonctions utilisées dans le modèle peut modifier entièrement le comportement observé." ], [ "Visualisation de l'attracteur en 3 dimensions", "On représente $(N(t),N(t+1),N(t+2)$ pour les valeurs entières de $t \\in [t_{\\min };t_{\\max }]$ .", "Ces valeurs entières correspondent précisément à la fin de l'été dans le cas $\\epsilon _{ete}=0$ , au milieu de l'automne dans le cas général." ], [ "Intervalle de temps choisi", "Le plus souvent, on a choisi de se limiter à ${19001}\\le t \\le {20000}$ .", "La valeur maximale est choisie pour que le régime transitoire soit largement dépassé, et elle fixe la durée du calcul : avec 100 pas par an, un calcul sur 20000 ans se fait en à peu près 2 minutes avec un ordinateur récentPC avec un processeur Athlon 24 GHz et 768 Mo de mémoire vive.. Avec $t\\ge {19001}$ , on dispose de suffisamment de points pour distinguer clairement le type d'attracteur (orbite périodique, cycle, type Hénon, étrange, etc.)", "tout en évitant le régime transitoire.", "Ces deux valeurs ont été testées sur quelques exemples de valeurs des paramètres, en s'assurant que le régime transitoire est très largement passé.", "Lors des simulations suivantes, on a vérifiéCela se détecte facilement sur la représentation en trois dimensions.", "que c'était toujours le cas." ], [ "Visualisation de la mesure physique", "Pour limiter la taille des figures représentant l'attracteur $(0.15,0.30,8.25)$ , nous avons représenté une partie seulement des points que nous avons calculés, de telle sorte qu'il y ait de l'ordre de 20000 points sur chaque graphique.", "Certaines zones étant beaucoup plus denses que d'autres, cette limitation a été faite dans des proportions différentes suivant les huit zones de l'attracteur que nous avons définies.", "Il devrait donc y avoir environ deux fois plus de points dans les zones 2 à 5, sur tous les graphiques où figurent les 80 points de la carte.", "Cette transformation n'a ainsi pas été effectuée sur la figure REF , qui permet donc de se faire une idée plus exacte de la mesure physique sur l'attracteur." ], [ "Diagrammes de bifurcation", "Afin de comprendre le rôle que jouent les paramètres du modèle dans la dynamique, nous avons réalisé des animations représentant les graphiques en 3 dimensions, l'un des paramètres variant au cours du temps.", "Pour retranscrire une partie de ces informations sur un graphique bidimensionnel, nous avons dû tracer des diagrammes de bifurcation.", "Le principe est le suivant : on porte en abscisse l'un des paramètres du système et $N(t)$ en ordonnée, pour $t$ entier, ${19001} \\le t \\le {20000}$ .", "On s'est contenté de 1000 valeurs de $t$ , au lieu de 10000, car la projection unidimensionnelle de l'attracteur ne permet pas de bien saisir sa géométrie.", "On distingue tout au plus les ensembles finis, les ensembles fractaux et les ensembles continus.", "Une précision accrue n'aurait rien apporté." ], [ "Injectivité de la projection", "Pour évaluer l'injectivité de la projection $\\pi $  : $^{201}$ $\\rightarrow $ $^3$ , nous avons cherché à évaluer $\\sup _{t\\ne t^{\\prime } \\in }\\frac{{x_{201}(t)-x_{201}(t^{\\prime })}_{^{201}}}{{x_{3}(t)-x_{3}(t^{\\prime })}_{^3}}$ au voisinage des 80 points de la « carte » de l'attracteur (section REF ).", "En chaque point, nous avons déterminé les éléments de la boule $B$ de rayon $r = 0.1$ dans $L^p(^{201})$ centrée en ce point $x_{201}(t_0)$ .", "Nous avons alors calculé $\\sup _{t \\ne t_0 \\in B}\\frac{{x_{201}(t)-x_{201}(t^{\\prime })}_{^{201}}}{{x_{3}(t)-x_{3}(t^{\\prime })}_{^3}}$ .", "Nous avons indiqué une deuxième information sur les graphiques ainsi obtenus : le nombre de points dans chaque boule.", "En effet, certains points sont situés dans des zones beaucoup plus denses que d'autres (au sens de la mesure physique), et cela peut fausser les résultats obtenus puisque nous ne conservons que le maximum sur les points de la boule.", "Il est normal d'obtenir un résultat plus élevé si la boule contient plus de points, puisque l'on risque d'avoir des points très proches dans $R^3$ par « accident », au vu des approximations que nous sommes obligés de faire.", "Nous avons choisi les normalisations suivantes pour les normes $L^p$ dans $^N$ , $p < \\infty $  : ${(x_1, \\ldots ,x_N)}_{L^p}^p = \\frac{1}{N} \\sum _{i=1}^N x_i^p$ .", "La norme $L^{\\infty }$ est simplement le $\\sup $ des coordonnées.", "Notons que lorsque $\\delta _t$ est pris non-nul, le vecteur $x_{201}(t)$ est décalé d'autant.", "Les boules considérées ne sont donc pas le mêmes lorsque $\\delta _t$ est différent.", "Ceci permet de mieux comparer les différentes valeurs de $\\delta _t$ , et de ne pas privilégier $\\delta _t = 0$ .", "Pour résumer les nombreux graphiques obtenus, nous avons choisi d'extraire des 80 valeurs de $\\sup $ deux données : le maximum et la valeur médiane.", "Elles nous permettent d'avoir une assez bonne idée de la qualité de la projection.", "Enfin, lorsque nous avons considéré d'autres visualisations, par exemple avec une échelle logarithmique, nous avons procédé aux mêmes opérations, en remplaçant $x_3$ par $g(x_3)$ , $x_{201}$ étant inchangé.", "Il y a alors un inévitable changement d'échelle homothétique : par exemple, avec $g(x) = \\lambda x$ et $\\lambda $ très grand, le résultat serait artificiellement bon.", "Il faut donc rapporter les quantités calculées au diamètre de la projection de l'attracteur.", "Dans le cas de l'échelle logarithmique, cela ne fait que confirmer notre conclusion en sa défaveur." ], [ "Géométrie", "La représentation simplifiée de la figure REF est simplement un extrait de la variété instable, $n=18$ (voir annexe REF ).", "Le découpage a été fait morceau par morceau, et le choix des régions est purement visuel." ], [ "Décomposition spectrale, mélange", "Pour déterminer s'il y a ou non mélange topologique, nous avons utilisé plusieurs couleurs suivant la valeur de $t$ modulo $N$ , pour quelques valeurs de $N$ entre 2 et 10.", "Lorsque des zones bien séparées se sont distinguées, nous avons conclu à la $N$ -périodicité de la dynamique.", "Dans le cas contraire, si nous avions l'impression d'un mélange des couleurs pour toutes les valeurs de $N$ (comme à la figure REF ), nous avons conclu au mélange topologique.", "Il ne s'agit donc que d'impressions visuelles, et non de vérifications rigoureuses." ], [ "Calcul de la dimension fractale", "La dimension fractale est définie en annexe REF .", "Calculer la dimension fractale d'un ensemble dont on ne connaît qu'un nombre fini de points, avec une précision limitée, est loin d'être un problème facile.", "Nous avons donc dû faire de nombreuses approximations pour tenter d'obtenir une valeur approchée raisonnable." ], [ "Attracteur $(015; 030; 825)$", "Nous avons considéré l'attracteur dans $^3$ , et non dans $^{201}$ , avec la projection naturelle, l'origine des temps étant prise $\\delta _t=0$ .", "Il s'agit donc d'un ensemble $K$ d'environ $N_0={100000}$ points.", "Pour différentes valeurs de $\\epsilon $ , nous avons calculé le nombre $\\widetilde{N}_{\\epsilon }(K)$ de cubes $C_{i,j,k} = [i \\epsilon ;(i+1) \\epsilon ] \\times [j \\epsilon ; (j+1) \\epsilon ] \\times [k\\epsilon ; (k+1) \\epsilon ]$ qui contiennent au moins un point de $K$ .", "La figure REF représente ainsi $\\log _{10} \\widetilde{N}_{\\epsilon }(K)$ en fonction de $\\log _{10} \\epsilon $ .", "En théorie, la dimension fractale est l'opposé de la pente limite en $-\\infty $ de cette courbe (rigoureusement, on sait que $\\widetilde{N}_{\\epsilon }(K) \\ge N_{\\epsilon }(K) \\ge \\frac{\\widetilde{N}_{\\epsilon }(K)}{8}$ , la pente limite doit donc être la même).", "Mais dès que $\\epsilon $ est assez petit, $N_{\\epsilon }$ est constant égal à $N_0={100000}$ , car l'ensemble $K$ est fini.", "Nous avons donc considéré la pente (obtenue par régression linéaire) en nous restreignant à $N_{\\epsilon } < \\frac{N_0}{10} = N_{\\epsilon _0}$ et $\\epsilon > \\sqrt{\\epsilon _0}$ .", "Le choix de ce domaine a été fait au vu des données et correspond à la zone où les points sont bien alignés.", "En raison de ces nombreuses approximations, il ne faut pas accorder trop d'importance à la valeur précise que nous avons obtenue, seul l'ordre de grandeur (entre 1 et 15) a de bonnes raisons d'être exact." ], [ "Diagramme $(0.18,0.41,\\gamma )$", "Pour réaliser la figure REF , nous avons dû calculer la dimension fractale d'un grand nombre d'ensemble, via un processus automatisé.", "Au lieu de 100000 points, nous avons dû nous contenter de $N_0={10000}$ points.", "La précision de ces calculs est donc encore inférieure.", "Nous disposons probablement d'une légère sous-estimation des dimensions fractales de ces attracteurs." ], [ "Sensibilité aux conditions initiales", "On choisit un point sur l'attracteur, considéré dans $^{201}$ , et l'on détermine l'ensemble des points de l'attracteur situés à une distance inférieure à $r$ pour la norme $L^p$ (sur la figure REF , $r=0 04 $ et $p=\\infty $ ).", "Par point « de l'attracteur », on entend qu'il s'agisse d'un des 20000 points d'une orbite calculée précédemment et dont on a les coordonnées dans $^{201}$ .", "Ces points sont donc répartis selon la mesure physique.", "Pour chacun de ces points, on trace $N(t)-N^{\\star }(t)$ pour $t\\in [-20;20]$ , où $N^{\\star }(t)$ correspond au point initialement choisi.", "L'intervalle de temps de référence est $[0;2]$ , le graphique de la figure REF n'est donc pas exactement centré sur le présent." ], [ "Première localisation dans $^3$", "À la recherche d'éventuels points fixes, nous avons cherché à minimiser la distance $L^1$ (non-normalisée), dans $^3$ , entre $x_3(t)$ et $x_3(t+2)=T^2(x_3(t))$ .", "Sur les 50000 points considérés (nous nous sommes limités à la « grande » composante), deux sont écartés de leurs images de moins $4.10^{-3}$ .", "Ces points étant très proches l'un de l'autre, nous avons considéré qu'ils sont à proximité d'un unique équilibre.", "La visualisation de ce point de son image en 3 dimensions ont confirmé cette impression dans la mesure où l'image du point « fixe » en est très proche et sur le même filament (figure REF ).", "Figure: Les « points fixes »et leurs préimages par T 2 T^2.En effet, d'autres points sur l'attracteur sont également à une petite distance (inférieure à $2.10^{-2}$ ) de leur image par $T^2$ (à proximité du point 43), mais celle-ci est sur un filament bien distinct.", "Il n'y aurait donc qu'un seul point fixe sur l'attracteur." ], [ "Localisation plus précise, dans $^{201}$", "Afin de disposer d'une meilleure approximation de ce point fixe, aussi bien dans $^3$ que dans $^{201}$ , nous avons eu recours à une méthode un peu plus sophistiquée.", "Nous avons tout d'abord choisi deux points de part et d'autre de l'équilibre présumé, avec une marge assez importante, et nous avons tracé le segment rejoignant ces deux points.", "Nous avons calculé les images successives de ce segment par $T^2$ , en les tronquant de telle sorte que l'on reste dans un même voisinage de l'équilibre présumé.", "Après un nombre suffisant d'itérations de ce processus, nous disposons d'une bonne approximation de la variété instable locale à l'équilibre.", "C'est en effet ce que montre le $\\lambda $ -lemme sous certaines conditionsLe $\\lambda $ -lemme ne s'applique sans doute pas directement dans notre cadre non hyperbolique, en dimension infinie, mais nous donne de bonnes raisons de penser que sa conclusion est au moins approximativement vraie..", "Le segment initial ayant été choisi transverse à la variété stable, et la troncature nous permettant déviter la région de pli, cela devrait être vrai dans notre cas pratique.", "Disposant de la variété instable locale, son image par $T^{-2n}$ doit se contracter autour de l'équilibre pour $n$ assez grand.", "Nous avons ainsi commencé par calculer $T^{4n}$ de la variété instable, suffisamment tronquée pour éviter le pli, tout en gardant la mémoire du passé (les troncatures successives nous obligeant à rajouter des points sur la ligne brisée).", "Pour une valeur suffisamment grande de $n$ (de l'ordre de 10), nous avons obtenu, en appliquant $T^{-2n}$ à la variété instable locale obtenue, un morceau de variété instable très proche de léquilibre.", "En minimisant la distance $L^1$ entre les points de cette variété instable locale et leur image (dans $^{201}$ ), on obtient ainsi une très bonne approximation de l'équilibre.", "L'erreur $L^1$ que nous avons obtenue est en effet environ égale à $10^{-4}$ , ce qui est presque 3000 fois mieux que notre première approximationdans la mesure où la norme $L^1$ utilisée n'est pas normalisée." ], [ "Variété instable", "On part de la variété instable locale à l'équilibre.", "La variété instable globale est alors donnée par la définition REF .", "Comme il s'agit d'un équilibre, il s'agit de calculer les images successives par $T^2$ de la variété instable locale.", "Afin de garder une précision finale correcte, on a augmenté progressivement le nombre de points définissant la variété instable.", "On a rajouté des points intermédiaires entre deux sommets consécutifs de la ligne brisée dès que leur distance $L^2$ dans $^3$ est inférieure à $\\eta = 10^{-2}$ .", "On a alors deux manières de visualiser la croissance de la variété instable dans l'attracteur.", "D'une part, les images successives $T^{2n}(W^u_{loc}(x_{eq}))$ pour $1 \\le n \\le 18$ .", "D'autre part, en ne considérant que $n = 18$ , on peut faire grandir la variété instable en partant du voisinage de l'équilibre.", "L'ordre d'apparition des différentes zones est en principe le même.", "La première méthode est plus naturelle et plus simplement interprétable, mais la seconde donne plus de détails sur l'ordre dans lequel la variété instable se déploie dans l'attracteur." ], [ "Calcul de la différentielle et de ses valeurs\npropres", "Étant donné un point $x$ de $^{201}$ , il est aisé d'estimer la différentielle de $T^2$ en ce point : on fixe $\\epsilon =10^{-6}$ et on calcule $(T^2(x+\\epsilon e_i)-T^2(x))/\\epsilon $ .", "Cela nous donne la dérivée partielle dans la direction $e_i$ (de la base canonique de $^{201}$ .", "La matrice des dérivées partielles nous donne une expression de la différentielle.", "Le calcul des valeurs propres et des vecteurs propres est alors réalisé à l'aide des fonctions intégrées de Matlab$^$ .", "Cela signifie que les espaces tangents en $x$ et $T^2(x)$ sont égaux à l'espace euclidien $^{201}$ .", "La norme considérée est donc la norme $L^2(^{201})$ .", "Pour différentes valeurs de $\\epsilon $ (allant de $10^{-3}$ à $10^{-6}$ ), les valeurs des cinq premières valeurs propres changent très peu : $\\lambda _1 = -22942$ , $\\lambda _2\\approx 00433$ , $\\lambda _3 \\approx -00283$ , $\\lambda _{4,5} \\approx 00214 \\pm 00028 i$ .", "Une incertitude persiste cependant sur les valeurs propres $\\lambda _j$ , $j \\ge 2$ , dans la mesure où les algorithmes de calculs sont assez instables.", "Il serait utile de vérifier les résultats ci-dessus par un autre calcul de la différentielle, par exemple en linéarisant directement l'équation.", "Les vecteurs propres représentés sont normalisés pour que leur moyenne ($L^1$ ) sur $[0;2]$ soit égale à 1." ], [ "Hyperbolicité", "Pour le calcul de la différentielle et de son spectre, on a utilisé la méthode précédente avec $\\epsilon =10^{-3}$ .", "On a considéré qu'une valeur propre met en défaut l'hyperbolicité de l'attracteur lorsque (${\\log _{10}(\\lambda )} < \\epsilon _d=\\log _{10}(3/2)$ ).", "Notons que les valeurs propres dépendent fortement de la structure des espaces tangents d'arrivée et de départ.", "Ce n'est pas parce que la structure canonique du fibré tangent ne rend pas l'attracteur hyperbolique que celui-ci ne l'est pas.", "Pour certains points, les deux premières valeurs propres sont égales (en module) : il s'agit des points où l'on a deux valeurs propres complexes conjuguées de module maximal.", "Dans les autres cas, il s'agit toujours d'une valeur propre réelle." ], [ "Pli : Localisation", "La première étape dans la localisation du pli est arbitraire et un peu imprécise : on choisit deux points sur l'attracteur $x_1$ et $x_{1000}$ (figure REF a), à peu près sur le même filament et tels que $T^2(x_1)$ et $T^2(x_{1000})$ sont situés de part et d'autre du pli (figure REF b).", "On détermine ensuite un segment dans $^{201}$ joignant $T^{-10}(x_1)$ et $T^{-10}(x_{1000})$ , sur lequel on place 1000 points (y compris les extrémités).", "En réappliquant $T^{10}$ à ces 1000 points, on définit une ligne brisée $(x_j)_{j=1 \\ldots 1000}$  : c'est le filament non-plié de la figure REF a, qui correspond au temps $t=0$ .", "Le temps $t$ correspond ainsi à la ligne brisée $(T^t(x_j))_{j=1 \\ldots 1000}$ .", "Les détails concernant les calculs de courbure sont donnés à la section suivante.", "Les calculs des différentielles ont été effectués comme indiqué section REF .", "La différentielle a été calculée à deux instants distincts : avant pliage ($t=0$ ) et à l'emplacement du pli ($t=2$ ).", "Précisons enfin que lorsqu'on parle du filament à l'instant $t=0$ (ou de sa courbure, etc.", "), il s'agit des $x_j$ .", "Dans le cadre d'une visualisation d'un $x_j$ particulier, dans $^{201}$ , on note l'intervalle de temps $[-2;0]$  : c'est l'instant final qui est pris en compte.", "De même, le filament à l'instant $s$ est composé des $T^s(x_j)$ , et correspond à l'intervalle $[s-2;s]$ dans $^{201}$ ." ], [ "Deuxième tentative", "On a réalisé une deuxième tentative de localisation du pli, plus précise, devant en principe supprimer les irrégularités de courbure que nous avons constatées.", "Pour cela, on utilise la variété instable globale au point fixe calculée précédemment.", "Pour $n=13$ , une partie de $f^n(W^u(x_{eq}))$ est située au niveau du pli.", "On a alors déterminé sa préimage par $f^{13}$ et augmenté le nombre sommets de la ligne brisée pour disposer d'un filament de plus de 1000 points dans la zone de pliage (en recalculant son image par $f^{13}$ )." ], [ "Vecteur tangent, courbure d'une ligne brisée", "Soient $x_1 , \\ldots , x_N$ les sommets successifs de cette ligne brisée." ], [ "Vecteur tangent", "Le vecteur tangent $T_j$ en un point $x_j$ a été calculé tout simplement à l'aide de la formule suivante : $ T_j =\\frac{x_{j+1}-x_j}{{x_{j+1}-x_j}_{L^1}} \\times 201.", "$ $T_j$ est donc normalisé pour avoir une moyenne ($L^1$ ) égale à 1 (c'est un élément de $^{201}$ )." ], [ "Courbure", "En supposant les sommets de la ligne brisée assez proches, on peut évaluer la courbure de la courbe qu'elle approche à l'aide d'une version discrète des formules continues définissant la courbure : $ds_j &= {x_{j+1}-x_j}_{L^2} \\\\T_j &= \\frac{x_{j+1}-x_j}{ds_j} \\\\\\kappa _j &= {\\frac{T_{j+1}-T_j}{ds_j}}$ Les points $x_j$ sont considérés dans $^3$ et non dans $R^{201}$ .", "La norme $L^2$ utilisée pour estimer l'élément de longueur $ds_j$ n'est pas normalisé, mais cela ne change rien pour le calcul de $\\kappa _j$ .", "On estime de cette façon la valeur absolue de la courbure, et non la courbure proprement dite." ], [ "Rappels de systèmes dynamiques", "L'objectif de cette section est de rappeler les notions les plus importantes de théorie des systèmes dynamiques et que nous avons évoquées précédemment.", "Pour plus de détails, on se reportera par exemple à [3] ou [1].", "Pour les aspects plus spécifiquement hyperboliques, on se reportera à [7] et [11]." ], [ "Un peu de vocabulaire", "Un système dynamique continu est la donnée d'un espace $X$ et d'un groupe à un paramètre de transformation $(f^t)_{t\\in }$ (c'est-à-dire une famille d'applications continues $X\\rightarrow X$ telles que $f^{t+t^{\\prime }} = f^t \\circ f^{t^{\\prime }}$ ).", "Un système dynamique discret est la donnée d'un espace topologique $X$ (l'espace des phases) et d'une application $f$ continue $X \\rightarrow X$ .", "On peut se ramener du premier cas au second par le biais de l'application de retour de Poincaré $f^1$ .", "Dans la suite, sauf indication contraire, on se placera toujours dans le cas discret.", "[Orbite] Si $x$ est un point de $X$ , l'orbite (positive) de $x$ est l'ensemble $\\lbrace f^n(x) n \\ge 0 \\rbrace $ .", "Si $f$ est bijective, l'orbite de $x$ est $\\lbrace f^n(x)n \\in \\rbrace $ .", "La théorie des systèmes dynamiques s'intéresse particulièrement au comportement des orbites.", "Il est souvent utile de considérer aussi des « pseudo-orbites » au sens de la définition suivante.", "[$\\delta $ -pseudo-orbite] Soit $\\delta >0$ .", "Une suite $(x_n)_{n \\in }$ (ou $(x_n)_{n \\in }$ ) est une $\\delta $ -pseudo-orbite si $\\forall i \\in $ (ou $$ ), $d(f(x_i),x_{i+1}) < \\delta $ .", "Par exemple, une orbite simulée numériquement est une $\\delta $ -pseudo-orbite, puisque les calculs sont effectués avec une précision limitée.", "[Partie invariante] Une partie $A$ de $X$ est dite invarianteattention, cette définition est parfois remplacée par $f(A)\\subset A$ .", "par $f$ si $f(A) = A$ .", "Dans la suite, on supposera $X$ métrique compact.", "La notion de conjugaison topologique est extrêmement importantes.", "Elle traduit l'idée que deux systèmes dynamiques sont topologiquement équivalents.", "[Conjugaison topologique] Soit $r \\ge 0$ .", "Deux applications $C^r$ $f: X\\rightarrow X$ et $g: Y \\rightarrow Y$ sont topologiquement conjuguées lorsqu'il existe un homéomorphisme $h : X\\rightarrow Y$ tel que $f = h^{-1} \\circ g \\circ h$ .", "Lorsque $h$ est un $C^m$ difféomorphisme ($m \\le r$ ), on parle de conjugaison lisse.", "Parfois, on peut seulement trouver $h: X \\rightarrow Y$ continue surjective telle que $h \\circ f = g\\circ h$ .", "On parle alors de semi-conjugaison.", "Nous pouvons désormais définir la stabilité structurelle d'un système dynamique.", "[Stabilité structurelle] Une application $f$ $C^r$ est $C^m$ structurellement stable ($1 \\le m \\le r \\le \\infty $ ) s'il existe un voisinage $U$ de $f$ pour la $C^m$ topologie telle que toute application $g \\in U$ est topologiquement conjuguée à $f$ .", "Si de plus on peut choisir $h = h_g$ dans la conjugaison de $f$ et $g$ tel que $h_g$ et $h_g^{-1}$ convergent uniformément vers l'identité lorsque $g$ converge vers $f$ pour la topologie $C^m$ , alors on dit que $f$ est $C^m$ fortement structurellement stable." ], [ "Récurrence", "[Point périodique] Un point $x$ de $X$ est dit périodique lorsqu'il existe $n \\ge 1$ tel que $f^n (x) =x$ .", "On note $Per(f)$ l'ensemble des points périodiques.", "$ f(Per(f)) = Per(f) \\text{ et } f({Per(f)}) ={Per(f)}.", "$ [Point récurrent] Un point $x$ de $X$ est dit positivement récurrent (resp.", "négativement récurrent) si $x$ est un point d'accumulation de la suite $(f^n(x))_{n \\ge 0}$ (resp.", "$(f^{-n}(x))_{n \\ge 0}$ ).", "On note $R^+(f)$ l'ensemble des points positivement récurrents, $R^-(f)$ l'ensemble des points négativement récurrents et $R(f) =R^+(f) \\cup R^-(f)$ l'ensemble des points récurrents.", "Autrement dit, partant d'un point récurrent, on revient une infinité de fois dans son voisinage.", "On démontre : $Per(f) \\subset R^+(f) \\cap R^-(f) \\ne \\emptyset \\\\f(R^+(f))=R^+(f) \\text{ et } f(R^-(f))=R^-(f) $ [Point limite] Pour tout point $x$ , on note $\\omega (x)$ (ensemble $\\omega $ -limite de $x$) l'ensemble des points d'accumulation de $(f^n(x))_{n \\ge 0}$ et $\\alpha (x)$ (ensemble $\\alpha $ -limite de $x$)l'ensemble des points d'accumulation de $(f^n(x))_{n \\le 0}$ .", "On définit alors l'ensemble $\\omega $ -limite $L^+ (f) = \\overline{\\bigcup _{x\\in X} \\omega (x)}$ , l'ensemble $\\alpha $ -limite $L^- (f) =\\overline{\\bigcup _{x \\in X} \\alpha (x)}$ et l'ensemble limite $L(f) = L^+(f) \\cup L^-(f)$ .", "Un point de $\\omega (x)$ est un point dont l'orbite issue de $x$ visite le voisinage une infinité de fois.", "On a les propriétés suivantes : $ R^+(f) \\subset L^+(f) \\text{ et } R^-(f) \\subset L^-(f) \\\\ f(L^+(f))=L^+(f) \\text{ et } f(L^-(f))=L^-(f)$ [Point erranten anglais : wandering] Un point $x \\in X$ est dit errant s'il possède un voisinage $V$ tel que $f^n(V) \\cap V = \\emptyset $ pour tout $n\\ge 1$ .", "Sinon, on dit que $x$ est non-errant, et on note $\\Omega (f)$ l'ensemble des points non-errants.", "Un point est donc non-errant lorsque tout voisinage se recoupe au moins une fois dans le futur.", "On peut démontrer les résultats suivants : $ {L^+(f)} \\subset \\Omega (f) \\text{ et }{L^-(f)} \\subset \\Omega (f)\\\\ f(\\Omega (f)) = \\Omega (f) $ [Point récurrent par chaîneen anglais : chain-recurrent] Un point est récurrent par chaîne si pour tout $\\delta >0$ il existe une $\\delta $ -pseudo-orbite périodique issue de $x$ .", "On note $C(f)$ l'ensemble des points récurrents par chaîne.", "Un point récurrent par chaîne est un point qui peut revenir exactement en lui-même en autorisant des « erreurs » d'amplitude aussi petites que l'on veut.", "$f(C(f)) = C(f) \\text{ et } \\Omega (f) \\subset C(f) \\\\C(f_{|\\Lambda })= \\Lambda \\text{ si } \\Lambda \\in \\lbrace {Per(f)} , {R(f)} , {L(f)} , C(f) \\rbrace $ En résumé, on a : $ Per(f) \\subset R^{\\pm }(f) \\subset L^{\\pm }(f) \\subset L(f) \\subset \\Omega (f)\\subset C(f) $ et chacune de ces inclusions peut être stricte.", "De plus, chacun de ces ensembles (ainsi que leurs adhérences) est une partie invariante par $f$ , au sens de la définition REF .", "[Transitif] Un homéomorphisme $f$ est transitif si pour tout ouvert non-vide $U$ , $\\bigcup _{n \\in } f^n(U)$ est dense dans $X$ .", "Ceci équivaut à dire qu'il existe $x \\in X$ dont l'orbite est dense (i.e.", "$\\omega (x) = X$ ).", "[Mélange topologique] Un homéomorphisme $f$ d'un espace métrique compact $X$ est topologiquement mélangeant si pour tous $U,V$ ouverts non-vides, il existe $n_0 \\in $ tel que $\\forall n \\ge n_0$ , $f^n(U) \\cap V \\ne \\emptyset $ .", "Si $f$ est topologiquement mélangeant, alors $f^k$ est transitif pour tout entier $k \\ne 0$ .", "La réciproque est fausse (voir l'exemple de la rotation du cercle : REF ).", "Il n'y a en effet pas nécessairement de « mélange » de l'espace des phases sous l'action de la dynamique dans le cas d'un système transitif.", "Il y a ainsi transitivité de tout système restreint à une orbite périodique, mais jamais de mélange.", "[Minimalité] Une partie fermée $A \\subset X$ est minimale pour $f$ si elle est non-vide, invariante par $f$ et si $A$ ne contient pas de fermé non-vide invariant par $f$ autre que $A$ .", "Ceci équivaut à dire que l'orbite (positive) de tout point $x \\in A$ est dense dans $X$ .", "En particulier, tout point de $A$ est récurrent.", "Le cercle $S^1$ est minimal pour la rotation $R_{\\alpha }$  : $\\theta $ $\\rightarrow $ $\\theta + \\alpha $ (mod.", "1) si $\\alpha $ est irrationel.", "Ce système est transitif, mais pas topologiquement mélangeant.", "Le doublement de l'angle $\\theta $ $\\rightarrow $ $2 \\theta $ (mod.", "1) sur $S^1$ est topologiquement mélangeant, donc transitif.", "Si $X$ est métrique compact non-vide, $f$ continue $X\\rightarrow X$ , alors $X$ contient un fermé minimal pour $f$ .", "En particulier, $R(f) \\ne \\emptyset $ .", "Les définitions suivantes précisent les notions intuitives d'attracteur et de bassin d'attraction.", "[Attracteur] Une partie compacte $A \\subset X$ est un attracteur pour $f$ s'il existe un voisinage $V$ de $A$ et un entier $N \\in $ tel que $f^N (V) \\subset V$ et $A =\\bigcap _{n \\in N} f^n(V)$ .", "[Bassin d'attraction] Soit $A$ un attracteur.", "Le bassin d'attraction de $A$ , noté $B(A)$ , est l'ensemble des points $x \\in X$ tels que $\\omega (x) \\subset A$ ." ], [ "Dynamique hyperbolique", "Le cas particulier de la dynamique hyperbolique est extrêment important, notamment parce qu'elle est présente dans la plupart des systèmes dynamiques.", "Le comportement hyperbolique est, comme nous allons le voir, le comportement « typique » d'un système dynamique.", "Nous parlerons ici de systèmes dynamiques uniformément hyperboliques." ], [ "Définitions", "Commençons par traiter le cas d'une application linéaire.", "[Application linéaire hyperbolique] Une bijection linéaire $T : E=^n \\rightarrow ^n$ est dite hyperbolique s'il existe une décomposition $E = E_s\\oplus E_u$ en somme directe de sous-espaces $T$ -invariants (i.e.", "$T E_s = E_s$ et $T E_u = E_u$ ) tels que, en notant $S = T |_{E_s}$ et $U = T|_{E_u}$ , il existe $n \\ge 1$ tel que ${S^n} <1$ et ${U^{-n}}<1$ .", "Cette définition est indépendante du choix de la norme, et est équivalente à dire que $T$ n'a pas de valeur propre de module 1.", "On dit que ${\\cdot }$ est adaptée à $T$ lorsque $n=1$ convient dans la définition précédente et si $\\forall x_s\\in E_s$ et $x_u \\in E_u$ , on a ${x_s + x_u} = \\max \\lbrace {x_s} , {x_u} \\rbrace $ .", "On appelle alors constante d'hyperbolicité la constante $ ch(T) = \\max \\lbrace {S} ,{U^{-1}} \\rbrace <1.", "$ Dans le reste de cette section, on considérera $X=M$ une variété lisse, munie d'une distance $d$ , $U$ un ouvert de $M$ et $f : U \\rightarrow M$ un $C^1$ difféomorphisme sur son image.", "[Point périodique hyperbolique] Un point périodique $p$ de $f$ , de période $n$ , est hyperbolique si $Df^n_p:T_p M \\rightarrow T_p M$ est une application linéaire hyperbolique.", "Son orbite est appellée orbite périodique hyperbolique.", "Sans perte de généralité, on peut se limiter au cas d'un point fixe.", "Le théorème de Grobman-Hartman affirme alors qu'au voisinage d'un point fixe hyperbolique, un difféomorphisme est topologiquement conjugué à sa différentielle.", "[Grobman-Hartman] Soit $\\Omega $ un ouvert de $^N$ , $f: \\Omega \\rightarrow ^N$ un $C^1$ -difféomorphisme local, $x_0$ un point fixe hyperbolique de $f$ et $T=Df(x_0)$ la différentielle de $f$ en $x_0$ .", "Alors il existe des voisinages ouverts $U$ de 0 dans $^N$ et $V$ de $x_0$ dans $\\Omega $ , et un homéomorphisme $H : U \\rightarrow V$ tel que, pour tout $x$ dans $U$ avec $T(x)$ dans $U$ , on a $ H \\circ T(x) = f \\circ H(x) .$ Généralisons cette notion au cas d'un ensemble invariant quelconque.", "[Ensemble hyperbolique] Une partie $f$ -invariante $\\Lambda \\subset U$ est hyperbolique si pour tout $x \\in \\Lambda $ il existe une décomposition $TM_x = E^s_x\\oplus E^u_x$ et des constantes $\\lambda < 1 < \\mu $ et une métrique Riemannienne sur $M$ vérifiant les propriétés suivantes : $\\forall x \\in \\Lambda $ , $Tf_x(E^s_x) = E^s_{f(x)}$ et $Tf_x(E^u_x) = E^u_{f(x)}$ .", "$\\forall x \\in \\Lambda $ , ${Tf_{x \\mid E^s_x}} \\le \\lambda $ et ${Tf^{-1}_{x \\mid E^u_x}} \\le \\mu ^{-1}$ (les normes étant induites par la métrique de $M$ ).", "On peut alors montrer que les sous-espaces $E^s_x$ et $E^u_x$ dépendent continûment de $x$ , ont des dimensions localement constantes, et sont uniformément transverses (il existe $\\alpha _0 >0$ tel que pour tous $x \\in \\Lambda $ , $\\xi \\in E^s_x$ , $\\eta \\in E^u_x$ , l'angle entre $\\xi $ et $\\eta $ est au moins $\\alpha _0$ ).", "[Anosov] Un $C^1$ difféomorphisme $f : M \\rightarrow M$ d'une variété compacte $M$ est appelé un difféomorphisme Anosov si $M$ est hyperbolique pour $f$ .", "L'ensemble des difféomorphismes Anosov sur $M$ est un ouvert de $C^1(M,M)$ .", "Pour déterminer si un ensemble est hyperbolique, en général, on regarde s'il vérifie la condition de cône suivante.", "[Condition de cône] Soit $\\Lambda $ une partie $f$ -invariante, $U$ un voisinage de $\\Lambda $ , $\\alpha > 1$ .", "On suppose qu'il existe en tout point $x \\in U$ une décomposition $TM_x = E^1_x \\oplus E^2_x$ .", "Supposons que $\\forall x\\in U$ , $\\forall v \\in T_x M$ , en posant $v=v_1+v_2$ et $w = T_x f(v) = w_1+w_2$ (décomposition sur $E^1_x$ et $E^2_x$ ), on a : $ {v_2} \\ge {v_1} \\Rightarrow {w_2} \\ge \\alpha {w_1} \\text{ et } {w_2} \\ge \\alpha {v_2}$ $ {w_2} \\ge {w_1} \\Rightarrow {v_1} \\ge \\alpha {w_1} \\text{ et } {v_1} \\ge \\alpha {v_2}.$ Sous ces conditions, $\\Lambda $ est hyperbolique.", "Bien sûr, cette condition est suffisante, mais pas nécessaire." ], [ "Propriétés fondamentales", "[Expansivité] Un homéomorphisme $f : X \\rightarrow X$ est expansif s'il existe une constante $\\delta _0 >0$ telle que pour tous $x,y \\in X$ , $x \\ne y$ , il existe $n \\in $ tel que $d(f^n(x),f^n(y))>\\delta _0$ .", "[Expansivité] La restriction d'un difféomorphisme à un ensemble hyperbolique est expansive.", "Le lemme de pistage est fondamental pour justifier la validité de simulations numériques, où l'on fait des calculs avec une précision limitée sur un système dynamique chaotique.", "En revanche, il ne garantit pas que les orbites pistant les pseudo-orbites sont typiques.", "Ainsi, pour l'application $f: x\\rightarrow 2x$ (mod.", "1), une orbite calculée par ordinateur s'achevera toujours en 0, car la condition initiale est donnée par un nombre fini de bits.", "L'ordinateur calcule ainsi toujours une vraie orbite, mais systématiquement attirée par 0, ce qui n'est pas le comportement typique du système.", "[Lemme de pistageshadowing lemma] Si $\\Lambda $ est un compact hyperbolique pour $f$ , alors il existe un voisinage $U(\\Lambda )$ de $\\Lambda $ tel que pour tout $\\delta >0$ , il existe $\\epsilon >0$ , $\\forall (x_n)_{n \\in }$ $\\epsilon $ -pseudo-orbite contenue dans $U(\\Lambda )$ , il existe $x \\in X$ vérifiant $\\forall n \\in $ , $d(f^n(x),x_n) < \\delta $ .", "Les orbites sous l'action d'une petite perturbation de $f$ sont ainsi proches des « vraies » orbites de $f$ , ce qui nous amène à la question de la stabilité structurelle.", "En fait, on peut même montrer qu'en un certain sens, la stabilité structurelle est équivalente à la notion d'hyperbolicité.", "[Stabilité structurelle] Si $\\Lambda $ est hyperbolique pour $f : U \\rightarrow M$ , alors pour tout voisinage $V \\subset U$ de $\\Lambda $ et tout $\\delta >0$ , il existe $\\epsilon >0$ tel que si $f^{\\prime } : U \\rightarrow X$ et $d_{C^1} (f _{| V}, f^{\\prime }) < \\epsilon $ , il existe un ensemble hyperbolique $\\Lambda ^{\\prime } =f^{\\prime }(\\Lambda ^{\\prime }) \\subset V$ pour $f^{\\prime }$ et un homéomorphisme $h : \\Lambda ^{\\prime } \\rightarrow \\Lambda $ avec $d_{C^0} (Id, h) + d_{C^0} (Id, h^{-1}) < \\delta $ tel que le diagramme suivant commute : ${\\begin{matrix}\\Lambda ^{\\prime } &\\xrightarrow{}& \\Lambda ^{\\prime }\\\\{\\scriptstyle h}\\downarrow \\mathbox{mphantom}{\\scriptstyle h}&& \\mathbox{mphantom}{\\scriptstyle h}\\downarrow {\\scriptstyle h}&&\\\\\\Lambda &\\xrightarrow{}& \\Lambda \\end{matrix}} $ De plus, $h$ est unique si $\\delta $ est assez petit.", "En particulier, les difféomorphismes Anosov sont fortement structurellement stables (voir définition REF )." ], [ "Variétés stables et instables", "La définition d'un ensemble hyperbolique dégage au voisinage de chaque point deux directions, l'une stable, l'autre instable, provenant de l'étude du système linéarisé.", "Intuitivement (le théorème de Grobman-Hartman fait déjà un pas dans ce sens), le système non-linéarisé devrait présenter le même type de décomposition, au moins localement : cela conduit à définir les variétés stables et instables.", "Le théorème suivant justifie leur définition, dans un cadre local.", "Soit $\\lambda $ hyperbolique pour un $C^1$ difféomorphisme $f : V \\rightarrow M$ , avec des constantes $\\lambda < 1 < \\mu $ .", "Alors, pour tout point $x \\in \\Lambda $ , il existe $W^s(x)$ et $W^u(x)$ , images de disques par des $C^1$ -plongements, appelés variétés stables et instables locales en $x$ , telles que $T_xW^s(x) = E^s_x$ et $T_x W^u(x) = E^u_x$ .", "$f(W^s(x)) \\subset W^s(f(x))$ et $f^{-1}(W^u(x)) \\subset W^u(f^{-1}(x))$ .", "pour tout $\\delta >0$ , il existe $C(\\delta )$ telle que pour tout $n \\in $ , $ \\forall y \\in W^s(x), \\, d(f^n(x),f^n(y)) <C(\\delta ) (\\lambda + \\delta )^n d(x,y) \\\\ \\forall y \\in W^u(x), \\,d(f^{-n}(x),f^{-n}(y)) < C(\\delta ) (\\mu - \\delta )^{-n} d(x,y).$ il existe $\\beta >0$ et une famille de voisinages $O_x$ contenant une boule autour de $x \\in \\Lambda $ de rayon $\\beta $ telle que $ W^s(x) &= \\lbrace y | f^n(y) \\in O_{f^n(x)} , n \\in \\rbrace \\\\ W^u(x) &= \\lbrace y | f^{-n}(y) \\in O_{f^{-n}(x)} , n \\in \\rbrace $ Les variétés locales stables et instables ne sont pas uniques, mais l'intersection de $W^s_1(x)$ et $W^s_2(x)$ contient toujours un voisinage de $x$ .", "On peut ainsi définir indépendamment du choix des variétés locales les variétés stables et instables globales.", "[Variétés stables et instables globales] $ \\widetilde{W}^s(x) &= \\bigcup _{n=0}^{\\infty }f^{-n}(W^s(f^n(x))) \\\\ \\widetilde{W}^u(x) &=\\bigcup _{n=0}^{\\infty } f^n(W^u(f^{-n}(x)))$ On a alors une caractérisation topologique des variétés stables et instables correspondant à la notion intuitive.", "[Variétés stables et instables] $ \\widetilde{W}^s(x) &= \\lbrace y \\in U d(f^n(x), f^n(y)) {} 0 \\rbrace \\\\ \\widetilde{W}^u(x) &= \\lbrace y \\in Ud(f^{-n}(x), f^{-n}(y)) {} 0 \\rbrace $ Ainsi, deux variétés stables (resp.", "instables) globales sont disjointes ou égales." ], [ "Produit local, ensemble localement maximal", "Lorsque des variétés stables et instables globales en un point s'intersectent, il est intéressant de considérer leurs points d'intersection.", "Ceci amène à définir le produit local, à l'aide de la proposition suivante (qui découle en partie du théorème REF ).", "Soit $x \\in \\Lambda $ .", "On note $W^s_{\\epsilon } (x)$ et $W^u_{\\epsilon } (x)$ les boules de rayon $\\epsilon $ dans $\\widetilde{W}^s(x)$ et $\\widetilde{W}^u(x)$ .", "Alors il existe $\\epsilon >0$ tel que pour tous $x \\in \\Lambda $ , $W^s_{\\epsilon } (x) \\cap W^u_{\\epsilon } (y)$ contient au plus un point $[x, y]$ , et il existe $\\delta >0$ tel que si $d(x,y) <\\delta $ avec $x,y \\in \\Lambda $ , alors $W^s_{\\epsilon } (x) \\cap W^u_{\\epsilon } (y) \\ne \\emptyset $ .", "[Produit local] On dit qu'un ensemble hyperbolique $\\Lambda $ a une structure de produit local si pour $\\epsilon >0$ assez petit, le point $[x,y]$ défini par la proposition REF appartient à $\\Lambda $ .", "En fait, cette propriété est équivalente à la notion suivante (que l'on distinguera bien de la notion d'attracteur, définition REF ).", "[Localement maximal] Soit $\\Lambda $ hyperbolique pour $f: U \\rightarrow M$ .", "S'il existe un voisinage ouvert $V$ de $\\Lambda $ tel que $\\Lambda = \\Lambda ^f_V := \\bigcap _{n \\in }f^n(\\overline{V})$ , on dit que $\\Lambda $ est localement maximal.", "Dans ce cas, on montre que les points périodiques de $f$ sont denses dans l'ensemble des points non-errants de $f_{|\\Lambda }$ .", "Soit $\\Lambda $ un ensemble compact hyperbolique.", "Il a une structure de produit local si et seulement si il est localement maximal.", "Une autre notion importante reliée au produit local est celle d'intersection homocline.", "[Intersection homocline] Soit $p$ un point fixe hyperbolique.", "Un point $q\\ne p$ est homocline à $p$ si $q \\in \\widetilde{W}^s(p) \\cap \\widetilde{W}^u(p)$ .", "Cette intersection est dite transverse homocline si les deux variétés stables et instables s'intersectent transversalement.", "Les exemples les plus importants sont le pendule (en temps continu) et le fer-à-cheval (en temps discret).", "En général, un système possédant une intersection homocline présente une dynamique très complexe.", "À ce sujet, on pourra consulter [7]." ], [ "Décomposition spectrale", "Il s'agit d'un résultat important qui permet d'étudier la récurrence des orbites dans le cas d'ensemble localement maximaux hyperboliques.", "[Décomposition spectrale] Soit $\\Lambda $ compact hyperbolique localement maximal pour un difféomorphisme $f : U \\rightarrow M$ .", "Alors il existe une famille finie des compacts invariants disjoints $\\Lambda _1,\\ldots , \\Lambda _m$ telle que $\\Omega (f_{| \\Lambda }) =\\bigcup _{i=1}^m \\Lambda _i$ .", "Les ensembles $\\Lambda _i)=\\Lambda _{i}$ sont « irréductibles » au sens où $f_{|\\Lambda _i}$ est transitif.", "De plus, $\\Lambda _i =\\bigcup _{j=1}^{m_i} \\Lambda _{i,j}$ avec $f(\\Lambda _{i,j}) =\\Lambda _{i,j+1}$ , et $f^{M_i}_{| \\Lambda _{i,1}}$ est topologiquement mélangeante.", "Si de plus $\\Lambda = \\overline{C(f_{|\\Lambda })}$ (i.e.", "$\\Lambda $ est récurrent par chaîne) , alors le théorème suivant montre que $\\Lambda =\\overline{\\Omega (f_{|\\Lambda })}$ , et donc la décomposition spectrale s'applique directement à $\\Lambda $ .", "Supposons que $\\Lambda = {Per(f)}$ , ${R(f)}$ , ${L(f)}$ ou ${C(f)}$ est hyperbolique.", "Alors cet ensemble est localement maximal et récurrent par chaîne.", "De plus, cet ensemble est égal à ${Per(f)}$ .", "Si $\\Lambda $ est compact invariant par $f$ , les ensembles stables et instables de $K$ sont $W^s(\\Lambda ) &= \\lbrace y \\in X \\omega (y) \\subset \\Lambda \\rbrace \\\\ W^u(\\Lambda ) &= \\lbrace y \\in X \\alpha (y) \\subset \\Lambda \\rbrace .", "$ Cette définition est cohérente avec les notions précédentes de variétés stables et instables globales, comme le justifie la proposition suivante.", "Si $\\Lambda $ est un compact invariant hyperbolique localement maximal, alors $W^s(\\Lambda ) = \\bigcup _{x \\in \\Lambda }\\widetilde{W}^s(x)$ et $W^u(\\Lambda ) = \\bigcup _{x \\in \\Lambda } \\widetilde{W}^u(x)$ .", "On appelle ensemble basique l'un des $\\Lambda _i$ de la décomposition spectrale.", "Trois situations peuvent alors se présenter : si $W^s(\\Lambda _i)$ est ouvert, on dit que $\\Lambda _i$ est un attracteur.", "si $W^u(\\Lambda _i)$ est ouvert, on dit que $\\Lambda _i$ est un répulseurrepeller en anglais.", "si aucun de ces deux ensembles n'est ouvert, alors $\\Lambda _i$ est de type-selle." ], [ "Exemple de dynamique uniformément hyperbolique : le\nsolénoïde", "Exemple du solénoïde Soit le tore $T = \\lbrace (\\theta , z) \\theta \\in /, z \\in {z} \\le 1 \\rbrace $ et l'application $ f : {\\begin{matrix} T &\\xrightarrow{}& T \\\\(\\theta ,z) &\\xrightarrow{}& \\left(2\\theta \\text{ mod.", "1} , \\frac{1}{2} e^{2\\pi i \\theta } + \\frac{1}{10} z \\right) \\end{matrix}}$ Le tore et son image par $f$ sont représentés figure REF .", "L'ensemble maximal invariant de $f$ est $\\Lambda = \\bigcap _{n \\in } f^n(T)$ , et il est hyperboliqueLes sous-espaces $E^s_x$ sont les plans $\\theta = cte$ , les $E^u_x$ sont de dimension 1, tangents à $\\Lambda $ ., on l'appelle attracteur de Smale (figure REF a).", "Localement, $\\Lambda $ est le produit d'un ensemble de Cantor dyadique (inclus dans le disque unité de $^2$ ) et d'une droite (figure REF b).", "Figure: Le solénoïde : TT etf(T)f(T).Figure: Le solénoïde.Les variétés stables sont les sections $C_{\\theta _0} =\\lbrace (\\theta ,z) \\in \\Lambda | \\theta = \\theta _0 \\rbrace $ .", "Les variétés instables sont plus difficiles à décrire, et on peut montrer que chacune est dense dans $\\Lambda $ .", "Dans le cas du solénoïde, il est également possible de montrer l'existence d'une mesure physique (définition REF ) qui donne la répartition statistique dans $\\Lambda $ des points de presque toutes les orbites." ], [ "Dynamique non-uniformément hyperbolique", "Considérons l'exemple de l'attracteur de Hénon.", "Soit l'application $ H = H_{b,c} : {\\begin{matrix} ^2 &\\xrightarrow{}& ^2 \\\\(x,y) &\\xrightarrow{}& (x^2+c-by,x) \\end{matrix}}$ avec $0<b\\ll 1$ et $c$ un peu plus grand que -2.", "Il existe un rectangle $R$ tel que $H(R) \\subset R$ et $H(R)$ ressemble à un arc de parabole « épaissi » (figure REF a) : le rectangle est fortement pincé, étiré, et plié (dans le cas du solénoïde, il n'y avait pas de pli).", "L'attracteur est $\\Lambda = \\bigcup _{n \\in }H^n(R)$ , et possède en presque tout point une structure de Cantor $\\times $ une droite (comme le solénoïde) (figure REF a).", "En revanche, il existe un ensemble de points (dense dans $\\Lambda $ ) où ce n'est pas le cas : ce sont les « pointes ».", "Ainsi, figure REF c, on visualise une petite zone de l'attracteur, qui semblait rectiligne sur la vue d'ensemble, et on distingue une pointe, i.e.", "un filament qui ne se poursuit pas vers la gauche.", "Figure: L'application de Hénon et sonattracteur (b=0.02b=0.02, c=-1.8c=-1.8).On a alors : une décomposition $^2 =E^s_x \\oplus E^u_x$ en presque tous les points $x \\in \\Lambda $ , la contraction et l'expansion n'étant pas uniformes, et les espaces $E^s_x$ et $E^u_x$ ne dépendent pas continûment de $x$.", "sensibilité aux conditions initiales.", "le lemme de pistage reste valide pour presques toutes les pseudo-orbites (mais pas toutes).", "pas de stabilité structurelle : avec des perturbations arbitrairement petites de $c$ , on peut obtenir une dynamique complètement différente.", "Cependant, pour presque toute perturbation, on a un attracteur du même type.", "la mesure physique existe." ], [ "Dynamique symbolique", "La dynamique symbolique fournit un exemple de système dynamique très important car il permet de modéliser la dynamique de très larges classes de systèmes.", "Nous la décrivons ici très brièvement.", "On trouvera une introduction plus complète dans [1] et [3].", "Soit $\\mathcal {A}$ un alphabet (i.e.", "un ensemble) fini, muni de la topologie discrète, et $X = \\mathcal {A}^{}$ muni de la topologie produit, l'ensemble des mots sur $\\mathcal {A}$ .", "C'est un espace métrique compact, muni de la distance $ d(\\omega ,\\omega ^{\\prime }) = \\sup _{i \\in } \\left( 2^{-{i}} _{\\omega _i\\ne \\omega ^{\\prime }_i} \\right).", "$ On note $\\sigma : X\\rightarrow X$ et on appelle décalage à gaucheshift en anglais l'application définie par $\\sigma (\\omega _i) = \\omega _{i+1}$ .", "Le système dynamique $(X,\\sigma )$ est appelé système de Bernoulli ou système symbolique.", "Soit $\\mathcal {B} \\subset \\mathcal {A}^2$ un ensemble de règles de compatibilité.", "On appelle sous-décalage de type fini une partie de $X$ $ \\Sigma = \\lbrace (\\omega _i)_{i \\in } (\\omega _i,\\omega _{i+1}) \\in \\mathcal {B}, \\, \\forall i \\in \\rbrace .$ $\\Sigma $ est invariante par $\\sigma $ .", "Ces notions permettent de coder de façon combinatoire la dynamique de certains systèmes, à l'aide de partitions de Markov.", "Une partition de Markov d'un ensemble invariant compact hyperbolique localement maximal $\\Lambda $ est un recouvrement fini par des rectangles propresUn rectangle $R$ est une partie de $\\Lambda $ de diamètre plus petit que $\\epsilon /10$ ($\\epsilon $ étant donné par la proposition REF ) et telle que $[x,y] \\in R$ si $x,y \\in R$ .", "Il est propre si $R = {{R}}$ .", "$(R_a)_{a \\in \\mathcal {A}}$ d'intérieurs disjoints et tels que si $x \\in {R_{a}}$ et $f(x) \\in {R_b}$ , alors $f(W^s_{R_a}(x)) \\subset W^s_{R_b}(f(x))$ et $W^u_{R_b}(f(x))\\subset f(W^u_{R_a}(x))$ .", "On définit alors un ensemble de transitions $ \\mathcal {B} = \\lbrace (a,b) \\in \\mathcal {A}^2 f({R_a}) \\cap {R_b} \\ne \\emptyset \\rbrace $ et $\\Sigma $ le sous-décalage de type fini associé.", "Le théorème suivant précise dans quelle mesure $(\\Sigma ,\\sigma )$ « code » la dynamique de $f$ sur $\\Lambda $ .", "Soient $\\mathcal {R} = (R_a)_{a \\in \\mathcal {A}}$ une partition de Markov de $\\Lambda $ pour $f$ et $(\\Sigma ,\\sigma )$ le sous-décalage de type fini associé.", "Pour tout $\\omega \\in \\Sigma $ , l'intersection $\\bigcap _{n \\in }f^{-n}(R_{\\omega _n})$ est réduite à un point $\\pi (\\omega )$ .", "L'application $\\pi : \\Sigma \\rightarrow \\Lambda $ est continue, surjective, et le diagramme suivant commute : ${\\begin{matrix} \\Sigma &\\xrightarrow{}& \\Sigma \\\\{\\scriptstyle \\pi }\\downarrow \\mathbox{mphantom}{\\scriptstyle \\pi }&& \\mathbox{mphantom}{\\scriptstyle \\pi }\\downarrow {\\scriptstyle \\pi }&& \\\\ \\Lambda &\\xrightarrow{}& \\Lambda \\end{matrix}} $ Pour toute mesure de probabilité $\\sigma $ -invariante et ergodique $\\mu $ , de support $\\Sigma $ , on a $ \\mu \\left( \\lbrace \\omega \\in \\Sigma \\pi ^{-1}(\\pi (\\omega ))>1 \\rbrace \\right) = 0.$ Un ensemble basique $\\Lambda _i$ de la décomposition spectrale (Théorème REF ) possède une partition de Markov de diamètre arbitrairement petit (voir [3])." ], [ "Chaos", "La notion de « chaos » en systèmes dynamiques, contrairement à sa signification usuelle de désordre total, se réfère à une situation où les orbites ne convergent pas vers une orbite périodique ou quasi-périodique, et où l'évolution des orbites est imprévisible à un certain point, ou leur comportement est sensible aux conditions initiales.", "Les premiers exemples étudiés furent — entre autres — l'attracteur de Lorenz, l'application logistique et l'application de Hénon.", "[Orbite chaotique] L'orbite de $x$ , $\\lbrace f^n(x) / n \\ge 0 \\rbrace $ , est sensible (ou chaotique) s'il existe une constante $C>0$ telle que $ \\begin{split} \\forall q \\in \\omega (x), \\, \\forall \\epsilon >0 , \\, \\exists n_1, n_2, n > 0 / \\, & d(f^{n_1}(x),q)<\\epsilon , \\\\ d(f^{n_2}(x),q)< \\epsilon &\\text{ et }d(f^{n_1+n}(x),f^{n_2+n}(x))> C.\\end{split} $ Une orbite asymptotique à une orbite périodique ou quasi-périodique n'est pas chaotique au sens où si $f^{n_1}(x)$ et $f^{n_2}(x)$ sont proches, alors $f^{n_1+n}(x)$ et $f^{n_2+n}(x)$ restent proches pour tout $n \\ge 0$ .", "Une orbite sensible est également imprévisible dans la mesure où savoir qu'un point $y$ de l'orbite est extrêmement proche de $q \\in \\omega (x)$ n'est pas suffisant pour prédire le futur de $y$ à une distance $C$ près.", "Dans l'ensemble stable d'un attracteur hyperbolique non-trivial, de même que l'on a une forte sensibilité aux conditions initialesComme l'indique la propriété d'expansivité REF, on peut montrer que l'ensemble des points ayant une orbite chaotique a une mesure de Lebesgue totale.", "[Dynamique chaotique] Un système dynamique $(X,f)$ est sensible (ou a une dynamique chaotique) lorsque l'ensemble des points ayant une orbite chaotique a une mesure de Lebesgue non-nulleCette définition n'a de sens que lorsque $X$ est une variété, pour que les ensembles de mesure de Lebesgue nulle soient définis.. Cependant, le chaos ainsi défini ne doit pas être interprété comme une totale imprédictibilité.", "En effet, on observe numériquement, pour certains systèmes chaotiques, que pour toute condition initiale prise dans un certain ouvert, on obtient le même ensemble $\\omega $ -limite.", "Ceci conduit à la notion d'attracteur étrange.", "[Attracteur étrange] Une partie compacte $A$ de $X$ est un attracteur étrange s'il existe un ouvert $U$ et $N\\subset U$ de mesure de Lebesgue nulle tel que $\\forall x \\in U\\backslash N$ , $\\omega (x) = A$ et l'orbite de $x$ est chaotique.", "Un exemple d'attracteur étrange est l'attracteur de Hénon (section REF ).", "On appelle parfois également attracteur étrange un attracteur $A$ tel que $f$ a une dépendance sensible aux conditions initiales avec probabilité totale sur $B(A) \\times B(A)$ (où $B(A)$ est le bassin d'attraction de $A$  : voir définition REF ).", "Une dernière notion importante est celle de dynamique chaotique persistante, qui traduit que de petites perturbations de $f$ ont, avec une probabilité positive, une dynamique chaotique.", "Cette définition a un sens lorsque par exemple $f =f_{\\alpha }$ est paramétrée par $\\alpha \\in ^n$ , car alors on dispose de la mesure de Lebesgue sur l'espace des paramètre $\\alpha $ .", "De façon plus restrictive, on peut demander la persistance d'une dynamique chaotique dans un voisinage ouvert de $f$ .", "Une notion que nous avons déjà introduite est étroitement reliée au chaos.", "Il s'agit de celle d'intersection homocline (définition REF ).", "Il y a équivalence entre l'existence d'une orbite chaotique (voir section REF , définition REF ) et l'existence d'une orbite homocline.", "En revanche, cela n'entraîne pas forcément que la dynamique est chaotique (définition REF )." ], [ "Chaos et simulations numériques", "Il est problématique de vouloir observer ou même caractériser un comportement chaotique lors d'une simulation numérique.", "Comment en effet mettre en évidence un tel phénomène malgré la précision finie d'un ordinateur ?", "Celle-ci a plusieurs conséquences majeures.", "Tout d'abord, les erreurs d'arrondi font que l'on n'observe que des pseudo-orbites.", "Si le système étudié possède une propriéte de pistage, comme c'est le cas avec les systèmes uniformément hyperboliques, on a de quoi être partiellement rassuré.", "Il reste cependant des cas (par exemple le doublement de l'angle) où les orbites qu'un ordinateur peut pister ne sont pas des orbites typiques du système.", "De même, lorsque les orbites calculées sont bornées, toutes les pseudo-orbites observées sont périodiques (même si la période est très longue), en raison du nombre fini de décimales que l'on peut calculer.", "Il faut donc fixer (arbitrairement) un seuil pour séparer orbites périodiques et non-périodiques.", "Un deuxième effet est que l'on ne peut observer que le comportement en temps fini.", "Comment alors être sûrs qu'il s'agit bien du comportement stationnaire, et non d'un régime transitoire très long ?", "Il nous faut en effet fixer un seuil à partir duquel on observe la dynamique « à l'infini ».", "Le choix de ce seuil est crucial pour éviter des erreurs, tout en limitant la durée des calculs.", "Enfin, lorsque l'on étudie un système dépendant de paramètres réels, il faut garder à l'esprit que l'on ne peut observer celui-ci que sur un ensemble de mesure nul, l'ensemble des rationnels.", "C'est tout l'intérêt de considérer la persistance de la dynamique dans un voisinage ouvert, $$ étant dense dans $$ .", "Ce problème peut cependant se ramener à celui du lien entre pseudo-orbites et vraies orbites si la famille $(f_{\\alpha })_{\\alpha \\in }$ dépend continûment de $\\alpha $ pour la topologie de la convergence uniforme sur $X$ , car alors une orbite sous $f_{\\alpha +\\epsilon }$ est une pseudo-orbite sous $f_{\\alpha }$ si $\\epsilon $ est assez petit." ], [ "Bifurcations", "Considérons une famille de systèmes dynamiques dépendant d'un ou plusieurs paramètres.", "Même si pour presque toutes les valeurs des paramètres, le système a un comportement transverse (par exemple structurellement stable), il peut y avoir des valeurs particulières de ceux-ci où se produit une transition entre deux différents types d'orbites.", "De tels changements sont appelés bifurcations.", "Leur étude — qui est une branche à part entière de la théorie des systèmes dynamiques — est fondamentale pour comprendre les propriétés d'un système typique car les bifurcations montrent comment différents comportements transverses peuvent apparaître.", "Nous ne parlons ici que de quelques cas simples de bifurcations, en petite dimension.", "Il en existe bien sûr beaucoup d'autres types.", "Nous nous limitons de plus à des bifurcations locales, c'est-à-dire pouvant être définies seulement au voisinage d'un point, par opposition aux bifurcations globales.", "Nous considérons plus particulièrement le cas des bifurcations structurellement stables, définies de la manière suivante dans le cas de systèmes discrets.", "[Bifurcation structurellement stable] Une famille $\\lbrace f_{\\tau } \\rbrace $ de $C^{\\infty }$ difféomorphismes définis localement a une bifurcation structurellement stable à $\\tau = \\tau _0$ si $f_{\\tau _0}$ n'est pas localement structurellement stable et si pour toute famille $\\lbrace g_{\\tau } \\rbrace $ de $C^{\\infty }$ difféomorphismes définis localement suffisamment $C^2$ -proche de $\\lbrace f_{\\tau } \\rbrace $ , il existe une reparamétrisation $\\phi (\\tau )$ de $\\lbrace g_{\\tau } \\rbrace $ et une famille continue $\\lbrace h_{\\tau } \\rbrace $ d'homéomorphismes définis localement telle que $ g_{\\phi (\\tau )} = h_{\\tau }^{-1} \\circ f_{\\tau } \\circ h_{\\tau } $ partout où cela est défini." ], [ "Diagramme de bifurcations", "Il existe un moyen simple de visualiser une bifurcation, appelé diagramme de bifurcation.", "On trace l'ensemble $\\omega $ -limite $L^+(f_{\\epsilon })$ pour les différentes valeurs du paramètre $\\epsilon $ , que l'on porte sur l'axe des abscisses.", "Un tel diagramme peut aisément être tracé numériquement, en prenant pour ensemble $\\omega $ -limite les valeurs de $f^n_{\\epsilon }(x)$ pour $n$ « grand » et pour un ou plusieurs $x$ choisis aléatoirement.", "Il y a cependant une différence entre un diagramme obtenu par simulations et un diagramme théorique : les objets instables, ou de « petit » bassin d'attraction, n'apparaissent que dans le second cas.", "Il n'est ainsi pas forcément simple de déterminer la nature d'une bifurcation en comparant son diagramme empirique avec les diagrammes théoriques des bifurcations classiques." ], [ "Cas discret, dimension 1", "En dimension 1, on peut classifier les bifurcations structurellement stables autour d'un point d'équilibre $p$ .", "En effet, dans ce cas, la dérivée de $f_{\\tau _0}$ en $p$ doit valoir $\\lambda = \\pm 1$ .", "Commençons par le cas $\\lambda = 1$ .", "La famille $(f_{(+1),\\tau })_{\\tau \\in }$ , définie par $ \\forall x \\in , \\,f_{(+1),\\tau } (x) = x + x^2 + \\tau $ a une bifurcation structurellement stable en $\\tau _0 = 0$ , avec dérivée 1, et est caractéristique de cette situation.", "La bifurcation de la famille (REF ) en $\\tau _0 = 0$ est structurellement stable, et toute bifurcation locale structurellement stable en dimension 1 ayant lieu en un point fixe avec dérivée 1 est (topologiquement) équivalente (après reparamétrisation) à cette bifurcation.", "Figure: Bifurcation de la famillef (+1),τ (x)f_{(+1),\\tau }(x).Figure: Diagramme de bifurcation de lafamille f (+1),τ (x)f_{(+1),\\tau }(x), autour de τ 0 =0\\tau _0 = 0.Ainsi, pour $\\tau < \\tau _0$ , $f_{(+1),\\tau }$ possède un point fixe stable $-\\sqrt{-\\tau }$ et un point fixe instable $\\sqrt{-\\tau }$  ; pour $\\tau =\\tau _0$ , ces deux points fixes sont confondus, et l'équilibre qui en résulte est semi-stable ; enfin, dès que $\\tau > \\tau _0$ , $f_{(+1),\\tau }$ n'a plus de point fixe (figure REF ).", "Le diagramme de bifurcation correspondant est représenté à la figure REF .", "Dans le cas où $\\lambda = -1$ , le point fixe $p$ est transverse et donc persistent.", "La valeur de la dérivée en $p$ en supérieure à $-1$ pour $\\tau < \\tau _0$ et inférieure à $-1$ pour $\\tau > \\tau _0$ , le point fixe restant isolé.", "Cela s'accompagne de la création d'une orbite stable de période 2, tandis que le point fixe devient instable.", "On parle de bifurcation par doublement de période, dont l'exemple typique est le suivant : $f_{(-1),\\tau }(x) = -\\tau x + x^2 $ au voisinage de $x_0=0$ , $\\tau _0 = 1$ .", "On montre alors une proposition similaire à la proposition REF , ce qui achève la classification dans le cas de la dimension 1.", "Pour visualiser cette bifurcation, on peut tracer $f_{(-1),\\tau }$ (figure REF ), mais aussi $f^2_{(-1),\\tau }$ (figure REF ) pour mieux comprendre les orbites de période 2.", "Le diagramme de cette bifurcation est représenté figure REF .", "Figure: Bifurcation subie parf (-1),τ (x)f_{(-1),\\tau }(x).Figure: Bifurcation subie parf (-1),τ 2 (x)f^{2}_{(-1),\\tau }(x).Figure: Diagramme de bifurcation de lafamille f (-1),τ (x)f_{(-1),\\tau }(x), autour de τ 0 =1\\tau _0 = 1." ], [ "Bifurcation selle-nœud", "En dimension supérieure, les bifurcations structurellement stables se produisent lorsqu'une valeur propre vaut $\\pm 1$ et les autres sont en-dehors du cercle unité.", "Un exemple classique, en dimension deux ou plus, est le suivant : deux point fixes, un noeud (point fixe attractif) et une selle (attractif dans une direction, répulsif dans une autre) se rencontrent.", "Après bifurcation, il n'y a plus aucun point fixe (localement).", "Une telle bifurcation est appelée selle-nœudsaddle-node en anglais.", "Le système différentiel suivant donne un exemple de bifurcation selle-nœud : $\\left\\lbrace \\begin{aligned}\\frac{dx}{dt} &= x^2 - \\mu \\\\\\frac{dy}{dt} &= - y\\end{aligned} \\right.", "$ Il ne s'agit en fait que d'une légère modification par rapport à la bifurcation REF , qui se produit sur la première coordonnée de ce système.", "La deuxième coordonnée est là pour que le point fixe instable devienne une selle (il ne peut pas y avoir de selle en dimension 1).", "Le diagramme de bifurcation est donc exactement le même que celui de la figure REF .", "L'espace des phases de part et d'autre de la bifurcation ($\\mu = 0$ ) est représenté figure REF .", "Figure: Bifurcation selle-noeud :espace des phases de part et d'autre de μ 0 =0\\mu _0=0." ], [ "Étude d'un exemple dans $^2$", "Considérons l'exemple de la famille de systèmes dynamiques continus suivante : $\\left\\lbrace \\begin{aligned}\\frac{dx}{dt} &= - \\lambda y + \\epsilon x - a x (x^2 + y^2) \\\\\\frac{dy}{dt} &= \\lambda x + \\epsilon y - a y (x^2 + y^2)\\end{aligned} \\right.", "$ où $\\lambda $ et $a$ sont des constantes strictement positives.", "Pour tout $\\epsilon $ , $(0,0)$ est un équilibre du système, les valeurs propres de la dérivée en 0 sont $\\mu _{\\epsilon } = i\\lambda + \\epsilon $ et $\\overline{\\mu _{\\epsilon }}$ .", "L'équilibre est donc stable si $\\epsilon <0$ et instable si $\\epsilon >0$ .", "En coordonnées « polaires » (un peu modifiées), $R = x^2 +y^2$ et $\\theta = \\arctan \\frac{y}{x}$ , (REF ) devient : $ \\left\\lbrace \\begin{aligned}\\frac{dR}{dt} &= 2R (\\epsilon - a R) \\\\\\frac{d\\theta }{dt} &= \\lambda \\end{aligned} \\right.", "$ Figure: Bifurcation de Hopf : orbites de partet d'autre de ϵ 0 =0\\epsilon _0=0.Ce système se résout explicitement (voir des exemples d'orbites figure REF , dans le cas $a=1$ , $\\lambda =2\\pi $ ), d'où si $\\epsilon <0$ , toutes les solutions convergent vers l'équilibre.", "si $\\epsilon >0$ , toutes les solutions (sauf la solution constante nulle) convergent vers l'orbite périodique $ \\left\\lbrace \\begin{aligned}R &= \\frac{\\epsilon }{a} \\\\\\dot{\\theta } &= \\lambda \\end{aligned} \\right.", "$ La figure REF représente le diagramme de bifurcations de cette famille de systèmes dynamiques en $\\epsilon _0=0$ .", "Figure: Diagramme d'une bifurcation deHopf." ], [ "Description du phénomène général", "La bifurcation décrite au paragraphe précédent est une bifurcation de Hopf.", "Plus généralement, considérons la famille à un paramètre d'équations différentielles dans $^N$ $ \\frac{dx}{dt} =F_{\\epsilon } (x) .$ Nous faisons l'hypothèse $\\mathbf {(H_0)}$  : $F_0(0) = 0$ et $D_0(F_0)$ n'a que des valeurs propres de partie réelle strictement négative, sauf deux qui sont imaginaires pures et non-nulles : $\\mu _0 = i \\lambda $ et $\\overline{\\mu _0} = - i \\lambda $ , avec $\\lambda >0$ .", "Dans un voisinage de l'origine, le système peut se réecrire (après changement de variable), à des termes négligeables près : $\\left\\lbrace \\begin{aligned}\\frac{dx_0}{dt} &= - \\lambda x_1 - a x_0 (x_0^2 + x_1^2) \\\\\\frac{dx_1}{dt} &= \\lambda x_0 - a x_1 (x_0^2 + x_1^2) \\\\\\frac{dx^{\\prime }}{dt} &= A x^{\\prime }\\end{aligned} \\right.", "$ Nous supposerons désormais $\\mathbf {(H_1)}$  : $a>0$ .", "Nous avons enfin besoin d'une dernière hypothèse relative à la dépendence en $\\epsilon $ .", "Dans un voisinage de 0, on peut suivre l'équilibre et les valeurs propres $\\mu _{\\epsilon }$ , $\\overline{\\mu _{\\epsilon }}$ proches de l'axe imaginaire.", "On suppose $\\mathbf {(H_2)}$  : $\\frac{\\partial }{\\partial \\epsilon } \\Re \\mu _{\\epsilon } > 0$ en $\\epsilon = 0$ .", "Sous les hypothèses $\\mathbf {(H_0)}$ , $\\mathbf {(H_1)}$ et $\\mathbf {(H_2)}$ , la dynamique de l'équation (REF ) présente une bifurcation de Hopf au voisinage de l'origine en $\\epsilon = 0$  : pour $\\epsilon <0$ petit, il y a un équilibre stable.", "pour $\\epsilon =0$ , l'équilibre reste stable mais plus faiblement.", "pour $\\epsilon >0$ petit, l'équilibre est instable, mais une orbite périodique quasi-circulaire de rayon $\\simeq \\sqrt{\\epsilon /a}$ est stable." ], [ "Cas des difféomorphismes", "Un phénomène semblable peut se produire pour des systèmes dynamiques discrets $x \\mapsto f_{\\epsilon }(x)$ , $x \\in ^N$ .", "On fait les hypothèses suivantes : $f_0(0)=0$ , et les valeurs propres de $D_0 f_0$ ont toutes un module strictement inférieur à 1 sauf deux, $\\mu _0$ et $\\overline{\\mu _0}$ pour lesquelles ${\\mu _0}=1$ .", "Pour $k=1,2,3,4$ , $\\mu _0^k \\ne 1$ , i.e.", "$\\mu _0 \\notin \\lbrace \\pm 1, \\pm i , \\pm j \\rbrace $ .", "$\\mathbf {(H_1^{\\prime })}$ et $\\mathbf {(H_2^{\\prime })}$ comme dans le paragraphe précédent.", "La dynamique pour $\\epsilon $ proche de 0 est alors la même que dans le cas précédent.", "Un exemple de tel difféomorphisme est donné par $ f_{\\epsilon } (z) = \\lambda (1+\\epsilon ) z - a z {z}^2 ,\\, z \\in {equation} avec{\\lambda }=1, \\lambda \\ne \\pm 1, a>0.", "L^{\\prime }équilibre 0 eststable pour \\epsilon <0, faiblement stable pour\\epsilon =\\epsilon _0=0, instable pour \\epsilon > 0 et alors lecercle {z} = \\left(\\frac{\\epsilon }{a}\\right)^{1/2} estinvariant et attire toutes les orbites proches de 0 saufl^{\\prime }équilibre lui-même.$ Remarquons également que si la dynamique sur la courbe invariante est proche d'une rotation, elle ne se comporte pas toujours comme une rotation.", "C'est le cas pour presque tous les paramètres, mais pas nécessairement pour tous." ], [ "Cas des orbites périodiques", "On se ramène en fait aux difféomorphismes via l'application de retour de Poincaré.", "En effet, soit l'équation différentielle $\\frac{dx}{dt} = F_0(x)$ dans $^N$ possédant une solution périodique $x_0$ .", "Considérons une section $\\Sigma $ transverse à l'orbite $x_0$ en $x_0(t_0)$ .", "Une condition initiale suffisamment proche de $x_0(t_0)$ retourne sur $\\Sigma $ en un temps fini, ce qui définit (dans un voisinage de $x_0(t_0)$ ) un difféomorphisme $f_0$ de $\\Sigma $ .", "La même opération pouvant être faite pour une petite perturbation $F_{\\epsilon }$ de $F_0$ , cela définit une famille $f_{\\epsilon }$ de difféomorphismes, comme dans le paragraphe précédent." ], [ "Autres bifurcations", "Nous n'avons bien sûr pas abordé ici toutes les bifurcations possibles, même en nous limitant a priori à un cadre restreint.", "Un exemple particulièrement intéressant est celui de la bifurcation homocline, reliée à celle d'intersection homocline (définition REF ) : deux intersections transverses homoclines se rencontrent, forment une tangence à cet instant, puis disparaissent.", "Une référence à ce sujet est [7]." ], [ "Dynamique des polynômes quadratiques", "La référence pour cette section est [12].", "On considère la famille d'applicationsTout polynôme complexe de degré deux est conjugué par une application affine à une application de cette forme.", "C'est en particulier le cas de la famille logistique $x \\mapsto rx(1-x)$ , bien connue en dynamique des populations.", "$P_c : z \\mapsto z^2 +c$ pour $z \\in et $ c .", "Cette famille de systèmes dynamiques est l'une des plus simples qui, en dimension 1, peut générer un comportement chaotique.", "Son étude est de plus particulièrement intéressante car on y observe des phénomènes que l'on retrouve dans de nombreux autres cas." ], [ "Ensembles de Julia et de Mandelbrot", "Il est intéressant de se placer dans $ au lieu de $$ car onpeut alors utiliser de nombreux résultats d^{\\prime }analyse complexe.Nous reviendrons ensuite au cas réel.$ Pour $c \\in , on considère l^{\\prime }\\emph {ensemble de Julia rempli}(figure~\\ref {fig:julia})$$ K_c = \\lbrace z \\in P_c^n(z) \\text{ est borné} \\rbrace $$ que l^{\\prime }onpeut également écrire $$ K_c = \\bigcap _{n \\ge 0}P_c^{-n}({\\mathbb {D}(O,R)}) $$ en ayant posé $ R = (1+ 1+4c)/2$.", "Ainsi, $ Kc$ est : \\begin{itemize} \\item compact, \\item non-vide et il contient tous les pointspériodiques de P_c, \\item totalement invariant, {i.e.", "}P_c(K_c) = K_c = P_c^{-1}(K_c), \\item plein, {i.e.}", "K_c est connexe.", "\\end{itemize}$ Figure: Ensemble de Julia rempli K c K_c.Le bord $J_c = \\partial K_c$ est l'ensemble de Julia.", "D'après un théorème montré indépendamment par Julia et Fatou, c'est aussi l'adhérence de l'ensemble des points périodiques répulsifs.", "Un théorème de Fatou (1919) montre que $0 \\in K_c$ si et seulement si $K_c$ est connexe.", "Dans l'espace des paramètres, on considère l'ensemble de Mandelbrot (figure REF ) $ M = \\lbrace c \\in K_c \\text{ est connexe} \\rbrace .", "$ On montre que $M = \\lbrace c {P_c^n(0)}\\le 2 , \\, \\forall n>0 \\rbrace $ et donc $M$ est compact.", "De plus, $M$ est plein, symétrique par rapport à l'axe réel qu'il coupe suivant l'intervalle $[-2,1/4]$ .", "Sur la figure REF , on distingue des îlots disjoints de la grande composante de $M$ .", "Un calcul plus poussé montrerait qu'ils lui sont en réalité reliés par des filaments extrêmement fins.", "Figure: L'ensemble de Mandelbrot :approximation numérique.Si $c \\notin M$ , $K_c = J_c$ est un ensemble de Cantor (figure REF .c).", "La dynamique est de type décalage, et $P_c$ est hyperbolique.", "Si $c \\in M$ , la présence d'orbites périodiques attractives (i.e.", "$z_0$ tel que $P_c^m(z_0)=z_0$ et ${(P_c^m)^{\\prime }(z_0)} <1$ ) est déterminante pour la structure de $K_c$ .", "Douady a montré (1982) que $P_c$ a au plus une orbite périodique attractive.", "Lorsque c'est le cas, le bassin d'attraction $W = \\lbrace z \\lim _{n \\rightarrow \\infty }d(P_c^n(z),O(z_0))=0 \\rbrace $ de l'orbite est l'intérieur de $K_c$ , et $P_{c|J_c}$ est hyperbolique.", "Un tel $c$ est alors dans l'intérieur de $M$ , et la composante connexe de l'intérieur de $M$ contenant $c$ est appelée composante hyperbolique de $M$ .", "Par exemple, l'ensemble des $c$ tels que $P_c$ possède un point fixe attractif est l'intérieur d'une cardioïde dite principale contenant 0.", "Pour un tel $c$ , l'intérieur de $K_c$ a une seule composante et $J_c$ est un quasi-cercle (figure REF .b).", "La figure REF .a donne un exemple d'ensemble de Julia rempli lorsque $c$ est dans une autre composante hyperbolique.", "L'intérieur de $M$ est dense dans $M$ et contient toutes les composantes hyperboliques.", "La conjecture d'hyperbolicité dit que l'union des composantes hyperboliques est en fait exactement l'intérieur de $M$ .", "Parmi les résultats partiels obtenus dans cette direction, on a montré que les composantes hyperboliques de $M$ rencontrent $M \\cap = [-2,1/4]$ suivant un ensemble dense.", "En revanche, cette intersection n'est pas de mesure totale, comme le montre le théorème de Jakobson [13]." ], [ "Dynamique sur la droite réelle", "Supposons $c \\in m \\cap = [-2,1/4]$ .", "Nous venons de voir que pour un ensemble dense (mais pas de mesure totale) de valeurs de $c$ , $c$ est dans une composante hyperbolique et donc il existe une unique orbite périodique attractive, et son bassin d'attraction est l'intérieur de $K_c$ .", "Partons de $c=1/4$ et faisons diminuer $c$ (voir le diagramme de bifurcations, figure REF ).", "On a tout d'abord un point fixe attractif, puis une orbite attractive de période 2 (après une bifurcation doublement de période en $c=c^{(1)}$ ).", "Les bifurcations doublement de période se succèdent ainsi jusqu'à atteindre $c = c^{(\\infty )}$ où il n'y a plus d'orbite périodique attractive.", "Cette succession de bifurcations est appelée cascade sous-harmonique directe.", "Figure: Dynamique de P c P_c pour c∈[-2;1 4]c\\in [-2;\\frac{1}{4}].Le point $c = c^{(\\infty )}$ est appelé le point de Feigenbaum, où la dynamique peut encore être décrite assez simplement.", "Un intervalle $I$ est stable, et possède deux sous-intervalles $I_0$ et $I_1$ disjoints tels que $P_c(I_0)\\subset I_1$ et $P_c(I_1) \\subset I_0$ .", "Dans chacun de ces intervalles $I_{\\alpha }$ , on trouve deux sous-intervalles $I_{\\alpha ,0}$ et $I_{\\alpha ,1}$ disjoints tels que $P_c^2(I_{\\alpha ,0}) \\subset I_{\\alpha ,1}$ et $P_c^2(I_{\\alpha ,1})\\subset I_{\\alpha ,0}$ .", "On retrouve cette dynamique en faisant un double changement déchelle, spatial et temporel.", "L'ensemble limite a ainsi une structure d'ensemble de Cantor.", "Cependant, la dynamique de $P_c$ sur cet ensemble n'est pas chaotique.", "Au delà du point de Feigenbaum, la dynamique devient chaotique.", "Il se produit alors un processus, miroir de la cascade directe, de regroupement par bandes : les composantes connexes de l'attracteur fusionnent successivement.", "On parle alors de cascade inverse.", "La complexité du comportement du système lorsque $c$ varie provient de l'alternance de régimes périodiques et chaotiques.", "En effet, la situation présentée avec une période 1 initiale se reproduit pour toutes les valeurs de périodes impaires.", "Ainsi, des fenêtres de périodicité s'installent brutalement à la suite de régimes chaotiques, s'achèvent par une cascade sous-harmonique suivie d'une cascade inverse et d'un régime chaotique.", "Ces fenêtres de périodicité sont denses dans $[-2; 1/4]$ , ce qui montre bien toute la complexité de la dynamique dans cette région de l'espace des paramètres.", "De plus, le complémentaire de cet ensemble ayant une mesure non-nulle, il reste possible d'observer un comportement chaotique en choisissant le paramètre $c$ aléatoirement suivant la mesure de Lebesgue.", "Une étude plus détaillée (et plus expérimentale) est faite dans [6]." ], [ "Comportement statistique des orbites", "L'étude des systèmes dynamiques mesurables est l'objet de la théorie ergodique, qui est notamment introduite dans [1].", "Nous ne donnons ici que quelques définitions utiles pour notre étude.", "[Mesure invariante] Une mesure $\\mu $ est invariante par l'application (mesurable) $f$ si pour toute partie mesurable $A$ $ \\mu (f^{-1}(A)) = \\mu (A).", "$ [Ergodicité] Une application $f : (X,\\mathcal {B},\\mu )\\rightarrow (X,\\mathcal {B},\\mu )$ qui préserve $\\mu $ est ergodique si $ \\forall A \\in \\mathcal {B}, \\, f^{-1}(A) = A\\Rightarrow \\mu (A) \\in \\lbrace 0 , 1 \\rbrace .$ Une application $f$ est ergodique si et seulement si toute application $\\phi : X \\rightarrow mesurable, telle que$ f = $ presque partout, est presque partoutconstante.", "$ On peut remplacer dans cette proposition « mesurable » par $L^1(X,\\mu )$ ou $L^2(X,\\mu )$ .", "Les rotations d'angle $\\alpha $ irrationnel et le doublement de l'angle sont ergodiques sur le cercle $S^1$ , pour la mesure de Lebesgue sur le cercle.", "La proposition suivante fait le lien avec la notion physique d'ergodicité.", "Si $X$ est un espace métrique séparable, $\\mu $ une probabilité borélienne sur $X$ , $f : X \\rightarrow X$ continue préservant $X$ .", "Si $f$ est ergodique, alors $\\mu $ -presque toute orbite est dense dans $X$ .", "On a alors une estimation quantitative de la « densité » des orbites : pour toute partie mesurable $A$ , la proportion de temps passée dans $A$ par presque toutes les orbites est égale à $\\mu (A)$ .", "[Théorème ergodique de Birkhoff] Soit $(X,\\mathcal {B},\\mu )$ un espace mesuré, $f : X \\rightarrow X$ mesurable préservant $\\mu $ .", "Pour tout $\\phi $ dans $L^1(X,\\mu )$ , on note $ S_n \\phi (x)= \\frac{1}{n} \\sum _{k=0}^{n-1} \\phi (f^k(x)) \\text{ (somme deBirkhoff de $\\phi $).}", "$ La limite $\\widetilde{\\phi }(x) = \\lim _{n\\rightarrow \\infty } S_n \\phi (x)$ existe pour $\\mu $ -presque tout $x$ , $\\widetilde{\\phi } \\circ f = \\widetilde{\\phi }$ presque partout.", "Pour toute partie $f$ -invariante $A$ mesurable, de mesure finie, on a $ \\int _A \\phi d\\mu = \\int _A \\widetilde{\\phi } d\\mu .", "$ En particulier, si $\\mu $ est une mesure de probabilité ergodique, alors $ \\widetilde{\\phi }(x) = \\int _X \\phi d\\mu $ pour $\\mu $ -presque tout $x$ .", "[Mesure physique, Ruelle–Bowen] C'est une mesure de probabilité $\\mu $ invariante par $f$ , telle que pour toute application $\\phi $ continue sur $X$ , pour $\\lambda $ -presque tout $x \\in X$ , $\\widetilde{\\phi }(x) = \\int _X\\phi d\\mu $ .", "Cette condition est bien plus forte que l'ergodicité, puisque contrairement au résultat du théorème de Birkhoff, le résultat de convergence est vrai $\\lambda $ -p.p.", "($\\lambda $ est la mesure de Lebesgue sur $X$ , dont le support est $X$ tout entier), et non $\\mu $ -p.p., $\\mu $ pouvant avoir un support bien moins grand que $X$ tout entier.", "En particulier, si le support de $\\mu $ a une mesure de Lebesgue nulle, le théorème ergodique de Birkhoff énonce un résultat que l'on n'observera jamais (p.s.)", "si l'on choisit une condition initiale $x$ aléatoirement suivant $\\lambda $ .", "De plus, lorsque la mesure physique existe (cela a été prouvé dans le cas du solénoïde), elle est unique (ce n'est pas toujours le cas pour les mesures ergodiques).", "La mesure physique (lorsqu'elle existe) donnant la densité de $\\lambda $ -presque toute orbite, c'est elle que l'on observe empiriquement au cours des simulations numériques." ], [ "Dimension fractale", "Certains des attracteurs que nous avons évoqués ont — au moins partiellement — une structure d'ensemble de CantorNotamment le solénoïde, section REF , et l'attracteur de Hénon, section REF ., de dimension non-entière.", "Nous allons donner un sens à cette affirmation, en définissant la dimension fractale d'un compact $K$ .", "Il existe plusieurs autres notions de dimension non-entière (reliées les unes aux autres), par exemple la dimension de Hausdorffon montre en général que la dimension de Hausdorff $HD(K)$ est inférieure où égale à $D_f(K)$ .", "Il y a égalité pour des classes assez générales d'ensemble, par exemple pour l'exemple d'ensemble de Cantor décrit dans ce paragraphe.", "On pourra se référer à [7] pour le cas des ensembles de Cantor définis dynamiquement.", "; la dimension fractale possède l'avantage d'être la plus simple à évaluer numériquement.", "[Dimension fractale] Soit $K$ un compact d'un espace métrique $(X,d)$ .", "Pour tout $\\epsilon >0$ , on note $N_{\\epsilon }(K)$ le nombre minimal de boules de rayon $\\epsilon $ nécessaires pour recouvrir $K$ .", "La capacité limite ou dimension fractale de $K$ est définie par $ D_f (K) = \\limsup _{\\epsilon \\rightarrow 0}\\frac{\\log N_{\\epsilon }(K)}{- \\log \\epsilon } $ Lorsque $K$ est une sous-variété de dimension finie, la dimension fractale est égale à la dimension topologique.", "Un autre cas classique est celui des ensembles de Cantor.", "Considérons un exemple où la dimension se calcule facilement : pour $I=[a,b]$ intervalle, on note $f(I)=[a,a+\\frac{b-a}{3}] \\cup [a+\\frac{2(b-a)}{3},b]$ .", "Le compact $K = \\bigcap _{n \\in } f^n([0,1])$ est un ensemble de Cantor.", "On a $N_{3^n}(K) = 2^n$ , et $\\epsilon \\mapsto N_{\\epsilon }(K)$ est croissante, donc si $3^n\\le \\epsilon \\le 3^{n+1}$ , $ \\frac{n \\log 2}{(n+1) \\log 3} \\le \\frac{\\log N_{\\epsilon }(K)}{- \\log \\epsilon } \\le \\frac{(n+1)\\log 2}{n \\log 3}.", "$ On en déduit que $D_f(K) = \\frac{\\log 2}{\\log 3}.$" ], [ "Théorème de Whitney", "Pour visualiser les résultats des simulations numériques, nous avons projeté en dimension 3 les points de $^{N}$ ($N$ grand) que nous avions calculés.", "L'une des justifications a posteriori de la validité de la méthode est théorique et passe par le théorème de Whitney.", "En effet, l'attracteur semblant avoir une dimension (fractale) strictement inférieure à 15, il est possible de le plonger dans $^3$ .", "[Whitney] Toute variété compacte lisse de dimension $n \\in $ se plonge dans $^{2n+1}$ .", "Ce résultat est démontré dans [4].", "Il se généralise au cas d'un compact de dimension fractale $d$ , qui se plonge dans $^N$ dès que $N > 2d$ ." ], [ "Résultats détaillés", "10 21 32 43 54 65 76 87" ] ]

[ [ "Comment on the narrow structure reported by Amaryan et al" ], [ "Abstract The CLAS Collaboration provides a comment on the physics interpretation of the results presented in a paper published by M. Amaryan et al.", "regarding the possible observation of a narrow structure in the mass spectrum of a photoproduction experiment." ], [ "Comment on the narrow structure reported by Amaryan et al.", "13.40.Rj,14.40.Ak,24.85.+p,25.20.Lj Argonne National Laboratory, Argonne, Illinois 60439 Arizona State University, Tempe, Arizona 85287-1504 University of California at Los Angeles, Los Angeles, California 90095-1547 California State University, Dominguez Hills, Carson, CA 90747 Canisius College, Buffalo, NY Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Catholic University of America, Washington, D.C. 20064 CEA, Centre de Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-sur-Yvette, France Christopher Newport University, Newport News, Virginia 23606 University of Connecticut, Storrs, Connecticut 06269 Fairfield University, Fairfield, CT 06824 Florida International University, Miami, Florida 33199 Florida State University, Tallahassee, Florida 32306 Universit$\\grave{a}$ di Genova, 16146 Genova, Italy The George Washington University, Washington, DC 20052 Idaho State University, Pocatello, Idaho 83209 INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy INFN, Sezione di Genova, 16146 Genova, Italy Institut de Physique Nucléaire ORSAY, Orsay, France Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia James Madison University, Harrisonburg, Virginia 22807 Kyungpook National University, Daegu 702-701, Republic of Korea University of New Hampshire, Durham, New Hampshire 03824-3568 Ohio University, Athens, Ohio 45701 Northern Illinois University, Dekalb, IL 60115 Rensselaer Polytechnic Institute, Troy, New York 12180-3590 University of Richmond, Richmond, Virginia 23173 Skobeltsyn Nuclear Physics Institute at Moscow State University, 119899 Moscow, Russia University of South Carolina, Columbia, South Carolina 29208 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 Union College, Schenectady, NY 12308 Universidad Técnica Federico Santa María, Casilla 110-V Valparaíso, Chile University of Glasgow, Glasgow G12 8QQ, United Kingdom Washington & Jefferson College, Washington, PA 15301 College of William and Mary, Williamsburg, Virginia 23187-8795 M. Anghinolfi INFN, Sezione di Genova, 16146 Genova, Italy J.", "Ball CEA, Centre de Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-sur-Yvette, France N.A.", "Baltzell Argonne National Laboratory, Argonne, Illinois 60439 University of South Carolina, Columbia, South Carolina 29208 M. Battaglieri INFN, Sezione di Genova, 16146 Genova, Italy I. Bedlinskiy Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia M. Bellis Northern Illinois University, Dekalb, IL 60115 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 A.S. Biselli Fairfield University, Fairfield, CT 06824 C. Bookwalter Florida State University, Tallahassee, Florida 32306 S. Boiarinov Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia P. Bosted Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 V.D.", "Burkert Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 D.S.", "Carman Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 A. Celentano INFN, Sezione di Genova, 16146 Genova, Italy S.  Chandavar Ohio University, Athens, Ohio 45701 P.L.", "Cole Idaho State University, Pocatello, Idaho 83209 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 V. Crede Florida State University, Tallahassee, Florida 32306 R. De Vita INFN, Sezione di Genova, 16146 Genova, Italy E. De Sanctis INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy B. Dey Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 R. Dickson Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 D. Doughty Christopher Newport University, Newport News, Virginia 23606 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 M. Dugger Arizona State University, Tempe, Arizona 85287-1504 R. Dupre Argonne National Laboratory, Argonne, Illinois 60439 H. Egiyan Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 College of William and Mary, Williamsburg, Virginia 23187-8795 A. El Alaoui Argonne National Laboratory, Argonne, Illinois 60439 L. El Fassi Argonne National Laboratory, Argonne, Illinois 60439 L. Elouadrhiri Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 P. Eugenio Florida State University, Tallahassee, Florida 32306 G. Fedotov University of South Carolina, Columbia, South Carolina 29208 M.Y.", "Gabrielyan Florida International University, Miami, Florida 33199 M. Garcon CEA, Centre de Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-sur-Yvette, France G.P.", "Gilfoyle University of Richmond, Richmond, Virginia 23173 K.L.", "Giovanetti James Madison University, Harrisonburg, Virginia 22807 F.X.", "Girod Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 J.T.", "Goetz University of California at Los Angeles, Los Angeles, California 90095-1547 E. Golovatch Skobeltsyn Nuclear Physics Institute at Moscow State University, 119899 Moscow, Russia M. Guidal Institut de Physique Nucléaire ORSAY, Orsay, France L. Guo Florida International University, Miami, Florida 33199 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 K. Hafidi Argonne National Laboratory, Argonne, Illinois 60439 H. Hakobyan Universidad Técnica Federico Santa María, Casilla 110-V Valparaíso, Chile D. Heddle Christopher Newport University, Newport News, Virginia 23606 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 K. Hicks Ohio University, Athens, Ohio 45701 M. Holtrop University of New Hampshire, Durham, New Hampshire 03824-3568 D.G.", "Ireland University of Glasgow, Glasgow G12 8QQ, United Kingdom B.S.", "Ishkhanov Skobeltsyn Nuclear Physics Institute at Moscow State University, 119899 Moscow, Russia E.L. Isupov Skobeltsyn Nuclear Physics Institute at Moscow State University, 119899 Moscow, Russia H.S.", "Jo Institut de Physique Nucléaire ORSAY, Orsay, France K. Joo University of Connecticut, Storrs, Connecticut 06269 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 P. Khetarpal Florida International University, Miami, Florida 33199 A. Kim Kyungpook National University, Daegu 702-701, Republic of Korea W. Kim Kyungpook National University, Daegu 702-701, Republic of Korea V. Kubarovsky Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 S.V.", "Kuleshov Universidad Técnica Federico Santa María, Casilla 110-V Valparaíso, Chile Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia H.Y.", "Lu Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 I.J.D.", "MacGregor University of Glasgow, Glasgow G12 8QQ, United Kingdom N. Markov University of Connecticut, Storrs, Connecticut 06269 M.E.", "McCracken Washington & Jefferson College, Washington, PA 15301 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 B. McKinnon University of Glasgow, Glasgow G12 8QQ, United Kingdom M.D.", "Mestayer Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 C.A.", "Meyer Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 M. Mirazita INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy V. Mokeev Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 Skobeltsyn Nuclear Physics Institute at Moscow State University, 119899 Moscow, Russia K. Moriya [Current address:]Indiana University, Bloomington, IN 47405 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 B. Morrison Arizona State University, Tempe, Arizona 85287-1504 A. Ni Kyungpook National University, Daegu 702-701, Republic of Korea S. Niccolai Institut de Physique Nucléaire ORSAY, Orsay, France G. Niculescu James Madison University, Harrisonburg, Virginia 22807 Ohio University, Athens, Ohio 45701 I. Niculescu James Madison University, Harrisonburg, Virginia 22807 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 The George Washington University, Washington, DC 20052 M. Osipenko INFN, Sezione di Genova, 16146 Genova, Italy A.I.", "Ostrovidov Florida State University, Tallahassee, Florida 32306 K. Park Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 Kyungpook National University, Daegu 702-701, Republic of Korea S. Park Florida State University, Tallahassee, Florida 32306 S.  Anefalos Pereira INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy S. Pisano INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy O. Pogorelko Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia S. Pozdniakov Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia J.W.", "Price California State University, Dominguez Hills, Carson, CA 90747 G. Ricco Universit$\\grave{a}$ di Genova, 16146 Genova, Italy M. Ripani INFN, Sezione di Genova, 16146 Genova, Italy B.G.", "Ritchie Arizona State University, Tempe, Arizona 85287-1504 P. Rossi INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy D. Schott Florida International University, Miami, Florida 33199 R.A. Schumacher Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 E. Seder University of Connecticut, Storrs, Connecticut 06269 Y.G.", "Sharabian Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 E.S.", "Smith Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 D.I.", "Sober Catholic University of America, Washington, D.C. 20064 S.S. Stepanyan Kyungpook National University, Daegu 702-701, Republic of Korea P. Stoler Rensselaer Polytechnic Institute, Troy, New York 12180-3590 W.  Tang Ohio University, Athens, Ohio 45701 M. Ungaro Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 Rensselaer Polytechnic Institute, Troy, New York 12180-3590 University of Connecticut, Storrs, Connecticut 06269 B .Vernarsky Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 M.F.", "Vineyard Union College, Schenectady, NY 12308 University of Richmond, Richmond, Virginia 23173 D.P.", "Weygand Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 M.H.", "Wood Canisius College, Buffalo, NY University of South Carolina, Columbia, South Carolina 29208 N. Zachariou The George Washington University, Washington, DC 20052 B. Zhao College of William and Mary, Williamsburg, Virginia 23187-8795 The CLAS Collaboration In Ref.", "[1], the authors claim to observe a narrow structure in the mass spectrum constructed from the $(pK_L)$ system using data from the CLAS detector.", "The interpretation of this narrow structure given in Ref.", "[1] is: \"It may be due to the photoproduction of the $\\Theta ^+$ pentaquark or some unknown $\\Sigma ^*$ resonance.\"", "They go on to say that \"it is unlikely for the observed structure to be due to a $\\Sigma ^*$ resonance\".", "This analysis was reviewed by the CLAS Collaboration, following the established procedures for all CLAS papers, and did not receive approval.", "The purpose of this note is to explain the reasons why that analysis was not approved for publication.", "An extensive review of the analysis in Ref.", "[1] was carried out by two separate committees of the Hadron Spectroscopy Physics Working Group in the CLAS Collaboration.", "In both cases, the committees came to the same conclusion: the physics claims of Ref.", "[1] could not be supported.", "The reasons for this conclusion are many-fold, but a primary concern is the lack of justification for the kinematic cuts used in that analysis.", "The review committees reported that the narrow structure appears only within a specific range of values of the kinematic cuts.", "Here, the details are important (what cuts were varied and by how much) but this would require more space to document than a simple comment letter will allow.", "We give only one example below, but note that the CLAS committees conducted an extensive review of the sensitivity of the narrow structure to what they considered reasonable variations of the cuts [2].", "As an example, the cut on the $t_\\Theta $ variable (defined in Ref.", "[1]) was restricted to a small region of the total phase space ($-t_\\Theta < 0.45$ GeV$^2$ ).", "Without this cut, the narrow structure is not statistically significant.", "By examining Fig.", "8 of Ref.", "[1], one can see that the structure is not really visible in the top spectrum (Fig.", "8a), and only appears in Fig.", "8c.", "When the cut value is increased by 20% ($-t_\\Theta <0.55$ ) as shown by Fig.", "8b, or decreased by 10% ($t_\\Theta <0.4$ ) as shown by Fig.", "8d, then the purported structure at a mass of 1.54 GeV is consistent in size with other fluctuations in those spectra.", "While the authors of Ref.", "[1] make an argument about why the $t_\\Theta $ cut was necessary, the CLAS Collaboration was not convinced.", "For example, it is possible that an interference between the narrow structure and the background is dependent on the $t_\\Theta $ variable, but this assumption is difficult to prove.", "The analysis of Ref.", "[1] did not provide any evidence of interference phases.", "It is not uncommon to use kinematic cuts to reduce background and hence improve the signal-to-background ratio for known particles, but other studies [3] have shown that one must be careful to apply kinematic cuts which can create spurious fluctuations.", "We could argue whether the kinematic cuts used in Ref.", "[1] are justified, but the fact remains that the CLAS Collaboration as a whole was not convinced that the narrow structure of Ref.", "[1] corresponds to a real physical entity.", "At the request of the lead author of Ref.", "[1], presentations were made at a CLAS Collaboration meeting by both the authors and the review committee, followed by discussions and a vote on whether to publish that result as a collaboration paper.", "The outcome of the vote was to not publish this analysis.", "In the end, the validity of the narrow structure claimed by Ref.", "[1] will be determined by future experiments.", "If it is physical resonance, as suggested by Ref.", "[1], then it should be reproducible.", "The evidence presented in Ref.", "[1] was not sufficient to convince the CLAS Collaboration of the physics conclusions of that analysis." ] ]

[ [ "Controlled G-Frames and Their G-Multipliers in Hilbert spaces" ], [ "Abstract Multipliers have been recently introduced by P. Balazs as operators for Bessel sequences and frames in Hilbert spaces.", "These are operators that combine (frame-like) analysis, a multiplication with a fixed sequence (called the symbol) and synthesis.", "Weighted and controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator Also g-frames are the most popular generalization of frames that include almost all of the frame extensions.", "In this manuscript the concept of the controlled g-frames will be defined and we will show that controlled g-frames are equivalent to g-frames and so the controlled operators C and C0 can be used as preconditions in applications.", "Also the multiplier operator for this family of operators will be introduced and some of its properties will be shown." ], [ "Introduction", "In [31], R. Schatten provided a detailed study of ideals of compact operators by using their singular decomposition.", "He investigated the operators of the form $\\sum _{k}\\lambda _{k}\\varphi _{k}\\otimes \\overline{\\psi _{k}}$ where $(\\phi _{k})$ and $(\\psi _{k})$ are orthonormal families.", "In [3] the orthonormal families were replaced with Bessel and frame sequences to define Bessel and frame multipliers.", "Definition 1.1 Let $\\mathcal {H}_{1}$ and $\\mathcal {H}_{2}$ be Hilbert spaces, let $(\\psi _{k})\\subseteq \\mathcal {H}_{1}$ and $(\\phi _{k})\\subseteq \\mathcal {H}_{2}$ be Bessel sequences.", "Fix $m=(m_k)\\in l^{\\infty }$ .", "The operator ${\\bf M}_{m, (\\phi _k),(\\psi _k)} : \\mathcal {H}_{1} \\rightarrow \\mathcal {H}_{2}$ defined by $ {\\bf M}_{m, (\\phi _k), ( \\psi _k )} (f) = \\sum \\limits _k m_k \\langle f,\\psi _k\\rangle \\phi _k $ is called the Bessel multiplier for the Bessel sequences $(\\psi _{k})$ and $(\\phi _{k})$ .", "The sequence $m$ is called the symbol of M. Several basic properties of these operators were investigated in [3].", "Multipliers are not only interesting from a theoretical point of view, see e.g.", "[3], [18], [13], but they are also used in applications, in particular in the field of audio and acoustic.", "They have been investigated for fusion frames [1], for generalized frames , $p$ -frames in Banach spaces and for Banach frames , [17].", "In signal processing they are used for Gabor frames under the name of Gabor filters [23], in computational auditory scene analysis they are known by the name of time-frequency masks [24].", "In real-time implementation of filtering system they approximate time-invariant filters [7].", "As a particular way to implement time-variant filters they are used for example for sound morphing [12] or psychoacoustical modeling [8].", "$G$ -frames, introduced by W. Sun in and improved by the first author , are a natural generalization of frames which cover many other extensions of frames, e.g.", "bounded quasi-projectors [19], , pseudo-frames , frame of subspaces or fusion frames , outer frames , oblique frames , , and a class of time-frequency localization operators .", "Also it was shown that $g$ -frames are equivalent to stable spaces splitting studied in .", "All of these concepts are proved to be useful in many applications.", "Multipliers for $g$ -frames introduced in and some of its properties investigated.", "Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator on abstract Hilbert spaces [6], however they are used earlier in [9] for spherical wavelets.", "In this manuscript the concept of controlled $g$ - frame will be defined and we will show that any controlled $g$ -frame is equivalent a $g$ -frame and the role of controller operators are like the role of preconditions matrices or operators in linear algebra.", "Furthermore the multiplier operator for these family will be investigated.", "The paper is organized as follows.", "In Section 2 we fix the notation in this paper, summarize known and prove some new results needed for the rest of the paper.", "In Section 3 we will define the concept of controlled $g$ -frames and we will show that a controlled $g$ -frame is equivalent to a $g$ -frame and so the controlling operators can be used as precondition matrices in the problems related to applications.", "In section 4 we will define multipliers of controlled $g$ -frame operators and we will prove some of its properties." ], [ "Preliminaries", "Now we state some notations and theorems which are used in the present paper.", "Through this paper, $ and $ K$ are Hilbert spaces and $ { i:iI}$ is a sequence of Hilbert spaces, where $ I$ is a subset of $ Z.$ $ L(K) $ and $ L($ is the collection of all bounded linear operators from $ into $\\mathcal {K}$ and $ respectively.$ A bounded operator $T$ is called positive (respectively non-negative), if $\\langle Tf,f\\rangle >0$ for all $f\\ne 0$ (respectively $\\langle Tf,f\\rangle \\ge 0$ for all $f$ ).", "Every non-negative operator is clearly self-adjoint.", "For $T_{1},T_{2} \\in \\mathcal {L}($ , we write $ T_{1} \\leqslant T_{2} $ whenever $ \\langle T_{1}(f) , f \\rangle \\leqslant \\langle T_{2}(f) , f \\rangle , \\quad \\forall f \\in $ If $U\\in \\mathcal {L}($ is non-negative, then there exists a unique non-negative operator $V$ such that $V^2=U$ .", "Furthermore $V$ commutes with every operator that commute with $U$ .", "This will be denoted by $V=U^{\\frac{1}{2}}$ .", "Let $\\mathcal {G}\\mathcal {L}($ be the set of all bounded operators with a bounded inverse and $\\mathcal {G}\\mathcal {L}^+($ be the set of positive operators in $\\mathcal {G}\\mathcal {L}($ .", "For $U \\in \\mathcal {L}($ , $U \\in \\mathcal {G}\\mathcal {L}^+($ if and only if there exists $0<m \\leqslant M<\\infty $ such that $ m \\leqslant U \\leqslant M .$ For $U^{-1}$ we have $ M^{-1} \\leqslant U^{-1} \\leqslant m^{-1} .$ The following theorem can be found in .", "Theorem 2.1 Let $T_{1},T_{2}, T_{3} \\in \\mathcal {L}($ and $ T_{1} \\le T_{2} $ .", "Suppose $T_{3} > 0$ commutes with $T_{1}$ and $T_{2}$ then $ T_{1} T_{3} \\le T_{2} T_{3} .$ Recall that if $T$ is a compact operator on a separable Hilbert space $\\mathcal {H}$ , then in it is proved that there exist orthonormal sets $\\lbrace e_{n}\\rbrace $ and $\\lbrace \\sigma _{n}\\rbrace $ in $\\mathcal {H}$ such that $Tx=\\sum _{n}\\lambda _{n}\\langle x, e_{n}\\rangle \\sigma _{n}, $ for $x\\in \\mathcal {H}$ , where $\\lambda _{n}$ is the $n$ -th singular value of $T$ .", "Given $0<p<\\infty $ , the Schatten $p$ -class of $\\mathcal {H}$ , denoted $\\mathcal {S}_{p}$ , is the space of all compact operators $T$ on $\\mathcal {H}$ with the singular value sequence $\\lbrace \\lambda _{n}\\rbrace $ belonging to $\\ell ^{p}$ .", "It was shown that , $\\mathcal {S}_{p}$ is a Banach space with the norm $\\Vert T\\Vert _{p}=[\\sum _{n}|\\lambda _{n}|^{p}]^{\\frac{1}{p}}.$ $\\mathcal {S}_{1}$ is called the trace class of $\\mathcal {H}$ and $\\mathcal {S}_{2}$ is called the Hilbert-Schmidt class.", "$T\\in \\mathcal {S}_p$ if and only if $T^{p}\\in \\mathcal {S}_1$ .", "Moreover $\\Vert T\\Vert ^{p}_{p}=\\Vert T^{p}\\Vert _{1}$ .", "Also, $T\\in \\mathcal {S}_{p}$ if and only if $|T|^p=(T^*T)^\\frac{p}{2}\\in \\mathcal {S}_{1}$ if and only if $T^{*}T\\in \\mathcal {S}_{\\frac{p}{2}}$ .", "Moreover, $\\Vert T\\Vert ^{p}_{p}=\\Vert T^{*}\\Vert ^{p}_{p}=\\Vert |T|\\Vert ^{p}_{p}=\\Vert |T|^{p}\\Vert _{1}=\\Vert T^{*}T\\Vert _{\\frac{p}{2}}.$" ], [ "For any sequence $\\lbrace \\,{i}:{i\\in I}\\rbrace ,$ we can assume that there exits a Hilbert space $\\mathcal {K}$ such that for all $i\\in I, {i}\\subseteq \\mathcal {K}$ ( for example $\\mathcal {K}={(\\bigoplus _{i\\in I}{i})}_{\\ell ^2}$ ).", "Definition 2.1 A sequence $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is called generalized frame, or simply a $g$ -frame, for $ with respect to $ {i: iI}$ if there exist constants $ A>0$ and $ B<$ such that\\begin{equation}A\\Vert f\\Vert ^{2}\\leqslant \\sum _{i\\in I}\\Vert \\Lambda _{i}f\\Vert ^{2} \\leqslant B\\Vert f\\Vert ^{2},\\quad \\forall f\\in \\end{equation}The numbers $ A$ and $ B$ are called $ g$-frame bounds.$ $\\Lambda =\\lbrace \\Lambda _{i}: i\\in I\\rbrace $ is called tight $g$ -frame if $A = B$ and Parseval $g$ -frame if $A=B=1$ .", "If the second inequality in () holds, the sequence is called $g$ -Bessel sequence.", "$\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is called a $g$ -frame sequence, if it is a $g$ -frame for $\\overline{\\text{span}}\\lbrace \\Lambda _{i}^{*}({i})\\rbrace _{i\\in I}$ .", "It is easy to see that, if $\\lbrace f_i\\rbrace _{i\\in I}$ is a frame for $ with bounds $ A$ and $ B$, then by putting $ i=C$ and $ i()=, fi$, the family $ {i: iI}$ is a $ g$-frame for $ with bounds $A$ and $B$ .", "Let $\\Big (\\bigoplus _{i\\in I}{i}\\Big )_{\\ell _{2}}=\\Big \\lbrace \\,\\lbrace f_{i}\\rbrace _{i\\in I}\\mid f_{i}\\in {i},\\,\\forall i\\in I\\ \\ \\text{and}\\ \\ \\sum _{i\\in I}\\Vert f_{i}\\Vert ^{2}<+\\infty \\Big \\rbrace .$ Proposition 2.2 [25] $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is a $g$ -Bessel sequence for $ with bound $ B$, if and only if the operator$$T_{ \\Lambda }:\\Big (\\bigoplus _{i\\in I}{i}\\Big )_{\\ell _{2}}\\longrightarrow defined by$$T_{ \\Lambda }(\\lbrace f_{i}\\rbrace _{i\\in I})=\\sum _{i\\in I}\\Lambda _{i}^{*}(f_{i})$$is a well-defined and bounded operator with $ TB$.", "Furthermore$$T^{*}_{ \\Lambda }:\\Big (\\bigoplus {i}\\Big )_{\\ell _{2}}$$$$T^{*}_{ \\Lambda }(f)=\\lbrace \\Lambda _{i}f\\rbrace _{i\\in I}.$$$ If $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is a $g$ -frame, the operators $T_{ \\Lambda }$ and $T^{*}_{ \\Lambda }$ in Proposition REF are called synthesis operator and analysis operator of $ \\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace ,$ respectively.", "Proposition 2.3 $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is a $g$ -frame for $ if and only if the synthesis operator$ T$ is well-defined, bounded and onto.$ The ordinary version of the next theorem which is proved in [16], can be extended easily to the general case.", "Theorem 2.2 [16] $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is a $g$ -Bessel sequence for $ if and only if the operator\\begin{equation}S_{\\Lambda }: \\quad S_{\\Lambda }=\\sum _{i\\in I}\\Lambda ^{\\ast }_{i}\\Lambda _{i}f ,\\end{equation}is a welldefined operator.", "In this case $ S$ is bounded.$ Theorem 2.3 [17] $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is a $g$ -frame for $ if and only if the operator$$ S_{\\Lambda }: \\quad S_{\\Lambda }=\\sum _{i\\in I}\\Lambda ^{\\ast }_{i}\\Lambda _{i}f ,$$is a welldefined invertible operator.", "In this case $ S$ is bounded.$ $ S_{\\Lambda }$ is called the $g$ -frame operator of $\\Lambda =\\lbrace \\,\\Lambda _{i}: i\\in I\\,\\rbrace $ and it is known [25] that $S_{\\Lambda }$ is a positive and $AI\\leqslant S_{\\Lambda }\\leqslant BI,$ where A and B are the frame bounds.", "Every $f\\in has anexpansion $ f=ii*iS-1f$.One of the most important advantages of $ g$-frames is aresolution of identity $ ii*iS-1 =I$.", "\\vspace{11.38092pt}\\vspace{22.76228pt}\\subsection {\\bf Multipliers of g-frames}\\vspace{22.76228pt}The concept of multipliers for $ g$-Bessel sequences introduced by the first author in \\cite {2011 A Rahimi Multipliers of Genralized frames (5)} and some of their properties will be shown.$ Definition 2.4 Let $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ and $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be $g$ -Bessel sequences.", "If for $m \\subset \\mathbb {C}$ , the operator $ \\mathbf {M}=\\mathbf {M}_{m,\\Lambda ,\\Theta }:\\begin{equation}\\mathbf {M}(f)=\\sum _{i}m_{i}\\Lambda ^{*}_{i}\\Theta _{i}f\\end{equation}is welldefined, then $ M$ is called the \\textbf {$ g$-multiplier} of $$,$$ and $ m$.$ If $m=(m_i)=(1, 1, 1,...)$ and $\\mathbf {M}=I$ , $(\\Lambda ,\\Theta )$ is called a pair dual (i.e.", "$I=\\sum _{i\\in I}\\Lambda ^*_i\\Theta _i$ ).", "Let $\\lbrace \\lambda _{i}\\rbrace $ and $\\lbrace \\varphi _{i}\\rbrace $ be Bessel sequences and $m\\in \\ell ^{\\infty }$ , consider the corresponding $g$ -Bessel sequences $ \\Lambda _{i}\\cdot =\\langle \\cdot ,\\lambda _{i}\\rangle $ and $\\Theta _{i}\\cdot =\\langle \\cdot ,\\varphi _{i}\\rangle $ .", "For any $f\\in we have:\\begin{equation*}\\begin{aligned}\\mathbf {M}_{m,\\Lambda ,\\Theta }(f) ={\\bf M}_{m, ( \\lambda _k ),(\\phi _k) } (f)= \\sum _i m_i \\langle f ,\\varphi _i \\rangle \\lambda _i.\\end{aligned}\\end{equation*}$ It is easy to show that the adjoint of $\\mathbf {M}_{m,\\Lambda ,\\Theta }$ is $\\mathbf {M}_{\\overline{m},\\Theta ,\\Lambda }$ .", "Lemma 2.5 [29] If $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is a $g$ -Bessel sequence with bound $B_{\\Theta }$ and $m=(m_i)\\in \\ell ^{\\infty }$ , then $\\lbrace m_{i}\\Theta _{i}\\rbrace _{i\\in I}$ is a $g$ -Bessel sequence with bound $\\Vert m\\Vert _{\\infty }B_{\\Theta }$ .", "Like weighted frames [6], $\\lbrace m_{i}\\Theta _{i}\\rbrace _{i\\in I}$ can be called weighted $g$ - frame ($g$ -Bessel).", "By using the synthesis and the analysis operators of $\\Lambda $ and $ m \\Theta $ , respectively, we can write $\\mathbf {M}_{m,\\Lambda ,\\Theta }f=\\sum _{i}m_{i}\\Lambda _{i}^{*}\\Theta _{i}f=\\sum _{i}\\Lambda _{i}^{*}(m_{i}\\Theta _{i})f =T_{\\Lambda }\\lbrace m_i \\Theta _i f\\rbrace =T_{\\Lambda }T^{*}_{m\\Theta }f.$ So $\\mathbf {M}_{m,\\Lambda ,\\Theta }=T_{\\Lambda }T^{*}_{m \\Theta }.$ If we define the diagonal operator $ D_m:\\Big (\\bigoplus {i}\\Big )_{\\ell _{2}}\\rightarrow \\Big (\\bigoplus {i}\\Big )_{\\ell _{2}},$ $D_m((\\xi _{i}))=(m_i \\xi _i)_{i\\in I}$ then $\\mathbf {M}_{m,\\Lambda ,\\Theta }=T_{\\Lambda }D_m T^{*}_{\\Theta }.$ The notations in (REF ), (REF ) and (REF ) were used for proving the following propositions in [29] .", "Proposition 2.6 [29] Let $m\\in \\ell ^{\\infty }$ , $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be a $g$ -Riesz base and $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be a $g$ -Bessel sequence.", "The map $m\\rightarrow \\mathbf {M}_{m,\\Lambda ,\\Theta }$ is injective.", "Proposition 2.7 [29] Let $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ and $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be $g$ -Bessel sequences for $.", "If $ m=(mi)c0$ and $ (rank i)$,then $ Mm,,$ is compact.$ Proposition 2.8 [29] Let $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ and $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be $g$ -Bessel sequences for $.", "If $ m=(mi)p$ and $ (dimi)iI$, then $ Mm,,$ is a Schatten $ p$-class operator.$ Corollary 2.9 [29] Let $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ and $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be $g$ -Bessel sequences for $.\\begin{enumerate}\\item If m=(m_i)\\in \\ell ^{1} and (dimi)_{i\\in I}\\in \\ell ^{\\infty }, then \\mathbf {M}_{m,\\Lambda ,\\Theta } is a trace- class operator.\\item If m=(m_i)\\in \\ell ^{2} and (dimi)_{i\\in I}\\in \\ell ^{\\infty }, then \\mathbf {M}_{m,\\Lambda ,\\Theta } is a Hilbert-Schmit operator.\\end{enumerate}$" ], [ "Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator.", "In [6], it was shown that the controlled frames are equivalent to standard frames and it was used in the sense of preconditioning.", "In this section, the concepts of controlled frames and controlled Bessel sequences will be extended to $g$ -frames and we will show that controlled $g$ -frames are equivalent $g$ -frames.", "Definition 3.1 Let $C,C^{\\prime }\\in \\mathcal {G}\\mathcal {L}^+($ .", "The family $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ will be called a $(C,C^{\\prime })$ -controlled $g$ -frame for $, if $$ is a $ g$-Bessel sequence andthere exists constants $ A>0$ and $ B<$ such that\\begin{equation}A\\Vert f\\Vert ^{2}\\leqslant \\sum _{i\\in I} \\langle \\Lambda _{i}Cf,\\Lambda _{i}C^{^{\\prime }} f \\rangle \\leqslant B\\Vert f\\Vert ^{2}, \\quad \\forall f\\in \\end{equation}$ A and B will be called controlled frame bounds.", "If $C^{\\prime }=I$ , we call $\\Lambda =\\lbrace \\,\\Lambda _{i}\\rbrace $ a $C$ -controlled $g$ -frame for $ with bounds $ A$ and $ B$.", "If the second part ofthe above inequality holds, it will be called\\textbf {$ (C,C')$-controlled $ g$-Bessel sequence} with bound $ B$.$ The proof of the following lemmas is straightforward.", "Lemma 3.1 Let $C\\in \\mathcal {G}\\mathcal {L}^{+}($ .", "The $g$ -Bessel sequence $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is $(C,C)$ -controlled Bessel sequence (or $(C,C)$ -controlled $g$ -frame) if and only if there exists constant $B<\\infty $ (and $A>0$ )such that $ \\, \\sum _{i\\in I} \\Vert \\Lambda _{i}Cf \\Vert ^2 \\leqslant B\\Vert f\\Vert ^{2}, \\quad \\forall f\\in $ $ (\\, or \\, A\\Vert f\\Vert ^{2}\\leqslant \\sum _{i\\in I} \\Vert \\Lambda _{i}Cf \\Vert ^2 \\leqslant B\\Vert f\\Vert ^{2}, \\quad \\forall f\\in .$ We call the $(C,C)$ -controlled Bessel sequence and $(C,C)$ -controlled $g$ -frame, $C^2$ -controlled Bessel sequence and $C^2$ -controlled $g$ -frame with bounds $A,B$ .", "Lemma 3.2 For $C,C^{\\prime }\\in \\mathcal {G}\\mathcal {L}^+($ , the family $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is a $(C,C^{\\prime })$ -controlled Bessel sequence for $ if and only if the operator$$ L_{C \\Lambda C^{\\prime }}: \\quad L_{C \\Lambda C^{\\prime }} f:=\\sum _{i\\in I}C^{\\prime }\\Lambda _{i}^{*}\\Lambda _{i}Cf, $$is well defined and there exists constant $ B<$ such that$$\\sum _{i\\in I} \\langle \\Lambda _{i}Cf, \\Lambda _{i}C^{^{\\prime }} f \\rangle \\leqslant B\\Vert f\\Vert ^{2}, \\quad \\forall f\\in $$$ The operator $ L_{C \\Lambda C^{\\prime }}: \\quad L_{C \\Lambda C^{\\prime }} f:=\\sum _{i\\in I}C^{\\prime }\\Lambda _{i}^{*}\\Lambda _{i}Cf,$ is called the $(C,C^{\\prime })$ -controlled Bessel sequence operator, also $ L_{C \\Lambda C^{\\prime }} = C S_{\\Lambda } C^{\\prime }$ .", "It follows from the definition that for a $g$ -frame, this operator is positive and invertible and $ AI\\le L_{C \\Lambda C^{\\prime }}\\le BI.$ Also, if $C$ and $C^{\\prime }$ commute with each other, then $C^{\\prime } ,C^{\\prime -1},C ,C^{-1}$ commute with $ L_{C \\Lambda C^{\\prime }}, CS_{\\Lambda },S_{\\Lambda }C^{\\prime }$ .", "The following proposition shows that any $g$ -frame is a controlled $g$ -frame and versa.", "This is the most important advantage of weighted and controlled $g$ -frame in the sense of precondition.", "Proposition 3.3 Let $C \\in \\mathcal {G}\\mathcal {L}^{+}($ .", "The family $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is a $g$ -frame if and only if $\\Lambda $ is a $C^2$ -controlled $g$ -frame.", "Proof.", "Suppose that $\\Lambda $ is a $C^2$ -controlled $g$ -frame with bounds $A,B$ .", "Then $ \\,A\\Vert f\\Vert ^{2} \\leqslant \\sum _{i\\in I} \\Vert \\Lambda _{i}C f \\Vert ^2 \\leqslant B\\Vert f\\Vert ^{2}, \\quad \\forall f\\in $ For $f \\in $$ \\begin{split}A\\Vert f\\Vert ^{2} = A\\Vert CC^{-1} f\\Vert ^{2} \\leqslant A\\Vert C \\Vert ^{2} \\Vert C^{-1} f\\Vert ^{2} \\leqslant \\Vert C \\Vert ^{2} \\sum _{i\\in I} \\Vert \\Lambda _{i}C C^{-1} f \\Vert ^2 \\\\= \\Vert C \\Vert ^{2} \\sum _{i\\in I} \\Vert \\Lambda _{i} f \\Vert ^2.\\end{split} $$Hence$$ A \\Vert C \\Vert ^{-2} \\Vert f\\Vert ^{2} \\leqslant \\sum _{i\\in I} \\Vert \\Lambda _{i} f \\Vert ^2, \\quad \\forall f \\in $$On the other hand for every $ f , $ \\begin{split}\\sum _{i\\in I} \\Vert \\Lambda _{i} f \\Vert ^2=\\sum _{i\\in I} \\Vert \\Lambda _{i} CC^{-1} f \\Vert ^2 \\leqslant B\\Vert C^{-1}f\\Vert ^{2}\\leqslant B\\Vert C^{-1}\\Vert ^{2} \\Vert f\\Vert ^{2} .\\end{split} $ These inequalities yields that $\\Lambda $ is a $g$ -frame with bounds $ A \\Vert C \\Vert ^{-2} , B\\Vert C^{-1}\\Vert ^{2} $ .", "For the converse assume that $\\Lambda $ is $g$ -frame with bounds $ A^{\\prime } , B^{\\prime } $ .", "Then for all $f \\in ,$$ A^{\\prime } \\Vert f\\Vert ^{2} \\leqslant \\sum _{i\\in I} \\Vert \\Lambda _{i} f \\Vert ^2 \\leqslant B^{\\prime } \\Vert f\\Vert ^{2}.$$So for $ f , $ \\sum _{i\\in I} \\Vert \\Lambda _{i} Cf \\Vert ^2 \\leqslant B^{\\prime } \\Vert Cf\\Vert ^{2} \\leqslant B^{\\prime } \\Vert C \\Vert ^{2} \\Vert f\\Vert ^{2}.$ For lower bound, the $g$ -frameness of $\\Lambda $ shows that for any if $f \\in ,$$ \\begin{split} A^{\\prime } \\Vert f\\Vert ^{2} = A ^{\\prime } \\Vert C^{-1} C f\\Vert ^{2} \\leqslant A^{\\prime } \\Vert C^{-1} \\Vert ^{2} \\Vert C f\\Vert ^{2}\\leqslant \\Vert C^{-1} \\Vert ^{2}\\sum _{i\\in I} \\Vert \\Lambda _{i} C f \\Vert ^2.\\end{split} $$ Therefore $$ is a $ C2$-controlled$ g$-frame with bounds $ A' C-1 -2 ,B' C 2$.$ $ \\vspace{11.38092pt}\\begin{prop}Assume that \\Lambda =\\lbrace \\,\\Lambda _{i}: i\\in I\\,\\rbrace is ag-frame and C,C^{\\prime } \\in \\mathcal {G}\\mathcal {L}^+(, which commute with eachother and commute with S_{\\Lambda }.", "Then\\Lambda =\\lbrace \\,\\Lambda _{i}: i\\in I\\,\\rbrace is a (C,C^{\\prime })-controlledg-frame.\\end{prop}{\\bf Proof.}", "Let $$ be g-frame with bounds $ A,B$ and $ m , m'>0, M, M'<$ so that$$m \\leqslant C \\leqslant M, \\quad m^{\\prime } \\leqslant C^{\\prime } \\leqslant M^{\\prime } .", "$$Then$$ m A \\leqslant C S_{\\Lambda } \\leqslant M B ,$$because $ C$ commute with $ S$.", "Again $ C'$ commutes with$ C S$ and then$$ mm^{\\prime }A \\leqslant L_{C \\Lambda C^{\\prime }} \\leqslant MM^{\\prime }B .$$$$ \\Box $" ], [ "Extending the concept of multipliers of frames, in this section, we will define controlled $g$ -frame's multiplier for $C$ -controlled $g$ -frames in Hilbert spaces.", "The definition of general case $( C, C^{\\prime })$ -controlled $g$ -frames goes smooth.", "Lemma 4.1 Let $C,C^{\\prime }\\in \\mathcal {G}\\mathcal {L}^+($ and $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ , $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}({i}): i\\in I\\,\\rbrace $ be ${C^{\\prime }}^2$ and $C^2$ -controlled $g$ -Bessel sequences for $, respectively.", "Let $ m$.", "Theoperator$$\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}:$ defined by $ \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} f:=\\sum _{i\\in I}m_i C \\Theta _i^* \\Lambda _i C^{\\prime } f $ is a well-defined bounded operator.", "Proof.", "Assume $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}):i\\in I\\,\\rbrace $ , $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be ${C^{\\prime }}^2$ and $C^2$ -controlled $g$ -Bessel sequences for $ with bounds $ B,B'$, respectively.", "For any $ f,g and finite subset $J \\subset I$ , $\\Vert \\sum _{i \\in J}m_i C \\Theta _i^* \\Lambda _i C^{\\prime } f\\Vert = & \\sup _{g \\in \\Vert g\\Vert =1} \\Vert \\sum _{i \\in J}m_i \\langle \\Lambda _i C^{\\prime } f,\\Theta _i C^* g \\rangle \\Vert \\\\& \\leqslant \\sup _{g \\in \\Vert g\\Vert =1}\\sum _{i \\in J} |m_i | \\Vert \\Lambda _i C^{\\prime } f\\Vert \\Vert \\Theta _i C^* g \\Vert \\\\& \\leqslant \\sup _{g \\in \\Vert g\\Vert =1}\\Vert m\\Vert _{\\infty } \\big (\\sum _{i \\in I} \\Vert \\Theta _i C^* g \\Vert ^2\\big )^{\\frac{1}{2}} \\big (\\sum _{i \\in J}\\Vert \\Lambda _i C^{\\prime } f\\Vert ^2 \\\\& \\leqslant \\Vert m\\Vert _{\\infty }\\sqrt{BB^{\\prime }}\\Vert f\\Vert $ This shows that $\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}$ is well-defined and $\\Vert \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}\\Vert \\le \\Vert m\\Vert _{\\infty }\\sqrt{BB^{\\prime }}.$ $ \\Box $ Above Lemma ia a motivation to define the following definition.", "Definition 4.2 Let $C,C^{\\prime }\\in \\mathcal {G}\\mathcal {L}^+($ and $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ , $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}({i}): i\\in I\\,\\rbrace $ be ${C^{\\prime }}^2$ and $C^2$ -controlled $g$ -Bessel sequences for $, respectively.", "Let $ m$.", "Theoperator$$ \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}:$ defined by $\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} f:=\\sum _{i\\in I}m_i C \\Theta _i^* \\Lambda _i C^{\\prime } f,$ is called the $(C,C^{\\prime })$ -controlled multiplier operator with symbol $m$ .", "By using representations (REF ) and (REF ), we have $\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}=C \\textbf {M}_{m \\Theta \\Lambda } C^{\\prime }=CT_ \\Theta D_m T^*_\\Lambda C^{\\prime }.", "$ The proof of Proposition 4.7. of [29] shows that if $m=(m_i)\\in \\ell ^{p}$ and $(dimi)_{i\\in I}\\in \\ell ^{\\infty }$ , then the diagonal operator $D_m$ is a Schatten $p$ -class operator.", "Since $\\mathcal {S}_p$ is a $*$ -ideal of $\\mathcal {L}($ so we have: Theorem 4.3 Let $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ and $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be controlled $g$ -Bessel sequences for $.", "If $ m=(mi)p$ and $ (dimi)iI$, then $Mm C C' $ is a Schatten $ p$-class operator.$ And Corollary 4.4 Let $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ and $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be controlled $g$ -Bessel sequences for $.\\begin{enumerate}\\item If m=(m_i)\\in \\ell ^{1} and (dimi)_{i\\in I}\\in \\ell ^{\\infty }, then \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} is a trace- class operator.\\item If m=(m_i)\\in \\ell ^{2} and (dimi)_{i\\in I}\\in \\ell ^{\\infty }, then \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} is a Hilbert-Schmit operator.\\end{enumerate}$ Acknowledgment: Some of the results in this paper were obtained during the first author visited the Acoustics Research Institute, Austrian Academy of Sciences, Austria, he thanks this institute for their hospitality." ], [ "Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator.", "In [6], it was shown that the controlled frames are equivalent to standard frames and it was used in the sense of preconditioning.", "In this section, the concepts of controlled frames and controlled Bessel sequences will be extended to $g$ -frames and we will show that controlled $g$ -frames are equivalent $g$ -frames.", "Definition 3.1 Let $C,C^{\\prime }\\in \\mathcal {G}\\mathcal {L}^+($ .", "The family $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ will be called a $(C,C^{\\prime })$ -controlled $g$ -frame for $, if $$ is a $ g$-Bessel sequence andthere exists constants $ A>0$ and $ B<$ such that\\begin{equation}A\\Vert f\\Vert ^{2}\\leqslant \\sum _{i\\in I} \\langle \\Lambda _{i}Cf,\\Lambda _{i}C^{^{\\prime }} f \\rangle \\leqslant B\\Vert f\\Vert ^{2}, \\quad \\forall f\\in \\end{equation}$ A and B will be called controlled frame bounds.", "If $C^{\\prime }=I$ , we call $\\Lambda =\\lbrace \\,\\Lambda _{i}\\rbrace $ a $C$ -controlled $g$ -frame for $ with bounds $ A$ and $ B$.", "If the second part ofthe above inequality holds, it will be called\\textbf {$ (C,C')$-controlled $ g$-Bessel sequence} with bound $ B$.$ The proof of the following lemmas is straightforward.", "Lemma 3.1 Let $C\\in \\mathcal {G}\\mathcal {L}^{+}($ .", "The $g$ -Bessel sequence $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is $(C,C)$ -controlled Bessel sequence (or $(C,C)$ -controlled $g$ -frame) if and only if there exists constant $B<\\infty $ (and $A>0$ )such that $ \\, \\sum _{i\\in I} \\Vert \\Lambda _{i}Cf \\Vert ^2 \\leqslant B\\Vert f\\Vert ^{2}, \\quad \\forall f\\in $ $ (\\, or \\, A\\Vert f\\Vert ^{2}\\leqslant \\sum _{i\\in I} \\Vert \\Lambda _{i}Cf \\Vert ^2 \\leqslant B\\Vert f\\Vert ^{2}, \\quad \\forall f\\in .$ We call the $(C,C)$ -controlled Bessel sequence and $(C,C)$ -controlled $g$ -frame, $C^2$ -controlled Bessel sequence and $C^2$ -controlled $g$ -frame with bounds $A,B$ .", "Lemma 3.2 For $C,C^{\\prime }\\in \\mathcal {G}\\mathcal {L}^+($ , the family $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is a $(C,C^{\\prime })$ -controlled Bessel sequence for $ if and only if the operator$$ L_{C \\Lambda C^{\\prime }}: \\quad L_{C \\Lambda C^{\\prime }} f:=\\sum _{i\\in I}C^{\\prime }\\Lambda _{i}^{*}\\Lambda _{i}Cf, $$is well defined and there exists constant $ B<$ such that$$\\sum _{i\\in I} \\langle \\Lambda _{i}Cf, \\Lambda _{i}C^{^{\\prime }} f \\rangle \\leqslant B\\Vert f\\Vert ^{2}, \\quad \\forall f\\in $$$ The operator $ L_{C \\Lambda C^{\\prime }}: \\quad L_{C \\Lambda C^{\\prime }} f:=\\sum _{i\\in I}C^{\\prime }\\Lambda _{i}^{*}\\Lambda _{i}Cf,$ is called the $(C,C^{\\prime })$ -controlled Bessel sequence operator, also $ L_{C \\Lambda C^{\\prime }} = C S_{\\Lambda } C^{\\prime }$ .", "It follows from the definition that for a $g$ -frame, this operator is positive and invertible and $ AI\\le L_{C \\Lambda C^{\\prime }}\\le BI.$ Also, if $C$ and $C^{\\prime }$ commute with each other, then $C^{\\prime } ,C^{\\prime -1},C ,C^{-1}$ commute with $ L_{C \\Lambda C^{\\prime }}, CS_{\\Lambda },S_{\\Lambda }C^{\\prime }$ .", "The following proposition shows that any $g$ -frame is a controlled $g$ -frame and versa.", "This is the most important advantage of weighted and controlled $g$ -frame in the sense of precondition.", "Proposition 3.3 Let $C \\in \\mathcal {G}\\mathcal {L}^{+}($ .", "The family $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ is a $g$ -frame if and only if $\\Lambda $ is a $C^2$ -controlled $g$ -frame.", "Proof.", "Suppose that $\\Lambda $ is a $C^2$ -controlled $g$ -frame with bounds $A,B$ .", "Then $ \\,A\\Vert f\\Vert ^{2} \\leqslant \\sum _{i\\in I} \\Vert \\Lambda _{i}C f \\Vert ^2 \\leqslant B\\Vert f\\Vert ^{2}, \\quad \\forall f\\in $ For $f \\in $$ \\begin{split}A\\Vert f\\Vert ^{2} = A\\Vert CC^{-1} f\\Vert ^{2} \\leqslant A\\Vert C \\Vert ^{2} \\Vert C^{-1} f\\Vert ^{2} \\leqslant \\Vert C \\Vert ^{2} \\sum _{i\\in I} \\Vert \\Lambda _{i}C C^{-1} f \\Vert ^2 \\\\= \\Vert C \\Vert ^{2} \\sum _{i\\in I} \\Vert \\Lambda _{i} f \\Vert ^2.\\end{split} $$Hence$$ A \\Vert C \\Vert ^{-2} \\Vert f\\Vert ^{2} \\leqslant \\sum _{i\\in I} \\Vert \\Lambda _{i} f \\Vert ^2, \\quad \\forall f \\in $$On the other hand for every $ f , $ \\begin{split}\\sum _{i\\in I} \\Vert \\Lambda _{i} f \\Vert ^2=\\sum _{i\\in I} \\Vert \\Lambda _{i} CC^{-1} f \\Vert ^2 \\leqslant B\\Vert C^{-1}f\\Vert ^{2}\\leqslant B\\Vert C^{-1}\\Vert ^{2} \\Vert f\\Vert ^{2} .\\end{split} $ These inequalities yields that $\\Lambda $ is a $g$ -frame with bounds $ A \\Vert C \\Vert ^{-2} , B\\Vert C^{-1}\\Vert ^{2} $ .", "For the converse assume that $\\Lambda $ is $g$ -frame with bounds $ A^{\\prime } , B^{\\prime } $ .", "Then for all $f \\in ,$$ A^{\\prime } \\Vert f\\Vert ^{2} \\leqslant \\sum _{i\\in I} \\Vert \\Lambda _{i} f \\Vert ^2 \\leqslant B^{\\prime } \\Vert f\\Vert ^{2}.$$So for $ f , $ \\sum _{i\\in I} \\Vert \\Lambda _{i} Cf \\Vert ^2 \\leqslant B^{\\prime } \\Vert Cf\\Vert ^{2} \\leqslant B^{\\prime } \\Vert C \\Vert ^{2} \\Vert f\\Vert ^{2}.$ For lower bound, the $g$ -frameness of $\\Lambda $ shows that for any if $f \\in ,$$ \\begin{split} A^{\\prime } \\Vert f\\Vert ^{2} = A ^{\\prime } \\Vert C^{-1} C f\\Vert ^{2} \\leqslant A^{\\prime } \\Vert C^{-1} \\Vert ^{2} \\Vert C f\\Vert ^{2}\\leqslant \\Vert C^{-1} \\Vert ^{2}\\sum _{i\\in I} \\Vert \\Lambda _{i} C f \\Vert ^2.\\end{split} $$ Therefore $$ is a $ C2$-controlled$ g$-frame with bounds $ A' C-1 -2 ,B' C 2$.$ $ \\vspace{11.38092pt}\\begin{prop}Assume that \\Lambda =\\lbrace \\,\\Lambda _{i}: i\\in I\\,\\rbrace is ag-frame and C,C^{\\prime } \\in \\mathcal {G}\\mathcal {L}^+(, which commute with eachother and commute with S_{\\Lambda }.", "Then\\Lambda =\\lbrace \\,\\Lambda _{i}: i\\in I\\,\\rbrace is a (C,C^{\\prime })-controlledg-frame.\\end{prop}{\\bf Proof.}", "Let $$ be g-frame with bounds $ A,B$ and $ m , m'>0, M, M'<$ so that$$m \\leqslant C \\leqslant M, \\quad m^{\\prime } \\leqslant C^{\\prime } \\leqslant M^{\\prime } .", "$$Then$$ m A \\leqslant C S_{\\Lambda } \\leqslant M B ,$$because $ C$ commute with $ S$.", "Again $ C'$ commutes with$ C S$ and then$$ mm^{\\prime }A \\leqslant L_{C \\Lambda C^{\\prime }} \\leqslant MM^{\\prime }B .$$$$ \\Box $" ], [ "Extending the concept of multipliers of frames, in this section, we will define controlled $g$ -frame's multiplier for $C$ -controlled $g$ -frames in Hilbert spaces.", "The definition of general case $( C, C^{\\prime })$ -controlled $g$ -frames goes smooth.", "Lemma 4.1 Let $C,C^{\\prime }\\in \\mathcal {G}\\mathcal {L}^+($ and $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ , $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}({i}): i\\in I\\,\\rbrace $ be ${C^{\\prime }}^2$ and $C^2$ -controlled $g$ -Bessel sequences for $, respectively.", "Let $ m$.", "Theoperator$$\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}:$ defined by $ \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} f:=\\sum _{i\\in I}m_i C \\Theta _i^* \\Lambda _i C^{\\prime } f $ is a well-defined bounded operator.", "Proof.", "Assume $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}):i\\in I\\,\\rbrace $ , $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be ${C^{\\prime }}^2$ and $C^2$ -controlled $g$ -Bessel sequences for $ with bounds $ B,B'$, respectively.", "For any $ f,g and finite subset $J \\subset I$ , $\\Vert \\sum _{i \\in J}m_i C \\Theta _i^* \\Lambda _i C^{\\prime } f\\Vert = & \\sup _{g \\in \\Vert g\\Vert =1} \\Vert \\sum _{i \\in J}m_i \\langle \\Lambda _i C^{\\prime } f,\\Theta _i C^* g \\rangle \\Vert \\\\& \\leqslant \\sup _{g \\in \\Vert g\\Vert =1}\\sum _{i \\in J} |m_i | \\Vert \\Lambda _i C^{\\prime } f\\Vert \\Vert \\Theta _i C^* g \\Vert \\\\& \\leqslant \\sup _{g \\in \\Vert g\\Vert =1}\\Vert m\\Vert _{\\infty } \\big (\\sum _{i \\in I} \\Vert \\Theta _i C^* g \\Vert ^2\\big )^{\\frac{1}{2}} \\big (\\sum _{i \\in J}\\Vert \\Lambda _i C^{\\prime } f\\Vert ^2 \\\\& \\leqslant \\Vert m\\Vert _{\\infty }\\sqrt{BB^{\\prime }}\\Vert f\\Vert $ This shows that $\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}$ is well-defined and $\\Vert \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}\\Vert \\le \\Vert m\\Vert _{\\infty }\\sqrt{BB^{\\prime }}.$ $ \\Box $ Above Lemma ia a motivation to define the following definition.", "Definition 4.2 Let $C,C^{\\prime }\\in \\mathcal {G}\\mathcal {L}^+($ and $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ , $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}({i}): i\\in I\\,\\rbrace $ be ${C^{\\prime }}^2$ and $C^2$ -controlled $g$ -Bessel sequences for $, respectively.", "Let $ m$.", "Theoperator$$ \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}:$ defined by $\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} f:=\\sum _{i\\in I}m_i C \\Theta _i^* \\Lambda _i C^{\\prime } f,$ is called the $(C,C^{\\prime })$ -controlled multiplier operator with symbol $m$ .", "By using representations (REF ) and (REF ), we have $\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}=C \\textbf {M}_{m \\Theta \\Lambda } C^{\\prime }=CT_ \\Theta D_m T^*_\\Lambda C^{\\prime }.", "$ The proof of Proposition 4.7. of [29] shows that if $m=(m_i)\\in \\ell ^{p}$ and $(dimi)_{i\\in I}\\in \\ell ^{\\infty }$ , then the diagonal operator $D_m$ is a Schatten $p$ -class operator.", "Since $\\mathcal {S}_p$ is a $*$ -ideal of $\\mathcal {L}($ so we have: Theorem 4.3 Let $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ and $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be controlled $g$ -Bessel sequences for $.", "If $ m=(mi)p$ and $ (dimi)iI$, then $Mm C C' $ is a Schatten $ p$-class operator.$ And Corollary 4.4 Let $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ and $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be controlled $g$ -Bessel sequences for $.\\begin{enumerate}\\item If m=(m_i)\\in \\ell ^{1} and (dimi)_{i\\in I}\\in \\ell ^{\\infty }, then \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} is a trace- class operator.\\item If m=(m_i)\\in \\ell ^{2} and (dimi)_{i\\in I}\\in \\ell ^{\\infty }, then \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} is a Hilbert-Schmit operator.\\end{enumerate}$ Acknowledgment: Some of the results in this paper were obtained during the first author visited the Acoustics Research Institute, Austrian Academy of Sciences, Austria, he thanks this institute for their hospitality.", "Extending the concept of multipliers of frames, in this section, we will define controlled $g$ -frame's multiplier for $C$ -controlled $g$ -frames in Hilbert spaces.", "The definition of general case $( C, C^{\\prime })$ -controlled $g$ -frames goes smooth.", "Lemma 4.1 Let $C,C^{\\prime }\\in \\mathcal {G}\\mathcal {L}^+($ and $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ , $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}({i}): i\\in I\\,\\rbrace $ be ${C^{\\prime }}^2$ and $C^2$ -controlled $g$ -Bessel sequences for $, respectively.", "Let $ m$.", "Theoperator$$\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}:$ defined by $ \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} f:=\\sum _{i\\in I}m_i C \\Theta _i^* \\Lambda _i C^{\\prime } f $ is a well-defined bounded operator.", "Proof.", "Assume $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}):i\\in I\\,\\rbrace $ , $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be ${C^{\\prime }}^2$ and $C^2$ -controlled $g$ -Bessel sequences for $ with bounds $ B,B'$, respectively.", "For any $ f,g and finite subset $J \\subset I$ , $\\Vert \\sum _{i \\in J}m_i C \\Theta _i^* \\Lambda _i C^{\\prime } f\\Vert = & \\sup _{g \\in \\Vert g\\Vert =1} \\Vert \\sum _{i \\in J}m_i \\langle \\Lambda _i C^{\\prime } f,\\Theta _i C^* g \\rangle \\Vert \\\\& \\leqslant \\sup _{g \\in \\Vert g\\Vert =1}\\sum _{i \\in J} |m_i | \\Vert \\Lambda _i C^{\\prime } f\\Vert \\Vert \\Theta _i C^* g \\Vert \\\\& \\leqslant \\sup _{g \\in \\Vert g\\Vert =1}\\Vert m\\Vert _{\\infty } \\big (\\sum _{i \\in I} \\Vert \\Theta _i C^* g \\Vert ^2\\big )^{\\frac{1}{2}} \\big (\\sum _{i \\in J}\\Vert \\Lambda _i C^{\\prime } f\\Vert ^2 \\\\& \\leqslant \\Vert m\\Vert _{\\infty }\\sqrt{BB^{\\prime }}\\Vert f\\Vert $ This shows that $\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}$ is well-defined and $\\Vert \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}\\Vert \\le \\Vert m\\Vert _{\\infty }\\sqrt{BB^{\\prime }}.$ $ \\Box $ Above Lemma ia a motivation to define the following definition.", "Definition 4.2 Let $C,C^{\\prime }\\in \\mathcal {G}\\mathcal {L}^+($ and $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ , $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}({i}): i\\in I\\,\\rbrace $ be ${C^{\\prime }}^2$ and $C^2$ -controlled $g$ -Bessel sequences for $, respectively.", "Let $ m$.", "Theoperator$$ \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}:$ defined by $\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} f:=\\sum _{i\\in I}m_i C \\Theta _i^* \\Lambda _i C^{\\prime } f,$ is called the $(C,C^{\\prime })$ -controlled multiplier operator with symbol $m$ .", "By using representations (REF ) and (REF ), we have $\\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }}=C \\textbf {M}_{m \\Theta \\Lambda } C^{\\prime }=CT_ \\Theta D_m T^*_\\Lambda C^{\\prime }.", "$ The proof of Proposition 4.7. of [29] shows that if $m=(m_i)\\in \\ell ^{p}$ and $(dimi)_{i\\in I}\\in \\ell ^{\\infty }$ , then the diagonal operator $D_m$ is a Schatten $p$ -class operator.", "Since $\\mathcal {S}_p$ is a $*$ -ideal of $\\mathcal {L}($ so we have: Theorem 4.3 Let $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ and $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be controlled $g$ -Bessel sequences for $.", "If $ m=(mi)p$ and $ (dimi)iI$, then $Mm C C' $ is a Schatten $ p$-class operator.$ And Corollary 4.4 Let $\\Lambda =\\lbrace \\,\\Lambda _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ and $\\Theta =\\lbrace \\,\\Theta _{i}\\in \\mathcal {L}( {i}): i\\in I\\,\\rbrace $ be controlled $g$ -Bessel sequences for $.\\begin{enumerate}\\item If m=(m_i)\\in \\ell ^{1} and (dimi)_{i\\in I}\\in \\ell ^{\\infty }, then \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} is a trace- class operator.\\item If m=(m_i)\\in \\ell ^{2} and (dimi)_{i\\in I}\\in \\ell ^{\\infty }, then \\textbf {M}_{m C \\Theta \\Lambda C^{\\prime }} is a Hilbert-Schmit operator.\\end{enumerate}$ Acknowledgment: Some of the results in this paper were obtained during the first author visited the Acoustics Research Institute, Austrian Academy of Sciences, Austria, he thanks this institute for their hospitality." ] ]

[ [ "The QCD equation of state and the effects of the charm" ], [ "Abstract We present an update on the QCD equation of state of the Wuppertal-Budapest Collaboration, extending our previous studies [JHEP 0601 (2006) 089, JHEP 1011 (2010) 077].", "A Symanzik improved gauge and a stout-link improved staggered fermion action is utilized.", "We discuss partial quenching and present preliminary results for the fully dynamical charmed equation of state." ], [ "Introduction", "The properties of the Quark-Gluon-Plasma (QGP) are the focus of a large number of heavy ion experiments, such as ALICE at LHC, CERN SPS, RHIC at BNL, and, in the future, FAIR at GSI Darmstadt.", "Hydrodynamical models, based on the observation that the QGP is well approximated by an ideal liquid [3], [4], [5], require as an input parameter the equation of state (EOS) of Quantum Chromodynamics (QCD).", "Computing the EOS from QCD directly is possible through simulations of the lattice regularized theory, Lattice QCD.", "As algorithms and computers improve, such simulations aiming to understand the features of the QGP are reaching unprecedented levels of accuracy both for mapping the phase diagram [6], [7], [8], [9], [10] or studying bulk obervables [1], [2], [11], [12], [13], [14], [15], [16], [17].", "Here, we provide a status report on our efforts to improve upon our result of [2] for the $N_f=2+1$ EOS by performing a controlled continuum limit extrapolation and by studying the effects of an additional sea charm quark ($N_f=2+1+1$ EOS)." ], [ "Continuum estimate for the $N_f=2+1$ EOS", "In order to perform a continuum extrapolation for the EOS, we interpolated the results for any given $N_t$ using a standard cubic spline ansatz.", "Furthermore, we augmented our dataset from [2] by adding a set of $N_t=12$ points, thereby making a spline interpolation possible.", "We then varied the details of the spline interpolation to account for the statistical uncertainties and, by adding or removing spline points, to estimate the systematic effects due to this particular interpolation procedure.", "The continuum limit was then taken with an $O(a^2)$ fit ansatz.", "The result for the trace anomaly is depicted in Figure REF .", "While the data points depicted in Figure REF have been corrected for tree level effects, no such correction was applied prior to computing the continuum extrapolation.", "As already discussed in [2], these improvement factors would not influence the continuum extrapolation procedure.", "This is also nicely visible in Figure REF , since the tree level improved data points agree perfectly with the continuum estimate.", "Furthermore, excellent agreement of the lattice data with the prediction of the Hadron Resonance Gas (HRG) model was found, strengthening our assessment that the results of [2] are accurate.", "Figure: Finite volume study for our EOS.", "Shown is the trace anomaly for two different volumes at N t =8N_t=8.", "Complete agreement between the two volumes is found, extending our evidence in , where the same outcome was found for N t =6N_t=6.Finally, in order to check for finite size effects, we have repeated our analysis at $N_t=8$ with two different aspect ratios, shown in Figure REF .", "As in [2], where the same analysis was done for $N_t=6$ , the results agree perfectly.", "Therefore, we believe our volumes are more than adequate and our results are free of finite size effects." ], [ "Partially quenched and dynamical charm effects on the equation of state", "A straightforward approach in estimating charm quark effects is to include the charm quark in the valence sector only.", "Estimates of charm quark effects computed with this ansatz are discussed in [2], [19], [20].", "Here, the charm quark becomes relevant already for temperatures around 200 MeV, which is incompatible with perturbative estimates [18], see Figure REF .", "Figure: Subtracted negative gauge action (〈-s g 〉 sub \\langle -s_g\\rangle ^{sub} in the language of ), used as input in the integral technique for calculating the EOS.", "Shown is the gauge action from the N f =2+1N_f=2+1 set, as it would apply to a partially quenched calculation, compared to the preliminary results for N f =2+1+1N_f=2+1+1.", "In terms of physical units, no sizable effect is visible.Figure: In terms of physical units, the effects of the dynamical charm quark are small.", "Left panel: Subtracted strange condensate.", "Right panel: Quark number susceptibilities.This is analyzed more closely in Figure REF .", "Here, the gauge action is plotted in terms of the temperature as defined through the LCP for both $N_f=2+1$ and $N_f=2+1+1$ sea quarks.", "Note that the gauge action enters into the calculation of the equation of state through the integral technique (here, $p$ refers to the pressure, and $\\langle \\rangle ^{sub}$ to subtracted/renormalized quantities; our notation is explained in [2]): $\\frac{p(T)}{T^4}-\\frac{p(T_0)}{T_0^4}=N_t^4 \\int ^{(\\beta ,m_q)}_{(\\beta _0,m_{q0})}\\left(d\\beta \\langle -s_g \\rangle ^{\\rm sub} +\\sum _q d m_q \\langle \\bar{\\psi }_q\\psi _q\\rangle ^{\\rm sub}\\right).$ In terms of physical units, no sizable effects from the charm quark in the sea are visible, as is the case for other quantities as well (see Figure REF ).", "The large deviation between the results shown in Figure REF is, therefore, likely due to a shift in the LCP, which enters into eq.", "REF through the measure.", "It is conceivable to try to correct for this shift using an $N_f=2+1+1$ LCP, but in the full result some effects will remain.", "Hence, we prefer to perform a fully dynamical simulation of the charmed EOS." ], [ "Charmed equation of state for QCD", "Using a new $N_f=2+1+1$ LCP, we computed the EOS for QCD for $N_t=6,8$ , and 10, as shown in Figure REF .", "As already mentioned earlier, the fully dynamical charmed EOS agrees with the $N_f=2+1$ EOS up to higher temperatures than the partially quenched one.", "Similarly to the perturbative estimate, the temperature where charm effects become sizeable is close to 300 MeV.", "Furthermore, there do not seem to be sizeable discretization artifacts due to the heavy charm.", "At low temperatures, where the lattice spacing is coarse, the data points for the different $N_t$ agree with each other, as well as with the $N_f=2+1$ EOS.", "As the temperature increases, and the lattice spacing becomes increasingly finer, potential discretization effects should become smaller.", "At our present level of precision, we do not see sizable deviations between the different $N_t$ from low to high temperatures; therefore, we presently believe that discretization effects due to the heavy charm quark will not be significant, or at least minor." ], [ "Conclusion", "We have presented a first attempt to provide a continuum extrapolated equation of state for QCD with $N_f=2+1$ flavors of quarks, and have shown that it is free of finite size effects.", "We intend to supplement this calculation with additional $N_t=12$ and very few $N_t=16$ data points.", "In addition, we have addressed the issue of partially quenching for the (charmed) $N_f=2+1+1$ equation of state, and have argued that the difference of partially quenched and fully dynamical results is caused by a shift in the line of constant physics.", "Finally, we have shown first fully dynamical results for the charmed equation of state." ], [ "Acknowledgements", "Computations were performed on the Blue Gene supercomputers at Forschungszentrum Jülich, on clusters [21] at Wuppertal and Eötvös University, Budapest and on the QPACE at Forschungszentrum Jülich and Wuppertal University.", "This work is supported in part by the Deutsche Forschungsgemeinschaft grants FO 502/2 and SFB- TR 55 and by the EU (FP7/2007-2013)/ERC no.", "208740." ] ]

[ [ "An Analytical Study on the Multi-critical Behaviour and Related\n Bifurcation Phenomena for Relativistic Black Hole Accretion" ], [ "Abstract We apply the theory of algebraic polynomials to analytically study the transonic properties of general relativistic hydrodynamic axisymmetric accretion onto non-rotating astrophysical black holes.", "For such accretion phenomena, the conserved specific energy of the flow, which turns out to be one of the two first integrals of motion in the system studied, can be expressed as a 8$^{th}$ degree polynomial of the critical point of the flow configuration.", "We then construct the corresponding Sturm's chain algorithm to calculate the number of real roots lying within the astrophysically relevant domain of $\\mathbb{R}$.", "This allows, for the first time in literature, to {\\it analytically} find out the maximum number of physically acceptable solution an accretion flow with certain geometric configuration, space-time metric, and equation of state can have, and thus to investigate its multi-critical properties {\\it completely analytically}, for accretion flow in which the location of the critical points can not be computed without taking recourse to the numerical scheme.", "This work can further be generalized to analytically calculate the maximal number of equilibrium points certain autonomous dynamical system can have in general.", "We also demonstrate how the transition from a mono-critical to multi-critical (or vice versa) flow configuration can be realized through the saddle-centre bifurcation phenomena using certain techniques of the catastrophe theory." ], [ "Introduction", "In order to satisfy the inner boundary conditions imposed by the event horizon, accretion onto astrophysical black holes exhibit transonic properties in general [1].", "A physical transonic accretion solution can mathematically be realized as critical solution on the phase portrait [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].", "Multi-critical accretion may be referred to the specific category of accretion flow configuration having multiple critical points accessible to the accretion solution.", "For certain astrophysically relevant values of the initial boundary conditions, low angular momentum sub-Keplerian axisymmetric black hole accretion can have at most three critical points all together – where two saddle type critical points accommodate one centre type critical point in between them [1], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34].", "Transonic solution passing through the aforementioned two critical points can be joined through a stationary shock generated as a consequence of the presence of the angular momentum barrier [18], [35], [29], [36], [32], [33].", "The existence of such weakly rotating accretion in realistic astrophysical environment have also been observed [37], [38], [39], [40], [41], [42].", "A complete investigation of the multi-critical shocked accretion flow around astrophysical black holes necessitates the numerical integration of the nonlinear stationary equations describing the velocity phase space behaviour of the flow.", "However, for all the importance of transonic flows, there exists as yet no general mathematical prescription allowing one a direct analytical understanding of the nature of the multi-criticality without having to take recourse to the existing semi-analytic approach of numerically finding out the total number of physically acceptable critical points the accretion flow can have.", "This is precisely the main achievement of our work presented in this paper.", "Using the theory of algebraic polynomials, we developed a mathematical algorithm capable of finding the number of physically acceptable solution a polynomial can have, for any arbitrary large value of $n$ ($n$ being the degree of the polynomial).", "For a specified set of values of the initial boundary conditions, we mathematically predict whether the flow will be multi-critical (more than one real physical roots for the polynomial) or not.", "This paper, thus, purports to address that particular issue of investigating the transonicity of a general relativistic flow structure around non rotating black holes without encountering the usual semi-analytic numerical techniques, and to derive some predictive insights about the qualitative character of the flow, and in relation to that, certain physical features of the multi-criticality of the flow will also be addressed.", "In our work, we would like to develop a complete analytical formalism to investigate the critical behaviour of the general relativistic low angular momentum inviscid axisymmetric advective hydrodynamic accretion flow around a non rotating black hole.", "To accomplish the aforementioned task, we first construct the equation describing the space gradient of the dynamical flow velocity of accreting matter.", "Such equation is isomorphic to a first order autonomous dynamical system.", "Application of the fixed point analysis enables to construct an 8th degree algebraic equation for the space variable along which the flow streamlines are defined to possess certain first integrals of motion.", "The constant coefficients for each term in that equation are functions of astrophysically relevant initial boundary conditions.", "Such initial boundary conditions span over a certain domain on the real line $\\mathbb {R}$ – effectively, as individual sub-domain of ${\\mathbb {R}}{\\times }\\mathbb {R}{\\times }\\mathbb {R}$ for the polytropic accretion.", "The solution of aforesaid equation would then provide the critical (and consequently, the sonic) point $r_c$ .", "The critical points itself are permissible only within a certain open interval $\\left]r_g,L_{{\\rightarrow }{\\infty }}\\right[$ , where $r_g$ is the radius of the event horizon and $L_{{\\rightarrow }{\\infty }}$ is the physically acceptable maximally allowed limit on the value of a critical point.", "Since for polynomials of degree $n>4$ , analytical solutions are not available, we use the Sturm's theorem (a corollary of the Sylvester's theorem), to construct the Sturm's chain algorithm, which can be used to calculate the number of real roots (lying within a certain sub-domain of $\\mathbb {R}$ ) for a polynomial of any countably finite arbitrarily large integral $n$ , subjected to certain sub-domains of constant co-efficients.", "The problem now reduces to identify the polynomials in $r_c$ with the Sturm's sequence, and to find out the maximum number of physically acceptable solution an accretion flow with certain geometric configuration, space-time metric, and equation of state can have, and thus to investigate its multi-critical properties completely analytically, for accretion flow in which the critical points can not be computed analytically.", "Our work, as we believe, has significant importance, because for the first time in the literature, we provide a purely analytical method, by applying certain theorem of algebraic polynomials to check whether certain astrophysical hydrodynamic accretion may undergo more than one sonic transitions.", "We further demonstrate how the transition of number of critical points may be taken into account considering the bifurcation phenomenon in the parameter space.", "The transition of number of critical points in this case is associated with the merging and destruction (or emergence and separating apart, viewing in the other way round) of a saddle-centre pair, i.e.", "a saddle-centre bifurcation common in conservative systems, which may be tracked down using technique of catastrophe theory.", "The bifurcation lines in the parameter space exactly conform with the transition boundaries of the across which the number of critical points changes." ], [ "First Integral of Motion as a Polynomial in Critical Radius", "Following standard literature, we assume that the axisymmetric accretion flow has a radius dependent local thickness $H(r)$ , and its central plane coincides with the equatorial plane of the black hole.", "It is common practice in accretion disc theory ([43], [44], [45], [46], [26], [47], [48], [49], [50], [51]) to use the vertically integrated model in describing the black hole accretion discs where the equations of motion apply to the equatorial plane of the black hole assuming the flow to be in hydrostatic equilibrium along transverse direction.", "We follow the same procedure here.", "The thermodynamic flow variables are averaged over the disc height, i.e.", "a thermodynamic quantity $y$ used in our model is vertically integrated over the disc height and averaged as $\\bar{y}=\\int ^{H(r)}_0y dh/\\int ^{H(r)}_0dh$ .", "We follow [52] to derive an expression for the disc height $H(r)$ in our geometry since the relevant equations in [52] are non-singular on the horizon and can accommodate both axial and quasi spherical flow geometry.", "The disc height comes out to be [32], $H(r)=\\frac{c_sr}{\\lambda }\\sqrt{\\frac{2(\\gamma -1)(1-u^2)[r^3-\\lambda ^2(r-2)]}{\\gamma [\\gamma -(1+c_s^2)](r-2)}}$ where $\\lambda $ and $\\gamma $ are the specific flow angular momentum and the adiabatic index of the flow, respectively.", "$u$ and $c_s$ being the dynamical flow velocity and the speed of propagation of the acoustic perturbation (adiabatic sound speed) embedded within the accretion flow.", "In this work, we employ polytropic accretion.", "However, polytropic accretion is not the only choice to describe the general relativistic axisymmetric black-hole accretion.", "Equations of state other than the adiabatic one, such as the isothermal equation [27] or two temperature plasma [53] have also been used to study the black-hole accretion flow.", "For accretion flow of aforementioned category, two first integrals of motion along the streamline, viz, the dimensionless conserved specific flow energy i.e., the energy per unit mass which actually is scaled by the rest mass of the flow ${\\cal E}$ , and the mass accretion rate ${\\dot{M}}$ , may be obtained as (the radial distance $r$ here is actually scaled by the factor $GM_{BH}/c^2$ , and all the velocities, both $u$ as well as $c_s$ have been scaled by the velocity of light $c$ in vacuum.", "$M_{BH}$ is the mass of the black hole.", "Natural geometric unit has been used where the values of all fundamental constants have been taken to be unity, see, e.g., [32] for further detail) ${\\cal E}=\\left[ \\frac{(\\gamma -1)}{\\gamma -(1+c^{2}_{s})}\\right]r\\sqrt{\\frac{r-2}{r^3-\\lambda ^2\\left(r-2\\right)}}\\frac{1}{\\sqrt{1-u^2}},$ ${\\dot{M}}=\\frac{4{\\pi }{\\rho }c_sr^{\\frac{3}{2}}u}{\\lambda }\\sqrt{\\frac{2\\left(\\gamma -1\\right)\\left[r^3-\\lambda ^2\\left(r-2\\right)\\right]}{\\gamma \\left[\\gamma -\\left(1+c_s^2\\right)\\right]}}\\, ,$ where $\\rho $ is the mass density.", "The expression for ${\\cal E}$ is obtained by integrating the stationary part of the Euler equation and the expression for ${\\dot{M}}$ is obtained by integrating the stationary part of the continuity equation (by properly taking care of the flow thickness).", "The conserved specific entropy accretion rate ${\\dot{\\cal M}}$ is computed as a quasi constant multiple of ${\\dot{M}}$ as: ${\\dot{\\cal M}}=4\\pi \\left( \\frac{1}{\\lambda } \\sqrt{\\frac{}{}}{2}{\\gamma } \\right)\\left[\\frac{c_{s}}{\\left(1-\\frac{c_{s}^2}{\\gamma -1}\\right)^{\\frac{1}{2}}}\\right]^{\\frac{\\gamma +1}{\\gamma -1}} u r \\left[r^4-\\lambda ^2 r(r-2)\\right]^\\frac{1}{2},$ We thus have two primary first integrals of motion along the streamline – the specific energy of the flow ${\\cal E}$ and the mass accretion rate ${\\dot{M}}$ .", "Even in the absence of creation or annihilation of matter, the entropy accretion rate ${\\dot{\\cal M}}$ is not a generic first integral of motion.", "As the expression for ${\\dot{\\cal M}}$ contains the quantity $K{\\equiv }p/{\\rho }^{\\gamma }$ ($p$ being the flow pressure), which is a measure of the specific entropy of the flow, the entropy accretion rate ${\\dot{\\cal M}}$ remains constant throughout the flow only if the entropy per particle remains locally invariant.", "This condition may be violated if the accretion is accompanied by a shock.", "Thus ${\\dot{\\cal M}}$ is conserved for shock free polytropic accretion and becomes discontinuous (actually, increases) at the shock location, if such a shock is formed.", "The gradient of the acoustic velocity $c_s$ as well as the dynamical velocity $u$ can be obtained by differentiating the expression for the entropy accretion rate and the mass accretion rate respectively: $\\frac{dc_{s}}{dr}=-\\frac{c_{s}(\\gamma -1)\\left[\\gamma -(1+c^{2}_{s}\\right)]}{(\\gamma +1)}\\left[ \\frac{1}{u} \\frac{du}{dr} + {f_{1}}(r,\\lambda ) \\right],$ where ${f_{1}}(r,\\lambda ) = \\frac{3r^{3}-2\\lambda ^{2}r+3{\\lambda ^2}}{r^{4}-\\lambda ^{2}r(r-2)}\\, .$ $\\frac{du}{dr} = \\frac{(\\frac{2}{\\gamma +1})c^{2}_{s} {f_{1} } (r,\\lambda )- {f_{2} } (r,\\lambda )}{\\frac{u}{1-u^{2}} - \\frac{2c^{2}_{s}}{u(\\gamma +1)}}=\\frac{{\\cal N}\\left(r,\\lambda ,c_s\\right)}{{\\cal D}\\left(u,c_s\\right)}\\, ,$ where ${f_{2} } (r,\\lambda ) = \\frac{2r-3}{r(r-2)} -\\frac{2r^{3}-\\lambda ^{2}r+\\lambda ^{2}}{r^{4}-\\lambda ^{2}r(r-2)}\\, .$ A real physical transonic flow must be smooth everywhere, except possibly at a shock.", "Hence, if the denominator ${{\\cal D}\\left(u,c_s\\right)}$ of Eq.", "(REF ) vanishes at a point, the numerator ${{\\cal N}\\left(r,\\lambda ,c_s\\right)}$ must also vanish at that point to ensure the physical continuity of the flow.", "One therefore arrives at the critical point conditions by making ${{\\cal D}\\left(u,c_s\\right)}$ and ${{\\cal N}\\left(r,\\lambda ,c_s\\right)}$ of Eq.", "(REF ) simultaneously equal to zero.", "We thus obtain the critical point conditions as $u_c=\\pm \\sqrt{\\frac{{f_{2}}(r_c,\\lambda )}{{{f_{1}}(r_c,\\lambda )}+{{f_{2}}(r_c,\\lambda )}}}; \\;\\;\\;\\;c_c=\\pm \\sqrt{\\frac{\\gamma +1}{2}\\left[\\frac{{f_{2}}(r_c,\\lambda )}{{f_{1}}(r_c,\\lambda )}\\right]};$ where $u_c\\equiv u({r_c})$ and $c_c\\equiv c_s (r_c)$ , $r_c$ being the location of the critical point.", "$f_1(r_c,\\lambda )$ and $f_2(r_c,\\lambda )$ are defined as: ${f_{1}}(r_c,\\lambda ) = \\frac{3r_c^{3}-2\\lambda ^{2}r_c+3{\\lambda ^2}}{r_c^{4}-\\lambda ^{2}r_c(r_c-2)},~{f_{2} } (r_c,\\lambda ) = \\frac{2r_c-3}{r_c(r_c-2)} -\\frac{2r_c^{3}-\\lambda ^{2}r_c+\\lambda ^{2}}{r_c^{4}-\\lambda ^{2}r_c(r_c-2)}$ Clearly, the critical points are not coincident with the sonic points since $M_c=\\left({u_c}/{c_c}\\right)< 1$ .", "This is a consequence of the choice of the equation of state.", "The adiabatic equation of state used in this work produces non constant (with respect to the radial space direction) sound speed.", "Since the disc height contains the sound speed and the thermodynamic quantities calculated in the accretion flow have been averaged over the flow thickness, non constant sound speed accounts for the non-isomorphism of the critical points and the sonic points.", "If one uses the sound speed obtained from isothermal equation of state, or a flow geometry different from the configuration in the vertical equilibrium as has been assumed here, the critical points will coincide with the sonic points, see, e. g., [31], [34] for further detail.", "We substitute the explicit value of $u_c$ and $c_c$ from Eq.", "(REF ) to the expression for the specific energy ${\\cal E}$ in Eq.", "(REF ) to derive the explicit form of the energy first integral polynomial in $r_c$ as: $&&r_c^{8} \\lbrace -36\\left( -1 + {\\cal E}^2 \\right) {\\left( -1 + \\gamma \\right) }^2 \\rbrace + r_c^{7} \\lbrace 12\\left( -1 + \\gamma \\right)\\left( -17\\left( -1 + \\gamma \\right) +{\\cal E}^2\\left( -11 + 13\\gamma \\right) \\right) \\rbrace +\\nonumber \\\\ &&r_c^{6} \\lbrace -24{\\left( -1 + \\gamma \\right) }^2\\left( -16 + {\\lambda }^2 \\right) +{\\cal E}^2\\left( -121 + 60{\\lambda }^2 +\\gamma \\left( 286 - 96{\\lambda }^2 \\right) +{\\gamma }^2\\left( -169 + 36{\\lambda }^2 \\right) \\right) \\rbrace \\nonumber \\\\ &&+ r_c^{5} \\lbrace -2\\left( 120 + \\left( -86 + 163{\\cal E}^2 \\right) {\\lambda }^2 +{\\gamma }^2\\left( 120 + \\left( -86 + 99{\\cal E}^2 \\right) {\\lambda }^2\\right) - 2\\gamma \\left( 120 +\\left( -86 + 133{\\cal E}^2 \\right) {\\lambda }^2 \\right) \\right) \\rbrace \\nonumber \\\\ && \\;\\;+ r_c^{4}\\lbrace {\\lambda }^2\\left( -460{\\left( -1 + \\gamma \\right) }^2 +{\\cal E}^2\\left( 588 - 25{\\lambda }^2 +{\\gamma }^2\\left( 356 - 9{\\lambda }^2 \\right) +\\gamma \\left( -976 + 30{\\lambda }^2 \\right) \\right) \\right) \\rbrace \\nonumber \\\\ &&\\;\\;\\; \\;+ r_c^3 \\lbrace 4{\\lambda }^2\\left( 136{\\left( -1 + \\gamma \\right) }^2 +{\\cal E}^2\\left( -88 + 45{\\lambda }^2 +\\gamma \\left( 148 - 52{\\lambda }^2 \\right) +{\\gamma }^2\\left( -52 + 15{\\lambda }^2 \\right) \\right) \\right) \\rbrace \\nonumber \\\\ && \\;\\;\\; \\;+ r_c^2 \\lbrace -4{\\lambda }^2\\left( 60 + 121{\\cal E}^2{\\lambda }^2 +{\\gamma }^2\\left( 60 + 37{\\cal E}^2{\\lambda }^2 \\right) -2\\gamma \\left( 60 + 67{\\cal E}^2{\\lambda }^2 \\right) \\right) \\rbrace \\nonumber \\\\ && \\;\\;\\; \\;\\;+ r_c \\lbrace 32{\\cal E}^2\\left( 18 - 19\\gamma + 5{\\gamma }^2 \\right) {\\lambda }^4\\rbrace +\\lbrace -64{\\cal E}^2{\\left( -2 + \\gamma \\right) }^2{\\lambda }^4\\rbrace =0$ The above equation, being an $n=8$ polynomial, is non analytically solvable.", "Being equipped with the details of the Sturm theorem and its appropriate application in the next section (§), in § we will demonstrate how we can analytically find out the number of physically admissible real roots for this polynomial, and can investigate the transonicity of the flow." ], [ "Sturm theorem and generalized sturm sequence (chain)", "In this section we will elaborate the idea of the generalized Strum sequence/chain, and will discuss its application in finding the number of roots of a algebraic polynomial equations with real co-efficients.", "Since the central concept of this theorem is heavily based on the idea of the greatest common divisor of a polynomial and related Euclidean algorithm, we start our discussion by clarifying such concept in somewhat great detail for the convenience of the reader." ], [ "Greatest common divisor for two numbers", "Given two non-zero integers $z_1$ and $z_2$ , one defines that $z_1$ divides $z_2$ , if and only if there exists some integer $z_3{\\in {\\mathbb {Z}}}$ such that: $z_2=z_3z_1$ The standard notation for the divisibility is as follows: $z_1{{\\vert }}z_2 \\text{~means~`} z_1 \\text{~divides~} z_2 \\text{'}$ The concept of divisibility applies to the polynomials as well, we treat such situations in the subsequent paragraphs.", "Now consider two given integers $z_1$ and $z_2$ , with at least one of them being a non-zero number.", "The `greatest common divisor' (or the `greatest common factor' or the `highest common factor') of $z_1$ and $z_2$ , denoted by $gcd(z_1,z_2)$ , is the positive integer $z_d{\\in }{\\mathbb {Z}}$ , which satisfies: ${\\rm i)} z_d{\\vert }z_1 ~{\\rm and}~ z_d{\\vert }z_2.\\nonumber \\\\{\\rm ii)} {\\rm For~any~other}~z_c{\\in }{\\mathbb {Z}},~if~z_c{\\vert }z_1~{\\rm and} z_c{\\vert }z_2\\nonumber \\\\{\\rm then} z_c{\\vert }z_d$ In other words, the greatest common divisor $gcd(z_1,z_2)$ of two non zero integers $z_1$ and $z_2$ is the largest possible integer that divides both the integers without leaving any remainder.", "Two numbers $z_1$ and $z_2$ are called `co-prime] (alternatively, `relatively prime'), if: $gcd(z_1,z_2)=1$ The idea of a greatest common divisor can be generalized by defining the greater common divisor of a non empty set of integers.", "If ${\\cal S_{\\mathbb {Z}}}$ is a non-empty set of integers, then the greatest common divisor of ${\\cal S_{\\mathbb {Z}}}$ is a positive integer $z_d$ such that: ${\\rm i)~If}~z_d{\\vert }z_1{\\rm for~all}~z_1{\\in }{\\cal S_{\\mathbb {Z}}}\\nonumber \\\\{\\rm ii) If}~z_2{\\vert }z_1,~{\\rm for~all}~z_1{\\in } {\\cal S_{\\mathbb {Z}}},~{\\rm then}~z_2{\\vert }z_d$ then we denote $z_d=gcd({\\cal S_{\\mathbb {Z}}})$ ." ], [ "Euclidean algorithm", "Euclidean algorithm (first described in detail in Euclid's `Elements' in 300 BC, and is still in use, making it the oldest available numerical algorithm still in common use) provides an efficient procedure for computing the greatest common divisor of two integers.", "Following Stark [54], below we provide a simplified illustration of the Euclidean algorithm for two integers: Let us first set a `counter' $i$ for counting the steps of the algorithm, with initial step corresponding to $i=0$ .", "Let any $i$ th step of the algorithm begins with two non-negative remainders $r_{i-1}$ and $r_{i-2}$ with the requirement that $r_{i-1}<r_{i-2}$ , owing to the fact that the fundamental aim of the algorithm is to reduce the remainder in successive steps, to finally bring it down to the zero in the ultimate step which terminates the algorithm.", "Hence, for the dummy index $i$ , at the first step we have: $r_{-2}=z_2~{\\rm and}~r_{-1}=z_1$ the integers for which the greatest common divisor is sought for.", "After we divide $z_2$ by $z_1$ (operation corresponds to $i=1$ ), since $z_2$ is not divisible by $z_1$ , one obtains: $r_{-2} = q_0 r_{-1} + r_{0}$ where $r_0$ is the remainder and $q_0$ be the quotient.", "For any arbitrary $i$ th step of the algorithm, the aim is to find a quotient $q_j$ and remainder $r_i$ , such that: $r_{i-2}=q_ir_{i-1}+r_i,~{\\rm where~r_i<r_{i-1}}$ at some step $i=j$ (common sense dictates that $j$ can not be infinitely large), the algorithm terminates because the remainder becomes zero.", "Hence the final non-zero remainder $r_{j-1}$ will be the greatest common divisor of the corresponding integers.", "We will now illustrate the Euclidean algorithm for finding the greatest common divisor for two polynomials." ], [ "Greatest common divisor and related Euclidean algorithm for polynomials", "Let us first define a polynomial to be `monic' if the co-efficient of the term for the highest degree variable in the polynomial is unity (one).", "Let us now consider $p_1(x)$ and $p_2(x)$ to be two nonzero polynomials with co-efficient from a field ${\\mathbb {F}}$ (field of real, complex, or rational numbers, for example).", "A greatest common divisor of $p_1(x)$ and $p_2(x)$ is defined to the the monic polynomial $p_d(x)$ of highest degree such that $p_d(x)$ divides both $p_1(x)$ and $p_2(x)$ .", "It is obvious that ${\\mathbb {F}}$ be field and $p_d(x)$ be a monic, are necessary hypothesis.", "In more compact form, a greatest common divisor of two polynomials $p_1,p_2{\\in }{\\mathbb {R}}[{\\mathbb {X}}]$ is a polynomial $p_d{\\in }{\\mathbb {R}}[{\\mathbb {X}}]$ of greatest possible degree which divides both $p_1$ and $p_2$ .", "Clearly, $p_d$ is not unique, and is only defined upto multiplication by a non zero scalar, since for a non zero scalar $c{\\in }{\\mathbb {R}}$ , if $p_d$ is a $gcd(p_1,p_2{\\in }{\\mathbb {R}}[{\\mathbb {X}}]$ ), so as $cp_d$ .", "Given polynomials $p_1,p_2{\\in }{\\mathbb {R}}[{\\mathbb {X}}]$ , the division algorithm provides polynomials $p_3,p_4{\\in }{\\mathbb {R}}[{\\mathbb {X}}]$ , with $deg(p_4)<deg(p_3)$ such that $p_1=p_3p_2+p_4$ Then, if $p_d$ is $gcd(p_1,p_2)$ , if and only if $p_d$ is $gcd(p_2,p_4)$ as is obvious.", "One can compute the $gcd$ of two polynomials by collecting the common factors by factorizing the polynomials.", "However, this technique, although intuitively simple, almost always create a serious practical threat while making attempt to factorize the large high degree polynomials in reality.", "Euclidean algorithm appears to be relatively less complicated and a faster method for all practical purposes.", "Just like the integers as shown in the previous subsection, Euclid's algorithm can directly be applied for the polynomials as well, with decreasing degree for the polynomials at each step.", "The last non-zero remainder, after made monic if necessary, comes out to be the greatest common divisor of the two polynomials under consideration.", "Being equipped with the concept of the divisibility, $gcd$ and the Euclidean algorithm, we are now in a position to define the Strum theorem and to discuss its applications." ], [ "The Sturm Theorem: The purpose and the definition", "The Sturm theorem is due to Jacaues Charles Francois Sturm, a Geneva born French mathematician and a close collaborator of Joseph Liouville.", "The Sturm theorem, published in 1829 in the eleventh volume of the `Bulletin des Sciences de Ferussac' under the title `Memoire sur la resolution des equations numeriques' According to some historian, the theorem was originally discovered by Jean Baptist Fourier, well before Sturm, on the eve of the French revolution..", "The Sturm theorem, which is actually a root counting theorem, is used to find the number of real roots over a certain interval of a algebraic polynomial with real co-efficient.", "It can be stated as: The number of real roots of an algebraic polynomial with real coefficient whose roots are simple over an interval, the endpoints of which are not roots, is equal to the difference between the number of sign changes of the Sturm chains formed for the interval ends.", "Hence, given a polynomial $p{\\in }{\\mathbb {R}}[{\\mathbb {X}}]$ , if we need to find the number of roots it can have in a certain open interval $]a,b[$ , $a$ and $b$ not being the roots of $f$ , we then construct a sequence, called `Sturm chain', of polynomials, called the generalized strum chains.", "Such a sequence is derived from $p$ using the Euclidean algorithm.", "For the polynomial $p$ as described above, the Sturm chain $p_0,p_1...$ can be defined as: $p_0 = p\\nonumber \\\\p_1 = p^{\\prime }\\nonumber \\\\p_n=-{\\rm rem}\\left(p_{n-2},p_{n-1}\\right), n{\\ge }2$ where $rem\\left(p_{n-2},p_{n-1}\\right)$ is the remainder of the polynomial $p_{n-2}$ upon division by the polynomial $p_{n-1}$ .", "The sequence terminates once one of the $p_i$ becomes zero.", "We then evaluate this chain of polynomials at the end points $a$ and $b$ of the open interval.", "The number of roots of $p$ in $]a,b[$ is the difference between the number of sign changes on the chain of polynomials at the end point $a$ and the number of sign changes at the end point $b$ .", "Thus, for any number $t$ , if $N_{p(t)}$ denotes the number of sign changes in the Sturm chain $p_0(t),p_1(t),...$ , then for real numbers $a$ and $b$ that (both) are not roots of $p$ , the number of distinct real roots of $p$ in the open interval $]a,b[$ is $\\left[N_{p(a)}-N_{p(b)}\\right]$ .", "By making $a{\\rightarrow }{-\\infty }$ and $b{\\rightarrow }{+\\infty }$ , one can find the total number of roots $p$ can have on the entire domain of ${\\mathbb {R}}$ .", "A more formal definition of the Strum theorem, as a corollary of the Sylvester's theorem, is what follows: Definition Let $R$ be the real closed field, and let $p$ and $P$ be in $R[X]$ .The Sturm sequence of $p$ and $P$ is the sequence of polynomials $({p_0},{p_1},...,{p_k})$ defined as follows: ${p_0}=p$ , ${p_1}=p^{\\prime }P$ ${p_i}={p_{i-1}}{q_i}-{p_{i-2}}$ with ${q_i}\\in R[X]$ and deg$({p_i})<deg({p_{i-1}})$ for $i=2,3,...,k$ , ${p_k}$ is a greatest common divisor of $p$ and $p^{\\prime }P$ .", "Given a sequence $({a_0},...,{a_k})$ of elements of $R$ with ${a_0} \\ne 0$ , we define the number of sign changes in the sequence $({a_0},...{a_k})$ as follows: count one sign change if ${a_i}{a_l}<0$ with $l=i+1$ or $l>i+1$ and ${a_j}=0$ for every $j$ , $i<j<l$ .", "If $a \\in R$ is not a root of $p$ and $({p_0},...,{p _k})$ is the Sturm sequence of $p$ and $P$ , we define $v(p,P;a)$ to be the number of sign changes in $({p_0}(a),...{p_k}(a))$ .", "(Sylvester's TheoremAs stated in [55].)", "Let $R$ be a real closed field and let $p$ and $P$ be two polynomials in $R[X]$ .", "Let $a,b \\in R$ be such that $a < b$ and neither $a$ nor $b$ are roots of $p$ .", "Then the difference between the number of roots of $p$ in the interval $]a,b[$ for which $P$ is positive and the number of roots of $p$ in the interval $]a,b[$ for which $P$ is negative, is equal to $v(p,P;a) - v(p,P;b)$ (Sturm's Theorem): Let $R$ be a real closed field and $p\\in R[X]$ .", "Let $a,b\\in R$ be such that $a<b$ and neither $a$ nor $b$ are roots of $p$ .", "Then the number of roots of $p$ in the interval $]a,b[$ is equal to $v(p,1;a)-v(p,1;b)$ .", "The proof of these two theorems are given in the Appendix I.", "We first write down the complete expression for the Sturm chains.", "Then for a suitable parameter set $\\left[{\\cal E},\\lambda ,\\gamma \\right]$ , we can find the difference of the sign change of the Sturm chains at the open interval left boundary, i.e., at the event horizon and at the right boundary, i.e., at some suitably chosen large distance, say, $10^8$ gravitational radius (which is such a large distance that beyond which practically no critical point is expected to form unless the specific flow energy has an extremely low value, i.e., very cold accretion flow), to find the number of critical points the accretion flow can have.", "The form of the original polynomial has already been explicitly expressed using left hand side of Eq.", "REF .", "We now construct the Sturm chains as: $p_0(r) &=& a_8 r^8 + a_7 r^7 + a_6 r^6 + a_5 r^5 + a_4 r^4 + a_3 r^3 + a_2 r^2 + a_1 r + a_0\\nonumber \\\\p_1(r) &=& 8 a_8 r^7 + 7a_7 r^6 + 6 a_6 r^5 + 5 a_5 r^4 + 4 a_3 r^3 + 3 a_3 r^2 + 2 a_2 r + a_1\\nonumber \\\\p_2(r) &=& - rem(p_0/p_1)= c_6 r^6 + c_5 r^5 + c_4 r^4 + c_3 r^3 + c_2 r^2 + c_1 r + c_0\\\\ &&\\textrm {(the negative of the remainder of division ofp_0 by p_1)}\\nonumber \\\\p_3(r) &=& -rem (p_1/p_2) = d_5 r^5 + d_4 r^4 + d_3 r^3 + d_2 r^2 + d_1 r + d_0 \\nonumber \\\\p_4(r) &=& -rem(p_2/p_3) = e_4 r^4 + e_3 r^3 + e_2 r^2 + e_1 r + e_0 \\nonumber \\\\p_5(r) &=& -rem(p_3/p_4) = f_3 r^3 + f_2 r^2 + f_1 r + f_0\\nonumber \\\\p_6(r) &=& -rem(p_4/p_5) = g_2 r^2 + g_1 r + g_0 \\nonumber \\\\p_7(r) &=& -rem(p_5/p_6) = h_1 r + h_0\\nonumber \\\\p_8 (r) &=& - rem(p_6/p_7) = i_0\\nonumber \\\\$ Where the explicit expression of the corresponding co-efficients $a_i, c_i, d_i ...$ has been provided in the equation (REF ) and in the Appendix - II.", "If one needs to figure out the number of roots of $p_0$ in $[a,b]$ , the number of sign changes in the sequence $p_0(a), p_1(a), p_2(a), p_3(a), p_4(a), p_5(a), p_6(a), p_7(a), p_8(a)$ is to be counted and let us call it $v(p_0, a)$ .", "Similarly, the count the number of sign changes in the sequence $p_0(b), p_1(b), p_2(b),p_3(b), p_4(b), p_5(b), p_6(b), p_7(b), p_8(b)$ is to be called as $v(p_0, b)$ .", "Then, the number of roots of $p_0$ in $[a,b]$ is $v(p_0, a) - v(p_0, b).$ It is important to note that direct application of the Sturm's theorem may not always be sufficient since some of the roots may yield a negative energy for ${\\cal E}$ (since the ${\\cal E}$ equation was squared to get the polynomial).", "Since we are interested in accretion with the positive positive Bernoulli's constant, to get positive values of the energy, we must impose the condition that $\\gamma -(1+ c_{s}^{2}) \\ge 0,$ which is the term present in ${\\cal E}$ which could go negative.", "This introduces the condition that $\\displaystyle \\frac{p(r)}{q(r)} \\ge 0$ , where $p(r)$ and $q(r)$ are 4th order polynomials given by, $p(r) &=& 6(\\gamma -1)r^4-(11\\gamma -13)r^3\\nonumber \\\\&&-(5\\gamma -3)\\lambda ^2r^2+2(9\\gamma -5)\\lambda ^2r-8(2\\gamma -1)\\lambda ^2,\\\\q(r) &=& 6r^4-12r^3-4\\lambda ^2r^2+14\\lambda ^2r-12\\lambda ^2.$ To find the region where this happens, one has to find the 4 roots of each of $p(r)$ and $q(r)$ – which is analytically possible since roots of quartics are analytically solvable.", "Once the roots are obtained it is a trivial matter to check for what regions the rational function is positive.", "Figure: The lighter region (online version red) corresponds to 3 roots and the darker shade (online version blue) indicates 1 root only.", "The value of γ\\gamma is 4/34/3.A simplified version for the above mentioned procedure to find the positivity condition is as follows: We would like to find out the intervals in which $p(r)/q(r) >0$ where $p(r)$ and $q(r)$ are quartic polynomials.", "We factorize $p(r) =(r-r_1)(r-r_2)(r-r_3)(r-r_3)$ and $q(r)= (r-s_1)(r-s_2)(r-s_3)(r-s_4)$ using the algorithm for finding roots of a quartic.", "If the roots are all real, we note down the sign changes of each factor to the right and left of each root and find out the intervals where the rational function is positive.", "If there are complex roots, they come in complex conjugates, since the coefficients of the polynomials are real.", "Say, if $r_3$ is complex and $r_4$ is its complex conjugate, then the part $(r-r_3)(r-r_4) = r^2 - (r_3+r_4)r + r_3 r_4$ does not change sign since it is non-zero on the real line.", "It is easy to determine its sign.", "To demonstrate the procedure described above, the number of roots of the 8th order polynomial $p_0$ (in the Strum sequence) within the admissible range of $\\cal {E}, \\lambda $ and $\\gamma $ (usually by keeping the value of $\\gamma $ to be fixed to obtain a two dimensional parameter space) are evaluated explicitly and that shows two distinct regions in $\\cal {E} -\\lambda $ space (see Fig.", "REF ).", "The wedge shaped region corresponds to 3 roots implying 3 critical points and the rest of the parametric space corresponds to single root implying only one critical point.", "This feature emerging from the above mentioned algorithm, exactly conforms with the numerical results (using the explicit root finding methods) available in the current literature [32].", "It may be worthwhile to mention here that in addition to these roots there exists another root for the the whole range of parameter space shown in the Fig.", "REF that is located very near to the event horizon (i.e.", "within 1–1.5 times Schwarzchild radius), but being a centre it is physically untenable to be a sonic point (a critical point through which a physical accretion solution, connecting the event horizon with to infinity, can pass) and hence has always been justifiably ignored in the literature.", "Figure: Boundary of transition: Contour line det(S)=0\\det {(S)}=0 (for γ=4/3\\gamma = 4/3).The transition boundaries from $n_1$ number of roots to $n_2$ number of roots, in the parameter space, can be more easily obtained using catastrophe theory.", "The boundaries of the region in the parameter space permitting transition of number of critical points in this case are associated with saddle-centre bifurcation or merging of a pair of roots of the equation (Eq.REF ).", "Now all these equations are polynomial equations.", "As a general rule the discriminant of a polynomial, $P_n(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots +a_1x+a_0,$ can be expressed as in terms of its roots, $x_i$ 's, as $D=a_n^{n-2}\\prod _{i<j}{(x_i-x_j)^2}.$ The discriminant may be expressed as the determinant of a matrix called the Sylvester matrix (see, e.g., http://mathworld.wolfram.com/PolynomialDiscriminant.html, and references therein), $S=\\left[\\begin{array}{lllllllll}\\multicolumn{1}{c}{a_n} & \\multicolumn{1}{c}{a_{n-1}} & \\multicolumn{1}{c}{a_{n-2}} & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{a_1} & \\multicolumn{1}{c}{a_0} & \\multicolumn{1}{c}{0\\ldots } & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{0} \\\\\\multicolumn{1}{c}{0} & \\multicolumn{1}{c}{a_n} & \\multicolumn{1}{c}{a_{n-1}} & \\multicolumn{1}{c}{a_{n-2}} & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{a_1} & \\multicolumn{1}{c}{a_0} & \\multicolumn{1}{c}{0\\ldots } & \\multicolumn{1}{c}{0} \\\\\\multicolumn{1}{c}{\\vdots } & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{\\vdots } \\\\\\multicolumn{1}{c}{0} & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{0} & \\multicolumn{1}{c}{a_n} & \\multicolumn{1}{c}{a_{n-1}} & \\multicolumn{1}{c}{a_{n-2}} & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{a_1} & \\multicolumn{1}{c}{a_0} \\\\\\multicolumn{1}{c}{na_n} & \\multicolumn{1}{c}{(n-1)a_{n-1}} & \\multicolumn{1}{c}{(n-2)a_{n-2}} & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{1a_1} & \\multicolumn{1}{c}{0} & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{0} \\\\\\multicolumn{1}{c}{0} & \\multicolumn{1}{c}{na_n} & \\multicolumn{1}{c}{(n-1)a_{n-1}} & \\multicolumn{1}{c}{(n-2)a_{n-2}} & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{1a_1} & \\multicolumn{1}{c}{0} & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{0} \\\\\\multicolumn{1}{c}{\\vdots } & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{\\vdots } \\\\\\multicolumn{1}{c}{0} & \\multicolumn{1}{c}{0} & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{0} & \\multicolumn{1}{c}{na_n} & \\multicolumn{1}{c}{(n-1)a_{n-1}} & \\multicolumn{1}{c}{(n-2)a_{n-2}} & \\multicolumn{1}{c}{\\ldots } & \\multicolumn{1}{c}{1a_1} \\\\\\end{array}\\right],$ up to a factor.", "Putting $n=8$ , $\\det {(S)}$ will be zero on the above mentioned boundaries and actually it is so.", "Here the plot of $\\det {(S)}=0$ for the polytropic flow (i.e.", "for the polynomial in $r_c$ in Eq.REF ) in $\\cal {E}$ –$\\lambda $ space is shown in Fig.REF .", "The curve exactly conforms with the corresponding boundary curve in Fig.REF , drawn on the basis of the previous method.", "So this procedure may be thought of as a much easier alternative to find the multi-critical parametric values; though this method cannot give the exact number of critical points in each region of the parameter space." ], [ "Discussion", "Our methodology is based on the algebraic form of the first integral obtained by solving the radial momentum equation (the Euler equation to be more specific, since we are confined to the inviscid flow only).", "The structure for such a first integral has to be a formal polynomial with appropriate constant co-efficients.", "For general relativistic accretion in the Kerr metric, the expression for the energy first integral can not be reduced to such a polynomial form (see, e.g., [33] for the detail form of such algebraic expression).", "Hence, the Sturm's generalized chain can not be constructed for such accretion flow.", "Alternative methodology are required to investigate the multi-critical behaviour for such kind of accretion.", "Using the method illustrated in this work, it is possible to find out how many critical points a transonic black hole accretion flow can have.", "It is thus possible to predict whether such accretion flow can have multi-critical properties for a certain specific value/domain of the initial boundary conditions.", "It is, however, not possible to investigate, using the eigenvalue analysis as illustrated in  [8], [12], the nature of such critical points - i.e., whether they are of saddle type or are of centre type, since such prediction requires the exact location of the critical points (the value of the roots of the polynomial).", "However, the theory of dynamical systems ensures that no two consecutive critical points be of same nature (both saddle or both centre).", "On the other hand, our experience predicts (it is rather a documented fact) that for all kind of multi-critical black hole accretion, irrespective of the equation of state, the space time geometry or the flow configuration used, one has two saddle type critical points and one centre type critical point flanked by them (see, e.g., [33] and [34] for further detail).", "Hence if the application of the Sturm's generalized chain ensures the presence of three critical points, we can say that out of those three critical points, accretion flow will have two saddle type critical points, hence a specific subset of the solution having three roots corresponding to the first integral polynomial, can make transonic transition for more than one times, if appropriate conditions for connecting the flow through the outer critical point and for flow through the inner critical points are available, see, e.g., [33] for further discussion.", "In this work we have considered only inviscid accretion.", "Our methodology of investigating the multi-critical properties, however, is expected to be equally valid for the viscous accretion disc as well.", "For the viscous flow, the radial momentum conservation equation involving the first order space derivative of the dynamical flow velocity will certainly provide a first integral of motion upon integration.", "Because of the fact that a viscous accretion disc is not a non-dissipative system, such constant of motion, however, can never be identified with the specific energy of the flow.", "The integral solution of the radial momentum equation would then be an algebraic expression of various flow variables and would perhaps involve certain initial boundary conditions as well.", "Such an algebraic expression would actually be a constant of motion.", "What exactly would that expression physically signify, would definitely be hard to realize.", "However, one may perhaps arbitrarily parameterize that conserved algebraic expression using some astrophysically relevant outer boundary conditions, and if such algebraic expressions can finally be reduced, using the appropriate critical point conditions, to an algebraic polynomial form of the critical points, construction of a generalized Sturm chain can be made possible to find out how many critical points such an accretion flow can have subjected to the specific initial boundary condition.", "Since for accretion onto astrophysical black holes, having multiple critical points is a necessary (but not sufficient) condition to undergo shock transition, one can thus analytically predict, at least to some extent, which particular class of viscous accretion disc are susceptible for shock formation phenomena.", "Our work, as we believe, can have a broader perspective as well, in the field of the study of dynamical systems in general.", "For a first order autonomous dynamical system, provided one can evaluate the critical point conditions, the corresponding generalized $n$ th degree algebraic equation involving the position co-ordinate and one (or more) first integral of motion can be constructed.", "If such algebraic equation can finally be reduced to a $n$ th degree polynomial with well defined domain for the constant co efficient, one can easily find out the maximal number of fixed points of such dynamical systems." ], [ "Acknowledgments", "This research has made use of NASA's Astrophysics Data System as well as various online encyclopedia.", "SA and SN would like to acknowledge the kind hospitality provided by HRI and by astrophysics project under the XI th plan at HRI., Allahabad, India.", "The work of TKD is partially supported by the grant NN 203 380136 provided by the Polish academy of sciences and by astrophysics project under the XI th plan at HRI.", "RD acknowledges useful discussions with S. Ramanna." ], [ "Appendix - I : Proof of the Sylvester's theorem:", "First note that the Sturm sequence $({f_0},...{f_k})$ is (up to signs) equal to the sequence obtained from the Euclidean algorithm.", "Define a new sequence $({g_0},...,{g_k})$ by ${g_i}={f_i}/{f_k}$ for $i \\in \\lbrace 0,...,k \\rbrace $ .", "Note that the number of sign changes in $({f_0}(x),{f_1}(x))$ $($ resp.", "$(f_{i-1}(x),f_i(x),f_{i+1}(x)))$ and the number of sign changes in $({g_0}(x),{g_1}(x))$ $($ resp.", "$({g_{i-1}}(x),{g_i}(x),{g_{i+1}}(x)))$ coincide for any $x$ which is not a root of $f$ .", "Note also that the roots of ${g_0}$ are exactly the roots of $f$ which are not roots of $g$ .", "Observe that for $i\\in {{0,...,k}}$ ,${g_{i-1}}$ and ${g_i}$ are relatively prime.", "We consider, now, how $v(f,g;x)$ behaves when $x$ passes through a root $c$ of a polynomial ${g_i}$ .", "If $c$ is a root of ${g_0}$ , then it is not a root of ${g_1}$ .", "We write $f^{\\prime }(c_)>0$ ($resp.", "<0$ ) if $f^{\\prime }$ is positive $($ resp.", "negative $)$ immediately to the left of $c$ .", "The sign of $f^{\\prime }(c_{+})$ is defined similarly.", "Now we recall the following result: if $R$ is a real closed field, $f\\in R[X], a,b\\in R$ with $a<b$ and if the derivative $f^{\\prime }$ is positive (resp.", "negative) on $]a,b[$ , then $f$ is strictly increasing (resp.", "strictly decreasing) on $[a,b].$ Then, according to the signs of $g(c), f^{\\prime }(c_{-})$ and $f^{\\prime }(c_+)$ we have the following 8 cases: $g(c)>0, f^{\\prime }(c_-)>0, f^{\\prime }(c_+)>0$ Table: NO_CAPTION$g(c)<0, f^{\\prime }(c_-)>0, f^{\\prime }(c_+)>0$ Table: NO_CAPTION$g(c)>0, f^{\\prime }(c_-)<0, f^{\\prime }(c_+)>0$ Table: NO_CAPTION$g(c)<0, f^{\\prime }(c_-)<0, f^{\\prime }(c_+)>0$ Table: NO_CAPTION$g(c)>0, f^{\\prime }(c_-)>0, f^{\\prime }(c_+)<0$ Table: NO_CAPTION$g(c)<0, f^{\\prime }(c_-)>0, f^{\\prime }(c_+)<0$ Table: NO_CAPTION$g(c)>0, f^{\\prime }(c_-)<0, f^{\\prime }(c_+)<0$ Table: NO_CAPTION$g(c)<0, f^{\\prime }(c_-)<0, f^{\\prime }(c_+)<0$ Table: NO_CAPTIONIn every as $x$ passes through $c$ , the number of sign changes in $(f_0(x),f_1(x))$ decreases by 1 if $g(c)>0$ , and increases by 1 if $g(c)<0$ .", "If $c$ is a root of $g_i$ with $i=1,...k$ , then it is neither a root of $g_{i-1}$ nor a root of $g_{i+1}$ , and $g_{i-1}(c)g_{i+1}(c)<0$ , by the definition of the sequence.", "Passing through $c$ does not lead to any modification of the number of sign changes in $(f_{i-1}(x),f_i(x),f_{i+1}(x))$ in this case.", "Proof of the Sturm's theorem: Using $g=1$ in previous theorem." ] ]

[ [ "Thermal rounding exponent of the depinning transition of an elastic\n string in a random medium" ], [ "Abstract We study numerically thermal effects at the depinning transition of an elastic string driven in a two-dimensional uncorrelated disorder potential.", "The velocity of the string exactly at the sample critical force is shown to behave as $V \\sim T^\\psi$, with $\\psi$ the thermal rounding exponent.", "We show that the computed value of the thermal rounding exponent, $\\psi = 0.15$, is robust and accounts for the different scaling properties of several observables both in the steady-state and in the transient relaxation to the steady-state.", "In particular, we show the compatibility of the thermal rounding exponent with the scaling properties of the steady-state structure factor, the universal short-time dynamics of the transient velocity at the sample critical force, and the velocity scaling function describing the joint dependence of the steady-state velocity on the external drive and temperature." ], [ "Introduction", "The understanding of the static and dynamic properties of elastic interfaces in disordered media has direct impact on different fields on condensed matter physics.", "Among a large variety of systems one can mention magnetic [1], [2], [3], [4] or ferroelectric [5], [6] domain walls, contact lines [7], fractures [8], [9], vortex lattices [10], [11], [12], charge density waves [13], and Wigner crystals [14], as paradigmatic examples.", "Since the effect of the disordered media in all these systems is non-trivial, an important question is how these elastic objects respond to an external drive.", "When the temperature is zero, there exists a critical force value $F_c$ such that the steady state velocity of the center of mass of the interface is zero below $F_c$ and is finite above it.", "This is due to the complex interplay between disorder and external force: the interface accommodates within the disorder energy landscape and a finite energy barrier must be overcome by the external force in order to generate a net movement.", "Therefore a finite force value has to be set to have an infinitesimally small finite velocity.", "This is the so called depinning transition.", "If the critical force value is approached from above, the velocity vanishes as $V \\sim (F-F_c)^\\beta $ for a thermodynamic system, with $\\beta $ the depinning exponent.", "Concomitant with the power-law decrease of the velocity is the divergence of a characteristic length as $\\xi \\sim (F-F_c)^{-\\nu }$ , with $\\nu $ the correlation length exponent.", "This depinning correlation length gives the typical size of the correlated displacement (or avalanche) that makes the interface advance in the direction of the external force.", "The finite force threshold, the critical decrease of the velocity order parameter and the divergence of the typical length scale led to propose a description of the depinning transition using tools from standard critical phenomena [15].", "More recently however, the analysis of the low-temperature averaged steady-state geometry has shown that no divergent steady-state correlation length-scale exists approaching the critical force from below, thus breaking the naive analogy with standard phase transitions, where two divergent length-scales are expected above and below the critical point [16], [17].", "When the temperature is finite there is no sharp transition between zero and finite velocity regimes.", "Even at forces much smaller that the critical value the interface is able to move since thermal activation is enough to overcome the effective energy barriers generated by the disorder.", "This regime, $F \\ll F_c$ , is the creep regime, and it is characterized by a stretched exponential dependence of the velocity with the inverse of the external force [18], [19], [20], [21], [22], [23].", "On the other hand, at forces around the critical value, $F \\approx F_c$ , a finite temperature value smears out the transition, which is no longer abrupt.", "This thermal rounding of the depinning transition can be characterized, exactly at the critical force $F=F_c$ , by a power-law vanishing of the velocity with the temperature as $V \\sim T^\\psi $ , with $\\psi $ the thermal rounding exponent [24], [25], [26], [27], [28], [29], [30].", "The values of the different exponents characterizing the depinning transition are universal in the sense that their values depend on few parameters of the system such as the range of the intrinsic elasticity, the dimensionality of the problem, and the correlated structure of the disorder.", "For the experimentally relevant case of $1+1$ dimensional elastic interfaces moving in a random-bond disorder environment with short-range correlations and short-range elasticity, we have recently reported the value $\\psi = 0.15 \\pm 0.01$ using Langevin dynamics numerical simulations [30].", "This value compares well with the value $\\psi = 0.16$ reported in Ref.", "[25] based in numerical simulations.", "However, these values are smaller than the value $\\psi =0.24$ obtained using an artificial extremal activated dynamics [28], which might indeed be in a different universality class.", "The value $\\psi = 0.2$ was obtained using numerical simulations of domain wall motion with the random-field Ising model [26], [27].", "Although it is expected that for $T>0$ and around the depinning transition the characteristic exponents do not depend on the random-bond or random-field character of the disorder, this slightly larger value might be possibly ascribed to the anharmonic corrections to the elasticity present in the random-field Ising model.", "On the other hand, functional renormalization group equations at the depinning [23] allow in principle to extract the thermal rounding exponents.", "However, in practice there are, up to now, no analytical estimates of $\\psi $ , unlike the other critical exponents which have been computed using functional renormalization group up to two loops [31].", "The very existence of a thermal rounding, obeying a power law scaling is not rigorously proven, and there are indeed some models of depinning which exhibit at finite temperature a totally different type of thermal rounding [32].", "It is thus crucial, given the uncertainty on the very type of thermal rounding and certainly on the value of the thermal rounding exponent, to develop new methods to determine $\\psi $ , and to check the robustness, consistency, and expected universality of the phenomenological scaling arguments.", "Experimentally, access to the full force range relevant to the depinning transition has been reported in ultrathin ferromagnetic layers [4], [33], [34].", "In this case, the thermal rounding of the depinning transition is generated through an effective temperature dependence controlled by the relative disorder intensity among different samples.", "Indeed, it has been shown that thermal effects on the velocity-force characteristics can be well described using the value $\\psi =0.15 \\pm 0.10$  [33].", "The aim of the present work is to give further numerical support to the reported value $\\psi =0.15$ , by checking the robustness and consistency of the scaling arguments applied to different observables.", "To this end, we show how this value allows to describe different measures characterizing the critical behavior of the depinning transition: an analysis of the finite temperature structure factor, a short-time dynamics analysis, and the analysis of the scaling function describing the velocity dependence on force and temperature around depinning for different disorder intensities." ], [ "Model system and numerical simulations", "In order to model the dynamics of one-dimensional interfaces in disordered media we use a short-range elastic string, as described in the following.", "The string is defined by a single valued function $u(z,t)$ , giving its transverse position $u$ in the $z$ axis.", "The time evolution of the string is given by the overdamped equation of motion $\\gamma \\, \\partial _t u(z,t) = c \\, \\partial ^2_z u(z,t) + F_p(u,z) + F +\\eta (z,t),$ where $\\gamma $ is the friction coefficient and $c$ the elastic constant.", "The pinning force comes from the derivative of the random-bond pinning potential $U(u,z)$ , i.e.", "$F_p(u,z) = - \\partial _u U(u,z)$ , whose sample to sample fluctuations are given by $\\overline{\\left[ U(u,z)- U(u^{\\prime },z^{\\prime }) \\right]^2} = \\delta (z-z^{\\prime }) \\, R^2(u-u^{\\prime })$ where $R(u)$ stands for a correlator of range $r_f$  [22], and the overline indicates average over disorder realizations.", "Thermal fluctuations are included through the thermal noise term $\\eta (z,t)$ which satisfies $\\langle \\eta (z,t) \\rangle &=&0, \\nonumber \\\\\\langle \\eta (z,t) \\eta (z^{\\prime },t^{\\prime }) \\rangle &=& 2 \\gamma T \\delta (t-t^{\\prime }) \\delta (u-u^{\\prime }),$ where $T$ is the temperature (with Boltzmann constant set to unity, $k_B = 1$ ) and the angular brackets denote a thermal average.", "Finally, the force $F$ on Eq.", "(REF ) corresponds to a uniform and constant external field which drives the string in the $u$ direction.", "The evolution Eq.", "(REF ) is numerically solved.", "The $z$ direction is discretized in $L$ segments of size $\\delta z=1$ , i.e.", "$z \\rightarrow j=0,...,L-1$ , while keeping $u_j(t)$ as a continuous variable.", "This sets the longitudinal finite system size $L$ .", "The equation is integrated using the Euler method with a time step $\\delta t=0.01$ .", "The pinning potential is modeled by performing a cubic spline passing through $M$ regularly spaced uncorrelated Gaussian numbers points [35], [36], which sets the transverse finite system size $M$ .", "Numerical simulations are performed using periodic boundary conditions in both directions and using the parameters $\\gamma =1$ , $c=1$ , and $r_f=1$ .", "The strength of the disorder is given by $R_0=R(0)$ .", "For each disorder realization, i.e.", "for each finite size sample, the critical force $F_c$ can be accurately obtained using an exact algorithm, which also gives the critical pinned configuration of the string $u_c(z)$  [35].", "The results presented in the following sections were obtained by typically averaging over 100 disorder configurations; the error bars being typically of the order of the size of the data points." ], [ "Velocity-force characteristics: scaled variables", "In this section we will present the general features of the velocity-force characteristics, allowing us to define the critical region and the scaled variables that will be used throughout the rest of this work.", "Figure REF (a) shows typical velocity-force curves at finite temperature, as obtained with the present model for $L=1024$ and $M = 5792 \\approx L^{\\zeta _{\\mathrm {dep}}}$ , with $\\zeta _{\\mathrm {dep}}=1.25$  [37], [38] the depinning roughness exponent (see below).", "Given a fixed force $F$ , the velocity is computed in the steady state, which is typically reached within one sweep over the transverse size $M$ (as detailed below, the transverse size $M$ will be varied following a scaling relation with the string length $L$ ).", "Then, of the order of five sweeps over $M$ are used to compute the velocity $V = \\langle \\partial _t u(z,t) \\rangle .$ The thermal average is taken by computing 200 values of the velocity with independent thermal noise realizations within this steady state regime.", "Different curves correspond to the same single disorder realization with intensity $R_0=2$ and increasing temperature.", "The characteristic critical force is indicated in the key.", "The lower curve, corresponding to $T=0$ , clearly presents the typical abrupt depinning transition: the velocity is strictly zero for $F < F_c$ , while it increases as $(F - F_c)^\\beta $ for $F > F_c$ , where $\\beta < 1$ is the velocity exponent.", "As observed, by increasing the temperature the $T=0$ sharp transition is smeared out.", "Although at very small temperatures the curves still present the curvature corresponding to $(F - F_c)^\\beta $ , at higher temperatures there are no clear signatures of the underlying $T=0$ depinning transition.", "Finally, at very high temperatures, when the thermal energy is larger than the typical pinning energy, the velocity tends to increase linearly with the force, $V = mF$ , with the mobility $m=1/\\gamma $ , corresponding to the dashed line in Fig.", "REF (a).", "In Fig.", "REF (b) the disorder intensity effects on the velocity-force characteristic can be observed.", "The critical forces for the corresponding disorder realizations for each intensity are quoted.", "Since around the depinning transition the velocity strongly depends on the sample critical force value, along the present work we will use scaled variables for velocity and force.", "The scaled velocity is given by $v = \\overline{\\left(\\frac{\\langle \\partial _t u(z,t) \\rangle }{m F_c}\\right)} =\\overline{\\left(\\frac{V}{m F_c}\\right)},$ which defines a systematics to average over disorder realizations.", "Besides, we use as the control parameter the scaled force $f = \\frac{F-F_c}{F_c},$ which measures the scaled distance to the critical force for each disorder realization.", "These definitions of scaled variables are different than the scaled variables used in standard critical phenomena.", "In our case, we are using the critical force of each disorder realization in order to measure how close the system is to the critical point, instead of using the disorder averaged value $\\overline{F_c}$ .", "Besides, we also incorporate into the definition of the order parameter $v$ a non-trivial disorder average when using the disorder realization dependent value $F_c$ .", "Although a temperature-dependent critical force can be considered for studying thermal properties at depinning [10], we are using here the zero-temperature value.", "From a practical point of view the temperature-dependent critical force can be defined as the inflexion point of the velocity-force curves.", "In fact, in the temperature range we are studying here this temperature-dependent critical force does not deviate much from the zero-temperature value.", "Instead of using a temperature-dependent critical force, we adhere here to the idea that the important quantity given by the disorder potential is the zero-temperature critical force and that the small temperature data can be interpreted using this quantity.", "Besides, the zero-temperature critical force strictly depends only on the disorder configuration and therefore permits the computation of the average velocity using Eq.", "(REF ) with disorder and temperature averages independently realized.", "The scaled variables, Eqs.", "(REF ) and (REF ), are natural for an overdamped particle driven in a periodic potential $U(x)=R_0 \\cos (x/\\lambda )$ , $\\gamma dx/dt = -dU(x)/dx + F$ , where one can readily obtain that $F_c = R_0/\\lambda $ and $\\gamma V/F_c \\approx \\sqrt{(F-F_c)/F_c}$ close to $F_c$ .", "They also arise in functional renormalization group calculations for center of mass velocity of an elastic manifold, $\\tilde{\\gamma }V/F_c \\sim [(F-F_c)/F_c]^\\beta $ but with $\\tilde{\\gamma }$ an effective friction coefficient [23].", "On the other hand, from a practical point of view, it was shown that the scaled variables defined in Eqs.", "(REF ) and (REF ), applied to each particular sample, serve to diminish sample-to-sample fluctuations when studying the depinning transition of a string [30], [37].", "The inset of Fig.", "REF (b) shows the same data as in the main panel but in scaled form for a single realization.", "The difference between these curves close to the threshold is due to the fact that the full function $V(F,T)$ depends on the disorder intensity not only through the value of the critical force, but one also needs to consider both extra disorder-dependent temperature scale and friction coefficient.", "This interesting issue will not be crucial for our present study however (see discussion in Sec.", "below)." ], [ "Temperature dependence of the velocity at the critical force", "Here we show the finite temperature response of the elastic string exactly at the critical force and discuss some finite size scaling effects, in particular the crossover to single particle dynamics.", "Figure REF presents velocity-temperature curves for different system sizes.", "All the curves were computed at exactly the sample critical force, using the scaled force variable $f=0$ , and then averaged over disorder realizations.", "The disorder intensity is $R_0 = 0.5$ and the results are qualitatively similar to those reported for $R_0 = 1$ in Ref. [30].", "At very high temperatures, $T \\gg R_0$ , the system enters the fast flow regime and the velocity practically equals the force; therefore the reduced velocity (which incorporates the critical force) tends to unity.", "At intermediate temperatures, the velocity reduces and the curves tend to display the critical behavior $v\\sim T^\\psi $ .", "This power-law behavior is however interrupted by finite-size effects at smaller temperatures, when the dynamic characteristic length $\\xi $ equals the system size $L$ .", "At very small temperatures a crossover to single particle dynamics [37], [39] is observed as shown by the $L=32$ curve.", "A simple ad-hoc model to rationalize this crossover has been given by Duemmer and Krauth [37] while numerically studying the zero-temperature depinning transition.", "Within this model, one can write the velocity in the very small temperature regime as $v = \\frac{M}{\\tau _0+\\tau _1(T)},$ where $\\tau _1(T)$ is the temperature-dependent time the interface spends near the critical configuration and $\\tau _0$ is the rest of the time needed to cover the transverse spatial period $M$ of the computational box.", "In this simple model, $\\tau _0$ is approximated to be temperature independent at very low temperatures.", "Using the temperature dependence of the escape rate for a particle in a random potential [40], one can propose that $\\tau _1 = a T^{-1/3}$ .", "In Fig.", "REF we show with a dotted line that the very small temperature regime for $L=32$ is well fitted with Eq.", "(REF ).", "For $L=32$ and $M=304 \\approx 4L^{\\zeta _{\\mathrm {dep}}}$ we fitted the $T<10^{-3}$ regime using Eq.", "(REF ) and we found the fitting parameters $\\tau _0=779.5$ and $a=3.42$ .", "This is a simplified model allowing to rationalize the crossover to one-particle dynamics and should be further tested.", "Figure: (Color online)Velocity-temperature curves for differentsystems sizes LL, as indicated, while keeping M=4L ζ dep M=4\\,L^{\\zeta _{\\mathrm {dep}}},with ζ dep =1.25\\zeta _{\\mathrm {dep}}=1.25 , the depinningroughness exponent.", "The disorder intensity is R 0 =0.5R_0 = 0.5.", "All the data werecomputed at exactly the sample critical force and then averaged over disorderrealizations.", "The dashed line corresponds to the expected power-law behavior.The dotted line describes the crossover to the one-particle regime as discussedin the text.Figure: (Color online) Finite size scaling of thevelocity-temperature curves for different LL values according toEq. ().", "The disorder strength is R 0 =0.5R_0 = 0.5 and the transversesize has been kept at M=4L ζ dep M=4\\,L^{\\zeta _{\\mathrm {dep}}}.", "We show points for T≤0.02T \\le 0.02 and L≥64L \\ge 64, since for L=32L=32 the single particle regime is present atsmall temperatures.", "The dashed line corresponds to the power-law behavior withψ=0.15\\psi = 0.15.", "We also used the values β=0.33\\beta =0.33  andν=1.33\\nu =1.33 .The finite size effects displayed by the velocity-temperature curves in Fig.", "REF are not easily accounted for by standard finite size scaling arguments.", "In fact, assuming finite-size scaling as in standard critical phenomena, the velocity should be described by universal functions as in $v = L^{-\\beta /\\nu } g\\left(L^{\\beta /\\nu \\psi }\\, T \\right),$ with $g(x) \\sim 1$ for $x \\ll 1$ and $g(x) \\sim x^\\psi $ for $x \\gg 1$ .", "As mentioned in Ref.", "[30] strong corrections-to-scaling effects are present in these results.", "In order to show that, Fig.", "REF presents an attempt to use the standard finite size scaling correction scaling, Eq.", "(REF ), with the bare data in Fig.", "REF .", "One can observe strong finite size corrections and this can also be observed with other values of $R_0$ .", "In addition, the collapse of the data does not improve significantly when using other values of the scaling exponents.", "Despite these strong finite-size effects exhibited by the velocity at critical force, the power-law regime characterized by the thermal rounding exponent $\\psi =0.15$ does not suffer from strong finite size effects, as shown in the following sections." ], [ "Structure factor analysis", "In this section we turn to the complementary geometrical analysis of the structure factor, which contains information on the geometry of the string at different length scales.", "The results presented in this section complements those reported in Refs.", "[30], [42] by including different disorder strengths.", "From the numerical simulations, the steady state structure factor is defined as $S_q = \\frac{1}{L}\\overline{\\left\\langle \\left| \\sum _{j=0}^{L-1} u_j\\, e^{iqj}\\right|^2 \\right\\rangle },$ where $q=2 \\pi n/L$ , with $n = 1,...,L-1$ .", "One can show using dimensional analysis that when the width $w$ of a self-affine interface of size $L$ is described through a roughness exponent $\\zeta $ , i.e.", "$w \\sim L^\\zeta $ , then the structure factor behaves as $S_q \\sim q^{-(1+2\\zeta )}$ in $1+1$ dimensions.", "Figure: (Color online) (Color online) Structure factor and itsscaling function for R 0 =0.5R_0=0.5.", "The curves in (a) correspond to the criticalforce and different temperatures.", "The size of the sample is given by L=1024L=1024and M=4L ζ dep M=4L^{\\zeta _{\\mathrm {dep}}}.", "(b) Scaled curves showing the crossoverbetween the depinning regime at small length scales (x=qT -ψν/β ≫1x=qT^{-\\psi \\nu /\\beta } \\gg 1) and the thermal regime atlarge length scales (x=qT -ψν/β ≪1x=qT^{-\\psi \\nu /\\beta } \\ll 1).", "Together with ψ=0.15\\psi =0.15, the valuesβ=0.33\\beta =0.33 , ν=1.33\\nu =1.33  andζ dep =1.25\\zeta _{\\mathrm {dep}}=1.25 , were also used.Figure: Structure factor and itsscaling function for R 0 =5R_0=5.", "The curves in (a) correspond to the critical forceand different temperatures.", "The size of the sample is given by L=1024L=1024 andM=L ζ dep M=L^{\\zeta _{\\mathrm {dep}}}.", "(b) Scaled curves showing the crossover betweenthe depinning regime at small length scales (x=qT -ψν/β ≫1x=qT^{-\\psi \\nu /\\beta } \\gg 1) and the thermal regime at largelength scales (x=qT -ψν/β ≪1x=qT^{-\\psi \\nu /\\beta } \\ll 1).", "The very large length scale random-periodic fast-flow regime hasbeen discarded (see the text).", "Together with ψ=0.15\\psi =0.15, the valuesβ=0.33\\beta =0.33 , ν=1.33\\nu =1.33  andζ dep =1.25\\zeta _{\\mathrm {dep}}=1.25 , were also used.At small length scales, $q\\gg 1/\\xi $ , the structure factor shows the typical roughness regime associated to depinning, i.e.", "$S_q \\sim q^{-(1+2\\zeta _{\\mathrm {dep}})}$ , while at large length scales, $q \\ll 1/\\xi $ , fluctuations are dictated by effective thermal fluctuations induced by the disorder, i.e.", "$S_q \\sim q^{-(1+2 \\zeta _{\\mathrm {th}})}$ .", "The thermal and depinning roughness exponents are, respectively, $\\zeta _{\\mathrm {th}} = 1/2$ and $\\zeta _{\\mathrm {dep}} = 1.25$  [38], [37].", "In the critical region the depinning correlation length is given by the velocity as $\\xi \\sim v^{-\\nu /\\beta }$ .", "Thus, the depinning correlation length depends on the temperature only through the velocity and in the thermal rounding regime $\\xi \\sim T^{-\\psi \\nu /\\beta }$  [30].", "With this information one can write for the structure factor that $S_q = T^{-\\psi \\nu (1+2 \\zeta _{\\mathrm {dep}})/\\beta } s\\left( qT^{-\\psi \\nu /\\beta } \\right),$ where the scaling function $s(x) \\sim x^{-(1+2\\zeta _{\\mathrm {th}})}$ for $x \\ll 1$ and $s(x) \\sim x^{-(1+2\\zeta _{\\mathrm {dep}})}$ for $x \\gg 1$ .", "In Ref.", "[30] we showed that the structure factor scales with the previous form using $L=1024$ and $M=L^{\\zeta _{\\mathrm {dep}}}$ for the disorder intensity $R_0 =1$ .", "Here, we show in Fig.", "REF (a) the temperature dependence of the structure factor corresponding to $R_0 = 0.5$ , $L=1024$ and $M=4L^{\\zeta _{\\mathrm {dep}}}$ .", "For these parameters the presented data do not show transverse finite size effects [42].", "Figure REF (b) shows the scaling of the structure factor according to Eq.", "(REF ) and using $\\psi = 0.15$ , which shows a very satisfactory data collapse.", "In order to reach the steady-state for the same temperatures of Fig.", "REF (a) but larger disorder intensities it is necessary to equilibrate the system for longer times.", "Since this equilibration time scales with the transverse system size we can reduce the simulation time by using $M=L^{\\zeta _{\\mathrm {dep}}}$ for $R_0=5$ .", "The resulting data, shown in Fig.", "REF (a), presents the small length scale depinning regime and the large scale effective thermal regime described above, but also clearly show a larger length scale regime where finite transverse size effects are present.", "In this regime the roughness exponent is the one corresponding to a random-periodic system in the fast-flow regime, $\\zeta _{\\mathrm {per}} =3/2$  [42].", "Hence, we can detect and discard the data corresponding to this random-periodic regime in order to get a curve that can be scaled again using Eq.", "(REF ) and the known random-manifold exponents, as shown in Fig.", "REF (b), getting again a very satisfactory collapse.", "Therefore, we have presented here data of the structure factor for different disorder intensities which shows that the quoted thermal rounding exponent is disorder independent.", "Furthermore, we have shown how the thermal rounding exponent gives the temperature dependence of the depinning correlation length, $\\xi \\sim T^{-\\psi \\nu /\\beta }$ , from a steady-state geometry analysis." ], [ "Short-time dynamics analysis", "One possible way to get rid of finite size effects is to analyze the short time dynamics.", "Starting from a given non-steady initial condition at fixed force $F$ and temperature $T$ , the velocity begins to evolve with time until it reaches the steady-state value corresponding to the values of $F$ and $T$ .", "This transient dynamics is controlled, at short times, by a single growing correlation length, $\\xi (t)$ , which at longer times saturates to the steady-state correlation length above threshold, $\\xi \\sim v^{-\\nu /\\beta }$ .", "Since the transient correlation length grows as $\\xi (t) \\sim t^{1/z}$ , with $z \\approx 3/2$  [41] the depinning dynamical exponent, scaling arguments show that the velocity decreases with time as $v(t) \\sim \\xi (t)^{-\\beta /\\nu } \\sim t^{-\\beta /z\\nu }$  [41] before saturating to the steady-state value above threshold, given by $v(t \\rightarrow \\infty ) \\sim f^\\beta $ at $T=0$ or by $v(t \\rightarrow \\infty ) \\sim T^\\psi $ at $f=0$ .", "Figure REF (a) shows the time evolution of the velocity exactly at the critical force and for different temperature values as indicated.", "The dashed line corresponds to the expected short-time critical behavior $v(t) \\sim t^{-\\beta /z\\nu }$ .", "Discarding the very short time regime, $t \\le 20$ , which contains information about the microscopic non-universal dynamics [41], the curves in Fig.", "REF (a) can be recast into a universal form using the scaling function $v(t) = T^{\\psi } h\\left( t T^{\\psi z \\nu /\\beta } \\right),$ with $h(x) \\sim x^{-\\beta / z \\nu }$ for $x \\ll 1$ and $h(x) \\sim 1$ for $x \\gg 1$ .", "The data collapse shown in Fig.", "REF (b) uses the previously known depinning exponents $\\beta =0.33$  [37], $\\nu =1.33$  [41], $z = 3/2$  [41], together with the thermal rounding exponent $\\psi = 0.15$ .", "Since the data collapse is good with no need of adjustable parameters we can conclude that the value of the thermal rounding exponent obtained in Ref.", "[30] is consistent and does not suffer from strong finite-size effects." ], [ "Velocity scaling function around depinning", "In this section we turn to the analysis of the universal behavior of the force and temperature dependent velocity function, $v(f,T)$ .", "Focusing on testing the robustness of the thermal rounding exponent $\\psi $ , parameter values around the critical point given by $f=0$ and $T=0$ are tested.", "If there were not strong finite size effects, in the vicinity of the critical region the velocity should scale as $v T^{-\\psi } \\sim H\\left( f T^{-\\psi /\\beta } \\right),$ with $H(x) \\sim x^\\beta $ for $x\\gg 1$ .", "Figure REF (a) shows velocity-force curves for different temperatures and for $R_0=1$ .", "The numerical data is split into two sets: on the one hand data points correspond to given parameters which are “inside” the thermal rounding region, and on the other hand continuous lines represent data “outside” the thermal rounding region.", "The data are outside the critical region either because temperature is too high, $T>0.02$ in the present case, or because the force is far away from the critical value, $|f| \\gg 1$ ($f \\gg 1$ and $f \\ll -1$ corresponding to the fast-flow and creep regimes, respectively).", "In addition, to avoid finite-size effects, data points are also considered “outside” the critical thermal rounding region if they correspond to velocities smaller than the crossover at $v \\sim v_{min}$ to single-particle behavior for each size $L$ .", "Since in the critical region $\\xi \\sim v(f,T)^{-\\nu /\\beta }$ , as shown from the structure factor analysis, we roughly have $v_{min} \\sim L^{-\\beta /\\nu }$ .", "According to such criteria, the selected data is finally presented in the scaled form Eq.", "(REF ) in Fig.", "REF (b) for $f>0$ .", "The dashed line indicates the expected asymptotic $x^\\beta $ form, corresponding to the scaling function $H$ around the critical region.", "The collapse into a single curve for different $T$ and $f$ confirms numerically that the data set used is inside the critical scaling region.", "The scaling function $H$ is not yet universal as it also depends on the disorder intensity.", "Figure REF shows the critical region and the form of $H$ for $R_0=1$ .", "In Fig.", "REF we show the velocity scaling function $H$ for different disorder intensities, $R_0=1,2$ and 5, for the full force range within the thermal rounding region.", "In Fig.", "REF the same data is presented in a double logarithmic scale.", "As can be observed all curves display the asymptotic power-law form $H \\sim x^\\beta $ for $f \\gg T^{\\psi /\\beta }$ , but with different prefactors for each disorder intensity.", "At this point, in order to properly include the disorder intensity on the scaling velocity function and obtain the universal function, a disorder-dependent temperature scale $T_c$ is needed.", "Again, for the simple example of the depinning of a particle in a periodic potential $U(x)=R_0 \\cos (x/\\lambda )$ , $\\gamma dx/dt = -dU(x)/dx + F + \\eta (t)$ , with $\\eta (t)$ a Langevin noise at temperature $T$ , it is easy to see, from pure dimensional analysis, that $T_c \\sim R_0$ , and therefore $\\gamma V/F_c = h[(F-F_c)/F_c, T/R_0]$ , where $F_c = R_0/\\lambda $ .", "Such naive approach does not work for the elastic string, as the characteristic energy scale is not simply $R_0$ as for the particle, but it arises from the interplay of disorder and elasticity.", "Although it is not obvious that it should work at depinning, one is tempted to use the scaled temperature $\\tau =T/T_c$ , where $T_c=U_c/k_B$ gives the characteristic energy scale in the creep regime at small forces [18], [19].", "This energy scale is however not universal and depends on microscopic details of the disorder  [18], [19], [43].", "The assessment of the full dependence of $U_c$ on microscopic parameters is thus not straightforward and from a pragmatic point of view one could directly fit it from the creep law.", "As shown with numerical simulations within the creep regime, at larger temperatures than the one used here, $U_c$ can be fitted from the creep law, but it dependence on $R_0$ is not trivial [36].", "We do not have access to $T_c$ with the present numerical results, which focused in the force region around the critical depinning threshold.", "Therefore, we can not incorporate at this stage the influence of the disorder intensity in the velocity function.", "In spite of this, the data displayed in Fig.", "REF clearly show that the velocity can be represented in a scaled form, with identical critical exponents, for different disorder intensities.", "More importantly, these data supports the disorder independent value $\\psi =0.15$ tested here.", "Figure: (Color online) Same data as in Fig.", "butin double-logarithmic representation.", "The upper and lower curves correspond tof>0f>0 and f<0f<0, respectively.", "We also show fitting curves usinga 1 +a 2 x β a_1+a_2x^\\beta and b 1 exp[-b 2 |x| β/ψ ]b_1\\exp [-b_2|x|^{\\beta /\\psi }], as suggested by thescaling functions Eqs.", "() and ().Finally, it is worth relating our results with the universal scaling function proposed by Nattermann, Pokrovsky and Vinokur [43] using a phenomenological interpolating form for the full force and temperature dependence of the velocity of a domain wall in a random medium.", "This form includes the thermal rounding regime around $F_c$ and the $F \\ll F_c$ creep regime, thus depending also on the universal creep exponent $\\mu =1/4$ (for a one dimensional elastic interface).", "The proposed functional form in Ref.", "[43] is different below and above the critical force and can be written as $V = m F \\frac{\\exp \\left[ -\\frac{T_c}{T} \\left( 1-\\frac{F}{F_c}\\right)^{\\beta /\\psi } \\left( \\frac{F_c}{F} \\right)^\\mu \\right]}{1+\\left[\\frac{T_c}{T} \\left( \\frac{F_c}{F} \\right)^\\mu \\right]^\\psi },$ for $F<F_c$ and $V = \\frac{m F}{1+\\left[ \\frac{T_c}{T} \\left( \\frac{F_c}{F} \\right)^\\mu \\right]^\\psi } + m F \\left( 1-\\frac{F_c}{F} \\right)^\\beta ,$ for $F>F_c$ .", "It can be shown that close to the depinning region, i.e.", "above the creep regime, where $f = (F-F_c)/F_c \\ll 1$ and $\\tau = T/T_c \\ll 1$ this phenomenological form can be reduced to the scaling form Eq.", "(REF ), with $H(x) = H^-(x)$ for $f<0$ and $H(x) = H^+(x)$ for $f>0$ .", "The corresponding limit functions are $H^-\\left( f \\tau ^{-\\psi /\\beta } \\right) = e^{-(-f\\tau ^{-\\psi /\\beta })^{\\beta /\\psi }}, \\\\H^+\\left( f \\tau ^{-\\psi /\\beta } \\right) = 1+(f \\tau ^{-\\psi /\\beta })^\\beta .$ Since we do not have the temperature scale $T_c$ from the creep law, we have directly fitted the data for the velocity scaling function to the universal forms suggested by Eqs.", "(REF ) and ().", "The $f<0$ and $f>0$ ranges have been fitted separately using $a_1+a_2x^\\beta $ and $b_1\\exp [-b_2|x|^{\\beta /\\psi }]$ , respectively, obtaining four fitting parameters for each disorder intensity.", "The results are shown in Fig.", "REF .", "In all cases the fit is better in the $f<0$ region.", "Furthermore, one can observe that the obtained curves interpolate badly around $f=0$ .", "In fact, enforcing $a_1=b_1$ makes the fitting considerably worse.", "We therefore conclude that the data can not be satisfactorily fitted using this phenomenological form, particularly above threshold, hence evidently pointing to the need of a more accurate description of the thermal rounding of the depinning transition.", "The phenomenological functional forms, Eqs.", "(REF ) and  (REF ), give a potentially important tool which allows to directly fit experimental data.", "This was directly used in Ref.", "[33], where the velocity-force characteristic below threshold for ultrathin ferromagnetic layers was fitted using Eq.", "(REF ).", "By fitting just one experimental curve below threshold, the value $\\psi = 0.15 \\pm 0.10$ was obtained for the thermal rounding exponent.", "Since several fitting parameters were used and due to the large error bar, this value can only be compared with our numerical value with extreme caution.", "Anyway, the experimental value is consistent with our numerical simulations." ], [ "Summary", "We have presented extensive numerical simulations to test the validity of the thermal rounding exponent of the depinning transition.", "We analyzed the direct scaling of the steady-state velocity-force characteristics, the steady-state structure factor and the short-time transient dynamics.", "The existence of a critical (power law) thermal rounding of the depinning transition is consistent with all our results, together with the existence of a unique divergent length scale, dependent on temperature and/or distance to the critical pinning force, but ultimately controlled by the velocity as in the zero temperature depinning transition.", "The results are all consistent with a value of the thermal rounding exponent of $\\psi = 0.15$ in agreement with our previously reported value [30].", "This exponent describes the power-law vanishing of the velocity with temperature exactly at the critical depinning force, $V \\sim T^\\psi $ , for the universality class of one dimensional elastic interfaces with short-range elasticity and short-range correlations in the disorder.", "Although the value of the thermal rounding exponent have been previously obtained with larger system sizes, where finite size corrections are still observable, we have shown here that this value is also consistent with short-time dynamics results which do not suffer from severe finite size effects.", "Besides, $\\psi = 0.15$ also describes the scaling properties of the structure factor for various disorder strength values, connecting this value with a geometrical roughness crossover in the interface.", "Finally, we have shown that it is consistent with a scaling function describing the velocity-force characteristics as a function of temperature and force.", "Experimental confirmation of our results, directly targeting the thermal rounding regime and allowing to test the value of the thermal rounding exponent, would be welcome.", "This work was supported in part by the Swiss National Science Foundation under MaNEP and Division II.", "SB and ABK acknowledge financial support from ANPCyT Grant No.", "PICT2007886 and CONICET Grant No.", "PIP11220090100051.", "ABK acknowledges Universidad de Barcelona, Ministerio de Ciencia e Innovación (Spain) and Generalitat de Catalunya for partial support through I3 program." ] ]

[ [ "Formes modulaires modulo 2 : structure de l'alg\\`ebre de Hecke" ], [ "Abstract Modular forms mod 2 : structure of the Hecke ring We show that the Hecke algebra for modular forms mod 2 of level 1 is isomorphic to the power series ring F2[[x, y]], where x = T3 and y = T5." ], [ "Notations", "Nous conservons les notations de la Note précédente [2].", "En particulier, nous notons $\\Delta $ l'élément de ${\\bf F}_2[[q]]$ défini par :    $ \\Delta = \\sum _{n=1}^\\infty \\tau (n)q^n = \\sum _{m=1}^\\infty q^{(2m+1)^2},$ et $\\mathcal {F}$ désigne le sous-espace vectoriel de ${\\bf F}_2[[q]]$ engendré par les puissances impaires de $\\Delta $  :    $\\mathcal {F}= \\ < \\!", "\\!", "\\Delta , \\Delta ^3, \\Delta ^5, ... \\!", "\\!>.$ L'espace $\\mathcal {F}$ est stable par les opérateurs de Hecke $T_p$ , $p$ premier $\\ne 2.$" ], [ "Les espaces $\\mathcal {F}(n)$ et les algèbres {{formula:47f1f087-19da-422f-98b4-b822d6fd12d7}}", "Soit $n$ un entier $> 0$ .", "Soit $\\mathcal {F}(n)$ le sous-espace de $\\mathcal {F}$ de base $\\lbrace \\Delta , \\Delta ^3, ..., \\Delta ^{2n-1}\\rbrace $ .", "On a $\\mathop {\\rm dim}\\mathcal {F}(n) = n$ .", "Soit $A(n)$ la ${\\bf F}_2$ -sous-algèbre de $\\mathop {\\rm End}(\\mathcal {F}(n))$ engendrée par ${\\bf F}_2$ et les $T_p$ .", "On a $A(n) = {\\bf F}_2 \\oplus {\\mathfrak {m}}(n)$ , où ${\\mathfrak {m}}(n)$ est l'unique idéal maximal de $A(n)$ (à savoir le sous-espace vectoriel de $A(n)$ engendré par les $T_p$ et leurs produits) ; cet idéal est nilpotent.", "Soit $\\mathcal {F}(n)^*$ le dual de l'espace vectoriel $\\mathcal {F}(n)$ , muni de sa structure naturelle de $A(n)$ -module, et soit $e_n$ l'élément de $\\mathcal {F}(n)^*$ défini par :    $< \\!", "e_n,\\Delta \\!> \\ = 1$    et    $ <\\!", "\\!e_n,\\Delta ^{2i+1} \\!", "\\!", "> \\ = 0$ si $1 \\leqslant i < n$ .", "Si $f = \\sum a_m(f) q^m$ est un élément de $\\mathcal {F}(n)$ , on a :    $<\\!", "\\!e_n,f\\!", "\\!> \\ = \\ a_1(f)$    et    $ < \\!", "T_pe_n,f\\!", "\\!> \\ = \\ a_p(f)$ pour tout $p$ .", "On en déduit par récurrence sur $r$ la formule : $< \\!T_{p_1}...T_{p_r}e_n,f \\!> \\ = \\ a_{p_1...p_r}(f),$ où les $p_i$ sont des nombres premiers $\\ne 2$ .", "Lemme 2.1 - Soit $f \\in \\mathcal {F}(n), f \\ne 0$ .", "Il existe $u \\in A(n)$ tel que $<\\!", "\\!e_n,uf\\!", "\\!> \\ = \\ 1$ .", "Démonstration.", "Ecrivons $f$ sous la forme $f = q^m + \\sum _{i>m}a_iq^i$ et soit $m = p_1...p_r$ une décomposition de $m$ en produit de nombres premiers.", "Comme $m$ est impair, il en est de même des $p_i$ .", "Soit $u = T_{p_1}...T_{p_r}.$ La formule (REF ) montre que $<\\!", "\\!ue_n,f\\!", "\\!> = 1$ .", "Comme $<\\!", "\\!ue_n,f\\!", "\\!> = <\\!", "\\!e_n,uf\\!", "\\!>$ , cela démontre le lemme." ], [ "Quelques propriétés de $\\mathcal {F}(n)$ et de {{formula:a47cab33-09be-4528-8ce6-e1da8d6dd0c0}}", "Proposition 3.1 - Le $A(n)$ -module $\\mathcal {F}(n)^*$ est libre de base $e_n$ .", "Démonstration.", "Soit $E$ le sous-$A(n)$ -module de $\\mathcal {F}(n)^*$ engendré par $e_n$ .", "Si $E$ était distinct de $\\mathcal {F}(n)^*$ , il existerait $f \\in \\mathcal {F}(n), f \\ne 0$ , tel que $<\\!", "\\!ue_n,f\\!", "\\!> = 0$ pour tout $u \\in A(n)$ , ce qui contredirait le lemme 1.", "On a donc $E = \\mathcal {F}(n)^*$ , ce qui montre que $\\mathcal {F}(n)^*$ est engendré par $e_n$ .", "D'où la proposition.", "Remarque.", "Si $n > 2$ , le $A(n)$ -module $\\mathcal {F}(n)$ n'est pas un module libre.", "Corollaire 3.2 - L'application $A(n) \\rightarrow \\mathcal {F}(n)^*$ donnée par $u \\mapsto ue_n$ est bijective.", "Ce n'est qu'une reformulation de la proposition.", "Noter que, par dualité, on obtient ainsi une bijection de $\\mathcal {F}(n)$ sur le dual $A(n)^*$ de l'espace vectoriel $A(n)$ .", "Corollaire 3.3 - On a $\\mathop {\\rm dim}A(n) = n$ .", "Cela résulte du Corollaire précédent et du fait que $\\mathop {\\rm dim}\\mathcal {F}(n) = n$ .", "Corollaire 3.4 - Le commutant de $A(n)$ dans $\\mathop {\\rm End}(\\mathcal {F}(n))$ est égal à $A(n)$ .", "Par dualité, cela revient à dire que le commutant de $A(n)$ dans $\\mathop {\\rm End}(\\mathcal {F}(n)^*)$ est égal à $A(n)$ , ce qui résulte de la proposition.", "Proposition 3.5 - L'algèbre $A(n)$ est engendrée par $T_3$ et $T_5$ .", "Démonstration.", "Soit $A^{\\prime }$ la sous-algèbre de $A(n)$ engendrée par $T_3$ et $T_5$ .", "C'est une algèbre locale ; soit ${\\mathfrak {m}}^{\\prime }$ son idéal maximal.", "Supposons que $A^{\\prime } \\ne A(n)$ , i.e.", "$\\mathop {\\rm dim}A^{\\prime } < n$ .", "Le $A^{\\prime }$ -module $\\mathcal {F}(n)^*$ n'est pas monogène : sinon, sa dimension serait $<n$ .", "D'après le lemme de Nakayama, cela signifie que le quotient $V = \\mathcal {F}(n)^*/{\\mathfrak {m}}^{\\prime }\\mathcal {F}(n)^*$ est de dimension $> 1$ .", "Par dualité, cela équivaut à dire que le sous-espace de $\\mathcal {F}(n)$ annulé par ${\\mathfrak {m}}^{\\prime }$ est de dimension $>1$ .", "Il existe donc $f \\in \\mathcal {F}(n)$ , avec $f \\ne 0, \\Delta $ , tel que $T_3f = T_5f = 0$ , ce qui contredit le cor.5.3 au th.5.1 de [2]." ], [ "Passage à la limite : l'algèbre $A$", "On a $\\mathcal {F}(n) \\subset \\mathcal {F}(n+1)$ et la restriction à $\\mathcal {F}(n)$ d'un élément de $A(n+1)$ appartient à $A(n)$ .", "On obtient ainsi un homomorphisme surjectif $A(n+1) \\rightarrow A(n)$ .", "D'où un système projectif $ ... \\rightarrow A(n+1) \\rightarrow A(n) \\rightarrow \\ \\ ... \\ \\ \\rightarrow A(2) \\rightarrow A(1) ={\\bf F}_2.$ Nous noterons $A$ la limite projective de ce système.", "C'est un anneau local commutatif ; il est compact pour la topologie limite projective.", "Son idéal maximal ${\\mathfrak {m}}$ est la limite projective des ${\\mathfrak {m}}(n)$ .", "L'anneau $A$ opère de façon naturelle sur $\\mathcal {F}$ .", "Soient $x$ et $y$ deux indéterminées.", "Pour chaque $n$ , il existe un unique homomorphisme $\\psi _n:{\\bf F}_2[x,y] \\rightarrow A(n)$ tel que $\\psi _n(x) = T_3$ et $\\psi _n(y) = T_5$ .", "Par passage à la limite, on en déduit un homomorphisme $\\psi : {\\bf F}_2[[x,y]] \\ \\rightarrow \\ A$ tel que $\\psi (x) = T_3$ et $\\psi (y) = T_5$ .", "Théorème 4.1 - L'homomorphisme $\\psi $ défini ci-dessus est un isomorphisme.", "Démonstration.", "La surjectivité de $\\psi $ résulte de la prop.", "REF .", "Pour prouver l'injectivité, il suffit de montrer que, pour tout élément $u = \\sum \\lambda _{ij}x^iy^j$ non nul de ${\\bf F}_2[[x,y]]$ , il existe $f\\in \\mathcal {F}$ tel que : $\\sum \\lambda _{ij}T_3^i T_5^j f = \\Delta .$ [Noter que la somme est une somme finie, car $T_3^i T_5^j f = 0$ quand $i+j$ est assez grand (par exemple $i+j > \\deg f$ ).]", "Si $\\lambda _{00} = 1$ on prend $f=\\Delta $ .", "Supposons donc $\\lambda _{00} = 0$ .", "Soit $\\Sigma $ l'ensemble des couples $(i,j)$ avec $\\lambda _{ij} =1$  ; considérons ceux pour lesquels l'entier $i+j$ est minimal, et parmi ceux-là, soit $(a,b)$ le couple où $a$ est maximum.", "Soit $k$ l'entier impair de code $[a,b]$ , au sens de [2] et soit $f = \\Delta ^k$ .", "On montre, en utilisant les Propositions 4.3 et 4.4 de [2], que l'on a $T_3^aT_5^bf = \\Delta $ et $T_3^iT_5^jf = 0$ pour tout $(i,j) \\in \\Sigma $ distinct de $(a,b)$ .", "D'où (REF ).", "Corollaire 4.2 - L'algèbre $A$ est un anneau local régulier de dimension 2.", "En particulier, c'est un anneau intègre.", "A partir de maintenant, nous identifierons les algèbres $A$ et ${\\bf F}_2[[x,y]]$ au moyen de $\\psi $  ; cela nous permettra d'écrire $x$ et $y$ à la place de $T_3$ et $T_5$ ." ], [ "Structure des $A$ -modules {{formula:9db3c139-5088-4aaf-b5e6-5c6921efcc9d}} et {{formula:1a9fd0c9-f1bd-46d0-8393-d199ea95b3ad}}", "L'algèbre $A$ opère sur $\\mathcal {F}$ .", "Par dualité, elle opère aussi sur le dual $\\mathcal {F}^*$ de $\\mathcal {F}$ , qui est la limite projective des $\\mathcal {F}(n)^*$ .", "Soit $e \\in \\mathcal {F}^*$ la forme linéaire sur $\\mathcal {F}$ définie par : $< \\!", "e,f \\!> = a_1(f)$ pour tout $f\\in \\mathcal {F}$ , où $a_1(f)$ désigne le coefficient de $q$ dans $f$ .", "Théorème 5.1 - a) Le $A$ -module $\\mathcal {F}^*$ est libre de base $e$ .", "b) Le $A$ -module $\\mathcal {F}$ est isomorphe à l'espace $A^*_{\\rm cont}$ des formes linéaires continues sur $A$ .", "[Une forme linéaire sur $A$ est continue si et seulement si elle s'annule sur une puissance de l'idéal maximal de $A$ .]", "Démonstration.", "L'assertion a) résulte de la prop.REF par dualité ; il en est de même de b) car $A^*_{\\rm cont} = \\cup _{n \\geqslant 1} A(n)^*$ .", "Corollaire 5.2 - Le $A$ -module $\\mathcal {F}$ est divisible$:$ pour tout $u\\in A,u \\ne 0$ , la multiplication par $u$ est un endomorphisme surjectif de $\\mathcal {F}$ .", "En particulier, les endomorphismes $T_p: \\mathcal {F}\\rightarrow \\mathcal {F}$ sont surjectifs.", "Démonstration.", "Par dualité, cela revient à dire que $u : \\mathcal {F}^* \\rightarrow \\mathcal {F}^*$ est injectif, ce qui est clair puisque $A$ est un anneau intègre.", "Remarque.", "D'après [3], $\\mathcal {F}$ est un $A$ -module injectif, à savoir l'enveloppe injective du corps résiduel ${\\bf F}_2$ de $A$ .", "C'est là une propriété plus forte que la propriété de divisibilité." ], [ "Une base de $\\mathcal {F}$ adaptée à {{formula:ee7f5d5d-135f-419c-a16a-5efb537a0bec}} et {{formula:61bd55d0-7e46-448b-9735-12d23776fc15}}", "Théorème 6.1 - Il existe une base $m(a,b)_{a,b \\geqslant 0}$ de $\\mathcal {F}$ et une seule qui a les quatre propriétés suivantes$~ :$ i) $m(0,0) = \\Delta .$ ii) $ < \\!", "e,m(a,b) \\!", "> \\ = \\ 0 $ si $a+b > 0$ .", "iii) $ T_3|m(a,b) = \\left\\lbrace \\begin{array}{ll}\\!", "\\!", "m(a \\!", "-1,b) \\ \\ \\ {\\it si} \\ a > 0 \\\\\\end{array}\\!", "\\!", "0 \\quad \\quad \\quad \\quad \\quad {\\it si} \\ a = 0.\\right.$ .", "$$ iv) $ T_5|m(a,b) = \\left\\lbrace \\begin{array}{ll}\\!", "\\!", "m(a,b \\!", "- \\!", "1) \\ \\ \\ {\\it si} \\ b > 0 \\\\\\end{array}\\!", "\\!", "0 \\quad \\quad \\quad \\quad \\quad {\\it si} \\ b = 0.\\right.$ .", "$$ Démonstration.", "D'après le th.REF , il suffit de prouver le même énoncé pour le $A$ -module $A^*_{\\rm cont}$ , et dans ce cas on définit $m(a,b)$ comme la forme linéaaire sur $A$ donnée par : $ \\quad \\sum n_{ij}x^iy^j \\ \\mapsto \\ n_{ab}.$ Les propriétés i) à iv) sont évidentes.", "L'unicité se démontre par récurrence sur $a+b$ .", "Exemples (cf.", "[5]) : $ m(0,0)= \\Delta ; \\ m(1,0)= \\Delta ^3; \\ m(0,1)=\\Delta ^5;$ $m(2,0)= \\Delta ^9; \\ m(1,1)=\\Delta ^7; \\ m(0,2)=\\Delta ^{17}; $ $m(3,0)=\\Delta ^{11}; \\ m(2,1)=\\Delta ^{13}; \\ m(1,2)= \\Delta ^{11}+\\Delta ^{19}; \\ m(0,3)= \\Delta ^{13}+\\Delta ^{21};$ $m(2^r,0)=\\Delta ^{1+2^{2r+1}},\\ m(2^r \\!", "- \\!", "1,0)= \\Delta ^{(1+2^{2r+1})/3}$ et $m(0,2^r)= \\Delta ^{1+2^{2r+2}}$ .", "Remarques.", "1) L'exposant dominant de $m(a,b)$ au sens de [2] est l'entier impair de code $(a,b)$  ; cela se déduit des résultats énoncés dans [2].", "En particulier, l'ordre de nilpotence de $m(a,b)$ est égal à $a+b+1$ .", "2) D'après Macaulay ([1], voir aussi [3]) il est commode de noter les $m(a,b)$ comme des monômes $x^{-a}y^{-b}$ , avec la convention que $x^{-a}y^{-b}=0$ si $a$ ou $b$ est $<0$ .", "Les formules du th.REF s'écrivent alors simplement $ x.x^{-a}y^{-b} = x^{1-a}y^{-b}$ et $y.x^{-a}y^{-b} = x^{-a}y^{1-b}$ ." ], [ "Développement des $T_p$ comme séries en {{formula:7111c357-c6ec-437b-bb13-90f226992e89}} et {{formula:8c014d30-71ce-4c6c-9e9d-2564e92851bf}}", "D'après le th.REF , tout $T_p$ peut s'écrire comme une série formelle en $x=T_3$ et $y=T_5$  : $T_p = \\sum _{i+j\\geqslant 1} a_{ij}(p)x^iy^j, {\\rm \\; \\;avec\\;\\;} \\ a_{ij}(p) \\in {\\bf F}_2.$ De façon plus précise, on a : $T_p \\in {\\bf F}_2[[x^2,y^2]] \\ \\ \\ \\ \\;\\;{\\rm si\\;} p\\equiv 1\\pmod {8},$ $T_p \\in x.", "{\\bf F}_2[[x^2,y^2]] \\ \\ \\;\\; {\\rm si\\;} p\\equiv 3\\pmod {8},$ $T_p \\in y.", "{\\bf F}_2[[x^2,y^2]] \\ \\ \\;\\;{\\rm si\\;} p\\equiv 5\\pmod {8},$ $T_p \\in xy.", "{\\bf F}_2[[x^2,y^2]] \\ \\;\\;{\\rm si\\;} p\\equiv 7\\pmod {8}.$ Exemples (cf.", "[5]) : $T_{17}=x^2+y^2+x^2y^2+x^6+x^4y^2+y^6+x^6y^2+x^4y^4+x^2y^6+x^{10}+x^{10}y^2+x^6y^6+x^4y^8+x^2y^{10}+\\ldots $ $T_{11}= x(1+x^2+y^2+x^4+x^2y^2+y^4+x^2y^4+y^6+x^6y^2+x^8y^2+x^6y^4+x^2y^8+y^{10}+x^{10}y^2+\\ldots )$ $T_{13}= y(1+x^2+y^2+x^4+y^4+x^6+x^4y^2+x^2y^4+x^6y^2+x^2y^6+y^8+x^{10}+x^8y^2+x^6y^4+y^{10}+\\ldots )$ $T_7 = xy(1 + x^2 + x^4+x^2y^2+y^6+x^6y^2+y^8+x^{10}+x^8y^2+x^6y^4+x^{12}+x^4y^8+x^2y^{10}+\\ldots )$ Dans des cas simples, on peut donner explicitement la valeur du coefficient $ a_{ij}(p)$ .", "Par exemple : $a_{10}(p) = 1 \\Longleftrightarrow \\ p\\equiv 3\\;\\pmod {8}$ $a_{01}(p) = 1 \\Longleftrightarrow \\ p\\equiv 5\\;\\pmod {8}$ $a_{11}(p) = 1 \\Longleftrightarrow \\ p\\equiv 7\\pmod {16}$ $a_{20}(p)= 1 \\Longleftrightarrow p {\\rm \\ est\\; de\\; la\\; forme\\;\\;} a^2+8b^2 {\\rm \\;avec\\;\\;} a,b \\in {\\bf Z}, \\; b {\\rm \\; impair}$ $a_{02}(p)= 1 \\Longleftrightarrow p {\\rm \\ est\\; de\\; la\\; forme\\;\\;} a^2+16b^2 {\\rm \\;avec\\;\\;} a,b \\in {\\bf Z}, b {\\rm \\;impair}.$ Les formules (REF ) et (REF ) montrent que, si $ p\\equiv 3\\pmod {8}$ , alors $T_p$ est le produit de $x$ par une série inversible en $x^2$ et $y^2$  ; en particulier, $T_p$ et $T_3$ ont le même noyau.", "Même chose si $ p\\equiv 5\\pmod {8}$ avec $x$ et $T_3$ remplacé par $y$ et $T_5$ .", "On en déduit que l'algèbre $A$ est topologiquement engendrée par n'importe quel couple $(T_p, T_{p^{\\prime }}) $ avec $ p\\equiv 3\\pmod {8}$ et $ p^{\\prime }\\equiv 5\\pmod {8}$ .", "Notons aussi que la proposition 4.3 (resp.", "4.4) de [2] reste valable si l'on remplace $T_3$ par $T_p$ avec $p\\equiv 3 \\pmod {8}$ (resp.", "$T_5$ par $T_{p^{\\prime }}$ avec $p^{\\prime }\\equiv 5 \\pmod {8}$ ).", "Remarques.", "1) Pour $i$ et $j$ fixés, la fonction $p \\mapsto a_{ij}(p)$ est frobénienne au sens de [4].", "De façon plus précise, sa valeur ne dépend que de la substitution de Frobenius de $p$ dans une certaine extension galoisienne finie de ${\\bf Q}$ , qui est non ramifiée en dehors de $\\lbrace 2\\rbrace $ et dont le groupe de Galois est un 2-groupe.", "Dans les deux premiers exemples ci-dessus, on peut prendre pour extension galoisienne le corps ${\\bf Q}(\\mu _8)$ des racines huitièmes de l'unité ; dans les trois autres, les corps ${\\bf Q}(\\mu _8,\\sqrt{uv})$ , ${\\bf Q}(\\mu _8,\\sqrt{u})$ et ${\\bf Q}(\\mu _8,\\sqrt{v})$ avec $u=1+i$ et $v=\\sqrt{2}$  ; le premier de ces corps est le corps ${\\bf Q}(\\mu _{16})$ des racines 16-ièmes de l'unité ; les deux autres ont des groupes de Galois sur ${\\bf Q}$ qui sont diédraux d'ordre 8.", "2) Si $p > 5$ , on peut se demander si la série donnant $T_p$ peut être un polynôme en $x$ et $y$ .", "La réponse est “non” : d'après un résultat récent de J. Bellaïche, les $T_p$ sont algébriquement indépendants sur ${\\bf F}_2$ ." ], [ "Séries thêta associées à ${\\bf Q}(\\sqrt{-2})$", "Soient $n$ un entier $\\geqslant 1$ et soit $t \\in {\\bf Z}/2^n{\\bf Z}$ .", "Soit ${\\theta }_{t,n} \\in {\\bf F}_2[[q]]$ la série définie par : $ {\\theta }_{t,n} = \\sum _{a \\ {\\rm impair} \\, > \\, 0} \\ \\ \\sum _{b \\equiv ta \\!\\!\\!\\pmod {2^n}}q^{a^2+2b^2}.$ On a : ${\\theta }_{0,n} = \\Delta , \\ \\ {\\theta }_{t,n} = {\\theta }_{-t,n}, \\ \\ {\\theta }_{2^{n-1},n}= 0, \\ \\ {\\theta }_{t,n}+{\\theta }_{2^{n-1}-t,n} = {\\theta }_{t,n-1},$ et $ {\\theta }_{2^{n-2},n} = \\Delta ^{1+2^{2n-3}}$ si $n \\geqslant 2$ .", "Les séries ${\\theta }_{t,n}$ appartiennent à $\\mathcal {F}$ .", "De façon plus précise : Théorème 8.1 - Pour $n > 0$ fixé, les ${\\theta }_{t,n}$ engendrent le même sous-espace vectoriel de $\\mathcal {F}$ que les formes $m(a,0)$ avec $0 \\leqslant a < 2^{n-1}$ .", "[Pour la définition des $m(a,b)$ , voir §.]", "Corollaire 8.2 - Soit $f = \\sum a_nq^n $ un élément de $\\mathcal {F}$ .", "Les propriétés suivantes sont équivalentes $:$ $1) \\ T_5|f = 0$ .", "$ 2)$ La série $f$ est de la forme $\\sum {\\theta }_{t_i,n_i}$ .", "$3) \\ a_n = 1 \\ \\ \\Rightarrow \\ \\ n$ est de la forme $a^2+2b^2$ , avec $a,b \\in {\\bf Z}$ .", "Exemples (la table des ${\\theta }_{t,n}$ pour $n\\le 6$ et $0\\le t\\le 2^{n-1}$ est sur le site [5]) : ${\\theta }_{0,1} = \\Delta ;$ ${\\theta }_{0,2} = \\Delta ; \\ {\\theta }_{1,2} = \\Delta ^3;$ ${\\theta }_{0,3} = \\Delta ; \\ {\\theta }_{1,3} = \\Delta ^3 + \\Delta ^{11}; \\ {\\theta }_{2,3} = \\Delta ^9; \\ {\\theta }_{3,3} = \\Delta ^{11}$ .", "Action des opérateurs de Hecke sur les ${\\theta }_{t,n}$.", "Si $p \\equiv 5 \\ {\\rm ou} \\ 7 \\pmod {8}$ , on a $T_p|{\\theta }_{t,n} = 0.$ Si $p \\equiv 1 \\ {\\rm ou} \\ 3 \\pmod {8}$ , on écrit $p$ sous la forme $p = a^2 + 2b^2$ , avec $a,b \\in {\\bf Z}$  ; on définit $t(p) \\in {\\bf Z}/2^n{\\bf Z}$ par $t(p) \\equiv b/a \\pmod {2^n}$ , et l'on pose $t^*(p) = -t(p)$ .", "On a : $T_p|{\\theta }_{t,n} = {\\theta }_{t\\bullet t(p),n} + {\\theta }_{t\\bullet t^*(p),n}$ où l'on a noté $x\\bullet y$ la loi de compositionCette loi munit ${\\bf Z}/2^n{\\bf Z}$ d'une structure de groupe abélien ; ce groupe est cyclique d'ordre $2^n$  ; on peut l'interpréter comme le groupe des classes de formes quadratiques binaires primitives de discriminant $-2^{2n+3}$ , ou encore comme le groupe Pic du sous-anneau de ${\\bf Z}[\\sqrt{-2}]$ de conducteur $2^n$ .", "sur ${\\bf Z}/2^n{\\bf Z}$ définie par la formule $x\\bullet y = (x+y)/(1-2xy)$ .", "On a en particulier $\\;{\\theta }_{2^{n-1}-t(p),n} = T_p|\\Delta ^{1+2^{2n-1}}$ ." ], [ "Séries thêta associées à ${\\bf Q}(i)$", "Les définitions et les résultats sont essentiellement les mêmes que ceux du §, à cela près que $a^2+2b^2, \\ T_5$ et $m(a,0)$ sont remplacés par $a^2+4b^2, \\ T_3$ et $m(0,b)$ .", "De façon plus précise, si $t$ et $n$ sont comme ci-dessus, on définit la série thêta d'indice $(t,n)$ par : $ {\\theta }_{t,n}^{\\prime } = \\sum _{a \\ {\\rm impair} \\, > \\, 0} \\ \\ \\sum _{b \\equiv ta \\!", "\\!", "\\!\\pmod {2^n}}q^{a^2+4b^2}.$ On a :    ${\\theta }_{0,n}^{\\prime } = \\Delta , \\quad {\\theta }_{t,n}^{\\prime } = {\\theta }_{-t,n}^{\\prime }, \\quad {\\theta }_{2^{n-1},n}^{\\prime }= 0, \\quad {\\theta }_{t,n}^{\\prime }+{\\theta }_{2^{n-1}-t,n}^{\\prime } = {\\theta }_{t,n-1}^{\\prime }, $ et $ {\\theta }_{2^{n-2},n}^{\\prime } = \\Delta ^{1+2^{2n-2}}$ si $n \\geqslant 2 $ .", "De plus : Théorème 9.1 - Pour $n > 0$ fixé, les ${\\theta }_{t,n}^{\\prime }$ engendrent le même sous-espace vectoriel de $\\mathcal {F}$ que les formes $m(0,b)$ avec $0 \\leqslant b < 2^{n-1}$ .", "Corollaire 9.2 - Soit $f = \\sum a_nq^n $ un élément de $\\mathcal {F}$ .", "Les propriétés suivantes sont équivalentes $:$ $1) \\ T_3|f = 0$ .", "$ 2)$ La série $f$ est de la forme $\\sum {\\theta }_{t_i,n_i}^{\\prime }$ .", "$3) \\ a_n = 1 \\ \\ \\Rightarrow \\ \\ n$ est de la forme $a^2+b^2$ , avec $a,b \\in {\\bf Z}$ .", "Exemples (la table des ${\\theta }_{t,n}^{\\prime }$ pour $n\\le 6$ et $0\\le t\\le 2^{n-1}$ est sur le site [5]) : ${\\theta }_{0,1}^{\\prime } = \\Delta ;$ ${\\theta }_{0,2}^{\\prime } = \\Delta ; \\ {\\theta }_{1,2}^{\\prime } = \\Delta ^5;$ ${\\theta }_{0,3}^{\\prime } = \\Delta ; \\ {\\theta }_{1,3}^{\\prime } = \\Delta ^5 + \\Delta ^{13}+\\Delta ^{21} ; \\ {\\theta }_{2,3}^{\\prime } =\\Delta ^{17} \\ ; \\ {\\theta }_{3,3}^{\\prime } = \\Delta ^{13}+\\Delta ^{21}$ .", "Action des opérateurs de Hecke sur les ${\\theta }_{t,n}^{\\prime }$.", "Si $p \\equiv 3 \\ {\\rm ou} \\ 7 \\pmod {8}$ , on a $T_p|{\\theta }_{t,n}^{\\prime } = 0.$ Si $p \\equiv 1 \\ {\\rm ou} \\ 5 \\pmod {8}$ , on écrit $p$ sous la forme $p = a^2 + 4b^2$ , avec $a,b \\in {\\bf Z}$  ; on pose $t(p)^{\\prime } \\equiv b/a \\pmod {2^n}$ et $t^*(p)^{\\prime } = - t_1(p)^{\\prime }$ .", "On a : $T_p|{\\theta }_{t,n}^{\\prime } = {\\theta }_{t\\bullet ^{\\prime } t(p)^{\\prime },n}^{\\prime } + {\\theta }_{t\\bullet ^{\\prime } t^*(p)^{\\prime },n}^{\\prime }$ où l'on a noté $x\\bullet ^{\\prime } y$ la loi de composition sur ${\\bf Z}/2^n{\\bf Z}$ définie par la formule $x\\bullet ^{\\prime } y = (x+y)/(1-4xy)$ .", "On a en particulier $\\;{\\theta }^{\\prime }_{2^{n-1}-t(p)^{\\prime },n} = T_p|\\Delta ^{1+2^{2n}}$ ." ], [ "Développement des $T_p$ comme séries en {{formula:7111c357-c6ec-437b-bb13-90f226992e89}} et {{formula:8c014d30-71ce-4c6c-9e9d-2564e92851bf}}", "D'après le th.REF , tout $T_p$ peut s'écrire comme une série formelle en $x=T_3$ et $y=T_5$  : $T_p = \\sum _{i+j\\geqslant 1} a_{ij}(p)x^iy^j, {\\rm \\; \\;avec\\;\\;} \\ a_{ij}(p) \\in {\\bf F}_2.$ De façon plus précise, on a : $T_p \\in {\\bf F}_2[[x^2,y^2]] \\ \\ \\ \\ \\;\\;{\\rm si\\;} p\\equiv 1\\pmod {8},$ $T_p \\in x.", "{\\bf F}_2[[x^2,y^2]] \\ \\ \\;\\; {\\rm si\\;} p\\equiv 3\\pmod {8},$ $T_p \\in y.", "{\\bf F}_2[[x^2,y^2]] \\ \\ \\;\\;{\\rm si\\;} p\\equiv 5\\pmod {8},$ $T_p \\in xy.", "{\\bf F}_2[[x^2,y^2]] \\ \\;\\;{\\rm si\\;} p\\equiv 7\\pmod {8}.$ Exemples (cf.", "[5]) : $T_{17}=x^2+y^2+x^2y^2+x^6+x^4y^2+y^6+x^6y^2+x^4y^4+x^2y^6+x^{10}+x^{10}y^2+x^6y^6+x^4y^8+x^2y^{10}+\\ldots $ $T_{11}= x(1+x^2+y^2+x^4+x^2y^2+y^4+x^2y^4+y^6+x^6y^2+x^8y^2+x^6y^4+x^2y^8+y^{10}+x^{10}y^2+\\ldots )$ $T_{13}= y(1+x^2+y^2+x^4+y^4+x^6+x^4y^2+x^2y^4+x^6y^2+x^2y^6+y^8+x^{10}+x^8y^2+x^6y^4+y^{10}+\\ldots )$ $T_7 = xy(1 + x^2 + x^4+x^2y^2+y^6+x^6y^2+y^8+x^{10}+x^8y^2+x^6y^4+x^{12}+x^4y^8+x^2y^{10}+\\ldots )$ Dans des cas simples, on peut donner explicitement la valeur du coefficient $ a_{ij}(p)$ .", "Par exemple : $a_{10}(p) = 1 \\Longleftrightarrow \\ p\\equiv 3\\;\\pmod {8}$ $a_{01}(p) = 1 \\Longleftrightarrow \\ p\\equiv 5\\;\\pmod {8}$ $a_{11}(p) = 1 \\Longleftrightarrow \\ p\\equiv 7\\pmod {16}$ $a_{20}(p)= 1 \\Longleftrightarrow p {\\rm \\ est\\; de\\; la\\; forme\\;\\;} a^2+8b^2 {\\rm \\;avec\\;\\;} a,b \\in {\\bf Z}, \\; b {\\rm \\; impair}$ $a_{02}(p)= 1 \\Longleftrightarrow p {\\rm \\ est\\; de\\; la\\; forme\\;\\;} a^2+16b^2 {\\rm \\;avec\\;\\;} a,b \\in {\\bf Z}, b {\\rm \\;impair}.$ Les formules (REF ) et (REF ) montrent que, si $ p\\equiv 3\\pmod {8}$ , alors $T_p$ est le produit de $x$ par une série inversible en $x^2$ et $y^2$  ; en particulier, $T_p$ et $T_3$ ont le même noyau.", "Même chose si $ p\\equiv 5\\pmod {8}$ avec $x$ et $T_3$ remplacé par $y$ et $T_5$ .", "On en déduit que l'algèbre $A$ est topologiquement engendrée par n'importe quel couple $(T_p, T_{p^{\\prime }}) $ avec $ p\\equiv 3\\pmod {8}$ et $ p^{\\prime }\\equiv 5\\pmod {8}$ .", "Notons aussi que la proposition 4.3 (resp.", "4.4) de [2] reste valable si l'on remplace $T_3$ par $T_p$ avec $p\\equiv 3 \\pmod {8}$ (resp.", "$T_5$ par $T_{p^{\\prime }}$ avec $p^{\\prime }\\equiv 5 \\pmod {8}$ ).", "Remarques.", "1) Pour $i$ et $j$ fixés, la fonction $p \\mapsto a_{ij}(p)$ est frobénienne au sens de [4].", "De façon plus précise, sa valeur ne dépend que de la substitution de Frobenius de $p$ dans une certaine extension galoisienne finie de ${\\bf Q}$ , qui est non ramifiée en dehors de $\\lbrace 2\\rbrace $ et dont le groupe de Galois est un 2-groupe.", "Dans les deux premiers exemples ci-dessus, on peut prendre pour extension galoisienne le corps ${\\bf Q}(\\mu _8)$ des racines huitièmes de l'unité ; dans les trois autres, les corps ${\\bf Q}(\\mu _8,\\sqrt{uv})$ , ${\\bf Q}(\\mu _8,\\sqrt{u})$ et ${\\bf Q}(\\mu _8,\\sqrt{v})$ avec $u=1+i$ et $v=\\sqrt{2}$  ; le premier de ces corps est le corps ${\\bf Q}(\\mu _{16})$ des racines 16-ièmes de l'unité ; les deux autres ont des groupes de Galois sur ${\\bf Q}$ qui sont diédraux d'ordre 8.", "2) Si $p > 5$ , on peut se demander si la série donnant $T_p$ peut être un polynôme en $x$ et $y$ .", "La réponse est “non” : d'après un résultat récent de J. Bellaïche, les $T_p$ sont algébriquement indépendants sur ${\\bf F}_2$ ." ], [ "Séries thêta associées à ${\\bf Q}(\\sqrt{-2})$", "Soient $n$ un entier $\\geqslant 1$ et soit $t \\in {\\bf Z}/2^n{\\bf Z}$ .", "Soit ${\\theta }_{t,n} \\in {\\bf F}_2[[q]]$ la série définie par : $ {\\theta }_{t,n} = \\sum _{a \\ {\\rm impair} \\, > \\, 0} \\ \\ \\sum _{b \\equiv ta \\!\\!\\!\\pmod {2^n}}q^{a^2+2b^2}.$ On a : ${\\theta }_{0,n} = \\Delta , \\ \\ {\\theta }_{t,n} = {\\theta }_{-t,n}, \\ \\ {\\theta }_{2^{n-1},n}= 0, \\ \\ {\\theta }_{t,n}+{\\theta }_{2^{n-1}-t,n} = {\\theta }_{t,n-1},$ et $ {\\theta }_{2^{n-2},n} = \\Delta ^{1+2^{2n-3}}$ si $n \\geqslant 2$ .", "Les séries ${\\theta }_{t,n}$ appartiennent à $\\mathcal {F}$ .", "De façon plus précise : Théorème 8.1 - Pour $n > 0$ fixé, les ${\\theta }_{t,n}$ engendrent le même sous-espace vectoriel de $\\mathcal {F}$ que les formes $m(a,0)$ avec $0 \\leqslant a < 2^{n-1}$ .", "[Pour la définition des $m(a,b)$ , voir §.]", "Corollaire 8.2 - Soit $f = \\sum a_nq^n $ un élément de $\\mathcal {F}$ .", "Les propriétés suivantes sont équivalentes $:$ $1) \\ T_5|f = 0$ .", "$ 2)$ La série $f$ est de la forme $\\sum {\\theta }_{t_i,n_i}$ .", "$3) \\ a_n = 1 \\ \\ \\Rightarrow \\ \\ n$ est de la forme $a^2+2b^2$ , avec $a,b \\in {\\bf Z}$ .", "Exemples (la table des ${\\theta }_{t,n}$ pour $n\\le 6$ et $0\\le t\\le 2^{n-1}$ est sur le site [5]) : ${\\theta }_{0,1} = \\Delta ;$ ${\\theta }_{0,2} = \\Delta ; \\ {\\theta }_{1,2} = \\Delta ^3;$ ${\\theta }_{0,3} = \\Delta ; \\ {\\theta }_{1,3} = \\Delta ^3 + \\Delta ^{11}; \\ {\\theta }_{2,3} = \\Delta ^9; \\ {\\theta }_{3,3} = \\Delta ^{11}$ .", "Action des opérateurs de Hecke sur les ${\\theta }_{t,n}$.", "Si $p \\equiv 5 \\ {\\rm ou} \\ 7 \\pmod {8}$ , on a $T_p|{\\theta }_{t,n} = 0.$ Si $p \\equiv 1 \\ {\\rm ou} \\ 3 \\pmod {8}$ , on écrit $p$ sous la forme $p = a^2 + 2b^2$ , avec $a,b \\in {\\bf Z}$  ; on définit $t(p) \\in {\\bf Z}/2^n{\\bf Z}$ par $t(p) \\equiv b/a \\pmod {2^n}$ , et l'on pose $t^*(p) = -t(p)$ .", "On a : $T_p|{\\theta }_{t,n} = {\\theta }_{t\\bullet t(p),n} + {\\theta }_{t\\bullet t^*(p),n}$ où l'on a noté $x\\bullet y$ la loi de compositionCette loi munit ${\\bf Z}/2^n{\\bf Z}$ d'une structure de groupe abélien ; ce groupe est cyclique d'ordre $2^n$  ; on peut l'interpréter comme le groupe des classes de formes quadratiques binaires primitives de discriminant $-2^{2n+3}$ , ou encore comme le groupe Pic du sous-anneau de ${\\bf Z}[\\sqrt{-2}]$ de conducteur $2^n$ .", "sur ${\\bf Z}/2^n{\\bf Z}$ définie par la formule $x\\bullet y = (x+y)/(1-2xy)$ .", "On a en particulier $\\;{\\theta }_{2^{n-1}-t(p),n} = T_p|\\Delta ^{1+2^{2n-1}}$ ." ], [ "Séries thêta associées à ${\\bf Q}(i)$", "Les définitions et les résultats sont essentiellement les mêmes que ceux du §, à cela près que $a^2+2b^2, \\ T_5$ et $m(a,0)$ sont remplacés par $a^2+4b^2, \\ T_3$ et $m(0,b)$ .", "De façon plus précise, si $t$ et $n$ sont comme ci-dessus, on définit la série thêta d'indice $(t,n)$ par : $ {\\theta }_{t,n}^{\\prime } = \\sum _{a \\ {\\rm impair} \\, > \\, 0} \\ \\ \\sum _{b \\equiv ta \\!", "\\!", "\\!\\pmod {2^n}}q^{a^2+4b^2}.$ On a :    ${\\theta }_{0,n}^{\\prime } = \\Delta , \\quad {\\theta }_{t,n}^{\\prime } = {\\theta }_{-t,n}^{\\prime }, \\quad {\\theta }_{2^{n-1},n}^{\\prime }= 0, \\quad {\\theta }_{t,n}^{\\prime }+{\\theta }_{2^{n-1}-t,n}^{\\prime } = {\\theta }_{t,n-1}^{\\prime }, $ et $ {\\theta }_{2^{n-2},n}^{\\prime } = \\Delta ^{1+2^{2n-2}}$ si $n \\geqslant 2 $ .", "De plus : Théorème 9.1 - Pour $n > 0$ fixé, les ${\\theta }_{t,n}^{\\prime }$ engendrent le même sous-espace vectoriel de $\\mathcal {F}$ que les formes $m(0,b)$ avec $0 \\leqslant b < 2^{n-1}$ .", "Corollaire 9.2 - Soit $f = \\sum a_nq^n $ un élément de $\\mathcal {F}$ .", "Les propriétés suivantes sont équivalentes $:$ $1) \\ T_3|f = 0$ .", "$ 2)$ La série $f$ est de la forme $\\sum {\\theta }_{t_i,n_i}^{\\prime }$ .", "$3) \\ a_n = 1 \\ \\ \\Rightarrow \\ \\ n$ est de la forme $a^2+b^2$ , avec $a,b \\in {\\bf Z}$ .", "Exemples (la table des ${\\theta }_{t,n}^{\\prime }$ pour $n\\le 6$ et $0\\le t\\le 2^{n-1}$ est sur le site [5]) : ${\\theta }_{0,1}^{\\prime } = \\Delta ;$ ${\\theta }_{0,2}^{\\prime } = \\Delta ; \\ {\\theta }_{1,2}^{\\prime } = \\Delta ^5;$ ${\\theta }_{0,3}^{\\prime } = \\Delta ; \\ {\\theta }_{1,3}^{\\prime } = \\Delta ^5 + \\Delta ^{13}+\\Delta ^{21} ; \\ {\\theta }_{2,3}^{\\prime } =\\Delta ^{17} \\ ; \\ {\\theta }_{3,3}^{\\prime } = \\Delta ^{13}+\\Delta ^{21}$ .", "Action des opérateurs de Hecke sur les ${\\theta }_{t,n}^{\\prime }$.", "Si $p \\equiv 3 \\ {\\rm ou} \\ 7 \\pmod {8}$ , on a $T_p|{\\theta }_{t,n}^{\\prime } = 0.$ Si $p \\equiv 1 \\ {\\rm ou} \\ 5 \\pmod {8}$ , on écrit $p$ sous la forme $p = a^2 + 4b^2$ , avec $a,b \\in {\\bf Z}$  ; on pose $t(p)^{\\prime } \\equiv b/a \\pmod {2^n}$ et $t^*(p)^{\\prime } = - t_1(p)^{\\prime }$ .", "On a : $T_p|{\\theta }_{t,n}^{\\prime } = {\\theta }_{t\\bullet ^{\\prime } t(p)^{\\prime },n}^{\\prime } + {\\theta }_{t\\bullet ^{\\prime } t^*(p)^{\\prime },n}^{\\prime }$ où l'on a noté $x\\bullet ^{\\prime } y$ la loi de composition sur ${\\bf Z}/2^n{\\bf Z}$ définie par la formule $x\\bullet ^{\\prime } y = (x+y)/(1-4xy)$ .", "On a en particulier $\\;{\\theta }^{\\prime }_{2^{n-1}-t(p)^{\\prime },n} = T_p|\\Delta ^{1+2^{2n}}$ ." ] ]

[ [ "Integrability of higher pentagram maps" ], [ "Abstract We define higher pentagram maps on polygons in $P^d$ for any dimension $d$, which extend R.Schwartz's definition of the 2D pentagram map.", "We prove their integrability by presenting Lax representations with a spectral parameter for scale invariant maps.", "The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$-equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D.", "We also study in detail the 3D case, where we prove integrability for both closed and twisted polygons and describe the spectral curve, first integrals, the corresponding tori and the motion along them, as well as an invariant symplectic structure." ], [ "Introduction", "The pentagram map was defined by R. Schwartz in [13] on plane convex polygons considered modulo projective equivalence.", "Figure 1 explains the definition: for a polygon $P$ the image under the pentagram map is a new polygon $T(P)$ spanned by the “shortest\" diagonals of $P$ .", "Iterations of this map on classes of projectively equivalent polygons manifest quasiperiodic behaviour, which indicates hidden integrability [14].", "Figure: The image T(P)T(P) of a hexagon PP under the 2D pentagram map.The integrability was proved in [11] for the pentagram map on a larger class of the so called twisted polygons in 2D, which are piecewise linear curves with a fixed monodromy relating their ends.", "Closed polygons correspond to the monodromy given by the identity transformation.", "It turned out that there is an invariant Poisson structure for the pentagram map and it has sufficiently many invariant quantities.", "Moreover, this map turned out to be related to a variety of mathematical domains, including cluster algebras [2], [4], frieze patterns, and integrable systems of mathematical physics: in particular, its continuous limit in 2D is the classical Boussinesq equation [11].", "Integrability of the pentagram map for 2D closed polygons was established in [15], [12], while a more general framework related to surface networks was presented in [3].", "In this paper we extend the definition of the pentagram map to closed and twisted polygons in spaces of any dimension $d$ and prove its various integrability properties.", "It is worth mentioning that the problem of finding integrable higher-dimensional generalizations for the pentagram map attracted much attention after the 2D case was treated in [11].There seem to be no natural generalization of the pentagram map to polytopes in higher dimension $d\\ge 3$ .", "Indeed, the initial polytope should be simple for its diagonal hyperplanes to be well defined.", "In order to iterate the pentagram map the dual polytope has to be simple as well.", "Thus iterations could be defined only for $d$ -simplices, which are all projectively equivalent.", "The main difficulty in higher dimensions is that diagonals of a polygon are generically skew and do not intersect.", "One can either confine oneself to special polygons (e.g., corrugated ones, [3]) to retain the intersection property or one has too many possible choices for using hyperplanes as diagonals, where it is difficult to find integrable ones, cf.", "[9].", "Below, as an analog of the 2D shortest diagonals for a generic polygon in a projective space ${\\mathbb {RP}}^d$ we propose to consider a “short-diagonal hyperplane\" passing through $d$ vertices where every other vertex is taken starting with a given one.", "Then a new vertex is constructed as the intersection of $d$ consecutive diagonal hyperplanes.", "We repeat this procedure starting with the next vertex of the initial polygon.", "The higher (or $d$ -dimensional) pentagram map $T$ takes the initial polygon to the one defined by this set of new vertices.", "As before, the obtained polygon is considered modulo projective equivalence in ${\\mathbb {RP}}^d$ .", "We also describe general pentagram maps $T_{p,r}$ in ${\\mathbb {RP}}^d$ enumerated by two integral parameters $p$ and $r$ by considering $p$ -diagonals (i.e., hyperplanes passing through every $p$ th vertex of the polygon) and by taking the intersections of every $r$ th hyperplane like that.", "There is a curious duality between them: the map $T_{p,r}$ is equal to $T^{-1}_{r,p}$ modulo a shift in vertex indices.", "However, we are mostly interested in the higher pentagram maps, which correspond to $T:=T_{2,1}$ in ${\\mathbb {RP}}^d$ .", "We start by describing the continuous limit of the higher pentagram map as the evolution in the direction of the “envelope\" for such a sequence of short-diagonal planes as the number of vertices of the polygon tends to infinity.", "(More precisely, the envelope here is the curve whose osculating planes are limits of the short-diagonal planes.)", "Theorem A (= Theorems REF , REF) The continuous limit of the higher pentagram map in ${\\mathbb {RP}}^d$ is the $(2,d+1)$ -equation in the KdV hierarchy, which is an infinite-dimensional completely integrable system.", "This generalizes the Boussinesq equation as a limit of the pentagram map in ${\\mathbb {RP}}^2$ and this limit seems to be very robust.", "Indeed, the same equation appears for an almost arbitrary choice of diagonal planes.", "It also arises when instead of osculating planes one considers other possible definitions of higher pentagram maps (cf.", "e.g.", "[9]).", "However, the pentagram map in the above definition with short-diagonal hyperplanes exhibits integrability properties not only in the continuous limit, but as a discrete system as well.", "To study them, we define two coordinate systems for twisted polygons in 3D (somewhat similar to the ones used in 2D, cf.", "[11]), and present explicit formulas for the 3D pentagram map using these coordinates (see Theorem REF ).", "Then we describe the pentagram map as a completely integrable discrete dynamical system by presenting its Lax form in any dimension and studying in detail the 3D case (see Section ).", "For algebraic-geometric integrability we complexify the pentagram map.", "The corresponding 2D case was investigated in [15].", "The key ingredient of the algebraic-geometric integrability for a discrete dynamical system is a discrete Lax (or zero curvature) equation with a spectral parameter, which in our case assumes the following form: $L_{i,t+1}(\\lambda ) = P_{i+1,t}(\\lambda ) L_{i,t}(\\lambda ) P_{i,t}^{-1}(\\lambda ).$ Here the index $t$ represents the discrete time variable, the index $i$ refers to the vertex of an $n$ -gon, and $\\lambda $ is a complex spectral parameter.", "(For the pentagram map in ${\\mathbb {CP}}^d$ the functions $L_{i,t}(\\lambda )$ and $P_{i,t}(\\lambda )$ are matrix-valued of size $(d+1) \\times (d+1)$ .)", "The discrete Lax equation arises as a compatibility condition of an over-determined system of equations: ${\\left\\lbrace \\begin{array}{ll}L_{i,t}(\\lambda ) \\Psi _{i,t}(\\lambda ) = \\Psi _{i+1,t}(\\lambda )\\\\P_{i,t}(\\lambda ) \\Psi _{i,t}(\\lambda ) = \\Psi _{i,t+1}(\\lambda ),\\end{array}\\right.", "}$ for an auxiliary function $\\Psi _{i,t}(\\lambda )$ .", "Remark 1.1 Recall that for a smooth dynamical system the Lax form is a differential equation of type $\\partial _t L=[P,L]$ on a matrix $L$ .", "Such a form of the equation implies that the evolution of $L$ changes it to a similar matrix, thus preserving its eigenvalues.", "If the matrix $L$ depends on a parameter, $L=L(\\lambda )$ , then the corresponding eigenvalues as functions of parameter do not change and in many cases provide sufficiently many first integrals for complete integrability of such a system.", "Similarly, an analogue of the Lax form for differential operators of type $\\partial _x-L$ is a zero curvature equation $\\partial _tL-\\partial _x P=[P,L]\\,.$ This is a compatibility condition which provides the existence of an auxiliary function $\\psi =\\psi (t,x)$ satisfying a system of equations $\\partial _x \\psi = L\\psi $ and $\\partial _t \\psi = P\\psi \\,.$ The above Lax form and auxiliary system are discrete versions of the latter.", "In our case, the equivalence of formulas for the pentagram map to the dynamics defined by the Lax equation implies complete algebraic-geometric integrability of the system.", "More precisely, the following theorem summarizes several main results on the 3D pentagram map, which are obtained by studying its Lax equation.", "The dynamics is (generically) defined on the space ${\\mathcal {P}}_n$ of projectively equivalent twisted $n$ -gons in 3D, which we describe below, and has dimension $3n$ , while closed $n$ -gons form a submanifold of codimension 15 in it.", "Later on we will introduce the notion of spectral data which consists of a Riemann surface, called a spectral curve, and a point in the Jacobian (i.e., the complex torus) of this curve, as well as a notion of a spectral map between the space ${\\mathcal {P}}_n$ and the spectral data.", "Theorem B (= Theorems REF , REF , REF) A Zariski open subset of the complexified space ${\\mathcal {P}}_n$ of twisted $n$ -gons in 3D is a fibration whose fibres are Zariski open subsets of tori.", "These tori are Jacobians of the corresponding spectral curves and are invariant with respect to the space pentagram map.", "Their dimension is $3\\lfloor n/2\\rfloor $ for odd $n$ and $3(n/2)-3$ for even $n$ , where $\\lfloor n/2 \\rfloor $ is the integer part of $n/2$ .", "The pentagram dynamics on the Jacobians goes along a straight line for odd $n$ and along a staircase for even $n$ (i.e., the discrete evolution is either a constant shift on a torus, or its square is a constant shift).", "For closed $n$ -gons the tori have dimensions $3\\lfloor n/2\\rfloor -6$ for odd $n$ and $3(n/2)-9$ for even $n$ .", "Remark 1.2 One also has an explicit description of the the above fibration in terms of coordinates on the space of $n$ -gons.", "We note that the pentagram dynamics understood as a shift on complex tori does not prevent the corresponding orbits on the space ${\\mathcal {P}}_n$ from being unbounded.", "The dynamics described above takes place for generic initial data, i.e., for points on the Jacobians whose orbits do not intersect certain divisors.", "Points of generic orbits with irrational shifts can return arbitrarily close to such divisors.", "On the other hand, the inverse spectral map is defined outside of these special divisors and may have poles there.", "Therefore the sequences in the space ${\\mathcal {P}}_n$ corresponding to such orbits may escape to infinity.", "It is known that the pentagram map in 2D possesses an invariant Poisson structure [11], which can also be described by using the Krichever-Phong universal formula [15].", "Although we do not present an invariant Poisson structure for the pentagram map in 3D, we describe its symplectic leaves, as well as the action-angle coordinates.", "More precisely, we present an invariant symplectic structure (i.e., a closed non-degenerate 2-form), and submanifolds where it is defined (Theorem REF ).", "By analogy with the 2D case, it is natural to suggest that these submanifolds are symplectic leaves of an invariant Poisson structure, and that the inverse of our symplectic structure coincides with the Poisson structure on the leaves.", "An explicit description of this Poisson structure in 3D is still an open problem.", "Note that the algebraic-geometric integrability of the pentagram map implies its Arnold–Liouville complete integrability on generic symplectic leaves (in the real case).", "Namely, the existence of a (pre)symplectic structure coming from the Lax form of the pentagram map (see Section REF ), together with the generic set of first integrals, appearing as coefficients of the corresponding spectral curve, provides sufficiently many integrals in involution.", "(Note that proving independence of first integrals while remaining within the real setting is often more difficult than first proving the algebraic-geometric integrability, which in turn implies their independence in the real case.)", "Finally, in Section  we present a Lax form for the pentagram maps in arbitrary dimension (which implies their complete integrability) assuming their scaling invariance: Theorem C (= Theorem REF) The scale-invariant pentagram map in ${\\mathbb {CP}}^d$ admits a Lax representation with a spectral parameter.", "The scaling invariance of the pentagram maps is proved for all $d\\le 6$ , with some numerical evidence for higher values of $d>6$ as well.", "It would be interesting to establish it in full generality.", "There is a considerable difference between the cases of even and odd dimension $d$ , which can be already seen in the analysis of the 2D and 3D cases.", "Acknowledgments.", "We are grateful to M. Gekhtman and S. Tabachnikov for useful discussions.", "B.K.", "was partially supported by the Simonyi Fund and an NSERC research grant." ], [ "Review of the 2D pentagram map", "In this section we recall the main definitions and results in 2D (see [11]), which will be important for higher-dimensional generalizations below.", "We formulate the geometric results in the real setting, while the algebraic-geometric ones are presented for the corresponding complexification.", "First note that the pentagram map can be extended from closed to twisted polygons.", "Definition 2.1 Given a projective transformation $M\\in PSL(3,{\\mathbb {R}})$ of the plane ${\\mathbb {RP}}^2$ , a twisted $n$ -gon in ${\\mathbb {RP}}^2$ is a map $\\phi :{\\mathbb {Z}}\\rightarrow {\\mathbb {RP}}^2$ , such that $\\phi (k+n)=M\\circ \\phi (k)$ for any $k$ .", "$M$ is called the monodromy of $\\phi $ .", "Two twisted $n$ -gons are equivalent if there is a transformation $g\\in PSL(3,{\\mathbb {R}})$ such that $g\\circ \\phi _1=\\phi _2$ .", "Consider generic $n$ -gons, i.e., those that do not have any three consecutive vertices lying on the same line.", "Denote by ${\\mathcal {P}}_n$ the space of generic twisted $n$ -gons considered up to $PSL(3,{\\mathbb {R}})$ transformations.", "The dimension of ${\\mathcal {P}}_n$ is $2n$ .", "Indeed, a twisted $n$ -gon depends on $2n$ variables representing coordinates of vertices $v_k:=\\phi (k)$ for $k=1,...,n$ and on 8 parameters of the monodromy matrix $M$ , while the $PSL(3,{\\mathbb {R}})$ -equivalence reduces the dimension by 8.", "The pentagram map $T$ is generically defined on the space ${\\mathcal {P}}_n$ .", "Namely, for a twisted $n$ -gon vertices of its image are the intersections of pairs of consecutive shortest diagonals: $Tv_k:=(v_{k-1}, v_{k+1})\\cap (v_{k}, v_{k+2})$ .", "Such intersections are well defined for a generic point in ${\\mathcal {P}}_n$ .", "a) Results on integrability (in the twisted and closed cases).", "There is a Poisson structure on ${\\mathcal {P}}_n$ invariant with respect to the pentagram map.", "There are $2\\lfloor n/2 \\rfloor +2$ integrals in involution, which provide integrability of the pentagram map on ${\\mathcal {P}}_n$ .", "Its symplectic leaves have codimensions 2 or 4 in ${\\mathcal {P}}_n$ depending on whether $n$ is odd or even, and the invariant tori have dimensions $n-1$ or $n-2$ , respectively [11].", "Moreover, when restricted to the space ${\\mathcal {C}}_n$ of closed polygons ($\\dim {\\mathcal {C}}_n=2n-8=2(n-4)$ ), the map is still integrable and has invariant tori of dimension $n-4$ for odd $n$ and $n-5$ for even $n$ .", "Note that the space ${\\mathcal {C}}_n$ of closed polygons is not a Poisson submanifold in the space ${\\mathcal {P}}_n$ of twisted $n$ -gons, so the corresponding Poisson structure on ${\\mathcal {P}}_n$ cannot be restricted to ${\\mathcal {C}}_n$ .", "There is a Lax representation for the pentagram map.", "Coefficients of the corresponding spectral curve are the first integrals of the dynamics.", "The pentagram map defines a discrete motion on the Jacobian of the spectral curve.", "This motion is linear or staircase-like depending on the parity of $n$ , see [15].", "b) Coordinates on ${\\mathcal {P}}_n$ .", "The following two systems of coordinates on ${\\mathcal {P}}_n$ are particularly convenient to work with, see [11].", "Assume that $n$ is not divisible by 3.", "Then there exists a unique lift of points $v_k=\\phi (k)\\in {\\mathbb {RP}}^2$ to the vectors $V_k\\in {\\mathbb {R}}^3$ satisfying the condition $\\det |V_j, V_{j+1}, V_{j+2}|=1$ for each $j$ .", "Associate a difference equation to a sequence of vectors $V_k\\in {\\mathbb {R}}^3$ by setting $V_{j+3}=a_j V_{j+2} + b_j V_{j+1} + V_{j}$ for all $j\\in {\\mathbb {Z}}$ .", "The sequences $(a_j)$ and $(b_j)$ turn out to be $n$ -periodic, which is a manifestation of the fact that the lifts satisfy the relations $V_{j+n}=MV_j,\\; j \\in {\\mathbb {Z}},$ for a certain monodromy matrix $M\\in SL(3,{\\mathbb {R}})$ .", "The variables $a_j, b_j, \\, 0\\le j \\le n-1$ are coordinates on the space ${\\mathcal {P}}_n$ .", "There exists another coordinate system on the space ${\\mathcal {P}}_n$ , which is more geometric.", "Recall that the cross-ratio of 4 points in ${\\mathbb {P}}^1$ is given by $[t_1,t_2,t_3, t_4]=\\frac{(t_1-t_2)(t_3-t_4)}{(t_1-t_3)(t_2-t_4)},$ where $t$ is any affine parameter.", "Now associate to each vertex $v_i$ the following two numbers, which are the cross-ratios of two 4-tuples of points lying on the lines $(v_{i-2},v_{i-1})$ and $(v_{i+1},v_{i+2})$ respectively: $x_i=[v_{i-2},\\, v_{i-1},\\, ((v_{i-2}, v_{i-1})\\cap ( v_{i}, v_{i+1})),\\, ((v_{i-2}, v_{i-1})\\cap (v_{i+2}, v_{i+2}))]$ $y_i=[((v_{i-2}, v_{i-1})\\cap (v_{i+1}, v_{i+2})), (( v_{i-1}, v_{i}) \\cap (v_{i+1}, v_{i+2})),\\,v_{i+1},\\, v_{i+2}]$ In these coordinates the pentagram map has the form $T^*x_i=x_i\\frac{1-x_{i-1}y_{i-1}}{1-x_{i+1}y_{i+1}}\\,\\qquad T^*y_i=y_{i+1}\\frac{1-x_{i+2}y_{i+2}}{1-x_{i}y_{i}}\\,.$ One can see that the pentagram map commutes with the scaling transformation [11], [14]: $R_s: \\,(x_1,y_1, ..., x_n, y_n)\\rightarrow (sx_1, s^{-1}y_1, ..., sx_n, s^{-1} y_n)\\,.$ In these coordinates the invariant Poisson structure has a particularly simple form, see [11].", "c) Continuous limit: the Boussinesq equation.", "The $n\\rightarrow \\infty $ continuous limit of a twisted $n$ -gon with a fixed monodromy $M\\in PSL(3,{\\mathbb {R}})$ can be viewed as a smooth parameterized curve $\\gamma :{\\mathbb {R}}\\rightarrow {\\mathbb {RP}}^2$ satisfying $\\gamma (x+2\\pi )=M\\gamma (x)$ for all $x\\in {\\mathbb {R}}$ .", "The genericity assumption that every three consecutive points of an $n$ -gon are in general position corresponds to the assumption that $\\gamma $ is a non-degenerate curve in ${\\mathbb {RP}}^2$ , i.e., the vectors $\\gamma ^{\\prime }(x)$ and $\\gamma ^{\\prime \\prime }(x)$ are linearly independent for all $x\\in {\\mathbb {R}}$ .", "Figure: Constructing the envelope L ϵ (x)L_\\epsilon (x) in 2D.The space of such projectively equivalent curves is in one-to-one correspondence with linear differential operators of the third order: $L=\\partial ^3+u_1(x)\\partial +u_0(x)$ , where the coefficients $u_0$ and $u_1$ are periodic in $x$ .", "Namely, a curve $\\gamma (x)$ in ${\\mathbb {RP}}^2$ can be lifted to a quasi-periodic curve $G=\\lbrace G(x)\\rbrace $ in ${\\mathbb {R}}^3$ satisfying $\\det |G(x), G^{\\prime }(x), G^{\\prime \\prime }(x)|=1$ for all $x\\in {\\mathbb {R}}$ .", "The components of the vector function $G(x)=(G_1(x),G_2(x),G_3(x))$ are homogenous coordinates of $\\gamma (x)$ in ${\\mathbb {RP}}^2$ : $\\gamma (x)=(G_1:G_2:G_3)(x)\\in {\\mathbb {RP}}^2$ .", "The vector function $G(x)$ can be identified with a solution of the unique linear differential operator $L$ , i.e., the components of $G(x)$ are identified with three linearly independent solutions of the differential equation $Ly=0$ .", "A continuous analog of the pentagram map is obtained by the following construction.", "Given a non-degenerate curve $\\gamma (x)$ , we draw the chord $(\\gamma (x-\\epsilon ),\\gamma (x+\\epsilon ))$ at each point $x$ .", "Consider the envelope $\\ell _\\epsilon (x)$ of these chords.", "(Figure 2 shows their lifts: chords $(G(x-\\epsilon ),G(x+\\epsilon ))$ and their envelope $L_\\epsilon (x)$ .)", "Let $u_{1,\\epsilon }$ and $u_{0,\\epsilon }$ be the periodic coefficients of the corresponding differential operator.", "Their expansions in $\\epsilon $ have the form $u_{i,\\epsilon }=u_i +\\epsilon ^2 w_i +{\\mathcal {O}}(\\epsilon ^3)$ and allow one to define the evolution $du_i/dt:=w_i$ , $i=0,1$ .", "After getting rid of $u_0$ this becomes the classical Boussinesq equation on the periodic function $u=u_1$ , which is the $(2,3)$ -flow in the KdV hierarchy of integrable equations on the circle: $u_{tt}+2(u^2)_{xx}+u_{xxxx}=0$ .", "Below we generalize these results to higher dimensions." ], [ "Pentagram map in 3D", "First we extend the notion of a closed polygon to a twisted one, similar to the 2D case.", "We present the 3D case first, before giving the definition of the pentagram map in arbitrary dimension, since it is used in many formulas below.", "Definition 3.1 A twisted $n$ -gon in $ {\\mathbb {RP}}^3$ with a monodromy ${ M} \\in SL(4,{\\mathbb {R}})$ is a map $\\phi : {\\mathbb {Z}}\\rightarrow {\\mathbb {RP}}^3$ , such that $\\phi (k+n) = M \\circ \\phi (k)$ for each $k\\in {\\mathbb {Z}}$ .", "(Here we consider the natural action of $SL(4,{\\mathbb {R}})$ on the corresponding projective space ${\\mathbb {RP}}^3$ .)", "Two twisted $n$ -gons are (projectively) equivalent if there is a transformation $g \\in SL(4,{\\mathbb {R}})$ , such that $g \\circ \\phi _1 = \\phi _2$ .", "Note that equivalent $n$ -gons must have similar monodromies.", "Closed $n$ -gons (space polygons) correspond to the monodromies $M=\\mathrm {Id}$ and $-\\mathrm {Id}$ .", "Let us assume that vertices of an $n$ -gon are in general position, i.e., no four consecutive vertices belong to one and the same plane in ${\\mathbb {RP}}^3$ .", "Also, assume that $n$ is odd.", "Then one can show (see Section REF below and cf.", "Proposition 4.1 in [11]) that there exists a unique lift of the vertices $v_k:=\\phi (k) \\in {\\mathbb {RP}}^3$ to the vectors $V_k \\in {\\mathbb {R}}^4$ satisfying for all $j\\in {\\mathbb {Z}}$ the identities $\\det |V_j, V_{j+1}, V_{j+2}, V_{j+3}|=1$ and $V_{j+n}=MV_j$ , where $M\\in SL(4,{\\mathbb {R}})$ .This explains our choice of the group $SL(4,{\\mathbb {R}})$ rather than the seemingly more natural group $PSL(4,{\\mathbb {R}})$ : since $SL(4,{\\mathbb {R}})$ is a two-fold cover of $PSL(4,{\\mathbb {R}})$ , a twisted $n$ -gon would have two different lifts from ${\\mathbb {RP}}^3$ to $ {\\mathbb {R}}^4$ corresponding to two different lifts of the monodromy $M$ from the latter group.", "These vectors satisfy difference equations $V_{j+4} = a_j V_{j+3} + b_j V_{j+2} + c_j V_{j+1} - V_j, \\; j \\in {\\mathbb {Z}},$ with $n$ -periodic coefficients $(a_j , b_j, c_j)$ and we employ this equation to introduce the $(a,b,c)$ -coordinates on the space of twisted $n$ -gons.", "Define the pentagram map on the classes of equivalent $n$ -gons in such a way.", "Figure: In 3D the image Tv k Tv_k of the vertex v k v_k is the intersection of three “short-diagonal\" planes P k-1 ,P k ,P_{k-1}, P_k, and P k+1 P_{k+1}.Definition 3.2 Given an $n$ -gon $\\phi $ in ${\\mathbb {RP}}^3$ , for each $k\\in {\\mathbb {Z}}$ consider the two-dimensional “short-diagonal plane\" $P_k:=(v_{k-2}, v_k, v_{k+2})$ passing through 3 vertices $v_{k-2}, v_k, v_{k+2}$ .", "Take the intersection point of the three consecutive planes $P_{k-1}, P_k, P_{k+1}$ and call it the image of the vertex $v_k$ under the space pentagram map $T$ , see Figure 3.", "(We assume the general position, so that every three consecutive planes $P_k$ for the given $n$ -gon intersect at a point.)", "By lifting a two-dimensional plane $P_k$ from ${\\mathbb {RP}}^3$ to the three-dimensional plane through the origin in ${\\mathbb {R}}^4$ (and slightly abusing notation) we have $P_k =*(V_{k-2} \\wedge V_k \\wedge V_{k+2}) $ in terms of the natural duality $*$ between ${\\mathbb {R}}^4$ and ${\\mathbb {R}}^{4*}$ .", "The lift of $Tv_k$ to ${\\mathbb {R}}^4$ is proportional to $*[P_{k-1} \\wedge P_k \\wedge P_{k+1}]$ .", "Below we describe the properties of this space pentagram map in detail." ], [ "Pentagram map in any dimension", "Before defining the pentagram map in ${\\mathbb {RP}}^d$ , recall that $SL(d+1,{\\mathbb {R}})$ is a two-fold cover of $PSL(d+1,{\\mathbb {R}})$ for odd $d$ and coincides with the latter for even $d$ .", "Definition 3.3 A twisted $n$ -gon in ${\\mathbb {RP}}^d$ with a monodromy $M \\in SL(d+1,{\\mathbb {R}})$ is a map $\\phi : {\\mathbb {Z}}\\rightarrow {\\mathbb {RP}}^d$ , such that $\\phi (k+n) = M \\circ \\phi (k)$ for each $k\\in {\\mathbb {Z}}$ , and where $ M$ acts naturally on ${\\mathbb {RP}}^d$ .", "We define the $SL(d+1,{\\mathbb {R}})$ -equivalence of $n$ -gons as above, and assume the vertices $v_k:=\\phi (k)$ to be in general position, i.e., in particular, no $d+1$ consecutive vertices of an $n$ -gon belong to one and the same $(d-1)$ -dimensional plane in ${\\mathbb {RP}}^d$ .", "Remark 3.4 One can show that there exists a unique lift of the vertices $v_k=\\phi (k) \\in {\\mathbb {RP}}^d$ to the vectors $V_k \\in {\\mathbb {R}}^{d+1}$ satisfying $\\det |V_j, V_{j+1}, ..., V_{j+d}|=1$ and $ V_{j+n}=MV_j,\\; j \\in {\\mathbb {Z}},$ where $M\\in SL(d+1,{\\mathbb {R}})$ , if and only if the condition $gcd(n,d+1)=1$ holds.", "The corresponding difference equations have the form $V_{j+d+1} = a_{j,d} V_{j+d} + a_{j,d-1} V_{j+d-1} +...+ a_{j,1} V_{j+1} +(-1)^{d} V_j,\\quad j \\in {\\mathbb {Z}},$ with $n$ -periodic coefficients in the index $j$ .", "This allows one to introduce coordinates $\\lbrace a_{j,k} ,\\;0\\le j\\le n-1, \\; 1\\le k\\le d \\rbrace $ on the space of twisted $n$ -gons in ${\\mathbb {RP}}^d$ .", "For a generic twisted $n$ -gon in ${\\mathbb {RP}}^d$ one can define the “short-diagonal\" $(d-1)$ -dimensional plane $P_k$ passing through $d$ vertices of the $n$ -gon by taking every other vertex starting at the point $v_k$ , i.e., through the vertices $v_k, v_{k+2}, ..., v_{k+2d-2}$ .", "For calculations, however, it is convenient to have the set of vertices “centered\" at $v_k$ , and then the definition becomes slightly different in the odd and even dimensional cases.", "Namely, for odd dimension $d=2\\varkappa +1$ we consider the short-diagonal hyperplane $P_k$ through the $d$ vertices $P_k:=(v_{k-2\\varkappa }, v_{k-2\\varkappa +2},...,v_k, ..., v_{k+2\\varkappa })$ (thus including the vertex $v_k$ itself), while for even dimension $d=2\\varkappa $ we take $P_k$ passing through the $d$ vertices $P_k:=(v_{k-2\\varkappa +1}, v_{k-2\\varkappa +3}, ..., v_{k-1},v_{k+1}, ..., v_{k+2\\varkappa -1})$ (thus excluding the vertex $v_k$ ).", "Definition 3.5 The higher pentagram map $T$ takes a vertex $v_k$ of a generic twisted $n$ -gon in ${\\mathbb {RP}}^d$ to the intersection point of the $d$ consecutive short-diagonal planes $P_i$ around $v_k$ .", "Namely, for odd $d=2\\varkappa +1$ one takes the intersection of the planes $Tv_k:=P_{k-\\varkappa }\\cap P_{k-\\varkappa +1}\\cap ...\\cap P_k\\cap ...\\cap P_{k+\\varkappa }\\,,$ while for even $d=2\\varkappa $ one takes the intersection of the planes $Tv_k:=P_{k-\\varkappa +1}\\cap P_{k-\\varkappa +2}\\cap ...\\cap P_{k}\\cap ...\\cap P_{k+\\varkappa }\\,.$ The corresponding map $T$ is well defined on the equivalence classes of $n$ -gons in ${\\mathbb {RP}}^d$ .", "As usual, we invoke the generality assumption to guarantee that every $d$ consecutive hyperplanes $P_i$ intersect at one point in ${\\mathbb {RP}}^d$ .", "It turns out that the pentagram map defined this way has a special scaling invariance, which allows one to prove its integrability: Theorem C. (= Theorem REF ) The scale-invariant higher pentagram map is a discrete completely integrable system on equivalence classes of $n$ -gons in ${\\mathbb {RP}}^d$ .", "It has an explicit Lax representation with a spectral parameter.", "Remark 3.6 The pentagram map defined this way in 1D is the identity map.", "In the 2D case this definition was given in [13] and its integrability for twisted polygons was proved in [11], while for closed ones it was proved in [15], [12].", "Remark 3.7 One can also give an “asymmetric definition\" for planes $P_k$ , where more general sequences of $d$ vertices $v_{k_j}$ are used, and then $Tv_k$ is defined as the intersection of $d$ consecutive planes $P_k$ .", "It turns out, however, that exactly this “uniform\" definition of diagonal planes $P_k$ , where $P_k$ passes through every other vertex, leads to integrability of the pentagram map.", "One of possible definitions of the pentagram map discussed in [9] coincides with ours in 3D.", "In that definition one takes the intersection of a (possibly asymmetric, but containing the vertex $v_k$ ) plane $P_k$ and the segment $[v_{k-1}, v_{k+1}]$ : for our centered choice of $P_k$ this segment belongs to both planes $P_{k-1}$ and $P_{k+1}$ .", "The definitions become different in higher dimensions." ], [ "General pentagram maps and duality", "We define more general pentagram maps $T_{p,r}$ depending on two integral parameters in arbitrary dimension $d$ .", "These parameters specify the diagonal planes and which of them to intersect.", "Definition 3.8 For a generic twisted $n$ -gon in ${\\mathbb {RP}}^d$ one can define a $p$ -diagonal hyperplane $P_k$ as the one passing through $d$ vertices of the $n$ -gon by taking every $p$ th vertex starting at the point $v_k$ , i.e., $P_k:=(v_k, v_{k+p}, ..., v_{k+(d-1)p})\\,.$ The image of the vertex $v_k$ under the general pentagram map $T_{p,r}$ is defined by intersecting every $r$ th out of the $p$ -diagonal hyperplanes starting with $P_k$ : $T_{p,r}v_k:=P_{k}\\cap P_{k+r}\\cap ...\\cap P_{k+(d-1)r}\\,.$ The corresponding map $T_{p,r}$ is considered on the space of equivalence classes of $n$ -gons in ${\\mathbb {RP}}^d$ .", "For the higher pentagram map $T$ discussed in Section REF one has $p=2$ , $r=1$ , and the indices in the definition of $P_k$ are “centered\" at $v_k$ .", "In other words, $T=T_{2,1}\\circ Sh$ , where $Sh$ stands for some shift in the vertex index.", "Below we denote by $Sh$ any shift in the index without specifying the shift parameter.", "Note that $T_{p,p}=Sh$ .", "Theorem 3.9 There is a duality between the general pentagram maps $T_{p,r}$ and $T_{r,p}$ : $T_{p,r}=T^{-1}_{r,p}\\circ Sh\\,.$ For example, the map $T_{1,2}$ in 2D is defined by extending the sides of a polygon and intersecting them with the “second neighbouring\" sides.", "This corresponds exactly to passing from $T(P)$ back to $P$ in Figure 1, i.e., it is the inverse of $T$ modulo the numeration of vertices." ], [ "Proof.", "To prove this theorem we introduce the following “duality maps,” cf.", "[11].", "Definition 3.10 Given a generic sequence of points $\\phi (j) \\in {\\mathbb {RP}}^d, \\; j \\in {\\mathbb {Z}},$ and a nonzero integer $p$ we define the sequence $\\alpha _p(\\phi (j))\\in ({\\mathbb {RP}}^d)^*$ as the plane $\\alpha _p(\\phi (j)):=(\\phi (j), \\phi (j+p),..., \\phi (j+(d-1)p))\\,.$ The following proposition is straightforward for our definition of the general pentagram map.", "Proposition 3.11 For every nonzero $p$ the maps $\\alpha _p$ are involutions modulo index shifts (i.e., $\\alpha _p^2=Sh$ ), they commute with the shifts (i.e., $\\alpha _p\\circ Sh= Sh\\circ \\alpha _p$ ), and the general pentagram map is the following composition: $T_{p,r}=\\alpha _r\\circ \\alpha _p\\circ Sh$ .", "To complete the proof of the theorem we note that $T^{-1}_{r,p}=(\\alpha _p\\circ \\alpha _r\\circ Sh)^{-1}=Sh^{-1}\\circ \\alpha _r^{-1}\\circ \\alpha _p^{-1}=Sh \\circ \\alpha _r \\circ Sh\\circ \\alpha _p\\circ Sh=\\alpha _r \\circ \\alpha _p\\circ Sh =T_{p,r}\\circ Sh,$ since $\\alpha _p^{-1}=\\alpha _p\\circ Sh$ from the Proposition above, while $Sh^{-1}=Sh$ and $Sh\\circ Sh=Sh$ .", "$\\Box $ Below we construct a Lax form for the higher pentagram maps, i.e., for the maps $T_{2,1}$ (and hence for $T_{1,2}$ as well) for any $d$ .", "Integrability of the pentagram map on a special class of the so-called corrugated twisted polygons in ${\\mathbb {RP}}^d$ was proved in [3], which should imply the integrability of the pentagram map $T_{p,1}$ in 2D.", "Then the above duality would also give integrability of $T_{1,p}$ in ${\\mathbb {RP}}^2$ , defined as the intersection of a pair of polygon edges whose numbers differ by $p$ .", "(One should also mention that for $p$ and $r$ mutually prime with $n$ one can get rid of one of the parameters by appropriately renumbering vertices, at least for the closed $n$ -gon case, cf.", "[12] in 2D.", "This reduces the study to the map $T_{p,1}$ for some values of $n$ .)", "Complete integrability of general pentagram maps for other pairs $(p,r)$ in ${\\mathbb {RP}}^d$ is a wide open problem.", "Problem 3.12 Which of the general pentagram maps $T_{p,r}$ in ${\\mathbb {RP}}^d$ are completely integrable systems?" ], [ "Definition of the continuous limit", "In this section we consider the continuous limit of polygons and the pentagram map on them.", "In the limit $n\\rightarrow \\infty $ a twisted $n$ -gon becomes a smooth quasi-periodic curve $\\gamma (x)$ in ${\\mathbb {RP}}^d$ .", "Its lift $G(x)$ to ${\\mathbb {R}}^{d+1}$ is defined by the conditions: $i)$ the components of the vector function $G(x):=(G_1(x),...,G_{d+1}(x))$ provide homogeneous coordinates for $\\gamma (x)=(G_1:...:G_{d+1})(x)$ in ${\\mathbb {RP}}^d$ , $ii)$ $\\det |G,G^{\\prime },...,G^{(d)}|(x)=1$ for all $x\\in {\\mathbb {R}}$ , and $iii)$ $G(x+2\\pi )=MG(x)$ for a given $M\\in SL(d+1,{\\mathbb {R}})$ .", "Then $G$ satisfies a linear differential equation of order $d+1$ : $G^{(d+1)}+u_{d-1}(x)G^{(d-1)}+...+u_1(x)G^{\\prime }+u_0(x)G=0$ with periodic coefficients $u_i(x)$ .", "Here and below $^{\\prime }$ stands for $d/dx$ .", "Let us consider the case of odd $d=2\\varkappa +1$ .", "Fix a small $\\epsilon >0$ .", "A continuous analog of the hyperplane $P_k$ is the hyperplane $P_\\epsilon (x)$ passing through $d$ points $\\gamma (x-\\varkappa \\epsilon ),..., \\gamma (x),...,\\gamma (x+\\varkappa \\epsilon )$ of the curve $\\gamma $ .For a complete analogy with the discrete case, one could take the points $\\gamma (x-2\\varkappa \\epsilon ),\\gamma (x-2(\\varkappa -1)\\epsilon ),..., \\gamma (x),...,\\gamma (x+2\\varkappa \\epsilon )$ .", "However, one can absorb the factor 2 by rescaling $\\epsilon \\rightarrow 2 \\epsilon $ .", "Note that as $\\epsilon \\rightarrow 0$ the hyperplanes $P_\\epsilon (x)$ tend to the osculating hyperplane of the curve $\\gamma $ spanned by the vectors $\\gamma ^{\\prime }(x), \\gamma ^{\\prime \\prime }(x),...,\\gamma ^{(d-1)}(x)$ at the point $\\gamma (x)$ .", "Let $\\ell _\\epsilon (x)$ be the envelope curve for the family of hyperplanes $P_\\epsilon (x)$ for a fixed $\\epsilon $ .", "The envelope condition means that for each $x$ the point $\\ell _\\epsilon (x)$ and the derivative vectors $\\ell ^{\\prime }_\\epsilon (x),...,\\ell ^{(d-1)}_\\epsilon (x)$ belong to the plane $P_\\epsilon (x)$ .", "This means that the lift of $\\ell _\\epsilon (x)$ to $L_\\epsilon (x)$ in ${\\mathbb {R}}^{d+1}$ satisfies the system of $d=2\\varkappa +1$ equations (see Figure 4): $\\det | G(x-\\varkappa \\epsilon ), G(x-(\\varkappa -1)\\epsilon ), ... , G(x),..., G(x+\\varkappa \\epsilon ), L^{(j)}_\\epsilon (x) |=0,\\quad j=0,...,d-1.$ Figure: The envelope L ϵ (x)L_\\epsilon (x) in 3D.", "The point L ϵ (x)L_\\epsilon (x) and the vectors L ϵ ' (x)L_\\epsilon ^{\\prime }(x) and L ϵ '' (x)L_\\epsilon ^{\\prime \\prime }(x) belong to the plane (G(x-ϵ),G(x),G(x+ϵ))(G(x-\\epsilon ), G(x), G(x+\\epsilon )).Similarly, for even $d=2\\varkappa $ the lift $L_\\epsilon (x)$ satisfies the system of $d$ equations: $\\det | G(x-(2\\varkappa -1)\\epsilon ), G(x-(2\\varkappa -3)\\epsilon ), ... , G(x-\\epsilon ),G(x+\\epsilon ),...$ $...,G(x+(2\\varkappa -1)\\epsilon ), L^{(j)}_\\epsilon (x) |=0,\\quad j=0,...,d-1.$ The evolution of the curve $\\gamma $ in the direction of the envelope $\\ell _\\epsilon $ , as $\\epsilon $ changes, defines a continuous limit of the pentagram map $T$ .", "Namely, below we show that the expansion of $L_\\epsilon (x)$ has the form $L_\\epsilon (x)=G(x)+\\epsilon ^2 B(x)+{\\mathcal {O}} (\\epsilon ^4).$ The family of functions $L_\\epsilon (x)$ satisfies a family of differential equations: $L_\\epsilon ^{(d+1)}+u_{d-1,\\epsilon }(x)L_\\epsilon ^{(d-1)}+...+u_{1,\\epsilon }(x)L_\\epsilon ^{\\prime }+u_{0,\\epsilon }(x)L_\\epsilon =0,$ where $u_{j,0}(x)=u_{j}(x).$ Expanding the coefficients $u_{j,\\epsilon }(x)$ as $u_{j,\\epsilon }(x)=u_{j}(x)+\\epsilon ^2w_j(x)+{\\mathcal {O}}(\\epsilon ^4)$ , we define the continuous limit of the pentagram map $T$ as the system of evolution differential equations $du_j(x)/dt\\, =w_j(x)$ for $j=0,...,d-1$ , i.e., $\\epsilon ^2$ plays the role of time.", "Theorem A.", "The continuous limit $du_j(x)/dt\\, =w_j(x), \\, j=0,...,d-1$ for $x\\in S^1$ of the pentagram map is the $(2, d+1)$ -KdV equation of the Adler-Gelfand-Dickey hierarchy on the circle.", "This theorem is proved as a combination of Theorems REF and REF below.", "Remark 4.1 Recall the definition of the KdV hierarchy (after Adler-Gelfand-Dickey, [1]).", "Let $L$ be a linear differential operator of order $d+1$ : $L = \\partial ^{d+1} + u_{d-1}(x) \\partial ^{d-1} + u_{d-2}(x) \\partial ^{d-2} + ...+ u_1(x) \\partial + u_0(x)$ with periodic coefficients $u_j(x)$ , where $\\partial ^{k}$ stands for $d^k/dx^k$ .", "One can define its fractional power $L^{m/{d+1}}$ as a pseudo-differential operator for any positive integer $m$ and take its purely differential part $Q_m :=(L^{m/{d+1}})_+$ .", "In particular, for $m=2$ one has $Q_2= \\partial ^2 + \\dfrac{2}{d+1}u_{d-1}(x) $ .", "Then the $(m, d+1)$ -KdV equation is the following evolution equation on (the coefficients of) $L$ : $\\frac{d}{dt}{L} = [Q_m,L] \\,.$ Remark 4.2 a) For $d=1$ the discrete pentagram map is the identity map, hence the continuous limit is trivial, which is consistent with vanishing of the (2,2)-KdV equation.", "b) For $d=2$ the (2,3)-KdV equation is the classical Boussinesq equation, found in [11].", "c) Apparently, the $(2, d+1)$ -KdV equation is a very robust continuous limit.", "One obtains it not only for the pentagram map defined by taking every other vertex, but also for a non-symmetric choice of vertices for the plane $P_k$ , see Remark REF .", "Also, the limit remains the same if instead of taking the envelopes one starts with a map defined by taking intersections of various planes [9]." ], [ "Envelopes and the KdV hierarchy", "Theorem 4.3 For any dimension $d$ , the envelope $L_\\epsilon (x) $ has the expansion $L_\\epsilon (x) = G(x)+ {\\epsilon ^2} \\, C_d \\left( G^{\\prime \\prime }(x) +\\dfrac{2}{d+1}u_{d-1}G(x) \\right) +{\\mathcal {O}}(\\epsilon ^4)$ for a certain constant $C_d$ , as $ \\epsilon \\rightarrow 0$ .", "The $\\epsilon ^2$ -term of this expansion can be rewritten as $C_d\\left(\\partial ^2+ \\dfrac{2}{d+1}u_{d-1}(x) \\right)G(x).$ Consequently, it defines the following evolution of the curve $G(x)$ : $\\frac{d}{dt} G= \\left(\\partial ^2+ \\dfrac{2}{d+1}u_{d-1}\\right)G.$" ], [ "Proof.", "Since $L_\\epsilon $ approaches $G $ as $\\epsilon \\rightarrow 0$ , one has the expansion $L_\\epsilon =G+\\epsilon A+\\epsilon ^2 B+\\epsilon ^3 C+{\\mathcal {O}}(\\epsilon ^4).$ First we note that the expansion of $L_\\epsilon $ in $\\epsilon $ has only even powers of $\\epsilon $ , since the equations (REF ) and (REF ) defining $L_\\epsilon $ have the symmetry $\\epsilon \\rightarrow - \\epsilon $ .", "Therefore, we have $A=C=0$ and $L_\\epsilon =G+\\epsilon ^2 B+{\\mathcal {O}}(\\epsilon ^4).$ Notice that $G(x)$ with its first $d$ derivatives form a basis in ${\\mathbb {R}}^{d+1}$ for each $x$ .", "We express the vector coefficient $B$ in this basis: $B=b_0(x) G+b_1(x) G^{\\prime }+ ...+ b_d(x) G^{(d)}$ .", "Recall that, e.g., for odd $d=2\\varkappa +1$ the lift $L_\\epsilon (x)$ satisfies the system of $d$ equations: $\\det |\\, G(x-\\varkappa \\epsilon ), G(x-(\\varkappa -1)\\epsilon ), ... , G(x),...,G(x+\\varkappa \\epsilon ), L^{(j)}_\\epsilon (x) \\,|=0,\\quad j=0,...,d-1.$ Fix $x$ and expand all terms in $\\epsilon $ : e.g., $G(x+\\epsilon )=G(x)+\\epsilon G^{\\prime }(x)+\\frac{\\epsilon ^2}{2}G^{\\prime \\prime }+...$ , etc.", "In each equation consider the coefficients at the lowest power of $\\epsilon $ , being $2+{d(d-1)}/{2}$ here.", "The equation with $j=0$ gives $C_d\\det |G, G^{\\prime },...,G^{(d-1)}, B|=0 $ for some nonzero $C_d$ , which implies that there is no $G^{(d)}$ -term in the expansion of $B$ .", "Similarly, for $j=1$ we obtain $C_d\\det |G, G^{\\prime },...,G^{(d-1)}, B^{\\prime }|=0 $ , which means that there is no $G^{(d)}$ -term in the expansion of $B^{\\prime }$ , or, equivalently, there is no $G^{(d-1)}$ -term in the expansion of $B$ .", "Using this argument for $j=0,1,...,d-3,$ and $d-1$ , we deduce that $B$ contains no terms with $G^{(d)}, ..., G^{\\prime \\prime \\prime }$ and $G^{\\prime }$ .", "The equation with $j=d-2$ results in a different term in the expansion and gives $\\det |G, G^{\\prime },...,G^{(d-1)}, B^{(d-2)}|=C_d\\, \\det |G, G^{\\prime },...,G^{(d-1)}, G^{(d)}| \\,,$ which implies that $B=C_dG^{\\prime \\prime }+b(x)G$ for some function $b(x)$ .", "Finally, the normalization $\\det | \\,L_\\epsilon , L^{\\prime }_\\epsilon , ..., L^{(d)}_\\epsilon \\,|=1$ allows one to find $b(x)$ by plugging in it $L_\\epsilon =G+\\epsilon ^2 (C_dG^{\\prime \\prime }+b(x)G)+{\\mathcal {O}}(\\epsilon ^4).$ For the $\\epsilon ^2$ -terms one obtains $C_d\\det |\\,G,G^{\\prime },...,G^{(d-2)},G^{(d+1)},G^{(d)}\\,|+ C_d\\det |\\,G,G^{\\prime },...,G^{(d-2)},G^{(d-1)},G^{(d+2)}\\,|$ $+(d+1) b(x)\\det |\\,G,G^{\\prime },...,G^{(d-2)},G^{(d-1)},G^{(d)}\\,|=0\\,.$ By using the linear differential equation $G^{(d+1)}+u_{d-1}(x)G^{(d-1)}+...+u_0(x)G=0$ to express $G^{(d+1)}$ and $G^{(d+2)}$ via lower derivatives we see that the first and the second determinants are equal to $-u_{d-1}$ , while the last one is equal to 1.", "Thus one has $(d+1) b(x)-2C_d\\,u_{d-1}(x)=0$ , which gives $b(x)=C_d\\frac{2}{d+1}u_{d-1}(x)$ .", "Hence $L_\\epsilon =G+\\epsilon ^2 \\,C_d\\,(G^{\\prime \\prime }+\\frac{2}{d+1}u_{d-1}G)+{\\mathcal {O}}(\\epsilon ^4)$ , as required.", "$\\Box $ Remark 4.4 One can see that the only condition on vertices defining the hyperplane $P_k$ required for the proof above is that they are distinct.", "A different choice of vertices for the hyperplane $P_k$ changes the constant $C_d$ , but does not affect the evolution equation for $G$ .", "The above theorem for an envelope $L_\\epsilon (x) $ is similar to an analogous expansion in [9] for a curve defined via certain plane intersections.", "Theorem 4.5 In any dimension $d$ the continuous limit of the pentagram map defined by the evolution $\\frac{d}{dt} G= \\left(\\partial ^2+ \\dfrac{2}{d+1}u_{d-1} \\right)G$ of the curve $G$ coincides with the $(2,d+1)$ -KdV equation.", "Consequently, it is an infinite-dimensional completely integrable system." ], [ "Proof.", "Recall that the $(2,d+1)$ -KdV equation is defined as the evolution $\\dot{L}=[Q_2, L]:=Q_2L-LQ_2,$ where the linear differential operator $L$ of order $d+1$ is given by formula (REF ) and $Q_2=\\partial ^2+ \\dfrac{2}{d+1}u_{d-1} $ .", "Here $\\dot{L}$ stands for $dL/dt$ .", "By assumption, the evolution of the curve $G$ is described by the differential equation $\\dot{G} =Q_2G$ .", "We would like to find the evolution of the operator $L$ tracing it.", "For any $t$ , the curve $G$ and the operator $L$ are related by the differential equation $LG=0$ of order $d+1$ .", "Consequently, $\\dot{L} G + L \\dot{G}=0.$ Note that if the operator $L$ satisfies the $(2,d+1)$ -KdV equation and $G$ satisfies $\\dot{G} =Q_2G$ , we have the identity: $\\dot{L} G + L \\dot{G}=(Q_2L-LQ_2) G + L Q_2G= Q_2LG=0\\,.$ In virtue of the uniqueness of the linear differential operator of the form (REF ) for a given fundamental set of solutions, we obtain that indeed the evolution of $L$ is described by the $(2,d+1)$ -KdV equation.", "$\\Box $ Remark 4.6 The proof above is reminiscent of the one used in [10] to study symplectic leaves of the Gelfand-Dickey brackets.", "Note that the absence of the term linear in $\\epsilon $ is related to the symmetric choice of vertices for the hyperplane $P_k$ .", "For a non-symmetric choice the evolution would be defined by the linear term in $\\epsilon $ and given by the equation $\\dot{G}=G^{\\prime }$ .", "This is the initial, $(1,d+1)$ -equation of the corresponding KdV hierarchy, manifesting the fact that the space $x$ -variable can be regarded as the “first time\" variable.", "A natural question arises whether the whole KdV-hierarchy is hidden as an appropriate limit of the pentagram map.", "An evidence to this is given by noticing that the terms with higher powers in $\\epsilon $ lead to equations similar to the higher equations in the KdV hierarchy, see Appendix REF .", "Remark 4.7 One can see that the continuous limit of the general pentagram maps $T_{p,r}$ for various $p\\ne r$ in ${\\mathbb {RP}}^d$ defined via envelopes for a centered choice of vertices is the same $(2,d+1)$ -KdV flow, i.e., the limit depends only on the dimension.", "Indeed, an analog of the $p$ -diagonal is the plane $P_\\epsilon (x)$ passing through the points $G(x), G(x+\\epsilon p), ..., G(x+\\epsilon (d-1)p)$ .", "Rescaling $\\epsilon $ , we can assume the points to be $G(x), G(x+\\epsilon ), ..., G(x+\\epsilon (d-1))$ , which leads to the planes $P_\\epsilon (x)$ defined in Section REF after a shift in $x$ .", "Then the definition of $L_\\epsilon (x)$ via the envelope of such planes will give the same $(2,d+1)$ -KdV equation.", "It would be interesting to define the limit of the intersections of every $r$ th plane via some higher-order terms of the envelope, as mentioned in the above remark, so that it could lead to other $(m, d+1)$ -equations in the KdV hierarchy." ], [ "Two involutions", "Now we return to the 3D case.", "In this section we assume that $n$ is odd and consider $n$ -gons in ${\\mathbb {RP}}^3$ .", "Recall that in this case an $n$ -gon with a given monodromy $M\\in SL(4,{\\mathbb {R}})$ lifts uniquely to ${\\mathbb {R}}^4$ and is described by difference equations $V_{j+4} = a_j V_{j+3} + b_j V_{j+2} + c_j V_{j+1} - V_j,\\quad j \\in {\\mathbb {Z}},$ with $n$ -periodic coefficients $(a_j,b_j,c_j)$ .", "In other words, for odd $n$ the variables $(a_j,b_j,c_j), \\; 0 \\le j \\le n-1,$ provide coordinates on the space $\\mathcal {P}_n$ of twisted $n$ -gons in ${\\mathbb {RP}}^3$ considered up to projective equivalence (see Proposition REF ).", "In order to find explicit formulas for the pentagram map, we present it as a composition of two involutions $\\alpha $ and $\\beta $ , cf.", "Section REF , and then find the formulas for each of them separately.", "The same approach was used in [11] in 2D, although the formulas in 3D are more complicated.", "Definition 5.1 Given a sequence of points $\\phi (j) \\in {\\mathbb {RP}}^3, \\; j \\in {\\mathbb {Z}},$ define two sequences $\\alpha (\\phi (j))\\in ({\\mathbb {P}}^3)^*$ and $\\beta (\\phi (j))\\in ({\\mathbb {RP}}^3)^*$ , where $a)$ $\\alpha (\\phi (j))$ is the plane $(\\phi (j-1), \\phi (j), \\phi (j+1))$ ; $b)$ $\\beta (\\phi (j))$ is the plane $(\\phi ((j-2), \\phi (j), \\phi (j+2))$ .", "Proposition 5.2 The maps $\\alpha $ and $\\beta $ are involutions, i.e., $\\alpha ^2=\\beta ^2=\\rm {Id}$ , while the pentagram map is their composition: $T=\\alpha \\circ \\beta $ .", "Note that the indices which define the planes are symmetric with respect to $j$ .", "As a result, we do not have an extra shift of indices, cf.", "Proposition REF (unlike the 2D case and the general map $\\alpha _p$ ).", "Lemma 5.3 The involution $\\alpha : V_i \\rightarrow W_i = *(V_i \\wedge V_{i-1} \\wedge V_{i+1})$ maps equation (REF ) to the following difference equation: $W_{i+4} = c_{i+1} W_{i+3} + b_i W_{i+2} + a_{i-1} W_{i+1} - W_i.$ Lemma 5.4 The involution $\\beta : V_i \\rightarrow W_i = \\lambda _i *(V_i \\wedge V_{i-2} \\wedge V_{i+2})$ maps equation (REF ) to the difference equation $W_{i+4} = A_i W_{i+3} + B_i W_{i+2} + C_i W_{i+1} - W_i,$ where the coefficients $A_i,B_i,C_i$ are defined as follows: $A_i &= c_{i-1} (a_i a_{i+2} + a_{i+2} b_{i+1} c_i + c_i c_{i+2})^2 \\lambda _{i+1} \\lambda _{i+2} \\lambda _{i+4}^2,\\\\B_i &= ((a_{i-2}+b_{i-1}c_{i-2})(c_{i+2}+a_{i+2}b_{i+1})-a_{i+2}c_{i-2}) \\times \\\\&\\times (a_{i-1} a_{i+1} + a_{i+1} b_i c_{i-1} + c_{i-1} c_{i+1}) \\lambda _i \\lambda _{i+1} \\lambda _{i+3} \\lambda _{i+4},\\\\C_i &= a_{i+1} (a_i a_{i+2} + a_{i+2} b_{i+1} c_i + c_i c_{i+2})^2 \\lambda _{i+2} \\lambda _{i+3} \\lambda _{i+4}^2.$ The sequence $\\lambda _i,\\; i \\in {\\mathbb {Z}},$ is $n$ -periodic and is uniquely determined by the condition $\\lambda _i \\lambda _{i+1} \\lambda _{i+2} \\lambda _{i+3} =\\dfrac{1}{(a_{i-2} a_i + a_i b_{i-1} c_{i-2} + c_{i-2} c_i)(a_{i-1} a_{i+1} + a_{i+1} b_i c_{i-1} + c_{i-1} c_{i+1})}.$ The proofs of these lemmas are straightforward computations, which we omit.", "Combined together, these lemmas provide formulas for the pentagram map.", "They have, however, one drawback: one needs to solve a system of equations in $\\lambda _i,\\; i \\in {\\mathbb {Z}},$ which results in the non-local character of the formulas in $(a,b,c)$ -coordinates and their extreme complexity." ], [ "Cross-ratio type coordinates", "Similarly to the 2D case, there exist alternative coordinates on the space $\\mathcal {P}_n$ .", "They are defined for any $n$ , and the formulas for the pentagram map become local, i.e., involving the vertex $\\phi (j)$ itself and several neighboring ones.", "Definition 5.5 For odd $n$ the variables $x_j = \\dfrac{b_{j+1}}{a_j a_{j+1}}, \\quad y_j =\\dfrac{a_j}{b_{j+1} c_j}, \\quad z_j =\\dfrac{c_{j+1}}{a_{j+1} b_j}, \\quad 0 \\le j \\le n-1,$ provide coordinates on the space $\\mathcal {P}_n$ , where the $n$ -periodic variables $(a_j,b_j,c_j),\\; j \\in {\\mathbb {Z}},$ are defined by the difference equation (REF ).", "It turns out that the variables $(x_j,y_j,z_j),\\; 0 \\le j \\le n-1$ are well defined and independent for any $n$ , even or odd.", "Below we provide two (equivalent) ways to define them for even $n$ : a pure geometric (local) definition of these variables (see Proposition REF ) and the above definition extended to quasi-periodic sequences $(a_j,b_j,c_j)$ (see Section REF ).", "Theorem 5.6 In the coordinates $x_i, y_i, z_i$ the pentagram map for any (either odd or even) $n$ is given by the formulas: $T^*(x_i) &= x_{i+1} \\dfrac{1+y_{i-1}+z_{i+2}+y_{i-1}z_{i+2}-y_{i+1} z_i}{1+y_{i-1}+z_i},\\\\T^*(y_i) &= \\dfrac{x_{i-1} y_{i-1}z_i}{x_i z_{i-1}} \\dfrac{(1+y_{i+1}+z_{i+2})(1+y_{i-2}+z_{i-1})}{(1+y_i+z_{i+1})(1+y_{i-1}+z_{i+2}+y_{i-1} z_{i+2}-y_{i+1}z_i)},\\\\T^*(z_i) &= \\dfrac{x_{i+1} z_i}{x_i}\\dfrac{(1+y_{i+1}+z_{i+2})(1+y_{i-2}+z_{i-1})}{(1+y_{i-1}+z_i)(1+y_{i-2}+z_{i+1}-y_i z_{i-1}+y_{i-2}z_{i+1})}.$ Before proving this theorem we describe the $(x_i,y_i,z_i)$ coordinates in greater detail.", "It turns out that they may be defined completely independently of $(a_i,b_i,c_i)$ in the following geometric way.", "Recall that the $x,y$ coordinates for the 2D pentagram map are defined as cross-ratios for quadruples of points on the line $(V_i, V_{i+1})$ , where two points are these vertices themselves, and two others are intersections of this line with extensions of the neighbouring edges.", "Similarly, the next proposition describes the new coordinates via cross-ratios of quadruples of points, 2 of which are the vertices of an edge, and 2 others are the intersection of the edge extension with two planes.", "For instance, the variable $y_i$ is the cross-ratio of 4 points on the line $(V_i, V_{i+1})$ , two of which are $V_i$ and $V_{i+1}$ , while two more are constructed as intersections of this line with the planes via the triple $(V_{i+2}, V_{i+3}, V_{i+4})$ and with the plane via the triple $(V_{i+2}, V_{i+4}, V_{i+5})$ .", "More precisely, the following proposition holds.", "Proposition 5.7 The coordinates $x_i,y_i,z_i$ are given by the cross-ratios: $x_i &= -[V_{i+4},V_{i+5},\\Phi ^{45}_{012},\\Phi ^{45}_{123}],\\\\y_i &= -[V_i,V_{i+1},\\Phi ^{01}_{234},\\Phi ^{01}_{245}],\\\\z_i &= -[V_{i+4},V_{i+5},\\Phi ^{45}_{013},\\Phi ^{45}_{123}],$ where the point $\\Phi ^{j_1,j_2}_{m_1,m_2,m_3}$ for a given $i$ is the intersection of the line $(V_{i+j_1},V_{i+j_2})$ with the plane $(V_{i+m_1},V_{i+m_2},V_{i+m_3})$ .", "By the very definition these coordinates are projectively invariant." ], [ "Proof of proposition.", "If $*$ is the Hodge star operator with respect to the Euclidean metric in ${\\mathbb {R}}^4\\,(\\,\\mathrm { or~ }4)$ , then $\\Phi ^{j_1,j_2}_{m_1,m_2,m_3}=*\\left( *(V_{i+j_1} \\wedge V_{i+j_2}) \\wedge *(V_{i+m_1} \\wedge V_{i+m_2} \\wedge V_{i+m_3}) \\right).$ It suffices to prove the proposition in the case of an odd $n$ , because the formulas are local, and we can always add a vertex to change the parity of $n$ .", "Therefore, we may assume that $(a_j,b_j,c_j)$ are global coordinates and use them for the proof.", "A simple computation shows that $\\Phi ^{45}_{012} &= -(b_{i+1} + a_i a_{i+1}) V_{i+4} + a_i V_{i+5},\\\\\\Phi ^{01}_{234} &= V_i - c_i V_{i+1},\\\\\\Phi ^{01}_{245} &= -b_{i+1} V_i + (a_i+c_i b_{i+1})V_{i+1},\\\\\\Phi ^{45}_{013} &= (c_{i+1}+b_i a_{i+1})V_{i+4}-b_i V_{i+5},\\\\\\Phi ^{45}_{123} &= -a_{i+1} V_{i+4} + V_{i+5}.$ Recall (see Lemma 4.5 in [11]) that if 4 vectors $a,b,c,d \\in {\\mathbb {R}}^4\\,(\\mathrm { or~ }4)$ lie in the same 2-dimensional plane and are such that $c = \\lambda _1 a + \\lambda _2 b, \\quad d = \\mu _1 a + \\mu _2 b,$ then the cross-ratio of the lines spanned by these vectors in the plane is given by $[a,b,c,d] = \\dfrac{\\lambda _2 \\mu _1 - \\lambda _1 \\mu _2}{\\lambda _2 \\mu _1}.$ Comparing the cross-ratios with the original definition of the variables $x_i,y_i,z_i$ concludes the proof.", "$\\Box $ Now we are in a position to prove the explicit local formulas in Theorem REF .", "The proof is similar, but more involved than that of Proposition 4.11 in [11]." ], [ "Proof of theorem.", "Due to the local character of the formulas for the pentagram map $T$ in $(x_j,y_j,z_j)$ -coordinates, we may always add an extra vertex to make the number $n$ of vertices odd, and then use coordinates $(a_j,b_j,c_j)$ and Lemmas REF and REF for the proof.", "The pentagram map is a composition $T = \\alpha \\circ \\beta : V_i \\rightarrow U_i$ .", "Namely, $U_i = \\mu _i *\\left[ *(V_{i-3} \\wedge V_{i-1} \\wedge V_{i+1}) \\wedge *(V_{i-2} \\wedge V_i \\wedge V_{i+2}) \\wedge *(V_{i-1} \\wedge V_{i+1} \\wedge V_{i+3}) \\right],$ where the constants $\\mu _i$ are chosen so that $\\det |U_j, U_{j+1}, U_{j+2}, U_{j+3}|=1$ for all $j$ .", "At the level of the coordinates $(a_j,b_j,c_j)$ , we obtain: $T^*(x_i) = \\dfrac{B_{i+1}}{C_{i+1} C_{i+2}}, \\quad T^*(y_i) = \\dfrac{C_{i+1}}{B_{i+1} A_{i-1}}, \\quad T^*(z_i) = \\dfrac{A_i}{C_{i+2} B_i},$ where $A_i,B_i,C_i$ are defined in Lemma REF .", "Eliminating the variables $\\lambda _i$ with different $i$ by using the formula for the product $\\lambda _i \\lambda _{i+1} \\lambda _{i+2} \\lambda _{i+3}$ concludes the proof.", "$\\Box $" ], [ "Coordinates on twisted polygons: odd vs. even $n$", "In this section we compare how one introduces the coordinates on the space $\\mathcal {P}_n$ of twisted $n$ -gons for odd or even $n$ , and how this changes the statements above.", "Definition 5.8 Call a sequence $(a_j,b_j,c_j),\\; j \\in {\\mathbb {Z}}$ , $n$ -quasiperiodic if there is a sequence $t_j,\\; j \\in {\\mathbb {Z}}$ , satisfying $t_j t_{j+1} t_{j+2} t_{j+3}=1$ and such that $a_{j+n} = a_j \\dfrac{t_j}{t_{j+3}},\\quad b_{j+n} = b_j \\dfrac{t_j}{t_{j+2}},\\quad c_{j+n} = c_j \\dfrac{t_j}{t_{j+1}}$ for each $j \\in {\\mathbb {Z}}$ .", "Note that a sequence $t_j,\\; j \\in {\\mathbb {Z}},$ must be 4-periodic, and it is defined by three parameters, e.g., by $\\alpha :=t_0/t_3, \\,\\beta :=t_0/t_2,$ and $ \\gamma :=t_0/t_1$ with $\\alpha \\beta \\gamma >0$ , and hence $t_0=(\\alpha \\beta \\gamma )^{1/4}$ .", "Thus the space ${\\mathcal {Q}S}_n$ of $n$ -quasiperiodic sequences has dimension $3n+3$ , and $\\lbrace (a_j,b_j,c_j),\\; j =0,...,n-1\\rbrace \\times (\\alpha , \\beta , \\gamma )$ are coordinates on it.", "Now we associate a sequence of vectors $V_j \\in {\\mathbb {R}}^4,\\; j \\in {\\mathbb {Z}},$ and difference equations $V_{j+4} = a_j V_{j+3} + b_j V_{j+2} + c_j V_{j+1} - V_j,\\quad j \\in {\\mathbb {Z}},$ to each twisted $n$ -gon $v_j:=\\phi (j) \\in {\\mathbb {RP}}^3, j \\in {\\mathbb {Z}},$ with a monodromy $M\\in SL(4,{\\mathbb {R}})$ .", "This gives a correspondence between sequences $(a_j,b_j,c_j), \\; j \\in {\\mathbb {Z}},$ and twisted $n$ -gons.", "Proposition 5.9 There is a one-to-one correspondence between twisted $n$ -gons (defined up to projective equivalence) and three-parameter equivalence classes in the space ${\\mathcal {Q}S}_n$ of $n$ -quasiperiodic sequences $\\lbrace (a_j,b_j,c_j),\\; j =0,...,n-1\\rbrace \\times (\\alpha , \\beta , \\gamma )$ .", "If $n$ is odd, there exists a unique $n$ -periodic sequence $(a_j,b_j,c_j)$ in each class.", "If $n=4p$ , then the numbers $ \\alpha , \\beta , \\gamma $ are projective invariants of a twisted $n$ -gon.", "If $n=4p+2$ , then there is one projective invariant: $\\alpha \\gamma /\\beta $ .", "In other words, for odd $n$ the equivalence classes are “directed along\" the parameters $(\\alpha , \\beta , \\gamma )$ and one can chose a representative with $\\alpha =\\beta = \\gamma =1$ in each class.", "For $n=4p$ the classes are “directed across\" these parameters, and hence the latter are fixed for any given class.", "The case $n=4p+2$ is in between: in a sense, two of the $(\\alpha , \\beta , \\gamma )$ -parameters and one of the $(a,b,c)$ -coordinates can serve as coordinates on each equivalence class.", "This proposition can be regarded as an analogue of Proposition 4.1 and Remark 4.4 in [11] for $d=2$ ." ], [ "Proof.", "First, we construct the correspondence, and then consider what happens for different arithmetics of $n$ .", "For a given $n$ -gon $v_k=\\phi (k)$ we construct a sequence of vertices $V_j \\in {\\mathbb {R}}^4,\\; j \\in {\\mathbb {Z}},$ in the following way: choose the lifts $\\phi (0) \\rightarrow V_0, \\;\\phi (1) \\rightarrow V_1,\\; \\phi (2) \\rightarrow V_2$ arbitrarily, and then determine the vectors $V_j,\\; j > 2,$ and $V_j,\\; j < 0,$ recursively using the condition $\\det (V_j, V_{j+1}, V_{j+2}, V_{j+3})=1$ , which follows from equation (REF ).", "By definition of a twisted $n$ -gon, we have $\\phi (j+n) = M \\circ \\,\\phi (j)$ for each $j \\in {\\mathbb {Z}}$ , where $M\\in SL(4,{\\mathbb {R}})$ .", "Consequently, for each $j \\in {\\mathbb {Z}}$ there exists a number $t_j$ , such that $V_{j+n} = t_j M V_j$ , and the matrix $M$ is independent of $j$ .", "The equation $\\det (V_{j+n}, V_{j+1+n}, V_{j+2+n}, V_{j+3+n})=1$ implies that $t_j t_{j+1} t_{j+2} t_{j+3}=1$ and $t_{j+4}=t_j$ for each $j \\in {\\mathbb {Z}}$ .", "In other words, the whole sequence $\\lbrace t_j\\rbrace $ is determined by $t_1, t_2, t_3,$ and then $t_0=1/t_1t_2t_3$ .", "Quasiperiodic conditions (REF ) follow from the comparison of the equation $V_{j+4+n} = a_{j+n} V_{j+3+n} + b_{j+n} V_{j+2+n} + c_{j+n} V_{j+1+n} - V_{j+n}$ with equation (REF ).", "Now we rescale the initial three vectors: $V_0\\mapsto k_0V_0, \\; V_1\\mapsto k_1V_1, \\;V_2\\mapsto k_2V_2$ , where $k_0 k_1 k_2\\ne 0$ .", "A different lift of the three initial vectors corresponds to a different sequence $\\tilde{V}_j = k_j V_j$ , where the sequence $k_j, j \\in {\\mathbb {Z}},$ must also be 4-periodic and satisfy $k_0 k_1 k_2 k_3=1$ .", "This rescaling gives the action of $({\\mathbb {R}}^*)^3$ on the space ${\\mathcal {Q}S}_n$ of $n$ -quasiperiodic sequences.", "By construction, the corresponding orbits (or equivalence classes) of sequences are in a bijection with twisted $n$ -gons.", "The group $({\\mathbb {R}}^*)^3$ acts as follows: $t_j \\mapsto t_j (k_j/k_{j+n}),\\;a_j\\mapsto a_j (k_{j}/k_{j+3}), \\;b_j\\mapsto b_j (k_{j}/k_{j+2}), \\;c_j\\mapsto c_j (k_{j}/k_{j+1}).$ Now we have 3 cases: $n$ is odd.", "Then the above $({\\mathbb {R}}^*)^3$ -action on $t_0,\\,t_1,\\,t_2$ allows one to make them all equal to 1, which corresponds to the constant sequence $\\lbrace t_j=1,\\,j\\in {\\mathbb {Z}}\\rbrace $ and a periodic sequence $\\lbrace (a_j,b_j,c_j),\\; j \\in {\\mathbb {Z}}\\rbrace $ .", "Indeed, e.g., for $n=4p+3$ one has the system of 3 equations: $t_0(k_0/k_3)=1, \\,t_1(k_1/k_4)=1,\\,t_2(k_2/k_5)=1$ .", "Since $k_4=k_0$ , $k_5=k_1$ , and $k_3=1/k_0k_1k_2$ , we obtain a system of 3 equations on the unknowns $k_0,k_1,k_2$ , which has the unique solution.", "$n=4p+2$ .", "One can check that the $({\\mathbb {R}}^*)^3$ -action does not change the ratio $\\dfrac{t_0 t_{2}}{t_{1} t_{3}}=\\dfrac{\\alpha \\gamma }{\\beta }$ .", "$n=4p$ .", "The $({\\mathbb {R}}^*)^3$ -action does not change the three quantities $t_0/t_3, \\, t_0/t_2, \\, t_0/t_1$ .", "$\\Box $ Now we can introduce coordinates on the space ${\\mathcal {Q}S}_n$ of $n$ -quasiperiodic sequences using Definition REF and quasiperiodic variables $(a_j,b_j,c_j)$ .", "Proposition 5.10 For any $n$ the variables $(x_j,y_j,z_j),\\; 0 \\le j \\le n-1$ are independent and constant on the equivalence classes in ${\\mathcal {Q}S}_n$ , i.e., they are well-defined local coordinates on the space $\\mathcal {P}_n= {\\mathcal {Q}S}_n/\\sim $ ." ], [ "Proof.", "It is straightforward to check that the $({\\mathbb {R}}^*)^3$ -action defined above is trivial on the variables $(x_j,y_j,z_j)$ .", "For instance, $x_j =\\dfrac{b_{j+1}}{a_j a_{j+1}} \\rightarrow \\dfrac{b_{j+1}(k_{j+1}/k_{j+3})}{a_j (k_{j}/k_{j+3})\\,a_{j+1}(k_{j+1}/k_{j+4})}=x_j\\,.$ The independence of the new variables on ${\\mathcal {Q}S}_n$ follows from that for the original ones.", "Alternatively, it also follows from their local geometric definition (Proposition REF ).", "$\\Box $ Remark 5.11 In the $(a,b,c)$ -coordinates for even $n$ some of the $(\\alpha , \\beta , \\gamma )$ -parameters were needed to describe the equivalence classes in ${\\mathcal {Q}S}_n$ .", "In the $(x,y,z)$ -coordinates those parameters are functions of $x_j,y_j,z_j$ : $i)$ for $n=4p+2$ , $\\prod _{j=0}^{2p} \\dfrac{x_{2j}^2 y_{2j} z_{2j+1}}{ x_{2j+1}^2 y_{2j+1} z_{2j}}=\\dfrac{\\alpha \\gamma }{\\beta }\\;;$ $ii)$ for $n=4p$ , $\\prod _{j=0}^{p-1} \\dfrac{x_{4j} x_{4j+2} y_{4j+2} z_{4j+1}}{x_{4j+1} x_{4j+3} y_{4j+3} z_{4j+2}}= \\alpha ,\\quad \\prod _{j=0}^{p-1} \\dfrac{y_{4j+1} z_{4j}}{y_{4j+3} z_{4j+2}} =\\beta ,\\quad \\prod _{j=0}^{p-1} \\dfrac{y_{4j} z_{4j+3}}{y_{4j+2} z_{4j+1}} =\\frac{\\gamma }{\\alpha }\\,.$ These identities follow from Definitions REF and REF ." ], [ "Algebraic-geometric integrability of the 3D pentagram map", "In this section we complexify the pentagram map and assume that everything is defined over $.$" ], [ "Scaling transformations and a Lax function in 3D", "Recall that a discrete Lax equation with a spectral parameter is a representation of a dynamical system in the form $L_{i,t+1}(\\lambda ) = P_{i+1,t}(\\lambda ) L_{i,t}(\\lambda ) P_{i,t}^{-1}(\\lambda ),$ where $t$ stands for the discrete time variable, $i$ refers to the vertex index, and $\\lambda $ is a complex spectral parameter.", "The pivotal property responsible for algebraic-geometric integrability of all pentagram maps considered in this paper is the presence of a scaling invariance.", "In the 2D case, this means the invariance with respect to the transformations $a_j \\rightarrow a_j s, \\; b_j \\rightarrow b_j/s$ , where $s$ is an arbitrary number.", "In the 3D case, the pentagram map is invariant with respect to the transformations $a_j \\rightarrow a_j s, \\; b_j \\rightarrow b_j, \\; c_j \\rightarrow c_j s$ .", "In both cases the invariance follows from the explicit formulas of the map.", "Note that formally one can define other pentagram maps by choosing the intersection planes in many different ways.", "However, only very few of these maps possess any scaling invariance.", "Below we derive a Lax representation from the scaling invariance.", "First we do it for odd $n$ , when $(a_j,b_j,c_j),0 \\le j \\le n-1,$ are coordinates on the space ${\\mathcal {P}}_n$ .", "Theorem 6.1 The 3D pentagram map on twisted $n$ -gons with odd $n$ admits a Lax representation with the Lax function $L_j(\\lambda )$ given by $L_j(\\lambda ) =\\begin{pmatrix}c_j/\\lambda & 1/\\lambda & 0 & 0\\\\b_j & 0 & 1 & 0\\\\a_j/\\lambda & 0 & 0 & 1/\\lambda \\\\-1 & 0 & 0 & 0\\end{pmatrix} =\\begin{pmatrix}0 & 0 & 0 & -1\\\\\\lambda & 0 & 0 & c_j\\\\0 & 1 & 0 & b_j\\\\0 & 0 & \\lambda & a_j\\end{pmatrix}^{-1}$ in the coordinates $a_j,b_j,c_j$ .", "Its determinant is $\\det {L_j} \\equiv 1/\\lambda ^2$ .", "Note that we always consider a polygon and the corresponding Lax function at a particular moment of time.", "Whenever necessary we indicate the moment of time explicitly by adding the second index “$t$ ” to the Lax function (above $L_j:=L_{j,t}$ ), while if there is no ambiguity we keep only one index.", "Before proving this theorem we give the following Definition 6.2 The monodromy operators $T_{0,t},T_{1,t},...,T_{n-1,t}$ are defined as the following ordered products of the corresponding Lax functions: $&T_{0,t} = L_{n-1,t} L_{n-2,t} ... L_{0,t},\\\\&T_{1,t} = L_{0,t} L_{n-1,t} L_{n-2,t} ... L_{1,t},\\\\& ...\\\\&T_{i,t} = L_{i+n-1,t} L_{i+n-2,t} ... L_{i+1,t} L_{i,t},\\\\& ...\\\\&T_{n-1,t} = L_{n-2,t} L_{n-3,t} ... L_{0,t} L_{n-1,t},$ where the (integer) index $t$ represents the moment of time." ], [ "Proof of theorem.", "First observe that the Lax equation implies that the corresponding monodromy operators satisfy $T_{i,t+1}(\\lambda ) = P_{i,t}(\\lambda ) T_{i,t}(\\lambda ) P_{i,t}^{-1}(\\lambda ),$ i.e., $T_{i,t}(\\lambda )$ changes to a similar matrix when $t\\rightarrow t+1$ , and hence the eigenvalues of the matrices $T_{i,t}(\\lambda )$ as functions of $\\lambda $ are invariants of the map.", "Conversely, if some function $T_{i,t}(\\lambda )$ has this property, then there must exist a matrix $P_{i,t}(\\lambda )$ (defined up to a multiplication by a scalar function) satisfying the above equation.", "How to define such a monodromy depending on a parameter?", "The monodromy matrix associated with the difference equation $V_{j+4} = a_j V_{j+3} + b_j V_{j+2} + c_j V_{j+1} - V_j$ is $M = N_0 N_1 N_2... N_{n-1}$ , where $N_j =\\begin{pmatrix}0 & 0 & 0 & -1\\\\1 & 0 & 0 & c_j\\\\0 & 1 & 0 & b_j\\\\0 & 0 & 1 & a_j\\end{pmatrix}.$ For odd $n$ the variables $(a_j,b_j,c_j), 0 \\le j \\le n-1,$ are well-defined coordinates on the space of twisted $n$ -gons.", "These variables are periodic: for any $j$ we have $a_{j+n}=a_j,\\; b_{j+n}=b_j,\\; c_{j+n}=c_j$ .", "The vectors $V_j$ are quasi-periodic: $V_{j+n}=M V_j$ , and depend on the lift of the points from the projective space.", "This means that the pentagram map preserves the eigenvalues of the matrix $M$ , but not the matrix $M$ itself.", "Lemmas REF and REF imply that the pentagram map is invariant with respect to the scaling transformations: $a_j \\rightarrow s a_j, \\; c_j \\rightarrow s c_j$ .", "Therefore, the pentagram map also preserves the eigenvalues of the monodromy matrix $M(s)$ corresponding to the $n$ -gons scaled by $s$ .", "Namely, we have $M(s) = N_0(s) N_1(s) N_2(s)... N_{n-1}(s), \\text{ where }N_j(s) =\\begin{pmatrix}0 & 0 & 0 & -1\\\\1 & 0 & 0 & s c_j\\\\0 & 1 & 0 & b_j\\\\0 & 0 & 1 & s a_j\\end{pmatrix}.$ Now one can see that the matrix $N_j(s)$ can be chosen as a Lax function.", "For technical reasons (which will be clear later), we define the Lax matrix as $L_j^{-1}(\\lambda ) := (g^{-1} N_j(s) g)/s$ , where $g := \\text{diag}(1,s,1,s)$ , and $\\lambda := 1/s^2$ .", "This gives the required matrix $L_j(\\lambda )$ .$\\Box $ As we mentioned before, the formulas for the pentagram map are non-local in the variables $(a_j,b_j,c_j)$ .", "As a result, an explicit expression for the matrix $P_{i,t}(\\lambda )$ becomes non-local as well.", "On the other hand, one can use the variables $(x_j,y_j,z_j)$ (given by Definition REF ) to describe a Lax representation.", "Their advantage is that all formulas become local and are valid for any $n$ , both even and odd.", "Theorem 6.3 For any $n$ the equations for the 3D pentagram map are equivalent to the Lax equation $\\tilde{L}_{i,t+1}(\\lambda ) = \\tilde{P}_{i+1,t}(\\lambda ) \\tilde{L}_{i,t}(\\lambda ) \\tilde{P}_{i,t}^{-1}(\\lambda ),$ where $\\tilde{L}_{i,t}(\\lambda ) =\\begin{pmatrix}0 & 0 & 0 & -1\\\\\\lambda x_i y_i & 0 & 0 & 1\\\\0 & z_i & 0 & 1\\\\0 & 0 & \\lambda x_i & 1\\end{pmatrix}^{-1},$ $\\tilde{P}_{i,t}(\\lambda )=\\begin{pmatrix}0 & \\rho _i & 0 & -\\rho _i\\\\\\lambda \\sigma _i(1+z_i) & -\\rho _i & \\lambda \\sigma _i & \\rho _i\\\\y_{i-1} \\theta _i & \\dfrac{z_{i-1}}{\\tau _i} & -\\theta _i & \\dfrac{1+y_{i-2}}{\\tau _i}\\\\-\\dfrac{\\lambda y_{i-1}}{1+y_{i-1}+z_i} & 0 & \\dfrac{\\lambda }{1+y_{i-1}+z_i} & 0\\end{pmatrix},$ and the variables $\\rho _i, \\sigma _i, \\tau _i,$ and $\\theta _i$ stand for $\\rho _i &= \\dfrac{1}{x_i (1+y_i+z_{i+1})},\\\\\\sigma _i &= \\dfrac{x_{i-1} y_{i-1}(1+y_{i-2}+z_{i-1})}{x_i z_{i-1}(1+y_{i-1}+z_i)(1+y_i+z_{i+1})},\\\\\\tau _i &= x_i(1+y_{i-2}-y_i z_{i-1}+z_{i+1}+z_{i+1} y_{i-2}),\\\\\\theta _i &=\\dfrac{1+y_{i-2}+z_{i-1}}{\\tau _i (1+y_{i-1}+z_i)}.$" ], [ "Proof.", "The proof is a long but straightforward verification.", "$\\Box $ Remark 6.4 The Lax functions $L$ and $\\tilde{L}$ in the $(a_i,b_i,c_i)$ and $(x_i,y_i,z_i)$ variables are related to each other as follows: $\\tilde{L}_{i,t} = a_{i+1} \\left( h_{i+1}^{-1} L_{i,t} h_i\\right), \\text{ where } h_i := \\text{diag}(1,c_i,b_i,a_i).$" ], [ "Properties of the spectral curve", "Recall that the monodromy operators $T_{i,t}(\\lambda )$ satisfy the equation $T_{i,t+1}(\\lambda ) = P_{i,t}(\\lambda ) T_{i,t}(\\lambda ) P_{i,t}^{-1}(\\lambda ).$ It implies that the function of two variables $R(\\lambda ,k)=\\det {(T_{i,t}(\\lambda ) - k \\,\\text{Id})}$ is independent of $i$ and $t$ .", "Furthermore, $R(\\lambda ,k)=0$ is a polynomial relation between $\\lambda $ and $k$ : $R(\\lambda ,k)$ becomes a polynomial after a multiplication by a power of $\\lambda $ .", "Its coefficients are integrals of motion for the pentagram map.", "The zero set of $R(\\lambda ,k)=0$ is an algebraic curve in 2.", "A standard procedure (of adding the infinite points and normalization with a few blow-ups) makes it into a compact Riemann surface, which we call the spectral curve and denote by $\\Gamma $ .", "In this section we explore some of the properties of the spectral curve and, in particular, find its genus.", "Definition 6.5 For an odd $n$ define the spectral function $R(\\lambda ,k)$ as $R(\\lambda ,k):=\\det {(T_{i,t}(\\lambda ) - k \\,\\text{Id})},$ i.e., using the Lax function in the $(a,b,c)$ -coordinates from Theorem REF .", "The spectral curve $\\Gamma $ is the normalization of the compactification of the curve $R(\\lambda ,k)=0$ .", "We define the integrals of motion $I_j,J_j,G_j,\\; 0 \\le j \\le q=\\lfloor n/2 \\rfloor ,$ as the coefficients of the expansion $R(\\lambda ,k) = k^4 - k^3 \\left( \\sum _{j=0}^q G_j \\lambda ^{j-n} \\right) + k^2 \\left( \\sum _{j=0}^q J_j \\lambda ^{j-q-n} \\right)- k \\left( \\sum _{j=0}^q I_j \\lambda ^{j-2n} \\right) + \\lambda ^{-2n} = 0.$ When $n$ is even, the sequence $(a_j,b_j,c_j),\\; j \\in {\\mathbb {Z}}$ , is not $n$ -periodic, and the monodromy operator $T_{i,t}(\\lambda )$ cannot be defined.", "One should use the Lax function $\\tilde{L}_{i,t}(\\lambda )$ in the $(x,y,z)$ -coordinates from Theorem REF to define the monodromy operator $\\tilde{T}_{i,t}(\\lambda )$ and the corresponding spectral curve.", "Namely, first note that the integral of motion $I_0$ has the following explicit expression: $I_0 = \\prod _{i=0}^{n-1} a_i = \\left( \\prod _{i=0}^{n-1} x_i^2 y_i z_i \\right)^{-1/4}.$ Definition 6.6 For any $n$ (either even or odd), the spectral function is $R(\\lambda ,k) = \\tilde{R}(\\lambda , k I_0)/I_0^4,\\; \\text{ where }\\;\\tilde{R}(\\lambda ,k):=\\det {(\\tilde{T}_{i,t}(\\lambda ) - k \\,\\text{Id})}\\,,$ and the monodromy operator $\\tilde{T}_{i,t}(\\lambda )$ is defined using the Lax function $\\tilde{L}_{i,t}(\\lambda )$ from Theorem REF .", "The spectral function $R(\\lambda ,k)$ defined this way coincides with $\\det {(T_{i,t}(\\lambda ) - k \\,\\text{Id})}$ for odd $n$ , since $\\tilde{T}_{i,t} = I_0 \\left( h_i^{-1} T_{i,t} h_i \\right)$ , see Remark REF .", "It is convenient to have such a unified definition for computations of integrals of motion.", "Theorem 6.7 For generic values of the integrals of motion $I_j,J_j,G_j$ , the genus $g$ of the spectral curve $\\Gamma $ is $g = 3q$ for odd $n$ and $g = 3q-3$ for even $n$ , where $q=\\lfloor n/2 \\rfloor $ .", "To prove it, we first describe the singularities of $R(\\lambda ,k)=0$ by considering the formal series solutions (the so-called Puiseux series).", "Lemma 6.8 If $n$ is even, the equation $R(\\lambda ,k)=0$ has 4 formal series solutions at $\\lambda =0$ : $O_1: \\quad k_1 &= \\dfrac{1}{I_0} - \\dfrac{I_{1}}{I_0^2} \\lambda + {\\mathcal {O}}(\\lambda ^2),\\\\O_{2,3}: \\quad k_{2,3} &= \\dfrac{k_*}{\\lambda ^q} + {\\mathcal {O}}\\left(\\dfrac{1}{\\lambda ^{q-1}}\\right), \\quad \\text{ where $k_*$ satisfies} \\quad G_0 k_*^2 - J_0 k_* + I_0 = 0,\\\\O_4: \\quad k_4 &= \\dfrac{G_0}{\\lambda ^n} + \\dfrac{G_1}{\\lambda ^{n-1}}+\\dfrac{G_2}{\\lambda ^{n-2}}+{\\mathcal {O}}(\\lambda ^{3-n}),$ and 4 solutions at $\\lambda =\\infty $ : $W_{1,2,3,4}: \\quad k_{1,2,3,4}= \\dfrac{k_\\infty }{\\lambda ^q} + {\\mathcal {O}}\\left( \\dfrac{1}{\\lambda ^{q+1}} \\right), \\text{ where }k_\\infty ^4 - G_q k_\\infty ^3 + J_q k_\\infty ^2 - I_q k_\\infty + 1 = 0.$ If $n$ is odd, the equation $R(\\lambda ,k)=0$ has 4 formal series solutions at $\\lambda =0$ : $O_1: \\quad k_1 &= \\dfrac{1}{I_0} - \\dfrac{I_1}{I_0^2} \\lambda + {\\mathcal {O}}(\\lambda ^2),\\\\O_2: \\quad k_{2,3} &= \\pm \\dfrac{\\sqrt{-I_0/G_0}}{\\lambda ^{n/2}} + \\dfrac{J_0}{2 G_0 \\lambda ^{(n-1)/2}} + {\\mathcal {O}}\\left(\\dfrac{1}{\\lambda ^{(n-2)/2}}\\right),\\\\O_3: \\quad k_4 &= \\dfrac{G_0}{\\lambda ^n} + \\dfrac{G_1}{\\lambda ^{n-1}}+\\dfrac{G_2}{\\lambda ^{n-2}}+{\\mathcal {O}}(\\lambda ^{3-n}),$ and 4 solutions at $\\lambda =\\infty $ : $W_{1,2}: \\quad k_{1,2,3,4}= \\dfrac{k_\\infty }{\\lambda ^{n/2}} + {\\mathcal {O}}\\left( \\dfrac{1}{\\lambda ^{(n+1)/2}} \\right), \\text{ where }k_\\infty ^4 + J_q k_\\infty ^2 + 1 = 0.$ The remaining coefficients of the series are determined uniquely." ], [ "Proof of lemma.", "One finds the series coefficients recursively by substituting the series into the equation $R(\\lambda ,k)=0$ , which determines the spectral curve.", "$\\Box $ Now we can complete the proof of Theorem REF ." ], [ "Proof of theorem.", "As follows from the definition of the spectral curve $\\Gamma $ , it is a ramified 4-fold cover of ${\\mathbb {CP}}^1$ , since the $4\\times 4$ -matrix $\\tilde{T}_{i,t}$ (or ${T}_{i,t}$ ) has 4 eigenvalues.", "By the Riemann-Hurwitz formula the Euler characteristic of $\\Gamma $ is $\\chi (\\Gamma )=4\\chi ({\\mathbb {CP}}^1)-\\nu =8-\\nu $ , where $\\nu $ is the number of branch points.", "On the other hand, $\\chi (\\Gamma )=2-2g$ , and once we know $\\nu $ it allows us to find the genus of the spectral curve $\\Gamma $ from the formula $2-2g=8-\\nu $ .", "The number $\\nu $ of branch points of $\\Gamma $ on the $\\lambda $ -plane equals the number of zeroes of the function $\\partial _k R(\\lambda ,k)$ aside from the singular points.", "The function $\\partial _k R(\\lambda ,k)$ is meromorphic on $\\Gamma $ , therefore the number of its zeroes equals the number of its poles.", "One can see that for any $n$ the function $\\partial _k R(\\lambda ,k)$ has poles of total order $9n$ at $z=0$ , and it has zeroes of total order $6n$ at $z=\\infty $ .", "Indeed, substitute the local series from Lemma REF to the expression for $\\partial _k R(\\lambda ,k)$ .", "(E.g., for $n=2q$ at $O_4$ one has $k\\sim \\lambda ^{-n}$ .", "The leading terms of $\\partial _k R(\\lambda ,k)$ for the pole at $\\lambda =0$ are $k^3, k^2\\lambda ^{-n}, k\\lambda ^{-q-n}, \\lambda ^{-2n}$ .", "The first two terms, being of order $\\lambda ^{-3n}$ , dominate and give the pole order of $3n=6q$ .)", "The corresponding orders of the poles and zeroes of $\\partial _k R(\\lambda ,k)$ on $\\Gamma $ are summarized as follows: $\\begin{array}{||c|c||c|c||c|c||c|c||}\\hline n=2q & \\text{ pole } & & \\text{ zero } & n=2q+1 & \\text{ pole } & & \\text{ zero }\\\\ \\hline O_1 & 4q & W_1 & 3q & O_1 & 2n & W_1 & 3n\\\\ \\hline O_2 & 4q & W_2 & 3q & O_2 & 4n & W_2 & 3n\\\\ \\hline O_3 & 4q & W_3 & 3q & O_3 & 3n & & \\\\ \\hline O_4 & 6q & W_4 & 3q & & & &\\\\ \\hline \\end{array}$ $$ For instance, for $n=2q$ this gives the total order of poles: $4q+4q+4q+6q=18q=9n$ , while the total order of zeroes is $4\\times 3q=12q=6n$ .", "Therefore, the number of zeroes of $\\partial _k R(\\lambda ,k)$ at nonsingular points $\\lambda \\ne \\lbrace 0,\\infty \\rbrace $ is $\\nu =9n-6n=3n$ , and so is the total number of branch points of $\\Gamma $ in the finite part of the $(\\lambda ,k)$ plane.", "The difference between odd and even values of $n$ arises because $\\Gamma $ has 2 additional branch points at $\\lambda =\\infty $ , and 1 branch point at $\\lambda =0$ for odd $n$ , i.e., $\\nu =3n+3$ .", "Consequently, one has $2-2g=8-\\nu $ with $\\nu =3n$ for even $n$ and $\\nu =3n+3$ for odd $n$ .", "The required expression for the genus $g$ follows: $g=3q-3$ for $n=2q$ and $g=3q$ for $n=2q+1$ .", "$\\Box $ Remark 6.9 Now we describe a few integrals of motion using the coordinates $(a_j,b_j,c_j), 0 \\le j \\le n-1,$ when $n$ is odd.", "The description is similar to that in the 2D case (cf.", "Section 5.2 and Proposition 5.3 in [11]).", "Consider a code which is an ordered sequence of digits from 1 to 4.", "The number of digits in a code is $p,q,r,t$ , respectively.", "The code is called “admissible” if $p+2q+3r+4t=n$ .", "The number $p+r$ is called its “weight.” Each code expands in a “word” of $n$ characters in the following way: $1,2,3,4$ are replaced by “a”,“*b”,“**c”,“****”, respectively.", "Now we label the vertices of an $n$ -gon by $0,1,...,n-1$ , and associate letters in a word to them keeping the order.", "We obtain one monomial by taking the product of the variables $a_i$ ,$b_i$ , or $c_i$ that occur at the vertex $i$ .", "The letter “*” corresponds to “1”.", "The sign of the monomial is $(-1)^t$ .", "Next step is to permute the numbering of the vertices cyclically and take the sum of the monomials.", "Note, however, that if, for example, $n=9$ , then the code “333” corresponds to the sum $c_0 c_3 c_6+c_1 c_4 c_7 + c_2 c_5 c_8$ without the coefficient 3.", "Finally, we repeat this procedure for all admissible codes of weight $p+r$ and denote the total sum by $\\hat{I}_{p+r}$ .", "Additionally, we define the sum $\\hat{G}_{p+r}$ by substituting $a_i \\rightarrow c_{i+1}, \\; c_i \\rightarrow a_{i-1}$ in $\\hat{I}_{p+r}$ .", "Consider, for example, the case $n=7$ .", "Then all admissible codes of weight 1 are $142,124,1222,34,223$ .", "The corresponding sum is $\\hat{I}_1 = \\sum _{\\text{cyclic}} (-a_1 b_0-a_5 b_0+a_5 b_0 b_2 b_4-c_0+c_0 b_2 b_4).$ Proposition 6.10 For odd $n$ one has $I_0 = \\prod _{j=0}^{n-1} a_j\\,, \\quad I_q = \\hat{I}_1, \\quad I_i = \\hat{I}_{n-2i}\\,,$ $G_0 = \\prod _{j=0}^{n-1} c_j\\,, \\quad G_q = \\hat{G}_1, \\quad G_i = \\hat{G}_{n-2i}\\,.$" ], [ "Proof.", "The proof is analogous to the proof of Proposition 5.3 in [11].", "$\\Box $" ], [ "The spectral curve and invariant tori", "The spectral curve is a crucial component of algebraic-geometric integrability.", "Below we always assume it to be generic.", "(As everywhere in this paper, “generic” means values of parameters from some Zariski open subset in the space of parameters.)", "It has a natural torus, its Jacobian, associated with it.", "It turns out that one can recover a Lax function from the spectral curve and a point on the Jacobian, and vice versa: in our situation this correspondence is locally one-to-one.", "The dynamics of the pentagram map becomes very simple on the Jacobian.", "In this section, we construct this correspondence and describe the dynamics of the pentagram map.", "Definition 6.11 A Floquet-Bloch solution $\\psi _{i,t}$ of a difference equation $\\psi _{i+1,t} = \\tilde{L}_{i,t} \\psi _{i,t}$ is an eigenvector of the monodromy operator: $\\tilde{T}_{i,t} \\psi _{i,t} = k \\psi _{i,t}.$ The normalization $\\sum _{j=1}^4 \\psi _{0,0,j} \\equiv 1$ (i.e., the sum of all components of the vector $\\psi _{0,0}$ is equal to 1) determines all vectors $\\psi _{i,t}$ with $i,t \\ge 0$ uniquely.", "Denote the normalized vectors $\\psi _{i,t}$ by $\\bar{\\psi }_{i,t}$ , i.e., $\\bar{\\psi }_{i,t} = \\psi _{i,t}/\\left(\\sum _{j=1}^4 \\psi _{i,t,j}\\right)$ .", "(The vectors $\\psi _{0,0}$ and $\\bar{\\psi }_{0,0}$ are identical in this notation.)", "We also denote by $D_{i,t}$ the pole divisor of $\\bar{\\psi }_{i,t}$ on $\\Gamma $ .", "Remark 6.12 We use the Lax function $\\tilde{L}_{i,t}$ and the monodromy operator $\\tilde{T}_{i,t}$ in the above definition to allow for both even and odd $n$ .", "In the case of odd $n$ one can instead employ the Lax function $L_{i,t}$ and the monodromy operator $T_{i,t}$ , while all the statements and proofs below remain valid.", "Theorem 6.13 A Floquet-Bloch solution $\\bar{\\psi }_{i,t}$ is a meromorphic vector function on $\\Gamma $ .", "Generically its pole divisor $D_{i,t}$ has degree $g+3$ ." ], [ "Proof.", "The proof of the fact that the function $\\bar{\\psi }_{i,t}$ is meromorphic on the spectral curve $\\Gamma $ , as well as that its number of poles is $\\deg D_{i,t}=\\nu /2$ , is identical to the proof of Proposition 3.4 in [15].", "The number $\\nu $ of the branch points of $\\Gamma $ is different: in Theorem REF we found that $2-2g=8-\\nu $ , where $g$ is the genus of the spectral curve.", "This implies the required expression: $\\deg D_{i,t}=g+3$ .", "$\\Box $ Definition 6.14 Let $J(\\Gamma )$ be the Jacobian of the spectral curve $\\Gamma $ , and $[D_{0,0}]$ is the equivalence class of the divisor $D_{0,0}$ , the pole divisor of $\\psi _{0,0}$ , under the Abel map.", "The pair consisting of the spectral curve $\\Gamma $ (with marked points $O_i$ and $W_i$ ) and a point $[D_{0,0}] \\in J(\\Gamma )$ is called the spectral data.", "The spectral map $S$ associates to a given generic twisted $n$ -gon in ${\\mathbb {CP}}^3$ its spectral data $(\\Gamma , [D_{0,0}])$ .", "The algebraic-geometric integrability is based on the following theorem.", "Theorem 6.15 For any $n$ , the spectral map $S$ defines a bijection between a Zariski open subset of the space $\\mathcal {P}_n = \\lbrace (x_i,y_i,z_i),\\, 0 \\le i \\le n-1\\rbrace $ and a Zariski open subset of the spectral data.", "Corollary 6.16 For odd $n$ , the spectral map $S$ defines a bijection between a Zariski open subset of the space $\\mathcal {P}_n \\simeq {3n}= \\lbrace (a_i,b_i,c_i),\\, 0 \\le i \\le n-1\\rbrace $ and a Zariski open subset of the spectral data." ], [ "Proof of Corollary ", "The statement follows from Theorem REF and Definition REF relating the coordinates $(x_i,y_i,z_i) $ and $(a_i,b_i,c_i)$ for odd $n$ .", "$\\Box $ The proof of Theorem REF is based on Proposition REF (which completes the construction of the direct spectral map) and Proposition REF (an independent construction of the inverse spectral map), which are also used below to describe the corresponding pentagram dynamics.", "It will be convenient to introduce the following notation for divisors: $O_{pq}:=O_p+O_q$ and $W_{pq}:=W_p+W_q$ (e.g., $O_{12}:=O_1+O_2$ ).", "Proposition 6.17 The divisors of the coordinate functions $\\psi _{i,t,1},..., \\psi _{i,t,4}$ for $ 0 \\le i \\le n-1$ and any integer $t$ satisfy the following inequalities, provided that their divisors remain non-special up to time $t$ : For odd $n$ one has $(\\psi _{i,t,1}) \\ge -D +O_2 -i O_{23}+(i+1)W_{12}-t(W_{12}-O_{13});$ $(\\psi _{i,t,2}) \\ge -D +(1-i)O_{23}+iW_{12}-t(W_{12}-O_{13});$ $(\\psi _{i,t,3}) \\ge -D -i O_{23}+(i+1)W_{12}-t(W_{12}-O_{13});$ $(\\psi _{i,t,4}) \\ge -D +O_2+(1-i) O_{23}+iW_{12}-t(W_{12}-O_{13});$ For even $n$ one has $(\\psi _{i,t,1}) \\ge -D +O_2 + \\lfloor \\dfrac{i-t+2}{2} \\rfloor W_{12}+ \\lfloor \\dfrac{i-t+1}{2} \\rfloor W_{34}- \\lfloor \\dfrac{i+1}{2} \\rfloor O_{24}-\\lfloor \\dfrac{i}{2} \\rfloor O_{34}+tO_{14};$ $(\\psi _{i,t,2}) \\ge -D + \\lfloor \\dfrac{i-t+1}{2} \\rfloor W_{12}+ \\lfloor \\dfrac{i-t}{2} \\rfloor W_{34}- \\lfloor \\dfrac{i-1}{2} \\rfloor O_{24}-\\lfloor \\dfrac{i}{2} \\rfloor O_{34}+tO_{14};$ $(\\psi _{i,t,3}) \\ge -D + \\lfloor \\dfrac{i-t+2}{2} \\rfloor W_{12}+ \\lfloor \\dfrac{i-t+1}{2} \\rfloor W_{34}- \\lfloor \\dfrac{i+1}{2} \\rfloor O_{34}- \\lfloor \\dfrac{i}{2} \\rfloor O_{24}+tO_{14};$ $(\\psi _{i,t,4}) \\ge -D +O_2 + \\lfloor \\dfrac{i-t+1}{2} \\rfloor W_{12}+ \\lfloor \\dfrac{i-t}{2} \\rfloor W_{34}- \\lfloor \\dfrac{i-1}{2} \\rfloor O_{34}- \\lfloor \\dfrac{i}{2} \\rfloor O_{24}+tO_{14};$ where $D=D_{0,0}$ corresponds to the divisor at $t=0$ and is an effective divisor of degree $g+3$ , while $\\lfloor x \\rfloor $ is the floor (i.e., the greatest integer) function of $x$ ." ], [ "Proof.", "The proof is a routine comparison of power expansions in $\\lambda $ at the points $O_p, W_q$ for $k_i$ and $L_{i,t}$ and is very similar to the proof of Proposition 3.10 in the 2D case in [15], although the 3D explicit expressions are more involved.", "See more details in Appendix REF .", "$\\Box $ Proposition 6.18 For any $n$ , given a generic spectral curve with marked points and a generic divisor $D$ of degree $g+3$ one can recover a sequence of matrices $\\tilde{L}_{i,t}(\\lambda ) =\\begin{pmatrix}0 & 0 & 0 & -1\\\\\\lambda x_i y_i & 0 & 0 & 1\\\\0 & z_i & 0 & 1\\\\0 & 0 & \\lambda x_i & 1\\end{pmatrix}^{-1},$ for $ 0 \\le i \\le n-1 $ and any $ t.$ We describe the reconstruction procedure and prove this proposition in Appendix REF ." ], [ "Proof of Theorem ", "The proof consists of constructions of the spectral map $S$ and its inverse.", "The spectral map was described in Definition REF based on Theorem REF .", "We comment on an independent construction of the inverse spectral map now.", "Pick an arbitrary divisor $D$ of degree $g+3$ in the equivalence class $[D_{0,0}] \\in J(\\Gamma )$ and apply Proposition REF .", "“A Zariski open subset of the spectral data” is defined by spectral functions which may be singular only at the points $O_i,W_i$ and by such divisors $[D] \\equiv [D_{0,0}] \\in J(\\Gamma )$ that all divisors in Proposition 6.17 with $0 \\le i \\le n-1$ up to time $t$ are non-special.", "$\\Box $ The next theorem describes the time evolution of the pentagram map in the Jacobian of $\\Gamma $ .", "The difference between even and odd $n$ is very similar to the 2-dimensional case.", "Combined with Theorem REF , it proves the algebraic-geometric integrability of the 3D pentagram map.", "(It also implies that it is possible to obtain explicit formulas of the coordinates of the pentagram map as functions of time using the Riemann $\\theta $ -functions.)", "Theorem 6.19 The equivalence class $[D_{i,t}]\\in J(\\Gamma )$ of the pole divisor $D_{i,t}$ of $\\bar{\\psi }_{i,t}$ has the following time evolution: when $n$ is odd, $[D_{i,t}] = [D_{0,0} - tO_{13} + iO_{23} + (t-i)W_{12}] ,$ when $n$ is even, $[D_{i,t}] = \\left[D_{0,0} - tO_{14} + \\lfloor \\dfrac{i+1}{2} \\rfloor O_3 + \\lfloor \\dfrac{i}{2}\\rfloor O_2 + i O_4 -\\lfloor \\dfrac{i-t+1}{2}\\rfloor W_{12} - \\lfloor \\dfrac{i-t}{2}\\rfloor W_{34}\\right].$ where $\\deg {D_{i,t}} = g+3$ and $\\lfloor x\\rfloor $ is the floor function of $x$ , and provided that spectral data remains generic up to time $t$ .", "For an odd $n$ this discrete time evolution in $J(\\Gamma )$ takes place along a straight line, whereas for an even $n$ the evolution goes along a “staircase” (i.e., its square goes along a straight line)." ], [ "Proof.", "The vector functions $\\psi _{i,t}$ with $i,t \\ne 0$ are not normalized.", "The normalized vectors are equal to $\\bar{\\psi }_{i,t} = \\psi _{i,t}/f_{i,t},$ where $f_{i,t}=\\sum _{j=1}^4 \\psi _{i,t,j}$ .", "Proposition REF implies that the divisor of each function $f_{i,t}$ is: for odd $n$ , $(f_{i,t}) = D_{i,t} - D_{0,0} + tO_{13} - iO_{23}+ (i-t)W_{12},$ for even $n$ , $ (f_{i,t}) = D_{i,t} - D_{0,0} + tO_{14} - \\lfloor \\dfrac{i+1}{2}\\rfloor O_3 - \\lfloor \\dfrac{i}{2}\\rfloor O_2 - i O_4+\\lfloor \\dfrac{i-t+1}{2}\\rfloor W_{12}+\\lfloor \\dfrac{i-t}{2}\\rfloor W_{34}.$ Since the divisor of any meromorphic function is equivalent to zero, the result of the theorem follows.", "The staircase dynamics is related to alternating jumps in the terms $\\lfloor (i-t+1)/2\\rfloor $ and $\\lfloor (i-t)/2\\rfloor $ as $t$ increases over integers.", "$\\Box $ Note that although the pentagram map preserves the spectral curve, it exchanges the marked points.", "The “staircase” dynamics on the Jacobian appears after the identification of curves with different marking.", "One cannot observe this dynamics in the space of twisted polygons $\\mathcal {P}_n$ , before the application of the spectral map." ], [ "Closed polygons", "Closed polygons in ${\\mathbb {CP}}^3$ correspond to the monodromies $M=\\pm \\text{Id}$ in $SL(4,$ .", "They form a subspace of codimension $15=\\dim SL(4,$ in the space of all twisted polygons $\\mathcal {P}_n$ .", "The pentagram map on closed polygons in 3D is defined for $n \\ge 7$ .", "Theorem 7.1 Closed polygons in ${\\mathbb {CP}}^3$ are singled out by the condition that either $(\\lambda ,k)=(1,1)$ or $(\\lambda ,k)=(1,-1)$ is a quadruple point of $\\Gamma $ .", "Both conditions are equivalent to 9 independent linear constraints on $I_j,J_j,G_j$ .", "Generically, the genus of $\\Gamma $ drops to $g=3q-9$ when $n$ is even, and to $g=3q-6$ when $n$ is odd, where $q=\\lfloor n/2 \\rfloor $ .", "The dimension of the Jacobian $J(\\Gamma )$ drops by 6 for closed polygons for any $n$ .", "Theorem REF holds with this genus adjustment, and Theorems REF and REF hold verbatim for closed polygons (i.e., on the subspace of closed polygons $\\mathcal {C}_n \\subset \\mathcal {P}_n$ )." ], [ "Proof.", "For a twisted $n$ -gon its monodromy matrix at a moment $t$ is equal to $T_{0,t}(1)$ in the $(a,b,c)$ -coordinates or to $\\tilde{T}_{0,t}(1)$ in the $(x,y,z)$ -coordinates.", "An $n$ -gon is closed if and only if $T_{0,t}(1)=\\text{Id}$ or $T_{0,t}(1)=-\\text{Id}$ (respectively, $\\tilde{T}_{0,t}(1)=I_0 \\,\\text{Id}$ or $\\tilde{T}_{0,t}(1)=-I_0 \\,\\text{Id}$ ).", "For our definition of the spectral function, either of these conditions, $T_{0,t}(1)=\\pm \\text{Id}$ or $\\tilde{T}_{0,t}(1)=\\pm I_0 \\,\\text{Id}$ , implies that $(\\lambda ,k)=(1,\\pm 1)$ is a self-intersection point for $\\Gamma $ .", "The algebraic conditions implying that $(1,\\pm 1)$ is a quadruple point are: $R(1,\\pm 1) = 0$ , $\\partial _k R(1,\\pm 1) = \\partial _\\lambda R(1,\\pm 1) = 0$ , $\\partial _k^2 R(1,\\pm 1) = \\partial _\\lambda ^2 R(1,\\pm 1) = \\partial _{k \\lambda }^2 R(1,\\pm 1) = 0$ , $\\partial _k^3 R(1,\\pm 1) = \\partial _\\lambda ^3 R(1,\\pm 1) = \\partial _{kk \\lambda }^3 R(1,\\pm 1)= \\partial _{k \\lambda \\lambda }^3 R(1,\\pm 1) = 0$ .", "However, the function $R(\\lambda ,k)$ is special at the points $(1,\\pm 1)$ , because the following relation holds: $R(1,\\pm 1)=\\pm \\partial _k R(1,\\pm 1) - \\dfrac{1}{2} \\partial _k^2 R(1,\\pm 1) \\pm \\dfrac{1}{6} \\partial _k^3 R(1,\\pm 1).$ Consequently, the above 10 conditions are equivalent to only 9 independent linear equations on $I_j,J_j,G_j, 0 \\le j \\le q$ .", "The proofs of Theorems REF and REF apply, mutatis mutandis, to the periodic case.", "To define the Zariski open set of spectral data for closed polygons, we confine to spectral functions that can be singular only at the point $(\\lambda ,k)=(1,1)$ or $(1,-1)$ in addition to singularities at $O_i$ and $W_i$ and use the same restrictions on divisors $D$ as in the proof of Theorem REF .", "In the periodic case we also have to adjust the count of the number $\\nu $ of branch points of $\\Gamma $ and the corresponding calculation for the genus $g$ of $\\Gamma $ , cf.", "Theorem REF .", "Namely, as before, the function $\\partial _k R(\\lambda ,k)$ has poles of total order $9n$ over $\\lambda =0$ , and zeroes of total order $6n$ over $\\lambda =\\infty $ .", "Now since $R(\\lambda ,k)$ has a quadruple point $(1,\\pm 1)$ , $\\partial _k R(\\lambda ,k)$ has a triple zero at $(1,\\pm 1)$ .", "But $\\lambda =1$ is not a branch point of $\\Gamma $ .", "Consequently, $\\partial _k R(\\lambda ,k)$ has triple zeroes on 4 sheets of $\\Gamma $ over $\\lambda =1$ .", "The Riemann-Hurwitz formula is $2-2g=8-\\nu $ , where the number of branch points for even $n$ is $\\nu = 9n-6n-12=3n-12$ , while for odd $n$ it is $\\nu = 9n-6n-12+3=3n-9$ .", "Therefore, we have $g=3q-9$ for even $n$ , and $g=3q-6$ for odd $n$ .", "$\\Box $" ], [ "Invariant symplectic structure and symplectic leaves", "It was proved in [15] that in the 2D case an invariant symplectic structure on the space of twisted polygons ${\\mathcal {P}}_n$ provided by Krichever-Phong's universal formula [5], [6] coincides with the inverse of the invariant Poisson structure found in [11] when restricted to the symplectic leaves.", "We show that in 3D the same formula also provides an invariant symplectic structure defined on leaves described below.", "While we do not compute the symplectic structure explicitly in the coordinates $(a_i,b_i,c_i)$ or $(x_i,y_i,z_i)$ due to complexity of the formulas, the proofs are universal and applicable in the higher-dimensional case of ${\\mathbb {CP}}^d$ as well.", "Finding an explicit expression of the symplectic structure or of the corresponding Poisson structure is still an open problem.", "Definition 7.2 ([5], [6]) Krichever-Phong's universal formula defines a pre-symplectic form on the space of Lax operators, i.e., on the space $\\mathcal {P}_n$ .", "It is given by the expression: $\\omega = -\\dfrac{1}{2} \\sum _{\\lambda =0,\\infty } {\\text{res}} \\,\\text{Tr}\\left( \\Psi _0^{-1} \\tilde{T}_0^{-1} \\delta \\tilde{T}_0 \\wedge \\delta \\Psi _0\\right) \\dfrac{d\\lambda }{\\lambda }.$ The matrix $\\Psi _{0}:=\\Psi _{0,t}(\\lambda )$ is composed of the eigenvectors $\\psi _{0,t}$ on different sheets of $\\Gamma $ over the $\\lambda $ -plane, and it diagonalizes the monodromy matrix $\\tilde{T}_{0}:=\\tilde{T}_{0,t}(\\lambda )$ .", "(In this definition we drop the index $t$ , because all variables correspond to the same moment of time.)", "The leaves of the 2-form $\\omega $ are defined as submanifolds of $\\mathcal {P}_n$ , where the expression $\\delta \\ln {k}\\, (d\\lambda /\\lambda )$ is holomorphic.", "The latter expression is considered as a 1-form on the spectral curve $\\Gamma $ .", "Proposition 7.3 For even $n$ the leaves are singled out by 6 conditions: $\\delta I_0 = \\delta I_q = \\delta G_0 = \\delta G_q = \\delta J_0 = \\delta J_q = 0;$ For odd $n$ the leaves are singled out by 3 conditions: $\\delta G_0 = \\delta I_0 = \\delta J_q = 0.$" ], [ "Proof.", "These conditions follow immediately from the definition of the leaves and Lemma REF .", "For example, at the point $O_1$ we have $\\delta \\ln {k_1} \\dfrac{d\\lambda }{\\lambda } = \\left(\\dfrac{1}{\\lambda } \\dfrac{\\delta I_0}{I_0} + {\\mathcal {O}}(1) \\right) d\\lambda .$ This 1-form is holomorphic in $\\lambda $ if and only if $\\delta I_0 = 0$ .", "Similarly, we obtain $\\delta (I_0/G_0)=0$ at the point $O_2$ for odd $n$ .", "(One has to keep in mind that the local parameter around this point is $\\lambda ^{1/2}$ .)", "$\\Box $ Remark 7.4 The definition of a presymplectic structure $\\omega $ on $\\mathcal {P}_n$ uses $\\Psi _0$ and $\\tilde{T}_0$ and hence relies on the normalization of $\\Psi _0$ .", "When restricted to the leaves from Proposition REF , the 2-form $\\omega $ becomes independent of the normalization of the Floquet-Bloch solutions.", "Additionally, the form $\\omega $ becomes non-degenerate, i.e., symplectic, when restricted to these leaves, as we prove below.", "The symplectic form is invariant with respect to the evolution given by the Lax equation.", "The proof is very similar to that of Corollary 4.2 in [8] (cf.", "[5], [6] for other proofs).", "Theorem 7.5 The rank of the invariant 2-form $\\omega $ restricted to the leaves of Proposition REF is equal to $2g$ ." ], [ "Proof.", "Since the 1-form $\\delta \\ln {k} \\,d\\lambda /\\lambda $ is holomorphic on $\\Gamma $ , it can be represented as a sum of the basis holomorphic differentials: $\\delta \\ln {k} \\,\\frac{d\\lambda }{\\lambda } = \\sum _{i=1}^{g} \\delta U_i \\,d\\omega _i,$ where $g$ is the genus of $\\Gamma $ .", "The coefficients $U_i$ can be found by integrating the last expression over the basis cycles $a_i$ of $H_1(\\Gamma )$ : $U_i = \\oint _{a_i} \\ln {k} \\,\\frac{d\\lambda }{\\lambda }\\, .$ According to formula (5.7) in [7], we have: $\\omega =\\sum _{i=1}^{g+3} \\delta \\ln {k(p_i)} \\wedge \\delta \\ln {\\lambda (p_i)},$ where the points $p_i \\in \\Gamma ,\\; 1 \\le i \\le g+3,$ constitute the pole divisor $D_{0,0}$ of the normalized Floquet-Bloch solution $\\psi _{0,0}$ .", "After rearranging the terms, we obtain: $\\omega = \\delta \\left( \\sum _{s=1}^{g+3} \\int ^{p_s} \\delta \\ln {k} \\,\\frac{d\\lambda }{\\lambda } \\right)=\\delta \\left( \\sum _{s,i} \\int ^{p_s} \\delta U_i \\,d\\omega _i \\right) = \\sum _{i=1}^{g} \\delta U_i \\wedge \\delta \\varphi _i,$ where $\\varphi _i = \\sum _{s=1}^{g+3} \\int ^{p_s} d\\omega _i$ are coordinates on the Jacobian $J({\\Gamma })$ .", "The variables $U_i$ and $\\varphi _i$ are natural Darboux coordinates for $\\omega $ , which also turn out to be action-angle coordinates for the pentagram map.", "(The latter follows from the general properties of the Krichever-Phong universal form for a given Lax representation, cf.", "[5], [6].)", "Let us show that the functions $U_i$ are independent.", "Assume the contrary, then there exists a vector $v$ on the space ${\\mathcal {P}}_n$ , such that $\\delta U_i(v)=0$ for all $i$ .", "Then it follows from (REF ) that $\\partial _v k \\equiv 0$ .", "After applying the operator $\\partial _v$ to $R(\\lambda ,k)$ , we conclude that $k$ satisfies an algebraic equation of degree 3, which is impossible, since $\\Gamma $ is a 4-fold cover of the $\\lambda $ -plane.", "$\\Box $ Remark 7.6 In more details, there are the following two cases: even $n=2q$ .", "The dimension of the space $\\mathcal {P}_n$ is $6q$ .", "The codimension of the leaves is 6.", "Therefore, the dimension of the leaves matches the doubled dimension of the tori: $2g=6q-6$ .", "odd $n=2q+1$ .", "The dimension of the space $\\mathcal {P}_n$ is $6q+3$ .", "The codimension of the leaves is 3.", "Again, the dimension of the leaves matches the doubled dimension of the tori: $2g=6q$ .", "The algebraic-geometric integrability in the complex case implies Arnold-Liouville integrability in the real one.", "Indeed, the pre-symplectic form depends on entries of the monodromy matrix in a rational way, since it is independent of the permutation of sheets of the spectral curve $\\Gamma $ .", "Therefore, its restriction to the space of the real $n$ -gons provides a real pre-symplectic structure.", "One obtains invariant Poisson brackets on the space of polygons $\\mathcal {P}_n$ by inverting the real symplectic structure on the leaves, while employing invariants of Proposition REF as the corresponding Casimirs.", "Problem 7.7 Find an explicit formula for an invariant Poisson structure with the above symplectic leaves." ], [ "A Lax representation in higher dimensions", "The origin of the integrability of the pentagram map is the presence of its scaling invariance.", "Assume that $gcd(n,d+1)=1$ .", "The difference equation (REF ) $V_{j+d+1} = a_{j,d} V_{j+d} + a_{j,d-1} V_{j+d-1} +...+ a_{j,1} V_{j+1} + (-1)^d V_j$ allows one to introduce coordinates $a_{j,1},a_{j,2},...a_{j,d}, \\, 0\\le j\\le n-1$ , on the space of twisted $n$ -gons in any dimension $d$ .", "Proposition-conjecture 8.1 (The scaling invariance) The pentagram map on twisted $n$ -gons in ${\\mathbb {CP}}^d$ is invariant with respect to the following scaling transformations: for odd $d=2\\varkappa +1$ the transformations are $a_{j,1} \\rightarrow s a_{j,1},\\; a_{j,3} \\rightarrow s a_{j,3},\\;a_{j,5} \\rightarrow s a_{j,5},\\;...\\;, a_{j,d} \\rightarrow s a_{j,d}\\,,$ while other coefficients $a_{j,2l}$ with $l=1,...,\\varkappa $ do not change; for even $d=2\\varkappa $ the transformations are $a_{j,1} \\rightarrow s^{-\\varkappa }a_{j,1} ,\\; a_{j,3} \\rightarrow s^{1-\\varkappa }a_{j,3}, \\;...\\:, a_{j,d-1} \\rightarrow s^{-1}a_{j,d-1},$ $a_{j,2} \\rightarrow sa_{j,2} ,\\; a_{j,4} \\rightarrow s^{2}a_{j,4},\\;...\\:, a_{j,d} \\rightarrow s^{\\varkappa }a_{j,d}$ for all $s\\in *$ .We thank G.Mari-Beffa for correcting an error in the scaling for even $d$ in the first version of this manuscript, as well as in the short version [16].", "This error related to numerics with a different choice of vertices for the diagonal planes leads to another system, different from $T_{p,r}$ , which also turns out to be integrable and will be discussed elsewhere." ], [ "Proof.", "In any dimension $d$ the pentagram map is a composition of involutions $\\alpha $ and $\\beta $ , see Section REF .", "(More precisely, $\\alpha $ is not an involution for even $d$ , but its square $\\alpha ^2$ is a shift in the vertex index, see [11] for the 2D case.)", "One can prove that the involution $\\alpha : V_i \\rightarrow W_i $ in any dimension has the form $W_{j+d+1} = (-1)^{d+1}(a_{\\star ,1} W_{j+d} + a_{\\star ,2} W_{j+d-1} +...+ a_{\\star ,d} W_{j+1} - W_j)\\,,$ where $\\star $ stands for the first index, which is irrelevant for the scaling (Lemma REF proves the case $d=3$ ).", "We call this Proposition-conjecture because the proof of an analog of Lemma REF (for the map $\\beta $ ) in higher dimensions is computer assisted.", "One verifies that for a given dimension $d$ the coefficients consist of the terms that are consistent with the scaling.", "$\\Box $ We obtained explicit formulas, and hence a direct (theoretical) proof of the scaling invariance for the pentagram maps up to dimension $d\\le 6$ .", "This bound is related to computing powers to produce explicit formulas and might be extended.", "However, we have no general purely theoretical proof valid for all $d$ and it would be very interesting to find it.", "Problem 8.2 Find a general proof of the scaling invariance of the pentagram map in any dimension $d$ .", "Theorem 8.3 The scale-invariant pentagram map on twisted $n$ -gons in any dimension $d$ is a completely integrable system.", "It is described by the Lax matrix $L_j^{-1}(\\lambda ) =\\left(\\begin{array}{cccc|c}0 & 0 & \\cdots & 0 &(-1)^d\\\\ \\cline {1-5}\\multicolumn{4}{c|}{{4}*{D(\\lambda )}} & a_{j,1}\\\\&&&& a_{j,2}\\\\&&&& \\cdots \\\\&&&& a_{j,d}\\\\\\end{array}\\right),$ where $D(\\lambda )$ is the following diagonal matrix of size $d \\times d$ : for odd $d=2\\varkappa +1$ , one has $D(\\lambda ) = \\text{diag}(\\lambda ,1,\\lambda ,1,...,\\lambda )$ ; for even $d=2\\varkappa $ , one has one has $D(\\lambda ) = \\text{diag}(1,\\lambda ,1,\\lambda ,...,1, \\lambda )$ ." ], [ "Proof sketch.", "By using the scaling invariance of the pentagram map, one derives the Lax matrix exactly in the same way as in 3D, see Section REF .", "Namely, first construct the $(d+1) \\times (d+1)$ -matrix $N_j(s)$ depending on our scaling parameter $s$ , and then use the formula $L_j^{-1}(\\lambda ) = \\left(g^{-1} N_j(s) g \\right)/s^m$ with a suitable choice of the diagonal $(d+1) \\times (d+1)$ -matrix $g$ and an appropriate function of the parameter $s$ .", "For odd $d=2\\varkappa +1$ , we have $g = \\text{diag}(1,s,1,s,...,1,s)$ , $m=1$ , and $\\lambda \\equiv s^{-2}$ , whereas for even $d=2\\varkappa $ , we have $g = \\text{diag}(1, s^{-\\varkappa },s,s^{1-\\varkappa },s^2,..., s^{\\varkappa -1},s^{-1},s^{\\varkappa })$ , $m={\\varkappa }$ , and $\\lambda \\equiv s^{-d-1}$ .", "The Lax representation with a spectral parameter is constructed as we described above.", "Using the genericity assumptions similar to those used in the 2D and 3D cases, one constructs the spectral map and its inverse, which is equivalent to algebraic-geometric integrability of the pentagram map.", "Coefficients of the spectral curve form a maximal family of first integrals.", "Along with a (pre)symplectic structure defined by the Krichever–Phong formula, this provides the Arnold–Liouville integrability of the system on the corresponding symplectic leaves in the real case.", "$\\Box $ The scaling parameter has a clear meaning in the continuous limit: Proposition 8.4 For any dimension $d$ the continuous limit of the scaling transformations corresponds to the spectral shift $L\\rightarrow L+\\lambda $ of the differential operator $L$ ." ], [ "Proof.", "In 2D this was proved in [11].", "A continuous analog of the difference equation (REF ) is $G(x+(d+1)\\epsilon )=a_d(x,\\epsilon )G(x+d\\epsilon )+...+a_1(x,\\epsilon )G(x+\\epsilon )+(-1)^d G(x),$ where $G(x)$ satisfies the differential equation (REF ) with a differential operator $L$ of the form (REF ).", "Using the Taylor expansion for $G(x+j\\epsilon )$ and the expansion $a_j(x,\\epsilon )=a_j^0(x)+\\epsilon a_j^1(x)+...,$ we obtain expressions of $a_k^i$ in terms of the coefficients of $L$ , i.e., in terms of functions $u_j(x)$ and their derivatives.", "We find that the terms $a_k^0$ are constant, $a_k^1=0$ for all $k$ , while $a_k^i$ for $i\\ge 2$ are linear in $u_{d-i+1}$ and differential polynomials in the preceding coefficients $u_{d-1},...,u_{d-i+2}$ .", "The scaling parameter also has an expansion in $\\epsilon $ : $s=\\tau _0+\\epsilon \\tau _1+\\epsilon ^2 \\tau _2+...$ .", "We apply it to the coefficients $a_i(x,\\epsilon )$ and impose the condition that $a_k^0$ and $a_k^1=0$ are fixed, similarly to [11].", "By term-wise calculations (different in the cases of even and odd $d$ and using the “triangular form\" of the expressions for $a_k^i$ ), one successively obtains that $\\tau _0=1$ ,  $\\tau _1=...=\\tau _d=0$ , i.e., $s$ must have the form $s=1+\\tau _{d+1} \\epsilon ^{d+1}+{\\mathcal {O}}(\\epsilon ^{d+2})$ .", "Its action shifts only the last term of $L$ : $u_0\\rightarrow u_0+{\\rm const}\\cdot \\tau _{d+1} \\epsilon ^{d+1}$ , i.e., it is equivalent to the spectral shift $L\\rightarrow L+\\lambda $ .", "$\\Box $ Note that the spectral shift commutes with the KdV flows.", "Indeed, $d/dt (L+\\lambda )=d/dt\\, L=[Q_2,L]=[Q_2, L+\\lambda ]$ , since $Q_2(L):=\\partial ^2 +\\frac{2}{d+1}u_{d-1}=Q_2(L+\\lambda )$ for operators $L$ of degree $d+1\\ge 3$ .", "Equivalently, the pentagram map commutes with the scaling transformations in the continuous limit." ], [ "Continuous limit in the 3D case", "In this section we present explicit formulas manifesting Theorem REF on the continuous limit of the 3D pentagram map.", "Consider a curve $G(x)$ in ${\\mathbb {R}}^4$ given by the differential equation $G^{\\prime \\prime \\prime \\prime }+u(x)G^{\\prime \\prime }+v(x)G^{\\prime }+w(x)G=0$ with periodic coefficients $u(x),v(x),w(x)$ .", "To find the continuous limit, we fix $\\epsilon $ and consider a plane $P_\\epsilon (x)$ passing through the three points $G(x-\\epsilon ), G(x), G(x+\\epsilon )$ on this curve.", "We are looking for an equation of the envelope curve $L_\\epsilon (x)$ for these planes.", "This envelope curve $L_\\epsilon (x)$ satisfies the following system of equations: $&\\det | G(x), G(x+\\epsilon ), G(x-\\epsilon ), L_\\epsilon (x) |=0\\\\&\\det | G(x), G(x+\\epsilon ), G(x-\\epsilon ), L^{\\prime }_\\epsilon (x) |=0\\\\&\\det | G(x), G(x+\\epsilon ), G(x-\\epsilon ), L^{\\prime \\prime }_\\epsilon (x) |=0\\,.$ By considering the Taylor expansion and using the normalizations $\\det | L_\\epsilon , L^{\\prime }_\\epsilon , L^{\\prime \\prime }_\\epsilon , L^{\\prime \\prime \\prime }_\\epsilon |=1$ and $\\det | G, G^{\\prime }, G^{\\prime \\prime }, G^{\\prime \\prime \\prime } |=1$ we find that $L_\\epsilon (x) = G(x)+ \\dfrac{\\epsilon ^2}{6} \\left( G^{\\prime \\prime }(x)+\\dfrac{u}{2} G(x)\\right) +{\\mathcal {O}}(\\epsilon ^4)$ as $\\epsilon \\rightarrow 0$ .", "Now, the equation $L^{\\prime \\prime \\prime \\prime }_\\epsilon +u_\\epsilon L^{\\prime \\prime }_\\epsilon +v_\\epsilon L^{\\prime }_\\epsilon +w_\\epsilon L_\\epsilon =0$ implies that: $u_\\epsilon &= u + \\dfrac{\\epsilon ^2}{3}(v^{\\prime }-u^{\\prime \\prime })+{\\mathcal {O}}(\\epsilon ^4),\\\\v_\\epsilon &= v + \\dfrac{\\epsilon ^2}{6}(2w^{\\prime }+v^{\\prime \\prime }-uu^{\\prime }-2u^{\\prime \\prime \\prime })+{\\mathcal {O}}(\\epsilon ^4),\\\\w_\\epsilon &= w + \\dfrac{\\epsilon ^2}{12}(2w^{\\prime \\prime }-uu^{\\prime \\prime }-vu^{\\prime }-u^{\\prime \\prime \\prime \\prime })+{\\mathcal {O}}(\\epsilon ^4).$ These equations describe the $(2,4)$ -equation in the Gelfand-Dickey hierarchy: $\\dot{L} = [Q_2,L] \\Leftrightarrow {\\left\\lbrace \\begin{array}{ll}\\dot{u} = 2v^{\\prime }-2u^{\\prime \\prime },\\\\\\dot{v} = v^{\\prime \\prime }+2w^{\\prime }-uu^{\\prime }-2u^{\\prime \\prime \\prime },\\\\\\dot{w} = w^{\\prime \\prime } - \\dfrac{1}{2} vu^{\\prime }-\\dfrac{1}{2}uu^{\\prime \\prime } - \\dfrac{1}{2} u^{\\prime \\prime \\prime \\prime },\\end{array}\\right.", "}$ where $L = \\partial ^4 + u \\partial ^2 + v \\partial + w$ and $Q_2 = (L^{2/4})_+ = \\partial ^2 + \\dfrac{u}{2}$ .", "Remark 9.1 A different choice of the points defining the plane $P_\\epsilon (x)$ on the original curve leads to the same continuous limit.", "For instance, the choice of $G(x-3\\epsilon ),G(x+\\epsilon ),G(x+2\\epsilon )$ results in the same expression for $L_\\epsilon (x)$ , where in (REF ) instead of the coefficient $\\epsilon ^2/6$ one has $7\\epsilon ^2/6$ .", "This leads to the same evolution of the curve $G$ with a different time parameterization, cf.", "Remark REF ." ], [ "Higher terms of the continuous limit", "Recall that in the continuous limit for the pentagram map in ${\\mathbb {RP}}^d$ the envelope for osculating planes moves according to the $(2,d+1)$ -KdV equation (Theorem REF ).", "This evolution is defined by the $\\epsilon ^2$ -term of the expansion of the function $L_\\epsilon (x)$ .", "The same proof works in the following more general setting.", "Let $L$ be a differential operator (REF ) of order $d+1$ and $G$ a non-degenerate curve defined by its solutions: $LG=0$ .", "Proposition 9.2 Assume that the curve $G$ evolves according to the law $ \\dot{G}= Q_m G$ , where $Q_m:=(L^{m/(d+1)})_+$ is the differential part of the $m$ th power of the operator $Q=L^{1/(d+1)}$ .", "Then this evolution defines the equation $\\dot{L}=[Q_m, L]$ , which is the $(m,d+1)$ -equation in the corresponding KdV hierarchy of $L$ .", "Furthermore, one can define the simultaneous evolution of all terms in the $\\epsilon $ -expansion of $L_\\epsilon (x)$ using the following construction.", "For the pseudodifferential operator $Q:=L^{1/(d+1)}$ consider the formal series $\\exp (\\epsilon Q):=1+\\epsilon Q +\\frac{\\epsilon ^2}{2} Q^2+...$ and take its differential part: $\\left(\\exp (\\epsilon Q)\\right)_+=\\left(1+\\epsilon Q +\\frac{\\epsilon ^2}{2} Q^2+...\\right)_+=1+\\epsilon Q_1 +\\frac{\\epsilon ^2}{2} Q_2+...=\\sum _0^\\infty \\frac{\\epsilon ^m}{m!", "}Q_m\\,.$ For each power of $\\epsilon $ this is a multiple of the differential operator $Q_m$ , which is the differential part of the $m$ th power $Q^m$ of the operator $Q=L^{1/(d+1)}$ .", "Corollary 9.3 The formal evolution equation $ \\dot{G}= \\left(\\exp (\\epsilon Q)\\right)_+ G$ corresponds to the full KdV hierarchy $\\dot{L}=[\\left(\\exp (\\epsilon Q)\\right)_+ , L]$ , where the operator $L$ is of order $d+1$ and the $(m,d+1)$ -equation corresponds to the power $\\epsilon ^m$ .", "A natural question is which equations of this hierarchy actually appear as the evolution of the envelope $L_\\epsilon (x)$ .", "Recall that only even powers of $\\epsilon $ arise in the expansion of the function $L_\\epsilon (x)$ for the continuous limit of the pentagram map.", "The $\\epsilon ^2$ -term gives the $(2,d+1)$ -KdV equation.", "It turns out that the $\\epsilon ^4$ -term in the continuous limit of the 2D pentagram map results in the equation very similar to the $(4,3)$ -equation in the KdV hierarchy (which is a higher-order Boussinesq equation).", "Although the numerical coefficients in these differential equations are different, one may hope to obtain the exact equations of the KdV hierarchy for different $m$ by using an appropriate rescaling.", "This allows one to formulate Problem 9.4 Do higher $(m, d+1)$ -KdV flows appear as the $\\epsilon ^{m}$ -terms in the expansion of the envelope $L_\\epsilon (x)$ for the continuous limit of the pentagram map for any even $m>2$ ?" ], [ "Bijection of the spectral map", "In this appendix we sketch the proof of Proposition REF and prove Proposition REF , which allows one to reconstruct the $L$ -matrix from spectral data, and hence complete the proof of Theorem REF on the spectral map.", "Proposition 9.5 (= Proposition REF ) For any $n$ , given a generic spectral curve with marked points and a generic divisor $D$ of degree $g+3$ one can recover a sequence of matrices $\\tilde{L}_{i,t}(\\lambda ) =\\begin{pmatrix}0 & 0 & 0 & -1\\\\\\lambda x_i y_i & 0 & 0 & 1\\\\0 & z_i & 0 & 1\\\\0 & 0 & \\lambda x_i & 1\\end{pmatrix}^{-1},$ for $ 0 \\le i \\le n-1 $ and any $ t.$" ], [ "Proof.", "Without loss of generality we describe the procedure to reconstruct the matrices $L_i(\\lambda ):=\\tilde{L}_{i,0}(\\lambda ), $ for $ 0 \\le i \\le n-1$ and $t=0$ .", "First, we pick functions $\\psi _{i,j}:=\\psi _{i,0,j}$ for $\\; 0 \\le i \\le n-1,\\; 1 \\le j \\le 4,$ and $t=0$ , satisfying Proposition REF .", "Note that according to the Riemann-Roch theorem, the functions $\\psi _{i,1}$ and $\\psi _{i,4}$ are defined up to a multiplication by constants, whereas the functions $\\psi _{i,2}$ and $\\psi _{i,3}$ belong to 2-dimensional subspaces.", "The functions $\\psi _{i,1}$ and $\\psi _{i,4}$ belong to the same subspaces.", "We pick the pairs of functions $\\psi _{i,1},\\; \\psi _{i,3}$ and $\\psi _{i,2},\\; \\psi _{i,4}$ to be linearly independent.", "Observe that any sets of functions $\\psi _{i,1},...,\\psi _{i,4} $ satisfying Proposition REF are related by gauge transformations $\\psi _i \\rightarrow g_i^{-1} \\psi _i$ , where $g_i =\\begin{pmatrix}A_i & 0 & 0 & 0\\\\0 & B_i & 0 & E_i\\\\F_i & 0 & C_i & 0\\\\0 & 0 & 0 & D_i\\end{pmatrix},\\;g_i = g_{i+n},$ and $\\psi _i$ stands for $\\psi _i=(\\psi _{i,1},...,\\psi _{i,4})^T$ .", "We also define $\\psi _n$ to be $\\psi _n=I_0k\\psi _0$ for any $n$ in $(x,y,z)$ -variables.", "We find the unique matrix $L^{\\prime }_i$ satisfying the equation $\\psi _i = (L^{\\prime }_i)^{-1} \\psi _{i+1}$ : $L^{\\prime }_i(\\lambda ) =\\begin{pmatrix}0 & 0 & 0 & t_{i,1}\\\\\\lambda t_{i,5} & 0 & \\lambda t_{i,6} & t_{i,2}\\\\0 & t_{i,7} & 0 & t_{i,3}\\\\\\lambda t_{i,8} & 0 & \\lambda t_{i,9} & t_{i,4}\\end{pmatrix}^{-1}.$ One can check that there exists the unique choice of the matrices $g_i, \\; 0 \\le i \\le n-1,$ such that the equality $L_i(\\lambda ) = g_{i+1} L^{\\prime }_i(\\lambda ) g_i^{-1}$ is possible.", "The latter is equivalent to the following system of equations ($0 \\le i \\le n-1$ ): $\\dfrac{A_i t_{i,1}}{D_{i+1}}=-1; \\quad \\dfrac{B_i t_{i,2}+E_i t_{i,4}}{D_{i+1}}=1; \\quad \\dfrac{D_i t_{i,4}}{D_{i+1}}=1;$ $\\dfrac{B_{i+1} F_i t_{i,1}+B_{i+1} C_i t_{i,3}-C_i E_{i+1} t_{i,7}}{B_{i+1} D_{i+1}}=1; \\quad B_i t_{i,6} + E_i t_{i,9}=0; \\quad C_{i+1} t_{i,8} - F_{i+1} t_{i,9}=0.$ These equations decouple and may be solved explicitly.", "One only needs to check the solvability of $n$ equations $\\dfrac{D_i t_{i,4}}{D_{i+1}}=1, \\; 0 \\le i \\le n-1,$ for $n$ variables $D_i, \\; 0 \\le i \\le n-1$ .", "A non-trivial solution exists provided that $\\prod _{i=0}^{n-1} t_{i,4} = 1$ .", "It depends on an arbitrary constant, which corresponds to multiplication of all matrices $g_i$ by the same number and does not affect the Lax matrices.", "One can check that $t_{i,4} = \\dfrac{\\psi _{i+1,4}(O_1)}{\\psi _{i,4}(O_1)} \\text{ and } \\prod _{i=0}^{n-1} t_{i,4} = \\dfrac{\\psi _{n,4}(O_1)}{\\psi _{0,4}(O_1)}=I_0 k(O_1).$ By using Lemma REF we find the value $k(O_1) ={1}/{I_0}$ as required.", "Now the remaining variables $A_i,B_i,C_i,E_i,F_i,\\; 0 \\le i \\le n-1,$ are uniquely determined.", "$\\Box $ Corollary 9.6 For odd $n$ , given a generic spectral curve with marked points and a generic divisor $D$ one can recover a sequence of matrices $L_{i,t}(\\lambda ) =\\begin{pmatrix}0 & 0 & 0 & -1\\\\\\lambda & 0 & 0 & c_j\\\\0 & 1 & 0 & b_j\\\\0 & 0 & \\lambda & a_j\\end{pmatrix}^{-1}$ with $0 \\le i \\le n-1$ and any $t$ ." ], [ "Proof.", "The statement follows from Proposition REF and the fact that $(a_j,b_j,c_j),\\; 0 \\le i \\le n-1,$ are coordinates on the space $\\mathcal {P}_n$ for odd $n$ .", "$\\Box $ We complete the exposition with a sketch of the proof for Proposition REF for even $n$ (the case of odd $n$ is similar).", "Proposition 9.7 (= Proposition REF $^{\\prime }$ ) For even $n$ , the divisors of the coordinate functions $\\psi _{i,t,1},...,\\psi _{i,t,4}$ for $ 0 \\le i \\le n-1$ and any integer $t$ satisfy the following inequalities, provided that their divisors remain non-special up to time $t$ : $(\\psi _{i,t,1}) \\ge -D +O_2 + \\lfloor \\dfrac{i-t+2}{2} \\rfloor W_{12}+ \\lfloor \\dfrac{i-t+1}{2} \\rfloor W_{34}- \\lfloor \\dfrac{i+1}{2} \\rfloor O_{24}-\\lfloor \\dfrac{i}{2} \\rfloor O_{34}+tO_{14};$ $(\\psi _{i,t,2}) \\ge -D + \\lfloor \\dfrac{i-t+1}{2} \\rfloor W_{12}+ \\lfloor \\dfrac{i-t}{2} \\rfloor W_{34}- \\lfloor \\dfrac{i-1}{2} \\rfloor O_{24}-\\lfloor \\dfrac{i}{2} \\rfloor O_{34}+tO_{14};$ $(\\psi _{i,t,3}) \\ge -D + \\lfloor \\dfrac{i-t+2}{2} \\rfloor W_{12}+ \\lfloor \\dfrac{i-t+1}{2} \\rfloor W_{34}- \\lfloor \\dfrac{i+1}{2} \\rfloor O_{34}- \\lfloor \\dfrac{i}{2} \\rfloor O_{24}+tO_{14};$ $(\\psi _{i,t,4}) \\ge -D +O_2 + \\lfloor \\dfrac{i-t+1}{2} \\rfloor W_{12}+ \\lfloor \\dfrac{i-t}{2} \\rfloor W_{34}- \\lfloor \\dfrac{i-1}{2} \\rfloor O_{34}- \\lfloor \\dfrac{i}{2} \\rfloor O_{24}+tO_{14};$ where $D$ is an effective divisor of degree $g+3$ , and $\\lfloor x \\rfloor $ is the floor function of $x$ ." ], [ "Proof.", "First, we prove these inequalities for $t=0$ and $0 \\le i \\le n-1$ .", "For illustration we find the multiplicities of the components of the vector $\\psi _{i,0}$ at the point $O_2$ , while other points can be treated in a similar fashion.", "We employ the matrices $\\tilde{L}_{i,t}$ in the coordinates $x_i,y_i,z_i$ .", "Notice that a cyclic permutation of indices $(n-1,n-2,...,1,0)$ changes the monodromies $T_i \\rightarrow T_{i+1}$ and the Floquet-Bloch solutions $\\bar{\\psi }_i \\rightarrow \\bar{\\psi }_{i+1}$ .", "For even $n$ , it also permutes $\\bar{\\psi }_i(O_2) \\leftrightarrow \\bar{\\psi }_i(O_3)$ and $W_{12} \\leftrightarrow W_{34}$ , i.e., the corresponding pairs of the vectors $\\bar{\\psi }_i$ at the points $(W_1,W_2)$ and $(W_3,W_4)$ are swapped.", "Using the asymptotic expansion of $\\tilde{T}_{0,t}(\\lambda )$ at $\\lambda =0$ , the definition of the Floquet-Bloch solution, and the normalization condition, one can show that $\\psi _{0,0}=(O(\\lambda ),O(\\lambda ),1+O(\\lambda ),O(\\lambda ))^T$ as $\\lambda \\rightarrow 0$ at the point $O_2$ .", "Since $L_{1,0}(\\lambda )L_{0,0}(\\lambda ) =\\begin{pmatrix}1 & 1 & 0 & 0\\\\0 & 0 & 0 & 0\\\\y_1 & y_1 & 0 & 0\\\\0 & 0 & 0 & 0\\end{pmatrix}\\dfrac{1}{x_0 x_1 y_0 y_1 \\lambda ^2} + O\\left( \\dfrac{1}{\\lambda } \\right) \\text{ as } \\lambda \\rightarrow 0,$ and $\\psi _{2,0}=L_{1,0} L_{0,0} \\psi _{0,0}$ , generically one has $\\psi _{2,0}=(O(1),O(1),O(1/\\lambda ),O(1))^T$ at $O_2$ .", "By definition, the normalized vectors are $\\bar{\\psi }_{i,t} = f_{i,t} \\psi _{i,t}$ .", "Using a cyclic permutation, we find that $\\bar{\\psi }_{2k,0}=(O(\\lambda ),O(\\lambda ),1+O(\\lambda ),O(\\lambda ))^T$ and that $f_{2,0}(\\lambda ) = O(\\lambda )$ at $O_2$ .", "Using the permutation argument again, we derive that $f_{i+2,0}(\\lambda )/f_{i,0}(\\lambda ) = O(\\lambda )$ at $O_2$ for even $i$ .", "Therefore, one has $f_{2k,0}(\\lambda ) = O(\\lambda ^k)$ at $O_2$ .", "Now the required multiplicities for the vector $\\psi _{2k,0}$ at $O_2$ follow.", "Furthermore, since $\\psi _{2k+1,0} = L_{2k} \\psi _{2k,0}$ , one can check that generically $f_{2k+1,0}(\\lambda )/f_{2k,0}(\\lambda ) = O(1)$ and $\\bar{\\psi }_{2k+1,0}=(O(1),O(\\lambda ),O(1),O(1))^T$ at the point $O_2$ .", "This establishes also the multiplicities for the vector $\\psi _{2k+1,0}$ at $O_2$ .", "Having proved the proposition for $t=0$ , one can prove it for $t>0$ by using the formula $\\psi _{i,t+1} = \\tilde{P}_{i,t} \\psi _{i,t}$ .", "Note that it suffices to study the cases $t=0$ and $t=1$ only.", "Consider, for example, the multiplicity of the function $\\psi _{i,1,1}$ at the point $O_2$ .", "Since $\\psi _{i,1,1}=(\\psi _{i,0,2}-\\psi _{i,0,4})/(x_i(1+y_i+z_{i+1}))$ , one can check that the multiplicity of the right-hand side at $O_2$ is $1-k$ for $i=2k$ and it is equal to $-k$ for $i=2k+1$ , i.e., $\\psi _{i,1,1}$ and $\\psi _{i,0,1}$ have the same multiplicities at $O_2$ .", "Other cases are treated in a similar way.", "$\\Box $" ] ]

[ [ "Robust methods for LTE and WiMAX dimensioning" ], [ "Abstract This paper proposes an analytic model for dimensioning OFDMA based networks like WiMAX and LTE systems.", "In such a system, users require a number of subchannels which depends on their \\SNR, hence of their position and the shadowing they experience.", "The system is overloaded when the number of required subchannels is greater than the number of available subchannels.", "We give an exact though not closed expression of the loss probability and then give an algorithmic method to derive the number of subchannels which guarantees a loss probability less than a given threshold.", "We show that Gaussian approximation lead to optimistic values and are thus unusable.", "We then introduce Edgeworth expansions with error bounds and show that by choosing the right order of the expansion, one can have an approximate dimensioning value easy to compute but with guaranteed performance.", "As the values obtained are highly dependent from the parameters of the system, which turned to be rather undetermined, we provide a procedure based on concentration inequality for Poisson functionals, which yields to conservative dimensioning.", "This paper relies on recent results on concentration inequalities and establish new results on Edgeworth expansions." ], [ "Introduction", "Future wireless systems will widely rely on OFDMA (Orthogonal Frequency Division Multiple Access) multiple access technique.", "OFDMA can satisfy end user's demands in terms of throughput.", "It also fulfills operator's requirements in terms of capacity for high data rate services.", "Systems such as 802.16e and 3G-LTE (Third Generation Long Term Evolution) already use OFDMA on the downlink.", "Dimensioning of OFDMA systems is then of the utmost importance for wireless telecommunications industry.", "OFDM (Orthogonal Frequency Division Multiplex) is a multi carrier technique especially designed for high data rate services.", "It divides the spectrum in a large number of frequency bands called (orthogonal) subcarriers that overlap partially in order to reduce spectrum occupation.", "Each subcarrier has a small bandwidth compared to the coherence bandwidth of the channel in order to mitigate frequency selective fading.", "User data is then transmitted in parallel on each sub carrier.", "In OFDM systems, all available subcarriers are affected to one user at a given time for transmission.", "OFDMA extends OFDM by making it possible to share dynamically the available subcarriers between different users.", "In that sense, it can then be seen as multiple access technique that both combines FDMA and TDMA features.", "OFDMA can also be possibly combined with multiple antenna (MIMO) technology to improve either quality or capacity of systems.", "Figure: OFDMA principle : subcarriers are allocated according tothe required transmission rateIn practical systems, such as WiMAX or 3G-LTE, subcarriers are not allocated individually for implementation reasons mainly inherent to the scheduler design and physical layer signaling.", "Several subcarriers are then grouped in subchannels according to different strategies specific to each system.", "In OFDMA systems, the unit of resource allocation is mainly the subchannels.", "The number of subchannels required by a user depends on his channel's quality and the required bit rate.", "If the number of demanded subchannels by all users in the cell is greater than the available number of subchannel, the system is overloaded and suffer packet losses.", "The questions addressed here can then be stated as follows: how many subchannels must be assigned to a BS to ensure a small overloading probability ?", "Given the number of available subchannels, what is the maximum load, in terms of mean number of customers per unit of surface, that can be tolerated ?", "Both questions rely on accurate estimations of the loss probability.", "The objectives of this paper are twofold: First, construct and analyze a general performance model for an isolated cell equipped with an OFDMA system as described above.", "We allows several classes of customers distinguished by their transmission rate and we take into account path-loss with shadowing.", "We then show that for a Poissonian configuration of users in the cell, the required number subchannels follows a compound Poisson distribution.", "The second objective is to compare different numerical methods to solve the dimensioning problem.", "In fact, there exists an algorithmic approach which gives the exact result potentially with huge memory consumption.", "On the other hand, we use and even extend some recent results on functional inequalities for Poisson processes to derive some approximations formulas which turn to be rather effective at a very low cost.", "When it comes to evaluate the performance of a network, the quality of such a work may be judged according to several criteria.", "First and foremost, the exactness is the most used criterion: it means that given the exact values of the parameters, the real system, the performances of which may be estimated by simulation, behaves as close as possible to the computed behavior.", "The sources of errors are of three kinds: The mathematical model may be too rough to take into account important phenomena which alter the performances of the system, this is known as the epistemic risk.", "Another source may be in the mathematical resolution of the model where we may be forced to use approximate algorithms to find some numerical values.", "The third source lies in the lack of precision in the determination of the parameters characterizing the system: They may be hard, if not impossible, to measure with the desired accuracy.", "It is thus our point of view that exactness of performance analysis is not all the matter of the problem, we must also be able to provide confidence intervals and robust analysis.", "That is why, we insist on error bounds in our approximations.", "Resources allocation on OFDMA systems have been extensively studied over the last decade, often with joint power and subcarriers allocation, see for instance [1], [8], [14], [15].", "The problem of OFDMA planning and dimensioning have been more recently under investigation.", "In [7], the authors propose a dimensioning of OFDMA systems focusing on link outage but not on the other parameters of the systems.", "In [11], the authors give a general methodology for the dimensioning of OFDMA systems, which mixes a simulation based determination of the distribution of the signal-to-interference-plus-noise ratio (SINR) and a Markov chain analysis of the traffic.", "In [3], [9], the authors propose a dimensioning method for OFDMA systems using Erlang's loss model and Kaufman-Roberts recursion algorithm.", "In [4], the authors study the effect of Rayleigh fading on the performance of OFDMA networks.", "The article is organized as follows.", "In Section , we describe the system model and set up the problem.", "In Section , we examine four methods to derive an exact, approximate or robust value of the number of subchannels necessary to ensure a given loss probability.", "In Section , we apply these formulas to the particular situation of OFDMA systems.", "A new bound for the Edgeworth expansion is in Section and Section contains a new proof of the concentration inequality established for instance in [16]." ], [ "System Model", "In practical systems, such as WiMAX or 3G-LTE, resource allocation algorithms work at subchannel level.", "The subcarriers are grouped into subchannels that the system allocates to different users according to their throughput demand and mobility pattern.", "For example, in WiMAX, there are three modes available for building subchannels: FUSC (Fully Partial Usage of Subchannels), PUSC (Partial Usage of SubChannels) and AMC (Adaptive modulation and coding).", "In FUSC, subchannels are made of subcarriers spread over all the frequency band.", "This mode is generally more adapted to mobile users.", "In AMC, the subcarriers of a subchannel are adjacent instead of being uniformly distributed over the spectrum.", "AMC is more adapted to nomadic or stationary users and generally provides higher capacity.", "The grouping of subcarriers into subchannels raises the problem of the estimation of the quality of a subchannel.", "Theoretically channel quality should be evaluated on each subcarrier of the corresponding subchannel to compute the associated capacity.", "This work assumes that it is possible to consider a single channel gain for all the subcarriers making part of a subchannel (for example via channel gains evaluated on pilot subcarriers).", "We consider a circular cell $C$ of radius $R$ with a base station (BS for short) at its center.", "The transmission power dedicated to each subchannel by the base station is denoted by $P$ .", "Each subchannel has a total bandwidth $W$ (in kHz).", "The received signal power for a mobile station at distance $d$ from the BS can be expressed as $P(d)= \\frac{PK_\\gamma }{d^{\\gamma }}GF:=P_\\gamma G d^{-\\gamma },$ where $K_\\gamma $ is a constant equal to the attenuation at a reference distance, denoted by $d_{\\text{ref}}$ , that separates far field from near field propagation.", "Namely, $K_\\gamma =\\left(\\frac{c}{4\\pi f d_{\\text{ref}}}\\right)^2d_{\\text{ref}}^\\gamma ,$ where $f$ is the radio-wave frequency.", "The variable $\\gamma $ is the path-loss exponent which indicates the power at which the path loss increases with distance.", "Its depends on the specific propagation environment, in urban area, it is in the range from 3 to 5.", "It must be noted that this propagation model is an approximate model, difficult to calibrate for real life situations.", "In particular, it might be reasonable to envision models where $\\gamma $ depends on the distance so that the attenuation would be proportional to $d^{\\gamma (d)}$ .", "Because of the complexity of such a model, $\\gamma $ is often considered as constant but the path-loss is multiplied by two random variables $G$ and $F$ which represent respectively the shadowing, i.e.", "the attenuation due to obstacles, and the Rayleigh fading, i.e.", "the attenuation due to local movements of the mobile.", "Usually, $G$ is taken as a log-normal distribution: $G = 10^{S/10}$ , where $S\\sim \\mathcal {N}(\\kappa ,\\,v^2)$ .", "As to $F$ , it is customary to choose an exponential distribution with parameter 1.", "Both, the shadowing and the fading experienced by each user are supposed to be independent from other users' shadowing and fading.", "For the sake of simplicity, we will here treat the situation where only shadowing is taken into account, the computations would be pretty much like the forthcoming ones and the results rather similar should we consider Rayleigh fading.", "All active users in the cell compete to have access to some of the ${N_{\\text{avail}}}$ available subchannels.", "There are $K$ classes of users distinguished by the transmission rate they require: $C_k$ is the rate of class $k$ customers and $\\tau _k$ denotes the probability that a customer belongs to class $k$ .", "A user, at distance $d$ from the BS, is able to receive the signal only if the signal-to-interference-plus-noise ratio ${\\text{SNR}}=\\frac{P(d)}{I}$ is above some constant $\\beta _{min}$ where $I$ is the noise plus interference power and $P(d)$ is the received signal power at distance $d$ , see (REF ).", "If the ${\\text{SNR}}$ is below the critical threshold, then the user is said to be in outage and cannot proceed with his communication.", "To avoid excess demands, the operator may impose a maximum number ${N_{\\text{max}}}$ of allocated subchannels to each user at each time slot.", "According to the Shannon formula, for a user demanding a service of bit rate $C_k$ , located at distance $d$ from the BS and experiencing a shadowing $g$ , the number of requires subchannels is thus the minimum of ${N_{\\text{max}}}$ and of $N_{\\text{user}}={\\left\\lbrace \\begin{array}{ll}\\left\\lceil \\dfrac{C_k}{W\\log _2 \\left(1+P_\\gamma g d^{-\\gamma }/I\\right)}\\right\\rceil & \\text{ if } P_\\gamma g d^{-\\gamma }/I\\ge \\beta _{min},\\\\0 & \\text{ otherwise,}\\end{array}\\right.", "}$ where $\\left\\lceil x\\right\\rceil $ means the minimum integer number not smaller than $x$ .", "We make the simplifying assumption that the allocation is made at every time slot and that there is no buffering neither in the access point nor in each mobile station.", "All the users have independently from others a probability $p$ to have a packet to transmit at each slot.", "This means, that each user has a traffic pattern which follows a geometric process of intensity $p$ .", "We also assume that users are dispatched in the cell according to a Poisson process of intensity $\\lambda _0$ .", "According to the thinning theorem for Poisson processes, this induces that active users form a Poisson process of intensity $\\lambda =\\lambda _0p$ .", "This intensity is kept fixed over the time.", "That may result from two hypothesis: Either we consider that for a small time scale, users do not move significantly and thus the configuration does not evolve.", "Alternatively, we may consider that statistically, the whole configuration of active users has reached its equilibrium so that the distribution of active users does not vary through time though each user may move.", "From the previous considerations, a user is characterized by three independent parameters: his position, his class and the intensity of the shadowing he is experiencing.", "We model this as a Poisson process on $E=B(0,\\, R)\\times \\lbrace 1,\\,\\cdots ,\\, K\\rbrace \\times {\\mathbf {R}}^+$ of intensity measure $\\lambda \\text{ d}\\nu (x):=\\lambda (\\text{ d}x \\otimes \\text{ d}\\tau (k)\\otimes \\text{ d}\\rho (g))$ where $B(0,\\, R)=\\lbrace x\\in {\\mathbf {R}}^2, \\, \\Vert x\\Vert \\le R\\rbrace $ , $\\tau $ is the probability distribution of classes given by $\\tau (\\lbrace k\\rbrace )=\\tau _k$ and $\\rho $ is the distribution of the random variable $G$ defined above.", "We set $f(x,\\, k,\\, g)=\\min \\left({N_{\\text{max}}},\\right.\\\\\\left.", "{\\mathbf {1}}_{\\left\\lbrace P_\\gamma g \\Vert x\\Vert ^{-\\gamma }\\ge I\\beta _{min}\\right\\rbrace }\\left\\lceil \\frac{C_k}{W\\log _2\\left(1+P_\\gamma g\\Vert x\\Vert ^{-\\gamma }/I\\right)}\\right\\rceil \\right).$ With the notations of Section , ${N_{\\text{tot}}}=\\int _{\\text{cell}}f(x,\\, k,\\, g) \\ \\text{ d}\\omega (x,\\, k,\\, g).$ We are interested in the loss probability which is given by ${\\mathbf {P}}({N_{\\text{tot}}}\\ge {N_{\\text{avail}}}).$ We first need to compute the different moment of $f$ with respect to $\\nu $ in order to apply Theorem REF and Theorem REF .", "For, we set $l_k={N_{\\text{max}}}\\ \\wedge \\left\\lceil \\frac{C_k}{W\\log _2(1+\\beta _{\\min })}\\right\\rceil ,$ where $a\\wedge b=\\min (a,\\, b)$ .", "Furthermore, we introduce $\\beta _{k,\\,0}=\\infty $ , $\\beta _{k,\\, l}=\\frac{I}{P}\\left(2^{C_k/Wl}-1\\right), \\, 1\\le k\\le K,\\ 1\\le l\\le l_k-1,$ and $\\beta _{k,\\, l_k}=I\\beta _{\\min }/P.$ By the very definition of the ceiling function, we have $\\int _E f^p\\text{ d}\\nu \\\\= \\sum _{k=1}^K \\tau _k\\sum _{l=1}^{l_k}l^p\\ \\int _{\\text{cell}}\\int _{\\mathbf {R}}{\\mathbf {1}}_{[\\beta _{k,\\, l};\\, \\beta _{k,\\,l-1})}(g\\Vert x\\Vert ^{-\\gamma })\\text{ d}\\rho (g)\\text{ d}x.$ According to the change of variable formula, we have $\\int _{\\text{cell}} {\\mathbf {1}}_{[\\beta _{k,\\, l};\\, \\beta _{k,\\, l-1})}(g\\Vert x\\Vert ^{-\\gamma })\\text{ d}x\\\\=\\pi (\\beta _{k,\\, l}^{-2/\\gamma }\\wedge R^2 -\\beta _{k,\\,l-1}^{-2/\\gamma }\\wedge R^2)g^{2/\\gamma }.$ Thus, we have $\\int _{\\text{cell}}\\int _{\\mathbf {R}}{\\mathbf {1}}_{[\\beta _{k,\\, l};\\, \\beta _{k,\\, l-1})}(g\\Vert x\\Vert ^{-\\gamma })\\text{ d}\\rho (g)\\text{ d}x\\\\\\begin{aligned}&=\\pi (\\beta _{k,\\, l}^{-2/\\gamma }\\wedge R^2-\\beta _{k,\\, l-1}^{-2/\\gamma }\\wedge R^2){\\mathbf {E}}_{}\\left[10^{S/5\\gamma }\\right]\\\\&=\\pi (\\beta _{k,\\, l}^{-2/\\gamma }\\wedge R^2-\\beta _{k,\\,l-1}^{-2/\\gamma }\\wedge R^2)\\ 10^{(\\kappa +\\frac{v^2}{10\\gamma }\\ln 10)/5\\gamma }:=\\zeta _{k,\\, l}.\\end{aligned}$ We thus have proved the following theorem.", "Theorem 1 For any $p\\ge 0$ , with the same notations as above, we have: $\\int f^p\\text{ d}\\nu =\\sum _{k=1}^K \\tau _k\\sum _{l=1}^{l_k}l^p\\, \\zeta _{k,\\, l}.$" ], [ "Exact method", "Since $f$ is deterministic, ${N_{\\text{tot}}}$ follows a compound Poisson distribution: it is distributed as $\\sum _{k=1}^K\\sum _{l=1}^{l_k} l \\, N_{k,\\, l}$ where $(N_{k,\\, l},\\, 1\\le k\\le K, \\, 1\\le l\\le l_k)$ are independent Poisson random variables, the parameter of $N_{k,\\, l}$ is $\\lambda \\tau _k\\zeta _{k,\\, l}.$ Using the properties of Poisson random variables, we can reduce the complexity of this expression.", "Let $L=\\max (l_k, \\, 1\\le k\\le K)$ and for $l\\in \\lbrace 1,\\,\\cdots ,\\, L\\rbrace $ , let $K_l=\\lbrace k,\\, l_k\\ge l\\rbrace $ .", "Then, ${N_{\\text{tot}}}$ is distributed as $\\sum _{l=1}^{L} l \\, M_l$ where $(M_l, \\, 1\\le l\\le l_k)$ are independent Poisson random variables, the parameter of $M_l$ being $m_l:=\\sum _{k\\in K_l}\\lambda \\tau _k\\zeta _{k,\\, l}.$ For each $l$ , it is easy to construct an array which represents the distribution of $lM_l$ by the following rule: $p_{l}(w)={\\left\\lbrace \\begin{array}{ll}0 & \\text{ if } w \\mod {l} \\ne 0,\\\\\\exp (-m_l)m_l^q/q!", "&\\text{ if } w=ql.\\end{array}\\right.", "}$ By discrete convolution, the distribution of ${N_{\\text{tot}}}$ and then its cumulative distribution function, are easily calculable.", "The value of ${N_{\\text{avail}}}$ which ensures a loss probability below the desired threshold is found by inspection.", "The only difficulty with this approach is to determine where to truncate the Poisson distribution functions for machine representation.", "According to large deviation theory [6], ${\\mathbf {P}}(\\text{Poisson}(\\theta )\\ge a \\theta )\\le \\exp (-\\theta (a\\ln a +1-a)).$ When $\\theta $ is known, it is straightforward to choose $a(\\theta )$ so that the right-hand-side of the previous equation is smaller than the desired threshold.", "The total memory size is thus proportional to $\\max (m_la(m_l)l,\\, 1\\le l\\le l_k)$ .", "This may be memory (and time) consuming if the parameters of some Poisson random variables or the threshold are small.", "This method is well suited to estimate loss probability since it gives exact results within a reasonable amount of time but it is less useful for dimensioning purpose.", "Given ${N_{\\text{avail}}}$ , if we seek for the value of $\\lambda $ which guarantees a loss probability less than the desired threshold, there is no better way than trial and error.", "At least, the subsequent methods even imprecise may help to evaluate the order of magnitude of $\\lambda $ for the first trial." ], [ "Approximations", "We begin by the classical Gaussian approximation.", "It is clear that ${\\mathbf {P}}(\\int _E f\\text{ d}\\omega \\ge {N_{\\text{avail}}})={\\mathbf {P}}(\\int _E f_\\sigma (\\text{ d}\\omega -\\lambda \\text{ d}\\nu )\\ge N_\\sigma )\\\\={\\mathbf {E}}_{\\lambda \\nu }\\left[{\\mathbf {1}}_{[N_\\sigma ,\\, +\\infty )}(\\int _E f_\\sigma (\\text{ d}\\omega -\\lambda \\text{ d}\\nu ))\\right]$ where $N_\\sigma =({N_{\\text{avail}}}-\\int f\\lambda \\text{ d}\\nu )/\\sigma $ .", "Since the indicator function ${\\mathbf {1}}_{[N_\\sigma ,\\, +\\infty )}$ is not Lipschitz, we can not apply the bound given by Theorem REF .", "However, we can upper-bound the indicator by a continuous function whose Lipschitz norm is not greater than 1.", "For instance, taking $\\phi (x)=\\min (x^+, 1) \\text{ and } \\phi _N(x)=\\phi (x-N),$ we have ${\\mathbf {1}}_{[N_\\sigma +1,\\, +\\infty )}\\le \\phi _{N_\\sigma +1}\\le {\\mathbf {1}}_{[N_\\sigma ,\\, +\\infty )}\\le \\phi _{N_\\sigma -1}\\le {\\mathbf {1}}_{[N_\\sigma -1,\\, +\\infty )}.$ Hence, $1-Q(N_\\sigma +1)-\\frac{1}{2}\\sqrt{\\frac{2}{\\pi }}\\frac{m(3,\\, 1)}{\\sqrt{\\lambda }}\\\\\\le {\\mathbf {P}}(\\int _E f\\text{ d}\\omega \\ge {N_{\\text{avail}}})\\le \\\\1-Q(N_\\sigma -1)+\\frac{1}{2}\\sqrt{\\frac{2}{\\pi }}\\frac{m(3,\\,1)}{\\sqrt{\\lambda }},$ where $Q$ is the cumulative distribution function of a standard Gaussian random variable.", "According to Theorem REF , one can proceed with a more accurate approximation.", "Via polynomial interpolation, it is easy to construct a ${\\mathcal {C}}^3$ function $\\psi _N^l$ such that $\\Vert (\\psi _N^l)^{(3)}\\Vert _\\infty \\le 1\\text{ and } {\\mathbf {1}}_{[N_\\sigma +3.5,\\, +\\infty )}\\le \\psi _{N_\\sigma }^l\\le {\\mathbf {1}}_{[N_\\sigma ,\\, +\\infty )}$ and a function $\\psi _N^r$ such that $\\Vert (\\psi _N^r)^{(3)}\\Vert _\\infty \\le 1\\text{ and } {\\mathbf {1}}_{[N_\\sigma ,\\, +\\infty )} \\le \\psi _{N_\\sigma }^r\\le {\\mathbf {1}}_{[N_\\sigma -3.5,\\, +\\infty )}$ From (REF ), it follows that $1-Q(N_\\sigma +3.5)-\\frac{m(3,\\, 1)}{6\\sqrt{\\lambda }}Q^{(3)}(N_\\sigma +3.5)-E_\\lambda \\\\\\le {\\mathbf {P}}(\\int _E f\\text{ d}\\omega \\ge {N_{\\text{avail}}})\\le \\\\1-Q(N_\\sigma -3.5)+\\frac{m(3,\\,1)}{6\\sqrt{\\lambda }}Q^{(3)}(N_\\sigma -3.5)+E_\\lambda $ where $E_\\lambda $ is the right-hand-side of (REF ) with $\\Vert F^{(3)}\\Vert _\\infty =1$ .", "Going again one step further, following the same lines, according to (REF ), one can show that ${\\mathbf {P}}(\\int _E f\\text{ d}\\omega \\ge {N_{\\text{avail}}})\\le 1-Q(N_\\sigma -6.5)\\\\+\\frac{m(3,\\,1)}{6\\sqrt{\\lambda }}Q^{(3)}(N_\\sigma -6.5) +\\frac{m(3,1)^2}{72\\lambda }Q^{(5)}(N_\\sigma -6.5)\\\\+\\frac{m(4,1)}{24\\lambda }Q^{(3)}(N_\\sigma -6.5)+F_\\lambda $ where $F_\\lambda $ is bounded above in (REF ).", "For all the approximations given above, for a fixed value of ${N_{\\text{avail}}}$ , an approximate value of $\\lambda $ can be obtained by solving numerically an equation in $\\sqrt{\\lambda }$ ." ], [ "Robust upper-bound", "If we seek for robustness and not precision, it may be interesting to consider the so-called concentration inequality.", "We remark that in the present context, $f$ is non-negative and bounded by $L=\\max _k l_k$ so that we are in position to apply Theorem REF .", "We obtain that ${\\mathbf {P}}(\\int _E f\\text{ d}\\omega \\ge \\int _E f\\text{ d}\\nu + a)\\\\\\le \\exp \\left(-\\frac{\\int _E f^2\\lambda \\text{ d}\\nu }{L^2}g(\\frac{aL}{\\int _Ef^2\\lambda \\text{ d}\\nu })\\right),$ where $g$ is defined in Section ." ], [ "Applications to OFDMA and LTE", "In such systems, there is a huge number of physical parameters with a wide range of variations, it is thus rather hard to explore the while variety of sensible scenarios.", "For illustration purposes, we chose a circular cell of radius $R=300$ meters equipped with an isotropic antenna such that the transmitted power is 1 W and the reference distance is 10 meters.", "The mean number of active customers per unit of surface, denoted by $\\lambda $ , was chosen to vary between $0,001$ and $0.000\\, 1$ , this corresponds to an average number of active customers varying from 3 to 30, a realistic value for the systems under consideration.", "The minimum SINR is $0.3$  dB and the random variable $S$ defined above is a centered Gaussian with variance equal to 10.", "There are two classes of customers, $C_1=1,000$ kb/s and $C_2=400$ kb/s.", "It must be noted that our set of parameters is not universal but for the different scenarios we tested, the numerical facts we want to point out were always apparent.", "Since the time scale is of the order of a packet transmission time, the traffic is defined as the mean number of required subchannels at each slot provided that the time unit is the slot duration, that is to say that the load is defined as $\\rho =\\lambda \\int _{\\text{cell}}f\\text{ d}\\nu $ .", "Figure: Impact of γ\\gamma and τ\\tau on the loss probability(N avail =92{N_{\\text{avail}}}=92, λ=0.0001\\lambda =0.0001)Figure REF shows, the loss probability may vary up to two orders of magnitude when the rate and the probability of each class change even if the mean rate $\\sum _k \\tau _kC_k$ remains constant.", "Thus mean rate is not a sufficient parameter to predict the performances of such a system.", "The load $\\rho $ is neither a pertinent indicator as the computations show that the loads of the various scenarios differs from less than $3\\%$ .", "Comparatively, Figure REF shows that variations of $\\gamma $ have tremendous effects on the loss probability: a change of a few percents of the value of $\\gamma $ induces a variation of several order of magnitude for the loss probability.", "It is not surprising that the loss probability increases as a function of $\\gamma $ : as $\\gamma $ increases, the radio propagation conditions worsen and for a given transmission rate, the number of necessary subchannels increases, generating overloading.", "Beyond a certain value of $\\gamma $ (apparently around 3.95 on Figure REF ), the radio conditions are so harsh that a major part of the customers are in outage since they do not satisfy the ${\\text{SNR}}$ criterion any longer.", "We remark here that the critical value of $\\gamma $ is almost the same for all configurations of classes.", "Indeed, the critical value $\\gamma _c$ of $\\gamma $ can be found by a simple reasoning: When $\\gamma <\\gamma _c$ , a class $k$ customer uses less than the allowed $l_k$ subchannels because the radio conditions are good enough for $\\beta _{k,\\, j}^{1/\\gamma }\\ge R$ for some $j<l_k$ so that the load increases with $\\gamma $ .", "For $\\gamma >\\gamma _c$ , all the $\\beta _{k,\\ l}^{-1/\\gamma }$ are lower than $R$ and the larger $\\gamma $ , the wider the gap.", "Hence the number of customers in outage increases as $\\gamma $ increases and the load decreases.", "Thus, $\\gamma _c \\simeq \\inf \\lbrace \\gamma , \\beta _{s,\\, l_s-1}^{-1/\\gamma }\\le R\\rbrace \\text{ for } s=\\text{arg max}_k l_k.$ If we proceed this way for the data of Figure REF , we retrieve $\\gamma _c=3.95$ .", "This means that for a conservative dimensioning, in the absence of estimate of $\\gamma $ , computations may be done with this value of $\\gamma $ .", "For a threshold given by $\\epsilon =10^{-4}$ , we want to find ${N_{\\text{avail}}}$ such that ${\\mathbf {P}}({N_{\\text{tot}}}\\ge {N_{\\text{avail}}})\\le \\epsilon $ .", "As said earlier, the exact method gives the result at the price of a sometimes lengthy process.", "In view of REF , one could also search for $\\alpha $ such that $1-Q(\\alpha )+\\frac{1}{2}\\sqrt{\\frac{2}{\\pi }}m(3,\\lambda )=\\epsilon $ and then consider $\\lceil 1+\\int _E f\\text{ d}\\nu +\\alpha \\sigma \\rceil $ as an approximate value of ${N_{\\text{avail}}}$ .", "Unfortunately and as was expected since the Gaussian approximation is likely to be valid for large values of $\\lambda $ , the corrective term in (REF ) is far too large (between 30 and 500 depending on $\\gamma $ ) for (REF ) to have a meaning.", "Hence, we must proceed as usual and find $\\alpha $ such that $1-Q(\\alpha )=\\epsilon $ , i.e.", "$\\alpha \\simeq 3.71$ .", "The approximate value of ${N_{\\text{avail}}}$ is thus given by $\\lceil \\int _E f\\text{ d}\\nu +3.71\\sigma \\rceil $ .", "The consequence is that we do not have any longer any guarantee on the quality of this approximation, how close it is to the true value and even more basic, whether it is greater or lower than the correct value.", "In fact, it is absolutely impossible to choose a dimensioning value lower than the true value since there is no longer a guarantee that the loss probability is lower than $\\epsilon $ .", "As shows Figure REF , it turns out that the values returned by the Gaussian method are always under the true value.", "Thus this annihilates any possibility to use the Gaussian approximation for dimensioning purposes.", "Going one step further, according to (REF ), one may find $\\alpha $ such that $1-Q(\\alpha )-\\frac{m(3,\\, \\lambda )}{6}Q^{(3)}(\\alpha )+E_\\lambda =\\epsilon $ and then use $\\lceil {3.5+\\int _E f\\text{ d}\\nu +\\alpha \\sigma }\\rceil $ as an approximate guaranteed value of ${N_{\\text{avail}}}$ .", "By guaranteed, we mean that according to (REF ), it holds for sure that the loss probability with this value of ${N_{\\text{avail}}}$ is smaller than $\\epsilon $ even if there is an approximation process during its computation.", "Since the error in the Edgeworth approximation is of the order of $1/\\lambda $ , instead of $1/\\sqrt{\\lambda }$ for the Gaussian approximation, one may hope that this method will be efficient for smaller values of $\\lambda $ .", "It turns out that for the data sets we examined, $E_\\lambda $ is of the order of $10^{-7}/\\lambda $ , thus this method can be used as long as $10^{-7}/\\lambda \\ll \\epsilon $ .", "Otherwise, as for the Gaussian case, we are reduced to find $\\alpha $ such that $1-Q(\\alpha )-\\frac{m(3,\\, \\lambda )}{6}Q^{(3)}(\\alpha )=\\epsilon $ and consider $\\lceil {3.5+\\int _E f\\text{ d}\\nu +\\alpha \\sigma }\\rceil $ but we no longer have any guarantee on the validity of the value.", "As Figure REF shows, for the considered data set, Edgeworth methods leads to an optimistic value which is once again absolutely not acceptable.", "One can pursue the development as in (REF ) and use (REF ), thus we have to solve $1-Q(\\alpha )-\\frac{m(3,\\, \\lambda )}{6}Q^{(3)}(\\alpha )\\\\-\\frac{m(3,1)^2}{72\\lambda }Q^{(5)}(\\alpha )+\\frac{m(4,1)}{24\\lambda }Q^{(3)}(\\alpha )-F_\\lambda =\\epsilon .$ For the analog of REF to hold, we have to find $\\Psi $ a $\\mathcal {C}^5_b$ function greater than ${\\mathbf {1}}_{[x,\\,\\infty )}$ but smaller than ${\\mathbf {1}}_{[x-\\text{lag},\\, \\infty )}$ with a fifth derivative smaller than 1.", "Looking for $\\Psi $ in the set of polynomial functions, we can find such a function only if $\\text{lag}$ is greater than $6.5$ (for smaller value of the lag, the fifth derivative is not bounded by 1) thus the dimensioning value has to be chosen as: $\\lceil {6.5+\\int _E f\\text{ d}\\nu +\\alpha \\sigma }\\rceil .$ For the values we have, it turns out that $F_\\lambda $ is of the order of $10^{-9}\\lambda ^{-3/2}$ which is negligible compared to $\\epsilon =10^{-4}$ , so that we can effectively use this method for $\\lambda \\ge 10^{-4}$ .", "As it is shown in Figure REF , the values obtained with this development are very close to the true values but always greater as it is necessary for the guarantee.", "The procedure should thus be the following: compute the error bounds given by (REF ), (REF ) and (REF ) and find the one which gives a value negligible with respect to the threshold $\\epsilon $ , then use the corresponding dimensioning formula.", "If none is suitable, use a finer Edgeworth expansion or resort to the concentration inequality approach.", "Note that the Edgeworth method requires the computations of the first three (or five) moments, whose lengthiest part is to compute the $\\zeta _{k,\\, l}$ which is also a step required by the exact method.", "Thus Edgeworth methods are dramatically simpler than the exact method and may be as precise.", "However, both the exact and Edgeworth methods suffer from the same flaw: There are precise as long as the parameters, mainly $\\lambda $ and $\\gamma $ , are perfectly well estimated.", "The value of $\\gamma $ is often set empirically (to say the least) so that it seems important to have dimensioning values robust to some estimate errors.", "This is the goal of the last method we propose.", "According to (REF ), if we find $\\alpha $ such that $g(\\frac{\\alpha L}{\\int _E f^2\\lambda \\text{ d}\\nu })=-\\frac{\\log (\\epsilon )L^2}{\\int _E f^2\\lambda \\text{ d}\\nu }$ and ${N_{\\text{avail}}}=\\int _E f\\text{ d}\\nu +\\frac{\\alpha }{L^2}\\int _E f^2\\lambda \\text{ d}\\nu ,$ we are sure that the loss probability will fall under $\\epsilon $ .", "However, we do not know a priori how larger this value of ${N_{\\text{avail}}}$ than the true value.", "It turns out that the relative oversizing increases with $\\gamma $ from a few percents to $40\\%$ for the large value of $\\gamma $ and hence small values of ${N_{\\text{avail}}}$ .", "For instance, for $\\gamma =4.2$ , the value of ${N_{\\text{avail}}}$ given by (REF ) is 40 whereas the exact value is 32 hence an oversizing of $25\\%$ .", "However, for $\\gamma =4.12$ , which is $2\\%$ away from $4.2$ , the required number of subchannels is also 40.", "The oversizing is thus not as bad as it may seem since it may be viewed as a protection against traffic increase, epistemic risk (model error) and estimate error.", "Figure: Estimates of N avail {N_{\\text{avail}}} as a function of γ\\gamma by thedifferent methods" ], [ "Hermite polynomials", "Let $\\Phi $ be the Gaussian probability density function: $\\Phi (x)=\\exp (-x^2/2)/\\sqrt{2\\pi }$ and $\\mu $ the Gaussian measure on ${\\mathbf {R}}$ .", "Hermite polynomials $(H_k,\\, k\\ge 0)$ are defined by the recursion formula: $H_k(x)\\Phi (x)=\\frac{d^k}{dx^k}\\Phi (x).$ For the sake of completness, we recall that $H_0(x)=1,\\, H_1(x)=x,\\, H_2(x)=x^2-1,\\, H_3(x)=x^3-3x\\\\H_4(x)=x^4-6x^2+3,\\ H_5(x)=x^5-10x^3+15x.$ Thus, for $F\\in {\\mathcal {C}}^k_b$ , using integration by parts, we have $\\int _{\\mathbf {R}}F^{(k)}(x)\\text{ d}\\mu (x)= \\int _{\\mathbf {R}}F(x)H_k(x)\\text{ d}\\mu (x).$ Let $Q(x)=\\int _{-\\infty }^x \\Phi (u)\\text{ d}u=\\int _{\\mathbf {R}}{\\mathbf {1}}_{(-\\infty ;\\,x]}(u)\\Phi (u)\\text{ d}u$ .", "Then, $Q^\\prime =\\Phi $ and $\\int _{\\mathbf {R}}{\\mathbf {1}}_{(-\\infty ;\\, x]}(u)H_k(u)\\text{ d}\\mu (u) \\\\=\\int _{\\mathbf {R}}{\\mathbf {1}}_{(-\\infty ;\\, x]}(u)\\frac{d^{k+1}}{dx^{k+1}}Q(u)\\text{ d}u\\\\=Q^{(k)}(x)= H_{k-1}(x)\\Phi (x).$" ], [ "Edgeworth expansion", "For details on Poisson processes, we refer to [2], [5].", "For $E$ a Polish space equipped with a Radon measure $\\nu $ , $\\Gamma _E$ denotes the set of locally finite discrete measures on $E$ .", "The generic element $\\omega $ of $\\Gamma _E$ may be identified with a set $\\omega =\\lbrace x_n,\\, n\\ge 1\\rbrace $ such that $\\omega \\cap K$ has finite cardinal for any $K$ compact in $E$ .", "We denote by $\\int _E f\\text{ d}\\omega $ the sum $\\sum _{x\\in \\omega } f(x)$ provided that it exists as an element of ${\\mathbf {R}}\\cup \\lbrace +\\infty \\rbrace $ .", "A Poisson process of intensity $\\nu $ is a probability ${\\mathbf {P}}_\\nu $ on $\\Gamma _E$ , such that for any $f\\in {\\mathcal {C}}_K(E, \\; {\\mathbf {R}})$ , ${\\mathbf {E}}_{\\nu }\\left[\\exp (-\\int _E f\\text{ d}\\omega )\\right]=\\exp (-\\int _E 1-e^{-f(x)}\\text{ d}\\nu (x)).$ For $f\\in L^1(\\nu )$ , the Campbell formula states that ${\\mathbf {E}}_{\\nu }\\left[\\int f\\text{ d}\\omega \\right]=\\int f\\text{ d}\\nu .$ We introduce the discrete gradient $D$ defined by $D_xF(\\omega )=F(\\omega \\cup \\lbrace x\\rbrace )-F(\\omega ), \\text{ for all } x\\in E.$ In particular, for $f\\in L^1(\\nu )$ , we have $D_x\\int _Ef\\text{ d}\\omega =f(x).$ The domain of $D$ , denoted by ${\\text{Dom }}D$ is the set of functionals $F\\,:\\, \\Gamma _E\\rightarrow R$ such that ${\\mathbf {E}}_{\\nu }\\left[\\int _E |D_xF(\\omega )|^2\\text{ d}\\nu (x)\\right]<\\infty .$ The integration by parts then says that, for any $F\\in {\\text{Dom }}D$ , any $u\\in L^2(\\nu )$ , ${\\mathbf {E}}_{\\nu }\\left[F\\int _E u(x) (\\text{ d}\\omega (x)-\\text{ d}\\nu (x))\\right]\\\\={\\mathbf {E}}_{\\nu }\\left[\\int _ED_xF \\, u(x)\\text{ d}\\nu (x)\\right].$ We denote by $\\sigma =\\Vert f\\Vert _{L^2(\\nu )}\\sqrt{\\lambda }$ and $f_\\sigma =f/\\sigma $ .", "Note that $\\Vert f_\\sigma \\Vert _{L^2(\\nu )}=1/\\lambda $ and that $m(p,\\,\\lambda ):=\\int _E |f_\\sigma (x)|^p \\lambda \\text{ d}\\nu (x)=\\Vert f\\Vert _{L^2(\\nu )}^{-p}\\Vert f\\Vert ^p_{L^p(\\nu )}\\lambda ^{1-p/2}.$ The proof of the following theorem may be found in [5], [12], [13].", "Theorem 2 Let $f\\in L^2(\\nu )$ .", "For $\\lambda >0$ , let $N^\\lambda =\\int _E f_\\sigma (x)(\\text{ d}\\omega (x)-\\lambda \\text{ d}\\nu (x)).$ Then, for any Lipschitz function $F$ from ${\\mathbf {R}}$ to ${\\mathbf {R}}$ , we have $\\left| {\\mathbf {E}}_{\\lambda \\nu }\\left[F(N^\\lambda )\\right]-\\int _{\\mathbf {R}}F\\text{ d}\\mu \\right|\\le \\frac{1}{2}\\sqrt{\\frac{\\pi }{2}}\\ m(3,\\, \\lambda )\\,\\Vert F\\Vert _{{\\text{Lip}}}.$ To prove the Edgeworth expansion and its error bound, we introduce some notions of Gaussian calculus.", "For $F\\in {\\mathcal {C}}_b^2({\\mathbf {R}};\\,{\\mathbf {R}})$ , we consider $AF(x)=xF^\\prime (x)-F^{\\prime \\prime }(x), \\text{ for any } x\\in {\\mathbf {R}}.$ The Ornstein-Uhlenbeck semi-group is defined by $P_tF(x)=\\int _{\\mathbf {R}}F(e^{-t} x+\\sqrt{1-e^{-2t}}y)\\text{ d}\\mu (y) \\text{ for any } t\\ge 0.$ The infinitesimal generator $A$ and $P_t$ are linked by the following identity $F(x)-\\int _{\\mathbf {R}}F(y)\\text{ d}\\mu (y)=-\\int _0^\\infty AP_tF(x)\\text{ d}t.$ Theorem 3 For $F\\in {\\mathcal {C}}^3_b({\\mathbf {R}},\\, {\\mathbf {R}})$ , $\\left| {\\mathbf {E}}_{\\lambda \\nu }\\left[F(N^\\lambda )\\right]-\\int _{\\mathbf {R}}F(y)\\text{ d}\\mu (y)\\right.\\\\\\left.", "-\\frac{1}{6}\\ m(3,\\, \\lambda )\\ \\int _{\\mathbf {R}}F(y)H_3(y)\\text{ d}\\mu (y)\\vphantom{{\\mathbf {E}}_{lambda}\\left[\\right]}\\right| \\\\\\le \\left( \\frac{m(3,\\, 1)^2}{6}+\\frac{m(4,\\, 1)}{9}\\sqrt{\\frac{2}{\\pi }}\\right)\\frac{\\Vert F^{(3)}\\Vert _\\infty }{\\lambda }$ According to the Taylor formula, $D_x G(N^\\lambda )=G(N^\\lambda +f_\\sigma (x))-G(N^\\lambda )\\\\=G^\\prime (N^\\lambda )f_\\sigma (x)+\\frac{1}{2}f^2_\\sigma (x)\\, G^{\\prime \\prime }(N^\\lambda )\\\\+\\frac{1}{2}f_\\sigma (x)^3\\int _0^1 r^2G^{(3)}(rN^\\lambda +(1-r)f_\\sigma (x))\\text{ d}r.$ Hence, according to (REF ) and (REF ), ${\\mathbf {E}}_{\\lambda \\nu }\\left[N^\\lambda (P_tF)^\\prime (N^\\lambda )\\right]\\\\\\begin{aligned}&= {\\mathbf {E}}_{\\lambda \\nu }\\left[\\int _E f_\\sigma (x) D_x (P_tF)^\\prime (N^\\lambda )\\lambda \\text{ d}\\nu (x)\\right]\\\\&={\\mathbf {E}}_{\\lambda \\nu }\\left[(P_tF)^{\\prime \\prime }(N^\\lambda )\\right]\\\\&+ \\frac{1}{2} \\int _E f_\\sigma ^3(x)\\lambda \\text{ d}\\nu (x){\\mathbf {E}}_{\\lambda \\nu }\\left[(P_tF)^{(3)}(N^\\lambda )\\right]\\\\&+\\frac{1}{2} \\int _E f_\\sigma ^4(x)\\lambda \\text{ d}\\nu (x)\\\\&\\qquad \\times {\\mathbf {E}}_{\\lambda \\nu }\\left[\\int _0^1(P_tF)^{(4)}(rN^\\lambda +(1-r)f_\\sigma (x))r^2\\text{ d}r\\right] \\\\&=A_1+A_2+A_3.\\end{aligned}$ It is well known that for $F\\in {\\mathcal {C}}^k$ , $(x\\mapsto P_tF(x))$ is $k+1$ -times differentiable and that we have two expressions of the derivatives (see [10]): $(P_tF)^{(k+1)}(x)\\\\=\\frac{e^{-(k+1)t}}{\\sqrt{1-e^{-2t}}}\\int _{\\mathbf {R}}F^{(k)}(e^{-t}x+\\sqrt{1-e^{-2t}}y)y\\text{ d}\\mu (y).$ and $(P_tF)^{(k+1)}(x)=e^{-(k+1)t}P_tF^{(k)}(x)$ .", "The former equation induces that $\\Vert (P_tF)^{(k+1)}\\Vert _\\infty \\le \\frac{e^{-(k+1)t}}{\\sqrt{1-e^{-2t}}}\\Vert F^{(k)}\\Vert _\\infty \\int _{\\mathbf {R}}|y|\\text{ d}\\mu (y)\\\\= \\frac{e^{-(k+1)t}}{\\sqrt{1-e^{-2t}}}\\sqrt{\\frac{2}{\\pi }}\\ \\Vert F^{(k)}\\Vert _\\infty .$ Hence, $|A_3|\\le \\frac{e^{-4t}}{6\\sqrt{1-e^{-2t}}}\\sqrt{\\frac{2}{\\pi }}m(4,\\, \\lambda )\\ \\Vert F^{(3)}\\Vert _\\infty .$ Moreover, according to Theorem REF , $\\left| {\\mathbf {E}}_{\\lambda \\nu }\\left[(P_tF)^{(3)}(N^\\lambda )\\right]-\\int _{\\mathbf {R}}(P_tF)^{(3)}(x)\\text{ d}\\mu (x)\\right|\\\\\\begin{aligned}&\\le \\frac{1}{2}\\sqrt{\\frac{\\pi }{2}}\\,m(3,\\, \\lambda )\\Vert (P_tF)^{(4)}\\Vert _\\infty \\\\&= \\frac{1}{2}\\sqrt{\\frac{\\pi }{2}} \\,m(3,\\, \\lambda )e^{-3t}\\Vert (P_tF^{(3)})^\\prime \\Vert _\\infty \\\\&\\le \\frac{1}{2}\\, m(3,\\, \\lambda )\\frac{e^{-4t}}{\\sqrt{1-e^{-2t}}}\\Vert F^{(3)}\\Vert _\\infty .\\end{aligned}$ Then, we have, $|A_2-\\frac{1}{2}{m(3,\\, \\lambda )} \\int _{\\mathbf {R}}(P_tF)^{(3)}(x)\\text{ d}\\mu (x)|\\\\\\le \\frac{1}{4}\\, m(3,\\, \\lambda )^2\\frac{e^{-4t}}{\\sqrt{1-e^{-2t}}}\\Vert F^{(3)}\\Vert _\\infty .$ Hence, ${\\mathbf {E}}_{\\lambda \\nu }\\left[N^\\lambda (P_tF)^\\prime (N^\\lambda )-(P_tF)^{\\prime \\prime }(N^\\lambda )\\right]\\\\=\\frac{1}{2}m(3,\\, \\lambda )\\int _{\\mathbf {R}}(P_tF)^{(3)}(x)\\text{ d}\\mu (x)+R(t),$ where $R(t)\\le \\left( \\frac{m(3,\\, \\lambda )^2}{4}+\\frac{m(4,\\, \\lambda )}{6} \\sqrt{\\frac{2}{\\pi }}\\right)\\Vert F^{(3)}\\Vert _\\infty \\frac{e^{-4t}}{\\sqrt{1-e^{-2t}}}$ Now then, $\\int _{\\mathbf {R}}(P_tF)^{(3)}(x)\\text{ d}\\mu (x)\\\\\\begin{aligned}& = e^{-3t} \\int _R\\int _{\\mathbf {R}}F^{(3)}(e^{-t}x+\\sqrt{1-e^{-2t}}y)\\text{ d}\\mu (y)\\\\& =e^{-3t}\\int _{\\mathbf {R}}F^{(3)}(y)\\text{ d}\\mu (y)\\\\ &=e^{-3t}\\int _R F(y)H_3(y)\\text{ d}\\mu (y),\\end{aligned}$ since the Gaussian measure on ${\\mathbf {R}}^2$ is rotation invariant and according to (REF ).", "Remarking that $\\int _0^\\infty e^{-4t}(1-e^{-2t})^{-1/2}\\text{ d}t =2/3$ and applying (REF ) to $x=N^\\lambda $ , the result follows.", "This development is not new in itself but to the best of our knowledge, it is the first time that there is an estimate of the error bound.", "Following the same lines, we can pursue the expansion up to any order provided that $F$ be sufficiently differentiable.", "Namely, for $F\\in {\\mathcal {C}}^5_b$ , we have ${\\mathbf {E}}_{\\lambda \\nu }\\left[F(N^\\lambda )\\right]=\\int _{\\mathbf {R}}F(y)\\text{ d}\\mu (y)\\\\+\\frac{m(3,1)}{6\\sqrt{\\lambda }}\\int _{\\mathbf {R}}F^{(3)}(y)\\text{ d}\\mu (y)+\\frac{m(3,1)^2}{72 \\lambda }\\int _{\\mathbf {R}}F^{(5)}(y)\\text{ d}\\mu (y)\\\\+\\frac{m(4,\\, 1)}{24\\lambda }\\int _{\\mathbf {R}}F^{(4)}(y)\\text{ d}\\mu (y)+F_\\lambda \\Vert F^{(5)}\\Vert _\\infty .$ where $F_\\lambda \\le \\frac{m(3,1)}{\\lambda ^{3/2}}\\left(\\frac{2}{45}\\, m(3,1)^2\\right.\\\\\\left.+(\\frac{4}{135}+\\frac{\\pi ^2}{128})\\sqrt{\\frac{2}{\\pi }}\\ m(4,1)\\right).$" ], [ "Concentration inequality", "We are now interested in an upper bound, which is called concentration inequality.", "Theorem 4 Let $M,a>0$ .", "Assume that $\\vert f(z)\\vert \\le M$ $\\nu -$ a.s and $f\\in L^2(E,\\nu )$ , then ${\\mathbf {P}}(F>{\\mathbf {E}}\\left[F\\right]+a) \\le \\exp \\left\\lbrace -\\frac{M^2}{{\\mathbf {V}}\\left[F\\right]} g\\left(\\frac{a.M}{{\\mathbf {V}}\\left[F\\right]}\\right)\\right\\rbrace $ where $g(u)=(1+u)\\ln (1+u)-u$ .", "The above theorem can be directly derived from [16].", "However let us take this opportunity to prove this theorem in a very nice, simple and elementary fashion, exactly the same way as Bennett built his concentration inequality for the sum of $n$ i.i.d random variables.", "Using Chernoff's bound we have: P(F>E[F]+a) E[eF]/e(E[F]+a) = eE(ef(z)-1-f(z)) d(z)-a Now assume that $\\vert f(z)\\vert \\le M$ $\\nu -$ a.s .", "Observe that the function $({e^x-1-x})/{x^2}$ is increasing on ${\\mathbf {R}}$ (the value at 0 is $1/2$ ), we have that $e^{\\theta f(z)} -\\theta f(z) -1 \\le \\frac{e^{\\theta M}-1-\\theta M}{M^2}f^2(z) \\ \\nu \\text{ a.s.}$ Thus, ${\\mathbf {P}}(F>{\\mathbf {E}}\\left[F\\right]+a) \\\\\\le \\exp \\left\\lbrace \\int _{E}\\left(\\frac{e^{\\theta M}-\\theta M-1}{M^2}f^2(z)\\right)\\text{ d}\\nu (z)-\\theta a\\right\\rbrace \\\\= \\exp \\left\\lbrace \\frac{e^{\\theta M}-1-\\theta M}{M^2}{\\mathbf {V}}\\left[F\\right]-\\theta a\\right\\rbrace \\cdot $ We find that $\\theta =\\ln \\left(1+{aM}/{{\\mathbf {V}}\\left[F\\right]}\\right)/M$ minimizes the right-hand-side and thus we obtain ()." ] ]

[ [ "Search for Dark Matter and Large Extra Dimensions in pp Collisions\n Yielding a Photon and Missing Transverse Energy" ], [ "Abstract Results are presented from a search for new physics in the final state containing a photon and missing transverse energy.", "The data correspond to an integrated luminosity of 5.0 inverse femtobarns collected in pp collisions at sqrt(s) = 7 TeV by the CMS experiment.", "The observed event yield agrees with standard-model expectations for the photon-plus-missing-transverse-energy events.", "Using models for production of dark-matter particles (chi), we set 90% confidence level (C.L.)", "upper limits of 13.6--15.4 femtobarns on chi production in the photon-plus-missing-transverse-energy state.", "These provide the most sensitive upper limits for spin-dependent chi-nucleon scattering for chi masses between 1 and 100 GeV.", "For spin-independent contributions, the present limits are extended to chi masses below 3.5 GeV.", "For models with 3--6 large extra dimensions, our data exclude extra-dimensional Planck scales between 1.65 and 1.71 TeV at 95% C.L." ], [ "Acknowledgements", "We thank R. Harnik, P. J.", "Fox, and J. Kopp for help in modeling dark matter production.", "We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC machine.", "We thank the technical and administrative staff at CERN and other CMS institutes, and acknowledge support from: FMSR (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES (Croatia); RPF (Cyprus); MoER, SF0690030s09 and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NKTH (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); NRF and WCU (Korea); LAS (Lithuania); CINVESTAV, CONACYT, SEP, and UASLP-FAI (Mexico); MSI (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); MON, RosAtom, RAS and RFBR (Russia); MSTD (Serbia); MICINN and CPAN (Spain); Swiss Funding Agencies (Switzerland); NSC (Taipei); TUBITAK and TAEK (Turkey); STFC (United Kingdom); DOE and NSF (USA)." ], [ "The CMS Collaboration ", "Yerevan Physics Institute, Yerevan, Armenia S. Chatrchyan, V. Khachatryan, A.M. Sirunyan, A. Tumasyan Institut für Hochenergiephysik der OeAW, Wien, Austria W. Adam, T. Bergauer, M. Dragicevic, J. Erö, C. Fabjan, M. Friedl, R. Frühwirth, V.M.", "Ghete, J. Hammer1, N. Hörmann, J. Hrubec, M. Jeitler, W. Kiesenhofer, M. Krammer, D. Liko, I. Mikulec, M. Pernicka$^{\\textrm {\\dag }}$ , B. Rahbaran, C. Rohringer, H. Rohringer, R. Schöfbeck, J. Strauss, A. Taurok, F. Teischinger, P. Wagner, W. Waltenberger, G. Walzel, E. Widl, C.-E. Wulz National Centre for Particle and High Energy Physics, Minsk, Belarus V. Mossolov, N. Shumeiko, J. Suarez Gonzalez Universiteit Antwerpen, Antwerpen, Belgium S. Bansal, K. Cerny, T. Cornelis, E.A.", "De Wolf, X. Janssen, S. Luyckx, T. Maes, L. Mucibello, S. Ochesanu, B. Roland, R. Rougny, M. Selvaggi, H. Van Haevermaet, P. Van Mechelen, N. Van Remortel, A.", "Van Spilbeeck Vrije Universiteit Brussel, Brussel, Belgium F. Blekman, S. Blyweert, J.", "D'Hondt, R. Gonzalez Suarez, A. Kalogeropoulos, M. Maes, A. Olbrechts, W. Van Doninck, P. Van Mulders, G.P.", "Van Onsem, I. Villella Université Libre de Bruxelles, Bruxelles, Belgium O. Charaf, B. Clerbaux, G. De Lentdecker, V. Dero, A.P.R.", "Gay, T. Hreus, A. Léonard, P.E.", "Marage, T. Reis, L. Thomas, C. Vander Velde, P. Vanlaer Ghent University, Ghent, Belgium V. Adler, K. Beernaert, A. Cimmino, S. Costantini, G. Garcia, M. Grunewald, B. Klein, J. Lellouch, A. Marinov, J. Mccartin, A.A. Ocampo Rios, D. Ryckbosch, N. Strobbe, F. Thyssen, M. Tytgat, L. Vanelderen, P. Verwilligen, S. Walsh, E. Yazgan, N. Zaganidis Université Catholique de Louvain, Louvain-la-Neuve, Belgium S. Basegmez, G. Bruno, L. Ceard, C. Delaere, T. du Pree, D. Favart, L. Forthomme, A. Giammanco2, J. Hollar, V. Lemaitre, J. Liao, O. Militaru, C. Nuttens, D. Pagano, A. Pin, K. Piotrzkowski, N. Schul Université de Mons, Mons, Belgium N. Beliy, T. Caebergs, E. Daubie, G.H.", "Hammad Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil G.A.", "Alves, M. Correa Martins Junior, D. De Jesus Damiao, T. Martins, M.E.", "Pol, M.H.G.", "Souza Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil W.L.", "Aldá Júnior, W. Carvalho, A. Custódio, E.M. Da Costa, C. De Oliveira Martins, S. Fonseca De Souza, D. Matos Figueiredo, L. Mundim, H. Nogima, V. Oguri, W.L.", "Prado Da Silva, A. Santoro, S.M.", "Silva Do Amaral, L. Soares Jorge, A. Sznajder Instituto de Fisica Teorica, Universidade Estadual Paulista, Sao Paulo, Brazil T.S.", "Anjos3, C.A.", "Bernardes3, F.A.", "Dias4, T.R.", "Fernandez Perez Tomei, E. M. Gregores3, C. Lagana, F. Marinho, P.G.", "Mercadante3, S.F.", "Novaes, Sandra S. Padula Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria V. Genchev1, P. Iaydjiev1, S. Piperov, M. Rodozov, S. Stoykova, G. Sultanov, V. Tcholakov, R. Trayanov, M. Vutova University of Sofia, Sofia, Bulgaria A. Dimitrov, R. Hadjiiska, V. Kozhuharov, L. Litov, B. Pavlov, P. Petkov Institute of High Energy Physics, Beijing, China J.G.", "Bian, G.M.", "Chen, H.S.", "Chen, C.H.", "Jiang, D. Liang, S. Liang, X. Meng, J. Tao, J. Wang, J. Wang, X. Wang, Z. Wang, H. Xiao, M. Xu, J. Zang, Z. Zhang State Key Lab.", "of Nucl. Phys.", "and Tech.,  Peking University, Beijing, China C. Asawatangtrakuldee, Y.", "Ban, S. Guo, Y. Guo, W. Li, S. Liu, Y. Mao, S.J.", "Qian, H. Teng, S. Wang, B. Zhu, W. Zou Universidad de Los Andes, Bogota, Colombia C. Avila, B. Gomez Moreno, A.F.", "Osorio Oliveros, J.C. Sanabria Technical University of Split, Split, Croatia N. Godinovic, D. Lelas, R. Plestina5, D. Polic, I. Puljak1 University of Split, Split, Croatia Z. Antunovic, M. Dzelalija, M. Kovac Institute Rudjer Boskovic, Zagreb, Croatia V. Brigljevic, S. Duric, K. Kadija, J. Luetic, S. Morovic University of Cyprus, Nicosia, Cyprus A. Attikis, M. Galanti, G. Mavromanolakis, J. Mousa, C. Nicolaou, F. Ptochos, P.A.", "Razis Charles University, Prague, Czech Republic M. Finger, M. Finger Jr. Academy of Scientific Research and Technology of the Arab Republic of Egypt, Egyptian Network of High Energy Physics, Cairo, Egypt Y. Assran6, S. Elgammal7, A. Ellithi Kamel8, S. Khalil9, M.A.", "Mahmoud10, A. Radi9$^{, }$ 11 National Institute of Chemical Physics and Biophysics, Tallinn, Estonia M. Kadastik, M. Müntel, M. Raidal, L. Rebane, A. Tiko Department of Physics, University of Helsinki, Helsinki, Finland V. Azzolini, P. Eerola, G. Fedi, M. Voutilainen Helsinki Institute of Physics, Helsinki, Finland J. Härkönen, A. Heikkinen, V. Karimäki, R. Kinnunen, M.J. Kortelainen, T. Lampén, K. Lassila-Perini, S. Lehti, T. Lindén, P. Luukka, T. Mäenpää, T. Peltola, E. Tuominen, J. Tuominiemi, E. Tuovinen, D. Ungaro, L. Wendland Lappeenranta University of Technology, Lappeenranta, Finland K. Banzuzi, A. Korpela, T. Tuuva DSM/IRFU, CEA/Saclay, Gif-sur-Yvette, France M. Besancon, S. Choudhury, M. Dejardin, D. Denegri, B. Fabbro, J.L.", "Faure, F. Ferri, S. Ganjour, A. Givernaud, P. Gras, G. Hamel de Monchenault, P. Jarry, E. Locci, J. Malcles, L. Millischer, A. Nayak, J. Rander, A. Rosowsky, I. Shreyber, M. Titov Laboratoire Leprince-Ringuet, Ecole Polytechnique, IN2P3-CNRS, Palaiseau, France S. Baffioni, F. Beaudette, L. Benhabib, L. Bianchini, M. Bluj12, C. Broutin, P. Busson, C. Charlot, N. Daci, T. Dahms, L. Dobrzynski, R. Granier de Cassagnac, M. Haguenauer, P. Miné, C. Mironov, C. Ochando, P. Paganini, D. Sabes, R. Salerno, Y. Sirois, C. Veelken, A. Zabi Institut Pluridisciplinaire Hubert Curien, Université de Strasbourg, Université de Haute Alsace Mulhouse, CNRS/IN2P3, Strasbourg, France J.-L. Agram13, J. Andrea, D. Bloch, D. Bodin, J.-M. Brom, M. Cardaci, E.C.", "Chabert, C. Collard, E. Conte13, F. Drouhin13, C. Ferro, J.-C. Fontaine13, D. Gelé, U. Goerlach, P. Juillot, M. Karim13, A.-C.", "Le Bihan, P. Van Hove Centre de Calcul de l'Institut National de Physique Nucleaire et de Physique des Particules (IN2P3),  Villeurbanne, France F. Fassi, D. Mercier Université de Lyon, Université Claude Bernard Lyon 1,  CNRS-IN2P3, Institut de Physique Nucléaire de Lyon, Villeurbanne, France S. Beauceron, N. Beaupere, O. Bondu, G. Boudoul, H. Brun, J. Chasserat, R. Chierici1, D. Contardo, P. Depasse, H. El Mamouni, J. Fay, S. Gascon, M. Gouzevitch, B. Ille, T. Kurca, M. Lethuillier, L. Mirabito, S. Perries, V. Sordini, S. Tosi, Y. Tschudi, P. Verdier, S. Viret Institute of High Energy Physics and Informatization, Tbilisi State University, Tbilisi, Georgia Z. Tsamalaidze14 RWTH Aachen University, I. Physikalisches Institut, Aachen, Germany G. Anagnostou, S. Beranek, M. Edelhoff, L. Feld, N. Heracleous, O. Hindrichs, R. Jussen, K. Klein, J. Merz, A. Ostapchuk, A. Perieanu, F. Raupach, J. Sammet, S. Schael, D. Sprenger, H. Weber, B. Wittmer, V. Zhukov15 RWTH Aachen University, III.", "Physikalisches Institut A,  Aachen, Germany M. Ata, J. Caudron, E. Dietz-Laursonn, D. Duchardt, M. Erdmann, A. Güth, T. Hebbeker, C. Heidemann, K. Hoepfner, T. Klimkovich, D. Klingebiel, P. Kreuzer, D. Lanske$^{\\textrm {\\dag }}$ , J. Lingemann, C. Magass, M. Merschmeyer, A. Meyer, M. Olschewski, P. Papacz, H. Pieta, H. Reithler, S.A. Schmitz, L. Sonnenschein, J. Steggemann, D. Teyssier, M. Weber RWTH Aachen University, III.", "Physikalisches Institut B,  Aachen, Germany M. Bontenackels, V. Cherepanov, M. Davids, G. Flügge, H. Geenen, M. Geisler, W. Haj Ahmad, F. Hoehle, B. Kargoll, T. Kress, Y. Kuessel, A. Linn, A. Nowack, L. Perchalla, O. Pooth, J. Rennefeld, P. Sauerland, A. Stahl Deutsches Elektronen-Synchrotron, Hamburg, Germany M. Aldaya Martin, J. Behr, W. Behrenhoff, U. Behrens, M. Bergholz16, A. Bethani, K. Borras, A. Burgmeier, A. Cakir, L. Calligaris, A. Campbell, E. Castro, F. Costanza, D. Dammann, G. Eckerlin, D. Eckstein, D. Fischer, G. Flucke, A. Geiser, I. Glushkov, S. Habib, J. Hauk, H. Jung1, M. Kasemann, P. Katsas, C. Kleinwort, H. Kluge, A. Knutsson, M. Krämer, D. Krücker, E. Kuznetsova, W. Lange, W. Lohmann16, B. Lutz, R. Mankel, I. Marfin, M. Marienfeld, I.-A.", "Melzer-Pellmann, A.B.", "Meyer, J. Mnich, A. Mussgiller, S. Naumann-Emme, J. Olzem, H. Perrey, A. Petrukhin, D. Pitzl, A. Raspereza, P.M. Ribeiro Cipriano, C. Riedl, M. Rosin, J. Salfeld-Nebgen, R. Schmidt16, T. Schoerner-Sadenius, N. Sen, A. Spiridonov, M. Stein, R. Walsh, C. Wissing University of Hamburg, Hamburg, Germany C. Autermann, V. Blobel, S. Bobrovskyi, J. Draeger, H. Enderle, J. Erfle, U. Gebbert, M. Görner, T. Hermanns, R.S.", "Höing, K. Kaschube, G. Kaussen, H. Kirschenmann, R. Klanner, J. Lange, B. Mura, F. Nowak, N. Pietsch, D. Rathjens, C. Sander, H. Schettler, P. Schleper, E. Schlieckau, A. Schmidt, M. Schröder, T. Schum, M. Seidel, H. Stadie, G. Steinbrück, J. Thomsen Institut für Experimentelle Kernphysik, Karlsruhe, Germany C. Barth, J. Berger, T. Chwalek, W. De Boer, A. Dierlamm, M. Feindt, M. Guthoff1, C. Hackstein, F. Hartmann, M. Heinrich, H. Held, K.H.", "Hoffmann, S. Honc, U. Husemann, I. Katkov15, J.R. Komaragiri, D. Martschei, S. Mueller, Th.", "Müller, M. Niegel, A. Nürnberg, O. Oberst, A. Oehler, J. Ott, T. Peiffer, G. Quast, K. Rabbertz, F. Ratnikov, N. Ratnikova, S. Röcker, C. Saout, A. Scheurer, F.-P. Schilling, M. Schmanau, G. Schott, H.J.", "Simonis, F.M.", "Stober, D. Troendle, R. Ulrich, J. Wagner-Kuhr, T. Weiler, M. Zeise, E.B.", "Ziebarth Institute of Nuclear Physics \"Demokritos\",  Aghia Paraskevi, Greece G. Daskalakis, T. Geralis, S. Kesisoglou, A. Kyriakis, D. Loukas, I. Manolakos, A. Markou, C. Markou, C. Mavrommatis, E. Ntomari University of Athens, Athens, Greece L. Gouskos, T.J. Mertzimekis, A. Panagiotou, N. Saoulidou University of Ioánnina, Ioánnina, Greece I. Evangelou, C. Foudas1, P. Kokkas, N. Manthos, I. Papadopoulos, V. Patras KFKI Research Institute for Particle and Nuclear Physics, Budapest, Hungary G. Bencze, C. Hajdu1, P. Hidas, D. Horvath17, K. Krajczar18, B. Radics, F. Sikler1, V. Veszpremi, G. Vesztergombi18 Institute of Nuclear Research ATOMKI, Debrecen, Hungary N. Beni, S. Czellar, J. Molnar, J. Palinkas, Z. Szillasi University of Debrecen, Debrecen, Hungary J. Karancsi, P. Raics, Z.L.", "Trocsanyi, B. Ujvari Panjab University, Chandigarh, India S.B.", "Beri, V. Bhatnagar, N. Dhingra, R. Gupta, M. Jindal, M. Kaur, J.M.", "Kohli, M.Z.", "Mehta, N. Nishu, L.K.", "Saini, A. Sharma, J. Singh, S.P.", "Singh University of Delhi, Delhi, India S. Ahuja, B.C.", "Choudhary, A. Kumar, A. Kumar, S. Malhotra, M. Naimuddin, K. Ranjan, V. Sharma, R.K. Shivpuri Saha Institute of Nuclear Physics, Kolkata, India S. Banerjee, S. Bhattacharya, S. Dutta, B. Gomber, Sa.", "Jain, Sh.", "Jain, R. Khurana, S. Sarkar Bhabha Atomic Research Centre, Mumbai, India A. Abdulsalam, R.K. Choudhury, D. Dutta, S. Kailas, V. Kumar, A.K.", "Mohanty1, L.M.", "Pant, P. Shukla Tata Institute of Fundamental Research - EHEP, Mumbai, India T. Aziz, S. Ganguly, M. Guchait19, A. Gurtu20, M. Maity21, G. Majumder, K. Mazumdar, G.B.", "Mohanty, B. Parida, K. Sudhakar, N. Wickramage Tata Institute of Fundamental Research - HECR, Mumbai, India S. Banerjee, S. Dugad Institute for Research in Fundamental Sciences (IPM),  Tehran, Iran H. Arfaei, H. Bakhshiansohi22, S.M.", "Etesami23, A. Fahim22, M. Hashemi, H. Hesari, A. Jafari22, M. Khakzad, A. Mohammadi24, M. Mohammadi Najafabadi, S. Paktinat Mehdiabadi, B. Safarzadeh25, M. Zeinali23 INFN Sezione di Bari $^{a}$ , Università di Bari $^{b}$ , Politecnico di Bari $^{c}$ ,  Bari, Italy M. Abbrescia$^{a}$$^{, }$$^{b}$ , L. Barbone$^{a}$$^{, }$$^{b}$ , C. Calabria$^{a}$$^{, }$$^{b}$$^{, }$ 1, S.S. Chhibra$^{a}$$^{, }$$^{b}$ , A. Colaleo$^{a}$ , D. Creanza$^{a}$$^{, }$$^{c}$ , N. De Filippis$^{a}$$^{, }$$^{c}$$^{, }$ 1, M. De Palma$^{a}$$^{, }$$^{b}$ , L. Fiore$^{a}$ , G. Iaselli$^{a}$$^{, }$$^{c}$ , L. Lusito$^{a}$$^{, }$$^{b}$ , G. Maggi$^{a}$$^{, }$$^{c}$ , M. Maggi$^{a}$ , B. Marangelli$^{a}$$^{, }$$^{b}$ , S. My$^{a}$$^{, }$$^{c}$ , S. Nuzzo$^{a}$$^{, }$$^{b}$ , N. Pacifico$^{a}$$^{, }$$^{b}$ , A. Pompili$^{a}$$^{, }$$^{b}$ , G. Pugliese$^{a}$$^{, }$$^{c}$ , G. Selvaggi$^{a}$$^{, }$$^{b}$ , L. Silvestris$^{a}$ , G. Singh$^{a}$$^{, }$$^{b}$ , G. Zito$^{a}$ INFN Sezione di Bologna $^{a}$ , Università di Bologna $^{b}$ ,  Bologna, Italy G. Abbiendi$^{a}$ , A.C. Benvenuti$^{a}$ , D. Bonacorsi$^{a}$$^{, }$$^{b}$ , S. Braibant-Giacomelli$^{a}$$^{, }$$^{b}$ , L. Brigliadori$^{a}$$^{, }$$^{b}$ , P. Capiluppi$^{a}$$^{, }$$^{b}$ , A. Castro$^{a}$$^{, }$$^{b}$ , F.R.", "Cavallo$^{a}$ , M. Cuffiani$^{a}$$^{, }$$^{b}$ , G.M.", "Dallavalle$^{a}$ , F. Fabbri$^{a}$ , A. Fanfani$^{a}$$^{, }$$^{b}$ , D. Fasanella$^{a}$$^{, }$$^{b}$$^{, }$ 1, P. Giacomelli$^{a}$ , C. Grandi$^{a}$ , L. Guiducci, S. Marcellini$^{a}$ , G. Masetti$^{a}$ , M. Meneghelli$^{a}$$^{, }$$^{b}$$^{, }$ 1, A. Montanari$^{a}$ , F.L.", "Navarria$^{a}$$^{, }$$^{b}$ , F. Odorici$^{a}$ , A. Perrotta$^{a}$ , F. Primavera$^{a}$$^{, }$$^{b}$ , A.M. Rossi$^{a}$$^{, }$$^{b}$ , T. Rovelli$^{a}$$^{, }$$^{b}$ , G. Siroli$^{a}$$^{, }$$^{b}$ , R. Travaglini$^{a}$$^{, }$$^{b}$ INFN Sezione di Catania $^{a}$ , Università di Catania $^{b}$ ,  Catania, Italy S. Albergo$^{a}$$^{, }$$^{b}$ , G. Cappello$^{a}$$^{, }$$^{b}$ , M. Chiorboli$^{a}$$^{, }$$^{b}$ , S. Costa$^{a}$$^{, }$$^{b}$ , R. Potenza$^{a}$$^{, }$$^{b}$ , A. Tricomi$^{a}$$^{, }$$^{b}$ , C. Tuve$^{a}$$^{, }$$^{b}$ INFN Sezione di Firenze $^{a}$ , Università di Firenze $^{b}$ ,  Firenze, Italy G. Barbagli$^{a}$ , V. Ciulli$^{a}$$^{, }$$^{b}$ , C. Civinini$^{a}$ , R. D'Alessandro$^{a}$$^{, }$$^{b}$ , E. Focardi$^{a}$$^{, }$$^{b}$ , S. Frosali$^{a}$$^{, }$$^{b}$ , E. Gallo$^{a}$ , S. Gonzi$^{a}$$^{, }$$^{b}$ , M. Meschini$^{a}$ , S. Paoletti$^{a}$ , G. Sguazzoni$^{a}$ , A. Tropiano$^{a}$$^{, }$ 1 INFN Laboratori Nazionali di Frascati, Frascati, Italy L. Benussi, S. Bianco, S. Colafranceschi26, F. Fabbri, D. Piccolo INFN Sezione di Genova, Genova, Italy P. Fabbricatore, R. Musenich INFN Sezione di Milano-Bicocca $^{a}$ , Università di Milano-Bicocca $^{b}$ ,  Milano, Italy A. Benaglia$^{a}$$^{, }$$^{b}$$^{, }$ 1, F. De Guio$^{a}$$^{, }$$^{b}$ , L. Di Matteo$^{a}$$^{, }$$^{b}$$^{, }$ 1, S. Fiorendi$^{a}$$^{, }$$^{b}$ , S. Gennai$^{a}$$^{, }$ 1, A. Ghezzi$^{a}$$^{, }$$^{b}$ , S. Malvezzi$^{a}$ , R.A. Manzoni$^{a}$$^{, }$$^{b}$ , A. Martelli$^{a}$$^{, }$$^{b}$ , A. Massironi$^{a}$$^{, }$$^{b}$$^{, }$ 1, D. Menasce$^{a}$ , L. Moroni$^{a}$ , M. Paganoni$^{a}$$^{, }$$^{b}$ , D. Pedrini$^{a}$ , S. Ragazzi$^{a}$$^{, }$$^{b}$ , N. Redaelli$^{a}$ , S. Sala$^{a}$ , T. Tabarelli de Fatis$^{a}$$^{, }$$^{b}$ INFN Sezione di Napoli $^{a}$ , Università di Napoli \"Federico II\" $^{b}$ ,  Napoli, Italy S. Buontempo$^{a}$ , C.A.", "Carrillo Montoya$^{a}$$^{, }$ 1, N. Cavallo$^{a}$$^{, }$ 27, A.", "De Cosa$^{a}$$^{, }$$^{b}$ , O. Dogangun$^{a}$$^{, }$$^{b}$ , F. Fabozzi$^{a}$$^{, }$ 27, A.O.M.", "Iorio$^{a}$$^{, }$ 1, L. Lista$^{a}$ , S. Meola$^{a}$$^{, }$ 28, M. Merola$^{a}$$^{, }$$^{b}$ , P. Paolucci$^{a}$ INFN Sezione di Padova $^{a}$ , Università di Padova $^{b}$ , Università di Trento (Trento) $^{c}$ ,  Padova, Italy P. Azzi$^{a}$ , N. Bacchetta$^{a}$$^{, }$ 1, D. Bisello$^{a}$$^{, }$$^{b}$ , A. Branca$^{a}$$^{, }$ 1, R. Carlin$^{a}$$^{, }$$^{b}$ , P. Checchia$^{a}$ , T. Dorigo$^{a}$ , U. Dosselli$^{a}$ , F. Gasparini$^{a}$$^{, }$$^{b}$ , A. Gozzelino$^{a}$ , K. Kanishchev$^{a}$$^{, }$$^{c}$ , S. Lacaprara$^{a}$ , I. Lazzizzera$^{a}$$^{, }$$^{c}$ , M. Margoni$^{a}$$^{, }$$^{b}$ , A.T. Meneguzzo$^{a}$$^{, }$$^{b}$ , M. Nespolo$^{a}$$^{, }$ 1, L. Perrozzi$^{a}$ , N. Pozzobon$^{a}$$^{, }$$^{b}$ , P. Ronchese$^{a}$$^{, }$$^{b}$ , F. Simonetto$^{a}$$^{, }$$^{b}$ , E. Torassa$^{a}$ , M. Tosi$^{a}$$^{, }$$^{b}$$^{, }$ 1, S. Vanini$^{a}$$^{, }$$^{b}$ , P. Zotto$^{a}$$^{, }$$^{b}$ , G. Zumerle$^{a}$$^{, }$$^{b}$ INFN Sezione di Pavia $^{a}$ , Università di Pavia $^{b}$ ,  Pavia, Italy M. Gabusi$^{a}$$^{, }$$^{b}$ , S.P.", "Ratti$^{a}$$^{, }$$^{b}$ , C. Riccardi$^{a}$$^{, }$$^{b}$ , P. Torre$^{a}$$^{, }$$^{b}$ , P. Vitulo$^{a}$$^{, }$$^{b}$ INFN Sezione di Perugia $^{a}$ , Università di Perugia $^{b}$ ,  Perugia, Italy G.M.", "Bilei$^{a}$ , L. Fanò$^{a}$$^{, }$$^{b}$ , P. Lariccia$^{a}$$^{, }$$^{b}$ , A. Lucaroni$^{a}$$^{, }$$^{b}$$^{, }$ 1, G. Mantovani$^{a}$$^{, }$$^{b}$ , M. Menichelli$^{a}$ , A. Nappi$^{a}$$^{, }$$^{b}$ , F. Romeo$^{a}$$^{, }$$^{b}$ , A. Saha, A. Santocchia$^{a}$$^{, }$$^{b}$ , S. Taroni$^{a}$$^{, }$$^{b}$$^{, }$ 1 INFN Sezione di Pisa $^{a}$ , Università di Pisa $^{b}$ , Scuola Normale Superiore di Pisa $^{c}$ ,  Pisa, Italy P. Azzurri$^{a}$$^{, }$$^{c}$ , G. Bagliesi$^{a}$ , T. Boccali$^{a}$ , G. Broccolo$^{a}$$^{, }$$^{c}$ , R. Castaldi$^{a}$ , R.T. D'Agnolo$^{a}$$^{, }$$^{c}$ , R. Dell'Orso$^{a}$ , F. Fiori$^{a}$$^{, }$$^{b}$$^{, }$ 1, L. Foà$^{a}$$^{, }$$^{c}$ , A. Giassi$^{a}$ , A. Kraan$^{a}$ , F. Ligabue$^{a}$$^{, }$$^{c}$ , T. Lomtadze$^{a}$ , L. Martini$^{a}$$^{, }$ 29, A. Messineo$^{a}$$^{, }$$^{b}$ , F. Palla$^{a}$ , F. Palmonari$^{a}$ , A. Rizzi$^{a}$$^{, }$$^{b}$ , A.T. Serban$^{a}$$^{, }$ 30, P. Spagnolo$^{a}$ , P. Squillacioti1, R. Tenchini$^{a}$ , G. Tonelli$^{a}$$^{, }$$^{b}$$^{, }$ 1, A. Venturi$^{a}$$^{, }$ 1, P.G.", "Verdini$^{a}$ INFN Sezione di Roma $^{a}$ , Università di Roma \"La Sapienza\" $^{b}$ ,  Roma, Italy L. Barone$^{a}$$^{, }$$^{b}$ , F. Cavallari$^{a}$ , D. Del Re$^{a}$$^{, }$$^{b}$$^{, }$ 1, M. Diemoz$^{a}$ , C. Fanelli$^{a}$$^{, }$$^{b}$ , M. Grassi$^{a}$$^{, }$ 1, E. Longo$^{a}$$^{, }$$^{b}$ , P. Meridiani$^{a}$$^{, }$ 1, F. Micheli$^{a}$$^{, }$$^{b}$ , S. Nourbakhsh$^{a}$ , G. Organtini$^{a}$$^{, }$$^{b}$ , F. Pandolfi$^{a}$$^{, }$$^{b}$ , R. Paramatti$^{a}$ , S. Rahatlou$^{a}$$^{, }$$^{b}$ , M. Sigamani$^{a}$ , L. Soffi$^{a}$$^{, }$$^{b}$ INFN Sezione di Torino $^{a}$ , Università di Torino $^{b}$ , Università del Piemonte Orientale (Novara) $^{c}$ ,  Torino, Italy N. Amapane$^{a}$$^{, }$$^{b}$ , R. Arcidiacono$^{a}$$^{, }$$^{c}$ , S. Argiro$^{a}$$^{, }$$^{b}$ , M. Arneodo$^{a}$$^{, }$$^{c}$ , C. Biino$^{a}$ , C. Botta$^{a}$$^{, }$$^{b}$ , N. Cartiglia$^{a}$ , R. Castello$^{a}$$^{, }$$^{b}$ , M. Costa$^{a}$$^{, }$$^{b}$ , N. Demaria$^{a}$ , A. Graziano$^{a}$$^{, }$$^{b}$ , C. Mariotti$^{a}$$^{, }$ 1, S. Maselli$^{a}$ , E. Migliore$^{a}$$^{, }$$^{b}$ , V. Monaco$^{a}$$^{, }$$^{b}$ , M. Musich$^{a}$$^{, }$ 1, M.M.", "Obertino$^{a}$$^{, }$$^{c}$ , N. Pastrone$^{a}$ , M. Pelliccioni$^{a}$ , A. Potenza$^{a}$$^{, }$$^{b}$ , A. Romero$^{a}$$^{, }$$^{b}$ , M. Ruspa$^{a}$$^{, }$$^{c}$ , R. Sacchi$^{a}$$^{, }$$^{b}$ , A. Solano$^{a}$$^{, }$$^{b}$ , A. Staiano$^{a}$ , A. Vilela Pereira$^{a}$ , L. Visca$^{a}$$^{, }$$^{b}$ INFN Sezione di Trieste $^{a}$ , Università di Trieste $^{b}$ ,  Trieste, Italy S. Belforte$^{a}$ , F. Cossutti$^{a}$ , G. Della Ricca$^{a}$$^{, }$$^{b}$ , B. Gobbo$^{a}$ , M. Marone$^{a}$$^{, }$$^{b}$$^{, }$ 1, D. Montanino$^{a}$$^{, }$$^{b}$$^{, }$ 1, A. Penzo$^{a}$ , A. Schizzi$^{a}$$^{, }$$^{b}$ Kangwon National University, Chunchon, Korea S.G. Heo, T.Y.", "Kim, S.K.", "Nam Kyungpook National University, Daegu, Korea S. Chang, J. Chung, D.H. Kim, G.N.", "Kim, D.J.", "Kong, H. Park, S.R.", "Ro, D.C.", "Son, T. Son Chonnam National University, Institute for Universe and Elementary Particles, Kwangju, Korea J.Y.", "Kim, Zero J. Kim, S. Song Konkuk University, Seoul, Korea H.Y.", "Jo Korea University, Seoul, Korea S. Choi, D. Gyun, B. Hong, M. Jo, H. Kim, T.J. Kim, K.S.", "Lee, D.H.", "Moon, S.K.", "Park, E. Seo University of Seoul, Seoul, Korea M. Choi, S. Kang, H. Kim, J.H.", "Kim, C. Park, I.C.", "Park, S. Park, G. Ryu Sungkyunkwan University, Suwon, Korea Y. Cho, Y. Choi, Y.K.", "Choi, J. Goh, M.S.", "Kim, E. Kwon, B. Lee, J. Lee, S. Lee, H. Seo, I. Yu Vilnius University, Vilnius, Lithuania M.J. Bilinskas, I. Grigelionis, M. Janulis, A. Juodagalvis Centro de Investigacion y de Estudios Avanzados del IPN, Mexico City, Mexico H. Castilla-Valdez, E. De La Cruz-Burelo, I. Heredia-de La Cruz, R. Lopez-Fernandez, R. Magaña Villalba, J. Martínez-Ortega, A. Sánchez-Hernández, L.M.", "Villasenor-Cendejas Universidad Iberoamericana, Mexico City, Mexico S. Carrillo Moreno, F. Vazquez Valencia Benemerita Universidad Autonoma de Puebla, Puebla, Mexico H.A.", "Salazar Ibarguen Universidad Autónoma de San Luis Potosí,  San Luis Potosí,  Mexico E. Casimiro Linares, A. Morelos Pineda, M.A.", "Reyes-Santos University of Auckland, Auckland, New Zealand D. Krofcheck University of Canterbury, Christchurch, New Zealand A.J.", "Bell, P.H.", "Butler, R. Doesburg, S. Reucroft, H. Silverwood National Centre for Physics, Quaid-I-Azam University, Islamabad, Pakistan M. Ahmad, M.I.", "Asghar, H.R.", "Hoorani, S. Khalid, W.A.", "Khan, T. Khurshid, S. Qazi, M.A.", "Shah, M. Shoaib Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland G. Brona, K. Bunkowski, M. Cwiok, W. Dominik, K. Doroba, A. Kalinowski, M. Konecki, J. Krolikowski Soltan Institute for Nuclear Studies, Warsaw, Poland H. Bialkowska, B. Boimska, T. Frueboes, R. Gokieli, M. Górski, M. Kazana, K. Nawrocki, K. Romanowska-Rybinska, M. Szleper, G. Wrochna, P. Zalewski Laboratório de Instrumentação e Física Experimental de Partículas, Lisboa, Portugal N. Almeida, P. Bargassa, A. David, P. Faccioli, P.G.", "Ferreira Parracho, M. Gallinaro, P. Musella, J. Seixas, J. Varela, P. Vischia Joint Institute for Nuclear Research, Dubna, Russia I. Belotelov, P. Bunin, I. Golutvin, I. Gorbunov, V. Karjavin, V. Konoplyanikov, G. Kozlov, A. Lanev, A. Malakhov, P. Moisenz, V. Palichik, V. Perelygin, M. Savina, S. Shmatov, V. Smirnov, A. Volodko, A. Zarubin Petersburg Nuclear Physics Institute, Gatchina (St Petersburg),  Russia S. Evstyukhin, V. Golovtsov, Y. Ivanov, V. Kim, P. Levchenko, V. Murzin, V. Oreshkin, I. Smirnov, V. Sulimov, L. Uvarov, S. Vavilov, A. Vorobyev, An.", "Vorobyev Institute for Nuclear Research, Moscow, Russia Yu.", "Andreev, A. Dermenev, S. Gninenko, N. Golubev, M. Kirsanov, N. Krasnikov, V. Matveev, A. Pashenkov, D. Tlisov, A. Toropin Institute for Theoretical and Experimental Physics, Moscow, Russia V. Epshteyn, M. Erofeeva, V. Gavrilov, M. Kossov1, N. Lychkovskaya, V. Popov, G. Safronov, S. Semenov, V. Stolin, E. Vlasov, A. Zhokin Moscow State University, Moscow, Russia A. Belyaev, E. Boos, V. Bunichev, M. Dubinin4, L. Dudko, A. Gribushin, V. Klyukhin, O. Kodolova, I. Lokhtin, A. Markina, S. Obraztsov, M. Perfilov, S. Petrushanko, L. Sarycheva$^{\\textrm {\\dag }}$ , V. Savrin, A. Snigirev P.N.", "Lebedev Physical Institute, Moscow, Russia V. Andreev, M. Azarkin, I. Dremin, M. Kirakosyan, A. Leonidov, G. Mesyats, S.V.", "Rusakov, A. Vinogradov State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, Russia I. Azhgirey, I. Bayshev, S. Bitioukov, V. Grishin1, V. Kachanov, D. Konstantinov, A. Korablev, V. Krychkine, V. Petrov, R. Ryutin, 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H.C. Kaestli, S. König, D. Kotlinski, U. Langenegger, F. Meier, D. Renker, T. Rohe, J. Sibille38 Institute for Particle Physics, ETH Zurich, Zurich, Switzerland L. Bäni, P. Bortignon, M.A.", "Buchmann, B. Casal, N. Chanon, Z. Chen, A. Deisher, G. Dissertori, M. Dittmar, M. Dünser, J. Eugster, K. Freudenreich, C. Grab, P. Lecomte, W. Lustermann, A.C. Marini, P. Martinez Ruiz del Arbol, N. Mohr, F. Moortgat, C. Nägeli39, P. Nef, F. Nessi-Tedaldi, L. Pape, F. Pauss, M. Peruzzi, F.J. Ronga, M. Rossini, L. Sala, A.K.", "Sanchez, A. Starodumov40, B. Stieger, M. Takahashi, L. Tauscher$^{\\textrm {\\dag }}$ , A. Thea, K. Theofilatos, D. Treille, C. Urscheler, R. Wallny, H.A.", "Weber, L. Wehrli Universität Zürich, Zurich, Switzerland E. Aguilo, C. Amsler, V. Chiochia, S. De Visscher, C. Favaro, M. Ivova Rikova, B. Millan Mejias, P. Otiougova, P. Robmann, H. Snoek, S. Tupputi, M. Verzetti National Central University, Chung-Li, Taiwan Y.H.", "Chang, K.H.", "Chen, A.", "Go, C.M.", "Kuo, S.W.", 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"Butt, D.R.", "Claes, A. Dominguez, M. Eads, P. Jindal, J. Keller, I. Kravchenko, J. Lazo-Flores, H. Malbouisson, S. Malik, G.R.", "Snow State University of New York at Buffalo, Buffalo, USA U. Baur, A. Godshalk, I. Iashvili, S. Jain, A. Kharchilava, A. Kumar, S.P.", "Shipkowski, K. Smith Northeastern University, Boston, USA G. Alverson, E. Barberis, D. Baumgartel, M. Chasco, J. Haley, D. Trocino, D. Wood, J. Zhang Northwestern University, Evanston, USA A. Anastassov, A. Kubik, N. Mucia, N. Odell, R.A. Ofierzynski, B. Pollack, A. Pozdnyakov, M. Schmitt, S. Stoynev, M. Velasco, S. Won University of Notre Dame, Notre Dame, USA L. Antonelli, D. Berry, A. Brinkerhoff, M. Hildreth, C. Jessop, D.J.", "Karmgard, J. Kolb, K. Lannon, W. Luo, S. Lynch, N. Marinelli, D.M.", "Morse, T. Pearson, R. Ruchti, J. Slaunwhite, N. Valls, J. Warchol, M. Wayne, M. Wolf, J. Ziegler The Ohio State University, Columbus, USA B. Bylsma, L.S.", "Durkin, C. Hill, R. Hughes, P. Killewald, K. Kotov, T.Y.", "Ling, 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J. Gilmore, T. Kamon56, V. Khotilovich, R. Montalvo, I. Osipenkov, Y. Pakhotin, A. Perloff, J. Roe, A. Safonov, T. Sakuma, S. Sengupta, I. Suarez, A. Tatarinov, D. Toback Texas Tech University, Lubbock, USA N. Akchurin, J. Damgov, P.R.", "Dudero, C. Jeong, K. Kovitanggoon, S.W.", "Lee, T. Libeiro, Y. Roh, I. Volobouev Vanderbilt University, Nashville, USA E. Appelt, D. Engh, C. Florez, S. Greene, A. Gurrola, W. Johns, P. Kurt, C. Maguire, A. Melo, P. Sheldon, B. Snook, S. Tuo, J. Velkovska University of Virginia, Charlottesville, USA M.W.", "Arenton, M. Balazs, S. Boutle, B. Cox, B. Francis, J. Goodell, R. Hirosky, A. Ledovskoy, C. Lin, C. Neu, J.", "Wood, R. Yohay Wayne State University, Detroit, USA S. Gollapinni, R. Harr, P.E.", "Karchin, C. Kottachchi Kankanamge Don, P. Lamichhane, A. Sakharov University of Wisconsin, Madison, USA M. Anderson, M. Bachtis, D. Belknap, L. Borrello, D. Carlsmith, M. Cepeda, S. Dasu, L. Gray, K.S.", "Grogg, M. Grothe, R. Hall-Wilton, M. Herndon, A. Hervé, P. Klabbers, J. Klukas, A. Lanaro, C. Lazaridis, J. Leonard, R. Loveless, A. Mohapatra, I. Ojalvo, G.A.", "Pierro, I. Ross, A. Savin, W.H.", "Smith, J. Swanson †: Deceased 1:  Also at CERN, European Organization for Nuclear Research, Geneva, Switzerland 2:  Also at National Institute of Chemical Physics and Biophysics, Tallinn, Estonia 3:  Also at Universidade Federal do ABC, Santo Andre, Brazil 4:  Also at California Institute of Technology, Pasadena, USA 5:  Also at Laboratoire Leprince-Ringuet, Ecole Polytechnique, IN2P3-CNRS, Palaiseau, France 6:  Also at Suez Canal University, Suez, Egypt 7:  Also at Zewail City of Science and Technology, Zewail, Egypt 8:  Also at Cairo University, Cairo, Egypt 9:  Also at British University, Cairo, Egypt 10: Also at Fayoum University, El-Fayoum, Egypt 11: Now at Ain Shams University, Cairo, Egypt 12: Also at Soltan Institute for Nuclear Studies, Warsaw, Poland 13: Also at Université de Haute-Alsace, Mulhouse, France 14: Now at Joint Institute for Nuclear Research, Dubna, Russia 15: Also at Moscow State University, Moscow, Russia 16: Also at Brandenburg University of Technology, Cottbus, Germany 17: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary 18: Also at Eötvös Loránd University, Budapest, Hungary 19: Also at Tata Institute of Fundamental Research - HECR, Mumbai, India 20: Now at King Abdulaziz University, Jeddah, Saudi Arabia 21: Also at University of Visva-Bharati, Santiniketan, India 22: Also at Sharif University of Technology, Tehran, Iran 23: Also at Isfahan University of Technology, Isfahan, Iran 24: Also at Shiraz University, Shiraz, Iran 25: Also at Plasma Physics Research Center, Science and Research Branch, Islamic Azad University, Teheran, Iran 26: Also at Facoltà Ingegneria Università di Roma, Roma, Italy 27: Also at Università della Basilicata, Potenza, Italy 28: Also at Università degli Studi Guglielmo Marconi, Roma, Italy 29: Also at Università degli studi di Siena, Siena, Italy 30: Also at University of Bucharest, Bucuresti-Magurele, Romania 31: Also at Faculty of Physics of University of Belgrade, Belgrade, Serbia 32: Also at University of Florida, Gainesville, USA 33: Also at University of California, Los Angeles, Los Angeles, USA 34: Also at Scuola Normale e Sezione dell' INFN, Pisa, Italy 35: Also at INFN Sezione di Roma; Università di Roma \"La Sapienza\", Roma, Italy 36: Also at University of Athens, Athens, Greece 37: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom 38: Also at The University of Kansas, Lawrence, USA 39: Also at Paul Scherrer Institut, Villigen, Switzerland 40: Also at Institute for Theoretical and Experimental Physics, Moscow, Russia 41: Also at Gaziosmanpasa University, Tokat, Turkey 42: Also at Adiyaman University, Adiyaman, Turkey 43: Also at The University of Iowa, Iowa City, USA 44: Also at Mersin University, Mersin, Turkey 45: Also at Kafkas University, Kars, Turkey 46: Also at Suleyman Demirel University, Isparta, Turkey 47: Also at Ege University, Izmir, Turkey 48: Also at School of Physics and Astronomy, University of Southampton, Southampton, United Kingdom 49: Also at INFN Sezione di Perugia; Università di Perugia, Perugia, Italy 50: Also at University of Sydney, Sydney, Australia 51: Also at Utah Valley University, Orem, USA 52: Also at Institute for Nuclear Research, Moscow, Russia 53: Also at University of Belgrade, Faculty of Physics and Vinca Institute of Nuclear Sciences, Belgrade, Serbia 54: Also at Argonne National Laboratory, Argonne, USA 55: Also at Erzincan University, Erzincan, Turkey 56: Also at Kyungpook National University, Daegu, Korea" ] ]

[ [ "A commutative algebraic approach to the fitting problem" ], [ "Abstract Given a finite set of points $\\Gamma$ in $\\mathbb P^{k-1}$ not all contained in a hyperplane, the \"fitting problem\" asks what is the maximum number $hyp(\\Gamma)$ of these points that can fit in some hyperplane and what is (are) the equation(s) of such hyperplane(s).", "If $\\Gamma$ has the property that any $k-1$ of its points span a hyperplane, then $hyp(\\Gamma)=nil(I)+k-2$, where $nil(I)$ is the index of nilpotency of an ideal constructed from the homogeneous coordinates of the points of $\\Gamma$.", "Note that in $\\mathbb P^2$ any two points span a line, and we find that the maximum number of collinear points of any given set of points $\\Gamma\\subset\\mathbb P^2$ equals the index of nilpotency of the corresponding ideal, plus one." ], [ "Introduction", "Let $\\mathbb {K}$ be any field, and let $\\Gamma \\subseteq \\mathbb {P}_{\\mathbb {K}}^{k-1}$ be a finite reduced set of points, not all contained in a hyperplane.", "Let $hyp(\\Gamma )$ be the maximum number of points of $\\Gamma $ contained in some hyperplane.", "Computationally, the “fitting problem” (or “exact fitting problem”) asks for effective methods or algorithms to compute this number and to find the equation of the hyperplane.", "If $\\Gamma $ is $(k-2)-$ generic (i.e., any $k-1$ of the points span a hyperplane)In [3] and [8], the set of points $\\Gamma $ is in the affine space $\\mathbb {A}^{k-1}$ .", "Nevertheless we can embed it into $\\mathbb {P}^{k-1}$ by adding to each point of $\\Gamma $ an extra coordinate that equals to 1., by [3] (or [8], Algorithm MinN1), the problem of finding the hyperplane can be solved in $O(|\\Gamma |^{k-1})$ time.", "If one knows $hyp(\\Gamma )$ , in [8], Corollary 3.3 is presented an algorithm that finds the points in this hyperplane in $O(\\min \\lbrace \\frac{|\\Gamma |^{k-1}}{hyp(\\Gamma )^{k-2}} \\log (\\frac{|\\Gamma |}{hyp(\\Gamma )}),|\\Gamma |^{k-1}\\rbrace )$ time.", "The fitting problem is in direct connection with the computation of the minimum distance $d$ of the equivalence class of linear codes with generating matrix having as columns the coordinates of the points; the points can be placed as columns in any order, and homogeneous coordinates are the same up to multiplication by a nonzero constantFor background on linear codes we recommend [1]..", "The connection is that $d=|\\Gamma |-hyp(\\Gamma )$ .", "With this, [6], [9], [12], [13], [14], or [15] show that the minimum distance gives bounds for homological invariants of zero-dimensional projective schemes.", "We should mention also that the hyperplanes that contain $hyp(\\Gamma )$ points of $\\Gamma $ are in one-to-one correspondence with the (projective) codewords of minimum weight of the class of linear codes constructed from $\\Gamma $ : the coefficients of the linear form defining such a hyperplane are the coefficients of the linear combination of the rows of the generating matrix of the linear code that will give the codeword of minimum weight.", "Therefore, Lemma 2.2 in [13] allows us to determine these hyperplanes by finding the minimal primes of a certain ideal, which is very simple once we know $hyp(\\Gamma )$ .", "In this paper we link $hyp(\\Gamma )$ to the index of nilpotency of an ideal generated by products of linear forms.", "We can do this if $\\Gamma \\subset \\mathbb {P}^{k-1}$ is $(k-2)-$ generic.", "This restriction does not occur if $k-1=2$ , and therefore we give a new interpretation to the fitting problem of any set of points in the plane.", "The article is structured as follows.", "In the next section we generalize a result of Schenck ([11], Lemma 3.1) known so far to be true only for $\\mathbb {P}^2$ .", "This result with the results of Davis and Geramita (see [2]) was essential to show that the Orlik-Terao algebra of an arrangement of lines in $\\mathbb {P}^2$ is the homogeneous coordinate ring associated to a nef on the blowup of $\\mathbb {P}^2$ at the singularities of the arrangement (see [11] for more details).", "In the final part we discuss about the index of nilpotency of fat points and we prove our main result Theorem REF .", "The goal of this paper is to have an understanding of the fitting problem from a commutative algebraic point of view.", "Therefore our effort is directed towards presenting this abstract approach with as many details as possible also for a nonspecialist, leaving the analysis of the efficiency of the possible algorithms that may be created from our exposition to the experts in the field." ], [ "A fat point scheme constructed from hyperplane arrangements", "Let $\\mathcal {A}$ be a central essential hyperplane arrangement in $V=\\mathbb {K}^k$ .", "Suppose $\\mathcal {A}=\\lbrace H_1,\\ldots ,H_n\\rbrace ,$ and each $H_i$ is the vanishing of a linear form $L_i\\in Sym(V^*)=\\mathbb {K}[x_1,\\ldots ,x_k]$ .", "“Central” means that all the hyperplanes of $\\mathcal {A}$ pass through the origin, and “essential” means that the rank of $\\mathcal {A}$ is $k$ (i.e., $codim(H_1\\cap \\cdots \\cap H_n)=k$ ).", "For more details and background on hyperplane arrangements, we recommend [10].", "To any hyperplane arrangement one can associate the lattice of intersection $L(\\mathcal {A})$ , which is a lattice built on the intersections of hyperplanes, with levels $L_j(\\mathcal {A})$ : $X=H_{i_1}\\cap \\cdots \\cap H_{i_s}\\in L_j(\\mathcal {A})$ if and only if $codim(X)=j$ .", "$X$ is called a flat of rank $j$, and $s$ , which is the number of hyperplanes that contain $X$ , will be denoted $\\nu (X)$ .", "Flats of rank $k-1$ are called coatoms.", "We are interested in a special class of hyperplane arrangements.", "A central essential hyperplane arrangement $\\mathcal {A}\\subset \\mathbb {K}^k$ will be called $k-2$ generic if and only if any $k-1$ of the linear forms $L_i$ defining the hyperplanes of $\\mathcal {A}$ are linearly independent.", "Observe that all arrangements of rank 3 are $k-2$ generic (unless they are multiarrangements).", "For a hyperplane arrangement $\\mathcal {A}\\subset \\mathbb {K}^k$ , define $I_j(\\mathcal {A})=\\langle \\lbrace L_{i_1}\\cdots L_{i_j}:1\\le i_1<\\cdots <i_j\\le n\\rbrace \\rangle \\subset R:=\\mathbb {K}[x_1,\\ldots ,x_k],$ the ideal generated by all the distinct $j$ products of the linear forms defining the hyperplanes of $\\mathcal {A}$ .", "With all of these we can generalize the result of Schenck to arbitrary rank.", "But first, one more definition.", "Let $I$ be a homogeneous ideal in the polynomial ring $R:=\\mathbb {K}[x_1,\\ldots ,x_k]$ .", "The saturation of $I$ is the ideal $I^{sat}=\\lbrace f\\in R|f\\in I:\\langle x_1,\\ldots ,x_k\\rangle ^{n(f)}\\mbox{ for some }n(f)\\rbrace .$ Proposition 2.1 Let $\\mathcal {A}$ be a central essential $k-2$ generic arrangement of $n$ hyperplanes in $\\mathbb {K}^k$ .", "Then $I_{n-k+2}(\\mathcal {A})=\\bigcap _{X\\in L_{k-1}(\\mathcal {A})}I(X)^{\\nu (X)-k+2}.$ Following the same considerations as in Section 4 in [13], any minimal prime of the ideal $I_{n-k+2}(\\mathcal {A})$ is of the form $\\langle L_{i_1},\\ldots ,L_{i_{k-1}}\\rangle .$ So all the minimal primes have codimension exactly $k-1$ (since $\\mathcal {A}$ is $k-2$ generic), and they are the ideals of the coatoms of $\\mathcal {A}$ .", "From this we get: 1) the codimension of $I_{n-k+2}(\\mathcal {A})$ is $k-1$ , and 2) $I_{n-k+2}(\\mathcal {A})$ might have an embedded prime, the irrelevant ideal $\\langle x_1,\\ldots ,x_k\\rangle $ .", "So the primary decomposition of $I_{n-k+2}(\\mathcal {A})$ is $I_{n-k+2}(\\mathcal {A})=Q_1\\cap \\cdots \\cap Q_s\\cap J,$ where $Q_i$ are primary ideals of codimension $k-1$ and $J$ is a $\\langle x_1,\\ldots ,x_k\\rangle -$ primary ideal of codimension $k$ .", "Since for any two ideals $A,B\\subset R$ we have $(A\\cap B)^{sat}=A^{sat}\\cap B^{sat}$ , and because $J^{sat}=R$ we obtain that $I_{n-k+2}(\\mathcal {A})^{sat}=Q_1\\cap \\cdots \\cap Q_s.$ Here we used the fact that $Q_i$ are primary ideals of codimension $k-1$ and therefore $(Q_i)^{sat}=Q_i$ .", "If $f\\in (Q_i)^{sat}$ but $f\\notin Q_i$ , then $\\langle x_1,\\ldots ,x_k\\rangle ^{n(f)}\\cdot f\\subseteq Q_i$ , for some positive integer $n(f)$ .", "From the definition of the primary ideals, we obtain that $\\langle x_1,\\ldots ,x_k\\rangle ^u\\subseteq Q_i$ , for some power $u>0$ , which is in contradiction with $codim(Q_i)=k-1$ .", "To prove our assertion, first we show that $I_{n-k+2}(\\mathcal {A})^{sat}$ has the primary decomposition described in the statement, by localizations of $I_{n-k+2}(\\mathcal {A})$ at each of its minimal primes.", "And then we show that $I_{n-k+2}(\\mathcal {A})=I_{n-k+2}(\\mathcal {A})^{sat}$ .", "Let $X\\in L_{k-1}(\\mathcal {A})$ be a coatom.", "Denote $\\nu (X)=m$ , and assume that $X=H_1\\cap \\cdots \\cap H_m,$ with $I(X)=\\langle x_1,\\ldots ,x_{k-1}\\rangle \\subset R:=\\mathbb {K}[x_1,\\ldots ,x_k]$ .", "If we localize $R$ at $I(X)$ , we have that $L_{m+1},\\ldots ,L_n$ are invertible elements and therefore $I_{n-k+2}(\\mathcal {A})R_{I(X)}= \\langle \\lbrace L_{i_1}\\cdots L_{i_{m-k+2}}:1\\le i_1<\\cdots <i_j\\le m\\rbrace \\rangle R_{I(X)}.$ Consider now the linear code with generating matrix $A$ given by the coefficients of the linear forms $L_1,\\ldots ,L_m$ .", "These are linear forms in variables $x_1,\\ldots ,x_{k-1}$ .", "So this linear code has length $m$ and dimension $k-1$ .", "Since any $k-1$ of these linear forms are linearly independent, the maximum number of columns of $A$ that span a $k-2$ dimensional vector space is $k-2$ .", "So, by [15], Remark 2.3, the minimum distance of this code is $d=m-(k-2)=m-k+2.$ From [13], Theorem 3.1, we have that $\\langle \\lbrace L_{i_1}\\cdots L_{i_{m-k+2}}:1\\le i_1<\\cdots <i_j\\le m\\rbrace \\rangle =\\langle x_1,\\ldots ,x_{k-1}\\rangle ^{m-k+2}.$ So we got that $I_{n-k+2}(\\mathcal {A})R_{I(X)}=I(X)^{\\nu (X)-k+2}R_{I(X)},$ for all the minimal primes (which are the ideals $I(X)$ of all the coatoms $X$ ) of $I_{n-k+2}(\\mathcal {A})$ .", "This means that the primary decomposition of $I_{n-k+2}(\\mathcal {A})^{sat}$ is the one desired.", "Any $I_i(\\mathcal {A})$ can be generated by the maximal minors of a certain matrix.", "Let $M$ be a $i\\times n$ matrix with entries in $\\mathbb {K}$ such that all the $i\\times i$ minors are nonzero.", "There exists such a matrix since, by the way it is defined, the set of all these matrices is an open Zariski set.", "Then the maximal minors of the $i\\times n$ matrix with entries in $R_1$ $N=M\\cdot \\left[\\begin{array}{cccc}L_1&0&\\cdots &0 \\\\0&L_2&\\cdots &0 \\\\\\vdots &\\vdots & &\\vdots \\\\0&0&\\cdots &L_n\\end{array}\\right]$ are the generators of $I_i(\\mathcal {A})$ .", "When $i=n-k+2$ , and $\\mathcal {A}$ is $k-2$ generic, the codimension of $I_i(\\mathcal {A})$ is exactly $(i-i+1)(n-i+1)=k-1$ .", "So $R/I_{n-k+2}(\\mathcal {A})$ is a determinantal ring which are known to be Cohen-Macaulay ([4], Theorem 18.18).", "So, the projective dimension of $R/I_{n-k+2}(\\mathcal {A})$ is $pd(R/I_{n-k+2}(\\mathcal {A}))=codim(I_{n-k+2}(\\mathcal {A}))=k-1$ .", "This means that $Ext^k(R/I_{n-k+2}(\\mathcal {A}),R)=0$ and this gives that $I_{n-k+2}(\\mathcal {A})$ cannot have an associated prime of codimension $k$ .", "Therefore, $I_{n-k+2}(\\mathcal {A})=I_{n-k+2}(\\mathcal {A})^{sat}$ .", "For rank 3 arrangements, [11], Lemma 3.2 presents the graded minimal free resolution for $I_{n-1}(\\mathcal {A})$ .", "More generally, from the map $R^n\\longrightarrow R^i$ with matrix $N$ seen in the proof above, we get a complex of $R-$ modules, known as the Eagon-Northcott complex ([5], Chapter A2H, covers in full details this complex).", "If $depth(I_i(\\mathcal {A}),R)=n-i+1$ , then this complex is exact and therefore it provides a free resolution for $R/I_i(\\mathcal {A})$ .", "In our instance, $i=n-k+2$ and since $R/I_{n-k+2}(\\mathcal {A})$ is Cohen-Macaulay, $k-1$ is the smallest integer $r$ such that $Ext^r(R/I_{n-k+2}(\\mathcal {A}),R)\\ne 0$ .", "By [4], Proposition 18.4, $depth(I_{n-k+2}(\\mathcal {A}),R)=k-1=n-(n-k+2)+1,$ and we have exactness of the Eagon-Northcott complex for the $R-$ module $R/I_{n-k+2}(\\mathcal {A})$ ." ], [ "The fitting problem and the index of nilpotency", "Let $I$ be an ideal in $R=\\mathbb {K}[x_1,\\ldots ,x_k]$ .", "The index of nilpotency of $I$ , denoted $nil(I)$ , is the smallest integer $s$ such that $(\\sqrt{I})^s\\subseteq I.$ For a nice exposition about this invariant we recommend [16], Chapter 9.2.", "The next result determines the index of nilpotency of the ideal of a fat point scheme in $\\mathbb {P}^{k-1}$ .", "Proposition 3.1 Let $Z=m_1P_1+\\cdots +m_nP_n, m_i\\ge 1$ be a fat point scheme in $\\mathbb {P}_{\\mathbb {K}}^{k-1}$ .", "If $I_Z\\subset R=\\mathbb {K}[x_1,\\ldots ,x_k]$ is the ideal of $Z$ , then $nil(I_Z)=\\max \\lbrace m_1,\\ldots ,m_n\\rbrace .$ Let $X=\\lbrace P_1,\\ldots ,P_n\\rbrace \\subset \\mathbb {P}^{k-1}$ be the support of $Z$ .", "Let $I_X\\subset R$ be the ideal of $X$ .", "Then $\\sqrt{I_Z}=I_X.$ Denote with $s=nil(I_Z)$ , with $m=\\max \\lbrace m_1,\\ldots ,m_n\\rbrace $ , and with $I_{P_i}$ the ideal of the point $P_i$ .", "Suppose that $m=m_1$ and that $P_1=[0,\\ldots ,0,1]$ .", "We have that $I_Z=I_{P_1}^{m_1}\\cap \\cdots \\cap I_{P_n}^{m_n}$ and $I_X=I_{P_1}\\cap \\cdots \\cap I_{P_n}.$ Obviously, $I_X^m\\subseteq I_Z$ .", "Therefore $m\\ge s.$ Suppose $m\\ge 2$ .", "Otherwise, $I_Z=I_X$ and therefore $nil(I_Z)=1=m$ .", "Also, assume that $s\\le m-1$ , and let $f\\in I_X$ such that $f\\notin I_{P_1}^2$ .", "There must exist such an element, otherwise $I_{P_1}^2$ would become a primary component of $I_X$ which contradicts that $I_X$ is a radical ideal.", "Since $I_X^s\\subseteq I_Z$ , then $f^{m-1}\\in I_{P_1}^m.$ Because $f\\in I_{P_1}-I_{P_1}^2$ , and since $I_{P_1}=\\langle x_1,\\ldots ,x_{k-1}\\rangle $ we have that $f=x_1g_1+x_2g_2+\\cdots +x_{k-1}g_{k-1},$ with at least one of the polynomials $g_i$ not in $I_{P_1}$ .", "If $\\deg (f)=d+1$ , we can assume that $f=\\ell x_k^d+g,$ where $\\ell $ is a linear form in variables $x_1,\\ldots ,x_{k-1}$ , and $g\\in I_{P_1}^2$ .", "Then $f^{m-1}=\\ell ^{m-1}x_k^{d(m-1)}+h,$ where $h=\\sum _{b=1}^{m-1}{{m-1}\\atopwithdelims (){b}}\\ell ^{m-1-b}g^bx_k^{d(m-1-b)}$ .", "By the way we constructed $\\ell $ and $g$ , $\\ell ^{m-1-b}g^b\\in I_{P_1}^{m-1+b},$ and since $b\\ge 1$ , we have that the polynomial $h\\in I_{P_1}^m.$ We obtain $\\ell ^{m-1}x_k^{d(m-1)}\\in I_{P_1}^m$ , which is a contradiction: the leading monomial of $\\ell ^{m-1}x_k^{d(m-1)}$ under any monomial order $>$ with $x_1>\\cdots >x_{k-1}>x_k$ is $x_i^{m-1}x_k^{d(m-1)}$ for some $1\\le i\\le k-1$ , and should belong to the (monomial) ideal $\\langle x_1,\\ldots ,x_{k-1}\\rangle ^m$ .", "So $s\\ge m$ and therefore $s=m$ .", "Now we can put together the two propositions to obtain the main result of the notes.", "First, to a finite set of $n$ points $\\Gamma \\subset \\mathbb {P}^{k-1}$ , not all on a hyperplane, we can associate the central essential (dual) arrangement of $n$ hyperplanes $\\mathcal {A}_{\\Gamma }\\subset \\mathbb {K}^k$ defined by the vanishing of the linear forms with coefficients the coordinates of the points of $\\Gamma $ .", "Theorem 3.2 Let $\\Gamma \\subset \\mathbb {P}_{\\mathbb {K}}^{k-1}$ be a finite $(k-2)-$ generic set of $n$ points, not all contained in a hyperplane.", "Then $hyp(\\Gamma )=nil(I_{n-k+2}(\\mathcal {A}_{\\Gamma }))+k-2.$ If $hyp(\\Gamma )$ number of points lie on a hyperplane of equation $a_1x_1+\\cdots +a_kx_k=0$ , then, dually, the corresponding hyperplanes in $\\mathcal {A}_{\\Gamma }$ will intersect at the coatom $[a_1,\\ldots ,a_k]$ .", "So $hyp(\\Gamma )=\\max \\lbrace \\nu (X): X\\in L_{k-1}(\\mathcal {A}_{\\Gamma })\\rbrace .$ Immediate application of Proposition REF to Proposition REF gives the result.", "Example 3.3 We end with a simple example.", "Let $P_1=(1,0), P_2=(1,1), P_3=(3,-1), P_4=(-3,2)$ be four points in the real plane.", "Find the maximum number of collinear points, and the equation(s) of the line(s) where they are positioned.", "First we projectivize the problem by embedding the affine real plane into $\\mathbb {P}^2$ .", "So we add the extra coordinate $z=1$ to all the points to get $\\Gamma = \\lbrace Q_1=[1,0,1],Q_2=[1,1,1],Q_3=[3,-1,1],Q_4=[-3,2,1]\\rbrace \\subset \\mathbb {P}^2.$ To find $hyp(\\Gamma )$ , we create $\\mathcal {A}_{\\Gamma }$ defined by the linear forms $L_1=x+z, L_2=x+y+z, L_3=3x-y+z, L_4=-3x+2y+z,$ and build $I_3(\\mathcal {A}_{\\Gamma })=\\langle L_1L_2L_3,L_1L_2L_4,L_1L_3L_4,L_2L_3L_4\\rangle .$ Also consider $J=\\sqrt{I_3(\\mathcal {A}_{\\Gamma })}.$ With Macaulay 2 ([7]), observe that $I_3(\\mathcal {A}_{\\Gamma }):J=\\langle y+2z,x+z\\rangle \\mbox{ and }I_3(\\mathcal {A}_{\\Gamma }):J^2=R.$ From Theorem REF , this means that $hyp(\\Gamma )=3$ .", "Let $I=I_1^{m_1}\\cap \\cdots \\cap I_s^{m_s}$ be the ideal of a fat point scheme.", "Denote with $J=\\sqrt{I}$ and suppose that $m_1=\\cdots =m_p=m$ is the maximum multiplicity of any primary component of $I$ .", "Then $I:J^{m-1}=I_1\\cap \\cdots \\cap I_p.$ In the fitting problem setup, the points with ideals $I_1,\\ldots ,I_p$ correspond (dually) to the hyperplanes containing $hyp(\\Gamma )$ number of points of $\\Gamma $ .", "For our example if we do this operation we obtain $\\langle y+2z,x+z\\rangle $ , which is the ideal of the point $[-1,-2,1]$ .", "Dually, we obtained the projective line $-x-2y+z=0,$ which after dehomogeneization gives the line in the plane of equation $x+2y=1.$ Observe that the points $P_1,P_3$ and $P_4$ are collinear sitting on this line.", "Acknowledgements: I wish to thank Graham Denham and Hal Schenck for useful discussions.", "I am very grateful to the anonymous referee for the important terminology correction and for improving and simplifying the proof of Proposition REF that allows to drop the restriction on the characteristic of the field considered initially in this result as well as in Theorem REF ." ] ]

[ [ "Transition from subbarrier to deep subbarrier regimes in heavy-ion\n fusion reactions" ], [ "Abstract We analyze the recent experimental data of heavy-ion fusion cross sections available up to deep subbarrier energies in order to discuss the threshold incident energy for a deep subbarrier fusion hindrance phenomenon.", "To this end, we employ a one-dimensional potential model with a Woods-Saxon internuclear potential.", "Fitting the experimental data in two different energy regions with different Woods-Saxon potentials, we define the threshold energy as an intersect of the two fusion excitation functions.", "We show that the threshold energies so extracted are in good agreement with the empirical systematics as well as with the values of the Krappe-Nix-Sierk (KNS) potential at the touching point.", "We also discuss the asymptotic energy shift of fusion cross sections with respect to the potential model calculations, and show that it decreases with decreasing energies in the deep subbarrier region although it takes a constant value at subbarrier energies." ], [ "Transition from subbarrier to deep subbarrier regimes in heavy-ion fusion reactions Ei Shwe Zin Thein Department of Physics, Mandalay University, Myanmar N.W.", "Lwin Department of Physics, Mandalay University, Myanmar K. Hagino Department of Physics, Tohoku University, Sendai 980-8578, Japan We analyze the recent experimental data of heavy-ion fusion cross sections available up to deep subbarrier energies in order to discuss the threshold incident energy for a deep subbarrier fusion hindrance phenomenon.", "To this end, we employ a one-dimensional potential model with a Woods-Saxon internuclear potential.", "Fitting the experimental data in two different energy regions with different Woods-Saxon potentials, we define the threshold energy as an intersect of the two fusion excitation functions.", "We show that the threshold energies so extracted are in good agreement with the empirical systematics as well as with the values of the Krappe-Nix-Sierk (KNS) potential at the touching point.", "We also discuss the asymptotic energy shift of fusion cross sections with respect to the potential model calculations, and show that it decreases with decreasing energies in the deep subbarrier region although it takes a constant value at subbarrier energies.", "25.70.Jj,24.10.Eq Heavy-ion fusion reactions at low incident energies are intimately related to the quantum tunneling phenomena of many-body systems.", "Because of a strong cancellation between the repulsive Coulomb interaction and an attractive short range nuclear interaction between the colliding nuclei, a potential barrier, referred to as a Coulomb barrier, is formed, which has to be surmounted in order for fusion to take place.", "In heavy-ion reactions, because of a strong absorption inside the Coulomb barrier, it has been usually assumed that a compound nucleus is automatically formed once the Coulomb barrier has been overcome.", "The simplest approach to heavy-ion fusion reactions based on this idea, that is, a one-dimensional potential model has been successful in reproducing experimental fusion cross sections at energies above the Coulomb barrier[1].", "A one dimensional potential model fitted to reproduce fusion cross sections above the Coulomb barrier, however, have been found to underestimate fusion cross sections at lower energies.", "It has been well recognized by now that the sub-barrier fusion enhancement is caused by couplings of the relative motion between the colliding nuclei with other degrees of freedom, such as collective vibrational and rotational motions in the colliding nuclei[2], [3].", "The behavior of fusion cross sections at extremely low energies is a critical issue for estimating reaction rates of astrophysical interests.", "One of the currents interests in heavy-ion fusion reactions is a steep fall-off phenomenon of fusion cross sections at deep subbarrier energies.", "Recently, fusion cross sections for several colliding systems have been measured down to extremely low cross sections, up to several nb[4], [5], [6], [7], [8].", "These experimental data have shown that fusion cross sections fall off much more steeply at deep subbarrier energies as decreasing energies, compared to the expectation from the energy dependence of cross sections at subbarrier energies.", "Although a few theoretical models have been proposed[9], [10], the origin for the deep subbarrier fusion hindrance has not yet been fully understood.", "In Refs.", "[4], [5], [11], the deep subbarrier fusion hindrance has been analyzed using the astrophysical S-factor.", "It has been claimed [4], [5], [11] that the deep subbarrier fusion hindrance sets in at the energy at which the astrophysical S-factor takes the maximum.", "The authors of Refs.", "[4], [5], [11] even parametrized the threshold energy as $E_s= 0.356\\left(Z_1Z_2\\sqrt{\\frac{A_1A_2}{A_1+A_2}}\\right)^{2/3}~~~({\\rm MeV}).$ Notice that the S-factor representation provides a useful tool only when the penetration of the Coulomb repulsive potential is a dominant contribution, such as in fusion reactions of light systems at low energies.", "In fact, the relation between the threshold for the deep subbarrier hindrance and the maximum of the S-factor is not clear physically, and thus it is not trivial how to justify theoretically the identification of the threshold energy with the astrophysical S-factor.", "Nevertheless, it has turned out that the threshold energy so obtained closely follows the values of phenomenological internucleus potentials, such as the Krappe-Nix-Sierk (KNS)[12], the Bass[13], the proximity[14], and the Akyüz-Winther[15] potentials, at the touching configuration [16].", "This clearly implies that the dynamics which takes place after the colliding nuclei touch with each other is responsible for the deep subbarrier fusion hindrance, making at the same time the astrophysical S-factor decrease as the incident energy decreases.", "In this paper, we investigate the threshold energy for deep subbarrier fusion hindrance using an alternative method, which is physically more transparent than the definition with the maximum of S-factor.", "That is, we determine the threshold energies by fitting the experimental fusion cross sections in subbarrier and deep subbarrier energy regions separately using single-channel barrier penetration model calculations, and compare them with the systematics given by Eq.", "(REF ) as well as with the touching energy evaluated with the KNS potential.", "We also discuss the energy dependence of fusion cross sections at deep subbarrier energies in terms of an asymptotic energy shift proposed by Aguiar et al.[17].", "Figure: (Color online)Fusion excitation functions for the 64 ^{64}Ni+ 64 ^{64}Ni (the upper panel) and 16 ^{16}O+ 208 ^{208}Pb (the lower panel).The solid and the dashed lines are results of single-channelpotential model calculations which fit the experimental data in the subbarrier and thedeep subbarier energy regions, respectively.", "The experimental data are taken fromRefs.", ", .In order to illustrate our procedure, the upper and the lower panels of Fig.", "REF show fusion cross sections for $^{64}$ Ni+$^{64}$ Ni and $^{16}$ O+$^{208}$ Pb systems, respectively.", "We first define the subbarrier energy region as the one in which fusion cross sections take between $10^{-2}$ mb and $10^0$ mb.", "We fit the experimental data in this energy region with a potential model with a Woods-Saxon potential treating the three parameters of the potential, that is, the depth $V_0$ , the radius $R_0$ , and the surface diffuseness $a$ , as adjustable parameters.", "To this end, we numerically solve the Schrödinger equation without resorting to the parabolic approximation[18].", "The fusion cross sections calculated in this way are shown by the solid lines in the figure.", "Of course, these calculations do not account for the fusion cross sections at higher energies as the channel coupling effects are completely ignored.", "However, it is sufficient for our purpose, as we are interested only in the energy dependence of fusion cross sections at subbarrier energies, that is, the slope of fusion excitation functions.", "These calculations do not reproduce the experimental data at lower energies, either.", "In order to obtain a better fit in the lower energy region, the surface diffuseness parameter has to be increased, as has been noticed in Refs.", "[8], [18].", "We then define the deep subbarrier region as the one in which fusion cross sections take below $10^{-3}$ mb.", "The dashed lines in the figure show the fusion cross sections obtained by fitting to the experimental data in this energy region.", "See Table I for the actual values of the surface diffuseness parameter.", "From the two curves, we finally define the threshold energy for deep subbarrier fusion hindrance as the energy at which the two fusion excitation functions intersect with each other.", "Figure: (Color online) The threshold energy E s E_s for deep subbarrier hindrancefor several systems, determined with the two slope fit to theexperimental fusion cross sections, as a function of theparameter Z 1 Z 2 A 1 A 2 /(A 1 +A 2 )Z_1Z_2\\sqrt{A_1A_2/ (A_1+A_2)}.The solid curve is the empirical function given byEq.", "(), while the stars denote the“experimental” values defined as the maximum energy of the astrophysicalS-factors.Table: The threshold energy E s E_s for deep subbarrier fusion hindrance forseveral systems, obtained with the two slope fit to the experimentalfusion cross sections.", "a > a_> and a < a_< are the diffuseness parametersin the Woods-Saxon potential used to fit the subbarrier and the deepsubbarrier regions of fusion cross sections.ζ\\zeta is defined as ζ=Z 1 Z 2 A 1 A 2 /(A 1 +A 2 )\\zeta = Z_1Z_2\\sqrt{A_1A_2/ (A_1+A_2)},in which Z i Z_i and A i (i=1,2)A_i~(i=1,2) are the charge and the mass numbers ofthe nucleus ii.E s ( exp ) E_s^{\\rm (exp)} and E s ( emp ) E_s^{\\rm (emp)} are the “experimental” thresholdenergy and theempirical energies given by Eq.", "(), respectively.V KNS V_{\\rm KNS} is the potential energy at the touching configuration estimatedwith the KNS potential.All the energies are shown in units of MeV, while the lengths arein units of fm.Figure REF shows the threshold energies thus obtained as a function of $Z_1Z_2\\sqrt{A_1A_2/(A_1+A_1)}$ .", "The figure also shows the threshold energy for $^{28}$ Si+$^{64}$ Ni[20], $^{64}$ Ni+$^{64}$ Ni[4] $^{16}$ O+$^{208}$ Pb[8], $^{60}$ Ni+$^{89}$ Y[4], $^{90}$ Zr+$^{90}$ Zr[19], $^{90}$ Zr+$^{92}$ Zr[19], and $^{90}$ Zr+$^{89}$ Y [19]systems.", "For comparison, the figure also shows the empirical systematics given by Eq.", "(REF ) with the solid line and the “experimental” data defined as the maximum energy of the S-factor [11] by the crosses.", "These values are summarized in Table REF , together with the potential energy at the touching point [16] estimated with the KNS potential.", "One can see that the values of the threshold energy defined in our way are in good agreement with those defined as the maximum of the astrophysical S-factor as well as with the potential energy at the touching configuration.", "Let us next discuss briefly the asymptotic energy shift for deep subbarrier fusion reactions.", "This quantity was introduced by Aguiar et al.", "[17] as a measure of subbarrier enhancement of fusion cross sections.", "It was defined as an extra energy needed to fit the experimental fusion cross sections with respect to a single-channel potential model calculation.", "It has been argued that the calculated fusion cross sections have approximately the same exponential energy dependence as the experimental data in the subbarrier energy region, but are shifted in energy by a constant amount [17].", "In connection to the deep subbarrier fusion hindrance, it may be interesting to revisit this representation.", "Figure: (Color online)Fusion excitation functions for the 64 ^{64}Ni+ 64 ^{64}Ni system.The dashed and the solid lines are results of single-channelpotential model calculations which fit the experimental data in the subbarrier region andat energies above the Coulomb barrier, respectively.The experimental data are taken fromRefs.", ".In order to define the asymptotic energy shift, we first adjust the value of $V_0$ and $R_0$ in the Woods-Saxon potential, keeping the same value for the diffuseness parameter $a$ as the one which has been obtained to fit to the subbarrier fusion cross sections (see $a_>$ in Table I), so that the experimental fusion cross sections at high energies, that is, those above $\\sigma > 100$ mb, can be approximately reproduced (see Fig.", "REF ).", "We then define the asymptotic energy shift as a difference between the solid line in Fig.", "REF and the experimental data for a fixed value of fusion cross section.", "Figure REF shows the asymptotic energy shift so extracted for several systems as a function of corresponding fusion cross section.", "As one can see, the asymptotic energy shift is nearly constant in the range of 0.1 mb $\\sigma $ 1 mb, in accordance to the previous conclusion by Aguiar et al.[17].", "However, in the deep subbarrier region, the asymptotic energy shift start decreasing as the fusion cross sections decrease, reflecting the fact that the fusion cross sections have a different exponential slope from that in the subbarrier region, as shown in Fig.", "REF .", "Figure: (Color online)The asymptotic energy shift as a function of fusion cross section for 28 ^{28}Si+ 64 ^{64}Ni (the filled squares), 16 ^{16}O+ 208 ^{208}Pb (the filled triangles),and 64 ^{64}Ni+ 64 ^{64}Ni (the filled circles) systems.In summary, we have studied the energy dependence of heavy-ion fusion cross sections at deep subbarrier energies using the recent experimental data.", "To this end, we employed a one-dimensional potential model.", "We have shown that the asymptotic energy shift is almost a constant in the subbarrier region, but it decreases with decreasing energies in the deep subbarrier region.", "This is a clear manifestation of the hindrance phenomenon of deep sub-barrier fusion.", "In order to see at which energy the deep subbarrier hindrance takes place, we estimated the threshold energy with a two-slope fit procedure.", "That is, we defined the threshold energy as an intersect of two fusion excitation functions, which fit the experimental fusion cross sections either in the subbarrier energy region or in the deep subbarrier energy region.", "We have shown that the threshold energies so defined are in good agreement with those estimated from the maximum of astrophysical $S$ -factor.", "The definition for the threshold energy proposed in this paper is complementary to the one with the maximum of astrophysical S-factor.", "As we have shown in this paper, both the definitions provide a similar value of threshold energy as the potential energies at the touching configuration.", "This strongly suggests that the dynamics after the touching plays an important role in deep subbarrier fusion reactions, changing the exponential slope of fusion cross sections and at the same time making the astrophysical S-factor take the maximum, although it is an open question why and how the dynamics after the touching leads to the maximum of astrophysical S-factor.", "We thank T. Ichikawa for useful discussions.", "This work was supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology by Grant-in-Aid for Scientific Research under the program number (C) 22540262." ] ]

[ [ "The Suzaku X-ray spectrum of NGC 3147. Further insights on the best\n \"true\" Seyfert 2 galaxy candidate" ], [ "Abstract NGC 3147 is so far the most convincing case of a \"true\" Seyfert 2 galaxy, i.e.", "a source genuinely lacking the Broad Line Regions.", "We obtained a Suzaku observation with the double aim to study in more detail the iron line complex, and to check the Compton-thick hypothesis for the lack of observed optical broad lines.", "The Suzaku XIS and HXD/PIN spectra of the source were analysed in detail.", "The line complex is composed of at least two unresolved lines, one at about 6.45 keV and the other one at about 7 keV, most likely identified with Fe XVII/XIX, the former, and Fe XXVI, the latter.", "The high-ionization line can originate either in a photoionized matter or in an optically thin thermal plasma.", "In the latter case, an unusually high temperature is implied.", "In the photoionized model case, the large equivalent width can be explained either by an extreme iron overabundance or by assuming that the source is Compton-thick.", "In the Compton-thick hypothesis, however, the emission above 2 keV is mostly due to a highly ionized reflector, contrary to what is usually found in Compton-thick Seyfert 2s, where reflection from low ionized matter dominates.", "Moreover, the source flux varied between the XMM-Newton and the Suzaku observations, taken 3.5 years apart, confirming previous findings and indicating that the size of the emitting region must be smaller than a parsec.", "The hard X-ray spectrum is also inconclusive on the Compton-thick hypothesis.", "Weighting the various arguments, a \"true\" Seyfert 2 nature of NGC 3147 seems to be still the most likely explanation, even if the \"highly ionized reflector\" Compton-thick hypothesis cannot at present be formally rejected." ], [ "Introduction", "NGC 3147 ($z$ =0.009346) is at present the most convincing example of a “true” Seyfert 2 galaxy (Bianchi et al.", "2008; Shi et al.", "2010; Tran et al.", "2011).", "In such sources the lack of detection of broad emission lines in the optical/UV spectra cannot be explained by the obscuration of the Broad Line Regions (BLR) - as it is the case for “normal” Seyfert 2s in the so far very successful Unification models (Antonucci 1993) - but requires their absence.", "In fact, simultaneous X-ray and optical spectroscopy have demonstrated that in this source the BLR is lacking at the same time when the nucleus is seen unobscured (Bianchi et al.", "2008).", "The X-ray XMM-$Newton$ observation, albeit rather short (less than 20 ks), was good enough to put a tight constraint on the column density of any obscuring material along the line-of-sight.", "The same observation also indicated complex iron line emission, but the short exposure time did not permit a detailed analysis.", "It has been suggested (Nicastro 2000; Nicastro et al.", "2003; Elitzur & Shlosman 2006) that at low accretion rates (and thence low luminosities) the broad line regions (BLR) cannot form.", "Indeed, NGC 3147 is a low (even if not extremely so) luminosity source accreting at a low rate (bolometric luminosity $L\\sim 5\\times 10^{42}$ erg s$^{-1}$ , $L/L_{Edd}\\sim 10^{-4}$ , Bianchi et al.", "2008).", "A possible alternative explanation for the peculiar characteristics of NGC 3147 is that the source is Compton-thick.", "In fact, the simultaneous lack of BLR and absorption can be explained if the absorber is so thick to completely obscure the nuclear emission below 10 keV.", "In this case, the observed emission should be due to reflection off circumnuclear matter, but X-rays could pierce through the absorbers above 10 keV.", "The hard X-ray coverage offered by $Suzaku$ is the best tool at present to test it.", "With the dual goal of studying in more detail the iron line emission and to check the Compton-thick explanation for this source, we asked for, and obtained, a long (150 ks) $Suzaku$ observation.", "The paper is organized as follows.", "The observation and data reduction are described in Sec.", "2, while the data analysis is presented in Sec. 3.", "Results are discussed and summarized in Sec. 4.", "Figure: XIS0 light curve.", "Upper panel: count ratesin the source extraction region.", "Medium panel: count ratesin the background.", "Lower panel: background-subtracted sourcecount rates." ], [ "Observation and data reduction", "NGC 3147 was observed by Suzaku on 2010, May 24, for 150 ks (obsid 705054010).", "X-ray Imaging Spectrometer (XIS) and Hard X-ray Detector (HXD) event files were reprocessed with the latest calibration files available (2011-06-30 release), using ftools 6.11 and Suzaku software Version 18, adopting standard filtering procedures.", "Source and background spectra for all the three XIS detectors were extracted from circular regions with radius of 167 pixels ($\\simeq 175$ arcsec), avoiding the calibration sources.", "Response matrices and ancillary response files were generated using xisrmfgen and xissimarfgen.", "We downloaded the “tuned” non-X-ray background (NXB) files for our HXD/PIN data provided by the HXD teamsee ftp://legacy.gsfc.nasa.gov/suzaku/data/background/pinnxb_ver2.0_tuned/ and extracted source and background spectra using the same good time intervals.", "The PIN spectrum was then corrected for dead time, and the exposure time of the background spectrum was increased by a factor of 10, as required.", "Finally, the contribution from the cosmic X-ray background (CXB) was subtracted from the source spectrum, simulating it as suggested by the HXD team.", "The XIS spectra were fitted between 0.5 and 10 keV.", "The normalizations of XIS1 and XIS3 with respect to XIS0 were left free in the fits, and always resulted in agreement to within 3%.", "A constant factor of 1.16 was used instead between the PIN and the XIS0, as recommended for observations taken in the XIS nominal position.", "In the following, all the PIN fluxes are given with respect to the XIS0 flux scale, which is 1.16 times lower than the HXD absolute flux scale.", "In the following, quoted statistical errors correspond to the 90% confidence level for one interesting parameter ($\\Delta \\chi ^2=2.71$ ), unless otherwise stated.", "The adopted cosmological parameters are H$_{0}=70$ km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _\\Lambda =0.73$ , and $\\Omega _m=0.27$ (i.e., the default ones in xspec 12.7.0: Arnaud 1996).", "We use the Anders & Grevesse (1989) chemical abundances and the photoelectric absorption cross-sections by Balucinska-Church & McCammon (1992)." ], [ "Data analysis and results", "The light curve is shown in Fig REF .", "No significant variability is detected.", "For the spectral analysis, therefore, we used spectra integrated over the whole observation." ], [ "The XIS spectra", "We first analyzed the XIS spectra.", "A simple power law model (plus Galactic absorption, N$_{H,G}$ =3.64$\\times 10^{20}$ cm$^{-2}$ , Dickey & Lockman 1990) gives a poor fit ($\\chi ^2/d.o.f.$ =604.1/482).", "The most prominent features in the residuals are apparent at the energies of the iron line complex (see Fig.", "REF ).", "We therefore added two narrow emission lines, with energies free to vary around 6.4 and 7 keV (see next section), which improves significantly the quality of the fit ($\\chi ^2/d.o.f.$ =496.4/478).", "The power law index is 1.745$\\pm 0.014$ .", "The inclusion of either a warm absorber or a Compton reflection component is not required by the data ($\\Delta \\chi ^2$ =-2.4 and -2.8, respectively), while addition of a second power law improves the fit quality ($\\chi ^2/d.o.f.$ =481.3/476; the improvement is significant at the 99.94% confidence level, according to the F-test).", "The power law indices are 3.50($^{+0.42}_{-0.69}$ ) and 1.689($^{+0.022}_{-0.018}$ ).", "The 0.5-2 (2-10) keV flux is 8.39$\\times $ 10$^{-13}$ (1.64$\\times $ 10$^{-12}$ ) erg cm$^{-2}$ s$^{-1}$ , corresponding to a (absorption corrected) luminosity of 1.8$\\times $ 10$^{41}$ (3.2$\\times $ 10$^{41}$ ) erg s$^{-1}$ .", "Finally, no intrinsic absorption is detected.", "The upper limit to the column density of any neutral absorber at the redshift of the source is 5$\\times 10^{20}$ cm$^{-2}$ .", "Figure: Best fit model and residuals fitting the XIS spectrawith a simple power law plus Galactic absorption.We then compared the $Suzaku$ spectrum with the XMM-$Newton$ one (Bianchi et al.", "2008).", "XMM-$Newton$ observed NGC 3147 on 2006, October 6 (i.e.", "about three and half year earlier), with a net exposure time of 14 ks in the EPIC-pn instrument.", "We re-extracted the spectrum with the same procedure described in Bianchi et al.", "(2008), but using the latest version of SAS (11.0.1) and of the calibration files.", "According to Bianchi et al.", "(2008), we fitted the spectrum with a power law absorbed by both Galactic and intrinsic matter, and two narrow guassian lines.", "The fit is good ($\\chi ^2/d.o.f.$ =93.6/106).", "The power law index is 1.62$\\pm $ 0.05, the column density of the local absorber is $<$ 3.2$\\times 10^{20}$ cm$^{-2}$ , the energies of the two lines are 6.471($^{+0.089}_{-0.065}$ ) keV and 6.798($^{+0.083}_{-0.072}$ ) keV and their fluxes are 2.8($\\pm $ 1.6)$\\times 10^{-6}$ ph cm$^{-2}$ s$^{-1}$ (EW of 189 eV) and 2.4($\\pm $ 1.5)$\\times 10^{-6}$ ph cm$^{-2}$ s$^{-1}$ (EW of 172 eV), respectively.", "All these values are consistent within the errors with those of Bianchi et al.", "(2008).", "The 0.5-2 (2-10) keV flux is 5.78$\\times $ 10$^{-13}$ (1.43$\\times $ 10$^{-12}$ ) erg cm$^{-2}$ s$^{-1}$ .", "Comparing the XMM-$Newton$ and $Suzaku$ results, a variation, most prominent in the soft band, is found.", "This is shown in Fig REF , where the XMM-$Newton$ best fit model is superposed to the Suzaku/XIS1 spectrum.", "The variation can be explained either as a steepening of the power law component or, better, with different variations of the soft and hard X-ray component, the former varying most.", "An explanation purely in terms of a variation of the local absorber is, instead, not viable.", "This result confirms that the source is variable on time scales of years.", "In fact, the measured 2-10 keV flux, in units of 10$^{-12}$ erg cm$^{-2}$ s$^{-1}$ , has been measured as 1.6 in September 1993 (with ASCA: Ptak et al.", "1996), 2.3 in November 1997 (with BeppoSAX: Dadina 2007) and 3.7 in September 2001 (with Chandra: Terashima & Wilson 2003).", "That the variation cannot be due to a confusing source is demostrated by the fact that the highest flux has been measured by the best spatial resolution satellite.", "Figure: The SuzakuSuzaku/XIS1 spectrum and residuals when the best fitmodel for the XMM-NewtonNewton observation is superimposed." ], [ "The iron line complex", "We then studied in more detail the iron line complex.", "While the iron line properties in the XMM-$Newton$ and $Suzaku$ observations are roughly consistent, the much better statistics in the $Suzaku$ observation due to the longer exposure time allows for a more detailed analysis.", "To this end, we limited for simplicity the analysis to the XIS instruments, and the energy range to the 4-10 keV band.", "In Fig.", "REF , the data and best fit model are shown, after removal of any emission line from the fit.", "The presence of two emission lines is clear, one around 6.4 keV, to be attributed to K$\\alpha $ emission from neutral or low ionization iron, the other around 7 keV, likely due to either (or both) hydrogen-like (i.e.", "Fe XXVI) iron emission or K$\\beta $ emission from low ionization iron.", "We therefore fitted the spectrum with a model composed by a single power law plus three narrow ($\\sigma $ =0) emission lines, with rest-frame energies fixed at 6.4 keV (neutral iron, K$\\alpha $ ), 6.96 keV (hydrogen-like iron, K$\\alpha $ ), and 7.06 keV (neutral iron, K$\\beta $ ).", "The fit is good ($\\chi ^2/d.o.f.$ =116.2/120), but some wiggles are apparent in the residuals around the K$\\alpha $ neutral iron line.", "Letting the energy of that line free to vary, a significantly better fit is found ($\\chi ^2/d.o.f.$ =99.5/119), with a line energy of about 6.45 keV.", "No improvement, instead, is found letting the width of that line free to vary ($\\sigma <$ 63 eV), nor including the He-like iron line (upper limit to the flux of 5.5$\\times 10^{-7}$ ph cm$^{-2}$ s$^{-1}$ , corresponding to an equivalent width of 35 eV).", "Not surprisingly, given the narrowness of the line, no improvement is found with a relativistic profile (diskline model), and the inner disc radius is very large (hundreds of gravitational radii).", "Even in the relativistic line model, an intrinsic line energy at 6.45 keV is strongly preferred by the fit.", "The results are summarized in Table REF .", "The line fluxes are consistent within the errors to those derived from the XMM-$Newton$ spectra.", "While the hydrogen-like line is only marginally detected and the K$\\beta $ line is formally an upper limit, this of course does not mean that there is no significant line emission at that energy, but simply that the quality of the data is not good enough to accurately determine the parameters of the lines simultaneously.", "This is best seen in Fig.", "REF , where the contour plot of the fluxes of the two lines is shown: a simultaneous lack of emission for the two lines is in fact not allowed.", "Figure: Data and best fit model between 5 and 9 keV.", "The model is thebest fit one (see text), but without any emission lines.", "Line emissionaround 6.4 keV and 7 keV is apparent.Figure: Contour plot of the Kβ\\beta low-ionizationline flux vs. the hydrogen-like Kα\\alpha line flux." ], [ "The low ionization iron line", "The rest-frame energy of the low ionization K$\\alpha $ iron line is significantly larger than (and not consistent with) 6.4 keV, the energy for neutral iron.", "We are not aware of any major problem in the energy calibration of the XIS detectors and indeed, fitting the Mn K$\\alpha $ doublet and the K$\\beta $ in the calibration spectra, we found energies consistent with the intrinsic ones.", "Letting the energy of the line free to vary independently in the three XISs, similar values are found.", "We therefore conclude that the iron is truly ionized.", "According to House (1967), the best-fit energy corresponds to Fe XVII/XIX, where iron emission should be suppressed by resonant trapping (Ross & Fabian 1993, Matt et al.", "1993, 1996).", "To avoid this effect, the matter should be very optically thin, which is ruled out by the large EW (almost 200 eV), or very turbulent, so reducing the effective optical depth at the line core.", "Recent and more refined calculations by Garcia et al.", "(2011), however, show that suppression is not so efficient and that significant emission from the abovementioned ions is possible.", "In this case, we do not expect any K$\\beta $ line emission, because for those ions the M shell is no longer populated, and therefore the line emission at about 7 keV should be entirely due to the hydrogen-like iron line." ], [ "The high ionization iron line", "Setting the K$\\beta $ line to zero, the H-like line flux is 1.44($\\pm $ 0.59)$\\times $ 10$^{-6}$ ph cm$^{-2}$ s$^{-1}$ (see also Fig.", "REF ), with an equivalent width of about 100 eVAgain, assuming a relativistic profile a fit as good as the one with a gaussian line is obtained, but with a very large inner radius., and $\\chi ^2/d.o.f.$ =101.5/120.", "An upper limit of 35:100 to the ratio between helium- and hydrogen-like lines (as derived from the respective fluxes) implies high values of the ionization parameter, such that the EW of the hydrogen line should be, for a solar iron abundance, only a few eV with respect to the total continuum (Bianchi & Matt 2002).", "The required factor of (at least) 10-20 iron overabundance is rather extreme, even if not fully inconsistent with the EW of the low ionization line (which is a factor 2-3 larger than expected for solar abundances, in agreement with the calculations of Matt et al.", "1997 for optically thick matter with a similar iron overabundance).", "On the other hand, such large equivalent widths could be explained if the source is Compton-thick and the observed emission comes from reflection in two physically distinct mirrors, one at low and the other at high ionization, and no direct emission.", "To test this hypothesis, we fitted the spectrum with a power law (representing reflection from highly ionized matter), a Compton reflection component (to account for reflection from low ionized matter), plus two gaussian lines, one free to vary around 6.4 keV and the other fixed to 6.96 keV.", "The value of $R$ , the relative amount of Compton reflection with respect to the power law component, was fixed to 2, which is the value corresponding to the observed EW of the 6.4 keV line.", "This translates to a 2-10 keV flux from the high ionization reflector 7 times larger than the low ionization component, a quite unusual configuration in Compton-thick sources where it is the low ionization reflection which generally dominates.", "The fit is good ($\\chi ^2/d.o.f.$ =107.8/120), even if slightly worse than with a simple power law.", "The H-like line flux is now 0.97($^{+0.77}_{-0.60}$ )$\\times $ 10$^{-6}$ ph cm$^{-2}$ s$^{-1}$ , corresponding to an EW with respect to the high ionized reflection only of about 80 eV, a value consistent with what is expected from the large ionization parameter values implied by the absence of the He-like line (Bianchi & Matt 2002).", "Alternatively, this line may be emitted in a high temperature, optically thin thermal plasma.", "A 0.5-10 keV fit with the mekal model (iron abundance fixed to solar) instead of the 7 keV gaussian line is acceptable ($\\chi ^2/d.o.f.$ =485.5/476), giving a plasma temperature of 25($^{+23}_{-5}$ ) keV.", "The 2-10 keV luminosity of the thermal component is 1.2$\\times $ 10$^{41}$ erg/s.", "The emission measure, $\\sim \\int n_H^2dV$ , is about 7$\\times $ 10$^{63}$ cm$^{-3}$ .", "Assuming, for self-consistency, that the emitting matter is not optically thick to Thomson scattering, i.e.", "$n_HR\\sigma _T<$ 1 (with $R$ the radius of the emitting region, assumed spherical and with constant density), then a lower limit to $R$ of 7.4$\\times $ 10$^{14}$ cm (i.e.", "about 12 Schwarzschild radii if a value of the black hole mass of 2$\\times $ 10$^{8}$ solar masses is adopted, Dong & DeRobertis 2006), is obtained (corresponding to an upper limit to $n_H$ of 2$\\times $ 10$^{9}$ cm$^{-3}$ ).", "(If the mekal component is used instead of both the hard power law and the highly ionized iron line the fit is still good, the soft power law gets flatter, $\\Gamma $ =2.18($^{+0.13}_{-0.21}$ ) and the thermal plasma 2-10 keV luminosity is 2.2$\\times $ 10$^{41}$ erg s$^{-1}$ ).", "The origin and nature of this putative thermal plasma is however unclear.", "The accretion rate of NGC 3147 is low ($L/L_{Edd}\\sim 10^{-4}$ , Bianchi et al.", "2008), and therefore accretion could occur in a radiatively inefficient mode (ADAF, Narayan et al.", "1995; RIAF, Yuan et al.", "2003), where the X-ray emission is expected to be due to bremsstrahlung radiation.", "However, in such a mode the ion temperature is expected to be extremely high, and no visible line is expected.", "The thermal emission may be related to hot gas in a starburst region, but the implied luminosities are quite large, and there is no evidence for such an extreme starburst at other wavelengths.", "Alternatively, the plasma emitting region may be a compact one, in the innermost regions of the AGN.", "While there is so far no strong evidence of a significant contribution of such a plasma emission to the X-ray spectrum of Seyfert galaxies, it must be recalled that NGC 3147 is likely a peculiar source.", "Table: Best-fit parameters for the iron lines.Figure: XIS and HXD/PIN spectra fitted with the bestfit XIS-only model.", "A clear excess at hard X-rayis observe (left panel), which however disappears (right panel)once a 3.5% systematic increase in the background is included(see text for discussion)." ], [ "The hard X-ray emission", "One of the goals of the $Suzaku$ observation of NGC 3147 is to exploit its hard X-ray coverage to test the Compton-thick hypothesis for the unabsorbed X-ray spectrum of this Seyfert 2 galaxy.", "In fact, if the source is Compton-thick, the spectrum may appear unabsorbed because the primary emission below 10 keV would be completely hidden by the thick absorber, the observed emission being due to reflection from circumnuclear matter.", "In this scenario, the primary emission could emerge above 10 keV (if the absorber is not too thick, see e.g.", "Matt et al.", "1999) and then be observable with the HXD/PIN.", "As discussed in the previous section, the observed EWs of the iron lines are consistent with the Compton-thick hypothesis (although in a quite unusual configuration), even if other scenarios are also possible.", "In Fig.", "REF (left panel), the PIN spectrum is added to the already analyzed XIS spectra, and a clear excess with respect to the extrapolation of the XIS-only best fit model is apparent (the result is basically the same if a high temperature thermal plasma is used to account for the hydrogen-like iron line).", "No confusing source is known in the PIN field of view, according to the existing catalogs of bright X-ray sources.", "Assuming that the excess is due to the nuclear radiation piercing through a Compton-thick absorber, we included in the model the transmission and scattering components expected in such a case (Matt et al.", "1999).", "In practice, we added to the model described in the previous section (which in this case should represent the nuclear emission reflected mostly by highly ionized matter) the transmitted and scattered (by the torus) emission using the MYTorus model (Murphy & Yaqoob 2009; see also http://www.mytorus.com/).", "The inclination angle of the system was fixed to 90 degrees for simplicity.", "The values of the absorber column density and of the normalization (at 1 keV) of the intrinsic radiation are strongly correlated, and shown in Fig.", "REF .", "The normalization of the unabsorbed (reflected in this scenario) power law is about 4$\\times $ 10$^{-4}$ in the same units, which implies that at the 90% confidence level the ratio between primary and reflected components ranges from about 10 to 30, a value somewhat lower than usually found (e.g.", "Panessa et al.", "2006, Marinucci et al.", "2012).", "The 15-100 keV flux of the source is 1.2$\\times $ 10$^{-11}$ erg cm$^{-2}$ s$^{-1}$ .", "The 2-10/20-100 keV ratio is therefore more typical of Compton-thin sources, according to the diagnostic diagram of Malizia et al.", "(2007).", "This result is only marginally consistent with the upper limit of 1.3$\\times $ 10$^{-11}$ erg cm$^{-2}$ s$^{-1}$ to the 20-100 keV flux obtained by BeppoSAX (Dadina 2007) and with the upper limit of 7$\\times $ 10$^{-12}$ erg cm$^{-2}$ s$^{-1}$ to the 20-40 keV flux obtained by INTEGRAL on September/October 2009 (this work).", "It is inconsistent with the Swift-BAT observation of this source, which provides only an upper limit to the 15-150 keV flux of 4$\\times $ 10$^{-12}$ erg cm$^{-2}$ s$^{-1}$ (La Parola, private communication).", "It must be recalled that the PIN spectrum is obtained adopting the standard model for the background, which has an estimated average reproducibility of 3% at 1$\\sigma $ (Fukazawa et al.", "2009)The cosmic X-ray background is also highly variable, up to 10%, from place to place on scales of 1 sq degree (Barcons et al.", "2000).", "However, the CXB is only 5% of the total PIN background (Fukazawa et al.", "2009).. For individual observations, however, deviations of the background as high as 5% are sometimes observed (Pottschmidt, private communication).", "A background higher by 3.5% suffices to reduce the 15-100 keV flux to a value consistent with the Swift-BAT upper limit, and in this case no excess is apparent (Fig.", "REF , right panel).", "Unfortunately, this observation does not include a period of Earth occultation, so the real level of the background cannot be determined." ], [ "Conclusions", "Based on a short XMM-Newton observation, there were good arguments against the Compton-thick hypothesis for NGC 3147 (Bianchi et al.", "2008): the X-ray/[OIII] ratio is typical of unobscured objects, as well as the X-ray spectrum: in Compton-thick sources the 2-10 keV emission is usually much harder, being dominated by the Compton reflection component from low ionization matter and with a neutral iron line of about 1 keV EW.", "However, if reflection is from highly ionized matter the spectrum is steeper, but in this case emission from He- or H-like ions is also expected (Matt et al.", "1996).", "Our $Suzaku$ observation suggests the presence of a strong H-like line.", "Indeed, the overall line and continuum spectrum below 10 keV is consistent with a Compton-thick scenario in which reflection is mostly (but not entirely) due to a highly ionized mirror.", "Interestingly, if this is the case, the absorber may not be the torus, which in this source is claimed by Shi et al.", "(2010) to be seen almost face-on.", "Strong highly ionized reflectors in Compton-thick Seyfert 2s are rare, and even when observed, as in NGC 1068, their relative importance with respect to the low ionzation reflector is lower (Iwasawa et al.", "1997, Matt et.", "al.", "2004).", "NGC 3147 would therefore be rather extreme.", "It is worth noting that a larger-than-usual amount of highly ionized reflection may at least partly explain the unusually (for a Compton-thick source) high X-ray/[OIII] ratio, as well as the low primary-to-reflection ratio.", "On the other hand, it must also be stressed that alternative hypotheses, like e.g.", "large iron overabundance or a line emission from a very hot (temperature of about 20 keV), optically thin plasma cannot be ruled out.", "These characteristics would be unusual in Seyfert galaxies, but NGC 3147, if confirmed as a “true” Seyfert 2 galaxy, would be a peculiar source anyway.", "A strong argument against the Compton-thick scenario is provided by the observed flux variation between the $Suzaku$ and the XMM-$Newton$ observations, obtained 3.5 years apart.", "Flux variations on yearly time scales were also found comparing earlier observations with ASCA, BeppoSAX and Chandra.", "These variations indicate that we are looking at an emitting region with a size smaller than a parsec.", "On the other hand, the argument, albeit strong, is still not decisive.", "Variations on similar or even smaller time scales have been claimed for NGC 1068, the archetypal Compton-thick Seyfert 2 (Guainazzi et al.", "2000, Colbert et al.", "2002).", "Interestingly, in NGC 1068 the variations seems to concern the highly ionized reflector, which in that source is prominent only above about 2 keV (reflection from less ionized matter, possibly related to the Narrow Line Regions, dominates at lower energies).", "Moreover, there is evidence in some sources, most notably NGC 1365 (Risaliti et al.", "2005), of Compton-thick material very close to the central black hole.", "If the source is moderately Compton-thick, we expect that the direct emission would pierce through the absorber in hard X-rays.", "Using the standard background subtraction, an excess in the PIN is observed, which however disappears - becoming consistent with the SWIFT/BAT upper limit - assuming that the background is higher than usual by 3.5%, a not uncommon variation.", "Therefore, this observation is unfortunately not decisive in this respect.", "To conclude, after the new $Suzaku$ observation the “true” Seyfert 2 nature of NGC 3147 remains the most probable hypothesis.", "On the other hand, the ionized iron line is best explained in the “highly ionized reflector” Compton-thick scenario, which is still a viable option.", "Future sensitive hard X-ray observations with NuStar and/or high spectral resolution observations with Astro-H are needed to definitely settle the issue.", "Figure: Contour plot of the putative Compton-thick absorber column densityvs.", "normalization of the primary power-law." ], [ "Acknowledgements", "We thank the anonymous referee for her/his valuable suggestions.", "Katja Pottschmidt is gratefully acknowledged for her help on the PIN background, and Valentina La Parola for providing the SWIFT/BAT upper limit to the flux of the source.", "GM, SB and FP acknowledge financial support from ASI under grants ASI/INAF I/088/06/0 and I/009/10/0, FP also from grant INTEGRAL I/033/10/0.", "XB's research is funded by the Spanish Ministry of Economy and Competitivity through grant AYA2010-21490-C02-01." ] ]

[ [ "The baryonic Tully-Fisher Relation predicted by cold dark matter\n cosmogony" ], [ "Abstract Providing a theoretical basis for the baryonic Tully-Fisher Relation (BTFR; baryonic mass vs rotational velocity in spiral galaxies) in the LCDM paradigm has proved problematic.", "Simple calculations suggest too low a slope and too high a scatter, and recent semi-analytic models and numerical galaxy simulations typically fail to reproduce some aspects of the relation.", "Furthermore, the assumptions underlying one model are often inconsistent with those behind another.", "This paper aims to develop a rigorous prediction for the BTFR in the context of LCDM, using only a priori expected effects and relations, a minimum of theoretical assumptions, and no free parameters.", "The robustness of the relation to changes in key galactic parameters will be explored.", "I adopt a modular approach, taking each of the stand alone galaxy relations necessary for constructing the BTFR from up-to-date numerical simulations of dark halos.", "These relations -- and their expected scatter -- are used to describe model spirals with a range of masses, resulting in a band in the space of the BTFR that represents the current best guess for the LCDM prediction.", "Consistent treatment of expected LCDM effects goes a large way towards reconciling the naive slope-3 LCDM prediction with the data, especially in the range 10^9 M_sun < M_bar < 10^11 M_sun.", "The theoretical BTFR becomes significantly curved at M_bar > 10^11 M_sun, but this is difficult to test observationally due to the scarcity of extremely high mass spirals.", "Low mass gas-rich galaxies have systematically lower rotational velocity than the LCDM prediction, although the relation used to describe baryon mass fractions must be extrapolated in this regime.", "The fact that the BTFR slope derived here is significantly greater than in early predictions is a direct consequence of a corresponding increase in the expected sensitivity of baryon mass fraction to total halo mass." ], [ "The Tully-Fisher Relation in $\\Lambda $ CDM", "The Tully-Fisher Relation (TFR) was originally proposed as a correlation between 21 cm line width and optical luminosity in spiral galaxies [49], but it has subsequently become accepted that these are observational proxies for more fundamental physical properties, namely rotational velocity and stellar or baryonic mass [50], [33].", "The canonical approach towards deriving the TFR adopts the $\\Lambda $ CDM model of cosmology, in which the universe is flat and consists (by mass) of 4.6% baryons, 23% cold dark matter, and 73% dark energy [25].", "In this paradigm, visible galaxies are surrounded by roughly spherical halos of dark matter.", "Since it is total enclosed mass (baryonic plus dark) that determines a galaxy's rotational velocity at any particular radius, it is clear that the TFR describes a coupling between the baryons in a galaxy and the surrounding dark matter.", "The extent to which such a coupling is expected for cold, weakly interacting dark matter has recently been a hotly-debated subject, and consensus has not yet been reached [14], [30].", "The situation is complicated by the fact that many different forms for the TFR exist in the literature, using luminosities (at various wavelengths), stellar mass or baryonic mass as the dependent variable, and one of several different rotational velocity measures as the independent variable [2], [27].", "The earliest $\\Lambda $ CDM predictions for the TFR started with the expected relation between the total mass of a galaxy and its halo, $M_\\mathrm {vir}$ , and the corresponding characteristic velocity dispersion $V_\\mathrm {vir}$ , assuming virial equilbrium up to a radius $R_\\mathrm {vir}$ .", "Within this, the galaxy's average mass is taken to be a fixed multiple of the background mass density of the universe; simple manipulations then yield $M_\\mathrm {vir} \\propto V_\\mathrm {vir}^3$  [51].", "However, $M_\\mathrm {vir}$ and $V_\\mathrm {vir}$ cannot be measured directly: they are theoretical quantities pertaining to the putative dark matter halo.", "In order to make contact with the observed TFR, one must convert halo mass to observed mass or luminosity, and virial velocity to disk rotational velocity.", "The expectation was that the baryonic mass of a galaxy would be a fixed fraction of its total mass [51], [37], equal to the universal ratio of baryonic to total mass (now believed to be 0.17; [25]).", "Assuming further that galaxies' rotational velocities are proportional to their virial velocities, one derives a prediction for the baryonic mass TFR (BTFR): $M_\\mathrm {bar} \\propto V_\\mathrm {rot}^3$ .", "Such reasoning would provide a convenient explanation for a BTFR with a slope of 3 in log-log space.", "However, observational studies persistently find a slope greater than 3, typically in the range 3.5-4 [46].", "Several effects might be expected to invalidate the premises used to derive the slope-3 BTFR: There is no a priori reason why a galaxy's rotational velocity should be proportional to its virial velocity.", "A better approximation would be to use the density profile of a galaxy and halo to compute the overall rotation curve, and to take the velocity at some characteristic point on this as the TFR's independent variable.", "The maximum rotation velocity will be used in this paper.Selection effects arising from the use of the flat part of the rotation curve in some observational studies will be explored in Sect.", "REF .", "For example, dark halos are commonly parametrized by the NFW density profile [40]: $\\rho (r) = \\rho _\\mathrm {crit} \\frac{\\delta _0}{\\frac{r}{r_s} (1+\\frac{r}{r_s})^2},$ with $r_s$ the scale-length and $\\delta _0$ a characteristic overdensity.", "$\\delta _0$ may be written in terms of the concentration, $c = \\frac{R_\\mathrm {vir}}{r_s}$  [37]: $\\delta _0 = \\frac{\\Delta }{3} \\frac{c^3}{\\ln (1+c) - \\frac{c}{1+c}}.$ $\\Delta $ is the mean density of the galaxy, at the time of virialisation, in multiples of the critical density of the universe, $\\rho _\\mathrm {crit}$ .", "It will be set to 200 in this paper.$\\Delta $ determines the notional extent of the dark halo ($R_\\mathrm {vir}$ ), yet no consensus exists regarding exactly what value it should take (e.g.", "see [53]).", "Halos do not have a well-defined edge, but rather merge continuously with the background mass density of the universe.", "The ratio of maximum rotational velocity to virial velocity depends on the concentration [7], which is found in N-body simulations to be a function of virial mass [15], [53].", "After changing independent variable from virial velocity to maximum rotation velocity, this $M_\\mathrm {vir}$ -concentration relation is therefore expected to change the TFR's slope.", "So far, no mention has been made of the collapse of the baryons in galaxies into disks, yet both variables in the BTFR are properties of these disks.", "In $\\Lambda $ CDM, the particles in a newly-virialised galaxy acquire angular momentum through the action of cosmological torques [41].", "Dissipative processes cause the baryons to lose energy, and hence fall towards the centre, and their rotation forces them into a flattened disk.", "This modifies the rotation curve of the galaxy.", "Furthermore, the distribution of dissipationless dark matter is affected by baryonic collapse in a process known as adiabatic contraction (AC).", "The shells that make up the dark halo are pulled gravitationally towards the disk, with their final positions fixed by conservation of specific angular momentum [6], [19], [18].", "This effect is also expected to modify the galaxy's rotation curve, and hence the velocity used in the TFR.", "The proportion of a halo's mass that is in baryons is unlikely to be universal.", "In particular, baryons are expected to be more easily ejected (e.g.", "due to supernovae or stellar winds) from lower mass galaxies with shallower potential wells [16], [22].", "This would make the baryon mass fraction a rising function of total (virial) mass.", "As well as directly determining the baryonic mass that constitutes the dependent variable of the BTFR, this $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation governs the relative contributions of the disk and halo to the overall rotation curve, and hence affects the position of its maximum.", "Whilst the first two of these effects must be studied through numerical simulation of dark halos, the third is accessible observationally via weak lensing and satellite kinematics [26], [10].", "These techniques do indeed suggest that $M_\\mathrm {bar} / M_\\mathrm {vir}$ rises with $M_\\mathrm {vir}$ , but may only be used for galaxies with $M_\\mathrm {star} \\gtrsim 10^{9.5} M_{\\odot }$  [5] and must therefore be extended to lower mass to make contact with the majority of TFR measurements.", "A recent technique for doing this is called Halo Abundance Matching (HAM; [43], [5], [39], [21]).", "The premise of HAM is that the most massive halos harbour the most luminous galaxies, and thus galaxies observed in a particular survey may be uniquely assigned to dark halos formed in N-body simulations.", "In association with appropriate mass-to-light ratios for converting luminosity to stellar mass, and a relation between gas mass and stellar mass, the output of HAM is an $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation for the galaxies in the survey.", "Encouragingly, this technique agrees well with the direct methods at high $M_\\mathrm {star}$  [5].", "The way in which AC and the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ and $M_\\mathrm {vir}$ -concentration relations modify the “naïve” BTFR prediction is the primary concern of this paper.", "Before discussing the general approaches that have previously been employed in studies of this kind, it is important to describe an additional tension between observation and $\\Lambda $ CDM prediction that is exacerbated by the three effects listed above.", "It has been noted that the observed BTFR is very tight [50], [47], [32].", "Indeed, McGaugh argues in [31] that the scatter of individual spirals from a power-law TFR may be entirely accounted for by observational uncertainties, suggesting that the theoretical relation has zero intrinsic scatter.", "However, several factors are expected to create significant scatter in the $\\Lambda $ CDM prediction, including variations in the density profiles of dark halos caused by different mass aggregation histories (manifest in scatter in the concentration of halos of a given mass; e.g.", "see [13], [23]) and in baryon mass fractions [5], [37].", "These sources of scatter are described in more detail in Sect.", "REF .", "McGaugh argues that such effects will inevitably make the predicted scatter irreparably larger than that in the data, leading him to the conclusion that the BTFR is evidence against the $\\Lambda $ CDM paradigm itself.", "McGaugh proposes that the BTFR data are more consistent with the expectations of Modified Newtonian Dynamics (MOND), which implies a BTFR of the form $M_\\mathrm {bar} \\propto V_\\mathrm {rot}^4$  [36], [31].", "This appears to be in agreement with recent observations of gas-rich spiral galaxies in addition to many observations of star-dominated spirals ([32])." ], [ "The paths to an improved prediction", "Two main approaches exist for improving the “naïve” TFR prediction by taking into account the effects described above.", "The first is to perform a full numerical simulation of one or more spiral galaxies [42], [20], [45].", "Cosmological parameters and initial conditions are specified, and galaxies evolved stepwise through time according to general relativistic equations of motion and prescriptions for gas cooling, star formation, and stellar feedback.", "The baryonic or stellar masses of the resulting galaxies can then be measured in addition to their rotational velocities, enabling their positions on the TFR to be determined.", "In the future, this approach will likely provide the most complete and robust estimate for the TFR expected in $\\Lambda $ CDM.", "Currently, however, such simulations face several problems that limit their usefulness, including insufficient resolution for successful modelling of galaxy structure, insufficient computing power for the simulation of a statistically significant number of spirals, and uncertainties in our theoretical understanding of dissipative baryonic physics.", "These problems are manifest in significant discrepencies between simulated spirals and both the observed TFR [11] and expected galaxy formation efficiencies [21].", "Furthermore, this holistic approach risks masking the effects that individual correction factors have on the theoretical TFR: cosmological parameters are essentially fed into a `black box' describing a whole gamut of physical processes known to varying degrees of accuracy.", "Finally, approximations and uncertainties are introduced into the simulation that are not strictly required for prediction of (some versions of) the theoretical TFR.", "For example, star formation rates and thresholds are necessary for determining the stellar mass and luminosity of a galaxy – and hence its position on the luminosity or stellar mass TFR – but not the total baryonic mass.", "As another example, consider the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation.", "This may be predicted in numerical simulations from a prescription for the extent of baryon expulsion from a galaxy by supernovae and stellar winds, but may also be measured observationally (as documented in Sect.", "REF ).", "A disagreement would suggest a flaw in the simulations which would propagate into the predicted TFR, but the specific effect of this would not be readily visible.", "Using instead the observed $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation (or an extension thereof, such as that from HAM) would provide a more accurate prediction for the TFR, and would disentangle the result from any potential conflict between observed and simulated baryon mass fractions.", "The alternative to complete cosmological simulation is semi-analytic modelling [4].", "In this modular approach, the relationships between variables important for the TFR are determined empirically, where possible, and are otherwise given simple forms in accordance with the results of N-body simulations of dark halos.", "This allows the development of a TFR prediction with a minimum of assumptions and enables the individual effects of different components of the prediction to be investigated, thus helping to isolate the specific factors responsible for potential disagreement with observation.", "The disadvantage is that the various parameters used (those describing the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ and $M_\\mathrm {vir}$ -concentration relations, adiabatic contraction, and the halo density profile) are typically derived from different numerical simulations, which may use different initial conditions or halo virial parameters and therefore be marginally inconsistent.", "One of the first analyses of this nature was performed by [37], whose methodological framework provides a template for connecting the various pieces required for a prediction of the TFR.", "I adopt this template here.", "When the ingredient relations are not believed to be well-known, free parameters are often introduced which may be tuned to produce a desired result.", "An example of such an approach is [12], where agreement with an observational TFR data set is optimised by searching through a high-dimensional parameter space for the region that minimises a $\\chi ^2$ goodness of fit estimator.", "This method is useful for telling us what the existence of the TFR implies for the properties of spiral galaxies.", "However, one cannot be said to have explained the TFR if one has simply selected parameter values that reproduce it.", "A basic tenet of my methodology (which is described in more detail in Sects.", "REF and ) is that the introduction of free parameters in the context of the TFR is not necessary and can in some circumstances produce misleading conclusions.", "The reasons for this, drawing examples from the methods and results of D07, are as follows: It is necessary to assume some simple form for the relations in the first place in order to specify them with only a few parameters.", "However, these parametrizations themselves may lack physical motivation.", "For example, D07 use a power-law to describe the relation between $M_\\mathrm {bar}$ and $M_\\mathrm {vir}$ (a common approximation since the work of [37]), but recent Halo Abundance Matching studies observe a turnover at high masses [43], [39].", "These results will be inconsistent with those of D07 whatever parameter values are used.", "D07 tune their parameters to produce agreement with a power-law that is taken to fully describe the TFR.", "Yet no reason for adopting a power-law parametrization is inherent in the data – logically, such a form must instead be a property of a theoretical prediction.", "Indeed, [48] find agreement between the data and a theoretical line that is significantly curved, suggesting that it is not necessary for a prediction to be a power-law in order to be acceptable.", "Furthermore, the slope and intercept of the power-law fit to the TFR may not be very well defined: it may be possible to fit the TFR almost equally well using two power-laws with totally different slopes and intercepts [14].", "An observed TFR comprising data points with uncertainties is not uniquely equivalent to a power-law with uncertainties in its slope and intercept.", "Even if a power-law could fully describe the observed TFR, its parameters are obviously dependent on the data set used.", "There have been many studies into the TFR, all yielding slightly different results.", "Thus the results in D07 are strictly valid only for the particular observational data that they use to construct the TFR, intertwining theory and observation in a non-trivial way.", "In fact, different best fit slopes and intercepts can be given even for the same data set, by the use of different fitting methods (e.g.", "see MG12, Sect.", "2.6.1).", "Taking a particular power-law to be the correct description of the TFR requires the assumption that a particular fitting method is superior to all others.", "This is not the case: they are all just different approaches to an underdetermined statistical problem.", "It is not clear how changes to the parameters of the power-law TFR would affect the optimum values of the tunable parameters.", "Although it will be possible to find the global minimum of the $\\chi ^2$ value of the TFR fit, it may be that distant local minima exist in the many-dimensional parameter space with almost as low a $\\chi ^2$ value.", "Given the uncertain nature of the observed TFR, these could provide equally good descriptions of disk galaxies and may, in fact, be better physically-motivated.", "All of the free parameters are correlated in the context of the $\\chi ^2$ value that they produce, making it difficult to assign them unique and independent uncertainties.", "The relationships obtained after tuning the free parameters lack physical significance because they have become divorced from the motivations that originally existed for introducing them.", "At the end, the results must be compared with independent measurements to check consistency, but, to the extent that such independent measurements exist, one may as well use them from the outset." ], [ "Two important methodological issues", "In this section I discuss two further methodological issues that influence the nature of the TFR and the way in which it is presented.", "By clarifying them here I intend to illuminate the approach to TFR prediction that I believe to be most constructive." ], [ "Different versions of the Tully-Fisher Relation", "As alluded to in Sect.", "REF , several different versions of the TFR exist.", "In particular, one may use luminosity, stellar mass or baryonic mass as the dependent variable, and the significance that one attributes to these different TFRs depends on the perspective brought to the problem.", "McGaugh, for example, believes the baryonic TFR to be fundamental because of its power-law nature and low intrinsic scatter – and because it is such a relation that arises naturally in MOND ([31], MG12).", "Others (for example TG11 and [11]) consider the TFR to be a consequence of complicated physical processes occuring during hierarchical galaxy formation.", "As such, the TFR is no more significant than the luminosity TFR (LTFR).", "Since it is luminosity that is directly observed, these authors take the latter as the touchstone for their $\\Lambda $ CDM models.", "(The baryonic TFR is clearly of secondary importance in TG11, where its intrinsic scatter is not even considered.)", "However, regardless of the MOND-$\\Lambda $ CDM debate, it seems intuitively reasonable that mass should be more fundamental than luminosity when it comes to a relation with rotational velocity; luminosity depends on the details of stellar populations, which are not relevant for determining how fast a galaxy spins.", "Even in $\\Lambda $ CDM, one would expect $M_\\mathrm {vir}$ to be correlated more strongly with $M_\\mathrm {bar}$ than $M_\\mathrm {star}$ , and hence the BTFR is again the more “fundamental”.", "If one starts with a $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation yet wishes to construct a theoretical LTFR (as in D07, for example), two additional ingredients are required.", "First, a prescription for star formation is neccesary to convert baryonic mass to stellar mass.", "This is not required for predicting the BTFR – where stellar (and gas) masses are measured directly to constitute the observational data points – and unnecessarily introduces an extra degree of uncertainty.", "Put another way, comparison of the theoretical and observed LTFRs tests the hypothesis $\\Lambda $ CDM + star formation prescription, whilst comparing BTFRs tests $\\Lambda $ CDM alone.", "Disagreement with the expected LTFR may simply be due to errors in the star formation rate.", "The second required ingredient is a mass-to-light ratio to convert stellar mass to luminosity.", "Such a ratio is also required for consideration of the BTFR, but here for the observations as opposed to the theory.", "Reducing the number of assumptions and uncertain parametrizations that go into the theory allows it to be more directly compared with a range of observational data sets using different mass-to-light ratios.", "Nevertheless, there do exist uncertainties inherent in the BTFR that are absent from the LTFR.", "If gas masses in spirals are not well known, the BTFR may be subject to large systematic uncertainties that render impossible a fair comparison with any $\\Lambda $ CDM prediction.", "In this vein, [17] has argued that disagreement between BTFR observations and the $\\Lambda $ CDM prediction implies the existence of entirely unobserved baryons in spirals, which he believes to be in the form of warm ionised gas created by radiation from the inner stellar disk and the ionising cosmic background.", "Such claims are diametrically opposed to those of McGaugh, who argues that systematic errors such as missing gas mass should not be able to produce a tight BTFR correlation where none would otherwise exist.", "In other words, McGaugh uses his zero-intrinsic-scatter slope-4 BTFR (MG12, fig.", "1) as evidence that all baryonic mass has been accounted for: it would be extremely unlikely for omission of one baryonic component to turn the messy relation predicted by $\\Lambda $ CDM into the precise power-law relation that is observed.", "McGaugh also cites observational studies which find insufficient ionised gas to significantly affect the BTFR [1].", "In fact, it is argued in [31] that the uncertainties involved in determining the gas mass (due to imperfectly-known distances and uncertainties in measured line fluxes) are dwarfed by those associated with the conversion of luminosity to stellar mass (which stem from stellar population modelling and the Initial Mass Function).", "This suggests that of all the variants of Tully-Fisher Relation, the BTFR should provide the cleanest test of any theory.", "In any case, it is the BTFR that we must investigate if we are to make contact with McGaugh's claim that the slope and scatter of this relation constitute evidence against $\\Lambda $ CDM.", "One final point has great significance in the context of my work.", "Even if much ionised gas were to exist, this would affect neither the BTFR data points nor a theoretical prediction using baryon mass fractions obtained by Halo Abundance Matching.", "This is because the baryonic mass used in HAM is precisely the baryonic mass that we observe, and thus identical to that plotted in the BTFR.", "Use of an empirical $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation from HAM renders the issue of BTFR prediction entirely independent of the missing mass problem (observed baryon mass fractions significantly different from the “average” cosmological value of 0.17).", "The conjecture of [17] may shed light on the whereabouts of this missing mass, but has no ramifications for a BTFR built using HAM." ], [ "Inclusion of elliptical galaxies", "Both TG11 and [11] introduce ellipticals into their TFR, thereby turning it into a more general “luminosity-velocity” relation.", "By enlarging the scope of the “TFR” in this way, they are able to check their predictions against more observational data and explore the differences in the properties of early- and late-type galaxies.", "However, it is not clear to what extent the direct comparison of spirals and ellipticals in this way is appropriate.", "$\\sqrt{3} \\sigma _{los}$ is used as an elliptical equivalent of rotational velocity, where $\\sigma ^2_{los}$ is the stellar velocity dispersion along the line of sight to the galaxy.", "This assumes virial equilibrium and an isotropic stellar velocity distribution; the (poorly-known) geometry of ellipticals can cause differences of up to 20% [35].", "Further, one must be careful to measure $\\sigma _{los}$ at a radius comparable to the one at which the rotational velocities of spiral galaxies are measured.", "The introduction of elliptical galaxies clearly creates several uncertainties that need not be of concern if one is dealing specifically with the TFR.", "TG11 observe a difference in the position of spiral and elliptical galaxies in their mass-velocity plot (their fig.", "11).", "As they themselves note, this is likely to be due to a dependence on galaxy morphology of the relationships which act as ingredients of the theoretical BTFR, for example the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation.", "No distinction is made between spirals and ellipticals in the Halo Abundance Matching that TG11 use – both types of galaxy are assumed to have the same baryon mass fractions.", "Further, gas mass fractions are very much lower in ellipticals than they are in spirals (star formation has been completed in the former but is ongoing in the latter; e.g.", "see [43], fig.", "2), yet the empirical $M_\\mathrm {star}$ -$M_\\mathrm {gas}$ relation used in TG11 for both types of galaxy [3] is for spirals only.", "Thus their theoretical line strictly applies to neither spirals nor ellipticals on their own, but rather an amalgam of the two.Incorrect gas mass fractions will have a far larger effect on the BTFR than the LTFR.", "Part of the reason why this discrepency is not considered to be very important by TG11 is the subordinance of the BTFR to the LTFR in their work.", "Given the uncertainties introduced by ellipticals described in the paragraph above (and in light of the considerable amount of confusion that already exists surrounding the Tully-Fisher Relation itself) it seems sensible to remove ellipticals from the discussion.", "In the context of HAM, one should therefore match spiral galaxies in the observational survey with simulated halos that are believed would harbour spirals in reality.", "The latter may be identified by their environment and merger history, and matching of this type has, since the time of TG11's study, been performed in [43]." ], [ "Aim", "In view of the above issues, my approach will be the following.", "I aim to construct a theoretical BTFR, taking account, as completely as possible, of all relevent effects that are seen in up-to-date numerical simulations of dark matter halos.", "I will use only parameters and relations that are expected a priori from analytic and numerical studies of dark halos, and will not include any tunable quantities.", "Where applicable, these relations will be specifically for spirals.", "In addition, I will propagate into the BTFR the expected scatter in all relations that I use, investigate the effect of adiabatic contraction, and determine the robustness of my results to changes in the halo density profile.", "The result will be the expected BTFR in the $\\Lambda $ CDM paradigm, which may be fairly compared to a range of observational data sets.", "Two such data sets will be used to qualitatively illustrate the degree of agreement between theory and observation.", "Finally, I will investigate the extent to which the prediction would be modified by modelling two selection effects in the observational data.", "The structure of this paper is as follows.", "In Sect.", ", I describe the technical details of my calculation of the predicted BTFR.", "Section  explores the issues involved in selecting appropriate observational data for comparison with theory, and describes the data sets that will be used.", "The results will be presented and discussed in Sect. .", "Section  contains a comparison of my findings with those of three recent studies in the literature.", "Finally, Sect.", "summarises my conclusions and suggests fruitful avenues for further work." ], [ "Method", "The basic semi-analytic methodology employed in this work is adapted from M98 (Sect.", "2).", "As the starting point, I take a spiral galaxy with a particular stellar mass.", "The $M_\\mathrm {vir}$ -$M_\\mathrm {star}$ relation from the Halo Abundance Matching performed in RP11 (eq.", "5) is then used to calculate the virial mass of the surrounding dark halo.", "The corresponding gas mass is calculated from the $M_\\mathrm {star}$ -$M_\\mathrm {gas}$ relation described by RP11 eq.", "2, in order to derive an analytic form for the HAM $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation (RP11, fig.", "5).", "The total halo mass is then used to derive the virial radius, virial velocity and scale-length of the halo (assumed for the moment to have an NFW density profile; see M98, eq.", "2).", "Given these quantities, I find a self-consistent solution for the scale-length ($R_d$ ) of the spiral galaxy's disk (assumed exponential) that is formed by baryon collapse, and the total rotation curve.", "$R_d$ is found via calculation of the total energy of the halo and angular momentum of the disk (an integral over the rotation curve), and comparison with a dimensionless spin parameter $\\lambda $ (see M98, eq.", "9).", "This parameter is used because its distribution is well-constrained by numerical simulations, a fact that will become important in the discussion of scatter in Sect.", "REF .", "Before this, $\\lambda $ will be set equal to its mean value of 0.05.", "The response of the dark halo to disk formation follows the adiabatic contraction prescription laid out in [18].", "$R_d$ depends on the concentration of the halo, which will be derived from $M_\\mathrm {vir}$ according to the results of [53], fig.", "16.", "Starting with an estimate for $R_d$ , the adiabatic contraction equations may be solved[18] demonstrates that no single set of parameter values reliably specifies the effect of adiabatic contraction in all cases, and recommends considering a range.", "All plausible values are found to produce almost identical TFRs, so for simplicity I will use $A_0$ = 1.6, w = 0.8 [18] throughout this work.", "using the Newton-Raphson method to determine the final density profile of the dark halo and thence the rotation curve (a sum in quadrature of disk and halo contributions).", "The latter may be used to find a new estimate of the disk's angular momentum, and hence of $R_d$ .", "This updated scale-length is then adopted and the process iterated until the $R_d$ value generated from the rotation curve is identical to the value that produces that curve.", "(My convergence requirement is that $f_r$ [see M98, eq.", "29] change by less than 0.1% in the final iteration; reducing this is found to make a negligible difference to the results.)", "The self-consistent rotation curve is then sampled at 1000 radii logarithmically evenly spaced between $0.0001 \\times R_\\mathrm {vir}$ and $0.4 \\times R_\\mathrm {vir}$ , and 100 radii between $0.4 \\times R_\\mathrm {vir}$ and $R_\\mathrm {vir}$ , in order to determine the maximum rotation velocity of a model galaxy with this mass.", "Finally, this procedure is repeated for 200 stellar masses logarithmically evenly spaced in the range $10^5 M_{\\odot } \\le M_\\mathrm {star} \\le 10^{12} M_{\\odot }$ , allowing the predicted BTFR (maximum rotational velocity vs baryonic mass) to be plotted." ], [ "Observational data", "There have been numerous studies measuring the baryonic mass and rotational velocity of actual spiral galaxies, all producing results that vary to a greater or lesser extent from one to another.", "Reasons for this include (but are not limited to) differences in the following: 1) Wavelength used for luminosity measurements; 2) Mass-to-light ratio used for converting luminosity to stellar mass; 3) Method for accounting for mass of atomic, molecular and ionised gas; 4) Method for measuring rotational velocity (e.g.", "using line width or a resolved rotation curve); 5) Radius at which the rotational velocity is measured (for rotation curve studies); 6) Type of galaxy survey; 7) Criteria for deciding exactly which spirals should be included.", "The precise effects of variations in these factors are poorly-understood, but are certain to produce non-trivial systematic shifts in the resulting BTFRs (e.g.", "see [14], p. 3 and refs.", "therein).", "The nature of the observational data used for comparison with a BTFR prediction is therefore crucial for determining whether or not the prediction appears successful.", "This makes imperative the preliminary establishment of rigorous criteria for selecting observational data sets which have maximum compatibility with a given theoretical model.", "The first step in this endeavour is to investigate the approximations and assumptions that go into the theory.", "The Halo Abundance Matching used to derive the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation contains both a prescription for calculating stellar mass from luminosity, and gas mass from stellar mass.", "Thus the resulting BTFR prediction should be compared with observational data employing the same mass-to-light ratios and techniques for gas mass measurement.", "In addition, I use the maximum rotational velocity to describe my model galaxies; observed galaxies should use a similar rotation measure.", "By using only observations with identical assumptions to a theoretical prediction, unknown systematic effects associated with variation in the parameters listed above will be eliminated.", "As long as the observations are consistent among each other, cuts may be applied to the model galaxies to enhance compatibility with the data.", "In principle (if the observational uncertainties were sufficiently well-known) a hypothesis test such as $\\chi ^2$ could then provide a quantitative measure of the extent of agreement between theory and observation.", "From this perspective, it may be counterproductive to bin together the results of many different galaxy surveys without regard of the different assumptions and approximations on which they are based (as done for example in TG11, fig.", "11).", "The advantage of comparison with a large quantity of observational data is offset by a reduction in quality in the sense that the observational details of the surveys are lost.", "Further undesirable features of this method of data presentation are the following: The error bars show the scatter of the data points in each bin around their mean.", "Information concerning the uncertainties on the individual measurements is entirely lost, making it impossible to judge whether or not the points could actually be consistent with the theoretical line.", "It is not known how many data points are in each bin.", "Some bins may contain many more than the others, and should therefore be weighted more heavily when comparing to the theoretical line.", "This cannot be taken into account in a visual assessment.", "The mass and velocity range of the data points in each bin is unknown – the standard deviation is the only measure of dispersion that is retained.", "Projecting each data point onto one of a discrete set of rotational velocity values causes loss of accuracy.", "I will use two observational data sets for comparison with the BTFR prediction generated by the methodology of Sect. .", "I do not intend this to constitute a definitive test of theory, but rather a rough qualitative assessment of my results: a comprehensive literature review to find all data sets consistent with the principles above is beyond the scope of this paper.", "Indeed, my intention is to produce a prediction that includes a minimum of theoretical assumptions and may therefore be compared to a range of data sets as others see fit (and indeed as BTFR observations improve and proliferate).", "In the next two sub-sections I will describe the data that I use, and my reasons for selecting them." ], [ "Gas-rich spirals", "As discussed in Sect.", "REF , a major (if not the dominant) source of uncertainty in BTFR measurements is the mass-to-light ratio used to determine $M_\\mathrm {star}$ .", "Thus it makes sense to prioritise gas- as opposed to star-dominated spirals, for which the relative contribution of the uncertainty in $M_\\mathrm {star}$ to the error budget is low.", "MG12 presents a compilation of 47 spirals which have $M_\\mathrm {gas} > M_\\mathrm {star}$ and which moreover satisfy several quality criteria including a resolved rotation curve and consistent optical and HI inclinations.", "McGaugh uses the flat part of the rotation curve as the rotational velocity measure – this can be approximately accounted for in my modelling by discarding galaxies with non-flat rotation curves (see Sect.", "REF ).", "A further discrepancy between McGaugh's approximations and those of Sect.", "is that RP11's Halo Abundance Matching uses mass-to-light ratios derived from the [52] stellar mass function, whilst McGaugh assumes a Portinari population synthesis model and Kroupa Initial Mass Function.", "However, the gas-dominated nature of the MG12 galaxies renders them relatively insensitive to variations in the mass-to-light ratio." ], [ "The GHASP survey", "No gas-dominated spirals exist with $M_\\mathrm {bar} \\gtrsim 10^{10.5} M_\\mathrm {\\odot }$ .", "To extend the BTFR to higher-mass (e.g.", "to better constrain its slope), the data from [46] will be used.", "This is derived from a homogeneous galaxy survey (GHASP) undertaken using a scanning Fabry-Perot interferometer in France.", "Besides homogeneity, the GHASP survey offers the advantages of a completely resolved 2D velocity field (enabling rotation curves to be determined without uncertainties concerning position angle or inclination), observation of close galaxies (minimising distance uncertainties), and the adoption of $V_\\mathrm {max}$ as the rotational velocity measure (in accordance with my theoretical model).", "Mass-to-light ratios are obtained using a Bell population synthesis model, which is shown in RP11's fig.", "1 to produce a similar galaxy stellar mass function to the [52] results used in the Halo Abundance Matching." ], [ "The baryonic Tully-Fisher Relation predicted by $\\Lambda $ CDM", "In this subsection, scatter in the BTFR prediction is entirely suppressed.", "Fig.", "REF compares three theoretical BTFRs with the observational data described in Sect.", "(redOne of the galaxies from MG12 (DDO 210) was removed due to its extremely low baryonic mass.", "and blue points).The green data points are from TG11.", "A comparison with these will be presented in Sect.", "REF .", "In each case, an NFW halo density profile is assumed.", "The black line uses the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation for spirals only (RP11 table 1, column 3) and includes the effect of adiabatic contraction, and is therefore the primary result of this subsection.", "This prediction is contrasted with two others, one for which adiabatic contraction is switched off (i.e.", "halo unaffected by disk formation; cyan) and another which uses the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation for ellipticals (magenta).", "This takes $M_\\mathrm {vir}$ to be calculated from $M_\\mathrm {star}$ using the results in column of 4 RP11's table 1, and $M_\\mathrm {gas}$ from the red line in RP11, fig.", "2, and is displayed to illustrate the extent to which contamination of the HAM $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation with elliptical galaxies may be expected to distort the BTFR (see also Sect.", "REF ).", "Some observations from this plot: Figure: The black (magenta) line uses the M vir M_\\mathrm {vir}-M star M_\\mathrm {star} and M star M_\\mathrm {star}-M gas M_\\mathrm {gas} relations for spirals (ellipticals) – see RP11, eq.", "5 and fig.", "2.", "The cyan line is the analogue of the black line but with AC switched off.", "Red data points are from , blue from  and green from .", "The M vir M_\\mathrm {vir}-M bar M_\\mathrm {bar} relation must be extrapolated below the dotted horizontal line.", "Each line is almost a power-law up to $M_\\mathrm {bar} \\approx 10^{10.5} M_{\\odot }$ , beyond which it becomes much shallower.", "This is clearly due to the nature of the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation, in which a similar break occurs (RP11, fig.", "5, top right panel).", "At high virial masses, $M_\\mathrm {bar}$ rises very slowly with $M_\\mathrm {vir}$ .", "Thus the increase in rotational velocity is large relative to that in baryonic mass and the BTFR flattens out.", "The curvature becomes very obvious around $M_\\mathrm {bar} \\approx 10^{11} M_{\\odot }$ .", "However, this is approximately where the observational data ends.", "The data does not, therefore, provide compelling evidence for either a continuation of the power-law from lower masses (as expected for example in MOND) or for the decrease in slope predicted here.", "Observations of higher-mass spirals would be needed to decide between these two possibilities.", "There is quite good agreement with the entire GHASP sample.", "However, the predictions lie systematically below the data for the gas-dominated spirals.", "In this respect, the black line gives a better fit than the magenta line, suggesting that a failure to discriminate between spirals and ellipticals will reduce agreement with the data at low baryonic mass (as might be expected from the fact that the observations are specifically of spirals).This issue will be discussed further in Sect.", "REF .", "It is crucial to note, however, that the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation from RP11 must be extrapolated for $M_\\mathrm {bar} \\lesssim 10^{8.9} M_{\\odot }$ (see RP11, fig.", "5), increasing the uncertainty in the theoretical prediction in this region.", "This threshold is shown by the horizontal dotted line.", "The effect of AC is to slightly reduce $M_\\mathrm {bar}$ for spirals with $10^2~\\mathrm {km~s^{-1}} \\lesssim V_\\mathrm {rot} \\lesssim 10^{2.5}~\\mathrm {km~s^{-1}}$ .", "This may marginally improve agreement with the observational data.", "Fig.", "REF shows the results when the density profile of the model halos is switched from NFW to Einasto [15], Moore [38], or Burkert [8].", "To construct these lines, eqs.", "19, 21 and 23 of M98 were modified according to the change in $\\rho (r)$ .", "For the Moore and Burkert profiles, an identical $M_\\mathrm {vir}$ -concentration relation to the one used for NFW was adopted.Concentration depends on halo scale-length, which strictly is only defined for the NFW profile.", "For alternatives, the scale-length will be taken as the radius at which the density becomes proportional to $r^{-2}$ .", "Concentrations for the Einasto profile were derived from $M_\\mathrm {vir}$ using eq.", "6 of [15], a study working explicitly with this density profile.", "The Einasto shape parameter $\\alpha $ was calculated using [15], fig.", "2.", "The curvature in the BTFR prediction is largest using the Einasto profile and smallest using Burkert, but, in general, changing the density profile has relatively little effect.", "Whilst the Einasto result fits the data somewhat better at the highest and lowest baryonic masses, the NFW line appears near-optimal in the range $10^9 M_{\\odot } < M_\\mathrm {bar} < 10^{11} M_{\\odot }$ .", "Figure: Data points as in Fig. .", "See text for further details." ], [ "Intrinsic scatter", "Three sources of scatter in the theoretical BTFR are variability in halo spin, concentration and baryon mass fraction.", "The spin parameter $\\lambda $ is determined by cosmological torques and inter-galactic tidal interactions [13], and therefore varies with environment.", "Numerical simulations find $\\lambda $ following a log-normal probability distribution with mean 0.05 and standard deviation 0.5 [37].", "Galaxies of the same mass may have different concentrations (differently shaped density profiles) due to differences in their merger history and hence epoch of virialisation.", "Although no uncertainty is quoted in the source of the $M_\\mathrm {vir}$ -concentration relation used here [53], other studies have indicated that the concentration follows a log-normal distribution with standard deviation roughly independent of $M_\\mathrm {vir}$ and equal to 0.18 [23], [7].", "Scatter in the HAM $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation comes from a variety of sources (RP11, Sect.", "3.1.1; see also [5]).", "RP11 quote the scatter in $\\log M_\\mathrm {bar}$ as 0.23 dex at all virial masses.", "To model the effect that these sources of scatter have on the intrinsic scatter of the BTFR, I make the following modifications to the methods used to generate the black line of Fig.", "REF : 1) The number of $M_\\mathrm {star}$ values is increased from 200 to 500.", "2) For each $M_\\mathrm {star}$ , 500 different values of $\\lambda $ , $M_\\mathrm {bar}$ and concentration are randomly drawn from their respective probability distributions.", "The mean of the log-normal $M_\\mathrm {bar}$ and concentration distributions are the values of these parameters used in Sect.", "REF .", "3) A 150$\\times $ 150 element grid is constructed spanning the range $10^6~M_{\\odot } < M_\\mathrm {bar} < 10^{12}~M_{\\odot }$ , $10^{0.7}~\\mathrm {km~s^{-1}} < V_\\mathrm {max} < 10^{2.7}~\\mathrm {km~s^{-1}}$ .", "For each of the 500$\\times $ 500 = 250 000 input parameter sets, $M_\\mathrm {bar}$ and $V_\\mathrm {max}$ are calculated and identified with a particular element of this grid.", "The final number of points in each element is then outputted and colour-coded for comparison with the observational data.", "The resulting contour plot constitutes Fig.", "REF , and shows the band in which observational points are predicted to lie, taking into account the expected intrinsic scatter.", "A little under 250 000 model galaxies lie within the mass and velocity ranges shown in this figure, and a grid element is coloured if it contains at least 10 galaxy points.", "Thus it should be rare, in the $\\Lambda $ CDM model presented here, for galaxies to lie outside the coloured band.", "As with the black line in Fig.", "REF , there is good agreement with the data at $M_\\mathrm {bar} \\gtrsim 10^9 M_{\\odot }$ and poor agreement below (where the HAM $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation must be extrapolated).", "The amount of intrinsic scatter looks to be in reasonable accord with that in the data.", "Figure: Dependence of model galaxy count on region of the M bar M_\\mathrm {bar}-V max V_\\mathrm {max} plane, following the procedure of Sect. .", "The contours are at 10, 50, 100, 150, 200, 250, 300, 350, and 400.An additional quantity required for calculation of the maximum rotation velocity of each model galaxy is the fraction of the galaxy's total angular momentum that belongs to the baryons ($j_d$ ; see M98, eq.", "8).", "Thus far, $j_d$ has been set equal to $M_\\mathrm {bar}$ / $M_\\mathrm {vir}$ , the fiducial assumption of M98.", "However, this paper also cites evidence from numerical simulations (in Sect.", "2.2) that this may overestimate $j_d$ .", "To test the robustness of my results to a decrease in baryon angular momentum fraction, I plot in Fig REF the analogue of Fig.", "REF but with all $j_d$ values halved.", "The effect is seen to be a slight downward shift of the predicted band at $M_\\mathrm {bar} \\gtrsim 10^{9.5} M_{\\odot }$ (i.e.", "increased curvature) in addition to a small increase in scatter in this region.", "This decreases agreement with the high-mass data points, but to a sufficiently marginal extent for plausible changes in $j_d$ to be considered inconsequential.", "Figure: As Fig.", ", but with the baryon angular momentum fractions of all model galaxies halved." ], [ "Selection effects", "In Sect.", "REF , all allowed halo spins, concentrations and baryon mass fractions were used to construct model galaxies, and contributed to Fig.", "REF .", "However, it is unlikely that all such parameter values would yield spiral galaxies whose properties astronomers would measure and plot on the BTFR.", "Reasons for this fall in one of two categories: 1) Halos with extreme properties may be unstable and hence never form disk galaxies; 2) Astronomers typically require galaxies to fulfill certain selection criteria to be included in the TFR, with the intention of minimizing systematic uncertainties and ensuring some degree of consistency within the sample.", "A proper comparison of a theoretical BTFR with observational data should limit the input parameter space to the regions which generate stable spiral galaxies with properties passing all selection requirements of the data (see also Sect. ).", "In this section, I explore two such selection effects, one from each of the categories listed above.", "It is theoretically expected that low $\\lambda $ will make a spiral galaxy prone to the development of a bar instability, potentially causing either the total disruption of the disk or its transformation into an irregular galaxy which would not be plotted on the TFR [9].", "Removing the unstable model galaxies from the contour plot may therefore boost compatibility with the observational data.", "A simple prescription for the stability threshold is given by M98 (Sect.", "3.2) in terms of the mass and scale-length of the disk, and maximum rotational velocity of the halo.", "Since these quantites are calculated for each model galaxy as part of the BTFR calculation, removing unstable galaxies in this way is straightforward.", "Of the $\\sim $ 250 000 model galaxies, only 5 169 are rejected using the stability threshold $e_{m,crit}$ = 1, 10 475 for $e_{m,crit}$ = 1.1 and 17 512 adopting the the upper limit deemed plausible by M98, $e_{m,crit}$ = 1.2 (a larger value causes rejection of halos with larger spins; see M98 eqs.", "35 and 37).", "Thus we expect that the effect on the BTFR of rejecting galaxies unstable by this criterion is small, indicating that most of the model galaxies are in fact stable.", "This is illustrated in Fig.", "REF , a replica of Fig.", "REF but with galaxies unstable to bar formation at the $e_{m,crit}$ = 1.2 level removed.", "The majority of unstable galaxies have baryonic masses in the range $10^{10}$ – $10^{10.5} M_{\\odot }$ , and the theoretical intrinsic scatter in this region is minorly reduced by excluding them.", "At lower-masses, where disagreement with the data is most pronounced, there is virtually no effect.", "Figure: As Fig.", ", but excluding model galaxies with V max /GM bar /R d <1.2V_\\mathrm {max} / \\sqrt{GM_\\mathrm {bar}/R_d} < 1.2.", "See text and M98 (Sect.", "3.2) for further details.An important observational issue is the point on a galaxy's rotation curve at which velocity is measured.", "MG12 adopts $V_\\mathrm {flat}$ (due to earlier work suggesting that this minimizes the scatter in the TFR), and defines a rotation curve as flat if the difference between the velocity at 3 disk scale-lengths and at the last measured point of the rotation curve is less than 15% [44].", "This requirement is applied to many galaxies in the gas-rich data set, which are therefore not directly comparable to the data points for model galaxies with parameter values such that their rotation curves do not plateau.", "Here I apply a somewhat modified requirement to remove such galaxies, which should nevertheless produce similar results: the difference between the maximum velocity and that at 3 disk scale lengths must be less than 15%.The reason why McGaugh's flatness criterion cannot be applied directly is that the “last measured point” is not defined for the model galaxies.", "To improve the theoretical flatness requirement, one would require a radius marking the end of the observationally-resolved rotation curve.", "This radius would depend on the observational techniques and instruments used in the galaxy survey under consideration.", "Only 200 model galaxies are excluded for having non-flat rotation curves, resulting in a negligible change to the BTFR band.", "Since rotation curves are typically observed to be flat in the outer regions of spiral galaxies, this gives some confidence that the galaxies produced by the methods and relations of Sect.", "are realistic.", "These results also support the claim of [32] and [44] that $V_\\mathrm {flat}$ and $V_\\mathrm {max}$ are typically very similar." ], [ "Comparison with the literature", "In this section I relate my results to those of TG11, D07 and MG12." ], [ "Trujillo-Gomez et al. (2011)", "TG11 use the Bolshoi N-body simulation in association with baryon mass fractions from HAM to give a prediction for the BTFR which exhibits considerable curvature (their fig.", "11).", "This presumably derives from the precise $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation that is adopted (although this is not actually visible in TG11).", "The fact that no such curvature is evident in the data is pointed out in MG12, where it is used to quickly dismiss TG11's prediction; yet TG11 claim to find a “good fit” to the observational data.", "The extent to which curvature militates against a predicted BTFR depends on the status that the relation has within the paradigm under consideration.", "For TG11, the BTFR is simply an observed empirical relation that arises in some complicated way via galaxy formation, growth and merger history in addition to baryonic processes.", "The result is that, from this perspective, there is no reason for the BTFR to be linear in log-log space, and it is a coincidence if it approximately is.", "As long as the curved prediction is more or less consistent with each data point, the agreement may be considered adequate.", "The situation is very different, however, for McGaugh, who holds the BTFR to be a relation arising directly from a fundamental force law (MOND).", "From this persepective, the perfect linearity of the BTFR is one its defining characteristics, and hence inability to reproduce this is a major shortcoming.", "Furthermore, it can be seen in Fig.", "REF (or by comparison of MG12's fig.", "5 with TG11's fig.", "11) that although a power-law fit to both MG12's and TG11's data sets would have a best-fit slope of around 4, the intercept in the TG11 data is significantly lower.", "This increases agreement with their prediction.", "Aside from the issue of whether or not the linearity of the BTFR is one of its fundamental properties, it is clear from fig.", "11 in TG11 that the data points do lie above the predicted line at both low and high masses: the curvature does appear to be too great.", "This may be partially explained with the help of Fig.", "REF , in which the magenta line shows the analogue of the theoretical BTFR for elliptical galaxies.", "Contamination of TG11's HAM $M_\\mathrm {vir}$ -$M_\\mathrm {star}$ relation with ellipticals would be expected to give a result between the red and magenta lines, which might exhibit considerable curvature.However, the fact that spirals are far more abundant than ellipticals at low masses suggests that the HAM $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation for all galaxies – as used by TG11 – should be much closer to the spiral line than the elliptical line.", "One might also expect to be able to explain the behaviour at the high-mass end of fig.", "11 in this way.", "TG11's prediction lies below the data points describing spirals but is in agreement with those describing ellipticals.", "However, my Fig.", "REF shows that exchanging the expected spiral baryon mass fractions with those for ellipticals has very little effect on the predicted BTFR at high mass (if anything, the relation for ellipticals lies above that for spirals).", "A possible explanation is that not only does the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation depend on galaxy morphology, but other relations important for calculation of the BTFR do too (e.g.", "the $M_\\mathrm {vir}$ -concentration relation).", "This reinforces the point made in Sect.", "REF that the differences between the fundamental parameters of spirals and ellipticals are not sufficiently well-known to permit a fair comparison between them in baryonic mass-rotational velocity space.Of course, the theoretical difference between spirals and ellipticals in this space is known only to the extent that the halos hosting spirals and ellipticals may be accurately distinguished (as described in RP11, Sect.", "2.2.1).", "Methods for achieving this are still in their infancy and may be significantly refined in the future.", "A further difference between the Abundance Matching of TG11 and RP11 is that the former uses only the galaxies in their TFR sample, whilst the latter matches the theoretical halo mass function to the entirety of the Sloan Digital Sky Survey.", "Thus the $M_\\mathrm {vir}$ -$M_\\mathrm {star}$ relation used by TG11 is built from around 1 000 data points, whilst that described in RP11 contains of order 500 000 and should therefore enjoy greater precision.", "Underneath this difference runs the question of whether the BTFR is a fundamental relation applicable to spirals of any mass, composition and location (a view championed by McGaugh), or whether it is a product of complex evolution histories in a $\\Lambda $ CDM universe.", "In the latter scenario, the BTFR may well be different for different sets of galaxies – which may have been subject to significantly different conditions during their evolution – and hence it would be reasonable to limit the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation to those galaxies used in the TFR.", "However, if all types of spiral lie on the same BTFR, it is unsatisfactory to “explain” the BTFR for only a small subset of these.", "A universal $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation (as approximated for example by the results of RP11) is necessary to give the theoretical BTFR universal applicability." ], [ "Dutton et al. (2007)", "D07 use a semi-analytic approach, also based on M98, to investigate the LTFR in the $\\Lambda $ CDM paradigm, using simple parametrizations of the required relations which include tunable parameters.", "These are selected to fit the size-lumunosity (RL) relation in addition to the LTFR.", "D07 find that the slope and intercept of the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation can be tuned to fit the RL relation without significantly affecting the TFR.", "While it may be true that baryon mass fractions are of little importance to the luminosity TFR (the primary concern of D07), they are clearly of great importance to the baryonic TFR.", "For example, the BTFR described by the black line in Fig.", "REF is significantly steeper than early predictions for the BTFR adopting fixed baryon fractions (eg.", "[37], [7]) primarily because of the steep $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation predicted by HAM.", "Thus we see that successful prediction of the BTFR does not necessarily follow from successful prediction of the LTFR (cf.", "Sect.", "REF ).", "In a similar vein, D07 claim that scatter in baryon mass fraction does not significantly affect the scatter in the LTFR, and, in view of the over-budgeted scatter in the RL relation, they set this to zero.", "Not only does this lack physical motivation (if the low observed baryon mass fractions are due to expulsion of baryons by astrophysical phenomena, one would clearly expect statistical fluctuations between galaxies), but it again neglects the fact that scatter in baryonic mass has a large effect on the BTFR.", "Indeed, this contributes a significant fraction of the scatter in Fig.", "REF .", "The results in D07 are specific to the luminosity TFR, and many of the conclusions are unlikely to hold if the BTFR were considered also.", "Finally, D07 argue that adiabatic contraction must be replaced by expansion in order to successfully reproduce the LTFR and galaxy luminosity function with a realistic Initial Mass Function.", "Although I make no claim to “successfully reproduce” the BTFR data with my model, I do notice from Fig.", "REF that the effect of adiabatic contraction is very small and does, if anything, increase agreement with the data.", "D07's result may be specific to the parametrizations and parameter values produced by their particular optimisation procedure." ], [ "McGaugh (2012)", "McGaugh argues that the failure of $\\Lambda $ CDM models to reproduce the low intrinsic scatter and high slope ($\\sim $ 4) of a power-law fit to the BTFR without excessive fine tuning constitutes evidence against the paradigm.", "Given the existence of a Modified Gravity Model (MOND) which naturally explains these features, McGaugh further claims that the BTFR militates against the existence of dark matter itself.", "In this subsection I discuss the way in which my results impinge on two aspects of this argument.", "In MG12, McGaugh claims that natural $\\Lambda $ CDM predictions have a slope of approximately 3.", "This was indeed the case for the “naïve” derivation of Sect.", "REF in addition to several subsequent studies that introduced a dependence of halo concentration on $M_\\mathrm {vir}$  [7].", "However, the slope is significantly steepened by replacing the constant baryon mass fractions used in these works by more “realistic” values, such as those derived from Halo Abundance Matching.", "For example, the low-mass end of the black line in Fig.", "REF has a slope in excess of 5.", "On the one hand, the HAM $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation could be said to be a “natural” $\\Lambda $ CDM prediction in the sense that the halo mass function used in the Abundance Matching is determined from N-body simulations of dark halos in a $\\Lambda $ CDM cosmological background.", "On the other hand, a theoretical understanding of the way in which the appropriate number of baryons is ejected during galaxy formation (and where they went) remains wanting [34], [29].", "This is a more fundamental problem for $\\Lambda $ CDM than failure to accurately reproduce the observed BTFR.", "Indeed, it is possible that the remaining discrepency between the observed and predicted BTFRs in Figs.", "REF and REF is entirely attributable to errors in the adopted $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation.", "It is clear that this relation makes a huge difference to the BTFR (contrast my results with those of TG11 and [7]), yet the method of HAM is only just now being fully developed, and may be subject to major refinements in the future.", "We also see from Fig.", "REF that the BTFR prediction is significantly different to the data only in the region in which the HAM $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation must be extrapolated.", "Could this be due to the form of this relation changing at low halo masses?", "McGaugh claims that the scatter of gas-rich galaxies around a power-law BTFR may be fully accounted for by the uncertainties on the data points.", "Even if this were so,Significant controversy surrounds this issue.", "For example, [14] find McGaugh's gas rich data to be inconsistent with any zero-intrinsic scatter relation at the 95% confidence level, and many other observational studies of the BTFR find at least some evidence for a non-zero intrinsic scatter.", "it would not preclude the possibility of a non-zero intrinsic scatter in the underlying theoretical relation.", "Many potential BTFR predictions, with a range of non-zero intrinsic scatters, may be “compatible” with the data in the sense that they cannot be rejected at the 95% confidence level.", "McGaugh's argument succeeds in demonstrating that the data is consistent with MOND (which no doubt is worthy of consideration for providing a simple explanation of this and many other aspects of galaxy phenomenology), but not that it is inconsistent with $\\Lambda $ CDM.", "Indeed, the magnitude of the theoretical scatter in Fig.", "REF does not appear wildly discrepant with that in the data." ], [ "Summary and suggestions for further work", "This paper illustrates methodically the way in which the “naïve” slope-3 $\\Lambda $ CDM prediction for the BTFR can be improved by taking proper account of concentration, baryon mass fraction, and baryon disk collapse followed by adiabatic halo contraction.", "The slope of the relation is significantly increased by adopting baryon mass fractions from spiral-only Halo Abundance Matching, improving agreement with observational data at moderate to high baryonic masses.", "Predicted rotational velocities are systematically higher than those observed at the low-mass end, although plotting the theoretical BTFR in this region requires extrapolation of the $M_\\mathrm {vir}$ -$M_\\mathrm {bar}$ relation obtained by the Abundance Matching.", "In addition, curvature (in log-log space) is introduced into the prediction and becomes significant for $M_\\mathrm {bar} \\gtrsim 10^{11} M_{\\odot }$ .", "Although curvature is not evident in the data, spirals with such a high baryonic mass are rare, and the observational studies used here do not preclude a reduction in the slope of the BTFR at very high masses.", "The effect of adiabatic contraction is small, especially at low and very high baryonic masses.", "The BTFR prediction obtained here is reasonably robust against changes to the density profile used to describe the dark matter halos, and the angular momentum fraction posessed by the baryons.", "Accounting for the expected variability of the halo spin parameter, concentration, and baryon mass fraction, I find the theoretical BTFR to have an intrinsic scatter that is not clearly discrepant with that in the data, although may be somewhat larger than necessary to successfully account for the observations.", "Two simple selection requirements imposed on the model galaxies – a flat rotation curve and stability against bar formation – were found to have minimal effect on the predicted BTFR.", "This suggests that the spirals formed in the $\\Lambda $ CDM model presented here have rotation curves that plateau (as typically observed) and are predominantly stable.", "There are two main ways in which this study could be usefully supplemented.", "Only a subset of the effects causing intrinsic scatter in the TFR were considered here.", "Further sources of scatter include the possibility that dark halos have yet to virialise or are not well-fitted by an NFW profile [23], variations in galaxies' mass aggregation histories [13], and halo triaxiality [24].", "However, there exist also unconsidered selection factors which will reduce the scatter, such as the possibility that disk formation in halos with high baryon mass fraction is likely to create elliptical or S0 galaxies to which the TFR does not apply [28].", "The true theoretical intrinsic scatter can only be determined once all of these effects have been included in the style of Sect.", "REF .", "Proper implementation of these selection criteria is contingent on a solid theoretical understanding of galaxy formation.", "The comparison with observational data presented in Sect.", "is far from optimal.", "As discussed in Sect.", ", the ideal would be to amalgamate all data sets that employ compatible mass-to-light ratios, prescriptions for gas mass measurement, methods of rotational velocity measurement and selection criteria.", "By implementing similar constraints in the theoretical model, a comparison could be made between theory and observation that would be free from systematic errors.", "A simple hypothesis test could then be used to give a quantitative measure of the agreement between the BTFR and the $\\Lambda $ CDM paradigm as we currently understand it.", "The results of this study will be modified by advances in the techniques of Halo Abundance Matching and halo spin and concentration estimation from N-body simulations.", "The modular approach will make such updates easy to implement, and their effects on the BTFR readily visible.", "I would like to thank Subir Sarkar for guidance and encouragement, and Stacy McGaugh for helpful discussions of his work and mine." ] ]

[ [ "Suppression of f-Electron Itinerancy in CeRu2Si2 by a Strong Magnetic\n Field" ], [ "Abstract The valence state of Ce in a canonical heavy fermion compound CeRu2Si2 has been investigated by synchrotron X-ray absorption spectroscopy at 1.8 K in high magnetic fields of up to 40 T. The valence was slightly larger than for the pure trivalent state (Ce3+: f1), as expected in heavy fermion compounds, and it decreased toward the trivalent state as the magnetic field was increased.", "The field-induced valence reduction indicates that the itinerant character of the 4f electrons in CeRu2Si2 was suppressed by a strong magnetic field.", "The suppression was gradual and showed characteristic magnetic field dependence, which reflects the metamagnetism around Hm \\sim 8 T. The itinerant character persisted, even at 40 T (\\sim 5Hm), suggesting that the Kondo bound state is continuously broken by magnetic fields and that it should completely collapse at fields exceeding 200 T." ], [ "Suppression of $f$ -Electron Itinerancy in CeRu$_2$ Si$_2$ by a Strong Magnetic Field Y. H. Matsuda [email protected] Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan T. Nakamura Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan J. L. Her [Present address, ]Center for General Education, Chang Gung University, Taoyuan County 333, Taiwan (R.O.C.)", "Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan S. Michimura T. Inami Condensed Matter Science Division, Japan Atomic Energy Agency, Sayo, Hyogo 679-5148, Japan K. Kindo Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan T. Ebihara Department of Physics, Faculty of Science, Shizuoka University, Shizuoka 422-8529, Japan The valence state of Ce in a canonical heavy fermion compound CeRu$_2$ Si$_2$ has been investigated by synchrotron X-ray absorption spectroscopy at 1.8 K in high magnetic fields of up to 40 T. The valence was slightly larger than for the pure trivalent state (Ce$^{3+}$ : $f^1$ ), as expected in heavy fermion compounds, and it decreased toward the trivalent state as the magnetic field was increased.", "The field-induced valence reduction indicates that the itinerant character of the 4$f$ electrons in CeRu$_2$ Si$_2$ was suppressed by a strong magnetic field.", "The suppression was gradual and showed characteristic magnetic field dependence, which reflects the metamagnetism around $H_m \\sim $ 8 T. The itinerant character persisted, even at 40 T ( $\\sim 5H_m$ ), suggesting that the Kondo bound state is continuously broken by magnetic fields and that it may completely collapse at fields exceeding 200 T. The correlation between localized and itinerant electrons is one of the most intriguing subjects in condensed matter physics.", "Heavy electrons emerge in rare-earth and actinide compounds and give rise to exotic phenomena such as non-Fermi liquid behavior [1], [2], [3] and unconventional superconductivity [4], [5], [3] because of their strong correlation.", "The magnetic quantum critical point (QCP) in the Doniac model [6], [2] is one of the most important concepts for understanding the low-temperature physics of heavy fermion (HF) systems.", "However, another type of QCP, which is caused by the valence transition (VTQCP),[7], [8] has been proposed as the origin of the unusual superconductivity, such as that observed in CeCu$_2$ Si$_2$ [9] and also as the origin of the metamagnetism inYb-based HF systems [8], [10].", "Most of the HF systems contain Ce or Yb.", "The occupation number of the 4$f$ electron in the orbital ($n_f$ ) is close to 1 for Ce, and in Yb, the occupation number of the 4$f$ hole ($n_h$ ) is $\\sim 1$ .", "The 4$f$ electrons acquire an itinerant character because of the strong hybridization with the conduction electrons.", "The energy level of the correlated 4$f$ electrons is near the Fermi energy, because the significant electron-electron many-body interaction causes the electrons to become itinerant HFs.", "The degree of the itinerancy is determined by the strength of the electron interaction.", "The itinerant character manifests itself in the deviation of the occupation number of the $f$ electron or hole from unity: $1-n_f$ or $1-n_h$ .", "According to the Gützwiller approximation, the deviation $1-n_f$ can be expressed by $1-n_f = \\frac{kT_F^*}{\\Gamma }$ , where $kT_F^*$ and $\\Gamma $ correspond to the kinetic energy of the $f$ electrons with the electron interaction and that without the interaction, respectively.", "[11] In HF systems, $1-n_f$ must be very small but finite, indicating that the electrons are highly localized and only slightly itinerant.", "The direct observation of the valence of Ce and Yb in heavy fermion compounds in high magnetic fields is particularly desirable because the magnetic field dependence of $n_f$ and the change in the itinerant behavior of the electrons can be directly measured.", "However, it is not easy to determine the precise magnetic field variation of $1-n_f$ or $1-n_h$ experimentally, because they generally have values smaller than 0.1 in HF systems; the field dependence is expected to be of the order of 0.01 or smaller.", "Therefore, the change in the behavior of the heavy electrons caused by a magnetic field has rarely been studied in terms of the valence state.", "CeRu$_2$ Si$_2$ is a canonical HF system and has been attracting considerable attention for its metamagnetism around $H_m \\sim $ 8 T.[12] It was proposed that the transformation of the 4$f$ electrons from the itinerant to the localized state induced by a magnetic field (so-called the Kondo breakdown) could be observed based on the change from a large to small Fermi surface.", "[13] However, it was later found that the change in the Fermi surface is not necessarily concrete evidence of the transformation of the $f$ electrons [14], [15], [16]; the possible change in the behavior of the $f$ electrons is still controversial.", "[17] We have performed synchrotron X-ray absorption spectroscopy for CeRu$_2$ Si$_2$ in high magnetic fields of up to 40 T at 1.8 K. The magnetic field caused significant changes in the X-ray absorption spectrum near the Ce-$L_3$ edge, suggesting a change in the Ce valence toward the pure trivalent state.", "The suppression of the $f$ -electron itinerancy and the magnetic field dependence are discussed.", "The X-ray absorption experiment was carried out at beamline BL22XU of SPring-8.", "Pulsed high magnetic fields of up to 40 T were generated by a miniature pulsed magnet.", "[18] The duration of the magnetic field was  1 ms and the repetition rate was 4 pulses per hour for a 40 T measurement.", "The magnet and sample were cooled to 1.8 K by using an ILL-type He gas flow cryostat.", "The X-ray absorption spectrum was taken by a direct transmission method.", "The details of the experimental techniques have previously been described.", "[18] A single crystal of CeRu$_2$ Si$_2$ was grown by the Czochralski pulling method.", "The crystals were powdered and mixed with epoxy resin so that the effective thickness was appropriate for an X-ray absorption intensity of $\\mu t \\sim 1$ , where $\\mu $ and $t$ are the absorption coefficient and the thickness of the sample, respectively.", "The $c$ -axes of the powder crystals were aligned by means of a steady magnetic field of 14 T when the epoxy resin was solidified.", "The diameter of the powder was $\\mu $ m. Diluting fine powders in epoxy resin avoided Joule heating of the sample by a pulsed magnetic field.", "The high magnetic field magnetization data was obtained by the induction method using a conventional pulsed magnet.", "The X-ray absorption spectra (XAS) at 1.8 K in a zero magnetic field and at 25 T are shown in Fig.", "REF .", "The magnetic field was applied in parallel to the $c$ -axis of the sample.", "Although there is a clear magnetic field effect, which is discussed later, the spectrum at 25 T was very similar to that at 0 T. The absorption peak near 5.727 keV was the white line at the Ce $L_3$ absorption edge in the trivalent state (Ce$^{3+}$ : $f^1$ ).", "A small absorption band was observed in the spectrum around 5.735 keV, which can be attributed to the tetravalent state (Ce$^{4+}$ : $f^0$ ) [19].", "The spectrum shape fitting was performed using the standard Lorentz and arctangent functions with a linear background.", "The solid curve, dotted, and dot-dashed curves are the results of the spectrum fitting to the zero magnetic field curve.", "The dot-dashed and dotted curves represent the $f^1$ and $f^0$ states, respectively, and the solid curve represents the whole shape of the spectrum.", "The valence $v$ was directly determined by the relative intensity of the absorption bands, $v$ = 3+$I$ ($f^0$ )/($I$ ($f^0$ )+$I$ ($f^1$ )), where $I$ ($f^0$ ) denotes the absorption intensity of the $f^0$ absorption band and $I$ ($f^1$ ) denotes that of the $f^1$ absorption band.", "The observed $f^0$ and $f^1$ contributions are 0.053 and 0.947, respectively, and the Ce valence was calculated as $v=3.053 \\pm 0.02$ ($1-n_f = 0.053\\pm 0.02$ ) at zero magnetic fields.", "The $f^0$ component of CeRu$_2$ Si$_2$ at 20 K was reported to be 0.06 by the photoemission spectroscopy and the soft-X-ray absorption spectra with the analysis using the impurity Anderson model.", "[20], [21] Hence, the estimation of the $f^0$ component in our XAS study at 1.8 K seems to be in good agreement with the previous reported value.", "Since we did not take $f^2$ contribution into account in this work, we may overestimate $n_f$ .", "Actually $n_f$ was deduced to be 0.013 at 20 K if we assume that there is the $f^2$ contribution of 0.05.", "[20], [21] However,in terms of the Kondo breakdown scenario, the $f^0 \\rightarrow f^1$ transformation is the dominant magnetic field effect.", "Moreover, we can safely assume that the $f^2$ state is much less sensitive to magnetic field than the $f^1 \\leftrightarrow f^0$ transformation similarly to the pressure variation of Ce valence in CeCu$_2$ Si$_2$ .", "[9] Therefore, since $1- n_f$ obtained in this work corresponds to the $f^0$ component and should directly reflects the degree of the itinerancy, a small correction by the $f^2$ component is not important for evaluation of the magnetic field effect on the valence state in the present study.", "Figure: X-ray absorption spectra near the Ce L 3 L_3 edge at 0 and 25 T.The solid, dotted, and dot-dashed curves are the results of the spectrum fitting to the zero magnetic field spectrum.The change in the spectrum induced by the magnetic field was very small; therefore the spectrum at zero magnetic field was subtracted from the spectrum at finite magnetic fields in order to see the field evolution of the spectrum clearly.", "The difference XAS (dXAS) at different magnetic fields are shown in Fig.", "REF .", "The open circles are the experimental results and the solid curves are derived from the fitting curves.", "The positive and negative peak structures appeared around 5.725 and 5.735 keV, respectively; these energy positions correspond to the absorption bands from the $f^1$ and $f^0$ states shown in Fig.", "REF .", "This characteristic feature evolved as the magnetic fields increased, suggesting the $f^1$ state became more prominent, whereas the $f^0$ state diminished.", "The intensities of the $f^1$ and $f^0$ band were used as the fitting parameters and the fitting results accurately reproduced the characteristic features of the dXAS (Fig.", "REF ).", "Therefore, the valence decreases toward the trivalent state as the magnetic field increases.", "When the same experiment was performed with the magnetic field perpendicular to the $c$ -axis ($H \\perp c$ ), the dXAS was almost flat with no significant features even at 25 T, suggesting that there was no field dependence of the valence state when $H \\perp c$ .", "Since the magnetization for $H \\perp c$ is more than 10 times smaller than that for $H \\parallel c$ , it is expected that the magnetic energy gain for $H \\perp c$ is not sufficient to induce a suppression of the $f$ -electron itinerancy.", "The Ce valence was evaluated at different magnetic fields through the fitting analysis, and decreased as the magnetic field increased.", "The valences are shown in parentheses in Fig.", "REF .", "The experimental error in the relative change of the valence from the zero field value was about $\\pm 0.003$ , which is too large to allow for detailed discussion of the magnetic field dependence of the valence.", "Figure: Difference spectra obtained by subtracting the XAS at zero magnetic fields from that at finite magnetic fields.The open circles are the experimental results and the solid curves were obtained from the XAS fitting.", "The values in the parentheses are the valences deduced from the fitting analysis.", "The relative error from the zero field valence (v=3.053v = 3.053) was about ±0.003\\pm 0.003.To observe the magnetic field variation of the valence more precisely, we focused on two particular energy positions, 5.725 and 5.737 keV.", "The measurements for the magnetic field dependence of the absorption intensity were repeated 10 times for each energy and the average was taken (Fig.", "REF ).", "The change in the absorption intensities at the two energy positions should follow the change in the components of the $f^1$ and $f^0$ states.", "It is found that the change in the absorption intensity at low magnetic fields was small and the change rate increased under magnetic fields around 8 T. Figure: Change in the absorption intensity at 5.725 and 5.737 keV induced by magnetic fields.Assuming that only the intensity changed and that the shape and the energy shift of the $f^1$ and $f^0$ absorption bands were not altered, the magnetic field dependence of the valence was determined from the results shown in Fig.", "REF .", "In Fig.", "REF (a), the valence is plotted as a function of magnetic field, and the magnetization ($M$ ) and its magnetic field derivative ($dM/dH$ ) at 4.2 K are shown as a function of the magnetic field in Fig.", "REF (b) for comparison.", "In the $M$ and $dM/dH$ curves, the metamagnetic transition is clearly visible at 8 T ($H_m$ ).", "The valences were in good agreement with those obtained by XAS shown in Fig.", "REF within the experimental error.", "Because of the better signal to noise ratio of the results shown in Fig.", "REF , the unusual magnetic field dependence of the valence was visible.", "The valence decreased slightly in low magnetic fields and the rate of the decrease was larger when the magnetic field was higher than about 8 T; this magnetic field was defined as $H_v$ .", "The two dashed lines labeled (1) and (2) show the slope of the valence change in low and high magnetic fields around $H_v$ .", "It was clear that $H_v$ corresponded to the metamagnetic transition field, $H_m$ .", "Because $H_m$ shows almost no temperature dependence, [12] the temperature difference between 1.8 and 4.2 K is not important as we compare $H_v$ and $H_m$ .", "The valence gradually decreased and continued to change even at magnetic fields higher than $H_m$ .", "The valence decreased even at 40 T, although the metamagnetic transition was probably complete above 20 T. Figure: (a) Magnetic field dependence of the Ce valence and the corresponding f 0 f^0 component at 1.8 K.The dashed lines (1), (2) and (3) show the slope of the valencechange in low, medium and high magnetic field ranges, respectively.", "(b) Magnetic field dependence of the magnetization (MM) and the field derivative(dM/dHdM/dH) at 4.2 K.The significance of the decrease in the Ce valence when the magnetic field was applied parallel to the $c$ -axis of the crystal was examined.", "It is generally accepted that the HF state is broken if a strong enough magnetic field is applied.", "This is because the heavy fermion state is reached through the Kondo singlet bound state, and the singlet state can be broken if the Zeeman energy exceeds $k_BT_K$ , where the $k_B$ is the Boltzmann constant and $T_K$ is the Kondo temperature.", "For CeRu$_2$ Si$_2$ , shown in Fig.", "REF , the corresponding Zeeman energy is about 14 K which is comparable to the $T_K \\sim $ 24 K, [12] because the magnetization is roughly 1 $\\mu _B$ at 10 T. Hence, we propose that the Kondo bound state begins breaking at the metamagnetic transition.", "This qualitative explanation is the same as that given for the Fermi surface shrinkage at the metamagnetic transition.", "[13] However, if the $f$ electron becomes completely localized because the Kondo singlet state is broken, the Ce valence state should become purely trivalent and independent of magnetic fields.", "According to our x-ray absorption results, the $f^0$ component was still finite value and continued to decrease with the magnetic field, even at 40 T. Therefore, it is found that the itinerancy of the electrons is gradually suppressed by the magnetic field and the suppression becomes prominent at the metamagnetic transition field $H_m$ .", "The suppression rate becomes small again around 19 T where the lines (2) and (3) cross, and the metamagnetic transition also appears to finish according to the $dM/dH$ curve.", "The suppression continues even at magnetic fields as strong as $5H_m$ .", "The valence fluctuation phenomena around $H_m$ in CeRu$_2$ Si$_2$ may be related to the QCP of the valence transition (VTQCP), as has been suggested in other heavy fermion compounds.", "[22], [10] The Ce valence of CeRu$_2$ Si$_2$ was thought to be nearly trivalent and stable in magnetic fields.", "However, we have discovered that the valence depends on the applied magnetic field and changes by about 0.005 around the metamagnetic transition (8-18 T).", "The valence change was several times smaller than that of YbAgCu$_4$ [25] a heavy fermion material that exhibits metamagnetism, which is probably caused by the VTQCP.", "It is not clear at the present how far away CeRu$_2$ Si$_2$ is located from the VTQCP; [22], [10] and the mechanism of the metamagnetic transition in CeRu$_2$ Si$_2$ should be re-examined.", "Another interesting finding is that the valence change can quantitatively explain the large magnetovolume effect in CeRu$_2$ Si$_2$ [23].", "When 5.221 and 4.661 $\\textrm {Å}$ were used for the ionic radii ($r$ ) of Ce$^{3+}$ and Ce$^{4+}$ , respectively, [24] the relative change in the radius $\\frac{\\Delta r}{r}$ was $6.47 \\times 10^{-4}$ at 12 T using the valence shown in Fig.", "REF , which gave the relative volume change $\\Delta V /V \\sim 1.94 \\times 10^{-3}$ .", "This was consistent with the $\\Delta V /V \\sim 1.8 \\times 10^{-3}$ at around 12 T that was obtained from magnetostriction experiments.", "[23] This is the first clear evidence that the magnetovolume effect in CeRu$_2$ Si$_2$ is due to the field-induced valence change.", "The field variation of the valence observed in this study corresponds to the loss of itinerancy in terms of the Gützwiller approximation; magnetic fields reduce the value of $1-n_f$ .", "The relative reduction of the itinerancy was possibly evaluated as $ \\frac{\\Delta n_f }{1-n_f (0~T)} \\sim \\frac{0.005}{0.053} \\sim 0.09$ around the metamagnetic transition (8-18 T); $\\Delta n_f $ corresponds to the change in the valence in a magnetic field and $n_f = 4 - v$ .", "If a complete field-induced valence transition takes place, $\\frac{\\Delta n_f }{1-n_f (0~T)} $ should be close to 1.0.", "It is actually about 0.7 in the Yb-based heavy fermion compound, YbAgCu$_4$ .", "[25] The value of $ \\frac{\\Delta n_f }{1-n_f (0~T)} \\sim 0.09$ in CeRu$_2$ Si$_2$ suggests that the itinerancy is suppressed by only 9%.", "This is considerably smaller than the value expected for the complete valence transition.", "At 40 T, the value was $ \\frac{\\Delta n_f }{1-n_f (0~T)} \\sim \\frac{0.012}{0.053} \\sim 0.23$ , suggesting that the $f$ electrons were still far from being completely localized.", "This evaluation is not changed if we consider the $f^2$ contribution that we did not take into account in our analysis.", "It is because the localization of the $f$ -electron can be evaluated by the reduction of the $f^0$ component.", "At a very high magnetic field, $1-n_f = 0$ and the $f$ electrons should become completely localized.", "If the magnetic field, $H_L$ , where $1-n_f = 0$ is simply estimated using an extrapolation of line (3) in Fig.", "REF (a) to the position $v$ = 3.00, then $H_L \\sim $ 220 T. In conclusion, we have found that the Ce valence fluctuated at 1.8 K in CeRu$_2$ Si$_2$ and that it decreased toward the pure trivalent state as the magnetic field was increased.", "The valence reduction, which corresponds to the metamagnetism at $H_m \\sim $ 8 T, was around 0.005, and was too small for the metamagnetism to be considered as the complete localization of the electrons.", "The valence changes gradually and thus cannot be a first order transition as has previously been suggested [26], [16].", "The characteristic magnetic field dependence of $n_f$ , or the Ce valence, can be understood by the gradual break down of the Kondo bound state.", "However, the relationship between the valence change and the metamagnetism is still not clear.", "CeRu$_2$ Si$_2$ may be located close to the VTQCP, and it can be drawn closer by applying magnetic fields.", "Further theoretical studies are necessary to solve this problem.", "Y. H. M thanks Prof. S. Watanabe for useful discussions.", "This work is partly supported by a Grant-in-Aid for Scientific Research B (22340091) and for Scientific Research A (22244047) provided by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan." ] ]

[ [ "Quantum phase transitions in the exactly solved spin-1/2\n Heisenberg-Ising ladder" ], [ "Abstract Ground-state behaviour of the frustrated quantum spin-1/2 two-leg ladder with the Heisenberg intra-rung and Ising inter-rung interactions is examined in detail.", "The investigated model is transformed to the quantum Ising chain with composite spins in an effective transverse and longitudinal field by employing either the bond-state representation or the unitary transformation.", "It is shown that the ground state of the Heisenberg-Ising ladder can be descended from three exactly solvable models: the quantum Ising chain in a transverse field, the 'classical' Ising chain in a longitudinal field or the spin-chain model in a staggered longitudinal-transverse field.", "The last model serves in evidence of the staggered bond phase with alternating singlet and triplet bonds on the rungs of two-leg ladder, which appears at moderate values of the external magnetic field and consequently leads to a fractional plateau at a half of the saturation magnetization.", "The ground-state phase diagram totally consists of five ordered and one quantum disordered phase, which are separated from each other either by the lines of discontinuous or continuous quantum phase transitions.", "The order parameters are exactly calculated for all five ordered phases and the quantum disordered phase is characterized through different short-range spin-spin correlations." ], [ "Introduction", "Quantum spin models with competing interactions represent quite interesting and challenging topic for the current research [1], [2].", "Such models show many unusual features in their ground-state properties and are very sensitive to the approximative schemes applied to them.", "An existence of exact solutions is therefore quite important, since they provide a rigorous information about the complex behaviour of frustrated models.", "However, exact results for such systems are still limited (see [3] and references cited therein).", "The simplest examples of rigorously solved quantum spin models are the models with the known dimerized [3], [4], [5], [6] or even trimerized [7] ground states.", "Another example represents frustrated quantum spin antiferromagnets in high magnetic fields with the so-called localized-magnon eigenstates (see e.g.", "[8] for recent review).", "It should be nevertheless noticed that the aforementioned exact results are usually derived only under certain constraint laid on the interaction parameters and those models are not tractable quite generally.", "On the other hand, the exact solutions for some spin-$\\frac{1}{2}$ quantum spin ladders with multispin interaction allows to find, besides the ground state, also thermodynamic properties quite rigorously [9], [10].", "Exactly solvable models, which admit the inclusion of frustration, can also be constructed from the Ising models where the decorations of quantum spins are included [11], [12].", "The decoration-iteration procedure allows one to calculate exactly all thermodynamic properties of the decorated models when the exact solution for the corresponding Ising model is known.", "The distinctive feature of these models is that the quantum decorations are separated from each other, so that the Hamiltonian can be decomposed into the sum of commuting parts.", "The eigenstates of the total Hamiltonian are then simply factorized into a product of the eigenstates of its commuting parts.", "New findings of the exactly solvable frustrated quantum spin models with non-trivial (not simply factorizable) ground states are thus highly desirable.", "The present article deals with the frustrated quantum spin-$\\frac{1}{2}$ two-leg ladder with the Heisenberg intra-rung interaction and the Ising inter-rung interactions between nearest-neighbouring spins from either the same or different leg.", "Such a model can be regarded as an extension of the spin-$\\frac{1}{2}$ Heisenberg-Ising bond alternating chain, which was proposed and rigorously solved by Lieb, Schultz and Mattis [13].", "Alternatively, this model can also be viewed as the generalization of the exactly solved quantum compass ladder [14], [15], [16] when considering the Heisenberg rather than XZ-type intra-rung interaction and accounting for the additional frustrating Ising inter-rung interaction.", "Moreover, it is quite plausible to suspect that the exact results presented hereafter for the spin-$\\frac{1}{2}$ Heisenberg-Ising two-leg ladder may also bring insight into the relevant behaviour of the corresponding Heisenberg two-leg ladder, which represents a quite challenging and complex research problem in its own right [17], [18].", "The frustrated spin-$\\frac{1}{2}$ Heisenberg two-leg ladder have been extensively investigated by employing various independent numerical and analytical methods such as density-matrix renormalization group [19], [20], [21], [22], numerical diagonalization [22], [23], [24], [25], [26], series expansion [26], bosonization technique [27], [28], [29], strong- and weak-coupling analysis [30], [31], [32], [33], [34], valence-bond spin-wave theory [36], variational matrix-product approach [37], [38], and bond mean-field theory [39], [40], [41].", "Among the most interesting results obtained for this quantum spin chain, one could mention an existence of the columnar-dimer phase discussed in [32], [33] or a presence of the fractional plateau in the magnetization process examined in [22], [23], [24], [25], [34].", "The theoretical investigation of two-leg ladder models is motivated not only from the academic point of view, but also from the experimental viewpoint, because the two-leg ladder motif captures the magnetic structure of a certain class of real quasi-one-dimensional magnetic materials.", "The most widespread family of two-leg ladder compounds form cuprates, in which one finds both experimental representatives with the dominating intra-rung interaction like SrCu$_2$ O$_3$ [42], [43], Cu$_2$ (C$_5$ H$_{12}$ N$_2$ )$_2$ Cl$_4$ [44], [45], [46], [47], (C$_5$ H$_{12}$ N)$_2$ CuBr$_4$ [48], (5IAP)$_2$ CuBr$_4$ [49] as well as, the magnetic compounds with the dominating intra-leg interaction such as KCuCl$_3$ [50], [51], TlCuCl$_3$ [52], [53], [54], NH$_4$ CuCl$_3$ , KCuBr$_3$ .", "Understanding the low-temperature magnetism of two-leg ladder models also turns out to be crucial for an explanation of the mechanism, which is responsible for the high-temperature superconductivity of cuprates [55].", "Besides the cuprates, another experimental representatives of the two-leg ladder compounds represent vanadates (VO)$_2$ P$_2$ O$_7$ [56], CaV$_2$ O$_5$ and MgV$_2$ O$_5$ [57], as well as, the polyradical BIP-BNO [58], [59].", "The outline of the paper is as follows.", "In section  the model is defined and the pseudospin representation is considered.", "The ground-state properties of the model with and without external field are studied in section .", "The most important findings are summarized in section ." ], [ "Heisenberg-Ising two-leg ladder", "Let us define the spin-$\\frac{1}{2}$ Heisenberg-Ising ladder through the following Hamiltonian: $H&=&\\sum _{i=1}^{N} \\Big [ J_1({\\mathbf {s}}_{1,i} \\cdot {\\mathbf {s}}_{2,i})_{\\Delta }+J_2(s_{1,i}^z s_{1,i+1}^z+s_{2,i}^z s_{2,i+1}^z)\\nonumber \\\\&&+J_3 (s_{2,i}^z s_{1,i+1}^z+ s_{1,i}^z s_{2,i+1}^z)-h(s_{1,i}^z+s_{2,i}^z) \\Big ],$ where $({\\mathbf {s}}_{1,i} \\cdot {\\mathbf {s}}_{2,i})_{\\Delta }=s_{1,i}^x s_{2,i}^x + s_{1,i}^y s_{2,i}^y + \\Delta s_{1,i}^z s_{2,i}^z$ , $s_{l,i}^\\alpha $ are three spatial projections of spin-$\\frac{1}{2}$ operator, the first index denotes the number of leg, the second enumerates the site, $J_1$ is the $XXZ$ Heisenberg intra-rung interaction between nearest-neighbour spins from the same rung, $J_2$ is the Ising intra-leg interaction between nearest-neighbour spins from the same leg, $J_3$ is the crossing (diagonal) Ising inter-rung interaction between next-nearest-neighbour spins from different rungs, $h$ is the external magnetic field.", "We also imply the periodic boundary conditions ${\\mathbf {s}}_{1,N+1} \\equiv {\\mathbf {s}}_{1,1}$ and ${\\mathbf {s}}_{2,N+1} \\equiv {\\mathbf {s}}_{2,1}$ along legs.", "The coupling constants $J_2$ and $J_3$ can be interchanged by renumbering of the sites, as well as their signs can be simultaneously reverted by spin rotations.", "Therefore, the Hamiltonians $H(J_2,J_3)$ , $H(J_3,J_2)$ and $H(-J_2,-J_3)$ in zero field have equal eigenvalues and the corresponding models are thermodynamically equivalent.", "It can be checked that $z$ -projection of total spin on a rung $S_i^z=s_{1,i}^z+s_{2,i}^z$ commutes with the total Hamiltonian $[S_i^z,H]=0$ and hence, it represents a conserved quantity.", "For further convenience, it is therefore advisable to take advantage of the bond representation, which has been originally suggested by Lieb, Schultz and Mattis for the Heisenberg-Ising chain [13].", "Let us introduce the bond-state basis consisting of four state vectors: $&& |\\phi _{0,0}^{i}\\rangle =\\frac{1}{\\sqrt{2}}(|\\!", "\\downarrow _{1,i}\\uparrow _{2,i}\\rangle - |\\!", "\\uparrow _{1,i}\\downarrow _{2,i}\\rangle ),\\; \\;|\\phi _{1,0}^{i}\\rangle =\\frac{1}{\\sqrt{2}}(|\\!", "\\downarrow _{1,i}\\uparrow _{2,i}\\rangle + |\\!", "\\uparrow _{1,i}\\downarrow _{2,i}\\rangle ),\\nonumber \\\\&& |\\phi _{1,-}^{i}\\rangle =\\frac{1}{\\sqrt{2}}(|\\!", "\\uparrow _{1,i}\\uparrow _{2,i}\\rangle - |\\!", "\\downarrow _{1,i}\\downarrow _{2,i}\\rangle ),\\; \\;|\\phi _{1,+}^{i}\\rangle =\\frac{1}{\\sqrt{2}}(|\\!", "\\uparrow _{1,i}\\uparrow _{2,i}\\rangle + |\\!", "\\downarrow _{1,i}\\downarrow _{2,i}\\rangle ).$ These states are the eigenstates of the Heisenberg coupling between two spins located on the $i$ th rung, i.e.", "$|\\phi _{0,0}^{i}\\rangle $ is the singlet-bond state, $|\\phi _{1,0}^{i}\\rangle $ , $|\\phi _{1,\\pm }^{i}\\rangle $ are triplet states.", "Following [13] two subspaces can be singled out in the bond space: i) $|\\phi _{0,0}^{i}\\rangle $ , $|\\phi _{1,0}^{i}\\rangle $ and ii) $|\\phi _{1,+}^{i}\\rangle $ , $|\\phi _{1,-}^{i}\\rangle $ .", "The first and second subspaces corresponds to $(S_i^z)^2=0$ and $(S_i^z)^2=1$ respectively, and the Hamiltonian can be diagonalized separately in each subspace.", "We can introduce the index of subspace at $i$ -th site: $n_i=0 (1)$ if the given state is in the subspace with $(S_i^z)^2=0$ ($(S_i^z)^2=1$ ) spanned by states $|\\phi _{0,0}^{i}\\rangle $ , $|\\phi _{1,0}^{i}\\rangle $ ($|\\phi _{1,-}^{i}\\rangle $ , $|\\phi _{1,+}^{i}\\rangle $ ).", "According to [13], let us also call the bond states as purity (impurity) states if $n_i=0(1)$ .", "If one makes pseudospin notations for the states $|\\phi _{0,0}^{i}\\rangle =|\\!", "\\!", "\\!", "\\downarrow \\rangle _0^i$ , $|\\phi _{1,0}^{i}\\rangle =|\\!", "\\!", "\\!", "\\uparrow \\rangle _0^i$ , the action of the spin-$\\frac{1}{2}$ operators in the subspace $n_i=0$ can be expressed in terms of new raising and lowering operators: $&& s_{1,i}^z = -\\frac{1}{2}(a_i^+ + a_i)=-\\tilde{s}_i^x, \\; \\; \\;s_{2,i}^z = \\frac{1}{2}(a_i^+ + a_i)=\\tilde{s}_i^x, \\; \\; \\;\\nonumber \\\\&&({\\mathbf {s}}_{1,i} \\cdot {\\mathbf {s}}_{2,i})_{\\Delta } = a_i^+ a_i -\\frac{2+\\Delta }{4}=\\tilde{s}_i^z-\\frac{\\Delta }{4}.$ The operators $a_i^+$ , $a_j$ satisfy the Pauli algebra ($\\lbrace a_i,a_i^+\\rbrace =0$ , $\\lbrace a_i,a_i\\rbrace =\\lbrace a_i^+,a_i^+\\rbrace =0$ , $[a_i,a_j]=[a_i,a_j^+]=[a_i^+,a_j^+]=0$ for $i\\ne j$ ), and half of their sum can be identified as a new pseudospin operator $\\tilde{s}_i^x$ .", "Analogously, one can consider the pseudospin representation of the subspace $n_i=1$ ($|\\phi _{1,-}^{i}\\rangle =|\\!", "\\!", "\\downarrow \\rangle _1^i$ , $|\\phi _{1,+}^{i}\\rangle =|\\!", "\\!", "\\uparrow \\rangle _1^i$ ) and find the action of spin operators in it as follows: $&& s_{1,i}^z = \\frac{1}{2}(a_i^+ + a_i)=\\tilde{s}_i^x, \\; \\;s_{2,i}^z = \\frac{1}{2}(a_i^+ + a_i)=\\tilde{s}_i^x, \\; \\;({\\mathbf {s}}_{1,i} \\cdot {\\mathbf {s}}_{2,i})_{\\Delta } = \\frac{\\Delta }{4}.$ Combining (REF ), (REF ) we find the general expressions for the pseudospin representation of these operators, which are valid in both subspaces: $&& s_{1,i}^z = (2n_i-1)\\tilde{s}_i^x, \\; \\;s_{2,i}^z = \\tilde{s}_i^x, \\; \\;({\\mathbf {s}}_{1,i} \\cdot {\\mathbf {s}}_{2,i})_{\\Delta } =(1-n_i)\\tilde{s}_i^z+\\frac{\\Delta }{4}(2n_i-1).$ The effective Hamiltonian can be rewritten in terms of new operators as follows: $H&=&\\sum _{i=1}^{N} \\Big \\lbrace J_1 (1-n_i) \\tilde{s}_i^z- 2 h n_i \\tilde{s}_i^x+\\frac{J_1\\Delta }{4}(2n_i-1)\\nonumber \\\\&&+\\left[ 4J_2 n_i n_{i+1} -2(J_2-J_3)(n_i + n_{i+1} - 1) \\right]\\tilde{s}_i^x \\tilde{s}_{i+1}^x \\Big \\rbrace .$ It is noteworthy that equation (REF ) can be considered as some nonlinear spin transformation from $s_{1,i}^\\alpha $ , $s_{2,i}^\\alpha $ to new operators ${s^{\\prime }}_{1,i}^{\\alpha }$ , ${s^{\\prime }}_{2,i}^{\\alpha }$ , where e.g.", "${s^{\\prime }}_{1,i}^{z}=(n_i-\\frac{1}{2})$ , ${s^{\\prime }}_{2,i}^\\alpha ={\\tilde{s}}_{i}^\\alpha $ .", "The transformation can be generated by the following unitary operator: $U=\\prod _{i=1}^N\\exp \\left[-i\\frac{\\pi }{2}(s_{1,i}^x+s_{2,i}^x)\\right]\\exp \\left(i\\pi s_{1,i}^x s_{2,i}^x\\right)\\exp \\left(-i\\frac{\\pi }{2} s_{2,i}^y\\right)\\exp \\left(i\\pi s_{2,i}^z\\right).$ The crucial point is the second factor, which introduces the nonlinearity.", "Other terms are the rotation in spin space, they are used to adjust the transformed operators to the form of (REF ).", "Finally, the spin operators are transformed as follows: $U{s}_{1,i}^x U^+=s_{1,i}^x, &U{s}_{1,i}^y U^+=2s_{1,i}^y s_{2,i}^x, &U{s}_{1,i}^z U^+=2s_{1,i}^z s_{2,i}^x,\\nonumber \\\\U{s}_{2,i}^x U^+=2s_{1,i}^x s_{2,i}^z, &U{s}_{2,i}^y U^+=-2s_{1,i}^x s_{2,i}^y, &U{s}_{2,i}^z U^+=s_{2,i}^x.$ The transformed Hamiltonian has the form of the quantum Ising chain with composite spins in an effective longitudinal and transverse magnetic field: $U H U^+ &=&\\sum _{i=1}^N\\bigg \\lbrace \\frac{J_1}{2}\\left(1- 2s_{1,i}^z\\right)s_{2,i}^z+\\frac{J_1 \\Delta }{2} s_{1,i}^z+\\Big [J_2(4 s_{1,i}^z s_{1,i+1}^z +1)\\nonumber \\\\&+&2J_3( s_{1,i+1}^z + s_{1,i}^z) \\Big ]s_{2,i}^x s_{2,i+1}^x-h(1+2 s_{1,i}^z) s_{2,i}^x\\bigg \\rbrace .$ The straightforward correspondence $s_{1,i}^z=n_i-\\frac{1}{2}$ , $s_{2,i}^\\alpha =\\tilde{s}_{i}^\\alpha $ leads to the equivalence between the Hamiltonians (REF ) and (REF ).", "It should be noted that such kind of a representation remains valid also in case of the asymmetric ladder having both diagonal (crossing) Ising interactions different from each other." ], [ "Ground state of the Heisenberg-Ising two-leg ladder", "The Hamiltonian (REF ) can also be rewritten in the following more symmetric form: $H&{=}&\\sum _{i=1}^{N} \\Big \\lbrace J_1 (1-n_i) \\tilde{s}_i^z-2h n_i\\tilde{s}_i^x+\\frac{J_1\\Delta }{4}(2n_i-1)\\nonumber \\\\&&{+}\\left[ 2(J_2+J_3) n_i n_{i+1} +2(J_2-J_3)(1-n_i)(1-n_{i+1}) \\right]\\tilde{s}_i^x \\tilde{s}_{i+1}^x \\Big \\rbrace .$ This form serves in evidence that the effective model splits at $i$ th site into two independent chains provided that two neighbouring bonds are being in different subspaces, i.e.", "$n_i\\ne n_{i+1}$ .", "The uniform Hamiltonians, when all $n_i=0$ or all $n_i=1$ , read: $&&H^0=\\sum _{i=1}^N \\Big [ J_1\\Big (\\tilde{s}_i^z-\\frac{\\Delta }{4}\\Big )+2(J_2-J_3)\\tilde{s}_i^x \\tilde{s}_{i+1}^x \\Big ],\\\\&&H^1=\\sum _{i=1}^N \\Big [ 2(J_2+J_3)\\tilde{s}_i^x \\tilde{s}_{i+1}^x-2h\\tilde{s}_i^x + \\frac{J_1\\Delta }{4} \\Big ].$ If all bonds are in purity states ($n_i=0$ ), one obtains the effective Hamiltonian of the Ising chain in the transverse field that is exactly solvable within Jordan-Wigner fermionization [60], [61].", "If all bonds are in impurity states ($n_i=1$ ), one comes to the Ising chain in the longitudinal field that is solvable by the transfer-matrix method (see e.g.", "[62]).", "Accordingly, the ground-state energy of the model in $n_i=0$ subspace is given by [61]: $e_0^0=\\lim _{N\\rightarrow \\infty }\\frac{E_0^{0}(N)}{N}=-\\frac{(J_1+|J_2-J_3|)}{\\pi }{\\mathbf {E}}(\\sqrt{1-\\gamma ^2})-\\frac{J_1\\Delta }{4},$ where $\\gamma =\\frac{J_1-|J_2-J_3|}{J_1+|J_2-J_3|}$ and ${\\mathbf {E}}(\\kappa ) = \\displaystyle \\int _0^{\\frac{\\pi }{2}} \\!\\!\\!\\!", "{\\rm d} \\theta \\sqrt{1 - \\kappa ^2 \\sin ^2 \\theta }$ is the complete elliptic integral of the second kind.", "The ground-state energy of the model in $n_i=1$ subspace is given by the ground-state energy of the effective Ising chain in the longitudinal field: $e_0^1=\\lim _{N\\rightarrow \\infty }\\frac{E_0^{1}(N)}{N}=\\left\\lbrace \\begin{array}{ll}\\frac{J_1\\Delta }{4}+\\frac{J_2+J_3}{2}-|h|, & \\mbox{if}\\: |h|>(J_2+J_3),\\\\\\frac{J_1\\Delta }{4}-\\frac{J_2+J_3}{2}, & \\mbox{if}\\: |h|\\le (J_2+J_3).\\end{array}\\right.$ The upper (lower) case corresponds to the ferromagnetically (antiferromagnetically) ordered state being the ground state at strong (weak) enough magnetic fields." ], [ "Ground-state phase diagram in a zero field", "If the external field vanishes ($h=0$ ), it is sufficient to show that the inequality $E_0^0(N_1)+E_0^0(N_2)\\ge E_0^0(N_1+N_2)$ holds for any finite transverse Ising chain with free ends in order to prove that the ground state always corresponds to a uniform bond configuration.", "Here $E_0^0(N)$ denotes the ground state energy of the Hamiltonian: $H^0{(N)}&=& {2(J_2-J_3)}\\sum _{i=1}^{N-1}\\tilde{s}_{i}^x\\tilde{s}_{i+1}^x+ J_1\\sum _{i=1}^{N}\\Big (\\tilde{s}_i^z-\\frac{\\Delta }{4}\\Big ).$ Indeed, two independent chains of size $N_1$ and $N_2$ can be represented by the following Hamiltonian: $H^0{(N_1,N_2)}&=& {2(J_2-J_3)}\\sum _{i=1}^{N_1-1}\\tilde{s}_{i}^x\\tilde{s}_{i+1}^x+ 2(J_2-J_3)\\sum _{i=N_1+1}^{N_1+N_2-1}\\tilde{s}_{i}^x\\tilde{s}_{i+1}^x+ J_1\\sum _{i=1}^{N_1+N_2}\\Big (\\tilde{s}_i^z-\\frac{\\Delta }{4}\\Big )\\nonumber \\\\&=&H^0{(N_1+N_2)} - 2(J_2-J_3)\\tilde{s}_{N_1}^x\\tilde{s}_{N_1+1}^x.$ If $E_0^0(N)$ and $|\\psi _0^{N}\\rangle $ are the lowest eigenvalue and eigenstate of $H^0{(N)}$ , then $E_0^0(N_1,N_2)=E_0^0(N_1)+E_0^0(N_2)$ and $|\\psi _0^{N_1,N_2}\\rangle =|\\psi _0^{N_1}\\rangle |\\psi _0^{N_2}\\rangle $ are the lowest eigenvalue and eigenstate of $H^0{(N_1,N_2)}$ .", "Now, it is straightforward to show that $E_0^0(N_1)+E_0^0(N_2)&=&\\langle \\psi _0^{N_1,N_2}| H^0{(N_1+N_2)} |\\psi _0^{N_1,N_2}\\rangle \\nonumber \\\\&-& 2(J_2-J_3)\\langle \\psi _0^{N_1}|\\tilde{s}_{N_1}^x|\\psi _0^{N_1}\\rangle \\langle \\psi _0^{N_2}|\\tilde{s}_{N_1+1}^x |\\psi _0^{N_2}\\rangle \\ge E_0^0(N_1+N_2).$ Here, we have used that $\\langle \\psi _0^{N_1}|\\tilde{s}_{N_1}^x|\\psi _0^{N_1}\\rangle =0$ for any finite chain [13], [61], and that the lowest mean value of the operator $H^0{(N_1+N_2)}$ is achieved in its ground state.", "It is easy to show by straightforward calculation that the same property is valid for the Ising chain with free ends in zero longitudinal field, i.e.", "$E_0^1(N_1+N_2)=E_0^1(N_1)+E_0^1(N_2)+\\frac{|J_2+J_3|}{2}\\le E_0^1(N_1)+E_0^1(N_2).$ Now, let us prove that the ground state of the whole model may correspond only to one of uniform bond configurations.", "It can be readily understood from (REF ) that the effective Hamiltonian does not contain an interaction between spins from two neighbouring bonds if they are in different subspaces [63].", "It means that the effective model splits into two independent parts at each boundary ('domain wall') between the purity and impurity states.", "Thus, the Heisenberg-Ising ladder for any given configuration of bonds can be considered as a set of independent chains of two kinds and of different sizes.", "Then, the ground-state energy of any randomly chosen bond configuration will be as follows: $E&=&E_0^0(N_1)+E_0^0(N_2)+\\dots +E_0^1(M_1)+E_0^1(M_2)+\\dots \\nonumber \\\\&\\ge & E_0^0(N_1+N_2+\\dots ) + E_0^1(M_1+M_2+\\dots )$ It is quite evident from equations (REF ) and (REF ) that the one uniform configuration, which corresponds to the state with lower energy than the other one, must be according to inequality (REF ) the lowest-energy state (i.e.", "ground state).", "Therefore, the model may show in the ground state the first-order quantum phase transition when the lowest eigenenergies in both subspaces becomes equal $e_0^1=e_0^0$ .", "Besides, there also may appear the more striking second-order (continuous) quantum phase transition at $|J_1|=|J_2-J_3|$ in the ground state, which is inherent to the pure quantum spin chain [61].", "From this perspective, the Heisenberg-Ising ladder can show a variety of quantum phase transitions in its phase diagram.", "The ground-state phase diagram of the spin-$\\frac{1}{2}$ Heisenberg-Ising ladder in a zero magnetic field is shown in figure REF .", "Figure: Ground-state phase diagrams for the spin-1 2\\frac{1}{2} Heisenberg-Ising two-leg ladder in an absence of the external magnetic field (h=0h=0).Broken (solid) lines denote the lines of first-order (second-order) quantum phase transitions,dotted lines denote the rung singlet-dimer state.", "(a) J 1 =1J_1=1, Δ=0.0\\Delta =0.0(red), 1.01.0(blue), 1.51.5(green) (from thick to thin lines);(b) J 2 =1J_2=1, Δ=1\\Delta =1.", "Thick solid and broken (blue) lines are the ground-state boundaries of the Heisenberg-Ising ladder, the thin line with dots showsthe ground-state boundary of the Heisenberg ladder and straight broken lines the ground-state boundaries of the Ising ladder.It reflects the symmetry of the model, i.e.", "it is invariant under the exchange of the Ising interactions $J_2$ and $J_3$ , as well as, under the simultaneous change of the signs of $J_2$ and $J_3$ .", "Altogether, five different ground states can be recognised in figure REF : Quantum paramagnetic (QPM) state for $e_0^0<e_0^1$ , and $|J_1|>|J_2-J_3|$ : the equivalent transverse Ising chain (REF ) is in the gapped disordered state with no spontaneous magnetization $\\langle \\tilde{s}_i^x\\rangle =0$ and non-zero magnetization $\\langle \\tilde{s}_i^z\\rangle \\ne 0$ induced by the effective transverse field.", "For the initial Heisenberg-Ising ladder it means that the rung singlet dimers are dominating on Heisenberg bonds in the ground state.", "Although each bond may possess the antiferromagnetic order, the interaction along legs demolishes it and leads to the disordered quantum paramagnetic state.", "Stripe Leg (SL) state for $e_0^0<e_0^1$ , and $|J_1|<J_3-J_2$ : the equivalent transverse Ising chain exhibits the spontaneous ferromagnetic ordering with $\\langle \\tilde{s}_i^x\\rangle \\ne 0$ .", "Due to relations (REF ) and (REF ), one obtains for the Heisenberg-Ising ladder $\\langle s_{1,i}^z\\rangle =\\langle s_{1,i+1}^z\\rangle =-\\langle s_{2,i}^z\\rangle =-\\langle s_{2,i+1}^z\\rangle \\ne 0$ .", "This result is taken to mean that the Heisenberg-Ising ladder shows a ferromagnetic order along legs and antiferromagnetic order along rungs, i.e.", "the magnetizations of chains are opposite.", "The staggered magnetization as the relevant order parameter in this phase is non-zero and it exhibits evident quantum reduction of the magnetization given by: $m_{SL}^z=\\frac{1}{2N}\\sum _{i=1}^N(\\langle s_{1,i}^z\\rangle -\\langle s_{2,i}^z\\rangle )=\\frac{1}{2}[1-J_1^2/(J_2-J_3)^2]^{\\frac{1}{8}}$ .", "Here we used the result for the spontaneous magnetization of transverse Ising chain [61].", "Néel state for $e_0^0<e_0^1$ , and $|J_1|<J_2-J_3$ : the effective transverse Ising chain shows the spontaneous antiferromagnetic order with $\\langle \\tilde{s}_i^x\\rangle =(-1)^i m_x\\ne 0$ .", "For the Heisenberg-Ising ladder one consequently obtains $\\langle s_{1,i}^z\\rangle =-\\langle s_{1,i+1}^z\\rangle =-\\langle s_{2,i}^z\\rangle =\\langle s_{2,i+1}^z\\rangle \\ne 0$ .", "Hence, it follows that the nearest-neighbour spins both along legs as well as rungs exhibit predominantly antiferromagnetic ordering.", "The dependence of staggered magnetization as the relevant order parameter is quite analogous to the previous case $m_{AF}^z=\\frac{1}{2N}\\sum _{i=1}^N(-1)^{i}(\\langle s_{1,i}^z\\rangle -\\langle s_{2,i}^z\\rangle )=\\frac{1}{2}[1-J_1^2/(J_2-J_3)^2]^{\\frac{1}{8}}$ .", "Stripe Rung (SR) state for $e_0^1<e_0^0$ , $J_2>0$ , $J_3>0$ : the model shows classical ordering in this phase with the antiferromagnetically ordered nearest-neighbour spins along legs and the ferromagnetically ordered nearest-neighbour spins along rungs.", "Ferromagnetic (FM) state for $e_0^1<e_0^0$ , $J_2<0$ , $J_3<0$ : the ground state corresponds to the ideal fully polarized ferromagnetic spin state.", "The results displayed in figure REF (a) demonstrate that the Heisenberg-Ising ladder is in the disordered QPM phase whenever a relative strength of both Ising interactions $J_2$ and $J_3$ is sufficiently small compared to the Heisenberg intra-rung interaction $J_1$ .", "It is noteworthy, moreover, that the QPM phase reduces to a set of fully non-correlated singlet dimers placed on all rungs (the so-called rung singlet-dimer state) along the special line $J_2=J_3$ up to $e_0^0<e_0^1$ , which is depicted in figure REF by dotted lines.", "Under this special condition, the intra-rung spin-spin correlation represents the only non-zero pair correlation function and all the other short-ranged spin-spin correlations vanish and/or change their sign across the special line $J_2=J_3$ .", "It should be remarked that the completely identical ground state can also be found in the symmetric Heisenberg two-leg ladder with $J_2=J_3$ (see figure REF (b)).", "To compare with, the rung singlet-dimer state is being the exact ground state of the symmetric Heisenberg-Ising ladder with $\\Delta =1$ for $J_1>J_2$ , while the symmetric Heisenberg ladder displays this simple factorizable ground state for $J_1>1.4015 J_2$ [35], [36] (this horizontal line is for clarity not shown in figure REF (b) as it exactly coincides with the ground-state boundary of the Heisenberg-Ising ladder extended over larger parameter space).", "It should be stressed, however, that the short-range spin-spin correlations become non-zero in QPM whenever $J_2 \\ne J_3$ even if this phase still preserves its disordered nature with the prevailing character of the rung singlet-dimer state.", "To support this statement, the zero-temperature dependencies of the order parameters and the nearest-neighbour spin-spin correlation along legs are plotted in figure REF .", "The relevant order parameters evidently disappear in QPM, whereas the nearest-neighbour correlation function changes its sign when passing through the special line $J_2=J_3$ of the rung singlet-dimer state.", "Figure: Zero-temperature variations of: (a) the order parameters; (b) the nearest-neighbour spin-spin correlation along legs;as a function of the Ising intra-leg interaction J 2 J_2 for J 1 =1J_1=1, Δ=1\\Delta =1 and three different values of the crossing Isinginteraction J 3 =0,0.6,1.2J_3=0,0.6,1.2.Note furthermore that the Heisenberg-Ising ladder undergoes the second-order quantum phase transition from the disordered QPM phase to the spontaneously long-range ordered Néel or SL phase, which predominantly appear in the parameter region where one from both Ising couplings $J_2$ and $J_3$ is being antiferromagnetic and the other one is ferromagnetic.", "The significant quantum reduction of staggered order parameters implies an obvious quantum character of both Néel as well as SL phases in which the antiferromagnetic correlation on rungs still dominates (see figure REF ).", "Thus, the main difference between both the quantum long-range ordered phases emerges in the character nearest-neighbour correlation along legs, which is ferromagnetic in the SL phase but antiferromagnetic in the Néel phase.", "The continuous vanishing of the order parameters depicted in figure REF (a) provides a direct evidence of the second-order quantum phase transition between the disordered QPM phase and the ordered SL (or Néel) phase, which is also accompanied with the weak-singular behaviour of the nearest-neighbour correlation function visualized in figure REF (b) by black dots.", "Here it should be noted that the $zz$ correlation function of the Heisenberg-Ising ladder can be easily derived from the result of the $xx$ correlation function of the transverse Ising chain calculated in [61].", "Last but not least, the strong enough Ising interactions $J_2$ and $J_3$ may break the antiferromagnetic correlation along rungs and lead to a presence of the fully ordered ferromagnetic rung states FM or SR depending on whether both Ising interactions are ferromagnetic or antiferromagnetic, respectively.", "It should be noticed that this change is accompanied with a discontinuous (first-order) quantum phase transition on behalf of a classical character of both FM and SR phases, which can be also clearly seen in figure REF from an abrupt change of the order parameters as well as the nearest-neighbour correlation function.", "Finally, let us also provide a more detailed comparison between the ground-state phase diagrams of the Heisenberg, Heisenberg-Ising and Ising two-leg ladders all depicted in figure REF (b).", "One can notice that the displayed ground-state phase diagrams of the Heisenberg and Heisenberg-Ising two-leg ladders have several similar features.", "The phase boundary between the classically ordered SR phase and three quantum phases (QPM, SL, Néel) of the Heisenberg-Ising ladder follows quite closely the first-order phase boundary between the Haldane-type phase and the dimerized phase of the pure Heisenberg ladder obtained using the series expansion and exact diagonalization [26].", "Of course, the most fundamental difference lies in the character of SR and Haldane phases, because the former phase exhibits a classical long-range order contrary to a more spectacular topological order of the pure quantum Haldane-type phase with a non-zero string order parameter [31] even though the ferromagnetic intra-rung correlation is common for both phases.", "The difference between the ground states of the Heisenberg-Ising and pure Heisenberg two-leg ladder becomes much less pronounced in the parameter region with the strong intra-rung interaction $J_1$ , which evokes the quantum phases in the ground state of both these models.", "In the case of a sufficiently strong frustration $J_2\\sim J_3$ , the ground state of the Heisenberg-Ising ladder forms the disordered QPM phase, whereas the antiferromagnetic or ferromagnetic (Néel or SL) long-range order emerges along the legs if the diagonal coupling $J_3$ is much weaker or stronger than the intra-leg interaction $J_2$ .", "Note that the quantum Néel and SL phases have several common features (e.g.", "predominant antiferromagnetic correlations along rungs) with the disordered QPM phase from which they evolve when crossing continuous quantum phase transitions given by the set of equations: $J_3=J_2-J_1$ (for $J_3<J_2$ ) and $J_3=J_2+J_1$ (for $J_3>J_2$ ).", "It cannot be definitely ruled out whether or not these quantum ordered phases may become the ground state of the pure Heisenberg ladder, because they are also predicted by the bond-mean-field approximation [39], [40], [41] but have not been found by the most of the numerical methods.", "Further numerical investigations of the Heisenberg ladder are therefore needed to clarify this unresolved issue." ], [ "Ground-state phase diagram in a non-zero field", "When the external field $h$ is switched on, the inequality for the ground states of the Ising chain in the longitudinal field (REF ) is generally valid only for $J_1+J_2\\le 0$ or $J_1+J_2\\ge 2|h|$ .", "Hence, it is necessary to modify the procedure of finding the ground states inside the region where the relation (REF ) is broken.", "However, one may use with a success the method suggested by Shastry and Sutherland [64], [25] in order to find the ground states inside this parameter region.", "Let us represent our effective Hamiltonian (REF ) in the form: $&& H=\\sum _{i=1}^N H_{i,i+1},\\nonumber \\\\&& H_{i,i+1}=\\frac{1}{2}\\sum _{l=i}^{i+1} \\Big \\lbrace J_1 (1-n_l) \\tilde{s}_l^z-2h n_l\\tilde{s}_l^x+\\frac{J_1\\Delta }{4}(2n_l-1) \\Big \\rbrace \\nonumber \\\\&& +\\left[ 2(J_2+J_3) n_i n_{i+1} +2(J_2-J_3)(1-n_i)(1-n_{i+1}) \\right]\\tilde{s}_i^x \\tilde{s}_{i+1}^x.$ Then, one can employ the variational principle implying that $E_0\\ge \\sum _{l=1}^N E_0(l,l+1)$ , where $E_0(l,l+1)$ is the lowest eigenenergy of $H_{l,l+1}$ .", "Looking for the eigenenergies of $H_{l,l+1}$ it is enough to find the lowest eigenstate of each bond configuration: $n_i=n_{i+1}=0$ , $E_0^{0,0}(i,i+1)=-\\frac{1}{2}\\sqrt{J_1^2+(J_2-J_3)^2}-\\frac{J_1\\Delta }{4}$ ; $n_i=n_{i+1}=1$ , $E_0^{1,1}(i,i+1)=\\left\\lbrace \\begin{array}{ll}\\frac{J_1\\Delta }{4}+\\frac{J_2+J_3}{2}-|h|, & \\mbox{if}\\: |h|>(J_2+J_3) (\\rm \\bf {FM}),\\\\\\frac{J_1\\Delta }{4}-\\frac{J_2+J_3}{2}, & \\mbox{if}\\: |h|\\le (J_2+J_3) (\\rm \\bf {AF}).\\end{array}\\right.\\nonumber $ $n_i=0$ , $n_{i+1}=1$ ($n_i=1$ , $n_{i+1}=0$ ), $E_0^{0,1}(i,i+1)=E_0^{1,0}(i,i+1)=-\\frac{J_1}{4}-\\frac{|h|}{2}$ .", "The phase corresponding to the alternating bond configuration $n_i=0$ , $n_{i+1}=1$ will be hereafter referred to as the staggered bond (SB) phase.", "It should be mentioned that there does not exist in the SB phase any correlations between spins from different rungs and the overall energy comes from the intra-rung spin-spin interactions and the Zeeman's energy of the fully polarized rungs [63].", "It can be easily seen from a comparison of $E_0^{0,1}(i,i+1)$ and $E_0^{1,1}(i,i+1)$ that the eigenenergy of the SB phase $E_0^{0,1}(i,i+1)$ has always lower energy than the lowest eigenenergy of the fully polarized state $E_0^{1,1}(i,i+1)$ inside the stripe: $J_2+J_3 - \\frac{J_1(1+\\Delta )}{2}\\le |h| \\le J_2+J_3 + \\frac{J_1(1+\\Delta )}{2}.$ Note that the ferromagnetic ordering is preferred for external fields above this stripe $|h| \\ge J_2+J_3 + \\frac{J_1(1+\\Delta )}{2}$ , while the antiferromagnetic ordering becomes the lowest-energy state below this stripe $|h| \\le J_2+J_3 - \\frac{J_1(1+\\Delta )}{2}$ .", "If one compares the respective eigenenergies of the staggered bond phase $E_0^{0,1}(i,i+1)$ and the uniform purity phase $E_0^{0,0}(i,i+1)$ ($n_i=n_{i+1}=0$ ), one gets another condition implying that $E_0^{0,1}(i,i+1)$ becomes lower than $E_0^{0,0}(i,i+1)$ only if: $|h|\\ge \\sqrt{J_1^2+(J_2-J_3)^2}-\\frac{J_1(1-\\Delta )}{2}.$ This means that the SB phase with the overall energy $E_0^{0,1}=-N(J_1+2|h|)/4$ becomes the ground state inside the region confined by conditions (REF ) and (REF ).", "The lowest field, which makes the SB phase favourable, can be also found as: $|h_{min}|=\\frac{J_1(1+\\Delta )}{2}.$ Similarly, one may also find the condition under which two eigenenergies corresponding to the uniform impurity configuration $E_0^{1,1} (n_i=n_{i+1}=1)$ become lower than the respective eigenenergy of the uniform purity configuration $E_0^{0,0} (n_i=n_{i+1}=0)$ .", "The ferromagnetic state of the uniform impurity configuration becomes lower if the external field exceeds the boundary value: $|h| \\ge \\frac{1}{2}\\left[J_2+J_3+J_1\\Delta + \\sqrt{J_1^2+(J_2-J_3)^2}\\right],$ whereas the condition for the antiferromagnetic state of the uniform impurity configuration is independent of the external field: $J_1\\Delta +\\sqrt{J_1^2+(J_2-J_3)^2}\\le (J_2+J_3).$ It is worthy of notice that it is not possible to find the ground state outside the boundaries (REF ), (REF ), (REF ) using the variational principle.", "However, it is shown in the appendix that the bond configuration, which corresponds to the ground state, cannot exceed period two.", "Therefore, one has to search for the ground state just among the states that correspond to the following bond configurations: all $n_i=0$ ; all $n_i=1$ ; $n_{2i-1}=0$ , $n_{2i}=1$ ($n_{2i-1}=1$ , $n_{2i}=0$ ).", "In this respect, two phases are possible inside the stripe given by (REF ).", "The ground state of the Heisenberg-Ising ladder forms the SB phase if $e_0^{0,1}<e_0^{0}$ : $|h| \\ge \\frac{2(J_1+|J_2-J_3|)}{\\pi }{\\mathbf {E}}(\\sqrt{1-\\gamma ^2})-\\frac{J_1(1-\\Delta )}{2}.$ Figure: Ground-state phase diagrams for the Heisenberg-Ising two-leg ladderin the J 2 -J 3 J_2-J_3 plane for J 1 =1J_1=1, Δ=1\\Delta =1 and two different values of the external field:(a) h=1.0h=1.0; (b) h=1.1h=1.1.Figure: Ground-state phase diagrams for the Heisenberg-Ising two-leg ladderin the J 2 -hJ_2-h plane for J 1 =1J_1=1, Δ=1\\Delta =1 and two different values of thecrossing interaction: (a) J 3 =0J_3=0; (b) J 3 =1J_3=1.Several ground-state phase diagrams are plotted in figures REF -REF for a non-zero magnetic field.", "The most interesting feature stemming from these phase diagrams is that the SB phase may become the ground state for the magnetic field $h\\ge \\frac{J_1(1+\\Delta )}{2}$ , which is sufficiently strong to break the rung singlet-dimer state.", "It can be observed from figure REF that the SB phase indeed evolves along the line of the rung singlet-dimer state and replaces the QPM phase in the ground-state phase diagram.", "The external magnetic field may thus cause an appearance of another peculiar quantum SB phase with the translationally broken symmetry, i.e.", "the alternating singlet and fully polarized triplet bonds on the rungs of the two-leg ladder.", "Hence, it follows that the SB phase emerges at moderate values of the external magnetic field and it consequently leads to a presence of the intermediate magnetization plateau at a half of the saturation magnetization.", "It is quite apparent from figures REF (a), REF (a) that the Heisenberg-Ising ladder without the frustrating diagonal Ising interaction $J_3=0$ exhibits this striking magnetization plateau just for the particular case of the antiferromagnetic Ising intra-leg interaction $J_2>0$ .", "On the other hand, the fractional magnetization plateau inherent to a presence of the SB phase is substantially stabilized by the spin frustration triggered by the non-zero diagonal Ising interaction $J_3 \\ne 0$ and hence, the plateau region may even extend over a relatively narrow region of the ferromagnetic Ising intra-leg interaction $J_2<0$ as well (see figure REF (b)).", "It should be noted that the same mechanism for an appearance of the magnetization plateau has also been predicted for the spin-$\\frac{1}{2}$ Heisenberg two-leg ladder by making use of exact numerical diagonalization and DMRG methods [22], [23], [24], [25], [34].", "Let us therefore conclude our study by comparing the respective ground-state phase diagrams of the Heisenberg-Ising and Heisenberg two-leg ladders in a presence of the external magnetic field displayed in figure REF .", "Figure: Ground-state phase diagram for the Heisenberg-Ising two-leg ladderin the J 1 -hJ_1-h plane for J 2 =1J_2=1, Δ=1\\Delta =1and two different values of the crossing interaction: (a) J 3 =0J_3=0; (b) J 3 =1J_3=1.For a comparison, figure (b) depicts by red lines the ground-state phase diagramof the pure Heisenberg two-leg ladder adapted from .According to this plot, both models give essentially the same magnetization process in the strong-coupling limit of the Heisenberg intra-rung interaction $J_1 \\gtrapprox 1.5$ , where two subsequent field-induced transitions in the following order QPM-SB-FM can be observed and consequently, the magnetization exhibits two abrupt jumps at the relevant transition fields.", "On the other hand, the fundamental differences can be detected in the relevant magnetization process of the Heisenberg-Ising and Heisenberg ladders in the weak-coupling limit of the intra-rung interaction $J_1$ .", "At low fields, the quantum Haldane-like phase constitutes the ground state of the pure Heisenberg ladder in contrast to the classical SR phase, which is being the low-field ground state of the Heisenberg-Ising ladder.", "In addition, the Heisenberg-Ising ladder still exhibits a magnetization plateau corresponding to a gapped SB phase at intermediate values of the magnetic field in this parameter space, while a continuous increase of the magnetization can be observed in the pure Heisenberg ladder due to a presence of the gapless Luttinger-liquid phase at moderate fields.", "Finally, it is also noteworthy that the saturation fields towards FM phase are also quite different for the Heisenberg-Ising and Heisenberg ladders in the weak-coupling limit of the Heisenberg intra-rung interaction." ], [ "Conclusions", "The frustrated Heisenberg-Ising two-leg ladder was considered within the rigorous approach using that $z$ -projection of total spin on a rung is a conserved quantity.", "By means of the pseudospin representation of bond states, we have proved the exact mapping correspondence between the investigated model and some generalized spin-$\\frac{1}{2}$ quantum Ising chain with the composite spins in an effective longitudinal and transverse field.", "We have also found the unitary transformation which reproduces the analogous mapping between the models.", "While the bond-state representation has more transparent physical interpretation, the unitary transformation gives the complete relation between spin operators of both models, and it might be useful for searching quantum spin ladders which admit exact solutions.", "It has been shown that the ground state of the model under investigation must correspond to a regular bond configuration not exceeding the period two.", "Hence, the true ground state will be the lowest eigenstate of either the transverse Ising chain, the Ising chain in the longitudinal field, or the non-interacting spin-chain model in the staggered longitudinal-transverse field.", "The most interesting results to emerge from the present study are closely related to an extraordinary diversity of the constructed ground-state phase diagrams and the quantum phase transitions between different ground-state phases.", "In an absence of the external magnetic field, the ground-state phase diagram constitute four different ordered phases and one quantum paramagnetic phase.", "The disordered phase was characterized through short-range spin correlations, which indicate a dominating character of the rung singlet-dimer state in this phase.", "On the other hand, the order parameters have been exactly calculated for all the four ordered phases, two of them having classical character and two purely quantum character as evidenced by the quantum reduction of the staggered magnetization in the latter two phases.", "Last but not least, it has been demonstrated that the external magnetic field of a moderate strength may cause an appearance of the peculiar SB phase with alternating character of singlet and triplet bonds on the rungs of two-leg ladder.", "This latter finding is consistent with a presence of the fractional magnetization plateau at a half of the saturation magnetization in the relevant magnetization process.", "The authors are grateful to Oleg Derzhko for several useful comments and suggestions.", "T.V.", "was supported by the National Scholarship Programme of the Slovak Republic for the Support of Mobility of Students, PhD Students, University Teachers and Researchers.", "J.S.", "acknowledges the financial support under the grant VEGA 1/0234/12." ], [ "Ground-state bond configuration in magnetic field", "In general each state of the model can be represented as an array of alternating purity and impurity non-interacting clusters of different length.", "One can formally write the lowest energy of some configuration as $E=E_0(N_1,M_1)+E_0(N_2,M_2)+\\dots +E_0(N_L,M_L),$ where $E_0(N_i,M_i)=E_0^0(N_i)+E_0^1(M_i)$ is the lowest energy of the complex of two independent spin chains which correspond to the purity and impurity bond states, $N_1+M_1+N_2+M_2+\\dots N_L+M_L=N$ .", "If $e_0(N_i,M_i)=E_0(N_i,M_i)/(N_i+M_i)$ corresponds to the lowest energy per spin among other clusters, it is evident that the ground state configuration is alternating purity and impurity clusters of length $N_i$ and $M_i$ .", "Let us consider at first the case when $h$ is restricted by condition (REF ).", "If the configuration contains the cluster of more than 2 impurity bonds, its energy can be lowered by adding pure bond in-between.", "If condition (REF ) is valid, such a configuration can achieve a lower energy.", "Using this procedure we can reduce the number of sites in impurity clusters to 2, i.e.", "only $E_0(N,1)$ or $E_0(N,2)$ can correspond to the ground state.", "Define the energy per spin for each configuration as: $e_0(N,1)=\\frac{E_0^0(N)+E_0^1(1)}{N+1},\\\\e_0(N,2)=\\frac{E_0^0(N)+E_0^1(2)}{N+2}.$ If $e_0(1,1)<e_0(2,1)$ and so on ($e_0(N,1)<e_0(N+1,1)$ ) the ground state corresponds to the staggered bond configuration.", "Suppose that for some $N$ $\\Delta e_0(N-1)&=&e_0(N,1)-e_0(N-1,1)\\nonumber \\\\&=&\\frac{1}{N(N+1)}(NE_0^0(N)-(N+1)E_0^0(N-1)-E_0^1(1))<0.$ Consequently, it is clear that $\\Delta e_0(N)&=&\\frac{1}{(N+1)(N+2)}((N+1)E_0^0(N+1)-(N+2)E_0^0(N)-E_0^1(1))<0,$ if $E_0^0(N+1)-2E_0^0(N)+E_0^0(N-1)\\le 0$ , i.e.", "the ground state energy of finite transverse Ising chain is a convex function of $N$ .", "The last condition is ensured by non-increasing $\\Delta E_0^0(N)=E_0^0(N+1)-E_0^0(N)$ ($\\Delta E_0^0(N)\\ge \\Delta E_0^0(N+1)$ ) that can be shown by numerical calculations for finite chains (see figure REF ) using the numerical approach [65].", "For $J_2=J_3$ we get the free spin Hamiltonian and $E_0^0(N)$ depends linearly on $N$ .", "Figure: ΔE 0 0 (N)=E 0 0 (N+1)-E 0 0 (N)\\Delta E_0^0(N)=E_0^0(N+1)-E_0^0(N) as a function of system size NNfor the transverse Ising chain with free ends (): J 1 =1J_1=1, Δ=0\\Delta =0,|J 2 -J 3 |=0,0.2,0.4,⋯,4|J_2-J_3|=0,0.2,0.4,\\dots ,4 from top to bottom.Conditions (REF ),(REF ) together mean that $e_0(N,1)$ may have only one extremum and it is the maximum.", "Therefore, $e_0(N,1)$ may take the minimal value only in two limiting cases $N=1$ , $N\\rightarrow \\infty $ .", "The same result can be analogously obtained for $e_0(N,2)$ .", "Now we can compare the energies of $(N,1)$ and $(N,2)$ configurations.", "For $|h|\\ge \\frac{J_1+J_2}{2}$ $e_0(1,1)-e_0(1,2)=\\frac{1}{6}\\left(-\\frac{J_1(1+\\Delta )}{2}-(J_2+J_3)+|h|\\right)<0,$ due to the upper boundary of condition (REF ).", "To summarize, the ground state configuration in the region confined by (REF ) can correspond to either staggered bond or uniform purity configuration.", "Let us consider the ground state for $|h|>J_2+J_3+\\frac{J_1(1+\\Delta )}{2}$ .", "Similarly to the previous case we can prove that $e_0(N,M)$ is a convex function of $N$ and $M$ if the ground state energies of the uniform chains have the following properties: $E_0^0(N+1)-2E_0^0(N)+E_0^0(N-1)\\le 0,\\nonumber \\\\E_0^1(N+1)-2E_0^1(N)+E_0^1(N-1)\\le 0.$ One can find that $E_0^1(N+1)-E_0^1(N)=\\frac{|J_2-J_3|}{4}$ , and $E_0^1(N+1)-2E_0^1(N)+E_0^1(N-1)= 0$ .", "That is why the minimal value of the ground state can be achieved in one of three cases: staggered bond, purity and impurity configuration.", "The staggered bond configuration should be excluded from this list for $|h|>J_2+J_3+\\frac{J_1(1+\\Delta )}{2}$ , since it cannot be the ground state due to the variational principle." ] ]

[ [ "A Constrained Random Demodulator for Sub-Nyquist Sampling" ], [ "Abstract This paper presents a significant modification to the Random Demodulator (RD) of Tropp et al.", "for sub-Nyquist sampling of frequency-sparse signals.", "The modification, termed constrained random demodulator, involves replacing the random waveform, essential to the operation of the RD, with a constrained random waveform that has limits on its switching rate because fast switching waveforms may be hard to generate cleanly.", "The result is a relaxation on the hardware requirements with a slight, but manageable, decrease in the recovery guarantees.", "The paper also establishes the importance of properly choosing the statistics of the constrained random waveform.", "If the power spectrum of the random waveform matches the distribution on the tones of the input signal (i.e., the distribution is proportional to the power spectrum), then recovery of the input signal tones is improved.", "The theoretical guarantees provided in the paper are validated through extensive numerical simulations and phase transition plots." ], [ "Introduction", "Modern signal processing relies on the sampling of analog signals for discrete-time processing.", "The standard approach to sampling signals is based on the Shannon–Nyquist sampling theorem, which states that a bandlimited signal can be faithfully reconstructed from its samples collected uniformly at the Nyquist rate.", "However, this standard approach to sampling can be unwieldy for signals with very large bandwidths due to the physical constraints on modern Analog-to-Digital Converter (ADC) technology.", "The rule of thumb in ADC technology is that a doubling of the sampling rate causes a 1 bit reduction in resolution [1] or, more explicitly, $P = 2^Bf_s$ where $B$ is the effective number of bits (ENOB), a measure of resolution of an ADC, and $f_s$ is the sampling rate.", "This expression states that for a required sampling resolution, the sampling rate has a hard upper limit due to constraints on the ADC technology, and vice versa.", "The constant $P$ is dependent on the particular ADC architecture and has steadily increased over time as the technology has improved; the current state-of-the-art allows for sampling at 1 GHz with a resolution of approximately 10 ENOB[2], [3].", "Unfortunately, this increase tends to happen rather slowly compared to the advancement seen in other areas of technology, such as microprocessor technology following Moore's law.", "In particular, applications such as spectrum sensing for cognitive radios push modern ADC technology to its limit.", "Though Nyquist sampling is the standard approach to sampling, other schemes have been considered that require a lower sampling rate for analog-to-digital conversion.", "The key to the success of these schemes is leveraging additional prior information about the class of signals to be sampled (perhaps in addition to being bandlimited).", "One such class of signals corresponds to complex-valued signals comprising a relatively small number of tones ($S$ ) in a very large (two-sided) bandwidth ($W$ ): $S \\ll W$ .", "We say these signals have sparse spectral content.", "This class of signals is of significant interest in applications such as spectrum sensing, and is the one we will concentrate on for the rest of this paper.", "We refer the reader to Section  for a mathematically precise definition of this signal class.", "Two good architectures to sample such signals are the Non-Uniform Sampler (NUS) [4], [5] and the Random Demodulator (RD) [6].", "In this paper we concentrate exclusively on the RD because it offers a much more general framework for sub-Nyquist sampling.", "The block diagram of the RD architecture is presented in Fig.", "REF and will be reviewed in more detail laterWhile we focus exclusively on a single-channel system, the analysis can be easily extended to the multi-channel setting..", "The major results for the RD can be summarized as follows [6]: let $\\mathrm {C}$ be a positive, universal constant and let $W$ be the Nyquist rate.", "The constituent tones of signals sampled by the RD can be recovered with high probability if the sampling rate $R$ scales as ($i$ ) $R\\ge \\mathrm {C}[S\\log W + \\log ^3 W]$ for signals composed of $S$ randomly located tonesIt is worth noting here that the NUS is shown to have similar results [5].", "and ($ii$ ) $R\\ge \\mathrm {C}S\\log ^6 W$ for signals composed of arbitrary $S$ tones.", "Contrast these results to the Shannon–Nyquist sampling theorem, which guarantees recovery of the original signal from its samples if $R \\ge W$ .", "A building block of the RD is a white noise-like, bipolar modulating waveform $p_m(t)$ (see Fig.", "REF ).", "This waveform switches polarity at the Nyquist rate of the input signal.", "An implicit assumption is that this waveform, in the analog domain, is made up of perfect square pulses with amplitude either $+1$ or $-1$ .", "Hardware constraints, however, mean that a real waveform cannot switch polarity instantaneously and will encounter shape distortion.", "A non-zero time $\\tau $ is required to switch polarity and is dictated by the circuits encountered in ADC architectures[7].", "The transitions therefore occur over this time-scale, and the square waveform can be viewed as passing through a low-pass filter with a bandwidth proportional to $1/\\tau $ .", "One implication is a reduction of the energy captured in the measurements that depends on $\\tau $ and the number of transitions in the waveform.", "For a larger $\\tau $ , or for more transitions in the waveform, less energy is captured in the measurements.", "Over 30 years ago a similar problem affected the peak detection of binary signals written on magnetic media.", "In magnetic recording, data is recovered by passing a read head over the media; a higher recording density means there is greater interference between the read-back voltages of adjacent bits.", "To reduce distortion in the read back voltages, Tang and Bahl introduced Run-Length Limited (RLL) sequences [8].", "Run-length constraints specify the minimum separation $d$ and the maximum separation $k$ between transitions from one symbol to another (say $+1$ to $-1$ ).", "Tang and Bahl proposed using these RLL sequences to increase the number of bits written on the magnetic medium by a factor of $d+1$ without affecting the read-back fidelity.", "Note that RLL sequences, compared to unconstrained sequences, require a longer length to store the same amount of information.", "Tang and Bahl nonetheless observed that for certain RLL sequences the fractional increase in length is smaller than $d + 1$ , leading to a net increase in recording density because the allowed closer spacing of the physical bits (on the medium) overcomes the increase in bit-sequence length.", "The reader may refer to [9] for further details and a nice overview on this topic.", "Figure: Block diagram of the random demodulator: The input signal is multiplied by a waveform generated from a Rademacher chipping sequence, then low-pass filtered, and finally sampled at a sub-Nyquist rate R≪WR \\ll W." ], [ "Our Contributions: Constrained Random Demodulation", "In this paper, we make two major contributions to the area of sub-Nyquist sampling for signals with sparse spectral content.", "Our first contribution is to apply the lessons learned from magnetic recording to the RD.", "Specifically, we replace the modulating waveform of the RD with a $(d,k)$ -constrained waveform generated from an RLL sequence (see Fig.", "REF ).", "We refer to such a sampling system as a Constrained Random Demodulator (CRD).", "The use of an RLL sequence reduces the average number of transitions in the waveform by a factor of $d+1$ , which results in an increase in the signal energy captured by the hardware.", "From another viewpoint, if we fix the acceptable energy loss (or average number of transitions in the waveform), then using an RLL sequence allows a larger input signal bandwidth.", "We do, of course, pay a price: an RLL sequence introduces statistical dependence across the waveform.", "Our first major contribution is therefore establishing that the CRD still enjoys some theoretical guarantees for certain choices of waveform.", "In fact, we explicitly show that the power spectrum of the waveform is the key to understanding these guarantees and, hence, to choosing the best RLL sequence.", "Further, we outline a tradeoff in acquirable bandwidth versus sparsity of the input signal and show through numerical simulations that a $20\\%$ increase in the bandwidth can be handled by the CRD with a negligible decrease in average performance.", "Our work here builds upon our preliminary work in [10], [11] that was primarily limited to introducing the idea of the CRD along with Theorem REF (without proof).", "Figure: An RLL sequence with (d,k)=(1,4)(d,k)=(1,4)Remark 1 Heuristically, the theoretical guarantees in this paper rely on two things: (i) each (active) tone leaves an identifiable signature that can be extracted from the measurements and (ii) the measurements capture a significant amount of energy of each tone.", "We will show that the identifiability depends on the modulating sequence power spectrum.", "Once this is established, we would further like to maximize the captured energy.", "Since an RLL waveform leads to an increase in the captured energy because of the switching constraints previously discussed, its use in a hardware implementation will lead to improved performance as long as it satisfies the identifiability criterion.", "Our second contribution is laying down the foundations of a concept that we call Knowledge-Enhanced Compressive Sensing (KECoM) for sub-Nyquist sampling, which we preliminarily explored in [11] with limited numerical experiments.", "In the context of the CRD, the principle of KECoM assumes that some tones in the input signal are statistically more likely to appear than others.", "An immediate application of this is a spectrum sensing problem where some regions of the spectrum are assigned a higher detection priority than others, but none are deemed uninformative.", "We show through numerical simulations that the distribution of the tones in the input signal has a profound effect on the reconstruction of input signals from samples collected using a CRD.", "Specifically, we show with phase transition plots [12] that if the prior distribution over the tones matches the power spectrum of the RLL sequence used by the CRD, then the reconstruction performance improves when compared to a uniform distribution over the tones.", "Note that [13], [14] have also recently explored ideas along similar lines, albeit for a different class of sequences.", "In contrast to [13], [14], we provide a theoretical analysis and, additionally, a comprehensive numerical analysis of RLL sequences in the RD by examining the phase transition plots." ], [ "Other Sub-Nyquist Sampling Schemes", "The work of Rife and Boorstyn [15] in the mid-70's is an early example of a successful sub-Nyquist sampling scheme.", "Their goal was to take samples of a sinusoid at a sub-Nyquist rate and then perform parameter estimation to determine the amplitude, frequency, and phase of a single, unknown tone.", "They also extended their work to the case of multiple tones in a large bandwidth [16].", "Their work, however, becomes intractable when considering more than a couple tones.", "This is an early example of what has become known as compressed sensing of sparse signals.", "Compressed Sensing (CS) is the systematic exploration of sparsity as a prior model for input signals and recovery of these signals from a small number of linear measurements [17].", "It has produced many analytical tools and algorithms for signal recovery.", "In addition to the RD, several other sub-Nyquist sampling architectures have taken advantage of ideas from CS including Chirp Sampling [18] and Xampling [19].", "While the RD considers a bandlimited input signal model with few active tones, several other classes of signals have been considered in the literature with the goal of finding more efficient sampling methods.", "One such class contains signals with so-called “Finite Rates of Innovation\" [20].", "Signals belonging to this class can be described by a finite number of degrees of freedom in a given time interval, and it has been shown that they can be reconstructed from a small number of samples that is proportional to the degrees of freedom in that time interval.", "Another class constitutes signals in “shift-invariant subspaces.\"", "These signals are composed of a superposition of shifted `generator' functions (e.g., splines or wavelets); see [21] for a nice overview of this signal class.", "In [22] and [23], this signal model is shown to provide an alternative to the bandlimited signal model; in particular, it allows the reconstruction of signals belonging to Sobolev spaces with an approximation error that scales polynomially with the sampling period.", "One possible drawback to utilizing the RD for sampling is its assumed discrete-frequency signal model (cf. Section ).", "Specifically, the RD assumes that the input signal can be described by a discrete set of integral frequencies, while real-world signals are likely to contain tones off this grid.", "While this signal model might not entirely describe real-world signals, the effectiveness of the RD architecture has been successfully demonstrated in the lab [24], [25].", "To address signals with tones not conformant to the integral-frequency assumption, we consider energy leakage in the frequency domain.", "A tone that does not fall exactly on the assumed frequency grid will leak energy across several tones due to the inherent windowing.", "The result is that a signal which is $S$ -sparse in the analog domain becomes $(aS)$ -sparse after being sampled, where $a > 1$ .", "Other schemes, such as Xampling [19], offer an alternative approach assuming a different signal model; the pros and cons of both systems are examined in [26].", "While our focus in this paper is exclusively on the RD, we believe that our contributions could have implications for other sub-Nyquist architectures.", "Specifically, the Xampling architecture uses modulating sequences similar to the ones used in the RD/CRD, and we believe that RLL sequences could benefit the Xampling architecture as well.", "A detailed analysis is, however, beyond the scope of this paper.", "We would also like to point to a possible utility of RLL sequences in the NUS.", "The implementation described in [4] requires a minimum and maximum spacing between sample points while the analysis in [5] assumes the sample points are uniformly random without any constraints.", "The constraints in [4] can thus be described by an RLL sequence made up of 0's and 1's with 1's representing sampling points.", "We feel this interpretation of the limitations in [4] can help us mathematically analyze the architecture in [4], but a detailed investigation of this is beyond the scope of this paper." ], [ "Organization and Notation", "The remainder of the paper is organized as follows.", "We first provide some background on the RD in Section and then explain the challenges encountered by introducing RLL sequences into the RD architecture in Section .", "We then present our main theoretical results in Section and two examples of constrained sequences, one with bad results (Section ) and one with good results (Section ), to illustrate the effectiveness of our analysis.", "Finally, in Sections and we present numerical simulations to offer some verification of the theoretical results.", "In the following we denote matrices with upper case roman letters and vectors with lower case roman letters.", "Scalars are denoted with italic lower case letters.", "We write $^*$ for the conjugate transpose of a matrix, vector, or scalar.", "We reserve the letters $\\mathrm {C}$ and $\\mathrm {c}$ in roman font to denote universal constants that could change values at each instance.", "For a matrix, $\\mathrm {A}|_{\\Omega \\times \\Omega }$ denotes the principal submatrix of $\\mathrm {A}$ created from the columns/rows given in $\\Omega $ .", "We also use $||\\cdot ||$ for the spectral norm of a matrix and $||\\cdot ||_{\\max }$ for the maximum absolute entry of a matrix.", "For a random variable $\\mathrm {B}$ , let $\\mathbb {E}[\\mathrm {B}]$ be the expectation and $\\mathbb {E}^p\\mathrm {B} = (\\mathbb {E}|\\mathrm {B}|^p)^{1/p}$ .", "Let $\\mathbb {P}\\lbrace \\cdot \\rbrace $ denote the probability of an event.", "The short-hand $j\\sim r$ means $(r-1)W/R < j \\le rW/R$ for some $W$ and $R$ such that $R$ divides $W$ ." ], [ "Background: The Random Demodulator", "We start with a brief review of the RD architecture and highlight the key components that allow sampling of sparse, bandlimited signals and refer the reader to [6] for a thorough overview.", "To start, the RD takes samples at a sub-Nyquist rate $R$ while retaining the ability to reconstruct signals that are periodic, (two-sided) bandlimited to $W$ Hz, and completely described by a total of $S \\ll W$ tones.", "These conditions describe a large class of wide-band analog signals comprised of frequencies that are small in number relative to the total bandwidth but are at unknown locations.", "Formally, the input signal to a RD takes the following parametric form $f(t)=\\sum _{\\omega \\in \\Omega }a_{\\omega }e^{-2\\pi \\imath \\omega t},\\ t \\in [0,1)$ where $\\Omega \\subset \\lbrace 0,\\pm 1,...,\\pm W/2-1, W/2\\rbrace $We assume $W$ is even.", "An appropriate change of the set $\\Omega $ would cover the case of $W$ odd.", "is a set of $S$ integer-valued frequencies and $\\lbrace a_{\\omega } : \\omega \\in \\Omega \\rbrace $ is a set of complex-valued amplitudes.", "Fig.", "REF illustrates the actions performed by the RD.", "The input $f(t)$ is first multiplied by $p_m(t) = \\sum _{n=0}^{W-1}\\varepsilon _n 1_{\\left[\\frac{n}{W},\\frac{n+1}{W}\\right)}(t),$ where the discrete-time modulating sequence $\\varepsilon = [\\varepsilon _n]$ is a Rademacher sequence, a random sequence of independent entries taking values $\\pm 1$ equally likely.", "Next, the continuous-time product $f(t)p_m(t)$ is low-pass filtered using an “integrate and dump” filter.It can be easily shown that the frequency response of this filter tapers off at high frequencies.", "Hence, it is a low-pass filter.", "Finally, samples are taken at the output of the low-pass filter at a rate of $R \\ll W$ to obtain $y[n]$ ." ], [ "Matrix Representation of the Random Demodulator", "One of the major contributions of [6] is expressing the actions of the RD on a continuous-time, sparse, and bandlimited signal $f(t)$ in terms of the actions of an $R \\times W$ matrix $\\mathrm {\\Phi _{RD}}$ on a vector $\\mathrm {\\alpha } \\in \\mathbb {C}^W$ that has only $S$ nonzero entries.", "Specifically, let $\\mathrm {x} \\in \\mathbb {C}^W$ denote a Nyquist-sampled version of the continuous-time input signal $f(t)$ so that $\\mathrm {x}_n = f(\\frac{n}{W})$ , $n = 0,\\cdots ,W-1$ .", "It is then easy to conclude from (REF ) that $\\mathrm {x}$ can be written as $\\mathrm {x} = \\mathrm {F} \\alpha $ , where the matrix $\\mathrm {F} = \\frac{1}{\\sqrt{W}} \\left[e^{-2\\pi \\imath n\\omega /W}\\right]_{(n,\\omega )}$ denotes a (unitary) discrete Fourier transform matrix and $\\alpha \\in \\mathbb {C}^W$ has only $S$ nonzero entries corresponding to the amplitudes, $a_{\\omega }$ , of the nonzero frequencies in $f(t)$ .", "Next, the effect of multiplying $f(t)$ with $p_m(t)$ in continuous-time is equivalent in the discrete-time Shannon–Nyquist world to multiplying a $W \\times W$ diagonal matrix $\\mathrm {D} = \\text{diag}(\\varepsilon _0, \\varepsilon _1, \\cdots , \\varepsilon _{W-1})$ with $\\mathrm {x} = \\mathrm {F} \\alpha $ .", "Finally, the effect of the integrating filter on $f(t) \\cdot p_m(t)$ in the discrete-time Shannon–Nyquist setup is equivalent to multiplying an $R \\times W$ matrix $\\mathrm {H}$ , which has $W/R$ consecutive ones starting at position $rW/R+1$ in the $r^{th}$ row of $\\mathrm {H}$ and zeros elsewhere, with $\\mathrm {D}\\mathrm {F}\\alpha $ .Throughout this paper we assume that $R$ divides $W$ ; otherwise, a slight modification can be made to $\\mathrm {H}$ as discussed in [6].", "An example of $\\mathrm {H}$ for $R=3$ and $W=9$ is $\\mathrm {H}=\\begin{bmatrix}1&1&1&&&&&& \\\\&&&1&1&1&&& \\\\&&&&&&1&1&1\\end{bmatrix}$ The RD collects $R$ samples per second, and therefore, the $R$ samples collected over 1 second at the output of the RD can be collected into a vector $\\mathrm {y} \\in \\mathbb {C}^R$ .", "It follows from the preceding discussion that $\\mathrm {y} = \\mathrm {H}\\mathrm {D}\\mathrm {F}\\alpha = \\mathrm {\\Phi _{RD}}\\cdot \\alpha $ , where we have the complex-valued random demodulator matrix $\\mathrm {\\Phi _{RD}} = \\mathrm {H}\\mathrm {D}\\mathrm {F}$ ." ], [ "Signal Recovery", "Given the discrete-time representation $\\mathrm {y} = \\mathrm {\\Phi _{RD}}\\alpha $ , recovering the continuous-time signal $f(t)$ described in (REF ) is equivalent to recovering the $S$ -sparse vector $\\alpha $ from $\\mathrm {y}$ .", "In this regard, the primary objective of the RD is to guarantee that $\\alpha $ can be recovered from $\\mathrm {y}$ even when the sampling rate $R$ is far below the Nyquist rate $W$ .", "Recent theoretical developments in the area of CS provide us with greedy as well as convex optimization-based methods that are guaranteed to recover $\\alpha $ (or a good approximation of $\\alpha $ ) from $\\mathrm {y}$ (possibly in the presence of noise) as long as the sensing matrix $\\mathrm {\\Phi _{RD}}$ satisfies certain geometrical properties [17].", "Tropp et al.", "[6] uses two properties from the CS literature to analyze the sensing matrix.", "The first is the coherence.", "The coherence $\\mu $ of a matrix $\\mathrm {\\Phi }$ is the largest inner product between its (scaled to unit-norm) columns $\\phi _{\\omega }$ : $\\mu = \\max _{\\omega \\ne \\alpha }|\\langle \\phi _{\\omega },\\phi _{\\alpha }\\rangle |$ .", "Many recovery algorithms rely on the coherence of the sensing matrix being sufficiently small [27].", "The analysis in [6] in this regard also relies on the input signals conforming to a random signal model: given the signal model (REF ), the index $\\Omega $ is a set of $S$ tones drawn uniformly at random from the set of $W$ possible tones.", "Further, the coefficients $a_{\\omega }$ are drawn uniformly at random from the complex unit circle.", "Under this signal model, $S$ -sparse signals are recoverable with high probability if the sampling rate scales as $R \\ge \\mathrm {C}[S\\log W + \\log ^3 W]$[6].", "The second property used in [6] is the Restricted Isometry Property (RIP)[28].", "Definition 1 The RIP of order $S$ with restricted isometry constant $\\delta _S \\in (0,1)$ is satisfied for a matrix $\\Phi $ with unit-norm columns if $(1-\\delta _S)||\\mathrm {x}||_2^2 \\le ||\\mathrm {\\Phi }\\mathrm {x}||_2^2 \\le (1+\\delta _S)||\\mathrm {x}||_2^2$ or equivalently $\\left|\\frac{\\Vert \\Phi \\mathrm {x}\\Vert _2^2 - \\Vert \\mathrm {x}\\Vert _2^2}{\\Vert \\mathrm {x}\\Vert _2^2}\\right|\\le \\delta _S$ for every $\\mathrm {x}$ with $\\Vert \\mathrm {x}\\Vert _0 \\le S$ .", "Here, $||\\mathrm {x}||_0$ counts the number of non-zero entries in $\\mathrm {x}$ .", "Note that RIP-based analysis tends to be stronger than the coherence-based analysis because the RIP provides a better handle on worst-case performance as well as on performance in the presence of noise[29].", "It also provides stable recovery even if the signal is not exactly sparse, but is well-described by a sparse signal (so-called compressible signals)[29].", "We will therefore focus only on proving the RIP with the understanding that RIP automatically implies stable and robust recovery (see [17] and the references therein for a more extensive list of results).", "In this paper, we use the “triple-bar\" norm of [6] to describe the RIP condition.", "Given a matrix $\\mathrm {A}$ and set of indices $\\Omega \\subset \\lbrace 0,\\ldots , W-1\\rbrace $ , the triple-bar norm captures the least upper bound on the spectral norm of any $S\\times S$ principal submatrix of $\\mathrm {A}$ : $|||\\mathrm {A}||| = \\sup _{|\\Omega | \\le S}\\Vert \\mathrm {A}|_{\\Omega \\times \\Omega }\\Vert .$ It can easily be checked that $|||\\cdot |||$ is a norm and that (REF ) is satisfied if and only if $|||\\Phi ^*\\Phi - \\mathrm {I}||| \\le \\delta _S$ .", "The main result of [6] in this respect is that the RD matrix satisfies the RIP of order $S$ as long as the sampling rate $R$ scales as $R \\ge \\mathrm {C}S\\log ^6W$ ." ], [ "Constrained Random Demodulator", "As described in the previous section, the RD uses a random waveform generated from a Rademacher sequence with transition density of $\\frac{1}{2}$ (on average, one transition every 2 Nyquist periods).", "However, limitations of analog circuits imply that each transition in the waveform results in a loss of energy compared to a waveform with ideal square pulses[7].", "RLL sequences are an attractive way to generate waveforms with a reduced transition density of $\\frac{1}{d+2}$ .", "Additionally, we will later show that RLL sequences can also lead to superior performance for specific classes of input signals.", "We remind the reader that if an RLL sequence is used we call the resulting system a Constrained Random Demodulator (CRD) and denote the corresponding system matrix as $\\mathrm {\\Phi _{CRD}} = \\mathrm {HDF}$ where $\\mathrm {D}$ contains an RLL sequence $\\varepsilon $ instead of a Rademacher sequence.", "The properties of the Rademacher sequence, in particular independence, are central to the analysis of the RD in [6]; we therefore must carefully consider the impact of using an RLL sequence that is inherently correlated.", "The strength of [6] is that it shows that the RD matrix satisfies the RIP with high probability, allowing strong guarantees to be made about the recovery of signals sampled with the RD.", "The RIP is satisfied primarily because of three properties of the RD matrix: ($i$ ) the Gram matrix averages (over realizations of the modulating sequence) to the identity matrix, ($ii$ ) the rows are statistically independent, and ($iii$ ) the entries are uniformly bounded.", "All three properties rely on the independence of the modulating sequence.", "In the CRD, we have to deal with dependence across $\\varepsilon $ .", "Nevertheless, the last two properties are handled relatively easily.", "Specifically, if we can find some distance between entries in $\\varepsilon $ such that any two entries, when separated by this distance, are independent, then we can partition the rows of $\\mathrm {\\Phi _{CRD}}$ (or entries of $\\varepsilon $ ) into sets of independent rows (entries).", "We can then find bounds similar to those found in [6] for these sets and take a union bound over all the sets to obtain the desired properties." ], [ "Maximum Dependence Distance", "To make the previous discussion more concrete, recall that the $(r,\\omega )$ entry of $\\mathrm {\\Phi _{CRD}}$ is $\\varphi _{r\\omega } = \\sum _{j\\sim r}\\varepsilon _jf_{j\\omega }.$ If $\\varepsilon $ is an independent sequence, then each $\\varphi _{r\\omega }$ is a sum of independent random variables, and each row of $\\mathrm {\\Phi _{CRD}}$ is independent.", "However, if we use a correlated sequence then the rows may not be independent, and it is important to know the extent of the dependence within the sequence.", "Definition 2 The Maximum Dependence Distance (MDD), $\\ell $ , for a modulating sequence $\\varepsilon $ is the smallest $\\ell $ such that $\\mathbb {E}[\\varepsilon _j\\varepsilon _{j+k}] = 0$ for all $j$ and $|k| \\ge \\ell $ Note that this is a correlation distance, but that for the bipolar sequences of our concern, uncorrelated implies independent.. Now, if we define $\\rho = \\lceil \\frac{R}{W}(\\ell -1)\\rceil \\le (\\ell -1)$ , then any two rows of $\\mathrm {\\Phi _{CRD}}$ separated by at least $\\rho +1$ rows will be independent.", "Given $\\rho $ and $\\ell $ , we can now partition the rows of $\\mathrm {\\Phi _{CRD}}$ into $\\rho +1$ subsets where the rows in each subset are independent.We assume for convenience that $\\rho +1$ divides $R$ .", "This can be readily relaxed by adjusting the size of the last subset.", "Using this partitioning scheme, we can proceed with the analysis of independent rows and finally take a union bound over all subsets.", "Using $\\ell $ , we can similarly show that each entry of $\\mathrm {\\Phi _{CRD}}$ is uniformly bounded.", "The details are in Appendices and ." ], [ "The Gram Matrix", "Analysis of the Gram matrix of $\\mathrm {\\Phi _{CRD}}$ is a little more involved.", "To start, denote the columns of $\\mathrm {\\Phi _{CRD}}$ by $\\mathrm {\\phi }_{\\omega }$ and note that the $(r,\\omega )$ entry of $\\mathrm {\\Phi _{CRD}}$ is given by (REF ).", "The Gram matrix is a tabulation of the inner products between the columns and (as calculated in[6]) is given by $\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}} = \\mathrm {I} + \\mathrm {X}$ .", "Here, the $(\\alpha ,\\omega )$ entry of $\\mathrm {X}$ is the sum $x_{\\alpha \\omega } = \\sum _{j\\ne k}\\varepsilon _j\\varepsilon _k\\eta _{jk}f^*_{j\\alpha }f_{k\\omega }$ where $[\\varepsilon _0,\\cdots ,\\varepsilon _{W-1}] = \\varepsilon $ is the modulating sequence, $\\eta _{jk} = \\langle h_j,h_k\\rangle $ with $h_j$ being the $j$ th column of $\\mathrm {H}$ , and $f_{j\\alpha }$ is the $(j,\\alpha )$ entry of the (unitary) Fourier matrix $\\mathrm {F}$ .", "Expanding $\\eta _{jk}$ , we have that $\\eta _{jk} = {\\left\\lbrace \\begin{array}{ll} 1, & \\frac{W}{R}r \\le j,k < \\frac{W}{R}(r+1) \\\\ 0, & \\text{otherwise} \\end{array}\\right.", "}$ for each $r = 0,\\cdots ,R-1$ .", "We see that $\\eta _{jk}$ acts as a `windowing' function in the sum.", "In expectation, the Gram matrix is $\\mathbb {E}[\\mathrm {\\Phi _{CRD}^*}\\mathrm {\\Phi _{CRD}}] = \\mathrm {I} + \\mathbb {E}[\\mathrm {X}] = \\mathrm {I} + \\Delta $ where we have identified $\\Delta \\equiv \\mathbb {E}[\\mathrm {X}]$ with entries $\\Delta _{\\alpha \\omega } = \\sum _{j\\ne k}\\eta _{jk}f^*_{j\\alpha }f_{k\\omega }\\mathbb {E}[\\varepsilon _j\\varepsilon _k].$ Note that $\\Delta $ is completely determined by the autocorrelation of $\\varepsilon $ .", "If an independent $\\varepsilon $ is used (such as for the RD) then $\\mathbb {E}[\\varepsilon _j\\varepsilon _k] = 0$ for $j\\ne k$ , $\\Delta = \\mathrm {0}$ , and $\\mathbb {E}[\\mathrm {\\Phi _{RD}^*}\\mathrm {\\Phi _{RD}}] = \\mathrm {I}$ .", "In [6], this relation is taken to mean that the columns of $\\mathrm {\\Phi _{RD}}$ form an orthonormal system in expectation.", "This can of course never be true if $R<W$ , and the RIP is shown by bounding the deviation from this expectation in $|||\\cdot |||$ .", "If $\\varepsilon $ has non-zero correlations, however, then $\\Delta $ does not disappear and the expectation of the Gram matrix is not the identity matrix.", "To establish the RIP in this case, we still need to bound the deviation of the Gram matrix from the identity matrix, but now we must also contend with $\\Delta $ .", "Nevertheless, if this matrix is small in $|||\\cdot |||$ then our task is easier.", "Since the autocorrelation of $\\varepsilon $ determines $\\Delta $ , we want to choose a $\\varepsilon $ that produces small $|||\\Delta |||$ .", "In particular, recall that the RIP of order $S$ is satisfied if $|||\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}} - \\mathrm {I}||| \\le \\delta _S.$ Expressing $\\mathrm {I} = \\mathbb {E}[\\mathrm {\\Phi _{CRD}^*\\Phi _{CRD}}] - \\Delta $ , the left-hand side of (REF ) can be bounded as $|||&\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}} - \\mathrm {I}||| \\\\&= |||\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}} - \\mathbb {E}[\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}}] + \\mathrm {\\Delta }||| \\\\&\\le |||\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}} - \\mathbb {E}[\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}}]||| + |||\\mathrm {\\Delta }||| $ due to the triangle inequality.", "Therefore, to show the RIP we must upper bound the two terms in (REF ).", "The first term will be bounded using an argument very similar to that used in [6] but modified to deal with the correlations in $\\varepsilon $ .", "Since the second term, $|||\\Delta |||$ , is determined by the autocorrelation of $\\varepsilon $ , we will provide a bound on $|||\\Delta |||$ that directly relates to the choice of $\\varepsilon $ ." ], [ "Main Results", "The preceding discussion on $\\ell $ and $\\Delta $ enables us to make a statement about the RIP of a CRD that uses a correlated modulating sequence.", "Theorem 1 (RIP for the CRD) Let $\\mathrm {\\Phi _{CRD}}$ be an $R \\times W$ CRD matrix using a modulating sequence with maximum dependence distance $\\ell $ and $\\Delta $ (as defined by (REF )).", "Next, pick $\\delta , \\delta ^{\\prime } \\in (0,1)$ such that $\\delta ^{\\prime } < \\delta - |||\\Delta |||$ and suppose that $R$ divides $W$ , $\\ell $ divides $\\frac{W}{R}$ ,Throughout this paper, these requirements can be readily relaxed through meticulous accounting of terms in the analysis.", "and $R$ satisfies $R \\ge \\ell ^3\\delta ^{\\prime -2}\\mathrm {C}S\\log ^6(W)$ where $\\mathrm {C}$ is a positive constant.", "Then with probability $1-\\mathcal {O}(W^{-1})$ the CRD matrix $\\mathrm {\\Phi _{CRD}}$ satisfies the RIP of order $S$ with constant $\\delta _S\\le \\delta $ .", "The proof is provided in Appendix .", "As with the RD, the sampling rate $R$ must scale linearly with the sparsity $S$ of the input signal and (poly)logarithmically with the bandwidth $W$ .", "The sampling rate, however, also depends on the maximum correlation distance $\\ell $ and on the matrix $\\Delta $ .", "Both of these are determined by the choice of $\\varepsilon $ .", "If we choose an independent (i.e., unconstrained) $\\varepsilon $ , then $\\ell = 1$ , $\\Delta = 0$ and we get back the RD result of [6].", "For a constrained $\\varepsilon $ , we must restrict ourselves to sequences such that $\\Delta $ satisfies $|||\\Delta ||| < 1$ .", "Obviously we would like to find sequences for which both $\\ell $ and $|||\\Delta |||$ are as small as possible.", "With this criterion in mind, in the next two sections we will examine two classes of sequences to see how well they work in the CRD framework.", "In addition to the RIP, we also use the coherence of the sensing matrix to provide results for the random signal model described in Section .", "In the sequel, we use a matrix from [30] to capture the dependence in $\\varepsilon $ .", "For a sequence $\\varepsilon $ , define the triangular matrix $\\Gamma $ of “mixing coefficients\" as $\\Gamma = \\lbrace \\gamma _{ij}\\rbrace $ with $\\gamma _{ij} = {\\left\\lbrace \\begin{array}{ll} 0, & i>j \\\\1, & i=j \\\\\\big |\\mathbb {P}\\left(\\varepsilon _{j} = +1|\\varepsilon _i=-1\\right) \\\\ - \\mathbb {P}\\left(\\varepsilon _{j} = +1|\\varepsilon _i=+1\\right)\\big |, & i<j.\\end{array}\\right.", "}$ Theorem 2 (Recovery under the random signal model) Suppose that the sampling rate satisfies $R \\ge \\mathrm {C}\\ell ^2[S\\log W + \\log ^3 W]$ for some positive constant $\\mathrm {C}$ , and that $R$ divides $W$ and $\\ell $ divides $\\frac{W}{R}$ .", "Also suppose that $W$ satisfies $\\frac{\\log ^2 W}{\\sqrt{W}} \\le \\frac{\\mathrm {C}}{16\\sqrt{(\\ell -1)}||\\Gamma ||^2}.$ Now, let $\\mathrm {\\alpha }$ be a vector with $S$ non-zero components drawn according to the random signal model in Section REF , and let $\\mathrm {\\Phi _{CRD}}$ be an $R\\times W$ CRD matrix using a stationary modulating sequence with maximum dependence distance $\\ell $ .", "Let $\\mathrm {y} = \\mathrm {\\Phi _{CRD}} \\mathrm {\\alpha }$ be the samples collected by the CRD.", "The solution to the convex program $\\hat{\\mathrm {\\alpha }} = \\arg \\min _{\\mathrm {v}} ||\\mathrm {v}||_1 \\ \\text{subject to} \\ \\mathrm {\\Phi _{CRD}}\\mathrm {v} = \\mathrm {y}$ satisfies $\\hat{\\mathrm {\\alpha }} = \\mathrm {\\alpha }$ with probability $1-\\mathcal {O}(W^{-1})$ .", "The proof is given in Appendix .", "The bounds offered here are similar to those in [6] with the rate scaling linearly with the sparsity $S$ and logarithmically with the bandwidth $W$ but more tightly constrained by the factor of $\\ell ^2$ and the extra constraint on $W$ .", "Because the choice of modulating sequence plays such a pivotal role in our analysis of the CRD, a natural question is what types of sequences offer good performance and what types offer bad performance.", "In the sequel, we analyze two different types of sequences: one for which Theorems REF and REF (approximately) apply, and one for which they do not.", "Numerical experiments in Section then show that these results appear to be tight.", "Nevertheless, we must stress two points here.", "First, Theorems REF and REF are only sufficient conditions on the sampling rate and modulating sequence; a different analysis could offer stronger results.", "Second, the modulating sequences that are shown to work well numerically satisfy Theorems REF and REF in an approximate sense.", "From an engineering perspective, however, the approximation (discussed in Sec. )", "is well justified and validated further by the numerical experiments." ], [ "Repetition-coded Sequences", "We begin by analyzing sequences that satisfy the RLL constraints and have a small value of $\\ell $ but have a large $|||\\Delta |||$ and do not satisfy Theorems REF or REF .", "Definition 3 A repetition-coded sequence (RCS) is generated from a Rademacher sequence by repeating each element $d$ times.", "Let the repetition-coded sequence be denoted as $\\mathrm {\\varepsilon _{RCS}} = [\\varepsilon _0,\\ldots ,\\varepsilon _{W-1}]$ and let $[\\varepsilon _{(d+1)n}]$ , $0\\le n \\le \\frac{W}{d+1}-1$ be a Rademacher sequence.", "We then require for $1\\le i \\le d$ and each $n$ that $\\varepsilon _{(d+1)n} = \\varepsilon _{(d+1)n+i}.$ Such a sequence switches at a rate of $W/(d+1)$ .", "We discuss these sequences because they are one of the simplest forms of RLL sequences and also have very small MDD.", "To see this, notice that each group of repeated elements, $[\\varepsilon _{(d+1)n+i}]$ for $0\\le i \\le d$ , is completely dependent while independent of every other element in the sequence.", "The maximum dependence distance is $\\ell = d+1$ .", "Since the performance of the CRD also depends upon $|||\\Delta |||$ , we need to bound $|||\\Delta |||$ and understand its behavior.", "To start, assume that $R$ divides $W$ and $\\ell $ divides $\\frac{W}{R}$ and denote by $\\mathrm {\\varepsilon _{RCS}}$ an RCS.", "Let $\\mathrm {\\Phi _{RCS}}$ be a CRD matrix that uses $\\mathrm {\\varepsilon _{RCS}}$ as the modulating sequence: $\\mathrm {\\Phi _{RCS}} = \\mathrm {HDF}$ where $\\mathrm {D}$ contains $\\mathrm {\\varepsilon _{RCS}}$ on its diagonal.", "It is convenient to rewrite the entries of $\\Delta $ , given in (REF ), in this case as $\\Delta _{\\alpha \\omega } = \\sum _{j,k\\ne 0}\\eta _{j(j+k)}f^*_{j\\alpha }f_{(j+k)\\omega }\\mathbb {E}[\\varepsilon _j\\varepsilon _{j+k}].$ To calculate $|||\\Delta |||$ , it will be convenient to focus on the Gram matrix $\\mathrm {\\Lambda } = \\Delta ^*\\Delta $ , which has entries $\\mathrm {\\Lambda }_{\\alpha ,\\omega } = {\\left\\lbrace \\begin{array}{ll} \\frac{W}{\\ell }\\sum _{j=0}^{\\ell -1}e^{-\\pi \\imath qj}\\hat{F}(j,\\omega )\\hat{F}^*(j,\\alpha ), & \\omega -\\alpha = \\frac{W}{\\ell }q \\\\0, & \\text{otherwise} \\end{array}\\right.", "}$ where $\\hat{F}(j,\\omega ) = \\sum _{m\\ne 0}\\eta _{j(j+m)}f_{m\\omega }\\mathbb {E}[\\varepsilon _{j}\\varepsilon _{j+m}]$ and $q = 0,\\pm 1,...,\\pm (\\ell -1)$ .", "We bound $|||\\mathrm {\\Delta }|||$ by studying the entries of $\\mathrm {\\Lambda }$ .", "To do this, recall from the definition of the spectral norm that for a matrix $\\mathrm {A}$ we have $\\Vert \\mathrm {A}|_{\\Omega ^{\\prime }\\times \\Omega ^{\\prime }}\\Vert \\le \\Vert \\mathrm {A}|_{\\Omega \\times \\Omega }\\Vert $ for any $\\Omega ^{\\prime } \\subset \\Omega $ .", "We can therefore lower bound $|||\\Delta |||$ in this case by using $\\Omega $ such that $|\\Omega | = 1$ , i.e., $S=1$ .", "For $S=1$ , $|||\\Delta |||$ is the square root of the maximum entry on the diagonal of $\\mathrm {\\Lambda }$ .", "Applying (REF ) to the autocorrelation in (REF ), it is straightforward to show that $\\mathrm {\\Lambda }_{\\omega ,\\omega } = \\frac{W}{d+1}\\sum _{j=0}^{d}\\sum _{\\begin{array}{c}m=-j \\\\ m\\ne 0\\end{array}}^{d-j}\\sum _{\\begin{array}{c}k=-j \\\\ k\\ne 0\\end{array}}^{d-j}f_{m\\omega }^*f_{k\\omega },$ and that (REF ) is maximized by $\\omega =0$ .", "This results in $\\Lambda _{0,0} = d^2$ and, in the case of an RCS, for any $S$ that $|||\\Delta ||| \\ge 1$ .", "Finally, in the context of Theorem REF , recall (REF ) for the case of $\\alpha = \\omega $ .", "In this case, it is easy to see that $|x_{\\alpha \\omega }| \\ge d+1$ .", "Theorem REF , on the other hand, relies on bounding $||\\mathrm {X}||_{\\max }$ close to 0 (the details are in Appendix ) and this obviously cannot be done for an RCS.", "We see that Theorems REF and REF do not hold for $\\mathrm {\\Phi _{RCS}}$ .", "Although we do not have converses, we demonstrate the tightness of our theory for an RCS through numerical experiments.", "For this, we calculate the minimum and maximum singular values of the submatrices over an ensemble of matrices $\\mathrm {\\Phi _{RCS}}$ generated using an RCS with $d=1$ .", "The submatrices are chosen by picking $S=10$ columns at random from $\\mathrm {\\Phi _{RCS}}$ .", "The results are presented in Fig.", "REF , where we see the minimum singular values are often at or very near zero for some values of $R$ , indicating the RIP is either not satisfied or barely satisfied with an extremely small isometry constant.", "Further, we show through numerical experiments in Section that reconstruction performance is in general poor for $\\mathrm {\\Phi _{RCS}}$ .", "Figure: The singular values are bounded away from 0 and 2 indicating good conditioning of the submatrices of Φ MRS \\mathrm {\\Phi _{MRS}}." ], [ "Wide-Sense Stationary RLL Sequences", "We have seen in the previous section that $\\mathrm {\\Phi _{RCS}}$ does not satisfy the requirements for Theorems 1 or 2; Fig.", "REF offers further evidence that $\\mathrm {\\Phi _{RCS}}$ does not satisfy the RIP.", "We therefore do not expect it to perform well in the sampling and reconstruction of sparse signals.", "In this section, we show that a different class of RLL sequences[9], although more complicated than an RCS, produce measurement matrices with better conditioned submatrices and perform much better in the sampling and reconstruction of frequency-sparse signals.", "We begin by examining the RIP for a modulating sequence, $\\varepsilon $ , that is wide-sense stationary with autocorrelation function $R_{\\varepsilon }(m) = \\mathbb {E}[\\varepsilon _j\\varepsilon _{j+m}]$ .", "We assume the maximum dependence distance is $\\ell $ , so $R_{\\varepsilon }(m) = 0$ for $|m| \\ge \\ell $ .", "Under these assumptions, we want to upper bound $|||\\Delta |||$ .", "It will be easiest to focus on the Gram matrix (REF ).", "In this case, we can also rewrite (REF ) in terms of $R_{\\varepsilon }(m)$ : $\\hat{F}(j,\\omega ) = \\sum _{m\\ne 0}\\eta _{j(j+m)}f_{m\\omega }R_{\\varepsilon }(m)$ which we refer to as the “windowed\" spectrum because $\\eta _{j(j+m)}$ can be viewed as a “windowing\" operation on $R_{\\varepsilon }(m)$ .", "From (REF ), we see that the width of the window is $W/R$ , which will be quite large as $W$ increases (and $R$ scales as in (REF )).", "$\\hat{F}(j,\\omega )$ also looks very much like the power spectrum of $\\varepsilon $ : $F_{\\varepsilon }(\\omega ) = \\sum _{m}R_{\\varepsilon }(m)e^{-\\frac{2\\pi \\imath }{W}m\\omega }$ .", "Note that $F_{\\varepsilon }(\\omega )$ is real-valued.", "The significant differences in $\\hat{F}(j,\\omega )$ are the exclusion of $m=0$ in the sum, a scaling by $W^{-\\frac{1}{2}}$ from $f_{m\\omega }$ , and the windowing by $\\eta _{j(j+m)}$ .", "If $W/R \\gg \\ell $ then the windowing has negligible effect in $\\hat{F}(j,\\omega )$ because $R_{\\varepsilon }(m) = 0$ for $|m| \\ge \\ell $ ; $\\hat{F}(j,\\omega )$ and (REF ) both simplify greatly in this case.", "To see this, first notice that because $\\varepsilon $ is a bipolar sequence $R_{\\varepsilon }(0) = 1$ , and ${F}_{\\varepsilon }(\\omega ) = \\sum _{m\\ne 0}R_{\\varepsilon }(m)e^{-\\frac{2\\pi \\imath }{W}m\\omega } + 1$ where $\\tilde{F}_{\\varepsilon }(\\omega ) \\equiv \\sum _{m\\ne 0}R_{\\varepsilon }(m)e^{-\\frac{2\\pi }{W}m\\omega } = F_{\\varepsilon }(\\omega ) - 1.$ We call $\\tilde{F}_{\\varepsilon }(\\omega )$ the reduced spectrum of $\\varepsilon $ .", "Under the assumption that $W/R \\gg \\ell $ , $\\hat{F}(j,\\omega )$ reduces to $W^{-1/2}\\tilde{F}_{\\varepsilon }(\\omega )$ for all $j$ except $j$ satisfying $|rW/R+j| \\le \\ell $ for $r=0,\\cdots ,R-1$ (all but a fraction $2\\ell \\frac{R}{W}$ ).", "This fraction becomes increasingly small as $W$ grows.", "In this case, the entries of $\\Lambda $ are approximately $\\mathrm {\\Lambda }_{\\alpha ,\\omega } &\\approx \\frac{1}{W}\\sum _{j=0}^{W-1}e^{-\\frac{2\\pi \\imath }{W}(\\omega -\\alpha )j}\\tilde{F}_{\\varepsilon }(\\alpha )\\tilde{F}_{\\varepsilon }(\\omega ) \\\\&= \\delta _{\\alpha \\omega }\\tilde{F}_{\\varepsilon }(\\alpha )\\tilde{F}_{\\varepsilon }(\\omega ) $ where $\\delta _{\\alpha \\omega }$ is the Kronecker delta.", "In words, $\\Lambda $ is approximately a diagonal matrix with the square of the reduced spectrum on the diagonal: $\\Lambda \\approx \\text{diag}[(\\tilde{F}_{\\varepsilon }(\\omega ))^2]$ , and the eigenvalues of $\\mathrm {\\Lambda }$ are approximately $(\\tilde{F}_{\\varepsilon }(\\omega ))^2$ .", "Consequently, the singular values of $\\Delta $ are approximately $|\\tilde{F}_{\\varepsilon }(\\omega )|$ .", "We therefore have $||\\Delta || \\approx \\max _{\\omega }|\\tilde{F}_{\\varepsilon }(\\omega )|$ .", "Now, the spectral norm of a submatrix is upper bounded by the spectral norm of the matrix, so we finally obtain $|||\\Delta ||| \\le ||\\Delta || \\approx \\max _{\\omega }|\\tilde{F}_{\\varepsilon }(\\omega )|.$ We now have a way to estimate whether or not a stationary $\\varepsilon $ is well-suited for use within the CRD.", "A stationary $\\varepsilon $ whose spectrum is bounded within $(0,2)$ is good; one with $F_{\\varepsilon }(\\omega ) = 1$ $\\forall \\omega $ is best.", "We now present some examples to make this discussion clearer.", "First, consider an independent (unconstrained) $\\varepsilon $ , such as the one used in the RD.", "In this case, $F_{\\varepsilon }(\\omega ) = 1$ and $\\tilde{F}_{\\varepsilon }(\\omega ) = 0$ $\\forall \\omega $ .", "The Gram matrix exactly disappears ($\\Lambda = 0$ ) and $\\Delta = 0$ confirming our previous discussion.", "Next, we consider the RLL sequences described in [8] and [9].", "To understand how well these sequences will work in the CRD, we need to calculate the power spectrum of sequences generated from the Markov chain in Fig.", "REF ." ], [ "Power Spectrum of Markov Chain RLL Sequences", "To begin, we explicitly describe the RLL sequences in [9].", "Figure: State diagram of the Markov chain generating an MRS (see Definition ).", "The transition probabilities are symmetric in the sense that p (i+k)(j+k) =p ij p_{(i+k)(j+k)} = p_{ij} where the sum is taken modulo 2k2k.", "The top half outputs the symbol +1 while the bottom half outputs -1.Figure: Log-magnitude plot of the autocorrelation of an MRS.", "The autocorrelation experiences geometric decay as m→∞m\\rightarrow \\infty .", "The rate of decay is primarily dependent on dd.Definition 4 We call a $(d,k)$ -constrained RLL sequence that is generated from the Markov chain whose state diagram is found in Fig.", "REF a Markov-generated RLL Sequence (MRS).", "Denote such a sequence as $\\mathrm {\\varepsilon _{MRS}} = [\\varepsilon _0, \\cdots , \\varepsilon _{W-1}]$ with $\\varepsilon _k \\in \\lbrace +1,-1\\rbrace $ .", "The transition probabilities are defined by the matrix $\\mathrm {P} = [p_{ij}]$ where $p_{ij}$ is the probability of transitioning from state $i$ to state $j$ .", "The $p_{ij}$ also satisfy $p_{(i+k)(j+k)} = p_{ij}$ where the sum is modulo $2k$ .", "$\\mathrm {P}$ is of course a stochastic matrix with rows summing to 1.", "The average of the symbols output from each state $i$ are collected in the vector $\\mathrm {b} = \\lbrace b_i\\rbrace $ .", "The stationary distribution of the states is denoted by $\\pi = [\\pi _i]$ and satisfies $\\pi ^T = \\pi ^T\\mathrm {P}$ .", "Having defined these MRS, we have from [31] that their autocorrelation function is $R_{\\varepsilon }(m) = \\mathrm {a}^T\\mathrm {P}^m\\mathrm {b}$ where $\\mathrm {a}^T = \\mathrm {b}^T\\text{diag}[\\pi _1,\\cdots ,\\pi _{2k}]$ and $R_{\\varepsilon }(-m) = R_{\\varepsilon }(m)$ .", "To understand the performance of an MRS within the CRD, we need to understand the behavior of $R_{\\varepsilon }(m)$ as $m$ increases.", "Since $\\mathrm {P}$ is a stochastic matrix, we can make use of the theory of non-negative matrices to understand how $R_{\\varepsilon }(m)$ behaves.", "First note that $\\mathrm {b}$ is orthogonal to $\\mathrm {w}$ , where $\\mathrm {w} = [1, 1, \\cdots , 1]^T$ , and that $\\mathrm {a}^T\\mathrm {b} = 1$ .", "Since $\\mathrm {P}$ is a stochastic matrix, its second largest eigenvalue $\\lambda _2$ satisfies $\\lambda _2 < 1$ .", "Making use of [32], we can bound the autocorrelation (in magnitude) as $|R_{\\varepsilon }(m)| = |\\mathrm {a}^T\\mathrm {P}^m\\mathrm {b}| \\le \\lambda _2^m.$ We see that $|R_{\\varepsilon }(m)|$ experiences geometric decay, at a rate determined by $\\lambda _2$ .", "This is confirmed in Fig.", "REF where $10\\log _{10}|R_{\\varepsilon }(m)|$ is plotted for several pairs $(d,k)$ .", "Notice that the rate of decay (in magnitude) is smaller for larger values of $d$ and larger for larger values of $k$ , and the curve is roughly the same for $k=20$ and $k=\\infty $ .", "These facts can be directly tied to the eigenvalues of $\\mathrm {P}$ in each case.", "To evaluate the performance of an MRS within the CRD, we must evaluate the MDD and the matrix $\\Delta $ .", "Looking first at the MDD, we use (REF ) and the fact that $\\lambda _2 < 1$ to establish that $\\lim _{m\\rightarrow \\infty }|R_{\\varepsilon }(m)| = 0$ and, hence, for any $\\xi > 0$ , $|R_{\\varepsilon }(m)|$ $ < \\xi $ for all $m \\ge M$ where $M = M(\\xi ) < \\infty $ .", "Though we cannot guarantee that an MRS becomes completely uncorrelated for a finite $M$ , we can make $\\xi $ as small as we want so that the sequence is nearly uncorrelated for large enough $M$ .", "In this case, we can take the MDD to be $\\ell \\approx M(\\xi )$ for some small $\\xi $ .", "In other words, an MRS satisfies the setting of Theorem REF in an approximate sense.", "We believe this is justified from an engineering perspective because the correlation can be made very small; the numerical experiments in Section add further justification to this.", "Next, we estimate $|||\\Delta |||$ from the reduced spectrum of the MRS.", "Using (REF ) and $\\ell \\approx M(\\xi )$ from above, we have that $|||\\Delta ||| \\le ||\\Delta || \\approx \\max _{\\omega }|\\tilde{F}_{\\varepsilon }(\\omega )|$ where $\\tilde{F}_{\\varepsilon }(\\omega )$ is the reduced spectrum.", "We emphasize again that $\\xi $ can be made as small as we like at the expense of a larger $\\ell $ .", "Consequently, it can be argued that an MRS that satisfies $\\max _{\\omega }|\\tilde{F}_{\\varepsilon }(\\omega )| < 1$ leads to a matrix $\\mathrm {\\Phi _{MRS}}$ that approximately satisfies the RIP by virtue of Theorem REF .", "Turning to Theorem REF , we must show that $||\\Gamma ||$ is bounded independent of $W$ .", "It is easy to show that for $i < j$ , $\\gamma _{ij} = \\sqrt{|R_{\\varepsilon }(j-i)|/2} \\le \\sqrt{\\lambda ^{j-i}_2/2}$ for an MRS.", "It is then also straightforward to show (see, e.g., the discussion after [30]) that $||\\Gamma || \\le 1/\\sqrt{2}(1-\\lambda _2^{1/2}).$ Since $||\\Gamma ||$ is independent of $W$ , we can make $W$ large enough so that (REF ) is satisfied and Theorem REF is approximately satisfied.", "Figure: RCS with d=1d=1Our argument for the use of an MRS within the CRD makes use of some approximations.", "To demonstrate the validity of these approximations, we consider an MRS with $(d,k) = (1,20)$ .", "The spectrum of this MRS is shown in Fig.", "REF .", "From this figure, we see that $\\max _{\\omega }|\\tilde{F}_{\\varepsilon }(\\omega )| \\approx 0.9$ corresponding to $\\omega = \\pm 0.5$ .", "Our theory, therefore, predicts that the matrix $\\mathrm {\\Phi _{CRD}}$ in this case satisfies the RIP.", "To verify this, we calculate the average minimum and maximum singular values of the submatrices of $\\mathrm {\\Phi _{CRD}}$ and present the results in Fig.", "REF for submatrices containing 10 columns.", "We see that as $R$ decreases, the singular values approach 0 and 2 but remain bounded away from them.", "In Section , we carry out numerical reconstruction experiments to further validate our theory." ], [ "Random Demodulator vs. Constrained Random Demodulator: Numerical Results", "In this section we numerically contrast the performance of the RD with that of the CRD.", "In the case of the CRD, we focus on measurement matrices built using the RCS and MRS.", "The results here are obtained using the YALL1 software package, an $\\ell _1$ -solver using alternating direction algorithms[33].", "We first examine the use of an RCS and show that a CRD using these sequences gives unsatisfactory results.", "Figure: Probability of successful reconstruction over 1000 instances of Φ MRS \\mathrm {\\Phi _{MRS}} with d=1d = 1 and k=20k = 20Recall that we have argued in Section that $\\mathrm {\\Phi _{CRD}}$ using an RCS does not satisfy the RIP.", "Consequently, if we sample a sparse signal using such a measurement matrix and attempt to reconstruct it, we expect to get poor results.", "This is indeed the case in our numerical experiments as shown in Fig.", "REF .", "To produce these results, we hold the sampling rate constant at $R=50$ and vary the bandwidth $W$ .", "It is particularly noteworthy that sampling and reconstruction fail most of the time at $W=100$ and $W=200$ .", "Note that the RCS performs relatively better at $W = 150$ , owing to the splitting of some repeated entries of the RCS between successive rows of the composite matrix $\\mathrm {HD}$ .", "We then examine sampling with a CRD that uses an MRS with $d=1$ and $k=20$ and show that it produces results similar to those for the RD using a Rademacher sequence.", "Recall that we have argued in Section the usefulness of RLL sequences generated from the Markov chain of Fig.", "REF in the context of the CRD.", "Fig.", "REF validates this assertion and shows the empirical probability of reconstruction if we sample sparse signals with $\\mathrm {\\Phi _{CRD}}$ that uses these sequences.", "The baseline for comparison is of course the RD.", "The figure shows that the performance using an MRS is very similar to the performance using the Rademacher sequences of the RD.", "In fact, the CRD allows us to tradeoff between sparsity, bandwidth, and recovery success.", "In particular, if we concentrate on the RD curve at $W=250$ and the CRD curve at $W=300$ , we see that at a $90\\%$ success rate, we only pay a sparsity penalty of 2 ($\\approx 13\\%$ ) by using the CRD.", "At the same time, however, we have gained an advantage in bandwidth, $W$ , of $20\\%$ .", "Comparing the CRD curve at $W=300$ to the RD curve at $W=150$ we see that at a $90\\%$ success rate, we incur approximately a $28\\%$ sparsity penalty for a $100\\%$ increase in bandwidth.", "Other tradeoffs can be seen at different success rates, but it is reasonable to argue that most applications will operate A/D converters in the high success rate regions.", "At lower success rates, the advantage is even greater for the CRD.", "While our analysis concentrates on a high success rate, analysis at lower success rates could prove useful for future work." ], [ "Knowledge Enhanced Sub-Nyquist Sampling", "In this section, we argue that the performance of a CRD can be enhanced by leveraging a priori knowledge about the signal.", "We notice two operations in Fig.", "REF that are central to the functioning of the RD/CRD: the modulation by the random waveform and the subsequent low-pass filtering.", "The low-pass filtering operation allows the RD/CRD to operate at the sub-Nyquist rate $R$ , while modulation by the random waveform—which smears the input signal tones across the spectrum, including in the low-pass region—results in a unique signature of each tone within the low-pass region.", "Theorem REF states the sufficient conditions for uniqueness to hold for all possible input signals, and we explored in Section how the RIP depends on the power spectrum of the random sequence.", "In addition to uniqueness of each tone's signature in the low-pass region, the performance of the RD/CRD depends on the energy smeared into the low-pass region because tones with a low-energy signature will be harder to recover.", "Note that the modulation by the random waveform in time is equivalent to a convolution in the frequency domain.", "Therefore, the power spectrum of the random waveform tells us how much energy from each tone on average is smeared into the low-pass region (and thus collected in the measurements).", "Inspection of (REF ) tells us the RIP depends on the worst-case deviation from a flat spectrum.", "However, if we use an MRS within the CRD and the input signal is statistically more likely to contain low frequencies, then this additional knowledge about the signal can be leveraged to improve the reconstruction averaged over many signals and random waveform realizations.", "Note that this is a different “average case\" setup than the one in Theorem REF .", "Here, we impose a nonuniform distribution on tones in the input signal.", "We show in this setting that the CRD can perform better than the RD, provided the statistical distribution of the tones is matched to the power spectrum of the MRS, because the CRD in this case will on average smear and capture more energy from the input tones in the low-pass region of the spectrum.", "In addition to the case of possessing prior knowledge about the input signal distribution, the exposition in this section is also of interest in other scenarios.", "Consider, for example, a spectrum sensing application in which one assigns a higher priority of detection to some regions and a lower priority of detection to other regions.", "Similarly, consider the case where one possesses knowledge about colored noise or narrowband interference injected into the signal.", "In both these settings, the CRD can be tailored through the choice of the modulating waveform to perform better than either the RD, which treats all spectral regions the same way, or a pure passband system, which completely throws away information in some spectral regions.", "We term such usage of the CRD that exploits prior knowledge a knowledge-enhanced CRD.", "Note that somewhat similar ideas have been briefly explored in [13] and [14], but without the explicit examination of the uniqueness of tone signatures.", "Recent work on model-based compressed sensing also attempts to leverage additional a priori information in the signal model [34], but the focus there is exclusively on the reconstruction side instead of the sampling side." ], [ "Phase Transitions of Reconstruction Success", "To verify our understanding of the knowledge-enhanced CRD, we have conducted extensive numerical simulations to compare reconstruction performance for signals sampled by a CRD (using an MRS) against the RD (using a Rademacher sequence).", "Our focus here will be on two classes of input signals.", "The first class is generated by drawing a sparse set of tones uniformly at random; the second class is generated with a distribution on the tones that matches the power spectrum of an MRS with $(d,k)=(1,20)$ (see Fig.", "REF ).", "We also focus on two measurement matrices: the RD and the CRD using an MRS with $(d,k) =(1,20)$ .", "Recall, the RD uses an (unconstrained) Rademacher sequence.", "The sequence is comprised of independent terms, resulting in a flat spectrum (see Fig.", "REF ).", "Because the spectrum is flat, a Rademacher sequence will illuminate all tones equally well.", "That is to say, we expect good reconstruction performance for all sparse signals.", "On the other hand, the MRS used in the CRD has correlations between terms of the sequence that gives rise to the spectrum in Fig.", "REF .", "We see that the spectrum is close to 1 for the low frequencies (Region $\\textbf {1}$ ) and approximately $0.1$ at high frequencies (Region $\\textbf {2}$ ).", "If low-frequency tones are statistically more likely in the input signal, then we expect the CRD on average to capture more energy in the measurements and offer better reconstruction performance.", "Note, we do not consider the CRD using an RCS because we have shown in Section that the reconstruction performance is very poor.", "To understand why it is poor for an RCS, we can examine the spectra of these sequences.", "An RCS is not stationary but rather cyclo-stationary, so we calculate the spectrum by averaging over the cycle period.", "The resulting spectrum is shown in Fig.", "REF for $d=1$ .", "The spectrum approaches zero at high frequencies, so we expect the CRD in this case to capture very little energy from high frequency tones in the low-pass region.", "Consequently, we also expect poor reconstruction performance.", "The results are displayed in Fig.", "REF for the four combinations described above: two input signal classes and two measurement matrices.", "For these experiments, an RD or CRD matrix is generated using a random instance of the modulating sequence 3000 times for each point (i.e., pixel) on the plot.", "The matrix is used to sample a new randomly generated $S$ -sparse vector, and reconstruction of the original vector from its samples is carried out using the YALL1 software package.", "Success is defined as the two vectors being equal to each other to 6 decimal places in the $\\ell _{\\infty }$ norm.", "The results in Fig.", "REF show that the RD performs (almost) equally well for the two input signal classes.", "On the other hand, the CRD performs much better for the second class of input signals.", "Additionally, the CRD suffers more at very small $R/W$ ratios.", "Figure: MSE plot for a CRD for signals with a distribution on the tones that matches the spectrum in Fig.", "." ], [ "Reconstruction in the Presence of Noise", "The phase transitions of Fig.", "REF correspond to a noiseless setting, Here, we examine the results of reconstructing input signals from noisy samples, $\\mathrm {y} = \\mathrm {\\Phi }\\mathrm {\\alpha } + \\sqrt{p}\\mathrm {w}$ , where $\\mathrm {w}$ is white Gaussian noise and $p$ determines the noise powerThe model $\\mathrm {y} = \\mathrm {\\Phi }\\mathrm {(\\alpha + \\sqrt{p}\\mathrm {w})}$ yields similar results, but $\\mathrm {w}$ as colored noise could offer interesting future work.. We plot the mean-squared error (MSE) of the reconstruction as a function of $S/R$ and $W/R$ and use the SpaRSA software package, which solves an $\\ell _2/\\ell _1$ mixed-norm optimization termed lasso [35] for noisy reconstruction purposes [36]SpaRSA is better suited for noisy reconstruction than YALL1.", "For the regularization parameter, we used $1.9\\sqrt{2p\\log {W}}$ .", "; the results are shown in Fig.", "REF .", "Similar to the noiseless case, we see a sharp transition from low MSE to high MSE.", "The performance of the RD is also similar for each class of input signals while the CRD performs much better for the second class of input signals due to matching the prior to the power spectrum of the modulating sequence." ], [ "Reconstruction of Signals with Non-Integral Frequencies", "The signal model (REF ) assumes only integral-frequency tones.", "Real-world signals may contain non-integral frequency tones.", "These non-integral tones will `leak' energy to several integral tones based on the implicit windowing operation from the finite time assumption ($t\\in [0,1)$ ).", "The windowing produces a convolution of the input tones and the window in frequency[37] but does not invalidate the signal model (REF ).", "Rather, the result is a scaling of the sparsity factor from $S$ to $aS$ , where $a \\ge 1$ determines the extent of the leakage.", "Fig.", "REF shows reconstruction results if non-integral tones in the input signal are allowed.", "Tones are drawn at random from $[0,W)$ according to a distribution proportional to the spectrum in Fig.", "REF .", "The coefficients in the input at the integral tones are determined by a Hamming window (in the frequency domain) centered at the location of the tone.", "Now, compare Fig.", "REF with Fig.", "REF (for the RD) and Fig.", "REF with Fig.", "REF (for the CRD).", "Both plots look similar, but notice that Fig.", "REF has $S$ scaled by a factor of 16.", "This suggests that the penalty for considering leakage in (REF ) is roughly a factor of 16 in input signal sparsity.", "In the worst-case, this kind of `mismatch' can seriously degrade reconstruction performance[38].", "However, in our experiments we do not often see the worst case (a tone occurring halfway between two integral tones) and hence only see a manageable decrease in performance." ], [ "Conclusions", "In summary, we have proposed the use of RLL sequences in the RD because of hardware constraints on generating high-fidelity, fast-switching waveforms.", "We have shown both theoretically and numerically that for a fixed switching rate, certain classes of RLL sequences offer an increase in the observable bandwidth of the system.", "Specifically, we showed that an MRS works well and an RCS does not.", "Insight into why each sequence succeeds or fails is found in the power spectrum of the sequence.", "Further, we have argued that matching the distribution of tones in the input signal to the power spectrum of these RLL sequences improves performance, sometimes even beyond that of the RD.", "The most obvious future directions to take are a better theoretical understanding of knowledge-enhanced CRD and matching the modulating sequence to arbitrary distributions on the input tones.", "A more thorough understanding of the hardware system and the consideration of a more complex modulating waveform (e.g., with a pulse shape other than a square) would also be interesting and useful." ], [ "Restricted Isometry Property of the CRD", "To show that a CRD satisfies the RIP, we follow the proof technique of [6] for the RD with changes to account for correlations within $\\varepsilon $ in our case.", "We begin by bounding the entries of $\\mathrm {\\Phi _{CRD}}$ .", "Lemma 1 [A Componentwise Bound] Let $\\mathrm {\\Phi _{CRD}}$ be an $R \\times W$ CRD matrix, and let $\\ell $ be the maximum dependence distance of the corresponding modulating sequence.", "When $2 \\le p \\le 4\\log W$ , we have $\\mathbb {E}^p\\Vert \\mathrm {\\Phi _{CRD}}\\Vert _{\\max } \\le \\sqrt{\\frac{\\ell 6\\log W}{R}}$ and $\\mathbb {P}\\left\\lbrace \\Vert \\mathrm {\\Phi _{CRD}}\\Vert _{\\max } > \\sqrt{\\frac{\\ell 10\\log W}{R}}\\right\\rbrace \\le W^{-1}.$ We use the following Lemma of Tropp et al.", "[6].", "Lemma 2 [Bounded Entries – RD] Let $\\mathrm {\\Phi _{RD}}$ be an $R\\times W$ RD matrix.", "When $2 \\le p \\le 4\\log W$ , we have $\\mathbb {E}^p\\Vert \\mathrm {\\Phi _{RD}}\\Vert _{\\max } \\le \\sqrt{\\frac{6\\log W}{R}}$ and $\\mathbb {P}\\left\\lbrace \\Vert \\mathrm {\\Phi _{RD}}\\Vert _{\\max } > \\sqrt{\\frac{10\\log W}{R}}\\right\\rbrace \\le W^{-1}.$ We assume that $R$ divides $W$ and $\\ell $ divides $\\frac{W}{R}$ .", "We can write each entry of $\\mathrm {\\Phi _{CRD}}$ as $\\varphi _{r\\omega } &= \\sum _{j\\sim r}\\varepsilon _jf_{j\\omega } \\\\&= \\sum _{(j\\sim r)_0}\\varepsilon _jf_{j\\omega } + ... + \\sum _{(j\\sim r)_{\\ell -1}}\\varepsilon _jf_{j\\omega } \\\\&= \\varphi _{r\\omega }^{(0)} + ... + \\varphi _{r\\omega }^{(\\ell -1)} $ where $[\\varepsilon _j]$ is the modulating sequence, $[f_{j\\omega }] $ are the entries of the Fourier matrix $\\mathrm {F}$ , and $(j\\sim r)_m$ denotes all $j$ such that $j\\sim r$ and $(j \\mod {\\ell }) = m$ .", "Note that each $\\varphi _{r\\omega }^{(m)}$ in (REF ) is a Rademacher series containing $W/R\\ell $ terms, and we proceed by applying the triangle inequality to (REF ): $\\mathbb {E}^p\\varphi _{r\\omega } = \\mathbb {E}^p\\sum _{m=0}^{\\ell -1}\\varphi _{r\\omega }^{(m)} \\le \\sum _{m=0}^{\\ell -1}\\mathbb {E}^p\\varphi _{r\\omega }^{(m)}.$ Applying Lemma REF to each entry in the sum, we have $\\mathbb {E}^p ||\\mathrm {\\Phi _{CRD}}||_{\\max } \\le \\sum _{m=0}^{\\ell -1}\\sqrt{\\frac{6\\log W}{\\ell R}} = \\sqrt{\\frac{6\\ell \\log W}{ R}}.$ For the probability bound, we apply Markov's inequality.", "Let $M = \\Vert \\mathrm {\\Phi _{CRD}}\\Vert _{\\max }$ , then $\\mathbb {P}\\left\\lbrace M > u\\right\\rbrace = \\mathbb {P}\\left\\lbrace M^q > u^q\\right\\rbrace \\le \\left[\\frac{\\mathbb {E}^qM}{u}\\right]^q$ and choosing $u=e^{0.25}\\mathbb {E}^qM$ , we obtain $\\mathbb {P}\\left\\lbrace M > 2^{1.25}e^{0.25}\\sqrt{\\frac{\\ell \\log W}{R}} \\right\\rbrace \\le e^{-\\log W} = W^{-1}.$ Finally, a numerical bound yields the desired result.", "To complete the proof of Theorem REF , recall that the RIP of order $S$ with constant $\\delta _S \\in (0,1)$ holds if $|||\\mathrm {\\Phi _{CRD}}^*\\mathrm {\\Phi _{CRD}} - \\mathrm {I}||| < \\delta _S.$ Using (REF ), we want to show that $|||\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}} - \\mathbb {E}[\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}}]||| + |||\\Delta ||| < \\delta _S.$ We have already bounded $|||\\Delta |||$ in Section .", "We bound the first term by leveraging the results of [6] along with an argument similar to that used in [39] for proving the RIP of Toeplitz matrices.", "Before we continue, recall that the separation between two rows of $\\mathrm {\\Phi _{CRD}}$ required for independence between the rows is $\\rho = \\lceil \\frac{R}{W}(\\ell -1)\\rceil \\le (\\ell - 1)$ .", "In what follows, let $z_r^*$ denote the $r^{th}$ row of $\\mathrm {\\Phi _{RD}}$ or $\\mathrm {\\Phi _{CRD}}$ depending on the context.", "Note that $z_rz_r^*$ is a rank one matrix and that $\\mathrm {\\Phi ^*_{RD}}\\mathrm {\\Phi _{RD}} = \\sum _{r=1}^{R}z_rz_r^*.$ We now need the following proposition which is a corollary to [6].", "Proposition 1 Let $\\mathrm {\\Phi _{RD}}$ be an $R\\times W$ random demodulator matrix and $z_r^{\\prime }$ be an independent copy of $z_r$ .", "Define the random variable $Z_{\\mathrm {RD}} &= |||\\mathrm {\\Phi _{RD}}^*\\mathrm {\\Phi _{RD}} - \\mathbb {E}\\mathrm {\\Phi _{RD}}^*\\mathrm {\\Phi _{RD}}||| \\\\&= \\left|\\left|\\left|\\sum _r\\left(z_r z^*_r - \\mathbb {E}z^{\\prime }_r{z}^{\\prime *}_r\\right)\\right|\\right|\\right|.$ Then $Z_{\\mathrm {RD}}$ satisfies $\\mathbb {E}Z_{\\mathrm {RD}} \\le (\\mathbb {E}B^2)^{1/2}\\sqrt{{\\mathrm {C}S\\log ^4W}} \\le \\sqrt{\\frac{\\mathrm {C}S\\log ^5W}{R}} < \\delta $ , and $\\mathbb {P}\\lbrace Z_{\\mathrm {RD}} > \\delta \\rbrace \\le 8W^{-1}$ , provided $R \\ge \\mathrm {C}\\delta ^{-2}\\cdot S \\log ^6(W)$ .", "Note that $B = \\max _{r,\\omega }|\\varphi _{r\\omega }| \\le \\sqrt{\\frac{10\\log W}{R}}$ with probability exceeding $1-W^{-1}$ .", "To bound the first term in (REF ), we proceed as follows $Z_{\\mathrm {CRD}} &= |||\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}} - \\mathbb {E}\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}}||| \\\\&= \\left|\\left|\\left|\\sum _{r=1}^{R}z_r z_r^* - \\mathbb {E}\\sum _{r=1}^{R}z^{\\prime }_r {z}^{\\prime *}_r\\right|\\right|\\right| \\\\&= \\left|\\left|\\left| \\sum _{s = 1}^{\\rho +1}\\left(\\sum _{r\\in R_s}z_r z^*_r - \\mathbb {E}z^{\\prime }_r {z}^{\\prime *}_r\\right)\\right|\\right|\\right|$ where $R_s = \\lbrace (\\rho +1)n + s\\rbrace $ , $n= 0,1,...,\\frac{R}{\\rho +1} - 1$ .", "The triangle inequality tells us that $Z_{\\mathrm {CRD}} \\le \\sum _{s=1}^{\\rho +1}\\left|\\left|\\left| \\sum _{r\\in R_s}z_r z^*_r - \\mathbb {E}z^{\\prime }_r{z}^{\\prime *}_r \\right|\\right|\\right| = \\sum _{s=1}^{\\rho +1} Z_s.$ Each $Z_s$ is the norm of a sum of independent random variables, and we can apply Proposition REF to each of them.", "Using Lemma REF to obtain the value of $B$ needed in Proposition REF , we get $\\mathbb {E}Z_{\\mathrm {CRD}} &\\le \\sum _{s=1}^{\\rho +1}\\mathbb {E}Z_s \\le \\sum _{s=1}^{\\rho +1}\\sqrt{\\frac{\\mathrm {C}\\ell S\\log ^5W}{R}} \\\\&= (\\rho +1)\\sqrt{\\frac{\\mathrm {C}\\ell S\\log ^5W}{R}}.$ We require that $\\mathbb {E}Z_{\\mathrm {CRD}} < \\delta ^{\\prime }$ for $\\delta ^{\\prime } \\in (0,1)$ which is achieved as long as $R\\ge \\mathrm {C}\\ell (\\rho +1)^2(\\delta ^{\\prime })^{-2}S\\log ^5W.$ We can similarly appeal to the probability bound in Proposition REF to obtain $\\mathbb {P}\\lbrace Z_s > \\delta ^{\\prime }/(\\rho +1)\\rbrace \\le 8W^{-1}$ if $R\\ge \\mathrm {C}\\ell (\\rho +1)^2(\\delta ^{\\prime })^{-2}S\\log ^6W$ .", "Returning to (REF ), we have $|||\\mathrm {\\Phi ^*_{CRD}}\\mathrm {\\Phi _{CRD}} - \\mathrm {I}||| < \\delta $ if $\\delta ^{\\prime } < (\\delta - |||\\Delta |||)$ , and the RIP of order $S$ is satisfied with constant $\\delta _S \\le \\delta $ completing the proof of Theorem REF ." ], [ "Recovery under the Random Signal Model", "To prove Theorem REF , we must bound the coherence and column norms of the matrix $\\mathrm {\\Phi _{CRD}}$ .", "To bound the coherence, we bound the maximum absolute entry of $\\mathrm {X}$ (REF ): $\\max _{\\alpha ,\\omega }|x_{\\alpha ,\\omega }| = \\max _{\\alpha ,\\omega } \\left|\\sum _{j\\ne k}\\varepsilon _j\\varepsilon _k\\eta _{jk}f^*_{j\\alpha }f_{k\\omega }\\right|.$ If the sequence $\\varepsilon $ is not independent, but has maximum dependence distance $\\ell $ , then we need to break the sum up into smaller sums.", "Define the sets $J_a = \\lbrace n\\ell + a\\rbrace $ , $0 \\le a \\le \\ell -1$ , $0 \\le n \\le \\frac{W}{\\ell }-1$ and $K_j = \\lbrace j-(\\ell -1), ..., j+(\\ell -1)\\rbrace $ .", "We now apply the triangle inequality twice to $\\left|x_{\\alpha \\omega }\\right|$ : $& |x_{\\alpha \\omega }| = \\left|\\sum _{j\\ne k}\\varepsilon _j\\varepsilon _k\\eta _{jk}f^*_{j\\alpha }f_{k\\omega }\\right| \\\\&= \\left| \\sum _{j}\\left(\\sum _{\\begin{array}{c}k\\in K_j \\\\ k\\ne j\\end{array}}\\varepsilon _j\\varepsilon _k\\eta _{jk} f^*_{j\\alpha }f_{k\\omega }\\right) + \\left(\\sum _{k\\notin K_j}\\varepsilon _j\\varepsilon _k\\eta _{jk} f^*_{j\\alpha }f_{k\\omega }\\right) \\right| \\\\&\\le \\left| \\sum _{j}\\sum _{\\begin{array}{c}k\\in K_j \\\\ k\\ne j\\end{array}}\\varepsilon _j\\varepsilon _k\\eta _{jk} f^*_{j\\alpha }f_{k\\omega }\\right| + \\left|\\sum _{a=0}^{\\ell -1}\\left[\\sum _{\\begin{array}{c}j\\in J_a, \\\\ k\\notin K_j\\end{array}}\\varepsilon _j\\varepsilon _k\\eta _{jk} f^*_{j\\alpha }f_{k\\omega }\\right]\\right| \\\\&\\le \\left| \\sum _{j}\\sum _{\\begin{array}{c}k\\in K_j \\\\ k\\ne j\\end{array}}\\varepsilon _j\\varepsilon _k\\eta _{jk} f^*_{j\\alpha }f_{k\\omega }\\right| + \\sum _{a=0}^{\\ell -1}\\left|\\sum _{\\begin{array}{c}j\\in J_a, \\\\ k\\notin K_j\\end{array}}\\varepsilon _j\\varepsilon _k\\eta _{jk} f^*_{j\\alpha }f_{k\\omega }\\right| \\\\&= E + \\sum _{a=0}^{\\ell -1}M_a.$ Each $M_a$ is a second-order Rademacher chaos because of the indices of summation, $J_a$ and $K_j$ , and we need the following to deal with such a sum.", "Proposition 2 [6] Suppose that $R\\ge 2\\log W$ .", "Let $[\\varepsilon _j]$ be an independent modulating sequence and define $x_{\\alpha \\omega } = \\sum _{j\\ne k}\\varepsilon _j\\varepsilon _k\\eta _{jk}f^*_{j\\alpha }f_{k\\omega }$ and $\\mathrm {X} = [x_{\\alpha \\omega }]$ .", "Then $\\mathbb {E}^p[||\\mathrm {X}||_{\\max }] \\le 8\\mathrm {C}\\sqrt{\\frac{\\log W}{R}}$ and $\\mathbb {P}\\left\\lbrace ||\\mathrm {X}||_{\\max } > \\mathrm {C}\\sqrt{\\frac{\\log W}{R}} \\right\\rbrace \\le W^{-1}.$ Applying this proposition to each $M_a$ , we get $\\mathbb {E}^pM_a \\le 8\\mathrm {C}\\sqrt{\\frac{\\log W}{R}} \\Rightarrow \\mathbb {E}^p\\left[\\sum _{a=0}^{\\ell -1}M_a\\right] \\le 8\\mathrm {C}\\ell \\sqrt{\\frac{\\log W}{R}}.$ It follows from Markov's inequality that $\\mathbb {P}\\left\\lbrace \\sum _{a=0}^{\\ell -1}M_a > \\mathrm {C}\\ell \\sqrt{\\frac{\\log W}{R}} \\right\\rbrace \\le W^{-1}.$ Now we are left to deal with $E$ .", "Whenever $W/R \\ge \\ell $ we can drop $\\eta _{jk}$ because $\\eta _{jk} = 1$ over the index of summation.", "We can then rewrite $E$ in this case as follows: $E &= \\left| \\sum _{j}\\sum _{k\\in K_j, k\\ne j}\\varepsilon _j\\varepsilon _k\\eta _{jk} f^*_{j\\alpha }f_{k\\omega }\\right| \\\\&= \\left| \\sum _{j}\\varepsilon _j f^*_{j\\alpha } \\left(\\sum _{k\\in K_j, k\\ne j}\\varepsilon _k f_{k\\omega }\\right)\\right| \\\\&= \\left| \\sum _{j}\\varepsilon _j f^*_{j\\alpha } \\left|E^{(j)}_2\\right|\\exp \\left(\\imath \\text{phase}\\left(E^{(j)}_2\\right)\\right)\\right| \\\\& = \\left| \\sum _{j}\\varepsilon _j f^{\\prime }_{j\\alpha } \\left|E^{(j)}_2\\right|\\right|$ where $E^{(j)}_2 = \\sum _{k\\in K_j, k\\ne j}\\varepsilon _k f_{k\\omega },$ $\\text{phase}(\\cdot )$ is the phase angle of the complex argument, and $f^{\\prime }_{j\\alpha } = f^*_{j\\alpha }\\exp \\left(\\imath \\text{phase}\\left(E^{(j)}_2\\right)\\right)$ .", "In short order, we will bound $|E^{(j)}_2| \\le t_2$ $\\forall j$ with high probability so that $E$ can be bounded as $E \\le \\left| \\sum _{j}\\varepsilon _j f^{\\prime }_{j\\alpha } \\right| t_2 = E_1 t_2$ with high probability.", "To bound $E_1$ and to find $t_2$ , we turn to a result to bound the norm of a random series generated from a Markov chain.", "Proposition 3 [30] Let $\\varepsilon = [\\varepsilon _{j}]$ be a sequence of random variables generated from a Markov chain with $\\varepsilon _j \\in \\lbrace +1,-1\\rbrace $ equally likely.", "Let the matrix $\\Gamma $ be the matrix defined in Section REF .", "Let $b_i$ for $1\\le i\\le n$ be arbitrary complex numbers and let $f = \\left|\\sum _{i=1}^{n}\\varepsilon _i b_i \\right|$ .", "For every $t\\ge 0$ , $\\mathbb {P}\\left(|f-\\mathbb {E}[f]| \\ge t\\right) \\le \\exp \\left(-\\frac{t^2}{8\\sigma ^2||\\Gamma ||^2}\\right)$ where $\\sigma ^2 = \\sum _{i=1}^{n} |b_i|^2.$ We apply this proposition to both $E_1$ and $|E^{(j)}_2|$ with $t_1 = \\sqrt{\\log W 8\\sigma _1^2||\\Gamma ||^2}$ and $t_2 = \\sqrt{\\log W 16\\sigma _2^2||\\Gamma ||^2}$ respectively.", "As a result, $\\mathbb {P}\\left(E_1 \\ge t_1\\right) \\le \\exp (-\\log W) = W^{-1}$ and $\\mathbb {P}\\left(|E^{(j)}_2| \\ge t_2\\right) \\le W^{-2}$ $\\forall j$ .", "Finally, we have that $E \\le t_1t_2$ except with probability $2W^{-1}$ .", "To finish the calculation, note that $\\sigma _1^2 = \\sum _{j=0}^{W-1}|f_{j\\alpha }^*|^2 = 1$ and $\\sigma _2^2 = \\sum _{k\\in K_{\\ell }, k\\ne \\ell }|f_{k\\omega }|^2 = 2(\\ell -1)/W.$ Hence, $t_1t_2 &= \\sqrt{\\log W 8\\sigma _1^2||\\Gamma ||^2} \\sqrt{\\log W 16\\sigma _2^2||\\Gamma ||^2} \\\\&= \\log W 8\\sqrt{2}||\\Gamma ||^2 \\sqrt{\\sigma _1^2\\sigma _2^2} \\\\&= \\frac{\\log W}{\\sqrt{W}} 16\\sqrt{\\ell -1}||\\Gamma ||^2.$ Finally, we have the following for the matrix $\\mathrm {X}$ : $\\mathbb {P}\\left(||\\mathrm {X}||_{\\max } \\ge \\mathrm {C}\\ell \\sqrt{\\frac{\\log W}{R}} + t_1t_2 \\right) \\le 3W^{-1}.$ Note that $\\lim _{W\\rightarrow \\infty } (\\log W / \\sqrt{W}) = 0$ , so we can make the second term as small as we like by requiring a large enough $W$ .", "This leads us to the following statements about the coherence, $\\mu = \\max _{\\alpha \\ne \\omega }|\\langle \\phi _{\\alpha },\\phi _{\\omega }\\rangle |$ , and column norms of a CRD matrix: Lemma 3 [Coherence] Suppose that $R \\ge 2\\log W$ .", "An $R\\times W$ CRD matrix satisfies $\\mathbb {P}\\left(\\mu \\ge \\mathrm {C}\\ell \\sqrt{\\frac{\\log W}{R}} + \\frac{\\log W}{\\sqrt{W}} 16\\sqrt{\\ell -1}||\\Gamma ||^2 \\right) \\le 3W^{-1}.$ Lemma 4 [Column Norms] Suppose the sampling rate satisfies $R \\ge 4\\mathrm {C}\\ell ^2\\delta ^{-2}\\log W$ and that $W$ is large enough so that $\\frac{\\log (W)}{\\sqrt{W}} \\le \\frac{\\delta }{32\\sqrt{(\\ell -1)}||\\Gamma ||^2}.$ Then, an $R\\times W$ CRD matrix satisfies $\\mathbb {P}\\left\\lbrace \\max _{\\omega }\\left| ||\\phi _{\\omega }||_2^2 -1 \\right| \\ge \\delta \\right\\rbrace \\le 3W^{-1}.$ To prove recovery results, we finally use the following theorem.", "Theorem 3 [6] Suppose that the sampling rate satisfies $R \\ge \\mathrm {C}[S\\log W + \\log ^3 W].$ Draw an $R\\times W$ RD matrix such that $\\mathbb {P}\\left\\lbrace \\mu \\ge \\mathrm {C}\\sqrt{\\frac{\\log W}{R}}\\right\\rbrace \\le W^{-1}$ and $\\mathbb {P}\\left\\lbrace \\max _{\\omega }\\left|||\\phi _{\\omega }||^2_2 - 1 \\right| \\ge \\delta \\right\\rbrace \\le W^{-1}.$ Let $\\mathrm {s}$ be an $S$ -sparse vector drawn according to the random signal model in Section .", "The solution $\\hat{\\mathrm {s}}$ to the convex program (REF ) satisfies $\\hat{\\mathrm {s}} = \\mathrm {s}$ except with probability $8W^{-1}$ .", "Theorem REF is the result of applying Lemmata REF and REF to Theorem REF .", "The increased requirement on $R$ and the additional requirement on $W$ is needed to ensure the coherence and column norms are satisfactory to ensure recovery.", "Additionally, the probability of recovery failing increases slightly to $12W^{-1}$ ." ], [ "Uncorrelated implies Independence for Identically Distributed bipolar sequences", "Here we briefly show that if two entries in the modulating sequence are uncorrelated then they are independent for the sequences that arise in this paper.", "The sequences, denoted by $[\\varepsilon _j]$ for $j=1,...,W$ , that we are concerned with have two defining characteristics: ($i$ ) $\\varepsilon _j \\in \\lbrace +1,-1\\rbrace $ and ($ii$ ) $\\mathbb {P}\\lbrace \\varepsilon _j = +1\\rbrace = \\mathbb {P}\\lbrace \\varepsilon _j = -1\\rbrace = 1/2$ .", "The autocorrelation in this case can be expressed as: $\\mathbb {E}[\\varepsilon _j\\varepsilon _{j+k}] = \\mathbb {P}\\lbrace \\varepsilon _j = \\varepsilon _{j+k}\\rbrace - \\mathbb {P}\\lbrace \\varepsilon _j \\ne \\varepsilon _{j+k}\\rbrace .$ Now, given the maximum dependence distance $\\ell $ we have $\\mathbb {P}\\lbrace \\varepsilon _j = \\varepsilon _{j+k}\\rbrace = \\mathbb {P}\\lbrace \\varepsilon _j \\ne \\varepsilon _{j+k}\\rbrace $ for $|k| \\ge \\ell $ which implies that $\\mathbb {P}\\lbrace \\varepsilon _{j+k} = +1 | \\varepsilon _j = +1\\rbrace = \\mathbb {P}\\lbrace \\varepsilon _{j+k} = +1 | \\varepsilon _j = -1\\rbrace $ in this case.", "Characteristic (ii) also tells us that $\\mathbb {P}\\lbrace \\varepsilon _{j+k} = +1 | \\varepsilon _j = +1\\rbrace + \\mathbb {P}\\lbrace \\varepsilon _{j+k} = +1 | \\varepsilon _j = -1\\rbrace = 1,$ meaning we must have that $\\mathbb {P}\\lbrace \\varepsilon _{j+k} = +1 | \\varepsilon _j = +1\\rbrace = \\mathbb {P}\\lbrace \\varepsilon _{j+k} = +1 | \\varepsilon _j = -1\\rbrace = 1/2.$ The same argument applies to $\\varepsilon _{j+k} = -1$ , and the condition for independence results: $\\mathbb {P}\\lbrace \\varepsilon _{j+k} = a| \\varepsilon _j = b\\rbrace = \\mathbb {P}\\lbrace \\varepsilon _{j+k} = a\\rbrace $ for $a,b \\in \\lbrace +1,-1\\rbrace $ ." ] ]

[ [ "A proposal for impact-adjusted valuation: Critical leverage and\n execution risk" ], [ "Abstract The practice of valuation by marking-to-market with current trading prices is seriously flawed.", "Under leverage the problem is particularly dramatic: due to the concave form of market impact, selling always initially causes the expected leverage to increase.", "There is a critical leverage above which it is impossible to exit a portfolio without leverage going to infinity and bankruptcy becoming likely.", "Standard risk-management methods give no warning of this problem, which easily occurs for aggressively leveraged positions in illiquid markets.", "We propose an alternative accounting procedure based on the estimated market impact of liquidation that removes the illusion of profit.", "This should curb the leverage cycle and contribute to an enhanced stability of financial markets." ], [ "Introduction: The danger of marginal prices", "Mark-to-market or “fair value” accounting is standard industry practice.", "It consists in assigning a value to a position held in a financial instrument based on the current market price for this instrument or similar instruments.", "This is justified by the theory of efficient markets, which posits that at any given time market prices faithfully reflect all known information about the value of an asset.", "However, mark-to-market prices are only marginal prices, reflecting the value of selling an infinitesimal number of shares.", "Practitioners are typically concerned with selling more than an infinitesimal number of shares, and are intuitively aware that this practice is flawed.", "Selling has market impact, which depresses the price by an amount that increases with the quantity sold.", "The first piece will be sold near the current price, but as more is liquidated prices may drop substantially.", "This somewhat paradoxically implies the value of 10 % of a company is less than 10 times the value of 1 % of that company.", "We take advantage of what has been learned recently about market impact to propose a method for impact-adjusted valuation that results in better risk control than mark-to-market valuation.", "This is in line with other recent proposals that valuation should be based on liquidation prices [1], [2].", "Estimating liquidation prices requires a good understanding of market impact.", "In recent years there is been considerable progress in both theory and practice.", "For large trades there is growing evidence that market impact follows a universal functional form, see e.g.", "[3], [4], [5].", "By “large\" we mean trades that exceed the liquidity currently available in the order book; such trades need to be either broken up into pieces and executed incrementally or executed in a block market.", "Market impact is a concave function whose slope is infinite at the origin, which means that small trades have a disproportionally large impact.", "The need for a better alternative to marking to market is most evident with leverage, i.e.", "when assets are purchased with borrowed money.", "Leverage amplifies market impact.", "As a leveraged position is sold, a process we refer to here as deleveraging, the price tends to drop due to market impact.", "Counter-intuitively, due to the concave form of market impact, when a leveraged position is gradually unwound the depression in prices due to impact overwhelms the decrease in position size, and leverage initially rises rather than falls.", "When impact is concave, this is not unusual – the expected leverage as a sale begins always goes up, regardless of initial leverage, liquidity or position size.", "As more of the position is sold, provided the initial leverage and initial position are not too large, leverage eventually comes back down and the position retains some of its value upon liquidation.", "However, as we show here, if the initial leverage and initial position are too large, as the position is sold the leverage diverges, and the resulting liquidation value is less than zero, i.e.", "the debt to the creditors exceeds the resale value of the asset.", "The upshot is that under mark-to-market accounting a leveraged position that appears to be worth billions of dollars may predictably be worth less than nothing by the time it is liquidated.", "The above scenario assumes that positions are exited in an orderly fashion; under fire sale conditions or in very illiquid markets things are even worse.", "From the point of view of a regulator or a risk manager this makes it clear that an alternative to mark-to-market accounting is badly needed.", "Neglecting impact allows huge positions on illiquid instruments to appear profitable when it is actually not the case.", "We propose such an alternative based on the known functional form of market impact, and propose that valuations should be based on the expected liquidation value of assets.", "Under leverage this avoids the problems outlined above.", "Whereas mark-to-market valuation only indicates problems with excessive leverage after they have occurred, our method makes them clear before positions are entered.", "Thus our method gives clear indications about potential problems as they are developing, and makes such situations easier to avoid.", "This could be extremely useful for damping the leverage cycle and coping with pro-cyclical behaviors [6], [7], [8], [9], [10].", "In Section 2 we review the literature on market impact and present our proposal for impacted-adjusted valuation.", "In Section 3 we apply this to leverage and demonstrate that over-leveraging is a critical phenomenon, with a sharp transition where the problem of liquidating the position without bankruptcy becomes serious.", "In Section 4 we present some alternative formulas for estimating impact, run some numbers for typical assets, and show that this is not a serious problem for really liquid assets such as stocks, but it can occur at surprisingly low leverages for assets such as credit default swaps.", "Section 5 concludes, discussing the broader implications for the theory of market efficiency." ], [ "Market impact and liquidation accounting", "Accounting based on liquidation prices requires a quantitative model of market impact.", "Because market impact is very noisy, and because it usually requires proprietary data to study empirically, a good picture of market impact has emerged only gradually in the literature.", "In this section we review what is known about market impact and present our proposal for impact-adjusted valuation." ], [ "The emerging quantitative model of market impact", "Understanding the nature of market impact has now been the focus of a large number of empirical studies, both from academics and practitioners (for recent reviews, see [11], [12], [3], [4], [13], [14], [5]), and a consensus is beginning to emerge.", "Here we are particularly concerned with the liquidation of large positions, which must either be sold in a block market or broken into pieces and executed incrementallyOur interest in the impact of a single large trade that must be executed in pieces is in contrast to the impact of a single small trade in the orderbook, or the impact of the average order flow, both of which have different functional forms, see [12], [5]..", "These empirical studies now make it clear that the market impact $I = \\langle \\varepsilon \\cdot (p_{f} - p_{0})/p_0 \\rangle $ , defined as the expected shift in price from the price $p_0$ observed before a buy trade ($\\varepsilon =+1$ ) or a sell trade ($\\varepsilon =-1$ ) to the price $p_f$ at which the last share is executed, is a concave function of position size $Q$ normalized by the trading volume $V$ .", "When liquidation occurs in normal conditions, i.e.", "at a reasonable pace that does not attempt to remove liquidity too quickly from the order book, the expected impact $I$ due to liquidating $Q$ shares is ${I}(Q) = Y \\sigma \\, \\sqrt{\\frac{Q}{V}},$ where $\\sigma $ is the daily volatility, $V$ is daily share transaction volume, and $Y$ a numerical constant of order unity [5].", "We say more about how these parameters should be estimated when this formula is used for regulatory purposes in the next section.", "Note that we are defining the expected impact in terms of prices rather than log-prices.", "This is possible because for cases of interest the liquidation time $T$ is short enough for prices not to move significantly away from the initial price, and the impact itself is significantly less than the price itself, so that the difference between $p_f/p_0 - 1$ and $\\ln (p_f/p_0)$ is small and only has a minor effect on our conclusions.", "This means that the domain of validity for the formula requires that the impact $I(Q)$ not be too large, roughly less than $20\\%$ .", "The quantity above is the expected impact, in the sense that it is the average outcome of liquidating $Q$ shares.", "This is superimposed on the background price fluctuations due to the rest of the market.", "For typical small values of $Q/V$ allowing orderly execution, the realized market impact is very noisy, almost invisible to the naked eye.", "It is not uncommon that the realized impact is in the opposite direction of the average impact.", "The expected impact can be regarded either as the average impact or as a median price – $50\\%$ of prices will be above it, and $50\\%$ below it.", "We want to emphasize that here we have defined impact as the shift in prices caused by the execution of given order.", "Whether the long-term impact has a permanent component that remains embedded in prices long after the trade occurs, and how large such a component might be, remain controversial.", "Fortunately these questions do not need to be addressed for our purposes here, although they are highly relevant to understand how market prices move and how potentially destabilising feedback loops can occur (see e.g [15]).", "The earliest theory of market impact due to Kyle [16] predicted that expected impact should be linear.", "This was further supported by the work of Huberman and Stanzl [17], who argued that providing certain assumptions are met, such as lack of correlation in order flow, impact has to be linear in order to avoid arbitrage.", "However, more recent empirical studies have made it clear that these assumptions are not met [18], [19], [5], and the overwhelming empirical evidence that impact is concave has driven the development of alternative theories [20], [21], [22].", "For example, Farmer et al.", "[23] have proposed a theory based on a strategic equilibrium between liquidity demanders and liquidity providers, in which uncertainty about $Q$ on the part of liquidity providers dictates the functional form of the impact.", "Toth et al.", "[5], in contrast, derive a square root impact function within a stochastic order flow model.", "They impose that prices are diffusive, show that this implies a locally linear “latent order book”, and provide a proof-of-principle using a simple agent-based model.", "Both of these theories predict roughly square root impact, though with some differences.", "Both empirical studies and theory make it clear that the square-root law for expected impact under orderly execution also holds at intermediate points.", "That is, after a quantity $q \\le Q$ is executed, the average adverse price move is given by [14], [23], [5]: ${I}(q) = Y \\sigma \\, \\sqrt{\\frac{q}{V}},$ We should stress that the formulae above for market impact hold only under normal conditions, when execution is slow enough for the order book to replenish between successive trades (on this point, see e.g.", "[24], [25], [11]).", "If the execution schedule is so aggressive that $Q$ becomes comparable to $V$ , liquidity may dry up, in which case the parameters $\\sigma $ and $V$ can no longer be considered to be fixed, but themselves react to the trade, with an expected increase of the volatility and a decrease of the liquidity.", "Impact in such extreme conditions is expected to be much larger than the square-root formula above.", "The flash crash is a good example.", "In these cases the expected impact becomes less concave and it can become linear or even super-linear [22].", "For the above impact formula to be valid, the execution time $T$ needs to be large enough that $Q$ remains much smaller than $V$ ($20\\%$ is a typical upper limit).", "The execution time should not be too long either, otherwise impact is necessarily linear in $Q$ : beyond some “memory time” of the market, trades must necessarily become independent and impact must be additive, see [5]." ], [ "How should the impact parameters be estimated?", "When impact is estimated for regulatory purposes, for stability reasons it is important that the parameters should be computed over a long time horizon.", "For example one can take an exponential moving average of $\\sigma $ and $V$ over past values.", "If $\\sigma $ and $V$ are not measured over relatively long time scales impact-adjusted valuation could lead to an unstable feedback loop.", "Imagine, for example, an exogeneous shock (like the Japanese tsunami in March 2011) that leads to a sudden increase of volatility.", "If $\\sigma $ is measured over short-time scales, the expected impact $I(Q)$ also increases.", "This would cause a larger discount on the asset valuation, which could cause a systemic effect in which risk managers unload the asset, leading to plummeting market prices and further panic.", "Similarly in a temporary liquidity crisis a sudden drop of $V$ could lead to a mechanical reduction in asset values.", "In order to avoid these destabilising effects, the window over which $\\sigma $ and $V$ are computed should be chosen to be long, perhaps 6 months, and exclude the very recent past – e.g.", "the last week of trading." ], [ "Impact-adjusted accounting", "The establishment of a quantitative theory for expected impact makes it possible to do impact-adjusted accounting.", "Rather than using the mark-to-market price, which is the marginal price of an infinitesimal liquidation, we propose using the expected price under complete liquidation.", "For convenience we assume liquidation in $N$ equal sized increments of $v$ shares each, where $v$ is arbitrary but smallIn general it is possible that optimal liquidation might follow a different liquidation schedule.", "However, our feeling is that any gains from such a schedule are likely to be small, and in any case, empirical studies show that a uniform liquidation rate is a good approximation for the average investor [14]..", "The expected value $\\mathcal {V}$ of a position of $Q$ shares in a given asset with mark-to-market price $p_0$ that is liquidated in $N$ pieces of $v$ shares each is $\\mathcal {V} (Q) = \\sum _{t=1}^N v p_0 (1 - I(vt))$ Providing $Q$ is large, it is a good approximation to use the continuous limit where $dq = v$ is infinitesimal, in which case this can be written $\\mathcal {V} (Q) & = & \\int _0^{Q} p_0 (1 - I(q)) dq\\\\\\nonumber & = & p_0 Q(1 - \\frac{2}{3} Y \\sigma \\sqrt{Q/ V})\\\\\\nonumber & = & p_0 Q(1 - \\frac{2}{3} I(Q))$ It is sometimes also useful to use the average valuation price $\\tilde{p} = \\mathcal {V}/Q = p_0 (1 - \\frac{2}{3} I(Q))$ ." ], [ "The critical nature of leverage", "When leverage is used it becomes particularly important to take impact into account and value assets based on their expected liquidation prices.", "Consider an asset manager taking on liabilities $L$ to hold $Q$ shares of an asset with price $p$ .", "For simplicity we consider the case of a single asset.", "The leverage $\\lambda $ is given by the ratio of the value of the asset to the total equity, $\\lambda =\\frac{Qp}{Qp-L}.$ Holding $Q$ and $L$ constant, the leverage decreases when the price of the asset increases and vice versa when it decreases.", "Similarly, holding $p$ and $L$ constant, selling $q$ shares reduces leverage, $\\lambda =\\frac{Qp}{Qp-L} \\rightarrow \\lambda ^{\\prime } = \\frac{(Q-q)p}{(Q - q)p-L} < \\lambda \\quad {\\mbox{if}}\\quad q > 0,$ and vice versa for buying." ], [ "Deleveraging", "Now we take into account market impact and consider the case of deleveraging, i.e.", "exiting a leveraged position.", "Selling pushes current trading prices down, which under mark-to-market accounting decreases the value of the remaining unsold shares.", "As we show, this generally overwhelms the effect of selling the shares, increasing the leverage even as the overall position is reduced.", "After $q$ shares have been sold the amount of cash raised to offset the liabilites is $C(q)$ .", "Using the continuous approximation $C(q) \\approx \\int _0^q \\,{\\rm {d}}q^{\\prime } \\, p_0 \\left(1 - I(q^{\\prime })\\right) = p_0 q \\left(1 - \\frac{2}{3} {\\cal I} \\sqrt{\\frac{q}{Q}}\\right),$ where ${\\cal I} \\equiv I(Q)= Y \\sigma \\sqrt{Q/V}$ is the impact of selling the entire position, which can be large if the initial position is too big and/or the liquidity is too small.", "The leverage $\\lambda (q)$ after $q$ shares have been sold is $\\lambda (q) = \\frac{(Q-q)p(q)}{(Q-q)p(q)-L+C(q)},$ where $p(q)$ is the price after selling $q$ shares.", "Letting $x=q/Q$ be the fraction of assets that have been sold and $\\lambda _0$ be the initial leverage before selling begins, this can be rewritten in the form $\\lambda (x) = \\lambda _0 \\left( \\frac{(1-x)(1- {\\cal I} \\sqrt{x})}{1 - \\lambda _0 {\\cal I} \\sqrt{x} \\left(1 - x/3 \\right)} \\right).$ It is then easy to show that: For small $x$ , $\\lambda (x) \\approx \\lambda _0 \\left(1 + (\\lambda _0 -1) {\\cal I} \\sqrt{x} \\right)$ , which is larger than $\\lambda _0$ for $\\lambda _0 > 1$ , that is, whenever any leverage is used.", "This means, rather paradoxically, that when selling a leveraged position, the expected leverage under mark-to-market accounting always initially increases.", "If $\\lambda _0 {\\cal I} < 3/2$ the leverage $\\lambda (x)$ eventually reaches a maximum and decreases back to one for $x=1$ .", "The crossover point $x^{*}$ where the leverage drops below its starting value can be computed by solving Eq.", "(8) for $x$ with $\\lambda (q) = \\lambda _0$ , which gives $x^* = \\sqrt{\\frac{1 - \\sqrt{1 - \\frac{4}{3} (\\lambda _0-1)(3 - \\lambda _0) {\\cal I}^2}}{(2 - \\lambda _0/3){\\cal I}}}.$ It is easy to show that $x^* < 1$ whenever $\\lambda _0 {\\cal I} < 3/2$ .", "If $\\lambda _0 {\\cal I} > 3/2$ the leverage $\\lambda (x)$ diverges during liquidation.", "The leverage diverges when the value of the position is equal to the liability, i.e.", "$qp(q) = L$ .", "This occurs when the denominator of Eq.", "(9) becomes zero, which yields a cubic equation for the critical value $x = x_c$ .", "If $x_c < 1$ then the asset manager goes bankrupt before being able to take his position to zero.", "Three representative deleveraging trajectories $\\lambda (x)$ are illustrated in Fig.", "1, together with the trajectory obtained in absence of market impact.", "We assume a fixed starting mark-to-market leverage $\\lambda _0 = 9$ and show three cases corresponding to different values of the overall market impact parameter $\\mathcal {I}$ .", "For the two cases where the leverage is subcritical, i.e.", "with $\\lambda _0 {\\cal I} < 3/2$ , the manager unwinds the position without bankruptcy.", "However, due to the rise in leverage during the course of liquidation, he may get in serious trouble with his prime broker along the way.", "For example, in the case where ${\\cal I} = 0.15$ at its peak $\\lambda (x)$ is more than twice its starting value.", "The case where the leverage is allowed to become supercritical is a disaster.", "If $\\lambda _0 {\\cal I} > 3/2$ , which for $\\lambda _0 = 9$ implies ${\\cal I} > 0.16$ , the manager is trapped, and the likely outcome of attempting to deleverage is bankruptcy.", "(By bankruptcy we mean that the position ends up being worth less than the money borrowed to finance it, so that the manager ends up owing a debt for that position. )", "Figure: Possible deleveraging trajectories, showing the leverage λ(x)\\lambda (x) based on mark-to-market accounting as a function of the fraction xx of the position that has been liquidated.", "We hold the initial leverage λ 0 =9\\lambda _0 = 9 constant and show four trajectories for different values of the market impact parameter ℐ=I(Q)=YσQ/V{\\cal I} = I(Q) = Y \\sigma \\sqrt{Q/V}, i.e.", "ℐ=0{\\cal I} = 0 (black dashed line, corresponding to the no-impact case) ℐ={\\cal I} = 0.1 (green dotted line), 0.15 (blue solid line), and 0.19 (red dotted-dashed line).", "If the market impact is too high the leverage diverges before the position can be liquidated, implying that the position is bankrupt." ], [ "Leverage under impact-adjusted prices", "We now show how risk management is improved by impact-adjusted accounting.", "This is done by simply using the average impact-adjusted valuation price $\\tilde{p}$ in the formula for leverage, i.e.", "$\\tilde{\\lambda }(q) = \\frac{q\\tilde{p}(q)}{q\\tilde{p}(q) - L+C(q)}.$ Here $0 \\le q \\le Q$ is the number of shares held at any given time along the way to entering position $Q$ .", "We define the impact adjusted price for position $q$ as the liquidation price if buying were to stop and the current position $q$ were to be sold.", "Accordingly, when exiting a position we adopt the convention that the impact-adjusted price is based on complete liquidation of all $Q$ shares, i.e.", "we do not allow for the possibility of pausing along the wayThe square root law for market impact is inherently related to the market's memory [23], [5].", "Once liquidation begins the market has a memory – the response of prices to each successive sale is smaller and smaller as $Q - q$ gets bigger and bigger.", "An alternative definition of the impact-adjusted price while the position is being sold might be to assume a pause sufficiently long to break this memory, followed by subsequent liquidation of the remaining position $q$ .", "We have not adopted this because it requires an understanding of how impact decays in time, which we do not have, and in any case we do not believe this is necessary..", "Figure: Leverage as a function of position size for first entering and then exiting a position.", "The position q(t)q(t) varies from 0 to QQ in the left half of each panel and from QQ to 0 in the right half of each panel.", "Labels on the xx axes denote the number of shares held by the asset manager at the corresponding time.", "Three different measures are used for leverage.", "The dashed black line shows what the leverage would be if there were no impact and the price didn't change; the solid blue line shows the leverage including impact under mark-to-market accounting, and the dotted-dashed red line shows the leverage using impact-adjusted valuation.", "The left panel is a case in which QQ is small enough that the leverage never becomes critical; the right panel is a case where the leverage becomes super-critical.", "In this case the impact-adjusted leverage diverges as the position is entered, warning the manager of the impending disaster.", "The dashed red vertical line shows the critical position q c q_c.In figure REF we show how the leverage behaves when a manager first steadily assumes a position $0 \\le q(t) \\le Q$ and then steadily liquidates it.", "We compare three different notions of leverage: No impact leverage is represented by the dashed black line.", "This is the leverage that would exist if the price remained constant (on average).", "It rises and falls linearlyThe reason for linearity is that when the price is constant the denominator in Eq.", "(REF ) remains constant.", "This is because changes in cash cancel changes in asset value.", "proportional to the position $q(t)$ .", "Mark-to-market leverage is represented by the solid blue line.", "While the position is building it rises more slowly than linearly.", "This is because as the position is building impact causes the price to increase, lowering leverage and partially offsetting the increasing position size.", "This is dangerous because it artificially overestimates profits and therefore depresses leverage.", "When the position is exited, in contrast, the expected leverage initially shoots up.", "In the subcritical case it eventually returns to zero, but in the super-critical case it diverges, indicating (too late) that the position is bankrupt.", "Impact-adjusted leverage is represented by the dashed red line.", "It is always greater than either of the other two measures of leverage.", "It is particularly useful in the super-critical case – its rapid increase is a clear warning that a problem is developing, in contrast to the mark-to-market leverage.", "A sensible manager would thus easily avoid bankruptcy by buying less and avoiding the critical regime." ], [ "Taking noisy impact into account", "So far we have focused our attention on the expected impact, which can either be viewed as the average impact or as a median trajectory.", "In this section we show how impact-adjusted accounting can be used to compute the probability of adverse price movements.", "This improves on standard measures that fail to take impact into account and may dramatically underestimate the probability of bankruptcy in situations where impact is large.", "To illustrate this we estimate the probability of bankruptcy for positions of varying leverage.", "We make the simple assumption that the noisy component of impact is independent of the order being executed, and diffuses according to the volatility as the square root of time.", "Under the (admittedly crude) approximation that background price movements are normally distributed we model individual realizations of price trajectories as a discrete random walk with time varying drift.", "For convenience we measure the time $t$ in days.", "The evolution of the price during execution is given by $p(t+1) = p(t) + I(Q - q(t) - \\delta q) - I(Q - q(t)) + p_0 \\sigma n(t),$ where $p(t)$ is the price at time $t$ , $q(t)$ is the size of the position at time $t$ , and $n(t)$ is IID gaussian noise.", "The term $I(Q - q(t) - \\delta q) - I(Q - q(t))$ is the additional increment of impact between day $t$ and day $t+1$ , with $\\delta q$ is the volume traded in a given day.", "The total time $T$ needed to off-load the position is $T=Q/\\delta q$ .", "With this choice for the stochastic process we ensure that in absence of noise the price follows the deterministic trajectory predicted by the expected market impact, and that the price in absence of market impact undergoes an unbiased discrete random walk.", "For better risk analysis it is of course possible to use more sophisticated models of the background noise, incorporating factors such as clustered volatility, jump diffusions, or heavy tails as desired.", "The simple model above is sufficient to illustrate the basic idea of how such risk analysis can be done.", "To assign probabilities for a given event, in this case bankruptcy, we simulate realizations of the noise process using Eq.", "(REF ), keeping $\\sigma \\sqrt{T}$ small enough that the probability that the price becomes negative can be neglected.", "A typical result is shown in Fig.", "REF .", "Since the volatility of the final price scales as $\\sigma \\sqrt{T}$ , whereas the average impact scales as $\\sigma \\sqrt{Q/V}$ , the sharpness of the transition is determined by their ratio $\\eta \\equiv Q/VT$ .", "Using $T = Q/\\delta q$ we can write $\\eta = \\delta q/V$ , making it clear that $\\eta $ is an aggressivity parameter, often called the participation rate, measuring the fraction of daily volume used for trading.", "In Fig.", "REF we vary $Q$ and $T$ while keeping $\\eta $ constant.", "The probabilities of bankruptcy are dramatically higher than they are without impact, and as expected, the transition is centered at the critical point $ \\mathcal {I}_c = 3/(2 \\lambda _0)$ , which is independent of the volatility.", "The transition is sharp for aggressive trading schedules ($\\eta > 1$ ) and is blurred as $\\eta \\rightarrow 0$ .", "Figure: Bankruptcy probability as a function of the total impact ℐ\\mathcal {I}.Probability of bankruptcy as a function of impact I(Q)I(Q).", "In the left panel η=Q/(VT)=10\\eta = Q/(VT) = 10; QQ is varied from 0 to 10 5 10^5 while TT is varied to hold η\\eta constant.", "Red circles are bankruptcy probability with impact, and black triangles without impact.", "The vertical dashed red line is the critical value ℐ c \\mathcal {I}_c.", "The right panel is a similar plot for three different values of η\\eta .During liquidation it is possible for the position to temporarily become bankrupt and then recover.", "Whether or not a manager would be forced to default in such a situation will depend on her relationship with her creditors.", "Forcing bankruptcy if it occurs anywhere along the liquidation path slightly raises the probability of bankruptcy, depending on the time for execution." ], [ "Does leverage diverge in realistic situations?", "We have shown the dangers of mark-to-market accounting for understanding leverage, but the skeptical reader may wonder whether such extreme situations actually occur in practice.", "In this section we plug in some typical numbers and show that for large positions in illiquid stocks such problems are not uncommon." ], [ "Impact and bid-ask spread", "We have so far computed impact using the estimated volatility and volume.", "We now review results that connect these to the spread, which provide an alternative way to estimate the magnitude of liquidation effects, which might be more convenient in some circumstances.", "It is now well established empirically that the volatility is made up of two different effects: the size of the bid-ask spread $S$ on the one hand, and the number of transactions $\\phi $ per unit time on the other.", "For liquid markets the volatility $\\sigma _T$ on timescale $T$ can be written $\\sigma _T = b S \\sqrt{\\phi T},$ where $b \\approx 0.6-0.9$ is a constant of order unity, that weakly depends on the market [26].", "Suppose that the typical volume at the best prices is $v$ .", "If one assumes as before that the order is executed in increments of $v$ shares, the total number of transactions needed to liquidate $Q$ shares is $N = Q/v$ .", "Similarly the total volume $V_T$ in a time $T$ is $V_T= v \\phi T$ .", "The above impact formula can therefore be rewritten as:Note that the liquidation time $T$ drops out of the formula, which is one of the remarkable properties of the square-root impact law.", "$I = Y \\sigma _T \\sqrt{\\frac{Q}{V_T}} = Yb \\, S \\sqrt{N}, \\qquad N=\\frac{Q}{v}.$ This expression highlights the micro-structure origin of liquidity.", "As is intuitively clear, it is the spread $S$ and the available volume $v$ that determine the impact cost of a trade.", "The quantities $S$ and $v$ should again be estimated using moving averages using market data or broker quotes for OTC/illiquid markets." ], [ "Some examples", "Let us first give some orders of magnitude for stock markets.", "The daily volume of a typical stock is roughly $5 \\times 10^{-3}$ of its market cap (see e.g.", "[27], [28]), while its volatility is of the order $2 \\%$ per day.", "Suppose the portfolio to be liquidated owns $Q=5 \\%$ of the market cap of a given stock.", "Taking $Y = 0.5$ , the impact discount is ${I}(Q) \\approx 2 \\% \\times \\sqrt{\\frac{0.05}{0.005}} \\approx 6 \\%.$ A $6 \\%$ hair-cut on the value of a portfolio of very liquid stocks is already quite large, and it is obviously much larger for less liquid/more volatile markets.", "Let us now turn to the question of the critical leverage $\\lambda _c$ under mark-to-market accounting.", "From Section 3, the condition reads: $\\lambda _c \\mathcal {I} = \\frac{3}{2}.$ Substituting the two expressions for $\\mathcal {I} = I(Q)$ and rearranging gives $\\lambda _c & = & \\frac{3}{2 Y \\sigma } \\sqrt{V/Q}\\\\& = & \\frac{3}{2 Y b S \\sqrt{N}}$ To give a feeling for whether or not these conditions can be met, we present representative values for several different assets.", "For futures we assume $Q = V$ , implying that it would take five days to trade out of the position with $\\eta = 0.2$ .", "For stocks we assume $Q = 10V$ , which assuming the same participation rate implies a position that would take 50 trading days to unwind.", "Such positions might seem large, but they do occur for large funds; for instance, Warren Buffet was recently reported to have taken more than eight months to buy a $5.5\\%$ share of IBM.", "The results are given in Table REF .", "Table: Numerical values of the different parameters entering the two alternative impact formulae given in Eqs.", "() and ()and the corresponding estimates of impact and critical leverage.", "Except as otherwise noted, numbers are based on data for the first quarter of 2008.", "These are only rough orders of magnitude, intended for a qualitative discussion.☆\\star : Impact ℐ 1 =I(Q)\\mathcal {I}_1 = I(Q) based on volatility and volume, computed with Eq.", "(), with Y=1Y=1 and Q=VQ=V forfutures and Q=10VQ=10V for stocks.", "This corresponds to a position of roughly 5%5 \\% of the market capitalisation on stocks, and to a position equal to 3 Kervielson the BUND.", "♯\\sharp : Impact computed with Eq.", "(), with Y=1Y=1, b=0.6-0.9b=0.6-0.9 (depending on the market ) and the same values of QQ.†\\dagger : For futures, we refer to the nearest maturity; the numbers for the 10YUSNOTE are very similar to those for the BUND.", "Note that for liquid futures, the critical leverage level is very high (as expected).", "Still,a 1.5%1.5 \\% liquidity hair-cut on a position on the SP500 is by no means negligible.", "♦\\diamondsuit : Large cap US stocks: In this case, Q=10VQ=10V.", "Note that the two impact estimates are substantially different,with ℐ 1 >ℐ 2 {\\cal I}_1 > {\\cal I}_2.", "This maybe due to the fact that the volume at the best quote, vv, is highly skewed, i.e.", "the typical available volume is much smaller than the average volume.Furthermore, trades are usually only a fraction of the available volume.", "Therefore one expects N>Q/vN > Q/v.", "We have kept the more reliable formula Eq.", "() to compute λ c \\lambda _c.♥\\heartsuit This is Krispy Kreme Doughnuts, a small cap US stock.", "Q=10VQ=10 V where VV is now in M $ and vv in thousand $.", "The numbers correspond to March 2012.♣\\clubsuit Club Med is a small cap French stock.", "Q=10VQ=10 V, with VV is in M Euros.", "and vv in thousand Euros.", "The numbers correspond to 2002.♭\\flat : For CDS on single names, these are OTC markets for which we only have estimates.", "Daily transactions are very patchy and their number is typically in the range 1-201 - 20.We have chosen a reasonable value N=Q/v=10N =Q/v = 10, corresponding to a position of 10 to 100 M$.", "As expected, liquidity discount and potential deleveraging problems are very substantial here.We see that for liquid futures, such as the BUND or SP500, the critical leverage is large enough that the phenomenon we discuss here is unlikely to ever occur.", "As soon as we enter the world of equities, however, the situation looks quite different, whereas for OTC market the effect is certainly very real." ], [ "Conclusion", "The above discussion underscores the need to value positions based on liquidation prices rather than mark-to-market prices.", "For small, unleveraged positions in liquid markets there is no problem, but as soon as any of these conditions are violated, the problem can become severe.", "As we have shown, standard valuations, which do nothing to take impact into account, can be wildly over-optimistic.", "The solution that we have proposed accomplishes this goal by estimating liquidation prices based on recent advances in understanding market impact.", "The procedures that we suggest have the key virtue of being extremely easy to implement.", "They are based on quantities such as volatility, trading volume, or the spread, that are easy to measure.", "Risk estimates can be computed for the typical expected behavior or for the probability of a loss of a given magnitude – anything that can be done with standard risk measures can be easily replicated to take impact into account, with little additional effort.", "The worst negative side-effects of mark-to-market valuations occur when leverage is used.", "As we have shown here, when liquidity is low leverage can become critical.", "By this we mean that as a position is being entered there is a critical value of the leverage $\\lambda _c$ above which it becomes very likely that liquidation will result in bankruptcy, i.e.", "liquidation value less than money owed to creditors.", "This does not require bad luck or unusual price fluctuations – it is a nearly mechanical consequence of using too much leverage.", "Standard mark-to-market accounting gives no warning of this problem, in fact quite the opposite: Impact raises prices as a position is purchased, causing leverage to be underestimated.", "However, as a position is unwound the situation is reversed.", "The impact of unwinding causes leverage to rise, and if the initial leverage is critical, the leverage becomes infinite and the position is bankrupt.", "Under mark-to-market accounting this comes as a complete surprise.", "Under impact-adjusted accounting, in contrast, the warning is clear.", "As the critical point is approached the impact-adjusted leverage diverges, telling any sensible portfolio manager that it is time to stop buying.", "The method of valuation that we propose here could potentially be used both by individual risk managers as well as by regulators.", "Had such procedures been in place in the past, we believe that many previous disasters could have been avoided.", "As demonstrated in the previous section, the values where leverage becomes critical are not unreasonable compared to those used before, such as the leverages of 50 - 100 used by LTCM in 1998, or 30-40 used by Lehman Brothers and other investment banks in 2008.", "However, one should worry about other potentially destabilizing feedback loops that our impact-adjusted valuation could trigger.", "For example, in a crisis situation, spreads and volatilities increase while the liquidity of the market decreases, leading to a stronger discount on the asset valuation.", "But as was the case during the 2008 crisis, the write-down of the value of some books lead to further fire-sales, fueling more panic.", "So it is important to estimate the parameters entering the impact formula (volatility, spread and available volumes) using a slow moving average to avoid any over-reaction to temporary liquidity draughts.", "A key point underlying our discussion here is that market impact occurs for both informed and uninformed trades.", "Empirical studies make it clear that temporary market impact occurs even if trades are made for reasons, such as hedging or liquidity, that have nothing to do with underlying fundamentals.", "This should not be surprising: Typically the counterparty has no way of knowing whether the opposite side of the trade is “informed\" or “uninformed\"To be clear, there are two kinds of impact.", "The first is due to correctly anticipating price movements.", "On the timescales for completing a trade this is typically small.", "The second is due to influencing prices.", "Our point is that if my counterparty is anonymous I have no way of knowing how informed she is, and therefore must react in a generic manner.", "The failure of mark-to-market accounting can thus be viewed as a failure of the theory of efficient markets, or at the very least the need to take a liberal view of what it means.", "The fact that prices can change substantially due to random events that have nothing to do with fundamentals reflects a failure of prices to provide accurate valuations.", "Alternatively, one can take a generous view of what the word “accurate\" means, as Fisher Black did when he famously said, “we might define an efficient market as one in which price is within a factor of 2 of value $\\ldots $ By this definition I think almost all markets are efficient almost all of the time.", "`Almost all' means at least $90\\%$\" [29].", "The failure of marginal prices as a useful means of valuation is part of an emerging view of markets as dynamic, endogenously driven and self-referential [30], [15], as suggested long ago by Keynes [31] and more recently by Soros [32].", "For example, recent studies suggest that exogenous news play a minor role in explaining major price jumps [33], while self-referential feedback effects are strong [34].", "Market prices are molded and shaped by trading, just as trading is molded and shaped by prices, with intricate and sometimes destabilising feedback.", "Because the liquidity of markets is so low, the impact of trades is essential to understand why prices move [11]." ], [ "Acknowledgments", "This work was supported by the National Science Foundation under grant 0965673, by the European Union Seventh Framework Programme FP7/2007-2013 under grant agreement CRISIS-ICT-2011-288501, and by the Sloan Foundation.", "JPB acknowledges important discussions with X. Brokmann, J. Kockelkoren and B. Toth." ] ]

[ [ "Coherent States in Gravitational Quantum Mechanics" ], [ "Abstract We present the coherent states of the harmonic oscillator in the framework of the generalized (gravitational) uncertainty principle (GUP).", "This form of GUP is consistent with various theories of quantum gravity such as string theory, loop quantum gravity, and black-hole physics and implies a minimal measurable length.", "Using a recently proposed formally self-adjoint representation, we find the GUP-corrected Hamiltonian as a generator of the generalized Heisenberg algebra.", "Then following Klauder's approach, we construct exact coherent states and obtain the corresponding normalization coefficients, weight functions, and probability distributions.", "We find the entropy of the system and show that it decreases in the presence of the minimal length.", "These results could shed light on possible detectable Planck-scale effects within recent experimental tests." ], [ "Introduction", "The canonical quantization and the path integral quantization of gravity are two well-known but old proposals which attempted to quantize gravity and to unify the general relativity with the laws of quantum mechanics.", "However, from field theoretical approach, the theory of relativity is not renormalizable and results in the ultraviolet divergences.", "Indeed, beyond the Planck energy scale, the effects of gravity are so important which could lead to discreteness of the very spacetime.", "This is due to the fact that when we probe small distances with high energies, the spacetime structure will be significantly disturbed by the gravitational effects.", "However, we can solve the normalizability problem of gravity by introducing a minimal measurable length as an effective cutoff in the ultraviolet domain.", "Various candidates of quantum gravity such as string theory, loop quantum gravity and quantum geometry all agree on the existence of a minimum observable length.", "In the language of the string theory, a string cannot probe distances smaller than its length.", "Moreover, some Gedanken experiments in black-hole physics and noncommutative geometry imply a minimal length of the order of the Planck length $\\ell _{Pl}=\\sqrt{\\frac{G\\hbar }{c^3}}\\approx 10^{-35}m$ where $G$ is Newton's gravitational constant.", "For instance, since in string theory the mass of a string is proportional to its length, they expand in size when probed at sufficiently high energies.", "This suggests an additional momentum dependent uncertainty in the position of a string in the form of [1], [2] ($\\hbar =1$ ) $\\Delta X \\ge \\frac{1}{2\\Delta P } +k\\ell _{Pl}\\Delta P,$ where $k$ is a dimensionless constant.", "In order to incorporate the idea of the minimal length into quantum mechanics, we need to change the Heisenberg uncertainty principle to the so-called Generalized Uncertainty Principle (GUP).", "The introduction of this idea has attracted much attention in recent years and many papers have appeared in the literature to address the effects of GUP on various quantum mechanical systems [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26].", "In this paper, we are interested to find exact coherent states of the harmonic oscillator in the framework of the generalized commutation relation in the form $[X,P]=i\\hbar (1+\\beta P^{2})$ where $\\beta $ is the GUP parameter.", "Note that the problem of the GUP-corrected harmonic oscillator is exactly solvable in the momentum space and its exact energy eigenvalues and the eigenfunctions are obtained in Refs.", "[27], [28].", "Moreover, the perturbative construction of the corresponding coherent states is discussed in Refs.", "[29], [30] to first-order of the GUP parameter.", "Here, following Klauder's approach and using the formally self-adjoint representation of the deformed commutation relation, we take the Hamiltonian as a generator of the generalized Heisenberg algebra and find the exact form of the coherent states, weight functions, normalization coefficients, and probability distributions.", "We show that the entropy of the system reduces in the GUP scenario as a consequence of the minimal observable length and we explain the physical reasoning behind this phenomenon.", "The connection between our results and the recent progresses in probing Planck-scale physics with quantum optics is discussed finally." ], [ "The Generalized Uncertainty Principle", "The Heisenberg uncertainty relation asserts that we can measure the position and momentum of a particle separately with arbitrary precision.", "So if there is a absolute minimal value for the results of the measurements, the Heisenberg uncertainty relation should be modified.", "Here we consider a generalized uncertainty principle which implies a minimum observable length $\\Delta X \\Delta P \\ge \\frac{\\hbar }{2} \\left( 1 +\\beta \\left[(\\Delta P)^2+\\langle P\\rangle ^2 \\right]\\right),$ where $\\beta $ is the GUP parameter.", "We also have $\\beta =\\beta _0/(M_{Pl} c)^2$ where $M_{Pl}$ is the Planck mass and $\\beta _0$ is of the order of unity.", "GUPs of this type also arise from polymer quantization [31], [32], [33].", "Note that the deviation from the Heisenberg picture takes place in the high energy domain where the effects of gravity would be significant.", "Thus, for the energies much lower than the Planck energy $M_{Pl} c^2\\sim 10^{19}$ GeV, we recover the well-known Heisenberg uncertainty relation.", "It is easy to check that the above inequality relation (REF ) implies the existence of an absolute minimum observable length given by $(\\Delta X)_{min}=\\hbar \\sqrt{\\beta }$ .", "In the context of string theory, we can interpret this length as the length of the string and conclude that the string's length is proportional to the square root of the GUP parameter.", "In one spatial dimension, the above uncertainty relation can be obtained from the following deformed commutation relation $[X,P]=i\\hbar (1+\\beta P^2).$ As it is recently suggested in Ref.", "[28], [34], we can write $X$ and $P$ in terms of ordinary position and momentum operator as $X &=& x,\\\\ P &=&\\frac{\\tan \\left(\\sqrt{\\beta }p\\right)}{\\sqrt{\\beta }},$ where $[x,p]=i\\hbar $ and $X$ and $P$ are symmetric operators on the dense domain $S_{\\infty }$ with respect to the following scalar product: $\\langle \\psi |\\phi \\rangle =\\int _{-\\frac{\\pi }{2\\sqrt{\\beta }}}^{+\\frac{\\pi }{2\\sqrt{\\beta }}}\\mathrm {d}p\\,\\psi ^{*}(p)\\phi (p).$ With this definition, the commutation relation (REF ) is exactly satisfied.", "In this representation, the completeness relation and scalar product can be written as $\\langle p^{\\prime }|p\\rangle = \\delta (p-p^{\\prime }),\\\\\\int _{-\\frac{\\pi }{2\\sqrt{\\beta }}}^{+\\frac{\\pi }{2\\sqrt{\\beta }}}\\mathrm {d}p\\, |p\\rangle \\langle p|=1.$ Also the eigenfunctions of the position operator in momentum space is given by the solutions of the eigenvalue equation $X\\,u_x(p)=x\\,u_x(p),$ where $u_x(p)=\\langle p|x\\rangle $ .", "The normalized solution is $u_x(p)=\\sqrt{\\frac{\\sqrt{\\beta }}{\\pi }}\\exp \\left({-i\\frac{ p}{\\hbar }}x\\right),$ which can be used to check the scalar product relation (REF ).", "Now using () we find the wave function in coordinate space as $\\psi (x)=\\sqrt{\\frac{\\sqrt{\\beta }}{\\pi }}\\int _{-\\frac{\\pi }{2\\sqrt{\\beta }}}^{+\\frac{\\pi }{2\\sqrt{\\beta }}}e^{\\frac{i px}{\\hbar }}\\phi (p)\\mathrm {d} p.$ Note that this representation is equivalent with the seminal proposal by Kempf, Mangano and Mann (KMM) [27] $X &=& (1+\\beta p^2)x,\\\\P &=& p,$ through the following canonical transformation: $X&\\longrightarrow &\\left[1+\\arctan ^2\\left(\\sqrt{\\beta }P\\right)\\right]X,\\\\P&\\longrightarrow &\\arctan \\left(\\sqrt{\\beta }P\\right)/\\sqrt{\\beta },$ which transforms Eqs.", "(REF ) and () to Eqs.", "(REF ) and () subjected to Eq.", "(REF ).", "This representation (REF -) preserves the ordinary nature of the position operator and only affects the kinetic part of the Hamiltonian.", "Similar to KMM representation the momentum operator $P$ is self-adjoint, while the position operator $X$ is merely symmetric.", "This is due to the fact that the domain of $X^\\dagger $ is much larger than the domain of $X$ .", "However, this representation is formally self-adjoint, i.e., $A=A^\\dagger $ for $A\\in \\lbrace X,P\\rbrace $ (see [28] for details).", "Now $P$ and $p$ can be interpreted as follows: $p$ is the momentum operator at low energies ($p=-i\\hbar \\partial /\\partial {x}$ ) and $P$ is the momentum operator at high energies.", "Obviously, this procedure affects all Hamiltonians in the quantum mechanics, namely $H=\\frac{P^2}{2m} + V(X),$ where using Eqs.", "(REF ) and () can be rewritten as $H= \\frac{\\tan ^2\\left(\\sqrt{\\beta }p\\right)}{2\\beta m}+ V(x).$ Since this Hamiltonian is formally self-adjoint ($H^\\dagger =H$ ), we can use the general scheme of the coherent states to find the desirable solutions, whereas this property is absent in other representations [27], [35].", "In the quantum domain, this Hamiltonian results in the following generalized Schrödinger equation in coordinate space $-\\frac{\\hbar ^2}{2m}\\frac{\\partial ^2\\psi (x)}{\\partial x^2}+\\sum _{n=3}^\\infty \\frac{2^{2n} (2^{2n}-1)(2n-1)B_{2n}\\hbar ^{2(n-1)}\\beta ^{n-2}}{2m(2n)!", "}\\frac{\\partial ^{2(n-1)}\\psi (x)}{\\partial x^{2(n-1)}}+V(x)\\psi (x)=E\\,\\psi (x),\\hspace{28.45274pt}$ where $B_n$ is the $n$ th Bernoulli number." ], [ "GUP and the Harmonic oscillator", "For the harmonic oscillator, because of the quadratic form of the potential $V(x)=(1/2)\\,m\\omega ^2x^2$ , we obtain a second-order differential equation in the momentum space, namely [28] $-\\frac{\\partial ^2\\phi (p)}{\\partial p^2}+\\frac{\\tan ^2\\left(\\sqrt{\\gamma }p\\right)}{\\gamma }\\phi (p)=\\epsilon \\,\\phi (p),$ where $p\\rightarrow \\sqrt{m\\hbar \\omega }\\,p$ , $\\gamma =m\\hbar \\omega \\beta $ , and $\\epsilon =\\displaystyle \\frac{2E}{\\hbar \\omega }$ .", "In terms of the new variable $y=\\sqrt{\\gamma }p$ , it reads $\\left[-\\frac{\\partial ^2}{\\partial y^2}+\\nu (\\nu -1)\\tan ^2(y)-\\bar{\\epsilon }(\\nu )\\right]\\phi (y;\\nu )=0,$ where by definition $\\nu =\\frac{1}{2}\\left(1+\\sqrt{1+\\frac{4}{\\gamma ^2}}\\right),\\hspace{56.9055pt}\\bar{\\epsilon }=\\frac{\\epsilon }{\\gamma },$ and the boundary condition is $\\phi (y;\\nu )\\bigg |_{y=\\pm \\pi /2}=0.$ The above differential equation is exactly solvable and the eigenfunctions can be obtained in terms of the Gauss hypergeometric functions [28] $\\phi _{2k}(p\\,;\\gamma )&=&{\\cal A}_k(\\nu )\\left[\\cos (\\sqrt{\\gamma }p)\\right]^{\\left(1+\\sqrt{1+\\frac{4}{\\gamma ^2}}\\right)/2}\\nonumber \\\\&&\\times \\,{}_2F_1\\left(-k,\\nu +k; \\nu + \\frac{1}{2};\\cos ^2(\\sqrt{\\gamma }p)\\right),\\hspace{14.22636pt}k=0,1,2,\\ldots ,$ for even states and $\\phi _{2k+1}(p\\,;\\gamma )&=&{\\cal B}_k(\\nu )\\sin (\\sqrt{\\gamma }p)\\left[\\cos (\\sqrt{\\gamma }p)\\right]^{\\left(1+\\sqrt{1+\\frac{4}{\\gamma ^2}}\\right)/2}\\nonumber \\\\ &&\\times \\,{}_2F_1\\left(-k,\\nu +k+1; \\nu + \\frac{1}{2};\\cos ^2(\\sqrt{\\gamma }p)\\right),\\hspace{14.22636pt}k=0,1,2,\\ldots ,\\hspace{28.45274pt}$ for odd states.", "Moreover, the exact GUP-corrected energy spectrum is given by $\\epsilon _n=\\left(2n+1\\right)\\left(\\sqrt{1+\\frac{\\gamma ^2}{4}}+\\frac{\\gamma }{2}\\right)+\\gamma n^2,\\hspace{28.45274pt}n=0,1,2,\\ldots \\,.$ The generalization of this problem to arbitrary dimension is also discussed in Ref.", "[35]." ], [ "The Generalized Coherent States", "Coherent states were originally introduced by Schrödinger in 1926 [36] have applications in many areas of physics [37], [38], [39], [40], [41], [42].", "To construct coherent states, we follow Klauder's approach [43], [44] and use the version of the generalized Heisenberg algebra (GHA) [45], [46], [47], [48], [49] given in Refs.", "[48], [49].", "This version of GHA consists of the generators $J_0$ , $A$ , and $A^\\dagger $ that satisfy [48] $J_0A^\\dagger &=& A^\\dagger f(J_0),\\\\AJ_0 &=& f(J_0)A,\\\\\\left[A^{\\dagger },A\\right] &=& J_0-f(J_0).$ Here $A=(A^\\dagger )^\\dagger $ and $J_0=J_0^\\dagger $ is the Hamiltonian of the system.", "Moreover, $f(J_0)$ is an analytic function of $J_0$ and is called the characteristic function of the algebra.", "These generators also satisfy $J_0|m\\rangle &=& \\alpha _m|m\\rangle ,\\\\A^\\dagger |m\\rangle &=& N_m|m+1\\rangle ,\\\\A|m\\rangle &=& N_{m-1}|m-1\\rangle ,$ where $N_m^2=\\alpha _{m+1}-\\alpha _0$ .", "For instance, for the linear function $f(x)=x+1$ , we obtain the harmonic oscillator algebra, and for $f(x)=qx+1$ , the algebra in Eqs.", "(REF )-() becomes the deformed Heisenberg algebra [48].", "It is shown that the quantum systems having the energy spectrum $\\epsilon _{n+1}=f(\\epsilon _{n}),$ where $\\epsilon _{n}$ and $\\epsilon _{n+1}$ are successive energy levels and $f(x)$ is a distinct function for each physical system, are described by these generalized Heisenberg algebras.", "Also, This function is exactly the same function that appears in the construction of the algebra, i.e., the characteristic function of the algebra.", "In the algebraic langauge, $J_0$ is the Hamiltonian, $A$ is the annihilation operator, and $A^\\dagger $ is the creation operator.", "These operators are related to the Casimir operator of the GHA through the relation $C=A^\\dagger A-J_0=A A^\\dagger -f(J_0).$ Now the coherent states are given by [49] $|z\\rangle =N(z)\\sum _{n=0}^{\\infty }\\frac{z^n}{N_{n-1}!", "}|n\\rangle ,$ where $A|z\\rangle =z|z\\rangle $ , $N(z)$ is the normalization coefficient, by definition $N_{n}!\\equiv N_0N_1\\cdots N_n$ , and by consistency $N_{-1}!\\equiv 1$ .", "Note that Klauder's coherent states should satisfy the following minimal set of conditions [49]: normalizability condition $\\langle z|z\\rangle =1.$ continuity in the label $|z-z^{\\prime }|\\rightarrow 0,\\hspace{28.45274pt}|| |z\\rangle -|z^{\\prime }\\rangle ||\\rightarrow 0.$ completeness relation $\\int \\mathrm {d}^2z\\, w(z)\\, |z\\rangle \\langle z|=1.$ Figure: Normalization function for the GUP-correctedharmonic oscillator and for γ={0,0.1,1,10,100,1000}\\gamma =\\lbrace 0,0.1,1,10,100,1000\\rbrace .For the harmonic oscillator with the minimal length uncertainty, using Eq.", "(REF ) we obtain $\\epsilon _{n+1}&=&\\epsilon _{n}+2\\gamma (n+1)+\\sqrt{4+\\gamma ^2},\\nonumber \\\\&=&\\epsilon _{n}+2\\sqrt{\\gamma \\epsilon _{n}+1}+\\gamma .$ Thus, the characteristic function reads $f(x)=x+2\\sqrt{\\gamma x+1}+\\gamma .$ Since the above algebraic formalism implies $\\alpha _n=\\epsilon _{n}$ , we find $N^2_{n-1}&=&\\alpha _n-\\alpha _0=\\gamma \\left[n^2+\\left(\\sqrt{1+\\frac{4}{\\gamma ^2}}+1\\right)n\\right],\\nonumber \\\\&=&\\gamma \\left( n+2\\nu \\right)n,$ which results in $N_{n-1}!=\\gamma ^{n/2}\\sqrt{\\frac{n!", "(n+2\\nu )!", "}{(2\\nu )!", "}}.$ So the coherent states given in Eq.", "(REF ) can be written as $|z\\rangle =N(|z|)\\sqrt{(2\\nu )!", "}\\sum _{n=0}^{\\infty }\\frac{\\gamma ^{-n/2}z^n}{\\sqrt{n!", "}\\sqrt{(n+2\\nu )!", "}}|n\\rangle .$ The normalizability condition is given by $1=N^2(|z|)(2\\nu )!\\sum _{n=0}^{\\infty }\\frac{\\gamma ^{-n}|z|^{2n}}{n!", "(n+2\\nu )!", "}.$ Since we have $\\sum _{n=0}^{\\infty }\\frac{\\gamma ^{-n}|z|^{2n}}{n!", "(n+2\\nu )!", "}=\\frac{I_{2\\nu }\\left(\\frac{2|z|}{\\sqrt{\\gamma }}\\right)}{\\gamma ^{-\\nu }|z|^{2\\nu }},$ where $I_p(z)$ is the modified Bessel function of the first kind of order $p$ and $0\\le |z|<\\infty $ , the normalizability condition reads $N^2(|z|)=\\frac{\\gamma ^{-\\nu }|z|^{2\\nu }}{(2\\nu )!\\,I_{2\\nu }\\left(\\frac{2|z|}{\\sqrt{\\gamma }}\\right)}.$ In Fig.", "REF we have depicted $N(x)$ for various values of $\\gamma $ .", "Note that for $\\gamma \\rightarrow 0$ (harmonic oscillator without GUP), $N(x)$ goes to $e^{-x^2/4}$ and for $\\gamma \\rightarrow \\infty $ ($\\nu \\rightarrow 1$ ) (particle in a box), it tends to $\\displaystyle \\frac{x^2/\\gamma }{2I_2(2x/\\sqrt{\\gamma })}$ .", "Figure: Weight function for the GUP-corrected harmonicoscillator and for γ={0,0.1,1,10}\\gamma =\\lbrace 0,0.1,1,10\\rbrace .To satisfy the completeness relation, we need to find the adequate weight function $w(r)$ , $z=re^{i\\theta }$ , implying the equality $2\\pi \\sum _{n=0}^{\\infty }|n\\rangle \\langle n|\\frac{(2\\nu )!\\,\\gamma ^{-n}}{n!", "(n+2\\nu )!", "}\\int _0^{\\infty }\\mathrm {d}r\\,N^2(r)w(r) r^{2n+1}=1.$ If we take $x=r^2$ , we obtain $\\pi (2\\nu )!\\sum _{n=0}^{\\infty }|n\\rangle \\langle n|\\frac{2\\gamma ^{-n-\\nu -1}}{n!", "(n+2\\nu )!", "}\\int _0^{\\infty }\\mathrm {d}x\\,\\frac{\\gamma ^{\\nu +1}N^2(\\sqrt{ x})w(\\sqrt{ x})}{2x^\\nu }x^{n+\\nu }=1.$ So by taking $\\frac{\\pi (2\\nu )!\\,\\gamma ^{\\nu +1}N^2(\\sqrt{ x})w(\\sqrt{x})}{2x^\\nu }=K_{2\\nu }\\left(2\\sqrt{\\frac{x}{\\gamma }}\\right),$ where $K_p(x)$ is the modified Bessel function of the second kind of order $p$ and using $\\int _0^{\\infty }\\mathrm {d}x\\,K_{2\\nu }\\left(2\\sqrt{\\frac{x}{\\gamma }}\\right)x^{n+\\nu }=\\frac{1}{2}\\gamma ^{n+\\nu +1}n!", "(n+2\\nu )!,$ Eq.", "(REF ) is satisfied which gives the following weight function $w(\\sqrt{x})=\\frac{2x^\\nu K_{2\\nu }(2\\sqrt{x})}{\\pi (2\\nu )!\\,\\gamma ^{\\nu +1}N^2(\\sqrt{\\gamma x})},$ where can be finally expressed as $w(r)=\\frac{2}{\\pi \\gamma }I_{2\\nu }\\left(\\frac{2r}{\\sqrt{\\gamma }}\\right)K_{2\\nu }\\left(\\frac{2r}{\\sqrt{\\gamma }}\\right).$ In Fig.", "REF , the behavior of the weight function is shown for $\\gamma =\\lbrace 0,0.1,1,10\\rbrace $ .", "As we have expected, for $\\gamma \\rightarrow 0$ (harmonic oscillator without GUP), $w(r)$ tends to $\\displaystyle \\frac{1}{2\\pi }$ and for $\\gamma \\rightarrow \\infty $ ($\\nu \\rightarrow 1$ ) (particle in a box), it goes to $\\displaystyle \\frac{2}{\\pi \\gamma }I_{2}\\left(\\frac{2r}{\\sqrt{\\gamma }}\\right)K_{2}\\left(\\frac{2r}{\\sqrt{\\gamma }}\\right)$ .", "It is worth to mention that a potential application of our results is in quantum optics.", "Indeed the coherent states as the states of the light field can be used to approximately describe the output of a single-frequency laser well above the laser threshold.", "In the absence of GUP, the probability of detecting $n$ photons is given by Poisson distribution, namely $P(n;\\lambda )=|\\langle n|z\\rangle |^2=\\frac{e^{-\\lambda /2}}{n!", "}\\left(\\frac{\\lambda }{2}\\right)^n,$ where $\\lambda =|z|^2=\\langle z|A^\\dagger A|z\\rangle $ and $\\lambda /2$ is the average photon number $\\overline{n}$ in a coherent state for $\\gamma =0$ ($\\epsilon _n=2n+1$ ).In terms of $\\overline{n}$ we have $P(n)=\\frac{e^{-\\bar{n}}\\,\\bar{n}^n}{n!", "}$ .", "In the presence of the minimal length, the probability distribution is $P(n;\\lambda ,\\gamma )&=&|\\langle n|z\\rangle |^2=N^2(\\sqrt{\\lambda })\\frac{\\lambda ^n}{(N_{n-1}!", ")^2}, \\\\&=&\\frac{\\gamma ^{-n-\\nu }}{I_{2\\nu }\\left(2\\sqrt{\\frac{\\lambda }{\\gamma }}\\right)}\\frac{\\lambda ^{n+\\nu }}{n!", "(n+2\\nu )!", "},$ which encodes the effects of GUP on the harmonic oscillator statistics and satisfies $\\sum _{n=0}^\\infty P(n;\\lambda ,\\gamma )=1.$ Also we have $\\frac{\\lambda }{\\gamma }=\\overline{n^2}+2\\nu \\overline{n},$ which results in $P(n;\\overline{n},\\nu )=\\frac{\\left(\\overline{n^2}+2\\nu \\overline{n}\\right)^{n+\\nu }}{I_{2\\nu }\\left(2\\sqrt{\\overline{n^2}+2\\nu \\overline{n}}\\right)\\,n!", "(n+2\\nu )!", "}.$ For $\\gamma \\rightarrow 0$ , we have $N^2(\\sqrt{\\lambda })=e^{-\\lambda /2}$ (see Fig.", "REF ) and using $\\lim _{\\nu \\rightarrow \\infty }\\frac{\\nu ^{-n}\\,(n+2\\nu )!", "}{(2\\nu )!", "}=2^n,$ Eq.", "(REF ) reads $N_{n-1}!\\simeq \\sqrt{2^n\\,n!", "}\\,\\,.$ So Eq.", "(REF ) gives the Poisson distribution (REF ) at this limit.", "For $\\gamma \\rightarrow \\infty $ ($\\nu \\rightarrow 1$ ), Eq.", "() is expressed as $P(n;\\lambda ,\\gamma \\rightarrow \\infty )=\\frac{\\gamma ^{-n-1}}{I_{2}\\left(2\\sqrt{\\frac{\\lambda }{\\gamma }}\\right)}\\frac{\\lambda ^{n+1}}{n!(n+2)!", "},$ where $\\displaystyle \\frac{\\lambda }{\\gamma }\\simeq \\overline{n^2}+2\\overline{n}$ .", "Fig.", "REF shows the schematic behavior of the probability distribution for the GUP-corrected harmonic oscillator and for various values of the GUP parameter.", "As the figure shows, $n_{max}$ which gives the maximum probability, decreases as $\\gamma $ increases and is given by the root of the following equation: $H_{n_{max}}+H_{n_{max}}^{(2\\nu )}+\\ln \\frac{\\gamma }{\\lambda }-2\\xi =0,$ where $H_p$ is $p$ th harmonic number, $H_p^{(r)}$ is $p$ th harmonic number of order $r$ , and $\\xi $ is Euler's constant.", "Note that Eqs.", "() and (REF ) define the probability of detecting $n$ photons in a laser beam subject to (REF ) which is a result of the deformed generalized uncertainty relation (expected to be significant at very high energies where gravity is considerable).", "Therefore, for a fixed $\\lambda $ , the averaged number of photons decreases as the GUP parameter increases.", "Figure: The probability distribution for theGUP-corrected harmonic oscillator.", "We set λ=20\\lambda =20(n ¯=10\\overline{n}=10 for β=0\\beta =0) and γ={0,0.01,0.1,1}\\gamma =\\lbrace 0,0.01,0.1,1\\rbrace .Now we can define the entropy of this system as the logarithmic measure of the density of states: $S(\\lambda ;\\gamma )=-k_B\\sum _{n=0}^\\infty P(n;\\lambda ,\\gamma )\\,\\ln P(n;\\lambda ,\\gamma ),$ where $k_B$ is the Boltzmann constant.", "In Fig.", "REF , we have depicted the entropy of the GUP-corrected harmonic oscillator for $\\gamma =\\lbrace 0,0.1,1\\rbrace $ .", "As the figure shows, for fixed $\\lambda $ , the entropy decreases as $\\gamma $ increases and tends to the Poisson entropy for $\\gamma \\rightarrow 0$ $S(\\lambda ;0)=k_B\\left[\\frac{\\lambda }{2}\\left(1-\\ln \\frac{\\lambda }{2}\\right)+e^{-\\lambda /2}\\sum _{n=0}^{\\infty }\\frac{\\left(\\frac{\\lambda }{2}\\right)^n\\ln n!}{n!", "}\\right].$ The reason behind this behavior can be understood from the GUP commutation relation (REF ).", "Indeed, we can consider the right hand side of this equation effectively as the GUP-corrected Planck's constant which is always greater than $\\hbar $ .", "Therefore, in the language of the statistical mechanics, the size of the unit cell in the phase space increases and consequently the number of the accessible states and the entropy of the system decrease with respect to the absence of GUP.", "Figure: The entropy of the GUP-corrected harmonicoscillator for γ={0,0.1,1}\\gamma =\\lbrace 0,0.1,1\\rbrace .As we state before, a potential application of our calculations is in quantum optics.", "But the question is: could the relation between quantum optics and Plank-scale uncertainty relations have some detectable effects?", "To answer this question we should mention that in recent years various approaches are developed to test the effects of quantum gravity and to explore possible quantum gravitational phenomena.", "These attempts range from astronomical observations [50], [51] to table-top experiments [52].", "Amelino-Camelia and Lammerzahl proposed some laser interferometric setups to explain puzzling observations of ultrahigh energy cosmic rays in the context of quantum gravity modified laws of particle propagation [53].", "The implications of high intensity Laser projects for quantum gravity phenomenology are also discussed by Magueijo based on deformed special relativity [54].", "Recently, Pikovski et al.", "have introduced a scheme to experimentally test the existence of a minimal length scale as a modification of the Heisenberg uncertainty relation for various GUP scenarios in the context of quantum optics [52].", "They utilized quantum optical control to probe possible deviations from the quantum commutation relation at the Planck scale and showed that their scheme is within reach of current technology.", "In fact, the idea is direct measurement of the canonical commutator of a massive object using a quantum optical ancillary system that produces nonlinear enhancement without need for Planck-scale accuracy of position measurements.", "So the possible Planck-scale commutator deformations can be observed with very high accuracy by optical interferometric techniques that are within experimental reach.", "These experimental progresses support the implication of our calculations which could shed light on possible detectable Planck-scale effects with quantum optics." ], [ "Conclusions", "In this paper, we have studied the construction of the coherent states of the harmonic oscillator in the context of the generalized uncertainty principle which implies a minimal length uncertainty proportional to the Planck length.", "Following Klauder's approach, we constructed the generalized Heisenberg algebra where the Hamiltonian was its formally self-adjoint generator.", "Then, after finding the characteristic function of the algebra, we obtained the exact expression for the coherent states, weight functions, normalization coefficients, and probability distributions and studied their behavior in terms of the GUP parameter.", "We showed that because of the gravitational uncertainty relation the ordinary quantum description of laser light should be modified and the Poisson probability distribution will not be exactly preserved.", "Also, the entropy of the system decreased in the presence of the minimal length uncertainty due to the increase of the size of the unit cell in the phase space.", "Finally, we indicated that recent progresses to experimentally test the existence of a minimal length scale could reveal some evidence of possible quantum gravitational effects." ], [ "Acknowledgments", "I am very grateful to Kourosh Nozari for fruitful discussions and suggestions and for a critical reading of the manuscript." ] ]

[ [ "Measurement of psi(2S) meson production in pp collisions at sqrt(s)=7\n TeV" ], [ "Abstract The differential cross-section for the inclusive production of psi(2S) mesons in pp collisions at sqrt(s)=7 TeV has been measured with the LHCb detector.", "The data sample corresponds to an integrated luminosity of 36 pb-1.", "The psi(2S) mesons are reconstructed in the decay channels psi(2S) -> mu+ mu- and psi(2S) -> J/psi pi+ pi-, with the J/psi meson decaying into two muons.", "Results are presented both for promptly produced psi(2S) mesons and for those originating from b-hadron decays.", "In the kinematic range pT(psi(2S)) <= 16 GeV/c and 2 < y(psi(2S)) <= 4.5 we measure 1.44 +- 0.01 +- 0.12+0.2-0.4 mub for prompt psi(2S) production and 0.25 +- 0.01 +- 0.02 mub for psi(2S) from b-hadron decays, where the last uncertainty on the prompt cross-section is due to the unknown psi(2S) polarization.", "Recent QCD calculations are found to be in good agreement with our measurements.", "Combining the present result with the LHCb J/psi measurements we determine the inclusive branching fraction B(b -> psi(2S) X) = (2.73 +- 0.06 +- 0.16 +- 0.24) x 10^(-3), where the last uncertainty is due to the B(b -> J/psi X), B(J/psi -> mu+ mu-) and B(psi(2S) -> e+ e-) branching fraction uncertainties.", "All above results are corrected by an erratum included as an appendix." ], [ "Introduction", "Since its discovery, heavy quarkonium has been one of the most important test laboratories for the development of QCD at the border between the perturbative and non-perturbative regimes, resulting in the formulation of the nonrelativistic QCD (NRQCD) factorisation formalism [1], [2].", "However, prompt production studies carried out at the Tevatron collider in the early 1990s [3] made clear that NRQCD calculations, based on the leading-order (LO) colour-singlet model (CSM), failed to describe the absolute value and the transverse momentum ($p_{\\mathrm {T}}$ ) dependence of the charmonium production cross-section and polarization data.", "Subsequently, the inclusion of colour-octet amplitudes in the NRQCD model has reduced the discrepancy between theory and experiment, albeit at the price of tuning $ad~hoc$ some matrix elements [2].", "On the other hand, recent computations of the next-to-leading-order (NLO) and next-to-next-to-leading-order (NNLO) terms in the CSM yielded predictions in better agreement with experimental data, thus resurrecting interest in the colour-singlet framework.", "Other models have been proposed and it is important to test them in the LHC energy regime [4], [5].", "Heavy quarkonium is also produced from $b$ -hadron decays.", "It can be distinguished from promptly produced quarkonium exploiting its finite decay time.", "QCD predictions are based on the Fixed-Order-Next-to-Leading-Log (FONLL) approximation for the $b\\bar{b}$ production cross-section.", "The FONLL approach improves NLO results by resumming $p_{\\text{T}}$ logarithms up to the next-to-leading order [6], [7].", "To allow a comparison with theory, promptly produced quarkonia should be separated from those coming from $b$ -hadron decays and from those cascading from higher mass states (feed-down).", "The latter contribution strongly affects $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}$ production and complicates the interpretation of prompt $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}$ data.", "On the other hand, $\\psi (2S)$ charmonium has no appreciable feed-down from higher mass states and therefore the results can be directly compared with the theoretical predictions, making it an ideal laboratory for QCD studies.", "This paper presents a measurement of the $\\psi {(2S)}$ meson production cross-section in $pp$ collisions at the centre-of-mass energy $\\sqrt{s}$ = 7$\\mathrm {\\,Te\\hspace{-1.00006pt}V}$ .", "The data were collected by the LHCb experiment in 2010 and correspond to an integrated luminosity of 35.9$\\pm $ 1.3$\\mbox{\\,pb}^{-1}$ .", "The analysis is similar to that described in Ref.", "[8] for the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ production studies; in particular, the separation between promptly produced $\\psi (2S)$ and those originating from $b$ -hadron decays is based on the reconstructed decay vertex information.", "Two decay modes of the $\\psi {(2S)}$ meson have been used: $\\psi (2S) \\rightarrow \\mu ^+ \\mu ^- $ and $\\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^- $ followed by $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\rightarrow \\mu ^+ \\mu ^- $ .", "The $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ mode, despite a larger background and a lower reconstruction efficiency, is used to cross-check and average the results, and to extend the accessible phase space.", "The production of $\\psi {(2S)}$ meson at the LHC has also been studied at the CMS experiment [9]." ], [ "The LHCb detector and data sample", "The LHCb detector is a forward spectrometer [10], designed for precision studies of $C\\!P$ violation and rare decays of $b$ - and $c$ -hadrons.", "Its tracking acceptance covers approximately the pseudorapidity region $2 < \\eta < 5$ .", "The detector elements are placed along the beam line of the LHC starting with the vertex detector, a silicon strip device that surrounds the $pp$ interaction region and is positioned at 8$\\rm \\,mm$ from the beams during collisions.", "It provides precise measurements of the positions of the primary $pp$ interaction vertices and decay vertices of long-lived hadrons, and contributes to the measurement of particle momenta.", "Other detectors used for momentum measurement include a large area silicon strip detector located before a dipole magnet of approximately 4 Tm, and a combination of silicon strip detectors and straw drift chambers placed downstream.", "Two ring imaging Cherenkov detectors are used to identify charged hadrons.", "Further downstream an electromagnetic calorimeter is used for photon and electron detection, followed by a hadron calorimeter.", "The muon detection consists of five muon stations equipped with multi-wire proportional chambers, with the exception of the centre of the first station using triple-GEM detectors.", "The LHCb trigger system consists of a hardware level, based on information from the calorimeter and the muon systems and designed to reduce the frequency of accepted events to a maximum of 1${\\rm \\,MHz}$ , followed by a software level which applies a full event reconstruction.", "In the first stage of the software trigger a partial event reconstruction is performed.", "The second stage performs a full event reconstruction to further enhance the signal purity.", "The analysis uses events selected by single muon or dimuon triggers.", "The hardware trigger requires one muon candidate with a $p_{\\mathrm {T}}$ larger than 1.4${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ or two muon candidates with a $p_{\\mathrm {T}}$ larger than 560${\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c}$ and 480${\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c}$ .", "In the first stage of the software trigger, either of the two following selections is required.", "The first selection confirms the single muon trigger candidate and applies a harder cut on the muon $p_{\\mathrm {T}}$ at 1.8${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ .", "The second selection confirms the dimuon trigger candidate by requiring the opposite charge of the two muons and adds a requirement to the dimuon mass to be greater than 2.5${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c^2}$ .", "In the second stage of the software trigger, two selections are used for the $\\psi (2S) \\rightarrow \\mu ^+ \\mu ^-$ mode.", "The first tightens the requirement on the dimuon mass to be greater than 2.9${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c^2}$ and it applies to the firtst 8$\\mbox{\\,pb}^{-1}$ of the data sample.", "Since this selection was subsequently prescaled by a factor five, for the largest fraction of the remaining data (28$\\mbox{\\,pb}^{-1}$ ) a different selection is used, which in addition requires a good quality primary vertex and tracks for the dimuon system.", "For the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\pi ^+ \\pi ^-$ mode only one selection is used which requires the combined dimuon mass to be in a $\\pm $ 120${\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c^2}$ mass window around the nominal ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} $ mass.", "To avoid that a few events with high occupancy dominate the software trigger CPU time, a set of global event cuts is applied on the hit multiplicity of each subdetector used by the pattern recognition algorithms, effectively rejecting events with a large number of pile-up interactions.", "The simulation samples used for this analysis are based on the Pythia 6.4 generator [11] configured with the parameters detailed in Ref. [12].", "The prompt charmonium production processes activated in Pythia are those from the leading-order colour-singlet and colour-octet mechanisms.", "Their implementation and the parameters used are described in detail in Ref. [13].", "The EvtGen package [14] is used to generate hadron decays and the Geant4 package [15] for the detector simulation.", "The QED radiative corrections to the decays are generated using the Photos package [16]." ], [ "Signal yield", "The two modes, $\\psi (2S) \\rightarrow \\mu ^+ \\mu ^-$ and $\\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ , have different decay and background characteristics, therefore dedicated selection criteria have been adopted.", "The optimisation of the cuts has been performed using the simulation.", "A common requirement is that the tracks, reconstructed in the full tracking system and passing the trigger requirements, must be of good quality ($\\chi ^2/\\text{ndf}<4$ , where ndf is the number of degrees of freedom) and share the same vertex with fit probability $P(\\chi ^2) > 0.5 \\%$ ($\\psi (2S) \\rightarrow \\mu ^+ \\mu ^- $ ) and $P(\\chi ^2) > 5 \\%$ ($\\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^- $ ).", "A cut $p_{\\mathrm {T}}>$ 1.2${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ is applied for the muons from the $\\psi (2S) \\rightarrow \\mu ^+ \\mu ^- $ decay.", "For muons from $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}(\\mu ^+ \\mu ^-) \\pi ^+ \\pi ^-$ we require a momentum larger than 8${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ and $p_{\\mathrm {T}}>$ 0.7${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ .", "Finally the rapidity of the reconstructed $\\psi {(2S)}$ is required to satisfy the requirement $2 < y \\le 4.5$ .", "The $\\psi (2S) \\rightarrow \\mu ^+ \\mu ^- $ invariant mass spectrum for all selected candidates is shown in Fig.", "REF (a).", "The fitting function is a Crystal Ball [17] describing the signal plus an exponential function for the background.", "In total 90600$\\pm $ 690 signal candidates are found in the $p_{\\mathrm {T}}$ range 0–12${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ .", "The mass resolution is 16.01$\\pm $ 0.12${\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c^2}$ and the Crystal Ball parameters that account for the radiative tail are obtained from the simulation.", "For the $\\psi {(2S)} \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}(\\mu ^+ \\mu ^-) \\pi ^+ \\pi ^-$ decay, both pions are required to have $p_{\\mathrm {T}}>$  0.3${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ and the sum of the two-pion transverse momenta is required to be larger than 0.8${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ .", "The quantity $Q = M(J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+\\pi ^-) - M(\\pi ^+\\pi ^-) - M(\\mu ^+\\mu ^-)$ is required to be $\\le $ 200${\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c^2}$ and to improve the mass resolution the dimuon invariant mass $M_{\\mu ^+ \\mu ^-}$ is constrained in the fit to the nominal $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}$ mass value [18].", "Finally, both $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}$ and $\\psi {(2S)} $ candidates must have $p_{\\mathrm {T}}>$ 2${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ .", "The invariant mass spectrum is shown in Fig.", "REF (b) for all selected candidates.", "For this decay mode the peak is described by the sum of two Crystal Ball functions for the signal plus an exponential function for the background.", "The number of signal candidates is 12300$\\pm $ 200, the mass resolution is 2.10$\\pm $ 0.07${\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c^2}$ , and the Crystal Ball tail parameters are fixed to the values obtained from the simulation.", "The fits are repeated in each $\\psi {(2S)} $ $p_{\\mathrm {T}}$ bin to obtain the number of signal and background candidates for both decays.", "Figure: Invariant mass distribution for all ψ(2S)\\psi (2S) candidatespassing the selection cuts for the μ + μ - \\mu ^+ \\mu ^- decay (a) andthe J/ψ(μ + μ - )π + π - J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}(\\mu ^+ \\mu ^-) \\pi ^+ \\pi ^- decay (b)." ], [ "Cross-section measurement", "The differential cross-section for the inclusive $\\psi {(2S)}$ meson production is computed from $\\frac{d\\sigma }{dp_{\\mathrm {T}}}(p_{\\mathrm {T}})= \\frac{N_\\text{sig}(p_{\\mathrm {T}})}{\\mathcal {L} ~\\epsilon _{\\rm tot}(p_{\\mathrm {T}})~{\\cal B}~\\Delta p_{\\mathrm {T}}}$ where $d\\sigma /dp_{\\mathrm {T}}$ is the average cross-section in the given $p_{\\mathrm {T}}$ bin, integrated over the rapidity range $2 < y \\le 4.5$ , $N_\\text{sig}(p_{\\mathrm {T}})$ is the number of signal candidates determined from the mass fit for the decay under study, $\\epsilon _{\\rm tot}(p_{\\mathrm {T}})$ is the total detection efficiency including acceptance and trigger effects, ${\\cal B}$ denotes the relevant branching fraction and $\\Delta p_{\\mathrm {T}}$ is the bin size.", "All branching fractions are taken from Ref.", "[18]: $\\mathcal {B}(\\psi (2S) \\rightarrow e^+ e^-)$ = $(7.72 \\pm 0.17) \\times 10^{-3}$ , $\\mathcal {B}(\\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-)$ = $(33.6 \\pm 0.4) \\times 10^{-2}$ and $\\mathcal {B}(J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\rightarrow \\mu ^+ \\mu ^-)$ = $(5.93 \\pm 0.06) \\times 10^{-2}$ .", "Assuming lepton universality, we use the dielectron branching fraction $\\mathcal {B}(\\psi (2S) \\rightarrow e^+ e^-)$ in Eq.", "(REF ), since $\\mathcal {B}(\\psi (2S) \\rightarrow \\mu ^+ \\mu ^-)$ is less precisely known.", "$\\mathcal {L} $ is the integrated luminosity, which is calibrated using both Van der Meer scans [19], [20] and a beam-profile method [21].", "A detailed description of the two methods is given in Ref. [22].", "The knowledge of the absolute luminosity scale is used to calibrate the number of tracks in the vertex detector, which is found to be stable throughout the data taking period and can therefore be used to monitor the instantaneous luminosity of the entire data sample.", "The integrated luminosity of the data sample used in this analysis is determined to be 35.9$\\mbox{\\,pb}^{-1}$ .", "The total efficiency, $\\epsilon _{\\rm tot}(p_{\\mathrm {T}})$ , is a product of three contributions: the geometrical acceptance, the combined detection, reconstruction and selection efficiency, and the trigger efficiency.", "Each contribution has been determined using simulated events for the two decay channels.", "In order to evaluate the trigger efficiency, the trigger selection algorithms used during data taking are applied to the simulation.", "Figure: Total efficiency vs. p T p_{\\mathrm {T}} computed from simulation forunpolarized ψ(2S)\\psi {(2S)} mesons for ψ(2S)→μ + μ - \\psi (2S) \\rightarrow \\mu ^+\\mu ^- (a) andψ(2S)→J/ψ(μ + μ - )π + π - \\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}(\\mu ^+ \\mu ^-) \\pi ^+ \\pi ^- (b).The total efficiency vs. $p_{\\mathrm {T}}$ for the two channels, assuming the $\\psi (2S)$ meson unpolarized, is shown in Fig.", "REF .", "Extensive studies on dimuon decays of prompt $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}$  [8], $\\psi (2S)$ and $\\Upsilon $  [23] mesons have shown that the total efficiency in the LHCb detector depends strongly on the initial polarization state of the vector meson.", "This effect is absent for $\\psi {(2S)}$ mesons coming from $b$ -hadron decays.", "In fact for these events the natural polarization axis is the $\\psi {(2S)}$ meson flight direction in the $b$ -hadron rest frame, while the $\\psi {(2S)}$ meson appears unpolarized along its flight direction in the laboratory.", "Simulations [8] and measurements from CDF [24] confirm this.", "We do not measure the $\\psi {(2S)}$ meson polarization but we assign a systematic uncertainty to the unpolarized efficiencies in the case of prompt production.", "Events are generated with polarizations corresponding to the two extreme cases of fully transverse or fully longitudinal polarization and the efficiency is re-evaluated.", "The difference between these results and those with the unpolarized sample is taken as an estimate of the systematic uncertainty.", "A similar effect exists for the $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}$ meson emitted in the $\\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}(\\mu ^+ \\mu ^-) \\pi ^+ \\pi ^-$ decay.", "However, in this case, the $\\psi (2S)$ meson polarization is fully transferred to the $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}$ meson since, as measured by the BES collaboration [25], the two pions are predominantly in the $S$ -wave configurationThe small fraction of $D$ -wave measured in Ref.", "[25] has a negligible impact on our conclusion.", "and the dipion-$J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}$ system is also in a $S$ -wave configuration.", "This has been verified with data and is correctly reproduced by the simulation.", "Therefore the systematics due to polarization are fully correlated between the two channels and we use the systematic uncertainties computed for $\\psi (2S) \\rightarrow \\mu ^+ \\mu ^-$ also for the $\\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ decay.", "In order to separate prompt $\\psi (2S)$ mesons from those produced in $b$ -hadron decays, we use the pseudo-decay-time variable defined as $t=d_z (M/p_z)$ , where $d_z$ is the separation along the beam axis between the $\\psi (2S)$ decay vertex and the primary vertex, $M$ is the nominal mass of the $\\psi (2S)$ and $p_z$ is the component of its momentum along the beam axis.", "In case of multiple primary vertices reconstructed in the same event, that which minimises $|d_z|$ has been chosen.", "The prompt component is distributed as a Gaussian function around $t=0$ , with width corresponding to the experimental resolution, while for the $\\psi (2S)$ from $b$ -hadron decays the $t$ variable is distributed according to an approximately exponential decay law, smeared in the fit with the experimental resolution.", "The choice of taking the primary vertex which minimises $|d_z|$ could in principle introduce a background component in the pseudo-decay-time distribution arising from the association of the $\\psi (2S)$ vertex to a wrong primary vertex.", "The effect of such background is found to be of the order of 0.5% in the region around $t=0$ and has been neglected.", "The function used to fit the $t$ distribution in each $p_{\\mathrm {T}}$ bin is $F(t;f_\\text{p},\\sigma ,\\tau _b)=N_\\text{sig}\\left[f_\\text{p}\\delta (t)+(1-f_\\text{p})\\theta (t)\\frac{e^{-\\frac{t}{\\tau _b}}}{\\tau _b}\\right]\\otimes \\frac{e^{-\\frac{1}{2}(\\frac{t}{\\sigma })^2}}{\\sqrt{2\\pi }\\sigma }+N_\\text{bkg}f_\\text{bkg}(t;\\Theta )$ where $N_\\text{sig}$ and $N_\\text{bkg}$ are respectively the numbers of signal and background candidates obtained from the mass fit.", "The fit parameters are the prompt fraction, $f_\\text{p}$ , the standard deviation of the Gaussian resolution function, $\\sigma $ , and the lifetime describing the long-lived component of $\\psi (2S)$ mesons coming from $b$ -hadron decays, $\\tau _b$ .", "In principle, all fit parameters are dependent on $p_{\\mathrm {T}}$ .", "The function $f_\\text{bkg}(t;\\Theta )$ models the background component in the distribution and is defined as the sum of a $\\delta $ function and a Gaussian function for the prompt background, plus two exponential functions for the positive tail and one exponential function for the negative tail, all convolved with a Gaussian function to account for the detector resolution.", "The array of parameters $\\Theta $ is determined from a fit to the $t$ distribution of the events in the mass sidebands.", "Figure: Pseudo-decay-time distribution for ψ(2S)→μ + μ - \\psi (2S) \\rightarrow \\mu ^+ \\mu ^- (a)and ψ(2S)→J/ψπ + π - \\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^- (b) in the p T p_{\\mathrm {T}} range 4<p T ≤54<p_{\\mathrm {T}}\\le 5 Ge V/c{\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c},showing the background and prompt contributions.As an example, the pseudo-decay-time distributions for $\\psi (2S) \\rightarrow \\mu ^+ \\mu ^- $ and $\\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^- $ in the $p_{\\mathrm {T}}$ range $4<p_{\\mathrm {T}}\\le 5$${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ are presented in Fig.", "REF .", "The contributions of background and prompt $\\psi {(2S)}$ mesons are also shown.", "The values of the prompt fraction, $f_\\text{p}$  vs. $p_{\\mathrm {T}}$ in the rapidity range $2 < y \\le 4.5$ , obtained for the $\\mu ^+ \\mu ^-$ and the $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ modes, are in good agreement as shown in Fig.", "REF .", "Figure: Fraction of prompt ψ(2S)\\psi (2S) as a function of p T p_{\\mathrm {T}} for the μ + μ - \\mu ^+ \\mu ^-mode (solid squares) and the J/ψπ + π - J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^- mode (open squares).", "Error barsinclude the statistical uncertainties and the systematic uncertaintiesdue to the fitting procedure." ], [ "Systematic uncertainties on the cross-section measurement", "A variety of sources of systematic uncertainties affecting the cross-section measurement were taken into account and are summarised in Table REF .", "Table: Systematic uncertainties included in the measurement of the cross-section.Uncertainties labelled with aa are correlated between the μ + μ - \\mu ^+ \\mu ^- andJ/ψπ + π - J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^- mode, while bb indicates a correlationbetween ψ(2S)→μ + μ - \\psi (2S) \\rightarrow \\mu ^+ \\mu ^- and the J/ψ→μ + μ - J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\rightarrow \\mu ^+ \\mu ^-uncertainties .A thorough analysis of the luminosity scans yields consistent results for the absolute luminosity scale with a precision of 3.5% [22], this value being assigned as a systematic uncertainty.", "The statistical uncertainties from the finite number of simulated events on the efficiencies are included as a source of systematic uncertainty; this uncertainty varies from 0.4 to 2.2% for the $\\mu ^+ \\mu ^-$ mode and from 0.6 to 1% for the $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ mode.", "In addition, we assign a systematic uncertainty in order to account for the difference between the trigger efficiency evaluated on data by means of an unbiased $\\mu ^+ \\mu ^-$ sample, and the trigger efficiency computed from the simulation.", "This results in a bin-dependent uncertainty up to 8% for the $\\mu ^+ \\mu ^-$ mode and up to 7% for the $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ mode.", "This uncertainty is fully correlated between the two decay modes in the overlapping $p_{\\mathrm {T}}$ region.", "Finally, the statistical uncertainty on the global event cuts efficiency (2.1% for both modes) is taken as an additional systematic uncertainty [8].", "To assess possible systematic differences in the acceptance between data and simulation for the $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ mode, we have studied the dipion mass distribution.", "The LHCb simulation is based on the Voloshin-Zakharov model [26] which uses a single phenomenological parameter $\\lambda $ $\\frac{d\\sigma }{dm_{\\pi \\pi }} \\propto \\Phi (m_{\\pi \\pi })\\left[ m_{\\pi \\pi }^{2} - \\lambda m_{\\pi }^{2} \\right]^{2},$ where $\\Phi (m_{\\pi \\pi })$ is a phase space factor (see e.g.", "Ref.", "[25]) and in the simulation $\\lambda = 4$ is assumed.", "The dipion mass distribution obtained from the data is shown in Fig.", "REF .", "We obtain $\\lambda = 4.46 \\pm 0.07 (\\rm stat) \\pm 0.18 (\\rm syst)$ , from which we estimate a negligible systematic effect on the acceptance (0.25%).", "Our result is also in good agreement with the BES value $\\lambda = 4.36 \\pm 0.06 (\\rm stat) \\pm 0.17 (\\rm syst)$ [25].", "Figure: Dipion mass spectrum for the ψ(2S)→J/ψπ + π - \\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^- decay.", "The curve showsthe result of the fit with Eq.", "() corrected for the acceptance.To cross-check and assign a systematic uncertainty to the determination of the muon identification efficiency from simulation, the single track muon identification efficiency has been measured on data using a tag-and-probe method [27].", "This gives a correction factor for the dimuon of 1.025$\\pm $ 0.011, which we apply to the simulation efficiencies.", "The 1.1% uncertainty on the correction factor is used as systematic uncertainty.", "The efficiency of the selection requirement on the dipion identification has been studied on data and simulation and a difference of 1% has been measured between the two.", "Therefore, the simulation efficiencies are corrected for this difference and an additional systematic uncertainty of 0.5% is included.", "The $\\psi {(2S)} $ selection also includes a requirement on the track fit quality.", "The relative difference between the efficiency of this requirement in simulation and data is taken as a systematic uncertainty, resulting in an uncertainty of 0.5% per track.", "Tracking studies show that the ratio of the track-finding efficiencies between data and simulation is 1.09 for the $\\mu ^+ \\mu ^-$ mode and 1.06 for the $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ mode, with an uncertainty of 3.5% and 7.3% respectively; the simulation efficiencies are corrected accordingly and the corresponding systematic uncertainties are included.", "For the requirement on the secondary vertex fit quality, a relative difference of 1.6% for the $\\mu ^+ \\mu ^-$ mode and 2.6% for the $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ mode has been measured between data and simulation.", "The simulation efficiency is therefore corrected for this difference and a corresponding systematic uncertainty of 0.8% ($\\mu ^+ \\mu ^-$ ) and 1.3% ($J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ ) is assigned.", "The systematic uncertainty due to the unknown polarization is computed as discussed in Section .", "The study done for the two extreme polarization hypotheses gives an average systematic uncertainty between 15% and 26% for both modes, relative to the hypothesis of zero polarization, depending on the $p_{\\mathrm {T}}$ bin.", "These errors are fully correlated between the two decay modes and strongly asymmetric since the variations of the efficiency are of different magnitude for transverse and longitudinal polarizations.", "A systematic uncertainty from the fitting procedure has been estimated from the relative difference between the overall number of signal $\\psi {(2S)}$ and the number of signal candidates obtained by summing the results of the fits in the individual $p_{\\mathrm {T}}$ bins.", "A total systematic uncertainty of 1.1% for the $\\mu ^+ \\mu ^-$ mode and 0.5% for the $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ mode is assigned.", "Finally, to evaluate the systematic uncertainty on the prompt fraction from the $\\psi {(2S)}$ pseudo-decay-time fit we recompute $f_\\text{p}$ with $\\tau _b$ (see Eq.", "(REF )) fixed to the largest and smallest value obtained in the $p_{\\mathrm {T}}$ -bin fits.", "The relative variation is at most 2.7% and this value is assigned as a systematic uncertainty on $f_\\text{p}$ ." ], [ "Cross-section results", "The differential cross-sections for prompt $\\psi (2S)$ and $\\psi (2S)$ mesons from $b$ -hadron decays are shown in Fig.", "REF , where we compare the results obtained for the $\\psi (2S) \\rightarrow \\mu ^+ \\mu ^-$ and $\\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ channels separately for the prompt and $b$ -hadron decay components.", "Figure: Comparison of the differential cross-sections measuredfor prompt ψ(2S)\\psi (2S) (circles) and for ψ(2S)\\psi (2S) from bb-hadron decay (squares)in the ψ(2S)→μ + μ - \\psi (2S) \\rightarrow \\mu ^+ \\mu ^- (solid symbols) and ψ(2S)→J/ψπ + π - \\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^- (open symbols) modes.Only the uncorrelated uncertainties are shown.The values for the two cross-sections estimated using the different decay modes are consistent within $0.5~\\sigma $ .", "A weighted average of the two measurements is performed to extract the final result listed in Table REF .", "Table: Cross-section values for prompt ψ(2S)\\psi (2S) and ψ(2S)\\psi (2S) from bb-hadrons in different p T p_{\\mathrm {T}} binsand in the range 2<y≤4.52 < y \\le 4.5, evaluated as the weighted average of the μ + μ - \\mu ^+ \\mu ^- andJ/ψπ + π - J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^- channels.", "The first error is statistical, the second error is systematic, and the lastasymmetric uncertainty is due to the unknown polarization of the prompt ψ(2S)\\psi (2S) meson.The differential cross-section for promptly produced $\\psi (2S)$ mesons, along with a comparison with some recent theory predictions [28], [29], [30], [31] tuned to the LHCb acceptance, is shown in Fig.", "REF .", "In Ref.", "[28] and Ref.", "[29] the differential prompt cross-section has been computed up to NLO terms in nonrelativistic QCD, including colour-singlet and colour-octet contributions.", "In Ref.", "[30], [31] the prompt cross-section has been evaluated in a colour-singlet framework, including up to the dominant $\\alpha _s^5$ NNLO terms.", "Experimentally the large-$p_{\\mathrm {T}}$ tail behaves like $p_{\\mathrm {T}}^{-\\beta }$ with $\\beta = 4.2 \\pm 0.6$ and is rather well reproduced, especially in the colour-octet models.", "The differential cross-section for $\\psi {(2S)}$ produced in $b$ -hadron decays and the comparison with a recent theory prediction [32] based on the FONLL approach [6], [7] are presented in Fig.", "REF .", "The theoretical prediction of Ref.", "[32] uses as input the $b \\rightarrow \\psi (2S) X$ branching fraction obtained in the following section.", "Experimentally the $\\psi {(2S)}$ mesons resulting from $b$ -hadron decay have a slightly harder $p_{\\mathrm {T}}$ spectrum than those produced promptly: $\\beta = 3.6 \\pm 0.5$ .", "By integrating the differential cross-section for prompt $\\psi (2S)$ and $\\psi (2S)$ from $b$ -hadrons in the range $2<y\\le 4.5$ and $p_{\\mathrm {T}}\\le $ 16${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ , we obtain $\\sigma _{\\rm prompt}(\\psi (2S)) &= 1.44 \\pm 0.01~(\\text{stat}) \\pm 0.12~(\\text{syst})^{+0.20}_{-0.40}~(\\text{pol})~{\\rm b},\\nonumber \\\\\\sigma _{b}(\\psi (2S)) &= 0.25 \\pm 0.01~(\\text{stat}) \\pm 0.02~(\\text{syst})~{\\rm b}, \\nonumber $ where the systematic uncertainty includes all the sources listed in Table REF , except for the polarization, while the last asymmetric uncertainty is due to the effect of the unknown $\\psi (2S)$ polarization and applies only to the prompt component.", "Figure: Differential production cross-section vs. p T p_{\\mathrm {T}} for prompt ψ(2S)\\psi (2S).The predictions of three nonrelativistic QCD models are also shown for comparison.MWC  and KB  are NLO calculations including colour-singletand colour-octet contributions.", "AL , is a colour-singletmodel including the dominant NNLO terms.Figure: Differential production cross-section vs. p T p_{\\mathrm {T}} for ψ(2S)\\psi (2S) from bb-hadrons.The shaded band is the prediction of a FONLL calculation , , ." ], [ "Inclusive $b \\rightarrow \\psi (2S)X$ branching fraction measurement", "The inclusive branching fraction for a $b$ -hadron decaying to $\\psi {(2S)}$ is presently known with 50% precision: $\\mathcal {B}(b \\rightarrow \\psi (2S) X)$ = (4.8 $\\pm $ 2.4) $\\times 10^{-3}$  [18].", "Combining the present result for $\\sigma _{b}(\\psi (2S))$ with the previous measurement of $\\sigma _{b}(J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt})$ [8] we can obtain an improved value of the aforementioned branching fraction.", "To achieve this, it is necessary to extrapolate the two measurements to the full phase space.", "The extrapolation factors for the two decays have been determined using the LHCb simulation [12] and they have been found to be $\\alpha _{4\\pi }(J/\\psi )$ =5.88 [8] and $\\alpha _{4\\pi }(\\psi (2S))$ =5.48.", "Most of the theoretical uncertainties are expected to cancel in the ratio of the two factors $\\xi = \\alpha _{4 \\pi }(\\psi (2S))/\\alpha _{4 \\pi }(J/\\psi ) = 0.932$ , which is used in Eq.", "(REF ).", "A systematic uncertainty of $3.4 \\%$ is estimated for this correction and included in the final result below.", "Therefore $\\frac{\\mathcal {B}(b \\rightarrow \\psi (2S) X)}{\\mathcal {B}(b \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}X)} =\\xi \\frac{\\sigma _{b}(\\psi (2S))}{\\sigma _{b}(J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt})}.$ For $\\sigma _{b}(J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt})$ we rescale the value in [8] for the new determination of the integrated luminosity ($\\mathcal {L}$ = 5.49 $\\pm $ 0.19$\\mbox{\\,pb}^{-1}$ ).", "For $\\sigma _{b}(\\psi (2S))$ we use only the data from the $\\psi (2S)~\\rightarrow ~\\mu ^+\\mu ^-$ mode to cancel most of the systematic uncertainties in the ratio.", "Effects due to polarization are negligible for mesons resulting from $b$ -hadron decay.", "We obtain $\\frac{\\mathcal {B}(b \\rightarrow \\psi (2S) X)}{\\mathcal {B}(b \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}X)} = 0.235 \\pm 0.005~(\\text{stat})\\pm 0.015~(\\text{syst}),$ where the correlated uncertainties (Table REF ) between the two cross-sections are excluded.", "By inserting the value $\\mathcal {B}(b~\\rightarrow ~J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}X)=(1.16 \\pm 0.10)\\times 10^{-2}$ [18] we get $\\mathcal {B}(b \\rightarrow \\psi (2S)X) = (2.73 \\pm 0.06~(\\text{stat}) \\pm 0.16~(\\text{syst}) \\pm 0.24~(\\text{BF})) \\times 10^{-3},$ where the last uncertainty originates from the uncertainty of the branching fractions $\\mathcal {B}(b \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}X)$ , $\\mathcal {B}(\\psi (2S) \\rightarrow e^+ e^-)$ and $\\mathcal {B}(J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\rightarrow \\mu ^+ \\mu ^-)$ .", "The ratio of the $\\psi (2S) \\rightarrow \\mu ^+ \\mu ^- $ to $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\rightarrow \\mu ^+ \\mu ^- $ differential cross-sections is shown vs. $p_{\\mathrm {T}}$ in Fig.", "REF for prompt production ($R_{\\rm p}$ , Fig.", "REF (a)) and when the vector mesons originate from $b$ -hadron decays ($R_{b}$ , Fig.", "REF (b)).", "Since it is not known if the promptly produced $\\psi {(2S)}$ and ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ have similar polarizations [33], we do not assume any correlation of the polarization uncertainties when computing the uncertainties on $R_{\\rm p}$ .", "The increase of $R_{{\\rm p}(b)}$ with $p_{\\mathrm {T}}$ is similar to that measured in the central rapidity region by the CDF [24] and CMS [9] collaborations.", "Figure: Ratio of ψ(2S)→μ + μ - \\psi (2S) \\rightarrow \\mu ^+ \\mu ^- to J/ψ→μ + μ - J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\rightarrow \\mu ^+ \\mu ^- cross-sections for promptproduction (a) and for bb-hadron decay (b), as a function of p T p_{\\mathrm {T}}." ], [ "Conclusions", "We have measured the differential cross-section for the process $pp \\rightarrow \\psi (2S)X$ at the centre-of-mass energy of 7$\\mathrm {\\,Te\\hspace{-1.00006pt}V}$ , as a function of the transverse momentum in the range $p_{\\mathrm {T}}(\\psi (2S))\\le 16$${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ and $2<y(\\psi (2S))\\le 4.5$ , via the decay modes $\\psi (2S) \\rightarrow \\mu ^+ \\mu ^-$ and $\\psi (2S) \\rightarrow J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}\\pi ^+ \\pi ^-$ .", "The data sample corresponds to about 36$\\mbox{\\,pb}^{-1}$ collected by the LHCb experiment at the LHC.", "Results from the two decay modes agree.", "The $\\psi (2S)$ prompt cross-section has been separated from the cross-section of $\\psi (2S)$ from $b$ -hadrons through the study of the pseudo-decay-time and the two measurements have been averaged.", "In the above kinematic range we measure $\\sigma _{\\rm prompt}(\\psi (2S)) &=& 1.44 \\pm 0.01~(\\text{stat}) \\pm 0.12~(\\text{syst})^{+0.20}_{-0.40}~(\\text{pol})~{\\rm b}, \\\\\\sigma _{b}(\\psi (2S)) &=& 0.25 \\pm 0.01~(\\text{stat}) \\pm 0.02~(\\text{syst})~{\\rm b}.$ The measured $\\psi (2S)$ production cross-sections are in good agreement with the results of several recent NRQCD calculations.", "In addition, we obtain an improved value for the $b \\rightarrow \\psi (2S) X$ branching fraction by combining the two LHCb production cross-section measurements of the two vector mesons $J\\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}$ and $\\psi (2S)$ from $b$ -hadrons.", "The result, $\\mathcal {B}(b \\rightarrow \\psi (2S) X) = (2.73 \\pm 0.06~(\\text{stat}) \\pm 0.16~(\\text{syst}) \\pm 0.24~(\\text{BF}))\\times 10^{-3},$ is in good agreement with recent results from the CMS collaboration [9] and is a significant improvement over the present PDG average [18]." ], [ "Acknowledgments", "We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC.", "We thank the technical and administrative staff at CERN and at the LHCb institutes, and acknowledge support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS (Romania); MinES of Russia and Rosatom (Russia); MICINN, XUNGAL and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA).", "We also acknowledge the support received from the ERC under FP7 and the Region Auvergne.", "We thank B. Kniehl, M. Butenschön and M. Cacciari for providing theoretical predictions of $\\psi {(2S)}$ cross-sections in the LHCb acceptance range." ] ]

[ [ "The Fractional Quantum Hall Effect of Tachyons in a Topological\n Insulator Junction" ], [ "Abstract We have studied the tachyonic excitations in the junction of two topological insulators in the presence of an external magnetic field.", "The Landau levels, evaluated from an effective two-dimensional model for tachyons, and from the junction states of two topological insulators, show some unique properties not seen in conventional electrons systems or in graphene.", "The $\\nu=1/3$ fractional quantum Hall effect has also a strong presence in the tachyon system." ], [ "The Fractional Quantum Hall Effect of Tachyons in a Topological Insulator Junction Vadim M. Apalkov Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA Tapash Chakraborty$^\\ddag $ Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2 We have studied the tachyonic excitations in the junction of two topological insulators in the presence of an external magnetic field.", "The Landau levels, evaluated from an effective two-dimensional model for tachyons, and from the junction states of two topological insulators, show some unique properties not seen in conventional electrons systems or in graphene.", "The $\\nu =\\frac{1}{3}$ fractional quantum Hall effect has also a strong presence in the tachyon system.", "The surface state of recently discovered three-dimensional topological insulators [2] contains a single Dirac cone and as a result, the charge carriers on the surface are characterized as massless Dirac fermions.", "Some of the properties of these particles are well known from another topological system, the graphene [3].", "We have shown earlier [4] that the dispersion relation of the surface excitations in a junction between two such topological insulators (TIs) show several very unique properties.", "Most importantly, under certain conditions these excitations exhibit tachyon-like dispersion relation [5], [6] corresponding to superluminal propagation of the Dirac fermions along the interface of the two TIs.", "In addition to the tachyonic dispersion, the junction states can also support the usual massless relativistic dispersion relation of the Dirac fermions [7].", "Here we report on the properties of these tachyonic excitations in an external magnetic field.", "We discuss the unique nature of the Landau levels (LLs) of these tachyons and the interaction properties of tachyons in a strong magnetic field, which leads to the fractional quantum Hall effect (FQHE) of tachyons.", "Effective two-dimensional (2D) model of tachyonic excitations: The tachyons in the junction of two TIs can be described by the effective 2D matrix Hamiltonian of the Dirac fermions but with imaginary Fermi velocity [4] (instead of the imaginary proper mass commonly attributed to the tachyons [5], e.g., to the neutrinos [8]) ${\\cal H}^{}_{\\rm Tach} = \\left(\\begin{array}{cc}\\Delta ^{}_0 & {\\rm i} v^{}_{\\rm I} p^{}_{+} \\\\{\\rm i} v^{}_{\\rm I} p^{}_{-} & - \\Delta ^{}_0\\end{array}\\right) = \\Delta ^{}_0 \\sigma ^{}_z + {\\rm i} v^{}_{\\rm I}(\\vec{\\sigma } \\vec{p}),$ where $p^{}_{\\pm } = p^{}_x \\pm p^{}_y$ is the 2D momentum, $\\Delta ^{}_0$ is the effective mass of the tachyons, ${\\rm i}v^{}_{\\rm I}$ is the imaginary Fermi velocity, and $\\vec{\\sigma }$ are the Pauli matrices, corresponding to the spin degree of freedom of an electron (tachyon).", "The effective Hamiltonian thus described has the tachyon-like dispersion, $E^{}_{\\rm Tach} (p) = \\pm \\sqrt{\\Delta _0^2 - v^{}_{\\rm I} p^2}$ , where $p=\\left(p_x^2+p_y^2\\right)^{\\frac{1}{2}}$ .", "Typical values of the parameters are $\\Delta ^{}_0 \\sim 0.3$ eV and $v^{}_{\\rm I}\\sim 10^6$ m/s.", "We subject the tachyonic system to an external magnetic field, $B$ , pointing along the $z$ -direction, i.e., perpendicular to the junction between the TIs.", "The Hamiltonian describing such a system in a magnetic field can be found from the Hamiltonian (REF ) by replacing the tachyonic momentum, $(p^{}_x,p^{}_y)$ by the generalized momentum, $(\\pi ^{}_x,\\pi ^{}_y)$ and introducing the Zeeman energy, $\\Delta ^{}_z (B)=\\frac{1}{2} g^{}_s\\mu ^{}_B B$ .", "Here $\\mu ^{}_B$ is the Bohr magneton and $g^{}_s\\approx 8$ is the effective $g$ -factor of the tachyonic excitations, which we assume to be equal to the $g$ -factor of the TI surface states [9], [10].", "The effective tachyonic Hamiltonian in a magnetic field then takes the form ${\\cal H}^{}_{\\rm Tach} (B) = \\left[\\Delta ^{}_0 + \\Delta ^{}_z(B)\\right]\\sigma ^{}_z + {\\rm i} v^{}_{\\rm I} (\\vec{\\sigma }\\vec{\\pi }).$ The LL energy spectrum corresponding to the Hamiltonian [Eq.", "(REF )] is characterized by an integer number $n\\ge 0$ (the LL index), and is defined as $E^{}_{n=0}&=&\\Delta ^{}_0 + \\Delta ^{}_z (B) \\nonumber \\\\E^{}_{n\\ge 1,s} & = & s \\left[\\left[\\Delta ^{}_0 +\\Delta ^{}_z(B)\\right]^2 -2n \\left(\\hbar v^{}_{\\rm I}/\\ell ^{}_0\\right)^2\\right]^{\\frac{1}{2}}\\nonumber \\\\& = & s \\left[\\Delta ^{}_0 +\\Delta ^{}_z(B)\\right]\\left[1-nB/B^*(B)\\right]^{\\frac{1}{2}}.$ Here $s=\\pm 1$ , $\\ell ^{}_0=\\sqrt{e\\hbar /c B}$ is the magnetic length, and we introduced the effective magnetic field $B^*(B)=\\frac{e}{2\\hbar c}\\left[\\left(\\Delta ^{}_0 +\\Delta ^{}_z(B)\\right)/v^{}_{\\rm I}\\right]^2.$ The wave functions corresponding to the LLs [Eq.", "(REF )] can be expressed in terms of the functions $\\phi ^{}_{n,m}$ , which are wave functions of the conventional (`non-relativistic') LLs with index $n$ and the intra-Landau level index, $m$ , for example, the $z$ component of the angular momentum.", "The LL wave functions of the tachyons are $\\Psi ^{\\rm (Tach)}_{n=0}=\\left(\\begin{array}{c}\\phi ^{}_{0,m} \\\\0\\end{array}\\right), \\quad \\Psi ^{\\rm (Tach)}_{n\\ge 1,s}=\\left(\\begin{array}{c}\\cos \\alpha ^{}_{n,s}\\, \\phi ^{}_{n,m} \\\\\\sin \\alpha ^{}_{n,s}\\, \\phi ^{}_{n-1,m}\\end{array}\\right),$ where $\\alpha ^{}_{n,s}=\\sin ^{-1}\\left[\\frac{1}{2}\\left(1-s \\sqrt{1-n B/B^*}\\right)\\right]^{1/2}.$ Figure: (a) The LLs of the effective 2D tachyonic Hamiltonian () areshown as a function of the magnetic field.", "The numbers next to the lines arethe LL indices.", "The n=1n=1 LL wave functions at points A and B are identicalto conventional `nonrelativistic' LLs with indices 0 and 1, respectively.At point G, the LL wave function of the tachyon system is identical tothe n=1n=1 LL wave function of graphene.", "(b) ν=1 3\\nu =\\frac{1}{3}-FQHE gap is shownfor the n=0n=0 and n=1n=1 LLs of tachyons.", "The FQHE gap is calculated fora finite-size system with N=9N=9 particles in a spherical geometry with parameterS=12S = 12.", "The red and blue lines correspond to the n=1n=1 LL branches shownin panel (a).", "The FQHE gap is measured in units of the Coulomb energy,ϵ C =e 2 /κℓ 0 \\epsilon ^{}_{\\rm C} = e^2/\\kappa \\ell ^{}_0.", "The dashed curve is explainedin the text.The LL energy spectrum obtained from Eq.", "(REF ) is shown in Fig. 1.", "The energy spectrum for tachyons exhibits a few distinct features: (i) The LL energy spectrum is mainly restricted within the energy interval $-\\Delta ^{}_0< E^{}_{n,s} < \\Delta ^{}_0$ .", "(ii) At a given magnetic field $B$ , only the LL with index $n< B^*/B$ can be observed.", "Therefore, for a given magnetic field, only a few LLs exist in the system and, for a large enough magnetic field (e.g., $B > 32$ Tesla in Fig.", "1(a)), only the $n=0$ LL survives.", "This behavior of the LL of tachyonic excitations is totally different from that in the `non-relativistic' semiconductor systems with the LL energy spectrum $E^{}_n \\propto nB$ , and “relativistic\" graphene system with the LL energy spectrum $E^{}_n \\propto \\sqrt{nB}$ [3].", "However, there is one similarity between the tachyonic LL dispersion relations and that in graphene.", "In both cases there is one $n=0$ LL, the energy of which is independent of the strength of the magnetic field (without the Zeeman splitting).", "In graphene, the energy of this LL is $E^{}_{n=0}=0$ , while for the tachyonic excitations, $E^{}_{n=0}=\\Delta ^{}_0$ .", "In both cases the corresponding wave functions are $\\phi ^{}_{n=0,m}$ .", "To address the similarities and differences between the LL wave functions of the tachyonic system and those of graphene (or even conventional semiconductor systems), we consider the interaction properties of tachyonic excitations in a given LL.", "To characterize the strength of the inter-tachyonic interactions we study the strength of the FQHE, i.e., the magnitude of the FQHE gap.", "In the FQHE regime the electrons partially occupy a single LL and the properties of such a system are characterized by the inter-particle interactions within the corresponding LL [11].", "The interaction strength within a given LL is determined from the Haldane pseudopotentials, $V^{}_m$ [12], which are the interaction energies of two particles with relative angular momentum $m$ .", "The pseudopotentials are determined from [12] $V_m^{(n)} = \\int _0^{\\infty } \\frac{dq}{2\\pi } q V(q)\\left[F^{}_n(q) \\right]^2 L^{}_m (q^2)e^{-q^2},$ where $L^{}_m(x)$ are the Laguerre polinomials, $V(q) = 2\\pi e^2/(\\kappa \\ell ^{}_0q)$ is the Coulomb interaction in the momentum space, $\\kappa $ is the dielectric constant, and $F^{}_n(q)$ is the form factor for the $n$ -th Landau level.", "The form factor is determined by the structure of the LL wave functions.", "For the tachyonic system the form factor is given by $& & F^{}_{n=0}(q) = L^{}_0\\left({\\textstyle {1\\over 2}}q^2 \\right) \\\\& & F^{}_{n\\ge 1}(q) =\\cos ^2\\alpha ^{}_{n,s} L^{}_n \\left({\\textstyle {1\\over 2}}q^2\\right)+\\sin ^2\\alpha ^{}_{n,s} L^{}_{n-1} \\left({\\textstyle {1\\over 2}}q^2\\right).$ The form factor of the $n=0$ LL [Eq.", "(REF )] is identical to that of graphene and also to that of the non-relativistic systems.", "However, for $n\\ge 1$ the form factor of the tachyonic system becomes unique.", "For graphene and for the non-relativistic systems the corresponding form factors are $F_n^{\\rm (NR)} = L^{}_n$ and $F_n^{\\rm (Gr)} =\\frac{1}{2} (L^{}_n + L^{}_{n-1})$ , respectively.", "In both cases the form factors are independent of the magnetic field.", "For the tachyonic system, on the other hand, the form factor () depends on the magnetic field, through the effective angle $\\alpha ^{}_{n,s} (B)$ .", "With increasing magnetic field the tachyonic form factor, $F^{}_{n\\ge 1}$ , changes from the non-relativistic value, $F_n^{(NR)}$ [point B in Fig.", "1(a)] or $F_{n-1}^{(NR)}$ [point A in Fig.", "1(a)], in a small magnetic field, $B\\rightarrow 0$ , to the form factor of graphene, $F_n^{\\rm (Gr)}=\\frac{1}{2} (L^{}_n + L^{}_{n-1})$ [13], for $B=B^*$ .", "Figure: (a) The LL energy spectrum of a electron in a junction between two TIsis shown as a function of magnetic fieldfor a few lowest LLs.", "The numbers next to the lines are the LL indexes.The TIs have the same Hamiltonian parameters, except A 1 A^{}_1, whichtakes the values: A 1 =2.2A^{}_1 = 2.2 eV·\\cdot Å for TI-1 and A 1 =4.0A^{}_1 = 4.0eV·\\cdot Å for TI-2.", "(b) The width in the zz-direction of the LL wave functionsis shown for n=0n=0 and n=1n=1 LLs.", "The corresponding LL energy spectrum is shownin panel (a).The FQHE with an incompressible ground state can be observed only in the LL with strong short-range repulsion, i.e., a fast decay of the corresponding pseudopotentials, $V^{}_m$ , with $m$ .", "Such a strong repulsion is realized only in the LL with a strong admixture of $\\phi ^{}_{0,m}$ .", "Therefore, in a tachyonic system the FQHE is expected only in the $n=0$ and $n=1$ LLs.", "To study the strength of the corresponding FQHE we numerically evaluate the energy spectrum of a finite $N$ -electron system in a spherical geometry [12] with the radius of the sphere $\\sqrt{S}\\ell ^{}_0$ .", "Here $2S$ is the integer number of magnetic fluxes through the sphere in units of the flux quantum.", "For a given number of electrons, the radius of the sphere determines the filling factor of the system.", "For example, the $\\nu =1/q$ FQHE ($q$ is an odd integer) corresponds to the relation $S=(\\frac{q}{2})(N-1)$ [11].", "In Fig.", "1(b) the $\\nu = \\frac{1}{3}$ -FQHE gap is shown for $n=0$ and $n=1$ LLs of the tachyonis system.", "For the $n=0$ LL, the FQHE gap does not depend on $B$ and the gap exactly equals to the FQHE gap for the $n=0$ non-relativistic LL.", "This is because the $n=0$ tachyonic LL wave function consists of only the functions $\\phi ^{}_{0,m}$ .", "For the $n=1$ LL, the wave function is the $B$ -dependent mixture of $\\phi ^{}_{0,m}$ and $\\phi ^{}_{1,m}$ .", "As a result the FQHE gap depends on the magnetic field and changes from the $n=0$ non-relativistic LL value for $B\\rightarrow 0$ (point A) to the $n=1$ graphene LL value for $B= B^*$ (point G) and finally to $n=1$ non-relativistic LL value at point B (in the thermodynamic limit such a state becomes compressible).", "The maximum FQHE gap in a non-relativistic system corresponds to the green line in Fig.", "1(b), while the maximum FQHE gap in the graphene system corresponds to point G in Fig. 1(b).", "Therefore, comparing the data in Fig.", "1(b), we conclude that within some range of the magnetic fields (which is shown in Fig.", "1(b) by an oval curve), the FQHE gap in the model tachyonic system is the largest compared to other available 2D systems.", "The tachyonic dispersion relation provides an unique possibility to study, within a single tachyonic $n=1$ LL, the properties of the $n=0$ non-relativistic LL [point A in Fig.", "1(a)], $n=1$ graphene LL [point G in Fig.", "1(a)], and $n=1$ non-relativistic LL [point B in Fig. 1(a)].", "Three-dimensional (3D) model for the junction states between two TIs: Until now, we have discussed the magnetic field effects via an effective Hamiltonian for tachyons.", "The tachyonic dispersion can be realized in the junction of two TIs [4].", "The junction dispersion relation in this case is approximately described by the effective Hamiltonian (REF ).", "The realization of the tachyonic excitations as the junction states bring additional factors into consideration.", "For example, the junction states have a finite width in the $z$ -direction which can reduce the interaction strength in that system.", "We consider the junction between two TI insulators: TI-1 for $z<0$ and TI-2 for $z>0$ .", "Here $z=0$ corresponds to the junction surface.", "The electronic properties of both TIs are described by the same type of low-energy effective 3D Hamiltonian [9], [14] of the matrix form ${\\cal H}^{}_{\\rm TI}=\\epsilon (\\vec{k})+ \\left(\\begin{array}{cc}M(\\vec{p})\\sigma ^{}_z- {\\rm i} A^{}_1\\sigma ^{}_x \\partial ^{}_z & (A^{}_2/\\hbar )p^{}_{-}\\sigma ^{}_x \\\\(A^{}_2/\\hbar ) p^{}_{+}\\sigma ^{}_x & M(\\vec{p})\\sigma ^{}_z + {\\rm i}A^{}_1 \\sigma ^{}_x \\partial ^{}_z\\end{array}\\right),$ where $\\partial ^{}_z = \\partial /\\partial z$ , and $& & \\epsilon (\\vec{k})=C^{}_1 - D^{}_1\\partial _z^2+(D^{}_2/\\hbar ^2 )(p_x^2 + p_y^2),\\\\& & M(\\vec{k})= M^{}_0 + B^{}_1 \\partial _z^2 - (B^{}_2/\\hbar ^2 ) (p_x^2 + p_y^2).$ We assume that for TI-1 the constants in the above Hamiltonian are the same as for $\\mbox{Bi}^{}_2\\mbox{Se}^{}_3$ [14], while the for TI-2 the constants are different but close to the values for $\\mbox{Bi}^{}_2\\mbox{Se}^{}_3$ .", "We assume that only the constant $A^{}_1$ is different for the two TIs, i.e., $A^{}_1 = 2.2$ eV$\\cdot $ Å for TI-1 and $A^{}_1 = 4.0$ eV$\\cdot $ Å for TI-2.", "All other constants in the Hamiltonian (REF ) are kept the same for both TIs [4].", "For these parameters, the junction states exhibit the tachyonic dispersion [4].", "The four-component wave function corresponding to the Hamiltonian (REF ) determines the amplitudes of the wave functions at the positions of Bi and Se atoms: $(\\mbox{Bi}^{}_{\\uparrow }, \\mbox{Se}^{}_{\\uparrow },\\mbox{Bi}^{}_{\\downarrow }, \\mbox{Se}^{}_{\\downarrow })$ , where the arrows indicate the direction of the electron spin.", "The Hamiltonian of the TI in an external magnetic field, pointing along the $z$ -direction, can be obtained from the Hamiltonian (REF ) by replacing the 2D momentum $(p^{}_x,p^{}_y)$ by the generalized momentum $(\\pi ^{}_x, \\pi ^{}_y )$ [15] and introducing the Zeeman energy, $\\Delta ^{}_z=\\frac{1}{2} g^{}_s\\mu ^{}_B B$ .", "For the Hamiltonian (REF ) in a magnetic field, the wave function in the $n$ -th LL has the general form $\\Psi ^{\\rm (TI)}_{n\\ge 1} (z) =\\left( \\begin{array}{c}\\chi ^{(n)}_1 (z) \\phi ^{}_{n-1,m} \\\\\\chi ^{(n)}_2 (z) \\phi ^{}_{n-1,m} \\\\\\chi ^{(n)}_3 (z) \\phi ^{}_{n,m} \\\\\\chi ^{(n)}_4(z) \\phi ^{}_{n,m}\\end{array}\\right).$ Therefore, just as for the tachyonic states the wave function $\\Psi ^{\\rm (TI)}_{n\\ge 1}$ is the mixture of $n$ and $n-1$ non-relativistic LL functions.", "For the $n=0$ LL, only $\\chi ^{(n=0)}_3$ and $\\chi ^{(n=0)}_{4}$ are non-zero.", "To find the LL junction states we follow the same procedure as for the LL surface states of a TI [16], [17], [15].", "For each TI we find the general bulk solution of the Schrödinger equation in the form of $\\Psi \\propto e^{\\lambda ^{(m)} z}$ , where $\\lambda ^{(m)}$ is a complex constant, and $m=1$ and 2 for TI-1 and TI-2, respectively.", "This solution has a given energy, $E$ , and a given LL index, $n$ .", "The corresponding $\\lambda ^{(m)}$ are determined from a secular equation, $\\det \\left[ {\\cal {H}}^{(m)}_{TI} (\\vec{k}, \\lambda ^{(m)})- E\\right] = 0$ , for each TI.", "For each energy $E$ and the LL index $n$ , the secular equation provides eight values of $\\lambda ^{(m)}_j(n, E)$ , $j = 1, \\ldots , 8$ with the corresponding wave functions.", "Second, since we are looking for the localized LL junction states, we need to choose (for each TI) only four values of $\\lambda ^{(m)}_j$ out of eight with the properties: $\\mbox{Re} \\lambda ^{(m)}_j > 0 $ for TI-1 ($z<0$ ) and $\\mbox{Re} \\lambda ^{(m)}_j <0$ for TI-2 ($z>0$ ).", "We then choose the corresponding four wave functions (for each TI) as the basis and expand the solution for the LL junction state in this basis.", "Finally, the energy of the LL junction state is found from the condition of continuity of the wave function, $\\Psi ^{\\rm (TI)}_{n}(z)$ , and current $[\\delta {\\cal H}^{}_{\\rm TI} / \\delta k^{}_z]\\Psi ^{\\rm (TI)}_n$ in the junction.", "Figure: The ν=1 3\\nu =\\frac{1}{3} FQHE gap for n=0n=0 (green line) and n=1n=1 (red line) LLs in thejunction of two TIs.", "The corresponding LLs are shown in Fig.", "2(a) by green andred lines, respectively.", "The FQHE gap is calculated for a finite-size system withN=9N=9 particles in spherical geometry with parameter S=12S = 12.", "The FQHE gap ismeasured in units of Coulomb energy, ϵ C =e 2 /κℓ 0 \\epsilon ^{}_{\\rm C} = e^2 /\\kappa \\ell ^{}_0.The spectrum of the LL junction states corresponding to the tachyonic dispersion is shown in Fig. 2(a).", "The spectrum is qualitatively similar to that [see Fig.", "1(a)] obtained from the model 2D Hamiltonian.", "Both spectra have the finite range of magnetic fields and energies, where the LLs can be observed.", "At weak magnetic fields, the difference between the LL spectrum of the junction states and the 2D model is clearly visible.", "For a given LL index $n$ , there are no junction states for weak magnetic fields.", "These junction states are delocalized in the $z$ -direction.", "To illustrate this delocalization we show in Fig.", "2(b) the width of the $n=0$ and $n=1$ LL wave functions in the $z$ -direction.", "At a singular point of the LL spectrum, i.e., for $B=B^*$ , where the derivative of the LL energy with the magnetic field becomes infinitely large, the LL wave functions have the smallest width.", "This width increases with decreasing magnetic field and finally the LL junction states are delocalized in the $z$ -direction.", "A similar behavior is observed for $n=0$ LL, but there are no singular point in this case.", "Therefore, the LL energy spectrum of the junction states in the regime of tachyonic dispersion can be well described within 2D effective model near the singular point $B\\sim B^*$ .", "We have evaluated the FQHE gaps for the $n=0$ and $n=1$ junction LLs.", "We have used the same approach as for the 2D model of the tachyonic excitations, discussed above.", "The results are shown in Fig. 3.", "Quantitatively the behavior of the FQHE gap as a function of the magnetic field is similar to the 2D model of the tachyonic excitations [Fig. 1(b)].", "Due to a finite width of the LL wave functions in the TI junction, there is a reduction of the inter-electron interaction strength and correspondingly the FQHE gap.", "This reduction is visible for $n=0$ LL, where a smaller FQHE gap and the magnetic field dependence of the FQHE gap is shown in Fig. 3.", "Although for the 2D tachyonic model the FQHE gap is the largest for the $n=1$ LL, for the junction LLs the FQHE gap is the largest for the $n=0$ LL, due to the non-zero spin polarization of the $n=1$ junction LL.", "This spin polarization is clearly visible from the general property of the LL wave function [Eq.", "(REF )]; only the components $\\chi _3^{(n=0)}$ and $\\chi _4^{(n=0)}$ of $\\Psi _{n=0}^{\\rm (TI)}$ are non-zero and these components correspond to the spin-down polarization.", "The numerically found $n=1$ LL wave functions also show partial spin-down polarization.", "As a result, the LL wave function have larger contribution from the $\\phi ^{}_{n=1}$ non-relativistic LL function, which reduces the inter-electron interaction strength and the FQHE gap.", "To summarize: we have investigated the magnetic field effects of tachyonic excitations along the interface of two topological insulators.", "We used an effective two-dimensional model Hamiltonian for tachyons and the three-dimensional model for the junction states of the two TIs, both developed by us [4].", "The Landau levels in both these models show very similar behaviors.", "Unlike in graphene or in conventional electron systems, only a few LLs are found to exist for the tachyons.", "Only one LL ($n=0$ ) survives for large magnetic fields.", "The $\\nu =\\frac{1}{3}$ FQHE is the strongest (within a limited range of the magnetic field) when compared with that for conventional electron systems and graphene.", "Interestingly, the FQHE in the $n=1$ LLs for tachyons describes the FQHE of the $n=0, 1$ LLs of the non-relativistic electron system and that of the $n=1$ graphene LL in different regions of the magnetic field.", "Experimental confirmation of these properties of the Landau levels would provide strong evidence on the existence of elusive tachyons.", "The work has been supported by the Canada Research Chairs Program of the Government of Canada." ] ]

[ [ "Query Language for Complex Similarity Queries" ], [ "Abstract For complex data types such as multimedia, traditional data management methods are not suitable.", "Instead of attribute matching approaches, access methods based on object similarity are becoming popular.", "Recently, this resulted in an intensive research of indexing and searching methods for the similarity-based retrieval.", "Nowadays, many efficient methods are already available, but using them to build an actual search system still requires specialists that tune the methods and build the system manually.", "Several attempts have already been made to provide a more convenient high-level interface in a form of query languages for such systems, but these are limited to support only basic similarity queries.", "In this paper, we propose a new language that allows to formulate content-based queries in a flexible way, taking into account the functionality offered by a particular search engine in use.", "To ensure this, the language is based on a general data model with an abstract set of operations.", "Consequently, the language supports various advanced query operations such as similarity joins, reverse nearest neighbor queries, or distinct kNN queries, as well as multi-object and multi-modal queries.", "The language is primarily designed to be used with the MESSIF framework for content-based searching but can be employed by other retrieval systems as well." ], [ "Introduction", "Information has always been a valuable article but it has always been difficult to obtain.", "These days, we have an unprecedented advantage of having huge and rich data collections at our fingertips.", "On the other hand, we still need more efficient tools for data management to be able to locate the desired information in the vast amounts of resources.", "With the emergence of complex data types such as multimedia, traditional retrieval methods based on attribute matching are no longer satisfactory.", "Therefore, a new approach to searching has been proposed, exploiting the concept of similarity between complex objects.", "In recent years, we have witnessed intensive research in the field of indexing methods and search algorithms for similarity-based retrieval.", "As a result, state-of-the-art search systems already support quite complex similarity queries with a number of features that can be adjusted according to individual user's preferences.", "To communicate with such a system, it is either possible to employ low-level programming tools, or a higher-level communication interface that shields users from the implementation details employed by the particular search engine.", "As the low-level tools can only be used by a limited number of specialists, the high-level interface becomes a necessity when common users shall be allowed to issue advanced queries or adjust the parameters of the retrieval process.", "In this paper, we are proposing such high-level interface in a form of a structured query language that allows users to issue actual queries over complex data.", "The motivation to study query languages arose from the development of our own framework for similarity searching called MESSIF [6].", "The system currently supports a wide spectrum of retrieval algorithms and is used to support several multimedia search applications, such as large-scale image search, automatic image annotation, or gait recognition.", "So far, users are allowed only to select the query object via a graphical interface, and the choice of the actual search methods as well as its parameters and other settings are hard-coded into the system.", "To improve the usability of our systems, we decided to provide the framework with a query language that would allow advanced users to express their preferences without having to deal with the technical details.", "After a thorough study of existing solutions we came to a conclusion that none of them suits our specific needs.", "Therefore, we decided to propose a new language based on and extending the existing ones.", "At the same time, it was our desire to design the language in such a way that it could be also used by other systems.", "Consequently, we present here an SQL-based query language which can be used to formulate a wide range of similarity queries, as we demonstrate on examples from various application domains.", "Building on a thorough analysis of previous studies and our long-time experience with both theory and practice of similarity search systems, we have proposed its structure so that it supports all fundamental query types and can be easily extended.", "The language can be used by programmers or advanced users to issue queries in a standard declarative way, shielding them from the execution details.", "For less advanced users, we expect the language to be wrapped-up into a visual interface.", "The language is designed in a general way as to allow flexibility and extensibility.", "The paper is further organized as follows.", "First, we review the related work in Section 2.", "In the following section, we analyze the requirements for a multimedia query language, taking into account current trends in information retrieval research, lessons learned from other language proposals, and the functionality of the MESSIF framework.", "Next, we discuss the fundamental design decisions that determined the overall structure of the language in Section 4.", "Section 5 introduces both the theoretical model of the language and its syntax and semantics.", "Section 6 presents several real-world queries over multimedia data, formulated in our language.", "Finally, we outline the future work in Section 7." ], [ "Related Work", "The problem of defining a formal apparatus for similarity queries has been recognized and studied by the data processing community for more than two decades, with various research groups working on different aspects of the problem.", "Some of these studies focus on the underlying algebra, others deal with the query language syntax.", "Query languages can be further classified as SQL-based, XML-based, and others with a less common syntax.", "We shall briefly survey all these research directions.", "Similarity algebra as a tool for theoretical modeling and transformations of similarity queries was first introduced in [1].", "The authors define general abstractions for objects and similarity measures, present basic algebra operations and discuss their properties.", "Later works add new similarity operations [3] or study the integration of similarity-based querying into established data models, e.g.", "relational model [7].", "While these studies provide a valuable insight into the principles of similarity searching, the algebraic operations used to express the queries are not meant to be employed by users during a search session.", "The majority of the early proposals for practical query languages are based on SQL or its object-oriented alternative, OQL [8].", "Paper [15] describes MOQL, a multimedia query language based on OQL which supports spatial, temporal and containment predicates for searching in image or video.", "However, similarity-based searching is not supported in MOQL.", "The authors of [2] introduce new operators sim and match for object similarity and concept-object relevance, respectively.", "However, it is not possible to limit the similarity or define the way it is evaluated.", "In [12], a more flexible similarity operator for near and nearest neighbors is provided but it still does not allow to choose the similarity measure.", "Much more mature extensions of relational DBMS and SQL are presented in [5], [4], [13].", "The concept of [5], [4] enables to integrate similarity queries into SQL, using new data types with associated similarity measures and extended functionality of the select command.", "The authors also describe the processing of such extended SQL and discuss optimization issues.", "Even though the proposed SQL extension is less flexible than we need, the presented concept is sound and elaborate.", "The study [13] only deals with image retrieval but also presents an extension of the PostgreSQL database management system that enables to define feature extractors, create access methods and query objects by similarity.", "This solution is less complex than the previous one but on the other hand, it allows users to adjust the weights of individual features for the evaluation of similarity.", "Recently, we could also witness interest in XML-based languages for similarity searching.", "In particular, the MPEG committee has initiated a call for proposal for MPEG Query Format (MPQF).", "The objective is to enable easier and interoperable access to multimedia data across search engines and repositories.", "As described in [11], the MPQF consists of three fundamental parts – input query type, output query type, and query management tools.", "The format supports various query types (by example, by keywords, etc.", "), spatial-temporal queries and queries based on user preferences.", "It also supports result formatting and foresees service discovery functionality.", "From among various proposals we may highlight [21] which presents an MPEG-7 query language that also allows to query ontologies described in OWL syntax.", "Last of all, let us mention several efforts to create easy-to-use query tools that are not based on either XML or SQL.", "The authors of [17] propose to issue queries via filling table skeletons and issuing weights for individual clauses, with the complex queries being realized by specifying a (visual) condition tree.", "In [16], a simple language based on Lucene query syntax is proposed.", "Finally, [20] describes a rich ontological query language that works with structured English sentences but requires advanced image segmentation and domain knowledge." ], [ "Analysis of Requirements", "Our objective, as mentioned previously, is to create a query language that can be used to define advanced queries over multimedia or other complex data types.", "The language will be implemented on top of the MESSIF software, which is a framework for creating similarity-based retrieval systems.", "Naturally, we also want the language to be general and extensible, so that it can be employed in a wide range of applications.", "To achieve this, we first need to define the desired functionality of such a language.", "In this section, we study the following three issues that are closely related to the language design: (1) the current trends in multimedia information retrieval, which reveal the advanced features that should be supported by the language; (2) existing query languages and their philosophies, so that we can profit on previous work; and (3) the MESSIF framework architecture, which should be compatible with the language.", "After a thorough analysis of these sources we compose a structured list of requirements." ], [ "Current Trends in Multimedia Information Retrieval", "Contemporary science distinguishes two basic approaches to searching in digital data – the attribute-based searching [18] that is used in the traditional DBMS, and the similarity-based retrieval [22].", "In the first case, queries are defined by a set of strict conditions that are applied on attributes of data objects and the qualifying objects are returned.", "In similarity-based retrieval, queries are usually defined by an example object and objects most similar to it form the response.", "The similarity can be described by a distance function, the smallest distance representing the best similarity.", "Alternatively, the similarity can be expressed as a score where higher scores denote more similar objects.", "Since these two approaches are interchangeable, we will use the distance terminology from now on.", "The most commonly used similarity queries are the k-nearest neighbors query (kNN) and the range query; the first restricts the number k of the most similar objects to be retrieved, the second limits the search by the maximum distance of a qualifying object.", "However, there exist a number of other query types, such as various sorts of similarity join, reverse nearest neighbor query, skyline query, distinct kNN query, etc.", "[22] In order to enable efficient retrieval, any search method needs to be backed by a suitable data management structure.", "The indices used for attribute-based and similarity-based retrieval are substantially different.", "The traditional solutions used in relational databases employ index trees that organize data using the total ordering property of individual data domains.", "In content-based searching, the data domains frequently do not have this property and the objects need to be organized with respect to mutual distances only.", "In consequence, the indices for similarity searching usually cannot support attribute-based queries and vice-versa.", "Therefore, these two approaches to searching need to be considered as independent and complementary.", "The attribute-based approach is long-established and well-tuned but it is known to be unsuitable for complex data such as multimedia, since exact match queries can only find binary-identical content and the metadata is often not expressive enough or not available at all.", "Similarity-based methods enable to search the complex data in a more natural way but they also have some limitations.", "The retrieval methods typically employ low-level content descriptors, such as color histograms in case of an image, which are far from human understanding of the object.", "The discrepancy between the object descriptor level and human-perceived semantic level is often denoted as the semantic gap problem [19], which is one of the major challenges in multimedia retrieval nowadays.", "Recent works [10], [14] suggest that promising results can be achieved by combining the two above-mentioned approaches together.", "Attribute-based and simila-rity-based retrieval are orthogonal to each other and their composition can cover both the content of the object and its semantics.", "Let us consider the following query: Retrieve all information about a flower similar to this photo, which grows in the Alps and blossoms in spring, which includes both an example data object and strict conditions on some of its metadata.", "Such query can be evaluated in several ways – the system can first execute a content-based query and then filter the results, or start with the attribute restrictions, or evaluate several separate sub-queries and combine their results.", "Each of these execution plans may be suitable in different situations.", "Therefore, an advanced query interface should allow users to define how a combined query should be processed.", "Support for both types of searching, the various query types and their combinations needs to be part of a query language.", "An important issue connected with complex data searching is the formulation of a search task.", "Frequently it is not possible to define the query in a precise way.", "Instead, a user may describe the desired result by several conditions together with a specification of their importance.", "Typically, the individual conditions may have weights assigned to them.", "With the query-by-example paradigm, it is also often difficult to obtain a really representative query object.", "To overcome this, it is necessary to support queries with multiple examples as well as iterative searching with relevance feedback.", "Moreover, it is desirable to allow users to alter the definition of object similarity, as this may vary for different people and situations.", "There may also be additional parameters of the search process that users want to control, such as the cost/precision ratio for large data processing.", "Apart from including the features mentioned so far, which are perceived as necessary in most studies, the language should allow easy integration of other functionality that may be needed in applications, such as new query types or search algorithms." ], [ "State-of-the-art Query Languages", "In this section, we analyze the main requirements and functionality that can be encountered in various works on query languages surveyed in the Related work.", "Some of the requests were formulated explicitly, especially in the MPEG Query Format, others were picked from the design of the individual languages.", "The identified features fall into the following categories: Support for similarity queries: Many of the existing studies focus on introducing query language primitives for basic similarity queries – the kNN query, range query, and several types of similarity joins are mostly considered.", "Typically, a special primitive is designed for each query type.", "Different keywords are introduced in the individual languages.", "Integration of attribute-based and similarity queries: The need for combining the two approaches to searching is recognized in various proposals.", "Most often, the integration is performed by incorporating the similarity search algorithms into a relational DBMS.", "Support for spacio-temporal queries: Some of the languages, including the MPEG Query Format, give special attention to queries concerning spatial and temporal characteristics of a multimedia object.", "In [15], a set of operators is designed to support this type of queries.", "Adjustability of searching: There are a number of parameters of the search process that users may want to adjust.", "The ones that are most frequently supported in existing proposals are the weighting of search conditions and the definition of a distance function.", "Optimization issues: Optimization strategies strive to maximize the efficiency of query processing by evaluating the individual search operations in the most suitable order.", "To allow optimization, it is necessary to understand the priority of operators, their evaluation costs and the equivalences of expressions.", "Several optimization rules can be found in [5] concerning kNN, range and join query operators.", "As observed in [4], the more specialized operators we introduce, the more precise optimization rules can be defined and vice versa.", "Output formatting: In relational DBMS, output formatting options are limited to the choice of attributes and the ordering of tuples.", "Proposals of [11], [15] expand this with the result paging option and result layout specification, respectively.", "Service discovery: As the MPEG Query Format aims at creating a uniform access interface to various search services, it also provides functionality for service discovery.", "In particular, it allows to ask the search engine for supported query types, metadata, media types and expressions, and to inquire about system usage conditions." ], [ "MESSIF Architecture", "Metric Similarity Search Implementation Framework (MESSIF) [6] is a Java-based object-oriented library that eases the task of implementing metric similarity search systems.", "It provides various modules that are commonly needed by search engines such as memory and disk storage backends, network communication tools, statistics gathering and logging tools, and so on.", "The framework also offers an extensible way of defining data types and their associated metric similarity functions and provides implementations of several common data types and their typical distances, e.g.", "vectors with $L_p$ metrics.", "MESSIF-enabled indexing methods that utilize only the generic properties of the similarity functions are then applicable to any such data type.", "Finally, the framework offers generic hierarchy of data manipulation and querying operations.", "Typical engine operations such as insertion or range and kNN queries are of course implemented as well as various other queries including the similarity join or combined and multi-object queries.", "The definition of new operations is also possible and easy.", "When executing an operation, the framework automatically chooses the evaluation plan either by using an index structure that is able to answer the given query efficiently or by a sequential scan if there are no usable indexes.", "Moreover, the precise or approximate evaluation strategy (typically early-termination or pruning relaxation) can be specified for most queries and taken into consideration by the framework while evaluating the queries.", "Overall, the framework offers functionality of specifying the data type, the metric function, the type of similarity query and its evaluation strategy by means of programming API.", "By defining the query language we would allow to utilize this functionality without the need for actual Java coding." ], [ "Requirements Summary", "Obviously, there are a number of features that need to be considered in the design of a query language for advanced multimedia searching.", "Unfortunately, not all of them can be fully satisfied as it is hardly possible to provide a language that is general, extensible, and simple at the same time.", "In order to gain more insight into the problem, we try to identify the main involved parties and summarize their concerns: \"User interest\": The most obvious party is the end-users, who are often mainly interested in easy usability of the language.", "For a typical non-expert user, we should create a tool that allows to formulate any query they might need while keeping it simple.", "\"Application interest\": For the authors of a specific application, it is vital that the language supports the operations that are requested by the application.", "Apart from those, all other functionality is rather an obstacle as it makes the language unnecessarily complex to both implement and use.", "\"System interest\": The underlying search system is responsible for efficient evaluation of queries.", "For this purpose, it is advantageous that query reformulation and optimization strategies are available and the language philosophy complies with the underlying data structures and algorithms.", "The language needs to support all the functionality provided by the search system.", "\"Interoperability interest\": In many real-world-use scenarios it is necessary to combine information from several sources to get the desired knowledge.", "Therefore, it is desirable to have a tool that can be employed to query across multiple search services.", "A language designed for this purpose needs to be general and extensible.", "As we explained in the introduction, our primary objective is to create a communication interface to a retrieval system that is used in a number of diverse applications and supports a wide range of search settings.", "For this purpose, the system and user points of view are most important.", "Interoperability is desirable but not critical whereas the single-application viewpoint is not relevant at all.", "Most of all, we require the language to support all the functionality enabled by MESSIF.", "The usability and optimization issues are the second most important.", "We are aware of the fact that language suited to these priorities will not be the most convenient for amateur users.", "However, we are more interested in providing extended functionality for advanced users and rely on additional software to support beginners.", "Table REF summarizes the requirements identified earlier and the priority levels we assign to them.", "Table: NO_CAPTION Table: Ranked list of required language features.The fundamental decision in a query language design resides in the choice between the construction of a brand new query language and a modification of an existing one.", "In this section, we discuss our choice and its impact on the architecture of retrieval systems that would implement the language." ], [ "Overall Concept", "The desired functionality of the new language, as described in Table REF , comprehends the support for standard attribute-based searching which, while not being fully sufficient anymore, still remains one of the basic methods of data retrieval.", "A natural approach to creating a more powerful language therefore lies in extending some of the existing, well-established tools for query formulation, provided that the added functionality can be nested into it.", "Two advantages are achieved this way: only the extended functionality needs to be defined and implemented, and the users are not forced to learn a new syntax and semantics.", "The two most frequently used formalisms for attribute data querying are the relational data model with the SQL language, and the XML-based data modeling and retrieval.", "As we could observe in the related work, both these solutions have already been employed for multimedia searching.", "However, there are differences in their suitability for various use cases.", "The XML-based languages are well-suited for inter-system communication, but not practical for hand-typing queries because of the lengthy syntax.", "On the other hand, the SQL language was designed to facilitate user-friendly data access, with the query structure imitating English sentences.", "In addition, SQL is backed by a strong theoretical background of relational algebra, which is not in conflict with content-based data retrieval and offers promising possibilities with respect to query optimization.", "Therefore, we decided to base our approach on the SQL language, similar to existing proposals [5], [4], [13].", "By employing the standard SQL [18] we readily gain a very complex set of functions for attribute-based retrieval but no support at all for similarity-based searching.", "Since we aim at providing a wide and extensible selection of similarity queries, it is also not possible to employ any of the existing extensions to SQL, which focus only on a few most common query operations.", "Therefore, we created a new enrichment of both the relational data model and the SQL syntax so that it can encompass the general content-based retrieval as discussed in the Analysis section.", "The new features will be presented in detail in the following.", "In addition to attribute-based and content-based queries, some research papers distinguish a third type of retrieval – the spacio-temporal queries.", "While this sort of retrieval is definitely relevant for many applications, it does not require any functionality not available within the first two search paradigms.", "We consider spatial and temporal queries to be a special instance of either attribute-based or content-based query, depending on a particular spacio-temporal predicate: search for two time-overlapping actions would be an instance of the former, search for time-nearest action of the latter.", "Naturally, specialized predicates are needed to extract and evaluate the spacio-temporal information.", "Apart from the functionality directly related to query formulation, other features mentioned in Table REF comprise support for query reformulation and optimization, service management, and output formatting tools.", "As for query optimization, it is not possible to create a general and extensible framework with a definite set of optimization rules.", "However, we believe that the design of both the data model and operations that underlie the language itself allow to store all the necessary information that may be required by various optimization strategies of the individual search engines.", "The service management will be discussed shortly in the next section in connection with extensibility issues.", "Output formatting is not addressed in this study but may be easily added to the language." ], [ "System Architecture", "In the existing proposals for multimedia query languages based on SQL, it is always supposed that the implementing system architecture is based on RDBMS, either directly as in [13], or with the aid of a “blade” interface that filters out and processes the content-based operations [4] while passing the regular queries to the backing database.", "Both these solutions are valid for the proposed query language.", "Since we propose to extend the SQL language by adding some language constructs, they can be easily intercepted by a “blade”, evaluated by an external similarity search system, and passed back to the database where the final results are obtained.", "The integration into a RDBMS follows an inverse approach.", "The database SQL parser is updated to support the new language constructs and the similarity query is evaluated by internal operators.", "Of course, the actual similarity query evaluation is the corner stone in both approaches and similarity indexes are crucial for efficient processing.", "One of our priorities is creating a user-friendly tool for the MESSIF framework.", "It already supports a number of general data types and similarity operations and is easily extensible.", "The indexing algorithms can be plugged as needed to efficiently evaluate different queries and the framework automatically selects indexes according to the given query.", "The storage backend of the MESSIF utilizes a relational database and the functionality of the standard SQL is thus internally supported.", "The data and operation model of the proposed query language is designed in such a way that it is compatible with the framework." ], [ "Query Language Specification", "In this section we present the SimSeQL, an extension of the SQL query language which supports advanced multimedia searching in a flexible and extensible way.", "The language can be used as a communication interface to any retrieval system that complies with the abstract data model and operations described in Section REF and is able to parse and process the SQL syntax with the enrichment introduced in Section REF .", "In the end of the section we shortly discuss the extensibility of our design and the query processing procedure." ], [ "Data Model and Operations", "The core of any information management system is formed by data structures that store the information, and operations that allow to access and change it.", "To provide support for the content-based retrieval, we need to revisit the data model employed by the standard SQL and adjust it to the needs of complex data management.", "It is important to clarify here that we do not aim at defining a sophisticated algebra for content-based searching, which is being studied elsewhere.", "For the purpose of the query language, we only need to establish the basic building blocks.", "Our model is in fact a simplified version of the general framework presented in [1].", "Contrary to the theoretical algebra works, we do not study the individual operations and their properties but let these be defined explicitly by the underlying search systems.", "However, we introduce a more fine-grained classification of objects and operations to enable their easy integration into the query language.", "On the concept level, multimedia objects can be analyzed using standard entity-relationship (ER) modeling.", "In the ER terminology, a real-world object is represented by an entity, which is formed by a set of descriptive object properties – attributes.", "The attributes need to contain all information required by target applications.", "In contrast to common data types used in ER modeling, which comprise mainly text and numbers, attributes describing multimedia objects are often of more complex types, such as image or sound data, time series, etc.", "The actual attribute values form an n-tuple and a set of n-tuples of the same type constitute a relation.", "Relations and attributes (as we shall continue to call the elements of n-tuples) are the basic building blocks of the Codd's relational data model and algebra [9], upon which the SQL language is based.", "This model can also be employed for complex data retrieval but we need to introduce some extensions.", "A relation is traditionally defined as a subset of the Cartesian product of sets $D_1$ to $D_n$ , $D_i$ being the domain of attribute $A_i$ .", "The standard operations over relations (selection, projection, etc.)", "are defined using first-order predicate logic and can be readily applied on any data, provided the predicates can be reasonably evaluated over the data.", "To control this, we use the concept of data type that encapsulates both a specification of an attribute domain and the functions that can be applied on members of this domain.", "Let us note here that Codd used a similar concept of extended data type in [9], however he only worked with several special properties of the data type, in particular the total ordering.", "As we shall discuss presently, our approach is much more general.", "We allow for an infinite number of data types, as opposed to the traditional finite set of types that appear in most data management systems.", "The individual data types directly represent the objects (e.g.", "text, image, video, sound), or some derived information (e.g.", "color histogram vector).", "The translation of one data type into another can be realized by so called extractors, a special type of functions defined for each data type.", "According to the best-practices of data modeling [18], redundant data should not be present in the relations, which also concerns derived attributes.", "The rationale is that the derived information only requires extra storage space and introduces the threat of data inconsistency.", "Therefore, the derived attributes should only be computed when needed in the process of data management.", "In case of complex data, however, the computation (i.e.", "the extraction of derived data type) can be very costly.", "Thus, it is more suitable to allow storing some derived attributes in relations, especially when these are used for data indexing.", "Naturally, more extractors may be available to derive additional attributes when asked for.", "Figure REF depicts a possible representation of an image object in a relation.", "Figure: Transformation of image objectinto a relation.", "Full and dashed arrows on theright side depict materialized and available data type extractors, respectively." ], [ "Operations on Data Types", "As we already stated, each data type consists of a specification of a domain of values, and a listing of available functions.", "As some of the functions are vital for the formulation and execution of the algebra operations, we introduce several special classes of functions that may be associated with each data type.", "Comparison functions: Functions of this type define total ordering of the domain ($f_C:D \\times D \\rightarrow \\lbrace <, =, >\\rbrace $ ).", "When a comparison function is available, standard indexing methods such as B-trees can be applied and queries using value comparison can be evaluated.", "Comparison functions are typically not available for multimedia data types and the data types derived from them, where no meaningful ordering of values can be defined.", "Distance functions: In the context of datatypes we focus on basic distance functions that evaluate the dissimilarity between two values from a given data domain ($f_D:D \\times D \\rightarrow \\mathbb {R}_{0}^+$ ).", "The zero distance represents the maximum possible similarity – identity.", "We do not impose any additional restrictions on the behavior of $f_D$ in general, but there exists a way of registering special properties of individual functions that will be discussed later.", "More than one distance function can be assigned to a data type, in that case one of the functions needs to be denoted as default.", "When more distance functions are available for a given data type, a specification of the preferred distance can be part of relation definition.", "In case no distance function is provided, a trivial identity distance is associated to the data type, which assigns distance 0 to a pair of identical values and distance $\\infty $ to any other input.", "Extractors: Extractor functions transform values of one data type into the values of a different data type ($f_E:D_i \\rightarrow D_j$ ).", "Extractors are typically used on complex unstructured data types (such as binary image) to produce data types more suitable for indexing and retrieval (e.g.", "color descriptor).", "An arbitrary number of extractors can be associated to each data type.", "In addition to the declaration of functionality, each of the mentioned operations can be equipped by a specification of various properties.", "The list of properties that are considered worthwhile is inherent to a particular retrieval system and depends on the data management tools employed.", "For instance, many indexing and retrieval techniques for similarity searching rely on certain properties of distance functions, such as the metric postulates or monotonicity.", "To be able to use such a technique, the system needs to ascertain that the distance function under consideration satisfies these requirements.", "To solve this type of inquiries in general, the set of properties that may influence the query processing is defined, and the individual functions can provide values for those properties that are relevant for the particular function.", "To continue with our example, the Euclidean distance will declare that it satisfies the metric postulates as well as monotonicity, while the MinimumValue distance only satisfies monotonicity.", "Another property worth registering is a lower-bounding relationship between two distance functions, which may be utilized during query evaluation." ], [ "Operations on Relations", "The functionality of a search system is provided by the operations that can be evaluated over relations.", "In addition to standard selection and join operations, multimedia search engines need to provide operations for various types of similarity-based retrieval.", "Due to the diversity of possible approaches to searching, we do not introduce a fixed set of operations that need to be available in a search system, but expect each system to maintain its own list of operations.", "Each operation needs to specify its input, which consists of 1) number of input relations (one for simple queries, multiple for joins), 2) expected query objects (zero, singleton, or arbitrary set), 3) arbitrary number of operation-specific parameters, which may typically contain a specification of a distance function, distance threshold, or query operation execution parameters such as approximation settings.", "Apart from a special case discussed later the operations return relations, typically with the scheme of the input relation or the Cartesian product of input relations.", "In case of similarity-based operations the scheme is enriched with additional distance attribute which carries the information about the actual distance of a given result object with respect to the distance function employed by the search operation.", "Similar to operations on data types, operations on relations may also exhibit special properties that can be utilized with advantage by the retrieval system.", "In case of data retrieval operations, the properties are mainly related to query optimization.", "As debated earlier, it is not possible to define general optimization rules for a model with a variable set of operations.", "However, a particular retrieval system can maintain its own set of optimization rules together with the list of operations.", "A special subset of operations on relations is formed by functions that produce scalar values.", "Among these, the most important are the generalized distance functions that operate on relations and return a single number, representing the distance of objects more complex than values from a given attribute domain.", "The input of these functions contains 1) a relation representing the object for which the distance needs to be evaluated, 2) a relation with one or more query objects, and 3) additional parameters when needed.", "Similar to basic distance functions, generalized distance functions need to be treated in a special way since their properties often significantly influence the processing of a query.", "Depending on the architecture of the underlying search engine it may be beneficial to distinguish more types of generalized distance functions.", "For the MESSIF architecture in particular, we define the following two types: Set distance $f_{SD}: 2^D \\times D \\times (D \\times D \\rightarrow \\mathbb {R}_{0}^+) \\rightarrow \\mathbb {R}_{0}^+$: The set distance function allows to evaluate the similarity of object to a set of query objects of the same type, employing the distance function defined over the respective object type.", "In a typical implementation, such function may return the minimum of the distances to individual query objects.", "Aggregated distance $f_{AD}: (D_1 \\times ... \\times D_n) \\times (D_1 \\times ... \\times D_n)\\times ((D_1 \\times D_1 \\rightarrow \\mathbb {R}_{0}^+) \\times ... \\times (D_n \\times D_n \\rightarrow \\mathbb {R}_{0}^+)) \\rightarrow \\mathbb {R}_{0}^+$: The aggregation of distances is frequently employed to obtain a more complex view on object similarity.", "For instance, the similarity of images can be evaluated as a weighted sum of color- and shape-induced similarities.", "The respective weights of the partial similarities can be either fixed, or chosen by user for a specific query.", "Though we do not include the user-defined parameters into the definitions of the distances for easier readability, these are naturally allowed in all functions." ], [ "Data Indexing", "While not directly related to the data model, data indexing methods are a crucial component of a retrieval system.", "The applicability of individual indexing techniques is limited by the properties of the target data.", "To be able to control the data-index compatibility or automatically choose a suitable index, the search system needs to maintain a list of available indices and their properties.", "The properties can then be verified against the definition of the given data type or distance function (basic or generalized).", "Thus, metric index structures for similarity-based retrieval can only be made available for data with metric distance function, whereas traditional B-trees may be utilized for data domains with total ordering.", "It is also necessary to specify which search operations can be supported by a given query, as different data processing is needed e.g.", "for the nearest-neighbor and reverse-nearest-neighbor queries.", "Apart from the specialized indices, any search system inherently provides the basic Sequential Scan algorithm as a default data access method that can support any search operation." ], [ "SimSeQL Syntax and Semantics", "The SimSeQL language is designed to provide a user-friendly interface to state-of-the-art multimedia search systems.", "Its main contribution lies in enriching the standard SQL by new language constructs that enable to issue all kinds of content-based queries in a standardized manner.", "In accordance with the declarative paradigm of SQL, the new language constructs allow to describe the desired results while shielding users from the execution issues.", "On the syntactical level, the SimSeQL contributes mainly to the query formulation tools of SQL.", "Data modification and control commands are not discussed in this paper since their adaptation to the generalized data types and operations is straightforward.", "On the semantic level, however, the original SQL is significantly enriched by the introduction of the unlimited set of complex data types and operations over them.", "A SimSeQL query statement follows the same structure as standard SQL, being composed of the six basic clauses SELECT, FROM, WHERE, GROUP BY, HAVING, and ORDER BY, with their traditional semantics [18].", "The extended functionality is mainly provided by a new construct called SIMSEARCH, which is embedded into the FROM clause and allows to search by similarity, combine multiple sources of information, and reflect user preferences.", "Prior to a detailed description of the new primitives, we present the overall query syntax with the SIMSEARCH construct in the following scheme: Table: NO_CAPTIONIn general, there are two possible approaches to incorporating primitives for content-based retrieval into the SQL syntax.", "We can either make the similarity search results form a new information resource on the level of other data collections in the FROM clause (an approach used in [13]), or handle the similarity as another of the conditions applied on candidate objects in the WHERE clause (exercised in [4], [15], [2], [12]).", "However, the latter approach requires standardized predicates for various types of similarity queries, their parameters etc., which is difficult to achieve in case an extensible set of search operations and algorithms is to be supported.", "In addition, the similarity predicates are of different nature than attribute-based predicates and their efficient evaluation requires specialized data structures.", "Therefore, we prefer to handle similarity-based retrieval as an independent information source.", "For this, we only standardize the basic structure and expected output, which can be implemented by any number of search methods of the particular search engine.", "As anticipated, the similarity-based retrieval is wrapped into the SIMSEARCH language construct, which produces a standard relation and can be seamlessly integrated into the FROM clause.", "The SIMSEARCH expression is composed of the following parts: Specification of query objects: The selection of query objects follows immediately after the SIMSEARCH keyword.", "An arbitrary number of query objects can be issued, each object being in fact an attribute that can be compared to attributes of the target relations.", "The query object (attribute) can be represented directly by the attribute value, by a reference to object provided by an application, or by a nested query that produces the query object(s).", "The query objects need to be type-compatible with the attributes of target relation they are to be compared to.", "Often the extractor functions can be used with advantage on the query objects.", "Specification of a target relation: The keyword IN introduces the specification of one or more relations, elements of which are processed by the search algorithm.", "Naturally, each relation can be produced by a nested query.", "Specification of a distance function: An essential part of a content-based query is the specification of a distance function.", "The BY subclause offers three ways of defining the distance: calling a distance function associated to an attribute, referring directly to a distance function provided by the search engine, or constructing the function within the query.", "In the first case, it is sufficient to enter the name of attribute to invoke its default distance function.", "Non-default distance function of an attribute needs to be selected via the DISTANCE FUNCTION primitive that also allows to pass additional parameters for the distance function if necessary.", "The last case allows greater freedom of specifying the distance function by user, but both the attributes for which the distance is to be measured must be specified.", "A special function DISTANCE$(x,y)$ can be used to call the default distance function defined for the given data type of attributes $x,y$ .", "The nuances of referring to a distance function can be observed in the following: Table: NO_CAPTION Table: NO_CAPTION Specification of a search method: The final part of the SIMSEARCH construct specifies the search methods or, in other words, the query type.", "Users may choose from the list of methods offered by the search system.", "It can be reasonably expected that every system supports the basic nearest neighbor query, therefore this is considered a default method in case no other is specified with the METHOD keyword.", "The default nearest neighbor search returns all n-tuples from the target relation unless the number of nearest neighbors is specified in the SELECT clause by the TOP keyword.", "The complete SIMSEARCH phrase returns a relation with a scheme of the target relation specified by the IN keyword, or the Cartesian product in case of more source relations.", "Moreover, information about distance of each n-tuple of the result set computed during the content-based retrieval is available.", "This can be used in other clauses of the query, referenced either as DISTANCE, when only one distance evaluation was employed, or prefixed with the named data source in the clause when ambiguity should arise (e.g.", "ds.DISTANCE)." ], [ "Extensibility", "The extensibility of the SimSeQL language relies on the possibility to define a set of data types, functions, query operations, and index structures supported by each retrieval engine.", "The information about the system functionality should be maintained in special relations with standardized structure, which would allow automatic service discovery.", "The design of these relations will be subject of our future work." ], [ "Query Processing", "The query processing is a complex procedure that needs to be designed carefully with respect to the architecture of a given retrieval system.", "Nonetheless, the following succession of basic steps will always form the basic structure of the processing.", "Fist of all, a parser identifies the individual objects and operations contained in the query expression.", "Using registered properties, the query processing unit checks the compatibility.", "When successful, an evaluation plan is composed.", "For its construction, the system may use the available indices together with the registered properties of attributes, indices, and functions.", "The optimal evaluation plan is eventually executed and the results returned to the user." ], [ "Example Scenarios", "To illustrate the wide applicability of the SimSeQL language, we now present several query examples for various use-case scenarios found in image and video retrieval.", "Each of them is accompanied by a short comment on the interesting language features employed.", "For the examples, let us suppose that the following set of relations, data types and functions is available in the retrieval system: image relation: register of images Table: NO_CAPTION video_frame relation: list of video frames Table: NO_CAPTION keyword relation: a simple table of keywords which can be related to an image/video (e.g.", "web gallery tags) Table: NO_CAPTION image_keyword relation: keywords associated with an image Table: NO_CAPTION" ], [ "Query 1", "Retrieve 30 most similar images to a given example Table: NO_CAPTIONThis example presents the simplest possible similarity query.", "It employs the default nearest neighbor operation over the shape descriptor with its default distance function.", "User does not need any knowledge about the operations employed, only selects the means of similarity evaluation.", "The supplied parameter $queryImage$ represents the MPEG7_contour_shape type of a query image (provided by surrounding application).", "The output of the search is the list of identifiers of the most similar images as well as the distance measured between the query image and the respective image in the database." ], [ "Query 2", "Retrieve all variants of the word 'feather' with maximally two typos Table: NO_CAPTIONThis time, a simple range query is required.", "The user selected a non-default distance function for the evaluation of similarity, which is adjusted by user-defined weights for the edit, insert and remove actions.", "Depending on the underlying search system, the query can be either reformulated and evaluated as a range query, or evaluated as a nearest neighbor query with subsequent result object filtering." ], [ "Query 3", "Find all pairs of keywords that are within edit distance 1 (we may suppose that these are candidates for typos) Table: NO_CAPTIONIn this case, a similarity join with a threshold value 1 is required.", "The similarity join needs no query objects, is defined over two relations, and requires explicit reference to a distance function with the input parameters." ], [ "Query 4", "Retrieve images most similar to a set of examples (e.g.", "identifying a flower by supplying several photos) Table: NO_CAPTIONThis query represents an example of a multi-object query, input of which are binary images that are transformed to the required descriptors via extractors.", "Alternatively, the query objects could be provided as a result of a nested query.", "The minimum aggregation function employed for similarity evaluation would be formally defined on attributes and their respective distance functions.", "Here it is applied on the distances to individual objects only, as these are internally linked to the individual attributes and distance functions.", "Note that the default distance functions of the respective attributes are applied using DISTANCE$(x,y)$ construct." ], [ "Query 5", "Retrieve all videos where Obama and Bush appear Table: NO_CAPTIONThis query employs a join of two similarity search results, each of which uses a range query operation to retrieve objects very similar to the given example." ], [ "Query 6", "Retrieve all videos where the Vesuvius mountain appears (image similarity) and a commentator mentions volcanoes (speech/text similarity) within two minutes (time aggregation) Table: NO_CAPTIONIn this example, multiple modalities are combined to produce the result.", "In addition, the user selected a special search method that enables to set approximation in work (the second parameter of the search method is the maximum number of objects that may be visited in query processing)." ], [ "Query 7", "Retrieve text annotation for a given unknown image, exploiting the keywords associated to similar images Table: NO_CAPTIONThe final example presents a content-based search nested into a complex expression of the traditional SQL.", "The two main contributions of this paper are the analysis of requirements for a query language, and the proposal of a query language for retrieval over complex data domains.", "The presented language is backed by a general model of data structures and operations, which is applicable to a wide range of search systems that offer different types of content-based functionality.", "Moreover, the support for data indexing and query optimization is inherently contained in the model.", "The SimSeQL language extends the standard SQL by new primitives that allow to formulate content-based queries in a flexible way, taking into account the functionality offered by a particular search engine.", "The proposal of the language was influenced by the MESSIF framework that offers the functionality of executing complex similarity queries on arbitrary index structures but lacks a user-friendly interface for advanced querying.", "Having laid the formal foundations of the query interface here, we will proceed with the implementation of a language parser which will translate the query into MESSIF for the actual evaluation.", "In the future, we plan to research the possibilities of adapting the existing optimization strategies to utilize the reformulation capabilities of the proposed extension.", "Furthermore, we would like to create an intuitive (graphical) query formulation tool and, possibly, a conversion mechanism into the MPEG7 Query Format for inter-system communication.", "Also, the syntax for the various service discovery tools needs to be established." ], [ "Acknowledgments", " This work has been partially supported by Brno PhD Talent Financial Aid and by the national research projects GAP 103/10/0886 and VF 20102014004." ] ]

[ [ "Avoiding dark states in open quantum systems by tailored initial\n correlations" ], [ "Abstract We study the transport of excitations on a V-shaped network of three coupled two-level systems that are subjected to an environment that induces incoherent hopping between the nodes.", "Two of the nodes are coupled to a source while the third node is coupled to a drain.", "A common feature of these networks is the existence of a dark-state that blocks the transport to the drain.", "Here we propose a means to avoid this state by a suitable choice of initial correlations, induced by a source that is common to both coupled nodes." ], [ "Introduction", "The transport dynamics of energy or charge in many physical systems can be described by considering transport on a network of coupled 2-level systems.", "Depending on the physical system, the transport can range from being purely coherent to being purely incoherent.", "In the former case the dynamics can be described by continuous-time random walks [1], while in the latter case the dynamics follows from Schrödinger's equation, which for complex systems and certain choices of the Hamiltonian defines so-called continuous-time quantum walks [2], [3].", "Coupling of a quantum system with purely coherent dynamics to a bath of harmonic oscillators (for instance phonons) can lead to a mixture of coherent transport and incoherent hopping induced by the environment.", "The dynamics (of the reduced density matrix) can be described by a quantum master equation of Lindblad type [4], where a certain choice of Lindblad operators defines so-called quantum stochastic walks [5].", "In this work we consider transport on a V-shaped trimer configuration, resembling for instance a system of coupled quantum dots [6], [7], [8].", "The transport will be generated by connecting a (incoherent) single source to the end nodes of the V-shaped trimer and a (incoherent) drain to the middle node, see below.", "Most theoretical works have focused on the case where the source creates an excitation on a single node of the network [9], [10].", "One can also, however, consider the possibility where a source is connected to multiple nodes of the network.", "Such a source can then create an excitation that is in a superposition of these nodes, leading to initial correlations between these nodes.", "A common feature of these V-shaped, or circular, trimer configurations is the existence of a dark state [7], [11], [12], [13], [9], [14].", "Such a state causes the excitation to become trapped in the network and therefore leads to a blocking of the transport in the purely coherent case.", "To overcome this problem, one can either introduce an energetic disorder on the nodes [11], [9] or couple the system to a suitable environment where the decoherence process destroys the interference effects that lead to the dark state [9], [6].", "Here we propose a third method: a suitable choice of initial correlations that are induced by a single source which is coupled to the end nodes of the network creates an initial state which is orthogonal to the dark state.", "This then causes the absence of the dark state in the transport even in the purely coherent case, leading to a complete transport to the drain.", "The paper is organized as follows.", "In Sec.", "we introduce our model and provide a detailed discussion on the exact mathematical implementation of a source that induces initial correlations.", "In Sec.", "we discuss the transport efficiency of different initial configurations with the help of both analytical and numerical computations.", "We consider the dynamics of excitations on a trimer network.", "The coherent (quantum) dynamics on this network is described by the Schrödinger equation, or equivalently by the Liouville - von Neumann equation, with the general Hamiltonian $\\mathbf {H}_0 = \\left( \\begin{array}{ccc}E_1 & V_{12} & V_{13} \\\\V_{12} & E_2 & V_{23} \\\\V_{13} & V_{23} & E_3\\end{array}\\right).$ Here, $E_k$ is the site energy of node $\\mathinner {|{k}\\rangle }$ and $V_{kl}$ are the transfer rates between nodes $\\mathinner {|{k}\\rangle }$ and $\\mathinner {|{l}\\rangle }$ .", "Note that for certain choices of the site energies and the couplings there exists an eigenstate $\\mathinner {|{D}\\rangle } = (\\mathinner {|{1}\\rangle } - \\mathinner {|{2}\\rangle })/\\sqrt{2}$ of $\\mathbf {H}_0$ , having only an overlap with nodes 1 and 2 [7], [11], [12], [9].", "Now, if the system is in contact with an external environment, the total Hamiltonian takes the form $\\mathbf {H}_{\\text{tot}} = \\mathbf {H}_0 + \\mathbf {H}_E +\\mathbf {H}_{\\text{int}}$ , where $\\mathbf {H}_0$ is the Hamiltonian of the network, $\\mathbf {H}_E$ is the Hamiltonian of the environment and $\\mathbf {H}_{\\text{int}}$ specifies the interactions between the network and the environment.", "When the environmental correlation time is small compared to the relaxation time of the system, one can describe the dynamics on the network by a master equation in Lindblad form [4]: $\\frac{d \\mathbf {\\rho }_N(t)}{dt} = -i \\left[ \\mathbf {H}_0, \\mathbf {\\rho }_N(t) \\right] + \\sum _{k,l=1}^{3} \\lambda _{kl} \\mathcal {D}(\\mathbf {L}_{kl}, \\mathbf {\\rho }_N(t)) ,$ with the constants $\\lambda _{kl}\\ge 0$ for all $k$ and $l$ and $\\mathcal {D}(\\mathbf {L}_{kl}, \\mathbf {\\rho }_N(t)) = \\mathbf {L}_{kl}^{\\phantom{\\dagger }} \\mathbf {\\rho }_N(t) \\mathbf {L}_{kl}^\\dagger -\\frac{1}{2}\\left\\lbrace \\mathbf {L}_{kl}^\\dagger \\mathbf {L}_{kl}^{\\phantom{\\dagger }}, \\mathbf {\\rho }_N(t) \\right\\rbrace .$ For our model we assume the Lindblad operators $\\mathbf {L}_{kl}$ to be given by $\\mathbf {L}_{kl} = \\mathinner {|{k}\\rangle }\\mathinner {\\langle {l}|}$ .", "The term in Eq.", "(REF ) corresponding to the operator $\\mathbf {L}_{kl}$ models the incoherent excitation dynamics, induced by the environment, between the nodes $l$ and $k$ with rate $\\lambda _{kl}$ .", "In the purely incoherent limit we assume simple hopping dynamics to be described by a Pauli master equation (for the diagonal elements of $\\mathbf {\\rho }(t)$ only).", "The corresponding transition rates $\\lambda _{kl}$ , for $k\\ne l$ , can be phenomenologically estimated with Fermi's golden rule, e.g.", "$\\lambda _{kl}= \\Lambda |V_{kl}|^2$ .", "Here $\\Lambda $ is a constant that captures the particular details of the environment (for simplicity, we assume that $\\Lambda = 1$ ).", "We further assume additional pure dephasing, which is taken to be identical for all the nodes of our network, i.e.", "we have Lindblad operators $\\mathbf {L}_{kk}$ with $\\lambda _{kk} = \\lambda $ for all $k$ , where $\\lambda $ is the global dephasing rate [15].", "Since the dissipative terms in the Lindblad master equation induce incoherent hopping between the nodes, we can introduce a parameter $\\alpha $ , with $0\\le \\alpha \\le 1$ , that allows us to interpolate between purely coherent dynamics ($\\alpha =0$ ) and purely incoherent dynamics ($\\alpha =1$ ): $\\frac{d \\mathbf {\\rho }_N(t)}{dt} = (1-\\alpha ) \\mathcal {L}_{\\text{coh}}(\\mathbf {\\rho }_N(t)) + \\alpha \\mathcal {L}_{\\text{env}}(\\mathbf {\\rho }_N(t)) ,$ with $\\mathcal {L}_{\\text{coh}}(\\mathbf {\\rho }_N(t)) = -i \\left[ \\mathbf {H}_0,\\mathbf {\\rho }_N(t) \\right]$ and $ \\mathcal {L}_{\\text{env}}(\\mathbf {\\rho }_N(t))=\\mathcal {L}_{\\text{incoh}}(\\mathbf {\\rho }_N(t)) +\\mathcal {L}_{\\text{deph}}(\\mathbf {\\rho }_N(t))$ .", "This approach is also known as the quantum stochastic walk [5].", "The generator $\\mathcal {L}_{\\text{incoh}}$ corresponds to the terms in (REF ) that generate incoherent transfer between the nodes and the generator $\\mathcal {L}_{\\text{deph}}$ corresponds to the terms that generate pure dephasing.", "Figure: An illustration of the two different ways of connecting the source(s) to the trimer network: (I) corresponds tothe configuration with 2 independent transitions to the network and (II) corresponds to the configuration with a source that createsinitial correlations between nodes 1 and 2.Figure: Dependence of the EST η II (α)\\eta _{II}(\\alpha ) on the angle φ\\phi for λ=1\\lambda = 1 and Γ=0.5\\Gamma = 0.5.", "On the left we showthe general result for all values of φ\\phi and α\\alpha .", "The right figure shows the cross-sectional curves for φ=0,π/2\\phi = 0, \\pi /2 and φ=π\\phi =\\pi ,together with the result for η I (α)\\eta _{I}(\\alpha )." ], [ "Sources and drains", "A source is included in our system as an extra node $\\mathinner {|{0}\\rangle }$ that is incoherently coupled to the nodes of our network in order to make sure that there is only transport from the source to the network and not back.", "In general, transitions from the source to a general state $\\mathinner {|{\\psi }\\rangle } = \\sum _k a_k \\mathinner {|{k}\\rangle }$ of the network can be phenomenologically modelled by the Lindblad operator $\\mathbf {L}_s = \\mathinner {|{\\psi }\\rangle }\\mathinner {\\langle {0}|}$ , leading to the following additional term to the master equation: $\\mathcal {L}_{\\text{source}}(\\mathbf {\\rho }_{SN}(t)) = \\Gamma \\mathcal {D}( \\mathbf {L}_{s}, \\mathbf {\\rho }_{SN}(t)),$ with $\\Gamma $ representing the rate at which the excitation flows into the network.", "Note that for a $N$ -dimensional network with a source, the reduced density matrix $\\mathbf {\\rho }_{SN}$ is represented by a $(N+1)\\times (N+1)$ matrix.", "For an initial preparation in the source node, i.e.", "$\\mathbf {\\rho }_{SN}(0)= \\mathinner {|{0}\\rangle }\\mathinner {\\langle {0}|}$ , one can show that the density matrix can be written in the form [16]: $\\mathbf {\\rho }_{SN}(t) = \\left( \\begin{array}{cc}\\rho _{00}(t) & 0 \\\\0 & \\mathbf {\\rho }_{N}(t)\\end{array}\\right),$ where $\\mathbf {\\rho }_{N}(t)$ is the density matrix corresponding to the network nodes.", "In a similar fashion, we include a drain by coupling the state $\\mathinner {|{N+1}\\rangle }$ to the network.", "The incoherent transition from a state $\\mathinner {|{\\psi }\\rangle }$ of the network to the drain with rate $\\gamma $ can then be modelled by the term $\\mathcal {L}_{drain}(\\mathbf {\\rho }_{SND}(t)) = \\gamma \\mathcal {D}(\\mathbf {L}_{d}, \\mathbf {\\rho }_{SND}(t)),$ in the master equation, with $\\mathbf {L}_d = \\mathinner {|{N+1}\\rangle }\\mathinner {\\langle {\\psi }|}$ .", "Thus the final reduced density operator $ \\mathbf {\\rho }_{SND}(t) = \\mathbf {\\rho }(t)$ is represented by a $(N+2)\\times (N+2)$ matrix." ], [ "Creating initial correlations with a source", "There are now two interesting ways in which we can connect the source to the end nodes $\\mathinner {|{1}\\rangle }$ and $\\mathinner {|{2}\\rangle }$ of the trimer network: The source can either feed node 1 with rate $\\Gamma /2$ or node 2 with rate $\\Gamma /2$ .", "We assume these processes to be independent of each other.", "We can model this with two dissipators representing the two independent processes: $\\mathcal {L}_{\\text{source}}^{(1)}(\\mathbf {\\rho }(t)) &=& \\frac{\\Gamma }{2} \\mathcal {D}( \\mathinner {|{1}\\rangle }\\mathinner {\\langle {0}|}, \\mathbf {\\rho }(t)) + \\frac{\\Gamma }{2} \\mathcal {D}( \\mathinner {|{2}\\rangle }\\mathinner {\\langle {0}|},\\mathbf {\\rho }(t)) \\nonumber \\\\&=& \\frac{\\Gamma }{2}\\rho _{00}(t) \\left(\\mathinner {|{1}\\rangle }\\mathinner {\\langle {1}|} + \\mathinner {|{2}\\rangle }\\mathinner {\\langle {2}|} - 2\\mathinner {|{0}\\rangle }\\mathinner {\\langle {0}|}\\right) .$ The source can also feed a superposition state $\\mathinner {|{\\psi }\\rangle }$ between node 1 and node 2, which can in general be written as $\\mathinner {|{\\psi }\\rangle } = \\left( \\mathinner {|{1}\\rangle } + e^{i \\phi } \\mathinner {|{2}\\rangle }\\right)/\\sqrt{2}$ .", "We can model this process with one dissipator: $\\mathcal {L}_{\\text{source}}^{(2)}(\\mathbf {\\rho }(t)) &=& \\Gamma \\mathcal {D}( \\mathinner {|{\\psi }\\rangle }\\mathinner {\\langle {0}|}, \\mathbf {\\rho }(t)) \\\\&=& \\frac{\\Gamma }{2}\\rho _{00}(t) \\bigg [\\mathinner {|{1}\\rangle }\\mathinner {\\langle {1}|} + \\mathinner {|{2}\\rangle }\\mathinner {\\langle {2}|} - 2\\mathinner {|{0}\\rangle }\\mathinner {\\langle {0}|} \\nonumber \\\\&& + e^{-i\\phi } \\mathinner {|{1}\\rangle }\\mathinner {\\langle {2}|} + e^{i\\phi } \\mathinner {|{2}\\rangle }\\mathinner {\\langle {1}|}\\bigg ] .$ See Fig.", "REF for an illustration of these two configurations.", "The key difference between these two choices is that $\\mathcal {L}_{\\text{source}}^{(2)}(\\mathbf {\\rho }(t))$ creates initial correlations, depending on the phase $\\phi $ , between the two nodes, while $\\mathcal {L}_{\\text{source}}^{(1)}(\\mathbf {\\rho }(t))$ does not.", "How these initial correlations effect the transport properties will be adressed in the following section.", "In previous work we used the expected survival time (EST) $\\eta $ as a measure for the transport properties of the excitation in the network [16].", "Here we use it specifically to study the effects of initial correlations on the transport.", "This EST is defined as the average time it needs for the excitation to move completely from the source to the drain: $\\eta (\\alpha ) = \\int \\limits _0^\\infty dt \\ \\left( 1 - \\rho _{N+1,N+1}(t, \\alpha )\\right).$ The following representation of the EST in terms of the Laplace transforms $\\hat{\\rho }_{kk}(s)$ of the components of the density matrix, allows for a more convenient way to obtain analytical expressions for the EST [16]: $\\eta (\\alpha ) = \\lim _{s\\rightarrow 0} \\sum _{k=0}^N \\hat{\\rho }_{kk}(s,\\alpha ) .$ We denote the EST corresponding to a source feeding independently the two nodes 1 and 2, Eq.", "(REF ), as $\\eta _I(\\alpha )$ and the EST corresponding to the source feeding into an entangled state, Eq.", "(REF ), as $\\eta _{II}(\\alpha )$ .", "To illustrate the key effects, we assume, for simplicity, that $E_1 = E_2 = E_3 = 1$ , $V_{13} = V_{23} = 1$ , $\\gamma =1$ and $V_{12} = 0$ .", "That is, we focus on the situation when there is no bond between nodes 1 and 2.", "For these parameters, it follows from Eq.", "(REF ) that the ESTs $\\eta _I(\\alpha )$ and $\\eta _{II}(\\alpha )$ take the form $\\eta _I(\\alpha ) &=& 1/\\Gamma + f(\\alpha )/g(\\alpha ) \\\\\\eta _{II}(\\alpha ) &=& 1/\\Gamma + [ f(\\alpha ) - h(\\alpha )\\cos \\phi ]/g(\\alpha ) ,$ with $h(\\alpha ) = 4(1-\\alpha )^2$ and $f(\\alpha ) &=& 4 + \\alpha (17+13\\lambda )+2\\alpha ^2(\\lambda (\\lambda -8)-19) \\\\&& + 3\\alpha ^3(11+\\lambda (9+2\\lambda )) \\nonumber \\\\g(\\alpha ) &=& 4\\alpha (2+\\lambda ) - \\alpha ^2 (15+7\\lambda ) + \\alpha ^3 (11+\\lambda (9+2\\lambda )).", "\\nonumber $ The dependence of $\\eta _{II}(\\alpha )$ on the phase $\\phi $ is therefore proportional to $\\cos \\phi $ .", "Its amplitude $-h(\\alpha )/g(\\alpha )$ is a monotonically decreasing function of $\\alpha $ and vanishes when $\\alpha =1$ .", "Therefore $\\eta _{II}(\\alpha )$ converges to $\\eta _{I}(\\alpha )$ when $\\alpha \\rightarrow 1$ , where they both reach the value $4 + 1/\\Gamma $ .", "This also shows that for $\\phi \\in [0,\\pi /2)$ and $\\phi \\in (3\\pi /2, 2\\pi ]$ , $\\eta _{II}(\\alpha ) < \\eta _I(\\alpha )$ and that the converse result holds for $\\phi \\in (\\pi /2, 3\\pi /2)$ .", "In the limit $\\alpha \\rightarrow 0$ we find that $\\eta _{I}(0) = \\infty $ , while $\\lim _{\\alpha \\rightarrow 0}\\eta _{II}(\\alpha ) =\\left\\lbrace \\begin{array}{cc}\\displaystyle \\frac{1}{\\Gamma } + \\frac{25+13\\lambda }{8+4\\lambda } & \\mbox{for} \\quad \\phi = 0 \\\\\\infty & \\mbox{for} \\quad \\phi \\ne 0.", "\\end{array}\\right.$ To illustrate these analytical results we show in Fig.", "REF , for $\\lambda = 1$ , the dependence of the EST $\\eta _{II}(\\alpha )$ on the phase $\\phi $ and compare it to $\\eta _I(\\alpha )$ .", "One clearly observes the $\\cos \\phi $ -dependence of $\\eta _{II}(\\alpha )$ , see left panel, and an infinite EST for $\\alpha = 0$ and $\\phi \\ne 0$ .", "The infinite EST can be understood by noting that for these values of $\\alpha $ and $\\phi $ the dark state is not influenced by the drain and is a stationary state of the system, causing both EST's to diverge when $\\phi \\ne 0$ .", "For $\\phi = 0$ the state $\\mathinner {|{\\psi }\\rangle }$ is orthogonal to $\\mathinner {|{D}\\rangle }$ , causing the absence of the dark state in the full dynamics and leading to complete transfer to the drain.", "In Fig.", "REF we show for $\\phi = 0$ the dependence on the dephasing rate $\\lambda $ .", "We observe that for increasing values of $\\lambda $ the EST $\\eta _{I}(\\alpha )$ decreases, leading to faster transport.", "This resembles noise-assisted transport found in many other systems [15], [9].", "The EST $\\eta _{II}(\\alpha )$ , in contrast, increases for larger values of $\\lambda $ , leading to slower transport to the drain.", "Thus, here the optimal transport efficiency is obtained in the purely coherent case ($\\alpha = 0$ and $\\lambda = 0$ ).", "However, $\\eta _{II}(\\alpha )$ is always smaller than $\\eta _{I}(\\alpha )$ .", "We further observe that $\\eta _{II}(\\alpha )$ increases until a certain $\\alpha _c$ , after which it follows the curve of $\\eta _I(\\alpha )$ .", "The value of $\\alpha _c$ becomes smaller with increasing dephasing rates.", "This happens because the dephasing process destroys the coherences between nodes 1 and 2.", "Therefore after this point, the initial correlations do not significantly influence the transport properties anymore and $\\eta _{II}(\\alpha ) \\approx \\eta _I(\\alpha )$ .", "When $\\phi =0$ , a tractable analytical expression for $\\alpha _c$ is possible: $\\alpha _c =\\frac{4}{3+2\\lambda }\\left[\\sqrt{\\frac{(2+\\lambda ^3)}{11+9\\lambda +2\\lambda ^2}}-\\frac{1}{4}\\right].$ Our results show that initial correlations, induced by the source feeding the superposition state $\\mathinner {|{\\psi }\\rangle } = (\\mathinner {|{1}\\rangle } + \\mathinner {|{2}\\rangle })/\\sqrt{2}$ , leads to faster transport than for feeding any other state of the form $\\mathinner {|{\\psi }\\rangle } = (\\mathinner {|{1}\\rangle } + e^{i\\phi } \\mathinner {|{2}\\rangle })/\\sqrt{2}$ with $\\phi \\ne 0$ , even in the presence of dephasing.", "Additionally, if $\\cos \\phi >0$ one always has $\\eta _{II}(\\alpha ) < \\eta _I(\\alpha )$ , see Eqs.", "(REF ) and ().", "Therefore, when initial correlations are present, a smaller coupling to the environment (smaller values of $\\alpha $ ) is sufficient to avoid the dark state, compared to the situation without initial correlations." ], [ "Summary", "In conclusion, we have shown that by connecting a source to the two end nodes of the V-shaped network it is possible to induce initial correlations between the coupled nodes and that these initial correlations can overcome the detrimental effects of the dark state, leading to complete transfer to the drain.", "The source can also feed the two end nodes independently, i.e., without initial correlations between the two nodes.", "Then, the dark state inhibits complete transfer.", "When increasing the coupling to the environment, the differences in the expected survival times between the two types of sourcing processes diminish.", "We expect that the results obtained here also hold for larger networks that exhibit invariant subspaces, as for example described in [9] for fully connected networks, in [14] for larger ring-like structures or in [17] for Erdös-Rényi graphs.", "Furthermore, our results are also related to the study of electrical currents through a network of two-level systems, since the current is related to the long-time limit of the time derivative of the EST.", "We gratefully acknowledge support from the Deutsche Forschungsgemeinschaft (DFG grant MU2925/1-1).", "Furthermore, we thank A. Blumen, A. Anishchenko and L. Lenz for useful discussions." ] ]

[ [ "Resistive and magnetized accretion flows with convection" ], [ "Abstract We considered the effects of convection on the radiatively inefficient accretion flows (RIAF) in the presence of resistivity and toroidal magnetic field.", "We discussed the effects of convection on transports of angular momentum and energy.", "We established two cases for the resistive and magnetized RIAFs with convection: assuming the convection parameter as a free parameter and using mixing-length theory to calculate convection parameter.", "A self-similar method was used to solve the integrated equations that govern the behavior of the presented model.", "The solutions showed that the accretion and rotational velocities decrease by adding the convection parameter, while the sound speed increases.", "Moreover, by using mixing-length theory to calculate convection parameter, we found that the convection can be important in RIAFs with magnetic field and resistivity." ], [ "Introduction", "The existence of radiatively inefficient accretion flows (RIAFs) have been confirmed in low-luminosity state of X-ray binaries and nuclei of galaxies (Narayan et al.", "1996; Esin et al.", "1997; Di Matteo et al.", "2003; Yuan et al.", "2003).", "It was understood that RIAFs are likely to be convectively unstable in the radial direction due to the inward increase of the entropy of accreting gas (Narayan & Yi 1994).", "Moreover, hydrodynamical and magetohydrodynamical simulations of low-viscosity RIAFs have confirmed these flows are convectively unstable ( e. g. Igumenshchev et al.", "1996; Stone et al.", "1999; Machida et al.", "2001; Hawley & Balbus 2002; McKinney & Gammie 2002; Igumenshchev et al.", "2003).", "Self-similar or global solutions for convection-dominated accretion flows (CDAFs) were presented by several authors (e. g. Narayan et al.", "2000; Quataert & Gruzinov 2000; Abramowicz et al.", "2002; Lu et al.", "2004; Zhang & Dai 2008).", "Igumenshchev et al.", "(2003) studied the resistive MHD simulations of RIAFs onto black holes.", "They assumed two cases for the geometry of the injected magnetic field: pure toroidal field and pure poloidal field.", "They found that in the case of pure toroidal magnetic field, the accreting gas forms a nearly axisymmetric, geometrically thick, turbulent accretion disc.", "Moreover, their solutions represented that the flow resembles in many respects CDAFs found in previous numerical and analytical investigations of viscous hydrodynamic flows.", "Zhang & Dai (2008) investigated the effect of magnetic field on RIAFs with convection by a semi-analytically method.", "By exploit of $\\alpha $ -prescription for viscosity and convection, they used two methods to study of magnetized flows with convection, i.e.", "they take the convective coefficient $\\alpha _c$ as a free parameter to discuss the effects of convection for simplicity.", "They also established the $\\alpha _c$ -$\\alpha $ relation for magnetized flows using the mixing-length theory and compare this relation with the non-magnetized case.", "They found that the magnetic field makes the $\\alpha _c$ -$\\alpha $ relation be distinct from that of non-magnetized flows.", "Since the importance of toroidal magnetic field and resistivity in accretion flows have been confirmed observationally (see Faghei 2011 and references therein), Faghei (2011) considered the steady, radially self-similar solutions of accretion flows in the presence of the toroidal magnetic field and the resistivity.", "However, he ignored the effects of convection in his model.", "Generally semi-analytical studies of magnetized CDAFs are related to non-resistive magnetized CDAFs (e. g. Zhang & Dai 2008) and the resistive and magnetized CDAF was studied in MHD simulations (e. g. Igumenshchev & Narayan 2002; Hawley & Balbus 2002; Igumenshchev et al.", "2003).", "Thus, it will be interesting to study the effects of resistivity on RIAFs with convection.", "Here, we adopt the presented solutions by Narayan et al.", "(2000) and Faghei (2011).", "Similar to Narayan et al.", "(2000), we will discuss the effects of convection on angular momentum and energy equations.", "The paper is organized as follow.", "In section 2, the basic equations of constructing a model for quasi-spherical magnetized RIAFs with convection will be defined.", "In section 3, a self-similar method for solving equations which govern the behavior of the accreting gas was utilized.", "The summary of the model will appear in section 4." ], [ "Basic Equations", "Analytical theory of CDAF is based on a self-similar solution of a simplified set of equations describing RIAFs.", "We adopted the presented solutions by Narayan et al.", "(2000) and Faghei (2011).", "By using spherical coordinate ($r$ , $\\theta $ , $\\varphi $ ) centered on a accreting object, let us consider stationary, axisymmetric, quasi-spherical equations describing an accretion flow onto the black hole of mass $M$ .", "For the sake of simplicity, the general-relativistic effect has been neglected and the gravitational force on a fluid is characterized by Newtonian potential of a point mass, $\\psi =-G M / r$ .", "As magnetic fields, we consider only toroidal fields, $B_\\varphi $ .", "Under these assumptions, the continuity equation with mass loss is $\\frac{1}{r^{2}}\\frac{d}{d r}(r^{2}\\rho v_{r})=\\dot{\\rho },$ where $\\rho $ , $v_r$ and $\\dot{\\rho }$ are the density, the accretion velocity ($v_r < 0$ ), and the mass-loss per unit volume, respectively.", "The radial momentum equation is $v_{r}\\frac{d v_{r}}{d r} =r \\left( \\Omega ^{2}- \\Omega ^{2}_K\\right)-\\frac{1}{\\rho }\\frac{d }{d r}(\\rho c^2_s)-\\frac{c^2_A}{r}-\\frac{1}{2 \\rho }\\frac{d }{d r}(\\rho c^2_A),$ where $c_s$ is sound speed, which is defined as $c_s^2\\equiv p_{gas} / \\rho $ , with being $p_{gas}$ as the gas pressure, $\\Omega $ is the angular velocity, $\\Omega _K [=\\left( G M / r^3\\right)^{1/2}]$ is the Keplerian angular velocity, and $c_A$ is the alfven speed, which is defined as $c_A^2\\equiv B_\\varphi ^2 / 4 \\pi \\rho = 2 p_{mag} / \\rho $ , with being $p_{mag}$ as the magnetic pressure.", "The ram-pressure term $v_r d v_r / d r$ and last two terms due to the magnetic field in this equation were ignored in the self-similar CDAF model of Narayan et al.", "(2000), while we include them here in order to consider their effects.", "The angular momentum equations can be written in the form of the balance of advection and diffusion transport terms (Narayan et al.", "2000), $\\nonumber \\rho v_{r}\\frac{d}{d r}(r^{2}\\Omega )=\\frac{1}{r^{2}}\\frac{d}{d r}\\left[\\nu \\rho r^{4}\\frac{d \\Omega }{\\partial r}\\right]+ ~~~~~~~~~~\\\\\\frac{1}{r^{2}}\\frac{d}{d r}\\left[\\nu _{c} \\rho r^{(5+3 g)/2}\\frac{d }{d r} \\left( \\Omega r^{3(1-g)/2} \\right)\\right],$ where the two terms of right hand side represent the angular momentum transport by viscosity and convection.", "Here, $\\nu $ is the kinematic viscosity coefficient, $\\nu _c$ is the convective diffusion coefficient, and $g$ is the parameter to determine the condition of convective angular momentum transport.", "When $g=1$ , the flux of angular momentum due to convection is $\\dot{J}_c=-\\nu _{c}\\,\\rho \\,r^4\\,\\frac{d \\Omega }{d r}.$ The above equation implies that the convective angular momentum flux is oriented down the angular velocity gradient.", "For a quasi-Keplerian angular velocity, $\\Omega \\propto r^{-3/2}$ , angular momentum is transported outward.", "When $g=-1/3$ , the convective angular momentum flux can be written as $\\dot{J}_c=-\\nu _{c}\\,\\rho \\,r^2\\,\\frac{d \\left( \\Omega r^2 \\right)}{d r}.$ This equation represents that the convective angular momentum flux is oriented down the specific angular momentum gradient.", "For a quasi-Keplerian angular velocity, $\\Omega \\propto r^{-3/2}$ , angular momentum is transported inward.", "Generally, convection transports angular momentum inward (or outward) for $g<0$ (or $>0$ ), and the specific case $g=0$ corresponds to zero angular momentum transport (Narayan et al.", "2000).", "In this paper, we assume the kinematic coefficient of viscosity and the magnetic diffusivity due to turbulence in the accretion flow.", "So, we use these parameters in analogy to the $\\alpha $ -prescription of Shakura & Sunyaev (1973) for the turbulent, $\\nu = P_m \\eta = \\alpha \\frac{c_s^2}{\\Omega _{K}},$ where $P_m$ is the magnetic Prandtl number of the turbulence, which assumed to be a constant less than unity, $\\eta $ is the magnetic diffusivity, and $\\alpha $ is a free parameter less than unity.", "For the convective diffusion coefficient, $\\nu _c$ , we adopt the assumptions of Narayan et al.", "(2000) and Lu et al.", "(2004) that all transport phenomena due to convection have the same diffusion coefficient, which is defined as $\\nu _c=\\left(\\frac{L_M^2}{4}\\right)\\sqrt{-N_{eff}^2},$ where $L_M$ is the characteristic mixing length and $N_{eff}$ is the effective frequency of convective blobs.", "The characteristic mixing length $L_M$ in terms of the pressure scale height, $H_p$ , can be written as $L_M=2^{-1/4} l_M H_p, ~~~~ H_p=-\\frac{d r}{d \\ln p_{gas}},$ where $l_M$ is the dimensionless mixing-length parameter and its amount is estimated to be equal to $\\sqrt{2}$ in ADAFs (Narayan et al.", "2000; Lu et al.", "2004).", "the effective frequency of convective blobs, $N_{eff}$ , is given by $N_{eff}^2=N^2+\\kappa ^2,$ where $N$ is Brunt-Väisälä frequency, which is defined as $N^2=-{1\\over \\rho }{dp_{gas}\\over dr}{d\\over dr}\\ln \\left({p_{gas}^{1/\\gamma }\\over \\rho }\\right),$ and $\\kappa $ is epicyclic frequency, which is defined as $\\kappa ^2=2 \\Omega ^2 \\frac{d \\ln (\\Omega r^2)}{d \\ln r}.$ For a non-Keplerian flows $\\kappa \\ne \\Omega $ , while for a quasi-Keplerian ($\\Omega \\propto r^{-3/2}$ ), $\\kappa = \\Omega $ (Narayan et al.", "2000; Lu et al.", "2004).", "Convection appears in flows with $N_{eff}^2 < 0$ .", "We also write the convective diffusion coefficient in the form similar to usual viscosity of Shakura & Sunyaev (1973), $\\nu _c=\\alpha _c \\frac{c_s^2}{\\Omega _K}$ where $\\alpha _c$ is a dimensionless coefficient that describes the strength of convective diffusion.", "The $\\alpha _c$ coefficient can be obtained by equations (8) and (13) $\\alpha _c =\\frac{\\Omega _K}{c_s^2} \\left(\\frac{L_M^2}{4}\\right)\\sqrt{-N_{eff}^2}.$ The energy equation is $\\nonumber \\rho v_r T \\frac{d s}{d r}\\equiv \\rho v_r \\left[ \\frac{1}{\\gamma -1}\\frac{d c_s^2}{d r} - \\frac{c_s^2}{\\rho } \\frac{d \\rho }{d r} \\right]= \\\\Q_{diss}+Q_{conv}-Q_{rad},$ where $T$ is the temperature, $s$ is the specific entropy, $\\gamma $ is the ratio of specific heats, $Q_{diss}$ is dissipative heating rate, $Q_{rad}$ is the radiative cooling rate, and $Q_{conv}=-\\mathbf {\\nabla }\\cdot \\mathbf {F}_{conv}$ , with being $F_{conv}[=-\\rho \\nu _c T d s/d r]$ as the outward energy flux due to convection.", "For the right hand side of the energy equation, we can write $Q_{adv}=f Q_{diss}-\\frac{1}{r^2} \\frac{d}{dr}\\left(r^2 F_{conv} \\right),$ where $Q_{adv}$ is the advective transport of energy, and $f [= 1 - Q_{rad}/Q_{diss}]$ is the advection parameter.", "The parameter $f$ measures the degree to which the flow is advection-dominated (Narayan & Yi 1994).", "The dissipative heating rate can be written as $Q_{diss}=(\\nu +g \\nu _c)\\rho r^2 \\left(\\frac{\\partial \\Omega }{\\partial r}\\right)^2+\\frac{\\eta }{4\\pi } {\\bf J}^2,$ where the right-hand side terms are heating rate due to viscosity, convection, and resistivity, respectively.", "In above equation, ${\\mathbf {J}} [=\\nabla \\times {\\mathbf {B}}]$ is the current density, with being ${\\mathbf {B}}$ as the magnetic field.", "Finally, the induction equation with creation/escape of magnetic field can be written as $\\frac{1}{r}\\frac{d}{d r}\\left[r v_{r} B_{\\varphi }-\\eta \\frac{d}{d r}(r B_{\\varphi })\\right]=\\dot{B}_{\\varphi }.$ where $B_{\\varphi }$ is the toroidal component of magnetic field and $\\dot{B}_{\\varphi }$ is the field escaping/creating rate due to a magnetic instability or dynamo effect.", "This induction equation is rewritten as $\\nonumber \\dot{B}_{\\varphi }= ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\\frac{1}{r}\\frac{d}{d r}\\left[\\sqrt{4\\pi \\rho c^2_A}\\left(r v_{r} -\\frac{\\alpha }{4 \\chi P_m }\\frac{1}{ r \\rho \\Omega _K}\\frac{d}{d r}(r^2 \\rho c^2_A )\\right)\\right],$ where $\\chi $ is the ratio of the magnetic pressure to the gas pressure, which is defined by $\\chi =\\frac{p_{mag}}{p_{gas}}=\\frac{1}{2} \\left(\\frac{c_A}{c_s}\\right)^2.$" ], [ "Self-Similar Solutions", "We seek self-similar solutions in the following form (e.g.", "Narayan & Yi 1994; Akizuki & Fukue 2006) $v_r(r)=-c_1 \\alpha \\sqrt{\\frac{G M_*}{r}}$ $\\Omega (r)=c_2\\sqrt{\\frac{G M_*}{r^3}}$ $c^2_s(r)=c_3\\frac{G M_*}{r}$ $c^2_A(r)=\\frac{B^2_{\\varphi }}{4\\pi \\rho }=2 \\chi c_3\\frac{G M_*}{r}$ where $c_1$ , $c_2$ , and $c_3$ are dimensionless constant to be determined.", "We use a power-law relation for density $\\rho (r)=\\rho _0 r^\\lambda ,$ where $\\rho _0$ and $\\lambda $ are constant.", "Using equations (20)-(24), the mass-loss rate and the magnetic field escaping/creating rate can be written as $\\dot{\\rho }(r)=\\dot{\\rho }_0 r^{\\lambda -3/2},$ $\\dot{B}_{\\varphi }(r)=\\dot{B}_0 r^{\\frac{\\lambda -4}{2}},$ where $\\dot{\\rho }_0$ and $\\dot{B}_0$ are constant.", "Since we have not applied the effects of wind in the momentum and energy equations, we will assume a no wind case, $\\dot{\\rho }=0$ and $\\lambda =-3/2$ .", "In this case, $\\dot{B}_{\\varphi }\\propto r^{-11/4}$ , which implies that creation/escape of magnetic field increases with approaching to central object.", "This property is qualitatively consistent with previous studies of accretion flows (Machida et al.", "2006; Oda et al.", "2007; Faghei & Mollatayefeh 2012).", "Using the self-similar solutions in the continuity, radial momentum, angular momentum, convection parameter, energy, and induction equations [(1)-(3), (13), (14), and (18)], we can obtain the following relations: $\\dot{\\rho }_0=-\\left(\\lambda +\\frac{3}{2}\\right) \\alpha \\rho _0 c_1 \\sqrt{G M_*},$ $-\\frac{1}{2}c^2_1 \\alpha ^2 + 1 - c^2_2 + c_3 \\left[\\lambda -1+\\chi (1+\\lambda )\\right]=0 ,$ $\\alpha c_1 = 3 (\\alpha + g \\alpha _c) (\\lambda +2) c_3 ,$ $\\nonumber \\alpha c_1 \\left[\\frac{1}{\\gamma -1}+\\lambda \\right]=~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\\nonumber \\frac{9}{4}\\alpha f \\left[(1+\\frac{\\alpha _c}{\\alpha }g) c_2^2+\\frac{2\\chi }{9 P_m} c_3 (1+\\lambda )^2\\right]\\\\-\\alpha _c c_3 (\\lambda +\\frac{1}{2}) \\left[\\frac{1}{\\gamma -1}+\\lambda \\right]$ $\\alpha _c= \\frac{l_M^2}{4\\sqrt{2} c_3 (\\lambda -1)^2} \\sqrt{\\frac{c_3 (\\lambda -1)}{\\gamma } [\\lambda (1-\\gamma )-1]-c_2^2} ,$ $\\dot{B}_0=-\\frac{\\alpha \\lambda }{2} G M_* \\sqrt{2\\pi \\rho _0\\chi c_3}\\left[2 c_1 + \\frac{c_3}{P_m} (1+\\lambda ) \\right].$ Above equations express for $\\lambda =-3/2$ , there is no mass loss, while for $\\lambda > -3/2$ mass loss (wind) exists." ], [ "Results", "Here, similar to Zhang & Dai (2008), we will study the presence of convection in two cases: $\\alpha _c$ as a free parameter and $\\alpha _c$ as a variable.", "Figure: Physical variables as functions of χ\\chi for several values of convective viscosity.", "The input parameters are set toα=0.2\\alpha =0.2, γ=1.5\\gamma =1.5, P m =1/2P_m=1/2, f=1f=1, l=2l=\\sqrt{2}, g=-1/3g=-1/3, and λ=-3/2\\lambda =-3/2.", "The solid, dashed, and dotted lines representα c =0\\alpha _c=0, 0.050.05, and 0.10.1, respectively.Figure: Same as Figure 1, but α c =0.1\\alpha _c=0.1, and the solid, dashed, and dotted lines representP m =∞P_m=\\infty , 1.01.0, and 0.50.5, respectively.Figure: Physical variables as functions of χ\\chi for several values of magnetic Prandtl number.The input parameters are set toα=0.5\\alpha =0.5, γ=1.5\\gamma =1.5, f=1f=1, l=2l=\\sqrt{2}, g=-1/3g=-1/3, and λ=-3/2\\lambda =-3/2.", "The solid, dashed, and dotted lines representP m =∞P_m=\\infty , 1.01.0, and 0.50.5, respectively." ], [ "In this case, we take the convective coefficient $\\alpha _c$ as a free parameter to discuss the effects of convection for simplicity.", "Examples of such solutions are presented in Figures 1 and 2.", "In Figure 1, the self-similar coefficients $c_1$ , $c_2$ , and $c_3$ are shown as functions of the parameter $\\chi $ .", "By adding the parameter $\\chi $ which indicates the role of magnetic filed on the dynamics of accretion discs, we see the coefficients of radial and rotational velocities and sound speed decrease.", "This properties are qualitatively consistent with results of Faghei (2011).", "In Figure 1, we also studied the effect of convection parameter $\\alpha _c$ on the physical variables.", "The value of $\\alpha _c$ measures the strength of convective viscosity and a larger $\\alpha _c$ denotes a stronger turbulence due to convection.", "Figure 1 implies that for non-zero $\\alpha _c$ , the radial infall velocity is lower than the standard ADAF solution and for larger $\\alpha _c$ this reduction of radial infall velocity is more evident.", "It can be due to decrease of efficiency of angular momentum transport by adding the convection parameter $\\alpha _c$ (see equation 29).", "The profiles of angular velocity show that it decreases with the magnitude of $\\alpha _c$ , while the sound speed increases.", "These properties are in accord with results of Zhang & Dai (2008).", "In Figure 2, the physical variables are shown as functions of parameter $\\chi $ for several values of magnetic Prandtl number.", "Since inverse of magnetic Prandtl number is proportional to magnetic diffusivity, $P_m \\propto \\eta ^{-1}$ .", "Thus, reduce of magnetic Prandtl number denotes to increase of resistivity of the fluid.", "The solutions in Figure 2 imply that the accretion velocity and the sound speed both increase with the magnitude of resistivity, while the rotational velocity decreases.", "These properties qualitatively confirm the results of Faghei (2011)." ], [ "Here, we calculate the dimensionless coefficient $\\alpha _c$ by using the mixing-length theory.", "Because we used a steady self-similar method to derive $\\alpha _c$ , it becomes a constant throughout of the accreting gas.", "However, it is a function of position and time (e. g. Lu et al.", "2004).", "The amount of convection parameter $\\alpha _c$ is calculated by equation (31).", "Using this equation and equation (28)-(30), we can obtain the behavior of physical quantities in the presence of convection.", "Such solutions are shown in Figure 3.", "In Figure 3, the coefficients $c_1$ , $c_2$ , $c_3$ , and convection parameter $\\alpha _c$ are shown as functions of the degree of magnetic pressure.", "Similar to case 1, the accretion and rotational velocities, and sound speed decrease by adding the parameter $\\chi $ .", "While, the convection parameter $\\alpha _c$ increases for stronger toroidal magnetic field.", "This property is qualitatively consistent with result of Zhang & Dai (2008).", "In Figure 3, the physical variables are also studied for several values of magnetic Prandtl number.", "The profiles of convection parameter $\\alpha _c$ imply that it increases by adding the magnetic diffusivity.", "As for non-zero magnetic diffusivity, $\\alpha _c$ is larger than the standard CDAF solution and for larger magnetic diffusivity this increase of convection parameter $\\alpha _c$ is more evident." ], [ "Summary and Discussion", "The observational features of low-luminosity state of X-ray binaries and nuclei of galaxies can be successfully explained by the models of radiatively inefficient accretion flow (RIAF).", "The importance of convection in RIAFs was realized by semi-analytical and direct numerical simulation (e. g. Narayan et al.", "2000; Igumenshchev et al.", "2003).", "In this research, we considered the effects of convection on the presented model of Faghei (2011).", "Similar to Narayan et al.", "(2000), we assumed the convection affects on transports of angular momentum and energy.", "Using a radially self-similar approach, we studied the effects of convection on the model for several values of magnetic field and resistivity.", "The solutions showed that the accretion and rotational velocities, and sound speed decrease for stronger magnetic filed.", "Moreover, we found that the accretion velocity and sound speed increase with the magnitude of the resistivity, while the rotational velocity decreased.", "These properties are qualitatively consistent with results of Faghei (2011).", "We studied the effects of convection on a resistive and magnetized RIAF in two cases: assuming the convection parameter as a free parameter and using mixing length theory to calculate the convection parameter.", "In the first case, we found that by adding the convection parameter, the radial and rotational velocities decrease and the sound speed increases.", "In the second case, we found that the convection parameter increases by adding the magnetic filed and resistivity.", "These properties are in many aspects in accord with results of Zhang & Dai (2008).", "The present model have some limitations that can be modified in the future works.", "For example, the latitudinal dependence of physical variables have been ignored in this paper.", "While, two-dimensional and three-dimensional MHD simulations of RIAFs show that the disc geometry strongly depends on magnetic field configuration (e. g. Igumenshchev et al.", "2003).", "Thus, the study of present model in two/three dimensions can be an interesting subject for future research.", "Moreover, it has been understood the magnetic field can change the criterion for convective instability (e. g. Balbus & Hawley 2002).", "While, we igonred the effects of magnetic field on the instability criterion.", "Thus, the presented criterion in this paper can be modified in the future research." ], [ "Acknowledgements", "I wish to thank the anonymous referee for very useful comments that helped us to improve the initial version of the paper." ] ]

[ [ "Modulated reheating by curvaton" ], [ "Abstract There might be a light scalar field during inflation which is not responsible for the accelerating inflationary expansion.", "Then, its quantum fluctuation is stretched during inflation.", "This scalar field could be a curvaton, if it decays at a late time.", "In addition, if the inflaton decay rate depends on the light scalar field expectation value by interactions between them, density perturbations could be generated by the quantum fluctuation of the light field when the inflaton decays.", "This is modulated reheating mechanism.", "We study curvature perturbation in models where a light scalar field does not only play a role of curvaton but also induce modulated reheating at the inflaton decay.", "We calculate the non-linearity parameters as well as the scalar spectral index and the tensor-to-scalar ratio.", "We find that there is a parameter region where non-linearity parameters are also significantly enhanced by the cancellation between the modulated effect and the curvaton contribution.", "For the simple quadratic potential model of both inflaton and curvaton, both tensor-to-scalar ratio and nonlinearity parameters could be simultaneously large." ], [ "Introduction", "Cosmic inflation solves various problems in the standard Big Bang cosmology [1] and simultaneously provides the seed of large scale structure in our Universe from the quantum fluctuation of a light scalar field, e.g., inflaton field $\\phi $  [2].", "A single field inflation model predicts the density perturbation which is nearly scale-invariant and almost Gaussian.", "In other words, the scalar spectral index $n_s$ is close to unity and a non-linearity parameter $f_{\\rm NL}$ is much less than unity [3].", "This is consistent with the current limit on the local type non-linearity parameter $f_{\\rm NL}$ from the Wilkinson Microwave Anisotropy Probe (WMAP) seven-year data, $-10<f_{\\rm NL}<74$ at the 95% confidence level [4].", "However, besides canonical single field slow-roll inflation models, there are many possible mechanisms to generate density perturbation.", "By WMAP data, the scalar spectral index $n_s$ has been measured with a good accuracy, while the non-linearity parameters have been just weakly constrained as above.", "The sensitivity of the Planck satellite [5] to measure non-Gaussianity is as good as to probe $f_{\\rm NL}$ of ${\\cal O}(1)$ .", "The non-Gaussianity could be an important observable to discriminate between various mechanisms of density perturbation generation.", "For example, multi-field inflation models can show large non-Gaussianity with special conditions during inflation [6], [7], [8], [9], [10], [11], at the end of inflation [12], [13], [14], [15], [16], preheating [17], or deep in the radiation dominated era [18].", "The last case includes the “curvaton” scenario [19], [20], [21], [22], [23].", "A light scalar field, curvaton, has too little potential energy to drive inflationary expansion during inflation.", "At a later time when a curvaton decays, the isocurvature perturbation of the curvaton field becomes adiabatic or mixed with that from the inflaton field.", "If the curvaton energy density is subdominant at its decay time, the large non-Gaussianity is generated in general [24].", "Since an inflaton also generates density perturbations, inflaton and curvaton contributions to density perturbation could be comparable.", "This mixed inflaton-curvaton scenario has been also studied intensively [25].", "The quantum fluctuation of a subdominant light scalar field during inflation can modulate the efficiency of reheating by the inflaton decay [26], [27].", "This makes the reheating a spatially inhomogeneous process.", "The quasi-scale invariant perturbations of this field, which are isocurvature modes during inflation, may be converted into the primordial curvature perturbation during this process.", "Large non-Gaussianity also can be induced from modulated reheating [28], [29], [30], [31].", "For a review of modulated reheating after inflation, see for example Ref. [32].", "Again, in general, the density fluctuation can be generated by both inflaton and modulated reheating.", "Mixed inflaton-modulated reheating scenario also has been investigated [31].", "Here one may realize that a curvaton is a light scalar field and thus naturally can play the role of a scalar field which modulates the reheating by the inflaton decay.", "The inflaton field may have a coupling with a curvaton if that is small enough not to disturb dynamics of both an inflaton and a curvaton.", "Nevertheless this interaction modifies the decay rate of the inflaton field.", "Since the decay rate of inflaton becomes a function of the local value of a curvaton field $\\sigma ({\\bf x})$ , this gives rise to a perturbation in the decay rate of the inflaton field and thus in the reheating temperature which is responsible for the density perturbation after reheating.", "So far few attention has been paid to couplings between the inflaton and the curvaton [33], compared to self-interaction of curvatons [34], [35].", "In this study, we incorporate the modulated reheating effects in the curvaton scenario, taking the perturbation generated from the inflaton field also into account.", "The paper is organized as follows.", "After describing the model and its dynamics in section , we consider the curvature perturbation and calculate the power spectrum, the tensor-to-scalar ratio and non-linearity parameters in section .", "In section  we work on simple models and show how the modulated reheating effect by curvaton affects the density perturbation and parameter space of models can be constrained.", "We summarize our results in section ." ], [ "Dynamics", "Inflation is driven by the potential energy of the inflaton field, $\\phi $ .", "We assume that during inflation interaction terms of inflaton with other fields are negligible.", "However after inflation, the inflaton starts oscillating around the minimum and finally, via the interaction terms, decays into the standard model (SM) particles, which makes the hot thermal plasma in the standard Big Bang cosmology.", "One of the relevant terms for the decay of inflaton field including curvaton field $\\sigma $ could be given by ${\\cal L}_{\\rm int} = \\lambda |\\Phi |^2 \\phi \\sigma ,$ where $\\Phi $ is another light scalar field such as the SM Higgs field.", "Then, in the classical background of curvaton field, Eq.", "(REF ) induces the decay of inflaton into two Higgs scalars, with the curvaton expectation value dependent (CD) decay width $\\Gamma _{\\phi }^{CD}(\\sigma ) = \\frac{1}{8 \\pi m_{\\phi } } \\lambda ^2 \\sigma ^2 ,$ with $m_{\\phi }$ being the inflaton mass at the minimum.", "Together with the other curvaton independent interactions such as ${\\cal L} = {\\cal O}_{\\rm SM} \\frac{\\phi }{M_P} ,$ where ${\\cal O}_{\\rm SM}$ denotes a SM operator, induce the curvaton independent (CI) decay width of the inflaton, $\\Gamma _{\\phi }^{CI} $ .", "The total decay rate of the inflaton is given by $\\Gamma _{\\phi } (\\sigma )= \\Gamma _{\\phi }^{CI} + \\Gamma _{\\phi }^{CD}(\\sigma ) .$ For the very light curvaton field, $ \\Gamma _{\\phi } > m_{\\sigma }$ , with $m_\\sigma $ being curvaton mass, the curvaton starts to oscillate in the radiation-dominated epoch well after the inflaton decays, when the Hubble parameter becomes as small as $m_{\\sigma }$ .", "After the reheating by the inflaton $\\phi $ is completed, the energy density of the radiation from inflaton decay decreases as $\\rho _r = 3 M_P^2\\Gamma _{\\phi }^2 \\left(\\frac{a_{\\Gamma _{\\phi }}}{a}\\right)^4, $ where $a_{\\Gamma _{\\phi }}$ is a scale factor when inflaton decays, i.e.", "$H=\\Gamma _\\phi $ .", "The energy density of the curvaton after the onset of the oscillations decreases as $\\rho _{\\sigma } = \\frac{1}{2} m_{\\sigma }^2 \\sigma _*^2 \\left(\\frac{a_{m_{\\sigma }}}{a}\\right)^3,$ with $\\sigma _*$ being the expectation value of curvaton during inflation and $a_{m_{\\sigma }}$ is a scale factor when the curvaton start oscillation at $H=m_\\sigma $ .", "The curvaton decays at a later time and we call its decay rate $\\Gamma _{\\sigma }$ .", "Whether the Universe is curvaton dominated or radiation dominated at the moment of curvaton decay depends on the size of $\\Gamma _{\\sigma }$ and $\\sigma _*$ ." ], [ "Primordial curvature perturbation", "We consider that inflaton $\\phi $ and curvaton $\\sigma $ fields are relevant to the density perturbation in the early Universe.", "Their vacuum fluctuations are promoted to a classical perturbation around the time of horizon exit.", "During inflation the field trajectory is dominated by the inflaton field and thus the inflaton perturbation becomes adiabatic and that of the curvaton contributes to the isocurvature mode." ], [ "Modulated reheating from an interaction with a curvaton", "Reheating of the Universe is attained from the decay of the inflaton field $\\phi $ .", "Since the inflaton decay is modulated by the curvaton field $\\sigma $ , the curvature perturbation of the radiation produced from the decay of inflaton has two origins.", "One comes from the inflaton field itself in the standard picture of the generation of fluctuations.", "The other comes from the light scalar field (curvaton) $\\sigma $ during the reheating process due to the interaction between the inflaton and the curvaton field.", "Thus, as in the inflaton-modulated reheating mixed scenario, it is written [28], [31] as $\\zeta _r&=& \\frac{1}{M_P^2}\\frac{V}{V_\\phi }\\delta \\phi _* + \\frac{1}{2M_P^2} \\left( 1- \\frac{VV_{\\phi \\phi }}{V_\\phi ^2} \\right)\\delta \\phi _*^2+ \\frac{1}{6 M_P^2}\\left( -\\frac{V_{\\phi \\phi }}{V_{\\phi }} -\\frac{V V_{\\phi \\phi \\phi }}{V_{\\phi }^2}+2\\frac{V V_{\\phi \\phi }^2}{V_{\\phi }^3} \\right) \\delta \\phi _*^3 \\nonumber \\\\&& +Q_\\sigma \\delta \\sigma _* + \\frac{1}{2}Q_{\\sigma \\sigma }\\delta \\sigma _*^2+ \\frac{1}{6}Q_{\\sigma \\sigma \\sigma }\\delta \\sigma _*^3 +\\cdots ,$ where $Q$ is a function of $\\Gamma _\\phi (\\sigma )/H_c$ calculated at a time $t_c$ which is after several oscillations of the inflaton but well before the time of decay of inflaton.", "A quantity with $*$ is evaluated when the corresponding scale crosses the Hubble horizon during inflation.", "During inflation the energy density of $\\sigma $ field is negligible and the inflation is driven by the inflaton field $\\phi $ alone.", "The slow-roll parameters during inflation are defined by $\\begin{split}\\epsilon \\equiv \\frac{M_P^2}{2}{\\left(\\frac{V_\\phi }{V} \\right) }^2,\\qquad \\eta \\equiv M_P^2 \\frac{V_{\\phi \\phi }}{V},\\qquad \\xi ^2 \\equiv M_P^4 \\frac{V_\\phi V_{\\phi \\phi \\phi }}{V^2}.\\end{split}$ Using Eq.", "(REF ), the curvature perturbation of radiation $\\zeta _r$ is expressed as $\\begin{split}\\zeta _r= \\zeta _{r1} + \\frac{1}{2}\\zeta _{r2} + \\frac{1}{6}\\zeta _{r3} +\\cdots ,\\end{split}$ with $\\zeta _{r1} &=& \\frac{1}{M_P\\sqrt{2\\epsilon _*}} \\delta \\phi _* +Q_\\sigma \\delta \\sigma _* , \\\\\\zeta _{r2} &=& \\frac{1}{M_P^2} \\left( 1- \\frac{\\eta _*}{2\\epsilon _*} \\right)\\delta \\phi _*^2 + Q_{\\sigma \\sigma } \\delta \\sigma _*^2 , \\\\\\zeta _{r3} &=& \\frac{1}{ M_P^3 \\sqrt{2\\epsilon _*}}\\left( -\\eta - \\frac{\\xi ^2}{2 \\epsilon } + \\frac{\\eta ^2}{\\epsilon } \\right)\\delta \\phi _*^3+ Q_{\\sigma \\sigma \\sigma } \\delta \\sigma _*^3 .$" ], [ "After curvaton decay", "After inflaton decay, the energy density of radiation decreases, however the curvaton energy density stays the same for a while and starts to decrease when the mass of curvaton becomes larger than the Hubble expansion.", "Since the energy density of oscillating curvaton decreases slower than that of radiation, the curvaton becomes important well after the decay of inflaton.", "When the decay rate of curvaton $\\Gamma _\\sigma $ becomes comparable to the Hubble expansion rate, the curvaton $\\sigma $ decays quickly to radiation.", "After the decay of the curvaton, the remnant radiation is a mixture from the inflaton and curvaton decay products with different density perturbations.", "In this inflaton-curvaton mixed scenario, the curvature perturbation after the curvaton decay can be expressed as analytically with instant decay approximation by [36], [37], [38], $\\zeta = \\zeta _1+\\frac{1}{2}\\zeta _2+\\frac{1}{6}\\zeta _3+\\ldots ,$ where $\\begin{split}\\zeta _1=&(1-R)\\zeta _{r1} +R\\zeta _{\\sigma 1},\\\\\\zeta _2=&(1-R)\\zeta _{r2} +R\\zeta _{\\sigma 2}+R(1-R)(3+R)\\left(\\zeta _{r1}-\\zeta _{\\sigma 1}\\right)^2,\\\\\\zeta _3=&(1-R)\\zeta _{r3} +R\\zeta _{\\sigma 3}+3R(1-R)(3+R)\\left(\\zeta _{r1}-\\zeta _{\\sigma 1}\\right)\\left(\\zeta _{r2}-\\zeta _{\\sigma 2}\\right)\\\\+& R(1-R)(3+R)(-3+4 R +3 R^2)\\left(\\zeta _{r1}-\\zeta _{\\sigma 1}\\right)^3,\\end{split}$ with $R \\equiv \\left.\\frac{3 \\rho _{\\sigma } }{4 \\rho _r + 3 \\rho _{\\sigma }}\\right|_{H=\\Gamma _{\\sigma }} .", "$ Here $\\zeta _{r}$ is given in Eq.", "(REF ) and $\\zeta _{\\sigma }$ is the curvature perturbation from the curvaton field.", "$R$ parametrizes the dominance of the curvaton energy density when it decays.", "For the quadratic potential of curvaton, $\\zeta _\\sigma $ is given by [38] $\\zeta _\\sigma &=& \\zeta _{\\sigma 1} +\\frac{1}{2}\\zeta _{\\sigma 2} +\\frac{1}{6}\\zeta _{\\sigma 3} + \\cdots \\nonumber \\\\&=& \\frac{2}{3} \\frac{\\delta \\sigma _*}{\\sigma _*}- \\frac{1}{3} {\\left(\\frac{\\delta \\sigma _*}{\\sigma _*} \\right) }^2+\\frac{2}{9}{\\left(\\frac{\\delta \\sigma _*}{\\sigma _*} \\right) }^3 +\\cdots .$ Then from Eq.", "(REF ) with Eqs.", "(REF ) - (REF ) we obtain $\\zeta _1 &=& \\frac{1-R}{M_P\\sqrt{2\\epsilon _*}} \\delta \\phi _* +\\left( (1-R)Q_\\sigma + \\frac{2R}{3\\sigma _*} \\right) \\delta \\sigma _*, \\\\\\zeta _2 &=& (1-R)\\left\\lbrace \\frac{1}{M_P^2} \\left( 1- \\frac{\\eta _*}{2\\epsilon _*} \\right)\\delta \\phi _*^2+ Q_{\\sigma \\sigma } \\delta \\sigma _*^2\\right\\rbrace -\\frac{2R}{3}{\\left(\\frac{\\delta \\sigma _*}{\\sigma _*} \\right) }^2 \\nonumber \\\\&& +R(1-R)(3+R)\\left( \\frac{\\delta \\phi _* }{M_P\\sqrt{2\\epsilon _*}}+ Q_{\\sigma } \\delta \\sigma _* - \\frac{2}{3} \\frac{\\delta \\sigma _*}{\\sigma _*}\\right)^2,\\\\\\zeta _3 &=& \\frac{1-R}{M_P^3\\sqrt{2\\epsilon _*}} \\left( -\\eta - \\frac{\\xi ^2}{2 \\epsilon } + \\frac{\\eta ^2}{\\epsilon } \\right)\\delta \\phi _*^3+(1-R)Q_{\\sigma \\sigma \\sigma } \\delta \\sigma _*^3+\\frac{4R}{3}{\\left(\\frac{\\delta \\sigma _*}{\\sigma _*} \\right) }^3 \\nonumber \\\\&&+3R(1-R)(3+R)\\left( \\frac{\\delta \\phi _* }{M_P\\sqrt{2\\epsilon _*}}+ Q_{\\sigma } \\delta \\sigma _* - \\frac{2}{3} \\frac{\\delta \\sigma _*}{\\sigma _*}\\right) \\nonumber \\\\&&\\times \\left\\lbrace \\frac{1}{M_P^2} \\left( 1- \\frac{\\eta _*}{2\\epsilon _*} \\right)\\delta \\phi _*^2 + Q_{\\sigma \\sigma } \\delta \\sigma _*^2+\\frac{2}{3}{\\left(\\frac{\\delta \\sigma _*}{\\sigma _*} \\right) }^2\\right\\rbrace \\nonumber \\\\&& + R(1-R)(3+R)(-3+4 R +3 R^2)\\left( \\frac{\\delta \\phi _* }{M_P\\sqrt{2\\epsilon _*}}+ Q_{\\sigma } \\delta \\sigma _* - \\frac{2}{3} \\frac{\\delta \\sigma _*}{\\sigma _*}\\right)^3.$" ], [ "The power spectrum", "The power spectrum ${\\cal P}_\\zeta $ of the curvature perturbation is defined by $\\begin{split}\\langle \\zeta (k_1)\\zeta (k_2) \\rangle =(2\\pi )^3 \\delta ({\\bf k}_{1}+{\\bf k}_{2} )\\frac{2\\pi ^2}{k^3}{\\cal P}_\\zeta (k_1) , \\end{split}$ and the perturbations of the fields at the horizon exit satisfy $\\begin{split}\\langle \\delta \\phi _*(k_1)\\delta \\phi _*(k_2) \\rangle &= (2\\pi )^3 \\delta ({\\bf k}_{1}+{\\bf k}_{2} )\\frac{2\\pi ^2}{k^3}{\\cal P}_{\\delta \\phi _*} (k_1),\\\\\\langle \\delta \\sigma _*(k_1)\\delta \\sigma _*(k_2) \\rangle &= (2\\pi )^3 \\delta ({\\bf k}_{1}+{\\bf k}_{2} )\\frac{2\\pi ^2}{k^3}{\\cal P}_{\\delta \\sigma _*} (k_1),\\\\\\langle \\delta \\phi _*(k_1)\\delta \\sigma _*(k_2) \\rangle &= 0 , \\end{split}$ with $\\begin{split}{\\cal P}_{\\delta \\phi _*} (k)= {\\cal P}_{\\delta \\sigma _*} (k)= {\\left(\\frac{H_*}{2\\pi } \\right) }^2, \\end{split}$ which is determined at around horizon exit.", "Using Eq.", "(REF ) and Eqs.", "(REF ) - (REF ) , the power spectrum of the curvature perturbation is given by $\\begin{split}{\\cal P}_{\\zeta } =& \\frac{(1-R)^2}{2M_P^2\\epsilon _*}{\\cal P}_{\\delta \\phi _*} +\\left[ (1-R)Q_\\sigma + \\frac{2R}{3\\sigma _*} \\right]^2{\\cal P}_{\\delta \\sigma _*} \\\\=&\\frac{1}{2M_P^2\\epsilon _*} {\\left(\\frac{H_*}{2\\pi } \\right) }^2 (1-R)^2(1+{\\tilde{r}}),\\end{split}$ with $\\begin{split}{\\tilde{r}}\\equiv \\frac{2M_P^2 \\epsilon _*}{9\\sigma _*^2(1-R)^2} \\left[ 3Q_\\sigma \\sigma _*(1-R) + 2R \\right]^2.\\end{split}$ Here $0\\le R \\le 1$ defined in Eq.", "(REF ) parametrizes the contribution of the curvaton.", "For $R=1$ , which means that the curvaton dominates the background energy density when it decays, the power spectrum is given by $\\begin{split}{\\cal P}_{\\zeta ,0} \\equiv {\\left(\\frac{H_*}{3\\pi \\sigma _*} \\right) }^2.\\end{split}$ Therefore the contribution to the power spectrum from the curvaton $\\sigma $ dynamics scales $R^2$ , while those from both the inflaton $\\phi $ and the modulated reheating effect through the $\\sigma $ field by inflaton $\\phi $ decay scale $(1-R)^2$ .", "As one can see, in the limit of $R \\rightarrow 0 $ , the usual curvaton contribution disappears and it corresponds to the inflaton-modulated mixed scenario.", "The opposite limit with $R \\rightarrow 1 $ corresponds to the pure curvaton scenario.", "Whereas the $R$ parametrizes the $\\sigma $ contribution as the curvaton compared with the other two, the $\\sigma $ contribution via the modulated inflaton decay is parametrized by $Q_{\\sigma }$ .", "The parameter ${\\tilde{r}}$ compares the contribution to the Power spectrum from the $\\sigma $ field to that from inflaton $\\phi $ .", "In the limit of ${\\tilde{r}}\\rightarrow 0$ , the Power spectrum comes solely from the inflaton, while in the limit of ${\\tilde{r}}\\gg 1$ the $\\sigma $ field contributes dominantly through the modulated effect and/or the curvaton effect with the inflaton effect suppressed.", "One should notice that because the latter two contributions come from the single same source $\\sigma $ , there is a cross term of two.", "This cancellation between the modulation effect and the curvaton effect makes non-trivial features in the Power spectrum.", "Both contributions may cancel each other when $3 Q_{\\sigma } \\sigma _* \\simeq -2R/(1-R)$ and the inflaton contribution dominates.", "To quantify the amount of the cancellation, we define $\\delta $ as $\\begin{split}\\delta \\equiv 1+ \\frac{3Q_\\sigma \\sigma _*(1-R)}{2R}.", "\\end{split}$ This $\\delta $ is a measure of fine tuning of the cancellation and becomes $\\delta =0$ for the exact cancellation.", "The contour plot of $\\delta $ is shown in figure REF .", "With this, ${\\tilde{r}}$ can be written as $\\begin{split}{\\tilde{r}}= \\frac{8M_P^2 \\epsilon _*R^2}{9\\sigma _*^2(1-R)^2} \\delta ^2.", "\\end{split}$ Figure: The contour plot of δ\\delta defined in Eq. ().", "The cancellation happens along the line of δ=0\\delta =0.From Eq.", "(REF ), ignoring the negligible contribution from the curvature of the curveton potential $V_{\\sigma \\sigma }$ , the scalar spectral index $n_s$ is given by $n_s -1 \\equiv \\frac{d {\\cal P}_{\\zeta }}{d \\ln k} = -2\\epsilon _* +\\frac{-4\\epsilon _*+2\\eta _*}{1+\\tilde{r}} ,$ which we normalize to be $0.97$ through in our analysis.", "The tensor-to-scalar ratio is given by $r_T \\equiv \\frac{{\\cal P}_T}{{\\cal P}_{\\zeta }} = \\frac{16 \\epsilon }{(1-R)^2 (1+\\tilde{r})},$ where we used ${\\cal P}_T=8(H_*/2\\pi )^2$ .", "As you can see here, for small ${\\tilde{r}}$ , the observational limit $r_T< 0.36$ constrains the value of $R$ to be $R< 1-\\sqrt{16\\epsilon _*/0.36}\\simeq 0.53$ .", "In figure REF , we show the contour plot of ${\\tilde{r}}$ for $\\sigma _*=0.05\\, M_P$ (which we will later call Case B) with $\\epsilon _*\\simeq 0.005$ and $\\eta _*=0$ .", "There is a cancellation between curvaton effects and modulated effects around the dashed line (blue) which connect $(R,Q_\\sigma \\sigma _*)=(0.6,-1)$ and $(0,0)$ , where ${\\tilde{r}}$ vanishes.", "In this small ${\\tilde{r}}$ limit the inflaton contribution dominates the power spectrum.", "In the opposite region with a large ${\\tilde{r}}$ , the $\\sigma $ field dominates the Power spectrum.", "For different values of $\\sigma _*$ , the magnitude scales as $\\sigma _*^{-2}$ , since $\\epsilon _*$ does not change much.", "Figure: [Left window] : The contour plot of r ˜{\\tilde{r}} for Case B, σ * =0.05M P \\sigma _*=0.05M_P.For the other cases the magnitudes are scaled by σ * 2 \\sigma _*^2.", "Along the blue dashed line r ˜=δ=0{\\tilde{r}}=\\delta =0 .", "[Right window] : The tensor-to-scalar ratio r T r_T for Case B.", "The blue shaded region is ruled out using the constraint r T <0.36r_T<0.36.", "δ=0\\delta =0 along the blue dashed line.Figure: The contour plots on R-Q σ σ * R- Q_{\\sigma }\\sigma _* plane,of the power spectrum 𝒫 ζ {\\cal P}_{\\zeta } normalized by the value at R=1R=1 (upper-left), f NL f_{\\rm NL} (upper-right), τ NL \\tau _{\\rm NL} (lower-left)and g NL g_{\\rm NL} (lower-right) for Case A, σ * =10 -3 M P \\sigma _*=10^{-3}\\, M_P.", "We put η * =ξ * =0\\eta _*=\\xi _*=0.The red shaded region corresponds to too large non-Gaussianity to be consistent with the observation, -10<f NL <73-10<f_{\\rm NL}<73, τ NL <10 4 \\tau _{\\rm NL}< 10^4, and |g NL |<10 5 |g_{\\rm NL}| < 10^5.", "Along the blue dashed line r ˜=0{\\tilde{r}}=0 the cancellation happens.Figure: The contour plots on R-Q σ σ * R- Q_{\\sigma }\\sigma _* plane,of the power spectrum 𝒫 ζ {\\cal P}_{\\zeta } normalized by the value at R=1R=1 (upper-left), f NL f_{\\rm NL} (upper-right), τ NL \\tau _{\\rm NL} (lower-left)and g NL g_{\\rm NL} (lower-right) for Case B, σ * =0.05M P \\sigma _*=0.05\\, M_P.", "We put η * =ξ * =0\\eta _*=\\xi _*=0.The blue shaded region corresponds to too large tensor-to-scalar ratio to be consistent with the observation, r T <0.36r_T<0.36.", "Along the blue dashed line δ=0\\delta =0.Figure: The contour plots on R-Q σ σ * R- Q_{\\sigma }\\sigma _* plane,of the power spectrum 𝒫 ζ {\\cal P}_{\\zeta } normalized by the value at R=1R=1 (upper-left), f NL f_{\\rm NL} (upper-right), τ NL \\tau _{\\rm NL} (lower-left)and g NL g_{\\rm NL} (lower-right) for Case C, σ * =0.5M P \\sigma _*=0.5\\, M_P.", "We put η * =ξ * =0\\eta _*=\\xi _*=0.The blue shaded region corresponds to too large tensor-to-scalar ratio to be consistent with the observation, r T <0.36r_T<0.36.", "Along the blue dashed line δ=0\\delta =0.In the upper left panel of figures REF - REF , we show the power spectrum in the plane of $(R, Q_\\sigma \\sigma _*)$ plane in the upper-left window for each cases with different $\\sigma _*$ ; $10^{-3}\\, M_P$ (Case A), $0.05\\, M_P$ (Case B), $0.5\\, M_P$ (Case C).", "We have normalized the amplitude of the power spectrum by the value at the pure curvaton limit of $R=1$ , namely ${\\cal P}_{\\zeta }/{\\cal P}_{\\zeta ,0}$ .", "The slow-roll parameter $\\epsilon _*$ is calculated from Eq.", "(REF ) considering $n_s=0.97$ and we used for simplicity $\\eta _*=\\xi _*=0$ .", "We assume that the correct observational value can be attained using the residual parameter of $H_*$ .", "The three cases in figures REF - REF show the following features: Case A : $\\sigma _*=10^{-3}\\, M_P$ and ${\\tilde{r}}\\gg 1$ in most of the region.", "Modulated reheating and curvaton contributions are dominant and inflaton contribution to the curvature perturbation is subdominant, Case B : $\\sigma _*=0.05\\, M_P$ and ${\\tilde{r}}\\sim 1$ as shown in figure REF .", "All three contributions are effective, Case C : $\\sigma _*=0.5\\, M_P$ and ${\\tilde{r}}\\ll 1$ in most of the region.", "Inflaton contribution is dominant and it scales as $(1-R)^2$ .", "$R<0.45$ is allowed from the constraint on the tensor-to-scalar ratio.", "In the Case A (upper-left window in figure REF ), $\\delta \\sigma $ contribution is dominant in the overall region and the inflaton contribution is subdominant.", "In the cancellation region, the relative contribution from $\\delta \\sigma $ decreases.", "Of course, by increasing $H_*$ , the desired amplitude of ${\\cal P}_{\\zeta }$ can be recovered.", "The tensor-to-scalar ratio is always much smaller than 0.01 in the shown parameter range.", "In the Case B (upper-left window in figure REF ), the inflaton contribution is effective for cancellation region and modulation and/or curvaton effects are effective in the rest.", "The cancellation between the modulated reheating and the curvaton appears around $R\\sim 0.7$ for negative $Q_\\sigma \\sigma _*$ .", "The tensor-to-scalar ratio is comparable to the observation constraint and a region around $(R,Q_\\sigma \\sigma _*) \\sim (0.6,-1)$ is ruled out as shown in the right window of figure REF .", "In the Case C (upper-left window in figure REF ), the density perturbation dominantly comes from the inflaton field but the magnitude changes due to the effect of curvaton.", "For large $R$ the magnitude decreases.", "A portion of parameter space is excluded because of too large tensor-to-scalar ratio." ], [ "Non-Gaussianity", "When the curvaton field dominates the energy density when it decay, i.e.", "$R \\sim 1$ , the curvature perturbation is dominated by the pure curvaton and the non-Gaussianity is suppressed.", "However in the other case $0 \\lesssim R \\ll 1$ , there is a possibility to get large non-Gaussianity.", "The bispectrum $B_\\zeta $ is given by $\\begin{split}\\langle \\zeta _{\\vec{k}_1}\\zeta _{\\vec{k}_2}\\zeta _{\\vec{k}_3} \\rangle =(2\\pi )^3 B_\\zeta (k_1,k_2,k_3) \\delta (\\vec{k_1}+\\vec{k_2}+\\vec{k_3}) ,\\end{split}$ and the dimensionless non-linearity parameter for the bispectrum, $f_{\\rm NL}$ , is defined by $\\begin{split}B_{\\zeta }(k_1,k_2,k_3) =\\frac{6}{5} f_{\\rm NL}[P_\\zeta (k_1)P_\\zeta (k_2)+P_\\zeta (k_2)P_\\zeta (k_3)+P_\\zeta (k_3)P_\\zeta (k_1) ].", "\\end{split}$ From the trispectrum $T_\\zeta $ given by $\\begin{split}\\langle \\zeta _{\\vec{k}_1}\\zeta _{\\vec{k}_2}\\zeta _{\\vec{k}_3}\\zeta _{\\vec{k}_4} \\rangle _c =(2\\pi )^3 T_\\zeta (k_1,k_2,k_3,k_4) \\delta (\\vec{k_1}+\\vec{k_2}+\\vec{k_3}+\\vec{k_4}) ,\\end{split}$ the dimensionless non-linearity parameters $\\tau _{\\rm NL}$ and $g_{\\rm NL}$ are defined as [39] $\\begin{split}T_\\zeta (k_1,k_2,k_3,k_4)=&\\tau _{\\rm NL}[ P_\\zeta (k_{13})P_\\zeta (k_3)P_\\zeta (k_4)+ 11 \\text{ perms} ]\\\\& + \\frac{54}{25}g_{\\rm NL}[ P_\\zeta (k_{2})P_\\zeta (k_3)P_\\zeta (k_4) + 3 \\text{ perms} ].\\end{split}$ The current bounds have been derived by several groups [40], [41].", "For instance, Smidt et.", "al.", "reported as $-7.4 < g_{\\rm NL}10^{-5} < 8.2$ and $-0.6< \\tau _{\\rm NL} 10^{-4} <3.3$  [40].", "In figure REF , we show the contours of the non-linearity parameters $f_{\\rm NL}$ (upper-right window), $\\tau _{\\rm NL}$ (lower-left window) and $g_{\\rm NL}$ (lower-right window) for the Case A.", "For the contours we have taken account the relations, $Q_{\\sigma \\sigma }\\sigma _*^2 = Q_{\\sigma }\\sigma _* (6 Q_{\\sigma }\\sigma _* +1) $ and $Q_{\\sigma \\sigma \\sigma } \\sigma _*^3=12Q_\\sigma \\sigma _*Q_{\\sigma \\sigma } \\sigma _*^3$ , which are motivated from the specific example we will show in the next section.", "We can see clearly the enhancement of non-linearity parameters in the cancellation region of modulated reheating and curvaton.", "The pure curvaton limit is recovered along the line of $Q_\\sigma \\sigma _*=0$ .", "In figure REF for the Case B, we can see the enhancement of the non-linearity parameters along the cancellation region.", "The difference from the Case A is that now the inflaton becomes more important and diminishes the non-Gaussinaity.", "In both cases of A and B, the large non-Gaussianity is dominantly due to $\\delta \\sigma $ , while the power spectrum comes from both depending on ${\\tilde{r}}$ , as explained in the Appendix.", "The non-linearity parameters can be enhanced around the cancellation region ($\\delta \\ll 1$ and $\\delta \\ll {\\tilde{r}}$ ).", "In this region, a large $f_{\\rm NL}$ comes dominantly from $\\zeta _{2,\\sigma \\sigma }$ as defined in Eq.", "(REF ) of the Appendix and estimated to be $\\begin{split}f_{\\rm NL}\\simeq \\frac{5}{6}\\frac{{\\tilde{r}}^2}{(1+{\\tilde{r}})^2} \\frac{1}{\\delta ^2} \\frac{9(1-R)Q_{\\sigma \\sigma }\\sigma _*^2-6R +R(1-R)(3+R)(Q_\\sigma \\sigma _*-2)^2 }{4R^2}.\\end{split}$ Therefore small $\\delta $ (large cancellation due to fine-tuning) can induce larger $f_{\\rm NL}$ .", "However note that ${\\tilde{r}}$ is proportional to $\\delta ^2$ (so ${\\tilde{r}}^2\\propto \\delta ^4$ ), thus too small $\\delta $ makes ${\\tilde{r}}$ becomes smaller than $\\delta $ itself and reduces $f_{\\rm NL}$ .", "One can see this behavior clearly in the upper right figure of figure REF : $f_{\\rm NL}$ decreases when approaching the cancellation line (blue dashed line).", "In the same region $\\tau _{\\rm NL}$ is also dominated by $\\zeta _{2,\\sigma \\sigma }$ term and approximately the squared of $f_{\\rm NL}$ , $\\begin{split}\\tau _{\\rm NL}\\simeq {\\left(\\frac{1+{\\tilde{r}}}{{\\tilde{r}}} \\right) }\\left(\\frac{6}{5}f_{\\rm NL}\\right)^2.\\end{split}$ Large $g_{\\rm NL}$ is possible in the same cancellation region dominated by $\\zeta _{3,\\sigma \\sigma \\sigma }$ term and given by $\\begin{split}g_{\\rm NL}\\simeq \\frac{25}{54} &\\frac{{\\tilde{r}}^3}{(1+{\\tilde{r}})^3}\\frac{1}{\\delta ^3} \\frac{1}{8R^3} \\\\\\times [&9(1-R) Q_{\\sigma \\sigma \\sigma }\\sigma _*^3+9R(1-R)(3+R)(3Q_\\sigma \\sigma _*-2)(3Q_{\\sigma \\sigma }\\sigma _*^2+2) \\\\&+ 36R +R(1-R)(3+R) (-3+4R+3R^2)(3Q_\\sigma \\sigma _*-2)^3 ].\\end{split}$ As you can see here, $g_{\\rm NL}$ has different sign in the opposite side of the cancellation line due to the odd exponent '3' of $\\delta $ .", "Here one can see that there are two points in both sides of $3 Q_{\\sigma } \\sigma _* \\simeq -2R/(1-R)$ those give the same values of ${\\cal P}_{\\zeta }$ and $f_{\\rm NL}$ .", "This means that the measurements of only ${\\cal P}_{\\zeta }$ and $f_{\\rm NL}$ can not determine $R$ and $Q_{\\sigma }\\sigma _*$ uniquely.", "However, this degeneracy can be resolved by measuring $g_{\\rm NL}$ because one predicts $g_{\\rm NL}$ to be positive and the other does it to be negative.", "In figure REF for the Case C, the region of large non-linearity parameters are in the region of $R> 0.5$ but which is excluded out by large $r_T$ .", "As we have seen in the previous sections, if a light scalar curvaton field $\\sigma $ has an interaction with inflaton field $\\phi $ , the fluctuation of curvaton field $\\delta \\sigma $ can modulate reheating through the decay of inflaton and can affect the curvature perturbation besides the usual curvaton mechanism.", "The resultant power spectrum and the non-linearity parameters can be considerably affected.", "In this section, we examine these effects in a simple model.", "The inflaton $\\phi $ and the curvaton $\\sigma $ can have the following interactions in the scalar potential $V[\\phi ,\\sigma ] = \\frac{1}{2} m_{\\phi }^2 \\phi ^2 + \\frac{1}{2} m_{\\sigma }^2 \\sigma ^2+ \\frac{1}{2}\\lambda _{\\phi \\sigma } \\phi ^2 \\sigma ^2,$ as well as an interaction with another scalar field as given in Eq.", "(REF ).", "It is also possible that the inflaton has interaction with other fields independent from the curvaton.", "We consider that the inflation is dominantly driven by a single inflaton field with its quadratic mass term potential by assuming $m_{\\phi } \\gg m_{\\sigma }$ and $m_{\\phi }^2 \\gg \\lambda _{\\phi \\sigma }\\sigma _*^2.$ We consider cases of vanishing $\\lambda _{\\phi \\sigma }$ in subsection REF and REF , and mention the effect of nonvanishing $\\lambda _{\\phi \\sigma }$ in subsection REF .", "Its interactions with other fields are important during reheating or later.", "Therefore during inflation, field equations are reduced to $&& H^2 = \\frac{1}{6 M_P^2} m_{\\phi }^2 \\phi ^2, \\\\&& 3 H \\dot{\\phi } + m_\\phi ^2\\phi = 0 ,$ under the slow-roll condition $\\begin{split}\\epsilon \\equiv \\frac{M_P^2}{2} {\\left(\\frac{V_\\phi }{V} \\right) }^2\\simeq \\frac{2M_P^2}{\\phi ^2} \\ll 1, \\qquad \\eta \\equiv \\frac{M_P^2 V_{\\phi \\phi }}{V} \\ll 1 ,\\end{split}$ and the inflaton-domination condition $\\frac{1}{2} m_{\\phi }^2 \\phi ^2 \\gg \\frac{1}{2} m_{\\sigma }^2 \\sigma ^2 .$ By solving field equations, we obtain $\\phi _*^2 = ( 4 N_{\\rm inf} +2 ) M_P^2,$ where $N_{\\rm inf}$ is the number of e-fold at horizon exit from the end of inflation.", "The power spectrum is ${\\cal P}_{\\zeta } \\simeq \\frac{(1-R)^2}{6 (2 \\pi )^2} \\frac{m_{\\phi }^2}{M_P^2}( 2 N_{\\rm inf} +1 )^2 +\\left\\lbrace (1-R)Q_\\sigma \\sigma _* + \\frac{2R}{3} \\right\\rbrace ^2 \\frac{m_{\\phi }^2}{3(2\\pi \\sigma _*)^2}( 2 N_{\\rm inf} +1 ) .$ The inflaton mass $m_{\\phi }$ controls the amplitude of the density perturbation.", "For the observed ${\\cal P}_{\\zeta }$ we can estimate $m_{\\phi }$ .", "Figure: [Left window] : The contour plot of m φ m_{\\phi } of the Model Iin the unit of log 10 m φ M P \\log _{10} \\left(\\frac{m_{\\phi }}{M_P}\\right)where 𝒫 ζ {\\cal P}_{\\zeta } has been fixed by the observed value 𝒫 ζ =2.44×10 -9 {\\cal P}_{\\zeta }=2.44\\times 10^{-9}.In the left side of the red line radiation is dominated at the time of curvaton decay and in the right side curvaton dominates.The cancellation between curvaton and modulated reheating occurs along the blue line where r ˜=0{\\tilde{r}}=0.", "[Right window] : The contour plot of r ˜{\\tilde{r}}.", "The red and blue lines are the same as in the left window.It is natural to assume that inflaton decays at its oscillating stage after a while, so that $\\Gamma _{\\phi }(\\sigma )/m_{\\phi }$ is very small.", "In this case $Q$ is well approximated by [30] $\\begin{split}Q\\simeq -\\frac{1}{6} \\log {\\left(\\frac{\\Gamma _{\\phi }(\\sigma )}{H_c} \\right) }.\\end{split}$ The derivatives are expressed as $Q_\\sigma &=& -\\frac{1}{6}\\frac{\\partial _\\sigma \\Gamma _\\phi }{\\Gamma _\\phi }, \\\\Q_{\\sigma \\sigma } &=&-\\frac{1}{6} \\left( \\frac{\\partial _\\sigma ^2 \\Gamma _\\phi }{\\Gamma _\\phi }-\\frac{ (\\partial _\\sigma \\Gamma _\\phi )^2}{\\Gamma ^2_\\phi } \\right), \\\\Q_{\\sigma \\sigma \\sigma } &=&-\\frac{1}{6} \\left( \\frac{\\partial _\\sigma ^3 \\Gamma _\\phi }{\\Gamma _\\phi }-3\\frac{ \\partial _\\sigma \\Gamma _\\phi \\partial ^2_\\sigma \\Gamma _\\phi }{\\Gamma ^2_\\phi }+ 2\\frac{ (\\partial _\\sigma \\Gamma _\\phi )^3 }{\\Gamma ^3_\\phi } \\right) .$" ], [ "\nModel I : Inflaton decays through only the coupling in Eq. (", "First, let us consider the case that the inflaton $\\phi $ decays through only the coupling in Eq.", "(REF ).", "For this case, we obtain $(Q_{\\sigma }\\sigma _*, Q_{\\sigma \\sigma }\\sigma _*^2, Q_{\\sigma \\sigma \\sigma }\\sigma _*^3 )= \\left(\\begin{array}{ccc}-\\frac{1}{3} , & \\frac{1}{3} , & -\\frac{2}{3}\\end{array}\\right) .$ The parameter defined in Eq.", "(REF ) at the time of curvaton decay is can be obtained using Eqs.", "(REF ) - (REF ) by $R =\\frac{ \\sigma _*^2/M_P^2 }{8 (a_{m_{\\sigma }}/a_{\\Gamma _{\\sigma }})+ \\sigma _*^2/M_P^2},$ with $\\frac{a_{m_{\\sigma }}}{a_{\\Gamma _{\\sigma }}} =\\left\\lbrace \\begin{array}{l}\\left( \\frac{\\Gamma _{\\sigma }}{m_{\\sigma }} \\right)^{1/2}\\qquad \\qquad \\qquad {\\rm radiation-dominated} \\\\\\left( \\frac{ \\sigma _*^2 }{6 M_P^2}\\right)^{-1/3}\\left( \\frac{\\Gamma _{\\sigma }}{m_{\\sigma }} \\right)^{2/3} \\quad \\quad \\sigma {\\rm -dominated }\\end{array}\\right.", ".", "$ Here $\\Gamma _\\sigma $ is the decay rate of the curvaton field and $a_{m_{\\sigma }}/a_{\\Gamma _{\\sigma }}$ is expressed in different ways depending on whether it is radiation-dominated or curvaton-dominated when the curvaton decays at $H=\\Gamma _\\sigma $ .", "In this case, there are only three parameters $m_{\\phi }, \\sigma _*$ and $\\Gamma _{\\sigma }/m_{\\sigma }$ , since $Q_{\\sigma }\\sigma _*$ and so on are completely fixed.", "As mentioned above, from the normalization of the power spectrum, $m_{\\phi }$ can be expressed by the other two parameters as shown in the left window of figure REF .", "The largest value to $m_{\\phi }$ in figure REF seems to be $10^{-5.2} M_P$ which is the same as the results in the quadratic chaotic inflation.", "This is because the region corresponds to the cancellation, $3 Q_{\\sigma } \\sigma _* \\simeq -2R/(1-R)$ line (blue line and $R=1/3$ in this case), where ${\\tilde{r}}\\simeq 0$ and the contribution from $\\sigma $ field cancels, and the dominant contribution comes from $\\phi $ field.", "On the other hand, $\\sigma $ contribution is not negligible in other regions and hence a smaller $m_{\\phi }$ is needed to produce the observable ${\\cal P}_{\\zeta }$ .", "In the right window of figure REF , we showed a contour plot of ${\\tilde{r}}$ .", "We note that the condition for a negligible $\\lambda _{\\phi \\sigma }$ , Eq.", "(REF ), can be rewritten as $\\lambda _{\\phi \\sigma } \\ll 10^{-6}\\left(\\frac{m_{\\phi }}{10^{-5} M_P}\\right)^2 \\left( \\frac{10^{-2} M_P}{\\sigma _*} \\right)^2 .$ In figure REF , we showed the contour plot of $r_T$ .", "The present bound on the $r_T$ does not constrain this model, however future bound can rule out the parameter range around the cancellation region along the blue line.", "In figure REF , the non-linearity parameters $f_{\\rm NL}$ (left) and $g_{\\rm NL}$ (right) are shown.", "Figure: [Left window] : The contour plotof f NL f_{\\rm NL} in (σ * /M P ,Γ σ /m σ )(\\sigma _*/M_P, \\Gamma _{\\sigma }/m_{\\sigma } ) plainfor the Model I.The red shaded region corresponds to too large f NL f_{\\rm NL} to be consistent with the observation.", "[Right window] : The contour of g NL g_{\\rm NL} for the Model I." ], [ " Model II : Inflaton decays through the coupling in Eq. (", "Next, let us consider the case that the inflaton $\\phi $ has a nonvanishing $\\sigma $ independent decay modes and the total decay width is given by Eq.", "(REF ).", "For this case, we obtain $(Q_{\\sigma }\\sigma _*, Q_{\\sigma \\sigma }\\sigma _*^2, Q_{\\sigma \\sigma \\sigma }\\sigma _*^3 )= \\left(\\begin{array}{ccc}-\\frac{1}{3}Br , & \\frac{1}{3}Br (2 Br - 1) , & -\\frac{2}{3} Br^2 ( 4 Br -3 )\\end{array}\\right) ,$ with $Br \\equiv \\Gamma _{\\phi }^{(CD)}/\\Gamma _{\\phi } $ .", "Obviously, the $Br \\rightarrow 1$ limit reduces to the Model I in the previous subsection.", "In this case, we have four parameters $m_{\\phi }, \\sigma _*, \\Gamma _{\\sigma }/m_{\\sigma }$ and $Br$ .", "Again, $m_{\\phi }$ can be used for the normalization of the power spectrum.", "In figure REF , we show the contour plot of $f_{\\rm NL}$ for $Br=0.3$ (left) and $0.1$ (right).", "As we have seen, the cancellation happens at $3 Q_{\\sigma } \\sigma _* \\simeq -2R/(1-R)$ or $R\\simeq Br/(2+Br)$ .", "Figure: [Left window] : The contour plot of f NL f_{\\rm NL} with Br=0.3Br=0.3on (σ * /M P ,Γ σ /m σ )(\\sigma _*/M_P, \\Gamma _{\\sigma }/m_{\\sigma } ) plane for the Model II.The red shaded region corresponds to too large f NL f_{\\rm NL} to be consistent with the observation.", "[Right window] : The same as left window but with Br=0.1Br=0.1." ], [ "Model III : The effective inflaton mass with the coupling in Eq. (", "Finally, let us consider the case of nonvanishing $\\lambda _{\\phi \\sigma }$ .", "Then, through just only the coupling (REF ), in other words even without direct coupling (REF ), the inflaton decay width $\\Gamma \\sim M_{\\phi }^3/M_P^2$ with the effective inflaton mass $M_{\\phi }^2 = m_{\\phi }^2 + \\lambda _{\\phi \\sigma }\\sigma _*^2$ which is $\\sigma $ dependent.", "For this case, we obtain $(Q_{\\sigma }\\sigma _*, Q_{\\sigma \\sigma }\\sigma _*^2, Q_{\\sigma \\sigma \\sigma }\\sigma _*^3 )= \\left(\\begin{array}{ccc}-\\frac{1}{2} Fr , & \\frac{1}{2} Fr (2 Fr - 1) , & Fr^2 (3 - 4Fr )\\end{array}\\right) ,$ with $Fr \\equiv \\lambda _{\\phi \\sigma }\\sigma _*^2/M_{\\phi }^2 $ .", "The qualitative behavior of this model is the same as that of the Model II up to numerical factors, by replacing $Br$ with $Fr$ ." ], [ "Conclusion", "We have studied the case where a light scalar field $\\sigma $ induces the modulated reheating by the inflaton decay and also acts as the curvaton by its late time decay in the presence of the curvature perturbation generated from the inflaton field itself.", "In fact, the coupling in Eq.", "(REF ) is possible from the gauge invariance of the SM Of course, it is also possible that those terms are forbidden by additional symmetry such as a certain $Z_2$ -parity., provided both $\\phi $ and $\\sigma $ are gauge singlet as is often assumed to preserve the flatness of the potential.", "When $\\sigma $ field contributes to both inducing the modulated reheating by the inflaton decay and generating the density perturbation as the curvaton, there could be a cancellation between two contributions along the line $3 Q_{\\sigma } \\sigma _* \\simeq -2R/(1-R)$ .", "Around this cancellation region, $\\delta \\sigma $ contribution to the power spectrum of the density perturbation is subdominant and the inflaton contribution becomes dominant.", "Near such a parameter region, mostly the middle range of $R$ and a negative $Q_{\\sigma }\\sigma _*$ , non-linearity parameters tend to be large because of the cancellation between the modulated reheating and the curvaton originated from the same field $\\sigma $ .", "In this sense, this cancellation is a kind of mechanisms to generate a large non-Gaussianity As specific models, we have also studied a quadratic inflation and curvaton model and demonstrated how the parameter space of a given inflaton and curvaton model would be constrained by taking the interaction between them into account.", "The measurement of non-linearity and tensor-to-scalar ratio may probe the strength of (non-)interaction between the inflaton and the curvaton." ], [ "Acknowledgments", "K.-Y.C is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No.", "2011-0011083).", "K.-Y.C acknowledges the Max Planck Society (MPG), the Korea Ministry of Education, Science and Technology (MEST), Gyeongsangbuk-Do and Pohang City for the support of the Independent Junior Research Group at the Asia Pacific Center for Theoretical Physics (APCTP).", "This work of O.S.", "is in part supported by the scientific research grants from Hokkai-Gakuen.", "O.S.", "would like to thank the APCTP for warm hospitality during his stay where this work has been completed." ], [ "Large non-Gaussianity", "The curvature perturbation $\\zeta $ can be decomposed of the contributions from each field perturbations order by order as $\\zeta = \\zeta _1+\\frac{1}{2}\\zeta _2+\\frac{1}{6}\\zeta _3+\\ldots ,$ with $\\begin{split}\\zeta _1=& \\zeta _{1,\\phi }\\delta \\phi + \\zeta _{1,\\sigma }\\delta \\sigma ,\\\\\\zeta _2=& \\zeta _{2,\\phi \\phi }(\\delta \\phi )^2 +2 \\zeta _{2,\\phi \\sigma }(\\delta \\phi )(\\delta \\sigma )+ \\zeta _{2,\\sigma \\sigma }(\\delta \\sigma )^2,\\\\\\zeta _3=& \\zeta _{3,\\phi \\phi \\phi }(\\delta \\phi )^2 + 3\\zeta _{3,\\phi \\phi \\sigma }(\\delta \\phi )^2(\\delta \\sigma ) + 3\\zeta _{3,\\phi \\sigma \\sigma }(\\delta \\phi )(\\delta \\sigma )^2 + \\zeta _{3,\\sigma }(\\delta \\sigma )^3.\\\\\\ldots & \\end{split}$ Here we considered two-field case for simplicity and we suppressed $*$ which denotes the value at horizon exit, i.e.", "$\\delta \\phi =\\delta \\phi _*$ and $\\delta \\sigma =\\delta \\sigma _*$ .", "For the model of modulated reheating by curvaton, we can read each component from Eq. ().", "The power spectrum is given from Eq.", "(REF ) by $\\begin{split}{\\cal P}_\\zeta = (\\zeta _{1,\\phi }^2 + \\zeta _{1,\\sigma }^2) {\\left(\\frac{H_*}{2\\pi } \\right) }^2 = \\zeta _{1,\\phi }^2(1 + {\\tilde{r}}) {\\left(\\frac{H_*}{2\\pi } \\right) }^2,\\end{split}$ where we used Eq.", "(REF ) and ${\\tilde{r}}\\equiv \\zeta _{1,\\sigma }^2 / \\zeta _{1,\\phi }^2$ .", "From the definitions in Eq.", "(REF ) and Eq.", "(REF ), the non-Gaussianity parameters are given by $\\begin{split}f_{\\rm NL}=& \\frac{5}{6} \\frac{\\zeta _{1,\\phi }^2 \\zeta _{2,\\phi \\phi } + 2\\zeta _{1,\\phi }\\zeta _{1,\\sigma }\\zeta _{2,\\phi \\sigma } +\\zeta _{1,\\sigma }^2\\zeta _{2,\\sigma \\sigma } }{(\\zeta _{1,\\phi }^2 + \\zeta _{1,\\sigma }^2)^2},\\\\\\tau _{\\rm NL}=& \\frac{ \\sum _{a,b,c} \\zeta _{1,b}\\zeta _{1,c} \\zeta _{2,ab}\\zeta _{2,ac} }{(\\zeta _{1,\\phi }^2 + \\zeta _{1,\\sigma }^2)^3}, \\\\g_{\\rm NL}=&\\frac{25}{54} \\frac{ \\sum _{a,b,c} \\zeta _{1,a}\\zeta _{1,b}\\zeta _{1,c} \\zeta _{3,abc} }{(\\zeta _{1,\\phi }^2 + \\zeta _{1,\\sigma }^2)^3}.\\end{split}$ Here $a,b,c$ denotes $\\phi $ and $\\sigma $ .", "First we will investigate the condition for large $f_{\\rm NL}$ in the curvaton-modulated scenario.", "Using Eq.", "(), the denominator of $f_{\\rm NL}$ in Eq.", "(REF ) becomes $\\begin{split}(\\zeta _{1,\\phi }^2 + \\zeta _{1,\\sigma }^2)^2 = \\left[ \\frac{1}{2M_P^2\\epsilon _*}(1-R)^2 (1+{\\tilde{r}}) \\right]^2 \\sim \\frac{(1+{\\tilde{r}})^2}{M_P^4\\epsilon _*^2},\\end{split}$ where in the last equation we dropped $\\mathcal {O}(1)$ coefficient, $R$ and $Q_ \\sigma \\sigma $ .", "In the same way, the numerators are $\\begin{split}\\zeta _{1,\\phi }^2 \\zeta _{2,\\phi \\phi } =& \\frac{1}{4M_P^4\\epsilon _*^2}(1-R)^3[R(R+3)+2\\epsilon _* - \\eta _*] \\sim \\frac{1}{M_P^4\\epsilon _*^2},\\\\2\\zeta _{1,\\phi }\\zeta _{1,\\sigma }\\zeta _{2,\\phi \\sigma } =& \\frac{1}{9M_P^2\\epsilon _*\\sigma ^2} R(1-R)^2(3+R)[3Q_\\sigma \\sigma (1-R)+2R][3Q\\sigma - 2]\\sim \\frac{1}{M_P^4\\epsilon _*^2} \\frac{{\\tilde{r}}}{\\delta },\\\\\\zeta _{1,\\sigma }^2\\zeta _{2,\\sigma \\sigma } =& \\frac{1}{81\\sigma ^4}[3Q_\\sigma \\sigma (1-R)+2R]^2[9Q_{\\sigma \\sigma } \\sigma ^2 (1-R) -6R +R(1-R)(3+R) (3Q_\\sigma \\sigma -2)^2]\\\\\\sim & \\frac{1}{M_P^4\\epsilon _*^2} \\frac{{\\tilde{r}}^2}{\\delta ^2}.\\end{split}$ Therefore we find that large $f_{\\rm NL}$ can arise from the last term $\\zeta _{1,\\sigma }^2\\zeta _{2,\\sigma \\sigma }$ to give $\\begin{split}f_{\\rm NL}\\sim \\frac{{\\tilde{r}}^2}{(1+{\\tilde{r}})^2}\\frac{1}{\\delta ^2}.\\end{split}$ It is easy to see that $\\begin{split}f_{\\rm NL}\\sim \\frac{1}{\\delta ^2}\\qquad \\text{for}\\qquad {\\tilde{r}}\\gtrsim 1, \\qquad \\text{and} \\qquad \\delta \\ll 1.\\end{split}$ and $\\begin{split}f_{\\rm NL}\\sim \\frac{{\\tilde{r}}^2}{\\delta ^2}\\qquad \\text{for}\\qquad {\\tilde{r}}\\lesssim 1, \\qquad \\text{and} \\qquad \\delta \\ll {\\tilde{r}}.\\end{split}$ However ${\\tilde{r}}$ and $\\delta $ are not independent variables and ${\\tilde{r}}$ is proportional to $\\delta ^2$ in our case as in Eq.", "(REF ).", "Therefore for a given parameters, $f_{\\rm NL}$ vanishes in the limit of $\\delta \\rightarrow 0$ when ${\\tilde{r}}$ becomes vanishing too.", "The large $f_{\\rm NL}$ can arise whenl $\\delta $ is smaller than 1 and smaller than ${\\tilde{r}}$ .", "Similarly, large $\\tau _{\\rm NL}$ and $g_{\\rm NL}$ are obtained in the same region as that of $f_{\\rm NL}$ when $\\zeta _{2,\\sigma \\sigma }$ and $\\zeta _{3,\\sigma \\sigma \\sigma }$ term dominates and are estimated to be $\\begin{split}\\tau _{\\rm NL}\\sim \\frac{{\\tilde{r}}^3}{(1+{\\tilde{r}})^3} \\frac{1}{\\delta ^4},\\qquad \\text{and}\\qquad g_{\\rm NL}\\sim \\frac{{\\tilde{r}}^3}{(1+{\\tilde{r}})^3} \\frac{1}{\\delta ^3}.\\end{split}$ We find that $g_{\\rm NL}$ is smaller than $\\tau _{\\rm NL}$ in the large non-Gaussianity region." ] ]

[ [ "A survey of the Johnson homomorphisms of the automorphism groups of free\n groups and related topics" ], [ "Abstract This is a biased survey for the Johnson homomorphisms of the automorphism groups of free groups.", "We just exposit some well known facts and recent developments for the Johnson homomorphisms and its related topics." ], [ "Introduction", "In the 1980s, Dennis Johnson established a new remarkable method to investigate the group structure of the Torelli subgroup of the mapping class group of a surface in a series of works [22], [23], [24] and [25].", "In particular, he gave a finite set of generators of the Torelli group, and he constructed a homomorphism $\\tau $ to determine the abelianization of that group.", "Today, his homomorphism $\\tau $ is called the first Johnson homomorphism, and it is generalized to Johnson homomorphisms of higher degrees.", "Over the last two decades, good progress was made in the study of the Johnson homomorphisms of mapping class groups through the works of many authors including Morita [42], Hain [21] and others.", "As is well known, the mapping class group of a compact oriented surface with one boundary component can be embedded into the automorphism group of a free group by a classical work of Dehn and Nielsen in the 1910s and early 1920s.", "So far, a large number of theories and research techniques to study the mapping class group have been applied to investigate the automorphism group of a free group.", "The Johnson homomorphisms are one of these techniques.", "The definition of the Johnson homomorphisms were generalized not only for the automorphism group of a free group but also that of any group $G$ .", "In this paper, although we mainly consider the case where $G$ is a free group and a free metabelian group, we give the definition of the Johnson homomorphisms for the automorphism group of a general group $G$ .", "(See Section .)", "We discuss the case where $G$ is a free group of finite rank, the most basic and important case.", "To put it plainly, the Johnson homomorphisms are useful tools to study the graded quotients of a certain descending filtration of the automorphism group of a free group.", "To explain this, let us fix some notation.", "Let $F_n$ be a free group of rank $n \\ge 2$ , and $\\mathrm {Aut}\\,F_n$ the automorphism group of $F_n$ .", "We denote by $H:=H_1(F_n,\\mathbf {Z})$ the abelianization of $F_n$ .", "Let $\\rho : \\mathrm {Aut}\\,F_n \\rightarrow \\mathrm {Aut}\\,H$ be the natural homomorphism induced from the abelianization $F_n \\rightarrow H$ .", "The kernel of $\\rho $ is called the IA-automorphism group of $F_n$ , denoted by $\\mathrm {IA}_n$ .", "The letters I and A stands for “Identity\" and “Automorphism\" respectively.", "Bachmuth [5] called $\\mathrm {IA}_n$ the IA-automorphism group since that consists of automorphisms which induce identity automorphisms on the abelianized group $H$ of $F_n$ .", "The subgroup $\\mathrm {IA}_n$ reflects much richness and complexity of the structure of $\\mathrm {Aut}\\,F_n$ , and plays important roles in various studies of $\\mathrm {Aut}\\,F_n$ .", "Although the study of the IA-automorphism group has a long history since finitely many generators were obtained for that group by Magnus [36] in 1935, the group structure of $\\mathrm {IA}_n$ is still quite complicated.", "For instance, no presentation for $\\mathrm {IA}_n$ is known for $n \\ge 3$ .", "Nielsen [47] showed that $\\mathrm {IA}_2$ coincides with the inner automorphism group, hence, is a free group of rank 2.", "For $n \\ge 3$ , however, $\\mathrm {IA}_n$ is much larger than the inner automorphism group $\\mathrm {Inn}\\,F_n$ .", "Krstić and McCool [31] showed that $\\mathrm {IA}_3$ is not finitely presentable.", "For $n \\ge 4$ , it is not known whether $\\mathrm {IA}_n$ is finitely presentable or not.", "Because of the complexity of the group structure of $\\mathrm {IA}_n$ mentioned above, it is sometimes not suitable to handle all of $\\mathrm {IA}_n$ directly.", "In order to study $\\mathrm {IA}_n$ with a phased approach, we consider the Johnson filtration of $\\mathrm {Aut}\\,F_n$ .", "The Johnson filtration is one of descending central series $ \\mathrm {IA}_n = \\mathcal {A}_n(1) \\supset \\mathcal {A}_n(2) \\supset \\cdots $ consisting of normal subgroups of $\\mathrm {Aut}\\,F_n$ , whose first term is $\\mathrm {IA}_n$ .", "(For details, see Sections and .)", "The graded quotients $\\mathrm {gr}^k (\\mathcal {A}_n) := \\mathcal {A}_n(k)/\\mathcal {A}_n(k+1)$ naturally have $\\mathrm {GL}(n,\\mathbf {Z})$ -module structures, and they are considered to be a sequence of approximations of $\\mathrm {IA}_n$ .", "To understand the graded quotients $\\mathrm {gr}^k (\\mathcal {A}_n)$ more closely, we use the Johnson homomorphisms $ \\tau _k : \\mathrm {gr}^k (\\mathcal {A}_n) \\rightarrow H^* \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n).", "$ (For the definition, see Section .)", "One of the most fundamental properties of the Johnson homomorphism is that $\\tau _k$ is a $\\mathrm {GL}(n,\\mathbf {Z})$ -equivariant injective homomorphism for each $k \\ge 1$ .", "Hence, we can consider $\\mathrm {gr}^k (\\mathcal {A}_n)$ as a submodule of $H^* \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n)$ whose module structure is easy to handle.", "Using the Johnson homomorphisms, we can extract some valuable information about $\\mathrm {IA}_n$ .", "For example, $\\tau _1$ is just the abelianization of $\\mathrm {IA}_n$ by a result due to Andreadakis [1], and $\\tau _2$ was applied to determine the image of the cup product $\\cup _{\\mathbf {Q}} : \\Lambda ^2 H^1(\\mathrm {IA}_n,\\mathbf {Q}) \\rightarrow H^2(\\mathrm {IA}_n,\\mathbf {Q})$ by Pettet [51].", "In general, to determine the structure of the images, or equivalently the cokernels, of the Johnson homomorphisms is one of the most basic problems.", "In order to take advantage of representation theory, we consider the rationalization of modules.", "So far, for $1 \\le k \\le 3$ , the $\\mathrm {GL}(n,\\mathbf {Q})$ -module structure of the cokernel $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ of the rational Johnson homomorphism $\\tau _{k,\\mathbf {Q}}:= \\tau _k \\otimes \\mathrm {id}_{\\mathbf {Q}}$ has been determined.", "(See [1], [51] and [60] for $k=1$ , 2 and 3 respectively, and see also Subsection REF .)", "In general, however, it is quite a difficult problem to determine the $\\mathrm {GL}(n,\\mathbf {Q})$ -module structure of $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ for arbitrary $k \\ge 4$ .", "One reason for it is that we cannot obtain an explicit generating system of ${\\mathrm {gr}}^k (\\mathcal {A}_n)$ easily.", "To avoid this difficulty, we consider the lower central series $\\Gamma _{\\mathrm {IA}_n}(1)=\\mathrm {IA}_n$ , $\\Gamma _{\\mathrm {IA}_n}(2)$ , $\\dots $ of $\\mathrm {IA}_n$ .", "Since the Johnson filtration is central, we have $\\Gamma _{\\mathrm {IA}_n}(k) \\subset \\mathcal {A}_n(k)$ for any $k \\ge 1$ .", "It is conjectured that $\\Gamma _{\\mathrm {IA}_n}(k) = \\mathcal {A}_n(k)$ for each $k \\ge 1$ .", "The conjecture was made by Andreadakis who showed $\\Gamma _{\\mathrm {IA}_2}(k) = \\mathcal {A}_2(k)$ for each $k \\ge 1$ and $\\Gamma _{\\mathrm {IA}_3}(3) = \\mathcal {A}_3(3)$ in [1].", "Now, it is known that $\\Gamma _{\\mathrm {IA}_n}(2) = \\mathcal {A}_n(2)$ due to Bachmuth [6] and that $\\Gamma _{\\mathrm {IA}_n}(3)$ has at most finite index in $\\mathcal {A}_n(3)$ due to Pettet [51].", "For each $k \\ge 1$ , set $\\mathrm {gr}^k(\\mathcal {L}_{\\mathrm {IA}_n}) := \\Gamma _{\\mathrm {IA}_n}(k)/\\Gamma _{\\mathrm {IA}_n}(k+1)$ .", "Since $\\mathrm {IA}_n$ is finitely generated as mentioned above, each $\\mathrm {gr}^k(\\mathcal {L}_{\\mathrm {IA}_n})$ is also finitely generated as an abelian group.", "Then we can define a $\\mathrm {GL}(n,\\mathbf {Z})$ -equivariant homomorphism $ \\tau _k^{\\prime } : \\mathrm {gr}^k(\\mathcal {L}_{\\mathrm {IA}_n}) \\rightarrow H^* \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n) $ in the same way as $\\tau _k$ .", "We also call $\\tau _k^{\\prime }$ the Johnson homomorphism of $\\mathrm {Aut}\\,F_n$ .", "Then, we can directly obtain information about the cokernel of $\\tau _k^{\\prime }$ using finitely many generators of $\\mathrm {gr}^k(\\mathcal {L}_{\\mathrm {IA}_n})$ .", "Furthermore, if we consider the rational Johnson homomorphism $\\tau _{k,\\mathbf {Q}}^{\\prime } := \\tau _k^{\\prime } \\otimes \\mathrm {id}_{\\mathbf {Q}}$ , then using representation theory, we can consider $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ as a $\\mathrm {GL}(n,\\mathbf {Z})$ -submodule of $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime })$ .", "Hence, we can give an upper bound on $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ .", "In [64], we determined the cokernel of the rational Johnson homomorphism $\\tau _{k,\\mathbf {Q}}^{\\prime }$ in a stable range as follows.", "Theorem 1.1 (Satoh, [64]) ($=$ Theorem REF .)", "For any $k \\ge 2$ and $n \\ge k+2$ , $ \\mathrm {Coker}(\\tau _{k, \\mathbf {Q}}^{\\prime }) \\cong \\mathcal {C}_n^{\\mathbf {Q}}(k).", "$ Here $\\mathcal {C}_n(k)$ be a quotient module of $H^{\\otimes k}$ by the action of the cyclic group $\\mathrm {Cyc}_k$ of order $k$ on the components: $ \\mathcal {C}_n(k) := H^{\\otimes k} \\big {/} \\langle a_1 \\otimes a_2 \\otimes \\cdots \\otimes a_k - a_2 \\otimes a_3 \\otimes \\cdots \\otimes a_k \\otimes a_1 \\,|\\, a_i \\in H \\rangle , $ and $\\mathcal {C}_n^{\\mathbf {Q}}(k) := \\mathcal {C}_n(k) \\otimes _{\\mathbf {Z}} \\mathbf {Q}$ .", "The fact above is quite important if we consider an application of the study of the Johnson homomorphisms of $\\mathrm {Aut}\\,F_n$ to that of the mapping class group.", "Let $\\mathcal {M}_{g,1}$ be the mapping class group of a compact oriented surface $\\Sigma _{g,1}$ of genus $g$ with one boundary component.", "By an similar way as $\\tau _k$ , we can define an injective $\\mathrm {Sp}(2g,\\mathbf {Z})$ -equivariant homomorphism $ \\tau _k^{\\mathcal {M}} : \\mathrm {gr}^k (\\mathcal {M}_{g,1}) \\hookrightarrow \\mathrm {Hom}_{\\mathbf {Z}}(H, \\mathrm {gr}^{k+1}(\\mathcal {L}_{2g})), $ called the $k$ -th Johnson homomorphism of $\\mathcal {M}_{g,1}$ .", "Inspired by Johnson's works, Morita has studied various aspects of the Johnson homomorphisms of the mapping class groups over the last two decades, and gave many remarkable results.", "We also use $H$ for $H_1(\\Sigma _{g,1}, \\mathbf {Z})$ by abuse of the language.", "Identify $H$ with its dual module as an $\\mathrm {Sp}(2g,\\mathbf {Z})$ -module by Poincaré duality.", "Let $\\mathfrak {h}_{g,1}(k)$ be the kernel of a homomorphism $H \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_{2g}) \\rightarrow \\mathrm {gr}^{k+2}(\\mathcal {L}_{2g})$ , which is induced from the Lie bracket of the free Lie algebra.", "Then Morita [42] showed that $\\mathrm {Im}(\\tau _k^{\\mathcal {M}}) \\subset \\mathfrak {h}_{g,1}(k)$ for each $k \\ge 2$ .", "Hence, our main problem is to determine $\\mathrm {Coker}(\\tau _k^{\\mathcal {M}}):=\\mathfrak {h}_{g,1}(k)/\\mathrm {Im}(\\tau _k^{\\mathcal {M}})$ .", "At the present stage, the Sp-module structure of the cokernel of the rational Johnson homomorphisms $\\tau _{k,\\mathbf {Q}}^{\\mathcal {M}}$ are determined for $1 \\le k \\le 4$ by Johnson [22], Morita [40], Hain [21] and Asada-Nakamura [4] respectively.", "Furthermore, by using trace maps, Morita [42] showed that the symmetric tensor product $S^k H_{\\mathbf {Q}}$ injects into $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\mathcal {M}})$ for each odd $k \\ge 3$ .", "In 1996, in unpublished work, Hiroaki Nakamura showed that the multiplicity of $S^k H_{\\mathbf {Q}}$ in $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\mathcal {M}})$ is one for odd $k \\ge 3$ .", "Except for these results, there are few results for the cokernels of the Johnson homomorphisms of the mapping class groups.", "Recently, based on a work of Hain [21], Naoya Enomoto and the author proved Theorem 1.2 (Enomoto and Satoh [17]) ($=$ Theorem REF .)", "For an integer $k \\ge 5$ such that $k \\equiv 1$ mod 4, and $g \\ge k+2$ , the irreducible Sp-module $[1^k]$ appears in $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\mathcal {M}})$ with multiplicity one.", "See Section for details.", "This chapter is a survey on the Johnson homomorphisms of the automorphism group of a free group and related topics.", "We just exposit some well known facts and recent developments for them without proof.", "In Section , we give the definitions and some properties of the Andreadakis-Johnson filtration and the Johnson homomorphisms of the automorphism group of an arbitrary group $G$ .", "In Section , we consider the free group case.", "In particular, we show that the symmetric tensor product $S^k H_{\\mathbf {Q}}$ of $H_{\\mathbf {Q}} := H \\otimes _{\\mathbf {Z}} \\mathbf {Q}$ injects into the cokernel $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ using the Morita trace map.", "Moreover, we show that $\\mathrm {Coker}(\\tau _{k, \\mathbf {Q}}^{\\prime }) \\cong \\mathcal {C}_n^{\\mathbf {Q}}(k)$ for $k \\ge 2$ and $n \\ge k+2$ .", "In Section , we deal with the free metabelian group case.", "In particular, we show how the image and the cokernel of the Johnson homomorphisms of the automorphism group of a free metabelian group were completely determined by using the Morita trace maps.", "Furthermore we also show that as a Lie algebra, the abelianization of the derivation algebra of the free Lie algebra generated by $H$ was also determined by using the Morita trace maps.", "In the rest of this chapter, we consider some applications of the study of the Johnson homomorphisms of the automorphism group of a free group.", "In Section , we recall the definition of the braid group and the mapping class group of a surface with boundary, which are considered as subgroups of the automorphism group of a free group.", "In this section, we give only the definition and some properties of them as a preparation for later sections, and no topics related to the Johnson homomorphisms.", "In Section , we show a relation between the Johnson homomorphisms and the Magnus representation obtained by Morita.", "We also show the infinite generation of the kernel of the Magnus representation by using the first Johnson homomorphisms.", "In Section , we investigate the Johnson cokernels of the mapping class group of a surface.", "In particular, we show that we can detect new series $[1^k]$ in the Johnson cokernel of the mapping class group by using Theorem REF .", "In Section , we consider twisted cohomologies of the automorphism groups of a free group and a free nilpotent group.", "In particular, we compute $H^1(\\mathrm {Aut}\\,F_n, \\mathrm {IA}_n^{\\mathrm {ab}})$ , and show that the first Johnson homomorphism cannot be extended to the automorphism group of a free group as a crossed homomorphism.", "Moreover, for a free nilpotent group $N_{n,k}$ we show that $H^1(\\mathrm {Aut}\\,N_{n,k}, \\Lambda ^l H_{\\mathbf {Q}})=0$ for $k \\ge 3$ , $n \\ge k-1$ and $l \\ge 3$ .", "Then we see that the trace map $\\mathrm {Tr}_{[1^k]}$ for the exterior product $\\Lambda ^k H_{\\mathbf {Q}}$ defines a non-trivial second cohomology class in $H^2(\\mathrm {Aut}\\,N_{n,k}, \\Lambda ^k H_{\\mathbf {Q}})$ .", "Finally, in Section , we determine the abelianization of the congruence IA-automorphism group of a free group by using the first Johnson map defined by the Magnus expansion of a free group obtained by Kawazumi." ], [ "Notation and Conventions", "Throughout this chapter, we use the following notation and conventions.", "For a group $G$ , the abelianization of $G$ is denoted by $G^{\\mathrm {ab}}$ .", "For elements $x$ and $y$ of a group $G$ , the commutator bracket $[x,y]$ of $x$ and $y$ is defined to be $[x,y]:=xyx^{-1}y^{-1}$ .", "For subgroups $H$ and $K$ of $G$ , we denote by $[H,K]$ the commutator subgroup of $G$ generated by $[h, k]$ for $h \\in H$ and $k \\in K$ .", "For a group $G$ , and its quotient group $G/N$ , we also denote the coset class of an element $g \\in G$ by $g \\in G/N$ if there is no confusion.", "For a group $G$ , the group $\\mathrm {Aut}\\,G$ acts on $G$ from the right unless otherwise noted.", "For any $\\sigma \\in \\mathrm {Aut}\\,G$ and $x \\in G$ , the action of $\\sigma $ on $x$ is denoted by $x^{\\sigma }$ .", "If we need to consider the left action of $\\mathrm {Aut}\\,G$ on $G$ , we consider the usual action.", "Namely, $\\sigma \\cdot x := x^{\\sigma ^{-1}}$ for $\\sigma \\in \\mathrm {Aut}\\,G$ and $x \\in G$ .", "Similarly, for any right $\\mathrm {Aut}\\,G$ -module $M$ , we consider $M$ as a left $\\mathrm {Aut}\\,G$ -module by $\\sigma \\cdot m := m^{\\sigma ^{-1}}$ for $\\sigma \\in \\mathrm {Aut}\\,G$ and $m \\in M$ .", "For any $\\mathbf {Z}$ -module $M$ , we denote $M \\otimes _{\\mathbf {Z}} \\mathbf {Q}$ by the symbol obtained by attaching a subscript or superscript $\\mathbf {Q}$ to $M$ , like $M_{\\mathbf {Q}}$ or $M^{\\mathbf {Q}}$ .", "Similarly, for any $\\mathbf {Z}$ -linear map $f: A \\rightarrow B$ , the induced $\\mathbf {Q}$ -linear map $f \\otimes \\mathrm {id}_{\\mathbf {Q}} : A_{\\mathbf {Q}} \\rightarrow B_{\\mathbf {Q}}$ is denoted by $f_{\\mathbf {Q}}$ or $f^{\\mathbf {Q}}$ .", "For a group $G$ and a left $G$ -module $M$ , we set $\\begin{split}\\mathrm {Cros}(G,M) & := \\lbrace f : G \\rightarrow M \\,|\\, f : \\mathrm {crossed} \\,\\,\\, \\mathrm {homomorphism} \\rbrace .\\end{split}$ For any ring $R$ , we denote by $M(n, R)$ and $\\mathrm {GL}(n,R)$ the ring of matrices of rank $n$ over $R$ and the general linear group over $R$ respectively." ], [ "Johnson homomorphisms", "In this section, we recall the definition and some properties of the Johnson homomorphisms of the automorphism group of a group." ], [ "Graded Lie algebras associated to an N-series of a group $G$", "To begin with, we recall an N-series of a group and the graded Lie algebra associated to it, based on a classical work of Lazard [32].", "Definition 3.1 Let $G$ be a group, and $H_1, H_2, \\ldots $ a sequence of subgroups of $G$ such that $ G = H_1 \\supset H_2 \\supset \\cdots \\supset H_k \\supset \\cdots $ where $H_{k+1}$ is normal in $H_k$ , and $ [H_k, H_l] \\subset H_{k+l} $ for any $k, l \\ge 1$ .", "Such a sequence $\\mathcal {H} := \\lbrace H_k \\rbrace $ is called an N-series of $G$ .", "The most familiar example of an N-series of $G$ is its lower central series $\\lbrace \\Gamma _G(k) \\rbrace $ defined by $ \\Gamma _G(1):= G, \\hspace{10.0pt} \\Gamma _G(k+1) := [\\Gamma _G(k),G], \\hspace{10.0pt} k \\ge 1.", "$ The lower central series $\\lbrace \\Gamma _G(k) \\rbrace $ has the property that if $\\lbrace H_k \\rbrace $ is an N-series of $G$ then $\\Gamma _G(k) \\subset H_k$ for any $k \\ge 1$ .", "Obviously, each of the graded quotients $\\mathrm {gr}^k(\\mathcal {H}) := H_k/H_{k+1}$ is an abelian group for $k \\ge 1$ since $[H_k, H_k] \\subset H_{2k} \\subset H_{k+1}$ .", "We write the product in each of $\\mathrm {gr}^k(\\mathcal {H})$ additively.", "Namely, for any $x, y \\in H_k$ , if we denote their coset classes modulo $H_{k+1}$ by $[x]$ and $[y]$ then $ [x] + [y] = [xy].", "$ Consider the direct sum $ \\mathrm {gr}(\\mathcal {H}) := \\bigoplus _{k=1}^{\\infty } \\mathrm {gr}^k(\\mathcal {H}).", "$ We give the additive abelian group $\\mathrm {gr}(\\mathcal {H})$ a grading with the convention that the elements of $\\mathrm {gr}^k(\\mathcal {H})$ are homogeneous of degree $k$ .", "For any $k$ , $l \\ge 1$ , let us consider a bilinear alternating map $ [ \\, , \\,]_{\\mathrm {Lie}} : \\mathrm {gr}^k(\\mathcal {H}) \\times \\mathrm {gr}^l(\\mathcal {H}) \\rightarrow \\mathrm {gr}^{k+l}(\\mathcal {H}) $ defined by $[\\, [x], [y] \\, ]_{\\mathrm {Lie}} := [\\, [x, y] \\,]$ for any $[x] \\in \\mathrm {gr}^k(\\mathcal {H})$ and $[y] \\in \\mathrm {gr}^l(\\mathcal {H})$ where $[x, y]$ is a commutator in $G$ , and $[\\, [x, y] \\,]$ is a coset class of $[x, y]$ in $\\mathrm {gr}^{k+l}(\\mathcal {H})$ .", "Then $[ \\, , \\,]_{\\mathrm {Lie}}$ induces a graded Lie algebra structure of the graded sum $\\mathrm {gr}(\\mathcal {H})$ .", "Definition 3.2 The Lie algebra $\\mathrm {gr}(\\mathcal {H})$ is called the graded Lie algebra associated to the N-series $\\mathcal {H}$ .", "This is a generalization of the well-known construction introduced by Magnus [37] for the lower central series of a free group.", "(For example, see also Section 5.14 in [38] or Chapter VIII in [50] for basic materials concerning the graded Lie algebra associated to an N-series.)", "For a group $G$ and its lower central series $\\mathcal {L}_G := \\lbrace \\Gamma _G(k) \\rbrace $ , if generators of $G$ are given, then those of the graded quotient $\\mathrm {gr}^k(\\mathcal {L}_G) := \\Gamma _G(k)/\\Gamma _G(k+1)$ are obtained as follows.", "For any $g_1, \\ldots , g_k \\in G$ , a left-normed commutator $ [[ \\cdots [[ g_{1},g_{i}],g_{3}], \\cdots ], g_{k}] $ of weight $k$ is called a simple $k$ -fold commutator, denoted by $ [g_{1}, g_{2}, \\cdots , g_{k} ] $ for simplicity.", "Then we have Lemma 3.3 If a group $G$ is generated by $g_1, \\ldots , g_t$ , then each of the graded quotients $\\mathrm {gr}^k(\\mathcal {L}_G)$ is generated by the simple $k$ -fold commutators $ [g_{i_1},g_{i_2}, \\ldots , g_{i_k}], \\hspace{10.0pt} i_j \\in \\lbrace 1, \\ldots , t \\rbrace $ as an abelian group.", "(For a proof see Theorem 5.4 in [38], for example.)", "This shows that if $G$ is finitely generated then so is $\\mathrm {gr}^k(\\mathcal {L}_G)$ for any $k \\ge 1$ ." ], [ "The Andreadakis-Johnson filtration of $\\mathrm {Aut}\\,G$", "In this subsection, we consider a descending filtration of $\\mathrm {Aut}\\,G$ .", "Definition 3.4 For $k \\ge 1$ , the action of $\\mathrm {Aut}\\,G$ on each nilpotent quotient $G/\\Gamma _G(k+1)$ induces a homomorphism $ \\mathrm {Aut}\\,G \\rightarrow \\mathrm {Aut}(G/\\Gamma _G(k+1)).", "$ We denote its kernel by $\\mathcal {A}_G(k)$ .", "Then the groups $\\mathcal {A}_G(k)$ define a descending filtration $ \\mathcal {A}_G(1) \\supset \\mathcal {A}_G(2) \\supset \\cdots \\supset \\mathcal {A}_G(k) \\supset \\cdots $ of $\\mathrm {Aut}\\,G$ .", "We call $\\mathcal {A}_G := \\lbrace \\mathcal {A}_G(k) \\rbrace $ the Andreadakis-Johnson filtration of $\\mathrm {Aut}\\,G$ .", "The first term $\\mathcal {A}_G(1)$ is called the IA-automorphism group of $G$ , and is also denoted by $\\mathrm {IA}(G)$ .", "Namely, $\\mathrm {IA}(G)$ consists of automorphisms which act on the abelianization $G^{\\mathrm {ab}}$ of $G$ trivially.", "The Andreadakis-Johnson filtration of $\\mathrm {Aut}\\,G$ was originally introduced by Andreadakis [1] in the 1960s.", "The name “Johnson\" comes from Dennis Johnson who studied the Johnson filtration and the Johnson homomorphism for the mapping class group of a surface in the 1980s.", "(See Section for details.)", "In particular, Andreadakis showed that Theorem 3.5 (Andreadakis, [1])     For any $k$ , $l \\ge 1$ , $\\sigma \\in \\mathcal {A}_G(k)$ and $x \\in \\Gamma _G(l)$ , $x^{-1} x^{\\sigma } \\in \\Gamma _G(k+l)$ .", "For any $k$ and $l \\ge 1$ , $[\\mathcal {A}_G(k), \\mathcal {A}_G(l)] \\subset \\mathcal {A}_G(k+l)$ .", "If $\\displaystyle \\bigcap _{k \\ge 1} \\Gamma _G(k) =1$ , then $\\displaystyle \\bigcap _{k \\ge 1} \\mathcal {A}_G(k) =1$ .", "Part (2) of Theorem REF shows that the Andreadakis-Johnson filtration is an N-series of $\\mathrm {IA}(G)$ .", "Hence its $k$ -th term $\\mathcal {A}_G(k)$ contains $\\Gamma _{\\mathrm {IA}(G)}(k)$ , that of the lower central series of $\\mathrm {IA}(G)$ .", "It is a natural question to ask how different is $\\mathcal {A}_G(k)$ from $\\Gamma _{\\mathrm {IA}(G)}(k)$ .", "In general, however, it is quite a difficult problem to determine whether $\\mathcal {A}_G(k)$ coincides with $\\Gamma _{\\mathrm {IA}(G)}(k)$ or not.", "The next lemma immediately follows from Theorem REF .", "Lemma 3.6     For any $k$ , $l \\ge 1$ , $\\sigma \\in \\Gamma _{\\mathrm {IA}(G)}(k)$ and $x \\in \\Gamma _{\\mathrm {IA}(G)}(l)$ , $x^{-1} x^{\\sigma } \\in \\Gamma _G(k+l)$ .", "For any $k$ and $l \\ge 1$ , $[\\Gamma _{\\mathrm {IA}(G)}(k), \\Gamma _{\\mathrm {IA}(G)}(l)] \\subset \\Gamma _{\\mathrm {IA}(G)}(k+l)$ .", "If $\\displaystyle \\bigcap _{k \\ge 1} \\Gamma _G(k) =1$ , then $\\displaystyle \\bigcap _{k \\ge 1} \\Gamma _{\\mathrm {IA}(G)}(k) =1$ .", "Both of the graded quotients $\\mathrm {gr}^k(\\mathcal {A}_G) = \\mathcal {A}_G(k)/\\mathcal {A}_G(k+1)$ and $\\mathrm {gr}^k(\\mathcal {L}_{\\mathrm {IA}(G)}) = \\Gamma _{\\mathrm {IA}(G)}(k)/\\Gamma _{\\mathrm {IA}(G)}(k+1)$ are considered as sequences of approximations of $\\mathrm {IA}(G)$ .", "To clarify their structures plays an important role in various studies of $\\mathrm {IA}(G)$ ." ], [ "Actions of $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$", "In this subsection, we define actions of $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ on the graded quotients $\\mathrm {gr}^k(\\mathcal {L}_G)$ , $\\mathrm {gr}^k(\\mathcal {A}_G)$ and $\\mathrm {gr}^k(\\mathcal {L}_{\\mathrm {IA}(G)})$ .", "First, since each of $\\Gamma _G(k)$ is a characteristic subgroup of $G$ , the group $\\mathrm {Aut}\\,G$ naturally acts on $\\Gamma _G(k)$ from the right, and hence on $\\mathrm {gr}^k(\\mathcal {L}_G) = \\Gamma _G(k)/\\Gamma _G(k+1)$ for any $k \\ge 1$ .", "From Part (1) of Theorem REF , the restriction of this action of $\\mathrm {Aut}\\,G$ to $\\mathrm {IA}(G)$ is trivial.", "Thus, an action of the quotient group $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ on $\\mathrm {gr}^k(\\mathcal {L}_G)$ is well-defined.", "On the other hand, since each of $\\mathcal {A}_G(k)$ is a normal subgroup of $\\mathrm {Aut}\\,G$ , the group $\\mathrm {Aut}\\,G$ acts on $\\mathcal {A}_G(k)$ by conjugation from the right.", "Namely, for any $\\sigma \\in \\mathrm {Aut}\\,G$ and $\\tau \\in \\mathcal {A}_G(k)$ , the action of $\\sigma $ on $\\tau $ is given by $\\sigma ^{-1} \\tau \\sigma $ .", "Hence, $\\mathrm {Aut}\\,G$ also acts on each of $\\mathrm {gr}^k(\\mathcal {A}_G)$ for $k \\ge 1$ .", "From Part (2) of Theorem REF , the restriction of this action of $\\mathrm {Aut}\\,G$ to $\\mathrm {IA}(G)$ is trivial.", "Thus, an action of the quotient group $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ on $\\mathrm {gr}^k(\\mathcal {A}_G)$ is well-defined.", "Similarly, using Part (2) of Lemma REF , we can define an action of $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ on $\\mathrm {gr}^k(\\mathcal {L}_{\\mathrm {IA}(G)})$ .", "Throughout the paper, we fix these three actions of $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ on $\\mathrm {gr}^k(\\mathcal {L}_G)$ , $\\mathrm {gr}^k(\\mathcal {A}_G)$ and $\\mathrm {gr}^k(\\mathcal {L}_{\\mathrm {IA}(G)})$ ." ], [ "Johnson homomorphisms of $\\mathrm {Aut}\\,G$", "To study the $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ -module structures of the graded quotients ${\\mathrm {gr}}^k (\\mathcal {A}_G)$ and $\\mathrm {gr}^k(\\mathcal {L}_{\\mathrm {IA}(G)})$ , we use the Johnson homomorphisms of $\\mathrm {Aut}\\,G$ .", "We give their definitions and some of their properties with short proof since it seems that there are very few articles which discuss the definition of the Johnson homomorphisms of $\\mathrm {Aut}\\,G$ for any group $G$ .", "To begin with, we prepare some lemmas.", "For any $\\sigma \\in \\mathrm {Aut}\\,G$ and $x \\in G$ , set $s_{\\sigma }(x) :=x^{-1} x^{\\sigma } \\in G$ .", "By (1) of Theorem REF , we see that if $\\sigma \\in \\mathcal {A}_G(k)$ and $x \\in \\Gamma _G(l)$ , then $s_{\\sigma }(x) \\in \\Gamma _G(k+l)$ .", "Lemma 3.7 For any $\\sigma $ , $\\tau \\in \\mathrm {Aut}\\,G$ and $x$ , $y \\in G$ , we have $s_{\\sigma \\tau }(x) = s_{\\tau }(x) \\cdot s_{\\sigma }(x)^{\\tau } = s_{\\tau }(x) s_{\\sigma }(x) s_{\\tau }(s_{\\sigma }(x))$ .", "$s_{\\sigma }(xy) = y^{-1} s_{\\sigma }(x) y \\cdot s_{\\sigma }(y)=[y^{-1}, s_{\\sigma }(x)] s_{\\sigma }(x) s_{\\sigma }(y)$ .", "Proof.", "The equations immediately follow from $\\begin{split}s_{\\sigma \\tau }(x) & = x^{-1} x^{\\sigma \\tau } = x^{-1} x^{\\tau } \\cdot (x^{-1} x^{\\sigma })^{\\tau } \\\\& = x^{-1} x^{\\tau } \\cdot x^{-1} x^{\\sigma } \\cdot (x^{-1} x^{\\sigma })^{-1} \\cdot (x^{-1} x^{\\sigma })^{\\tau }, \\\\s_{\\sigma }(xy) & = y^{-1} x^{-1} x^{\\sigma } y^{\\sigma } = y^{-1} x^{-1} x^{\\sigma } y \\cdot y^{-1} y^{\\sigma }.\\end{split}$ $\\square $ As a corollary to Lemma REF , we obtain Corollary 3.8 For any $\\sigma $ , $\\tau \\in \\mathrm {Aut}\\,G$ and $x \\in G$ , $s_{\\mathrm {id}}(x) = 1$ , $s_{\\sigma ^{-1}}(x) = (s_{\\sigma }(x)^{-1})^{\\sigma ^{-1}}$ .", "$s_{[\\sigma ,\\tau ]}(x) = (s_{\\tau }(x)^{-1})^{\\tau ^{-1}} (s_{\\sigma }(x)^{-1})^{\\sigma ^{-1}\\tau ^{-1}}$ $s_{\\tau }(x)^{\\sigma ^{-1}\\tau ^{-1}} s_{\\sigma }(x)^{\\tau \\sigma ^{-1}\\tau ^{-1}}$ .", "Proof.", "Part (1) is trivial by the definition and (1) of Lemma REF .", "Part (2) is obtained from the observation $\\begin{split}s_{[\\sigma ,\\tau ]}(x) & = s_{\\sigma \\tau (\\tau \\sigma )^{-1}}(x) = s_{(\\tau \\sigma )^{-1}}(x) \\cdot s_{\\sigma \\tau }(x)^{\\sigma ^{-1} \\tau ^{-1}} \\\\& = (s_{\\tau \\sigma }(x)^{-1})^{\\sigma ^{-1} \\tau ^{-1}} (s_{\\sigma \\tau }(x))^{\\sigma ^{-1} \\tau ^{-1}}\\end{split}$ and (1) of Lemma REF .", "$\\square $ Furthermore, we see Lemma 3.9 For any $\\sigma \\in \\mathcal {A}_G(k)$ , $\\tau \\in \\mathcal {A}_G(l)$ and $x \\in G$ , $ s_{[\\sigma ,\\tau ]}(x) = s_{\\tau }(s_{\\sigma }(x)) - s_{\\sigma }(s_{\\tau }(x)) \\in \\mathrm {gr}^{k+l+1}(\\mathcal {L}_G).", "$ Proof.", "Since $s_{\\sigma }(x)^{-1} \\in \\Gamma _G(k+1)$ and $s_{\\tau }(x) \\in \\Gamma _G(l+1)$ , $ s_{\\sigma }(x)^{-1} s_{\\tau }(x) \\equiv s_{\\tau }(x) s_{\\sigma }(x)^{-1} \\pmod {\\Gamma _G(k+l+2)}, $ and hence $ (s_{\\sigma }(x)^{-1}s_{\\tau }(x))^{\\sigma ^{-1}\\tau ^{-1}} \\equiv (s_{\\tau }(x)s_{\\sigma }(x)^{-1})^{\\sigma ^{-1}\\tau ^{-1}} \\pmod {\\Gamma _G(k+l+2)}.", "$ Then, from (2) of Corollary REF , we have $\\begin{split}s_{[\\sigma ,\\tau ]}(x) & \\equiv (s_{\\tau }(x)^{-1})^{\\tau ^{-1}} s_{\\tau }(x)^{\\sigma ^{-1} \\tau ^{-1}} (s_{\\sigma }(x)^{-1})^{\\sigma ^{-1} \\tau ^{-1}}s_{\\sigma }(x)^{\\tau \\sigma ^{-1} \\tau ^{-1}} \\\\& = (s_{\\sigma ^{-1}}(s_{\\tau }(x)))^{\\tau ^{-1}} \\cdot (s_{\\tau }(s_{\\sigma }(x)))^{\\sigma ^{-1} \\tau ^{-1}} \\\\& = \\big {(} (s_{\\sigma }(s_{\\tau }(x)))^{-1} \\big {)}^{\\sigma ^{-1} \\tau ^{-1}} \\cdot (s_{\\tau }(s_{\\sigma }(x)))^{\\sigma ^{-1} \\tau ^{-1}}\\end{split}$ modulo $\\Gamma _G(k+l+2)$ .", "Since $\\sigma $ and $\\tau \\in \\mathrm {IA}(G)$ act on $\\mathrm {gr}^{k+l+1}(\\mathcal {L}_G)$ trivially, we obtain the required result.", "$\\square $ Now, for any $\\sigma \\in \\mathcal {A}_G(k)$ , consider a map $\\tilde{\\tau }_k(\\sigma ) :G^{\\mathrm {ab}} \\rightarrow \\mathrm {gr}^{k+1}({\\mathcal {L}}_{G})$ defined by $ x \\pmod {\\Gamma _G(2)} \\hspace{10.0pt} \\mapsto \\hspace{10.0pt} s_{\\sigma }(x) \\pmod {\\Gamma _G(k+2)} $ for any $x \\in G$ .", "This is well-defined.", "In fact, if $x \\equiv y \\pmod {\\Gamma _G(2)}$ , there exists some $c \\in \\Gamma _G(2)$ such that $y=xc$ .", "Then, by (2) of Lemma REF , we see $ s_{\\sigma }(y) = s_{\\sigma }(xc) =[c^{-1}, s_{\\sigma }(x)] s_{\\sigma }(x) s_{\\sigma }(c) \\equiv s_{\\sigma }(x) \\pmod {\\Gamma _G(k+2)}.", "$ Similarly, using (2) of Lemma REF , we see that $\\tilde{\\tau }_k(\\sigma )$ is a homomorphism between abelian groups.", "Furthermore, a map $\\tilde{\\tau }_k : \\mathcal {A}_G(k) \\rightarrow \\mathrm {Hom}_{\\mathbf {Z}}(G^{\\mathrm {ab}}, \\mathrm {gr}^{k+1}(\\mathcal {L}_G))$ defined by $ \\sigma \\hspace{5.0pt} \\mapsto \\hspace{5.0pt} \\tilde{\\tau }_k(\\sigma ) $ is a homomorphism.", "In fact, for any $\\sigma , \\tau \\in \\mathcal {A}_G(k)$ , by (1) of Lemma REF we have $\\begin{split}\\tilde{\\tau }_k(\\sigma \\tau )(x) & = s_{\\sigma \\tau }(x) = s_{\\tau }(x) s_{\\sigma }(x) s_{\\tau }(s_{\\sigma }(x)) \\\\& = s_{\\tau }(x) + s_{\\sigma }(x) = \\tilde{\\tau }_k(\\sigma )(x) + \\tilde{\\tau }_k(\\tau )(x)\\end{split}$ for any $x \\in G^{\\mathrm {ab}}$ .", "From the definition, it is easy to see that the kernel of $\\tilde{\\tau }_k$ is just $\\mathcal {A}_G(k+1)$ .", "Therefore $\\tilde{\\tau }_k$ induces an injective homomorphism $ \\tau _k : \\mathrm {gr}^k (\\mathcal {A}_G) \\hookrightarrow \\mathrm {Hom}_{\\mathbf {Z}}(G^{\\mathrm {ab}}, \\mathrm {gr}^{k+1}(\\mathcal {L}_G)).", "$ Definition 3.10 For each $k \\ge 1$ , we call the homomorphisms $\\tilde{\\tau }_k$ and $\\tau _k$ the $k$ -th Johnson homomorphisms of $\\mathrm {Aut}\\,G$ .", "Here we remark Lemma 3.11 For each $k \\ge 1$ , $\\tau _k$ is an $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ -equivariant homomorphism.", "Proof.", "For any $\\sigma \\in \\mathrm {Aut}\\,G$ and $\\tau \\in \\mathcal {A}_G(k)$ , we see $\\begin{split}\\tau _k(\\tau \\cdot \\sigma )(x) & = \\tau _k(\\sigma ^{-1} \\tau \\sigma )(x) = s_{\\sigma ^{-1} \\tau \\sigma }(x), \\\\(\\tau _k(\\tau ) \\cdot \\sigma )(x) & = ( \\tau _k(\\tau )(x^{\\sigma ^{-1}}) )^{\\sigma } = s_{\\tau }(x^{\\sigma ^{-1}})^{\\sigma } = ((x^{\\sigma ^{-1}})^{-1} x^{\\sigma ^{-1}\\tau })^{\\sigma } \\\\& = s_{\\sigma ^{-1} \\tau \\sigma }(x)\\end{split}$ for any $x \\in G^{\\mathrm {ab}}$ .", "Hence we have $\\tau _k(\\tau \\cdot \\sigma ) = \\tau _k(\\tau ) \\cdot \\sigma $ .", "This means $\\tau _k$ is an $\\mathrm {Aut}\\,G$ -equivariant homomorphism.", "Since $\\mathrm {IA}(G)$ acts on $\\mathrm {gr}^k (\\mathcal {A}_G)$ and $\\mathrm {Hom}_{\\mathbf {Z}}(G^{\\mathrm {ab}}, \\mathcal {L}_G(k+1))$ trivially, we obtain the required result.", "$\\square $ From the viewpoint of the study of the $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ -module structure of $\\mathrm {gr}^k(\\mathcal {A}_G)$ , it is a natural and basic problem to determine the image of $\\tau _k$ .", "In general, this is difficult even in the case where $G$ is a free group.", "(See Section .)", "Now, we recall the derivation algebra of a Lie algebra.", "(See Section 8 of Chapter II in [10] for details on the derivation algebra.)", "Definition 3.12 Let $\\mathfrak {g}$ be a Lie algebra over $\\mathbf {Z}$ .", "A $\\mathbf {Z}$ -linear map $f : \\mathfrak {g} \\rightarrow \\mathfrak {g}$ is called a derivation of $\\mathfrak {g}$ if $f$ satisfies $ f([a,b]) = [f(a),b]+ [a,f(b)] $ for any $a, b \\in \\mathfrak {g}$ .", "Let $\\mathrm {Der}(\\mathfrak {g})$ be a set of all derivations of $\\mathfrak {g}$ .", "Then $\\mathrm {Der}(\\mathfrak {g})$ has a Lie algebra structure over $\\mathbf {Z}$ .", "This Lie algebra $\\mathrm {Der}(\\mathfrak {g})$ is called the derivation algebra of $\\mathfrak {g}$ .", "We define a grading of $\\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_G))$ as follows.", "For $k \\ge 0$ , the degree $k$ part of $\\mathrm {Der}(\\mathcal {L}_G)$ is defined to be $ \\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_G))(k) := \\lbrace f \\in \\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_G)) \\,|\\, f(a)\\in \\mathrm {gr}^{k+1}(\\mathcal {L}_G), \\,\\,\\, a \\in G^{\\mathrm {ab}} \\rbrace .", "$ Then, we have $ \\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_G)) = \\bigoplus _{k \\ge 0} \\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_G))(k) $ as an $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ -module.", "For any derivation $f$ of $\\mathrm {gr}(\\mathcal {L}_G)$ , since the image of $f$ is completely determined by that of the degree 1 part $\\mathrm {gr}^1(\\mathcal {L}_G)=G^{\\mathrm {ab}}$ , the restriction of an element of $\\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_G))(k)$ to $\\mathrm {gr}^1(\\mathcal {L}_G)$ induces an $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ -equivariant embedding from $\\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_G))(k)$ into $\\mathrm {Hom}_{\\mathbf {Z}}(G^{\\mathrm {ab}}, \\mathrm {gr}^{k+1}(\\mathcal {L}_G))$ .", "In this paper, by this embedding, we consider $\\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_G))(k)$ as an $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ -submodule of $\\mathrm {Hom}_{\\mathbf {Z}}(G^{\\mathrm {ab}}, \\mathrm {gr}^{k+1}(\\mathcal {L}_G))$ for each $k \\ge 0$ .", "Let $\\mathrm {Der}^+(\\mathrm {gr}(\\mathcal {L}_G))$ be a graded Lie subalgebra of $\\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_G))$ with positive degree.", "Consider a direct sum of the Johnson homomorphisms $ \\bigoplus _{k \\ge 1} \\tau _k : \\mathrm {gr}(\\mathcal {A}_G) \\hookrightarrow \\mathrm {Der}^+(\\mathrm {gr}(\\mathcal {L}_G)).", "$ By Lemma REF , the map $\\bigoplus _{k \\ge 1} \\tau _k$ is a Lie algebra homomorphism.", "This is called the total Johnson homomorphism of $\\mathrm {Aut}\\,G$ .", "To end this section, we define the Johnson homomorphisms for the graded quotients $\\mathrm {gr}^k(\\mathcal {L}_{\\mathrm {IA}(G)})$ .", "Definition 3.13 The restriction of the homomorphism $\\tilde{\\tau }_k : \\mathcal {A}_G(k) \\rightarrow \\mathrm {Hom}_{\\mathbf {Z}}(G^{\\mathrm {ab}}$ , $\\mathrm {gr}^{k+1}(\\mathcal {L}_G))$ to $\\Gamma _{\\mathrm {IA}(G)}(k)$ induces a homomorphism $ \\tau _k^{\\prime } : \\mathrm {gr}^k (\\mathcal {L}_{\\mathrm {IA}(G)}) \\rightarrow \\mathrm {Hom}_{\\mathbf {Z}}(G^{\\mathrm {ab}}, \\mathrm {gr}^{k+1}(\\mathcal {L}_G)).", "$ We also call $\\tau _k^{\\prime }$ the $k$ -th Johnson homomorphism of $\\mathrm {Aut}\\,G$ .", "We remark that each of $\\tau _k^{\\prime }$ is an $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ -equivariant homomorphism, by the same argument as for $\\tau _k$ .", "The Andreadakis-Johnson filtration $\\mathcal {A}_G$ coincides with the lower central series $\\mathcal {L}_{\\mathrm {IA}(G)}$ of $\\mathrm {IA}(G)$ if and only if all of the Johnson homomorphisms $\\tau _k^{\\prime }$ are injective.", "By an argument similar to that of $\\bigoplus _{k \\ge 1} \\tau _k$ , we see that a direct sum $ \\bigoplus _{k \\ge 1} \\tau _k^{\\prime } : \\mathrm {gr}(\\mathcal {L}_{\\mathrm {IA}(G)}) \\rightarrow \\mathrm {Der}^+(\\mathrm {gr}(\\mathcal {L}_G)) $ of $\\tau _k^{\\prime }$ is an $\\mathrm {Aut}\\,G/\\mathrm {IA}(G)$ -equivariant Lie algebra homomorphism.", "We also call it the total Johnson homomorphism of $\\mathrm {Aut}\\,G$ .", "In this section we apply the argument in Section to the case where $G$ is a free group $F_n$ of rank $n$ with a basis $x_1, \\ldots , x_n$ .", "First, we denote the abelianization of $F_n$ by $H$ , and its dual group by $H^* :=\\mathrm {Hom}_{\\mathbf {Z}}(H,\\mathbf {Z})$ .", "If we fix the basis of $H$ as a free abelian group induced from the basis $x_1, \\ldots , x_n$ of $F_n$ , we can identify $\\mathrm {Aut}\\,F_n^{\\mathrm {ab}}=\\mathrm {Aut}(H)$ with the general linear group $\\mathrm {GL}(n,\\mathbf {Z})$ .", "By a classical work due to Nielsen, it is well-known that the homomorphism $\\rho : \\mathrm {Aut}\\,F_n \\rightarrow \\mathrm {GL}(n,\\mathbf {Z})$ induced from the abelianization of $F_n$ is surjective.", "(See Proposition 4.4 in [35], for example.)", "Hence we can identify $\\mathrm {Aut}(H)/\\mathrm {IA}(F_n)$ with $\\mathrm {GL}(n,\\mathbf {Z})$ .", "Namely, we have a group extension $ 1 \\rightarrow \\mathrm {IA}(F_n) \\rightarrow \\mathrm {Aut}\\,F_n \\rightarrow \\mathrm {GL}(n,\\mathbf {Z}) \\rightarrow 1.", "$ In the following, for simplicity, we omit the capital “$F$ \" from a symbol attached with $F_n$ if there is no confusion.", "For example, we write $\\Gamma _n(k)$ , $\\mathrm {gr}^k(\\mathcal {L}_n)$ , $\\mathrm {IA}_n$ , $\\mathcal {A}_n(k)$ and $\\mathrm {gr}^k(\\mathcal {A}_n)$ for $\\Gamma _{F_n}(k)$ , $\\mathrm {gr}^k(\\mathcal {L}_{F_n})$ , $\\mathrm {IA}(F_n)$ and $\\mathrm {gr}^k(\\mathcal {A}_{F_n})$ respectively.", "Although our ultimate purpose is to clarify the $\\mathrm {GL}(n,\\mathbf {Z})$ -module structures of $\\mathrm {gr}^k(\\mathcal {A}_n)$ , or equivalently those of $\\mathrm {Coker}(\\tau _k)$ , to take advantage of the representation theory for the general linear group $\\mathrm {GL}(n,\\mathbf {Q})$ , we consider the rationalization as follows.", "In general, $\\mathrm {GL}(n,\\mathbf {Z})$ can be considered as a subgroup of $\\mathrm {GL}(n,\\mathbf {Q})$ in a natural way.", "Then the actions of $\\mathrm {GL}(n,\\mathbf {Z})$ on $\\mathrm {gr}_{\\mathbf {Q}}^k(\\mathcal {A}_n)$ , $H_{\\mathbf {Q}}^* \\otimes _{\\mathbf {Q}} \\mathrm {gr}_{\\mathbf {Q}}^{k+1}(\\mathcal {L}_n)$ and $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ naturally extend to those of $\\mathrm {GL}(n,\\mathbf {Q})$ .", "We sum up several basic problems for the study of the Johnson homomorphisms of $\\mathrm {Aut}\\,F_n$ .", "Determine the $\\mathrm {GL}(n,\\mathbf {Q})$ -irreducible decompositions of $ \\mathrm {Im}(\\tau _{k,\\mathbf {Q}}) \\cong \\mathrm {gr}_{\\mathbf {Q}}^k(\\mathcal {A}_n) \\hspace{10.0pt} \\mathrm {and} \\hspace{10.0pt} \\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}).", "$ Furthermore, give a formula for the dimension of $\\mathrm {gr}_{\\mathbf {Q}}^k(\\mathcal {A}_n)$ , which is equal to the rank of $\\mathrm {gr}^k(\\mathcal {A}_n)$ .", "(Andreadakis's conjecture) Determine whether $\\mathcal {A}_n(k)$ coincides with $\\Gamma _{\\mathrm {IA}_n}(k)$ or not, for $n \\ge 3$ and $k \\ge 2$ .", "Determine whether each of $\\mathcal {A}_n(k)$ is finitely generated or not, for $k \\ge 2$ .", "Moreover, what about $H_1(\\mathcal {A}_n(k),\\mathbf {Z})$ ?" ], [ "Free Lie algebras and their derivations", "By a classical work of Magnus, it is known that each of $\\mathrm {gr}^k(\\mathcal {L}_n)$ is a free abelian group, and that the graded Lie algebra $\\mathrm {gr}(\\mathcal {L}_n)$ is isomorphic to the free Lie algebra generated by $H$ .", "(For example, see [38] or [54] for details for a free Lie algebra.)", "Witt [67] gave the rank of $\\mathrm {gr}^k(\\mathcal {L}_n)$ by $\\mathrm {rank}_{\\mathbf {Z}}(\\mathrm {gr}^k(\\mathcal {L}_n))=\\frac{1}{k} \\sum _{d | k} \\mu (d) n^{\\frac{k}{d}}$ where $\\mu $ is the M$\\ddot{\\mathrm {o}}$ bius function.", "The graded Lie algebra $\\mathrm {gr}(\\mathcal {L}_n)$ is considered as a Lie subalgebra of the tensor algebra generated by $H$ as follows.", "Let $ T(H):= \\mathbf {Z}\\oplus H \\oplus H^{\\otimes 2} \\oplus \\cdots $ be the tensor algebra of $H$ over $\\mathbf {Z}$ .", "Then $T(H)$ is the universal enveloping algebra of the free Lie algebra $\\mathrm {gr}(\\mathcal {L}_n)$ , and the natural map $\\iota : \\mathrm {gr}(\\mathcal {L}_n) \\rightarrow T(H)$ defined by $ [X,Y] \\mapsto X \\otimes Y - Y \\otimes X $ for $X$ , $Y \\in \\mathrm {gr}(\\mathcal {L}_n)$ is an injective graded Lie algebra homomorphism by the Poincaré-Birkoff-Witt's theorem.", "We denote by $\\iota _k$ the homomorphism which is the degree $k$ part of $\\iota $ , and consider $\\mathrm {gr}^k(\\mathcal {L}_n)$ as a $\\mathrm {GL}(n,\\mathbf {Z})$ -submodule $H^{\\otimes k}$ through $\\iota _k$ .", "Consider the derivation algebra $\\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_n))$ of the free Lie algebra $\\mathrm {gr}(\\mathcal {L}_n)$ .", "By the universality of the free Lie algebra $\\mathrm {gr}(\\mathcal {L}_n)$ , the embedding $ \\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_n))(k) \\hookrightarrow H^* {\\otimes }_{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n) $ as mentioned above is surjective.", "Namely, the degree $k$ -part $\\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_n))(k)$ is considered as $ \\mathrm {Hom}_{\\mathbf {Z}}(H,\\mathrm {gr}^{k+1}(\\mathcal {L}_n)) = H^* {\\otimes }_{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n) $ for each $k \\ge 1$ .", "(See Section 8 of Chapter II in [10] for details.)" ], [ "The IA-automorphism group of $F_n$", "Now, we consider the IA-automorphism group of $F_n$ .", "It is known by work of Nielsen [47] that $\\mathrm {IA}_2$ coincides with the inner automorphsim group $\\mathrm {Inn}\\,F_2$ of $F_2$ .", "Namely, $\\mathrm {IA}_2$ is a free group of rank 2.", "However, $\\mathrm {IA}_n$ for $n \\ge 3$ is much larger than the inner automorphism group $\\mathrm {Inn}\\,F_n$ .", "In fact, Magnus [36] showed Theorem 4.1 (Magnus, [36]) for any $n \\ge 3$ , the IA-automorphism group $\\mathrm {IA}_n$ of $F_n$ is finitely generated by automorphisms $ K_{ij} : {\\left\\lbrace \\begin{array}{ll}x_i &\\mapsto {x_j}^{-1} x_i x_j, \\\\x_t &\\mapsto x_t, \\hspace{40.0pt} (t \\ne i)\\end{array}\\right.", "}$ for distinct $i$ , $j \\in \\lbrace 1, 2, \\ldots , n \\rbrace $ and $ K_{ijk} : {\\left\\lbrace \\begin{array}{ll}x_i &\\mapsto x_i x_j x_k {x_j}^{-1} {x_k}^{-1}, \\\\x_t &\\mapsto x_t, \\hspace{40.0pt} (t \\ne i)\\end{array}\\right.", "}$ for distinct $i$ , $j$ , $k \\in \\lbrace 1, 2, \\ldots , n \\rbrace $ such that $j<k$ .", "For any $n \\ge 3$ , although a generating set of $\\mathrm {IA}_n$ is well known as above, no presentation for $\\mathrm {IA}_n$ is known.", "For $n=3$ , Krstić and McCool [31] showed that $\\mathrm {IA}_3$ is not finitely presentable.", "For $n \\ge 4$ , it is also not known whether $\\mathrm {IA}_n$ is finitely presentable or not." ], [ "Johnson homomorphisms of $\\mathrm {Aut}\\,F_n$", "In this subsection, we sum up some results for the Johnson homomorphisms of $\\mathrm {Aut}\\,F_n$ .", "In the following, we denote by $S^k M$ and $\\Lambda ^k M$ the symmetric tensor product and the exterior product of a $\\mathbf {Z}$ -module $M$ of degree $k$ respectively." ], [ "$\\tau _1$", "Andreadakis [1] calculated the images of the Magnus generators of $\\mathrm {IA}_n$ by the first Johnson homomorphism as $\\tau _1(K_{ij})= x_i^* \\otimes [x_i,x_j], \\hspace{10.0pt} \\tau _1(K_{ijk})= x_i^* \\otimes [x_j,x_k],$ and showed that $\\tau _1$ is surjective.", "Furthermore, recently, Cohen-Pakianathan [14], [15]�CFarb [18] and Kawazumi [27] independently showed that $\\tau _1$ induces the abelianization of $\\mathrm {IA}_n$ .", "Namely, for any $n \\ge 3$ , we have $\\mathrm {IA}_n^{\\mathrm {ab}} \\cong H^* \\otimes _{\\mathbf {Z}} \\Lambda ^2 H$ as a $\\mathrm {GL}(n,\\mathbf {Z})$ -module.", "Here we remark that using (REF ), we can recursively calculate $\\tau _k(\\sigma )=\\tau _k^{\\prime }(\\sigma )$ for any $\\sigma \\in \\Gamma _{\\mathrm {IA}_n}(k)$ .", "These computations are perhaps most easily explained with some examples, so we give two here.", "For the standard basis $x_1, \\ldots , x_n$ of $H$ induced from the generators of $F_n$ , let $x_1^*, \\ldots , x_n^*$ be the dual basis of $H^*$ .", "For distinct $1 \\le i, j, k, l \\le n$ , we have $\\begin{split}\\tau _2([K_{ij},K_{jik}]) &= x_i^* \\otimes ([s_{K_{jik}}(x_i), x_j] + [x_i, s_{K_{jik}}(x_j)]) \\\\& \\hspace{40.0pt} - x_j^* \\otimes ([s_{K_{ij}}(x_i),x_k] + [x_i, s_{K_{ij}}(x_k)] ), \\\\&= x_i^* \\otimes [x_i,[x_i,x_k]] - x_j^* \\otimes [[x_i,x_j],x_k]\\end{split}$ and $\\begin{split}\\tau _3 & ([K_{ij}, K_{jik},K_{il}]) \\\\&= x_i^* \\otimes ([s_{K_{il}}(x_i),[x_i,x_k]] + [x_i,[s_{K_{il}}(x_i),x_k]]+ [x_i,[x_i,s_{K_{il}}(x_k)]]), \\\\& \\hspace{20.0pt} - x_j^* \\otimes ([[s_{K_{il}}(x_i),x_j],x_k] + [[x_i,s_{K_{il}}(x_j)],x_k]+ [[x_i,x_j],s_{K_{il}}(x_k)]) \\\\& \\hspace{20.0pt} - x_i^* \\otimes ([s_{[K_{ij},K_{jik}]}(x_i),x_l]+ [x_i, s_{[K_{ij},K_{jik}]}(x_l)] ) \\\\&= x_i^* \\otimes [[x_i,x_l],[x_i,x_k]] + x_i^* \\otimes [x_i,[[x_i,x_l],x_k]] - x_j^* \\otimes [[[x_i,x_l],x_j],x_k] \\\\& \\hspace{20.0pt} - x_i^* \\otimes [[x_i,[x_i,x_k]],x_l].\\end{split}$" ], [ "$\\tau _2$ and {{formula:c4e22347-9169-4bd0-9b5a-fce24b38f3ec}}", "To begin with, we mention the difference between the Johnson filtration $\\lbrace \\mathcal {A}_n(k) \\rbrace $ and the lower central series $\\lbrace \\Gamma _{\\mathrm {IA}_n}(k) \\rbrace $ of $\\mathrm {IA}_n$ .", "In [1], Andreadakis showed that $\\mathcal {A}_2(k) = \\Gamma _{\\mathrm {IA}_2}(k)$ for any $k \\ge 1$ , and that $\\mathcal {A}_3(3) = \\Gamma _{\\mathrm {IA}_3}(3)$ .", "Since $\\tau _1$ is the abelianization of $\\mathrm {IA}_n$ as mentioned above, we see that $\\mathcal {A}_n(2) = \\Gamma _{\\mathrm {IA}_n}(2)$ for any $n \\ge 2$ .", "This fact was originally shown by Bachmuth in [6].", "On the other hand, Pettet [51] showed that $\\Gamma _{\\mathrm {IA}_n}(3)$ has at most finite index in $\\mathcal {A}_n(3)$ for $n \\ge 3$ .", "Conjecture (Andreadakis) $\\mathcal {A}_n(k) = \\Gamma _{\\mathrm {IA}_n}(k)$ for any $k \\ge 3$ .", "There are, however, few results for the difference between $\\mathcal {A}_n(k)$ and $\\Gamma _{\\mathrm {IA}_n}(k)$ .", "Based on the above facts, we can directly calculate the image of $\\tau _2$ and $\\tau _{3,\\mathbf {Q}}$ .", "Pettet [51] showed Theorem 4.2 (Pettet, [51]) For any $n \\ge 3$ , $ \\mathrm {Coker}(\\tau _{2,\\mathbf {Q}}) \\cong S^2 H_{\\mathbf {Q}}.", "$ More precisely, she gave the $\\mathrm {GL}(n,\\mathbf {Q})$ -decomposition of $\\mathrm {gr}_{\\mathbf {Q}}^2(\\mathcal {A}_n)$ .", "On the other hand, it is also known by work of Morita, mentioned in Subsection REF below, that $S^2 H_{\\mathbf {Q}}$ appears in $\\mathrm {Coker}(\\tau _{2,\\mathbf {Q}})$ .", "Then combining these two facts, we obtain Theorem REF .", "In [60], we show that the isomorphism above holds over $\\mathbf {Z}$ .", "Namely, we have $\\mathrm {Coker}(\\tau _{2}) \\cong S^2 H$ .", "In addition to the above, Pettet [51] also gave the $\\mathrm {GL}(n,\\mathbf {Q})$ -decomposition of the image of the rational cup product $ \\cup : \\Lambda ^2 H^1(\\mathrm {IA}_n,\\mathbf {Q}) \\rightarrow H^2(\\mathrm {IA}_n, \\mathbf {Q}).", "$ We remark that it is not determined whether $H^2(\\mathrm {IA}_n,\\mathbf {Q})$ coincides with $\\mathrm {Im}(\\cup )$ or not.", "In [60], using the fact that $\\Gamma _{\\mathrm {IA}_n}(3)$ has at most finite index in $\\mathcal {A}_n(3)$ , by work of Pettet mentioned above, we showed Theorem 4.3 (Satoh, [60]) For any $n \\ge 3$ , $ \\mathrm {Coker}(\\tau _{3,\\mathbf {Q}}) \\cong S^3 H_{\\mathbf {Q}} \\oplus \\Lambda ^3 H_{\\mathbf {Q}}.", "$ In general, for any $n \\ge 3$ and $k \\ge 4$ , the $\\mathrm {GL}(n,\\mathbf {Q})$ -module structures of ${\\mathrm {gr}}_{\\mathbf {Q}}^k(\\mathcal {A}_n)$ and $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ are not yet determined." ], [ "Morita's trace maps", "In the calculation of the cokernel of $\\tau _{k,\\mathbf {Q}}$ for $k=2$ and 3 above, we see that $S^k H_{\\mathbf {Q}}$ appears in its irreducible decomposition.", "It is known that $S^k H_{\\mathbf {Q}}$ appears in the irreducible decomposition of $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ for any $k \\ge 2$ , due to Morita.", "In this subsection, we review Morita's pioneer work to study $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ for a general $k \\ge 2$ .", "For each $k \\ge 1$ , let $\\varphi ^k : H^* {\\otimes }_{\\mathbf {Z}} H^{\\otimes (k+1)} \\rightarrow H^{\\otimes k}$ be the contraction map with respect to the first component, defined by $ x_i^* \\otimes x_{j_1} \\otimes \\cdots \\otimes x_{j_{k+1}} \\mapsto x_i^*(x_{j_1}) \\, \\cdot x_{j_2} \\otimes \\cdots \\otimes \\cdots \\otimes x_{j_{k+1}}.", "$ Using the natural embedding ${\\iota }_{k+1} : \\mathrm {gr}^{k+1}(\\mathcal {L}_n) \\rightarrow H^{\\otimes (k+1)}$ , we construct a $\\mathrm {GL}(n,\\mathbf {Z})$ -equivariant homomorphism $ \\Phi ^k = {\\varphi }^{k} \\circ ({id}_{H^*} \\otimes {\\iota }_{k+1}): H^* {\\otimes }_{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n) \\rightarrow H^{\\otimes k}.", "$ We also call $\\Phi ^k$ a contraction map.", "Definition 4.4 For each $k \\ge 2$ , the composition map of $\\Phi ^k$ and the natural projection $H^{\\otimes k} \\rightarrow S^k H$ is called Morita's trace map of degree $k$ , denoted by $ \\mathrm {Tr}_{[k]} : H^* {\\otimes }_{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n) \\rightarrow S^k H. $ We remark that Morita [42] originally defined Morita's trace using the Magnus representation of $\\mathrm {Aut}\\,F_n$ , and that Morita's original definition is equivalent to the above.", "(See Subsection REF .)", "Morita proved Theorem 4.5 (Morita, [45]) For any $n \\ge 3$ and $k \\ge 2$ , $\\mathrm {Tr}_{[k]}$ is surjective.", "$\\mathrm {Tr}_{[k]} \\circ \\tau _k \\equiv 0$ .", "As a corollary, we obtain Corollary 4.6 For any $n \\ge 3$ and $k \\ge 2$ , $S^k H_{\\mathbf {Q}}$ appears in the irreducible decomposition of $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ .", "We call each of $S^k H_{\\mathbf {Q}}$ the Morita obstruction.", "Here the term “obstruction\" means an obstruction for the surjectivity of the Johnson homomorphism $\\tau _k$ .", "At the present stage, there are few results on irreducible components of $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ for general $k \\ge 2$ except for the Morita obstruction.", "We remark that recently we [16] showed that the multiplicity of the Morita obstruction $S^k H_{\\mathbf {Q}}$ in $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ is one.", "(See also Theorem REF .)" ], [ "Cokernels of $\\tau _k^{\\prime }$", "The goal of this subsection is to give an upper bound on $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ as a $\\mathrm {GL}(n,\\mathbf {Q})$ -module.", "In order to do this, we consider the Johnson homomorphism $ \\tau _k^{\\prime } : \\mathrm {gr}^k(\\mathcal {L}_{\\mathrm {IA}_n}) \\rightarrow H^* \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n).", "$ Since $\\tau _k^{\\prime }$ factors through $\\tau _k$ , we see that $\\mathrm {Im}(\\tau _k^{\\prime }) \\subset \\mathrm {Im}(\\tau _k)$ .", "Hence using the representation theory for $\\mathrm {GL}(n,\\mathbf {Q})$ , we can consider $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ as a $\\mathrm {GL}(n,\\mathbf {Q})$ -submodule of $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime })$ .", "This means that $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime })$ can be regarded as an upper bound on $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ .", "There are at least three reasons to study $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime })$ .", "The first one is that we can directly compute $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime })$ using finitely many generators of $\\mathrm {gr}_{\\mathbf {Q}}^k(\\mathcal {L}_{\\mathrm {IA}_n})$ induced from the Magnus generators of $\\mathrm {IA}_n$ and Lemma REF .", "The second one is that this research would be applied to the study the Andreadakis conjecture.", "If the conjecture is true, we get $\\mathrm {Coker}(\\tau _{k}^{\\prime })=\\mathrm {Coker}(\\tau _{k})$ for all $k \\ge 1$ .", "Hence it seems to be important to determine the $\\mathrm {GL}(n,\\mathbf {Q})$ -module structure of $\\mathrm {Coker}(\\tau _{k, \\mathbf {Q}}^{\\prime })$ on the study of the difference between $\\lbrace \\mathcal {A}_n(k) \\rbrace $ and $\\lbrace \\Gamma _{\\mathrm {IA}_n}(k) \\rbrace $ .", "The final one is the most interesting motivation for a topological application.", "By using the structure of $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime })$ , we can study the cokernel of the Johnson homomorphisms of the mapping class group of a surface.", "This is discussed more precisely in Section later.", "For $1 \\le k \\le 3$ , we have $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime })=\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ , and hence they have been completely determined.", "In [62], we give the irreducible decomposition of $\\mathrm {Coker}(\\tau _{4,\\mathbf {Q}}^{\\prime })$ for $n \\ge 6$ .", "Furthermore, in [64], we determined $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime })$ in a stable range as follows.", "Let $\\mathcal {C}_n(k)$ be a quotient module of $H^{\\otimes k}$ by the action of the cyclic group $\\mathrm {Cyc}_k$ of order $k$ on the components: $ \\mathcal {C}_n(k) := H^{\\otimes k} \\big {/} \\langle a_1 \\otimes a_2 \\otimes \\cdots \\otimes a_k - a_2 \\otimes a_3 \\otimes \\cdots \\otimes a_k \\otimes a_1\\,|\\, a_i \\in H \\rangle .", "$ Then we have Theorem 4.7 (Satoh, [64]) For any $k \\ge 2$ and $n \\ge k+2$ , $ \\mathrm {Coker}(\\tau _{k, \\mathbf {Q}}^{\\prime }) \\cong \\mathcal {C}_n^{\\mathbf {Q}}(k).", "$ In our recent paper [16], which is a joint work with Naoya Enomoto, we gave a combinatorial description of the $\\mathrm {GL}(n,\\mathbf {Q})$ -irreducible decomposition of $\\mathcal {C}_n^{\\mathbf {Q}}(k)$ .", "We remark that, as a $\\mathrm {GL}(n,\\mathbf {Q})$ -module, $\\mathcal {C}_n^{\\mathbf {Q}}(k)$ is isomorphic to the invariant part $a_n(k):=(H_{\\mathbf {Q}}^{\\otimes k})^{\\mathrm {Cyc}_k}$ of $H_{\\mathbf {Q}}^{\\otimes k}$ by the action of the cyclic group $\\mathrm {Cyc}_k$ of order $k$ .", "Namely, the cokernel $\\mathrm {Coker}(\\tau _{k, \\mathbf {Q}}^{\\prime })$ is isomorphic to Kontsevich's $a_n(k)$ as a $\\mathrm {GL}(n,\\mathbf {Q})$ -module.", "We also remark that in our notation $a_n(k)$ is considered for any $n \\ge 2$ in contrast to Kontsevich's notation for even $n=2g$ .", "(See [29] and [30].)", "Now we give tables of the irreducible decompositions of $\\mathcal {C}_n^\\mathbf {Q}(k)$ and $\\mathrm {Im}(\\tau _{k,\\mathbf {Q}}^{\\prime })$ for $1 \\le k \\le 7$ .", "Table: NO_CAPTIONIn the table above, $(\\lambda )$ denotes the irreducible polynomial representation $L^{(\\lambda )}$ of $\\mathrm {GL}(n,\\mathbf {Q})$ corresponding to a Young tableau $\\lambda $ .", "(For the notation, see [16] for details.)", "${\\begin{array}{|c|c|c|}\\hline k & \\text{polynomial part of} \\ \\mathrm {Im}(\\tau _{k,\\mathbf {Q}}^{\\prime }) & \\text{non-polynomial part of} \\ \\mathrm {Im}(\\tau _{k,\\mathbf {Q}}^{\\prime }) \\\\\\hline 1 & (1) & (1,1) \\\\\\hline 2 & (1^2) & (2,1) \\\\\\hline 3 & 2(2,1) & (3,1) \\oplus (2,1^2) \\\\\\hline 4 & 3(3,1) \\oplus (2^2) \\oplus 2(2,1^2) \\oplus (1^4) & (4,1) \\oplus (3,2) \\oplus (3,1^2) \\\\& & \\oplus (2^2,1) \\oplus (2,1^3)\\\\\\hline 5 & 4(4,1) \\oplus 4(3,2) \\oplus 4(3,1^2) & (5,1) \\oplus (4,2) \\oplus 2(4,1^2) \\oplus (3^2) \\\\& \\oplus 4(2^2,1) \\oplus 4(2,1^3)& 3(3,2,1) \\oplus (3,1^3) \\oplus 2(2^2,1^2) \\oplus (2,1^4)\\\\\\hline 6 & 5(5,1) \\oplus 7(4,2) \\oplus 8(4,1^2) \\oplus 4(3^2) & (6,1) \\oplus 2(5,2) \\oplus 2(5,1^2) \\oplus 2(4,3) \\\\& \\oplus 14(3,2,1) \\oplus 9(3,1^3) \\oplus 3(2^3) & \\oplus 5(4,2,1) \\oplus 3(4,1^3) \\oplus 3(3^2,1) \\\\& \\oplus 8(2^2,1^2) \\oplus 4(2,1^4) \\oplus (1^6) & \\oplus 3(3,2^2) \\oplus 5(3,2,1^2) \\oplus 2(3,1^4) \\\\& & \\oplus 2(2^3,1) \\oplus 2(2^2,1^3) \\oplus (2,1^5)\\\\\\hline 7 & 6(6,1) \\oplus 12(5,2) \\oplus 12(5,1^2) \\oplus 12(4,3) & (7,1) \\oplus 2(6,2) \\oplus 3(6,1^2) \\oplus 4(5,3) \\\\& \\oplus 30(4,2,1) \\oplus 18(4,1^3) \\oplus 18(3^2,1) & \\oplus 8(5,2,1) \\oplus 4(5,1^3) \\oplus (4^2) \\oplus 9(4,3,1) \\\\& \\oplus 18(3,2^3) \\oplus 30(3,2,1^2) \\oplus 12(3,1^4) & \\oplus 6(4,2^2) \\oplus 12(4,2,1^2) \\oplus 4(4,1^4) \\oplus 6(3^2,2) \\\\& \\oplus 12(2^3,1) \\oplus 12(2^2,1^3) \\oplus 6(2,1^5) & \\oplus 9(3,2^2,1) \\oplus 8(3,2,1^3) \\oplus 3(3,1^5) \\oplus (2^4)\\\\& & \\oplus 4(2^3,1^2) \\oplus 2(2^2,1^4) \\oplus (2,1^6) \\\\\\hline \\end{array}}$ In the polynomial part of the table above, $(\\lambda )$ denotes the irreducible polynomial representation $L^{(\\lambda )}$ , and in the non-polynomial part, $(\\mu )$ denotes the irreducible non-polynomial representation $L^{\\lbrace \\mu ;(1)\\rbrace }$ of $\\mathrm {GL}(n,\\mathbf {Q})$ corresponding to Young tableaux $\\lambda $ and $\\mu $ .", "(For the notation, see [16] for details.)", "By Corollary REF , the irreducible module $S^k H_{\\mathbf {Q}} = (k)$ appears in $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime })$ for each $k \\ge 2$ .", "On the other hand, in [60] we showed that the irreducible module $\\Lambda ^k H_{\\mathbf {Q}} = (1^k)$ appears in $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime })$ for each odd $k \\ge 3$ .", "In [16], we showed that their multiplicities of them in $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime }) \\cong \\mathcal {C}_n^\\mathbf {Q}(k)$ are one.", "Namely, Theorem 4.8 (Enomoto and Satoh, [16]) For $k \\ge 2$ and $n \\ge k+2$ , $[\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime }): S^k H_{\\mathbf {Q}}]=1$ .", "$[\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime }): \\Lambda ^k H_{\\mathbf {Q}}]=1$ if $k$ is odd.", "From this theorem, we see that the multiplicity of the Morita obstruction $S^k H_{\\mathbf {Q}}$ in $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}})$ is also one.", "Hence, the Morita obstruction is essentially unique in the cokernel of the rational Johnson homomorphism $\\tau _{k,\\mathbf {Q}}$ .", "In this section, we consider a metabelian analogue of Section .", "The main purpose of this section is to give the irreducible decomposition of the images and the cokernels of the Johnson homomorphisms of the automorphism groups of free metabelian groups.", "The quotient group of the free group $F_n$ by its second derived group $[[F_n,F_n],[F_n,F_n]]$ is called a free metabelian group of rank $n$ , denoted by $F_n^M$ .", "Then, $(F_n^M)^{\\mathrm {ab}}=H$ , and hence $\\mathrm {Aut}\\,(F_n^M)^{\\mathrm {ab}}=\\mathrm {Aut}(H)=\\mathrm {GL}(n,\\mathbf {Z})$ .", "Since the surjective homomorphism $\\mathrm {Aut}\\,F_n \\rightarrow \\mathrm {GL}(n,\\mathbf {Z})$ factors through $\\mathrm {Aut}\\,F_n^M$ , a homomorphism $\\mathrm {Aut}\\,F_n^M \\rightarrow \\mathrm {GL}(n,\\mathbf {Z})$ induced from the abelianization of $F_n^M$ is surjective.", "Thus, we also identify $\\mathrm {Aut}\\,F_n^M/\\mathrm {IA}(F_n^M)$ with $\\mathrm {GL}(n,\\mathbf {Z})$ .", "In the following, for simplicity, we omit the capital “$F$ \" from a symbol attached with $F_n^M$ if there is no confusion.", "For example, we write $\\Gamma _n^M(k)$ , $\\mathrm {gr}^k(\\mathcal {L}_n^M)$ , $\\mathrm {IA}_n$ , $\\mathcal {A}_n^M(k)$ and $\\mathrm {gr}^k(\\mathcal {A}_n^M)$ for $\\Gamma _{F_n^M}(k)$ , $\\mathrm {gr}^k(\\mathcal {L}_{F_n^M})$ , $\\mathrm {IA}(F_n^M)$ and $\\mathrm {gr}^k(\\mathcal {A}_{F_n^M})$ respectively." ], [ "Free metabelian Lie algebras and their derivations", "By a remarkable work of Chen [12], it is known that each of $\\mathrm {gr}^k(\\mathcal {L}_n^M)$ is a free abelian group, and that the graded Lie algebra $\\mathrm {gr}(\\mathcal {L}_n^M)$ is isomorphic to the free metabelian Lie algebra generated by $H$ .", "Chen [12] also gave the rank of $\\mathrm {gr}^k(\\mathcal {L}_n^M)$ .", "That is, $\\mathrm {rank}_{\\mathbf {Z}}(\\mathrm {gr}^k(\\mathcal {L}_n^M))=(k-1) \\binom{n+k-2}{k}.$ Consider the derivation algebra $\\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_n^M))$ of the free Lie algebra $\\mathrm {gr}(\\mathcal {L}_n^M)$ .", "By the universality of the free metabelian Lie algebra $\\mathrm {gr}(\\mathcal {L}_n^M)$ , the embedding $ \\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_n^M))(k) \\hookrightarrow H^* {\\otimes }_{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n^M) $ as mentioned above is surjective.", "Namely, the degree $k$ -part $\\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_n^M))(k)$ is considered as $ \\mathrm {Hom}_{\\mathbf {Z}}(H,\\mathrm {gr}^{k+1}(\\mathcal {L}_n^M)) = H^* {\\otimes }_{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n^M) $ for each $k \\ge 1$ ." ], [ "The IA-automorphism group of $F_n^M$", "Consider a homomorphism $\\mathrm {Aut}\\,F_n \\rightarrow \\mathrm {Aut}\\,F_n^M$ induced from the action of $\\mathrm {Aut}\\,F_n$ on $F_n^M$ .", "Restricting it to $\\mathrm {IA}_n$ , we obtain a homomorphism $\\nu : \\mathrm {IA}_n \\rightarrow \\mathrm {IA}_n^M$ .", "Bachmuth and Mochizuki [7] showed that $\\nu $ is not surjective for $n =3$ , and that $\\mathrm {IA}_3^M$ is not finitely generated.", "They also showed in [8] that $\\nu $ is surjective for $n \\ge 4$ .", "Hence for $n \\ge 4$ , $\\mathrm {IA}_n^M$ is finitely generated by the images of the Magnus generators of $\\mathrm {IA}_n$ .", "It is, however, not known whether $\\mathrm {IA}_n^M$ is finitely presentable or not for $n \\ge 4$ ." ], [ "Johnson homomorphisms of $\\mathrm {Aut}\\,F_n^M$", "In the following, we consider the case where $n \\ge 4$ unless otherwise noted.", "To begin with, let $\\mathcal {K}_n$ be the kernel of the homomorphism $\\nu : \\mathrm {IA}_n \\rightarrow \\mathrm {IA}_n^M$ .", "Then by the definition of the Johnson filtration of $\\mathrm {Aut}\\,F_n$ , we see that $\\mathcal {K}_n \\subset \\mathcal {A}_n(2)=[\\mathrm {IA}_n, \\mathrm {IA}_n]$ .", "Hence, we obtain $ (\\mathrm {IA}_n^M)^{\\mathrm {ab}} \\cong \\mathrm {IA}_n^{\\mathrm {ab}} \\cong H^* \\otimes _{\\mathbf {Z}} \\Lambda ^2 H. $ Furthermore, the first Johnson homomorphism $\\tau _1$ of $\\mathrm {Aut}\\,F_n^M$ is just the abelianization of $\\mathrm {IA}_n^M$ .", "In [61], by showing that the Morita trace $\\mathrm {Tr}_{[k]} : H^* {\\otimes }_{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n) \\rightarrow S^k H$ factors through a homomorphism $\\mathrm {gr}^k(\\mathcal {A}_n) \\rightarrow \\mathrm {gr}^k(\\mathcal {A}_n^M)$ induced from the inclusion, we proved Theorem 5.1 (Satoh, [61]) For any $n \\ge 4$ and $k \\ge 2$ , $ 0 \\rightarrow {\\mathrm {gr}}^k (\\mathcal {A}_n^M) \\xrightarrow{} H^* \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n^M)\\xrightarrow{} S^k H \\rightarrow 0 $ is a $\\mathrm {GL}(n,\\mathbf {Z})$ -equivariant exact sequence.", "Moreover, recently, we gave the irreducible decomposition of $\\mathrm {gr}_{\\mathbf {Q}}^k(\\mathcal {A}_n^M)$ as follows: Theorem 5.2 (Enomoto and Satoh, [16]) For $k \\ge 2$ , $ \\mathrm {gr}_{\\mathbf {Q}}^k(\\mathcal {A}_n^M) \\cong L^{\\lbrace (k,1),(1)\\rbrace } \\oplus L^{\\lbrace (k-1,1),0\\rbrace }.", "$" ], [ "The abelianization of $\\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_n^M))$", "Here we show an application of Morita's trace maps $\\mathrm {Tr}_{[k]}$ .", "In [16], we determined the abelianization of the derivation algebra $\\mathrm {Der}^+(\\mathrm {gr}(\\mathcal {L}_n^M))$ as a Lie algebra using Morita's trace maps.", "That is, Theorem 5.3 (Enomoto and Satoh, [16]) For $n \\ge 4$ , we have $ (\\mathrm {Der}^+(\\mathcal {L}_n^M))^{\\mathrm {ab}} \\cong (H^* \\otimes _{\\mathbf {Z}} \\Lambda ^2 H ) \\oplus \\bigoplus _{k \\ge 2} S^k H. $ More precisely, we showed that this isomorphism is given by the degree one part of $\\mathrm {Der}^+(\\mathcal {L}_n^M)$ and Morita's trace maps $\\mathrm {Tr}_{[k]}$ for $k \\ge 2$ .", "Namely, consider a Lie algebra homomorphism $ \\Theta ^M := \\mathrm {id}_1 \\oplus \\bigoplus _{k \\ge 2} \\mathrm {Tr}_{[k]} : \\mathrm {Der}^+(\\mathcal {L}_n^M) \\rightarrow (H^* \\otimes _{\\mathbf {Z}} \\Lambda ^2 H) \\oplus \\bigoplus _{k \\ge 2} S^k H. $ Then we obtained an exact sequence $ 0 \\rightarrow \\bigoplus _{k \\ge 2} \\mathrm {gr}^k(\\mathcal {A}_n^M) \\xrightarrow{}\\mathrm {Der}^+(\\mathcal {L}_n^M) \\xrightarrow{} (H^* \\otimes _{\\mathbf {Z}} \\Lambda ^2 H ) \\oplus \\bigoplus _{k \\ge 2} S^k H \\rightarrow 0.", "$ of Lie algebras.", "We should remark that this result is the Chen Lie algebra version of Morita's conjecture for the free Lie algebra $\\mathrm {gr}(\\mathcal {L}_n)$ : Conjecture (Morita) For $n \\ge 3$ , $ (\\mathrm {Der}^+(\\mathcal {L}_n))^{\\mathrm {ab}} \\cong (H^* \\otimes _{\\mathbf {Z}} \\Lambda ^2 H ) \\oplus \\bigoplus _{k \\ge 2} S^k H $ Morita [45] showed that this conjecture is true up to degree $n(n-1)$ , based on a work of Kassabov [26]." ], [ "On the second cohomology of $\\mathrm {IA}_n^M$", "Due to Pettet [51], the image of the cup product $ \\cup : \\Lambda ^2 H^1(\\mathrm {IA}_n,\\mathbf {Q}) \\rightarrow H^2(\\mathrm {IA}_n,\\mathbf {Q}) $ is completely determined as mentioned above.", "In [61], we study a metabelian analogue of her work.", "Let $ \\cup ^M : \\Lambda ^2 H^1(\\mathrm {IA}_n^M,\\mathbf {Q}) \\rightarrow H^2(\\mathrm {IA}_n^M,\\mathbf {Q}) $ be the cup product of $H^1(\\mathrm {IA}_n^M,\\mathbf {Q})$ .", "Then, we proved Theorem 5.4 (Satoh, [61]) For $n \\ge 4$ , the natural homomorphism $\\nu : \\mathrm {IA}_n \\rightarrow \\mathrm {IA}_n^M$ induces an isomorphism $ \\nu ^* : \\mathrm {Im}(\\cup ^M) \\rightarrow \\mathrm {Im}(\\cup ).", "$ Observing the five term exact sequence of a group extension $ 1 \\rightarrow \\mathcal {K}_n \\rightarrow \\mathrm {IA}_n \\rightarrow \\mathrm {IA}_n^M \\rightarrow 1, $ we have $\\begin{split}0 & \\rightarrow H^1(\\mathrm {IA}_n^M,\\mathbf {Q}) \\rightarrow H^1(\\mathrm {IA}_n,\\mathbf {Q}) \\\\& \\hspace{20.0pt} \\rightarrow H^1(\\mathcal {K}_n,\\mathbf {Q})^{\\mathrm {IA}_n}\\rightarrow H^2(\\mathrm {IA}_n^M,\\mathbf {Q}) \\rightarrow H^2(\\mathrm {IA}_n,\\mathbf {Q}).\\end{split}$ Since $H^1(\\mathrm {IA}_n^M,\\mathbf {Q}) \\rightarrow H^1(\\mathrm {IA}_n,\\mathbf {Q})$ is isomorphism by $(\\mathrm {IA}_n^M)^{\\mathrm {ab}} \\cong \\mathrm {IA}_n^{\\mathrm {ab}}$ as mentioned above, we obtain $ 0 \\rightarrow H^1(\\mathcal {K}_n,\\mathbf {Q})^{\\mathrm {IA}_n}\\rightarrow H^2(\\mathrm {IA}_n^M,\\mathbf {Q}) \\rightarrow H^2(\\mathrm {IA}_n,\\mathbf {Q}).", "$ In [61], we showed using the third Johnson homomorphism $\\tau _3$ of $\\mathrm {Aut}\\,F_n^M$ that $H^1(\\mathcal {K}_n,\\mathbf {Q})^{\\mathrm {IA}_n}$ is non-trivial.", "Hence we see Theorem 5.5 (Satoh, [61]) For $n \\ge 4$ , $\\cup ^M$ is not surjective.", "Recently, we [63] showed that $\\mathcal {K}_n$ is not finitely generated.", "This will be mentioned more precisely in Subsection REF later." ], [ "Braid groups and Mapping class groups", "In this section, we recall well known facts of two important subgroups of the automorphism group of a free group.", "The first one is the braid group, and the other is the mapping class group of a surface.", "Needless to say, these groups have been studied by a large number of authors with a long history." ], [ "Braid groups", "Let $B_n$ be the braid group of $n$ strands.", "Artin [3] gave the first finite presentation of $B_n$ with generators $\\sigma _i$ for $1 \\le i \\le n-1$ subject to relations: $\\sigma _i \\sigma _{i+1} \\sigma _i = \\sigma _{i+1} \\sigma _{i} \\sigma _{i+1}$ for $1 \\le i \\le n-2$ , $[\\sigma _i, \\sigma _j]=1$ for $|i-j| \\ge 2$ .", "It is known that the braid group $B_n$ is isomorphic to the mapping class group of the $n$ -punctured disk $D^2 \\setminus \\lbrace p_1, \\ldots p_n \\rbrace $ .", "(See Theorem 1.10 in [9].)", "Then the action of $B_n$ on the fundamental group $\\pi _1(D^2 \\setminus \\lbrace p_1, \\ldots p_n \\rbrace ) \\cong F_n$ induces a faithful representation of $B_n$ as the automorphisms of $F_n$ .", "This representation, denoted by $\\psi : B_n \\hookrightarrow \\mathrm {Aut}\\,F_{n}$ , is given by $ \\sigma _i : x_i \\mapsto x_i x_{i+1} x_i^{-1}, \\hspace{10.0pt} x_{i+1} \\mapsto x_{i}, \\hspace{10.0pt} x_j \\mapsto x_j $ for $j \\ne i$ , $i+1$ .", "Furthermore, its image is completely determined as Theorem 6.1 (See Theorem 1.9 in [9]) $\\begin{split}\\lbrace \\sigma \\in \\mathrm {Aut}\\,F_{n} \\,\\, | \\,\\, x_i^{\\sigma }=A_i \\, x_{\\mu (i)} \\, A_i^{-1}, \\,\\,\\mu \\in \\mathfrak {S}_n, \\, A_i \\in F_n, \\,\\, \\omega ^{\\sigma }=\\omega \\rbrace \\end{split}$ where $\\mathfrak {S}_n$ is the symmetric group of degree $n$ , and $\\omega =x_1x_2 \\cdots x_n \\in F_n$ .", "Namely, $\\omega $ is a homotopy class of a simple closed curve on $D^2 \\setminus \\lbrace p_1, \\ldots p_n \\rbrace $ parallel to the boundary.", "Through this embedding, we consider $B_n$ as a subgroup of $\\mathrm {Aut}\\,F_{n}$ .", "Using this embedding, Artin solved the word problem for the braid group $B_n$ .", "(See [9] for details.)", "A subgroup $B_n \\cap \\mathrm {IA}_n$ is called the pure braid group , and is denoted by $P_n$ ." ], [ "Mapping class groups of surfaces", "Next, let us consider the mapping class groups of surfaces.", "(See also for example [46] by Morita for basic material for the mapping class groups.)", "For any integer $g \\ge 1$ , let $\\Sigma _{g,1}$ be the compact oriented surface of genus $g$ with one boundary component.", "The fundamental group $\\pi _1(\\Sigma _{g,1})$ of $\\Sigma _{g,1}$ is isomorphic to the free group $F_{2g}$ .", "Throughout the paper, we fix isomorphisms $\\pi _1(\\Sigma _{g,1}) \\cong F_{2g}$ and $H_1(\\Sigma _{g,1},\\mathbf {Z}) \\cong H$ .", "We denote by $\\mathcal {M}_{g,1}$ the mapping class group of ${\\Sigma }_{g,1}$ .", "Namely, $\\mathcal {M}_{g,1}$ is the group of isotopy classes of orientation preserving diffeomorphisms of ${\\Sigma }_{g,1}$ which fix the boundary pointwise.", "Then the action of $\\mathcal {M}_{g,1}$ on $\\pi _1(\\Sigma _{g,1})=F_{2g}$ induces a natural homomorphism $ \\phi : \\mathcal {M}_{g,1} \\hookrightarrow \\mathrm {Aut}\\,F_{2g}.", "$ By classical works due to Dehn and Nielsen, we have Theorem 6.2 (Dehn and Nielsen) For any $g \\ge 1$ , $\\phi $ is injective.", "Furthermore, its image is given by $ \\mathrm {Im}(\\phi ) = \\lbrace \\sigma \\in \\mathrm {Aut}\\,F_{2g} \\,\\, | \\,\\, \\zeta ^{\\sigma } = \\zeta \\rbrace $ where $\\zeta \\in F_{2g}$ is the homotopy class of a simple closed curve on $\\Sigma _{g,1}$ which is parallel to the boundary.", "By this embedding, we consider $\\mathcal {M}_{g,1}$ as a subgroup of $\\mathrm {Aut}\\,F_{2g}$ .", "The image of $\\mathcal {M}_{g,1}$ by $\\rho : \\mathrm {Aut}\\,F_{2g} \\rightarrow \\mathrm {GL}(2g,\\mathbf {Z})$ is the symplectic group $ \\mathrm {Sp}(2g,\\mathbf {Z}):=\\lbrace X \\in \\mathrm {GL}(2g,\\mathbf {Z}) \\ | \\ {}^{t}X J X = J \\rbrace \\,\\,\\, \\mathrm {for} \\,\\,\\, J= \\left(\\begin{array}{cc}0 & E_g \\\\-E_g & 0\\end{array}\\right) $ where $E_g$ is the identity matrix of degree $g$ .", "Set $\\mathcal {I}_{g,1} := \\mathcal {M}_{g,1} \\cap \\mathrm {IA}_n$ .", "Namely, $\\mathcal {I}_{g,1}$ is a subgroup of $\\mathcal {M}_{g,1}$ consisting of mapping classes which act on the first homology $H_1(\\Sigma _{g,1},\\mathbf {Z})$ of $\\Sigma _{g,1}$ trivially.", "The group $\\mathcal {I}_{g,1}$ is called the Torelli subgroup of $\\mathcal {M}_{g,1}$ , or the Torelli group of $\\Sigma _{g,1}$ .", "Then we have a commutative diagram including three group extensions: ${\\begin{matrix}1 &\\xrightarrow{}& P_n &\\xrightarrow{}& B_n &\\xrightarrow{}& \\mathfrak {S}_n &\\xrightarrow{}& 1 \\\\&& \\downarrow && \\mathbox{mphantom}{\\scriptstyle \\psi }\\downarrow {\\scriptstyle \\psi }&& \\downarrow && && \\\\1 &\\xrightarrow{}& \\mathrm {IA}_n &\\xrightarrow{}& \\mathrm {Aut}\\,F_n &\\xrightarrow{}& \\mathrm {GL}(n,\\mathbf {Z}) &\\xrightarrow{}& 1 \\\\&& {\\scriptstyle n=2g}\\uparrow \\mathbox{mphantom}{\\scriptstyle n=2g}&& \\mathbox{mphantom}{\\scriptstyle \\phi }{\\scriptstyle n=2g}\\uparrow {\\scriptstyle \\phi }\\mathbox{mphantom}{\\scriptstyle n=2g}&& {\\scriptstyle n=2g}\\uparrow \\mathbox{mphantom}{\\scriptstyle n=2g}&& && \\\\1 &\\xrightarrow{}& \\mathcal {I}_{g,1} &\\xrightarrow{}& \\mathcal {M}_{g,1} &\\xrightarrow{}& \\mathrm {Sp}(2g,\\mathbf {Z}) &\\xrightarrow{}& 1\\end{matrix}}$" ], [ "Magnus representations", "In this section we consider the Magnus representation of $\\mathrm {Aut}\\,F_n$ and $\\mathrm {Aut}\\,F_n^M$ .", "The main goal is to show a relation between the Johnson homomorphism and the Magnus representation due to Morita [42], and the fact that the kernel of the Magnus representation is not finitely generated.", "For basic material for the Magnus representation, see for example [9] or [55]." ], [ "Fox derivations", "To begin with, we recall some properties of the Fox derivations of $F_n$ .", "See also [55].", "Definition 7.1 For each $1 \\le i \\le n$ , let $ \\frac{\\partial }{\\partial x_i} : \\mathbf {Z}[F_n] \\rightarrow \\mathbf {Z}[F_n] $ be a map defined by $ \\frac{\\partial }{\\partial x_i}(w) =\\sum _{j=1}^{r} \\epsilon _j \\delta _{\\mu _j,i} x_{\\mu _1}^{\\epsilon _1} \\cdots x_{\\mu _j}^{\\frac{1}{2}(\\epsilon _j-1)} \\in \\mathbf {Z}[F_n] $ for a reduced word $w=x_{\\mu _1}^{\\epsilon _1} \\cdots x_{\\mu _r}^{\\epsilon _r} \\in F_n$ , $\\epsilon _j=\\pm 1$ .", "The maps $\\frac{\\partial }{\\partial x_i}$ are called the Fox derivations of $F_n$ .", "Definition 7.2 For a group $G$ , let $\\mathbf {Z}[G]$ be the integral group ring of $G$ over $\\mathbf {Z}$ .", "We denote the augmentation map by $\\epsilon : \\mathbf {Z}[G] \\rightarrow \\mathbf {Z}$ .", "The kernel $I_G$ of $\\epsilon $ is called the augmentation ideal of $\\mathbf {Z}[G]$ .", "Then the $k$ -times powers $I_G^k:=I_G \\times \\cdots \\times I_G$ of $I_G$ for all $k \\ge 1$ provide a descending filtration of $\\mathbf {Z}[G]$ , and the direct sum $ \\mathrm {gr}(\\mathbf {Z}[G]) := \\bigoplus _{k \\ge 1} \\, I_G^k/I_G^{k+1} $ naturally has a graded algebra structure induced from the multiplication of $\\mathbf {Z}[G]$ .", "It is called the graded algebra associated to the group ring $\\mathbf {Z}[G]$ .", "For $G=F_n$ , the free group of rank $n$ , it is classically known (by work of Magnus [38]) that each graded quotient $I_{F_n}^k/I_{F_n}^{k+1}$ is a free abelian group with basis $ \\lbrace (x_{i_1}-1)(x_{i_2}-1) \\cdots (x_{i_k}-1) \\,\\,|\\,\\, 1 \\le i_j \\le n \\rbrace , $ and a map $I_{F_n}^k/I_{F_n}^{k+1} \\rightarrow H^{\\otimes k}$ defined by $ (x_{i_1}-1)(x_{i_2}-1) \\cdots (x_{i_k}-1) \\mapsto x_{i_1} \\otimes x_{i_2} \\otimes \\cdots \\otimes x_{i_k} $ induces an associative algebra isomorphism from $\\mathrm {gr}(\\mathbf {Z}[F_n])$ to the tensor algebra $T(H) := \\bigoplus _{k \\ge 1} H^{\\otimes k}$ .", "In the following, we identify $I_{F_n}^k/I_{F_n}^{k+1}$ with $H^{\\otimes k}$ via this isomorphism.", "In $\\mathrm {gr}(\\mathbf {Z}[F_n])$ , the free Lie algebra $\\mathrm {gr}(\\mathcal {L}_n)$ is realized as follows.", "Theorem 7.3 (Fox, [19])     For any $k \\ge 1$ and $x \\in \\Gamma _n(k)$ , $x-1 \\in I_{F_n}^k$ .", "Furthermore, a map $\\Gamma _n(k) \\rightarrow I_{F_n}^k$ defined by $x \\mapsto x-1$ induces an injective homomorphism $ \\mathrm {gr}^k(\\mathcal {L}_n) \\hookrightarrow I_{F_n}^k/I_{F_n}^{k+1}.", "$ For any $k \\ge 1$ and $x \\in \\Gamma _n(k)$ , $\\frac{\\partial x}{\\partial x_i} \\in I_{F_n}^{k-1}$ for any $1 \\le i \\le n$ .", "Furthermore, $ x-1 = \\sum _{i=1}^n \\frac{\\partial x}{\\partial x_i} (x_i-1) \\in \\mathbf {Z}[F_n].", "$ The formula Part (2) above is called the fundamental formula of the Fox derivations.", "On the other hand, for $G=H$ , the free abelian group of rank $n$ , it is also known that each graded quotient $I_H^k/I_H^{k+1}$ is a free abelian group with basis $ \\lbrace (x_{i_1}-1)(x_{i_2}-1) \\cdots (x_{i_k}-1) \\,\\,|\\,\\, 1 \\le i_1 \\le i_2 \\le \\cdots \\le i_k \\le n \\rbrace , $ and the associated graded algebra $\\mathrm {gr}(\\mathbf {Z}[H])$ is isomorphic to the symmetric algebra $ S(H) := \\bigoplus _{k \\ge 1} S^k H $ of $H$ .", "(See [50], Chapter VIII, Proposition 6.7.)", "We also identify $I_H^k/I_H^{k+1}$ with $S^k H$ .", "Let $\\mathfrak {a} : F_n \\rightarrow H$ be the abelianization of $F_n$ .", "Then a homomorphism $I_{F_n}^k/I_{F_n}^{k+1} \\rightarrow I_H^k/I_H^{k+1}$ induced from the abelianization $\\mathfrak {a} : F_n \\rightarrow H$ is just the natural projection $H^{\\otimes k} \\rightarrow S^k H$ ." ], [ "Magnus representations of $\\mathrm {Aut}\\,F_n$", "Let $\\varphi : F_n \\rightarrow G$ be a group homomorphism.", "We also denote by $\\varphi $ the ring homomorphism $\\mathbf {Z}[F_n] \\rightarrow \\mathbf {Z}[G]$ induced from $\\varphi $ .", "For any matrix $A=(a_{ij}) \\in \\mathrm {GL}(n,\\mathbf {Z}[F_n])$ , let $A^{\\varphi }$ be the matrix $(a_{ij}^{\\varphi }) \\in \\mathrm {GL}(n,\\mathbf {Z}[G])$ .", "Definition 7.4 A map $r^{\\varphi } : \\mathrm {Aut}\\,F_n \\rightarrow \\mathrm {GL}(n,\\mathbf {Z}[G])$ defined by $ \\sigma \\mapsto \\biggl {(} \\frac{\\partial {x_i}^{\\sigma } }{\\partial x_j} {\\biggl {)}}^{\\varphi } $ for any $\\sigma \\in \\mathrm {Aut}\\,F_n$ , is called the Magnus representation of $\\mathrm {Aut}\\,F_n$ associated to $\\varphi $ .", "If $\\varphi = \\mathrm {id} : F_n \\rightarrow F_n$ , we write $r$ for $r^{\\mathrm {id}}$ for simplicity.", "For the abelianization $\\mathfrak {a} : F_n \\rightarrow H$ , the map $r^{\\mathfrak {a}}$ is not a homomorphism but a crossed homomorphism (with respect to right action).", "Namely, $r^{\\mathfrak {a}}$ satisfies $ r^{\\mathfrak {a}}(\\sigma \\tau ) = (r^{\\mathfrak {a}}(\\sigma ))^{\\tau ^*} \\cdot r^{\\mathfrak {a}}(\\tau ) $ for any $\\sigma , \\tau \\in \\mathrm {Aut}\\,F_n$ .", "Here $(r^{\\mathfrak {a}}(\\sigma ))^{\\tau ^*}$ denotes the matrix obtained from $r^{\\mathfrak {a}}(\\sigma )$ by applying the automorphism $\\tau ^* : \\mathbf {Z}[H] \\rightarrow \\mathbf {Z}[H]$ induced from $\\rho (\\tau ) \\in \\mathrm {Aut}(H)$ on each entry.", "Hence by restricting $r^{\\mathfrak {a}}$ to $\\mathrm {IA}_n$ , we obtain a group homomorphism $\\mathrm {IA}_n \\rightarrow \\mathrm {GL}(n,\\mathbf {Z}[H])$ , which is also denoted by $r^{\\mathfrak {a}}$ .", "Here we remark a relation between the Magnus representation and the Johnson homomorphism of $\\mathrm {Aut}\\,F_n$ .", "For any element $f \\in H^* \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n)$ , set $ \\begin{Vmatrix} f \\end{Vmatrix}:= \\biggl {(} \\frac{\\partial {({x_i}^{f})} }{\\partial x_j} {\\biggl {)}} \\in M(n, I_{F_n}^k/I_{F_n}^{k+1}) $ where we consider any lift of the element $ {x_i}^{f} \\in \\mathrm {gr}^{k+1}(\\mathcal {L}_n) = \\Gamma _n(k+1)/\\Gamma _n(k+2) $ to $\\Gamma _n(k+1)$ .", "We can see that the equivalence class of $\\frac{\\partial {({x_i}^{f})} }{\\partial x_j}$ in $I_{F_n}^k/I_{F_n}^{k+1}$ does not depend on the choice of the lift of ${x_i}^{f}$ .", "Then we have Theorem 7.5 (Morita, [42]) The Magnus representation $r:\\mathrm {Aut}\\,F_n \\rightarrow \\mathrm {GL}(n,\\mathbf {Z}[F_n])$ induces a homomorphism $ r_k : \\mathcal {A}_n(k) \\rightarrow \\mathrm {GL}(n,\\mathbf {Z}[F_n]/I_{F_n}^{k+1}).", "$ Moreover, it has the following form.", "For any $\\sigma \\in \\mathcal {A}_n(k)$ , we have $ r_k(\\sigma )= E_n + \\begin{Vmatrix} \\tau _k(\\sigma ) \\end{Vmatrix} $ where $E_n$ is the identity matrix of degree $n$ .", "Namely, the representation $r_k$ is essentially equal to the $k$ -th Johnson homomorphism.", "Finally, we remark Morita's original definition of the trace map $\\mathrm {Tr}_{[k]}$ .", "For any element $f \\in H^* \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_n)$ , take the trace $\\mathrm {Trace}(\\begin{Vmatrix} f \\end{Vmatrix}) \\in I_{F_n}^k/I_{F_n}^{k+1}$ of the matrix $\\begin{Vmatrix} f \\end{Vmatrix}$ .", "Then we have $ \\mathrm {Tr}_{[k]}(f) = (-1)^k (\\mathrm {Trace}(\\begin{Vmatrix} f \\end{Vmatrix}))^{\\mathfrak {a}} \\in I_H^k/I_H^{k+1} \\cong S^k H. $ Since $\\mathcal {A}_n(2)=\\Gamma _{\\mathrm {IA}_n}(2)$ by a result of Bachmuth [6], we have $\\mathrm {det} \\circ r^{\\mathfrak {a}}(\\sigma ) = 1$ for any $\\sigma \\in \\mathcal {A}_n(2)$ .", "This shows that $ 1 = \\mathrm {det} \\circ r_k^{\\mathfrak {a}}(\\sigma ) = 1 + \\mathrm {Tr}_{[k]}(\\tau _k(\\sigma )) \\in \\mathbf {Z}[H]/I_H^{k+1} $ for any $k \\ge 2$ and $\\sigma \\in \\mathcal {A}_n(k)$ .", "Hence $\\mathrm {Tr}_{[k]}(\\tau _k(\\sigma ))=0 \\in I_H^k/I_H^{k+1}$ .", "This means that the Morita trace $\\mathrm {Tr}_{[k]}$ vanishes on the image of $\\tau _k$ ." ], [ "The kernel of the Magnus representation of $r^{\\mathfrak {a}}$", "Here we consider the kernel $\\mathrm {Ker}(r^{\\mathfrak {a}})$ of the Magnus representation $r^{\\mathfrak {a}}$ .", "First, we recall a remarkable classical work of Bachmuth.", "Theorem 7.6 (Bachmuth, [5]) For any $n \\ge 2$ , the Magnus representation $r^{\\mathfrak {a}}$ factors through the natural homomorphism $\\nu : \\mathrm {IA}_n \\rightarrow \\mathrm {IA}_n^M$ .", "Moreover, the induced representation $ \\mathrm {IA}_n^M \\rightarrow \\mathrm {GL}(n,\\mathbf {Z}[H]) $ of $\\mathrm {IA}_n^M$ by $r^{\\mathfrak {a}}$ is faithful.", "This shows that $\\mathrm {Ker}(r^{\\mathfrak {a}}) = \\mathcal {K}_n$ , the kernel of $\\nu : \\mathrm {IA}_n \\rightarrow \\mathrm {IA}_n^M$ .", "Namely, the metabelianization of $F_n$ induces the injectivization of the Magnus representation $r^{\\mathfrak {a}}$ .", "Historically, the faithfulness of the Magnus representation restricted to two subgroups related to topology has been studied.", "One is the pure braid group $P_n$ , and the other is the Torelli group $\\mathcal {I}_{g,1}$ .", "The restriction $r^{\\mathfrak {a}}|_{P_n}$ of $r^{\\mathfrak {a}}$ to $P_n$ is called the Gassner representation.", "(cf.", "[55].)", "For $n = 2$ and 3, Magnus and Peluso [39] showed that $r^{\\mathfrak {a}}|_{P_n}$ is faithful.", "It is, however, still an open problem to determine whether $r^{\\mathfrak {a}}|_{P_n}$ is faithful or not for $n \\ge 4$ .", "On the other hand, it is known that $r^{\\mathfrak {a}}|_{\\mathcal {I}_{g,1}}$ is not faithful as follows: Theorem 7.7 (Suzuki, [66]) The Magnus representation $r^{\\mathfrak {a}}|_{\\mathcal {I}_{g,1}}$ restricted to the Torelli group $\\mathcal {I}_{g,1}$ is not faithful for $g \\ge 2$ .", "In [66], Suzuki gave a non-trivial element in $\\mathrm {Ker}(r^{\\mathfrak {a}}|_{\\mathcal {I}_{g,1}})$ .", "Recently, Church and Farb gave the following remarkable result for the infinite generation for $r^{\\mathfrak {a}}|_{\\mathcal {I}_{g,1}}$ .", "Theorem 7.8 (Church and Farb, [13]) $H_1(\\mathrm {Ker}(r^{\\mathfrak {a}}|_{\\mathcal {I}_{g,1}}),\\mathbf {Z})$ has infinite rank for $g \\ge 2$ .", "This shows that the above $r^{\\mathfrak {a}}|_{\\mathcal {I}_{g,1}}$ is not finitely generated for $g \\ge 2$ .", "They proved this theorem by using a result of Suzuki and a variant of the first Johnson homomorphism of the mapping class group.", "Now, it is a natural question to ask how large is $\\mathrm {Ker}(r^{\\mathfrak {a}})$ .", "Our answer is Theorem 7.9 (Satoh, [63]) For any $n \\ge 2$ , $H_1(\\mathrm {Ker}(r^{\\mathfrak {a}}),\\mathbf {Z})$ has infinite rank.", "To prove this, we consider some embeddings from $\\mathrm {IA}_n$ into $\\mathrm {IA}_m$ for various $m>n$ , which arise from the action of $\\mathrm {IA}_n$ on finite-index subgroups of $F_n$ .", "Then using the first Johnson homomorphisms on $\\mathrm {IA}_m$ which do not vanish on $\\mathrm {Ker}(r^{\\mathfrak {a}})$ , we can detect infinitely many independent elements in $H_1(\\mathrm {Ker}(r^{\\mathfrak {a}}),\\mathbf {Z})$ .", "We remark that we can also show the above theorem by using the techniques of Church and Farb for even $n \\ge 2$ ." ], [ "Johnson homomorphisms of $\\mathcal {M}_{g,1}$", "In this section, we consider the Johnson homomorphisms of the mapping class groups of surfaces.", "There is a broad range of results for the Johnson filtration and the Johnson homomorphisms of the mapping class groups.", "In this chapter, we concentrate on the study of the cokernel of the Johnson homomorphisms.", "By Dehn and Nielsen's classical work, we consider $\\mathcal {M}_{g,1}$ as a subgroup of $\\mathrm {Aut}\\,F_{2g}$ as above.", "Definition 8.1 For each $k \\ge 1$ , set $\\mathcal {M}_{g,1}(k) := \\mathcal {M}_{g,1} \\cap \\mathcal {A}_{2g}(k)$ .", "Then we have a descending filtration $ \\mathcal {I}_{g,1} = \\mathcal {M}_{g,1}(1) \\supset \\mathcal {M}_{g,1}(2) \\supset \\cdots \\supset \\mathcal {M}_{g,1}(k) \\supset \\cdots $ of the Torelli group $\\mathcal {I}_{g,1}$ .", "This filtration is called the Johnson filtration of $\\mathcal {M}_{g,1}$ .", "Set $\\mathrm {gr}^k (\\mathcal {M}_{g,1}) := \\mathcal {M}_{g,1}(k)/\\mathcal {M}_{g,1}(k+1)$ .", "For each $k \\ge 1$ , the mapping class group $\\mathcal {M}_{g,1}$ acts on $\\mathrm {gr}^k (\\mathcal {M}_{g,1})$ by conjugation.", "This action induces that of $\\mathrm {Sp}(2g,\\mathbf {Z})=\\mathcal {M}_{g,1}/\\mathcal {I}_{g,1}$ on it.", "By an argument similar to that of $\\mathrm {Aut}\\,F_n$ , the Johnson homomorphisms of $\\mathcal {M}_{g,1}$ are defined as follows.", "Definition 8.2 For $n=2g$ and $k \\ge 1$ , consider the restriction of $\\tilde{\\tau }_k : \\mathcal {A}_{2g}(k) \\rightarrow \\mathrm {Hom}_{\\mathbf {Z}}(H, \\mathrm {gr}^{k+1}(\\mathcal {L}_{2g}))$ to $\\mathcal {M}_{g,1}(k)$ .", "Then its kernel is just $\\mathcal {M}_{g,1}(k+1)$ .", "Hence we obtain an injective homomorphism $ \\tau _k^{\\mathcal {M}} : \\mathrm {gr}^k (\\mathcal {M}_{g,1}) \\hookrightarrow \\mathrm {Hom}_{\\mathbf {Z}}(H, \\mathrm {gr}^{k+1}(\\mathcal {L}_{2g}))= H^* \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_{2g}).", "$ The homomorphism $\\tau _k^{\\mathcal {M}}$ is $\\mathrm {Sp}(2g,\\mathbf {Z})$ -equivariant, and is called the $k$ -th Johnson homomorphism of $\\mathcal {M}_{g,1}$ .", "If we consider a $\\mathrm {GL}(2g,\\mathbf {Z})$ -module $H$ as an $\\mathrm {Sp}(2g,\\mathbf {Z})$ -module, then $H^* \\cong H$ by Poincaré duality.", "Hence, in the following, we canonically identify the target $H^* \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_{2g})$ of $\\tau _k^{\\mathcal {M}}$ with $H \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_{2g})$ .", "The Johnson filtration and the Johnson homomorphisms of $\\mathcal {M}_{g,1}$ begun to be studied by D. Johnson [22] in the 1980s who determined the abelianization of the Torelli group in [25] using the first Johnson homomorphism and the Birman-Craggs homomorphism.", "In particular, he showed Theorem 8.3 (Johnson, [25]) For $g \\ge 3$ , $\\mathrm {Im}(\\tau _1^{\\mathcal {M}}) \\cong \\Lambda ^3 H$ as an $\\mathrm {Sp}(2g,\\mathbf {Z})$ -module, and it gives the free part of $H_1(\\mathcal {I}_{g,1},\\mathbf {Z})$ .", "Now, let us recall the fact that the image of $\\tau _k^{\\mathcal {M}}$ is contained in a certain $\\mathrm {Sp}(2g,\\mathbf {Z})$ -submodule of $H \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_{2g})$ , by a result of Morita [42].", "In general, for any $n \\ge 1$ , let $H \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_{n}) \\rightarrow \\mathrm {gr}^{k+2}(\\mathcal {L}_{n})$ be a $\\operatorname{GL}(n,\\mathbf {Z})$ -equivariant homomorphism defined by $ a \\otimes X \\mapsto [a,X], \\hspace{10.0pt} \\mathrm {for} \\hspace{10.0pt} a \\in H, \\,\\,\\, X \\in \\mathrm {gr}^{k+1}(\\mathcal {L}_{n}).", "$ For $n=2g$ , we denote by $\\mathfrak {h}_{g,1}(k)$ the kernel of this homomorphism: $ \\mathfrak {h}_{g,1}(k) := \\mathrm {Ker}(H \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_{2g}) \\rightarrow \\mathrm {gr}^{k+2}(\\mathcal {L}_{2g})).", "$ Then Morita showed Theorem 8.4 (Morita, [42]) For $k \\ge 2$ , the image $\\mathrm {Im}(\\tau _k^{\\mathcal {M}})$ of $\\tau _k^{\\mathcal {M}}$ is contained in $\\mathfrak {h}_{g,1}(k)$ .", "Therefore, to determine how different is $\\mathrm {Im}(\\tau _k^{\\mathcal {M}})$ from $\\mathfrak {h}_{g,1}(k)$ is one of the most basic problems.", "Throughout theis chapter, the cokernel $\\mathrm {Coker}(\\tau _k^{\\mathcal {M}})$ of $\\tau _k^{\\mathcal {M}}$ always means the quotient $\\mathrm {Sp}(2g,\\mathbf {Z})$ -module $\\mathfrak {h}_{g,1}(k)/\\mathrm {Im}(\\tau _k^{\\mathcal {M}})$ .", "In order to take advantage of representation theory, we consider the rationalization of modules.", "Namely, we study the $\\mathrm {Sp}(2g,\\mathbf {Q})$ -module structures of $\\mathrm {Im}(\\tau _{k,\\mathbf {Q}}^{\\mathcal {M}})$ and $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\mathcal {M}})$ .", "So far, the $\\mathrm {Sp}$ -irreducible decompositions of $\\mathrm {Im}(\\tau _{k,\\mathbf {Q}}^{\\mathcal {M}})$ and $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\mathcal {M}})$ are determined for $1 \\le k \\le 4$ as follows.", "Table: NO_CAPTIONMorita [42] showed that the symmetric tensor product $S^k H_{\\mathbf {Q}}$ appears in the $\\mathrm {Sp}$ -irreducible decomposition of $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\mathcal {M}})$ for odd $k \\ge 3$ using the Morita trace map mentioned above.", "Moreover, Hiroaki Nakamura, in unpublished work, showed that its multiplicity is one.", "In general, however, to determine the cokernel of $\\tau _k^{\\mathcal {M}}$ is quite a difficult problem.", "Here, we recall a remarkable result of Hain.", "As an $\\mathrm {Sp}(2g,\\mathbf {Z})$ -module, we consider $\\mathfrak {h}_{g,1}(k)$ as a submodule of the degree $k$ part $\\mathrm {Der}(\\mathrm {gr}(\\mathcal {L}_n))(k)$ of the derivation algebra of $\\mathrm {gr}(\\mathcal {L}_n)$ .", "Moreover, the graded sum $ \\mathfrak {h}_{g,1} := \\bigoplus _{k \\ge 1} \\mathfrak {h}_{g,1}(k) $ naturally has a Lie subalgebra structure of $\\mathrm {Der}^+(\\mathrm {gr}(\\mathcal {L}_n))$ .", "Therefore we obtain a graded Lie algebra homomorphism $ \\tau ^{\\mathcal {M}} := \\bigoplus _{k \\ge 1} \\tau _k^{\\mathcal {M}} : {\\mathrm {gr}}(\\mathcal {M}_{g,1}) \\rightarrow \\mathfrak {h}_{g,1}, $ which is called the total Johnson homomorphism of $\\mathcal {M}_{g,1}$ .", "Then we have Theorem 8.5 (Hain [21]) The Lie subalgebra $\\mathrm {Im}(\\tau _{\\mathbf {Q}}^{\\mathcal {M}})$ is generated by the degree one part $\\mathrm {Im}(\\tau _{1,\\mathbf {Q}}^{\\mathcal {M}}) = \\Lambda ^3 H_{\\mathbf {Q}}$ as a Lie algebra.", "Now, we consider the lower central series of the Torelli group, and reformulate Hain's result above.", "For the lower central series $\\lbrace \\Gamma _{\\mathcal {I}_{g,1}}(k) \\rbrace $ of $\\mathcal {I}_{g,1}$ , consider its graded quotients $\\mathrm {gr}^k(\\mathcal {L}_{\\mathcal {I}_{g,1}}) := \\Gamma _{\\mathcal {I}_{g,1}}(k)/\\Gamma _{\\mathcal {I}_{g,1}}(k+1)$ for $k \\ge 1$ .", "Let $ {\\tau ^{\\prime }_{k}}^{\\mathcal {M}} : \\mathrm {gr}^k(\\mathcal {L}_{\\mathcal {I}_{g,1}}) \\rightarrow H \\otimes _{\\mathbf {Z}} \\mathrm {gr}^{k+1}(\\mathcal {L}_{2g}) $ be an $\\mathrm {Sp}$ -equivariant homomorphism induced from the restriction of $\\tilde{\\tau }_k$ to $\\Gamma _{\\mathcal {I}_{g,1}}(k)$ .", "Then, from Theorem REF , we have Proposition 8.6 (Hain, [21]) $\\mathrm {Im}(\\tau _{k,\\mathbf {Q}}^{\\mathcal {M}})=\\mathrm {Im}({\\tau ^{\\prime }_{k,\\mathbf {Q}}}^{\\hspace{-9.95845pt}\\mathcal {M}})$ for each $k \\ge 1$ .", "Combining Hain's result above and the fact that $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime }) \\cong \\mathcal {C}_n^{\\mathbf {Q}}(k)$ for $n \\ge k+2$ by Theorem REF , we can detect non-trivial Sp-irreducible components in $\\mathrm {Coker}(\\tau _k^{\\mathcal {M}})$ .", "In [17], we showed Theorem 8.7 (Enomoto and Satoh, [17]) For any $k \\ge 5$ such that $k \\equiv 1$ mod 4, and $g \\ge k+2$ , the irreducible Sp-module $[1^k]$ appears in $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\mathcal {M}})$ with multiplicity one.", "We wrote down a maximal vector of $[1^k]$ in [17].", "In [49], Nakamura and Tsunogai gave a table of the Sp-irreducible decompositions of $\\mathfrak {h}_{g,1}^{\\mathbf {Q}}(k)$ for $1 \\le k \\le 15$ , which are calculated by a computer.", "In the table, we can check that Sp-irreducible components $[1^k]$ have multiplicity one for $k = 5, 9, 13$ and $k = 6, 10, 14$ .", "In 2004, Nakamura communicated to the author the possibility that the multiplicities of $[1^{4m+1}]$ in $\\mathfrak {h}_{g,1}(4m+1)$ is exactly one for $m \\ge 1$ , and that they survive in the Johnson cokernels.", "The theorem above is the affirmative answer to his conjecture." ], [ "Twisted cohomology groups", "In this section, we consider some twisted first cohomology groups of $\\mathrm {Aut}\\,F_n$ and the automorphism group of a free nilpotent group, which are closely related to the study of the Johnson homomorphisms and trace maps." ], [ "$H^1(\\mathrm {Aut}\\,F_n, \\mathrm {IA}_n^{\\mathrm {ab}})$", "First, we consider a twisted first cohomology groups of $\\mathrm {Aut}\\,F_n$ with coefficients in $V:=\\mathrm {IA}_n^{\\mathrm {ab}} = H^* \\otimes _{\\mathbf {Z}} \\Lambda ^2 H$ .", "In [65], we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively.", "As a corollary, we see that the first Johnson homomorphism does not extend to $\\mathrm {Aut}\\,F_n$ as a crossed homomorphism for $n \\ge 5$ .", "Set $ \\mathbf {e}_{j,k}^i := e_i^* \\otimes e_j \\wedge e_k \\in V $ for any $1 \\le i, j, k \\le n$ , and fix them as a basis of $V$ .", "In [65], using Nielsen's finite presentation for $\\mathrm {Aut}\\,F_n$ , we computed Theorem 9.1 (Satoh, [65]) For $n \\ge 5$ , $ H^1(\\mathrm {Aut}\\,F_n, V) = {\\mathbf {Z}}^{\\oplus 2}.", "$ This computation is a free group analogue of Morita's work [41] for the mapping class group $\\mathcal {M}_{g,1}$ .", "In [41], Morita computed the first cohomology group of $\\mathcal {M}_{g,1}$ with coefficients in $\\Lambda ^3 H$ , the free part of the abelianization of the Torelli group $\\mathcal {I}_{g,1}$ , and showed Theorem 9.2 (Morita, [41]) For $g \\ge 3$ , $ H^1(\\mathcal {M}_{g,1},\\Lambda ^3 H)={\\mathbf {Z}}^{\\oplus 2}.", "$ We give a description of a generators of $H^1(\\mathrm {Aut}\\,F_n, V)$ .", "To begin with, we construct a crossed homomorphism from $\\mathrm {Aut}\\,F_n$ into $V$ using the Magnus representation due to Morita [42].", "We consider a left action of $\\mathrm {Aut}\\,F_n$ .", "Namely, define $ \\bar{r}^{\\mathfrak {a}} : \\mathrm {Aut}\\,F_n \\longrightarrow \\mathrm {GL}(n,\\mathbf {Z}[H]) $ by $\\bar{r}^{\\mathfrak {a}}(\\sigma ) := r^{\\mathfrak {a}}(\\sigma ^{-1})$ for any $\\sigma \\in \\mathrm {Aut}\\,F_n$ .", "Then we have $ \\bar{r}^{\\mathfrak {a}}(\\sigma \\tau ) = \\bar{r}^{\\mathfrak {a}}(\\sigma ) \\cdot ( \\sigma \\cdot \\bar{r}^{\\mathfrak {a}}(\\tau )) $ for any $\\sigma $ , $\\tau \\in \\mathrm {Aut}\\,F_n$ , and hence $\\bar{r}^{\\mathfrak {a}}$ is also a crossed homomorphism (with respect to the left action).", "Let us recall Nielsen's finite presentation for $\\mathrm {Aut}\\,F_n$ .", "Let $P$ , $Q$ , $S$ and $U$ be automorphisms of $F_n$ given by specifying their images of the basis $x_1, \\ldots , x_n$ as follows: Table: NO_CAPTIONIn 1924, Nielsen [48] showed that the four elements above generate $\\mathrm {Aut}\\,F_{n}$ .", "Furthermore, he obtained the first finite presentation for $\\mathrm {Aut}\\,F_{n}$ .", "(For details, see also [38].)", "By observing the images of Nielsen's generators by $\\mathrm {det} \\circ \\bar{r}^{\\mathfrak {a}}$ , we verify that $\\mathrm {Im}(\\mathrm {det} \\circ \\bar{r}^{\\mathfrak {a}})$ is contained in a multiplicative abelian subgroup $\\pm H$ of $\\mathbf {Z}[H]$ .", "In order to modify the image of $\\mathrm {det} \\circ \\bar{r}^{\\mathfrak {a}}$ , we consider the signature of $\\mathrm {Aut}\\,F_n$ .", "For any $\\sigma \\in \\mathrm {Aut}\\,F_n$ , set $\\mathrm {sgn}(\\sigma ) := \\mathrm {det}(\\rho (\\sigma )) \\in \\lbrace \\pm 1 \\rbrace $ , and define a map $f_M : \\mathrm {Aut}\\,F_n \\longrightarrow \\mathbf {Z}[H]$ by $ \\sigma \\mapsto \\mathrm {sgn}(\\sigma ) \\,\\, \\mathrm {det}(\\bar{r}^{\\mathfrak {a}}(\\sigma )).", "$ Then the map $f_M$ is also a crossed homomorphism whose image is contained in a multiplicative abelian subgroup $H$ in $\\mathbf {Z}[H]$ .", "In the following, we identify the multiplicative abelian group structure of $H$ with the additive one.", "In [57], we computed a twisted first cohomology group of $\\mathrm {Aut}\\,F_n$ with coefficients in $H$ using Gersten's presentation for $\\mathrm {Aut}\\,F_n$ .", "That is, Theorem 9.3 (Satoh, [57]) For $n \\ge 2$ , $ H^1(\\mathrm {Aut}\\,F_n, H) = \\mathbf {Z}.", "$ We showed that the crossed homomorphism $f_M$ generates $H^1(\\mathrm {Aut}\\,F_n, H)$ .", "This result is also a free group analogue of Morita's work in [40] for the mapping class group.", "That is, Theorem 9.4 (Morita, [40]) For $g \\ge 2$ , $ H^1(\\mathcal {M}_{g,1}, H) = \\mathbf {Z}.", "$ Furthermore, the restriction of the crossed homomorphism $f_M$ to $\\mathcal {M}_{g,1}$ generates $H^1(\\mathcal {M}_{g,1}, H)$ .", "This shows that the Dehn-Nielsen embedding $\\mathcal {M}_{g,1} \\hookrightarrow \\mathrm {Aut}\\,F_{2g}$ induces an isomorphism $ H^1(\\mathrm {Aut}\\,F_{2g},H) \\xrightarrow{} H^1(\\mathcal {M}_{g,1},H).", "$ See also Section 6 in [28] by Kawazumi for details of a relation between the above twisted cohomology classes of $\\mathcal {M}_{g,1}$ and the Morita-Mumford classes.", "Now, let us turn back to the discussion of a generator of $H^1(\\mathrm {Aut}\\,F_n, \\mathrm {IA}_n^{\\mathrm {ab}})$ .", "Consider the inner automorphism group $\\mathrm {Inn}\\,F_n$ of $F_n$ .", "Since $\\mathrm {Inn}\\,F_n$ is canonically isomorphic to $F_n$ , its abelianization can be identified with $H$ .", "By composing a homomorphism $H \\rightarrow V$ induced from the inclusion $\\mathrm {Inn}\\,F_n \\hookrightarrow \\mathrm {IA}_n$ and the crossed homomorphism $f_M$ , we obtain an element in $\\mathrm {Cros}(\\mathrm {Aut}\\,F_n, V)$ .", "We also denote it by $f_M$ .", "Next, we construct another crossed homomorphism from $\\mathrm {Aut}\\,F_n$ into $V$ using the Magnus expansion of $F_n$ by work of Kawazumi [27].", "(For basic material for the Magnus expansion, see Chapter 2 in [10], and also Section 7 in [28].)", "Let $\\widehat{T}$ be the complete tensor algebra generated by $H$ .", "For any Magnus expansion $\\theta : F_n \\rightarrow \\widehat{T}$ , Kawazumi define a map $ \\tau _1^{\\theta } : \\mathrm {Aut}\\,F_n \\rightarrow H^* \\otimes _{\\mathbf {Z}} H^{\\otimes 2} $ called the first Johnson map induced by the Magnus expansion $\\theta $ .", "The map $\\tau _1^{\\theta }$ satisfies $ \\tau _1^{\\theta }(\\sigma )([x]) = \\theta _2(x) - |\\sigma |^{\\otimes 2} \\theta _2(\\sigma ^{-1}(x)) $ for any $x \\in F_n$ , where $[x]$ denotes the coset class of $x$ in $H$ , $\\theta _2(x)$ is the projection of $\\theta (x)$ in $H^{\\otimes 2}$ , and $|\\sigma |^{\\otimes 2}$ denotes the automorphism of $H^{\\otimes 2}$ induced by $\\sigma \\in \\mathrm {Aut}\\,F_n$ .", "This shows that $\\tau _1^{\\theta }$ is a crossed homomorphism from $\\mathrm {Aut}\\,F_n$ to $H^* \\otimes _{\\mathbf {Z}} H^{\\otimes 2}$ .", "In [27], Kawazumi also showed that $\\tau _1^{\\theta }$ does not depend on the choice of the Magnus expansion $\\theta $ , and that the restriction of $\\tau _1^{\\theta }$ to $\\mathrm {IA}_n$ is a homomorphism satisfying $\\begin{split}\\tau _1^{\\theta }(K_{ij}) & = e_i^* \\otimes e_i \\otimes e_j - e_i^* \\otimes e_j \\otimes e_i, \\\\\\tau _1^{\\theta }(K_{ijk}) & = e_i^* \\otimes e_j \\otimes e_k - e_i^* \\otimes e_k \\otimes e_j.\\end{split}$ Now, compose a natural projection $H^* \\otimes _{\\mathbf {Z}} H^{\\otimes 2} \\rightarrow H^* \\otimes _{\\mathbf {Z}} \\Lambda ^2 H =V$ and $\\tau _1^{\\theta }$ .", "Then we obtain an element in $\\mathrm {Cros}(\\mathrm {Aut}\\,F_n,V)$ .", "We denote it by $f_K$ .", "From the result of Kawazumi mentioned above, we see that the restriction of $f_K$ to $\\mathrm {IA}_n$ coincides with the double of the first Johnson homomorphism $\\tau _1$ .", "Namely, we have $ f_K(K_{ij}) = 2 \\mathbf {e}_{i,j}^i = 2\\tau _1(K_{ij}), \\hspace{10.0pt} f_K(K_{ijk}) = 2 \\mathbf {e}_{j,k}^i=2\\tau _1(K_{ijk}).", "$ Here, we consider the images of the crossed homomorphisms $f_M$ and $f_K$ .", "From the definition, we have $ f_M(\\sigma ) := {\\left\\lbrace \\begin{array}{ll}-(\\mathbf {e}_{1,2}^2 + \\mathbf {e}_{1,3}^3 + \\cdots + \\mathbf {e}_{1,n}^n), \\hspace{10.0pt} & \\sigma =S, \\\\0, \\hspace{10.0pt} & \\sigma =P, Q, U\\end{array}\\right.", "}$ and $ f_K(\\sigma ) :={\\left\\lbrace \\begin{array}{ll}-\\mathbf {e}_{1,2}^1, \\hspace{10.0pt} & \\sigma =U, \\\\0, \\hspace{10.0pt} & \\sigma =P, Q, S.\\end{array}\\right.", "}$ Moreover, in [65], we showed that $f_M$ and $f_K$ generate $H^1(\\mathrm {Aut}\\,F_n, V_L)$ for $n \\ge 5$ .", "This shows that the first Johnson homomorphism $\\tau _1$ does not extend to a crossed homomorphism on $\\mathrm {Aut}\\,F_n$ for $n \\ge 5$ .", "It has already been known by Morita [43] that the first Johnson homomorphism $ \\tau _1 : \\mathcal {I}_{g,1} \\rightarrow \\Lambda ^3 H $ of the mapping class group does not extend to $\\mathcal {M}_{g,1}$ as a crossed homomorphism for $g \\ge 3$ ." ], [ "$H^2(\\mathrm {Aut}\\,N_{n,k}, \\Lambda ^k H_{\\mathbf {Q}})$ and a trace map {{formula:10183f25-90cf-4a53-99f6-903a35584db6}}", "Set $ \\mathrm {Tr}_{[1^k]} := f_{[1^k]} \\circ \\Phi _{1}^k : H^* {\\otimes }_{\\mathbf {Z}} \\mathcal {L}_n(k+1) \\rightarrow {\\Lambda }^k H. $ We call it the trace map for the exterior product $\\Lambda ^k H$ .", "In [60], we show that $\\Lambda ^k H_{\\mathbf {Q}}$ appears in $\\mathrm {Coker}(\\tau _{k,\\mathbf {Q}}^{\\prime })$ for odd $k$ and $3 \\le k \\le n$ , and we determine $\\mathrm {Coker}(\\tau _{3,\\mathbf {Q}})$ using $\\mathrm {Tr}_{[3]}$ and $\\mathrm {Tr}_{[1^3]}$ .", "In [16], we prove that the trace map $\\mathrm {Tr}_{[1^k]}$ defines a non-trivial twisted second cohomology class of the automorphism group $\\mathrm {Aut}\\,N_{n,k}$ of a free nilpotent group $N_{n,k}:=F_n/\\Gamma _n(k+1)$ with coefficients in $\\Lambda ^k H_{\\mathbf {Q}}$ for any $k \\ge 2$ and $n \\ge k$ .", "Here we give a brief strategy to prove it.", "First, we recall Bryant and Gupta's generators of $\\mathrm {Aut}\\,N_{n,k}$ .", "Let $\\mathrm {Aut}\\,F_n \\rightarrow \\mathrm {Aut}\\,N_{n,k}$ be a homomorpshim induced from the natural projection $F_n \\rightarrow N_{n,k}$ .", "Andreadakis [1] showed that it is surjective if $k=2$ , and not if otherwise.", "For any $\\sigma \\in \\mathrm {Aut}\\,F_n$ , we also denote its image in $\\mathrm {Aut}\\,N_{n,k}$ by $\\sigma $ .", "Under this notation, $\\mathrm {Aut}\\,N_{n,2}$ is generated by Nielsen's generators $P$ , $Q$ , $S$ and $U$ .", "Let $\\theta $ be an automorphism of $N_{n,k}$ , defined by $ \\theta : x_t \\mapsto {\\left\\lbrace \\begin{array}{ll}[x_1, [x_2,x_1]]x_1, & t=1, \\\\x_t, & t \\ne 1.\\end{array}\\right.", "}$ Then we have Theorem 9.5 (Bryant and Gupta [11]) For $k \\ge 3$ and $n \\ge k-1$ , the group $\\mathrm {Aut}\\,N_{n,k}$ is generated by $P$ , $Q$ , $S$ , $U$ and $\\theta $ .", "We remark that in 1984, Andreadakis [2] showed that $\\mathrm {Aut}\\,N_{n,k}$ is generated by $P$ , $Q$ , $S$ , $U$ and $k-2$ other elements for $n \\ge k \\ge 2$ .", "No presentation for $\\mathrm {Aut}\\,N_{n,k}$ is known except for $\\mathrm {Aut}\\,N_{2,k}$ for $k= 1, 2$ and 3 due to Lin [34].", "Using Bryant and Gupta's generators and some relations among them, we showed Proposition 9.6 (Enomoto and Satoh, [16]) For $k \\ge 3$ , $n \\ge k-1$ and $l \\ge 3$ , $H^1(\\mathrm {Aut}\\,N_{n,k}, \\Lambda ^l H_{\\mathbf {Q}})$ is trivial.", "Then observing a group extension $ 0 \\rightarrow \\mathrm {Hom}_{\\mathbf {Z}}(H, \\mathcal {L}_n(k+1)) \\rightarrow \\mathrm {Aut}\\, N_{n,k+1} \\rightarrow \\mathrm {Aut}\\,N_{n,k} \\rightarrow 1 $ introduced by Andreadakis [1], (See also Proposition 2.3 in Morita's paper [41].", "), and its cohomological five term exact sequence, we obtain $\\begin{split}0 & \\rightarrow H^1(H^* \\otimes _{\\mathbf {Z}} \\mathcal {L}_n(k+1), \\Lambda ^k H_{\\mathbf {Q}})^{\\mathrm {GL}(n,\\mathbf {Z})} \\\\& \\rightarrow H^2(\\mathrm {Aut}\\,N_{n,k}, \\Lambda ^k H_{\\mathbf {Q}}) \\rightarrow H^2(\\mathrm {Aut}\\,N_{n,k+1}, \\Lambda ^k H_{\\mathbf {Q}})\\end{split}$ for $k \\ge 3$ and $n \\ge k-1$ .", "Since the trace map $\\mathrm {Tr}_{[1^k]} \\in H^1(H^* \\otimes _{\\mathbf {Z}} \\mathcal {L}_n(k+1), \\Lambda ^k H_{\\mathbf {Q}})^{\\mathrm {GL}(n,\\mathbf {Z})}$ is surjective for any $3 \\le k \\le n$ , we have Proposition 9.7 For $k \\ge 3$ and $n \\ge k$ , we see $ 0 \\ne \\mathrm {tg}(\\mathrm {Tr}_{[1^k]}) \\in H^2(\\mathrm {Aut}\\,N_{n,k}, \\Lambda ^k H_{\\mathbf {Q}}) $ where $\\mathrm {tg}$ is the transgression map.", "In [16], we considered not only $\\mathrm {Aut}\\,N_{n,k}$ but also the image of the natural homomorphism $\\mathrm {Aut}\\,F_n \\rightarrow \\mathrm {Aut}\\,N_{n,k}$ , denoted by $T_{n,k}$ .", "The group $T_{n,k}$ is called the tame automorphism group of $N_{n,k}$ .", "Similarly to $\\mathrm {Aut}\\,N_{n,k}$ , observing the cohomological five term exact sequence of a group extension $ 0 \\rightarrow \\mathrm {gr}^k(\\mathcal {A}_n) \\rightarrow T_{n,k+1} \\rightarrow T_{n,k} \\rightarrow 1, $ we showed Proposition 9.8 For even $k$ and $2 \\le k \\le n$ , we see $ 0 \\ne \\mathrm {tg}(\\mathrm {Tr}_{[1^k]} \\circ \\tau _{k}) \\in H^2(T_{n,k}, \\Lambda ^k H_{\\mathbf {Q}}) $ where $\\mathrm {tg}$ is the transgression map.", "In [16], we showed that $ \\mathrm {Hom}_{\\mathbf {Z}}(H_{\\mathbf {Q}}^* \\otimes _{\\mathbf {Z}} \\mathcal {L}_{n,\\mathbf {Q}}(k+1), \\Lambda ^k H_{\\mathbf {Q}})^{\\mathrm {GL}(n,\\mathbf {Q})}\\cong \\mathbf {Q}$ for $n \\ge 2k+2$ , and that $ \\mathrm {Hom}_{\\mathbf {Z}}(\\mathrm {gr}_{\\mathbf {Q}}^k(\\mathcal {A}_n), \\Lambda ^k H_{\\mathbf {Q}})^{\\mathrm {GL}(n,\\mathbf {Q})} \\cong \\mathbf {Q}$ for even $k$ and $n \\ge 2k+2$ .", "At the present stage, however, it seems too difficult to determine the precise structures of $H^1(H^* \\otimes _{\\mathbf {Z}} \\mathcal {L}_n(k+1), \\Lambda ^k H_{\\mathbf {Q}})^{\\mathrm {GL}(n,\\mathbf {Z})}$ and $H^1(\\mathrm {gr}^k(\\mathcal {A}_n), \\Lambda ^k H_{\\mathbf {Q}})^{\\mathrm {GL}(n,\\mathbf {Z})}$ in general." ], [ "Congruence IA-automorphism groups of $F_n$", "For $n \\ge 2$ and $d \\ge 2$ , let $\\mathrm {GL}(n,\\mathbf {Z}) \\rightarrow \\mathrm {GL}(n,\\mathbf {Z}/d\\mathbf {Z})$ be the natural homomorphism induced by the mod reduction $d$ .", "The kernel of this homomorphism is called the congruence subgroup of $\\mathrm {GL}(n,\\mathbf {Z})$ of level $d$ , and is denoted by $\\Gamma (n,d)$ .", "Let $\\mathrm {IA}_{n,d}$ be the kernel of the composite homomorphism $\\mathrm {Aut}\\,F_n \\xrightarrow{} \\mathrm {GL}(n,\\mathbf {Z}) \\rightarrow \\mathrm {GL}(n,\\mathbf {Z}/d\\mathbf {Z})$ .", "We call $\\mathrm {IA}_{n,d}$ the congruence IA-automorphism group of $F_n$ of level $d$ .", "Then we have a group extension $1 \\rightarrow \\mathrm {IA}_n \\rightarrow \\mathrm {IA}_{n,d} \\rightarrow \\Gamma (n,d) \\rightarrow 1.$ In this section, we give a description of the abelianization of $\\mathrm {IA}_{n,d}$ with those of $\\mathrm {IA}_n$ and $\\Gamma (n,d)$ by using the first Johnson map." ], [ "The abelianization of $\\mathrm {IA}_{n,d}$", "In [58] and [59], using the first Johnson map over $\\mathbf {Z}/d\\mathbf {Z}$ defined by the Magnus expansion due to Kawazumi [27], we showed Theorem 10.1 (Satoh, [58] and [59]) For $n \\ge 2$ and $d \\ge 2$ , $ \\mathrm {IA}_{n,d}^{\\mathrm {ab}} \\cong (\\mathrm {IA}_n^{\\mathrm {ab}} \\otimes _{\\mathbf {Z}} \\mathbf {Z}/d\\mathbf {Z}) \\, \\bigoplus \\, {\\Gamma (n,d)}^{\\mathrm {ab}}.", "$ We give the outline of the proof.", "For simplicity, we consider the case where $d$ is odd.", "Observing the homological five term exact sequence of the group extension (REF ), we obtain an exact sequence $\\begin{split}\\cdots & \\rightarrow H_1(\\mathrm {IA}_n, \\mathbf {Z})_{\\Gamma (n,d)} \\xrightarrow{} H_1(\\mathrm {IA}_{n,d},\\mathbf {Z})\\rightarrow H_1(\\Gamma (n,d),\\mathbf {Z}) \\rightarrow 0.\\end{split}$ Then we show that $ H_1(\\mathrm {IA}_n, \\mathbf {Z})_{\\Gamma (n,d)} \\cong H_1(\\mathrm {IA}_n,\\mathbf {Z}) \\otimes _{\\mathbf {Z}} \\mathbf {Z}/d\\mathbf {Z}, $ and that the first Johnson map $ \\tau ^{\\theta } : H_1(\\mathrm {IA}_{n,d},\\mathbf {Z}) \\rightarrow H_1(\\mathrm {IA}_n,\\mathbf {Z}) \\otimes _{\\mathbf {Z}} \\mathbf {Z}/d\\mathbf {Z}$ satisfies $i \\circ \\tau ^{\\theta } = \\mathrm {id}$ .", "This means that $ 0 \\rightarrow H_1(\\mathrm {IA}_n, \\mathbf {Z})_{\\Gamma (n,d)} \\xrightarrow{} H_1(\\mathrm {IA}_{n,d},\\mathbf {Z})\\rightarrow H_1(\\Gamma (n,d),\\mathbf {Z}) \\rightarrow 0 $ is a split exact sequence.", "This shows Theorem REF .", "If $d$ is even, we need more complicated arguments since the image of $\\tau ^{\\theta }$ is not contained in $H_1(\\mathrm {IA}_n,\\mathbf {Z}) \\otimes _{\\mathbf {Z}} \\mathbf {Z}/d\\mathbf {Z}$ .", "(See [59] for details.)", "If $d$ is odd prime $p$ , then the structure of $\\Gamma (n,p)^{\\mathrm {ab}}$ is well-known due to Lee and Szczarba.", "Set $ \\mathfrak {sl}(n, \\mathbf {Z}/p\\mathbf {Z}) := \\lbrace A \\in M(n, \\mathbf {Z}/p\\mathbf {Z}) \\,|\\, \\mathrm {Trace}\\,A = 0 \\rbrace .", "$ Then we have Theorem 10.2 (Lee and Szczarba [33]) For $n \\ge 3$ and odd prime $p$ , $\\Gamma (n,p)^{\\mathrm {ab}} \\cong \\mathfrak {sl}(n,\\mathbf {Z}/p\\mathbf {Z})$ as an $\\mathrm {SL}(n,\\mathbf {Z}/p\\mathbf {Z})$ -module.", "From this theorem, we see that for any odd prime $p$ , the abelianization of $\\mathrm {IA}_{n,p}$ is isomorphic to $(\\mathbf {Z}/p\\mathbf {Z})^{\\oplus \\frac{1}{2}(n-1)(n^2+2n+2)}$ as an abelian group.", "We remark that a mapping class group analogue of the above was shown independently by Putman [52] and Sato [56].", "Let $\\mathcal {I}_{g,1}(d)$ be the kernel of the composite homomorphism $\\mathcal {M}_{g,1} \\rightarrow \\mathrm {Sp}(2g,\\mathbf {Z}) \\rightarrow \\mathrm {Sp}(2g,\\mathbf {Z}/d\\mathbf {Z})$ .", "Set $ \\mathfrak {sp}(2g, \\mathbf {Z}/d\\mathbf {Z}) := \\lbrace A \\in M(2g, \\mathbf {Z}/d\\mathbf {Z}) \\,|\\, {}^t A J + JA = O \\rbrace .", "$ Putman and Sato showed Theorem 10.3 (Putman [52] and [53], and Sato [56]) For any $g \\ge 3$ and odd $d \\ge 3$ , $ \\mathcal {I}_{g,1}(d)^{\\mathrm {ab}} \\cong (\\mathcal {I}_{g,1}^{\\mathrm {ab}} \\otimes _{\\mathbf {Z}} \\mathbf {Z}/d\\mathbf {Z}) \\, \\bigoplus \\, \\mathfrak {sp}(2g, \\mathbf {Z}/d\\mathbf {Z}).", "$ This result is generalized by Putman [53] for an even $d \\ge 2$ which is not divisible by 4.", "Sato [56] also proved the above for $d=2$ independently.", "That is, Theorem 10.4 (Putman [53] and Sato [56]) For any $g \\ge 3$ , $ \\mathcal {I}_{g,1}(2)^{\\mathrm {ab}} \\cong (\\mathbf {Z}/2\\mathbf {Z})^{\\oplus \\binom{2g}{3}} \\bigoplus (\\mathbf {Z}/4\\mathbf {Z})^{\\oplus \\binom{2g}{2}} \\bigoplus (\\mathbf {Z}/8\\mathbf {Z})^{\\oplus \\binom{2g}{1}}.", "$" ], [ "The case of $n=2$", "For $n=2$ and any odd prime $p$ , the group structure of $\\mathrm {IA}_{2,p}$ is much easier to handle than those for $n \\ge 3$ .", "We have a group extension $1 \\rightarrow \\mathrm {Inn}\\,F_2 \\rightarrow \\mathrm {IA}_{2,p} \\rightarrow \\Gamma (2,p) \\rightarrow 1.$ By a classical result by Frasch, we have Theorem 10.5 (Frasch, [20]) For any odd prime $p$ , the congruence subgroup $\\Gamma (2,p)$ is a free group of rank $\\alpha (p):= 1+ p(p^2-1)/12$ .", "Hence, $\\mathrm {IA}_{2,p}$ is a group extension of free groups of finite rank.", "In [58], we compute the integral homology groups of $\\mathrm {IA}_{2,p}$ for an odd prime $p$ , using the Lyndon-Hochshild-Serre spectral sequence of the extension (REF ).", "Proposition 10.6 (Satoh, [58]) For any prime $p$ , $ H_q(\\mathrm {IA}_{2,p},\\mathbf {Z}) ={\\left\\lbrace \\begin{array}{ll}\\mathbf {Z}& \\quad \\mathrm {if} \\,\\,\\, q=0, \\\\{\\mathbf {Z}}^{\\oplus \\alpha (p)} \\oplus (\\mathbf {Z}/p\\mathbf {Z})^{\\oplus 2} & \\quad \\mathrm {if} \\,\\,\\, q=1,\\\\{\\mathbf {Z}}^{\\oplus (2\\alpha (p)-2)} & \\quad \\mathrm {if} \\,\\,\\, q=2,\\\\0 &\\quad \\mathrm {if} \\,\\,\\, q \\ge 3.\\end{array}\\right.}", "$" ], [ "Acknowledgments", "The author would like to thank Professor Athanase Papadopoulos and Professor Nariya Kawazumi for giving me an opportunity to write this chapter, and to contribute to the Handbook of Teichm$\\ddot{\\mathrm {u}}$ ller Theory.", "He thanks Professor Athanase Papadopoulos for careful reading and valuable comments on this chapter.", "He also thanks to Andrew Putman for communicating his recent works with respect to the abelianization of $\\mathcal {I}_{g,1}(d)$ ." ] ]

[ [ "Phase separation and rotor self-assembly in active particle suspensions" ], [ "Abstract Adding a non-adsorbing polymer to passive colloids induces an attraction between the particles via the `depletion' mechanism.", "High enough polymer concentrations lead to phase separation.", "We combine experiments, theory and simulations to demonstrate that using active colloids (such as motile bacteria) dramatically changes the physics of such mixtures.", "First, significantly stronger inter-particle attraction is needed to cause phase separation.", "Secondly, the finite size aggregates formed at lower inter-particle attraction show unidirectional rotation.", "These micro-rotors demonstrate the self assembly of functional structures using active particles.", "The angular speed of the rotating clusters scales approximately as the inverse of their size, which may be understood theoretically by assuming that the torques exerted by the outermost bacteria in a cluster add up randomly.", "Our simulations suggest that both the suppression of phase separation and the self assembly of rotors are generic features of aggregating swimmers, and should therefore occur in a variety of biological and synthetic active particle systems." ], [ "Experiments on bacteria-polymer mixtures", "We studied suspensions of motile E. coli bacteria (`smooth swimming' strain HCB437 [11]) in motility buffer, and added various concentrations of sodium polystyrene sulfonate (NaPSS, molecular weight = 64,700 g/mol).", "(See Materials and Methods for details.)", "Previous work has shown that NaPSS is non-adsorbing to K-12 derived E. coli in motility buffer, and causes depletion-driven phase separation at high enough concentrations [9], [10].", "The range of the depletion attraction is $\\approx 35$  nm [10].", "Figure 1A shows samples with identical cell concentrations but different NaPSS concentrations ($c_p$ ) 2 hours after preparation.", "At each $c_p$ we show samples with non-motile (NM) and motile (M) cells, where the former were obtained from the latter by intense vortexing to break off their flagella.Experiments using NM cells of other origin (e.g.", "killing M cells by heating to 60$^\\circ $ C for 1 hour, deletion of key flagella synthesis gene FliF) gave the same results.", "Differential dynamic microscopy (DDM) [12] returned an average swimming speed of $7-10 \\mu $ m/s for the M cells, and confirmed that the NM cells were purely Brownian in their dynamics.", "In the images shown in Fig.", "1A, the brightness increases with cell concentration.", "Thus, in both NM and M samples with $c_p = 0$ and $0.1~\\%$ , cells remain homogeneously distributed after 2 hours.DDM showed that the motility of cells remained more or less constant over this duration.", "While this remains true of the M sample at $c_p = 0.2~\\%$ , the NM sample in this case has started to separate into upper and lower phases with low and high concentrations of cells (`vapor' and `liquid' respectively).", "This phase separation process is complete in the NM sample with $c_p = 0.5 \\%$ , while it is only just starting at this $c_p$ for the M cells.", "Phase separation is nearly complete at the next two higher $c_p$ (1% and 2%) in the M samples.", "Similar samples series at other cell concentrations enabled us to construct a phase diagram, Fig. 1B.", "The phase boundary and phase separation kinetics of NM cells correspond to that reported previously [9], [10].", "Our first important finding is that the phase boundary for the M cells is shifted significantly upwards: more polymer, or equivalently, a stronger depletion attraction, is needed to cause phase separation when the particles are active.", "Figure: A. Cuvettes of E. coli + NaPSS mixtures with the same cell concentration (4×10 10 4 \\times 10^{10} cfu/ml; cfu = colony forming unit) but different amount of polymer (as labeled, in wt.%) 2 hours after preparation.", "Pairs of cuvettes with non-motile (NM) and motile (M) cells are shown at each polymer concentration.", "B.", "Phase diagram showing data from NM (red circles) and M (black squares) cells; filled (open) symbols = single phase (phase separated).", "Approximate positions of the phase boundary are indicated by lines.", "Note that at a cell concentration of 10 10 10^{10} cfu/ml, the cell bodies occupy about 1% of the sample volume.We next imaged NM and M samples in the `pre-transition' region of the phase diagram, i.e.", "at polymer concentrations just below the respective phase boundaries.", "In this region of a classical atomic or molecular system, we expect to find transient clusters (or `liquid droplets').", "We found such pre-transition clusters in previous simulations of spherocylinder-polymer mixtures designed to mimic our experimental mixtures of NM bacteria and polymer [9], [10].", "We observe pre-transition clusters in both NM and M samples: the static snapshots are rather similar (an example from an M sample is shown in Fig. 2a).", "Movies of the clusters, however, reveal a dramatic difference between NM and M clusters.", "NM clusters undergo what is recognizably Brownian motion in both translational and rotation (Supplementary movie S1).", "Remarkably, however, the clusters self-assembled from motile cells translate with speeds approaching that of single cells and also rotate (Supplementary movies S2 and S3).", "The clusters persist over the entire duration of observation (minutes), during which the rotation is essentially unidirectional in a frame of reference in which the axis of the rotation is stationary (see Supplementary movies S2 and S3), although the axis of rotation itself seems to drift randomly.", "Occasional interruptions in this unidirectional rotation do occur, apparently due to collision with single motile cells or other clusters.", "The sense of the rotation appears to be random.", "This, then, is our second important finding.", "The interplay of motility and depletion attraction leads to the self-assembly of spontaneously rotating clusters, or micro-rotors.", "Note that our self-assembled micro-rotors are qualitatively different from the bacteria-driven rotation of externally added micro-lithographed gear wheels with a built-in chirality [1], [2], [3].", "Our micro-rotors are not only powered by bacteria, but self-assembled by and from them.", "The only externally-added agents are non-adsorbing polymers.", "Figure: A.", "Bright field microscopy image of a motile bacteria-polymer mixture (2.4×10 9 2.4 \\times 10^9 cells/ml, 0.2% NaPSS).", "(b) Snapshot of simulation of active dumbbells with parameters chosen to be similar to the experiments shown in A." ], [ "Effective potential theory for active phase separation", "We first propose a phenomenological theory for the observed suppression of phase separation.", "Our theoretical framework should be applicable to phase separation caused by inter-particle attraction of any origin in a system of self-propelled particles of any shape.", "For analytical tractability, we model individual bacteria as hard spheres of diameter $\\sigma $ in the presence of non-adsorbing polymers.", "The latter are modeled as mutually inter-penetrable spheres of diameter $\\xi \\sigma $ that are impenetrable relative to the hard spheres.", "In this Asakura-Oosawa (AO) model for colloid-polymer mixtures, the depletion attraction between two spheres at $\\xi \\ll 1$ can be approximated by [13]: $U(r) = \\left\\lbrace \\begin{array}{ll}\\infty & \\qquad r<\\sigma , \\\\-\\frac{\\epsilon }{\\xi ^2}\\left[\\frac{r}{\\sigma }-1-\\xi \\right]^2 & \\qquad \\sigma \\le r \\le \\sigma \\left(1+\\xi \\right), \\\\0 & \\qquad r > \\sigma \\left(1+\\xi \\right).", "\\end{array} \\right.$ Here $\\epsilon = \\frac{3}{2} \\phi _p k_B T \\left(1+\\xi \\right)/\\xi $ is the attractive well depth, and $\\phi _p$ is the volume fraction of polymer coils.Strictly speaking, this is the polymer concentration in a polymer bath in osmotic equilibrium with the sample, and not the concentration of polymers in the sample itself; but at the sort of particle concentrations we are working at, the two quantities are nearly equal.", "Under our experimental conditions, the polymer concentration around a swimming bacterium is constant.", "Figure: (A) A schematic of the depletion potentials for a passive (blue) and an active (red) system (with the range grossly exaggerated).", "(B) A typical fitted probability distribution function of the swimming speed of our E. coli from DDM .", "The dashed lines indicate the typical velocity values used to estimate F a F_a.", "(C) The calculated phase boundary for a passive (green) and active (red and blue) bacteria-polymer mixtures interacting via the depletion potential.", "The colours correspond to the propulsion velocity values in (B) used to estimate the active force F a F_a.In the absence of activity, two particles stay bound at separation $\\sigma $ until a thermal fluctuation increases this distance to beyond $\\sigma \\left(1+\\xi \\right)$ , at which point the particles cease to interact and become free.", "To estimate the effect of activity on depletion-driven phase separation, we consider two active particles pushing in the opposite directions with an active force $\\pm F_a$ while being held stationary by the depletion force.", "The particles come free at separation $r_*$ , where the total force acting on the particles becomes zero: $- \\frac{\\partial U(r_*)}{\\partial r} + F_a = 0,$ giving $r_*/\\sigma = 1+\\xi - \\xi k_b T f/2\\epsilon $ , with $f=F_a\\xi \\sigma /k_B T$ .", "We take the effective potential confining two active particles to be $U_{\\rm eff} (r) = U(r) - F_a r + U_0,$ where $U_0$ is chosen such that $U_{\\rm eff}(r_*) = 0$ (see Supplementary Material), Fig. 3A.", "Compared to the bare depletion potential, $U_{\\rm eff}(r)$ is shallower, and has slightly shorter range.", "To estimate the phase boundary for particles interacting via $U_{\\rm eff}(r)$ , we use the law of `extended corresponding states' [14], which states that the vapor-liquid phase boundaries (or `binodals') of various systems of attractive particles collapse onto a single master curve if the attraction is characterized by its reduced second virial coefficient, $b_2 = B_2/B_2^{\\rm HS}$ , where $B_2$ and $B_2^{\\rm HS}$ are the second virial coefficients of the attractive particles and of equivalent hard spheres respectively.", "The law is known to work, e.g., in relating the phase behavior of protein solutions and colloid-polymer mixtures [15].", "We therefore estimate the binodal in our system by mapping onto that of Baxter `adhesive hard spheres' [16] — hard spheres with an infinitely short range attraction (see Supplementary Material).", "We find that the shift in the VL binodal strongly depends on the value of the active force $F_a$ with which a stationary bacterium is pushing against the depletion potential.", "This force is approximately twice the force produced by the flagella of a free-swimming bacterium (see [22] and Supplementary Material for more details), which is proportional to the swimming speed, $v$ , of the bacteria.", "The measured probability density of the latter, $P(v)$ , can be fitted by a Schulz distribution [12], see Fig.", "3B, where we indicate by dashed lines the range of speeds used for estimating the value of $F_a$ .", "In Fig.", "3C we plot the contact strength of the depletion potential $\\epsilon /k_B T$ required to trigger phase separation for the passive and active bacteria-polymer mixtures as a function of cell volume fraction.", "We predict that activity shifts the binodal upwards by a factor of $\\approx 1.4-1.9$ , depending on the swimming speed used.", "Since $\\epsilon /k_B T$ is proportional to $c_p$ in our system, this should translate into a corresponding shift in the experimental phase boundary.", "The predicted shift of the phase boundary by a factor of $\\lesssim 2$ upwards has the same sign and a similar value as the observed upwards shift by a factor of $\\lesssim 3$ .", "The latter is likely an overestimate, since sedimentation of pre-transition clusters renders the experimental phase boundary for NM cells a lower bound.", "(The motility of the active pre-transition clusters means that sedimentation has minimal effect on the observed phase boundary for M cells.)", "Considering the crudeness of the approximations involved in our theory, we take such a degree of correspondence between experiment and theory as a validation of the basic physical content of our approach, viz., that activity renders it easier for thermal fluctuations to `free' cells bound by interparticle attraction, an effect that we model by a shallower effective potential, Fig. 3A.", "Note that we have above neglected hydrodynamic forces.", "These create an attraction between two particles as they move apart, which is proportional to their relative velocity and therefore vanishes at the balance point between the active and polymer-induced forces.", "Moreover, hydrodynamic interactions have no effect on barrier heights or phase boundaries in equilibrium systems.", "Their neglect therefore appears justified within our quasi-equilibrium evalation of the interaction potential." ], [ "Simulating active phase separation", "The simple physical ingredients entering our effective potential theory suggest that the suppression of phase separation by activity should be generic.", "To test this, we performed Brownian dynamics (BD) simulations intentionally preserving only what we think are the essential features of our experiments: motility and a short-range attraction (depth $\\epsilon $ ) but without hydrodynamics.", "Specifically, we simulated a collection of Brownian dumbbells (each with local frictional drag in the lab frame) interacting via a non-specific short-range attraction; a constant force applied along the axis of each dumbbell renders it motile (see Materials and Methods and Supplementary Material).", "While numerical parameters were chosen to resemble our experiments, our qualitative conclusions are valid for a much wider parameter space.", "Our simulated system is too small to allow reliable direct estimate of the effect of activity on the position of the binodal.", "Instead, we study how activity renormalizes the $\\epsilon /k_B T$ axis by quenching otherwise identical starting systems of active and passive attractive dumbbells to low temperatures (large $\\epsilon /k_B T$ ), at densities where the system forms a space-spanning gel.", "Starting from this configuration, we performed runs at decreasing $\\epsilon $ to locate gel melting as the point at which the potential energy per particle, $U$ , abruptly increased, Fig. 4.", "Intriguingly, the melting transition in the active system appears sharper than in a passive one.", "More importantly for our purposes, motility increases the value of $\\epsilon $ which is needed to stabilize the gel by a factor of $\\approx 3$ .", "Choosing different initial conditions and densities does not much affect this result (see Supplementary Materials for more details).", "The substantial agreement between experiment, Fig.", "1B, theory, Fig.", "3C, and simulations, Fig.", "4, on the effect of activity in renormalizing the $\\epsilon /k_B T$ axis, and the absence of system-specific features in either our theory or simulations, suggest that phase separation suppression by activity is likely generic for attractive particles.", "Figure: The energy of a gel in a simulated system of attractive dumbbells measured in units of the contact potential, U/ϵU/\\epsilon , as the temperature, measured as ϵ/k B T\\epsilon /k_B T, is changed, for active (bold) and non-active (thin) particles.", "The energy abruptly increases as the gel melts." ], [ "Self-assembled rotors", "We have observed coherently-rotating pre-transition active clusters in both experiments (Supplementary movies S2 and S3) and simulations (Supplementary movie S4).", "The existence of pre-transition clusters per se is unsurprising.", "However, their persistence (over at least minutes in experiments) and unidirectional rotation require explanation.", "First, clusters of short-range attractive particles are long-lived due to entropic reasons (Willem Kegel, private communication).", "Clustering in atomic systems interacting via van der Waals attraction is expected at a well depth of $\\epsilon \\sim k_B T$ .", "A shorter range attraction entails higher loss of entropy upon bonding, which therefore requires a larger $\\epsilon $ .", "Cluster life time is controlled by the escape rate of a single cell bonded to its neighbors, which itself is dominated by a Boltzmann factor, $e^{\\epsilon /k_B T}$ .", "A quantitative estimate using the Kramers formalism (see Supplementary Material) returns a life time of many minutes to months (depending on the average number of neighbors per cell we use in the calculation), consistent the observation of persistent clusters.", "Secondly, since we found self-assembled rotors in our BD simulations, which did not include fluid-mediated interactions, we may rule out specifically hydrodynamic explanations for the phenomenon, and the self assembly of micro-rotors is likely generic.", "We now show that the quantitative features of the micro-rotor motion do depend on system-specific details.", "The angular speed, $\\Omega $ , of our rotating bacterial clusters (see Materials and Methods) decreases with cluster size, $R$ , taken as half the arithmetic average of the longest and shortest dimensions of each cluster, Fig. 5.", "The data is consistent with $\\Omega \\sim R^{-1}$ , with typical $\\Omega $ in the range 1-20 rad/s.", "(See Supplementary Material for a discussion of the range of cluster sizes represented in our data.)", "The physical origin of cluster rotation is clear.", "The forces generated by the flagella bundle of each bacterium (which hereafter we refer to as a single (effective) flagellum) in a cluster do not cancel, giving a residual net torque about the cluster center.", "We assume that only bacteria on the surface of the cluster exert torques, implying that the total number of bacteria, $N$ , participating in propulsion and rotation of the cluster is equal to $N=4\\pi R^2/A_0$ , with each bacterium occupying $A_0 \\sim 2$ $\\mu $ m$^2$ .", "If each bacterium exerts a torque of magnitude $T_0$ on the cluster and we assume that these torques are randomly oriented, then the total torque on the cluster, $T_{\\rm tot}$ , is given by $T_{\\rm tot} \\sim \\sqrt{N} T_0$ , since the sum of $N$ random vectors of equal length scales as $\\sqrt{N} \\sim R$ .", "$T_{\\rm tot}$ is balanced by the rotational friction, which we take to be $8\\pi \\eta R^3 \\Omega $ , where $\\eta \\sim 1$  cP is the solvent viscosity.", "Thus, we predict that $\\Omega = \\frac{1}{4\\eta \\sqrt{\\pi A_0}R^2}T_0.$ If we take each bacterium to be a force monopole of magnitude $F_a$ that contributes a torque $T_0 \\sim F_a R$ , then Eq.", "REF gives $\\Omega \\sim R^{-1}$ , which is close to our observations.", "However, the total force exerted by a swimming bacterium on the fluid must sum up to zero, so that the lowest order approximation with the appropriate symmetry is not a monopole, but a dipole [17], [18].", "For simplicity, we assume that bacterial flagella lie tangentially to the cluster surface.", "A small portion of the flagellum at some distance from the cluster generates a propulsion force that is transmitted through the whole flagellum to the cluster, while a force of the same magnitude and opposite direction is applied to the fluid locally.", "The former corresponds to the monopole contribution already discussed.", "The latter force generates a fluid flow that has to vanish at the surface of the cluster, which exerts an extra drag force acting on the cluster that partially cancels the `direct' force transmitted through the flagellum.", "Summing up the contributions from different parts of the flagellum we find (see Supplementary Material) that the torque exerted per bacterium on the surface is $T_0 = F_a R\\left[ 1- R(l^2 + R^2)^{-0.5} \\right],$ with $l \\sim 10 \\mu $ m being the length of a flagellum.", "The right hand side consists of a monopole term $F_a R$ and a dipolar correction.", "We estimate $F_a$ as the force necessary to propel a $1 \\mu $ m diameter sphere at speeds in the range 5-$15\\mu $ m/s, Fig.", "3B, giving $F_a \\sim 0.05-0.15$  pN.", "The prediction of Eqs.", "REF and REF for various values of $F_a$ , Fig.", "5, is compatible with the data.", "Note that when $R < l$ , as in our experiments (Fig.", "5), the dipolar correction in Eq.", "REF does not effectively change the scaling, so that we may expect $\\Omega \\sim R^{-1}$ .", "We have assumed that the forces in a cluster add randomly and that their relative disposition is fixed.", "The alternative assumption that they add coherently predicts an angular velocity independent of cluster size, clearly at odds with our data in Fig. 5.", "While it is non-trivial to test experimentally our `quenched disorder' assumption, it is supported by an analysis of the ordering of active dumbbells within a rotating cluster in our BD simulations.", "These show that orientational ordering is largely absent, and that there is little rearrangement of the dumbbells after the clusters form (see Supplementary Material).", "The neglect of hydrodynamics in our simulations is not a major shortcoming in this respect, as in the micro-rotor phase we expect intra-cluster interactions to be dominated by excluded volume, rather than hydrodynamics.", "Figure: Plot of the angular velocity as a function of cluster radius.", "Different kinds of points refer to independent experimental data sets.", "The green, red and blue solid lines are predictions of Eqs.", "3 and 4 with F a F_a being 0.050.05, 0.10.1 and 0.150.15pN, correspondingly." ], [ "Discussion and Conclusions", "We have studied phase transition and pre-transition clustering in a mixture of bacteria and non-adsorbing polymers, viewed as an active suspensions with inter-particle attraction.", "We find that activity suppresses phase separation compared to the case of non-motile bacteria [9], [10].", "Addition of $\\sim 3$ times more polymer was needed to cause phase separation in a suspension of motile cells.", "We rationalize this by modeling the interaction between self-propelled cells by a shallower effective potential, determined by balancing the depletion force and the self propulsion.", "Quantitative prediction of this upward shift of the phase boundary follows from the extended law of corresponding states [14] via mapping second virial coefficients.", "An alternative interpretation of our results is to ascribe a higher effective temperature, $T_{\\rm eff} \\approx 3T$ , to our active particle system while leaving the inter-particle potential unchanged.", "The idea of a higher effective temperature due to activity has recently been applied to characterise the transport properties of a suspension of synthetic self-propelled colloids [7].", "Each active particle undergoes a random walk in the long-time limit because the direction of its self-propelled motion is subject to thermal fluctuations (controlled by the thermodynamic temperature $T$ ) [19].", "This random walk is described by an effective diffusion coefficient $D_{\\rm eff} = v^2\\tau _R/4$ , where $v$ is the self-propulsion speed and $\\tau _R = 4L^2/3D_0$ is the (Brownian) rotational relaxation time of an object of radius $L$ with (Brownian) translational diffusion coefficient $D_0$ .", "Palacci et al.", "found that $D_{\\rm eff} = k_B T_{\\rm eff}/6\\pi \\eta L$ .", "For our smooth swimming E. coli strain, we may estimate $D_{\\rm eff} \\lesssim 10^4 \\mu $ m$^2$ /s [20],Note that we have taken $2L \\sim 10 \\mu $ m to be an estimate of the `cell body + flagella bundle' unit; this is the size that controls Brownian rotation.", "giving $T_{\\rm eff} \\sim 10^4 T$ .", "This latter effective temperature for characterizing diffusive transport in the bacterial suspension is four orders of magnitude higher than the effective temperature that we may use to characterize a suspension of the same motile bacteria in the context of phase separation driven by inter-particle attraction.", "This `discrepancy' highlights an important point: there is no single parameter, `the effective temperature', that is appropriate for systems away from equilibrium.", "Indeed, the whole concept of effective temperature may be more appropriate for some situations than others.", "In a system such as ours, we suggest that it is more appropriate to talk of an effective potential between particles, which is explored by fluctuations that are characterized by the thermodynamic temperature.", "Remarkably, when the inter-particle attraction is just too weak to cause phase separation, we observe finite clusters that individually break chiral symmetry and rotate unidirectionally.", "The spontaneous formation of these micro-rotors constitutes a clear demonstration of the self assembly of functional structures from active suspensions.", "We accounted for the observed scaling of cluster rotation speed with cluster size using a hydrodynamic theory.", "These `active clusters' show minimal sedimentation: their motion confers a high effective temperature in the diffusive sense [7].", "We have also simulated Brownian dumbbells each subject to a body force directed along its axis and with a short-range inter-particle attraction, with parameters chosen to mimic our experimental bacteria-polymer mixtures.", "Our theoretical framework is also able to account for the melting of gel states in the simulated system, which we take to be a surrogate for the phase boundary in our small simulations.", "Moreover, clusters in the melted state also show spontaneous rotation.", "Our experimental system is a very specific one: living bacteria propelled by rotating flagella bundles operating via low-Reynolds number hydrodynamics in a chemically-complex buffer with added non-adsorbing polymer; the latter causes a short-range inter-particle attraction via the depletion mechanism.", "Our phenomenological theory and simulations strip away many of these system-specific features.", "The success of the theory and simulations in reproducing experimentally observed phase separation suppression and the self assembly of rotating clusters suggest strongly that these are generic features in attractive active particle systems.", "It would therefore be interesting to look for such effects not only in suspensions of other bacteria, but also in suspensions of self-propelled `Janus' particles [19].", "In the latter case, the competition for `fuel' among neighboring particles may act to limit cluster sizes.", "Finally, we point out that our bacterial suspensions contain a fraction of non-motile organisms [12].", "In the presence of polymer, these cells should be the first to aggregate, producing `nuclei' for the subsequent aggregation of motile cells.", "Preliminary BD simulations of mixtures of motile and non-motile attractive dumbbells support this picture, which motivates our assumption that cluster rotation and translation are powered by motile cells on the outside.", "It also points to possible relevance of our work for aquatic and marine ecology: bacteria aggregating in the presence of non-living organic particulate matter (detritus) can trap the latter and keep it suspended for the water column food chain [23]." ], [ "Culture conditions for motile bacteria", "We used E. coli strain HCB437, a smooth-swimming mutant.", "Overnight cultures ($\\sim 18$  h) were grown in 10 ml Luria Broth (LB, tryptone 10.0 g/l, yeast extract 5.0 g/l, NaCl 5.0 g/l) starting from a single colony on LB agar (tryptone 10.0 g/l, yeast extract 5.0 g/l, NaCl 5.0 g/l, agar 15 g/l) using an orbital shaker at 30$^\\circ $ C and 200 rpm.", "A fresh culture was inoculated as 1:100 dilution of overnight grown cells in 35ml tryptone broth (TB, tryptone 10.0g/l, NaCl 5.0g/l) and grown for 4 h (late exponential phase).", "These cells were treated with the following two protocols to achieve motile and non-motile cells, respectively." ], [ "Preparation of motile cells", "Cells were washed three times with motility buffer (MB, pH = 7.0) containing 6.2 mM K$_2$ HPO$_4$ , 3.8 mM KH$_2$ PO$_4$ , 67 mM NaCl and 0.1 mM EDTA by careful filtration (0.45 $\\mu $ m HATF filter, Millipore) to minimize flagellar damage.", "The final volume of the washed bacterial samples was 1-2 ml therefore suspensions could be concentrated up to 25-fold compared to the original culture." ], [ "Preparation of non-motile cells", "Cells were washed by centrifugation (10 min, 2700g Hermle Z323K).", "After removing the supernatant cell pellets were re-suspended in 1 ml MB by vortexing for 2 min.", "In total, three washing steps were completed, with the final sample being suspended in 1 ml MB." ], [ "Characterization of bacterial samples", "Optical density (OD) measurements at 600 nm (Cary 1E, Varian) normalized by viable plate counts on LB agar of serial diluted samples ($1.55 \\times 10^9$  cfu/ml $\\equiv $ OD600nm = 1) were used to determine cell densities.", "Motility was characterized by DDM [12]." ], [ "Polymer", "Sodium polystyrene sulfonate (Aldrich) was used (NaPSS, M$_{\\rm w}$ = 64,700 g/mol, M$_{\\rm w}$ /M$_{\\rm n}$ = 3.1).", "The molecular weights and polydispersities were determined by gel-permeation chromatography (GPC) against PSS standards.", "We previously estimated a radius of gyration in MB of 17.5 nm [9], [10].", "Polymer stock solutions were prepared at 20% (w/v) in MB and filtered through a $0.2\\mu $ m disposable syringe filter prior to use." ], [ "Study of phase separation", "Observations were made in closed 1.6 ml disposable cuvettes with a total sample volume of 1 ml.", "Polymer solution, cell suspensions and MB were mixed in different ratios to achieve cell concentrations in the range of $10^9$ to $10^{11}$ cfu/ml and polymer concentrations in the range of 0 to 2 wt%.", "Samples were homogenized by thorough, careful mixing, placed inside an incubator at 22°C (MIR-153, Sanyo) and observed using a camera (QImaging, Micro-publisher 3.3RTV) controlled by QCapture pro 5.0 software.", "Images were captured for varying periods up to 24 h." ], [ "Rotation of motile clusters", "Two types of sample cells were used: capillary cells (dimensions $8\\times 50 \\times 0.4$ mm, CMS) completely filled with $190 \\mu $ l sample (sealed with Vaseline), and 8-well chambered microscopic cover glass cells (dimensions $8\\times 8\\times 8$ mm, Lab-Tek, Nunc) half-filled with $400 \\mu $ l test sample covered with a lid.", "Samples were prepared at a cell concentration OD600 =0.3 ($5\\times 10^8$ cells/ml) and 1wt% NaPSS.", "Clustering of bacteria was observed using either bright-field or phase-contrast microscopy at $90\\times $ magnification using two different microscope-camera combinations: 1) Nikon Eclipse TE2000-U inverted microscope and Marlin F145B2 camera (17 fps) and 2) Nikon Eclipse Ti inverted microscope and Cool-Snap HQ2 camera (11 fps).", "Videos were recorded for 30 to 180 s. Clusters were observed during a period when the axis of rotation was perpendicular to the imaging plane.", "The angle of rotation was measured using ImageJ angle tool.", "Measurements were made 50-150$\\mu $ m above the surface at different times during the first 3 h after polymer addition." ], [ "Dumbbell simulations", "We simulated a system of 1000 dumbbells, each composed by two spheres of radius $\\sigma $ and kept together by a stiff harmonic spring.", "Simulations were performed by using the LAMMPS code [24] in the (overdamped) Brownian dynamics mode, and we employed the following additional force fields.", "Depletion-induced attraction was modelled by means of a truncated and shifted Lennard-Jones potential, where the depth and the size are matched to the interaction strength and range of NaPSS-induced depletion.", "Motility was achieved by applying a force directed from the rear to the front bead of each of the dumbbell, and applied in its centre of mass.", "Finally, the random force in the Brownian dynamics was chosen so as to satisfy the fluctuation-dissipation theorem in the passive limit.", "More details of the methods and parameter values are given in the Supplementary Material.", "We thank H. Berg for the smooth swimming E. coli, V. Martinez for assistance with motility measurements, and E. Sanz and P. B. Warren for discussions.", "The UK work was funded by the EPSRC (EP/D071070/1, EP/E030173 and EP/I004262/1) and the Royal Society.", "AC was funded by NSF Career Grant No.", "DMR-0846426.", "CV was funded by a Marie Curie Intra-European Fellowship.", "Supporting Information" ], [ "Phase separation of active colloids", "Here we explain how we estimate the phase boundary for active particles interacting via the depletion potential.", "First, we incorporate the effect of activity by defining an `effective potential' based on force balance.", "Then we invoke the extended law of corresponding states [14] and match the second virial coefficients along the phase boundary of the parent depletion potential and of our effective potential via the known phase boundary of Baxter's adhesive hard spheres [25], [16].", "The result will show that activity significantly postpones the transition." ], [ "Depletion potential between active particles", "We model bacteria as hard spheres of diameter $\\sigma $ .", "The interaction potential between two hard spheres in the presence of non-adsorbing polymer of diameter $\\xi \\sigma $ with $\\xi \\ll 1$ can be approximated by [13] $U(r) = \\left\\lbrace \\begin{array}{ll}\\infty & \\qquad r<\\sigma , \\\\-\\frac{\\epsilon }{\\xi ^2}\\left[\\frac{r}{\\sigma }-1-\\xi \\right]^2 & \\qquad \\sigma \\le r \\le \\sigma \\left(1+\\xi \\right), \\\\0 & \\qquad r > \\sigma \\left(1+\\xi \\right).\\end{array} \\right.$ Here, $\\epsilon = \\frac{3}{2} \\eta _p k_B T \\frac{1+\\xi }{\\xi }$ is the magnitude of the potential at contact, $U(\\sigma )=-\\epsilon $ , and $\\eta _p$ is the polymer volume fraction.Strictly, this is the polymer volume fraction in a reservoir of pure polymers in osmotic equilibrium with the bacteria-polymer mixture.", "However, at the bacterial concentrations we work at, this distinction is unimportant.", "In the absence fluctuations, the lowest-energy configuration for two particles is touching, i.e.", "at a center-to-center separation of $\\sigma $ .", "We model an active particle as a sphere subject to a propulsive force, and quantify the competition between polymer-induced attraction and activity for the case of two active particles with equal and opposite propulsive forces.Note that any other orientation of active forces is unstable and will convert into this one, with the active forces pointing in opposite direction.", "The force generated by the depletion potential is given by $F(r) = -\\frac{\\partial U(r)}{\\partial r} = \\frac{2\\epsilon }{\\xi ^2\\sigma }\\left( \\frac{r}{\\sigma }-1-\\xi \\right),$ and is maximal at contact: $F_c=-2\\epsilon /\\xi \\sigma $ .", "To cause phase separation, $\\epsilon \\gtrsim k_B T$ .", "Our polymer diameter is $\\xi \\sigma \\sim 35$ nm, so that $F_c\\gtrsim 0.2$  pN.", "The propulsive force generated by a free-swimming bacterium can be estimated by the drag force on the cell body, $F_{\\rm prop} \\sim 3\\pi \\eta \\sigma v$ with $\\eta =10^{-3}$ Pa$\\cdot $ s, $v\\sim 10\\mu $ m/s, and $\\sigma \\sim 1 \\mu $ m, so that $F_{\\rm prop}\\sim 0.1$ pN.", "These have similar orders of magnitude, so that we expect activity to be a significant perturbation on phase separation.", "Two passive particles stay bound at distance $\\sigma $ until a thermal fluctuation increases the inter-particle separation beyond $\\sigma \\left(1 + \\xi \\right)$ , at which point the particles become free.", "To see how activity changes this scenario, we first note that the total force on each active particle is given by the sum of the depletion and active forces: $F_{\\rm eff}(r) = F(r) + F_{\\rm prop}.$ This bound pair breaks up when a thermal fluctuation increases the inter-particle distance to some value $r_*$ such that $F_{\\rm eff}(r_*)=0$ , which, from Eq.", "[REF ], is $\\frac{r_*}{\\sigma } = 1+ \\xi - \\frac{\\xi }{2}\\frac{k_B T}{\\epsilon }f,$ where $f=F_{\\rm prop}\\xi \\sigma /k_B T$ is the typical work done by the active force in separating the particles by the range of the depletion potential in units of the thermal energy.", "Note that since $f \\propto F_{\\rm prop}$ and $F_{\\rm prop} \\propto v$ , $f$ increases linearly with swimming speed.", "We obtain the effective potential of interaction, $U_{\\rm eff}(r)$ , between two particles in the presence of depletion and activity by requiring that $-\\frac{\\partial U_{\\rm eff}(r)}{\\partial r} = F_{\\rm eff}(r),$ which after integration gives $U_{\\rm eff}(r) = U(r) - F_{\\rm prop} r + U_0,$ where the integration constant $U_0$ is chosen so that $U_{\\rm eff}(r_*)=0$ , since the particles become free after being separated beyond the distance $r_*$ .", "This procedure yields $U_{\\rm eff}(r) = \\left\\lbrace \\begin{array}{ll}\\infty & \\qquad r<\\sigma , \\\\-\\epsilon \\left(\\frac{r-r_*}{\\xi \\sigma }\\right)^2 & \\qquad \\sigma \\le r \\le r_*, \\\\0 & \\qquad r > r_*.\\end{array} \\right.$ In Fig.", "S1 we sketch the bare depletion potential $U(r)$ in the absence of activity and the `active' depletion potential $U_{\\rm eff}(r)$ .", "We observe that the active force decreases both the range of the potential and its slope.", "Note that this procedure is widely applied in modeling the effect of an applied force on ligand-receptor binding [28].", "Figure: Potential of interaction between colloidal particles.", "Blue:depletion potential U(r)U(r) in the absence of activity; red:sketch of the effective potential U eff (r)U_{\\rm eff}(r) for self-motileparticles; r * r_* marks the point where the total binding force due to depletion and activity vanishes.The second virial coefficient for the `active' depletion potential is given by $B_2^{(\\rm adp)} & = & \\frac{1}{2}\\int _0^\\infty \\left[ 1-e^{-\\frac{U_{\\rm eff}(r)}{k_B T} }\\right]4\\pi r^2 dr,\\\\& = & \\frac{2\\pi }{3} r_*^3 - 2\\pi \\xi ^3 \\sigma ^3 g\\left(x,r_*,\\xi \\right).$ Here $&&g\\left(x,r_*,\\xi \\right) = \\frac{r_*}{\\xi \\sigma x} -\\frac{r_*+\\sigma }{2x\\xi \\sigma } e^{\\frac{x(r_*-\\sigma )^2}{\\xi ^2\\sigma ^2}}\\nonumber \\\\&& \\qquad + \\frac{1}{\\sqrt{x}} \\left( \\frac{r_*^2}{\\xi ^2\\sigma ^2} - \\frac{1}{2x}\\right) \\int _0^{\\frac{\\sqrt{x}}{\\xi \\sigma }(r_*-\\sigma )}e^{t^2}dt,$ where we have introduced $x=\\epsilon /k_B T$ ." ], [ "Baxter potential", "The Baxter adhesive hard sphere potential [25] is defined by $\\frac{U_B(r)}{k_B T} = \\left\\lbrace \\begin{array}{ll}\\infty & \\qquad r<\\sigma , \\\\\\ln \\left[12\\tau \\left( \\frac{d-\\sigma }{d} \\right) \\right] & \\qquad \\sigma \\le r \\le d, \\\\0 & \\qquad r > d,\\end{array} \\right.$ in the limit $d\\rightarrow \\sigma $ .", "Here $d-\\sigma $ is the range of the attractive potential and the Baxter effective temperature $\\tau $ sets its strength.", "Its second virial coefficient is: $&& B_2^{(\\rm bp)} = \\lim _{d\\rightarrow \\sigma }\\frac{1}{2}\\int _0^\\infty \\left[ 1-e^{-\\frac{U_B(r)}{k_B T} }\\right]4\\pi r^2 dr \\nonumber \\\\&& \\qquad = \\frac{2\\pi }{3}\\sigma ^3\\left[ 1-\\frac{1}{4\\tau }\\right].$ The phase diagram for this system was calculated by Miller and Frenkel [16]." ], [ "The extended law of corresponding states", "We map a system of active particles interacting via the depletion potential onto the Baxter model by matching their second virial coefficients along the gas-liquid binodal.", "This mapping yields a relation between the Baxter effective temperature $\\tau $ and the parameters of the active suspension.", "Solving $B_2^{(\\rm bp)}=B_2^{(\\rm adp)}$ gives $\\tau ^{-1} = 4\\left[ 1 -\\frac{r_*^3}{\\sigma ^3} + 3\\,\\xi ^3 g(x,r_*,\\xi )\\right].$ This expression is used to calculate the phase boundary for the active depletion potential.", "For any value of the particle volume fraction $\\phi $ , we use the position of the phase boundary, $\\tau (\\phi )$ , found by Miller and Frenkel [16] and find from Eq.", "(REF ) the corresponding depth of the depletion potential $x=\\epsilon /k_B T$ at phase separation.", "We repeat this procedure for different values of the dimensionless active force $f$ to study how activity influences the phase diagram." ], [ "Estimating the propulsion force", "Here we provide a more detailed estimate for the propulsion force $F_{\\rm prop}$ .", "Note first that there is a distribution of swimming speeds in our bacterial populations.", "A representative swimming speed distribution from fitted differential dynamic microscopy [12] data is shown in Fig.", "3B in the main text.", "Since the average speed in this population is about $\\bar{v}\\sim 10\\mu $ m/s, a significant fraction of organisms swim at $> 10 \\mu $ m/s.", "Within the resistance matrix framework of a free swimming E. coli [30], [31], the propulsion force $F_{\\rm prop}$ generated by the flagella bundle and the motor torque $N_{m}$ required to rotate it are expressed in terms of the propulsion velocity $v$ and the rotation speed of the flagella bundle $\\omega $ : $&& F_{\\rm prop} = -A\\,v + B\\,\\omega ,\\\\&& \\,\\,\\,\\,\\,N_m = -B\\,v + D\\,\\omega ,$ where $A$ , $B$ and $D$ are the resistance matrix coefficients related to the bundle geometry.", "For a non-tumbling strain of E. coli swimming with $v=20\\mu $ m/s and $\\omega =780$ rad/s, Chattopadhyay et al.", "[32] found that $&& A = 1.48 \\times 10^{-8}\\;\\mbox{Ns/m}, \\nonumber \\\\&& B = 7.9 \\times 10^{-16} \\;\\mbox{Ns} , \\\\&& D = 7.0 \\times 10^{-22} \\;\\mbox{Nsm} \\nonumber $ For a free swimmer, these values give $F_{\\rm prop}\\sim 0.3$ pN, which is slightly higher than the estimate used above.", "Now consider a bacterium with free-swimming speed $v = 20\\mu $ m/s that is held stationary by a depletion potential.", "It is easily verified that for all $v \\lesssim 20 \\mu $ m/s, $B\\, v \\ll D\\omega $ in Eq. ().", "Thus, for our stationary bacterium with $v =0$ , $N_m$ , and hence $\\omega $ , remain approximately equal to their free-swimming values.", "On the other hand, the propulsion force, Eq.", "(REF ), is significantly increased to $F_{\\rm prop}=B\\,\\omega \\sim 0.6$ pN, leading to $f\\sim 5$ .", "In what follows, we study the effect of activity on phase separation for $f \\lesssim 5$ ." ], [ "Phase diagram for the active depletion potential", "Our results are plotted in Fig.", "S2, which corresponds to Fig.", "3C in the main text.", "It is clear that activity significantly suppresses phase separation, by up to a factor of $\\sim 1.9$ , which corresponds to $f = 5$ (or a swimming speed of $v = 20 \\mu $ m/s).", "Figure: The depth of the depletion potential at whichphase separation occurs as a function of particle volume fraction, for differentvalues of activity (from bottom to top): green: passive suspension (f=0f = 0), red: f=2f=2 (corresponding to the average speed of v=10μv = 10\\mu m/s), blue:f=5f=5 (corresponding to v=20μv = 20 \\mu m/s, close to the maximum speed observed in our bacterial populations).", "The shaded area therefore represents the shifts that can be expected from the faster half of our cell population.Note that we have used a similar approach previously to explain how oscillatory shear drives the crystallization of depletion-induced colloidal gels [33].", "Consider two particles bound by depletion in an initial configuration such that at the extremes of each oscillatory cycle, these two particles are maximally separated by the imposed shear.", "In this new configuration, the particles experience a shallower effective potential.", "Estimating the particles’ Kramers escape time out of this shallower effective potential provided a semi-quantitative explanation for the frequency and amplitude dependence of shear-induced crystallization.", "The details of the treatment differ because [33] involves an imposed strain, and the present work involves an imposed stress.", "Here we expand on our explanation for the persistence of pre-transition active clusters.", "A Brownian particle (diffusion coefficient $D$ ) trapped in a depletion potential of depth $\\epsilon $ and range $\\delta $ approximated as a ramp of the same depth and range has a Kramers escape time given by [33] $t_K = \\frac{\\delta ^2}{6Dx^2}e^x, $ where $x = \\epsilon /k_B T$ .", "Differential dynamic microscopy [12] measures $D \\approx 0.3 \\mu \\mbox{m}^2$ /s for our bacterium.", "The range is estimated by $\\delta = r^\\ast - \\sigma $ , Fig.", "S1 and Eq.", "(REF ), which, like $x$ , is a function of the propulsion speed via the dimensionless parameter $f$ .", "At the average swimming speed of $v = 10 \\mu $ m/s, $f = 2$ , from which we obtain $\\delta \\approx 30$ nm (using $\\sigma = 35$ nm, which is the size of our polymer) and $x \\approx 7$ at the phase boundary.", "Each cell on the surface of the cluster is in contact with $n$ other cells, so that its escape time, and therefore a `cluster life time' can be estimate using Eq.", "(REF ) with $x$ replaced by $nx$ .", "Using $n = 3$ we obtain $t_K \\approx 1500$ s (or $\\sim 25$ minutes); if we take $n = 4$ , this increases to $t_K \\approx 26 \\times 10^6$ s (or just under 10 months).", "These long escape times explain why our pre-transition clusters do not appear to break up and reform." ], [ "Brownian Dynamics (BD) simulations", "Here we detail the model and methods used to simulate a suspension of active, self-propelled, Brownian dumbbells, discuss the mapping from simulation parameters to experiments, and spell out some technical aspects of the data analysis." ], [ "Simulation details", "In our BD simulations, we study the evolution in three-dimensional space of a system of $N=1000$ hard-dumbbells in an NVT ensemble using cubic periodic boundary conditions.", "A dumbbell consists of two spheres with the same diameter $\\sigma $ .", "Any pair of spheres in the simulation interact via a truncated and shifted Lennard-Jones potential $V_{LJ}(r) & = & 4 \\epsilon \\left\\lbrace \\left[ \\left( \\frac{\\sigma }{r}\\right)^{12} - \\left(\\frac{\\sigma }{r}\\right)^6\\right] \\right.", "\\nonumber \\\\& & - \\left.\\left[ \\left( \\frac{\\sigma }{r_c}\\right)^{12}-\\left(\\frac{\\sigma }{r_c}\\right)^6\\right] \\right\\rbrace $ for $r\\le r_c$ ($r=r_i-r_j$ , with $i,j=1,...,2N$ ), whereas $ V_{LJ}(r)=0$ for $r > r_c$ .", "Choosing a cutoff of $r_c=1.2 \\sigma $ gives a very short range attraction that mimicks the depletion potential.", "The strength of $V_{LJ}$ is controlled by $\\epsilon $ , which is varied to simulate a change in the concentration of the polymer inducing the depletion attraction.", "Finally, the two spheres in a dumbbell are `glued' together by means of a stiff harmonic potential $V_{H}(r)=\\kappa (r-\\sigma )^2$ , where $\\kappa = 700k_B T/\\sigma ^2$ is the spring constant.", "Our simulations are carried out using the open source LAMMPS Molecular Dynamics package [34].", "The motion of the spheres are governed by the following under-damped Langevin equations of motion: $m \\frac{d^2 \\mathbf {r}_i}{dt^2} = - \\zeta \\frac{d \\mathbf {r}_i}{dt} - \\frac{d V }{d \\mathbf {r}_i} + \\mathbf {F_r} + \\mathbf {F_a}$ where $m$ is the mass of a sphere, $\\zeta $ is the friction coefficient ($\\zeta =m \\gamma $ with damping coefficient $\\gamma $ ), $V$ is the total conservative potential acting on each particle ($V=V_{LJ}+V_H$ ) and $\\mathbf {F_r}$ the random force due to the solvent at temperature $T$ .", "Activity via self-propulsion is introduced through an extra force ($\\mathbf {F_a}$ ) acting on each sphere, with constant magnitude and directed along the vector joining the front bead of a dumbbell to its rear one (front and rear beads are randomly chosen at the start of the simulation).", "We take $F_r = \\sqrt{k_B T \\zeta }R(t)$ with $R(t)$ a stationary Gaussian noise with zero mean and variance $\\langle R(t) R(t^{\\prime }) \\rangle = \\delta (t-t^{\\prime })$ , so that the fluctuation-dissipation theorem holds in the passive limit ($\\mathbf {F_a}=0$ ).", "Note that Brownian dynamics neglects hydrodynamic interactions.", "In our simulations, we set $\\gamma =2 \\tau ^{-1}$ ($\\tau = \\left(m\\sigma /\\epsilon \\right)^{1/2}$ is the time unit, which is set to 1 in our simulations).", "The total simulation time is typically $10^3 \\tau $ (we chose a time step $\\delta t=10^{-3} \\tau $ ).", "The value of $\\gamma $ is relatively large to ensure an effectively overdamped motion on the length scale of the particle size, which is the relevant regime for the bacteria in our experiments.", "Finally, the magnitude of the active force, $F_a$ is chosen so as to correspond to a propulsion velocity of $\\sim 9$ $\\mu $ m/s, close to the peak in the velocity distribution in Fig.", "3B in the main text.", "The relevant dimensionless number associated with the propulsion velocity is $F_a\\sigma /k_BT$ , which is equal to 20 in our simulations and may be thought of as an `active Péclet number'.This is because $F_a\\sigma /k_BT =v\\sigma /D$ where $v$ is the propulsion speed and $D=k_BT/\\gamma $ is the diffusion coefficient of a passive sphere of size $\\sigma $ .", "Another dimensionless number relevant to our corresponding state theory is the analogous of the previously defined quantity $f$ , which can be computed here as $F_a(r_c-r_{\\rm min})/k_BT \\sim 1.6$ , where $r_{\\rm min}=2^{1/6}\\sigma $ is the point at which the Lennard-Jones potential is minimum." ], [ "Stability of active and passive gels and\ndependence on the initial condition", "In order to study aggregation of the dumbbells into rotating clusters (cf.", "Fig.", "2B in the main text), we started the simulations from an initial configuration in which particles are randomly positioned at a (number) density of $\\rho = 5 \\times 10^{-3} \\sigma ^{-3}$ , corresponding to a concentration of about $5 \\times 10^{9}$ cells/ml.", "To compare with confocal micrographs, Fig.", "2B shows a 2$\\sigma $ -thick slab of the simulation box.", "In order to study the effect of activity on phase separation (cf.", "Fig.", "4 in the main text), we prepared an initial configuration with a gel-like structure ($\\rho =0.1 \\sigma ^{-3}$ ).", "We then performed two series of simulations, considering first a system of passive dumbbells, and then a solution of self-propelled ones (with $F_a$ chosen as detailed in the previous section).", "All the simulations in both series started from the same gel-like initial condition, and in both cases we varied $\\epsilon $ , which controls the depth of the depletion attractive potential, which in experiments depends on the concentration of NaPSS.", "We then measured in each run the potential energy per particle so as to estimate the minimum value of $\\epsilon $ which is required to not melt the initial gel.", "Figure 4 in the main text shows that such minimal values are equal to $\\sim 22$ for the passive case and to $\\sim 68$ for the active one.", "Figure: Plot of the potential energy per particle (normalised by ϵ\\epsilon ) of a system of NN dumbbells, subject to an attractive interaction ϵ\\epsilon .", "Red and blue curves refer to active and passive dumbbells respectively.Solid line correspond to a passive gel-like initial condition as in the text, whereas dashed lines correspond to a passive gas-like initial configuration.To estimate the effect of the initial configuration, we performed a second batch of BD simulations, again considering both passive and active systems separately, but now starting from a gas of passive dumbbells.", "The resulting curves for the potential energy per particle as a function of $\\epsilon $ relative to the two initial conditions are shown together in Fig.", "S3.", "In the passive case, we find no effect of the starting configuration; but in the active case we find hysteresis.", "When the initial configuration is a gas, a stronger attraction is needed to form a gel than to keep a gel stable.", "Finally, we have repeated these simulations (starting from a passive gel) for a range of densities ($0.005\\sigma ^{-3} <\\rho <0.25\\sigma ^{-3}$ ) and computed the critical values of $\\epsilon /k_BT$ , which we call $x^*$ , after which the gel is stable.", "In all cases, $x^*$ is larger in the active case.", "The ratio between the active and passive $x^*$ 's depends very weakly on the density of the system throughout the range we have simulated (cf.", "the theoretical predictions shown in Fig.", "S2), and closely approaches $\\sim 3$ , as also found in our experiments." ], [ "Torque exerted by a bacterium on the cluster", "Here we estimate the torque exerted by a single bacterium on the surface of a cluster, Eq.", "(4) of the main text.", "For convenience, we assume that the flagellum is oriented tangentially to the cluster.", "We will further neglect that the flagellum is a helix, since its radius is typically much smaller than its length ($\\sim 0.2\\mu $ m vs $\\sim 10\\mu $ m [32]), and replace it by a thin cylinder of length $l$ .", "Note that helicity is needed to generate a non-zero $F_{\\rm prop}$ but results only in a small correction to what follows.", "Figure: Sketch of the geometry used to compute thetorque generated by a bacterium on the surface of a cluster to which it belongs.", "The cluster isrepresented as a sphere of radius RR.We also show the force dipole (±f\\pm f) applied at the cell body tangentiallyto the cluster and at some distance zz along the flagellum.The force generated by a small portion of the flagellum of length $dz$ can be approximated by $F_{\\rm prop}dz/l$ , while a force of the same magnitude and opposite direction is applied locally to the fluid.", "The former force propagates along the flagellum and applies a torque on the cluster of magnitude $F_{\\rm prop}Rdz/l$ , where $R$ is the radius of the cluster.", "The latter force, applied by the flagellum to the fluid, generates a hydrodynamic flow that results in a torque of the opposite direction applied to the cluster.", "For a force applied to the fluid at some distance $z$ along the flagellum, see Fig.", "S4, the magnitude of this torque is given by [35] $dT^{\\rm hyd}=-\\frac{R^3}{\\Delta ^2}f_{\\perp },$ where $\\Delta $ is the distance along the line connecting the centre of the cluster and the point where the force is applied, and $f_{\\perp }$ is the component of the applied force $F_{\\rm prop}dz/l$ perpendicular to that line.", "From simple geometry we have $T^{\\rm hyd} =-\\int _{0}^{l}\\frac{F_{\\rm prop}R^4}{\\left(R^2+z^2\\right)^{3/2}}\\frac{dz}{l}= -\\frac{F_{\\rm prop}R^2}{\\sqrt{R^2+l^2}}.$ The total torque applied to the cluster is thus the sum of the “direct” and “hydrodynamic” torques $T_0 = F_{\\rm prop} R \\left[ 1 - \\frac{R}{\\sqrt{R^2+l^2}}\\right],$ which is Eq.", "(5) of the main text." ], [ "Cluster sizes", "Note that a significant range of cluster sizes is represented in the data presented in Figure 5 of the main text: If we had used the alternative (equally valid) variable of the number of bacteria in a cluster, $N = V/v = (R/\\sigma )^3$ , where $V$ and $v$ are the volumes of the cluster and an individual bacterium, and $R$ and $\\sigma \\sim 1 \\mu $ m are their corresponding radii, then our x-axis would have spanned 2 to $\\sim 50$ .", "Two factors probably control the size of the biggest clusters we observe.", "First, in the equivalent equilibrium system, the size distribution of pre-transition clusters is exponential (see refs.", "[9,10] in the main text), with the average increasing as one gets closer to the phase boundary.", "The success of our quasi-equilibrium approach suggest that this may also be true in our active system.", "On the other hand, the flagella of the surface bacteria create a ‘corona’ around the cluster that prevents other bacteria from joining it.", "At small cluster sizes, this corona will be very sparse, while at larger $R$ most of the space around the cluster will be occupied by the flagella thus screening the cluster from other bacteria and preventing it from growing beyond some critical size." ] ]

[ [ "Localisation and ageing in the parabolic Anderson model with Weibull\n potential" ], [ "Abstract The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential $\\xi$.", "We consider the case when $\\{\\xi(z):z\\in\\mathbb{Z}^d\\}$ is a collection of independent identically distributed random variables with Weibull distribution with parameter $0<\\gamma<2$, and we assume that the solution is initially localised in the origin.", "We prove that, as time goes to infinity, the solution completely localises at just one point with high probability, and we identify the asymptotic behaviour of the localisation site.", "We also show that the intervals between the times when the solution relocalises from one site to another increase linearly over time, a phenomenon known as ageing." ], [ "Parabolic Anderson model", "We consider the heat equation with random potential on the integer lattice $\\mathbb {Z}^d$ and study the Cauchy problem with localised initial condition, $ \\begin{array} {@{}rcl@{}}\\partial _t u(t,z) & = & \\Delta u(t,z)+\\xi (z)u(t,z), \\qquad (t,z)\\in (0,\\infty )\\times \\mathbb {Z}^d,\\\\[5pt]u(0,z) & = & {1}_{\\lbrace 0\\rbrace }(z), \\qquad z\\in \\mathbb {Z}^d, \\end{array}$ where $(\\Delta f) (z)=\\sum _{y\\sim z} \\bigl [f(y)-f(z) \\bigr ],\\qquad z\\in \\mathbb {Z}^d, f\\mathbb {Z}^d\\rightarrow \\mathbb {R}$ is the discrete Laplacian, and the potential $ \\lbrace \\xi (z) z\\in \\mathbb {Z}^d \\rbrace $ is a collection of independent identically distributed random variables.", "The problem (REF ) and its variants are often called the parabolic Anderson model.", "The model originates from the seminal work [1] of the Nobel laureate P. W. Anderson, who used the Hamiltonian $\\Delta +\\xi $ to describe electron localisation inside a semiconductor, a phenomenon now known as Anderson localisation.", "The parabolic version of the model appears naturally in the context of reaction–diffusion equations; see [5], [14], describing a system of noninteracting particles diffusing in space according to the Laplacian $\\Delta $ and branching at rate $\\xi (z)\\,dt$ at any given point $z$ .", "It turns out that the solution $u(t,z)$ gives the average number of such particles at time $t$ at location $z$ ." ], [ "Intermittency and localisation", "A lot of mathematical attention to the parabolic Anderson model over the last 30 years has been due to the fact that it exhibits the intermittency effect.", "In general, a random model is said to be intermittent if its long-term behaviour cannot be described using an averaging principle; see [18].", "In the context of the parabolic Anderson model, this means that, for large times $t$ , the solution $u(t,z)$ is mainly concentrated on a small number of remote random islands; see [7] for a survey.", "The long-term behaviour of the parabolic Anderson model is determined by the upper tail of the underlying distribution of the potential $\\xi $ , and it is believed that the intermittency is more pronounced for heavier tails.", "However, an initial approach to understanding intermittency was proposed for light-tailed potentials (those with finite exponential moments).", "It was suggested to study large time asymptotics of the moments of the total mass of the solution $U(t)=\\sum _{z\\in \\mathbb {Z}^d} u(t,z),$ which are finite for such potentials.", "The model was defined as intermittent if higher moments exhibited a faster growth rate, and it was proved in [9] that the parabolic Anderson model is intermittent in this sense.", "This method, however, does not work for heavy-tailed potentials (those with infinite exponential moments), as for them the moments of $U(t)$ are infinite.", "Such distributions include the exponential distribution and all heavier-tailed distributions.", "In order to understand the intermittent picture in more detail, it proved to be useful to study various large-time asymptotics of the total mass $U(t)$ , as they provided some insight into the geometry of the intermittent islands.", "It was shown in [16] that there are four types of behaviour the parabolic Anderson model can exhibit depending on the tail of the underlying distribution.", "The prime examples from each class are the following distributions: Weibull distribution with parameter $\\gamma >1$ , that is, $F(x)=1-e^{-x^{\\gamma }}$ .", "Double-exponential distribution with parameter $\\rho >0$ , that is, $F(x)=1-e^{-e^{x/\\rho }}$ .", "“Almost bounded” distributions, including some unbounded distributions with tails lighter than double-exponential and some bounded distributions.", "Other bounded distributions.", "The asymptotics of the total mass $U(t)$ was studied in [10] for cases (1) and (2), in [16] for case (3) and in [4] for case (4).", "Heuristics based on the asymptotics of $U(t)$ suggests that the intermittent islands will be single lattice points in case (1), bounded regions in case (2) and of size growing to infinity in cases (3) and (4).", "However, a rigorous geometric picture of intermittency has not been well understood.", "In particular, it is not clear how many intermittent islands are needed to carry the total mass of the solution, and where those islands are located.", "Moreover, the four classes above only cover light-tailed potential, and the class of all heavy-tailed distributions should be included to complete the picture.", "The prime examples of such distributions are Pareto distributions, that is, $F(x)=1-x^{-\\alpha }$ , $\\alpha >d$ ; Weibull potentials with parameter $\\gamma \\le 1$ .", "Heavy-tailed potentials were first studied in [17], and it turned out that the asymptotics of $U(t)$ in this case becomes nondeterministic and difficult to control.", "It was suggested to study the nondeterministic nature of $U(t)$ using extreme value theory and point processes techniques.", "This approach was further developed in [12], where the intermittency was fully described in its original geometric sense for Pareto potentials (0a).", "Polynomial tails are the heaviest tails for which the solution of the parabolic Anderson model still exists (see [9]), and one expected the localisation islands to be small and not numerous.", "It was proved that the extreme form of this conjecture is true, namely, that there is only one localisation island consisting of only one site.", "In other words, at any time the solution is localised at just one point with high probability, a phenomenon called complete localisation.", "It is a challenging problem to describe geometric intermittency for lighter tails.", "In [8], intermittent islands were described for potentials from classes (1) and (2), but the question about the number of islands remained open.", "Case (0b) was studied in [13], and it was shown that the solution is localised on an island of size $o(\\frac{t (\\log t)^{1/\\gamma -1}}{\\log \\log t})$ .", "However, it was believed that a much smaller region should actually contribute to the solution.", "In this paper, we assume that the potential has Weibull distribution with parameter $\\gamma >0$ , that is, the distribution function of each $\\xi (z)$ is $ F(x)=\\operatorname{Prob}\\bigl \\lbrace \\xi (z)<x \\bigr \\rbrace =1-e^{-x^{\\gamma }},\\qquad x\\ge 0.$ We focus on $0<\\gamma <2$ , which covers case (0b) and partly case (1).", "We prove that for such potentials the solution of the parabolic Anderson model completely localises at just one single site, exhibiting the strongest form of intermittency similar to the Pareto case (0a).", "This was plausible for $0<\\gamma <1$ as in this case the spectral gap of the Anderson Hamiltonian $\\Delta +\\xi $ in a relevant $t$ -dependent large box tends to infinity, but is quite surprising for the exponential distribution ($\\gamma =1$ ) where the spectral gap is bounded, and even more so for $1<\\gamma <2$ where the spectral gap tends to zero.", "We identify the localisation site explicitly in terms of the potential $\\xi $ and describe its scaling limit.", "For all sufficiently large $t$ (so that $\\log \\log t$ is well defined), denote $ \\Psi _t(z)=\\xi (z)-\\frac{|z|}{\\gamma t}\\log \\log t,\\qquad z\\in \\mathbb {Z}^d,$ and let $Z_t^{{{({1}})}}$ be such that $\\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )=\\max _{z\\in \\mathbb {Z}^d} \\Psi _t(z).$ The existence of $Z_t^{{{({1}})}}$ will be proved in Lemma REF .", "Denote by $|x|$ the $\\ell ^1$ -norm of $x\\in \\mathbb {R}^d$ , and denote by $\\Longrightarrow $ weak convergence.", "Theorem 1.1 ((Complete localisation)) Let $0<\\gamma <2$ .", "As $t\\rightarrow \\infty $ , $\\lim _{t\\rightarrow \\infty }\\frac{u(t,Z_t^{{{({1}})}})}{U(t)}= 1 \\qquad \\mbox{in probability}.$ It is easy to see that the solution cannot be localised at one point for all large times $t$ since occasionally it has to relocalise continuously from one site to another, and at those periods the solution will be concentrated at more than one point.", "It was shown in [12] that for Pareto potentials the solution in fact remains localised at just two points at all large times $t$ almost surely.", "We conjecture that the same is true for Weibull potentials with $0<\\gamma <2$ .", "There is a chance that our proof could be adjusted to the case $\\gamma =2$.", "However, new ideas are required to deal with $\\gamma >2$ , and there is a high chance that complete localisation will simply fail in that case.", "The technical reasons why our proof breaks down for $\\gamma \\ge 2$ are explained in Remark  and Remark  in Sections  and , respectively.", "Theorem 1.2 ((Scaling limit for the localisation site)) Let $\\gamma >0$ .", "Then $\\frac{Z_t^{{{({1}})}}}{r_t} \\Longrightarrow X^{{{({1}})}},$ as $t\\rightarrow \\infty $ where $ r_t=\\frac{t(\\log t)^{1/\\gamma -1}}{\\log \\log t}{\\goodbreak }$ and $X^{{{({1}})}}$ is an $\\mathbb {R}^d$ -valued random variable with independent exponentially distributed coordinates with parameter $d^{1-1/\\gamma }$ and uniform random signs, that is, with density $p^{{{({1}})}}(x) =\\frac{d^{d(1-1/\\gamma )}}{2^d}\\exp \\bigl \\lbrace -d^{1-1/\\gamma }|x| \\bigr \\rbrace ,\\qquad x\\in \\mathbb {R}^d.$ Although we prove Theorem REF for all $\\gamma >0$ , it only describes the scaling limit for the concentration site for $0<\\gamma <2$ as otherwise the solution may not be localised at $Z_t^{{{({1}})}}$ .", "This scaling limit agrees with the scaling limit for the centre of the intermittent island obtained in [13] for $0<\\gamma \\le 1$ .", "However, according to Theorem REF , this island is now of radius zero (being a single point) rather than $o(r_t)$ , and the result holds for the wider range $0<\\gamma <2$ ." ], [ "Ageing", "The notion of ageing is a key paradigm in studying the long-term dynamics of large disordered systems.", "A system exhibits ageing if, being in a certain state at time $t$ , it is likely to remain in this state for some time $s(t)$ which depends increasingly, and often linearly, on the time $t$ .", "Roughly speaking, the system becomes increasingly more conservative and reluctant to change.", "The ageing phenomenon has been extensively studied for disordered systems such as trap models and spin glasses; see [3] and references therein.", "In the context of the parabolic Anderson model, a certain form of ageing based on correlations was studied for some time-dependent potentials in [2], [6], and it was shown that such systems exhibit no ageing.", "The recent paper [11] dealt with potentials from class (1) and studied the correlation ageing (which gives only indirect information about the evolution of localisation) and more explicit annealed ageing (which, in contrast to the quenched setting, is based on the evolution of the islands contributing to the solution averaged over the environment).", "It was shown that these two forms of ageing are similar, and somewhat surprisingly, ageing was observed for Weibull potentials with parameter $\\gamma >2$ but not for heavier-tailed Weibull potentials with parameter $1< \\gamma \\le 2$ .", "The explicit ageing in the quenched setting has so far only been observed for Pareto potentials; see [15].", "In that case, the solution completely localises at just one point and ageing of the parabolic Anderson model is equivalent to ageing of the concentration site process.", "In this paper, we use a similar approach to show that the parabolic Anderson model with Weibull potential with parameter $0<\\gamma <2$ exhibits ageing as well.", "Notice that, remarkably, this is in sharp contrast to the absence of annealed and correlation ageing observed for $\\gamma >1$ in [11].", "For each $t>0$ , denote $T_t=\\inf \\bigl \\lbrace s>0Z_{t+s}^{{{({1}})}}\\ne Z_t^{{{({1}})}} \\bigr \\rbrace .$ Theorem 1.3 ((Ageing)) Let $\\gamma >0$ .", "As $t\\rightarrow \\infty $ $\\frac{T_t}{t}\\Longrightarrow \\Theta ,$ where $\\Theta $ is a nondegenerate almost surely positive random variable.", "In the proof of Theorem REF , we identify the distribution function of $\\Theta $ as a certain integral over $\\mathbb {R}^d\\times \\mathbb {R}$ .", "Although we prove Theorem REF for all $\\gamma >0$ , it only characterises the ageing behaviour of the parabolic Anderson model for $0<\\gamma <2$ as otherwise the solution may not be localised at $Z_t^{{{({1}})}}$ ." ], [ "Outline of the proofs", "It follows from [9], Theorem 2.1, that the parabolic Anderson model with Weibull potential possesses a unique nonnegative solution $u(0,\\infty )\\times \\mathbb {Z}^d\\rightarrow [0,\\infty )$ , which has a Feynman–Kac representation $u(t,z)=\\mathbb {E}_0 \\biggl [\\exp \\biggl \\lbrace \\int _0^t\\xi (X_s)\\,ds \\biggr \\rbrace {1}\\lbrace {X_t=z\\rbrace } \\biggr ],\\qquad (t, z)\\in (0,\\infty )\\times \\mathbb {Z}^d,$ where $(X_s s\\ge 0)$ is a continuous-time simple random walk on the lattice $\\mathbb {Z}^d$ with generator $\\Delta $ , and $\\mathbb {P}_z$ and $\\mathbb {E}_z$ denote the corresponding probability and expectation given that the random walk starts at $z\\in \\mathbb {Z}^d$ .", "The Feynman–Kac formula suggests that the main contribution to the solution $u$ at time $t$ comes from paths $(X_s)$ spending a lot of time at sites $z$ where the value $\\xi (z)$ of the potential is high but which are reasonably close to the origin so that the random walk would have a fair chance of reaching them in time $t$ .", "It turns out that the functional $\\Psi _t$ defined in (REF ) captures this trade-off, being the difference of the energetic term $\\xi (z)$ and an entropic term responsible for the cost of going to a point $z$ in time $t$ and staying there.", "Furthermore, the maximiser $Z_t^{{{({1}})}}$ of $\\Psi _t$ turns out to be the site where the solution $u$ is localised at time $t$ .", "In order to prove this, we decompose the solution $u$ into the sum ${u(t,z)=u_1(t,z)+u_2(t,z)}$ according to two groups of paths ending at $z$ : paths visiting $Z_t^{{{({1}})}}$ before time $t$ and staying in the ball $B_t$ centred in the origin with radius $|Z_t^{{{({1}})}}|(1+\\rho _t)$ , where $\\rho _t$ is a certain function tending to zero; all other paths.", "We show that $u_1$ localises around $Z_t^{{{({1}})}}$ and that the total mass of $u_2$ is negligible.", "To prove the localisation of $u_1$ , we use spectral analysis of the Anderson Hamiltonian $\\Delta +\\xi $ in the ball $B_t$ .", "In order to do so, we show that, although the spectral gap tends to zero for $\\gamma >1$ , it is still reasonably large.", "We suggest a new technique which allows us to show that the principal eigenfunction just manages to localise at $Z_t^{{{({1}})}}$ .", "Then we use a result from [8] to show that this is sufficient for the localisation of $u_1$ .", "In order to prove that the total mass of $u_2$ is negligible, we notice that the paths from the second group fall into one of the following three subgroups: paths having the maximum of the potential at the point $Z_t^{{{({1}})}}$ but making more than $|Z_t^{{{({1}})}}|(1+\\rho _t)$ steps; paths having the maximum of the potential not at the point $Z_t^{{{({1}})}}$ , with the maximum being reasonably large; paths missing all high values of the potential.", "In Section , we show that the total mass of the paths corresponding to each group is negligible.", "In all cases, this is due to an imbalance between the energetic forces (which do not contribute enough if the site $Z_t^{{{({1}})}}$ is not visited) and entropic forces (as the probabilistic cost is too high if a path is too long), as well as to the fact that the gap between $\\Psi _t(Z_t^{{{({1}})}})$ and the second largest value of $\\Psi _t$ is too large.", "Denote by $Z_t^{{{({2}})}}$ a point where the second largest value of $\\Psi _t$ is attained, that is, $\\Psi _t \\bigl (Z_t^{{{({2}})}} \\bigr )=\\max \\bigl \\lbrace \\Psi _t(z)z\\in \\mathbb {Z}^d, z\\ne Z_t^{{{({1}})}} \\bigr \\rbrace .$ In order to find the scale of growth of $\\Psi _t(Z_t^{{{({1}})}})-\\Psi _t(Z_t^{{{({2}})}})$ as well as of $Z_t^{{{({1}})}}$ and $Z_t^{{{({2}})}}$ we extend the point processes techniques developed in [17] and [12].", "For sufficiently large $t$ , we denote $a_t=(d\\log t)^{1/\\gamma } \\quad \\mbox{and}\\quad d_t=(d\\log t)^{1/\\gamma -1}.$ Further, for all $z\\in \\mathbb {Z}^d$ and all sufficiently large $t$ , we denote $ Y_{t,z}=\\frac{\\Psi _t(z)-a_{r_t}}{d_{r_t}},$ where $r_t$ is defined by (REF ), and define a point process $ \\Pi _t=\\sum _{z\\in \\mathbb {Z}^d}\\varepsilon _{(zr_t^{-1},Y_{t,z})},$ where we write $\\varepsilon _x$ for the Dirac measure in $x$ .", "In Section , we show that the point processes $\\Pi _t$ are well defined on a carefully chosen domain, and that they converge in law to a Poisson point process with certain density.", "This allows us to analyse the joint distribution of the random variables $Z_t^{{{({1}})}}$ , $Z_t^{{{({2}})}}$ , $\\Psi _t(Z_t^{{{({1}})}})$ , $\\Psi _t(Z_t^{{{({2}})}})$ and, in particular, prove Theorem REF .", "Finally, to prove ageing, we argue that due to the form of the functional $\\Psi _t$ the probability of $\\lbrace Z^{{{({1}})}}_{t+wt}= Z_t^{{{({1}})}}\\rbrace $ , for each $w>0$ , is roughly equal to $ \\int _{\\mathbb {R}^d\\times \\mathbb {R}}\\operatorname{Prob}\\bigl \\lbrace \\Pi _t(dx\\times dy)=1, \\Pi _t \\bigl (D_{w}(x,y) \\bigr )=0\\bigr \\rbrace ,$ where $ D_{w}(x,y)&=& \\biggl \\lbrace (\\bar{x},\\bar{y})\\in \\mathbb {R}^d\\times \\mathbb {R}y+\\frac{w\\theta |x|}{1+w}\\le \\bar{y}+\\frac{w\\theta |\\bar{x}|}{1+w}\\biggr \\rbrace \\nonumber \\\\[-8pt]\\\\[-8pt]&&{}\\cup \\bigl (\\mathbb {R}^d\\times [y,\\infty ) \\bigr ),\\nonumber $ and $ \\theta =\\gamma ^{-1}d^{1-1/\\gamma }.$ In particular, the integral in (REF ) converges to the corresponding finite integral with respect to the Poisson point process $\\Pi $ as $t\\rightarrow \\infty $ .", "This proves Theorem REF since that integral is a continuous function of $w$ decreasing from one to zero as $w$ varies from zero to infinity and so it is the tail of a distribution function.", "The paper is organised as follows.", "In Section , we introduce notation and prove some preliminary results.", "In Section , we develop a point processes approach, analyse the joint distribution of $Z_t^{{{({1}})}}$ , $Z_t^{{{({2}})}}$ , $\\Psi _t(Z_t^{{{({1}})}})$ , $\\Psi _t(Z_t^{{{({2}})}})$ and prove Theorem REF .", "In Section , we deal with the total mass corresponding to the paths from groups (1)–(3) and show that it is negligible.", "In Section , we discuss the localisation of $u_1$ and prove Theorem REF .", "Finally, in Section , we study ageing and prove Theorem REF ." ], [ "Preliminaries", "We focus on potentials with Weibull distribution (REF ) with parameter $0<\\gamma <2$ .", "However, most of our point processes results can be obtained for all $\\gamma >0$ at no additional cost.", "Therefore, we will assume $\\gamma >0$ in Sections , and , and restrict ourselves to the case $0<\\gamma <2$ in Sections  and ." ], [ "Extreme value notation and preliminary results", "We denote the upper order statistics of the potential $\\xi $ in the centred ball of radius $r>0$ by $\\xi _r^{{{({1}})}}=\\max _{|z|\\le r}\\xi (z)$ and $\\xi ^{{{({i}})}}_r=\\max \\bigl \\lbrace \\xi (z)|z|\\le r, \\xi (z)< \\xi _r^{{{({i-1}})}} \\bigr \\rbrace $ for $2\\le i\\le \\ell _r$ , where $\\ell _r$ is the number of points in the ball.", "Observe that throughout the paper we use the $\\ell ^1$ -norm.", "Let $0<\\rho <\\sigma <1/2$ and for all sufficiently large $r$ let $F_r&=& \\bigl \\lbrace z\\in \\mathbb {Z}^d|z|\\le r, \\exists i\\le r^{\\rho } \\mbox{ such that }\\xi (z)=\\xi _{r}^{{{({i}})}}\\bigr \\rbrace ,\\\\G_r&=& \\bigl \\lbrace z\\in \\mathbb {Z}^d|z|\\le r, \\exists i\\le r^{\\sigma } \\mbox{ such that }\\xi (z)=\\xi _{r}^{{{({i}})}}\\bigr \\rbrace .$ The sets $F_r$ and $G_r$ contain the sites in the centred ball of radius $r$ where the highest $\\lfloor r^{\\rho }\\rfloor $ and $\\lfloor r^{\\sigma }\\rfloor $ values of the potential $\\xi $ are achieved, respectively.", "Lemma 2.1 Almost surely $\\xi _{r}^{{{({1}})}}\\sim (d\\log r)^{1/\\gamma } \\qquad \\mbox{as }r\\rightarrow \\infty .$ This result was proved in [17] for the case $0<\\gamma \\le 1$ but it can be easily extended to all $\\gamma >0$ by observing that $\\zeta (z)=\\xi (z)^{\\gamma }, z\\in \\mathbb {Z}$ , are exponential identically distributed random variables.", "Denote the maximum of the potential $\\zeta $ by $\\zeta _r^{{{({1}})}}=\\max _{|z|\\le r}\\zeta (z).$ Since $\\xi _r^{{{({1}})}}= (\\zeta _r^{{{({1}})}} )^{1/\\gamma }$ and $\\zeta _r^{{{({1}})}}\\sim d\\log r$ by [17], Lemma 4.1, with $\\gamma =1$ , we obtain the required asymptotics.", "For all $c\\in \\mathbb {R}$ , $z\\in \\mathbb {Z}^d$ , and all sufficiently large $t$ define $\\Psi _{t,c}(z)=\\Psi _t(z)+\\frac{c|z|}{t}.$ Denote by $Z_t^{{{({1,c}})}}$ and $Z_t^{{{({2,c}})}}$ points where the first and second largest values of the functional $\\Psi _{t,c}$ are achieved, that is, $ \\begin{array} {@{}rcl@{}}\\Psi _{t,c} \\bigl (Z_t^{{{({1,c}})}} \\bigr )&=&\\max \\bigl \\lbrace \\Psi _{t,c}(z) z\\in \\mathbb {Z}^d \\bigr \\rbrace ,\\\\[5pt]\\Psi _{t,c} \\bigl (Z_t^{{{({2,c}})}} \\bigr )&=&\\max \\bigl \\lbrace \\Psi _{t,c}(z)z\\in \\mathbb {Z}^d, z\\ne Z_t^{{{({1,c}})}} \\bigr \\rbrace .", "\\end{array} $ Observe that $\\Psi _{t}=\\Psi _{t,0}$ and so $Z_t^{{{({1}})}}=Z_t^{{{({1,0}})}}$ and $Z_t^{{{({2}})}}=Z_t^{{{({2,0}})}}$ .", "We are mostly interested in the case $c=0$ , but some understanding of the general case is needed for Lemma REF .", "This is explained more carefully in Remark  in Section .", "Lemma 2.2 For each $c$ , the maximisers $Z_t^{{{({1,c}})}}$ and $Z_t^{{{({2,c}})}}$ (and, in particular, $Z_t^{{{({1}})}}$ and $Z_t^{{{({2}})}}$ ) are well defined for all sufficiently large $t$ almost surely.", "Observe that $\\Psi _{t,c}(0)>0$ and $\\Psi _{t,c}(1)>0$ almost surely if $t$ is large enough.", "On the other hand, by Lemma REF for all sufficiently large $t$ there exists a random radius $\\rho (t)>0$ such that, almost surely, $\\xi (z)\\le \\xi _{|z|}^{{{({1}})}}\\le \\bigl (2d\\log |z|\\bigr )^{1/\\gamma } \\le \\frac{|z|}{\\gamma t}\\log \\log t-\\frac{c|z|}{t}\\qquad \\mbox{for all }|z|>\\rho (t).$ Hence, $\\Psi _{t,c}(z)\\le 0$ for all $|z|>\\rho (t)$ and so $\\Psi _{t,c}$ takes only finitely many positive values.", "This implies that the maxima in (REF ) exist for all $c$ .", "The existence of $Z_t^{{{({1}})}}$ and $Z_t^{{{({2}})}}$ follows as a particular case when $c=0$ .", "Choose $\\left\\lbrace \\begin{array}{ll}{ \\beta \\in (1-1/\\gamma ,1/\\gamma )&\\quad \\mbox{if }1\\le \\gamma <2,\\cr \\beta =0 &\\quad \\mbox{if }0< \\gamma <1.", "}Observe that \\beta \\ge 0and define\\begin{equation} \\mu _r=(\\log r)^{-\\beta }\\end{equation}for all r large enough.For 0< \\gamma <1, the gaps between higher order statistics of thepotential get larger (as r\\rightarrow \\infty ) andthe auxiliary scaling function \\mu _r is not needed (so that we cansimply set \\mu _r=1 as above).", "For \\gamma =1, the gaps are of finiteorder, and for \\gamma >1 they tend to zero, and an extraeffort is required to control this effect.", "This is done by thecorrection term \\mu _r.It is essential for the choice of \\mu _r that, on the one hand, it isnegligible with respect to d_rand so with respect to the gap \\Psi _t(Z_t^{{{({1}})}})-\\Psi _t(Z_t^{{{({2}})}}) (which isachieved by the condition \\beta >1-1/\\gamma )and on the other hand -\\log \\mu _r must be smaller than \\log \\xi _r^{{{({1}})}} (which is guaranteed by \\beta <1/\\gamma ).However, this method only worksfor \\gamma <2 as the interval (-1/\\gamma +1,1/\\gamma ) is emptyotherwise.", "This is explained in more detailin Remark~\\ref {rem1} in Section~\\ref {s_neg}.\\end{array}We introduce four auxiliary positive scaling functions\\right.$ t0$, $ gt$, $ t0$, $ t0$ satisfyingthe following conditions as $ t$:\\begin{eqnarray} &&\\mbox{(a)} \\quad f_t^{-1},g_t, \\lambda _t^{-1}, \\rho _t^{-1}\\mbox{ are } o(\\log \\log t),\\\\ &&\\mbox{(b)} \\quad g_t\\rho _t\\lambda _t^{-1}\\rightarrow 0.", "\\end{eqnarray}$ Further, we define $k_t= \\bigl \\lfloor (r_tg_t)^{\\rho }\\bigr \\rfloor \\quad \\mbox{and}\\quad m_t= \\bigl \\lfloor (r_tg_t)^{\\sigma }\\bigr \\rfloor .$ For any $c\\in \\mathbb {R}$ , we introduce the event $ &&\\mathcal {E}_c(t)= \\bigl \\lbrace r_tf_t<\\bigl |Z_t^{{{({1}})}}\\bigr |<r_tg_t,\\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )-\\Psi _t\\bigl (Z_t^{{{({2}})}}\\bigr )>d_t\\lambda _t,\\nonumber \\\\&&\\phantom{\\mathcal {E}_c(t)= \\bigl \\lbrace } \\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )>a_{r_t}-d_tg_t, \\Psi _t\\bigl ( Z_t^{{{({2}})}}\\bigr )>a_{r_t}-d_t g_t,\\\\&&\\phantom{\\hspace*{128.0pt}}\\bigl |Z^{{{({1,c}})}}_t\\bigr |<r_tg_t,\\bigl |Z^{{{({2,c}})}}_t\\bigr |<r_tg_t \\bigr \\rbrace .\\nonumber $ For any $x,y\\in \\mathbb {R}$ , we denote by $x \\wedge y$ and $x\\vee y$ the minimum and the maximum of $x$ and $y$ , respectively, and we denote $x_{-}=-x\\vee 0$ ." ], [ "Geometric paths on the lattice", "For each $n\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace $ denote by $\\mathcal {P}_n= \\bigl \\lbrace y=(y_0,\\ldots ,y_n)\\in \\bigl (\\mathbb {Z}^d \\bigr )^{n+1}|y_i-y_{i-1}|=1\\mbox{ for all }1\\le i\\le n \\bigr \\rbrace $ the set of all geometric paths in $\\mathbb {Z}^d$ .", "Define $q(y)=\\max _{0\\le i\\le n}\\xi (y_i) \\quad \\mbox{and}\\quad p(y)=\\max _{0\\le i\\le n} |y_i-y_0|,$ and denote by $z(y)$ a point $y_i$ of the path $y$ such that $\\xi (y_i)=q(y)$ .", "Let $(\\tau _i)$ , $i\\ge 0$ , be waiting times of the random walk $(X_s)$ , which are independent exponentially distributed random variables with parameter $2d$ .", "Denote by $\\mathsf {E}$ the expectation with respect to $(\\tau _i)$ .", "For each $y\\in \\mathcal {P}_n$ , denote by $&&P(t,y)= \\lbrace X_0=y_0, X_{\\tau _0+\\cdots +\\tau _{i-1}}=y_i\\mbox{ for all }1\\le i\\le n,\\\\&&\\phantom{\\hspace*{98.0pt}} \\mbox{and } t-\\tau _n\\le \\tau _0+\\cdots +\\tau _{n-1}< t \\rbrace $ the event that the random walk has the trajectory $y$ up to time $t$ .", "Here, we assume that the random walk is continuous from the right.", "Denote by $ U(t,y)=\\mathbb {E}_0 \\biggl [\\exp \\biggl \\lbrace \\int _0^t\\xi (X_s)\\,ds \\biggr \\rbrace {1}_{P(t,y)} \\biggr ]$ the contribution of the event $P(t,y)$ to the total mass of the solution $u$ of the parabolic Anderson model.", "For any set $A\\subset \\mathbb {Z}^d$ and any geometric path $y\\in \\mathcal {P}_n$ denote $n_{+}(y,A)=\\bigl | \\lbrace 0\\le i\\le ny_i\\in A \\rbrace \\bigr | \\quad \\mbox{and}\\quad n_{-}(y,A)=\\bigl | \\lbrace 0\\le i\\le ny_i\\notin A \\rbrace \\bigr |.$ We call a set $A\\subset \\mathbb {Z}^d$ totally disconnected if $|x-y|\\ne 1$ whenever $x,y\\in A$ .", "Lemma 2.3 Let $A$ be a totally disconnected finite subset of $\\mathbb {Z}^d$ , and $y\\in \\mathcal {P}_n$ for some $n$ .", "Then $n_+(y,A)\\le \\frac{n-p(y)}{2}+|A|\\wedge \\biggl \\lceil \\frac{p(y)+1}{2} \\biggr \\rceil .", "$ Let $i(y)=\\min \\lbrace i|y_i-y_0|=p(y)\\rbrace $ and denote $z=y_{i(y)}$ .", "Similarly to [12], page 371, we first erase loops that the path $y$ may have made before reaching $z$ for the first time and extract from $(y_0,\\ldots ,y_{i(y)})$ a self-avoiding path $(y_{i_0},\\ldots ,y_{i_{p(y)}})$ starting at $y_0$ of length $p(y)$ , where we take $i_0=0$ and $i_{j+1}=\\min \\bigl \\lbrace iy_l\\ne y_{i_j}\\ \\forall l\\in \\bigl [i,i(y) \\bigr ] \\bigr \\rbrace .$ Since this path is self-avoiding and has length $p(y)$ , at most $|A|\\wedge \\lceil \\frac{p(y)+1}{2}\\rceil $ of its points belong to $A$ .", "Next, for each $0\\le j\\le p(y)-1$ , we consider the path $(y_{i_j+1},\\ldots ,y_{i_{j+1}-1})$ , which was removed during erasing the $j$ th loop.", "It contains an even number $i_{j+1}-{i_j}-1$ of steps and at most half of them belong to $A$ since $A$ is totally disconnected.", "Finally, the remaining piece $(y_{i_{p(y)}+1},\\ldots ,y_n)$ consists of $n-i_{p(y)}$ points, and at most half of them lie in $A$ for the same reason.", "We obtain $n_+(y,A) &\\le &|A|\\wedge \\biggl \\lceil \\frac{p(y)+1}{2} \\biggr \\rceil +\\sum _{j=0}^{p(y)-1}\\frac{{i_{j+1}}-{i_j}-1}{2}+\\frac{n-i_{p(y)}}{2}\\\\&=&|A|\\wedge \\biggl \\lceil \\frac{p(y)+1}{2} \\biggr \\rceil +\\frac{n-p(y)}{2}$ as required." ], [ "A point processes approach", "In this section, we use point processes techniques to understand the joint scaling limit of the random variables $Z_t^{{{({1, c}})}}$ , $Z_t^{{{({2, c}})}}$ , $\\Psi _{t,c}(Z_t^{{{({1, c}})}})$ , $\\Psi _{t,c}(Z_t^{{{({2, c}})}})$ for each $c$ and, in particular, that of $Z_t^{{{({1}})}}$ , $Z_t^{{{({2}})}}$ , $\\Psi _t(Z_t^{{{({1}})}})$ , $\\Psi _t(Z_t^{{{({2}})}})$ .", "We show that $Z_t^{{{({1, c}})}}$ and $Z_t^{{{({2, c}})}}$ grow at scale $r_t$ and that $\\Psi _{t,c}(Z_t^{{{({1, c}})}})-a_{r_t}$ and $\\Psi _{t,c}(Z_t^{{{({2, c}})}})-a_{r_t}$ grow or decay at scale $d_t$ (which goes to infinity for $\\gamma <1$ , is a constant for $\\gamma =1$ , and tends to zero for $\\gamma >1$ ), and we find their joint scaling limit in Proposition REF .", "In particular, we show that the probability of the event $\\mathcal {E}_c(t)$ defined in (REF ) tends to one for any $c$ and so it suffices to prove complete localisation and ageing on the event $\\mathcal {E}_c(t)$ for a sufficiently large constant $c$ .", "This constant will be identified later in Proposition REF in Section .", "Finally, in the end of this section we prove Theorem REF .", "For all $z\\in \\mathbb {Z}^d$ and all sufficiently large $r$ , denote $X_{r,z}=\\frac{\\xi (z)-a_r}{d_r}$ and define $\\Sigma _r=\\sum _{z\\in \\mathbb {Z}^d}\\varepsilon _{(zr^{-1},X_{r,z})},$ where $\\varepsilon _x$ denotes the Dirac measure in $x$ .", "For each $\\tau \\in \\mathbb {R}$ and $q>0$ , let $H_{\\tau }^q= \\bigl \\lbrace (x,y)\\in \\dot{\\mathbb {R}}^d\\times (-\\infty ,\\infty ]y\\ge q|x|+\\tau \\bigr \\rbrace ,$ where $\\dot{\\mathbb {R}}^d$ denotes the one-point compactification of the Euclidean space.", "It was proved in [17], Lemma 4.3, that for $0<\\gamma \\le 1$ the restriction of each $\\Sigma _r$ to $H_{\\tau }^q$ is a point process and, as $r\\rightarrow \\infty $ , $\\Sigma _r|_{H_{\\tau }^q}$ converges in law to a Poisson point process $\\Sigma $ on $H_{\\tau }^q$ with intensity measure $\\eta (dx,dy)=dx\\otimes \\gamma e^{-\\gamma y}\\,dy.$ However, it is easy to check that the same proof works for all $\\gamma >0$ .", "Observe that we need to restrict $\\Sigma _r$ from $\\mathbb {R}^d\\times \\mathbb {R}$ to $H_{\\tau }^q$ in order to ensure that there are only finitely many points of $\\Sigma _r$ in every relatively compact set.", "This is achieved with the help of $q$ , and $\\tau $ makes it possible for the spaces $H_{\\tau }^q$ to capture the behaviour of $\\Sigma _r$ on the whole space $\\mathbb {R}^d\\times \\mathbb {R}$ as it can be chosen arbitrarily small.", "For each $\\tau \\in \\mathbb {R}$ and $\\alpha > -\\theta $ , let $\\hat{H}_{\\tau }^{\\alpha }= \\bigl \\lbrace (x,y)\\in \\dot{\\mathbb {R}}^{d+1}y\\ge \\alpha |x|+\\tau \\bigr \\rbrace ,$ where the hat over $H$ reflects the fact that the spaces $\\dot{\\mathbb {R}}^d\\times (-\\infty ,\\infty ]$ and $\\dot{\\mathbb {R}}^{d+1}$ have different topology.", "For all $c\\in \\mathbb {R}$ , $z\\in \\mathbb {Z}^d$ , and all sufficiently large $t$ define $Y_{t,z,c}=\\frac{\\Psi _{t,c}(z)-a_{r_t}}{d_{r_t}} \\quad \\mbox{and}\\quad \\Pi _{t,c}=\\sum _{z\\in \\mathbb {Z}^d}\\varepsilon _{(zr_t^{-1},Y_{t,z,c})}.$ Recall the definitions of $Y_{t,z}$ and $\\Pi _t$ from (REF ) and (REF ) and observe that $Y_{t,z,c}=Y_{t,z,0}$ and $\\Pi _t=\\Pi _{t,0}$ .", "Lemma 3.1 Let $c\\in \\mathbb {R}$ .", "For all sufficiently large $t$ , $\\Pi _{t,c}$ is a point process on $\\hat{H}_{\\tau }^{\\alpha }$ .", "As $t\\rightarrow \\infty $ , $\\Pi _{t,c}$ converges in law to a Poisson point process $\\Pi $ on $\\hat{H}_{\\tau }^{\\alpha }$ with intensity measure $\\nu (dx,dy) =dx\\otimes \\gamma \\exp \\bigl \\lbrace -\\gamma \\bigl (y+\\theta |x|\\bigr ) \\bigr \\rbrace \\,dy.$ Observe that $Y_{t,z,c}=\\frac{\\xi (z)-a_{r_t}}{d_{r_t}}-\\frac{|z|}{\\gamma td_{r_t}}\\log \\log t+\\frac{c|z|}{td_{r_t}} =\\frac{\\xi (z)-a_{r_t}}{d_{r_t}}- \\bigl (\\theta +o(1) \\bigr )\\frac{|z|}{r_t}.$ Choose $\\alpha ^{\\prime }$ and $q$ so that $-\\theta <\\alpha ^{\\prime }<\\alpha $ and $\\alpha ^{\\prime }+\\theta <q<\\alpha +\\theta $ .", "Then $ \\Pi _{t,c}|_{\\hat{H}^{\\alpha }_{\\tau }}= \\bigl (\\Sigma _{r_t} |_{H_{\\tau }^q}\\circ T_{t,c}^{-1}\\bigr ) \\big |_{\\hat{H}^{\\alpha }_{\\tau }},$ where $T_{t,c}H_{\\tau }^q\\rightarrow \\hat{H}_{\\tau }^{\\alpha ^{\\prime }}$ is such that $T_{t,c}(x,y)\\mapsto \\left\\lbrace \\begin{array}{ll}{ \\bigl (x,y-\\bigl (\\theta +o(1) \\bigr )|x| \\bigr ), &\\quad \\mbox{if }x\\ne \\infty \\mbox{ and }y\\ne \\infty ,\\cr \\infty , &\\quad \\mbox{otherwise}.", "}We define TH_{\\tau }^q\\rightarrow \\hat{H}_{\\tau }^{\\alpha ^{\\prime }} by\\vspace*{-1.0pt}T(x,y)\\mapsto \\left\\lbrace \\begin{array}{ll}{ \\bigl (x,y-\\theta |x|\\bigr ), & \\quad \\mbox{if }x\\ne \\infty \\mbox{ and }y\\ne \\infty ,\\cr \\infty , &\\quad \\mbox{otherwise}.", "}It was proved in \\cite {HMS08}, Lemma~2.5, that one can pass to the limitin (\\ref {pppref}) as t\\rightarrow \\infty simultaneouslyin the mapping T_{t,c} and the point process \\Sigma _{r_t} to get\\vspace*{-1.0pt}\\Pi _{t,c}|_{\\hat{H}^{\\alpha }_{\\tau }}\\Longrightarrow \\bigl (\\Sigma |_{H_{\\tau }^q}\\circ T^{-1} \\bigr ) \\big |_{\\hat{H}^{\\alpha }_{\\tau }}.Observe that the conditions of that lemma are satisfied as T iscontinuous, H_{\\tau }^q is compact,T_{t,c}\\rightarrow T uniformly on \\lbrace (x,y)\\in H_{\\tau }^q|x|\\ge n\\rbrace ast\\rightarrow \\infty for each n\\in \\mathbb {N}, and\\vspace*{-1.0pt}\\eta \\bigl \\lbrace (x,y)\\in H_{\\tau }^q|x|\\ge n \\bigr \\rbrace \\rightarrow 0\\qquad \\mbox{as }n\\rightarrow \\infty since \\eta (H_{\\tau }^q) is finite.Finally, it remains to noticethat (\\Sigma |_{H_{\\tau }^q}\\circ T^{-1} ) |_{\\hat{H}^{\\alpha }_{\\tau }} is a Poisson process withintensity measure \\eta \\circ T^{-1}=\\nu restricted on \\hat{H}^{\\alpha }_{\\tau }.\\end{array}\\right.\\begin{prop}Let c\\in \\mathbb {R}.\\begin{enumerate}[(b)]\\item [(a)]As t\\rightarrow \\infty ,\\vspace*{-1.0pt}\\begin{eqnarray*}&&\\biggl (\\frac{Z_t^{{{({1,c}})}}}{r_t}, \\frac{\\Psi _{t,c}(Z_t^{{{({1,c}})}})-a_{r_t}}{d_{r_t}}, \\frac{Z_t^{{{({2,c}})}}}{r_t},\\frac{\\Psi _{t,c}(Z_t^{{{({2,c}})}})-a_{r_t}}{d_{r_t}} \\biggr )\\\\&&\\qquad \\Longrightarrow \\bigl (X^{{{({1}})}},Y^{{{({1}})}}, X^{{{({2}})}}, Y^{{{({2}})}} \\bigr ),\\end{eqnarray*}where the limit random variable has density\\vspace*{-1.0pt}\\begin{eqnarray*}&&p(x_1,y_1, x_2, y_2)\\\\&&\\qquad =\\gamma ^2\\exp \\bigl \\lbrace -\\gamma \\bigl (y_1+y_2+\\theta |x_1|+\\theta |x_2| \\bigr )-2^d(\\gamma \\theta )^{-d} e^{-\\gamma y_2} \\bigr \\rbrace {1}_{\\lbrace y_1>y_2\\rbrace }.\\end{eqnarray*}\\end{enumerate}\\item [(b)]\\operatorname{Prob}\\lbrace \\mathcal {E}_c(t)\\rbrace \\rightarrow 1 as t\\rightarrow \\infty .\\end{prop}\\end{array}\\right.\\begin{pf}(a) Let A\\subset \\hat{H}_{\\tau }^0\\times \\hat{H}_{\\tau }^0 for some\\tau , andassume that \\operatorname{Leb}(\\partial A)=0.", "Since H_{\\tau }^0 is compact,we have by Lemma~\\ref {l_ppp}\\begin{eqnarray} &&\\operatorname{Prob}\\biggl \\lbrace \\biggl (\\frac{Z_t^{{{({1,c}})}}}{r_t}, \\frac{\\Psi _{t,c}(Z_t^{{{({1,c}})}})-a_{r_t}}{d_{r_t}},\\frac{Z_t^{{{({2,c}})}}}{r_t}, \\frac{\\Psi _{t,c}(Z_t^{{{({2,c}})}})-a_{r_t}}{d_{r_t}} \\biggr )\\in A \\biggr \\rbrace \\nonumber \\\\&&\\qquad =\\int _A {1}_{\\lbrace y_1>y_2\\rbrace }\\operatorname{Prob}\\bigl \\lbrace \\Pi _{t,c}(dx_1\\times dy_1)=\\Pi _{t,c}(dx_2\\times dy_2)=1,\\nonumber \\\\&&\\phantom{\\qquad =\\int _A {1}_{\\lbrace y_1>y_2\\rbrace }\\operatorname{Prob}\\bigl \\lbrace } \\Pi _{t,c} \\bigl (\\mathbb {R}^d\\times (y_1,\\infty )\\bigr )=\\Pi _{t,c} \\bigl (\\mathbb {R}^d\\times (y_2,y_1)\\bigr )=0 \\bigr \\rbrace \\\\&&\\qquad \\rightarrow \\int _A {1}_{\\lbrace y_1>y_2\\rbrace }\\operatorname{Prob}\\bigl \\lbrace \\Pi (dx_1\\times dy_1)=1 \\bigr \\rbrace \\operatorname{Prob}\\bigl \\lbrace \\Pi (dx_2\\times dy_2)=1 \\bigr \\rbrace \\nonumber \\\\&&\\phantom{\\qquad \\rightarrow \\int _A} {} \\times \\operatorname{Prob}\\bigl \\lbrace \\Pi \\bigl (\\mathbb {R}^d\\times (y_1,\\infty ) \\bigr )=0 \\bigr \\rbrace \\operatorname{Prob}\\bigl \\lbrace \\Pi \\bigl (\\mathbb {R}^d\\times (y_2,y_1) \\bigr )=0\\bigr \\rbrace \\nonumber \\\\&&\\qquad =\\int _A {1}_{\\lbrace y_1>y_2\\rbrace }\\nu (dx_1,dy_1)\\nu (dx_2,dy_2)\\exp \\bigl \\lbrace -\\nu \\bigl (\\mathbb {R}^d\\times (y_2,\\infty ) \\bigr ) \\bigr \\rbrace .\\nonumber \\end{eqnarray}Integrating we obtain\\begin{eqnarray} \\nu \\bigl (\\mathbb {R}^d\\times (y_2,\\infty ) \\bigr )&=& \\gamma \\int _{\\mathbb {R}^d}\\int _{y_2}^{\\infty }\\exp \\bigl \\lbrace -\\gamma y-\\gamma \\theta |x|\\bigr \\rbrace \\,dy\\,dx\\nonumber \\\\[-8pt]\\\\[-8pt]&=&2^d(\\gamma \\theta )^{-d}e^{-\\gamma y_2}.\\nonumber \\end{eqnarray}Substituting this, as well as the expressions for \\nu (dx_1,dy_1) and \\nu (dx_2,dy_2) into (\\ref {4den})we obtain\\begin{eqnarray*}&&\\lim _{t\\rightarrow \\infty }\\operatorname{Prob}\\biggl \\lbrace \\biggl (\\frac{Z_t^{{{({1,c}})}}}{r_t},\\frac{\\Psi _{t,c}(Z_t^{{{({1,c}})}})-a_{r_t}}{d_{r_t}}, \\frac{Z_t^{{{({2,c}})}}}{r_t}, \\frac{\\Psi _{t,c}(Z_{t,c}^{{{({2,c}})}})-a_{r_t}}{d_{r_t}} \\biggr \\rbrace \\in A \\biggr )\\\\&&\\qquad = \\int _A p(x_1,y_1,x_2,y_2)\\,dx_1\\,dy_1\\,dx_2\\,dy_2.\\end{eqnarray*}\\end{pf}It remains now to generalise this equality to all sets $ RdR$ with $ Leb(A)=0$.Since $$ can be arbitrarily small, to do so it suffices to showthat $ p$ integrates to one.", "We have\\begin{eqnarray} &&\\int _{\\mathbb {R}^d\\times \\mathbb {R}\\times \\mathbb {R}^d\\times \\mathbb {R}}p(x_1,x_2,y_1,y_2)\\,dx_1\\,dy_1\\,dx_2\\,dy_2\\nonumber \\\\&&\\qquad =2^{2d}(\\gamma \\theta )^{-2d}\\int _{-\\infty }^\\infty \\int _{y_2}^{\\infty } \\gamma ^2\\exp \\bigl \\lbrace -\\gamma (y_1+y_2)-2^d (\\gamma \\theta )^{-d}e^{-\\gamma y_2} \\bigr \\rbrace \\,dy_1\\,dy_2\\nonumber \\\\[-8pt]\\\\[-8pt]&&\\qquad =2^{2d}(\\gamma \\theta )^{-2d}\\int _{-\\infty }^\\infty \\gamma \\exp \\bigl \\lbrace -2\\gamma y_2-2^d (\\gamma \\theta )^{-d}e^{-\\gamma y_2} \\bigr \\rbrace \\,dy_2\\nonumber \\\\&&\\qquad =\\int _{0}^\\infty ue^{-u}\\,du=1,\\nonumber \\end{eqnarray}where in the last line we used the substitution $ u=2d ()-de-y2$.$ (b) This immediately follows from (a) since $d_{r_t}=d_t (1+o(1))$ and $f_t\\rightarrow 0$ , $g_t\\rightarrow \\infty $ , $\\lambda _t\\rightarrow 0$ .", "The reason why we need to study a general $c$ rather than $c=0$ is just to show that $|Z^{{{({1,c}})}}_t|<r_tg_t$ and $|Z^{{{({2,c}})}}_t|<r_tg_t$ with high probability, which is done in part (b) of the proposition above.", "This will be required later on in Lemma REF with some $c$ identified in Proposition REF .", "The full strength of the convergence result proved in the part (a) of the proposition will only be used for $c=0$ .", "Proof of Theorem REF The result follows from Proposition REF (a) with $c=0$ by integrating the density $p$ over all possible values of $x_2$ , $y_1$ , and $y_2$ .", "Similarly to (), we obtain $p^{{{({1}})}}(x) & =&\\int _{\\mathbb {R}\\times \\mathbb {R}^d\\times \\mathbb {R}}p(x,y_1,x_2,y_2)\\,dy_1\\,dx_2\\,dy_2\\\\&=&2^{d}(\\gamma \\theta )^{-d}\\exp \\bigl \\lbrace -\\gamma \\theta |x| \\bigr \\rbrace \\\\&&{} \\times \\int _{-\\infty }^\\infty \\int _{y_2}^{\\infty } \\gamma ^2\\exp \\bigl \\lbrace -\\gamma (y_1+y_2)-2^d (\\gamma \\theta )^{-d}e^{-\\gamma y_2} \\bigr \\rbrace \\,dy_1\\,dy_2\\\\&=&2^{-d}d^{d(1-1/\\gamma )}\\exp \\bigl \\lbrace -d^{1-1/\\gamma }|x| \\bigr \\rbrace $ as required." ], [ "Negligible paths of the random walk", "Throughout this section, we assume that $0<\\gamma <2$ .", "We introduce three groups of paths of the random walk $(X_s)$ informally described in the [intro]Introduction and show that their contribution to the total mass of the solution $u$ of the parabolic Anderson model is negligible.", "Denote by $J_t$ the number of jumps the random walk $(X_s)$ makes up to time $t$ and consider the following three groups of paths: $E_i(t)=\\left\\lbrace \\begin{array}{ll}{ \\Bigl \\lbrace \\displaystyle \\max _{0\\le s\\le t}{\\xi (X_s)}=\\xi \\bigl (Z_t^{{{({1}})}}\\bigr ), J_t > \\bigl |Z_t^{{{({1}})}}\\bigr |(1+\\rho _t)\\Bigr \\rbrace , &\\quad i=1,\\vspace*{2.0pt}\\cr \\Bigl \\lbrace \\xi _{r_tg_t}^{{{({k_t}})}} \\le \\displaystyle \\max _{0\\le s\\le t}{\\xi (X_s)}\\ne \\xi \\bigl (Z_t^{{{({1}})}}\\bigr ) \\Bigr \\rbrace , & \\quad i=2,\\vspace*{2.0pt}\\cr \\Bigl \\lbrace \\displaystyle \\max _{0\\le s\\le t}{\\xi (X_s)}< \\xi _{r_tg_t}^{{{({k_t}})}} \\Bigr \\rbrace , & \\quad i=3.", "}Denote byU_i(t)=\\mathbb {E}_0 \\biggl [\\exp \\biggl \\lbrace \\int _0^t\\xi (X_s)\\,ds \\biggr \\rbrace {1}_{E_i(t)} \\biggr ],\\qquad 1\\le i\\le 3their contributions to the total mass of the solution.The aim of this section is to show that all U_i(t) is negligible withrespect to U(t).\\end{array}We start with Lemma~\\ref {app} where we collect all asymptoticproperties of the environmentwhich we use later on.", "In Lemma~\\ref {l_lb}, we prove a simple lowerbound for the total mass\\right.$ (t)$.", "Then we prove Proposition~\\ref {Hh}, which is a crucial tool foranalysing $ U1(t)$ and $ U2(t)$as it gives a general upper bound on the total mass corresponding tothe paths reaching the maximumof the potential in a certain setand having a lower bound restriction on the number of jumps $ Jt$.Equipped with this result,we show that $ U1(t)$ and $ U2(t)$ are negligible in Lemmas~\\ref {l_u1}and~\\ref {l_u2}.Finally, Lemma~\\ref {l_u3} provides a simple proof of the negligibilityof $ U3(t)$.$ Observe that Proposition REF identifies the constant $c$ , which is then fixed and used throughout the paper afterward.", "Lemma 4.1 Almost surely, $\\xi _r^{{{({\\lfloor r^{\\rho }\\rfloor }})}}\\sim ((d-\\rho )\\log r)^{1/\\gamma }$ and $ \\xi _r^{{{({\\lfloor r^{\\sigma }\\rfloor }})}}\\sim ((d-\\sigma )\\log r)^{1/\\gamma }$ as $r\\rightarrow \\infty $ ; $\\xi _{r_tg_t}^{{{({k_t}})}}\\sim ((d-\\rho )\\log t)^{1/\\gamma }$ and $ \\xi _{r_tg_t}^{{{({m_t}})}}\\sim ((d-\\sigma )\\log t)^{1/\\gamma }$ as $t\\rightarrow \\infty $ ; $\\log (\\xi _r^{{{({\\lfloor r^{\\rho }\\rfloor }})}}-\\xi _r^{{{({\\lfloor r^{\\sigma }\\rfloor }})}})= \\frac{1}{\\gamma }\\log \\log r+O(1)$ as $r\\rightarrow \\infty $ ; $\\log (\\xi _{r_tg_t}^{{{({1}})}}-\\xi _{r_tg_t}^{{{({m_t}})}})=\\frac{1}{\\gamma }\\log \\log t+O(1)$ as $t\\rightarrow \\infty $ ; the set $G_p$ is totally disconnected eventually for all $p$ .", "Further, for all $c$ , $Z_t^{{{({1}})}}\\in F_{r_tg_t}$ on the event $\\mathcal {E}_c(t)$ eventually for all $t$ ; for all $c$ , $\\log \\xi (Z_t^{{{({1}})}})=\\frac{1}{\\gamma }\\log \\log t+O(1)$ on the event $\\mathcal {E}_c(t)$ as $t\\rightarrow \\infty $ ; there exists a constant $c_1>0$ such that $|z|>t^{c_1}$ for all $z\\in F_{r_tg_t}$ eventually for all $t$ almost surely.", "(a) It follows from the proof of [17], Lemma 4.7, that for each $\\kappa \\in (0,d)$ almost surely $\\xi _r^{{{({\\lfloor r^{\\kappa }\\rfloor }})}}\\sim \\bigl ((d-\\kappa )\\log r\\bigr )^{1/\\gamma }$ as $r\\rightarrow \\infty $ .", "It remains to substitute $\\kappa =\\rho $ and $\\kappa =\\sigma $ .", "(b) This follows from (a) since $k_t=\\lfloor (r_tg_t)^{\\rho }\\rfloor $ and $m_t=\\lfloor (r_tg_t)^{\\sigma }\\rfloor $ .", "(c) This follows from (a) since $\\rho \\ne \\sigma $ .", "(d) This follows from (a) and Lemma REF since $\\rho \\ne 0$ .", "(e) This was proved in [12], Lemma 2.2, for Pareto potentials (observe that the proof relies on $\\sigma <1/2$ which is the reason why we have imposed this restriction).", "It remains to notice that $\\xi (z)=(\\alpha \\log (\\zeta (z)))^{1/\\gamma }$ , where $\\lbrace \\zeta (z)z\\in \\mathbb {Z}^d\\rbrace $ is a Pareto-distributed potential with parameter $\\alpha $ .", "As the locations of upper order statistics for $\\zeta $ and $\\xi $ coincide, we obtain that $G_p$ is eventually totally disconnected for Weibull potentials as well.", "(f) Denote by $w_t$ the maximiser of $\\xi $ in the ball of radius $t$ .", "Using Lemma REF , we obtain $\\xi \\bigl (Z_t^{{{({1}})}}\\bigr ) &\\ge &\\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )\\ge \\Psi _t(w_t)=\\xi (w_t)-\\frac{|w_t|}{\\gamma t}\\log \\log t\\\\&\\ge &\\xi _t^{{{({1}})}}-\\frac{1}{\\gamma }\\log \\log t \\sim (d\\log t)^{1/\\gamma }.$ It remains to observe that $|Z_t^{{{({1}})}}|\\le r_tg_t$ on the event $\\mathcal {E}_c(t)$ and use (a) to get $\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )\\ge \\bigl ((d-\\rho )\\log t\\bigr )^{1/\\gamma }\\sim \\xi _{r_tg_t}^{{{({k_t}})}}.$ (g) It follows from (f) that $\\log \\xi _{r_tg_t}^{{{({k_t}})}}\\le \\log \\xi (Z_t^{{{({1}})}})\\le \\log \\xi _{r_tg_t}^{{{({1}})}}$ on the event $\\mathcal {E}_c(t)$ .", "It remains to observe that $\\log \\xi _{r_tg_t}^{{{({k_t}})}}=\\frac{1}{\\gamma }\\log \\log t+O(1)$ according to (a) and $\\log \\xi _{r_tg_t}^{{{({1}})}}=\\frac{1}{\\gamma }\\log \\log t+O(1)$ by Lemma REF .", "(h) Choose $c_1$ small enough so that $c_1(d+c_1)<d-\\rho -c_1$ .", "Then almost surely eventually $\\xi ^{{{({1}})}}_{t^{c_1}}\\le \\bigl ((d+c_1)\\log t^{c_1} \\bigr )^{1/\\gamma }< \\bigl ((d-\\rho -c_1)\\log t\\bigr )^{1/\\gamma }<\\xi _{r_tg_t}^{{{({k_t}})}},$ which implies the result.", "Lemma 4.2 For each $c$ , $ \\log U(t) \\ge t\\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )-2dt+O(r_tg_t)$ on the event $\\mathcal {E}_c(t)$ eventually for all $t$ .", "The idea of the proof is the same as of [17], Lemma 2.1, for Weibull potentials and [12], Proposition 4.2, for Pareto potentials.", "However, we need to estimate the error term more precisely.", "Let $\\rho \\in (0,1]$ and $z\\in \\mathbb {Z}^d$ , $z\\ne 0$ .", "Following the lines of [12], Proposition 4.2, we obtain $ U(t)\\ge \\exp \\biggl \\lbrace t(1-\\rho )\\xi (z)-|z|\\log \\frac{|z|}{e\\rho t}-2dt+O\\bigl (\\log |z|\\bigr ) \\biggr \\rbrace .$ Take $z=Z_t^{{{({1}})}}$ and $\\rho =|Z_t^{{{({1}})}}|/(t\\xi (Z_t^{{{({1}})}}))$ .", "Observe that on the event $\\mathcal {E}_c(t)$ this $\\rho $ belongs to $(0,1]$ eventually as $\\frac{|Z_t^{{{({1}})}}|}{t\\xi (Z_t^{{{({1}})}})}\\le \\frac{r_tg_t}{t\\xi _{r_tg_t}^{{{({k_t}})}}} =O \\biggl (\\frac{g_t}{\\log t\\cdot \\log \\log t}\\biggr )=o(1)$ by Lemma REF (f) and according to ().", "Substituting this into (REF ) and using Lemma REF (g) we obtain $\\log U(t) &\\ge & t\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-\\bigl |Z_t^{{{({1}})}}\\bigr |\\log \\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-2dt+O(\\log t)\\\\&=&t\\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )-2dt+O(r_tg_t)$ on the event $\\mathcal {E}_c(t)$ .", "For all sufficiently large $t$ , consider a set $M_t\\subset \\mathbb {Z}^d$ and a nonnegative function $h_t=O(r_tg_t)$ (which may both depend on $\\xi $ ).", "Denote by $z_t$ a point along the trajectory of $(X)_s$ , $s\\in [0,t]$ , where the value of the potential is maximal.", "Define $U_{M,h}(t)=\\mathbb {E}_0 \\biggl [\\exp \\biggl \\lbrace \\int _0^t\\xi (X_s)\\,ds \\biggr \\rbrace {1}\\Bigl \\lbrace \\max _{0\\le s\\le t}{\\xi (X_s)}\\ge \\xi _{r_tg_t}^{{{({k_t}})}}, z_t\\in M_t,J_t\\ge h_t \\Bigr \\rbrace \\biggr ].$ In the sequel, $U_{M,h}(t)$ will correspond to $U_1(t)$ if we choose $M_t=\\lbrace Z_t^{{{({1}})}}\\rbrace $ , ${h_t=|Z_t^{{{({1}})}}|(1+\\rho _t)}$ and to $U_2(t)$ if we choose $M_t=\\mathbb {Z}^d\\setminus \\lbrace Z_t^{{{({1}})}}\\rbrace $ , $h_t=0$ .", "Proposition 4.3 There is a constant $c$ such that $&&\\log U_{M,h}(t) \\le \\max \\biggl \\lbrace t\\Psi _t\\bigl (Z_t^{{{({2}})}}\\bigr ),\\\\&&\\phantom{\\log U_{M,h}(t) \\le \\max \\biggl \\lbrace } {}\\max _{z\\in M_t} \\biggl \\lbrace t\\Psi _{t,c}(z)-\\frac{(h_t-|z|)_+}{2}\\bigl (\\gamma ^{-1}-\\beta \\bigr )\\log \\log t \\biggr \\rbrace \\\\&&\\phantom{\\hspace*{257.0pt}}{}+O(r_tg_t)\\biggr \\rbrace \\\\&&\\phantom{\\log U_{M,h}(t) \\le }{}-2dt$ on the event $\\mathcal {E}_c(t)$ eventually for all $t$ .", "Consider the event $\\mathcal {E}_c(t)$ and suppose that $t$ is sufficiently large.", "Using the notation from Section REF , for each $n, p\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace $ and $t$ large enough, we denote $\\mathcal {P}_{n,p}(t)= \\bigl \\lbrace y\\in \\mathcal {P}_ny_0=0, p(y)=p, q(y)>\\xi _{r_tg_t}^{{{({k_t}})}}, z(y)\\in M_t \\bigr \\rbrace .$ Observe that $q(y)\\ge \\xi _{r_tg_t}^{{{({k_t}})}}$ implies by Lemma REF (h) that $p(y)>t^{c_1}$ , for some $c_1>0$.", "In particular, $ \\log \\log p(y)\\ge \\log \\log t+\\log c_1.$ We have $U_{M,h}(t)=\\sum _{n\\ge h_t}\\sum _{t^{c_1}<p\\le n}\\sum _{y\\in \\mathcal {P}_{n,p}(t)}U(t,y),$ where $U(t,y)$ has been defined in (REF ).", "Since the number of paths in the set $\\mathcal {P}_{n,p}(t)$ is bounded by $(2d)^n$ , we obtain $U_{M,h}(t) &\\le &\\sum _{p>t^{c_1}}\\sum _{n\\ge p\\vee h_t}(2d)^{-n}\\max _{y\\in \\mathcal {P}_{n,p}(t)} \\bigl \\lbrace (2d)^{2n}U(t,y) \\bigr \\rbrace \\\\&\\le &4\\max _{p>t^{c_1}}\\max _{n\\ge p\\vee h_t} \\max _{y\\in \\mathcal {P}_{n,z}(t)} \\bigl \\lbrace (2d)^{2n}U(t,y) \\bigr \\rbrace $ and so $ \\log U_{M,h}(t) \\le \\max _{p>t^{c_1}}\\max _{n\\ge p\\vee h_t} \\max _{y\\in \\mathcal {P}_{n,z}(t)} \\bigl \\lbrace 3n\\log (2d)+\\log U(t,y) \\bigr \\rbrace .$ Let $p>t^{c_1}$ , $n\\ge p\\vee h_t$ , and $y\\in \\mathcal {P}_{n,p}(t)$ .", "Denote $i(y)=\\min \\lbrace i\\xi (y_i)=q(y)\\rbrace $ and $ Q(p,y)=q(y)\\vee \\xi _p^{{{({\\lfloor p^{\\rho }\\rfloor }})}}+\\mu _p,$ where the correction term $\\mu _p$ has been defined in (REF ).", "Define $\\xi ^{y}_i=\\left\\lbrace \\begin{array}{ll}{ \\xi (y_i),&\\quad \\mbox{if }i\\ne i(y),\\cr Q(p,y),&\\quad \\mbox{if }i=i(y).", "}Since \\xi ^y_i\\ge \\xi (y_i) for all i, we have\\begin{eqnarray*}&&U(t,y)\\le (2d)^{-n}\\mathsf {E} \\Biggl [\\exp \\Biggl \\lbrace \\sum _{i=0}^{n-1}\\tau _i\\xi ^y_i+\\Biggl (t-\\sum _{i=0}^{n-1}\\tau _i\\Biggr )\\xi ^y_n \\Biggr \\rbrace \\\\&&\\phantom{\\hspace*{112.0pt}} {}\\times {1}\\Biggl \\lbrace \\sum _{i=0}^{n-1}\\tau _i<t,\\sum _{i=0}^{n}\\tau _i>t \\Biggr \\rbrace \\Biggr ].\\end{eqnarray*}This expectation has been bounded from above in (4.16) and (4.17) of\\cite {HMS08}.", "Substituting its bound, we obtain\\begin{eqnarray*}U(t,y) &\\le &\\exp \\bigl \\lbrace t\\xi _{i(y)}^{y}-2dt \\bigr \\rbrace \\prod _{i\\ne i(y)}\\frac{1}{\\xi _{i(y)}^{y}-\\xi _i^y}\\\\&=&\\exp \\bigl \\lbrace tQ(p,y)-2dt \\bigr \\rbrace \\prod _{i\\ne i(y)}\\frac{1}{Q(p,y)-\\xi (y_i)}\\end{eqnarray*}and hence\\begin{equation} \\log U(t,y)\\le tQ(p,y)-2dt-\\sum _{i\\ne i(y)}\\log \\bigl (Q(p,y)-\\xi (y_i) \\bigr ).\\end{equation}\\end{array}The set \\right.$ p$ consists of $ p$elements and is totally disconnected by Lem\\-ma~\\ref {app}(e).", "Hence, byLemma~\\ref {l_count} we have\\begin{equation} n_{+}(y,G_{p})\\le \\frac{n-p}{2}+p^{\\sigma }.\\end{equation}In each point $ yiGp$ we use (\\ref {o}) to estimate\\begin{equation} \\log \\bigl (Q(p,y)-\\xi (y_i) \\bigr ) \\ge \\log \\mu _p= -\\beta \\log \\log p.\\end{equation}On the other hand,\\begin{eqnarray} n_{-}(y,G_{p})&=&n+1-n_{+}(y,G_{p})\\nonumber \\\\[-8pt]\\\\[-8pt]&\\ge & n+1-\\frac{n-p}{2}-p^{\\sigma } =p-p^{\\sigma }+\\frac{n-p}{2}+1\\nonumber \\end{eqnarray}and in each point $ yiGp$ we obtain by Lemma~\\ref {app}(c)\\begin{equation} \\log \\bigl (Q(p,y)-\\xi (y_i) \\bigr )\\ge \\log \\bigl (\\xi _p^{{{({\\lfloor p^{\\rho }\\rfloor }})}}-\\xi _p^{{{({\\lfloor p^{\\sigma }\\rfloor }})}} \\bigr ) \\ge \\gamma ^{-1}\\log \\log p+c_2\\end{equation}with some constant $ c2$.Using (\\ref {o2}) and (\\ref {o4}) together with (\\ref {e}),we obtain\\begin{eqnarray*}\\log U(t,y) &\\le & tQ(p,y)-2dt +n_{+}(y,G_{p})\\beta \\log \\log p\\\\&&{}- \\bigl (n_{-}(y,G_p)-1 \\bigr ) \\bigl (\\gamma ^{-1}\\log \\log p+c_2 \\bigr ).\\end{eqnarray*}Substituting (\\ref {o1}) and (\\ref {o3}) and using $ ppn$, we obtain\\begin{eqnarray} && 3n\\log (2d)+\\log U(t,y)\\nonumber \\\\&&\\qquad \\le 3n\\log (2d)+tQ(p,y)-2dt\\nonumber \\\\[-8pt]\\\\[-8pt]&&\\qquad \\phantom{\\le } {} + \\biggl [\\frac{n-p}{2}+p^{\\sigma } \\biggr ]\\beta \\log \\log p - \\biggl [p-p^{\\sigma }+\\frac{n-p}{2} \\biggr ] \\bigl (\\gamma ^{-1}\\log \\log p+c_2 \\bigr )\\nonumber \\\\&&\\qquad \\le tQ(p,y)-\\frac{p}{\\gamma }\\log \\log p-2dt -\\frac{n-p}{2} \\bigl (\\gamma ^{-1}-\\beta \\bigr )\\log \\log p+c_3n\\nonumber \\end{eqnarray}with some constant $ c3$.$ Now we distinguish between the following two cases.", "Case 1.", "Suppose $q(y)\\ge \\xi _p^{{{({\\lfloor p^{\\rho }\\rfloor }})}}$ .", "Then $Q(p,y)=\\xi (z(y))+\\mu _p$ and estimating $p\\ge |z(y)|$ we get $3n\\log (2d)+\\log U(t,y) &\\le & t\\xi \\bigl (z(y) \\bigr )+t\\mu _p-\\frac{|z(y)|}{\\gamma }\\log \\log p-2dt\\\\&&{}-\\frac{n-|z(y)|}{2} \\bigl (\\gamma ^{-1}-\\beta \\bigr )\\log \\log p+c_3n.$ Observe that $t\\mu _p\\le t\\mu _{t^{c_1}}= t(c_1\\log t)^{-\\beta }=o(r_tg_t)$ since $\\beta >1-1/\\gamma $ and according to ().", "Using monotonicity in $n$ and $n\\ge |z(y)|\\vee h_t$ together with (REF ), we obtain $ &&3n\\log (2d)+\\log U(t,y)\\nonumber \\\\&&\\qquad \\le t\\Psi _t \\bigl (z(y) \\bigr )+c\\bigl |z(y)\\bigr |-2dt\\nonumber \\\\[-8pt]\\\\[-8pt]&&\\qquad \\phantom{\\le } {}-\\frac{(h_t-|z(y)|)_+}{2} \\bigl (\\gamma ^{-1}-\\beta \\bigr )\\log \\log t +c h_t+o(r_tg_t)\\nonumber \\\\&&\\qquad \\le \\max _{z\\in M_t} \\biggl \\lbrace t\\Psi _{t,c}(z)-\\frac{(h_t-|z|)_+}{2} \\bigl (\\gamma ^{-1}-\\beta \\bigr )\\log \\log t \\biggr \\rbrace -2dt+O(r_tg_t)\\nonumber $ with some constant $c$ .", "Case 2.", "Suppose $q(y)< \\xi _p^{{{({\\lfloor p^{\\rho }\\rfloor }})}}$ .", "Then $Q(p,y)= \\xi _p^{{{({\\lfloor p^{\\rho }\\rfloor }})}}+\\mu _p$ .", "Now () implies $3n\\log (2d)+\\log U(t,y) &\\le & t\\xi _p^{{{({\\lfloor p^{\\rho }\\rfloor }})}}+t\\mu _p-\\frac{p}{\\gamma }\\log \\log p-2dt\\\\&&{}-\\frac{n-p}{2} \\bigl (\\gamma ^{-1}-\\beta \\bigr )\\log \\log p+c_4n$ with some constant $c_4$ .", "Using monotonicity in $n$ and $n\\ge p$ , we get $3n\\log (2d)+\\log U(t,y) \\le t\\xi _p^{{{({\\lfloor p^{\\rho }\\rfloor }})}}+t(\\log p)^{-\\beta }-\\frac{p}{\\gamma }\\log \\log p-2dt+c_4p.$ By Lemma REF (a) and using $\\beta \\ge 0$ , we obtain that the second term is dominated by the first one, the fifth by the third one, and so $ \\quad 3n\\log (2d)+\\log U(t,y) \\le t \\bigl ((d-\\rho /2)\\log p\\bigr )^{1/\\gamma }-c_5p\\log \\log p-2dt$ with some constant $c_5>0$ .", "Differentiating, we obtain the following equation for the maximiser $p_t$ of the expression on the right-hand side of (REF ): $\\frac{t(d-\\rho /2) ((d-\\rho /2)\\log p_t)^{1/\\gamma -1}}{\\gamma p_t}-c_5\\log \\log p_t-\\frac{c_5}{\\log p_t}=0.$ Resolving this asymptotics, we obtain $p_t=r_t(d-\\rho /2)^{1/\\gamma } \\bigl (1+o(1) \\bigr ).$ Finally, substituting this into (REF ) yields $ 3n\\log (2d)+\\log U(t,y) &\\le & t \\bigl ((d-\\rho /3)\\log r_t \\bigr )^{1/\\gamma }-2dt\\nonumber \\\\&\\le & \\bigl (1-\\rho /(3d) \\bigr )^{1/\\gamma }ta_{r_t}-2dt\\\\&\\le & t\\Psi _t\\bigl (Z_t^{{{({2}})}}\\bigr )-2dt$ on the event $\\mathcal {E}_c(t)$ .", "It remains to substitute (REF ) and (REF ) into (REF ) to complete the proof.", "Observe that the scaling function $\\mu _p$ , being part of $Q(p,y)$ , appears both in the main and in the logarithmic term of (REF ).", "Being part of the main term, $t\\mu _p$ needs to be as small as $O(r_tg_t)$ in order to not imbalance the significant terms.", "This leads to the restriction $\\beta >1-1/\\gamma $ .", "However, as a part of the logarithmic term, $\\mu _p$ needs to be large enough so that the contribution $\\gamma ^{-1}\\log \\log p$ of “good” points $y_i\\notin G_p$ dominates over the contribution $\\beta \\log \\log p$ of “bad” points $y_i\\in G_p$ .", "This imposes the restriction $\\beta <1/\\gamma $ .", "The combination of these two conditions only allows to choose such $\\beta $ if $0<\\gamma <2$ .", "From now on, we assume that the constant $c$ is fixed and chosen according to Proposition REF .", "Lemma 4.4 Almost surely, $\\frac{U_1(t)}{U(t)}{1}_{\\mathcal {E}_c(t)}\\rightarrow 0 \\qquad \\mbox{as }t\\rightarrow \\infty .$ We use Proposition REF with $M_t=\\lbrace Z_t^{{{({1}})}}\\rbrace $ and $h_t=|Z_t^{{{({1}})}}|(1+\\rho _t)$ .", "Clearly $h_t=O(r_tg_t)$ on the event $\\mathcal {E}_c(t)$ .", "By Lemma REF (f), we have $Z_t^{{{({1}})}}\\in F_{r_tg_t}$ , which implies $U_{M,h}(t)=U_1(t)$ eventually for all $t$ .", "Since $|Z_t^{{{({1}})}}|\\le r_tg_t$ and so $t\\Psi _{t,c}(Z_t^{{{({1}})}})=t\\Psi _t(Z_t^{{{({1}})}})+O(r_tg_t)$ , we obtain $ \\log U_1(t) &\\le &\\max \\biggl \\lbrace t\\Psi _t\\bigl ( Z_t^{{{({2}})}}\\bigr ), t\\Psi _{t}\\bigl (Z_t^{{{({1}})}}\\bigr )-\\frac{|Z_t^{{{({1}})}}|\\rho _t}{2}(1/\\gamma -\\beta )\\log \\log t +O(r_tg_t) \\biggr \\rbrace \\nonumber \\\\[-5pt]\\\\[-8pt]&&{}-2dt.\\nonumber $ In order to show that $ \\log U_1(t)-\\log U(t)\\rightarrow -\\infty $ we consider the terms under the maximum in (REF ) separately.", "Using the lower bound for the total mass given by Lemma REF and taking into account that $\\Psi _t(Z_t^{{{({1}})}})-\\Psi _t(Z_t^{{{({2}})}})>d_t\\lambda _t$ on the event $\\mathcal {E}_c(t)$ , we get for the first term $ t\\Psi _t\\bigl (Z_t^{{{({2}})}}\\bigr )-2dt-\\log U(t) &\\le & t \\Psi _t\\bigl (Z_t^{{{({2}})}}\\bigr )- t\\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )+O(r_tg_t)\\nonumber \\\\[-8pt]\\\\[-8pt]&<&-td_t\\lambda _t+O(r_tg_t)\\rightarrow -\\infty \\nonumber $ according to ().", "For the second term, we again use the lower bound from Lemma REF and take into account that $|Z_t^{{{({1}})}}|\\ge r_tf_t$ on the event $\\mathcal {E}_c(t)$ .", "This implies $ &&t\\Psi _{t}\\bigl (Z_t^{{{({1}})}}\\bigr )-\\frac{|Z_t^{{{({1}})}}|\\rho _t}{2} (1/\\gamma - \\beta )\\log \\log t+O(r_tg_t)-2dt-\\log U(t)\\nonumber \\\\&&\\qquad \\le -\\frac{|Z_t^{{{({1}})}}|\\rho _t}{2}(1/\\gamma -\\beta )\\log \\log t+O(r_tg_t)\\\\&&\\qquad \\le -\\frac{r_t f_t \\rho _t}{2}(1/\\gamma -\\beta )\\log \\log t+O(r_tg_t)\\rightarrow -\\infty \\nonumber $ by ().", "Combining (REF ), (REF ) and (REF ) we get (REF ) on the event $\\mathcal {E}_c(t)$ .", "Lemma 4.5 Almost surely, $\\frac{U_2(t)}{U(t)}{1}_{\\mathcal {E}_c(t)}\\rightarrow 0 \\qquad \\mbox{as }t\\rightarrow \\infty .$ We use Proposition REF with $M_t=\\mathbb {Z}^d\\setminus \\lbrace Z_t^{{{({1}})}}\\rbrace $ and $h_t=0$ .", "In this case $U_{M,h}(t)=U_2(t)$ , and we have $ \\log U_2(t)\\le \\max \\Bigl \\lbrace t\\Psi _t\\bigl ( Z_t^{{{({2}})}}\\bigr ), t \\max _{z\\ne Z_t^{{{({1}})}}}\\Psi _{t,c}(z)+O(r_tg_t) \\Bigr \\rbrace -2dt.$ Since $|Z_t^{{{({1, c}})}}|\\le r_tg_t$ and $|Z_t^{{{({2, c}})}}|\\le r_tg_t$ on the event $\\mathcal {E}_c(t)$ , we have for $i\\in \\lbrace 1,2\\rbrace $ $t\\Psi _{t,c} \\bigl (Z_t^{{{({i, c}})}} \\bigr )=t\\Psi _t \\bigl (Z_t^{{{({i,c}})}} \\bigr )+c|Z_t^{{{({i, c}})}}|=t\\Psi _t \\bigl (Z_t^{{{({i, c}})}} \\bigr )+O(r_tg_t).$ Substituting this into (REF ) and observing that $z\\ne Z_t^{{{({1}})}}$ , we obtain $\\log U_2(t)\\le t\\Psi _t\\bigl (Z_t^{{{({2}})}}\\bigr )+O(r_tg_t)-2dt.$ Using the lower bound for the total mass given by Lemma REF and taking into account that $\\Psi _t(Z_t^{{{({1}})}})-\\Psi _t(Z_t^{{{({2}})}})>d_t\\lambda _t$ on the event $\\mathcal {E}_c(t)$ , we get $\\log U_2(t)-\\log U(t) &\\le & t\\Psi _t\\bigl (Z_t^{{{({2}})}}\\bigr )-t \\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )+O(r_tg_t)\\\\&\\le &-t d_t\\lambda _t +O(r_tg_t)\\rightarrow -\\infty $ according to () on the event $\\mathcal {E}_c(t)$ .", "Lemma 4.6 Almost surely, $\\frac{U_3(t)}{U(t)}{1}_{\\mathcal {E}_c(t)}\\rightarrow 0 \\qquad \\mbox{as }t\\rightarrow \\infty .$ We can estimate the integral in the Feynman–Kac formula for $U_3(t)$ by $t\\xi _{r_tg_t}^{{{({k_t}})}}$ and get $\\log U_3(t)\\le t\\xi _{r_tg_t}^{{{({k_t}})}}\\sim t \\bigl ((d-\\rho )\\log t \\bigr )^{1/\\gamma }\\le (1-\\delta ) t a_{r_t}$ with some $\\delta >0$ eventually for all $t$ by Lemma REF (b).", "Using the lower bound for $U(t)$ from Lemma REF , we have $\\log U_3(t)-\\log U(t) &\\le &(1-\\delta )ta_{r_t}-t\\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )+2dt+O(r_tg_t)\\\\&\\le &-\\delta ta_{r_t}+td_t g_t+2dt+O(r_tg_t)\\rightarrow -\\infty $ since $\\Psi _t(Z_t^{{{({1}})}})>a_{r_t}-d_t g_t$ on the event $\\mathcal {E}_c(t)$ ." ], [ "Localisation", "The aim of this section is to prove Theorem REF .", "We assume throughout this section that $0<\\gamma <2$ and we suppose that $c$ is chosen according to Proposition REF .", "Let $B_t= \\bigl \\lbrace z\\in \\mathbb {Z}^d|z|\\le \\bigl |Z_t^{{{({1}})}}\\bigr |(1+\\rho _t) \\bigr \\rbrace .", "$ For any set $A\\subset \\mathbb {Z}^d$ denote by $A^c=\\mathbb {Z}^d\\setminus A$ its complement and by $\\tau (A)$ the hitting time of $A$ by the random walk $(X_s)$ , and we write $\\tau (z)$ for $\\tau (\\lbrace z\\rbrace )$ for any point $z\\in \\mathbb {Z}^d$ .", "Let us decompose the solution $u$ into $u=u_1+u_2$ according to the two groups of paths (I) and (II) mentioned in the [intro]Introduction $u_1(t,z)&=&\\mathbb {E}_0 \\biggl [\\exp \\biggl \\lbrace \\int _0^t\\xi (X_s)\\,ds \\biggr \\rbrace {1}\\lbrace X_t=z\\rbrace {1}\\bigl \\lbrace \\tau \\bigl (Z_t^{{{({1}})}}\\bigr )\\le t,\\tau \\bigl (B_t^c \\bigr )>t \\bigr \\rbrace \\biggr ],\\\\u_2(t,z)&=&\\mathbb {E}_0 \\biggl [\\exp \\biggl \\lbrace \\int _0^t\\xi (X_s)\\,ds \\biggr \\rbrace {1}\\lbrace X_t=z\\rbrace {1}\\bigl \\lbrace \\tau \\bigl (Z_t^{{{({1}})}}\\bigr )> t\\mbox{ or } \\tau \\bigl (B_t^c \\bigr )\\le t \\bigr \\rbrace \\biggr ].$ In Lemma REF below, we use the results from Section  to prove that the total mass of $u_2$ is negligible.", "In order to prove that $u_1$ localises around $Z_t^{{{({1}})}}$ , we introduce the gap $\\mathfrak {g}_t=\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-\\max \\bigl \\lbrace \\xi (z)z\\in B_t \\setminus \\bigl \\lbrace Z_t^{{{({1}})}}\\bigr \\rbrace \\bigr \\rbrace $ between the value of the potential $\\xi $ at the point $Z_t^{{{({1}})}}$ and in the rest of the ball $B_t$ .", "In Lemma REF we find a lower bound for $\\mathfrak {g}_t$ .", "This bound tends to infinity for $\\gamma <1$ but is going to zero for $1\\le \\gamma <2$ .", "However, the lower bound turns out to be just large enough to provide localisation of the principal eigenfunction of the Anderson Hamiltonian $\\Delta +\\xi $ around $Z_t^{{{({1}})}}$ , which is proved in Lemma REF .", "This easily implies the localisation of $u_1$ around $Z_t^{{{({1}})}}$ and allows us to prove Theorem REF in the end of this section.", "Lemma 5.1 Almost surely, $\\biggl \\lbrace U(t)^{-1}\\sum _{z\\in \\mathbb {Z}^d}u_2(t,z)\\biggr \\rbrace {1}_{\\mathcal {E}_c(t)}\\rightarrow 0 \\qquad \\mbox{as }t\\rightarrow \\infty .$ We have $\\quad \\sum _{z\\in \\mathbb {Z}^d}u_2(t,z) =\\mathbb {E}_0 \\biggl [\\exp \\biggl \\lbrace \\int _0^t\\xi (X_s)\\,ds \\biggr \\rbrace {1}\\bigl \\lbrace \\tau \\bigl (Z_t^{{{({1}})}}\\bigr )> t\\mbox{ or } \\tau \\bigl (B_t^c \\bigr )\\le t \\bigr \\rbrace \\biggr ].$ Observe that if a path belongs to the set in the indicator function above then either it passes through $Z_t^{{{({1}})}}$ and reaches the maximum of the potential there but leaves the ball $B_t$ thus belonging to $E_1(t)$ , or it reaches the maximum of the potential not in $Z_t^{{{({1}})}}$ thus belonging to $E_2(t)$ or $E_3(t)$ , depending on whether the maximum of the potential over the path exceeds the value $\\xi _{r_tg_t}^{{{({k_t}})}}$ .", "Hence, we have on the event $\\mathcal {E}_c(t)$ $\\sum _{z\\in \\mathbb {Z}^d}u_2(t,z) &\\le &\\mathbb {E}_0 \\biggl [\\exp \\biggl \\lbrace \\int _0^t\\xi (X_s)\\,ds \\biggr \\rbrace {1}_{E_1(t)\\cup E_2(t)\\cup E_3(t)} \\biggr ]\\\\&=&U_1(t)+U_2(t)+U_3(t).$ The statement of the lemma now follows from Lemmas REF , REF and REF .", "Lemma 5.2 On the event $\\mathcal {E}_c(t)$ , the gap $\\mathfrak {g_t}$ is positive and, for any $\\varepsilon >0$ , $\\log \\mathfrak {g}_t> (1/\\gamma -1-\\varepsilon )\\log \\log t$ eventually for all $t$ .", "Let $z\\in B_t\\setminus \\lbrace Z_t^{{{({1}})}}\\rbrace $ .", "Then $\\Psi _t(z)\\le \\Psi _t(Z_t^{{{({2}})}})$ and we have on the event $\\mathcal {E}_c(t)$ $d_t\\lambda _t&\\le &\\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )- \\Psi _t\\bigl (Z_t^{{{({2}})}}\\bigr ) \\le \\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )-\\Psi _t(z)\\\\&=&\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-\\xi (z)+\\frac{|z|-|Z_t^{{{({1}})}}|}{\\gamma t}\\log \\log t. $ Since $|Z_t^{{{({1}})}}|<r_tg_t$ on the event $\\mathcal {E}_c(t)$ , the last term satisfies $\\frac{|z|-|Z_t^{{{({1}})}}|}{\\gamma t}\\log \\log t\\le \\frac{|Z_t^{{{({1}})}}|\\rho _t}{\\gamma t}\\log \\log t\\le \\frac{r_tg_t\\rho _t}{\\gamma t}\\log \\log t=O(d_tg_t\\rho _t).$ We obtain uniformly for all $z\\in B_t\\setminus \\lbrace Z_t^{{{({1}})}}\\rbrace $ $d_t\\lambda _t\\le \\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-\\xi (z)+O(d_tg_t \\rho _t)$ and so $\\mathfrak {g}_t\\ge d_t\\lambda _t+O(d_tg_t\\rho _t)=d_t\\lambda _t+o(d_t\\lambda _t)$ on according to ().", "This estimate implies the statement of the lemma since $\\log d_t\\sim (\\frac{1}{\\gamma }-1)\\log \\log t$ and $\\lambda _t$ is negligible according to ().", "Let $\\gamma _t$ and $v_t$ be the principal eigenvalue and eigenfunction of $\\Delta +\\xi $ with zero boundary conditions in the ball $B_t$ .", "We extend $v_t$ by zero to the whole space $\\mathbb {Z}^d$ and we assume that $v_t$ is normalised so that $v_t(Z_t^{{{({1}})}})=1$ .", "The eigenfunction $v_t$ has the following probabilistic representation $v_t(z)=\\mathbb {E}_z \\biggl [\\exp \\biggl \\lbrace \\int _0^{\\tau (Z_t^{{{({1}})}})} \\bigl (\\xi (X_s)-\\gamma _t \\bigr )\\,ds \\biggr \\rbrace {1}\\bigl \\lbrace \\tau \\bigl (Z_t^{{{({1}})}}\\bigr )<\\tau \\bigl ( \\mathbb {Z}^d\\setminus B_t \\bigr ) \\bigr \\rbrace \\biggr ].$ Lemma 5.3 Almost surely, $\\biggl \\lbrace \\Vert v_t\\Vert _2^2 \\sum _{z\\in B_t\\setminus \\lbrace Z_t^{{{({1}})}}\\rbrace } v_t(z) \\biggr \\rbrace {1} _{\\mathcal {E}_c(t)}\\rightarrow 0\\qquad \\mbox{as }t\\rightarrow \\infty .$ Consider the event $\\mathcal {E}_c(t)$ and suppose that $t$ is sufficiently large.", "For each $n, p\\in \\mathbb {N}$ and $z\\in B_t\\setminus \\lbrace Z_t^{{{({1}})}}\\rbrace $ denote $\\mathcal {P}_{n,p}(t,z)= \\bigl \\lbrace y\\in \\mathcal {P}_ny_0=z, y_n=Z_t^{{{({1}})}},y_{i}\\in B_t \\setminus Z_t^{{{({1}})}}\\ \\forall i< n, p(y)=p \\bigr \\rbrace .$ Integrating with respect to the waiting times $(\\tau _i)$ of the random walk, which are independent and exponentially distributed with parameter $2d$ and observing that the probability of the first $n$ steps of the random walk to follow a given geometric path is $(2d)^{-n}$ we get $v_t(z)&=&\\sum _{n\\ge |z-Z_t^{{{({1}})}}|}\\sum _{p\\le n}\\sum _{y\\in \\mathcal {P}_{n,p}(t,z)}(2d)^{-n}{\\mathsf {E}} \\Biggl [\\exp \\Biggl \\lbrace \\sum _{i=0}^{n-1}\\bigl (\\xi (y_i)-\\gamma _t \\bigr )\\tau _i \\Biggr \\rbrace \\Biggr ]\\\\&=&\\sum _{n\\ge |z-Z_t^{{{({1}})}}|}\\sum _{1\\le p\\le n}\\sum _{y\\in \\mathcal {P}_{n,p}(t,z)} \\prod _{i=0}^{n-1}\\int _0^{\\infty } \\exp \\bigl \\lbrace - \\bigl (\\gamma _t+2d-\\xi (y_i) \\bigr )t \\bigr \\rbrace \\,dt.$ The Rayleigh–Ritz formula implies $\\gamma _t &=&\\sup \\bigl \\lbrace \\bigl \\langle (\\Delta +\\xi )\\varphi , \\varphi \\bigr \\rangle \\varphi \\in \\ell ^2(B_t), \\varphi |_{\\partial B_t}=0,\\Vert \\varphi \\Vert _2=1 \\bigr \\rbrace \\\\&\\ge & \\bigl \\langle (\\Delta +\\xi ){1}_{\\lbrace Z_t^{{{({1}})}}\\rbrace }, {1}_{\\lbrace Z_t^{{{({1}})}}\\rbrace } \\bigr \\rangle =\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-2d$ and so for all $i$ $ \\gamma _t+2d-\\xi (y_i)\\ge \\xi \\bigl (Z_t^{{{({1}})}}\\bigr )- \\xi (y_i)\\ge \\mathfrak {g}_t.$ Since $\\mathfrak {g}_t>0$ eventually on the event $\\mathcal {E}_c(t)$ by Lemma REF , we use (REF ) to compute $ v_t(z)&=&\\sum _{n=|z-Z_t^{{{({1}})}}|}^{\\infty }\\sum _{p\\le n}\\sum _{y\\in \\mathcal {P}_{n,p}(t,z)}\\prod _{i=0}^{n-1}\\frac{1}{\\gamma _t+2d-\\xi (y_i)}\\nonumber \\\\&\\le &\\sum _{p\\ge |z-Z_t^{{{({1}})}}|}\\sum _{n\\ge p}\\sum _{y\\in \\mathcal {P}_{n,p}(t,z)} \\prod _{i=0}^{n-1}\\frac{1}{\\xi (Z_t^{{{({1}})}})-\\xi (y_i)}\\nonumber \\\\[-8pt]\\\\[-8pt]&\\le &\\sum _{p\\ge |z-Z_t^{{{({1}})}}|}\\sum _{n\\ge p}(2d)^{-n}\\max _{y\\in \\mathcal {P}_{n,p}(t,z)} \\Biggl \\lbrace (2d)^{2n}\\prod _{i=0}^{n-1}\\frac{1}{\\xi (Z_t^{{{({1}})}})-\\xi (y_i)} \\Biggr \\rbrace \\nonumber \\\\&\\le &\\sum _{p\\ge |z-Z_t^{{{({1}})}}|} \\exp \\max _{n\\ge p}\\max _{y\\in \\mathcal {P}_{n,p}(t,z)} \\Biggl \\lbrace 2n\\log (2d)-\\sum _{i=0}^{n-1}\\log \\bigl (\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-\\xi (y_i) \\bigr ) \\Biggr \\rbrace \\nonumber $ since $\\sum _{n\\ge p}(2d)^{-n}\\le 1$ for $p\\ge 1$ .", "Fix some positive $\\varepsilon \\in (\\frac{1}{\\gamma }-1, \\frac{1}{\\gamma }-\\frac{1}{2})$ .", "Notice that this is possible since $\\gamma <2$ and so $\\frac{1}{\\gamma }-\\frac{1}{2}>0$ .", "Let $p\\ge |z-Z_t^{{{({1}})}}|$ , $n\\ge p$ , and $y\\in \\mathcal {P}_{n,p}(t,z)$ .", "By Lemma REF (e), the set $G_{r_tg_t}$ is totally disconnected and so $ n_{+}(y,G_{r_tg_t})\\le \\biggl \\lceil \\frac{n+1}{2} \\biggr \\rceil \\le \\frac{n}{2}+1.", "$ In each point $y_i\\in G_{r_tg_t}$ , we can estimate by Lemma  REF $ \\log \\bigl (\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-\\xi (y_i) \\bigr )\\ge \\log \\mathfrak {g}_t> (1/\\gamma -1-\\varepsilon )\\log \\log t.$ On the other hand, $ n_{-}(y,G_{r_tg_t})=n+1-n_{+}(y,G_{r_tg_t})\\ge \\frac{n}{2}$ and in each point $y_i\\notin G_{r_tg_t}$ we get by Lemma REF (d) $ \\log \\bigl (\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-\\xi (y_i) \\bigr )=\\log \\bigl ( \\xi _{r_tg_t}^{{{({k_t}})}}-\\xi _{r_tg_t}^{{{({m_t}})}} \\bigr ) >(1/ \\gamma -\\varepsilon )\\log \\log t$ by Lemma REF .", "Using (REF ) and (REF ) and taking into account that the last point $Z_t^{{{({1}})}}$ of the path belongs to $G_{r_tg_t}$ but does not contribute to the sum, we obtain $&&2n\\log (2d)-\\sum _{i=0}^{n-1}\\log \\bigl (\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-\\xi (y_i) \\bigr )\\\\&&\\qquad \\le 2n\\log (2d)- \\bigl (n_{+}(y,G_{r_tg_t})-1 \\bigr ) (1/\\gamma -1-\\varepsilon )\\log \\log t\\\\&&\\qquad \\phantom{\\le }-n_{-}(y,G_{r_tg_t}) (1/\\gamma -\\varepsilon )\\log \\log t.$ Since $\\frac{1}{\\gamma }-1-\\varepsilon <0$ and $\\frac{1}{\\gamma }-\\varepsilon >0$ , we can estimate further using (REF ) and (REF ) $&&2n\\log (2d)-\\sum _{i=0}^{n-1}\\log \\bigl (\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-\\xi (y_i) \\bigr )\\\\&&\\qquad \\le 2n\\log (2d)-\\frac{n}{2}(1/\\gamma -1-\\varepsilon )\\log \\log t -\\frac{n}{2}(1/\\gamma -\\varepsilon )\\log \\log t\\\\&&\\qquad =2n\\log (2d)-n(1/\\gamma -1/2-\\varepsilon )\\log \\log t.$ Since $\\frac{1}{\\gamma }-\\frac{1}{2}-\\varepsilon >0$ , this function is decreasing in $n$ and can be estimated by its value at $n=p$ .", "This implies $&&2n\\log (2d)-\\sum _{i=0}^{n-1}\\log \\bigl (\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-\\xi (y_i) \\bigr )\\\\&&\\qquad \\le 2p\\log (2d)-p(1/\\gamma -1/2-\\varepsilon )\\log \\log t \\le -p\\delta \\log \\log t$ with some $\\delta >0$ .", "Substituting this into (REF ), we obtain $v_t(z)\\le \\sum _{p\\ge |z-Z_t^{{{({1}})}}|}(\\log t)^{-p\\delta } \\le 2(\\log t)^{-\\delta |z-Z_t^{{{({1}})}}|}.$ Since $v_t(z)$ decays geometrically in distance of $z$ from $Z_t^{{{({1}})}}$ , $(\\log t)^{-\\delta }\\rightarrow 0$ , and $v_t(Z_t^{{{({1}})}})=1$ , the statement of the lemma is now obvious.", "Observe that, similarly to the proof of Proposition REF , we have a competition of the positive and negative terms in the sum in (REF ), and we want the negative terms to dominate.", "The contribution of the positive terms is of order $(1/\\gamma -1)\\log \\log t$ and the contribution of the negative terms is roughly $(1/\\gamma )\\log \\log t$ .", "This leads to the condition $1-1/\\gamma <1/\\gamma $ , which restricts our proof to the case $0<\\gamma <2$ .", "Proof of Theorem REF We have $ 1-\\frac{u(t,Z_t^{{{({1}})}})}{U(t)} &=&U(t)^{-1}\\sum _{z\\ne Z_t^{{{({1}})}}}u(t,z)\\nonumber \\\\[-8pt]\\\\[-8pt]&\\le & U(t)^{-1}\\sum _{z\\ne Z_t^{{{({1}})}}}u_1(t,z)+U(t)^{-1}\\sum _{z\\in \\mathbb {Z}^d}u_2(t,z).\\nonumber $ The second term converges to zero on the event $\\mathcal {E}_c(t)$ by Lemma REF .", "The first term satisfies the conditions of [8], Theorem 4.1, with $B=B_t$ , $V=\\xi $ , and $\\Gamma =\\lbrace Z_t^{{{({1}})}}\\rbrace $ , which implies that, for all $z\\in B_t$ , $u_1(t,z)\\le u_1\\bigl (t,Z_t^{{{({1}})}}\\bigr )\\Vert v_t\\Vert _2^2 v_t(z).$ Observing that $U(t)\\ge u_1(t,Z_t^{{{({1}})}})$ and $u_1(t,z)=0$ for $z\\notin B_t$ , we obtain $U(t)^{-1}\\sum _{z\\ne Z_t^{{{({1}})}}}u_1(t,z)\\le \\Vert v_t\\Vert _2^2\\sum _{z\\in B_t\\setminus \\lbrace Z_t^{{{({1}})}}\\rbrace }v_t(z),$ which converges to zero on the event $\\mathcal {E}_c(t)$ by Lemma REF .", "As both terms in (REF ) converge to zero on the event $\\mathcal {E}_c(t)$ and $\\operatorname{Prob}\\lbrace \\mathcal {E}_c(t)\\rbrace \\rightarrow 1$ by Proposition REF (b), we obtain that $1-\\frac{u(t,Z_t^{{{({1}})}})}{U(t)}\\rightarrow 0 \\qquad \\mbox{as }t\\rightarrow \\infty $ in probability." ], [ "Ageing", "In this section, we discuss the ageing behaviour of the parabolic Anderson model.", "Throughout this section, we assume that $\\gamma >0$ .", "As we pointed out in the [intro]Introduction, although the results proved in this section hold for all $\\gamma >0$ , they only imply ageing of the parabolic Anderson model for $0<\\gamma <2$ as otherwise the solution $u$ may not be localised at $Z_t^{{{({1}})}}$ .", "We begin by showing that whenever the maximiser of $\\Psi $ has moved from one point to another, it cannot go back to the original point.", "Lemma 6.1 For $s>0$ , $\\lbrace T_t>s\\rbrace = \\lbrace Z_t^{{{({1}})}}=Z_{t+s}^{{{({1}})}} \\rbrace $ eventually for all $t$ .", "If $T_t>s$ , then $Z_t^{{{({1}})}}=Z_{t+s}^{{{({1}})}}$ by the definition of $T_t$ .", "Suppose $Z_t^{{{({1}})}}=Z_{t+s}^{{{({1}})}}$ but there is $u\\in (t,t+s)$ such that $Z_t^{{{({1}})}}\\ne Z_u^{{{({1}})}}$ .", "Consider an auxiliary function $\\varphi [t,t+s]\\rightarrow \\mathbb {R}$ given by $\\varphi (x)=\\Psi _x\\bigl (Z_t^{{{({1}})}}\\bigr )-\\Psi _x \\bigl (Z_u^{{{({1}})}} \\bigr )=\\xi \\bigl (Z_t^{{{({1}})}}\\bigr )-\\xi \\bigl (Z_u^{{{({1}})}}\\bigr )-\\frac{|Z_t^{{{({1}})}}|-|Z_u^{{{({1}})}}|}{\\gamma x} \\log \\log x.", "$ Observe that $\\varphi ^{\\prime }(x)=\\frac{|Z_t^{{{({1}})}}|-|Z_u^{{{({1}})}}|}{\\gamma x^2\\log x}(\\log x\\log \\log x-1)$ and so $\\varphi ^{\\prime }$ does not change the sign on the interval $[t,t+s]$ if $t$ is large enough.", "Hence, $\\varphi $ is strictly monotone on $[t,t+s]$ .", "However, this contradicts the observation that $\\varphi (t)\\ge 0$ (since $Z_t^{{{({1}})}}$ is the maximiser of $\\Psi _t$ and $Z_u^{{{({1}})}}\\ne Z_t^{{{({1}})}}$ ), $\\varphi (u)\\le 0$ (since $Z_u^{{{({1}})}}$ is the maximiser of $\\Psi _u$ and $Z_t^{{{({1}})}}\\ne Z_u^{{{({1}})}}$ ), and $\\varphi (t+s)\\ge 0$ (since $Z_t^{{{({1}})}}=Z_{t+s}^{{{({1}})}}$ is the maximiser of $\\Psi _{t+s}$ and $Z_{u}^{{{({1}})}}\\ne Z_t^{{{({1}})}}$ ).", "Now we are going to compute the probability of $ \\lbrace Z_t^{{{({1}})}}=Z_{t+wt}^{{{({1}})}} \\rbrace $ , $w>0$ , using the point processes $\\Pi _t\\equiv \\Pi _{t,0}$ studied in Section .", "However, we need to restrict them to a finite box growing to infinity to justify integration and passing to the limit.", "In order to do so, for each $n\\in \\mathbb {N}$ , we define the event $\\mathcal {A}(n,w,t)= \\bigl \\lbrace Y_{t,Z_t^{{{({1}})}}}\\ge -n, \\Psi _{t+wt}(z)\\le \\Psi _{t+wt}\\bigl (Z_t^{{{({1}})}}\\bigr )\\ \\forall z\\in \\mathbb {Z}^d \\mbox{ s.t.", "}Y_{t,z}\\ge -n \\bigr \\rbrace $ and show that $\\operatorname{Prob}\\lbrace Z_t^{{{({1}})}}=Z_{t+wt}^{{{({1}})}} \\rbrace $ is captured by the probabilities of these events.", "Lemma 6.2 For any $w>0$ , $\\lim _{t\\rightarrow \\infty }\\operatorname{Prob}\\bigl \\lbrace Z_t^{{{({1}})}}=Z_{t+wt}^{{{({1}})}}\\bigr \\rbrace =\\lim _{n\\rightarrow \\infty }\\lim _{t\\rightarrow \\infty }\\operatorname{Prob}\\bigl \\lbrace \\mathcal {A}(n,w,t) \\bigr \\rbrace ,$ provided the limit on the right-hand side exists.", "To obtain an upper bound, observe that $ \\operatorname{Prob}\\bigl \\lbrace Z_t^{{{({1}})}}=Z_{t+wt}^{{{({1}})}}\\bigr \\rbrace \\le \\operatorname{Prob}\\bigl \\lbrace \\mathcal {A}(n,w,t) \\bigr \\rbrace +\\operatorname{Prob}\\lbrace Y_{t,Z_t^{{{({1}})}}}\\le -n \\rbrace .$ By Proposition REF , $ \\lim _{n\\rightarrow \\infty } \\lim _{t\\rightarrow \\infty } \\operatorname{Prob}\\lbrace Y_{t,Z_t^{{{({1}})}}}\\le -n \\rbrace =\\lim _{n\\rightarrow \\infty }\\operatorname{Prob}\\bigl \\lbrace Y^{{{({1}})}}\\le -n \\bigr \\rbrace =0.$ For a lower bound, we have $ \\operatorname{Prob}\\bigl \\lbrace Z_t^{{{({1}})}}=Z_{t+wt}^{{{({1}})}}\\bigr \\rbrace \\ge \\operatorname{Prob}\\bigl \\lbrace \\mathcal {A}(n,w,t) \\bigr \\rbrace -\\operatorname{Prob}\\lbrace Y_{t,Z_{t+wt}^{{{({1}})}}}\\le -n \\rbrace .$ Observe that for all $z$ we have, as $t\\rightarrow \\infty $ , $ \\Psi _{t+wt}(z)&=&\\xi (z)-\\frac{|z|}{\\gamma (t+wt)}\\log \\log (t+wt)\\nonumber \\\\&=&\\Psi _t(z)+\\frac{w|z|}{(1+w)\\gamma t} \\bigl (\\log \\log t +o(1) \\bigr )\\\\&=&\\Psi _t(z)+d_{r_t}\\frac{w\\theta }{1+w}\\frac{|z|}{r_t} \\bigl (1 +o(1) \\bigr )\\nonumber $ and so the condition $Y_{t,Z_{t+wt}^{{{({1}})}}}\\le -n$ is equivalent to $ \\frac{\\Psi _{t+wt}(Z_{t+wt}^{{{({1}})}})-a_{r_t}}{d_{r_t}}-\\frac{w\\theta }{1+w}\\frac{|Z_{t+wt}^{{{({1}})}}|}{r_t}\\bigl (1+o(1) \\bigr )\\le -n.$ It is easy to see that $r_{t+wt}\\sim (1+w)r_t$ .", "This implies that $d_{r_{t+wt}}\\sim d_{r_t}$ and $a_{r_{t+wt}}-a_{r_t}\\sim d_{r_t}\\gamma ^{-1}d\\log (1+w).$ Now condition (REF ) is equivalent to $\\biggl [\\frac{\\Psi _{t+wt}(Z_{t+wt}^{{{({1}})}})-a_{r_{t+wt}}}{d_{r_{t+wt}}} +\\gamma ^{-1}d\\log (1+w)-w\\theta \\frac{|Z_{t+wt}^{{{({1}})}}|}{r_{t+wt}} \\biggr ] \\bigl (1+o(1) \\bigr )\\le -n$ and by Proposition REF we obtain $ &&\\lim _{n\\rightarrow \\infty } \\lim _{t\\rightarrow \\infty }\\operatorname{Prob}\\lbrace Y_{t,Z_{t+wt}^{{{({1}})}}}\\le -n \\rbrace \\nonumber \\\\&&\\qquad =\\lim _{n\\rightarrow \\infty }\\lim _{t\\rightarrow \\infty }\\operatorname{Prob}\\biggl \\lbrace \\biggl [Y_{t+wt,Z_{t+wt}^{{{({1}})}}}+\\gamma ^{-1}d\\log (1+w) -w\\theta \\frac{|Z_{t+wt}^{{{({1}})}}|}{r_{t+wt}} \\biggr ]\\nonumber \\\\[-8pt]\\\\[-8pt]&&\\phantom{\\hspace*{230.0pt}} {}\\times \\bigl (1+o(1) \\bigr ) \\le -n \\biggr \\rbrace \\nonumber \\\\&&\\qquad =\\lim _{n\\rightarrow \\infty }\\operatorname{Prob}\\bigl \\lbrace Y^{{{({1}})}}+\\gamma ^{-1}d\\log (1+w)-w\\theta \\bigl |X^{{{({1}})}}\\bigr |\\le -n \\bigr \\rbrace =0.\\nonumber $ Combining the bounds (REF ) and (REF ) with the convergence results (REF ) and (REF ), we obtain the required statement.", "Now we show that the probabilities of the events $\\mathcal {A}(n,w,t)$ converge to a finite explicit integral.", "Lemma 6.3 For any $w\\ge 0$ , $\\lim _{n\\rightarrow \\infty }\\lim _{t\\rightarrow \\infty }\\operatorname{Prob}\\bigl \\lbrace \\mathcal {A}(n,w,t) \\bigr \\rbrace =\\int _{\\mathbb {R}^d\\times \\mathbb {R}}\\exp \\bigl \\lbrace -\\nu \\bigl (D_{w}(x,y) \\bigr ) \\bigr \\rbrace \\nu (dx,dy)<\\infty ,$ where $D_w(x,y)$ has been defined in (REF ).", "We have $&&\\operatorname{Prob}\\bigl \\lbrace \\mathcal {A}(n,w,t) \\bigr \\rbrace \\\\&&\\qquad =\\int _{\\mathbb {R}^d\\times [-n,\\infty )} \\operatorname{Prob}\\bigl \\lbrace \\bigl (Z_t^{{{({1}})}}r_t^{-1},Y_{t,Z_t^{{{({1}})}}}\\bigr )\\in dx\\times dy,\\\\&&\\phantom{\\qquad =\\int _{\\mathbb {R}^d\\times [-n,\\infty )} \\operatorname{Prob}\\bigl \\lbrace }\\Psi _{t+wt}(z)\\le \\Psi _{t+wt}\\bigl (Z_t^{{{({1}})}}\\bigr )\\ \\forall z\\in \\mathbb {Z}^d \\mbox{ s.t.", "}Y_{t,z}\\ge -n \\bigr \\rbrace .$ Observe that according to (REF ) the condition $\\Psi _{t+wt}(z)\\le \\Psi _{t+wt}(Z_t^{{{({1}})}})$ is equivalent to $\\Psi _t(z)+d_{r_t}\\frac{w\\theta }{1+w}\\frac{|z|}{r_t}\\bigl (1 +o(1) \\bigr ) \\le \\Psi _t\\bigl (Z_t^{{{({1}})}}\\bigr )+d_{r_t} \\frac{w\\theta }{1+w}\\frac{|Z_t^{{{({1}})}}|}{r_t} \\bigl (1 +o(1)\\bigr ),$ that is, to $Y_{t,z}+\\frac{w\\theta }{1+w}\\frac{|z|}{r_t} \\bigl (1 +o(1) \\bigr ) \\le Y_{t,Z_t^{{{({1}})}}}+\\frac{w\\theta }{1+w}\\frac{|Z_t^{{{({1}})}}|}{r_t} \\bigl (1 +o(1) \\bigr ).$ Consider the point process $\\Pi _t$ on $\\hat{H}_{-n}^{-\\alpha }$ , where $\\alpha \\in (\\theta \\frac{w}{1+w}, \\theta )$ .", "The requirement $\\bigl \\lbrace \\bigl (Z_t^{{{({1}})}}r_t^{-1},Y_{t,Z_t^{{{({1}})}}}\\bigr )\\in dx\\times dy, \\Psi _{t+wt}(z)\\le \\Psi _{t+wt}\\bigl (Z_t^{{{({1}})}}\\bigr )\\ \\forall z\\in \\mathbb {Z}^d \\mbox{s.t.", "}Y_{t,z}\\ge -n \\bigr \\rbrace $ means that $\\Pi _t$ has one point in $dx\\times dy$ and no points in the domain $D_{n, w,t}(x,y)&=& \\bigl (\\mathbb {R}^d\\times [y,\\infty ) \\bigr )\\\\&&{}\\cup \\biggl \\lbrace (\\bar{x},\\bar{y})\\in \\mathbb {R}^d\\times [-n,\\infty )\\\\&&\\phantom{{}\\cup \\biggl \\lbrace }y+\\frac{w\\theta |x|}{1+w} \\bigl (1 +o(1) \\bigr )\\le \\bar{y}+\\frac{w\\theta |\\bar{x}|}{1+w} \\bigl (1 +o(1)\\bigr ) \\biggr \\rbrace .$ Hence, by Lemma REF , $&&\\lim _{t\\rightarrow \\infty }\\operatorname{Prob}\\bigl \\lbrace \\mathcal {A}(n,w,t) \\bigr \\rbrace \\\\&&\\qquad =\\lim _{t\\rightarrow \\infty }\\int _{\\mathbb {R}^d\\times [-n,\\infty )}\\operatorname{Prob}\\bigl \\lbrace \\Pi _t(dx\\times dy)=1, \\Pi _t\\bigl (D_{n, w,t}(x,y) \\bigr )=0 \\bigr \\rbrace \\\\&&\\qquad = \\int _{\\mathbb {R}^d\\times [-n,\\infty )}\\operatorname{Prob}\\bigl \\lbrace \\Pi (dx\\times dy)=1,\\Pi \\bigl (D_{n, w}(x,y) \\bigr )=0 \\bigr \\rbrace \\\\&&\\qquad =\\int _{\\mathbb {R}^d\\times [-n,\\infty )}\\exp \\bigl \\lbrace -\\nu \\bigl (D_{n, w}(x,y)\\bigr ) \\bigr \\rbrace \\nu (dx,dy),$ where $D_{n, w}(x,y)=D_w(x,y)\\cap \\bigl (\\mathbb {R}^d\\times [-n,\\infty ) \\bigr ).$ Taking the limit in this way is justified as $\\hat{H}_{-n}^{-\\alpha }$ is compact and contains $\\mathbb {R}^d\\times [-n,\\infty )$ .", "It remains to show that $ &&\\lim _{n\\rightarrow \\infty }\\int _{\\mathbb {R}^d\\times [-n,\\infty )}\\exp \\bigl \\lbrace -\\nu \\bigl (D_{n,w}(x,y) \\bigr ) \\bigr \\rbrace \\nu (dx,dy)\\nonumber \\\\[-8pt]\\\\[-8pt]&&\\qquad = \\int _{\\mathbb {R}^d\\times \\mathbb {R}}\\exp \\bigl \\lbrace -\\nu \\bigl (D_{w}(x,y)\\bigr ) \\bigr \\rbrace \\nu (dx,dy)<\\infty .\\nonumber $ Observe that $\\nu (D_{n,w}(x,y))\\ge \\nu (\\mathbb {R}^d\\times (y,\\infty ))$ for all $x\\in \\mathbb {R}^d$ and $y\\ge -n$ .", "Then ${1}_{\\mathbb {R}^d\\times [-n,\\infty )}(x,y)\\exp \\bigl \\lbrace -\\nu \\bigl (D_{n,w}(x,y) \\bigr )\\bigr \\rbrace \\le \\exp \\bigl \\lbrace -\\nu \\bigl (\\mathbb {R}^d\\times (y,\\infty ) \\bigr ) \\bigr \\rbrace .$ It is easy to see that $\\exp \\lbrace -\\nu (\\mathbb {R}^d\\times (y,\\infty ))\\rbrace $ is integrable with respect to the measure $\\nu $ on $\\mathbb {R}^d\\times \\mathbb {R}$ since using () and the substitution $u=e^{-\\gamma y}$ we get $ &&\\int _{\\mathbb {R}^d\\times \\mathbb {R}} \\exp \\bigl \\lbrace -\\nu \\bigl (\\mathbb {R}^d\\times (y,\\infty ) \\bigr ) \\bigr \\rbrace \\nu (dx,dy)\\nonumber \\\\&&\\qquad =\\int _{-\\infty }^{\\infty }\\int _{\\mathbb {R}^d}\\gamma \\exp \\bigl \\lbrace -\\gamma y -\\gamma \\theta |x|-2^d(\\gamma \\theta )^{-d}e^{-\\gamma y} \\bigr \\rbrace \\,dx \\,dy\\nonumber \\\\[-8pt]\\\\[-8pt]&&\\qquad =2^d(\\gamma \\theta )^{-d}\\int _{-\\infty }^{\\infty }\\gamma \\exp \\bigl \\lbrace -\\gamma y -2^d(\\gamma \\theta )^{-d}e^{-\\gamma y}\\bigr \\rbrace \\,dy\\nonumber \\\\&&\\qquad =2^d(\\gamma \\theta )^{-d}\\int _{0}^{\\infty }\\exp \\bigl \\lbrace -2^d(\\gamma \\theta )^{-d}u \\bigr \\rbrace \\,du=1.\\nonumber $ Now (REF ) follows from the dominated convergence theorem.", "Finally, we combine all results of this section to prove ageing.", "Proof of Theorem REF For any $w>0$ , we have by Lemmas REF , REF and REF , $F(w)&:=&\\lim _{t\\rightarrow \\infty }\\operatorname{Prob}\\lbrace T_t/t\\le w \\rbrace =1-\\lim _{t\\rightarrow \\infty }\\operatorname{Prob}\\bigl \\lbrace Z_t^{{{({1}})}}=Z_{t+wt}^{{{({1}})}}\\bigr \\rbrace \\\\&=&1-\\lim _{n\\rightarrow \\infty }\\lim _{t\\rightarrow \\infty }\\operatorname{Prob}\\bigl \\lbrace \\mathcal {A}(n,w,t) \\bigr \\rbrace \\\\&=&1-\\int _{\\mathbb {R}^d\\times \\mathbb {R}}\\exp \\bigl \\lbrace -\\nu \\bigl (D_{w}(x,y)\\bigr ) \\bigr \\rbrace \\nu (dx,dy).$ Observe that $\\exp \\lbrace -\\nu (D_{w}(x,y))\\rbrace \\le \\exp \\lbrace -\\nu (\\mathbb {R}^d\\times (y,\\infty ))\\rbrace $ which is integrable with respect to the measure $\\nu $ by (REF ).", "Since $\\nu (D_{w}(x,y))\\rightarrow \\nu (D_{w_0}(x,y))$ whenever $w\\rightarrow w_0\\in (0,\\infty )$ the function $F$ is continuous.", "If $w\\rightarrow 0{+}$ then $\\nu (D_{w}(x,y)\\rightarrow \\nu (\\mathbb {R}^d\\times (y,\\infty ))$ and by (REF ) we obtain $\\lim _{w\\rightarrow 0{+}}F(w)=1-\\int _{\\mathbb {R}^d\\times \\mathbb {R}}\\exp \\bigl \\lbrace -\\nu \\bigl (\\mathbb {R}^d\\times (y,\\infty ) \\bigr ) \\bigr \\rbrace \\nu (dx,dy)=0.$ Finally, if $w\\rightarrow \\infty $ then $\\nu (D_{w}(x,y))\\rightarrow \\nu (D_{\\infty }(x,y))$ , where $D_{\\infty }(x,y)= \\bigl \\lbrace (\\bar{x},\\bar{y})\\in \\mathbb {R}^d\\times \\mathbb {R}y+\\theta |x|\\le \\bar{y}+\\theta |\\bar{x}| \\bigr \\rbrace \\cup \\bigl (\\mathbb {R}^d\\times [y,\\infty ) \\bigr ).$ Compute $\\nu \\bigl (D_{\\infty }(x,y) \\bigr ) &\\ge &\\int _{|\\bar{x}|> |x|}\\int _{y+\\theta |x|-\\theta |\\bar{x}|}^{\\infty }\\gamma \\exp \\bigl \\lbrace -\\gamma \\bar{y} -\\gamma \\theta |\\bar{x}| \\bigr \\rbrace \\,d\\bar{y}\\,d\\bar{x}\\\\&=&\\exp \\bigl \\lbrace -\\gamma y-\\gamma \\theta |x| \\bigr \\rbrace \\int _{|\\bar{x}|>|x|}d\\bar{x}=\\infty .$ Hence, $F(w)\\rightarrow 1$ as $w\\rightarrow \\infty $ ." ] ]

[ [ "A formal definition and a new security mechanism of physical unclonable\n functions" ], [ "Abstract The characteristic novelty of what is generally meant by a \"physical unclonable function\" (PUF) is precisely defined, in order to supply a firm basis for security evaluations and the proposal of new security mechanisms.", "A PUF is defined as a hardware device which implements a physical function with an output value that changes with its argument.", "A PUF can be clonable, but a secure PUF must be unclonable.", "This proposed meaning of a PUF is cleanly delineated from the closely related concepts of \"conventional unclonable function\", \"physically obfuscated key\", \"random-number generator\", \"controlled PUF\" and \"strong PUF\".", "The structure of a systematic security evaluation of a PUF enabled by the proposed formal definition is outlined.", "Practically all current and novel physical (but not conventional) unclonable physical functions are PUFs by our definition.", "Thereby the proposed definition captures the existing intuition about what is a PUF and remains flexible enough to encompass further research.", "In a second part we quantitatively characterize two classes of PUF security mechanisms, the standard one, based on a minimum secret read-out time, and a novel one, based on challenge-dependent erasure of stored information.", "The new mechanism is shown to allow in principle the construction of a \"quantum-PUF\", that is absolutely secure while not requiring the storage of an exponentially large secret.", "The construction of a PUF that is mathematically and physically unclonable in principle does not contradict the laws of physics." ], [ "Aims and outline of this work", "“Physical unclonable functions” (PUFs) are electronic hardware devices that are hard to reproduce and can be uniquely identified [14], [8].", "They promise to enable qualitatively novel security mechanisms (see e.g.", "[2], [9], [10]) and have consequently become a “hot topic” in hardware security[5].", "The present work asks the question “What characteristics exactly define the qualitative novelty of the PUF concept?”.", "We hope that a precise answer will aid the security evaluation of existing PUFs and help to develop new ideas for PUF security mechanisms.", "We searched for a formal definition of the properties that are required from a hardware device to be called “PUF”, and a a formal definition of the criteria that have to be fulfilled to consider a PUF “unclonable”.", "The formal PUF definition should not suffer from weaknesses of previous definitions (see section REF ), encompass at least the large majority of the existing PUF constructions, and be as flexible as possible, i.e.", "does not restrict further progress in PUF development (e.g.", "by demanding constructional details, like the amount of stored information).", "This aim is achieved in section REF .", "After formulating a simple definition of PUF-security (based on Armknecht et al.", "[1]) in section REF we delineate PUFs from some closely related security concepts (section ) and outline the elements of a PUF-security evaluation (section ).", "In a second part of the paper we systematically analyse and classify PUF security mechanisms and calculate their quantitative security levels against attacks that attempt mathematical cloning (section ).", "The aims of this section are to give a quantitative answer to Maes & Verbauwhede's[13] question whether mathematically-unclonable PUFs are possible in principle, and to apply and thereby illustrate the PUF-definitions of the first part of the paper.", "In section we characterise the qualitative novelty of PUFs as a new primitive of physical cryptography and discuss the future use and development of PUFs." ], [ "Previous work on the definition of a PUF", "There have already been several proposal for the first definition of required PUF properties.", "Gassend et al.", "[8] who invented the term “PUF” (earlier work by Pappu was on the slightly different concept of a physical one-way function[14]) demand that the function must be “easy to evaluate”, i.e.", "it must efficiently yield a response value “R” for a challenge argument “C”.", "and “hard to predict (characterize)”.", "The latter property means that an attacker who has obtained a polynomial number of C – R pairs (CRPs) but has no longer physical access to the PUF can only extract a negligible amount of information about the R for a random C. Rührmair et al.", "[15] criticised this definition because the information content of finite physical objects is always polynomially bound, and therefore no PUF fulfilling this definition can exist.", "They propose an alternative formal definition in which the PUF must only be hard to predict for an attacker “who may execute any physical operation allowed by the current stage of technology”.", "Maes & Verbauwhede[13] chose to exclude unpredictability from their “least common property subset” of PUFs, because they put into question whether it is possible in principle to construct a mathematically unclonable PUF.", "They demand that a PUF is “easy to evaluate” (property “evaluatable”) and that it is “reproducible”, meaning that a C always leads to the same R within a small error.", "Moreover they demand “physical unclonability” i.e.", "that it must be “hard” for an attacker to construct a device that reproduces the behaviour of the PUF.", "However, PUFs that are mathematically clonable are also physically clonable because the mathematical algorithm for PF can then be implemented on a device that is then a functional physical clone of the PUF.", "Summarizing, a first generation of definitions roughly defined PUFs to be devices that are efficiently evaluatable and are mathematically and physically unclonable.", "They remain unsatisfactory for two reasons: Most of the devices currently called PUFs do not fulfill these definitions (according to Rührmair et al.", "[15] there are only some “candidates”), i.e.", "the definition does evidently not really capture the PUF concept.", "They combine the definition of a PUF with the definition of its security, i.e.", "points 1. and 2. above.", "A PUF is defined by its unclonability i.e.", "its security against attacks.", "This is problematic because an open-ended security analysis of a PUF clearly must have an “insecure PUF” as one a priori possible outcome.", "Based on the above definitions an “insecure PUF” is a paradox, PUFs would be secure by definition.", "These two problems were elegantly solved in a seminal paper by Armknecht et al.", "[1] who propose to formalize a PUF as “physical function (“PF”) - which is a physical device that maps bit-string-challenges “C” to bit-string-responses “R”.", "The unclonability is recognized by Armknecht et al.", "as only one crucial security property, that they further formally define in great detail.", "We will supply a simplified version of their general security definition in section REF below.", "Following Armknecht et al., the PUF definition 1. consists in an answer to the question: What are the required characteristics of PF() in order to be a PUF?", "Armknecht et al.", "do not demand any specific mathematical properties but only that a PF is a “probabilistic procedure” that maps a set of challenges to a set of responses and that internally PF is a combination of a physical component and an evaluation procedure that creates a response.", "Armknecht et al.", "explain that the responses rely heavily on the properties of the physical component but also on uncontrollable random noise (hence “probabilistic”).", "This definition of PF() still faces the following problem: Consider a standard authentication chip with a stored secret in a physically protected memory that calculates a response from the challenge and the secret.", "Such a chip must contain a “physical component” (the memory) and an evaluation procedure (its read-out) that fulfills Armknecht et al.", "'s definition because some (very small) uncontrollable random noise is unavoidable even in standard computer memories.", "There is also no reason why a well designed standard authentication chip cannot posess Armknecht et al.", "'s security properties.", "Therefore, even though Armknecht et al.", "'s definitions constitute great progress of lasting value, they still do not capture the distinctive properties of the PUF concept.", "In practice Armknecht et al.", "define all devices that run any challenge-response protocol as PUFs." ], [ "Formal definition of “PUF”", "In the following we assume Armknecht et al.", "'s model of a PUF as physical function PF() (see section REF ).", "We break up the physical function PF() into three steps (see fig.", "(REF )).", "C,S$_r$ ,S and R are bit strings.", "In the first “physical read-out” step PF$_1$ = S$_r$ , internal information S$_r$ (the “raw secret”) is physically read-out from the PUF in response to a challenge C foreseen by the system architecture.", "In an optional second step PF$_2$ (S$_r$ ) = S error correction and/or privacy amplification are performed, such that errors in the read-out are corrected and parts of S$_r$ which may be known by the attacker (e.g.", "by guessing parts of the challenge) are removed by privacy amplification algorithms.", "In an optional third step PF$_3$ (S) = R, some additional algorithm is performed with S as input to calculate the final response R. Typically PF$_3$ is some cryptographic protocol that proves the possession of S without disclosing it.", "In many existing PUF architectures the challenge C is an address of information inside the PUF which is output as the response R. E.g.", "in arbiter PUFs[11] C defines the choice of a set of delay switches whose cumulative delay path defines S (and from this R).", "Our idea is that the possibility for this mode of addressing, rather than its “unclonability”, defines a PUF.", "The challenge C can then be understood as a key required for physical access to the response R. R remains secret without access to C. Figure: Symbolic model of a PUF The box delineates the PUFthat receives a challenge C (shown with anexample bit string) and sends a response R that is determined inthree distinct steps.", "The first step is the physical readout,the second the correction of errors that can occur in the firststep and the third step includes all operations of mathematicalcryptography.A security architecture based on this idea requires certain properties of PF() which define the PUF concept: Formal Definition 1 of a PUF A hardware device is called “PUF” if: a. a physical function PF$_2$ (PF$_1$ ()) which is deterministic for a specific set of challenges $\\mathfrak {M}$ , can be evaluated with each challenge at least once and b. the value S = PF$_2$ (PF$_1$ (C)) changes with its argument, for all outside challenges C $\\in $ $\\mathfrak {M}$ , i.e.", "PF$_2$ (PF$_1$ (C)) = S is not a constant function.", "One difference to some previous PUF definitions is that PF() is not required to be easily evaluatable.", "An efficient evaluation of S is certainly a desirable design goal, but there is no reason why a device with inefficient read out cannot be a PUF by definition.", "Another difference to most previous definitions is that it allows a PUF to be clonable.", "As an example consider the following physical function that fulfills the above definition 1: PF$_1$ (any C with more 1s than 0s) = 1001101101 PF$_1$ (any C with more 0s than 1s or equal number of 1s and 0s) = 0001101000 Clearly a PUF with this PF$_1$ can be reproduced by a trivial algorithm, i.e.", "it is trivially mathematically clonable.", "This is a desirable property because “clonable PUFs” do exist in the real world and should not present a PUF definition with a paradox.", "“Unclonability” is then a property that is aimed for, rather than achieved by definition.", "Analogously “cryptography” aims for secrecy (crypto) rather than achieving it by definition.", "Even though it is a child's game to break it, the Cesar cipher is a valid cryptographical algorithm according to this definition.", "Consequently cryptographic algorithms are commonly defined to be “key-dependent injective” (rather than “unbreakable”) mappings”[20].", "Where does this leave PUF security?", "It is not possible in principle to extract the secret S from a PUF without knowing of the challenge.", "This is true even for the above insecure PUF.", "However in the example above it is easy to reproduce PF$_1$ , and therefore, as soon as the challenge becomes known, S becomes known.", "Therefore the crucial necessary objective for the security of a PUF is the unclonability of PF$_2$ (PF$_1$ ()).", "In the next section we make this insight more precise.", "A complete and quantitative set of security requirements (i.e.", "with requirements on their length $\\ell $ , the number of independent challenges N etc.)", "can only be made in the context of a concrete PUF architecture.", "One example is discussed further in section REF .", "Even though the response of a PUF can in principle be used for various purposes, we will conclude in section that one central PUF capability is the distribution of remote authentication secrets.", "If S is used for authentication purposes, an attacker must be able to fully predict it, i.e.", "a partial prediction of S=PF$_2$ (PF$_1$ (C)) for a given argument C will not be considered a successful attack in the following.", "Therefore, the natural “basic objective” of PUF security is that the attacker cannot predict a complete, correct bit string S for a given bit string C." ], [ "Attack models", "Security can only be defined relative to an attack model, that lays down the assumptions about the security environment.", "We assume in the following two models from the literature that seem realistic in practice.", "The first one models an attempt to break Armknecht et al.", "'s[1] selective unclonabilityRührmair et al.", "'s[15] PUF definition demanded that the original manufacturer of the PUF cannot produce two PUFs which are clones of each other (Armknecht et al.", "[1] demand this “existential unclonabiliy” only optionally.).", "“Selectively unclonability”[1] means that given physical access to the device an attacker cannot produce a clone.", "In practice existential unclonability would hardly enhance the security against a malicious manufacturer, for the following reason.", "She could produce “quasi-existential-PUF” devices that do not meet the PUF definition 1, but algorithmically simulate - e.g.", "with an keyed hash function - an output that cannot be discriminated from the one of an existential PUF.", "These quasi-existential-PUFs could be easily cloned by the malicious manufacturer, and could serve exactly the same purpose as clonable PUFs.", "As an alternative to existential unclonability we will propose a weaker “resistance-against-insider-attacks” security level in this section REF .", ".", "It does not put any restriction on the attack strategy, therefore adaptive choices of challenges are possibleTherefore strong unpredictability in the sense of Armknecht et al.", "[1] will be necessary to protect the PUF..", "The second one is an attempt to do the same with a certain reasonable amount of insider knowledge.", "Both models assume that the attacker has only access to one single PUF, i.e.", "attacks exploiting correlations between different PFs are excluded by assumption (see Armknecht et al.", "[1] for the general case).", "Attack model 1: “Outsider attack”: The attacker has physical access only to the attacked PUF only for a finite amount of time $\\Delta $ t$_a$ .", "After this access period, she tries to predict a secret S from the PUF to a challenge C, randomly chosen from the set of all challenges.", "She has no knowledge of the set of challenges and secrets that will be used during the active lifetime of the PUF or any further previous knowledge of the PUF.", "Attack model 2: “Insider attack”: The attacker has physical access only to the attacked PUF only for a finite amount of time $\\Delta $ t$_a$ .", "After this access period she tries to predict a secret S from the PUF to a challenge C, randomly chosen from the set of all challenges.", "She has no knowledge of the set of challenges and secrets that will be used during the active lifetime of the PUF but she has all other information that the manufacturer of the PUF has about the attacked individual PUF.", "The attack models assume that the attacker tries to predict S rather than R, because PF$_3$ might be protected with non-PUF security mechanisms, e.g.", "with a secure tamper-resistance scheme in combination with a secure crypto algorithm.", "Such a security mechanism shall remain out of our consideration because we aim to define the security of the PUF itself.", "Security against a model-2 attacker corresponds to unclonability against an attacker who has most of the inside knowledge about the PUF production, but who cannot directly manipulate the production process.", "This unclonability is weaker than “existential unclonability” (see footnote 1) but perhaps more relevant in practice." ], [ "Definition of a secure PUF", "The PUF-security definition now follows from the requirement that the attack shall be unsuccessful: Formal definition 2 of the PUF-security objective A PUF is secure against an attack of a model-1 (“selectively unclonable”) attacker if a model-1 attacker can compute or physically copy the function PF$_2$ (PF$_1$ (C)) = S for not more than a negligible fraction L of challenges from the set of all possible challenges.", "Here “compute” means via a computation independent of the PUF and corresponds to “mathematical cloning”.", "“Physically copy” means to create a device that functionally reproduces PF$_2$ (PF$_1$ (C)) and corresponds to “physical cloning”.", "Replacing the model-1 by a model-2 attacker defines a PUF that is “insider selectively unclonable”.", "L is the security level of a secure PUF, i.e.", "the probability for an attacker to successfully predict the secret S for a challenge C without being in posession of the PUF after the access period.", "A precise quantification of “negligible”, i.e.", "the decision which upper limit of L is required, cannot be made on the level of this general definition because it depends on the detailed security environment.", "L is analogous to the required probability p of a successful brute force attack in classical cryptography that depends on the key length.", "We propose as a reasonable upper limit on L that it is “negligible on a terrestrial scale” which has been estimated by Emile Borel as $<$ 10$^{-15}$[4].", "In this section we delineate the concept of a PUF as defined in section REF and REF from five closely related concepts." ], [ "PUFs and conventional unclonable functions (“CUFs”) are qualitatively different", "Let us first differentiate between a PUF and a conventional physical function that serves the same function as a PUF (called “conventional unclonable function” CUF in the following).", "A CUF contains secret information that is protected by tamper resistance, by anti side-channel- and fault-induction-attack measures and by a cryptographic algorithm that protects the secret from disclosure via the response.", "A CUF does not fulfill the PUF definition 1., because the secret does not depend on the challenge.", "In other words: The first physical secret readout step PF$_1$ (C) is a constant function in a CUF.", "PUF and CUF differ qualitatively in the way they protect the secret.", "In a PUF the lack of knowledge of the challenges protects the secret S in a similar sense that the lack of knowledge of a cryptographical key protects the clear text in a cipher text.", "There is no analogous “key” in a CUF.", "Its security mechanisms merely rely on physical barriers and arrangements that prevent access to secret information." ], [ "PUFs and physically obfuscated keys are independent concepts", "Devices that extract physical information with “non-standard” methods are currently called PUF even if there is no (or effectively a single fixed internal) challenge (e.g.", "in SRAM PUFs[10]).", "In this case PF$_1$ () is formally constant, so that such devices are no PUFs in the sense of our definition 1.", "We endorse Rührmair et al.", "'s suggestion[15] to call information extracted in this way in general “physically obfuscated keys” (POKs).", "This limit of N=1 is the only one where devices that are currently called PUFs, would no longer be classified as PUF under our proposed definition.", "We find this appropriate because while POKs can enable valuable tamper-resistance mechanisms (see below), they are not the qualitatively novel primitive of physical cryptography that PUFs promise to be (see section for further discussion of the nature of this primitive).", "The protection by obfuscation is valuable: it consists in the extra-time an attacker needs to learn the non-standard readout mechanism or position in a standard memory where an obfuscated key has to be stored at least temporarily.", "POKs are delineated from CUF only by the “non-standard” qualifier because stored information is always physical[12].", "The secrets of PUFs will usually be stored in a non-standard way, i.e.", "they will also be POKs.", "But this is no necessary requirement for a PUF.", "There is no fundamental reason why PUFs cannot have “standard” computer memories (see e.g.", "SHICs[17], a PUF using a standard crossbar memory).", "Physically obfuscated functions (POFs) may also appear in PUF architectures.", "They are defined as computation with non-standard physical processes, e.g.", "via scattering of light or folding of proteins." ], [ "Random number generators", "In both deterministic and physical random number generators the initial read-out step PF$_1$ (the read out of the seed) does not depend on a challenge C. In secure deterministic RNGs PF$_1$ (C) must be a constant function.", "In physical RNGs PF$_1$ is not constant but intrinsically random, i.e.", "not deterministic.", "Therefore, RNGs do not meet the PUF definition 1." ], [ "Controlled PUFs: a PUF with additional tamper\nresistance", "In controlled PUFs[7], [9] tamper-resistance measures prevent the attacker from obtaining C – S$_r$ pairs from the PUF.", "Only the C – R pairs - from which S$_r$ cannot be derived if PF$_3$ is a suitable, secure cryptographical algorithm - can be accessed by an attacker.", "It seems likely that PUFs e.g.", "used in smart cards will eventually all be controlled PUFs, because such an additional well understood security layer stands to reason.", "However the security of PUFs themselves should be analysed under the assumption of no such a control because if one trusts the control mechanism, mathematically clonable PUFs suffice anyway." ], [ "“Strong PUFs”: not the only path to strength", "Rührmair et al.", "[15] defined a PUF to be “strong” if it “has so many C – R pairs ... that an attack ... based on exhaustively measuring the C – R pairs has a negligible probability of success”.", "In our nomenclature a strong PUF is roughly a MRT-PUF that fulfills our second security requirement (see section REF below, for further explanation of MRT).", "It is thus appropriate to call them ”strong“, but there can be secure PUFs which are not “strong” in Rührmair et al.", "'s sense, e.g.", "EUR-PUFs(see section REF below for further explanation of EUR)." ], [ "Security evaluation of PUFs", "A main purpose of the present proposed formal PUF definitions 1. and 2. of the concept “secure PUF” is to establish a consistent basis for security evaluations and certifications of PUFs.", "What is the structure of an evaluation on this basis?", "If the proposed PUF fulfills definition 1, the basic informal questions of a security evaluation based on definition 2 are: Which form has PF$_1$ (C) and by which physical mechanism is S$_r$ extracted?", "What is the form of PF$_2$ (S$_r$ )=S and how is the function evaluated physically?", "What is the total information content in the set of all secrets S?", "For what fraction L of the allowed challenges can PF$_2$ (PF$_1$ (C)) be either mathematically computed or physically copied?", "Which comprehensible physical security mechanisms prevent an attacker to compute or copy PF$_2$ (PF$_1$ (C)) for more than a fraction L of all challenges?", "Answers to questions 1.", "- 4. allow to evaluate quantitative and comprehensible security levels against “mathematical-cloning brute force attacks” (see section ).", "Question 5 will have a more qualitative answer, similar to answers to the question whether a mathematical cryptographic algorithm is secure against non-brute force attacks." ], [ "Analysis of PUF security mechanisms", "The holy grail of PUF construction is to construct PUFs that are unclonable i.e.", "fulfill the security definition 2 (section REF ).", "If an attacker succeeds to access the PUF's internal secrets, she will usually be able to compute PF$_2$ .", "Because physical reproduction of a PUF without knowledge of its internal secrets will probably be hard in practiceBut not necessarily impossible.", "She could e.g.", "succeed to reproduce to clone a PUF exactly copying its production process., PUF security mechanisms must above all prevent the attacker from computing PF$_2$ .", "In other words: mathematical unclonability is the hardest nut.", "Therefore we will classify the known PUF security mechanism and calculate their security level against brute-force mathematical cloning attacks.", "Up to now all proposed and constructed PUFsIn the sense of this paper, i.e.", "excluding POKs.", "are based on a mechanism that we propose to call “minimum readout time”(MRT) and that is further discussed in subsection REF .", "All these existing PUFs turn out to fulfill our PUF-definition 1, i.e.", "they “remain” PUFs, even in case they have turned out to be clonable (see below).", "Because currently the MRT mechanism dominates the field, one might be tempted to equate the very concept of PUFs with it.", "However, the flexibility of our definition allows a completely different PUF security mechanism that we call “erasure upon read-out”(EUR) (see section REF ) for devices.", "One concrete EUR PUF, the quantum PUF will be introduced below.", "These examples show that our proposed definitions have achieved their aims: nearly all existing (MRT) PUFs can be included in its scope, but its flexibility allows to include completely novel PUF constructions (the EUR PUFs)." ], [ "“Minimum readout time” PUFs", "This well known PUF security mechanism is to store a large enough number N of C – S pairs on the PUF so that the total time $\\Delta t_t = \\Delta t_r \\times N$ to read them all out is much longer than the time $\\Delta $ t$_a$ during which an attacker possesses the PUF.", "$\\Delta t_r$ is the read-out time for one C – S pair.", "The maximal fraction of pairs the attacker can read-out is then $\\Delta $ t$_a$ /$\\Delta t_t$ = L$_{bf}$ .", "L$_{bf}$ is the security level against mathematical-cloning brute force attacks.", "Pappu's optical PUF[14], the arbiter PUF[8] and nearly all other current PUFs are MRT-PUFsThe only exception are “PUFs” with only one challenge which we propose to call only “POKs” in the future, see section REF ..", "These constructions are valid PUFs according to our definition because their values of PF$_2$ changes with the challenge.", "However, many of the existing PUFs are insecure according to our definition because Rührmair et al.", "[16] succeeded to employ machine-learning methods that allow to infer PF$_2$ (PF$_1$ ()) from a small fraction of all C – R for which only short $\\Delta $ t$_a$ is necessary[16].", "Because all C – S pairs can be thus predicted, the security level against machine-learning attacks L$_{ml}$ = 1 which is “not negligible” in general, i.e.", "the PUF must be considered mathematically clonable according to PUF-security definition 2.", "The exact form of PF() depends on the detailed architecture of the MRT PUF.", "In general MRT PUFs can be hardened against mathematical cloning if their PF$_2$ (PF$_1$ ) fulfills the following demands: Security requirements for the MRT-PUF N must satisfy: $N \\ge L^{-1} (\\Delta t_a/\\Delta t_r $ ) Suppose PF$_2$ (PF$_1$ (C$_n$ )) = S$_{n}$ with n = 1...N where both C$_n$ and S$_{n}$ contain $\\ell $ bits.", "Then the combined information content (entropy) I of all C$_n$ and S$_{n}$ must satisfy: I $\\ge $ 2 N $\\ell $ The set of challenges to be used in operation must not be contained in any form in the PUF.", "The lengths of the challenge $\\ell $ and response $\\ell _S$ must both fulfill: $\\ell $ ,$\\ell _S$ $\\ge $ log$_2$ (N).", "The first condition expresses that to prevent brute force mathematical-cloning attacks the number of stored C – R pairs N must be extremely large if L = 10$^{-15}$ , (see section REF on the choice of L).", "With representative values of $\\Delta t_a$ = 1 day and $\\Delta t_r$ = 1 second the required N would be on the order of 10$^{20}$ which is exponentially larger than e.g.", "storable in common data storage devices of much larger size than a typical PUF.", "This is the sense in which a secure MRT-PUF requires the storage of an “exponentially large” secret.", "The second condition expresses that in order to reliably ward successful machine-learning attacks PF$_2$ must be just an ordered list of C – S pairs with random values that cannot be represented in any more compact form.", "The third requirement prevents an attack in which only the set of challenges to be used in the field operation of a PUF (which is much smaller than $\\mathfrak {M}$ in secure MRT PUFs) are extracted in an attack.", "The fourth constraint is necessary to avoid a decrease in the the effective L." ], [ "“Erasure Upon Readout” PUFs – Quantum PUFs", "Consider a PUF with only a single C – S pair foreseen by the system architecture.", "Because there is at least one other non-foreseen C, there are then at least two possible C. A novel PUF security mechanism requires the following: Security requirements for “Erasure Upon Readout” (EUR) PUF The correct S is returned if the challenge C is correct (i.e.", "the one foreseen by the PUF's architecture) and S is erased and returns a random value if it is not.", "The length of the challenge $\\ell $ and response $\\ell _S$ must both fulfill $\\ell $ ,$\\ell _S$ $\\ge $ log$_2$ (1/L).", "The set of challenges to be used in operation must not be contained in any form in the PUF.", "EUR PUFs can fulfill the PUF-definition 1 if they are non-constant PFs that are deterministic for the foreseen set of challenges.", "For EUR PUFs - completely opposite to the MRT case (see section REF ) - the total number of challenges “N” can remain as small as 2 but still be secure because by way of the second and third security requirement the probability to guess the correct challenge is only L and challenging with the wrong challenge will erase S by the first requirement.", "N can be chosen to as many different challenges as are actually needed during the practical deployment of the PUFs.", "The only concrete “Erasure Upon Readout” architecture proposed up to now, is Wiesner's “quantum money” and “quantum unforgeable subway token”[21], [3] that can be described as an electronic hardware device running a challenge - response protocol (such a kind of “money” or “token” has to be) and that fulfill our definition 1 of a PUF.", "In such a “quantum-PUF” the secret information consists of $\\ell $ quantum-mechanical two-state systems (“qubits”) that are prepared either in one of the two quantum mechanical so called “Fock” states $|0\\rangle $ or $|1\\rangle $ (base $\\#$ 0) or in either one of the states $1 \\over \\sqrt{2}$ ($|0\\rangle $ + $|1\\rangle $ ) or $1 \\over \\sqrt{2}$ ($|0\\rangle $ - $|1\\rangle $ ) (base $\\#$ 1).", "$|0\\rangle $ and $1 \\over \\sqrt{2}$ ($|0\\rangle $ + $|1\\rangle $ ) encode a “0” secret bit and $|1\\rangle $ and $1 \\over \\sqrt{2}$ ($|0\\rangle $ - $|1\\rangle $ ) encode a “1” secret bit.", "The challenge bits indicate the correct chosen measurement bases.", "The raw secret S$_r$ is encoded with the choice of the state within a chosen basis according to the rule stated above.", "In order to decode or copy S$_r$ , it is necessary to know in which of the two bases $\\#$ 0 or $\\#$ 1 the $\\ell $ qubits for one challenge were prepared.", "If a qubit is read out in a wrong base, the laws of quantum mechanics determine that the read-out result is a perfect random number and additional read out attempts will again yield this random number, rather than the original, correct number.", "The physical function PF of the quantum-PUF is thus given as: Quantum-EUR PF$_1$ (): First read-out: PF$_1$ (correct base bit) = correct bit of S$_r$ PF$_1$ (incorrect base bit) = random bit.", "Any further read-out in the same base: PF$_1$ () = same bit as in first read-out Evidently in the first read-out PF$_1$ is not constant and deterministic for the foreseen C i.e.", "a quantum-PUF fulfills definition 1.", "Reading out a C – S pair more than once is possible, but after the first read-out, the information is no longer secure because the qubits are no longer in a quantum-mechanical superposition of states.", "In the most simple case without any read-out errors or inefficiencies (so that no further processing is done PF$_2$ (S$_r$ )= S$_r$ ) and implementation mistakes (an assumption that will be difficult to fulfill [18]) the only potentially successful attack is to guess the challenge.", "On average, for half of the bits the guess will be correct and the correct corresponding bits of S$_r$ will be output.", "For the other half the probability to get the correct output bit is 1/2.", "The total probability to get a correct output bit of S$_r$ is therefore 0.75 and L$_a$ = $({3 \\over 4})^{-\\ell }$ , which is the absolute (i.e.", "not only mathematical-cloning brute force) security level of a quantum-PUF against this attack.", "E.g.", "with a secret S$_r$ consisting of 128 qubits, L $<$ 10$^{-15}$ thus fulfilling the criterion for a secure PUF with Borel's estimate for an upper bound on L (see section REF ).", "Wiesner's quantum money, interpreted as a “quantum PUF”, thus proves that an absolutely unclonable PUF is not in contradiction to the laws of physics.", "The use of quantum-PUFs for authentication is beyond the reach of current technology because qubits are unavoidably read out on very short timescales (presently qubits cannot be isolated for longer than milliseconds[6]) by interactions with their environment.", "As explained above, quantum-PUFs are no longer secure after read-out.", "Quantum cryptography[18] can be described as sending a quantum-PUF in the form of a chain of photons in order to distribute its secret S for use as cryptographic key.", "In the laboratory such a “light-field” PUF remains in the initially prepared coherent state for no longer than about a millisecond." ], [ "Discussion", "The protection of secrets in hardware devices that need to access these secrets in their normal operation - a necessary condition for any authentication procedure - cannot be implemented with methods of mathematical cryptography alone.", "Some physical protection mechanism is needed.", "The conventional tamper resistance mechanisms (employed in CUFs see section REF ) rely on protecting the memory with physical barriers.", "CUFs withstand known, vigorous direct attacks typically for not longer than a few months[19].", "We showed that PUFs are a qualitatively novel alternative.", "The secret is protected by the absence of information from the device of where of where the challenge is stored.", "In CUFs and POKs this information must exist on the device because otherwise the response cannot be evaluated, even if it is protected by direct, physical barriers.", "Thereby PUFs protect the secret by a novel genuine primitive of physical cryptography.", "The possibility of realizing PUFs based on the principles of quantum mechanics demonstrates that in principle the laws of physics allow to construct absolutely secure PUFs.", "This situation motivates more security-related physics research on unclonable quantum-PUF and MRT-PUF, to invent entirely new PUF construction principles.", "The real PUF promise are PUFs that withstand any known, practical attack, period, i.e.", "provide a level of authenticity protection similar to the one provided by mathematical cryptography for confidentiality.", "In the future PUFs will probably authenticate hardware devices.", "If Alice knows the C – S$_r$ pairs of a PUF she gave to Bob (e.g.", "from the designer of the PUF) she can publicly broadcast a challenge and be sure that the correct response S can only be created on Bob's original PUF.", "Therefore effectively PUFs allow the remote distribution of authenticated secret entropy (the S for Bob) via sending the challenges (the C chosen and sent by Alice) over standard channels.", "These entropy could “update” the secrets in conventional unclonable functions.", "In this way existing architectures based on CUFs could be augmented by PUFs without the need for a completely new PUF security architecture.", "Acknowledgements.", "We thank R. Breithaupt, U. Gebhardt, M. Ullmann, C. Wieschebrink and anonymous referees at the TrustED 2011 and PILATES 2012 workshops for helpful discussion and criticism on earlier versions of this manuscript." ] ]

[ [ "Direct Minimization for Ensemble Electronic Structure Calculations" ], [ "Abstract We consider a direct optimization approach for ensemble density functional theory electronic structure calculations.", "The update operator for the electronic orbitals takes the structure of the Stiefel manifold into account and we present an optimization scheme for the occupation numbers that ensures that the constraints remain satisfied.", "We also compare sequential and simultaneous quasi-Newton and nonlinear conjugate gradient optimization procedures, and demonstrate that simultaneous optimization of the electronic orbitals and occupation numbers improve performance compared to the sequential approach." ], [ "Introduction", "Advances in computer power and numerical methods during the past few decades has dramatically increased the scope of electronic structure problems that can be computationally studied.", "Kohn-Sham density functional theory (DFT) methods can be used to reach precision comparable to experimental accuracy for insulators and semiconductors, while metallic systems remain more challenging.", "Metallic systems lack a gap between occupied and unoccupied electronic states in the energy spectrum, which leads to slower convergence compared to insulators and semiconductors.", "Smearing of the Fermi surface is often used to enable convergence of metallic systems as well as insulators at positive temperatures.", "Ensemble DFT permits direct computation of the occupation numbers of the orbitals based on the entropic term in the Helmholtz free energy.", "We consider an optimization problem where the target functional $A$ corresponds to the Helmholtz free energy and the variables $\\mathbf {X}$ and $ to the electronic orbitals and occupation numbers respectively.$ The optimization problem is therefore $\\mathrm {minimize}\\;A(\\mathbf {X},,$ subject to $\\mathbf {X}\\in \\mathcal {M}=\\lbrace \\mathbf {X}\\in \\mathbb {R}^{m\\times n}\\,|\\,\\mathbf {X}^T\\mathbf {X}= \\mathbf {I}\\rbrace .$ Furthermore $\\mathbb {R}^n$ with $\\sum _{i=1}^n f_i = n_e$ and $0 \\le f_i \\le 1$ , where $n_e\\in \\mathbb {N}$ is the number of electrons in the system and $n_e\\le n$ .", "We also assume that $\\nabla _\\mathbf {X}A(\\mathbf {X},$ and $\\nabla _Ȃ(\\mathbf {X},$ are available, but expensive to compute.", "However, due to the form of $A(\\mathbf {X},$ the price to compute $A$ , $\\nabla _\\mathbf {X}A$ , and $\\nabla _Ȃ$ simultaneously is comparable to computing one of them separately.", "Furthermore we assume that $m \\gg n$ , and that $m$ is sufficiently large as to make storage of and operation with full $m\\times m$ matrices prohibitively expensive.", "The orthogonality constraint on $\\mathbf {X}$ means that $\\mathcal {M}\\subset \\mathbb {R}^{m \\times n}$ defines the Stiefel manifold, which has the tangent space $\\mathcal {T}_{\\mathbf {X}}\\mathcal {M} = \\lbrace \\mathbf {Y}= \\mathbf {X}\\mathbf {A}+\\mathbf {Z}\\;|\\;\\mathbf {A}^T=-\\mathbf {A}\\;\\mathrm {and}\\;\\mathbf {Z}^T\\mathbf {X}=\\mathbf {0}\\rbrace ,$ where $\\mathbf {Y}, \\mathbf {Z}\\in \\mathbb {R}^{m\\times n}$ and $\\mathbf {A}\\in \\mathbb {R}^{n\\times n}$ .", "We use the standard inner product $(\\mathbf {X},\\mathbf {Y}) = \\mathrm {trace}(\\mathbf {X}^T\\mathbf {Y}).$ Given an arbitrary matrix $\\mathbf {W}\\in \\mathbb {R}^{m\\times n}$ we can orthogonally project it onto $\\mathcal {T}_{\\mathbf {X}}\\mathcal {M}$ with $\\mathbf {Y}= \\mathbf {P}_\\mathbf {X}(\\mathbf {W}) = (\\mathbf {I}- \\tfrac{1}{2}\\mathbf {X}\\mathbf {X}^T)\\mathbf {W}- \\tfrac{1}{2}\\mathbf {X}\\mathbf {W}^T\\mathbf {X}.$ Minimization approaches to non-temperature dependent DFT do not in general permit fractional occupation of electronic orbitals [15], [25], [26], [24], [28], [4].", "In contrast, explicit minimization with regards to occupation numbers permits fractional occupation based on the entropy functional of the Helmholtz free energy and can improve convergence, especially for metallic systems [19], [6], [13].", "It is also possible to transform Equation (REF ) into a nonlinear eigenvalue problem that can be solved through a self consistent field iteration [23], [15], [18], [24].", "The absence of well separated occupied and unoccupied orbitals make metallic systems challenging to compute, and broadening of the Fermi surface is used to facilitate convergence [27], [5].", "This broadening is often achieved by assigning the orbitals close to the Fermi level a fractional occupation number determined by the energy of the electronic orbital [20], [14], [21].", "Direct minimization on the other hand does not require the orbital energies to be computed at every step, and these broadening schemes are therefore not well suited for minimization methods.", "In [11] a framework for optimization methods on the Stiefel and Grassmann manifolds is presented, while [9] discusses a Newton-like iteration scheme on a more general manifold.", "Univariate optimization methods for the Stiefel manifold is presented in [7], where identity plus rank one Householder transforms are given as one possible choice for moving on the manifold.", "The choice of coordinates can also be based on a QR factorization and polar decompositions [8], [10] or Lie groups [16].", "An overview of geometric numerical integration techniques can be found in [17].", "In Section  we first recall the nonlinear conjugate gradient and the quasi-Newton methods adapted for use on the Stiefel manifold.", "We then present an optimization procedure for the occupation numbers and end the section by presenting a simultaneous orbital-occupation optimization strategies.", "Then, in Section  we numerically demonstrate the method on a model problem that includes nonlinearities similar to a DFT problem.", "The conclusions are finally presented in Section ." ], [ "Update and transport", "We ensure that $\\mathbf {X}_{k+1}$ satisfies the orthogonality constraint by using a unitary update operator $\\mathbf {U}$ which maps $\\mathcal {M}\\rightarrow \\mathcal {M}$ .", "A search direction $\\mathbf {Y}\\in \\mathcal {T}_{\\mathbf {X}}\\mathcal {M}$ given by an optimization procedure can be written $\\mathbf {Y}= \\mathbf {X}\\mathbf {A}+\\mathbf {Q}\\mathbf {R},$ where $\\mathbf {Q}\\in \\mathbb {R}^{m\\times n}$ , $\\mathbf {A},\\mathbf {R}\\in \\mathbb {R}^{n\\times n}$ , $\\mathbf {A}^T=-\\mathbf {A}$ , $\\mathbf {Q}^T\\mathbf {Q}= \\mathbf {I}$ , and $\\mathbf {Q}^T\\mathbf {X}=\\mathbf {0}$ .", "If the terms in Equation (REF ) are not full rank the size of the matrices can be adjusted accordingly.", "If we follow $\\mathbf {Y}$ to update $\\mathbf {X}$ along a Stiefel geodesic we obtain the update operator for $\\mathbf {X}$  [11] $\\mathbf {U}=\\begin{bmatrix}\\mathbf {X}&\\mathbf {Q}\\end{bmatrix}\\exp \\left(\\tau \\begin{bmatrix}\\mathbf {A}& -\\mathbf {R}^T\\\\\\mathbf {R}&\\mathbf {0}\\end{bmatrix}\\right)\\begin{bmatrix}\\mathbf {I}&\\mathbf {0}\\end{bmatrix}^T,$ with step length parameter $\\tau $ .", "The update operator generalized for an arbitrary matrix in span$(\\mathbf {X},\\mathbf {Q})$ is $\\mathbf {U}=\\begin{bmatrix}\\mathbf {X}&\\mathbf {Q}\\end{bmatrix}\\exp \\left(\\tau \\begin{bmatrix}\\mathbf {A}& -\\mathbf {R}^T\\\\\\mathbf {R}&\\mathbf {0}\\end{bmatrix}\\right)\\begin{bmatrix}\\mathbf {X}&\\mathbf {Q}\\end{bmatrix}^T,$ where the orthogonality of $\\mathbf {X}$ and $\\mathbf {Q}$ has been exploited.", "In order to use information gained from previous evaluations of $A$ and $\\nabla A$ we must take $\\mathcal {M}$ into account.", "This requires us to transport vectors $\\mathbf {Y}\\in \\mathcal {T}_{\\mathbf {X}}\\mathcal {M}$ to $\\mathcal {T}_{\\mathbf {U}\\mathbf {X}}\\mathcal {M}$ with the transport operator $\\mathbf {T}= \\mathbf {I}_m+\\begin{bmatrix}\\mathbf {X}&\\mathbf {Q}\\end{bmatrix}\\left(\\exp \\left(\\tau \\begin{bmatrix}\\mathbf {A}& -\\mathbf {R}^T\\\\\\mathbf {R}&\\mathbf {0}\\end{bmatrix}\\right)-\\mathbf {I}_{2n}\\right)\\begin{bmatrix}\\mathbf {X}&\\mathbf {Q}\\end{bmatrix}^T.$ Here $\\mathbf {I}_m\\in \\mathbb {R}^{m\\times m}$ and $\\mathbf {I}_{2n}\\in \\mathbb {R}^{2n\\times 2n}$ , and $\\mathbf {T}$ does not modify matrices $\\mathbf {Z}$ that satisfy $\\begin{bmatrix}\\mathbf {X}&\\mathbf {Q}\\end{bmatrix}^T\\mathbf {Z}=\\mathbf {0}$ .", "Remark 1: The closely related Grassmann manifold is identical to the Stiefel manifold with the addition of the homogeneity condition $A(\\mathbf {X}) = A(\\mathbf {X}\\mathbf {Q})$ , where $\\mathbf {Q}$ is orthogonal.", "The homogeneity condition is satisfied for orbitals with identical occupation numbers, but does not generally hold for ensemble DFT.", "A discussion of direct minimization with integer occupation numbers is presented in [1]." ], [ "Nonlinear conjugate gradients", "The conjugate gradient (CG) method can be viewed as an optimization method for a quadratic problem.", "Several generalizations of the CG method have been presented to solve optimization problems that are not quadratic [22].", "Below, we review a nonlinear CG (NLCG) method adapted to account for the curvature of the manifold [11].", "Given $\\mathbf {X}_0$ which satisfies $\\mathbf {X}_0^T\\mathbf {X}_0 = \\mathbf {I}$ , the gradient projected onto $\\mathcal {T}_{\\mathbf {X}_0}\\mathcal {M}$ is $\\mathbf {F}_0 = \\mathbf {P}_{\\mathbf {X}_0}(\\nabla _\\mathbf {X}A(\\mathbf {X}_0,0)),$ and the initial search direction is the direction of steepest descent $\\mathbf {Y}_0 = -\\mathbf {F}_0.$ On the manifold the NLCG method then proceeds by minimizing $A$ along the path defined by the search direction $\\mathbf {Y}_k$ .", "In practice we evaluate $A$ once along the search direction and construct a quadratic approximation that we minimize.", "The step length, $\\tau _k$ , that minimizes $A$ along the search direction is then used to update $\\mathbf {X}_k$ such that $\\mathbf {X}_{k+1} = \\mathbf {T}(\\tau _k)\\mathbf {X}_k,$ and the gradient and search directions are transported to $\\mathcal {T}_{\\mathbf {X}_{k+1}}\\mathcal {M}$ by $\\mathbf {T}(\\tau _k)$ .", "The new projected gradient $\\mathbf {F}_{k+1} = \\mathbf {P}_{\\mathbf {X}_{k+1}}(\\nabla _\\mathbf {X}A(\\mathbf {X}_{k+1},{k+1})),$ and search direction $\\mathbf {Y}_{k+1} = -\\mathbf {F}_{k+1} + \\gamma _k \\mathbf {T}(\\tau _k) \\mathbf {Y}_k,$ are then computed where $\\gamma _k = \\frac{(\\mathbf {F}_{k+1}-\\mathbf {T}(\\tau _k)\\mathbf {F}_k,\\mathbf {F}_{k+1})}{(\\mathbf {F}_k,\\mathbf {F}_k)}.$ The step length is determined by the minimizer of a quadratic approximation of $A$ along the search direction.", "The quadratic approximation is constructed by taking a trial step length $\\tau _e = \\tfrac{1}{10}\\max (\\tau _\\mathrm {min},\\tau _{k-1})$ , where $\\tau _\\mathrm {min}$ is a predefined minimum trial step length and computing $p(0) &= A(\\mathbf {X},,\\nonumber \\\\p(\\tau _e) &= A(\\mathbf {T}(\\tau _e)\\mathbf {X},,\\\\p^{\\prime }(0) &= (\\mathbf {Y},\\nabla _\\mathbf {X}A(\\mathbf {X},).\\nonumber $ Then solve $\\tau _k$ and limit it by $2\\tau _{k-1}$ , and construct the update $\\mathbf {T}(\\tau _k)$ .", "This approximate line search requires one extra evaluation of $A$ per step." ], [ "Quasi-Newton method", "The quasi-Newton (QN) method is similar to Newton's method, but replaces the inverse Hessian with an approximation.", "This is frequently possible even when the Hessian is not available, and can still be used to improve performance for a badly conditioned minimization problem.", "We base the QN method on Broyden's second or bad generalized update to construct the approximate inverse Hessian, $\\mathbf {G}$ , of $A$ at $\\mathbf {X}_k$ .", "While Broyden's second update does not construct a symmetric approximation, or ensure that the approximation is positive definite it is a robust update choice for electronic structure calculations [18], [2].", "Furthermore, $\\mathbf {X}$ and $\\nabla _\\mathbf {X}A$ are $\\mathbb {R}^{m\\times n}$ matrices, which we take into account when constructing the generalized Broyden update.", "The secant condition is then $\\mathbf {G}\\Delta \\mathbf {\\Phi }= \\Delta \\mathbf {\\Xi },$ where $\\Delta \\mathbf {\\Phi }$ and $\\Delta \\mathbf {\\Xi }$ are the collected orbital gradient and position differences projected onto the tangent space and transported to $\\mathcal {T}_{\\mathbf {X}_k}\\mathcal {M}$ .", "That is $\\Delta \\mathbf {\\Phi }= \\begin{bmatrix}\\Delta \\mathbf {F}_{k-1} & \\mathbf {T}(\\tau _{k-1})\\Delta \\mathbf {F}_{k-2} &\\ldots & \\mathbf {T}(\\tau _{k-1})\\ldots \\mathbf {T}(\\tau _{l+1})\\Delta \\mathbf {F}_{l}\\end{bmatrix},$ and $\\Delta \\mathbf {\\Xi }= \\begin{bmatrix}\\Delta \\mathbf {X}_{k-1} & \\mathbf {T}(\\tau _{k-1})\\Delta \\mathbf {X}_{k-2} &\\ldots & \\mathbf {T}(\\tau _{k-1})\\ldots \\mathbf {T}(\\tau _{l+1})\\Delta \\mathbf {X}_{l}\\end{bmatrix},$ for history length $k-l$ .", "Here the gradient differences projected onto $\\mathcal {T}_{\\mathbf {X}_{i+1}}\\mathcal {M}$ are $\\Delta \\mathbf {F}_i = \\mathbf {F}_{i+1} - \\mathbf {T}(\\tau _i)\\mathbf {F}_i,$ and $\\mathbf {F}_i$ is like in (REF ), $\\mathbf {F}_{i} = \\mathbf {P}_{\\mathbf {X}_{i}}(\\nabla _\\mathbf {X}A(\\mathbf {X}_{i},{i})).$ The projected occupation weighted orbital differences are $\\Delta \\mathbf {X}_i = \\mathbf {P}_{\\mathbf {X}_{i+1}}\\big (\\mathbf {X}_{i+1}\\,\\mathrm {diag}({i+1})-\\mathbf {X}_i\\,\\mathrm {diag}(i)\\big ),$ and the motivation for including the weight is that the unoccupied electronic orbitals do not contribute to the energy of the system.", "The no change condition is now $\\mathbf {Z}= \\mathbf {G}\\mathbf {Z}\\quad \\forall \\,\\mathbf {Z}\\;\\;\\mathrm {such\\;that}\\;\\;\\mathbf {Z}^T\\Delta \\mathbf {\\Phi }= \\mathbf {0}.$ The secant and no change condition together correspond to the generalized Broyden's second update where all single orbital secant conditions are simultaneously enforced for the entire history length.", "We can therefore use the generalized update formula [12] $\\mathbf {G}= \\mu \\mathbf {I}+ (\\Delta \\mathbf {\\Xi }- \\mu \\Delta \\mathbf {\\Phi }) (\\Delta \\mathbf {\\Phi }^T\\Delta \\mathbf {\\Phi })^{-1}\\Delta \\mathbf {\\Phi }^T,$ where dropping the empty orbitals ensure that $\\Delta \\mathbf {\\Phi }^T\\Delta \\mathbf {\\Phi }$ is nonsingular in practice.", "The search direction given by the QN method is then $\\mathbf {Y}= -\\mathbf {G}\\mathbf {F},$ and $\\mathbf {X}_{k+1} = \\mathbf {U}(\\tau _k)\\mathbf {X}_k,$ where $\\mathbf {Y}$ determines $\\mathbf {U}$ as in Section (REF ).", "The line search is identical to the one described for the NLCG method in Section REF with the addition of the constant underrelaxation $\\beta _\\mathbf {X}\\in \\;]0,1]$ that we have included in the step length $\\tau _k$ .", "In practice only the last few history steps contribute significantly to the rate of convergence.", "Consequently, we discard the oldest trial solutions and gradient information when a predetermined history length is reached." ], [ "Optimization of occupation numbers", "Given a set of electronic orbitals $\\mathbf {X}$ it is possible to further reduce $A$ by optimizing $.", "Forcing occupation towards a uniform distribution increases contributions to $ A$ from higher energy states, while simultaneously increasing the entropy which contributes to a reduction of $ A$ at nonzero temperatures.The relative strength of both of these effects determine the ground state of the system, and can lead to nonzero occupation of higher energy states at positive temperatures or due to nonlinear effects.$ Therefore, given $\\mathbf {X}$ , we want to find $ that minimizes $ A$.", "To keep the number of particles constant we determine the search direction $ y$ which is the vector closest $ -Ȃ(X,$ that ensures that the conditions $ i=1n fi = ne$ and $ 0 fi 1$ remain satisfied.", "To this end we solve\\begin{equation}\\mathrm {minimize}\\; \\Vert \\mathbf {y}+ \\nabla _Ȃ(\\mathbf {X},\\Vert ,\\end{equation}with the constraints $ i=1n yi = 0$, $ yi 0$ if $ fi = 1$, and $ yi 0$ if $ fi = 0$.The first constraint on $ y$ ensures that the minimization step conserves electrons while the second and third condition prohibits unphysical occupation numbers.In practice we use the {\\tt quadprog} routine available in MATLAB to solve this problem.Given the search direction $ y$ we minimize $ A$ by constructing a quadratic approximation similar to~(\\ref {eq:step}).$ After we have solved $\\mathbf {y}$ the occupation step length $\\sigma _k$ is determined like in Section REF with the addition of the constant underrelaxation $\\beta _\\;]0,1]$ included in $\\sigma _k$ .", "In addition, we ensure that the occupation remains physical by limiting $\\sigma _k$ with $\\sigma _{\\mathrm {M}}$ such that $0 \\le f_i+\\sigma _{\\mathrm {M}}y_i \\le 1$ for all $i$ .It is possible to take a longer step than $\\sigma _\\mathrm {M}$ by recomputing $\\mathbf {y}$ from Equation () with the updated boundary information when an entry in $ reaches the boundary of physical occupation, $ 0$ or $ 1$.", "However, convergence of occupation numbers is faster than orbital convergence, and the numbers of steps needed for convergence is therefore determined by the orbital convergence.Furthermore, if the occupation number of the least populated orbital has been less than $ 10-12$ on two consecutive iterations we drop the associated orbitals.$" ], [ "Simultaneous step size selection", "Typically an ensemble DFT problem is solved by sequentially optimizing the orbitals with fixed occupation numbers and then fixing the orbitals and optimizing the occupation numbers.", "This process is then repeated until a satisfactory solution is obtained.", "The cost of evaluating $A$ , $\\nabla _\\mathbf {X}A$ , and $\\nabla _Ȃ$ is comparable to evaluating one of them separately, and simultaneous optimization of $A$ with respect to $\\mathbf {X}$ and $ can for this reason potentially reduce computational effort.$ Given a pair of search directions $(\\mathbf {Y}$ , $\\mathbf {y})$ for the orbitals and occupation numbers respectively and starting guesses for step lengths, $\\tau _{k-1}$ and $\\sigma _{k-1}$ we evaluate $A$ and its gradients with the following trial step lengths $\\tau _e = \\tfrac{1}{10}\\max (\\tau _\\mathrm {min},\\tau _{k-1}),$ and $\\sigma _e = \\mathrm {min}(\\sigma _\\mathrm {M},\\tfrac{1}{10}\\max (\\sigma _\\mathrm {min},\\sigma _{k-1})).$ Here $\\tau _\\mathrm {min}$ and $\\sigma _\\mathrm {min}$ are minimum trial step lengths.", "With this we construct a quadratic surface approximation $p(\\tau , \\sigma ) = c_1\\tau ^2+c_2\\sigma ^2+c_3\\tau +c_4\\sigma +c_5,$ that we use to simultaneously update both $\\mathbf {X}$ and $ by evaluation in one trial point.", "This surface is determined by the system of equations{\\begin{@align}{1}{-1}p(0,0) &= A(\\mathbf {X},,\\nonumber \\\\p_\\tau (0,0) &= (\\mathbf {Y},\\nabla _\\mathbf {X}A(\\mathbf {X},), \\nonumber \\\\p_\\sigma (0,0) &= (\\mathbf {y}_0,\\nabla _Ȃ(\\mathbf {X},), \\\\p_\\tau (\\tau _e,\\sigma _e) &= (\\mathbf {Y},\\nabla _\\mathbf {X}A(\\mathbf {T}(\\tau _e)\\mathbf {X},\\sigma _e\\mathbf {y})),\\nonumber \\\\p_\\sigma (\\tau _e,\\sigma _e) &= (\\mathbf {y}_{\\sigma _e},\\nabla _Ȃ(\\mathbf {T}(\\tau _e)\\mathbf {X},\\sigma _e\\mathbf {y})).\\nonumber \\end{@align}}Solving this system and finding the minimums gives the optimal step lengths $ k$ and $ k$ for the quadratic approximation of the search directions.For the simultaneous NLCG method the step lengths are then $ k = k$ and $ k = k$ while the QN method uses $ k = Xk$ and $ k = k$, where $ X, ]0,1]$ are constant underrelaxation parameters.We then simultaneously update $ X$ and $ with $\\mathbf {X}_{k+1}=\\mathbf {T}(\\tau _k)\\mathbf {X}_k$ and ${k+1} = k+\\min (\\sigma _\\mathrm {M},\\sigma _k)\\mathbf {y}$ respectively.", "We then simultaneously update $X$ and $ with $ Xk+1=T(k)Xk$ and $ k+1 = k+(M,k)y$ respectively.$Remark 2: The surface (REF ) is determined by computing $\\nabla _\\mathbf {X}A$ and $\\nabla _Ȃ$ at the trial step.", "The system of equations () could alternatively be determined by computing both $A$ and $\\nabla _\\mathbf {X}A$ or $A$ and $\\nabla _Ȃ$ at $(\\mathbf {T}(\\tau _e)\\mathbf {X},\\sigma _e\\mathbf {y})$ .", "Remark 3: Inclusion of the $\\tau \\sigma $ cross term would require an extra trial evaluation point for system () to be linearly independent." ], [ "Numerical experiments", "We use a two dimensional model problem to compare the sequential and simultaneous NLCG and QN methods.", "This model problem is inspired by ensemble DFT, and corresponds to a three dimensional system constrained to two dimensions without spin effects and exchange-correlation terms while taking entropy into account.", "The model problem adapted from Reference [19] is $A(\\mathbf {X}, = -\\tfrac{1}{2}\\mathrm {tr}\\big (\\mathbf {X}^T\\mathbf {L}\\mathbf {X}\\,\\mathrm {diag}(\\big ) + \\mathbf {v}_\\mathrm {ext}^T\\mathbf {n}+\\tfrac{1}{2} \\mathbf {v}_\\mathrm {int}^T\\mathbf {n}-TS(.$ Here $\\mathbf {L}\\in \\mathbb {R}^{m\\times m}$ is the discretized Laplace operator, $\\mathbf {v}_\\mathrm {ext}\\in \\mathbb {R}^{m}$ the external potential, $\\mathbf {n}\\in \\mathbb {R}^m$ the electron density, $\\mathbf {v}_\\mathrm {int}= $ the Hartree potential corresponding to the electron density $\\mathbf {n}$ , $T$ to temperature, and $S$ is the entropy.", "The electron density is $\\mathbf {n}= (\\mathbf {X}\\circ \\mathbf {X})\\,$ where $\\circ $ is the entrywise, or Hadamard, product.", "The entropy term is $S(=-\\sum _{i=1}^n f_i\\ln (f_i+\\delta (1-f_i))+(1-f_i)\\ln (1-f_i+\\delta f_i),$ where $\\delta > 0$ is a small regularization parameter that ensures that the derivative of $S$ remains finite.", "To calculate the potentials we use $(\\mathbf {v}_\\mathrm {ext})_i = -\\sum _{j=1}^N \\frac{Z_j}{\\Vert \\mathbf {r}_i - \\mathbf {R}_j\\Vert +\\alpha },$ where the sum is over the nuclei with charge $Z_j$ and position $\\mathbf {R}_j$ .", "The position corresponding to the discretization point $i$ is $\\mathbf {r}_i$ , and the parameter $\\alpha $ is used to regularize the potential.", "$\\mathbb {R}^{m\\times m}$ is similarly given by ${ij} = \\frac{1}{\\Vert \\mathbf {r}_i-\\mathbf {r}_j\\Vert + \\alpha }.$ We solve the problem in the unit square with zero boundary conditions corresponding to an infinite potential well.", "We use a uniform finite difference discretization with $m$ inner points to obtain a system where $\\mathbf {X}\\in \\mathbb {R}^{m\\times n}$ .", "Here $n$ corresponds to the number of electronic orbitals in the calculation.", "As initial guess we use the solution of the quadratic problem using the first two terms of (REF ).", "The occupation numbers are initialized to $f_i = \\tfrac{n_e}{n} + \\tfrac{1}{2}\\Delta \\frac{n+1-2i}{n+1},$ where $\\Delta = \\min (n_e/n,1-n_e/n)$ and $n_e\\le n$ is the number of electrons.", "This choice ensures that the initial occupation of all orbitals is nonzero and emphasizes lower energy orbitals.", "We demonstrate the methods for three external potentials.", "For all models we use potential regularization $\\alpha = 5\\times 10^{-2}$ and entropy regularization $\\delta = 10^{-3}$ .", "The first model is a single nucleus with charge $Z=2$ centered at the center of the unit square with two electrons.", "For this system the second and third orbitals are degenerate.", "We calculate the model with 10 electronic orbitals and a first order finite difference discretization with 25 interior points in one dimension resulting in $m=625$ spatial degrees of freedom.", "We will refer to this system as $Z_2$ .", "The second model, which we name $Z_3$ -$Z_2$ , consists of two nuclei, with a nuclei of charge $Z = 3$ placed at $(\\tfrac{1}{3},\\tfrac{1}{3})$ and another with charge $Z=2$ placed at $\\tfrac{2}{3},\\tfrac{2}{3}$ and 5 electrons.", "This system has four well separated electronic orbitals, while the fifth and sixth are relatively close.", "The computation is initialized with 10 orbitals and 29 interior grid points in one dimension for $m = 841$ .", "The last model, $Z_4$ -$Z_3$ , consists of two nuclei, $Z = 4$ placed at $(\\tfrac{1}{3},\\tfrac{1}{3})$ , and $Z=3$ at the grid point closest to $(\\tfrac{2}{3},\\tfrac{13}{24})$ and 7 electrons.", "The off diagonal placement is chosen to break the symmetry of the system.", "This model initially has 14 orbitals and 29 interior grid points in one dimension $(m=841)$ .", "For the sequential QN orbital minimizer uses the parameters $\\beta _\\mathbf {X}= 0.4$ , $\\mu = 5\\times 10^{-5}$ , and history length 6.", "The sequential QN and NLCG methods minimum trial step length $\\tau _0 = 10^{-3}$ and we perform 6 orbital optimization steps before engaging the occupation number minimizer.", "Both sequential optimization routines use an identical SD routine with $\\beta _ 0.5$ , $\\mu =10^{-4}$ and $\\sigma _0 = 10^{-4}$ for occupation number optimization with two optimization steps.", "We have tried several different combinations of orbital and occupation optimization steps and observed that this combination offers a good compromise.", "For the SD method $\\mu $ only serves to scale the approximate line search.", "We measure convergence by the energy difference to a reference energy computed by running the simultaneous methods for 3000 steps and the sequential methods for 3000 optimization rounds.", "We then use the lowest energy obtained as the reference energy.", "Figure: Orbital energy levels with occupation for Z 2 Z_2 at varying temperatures.", "Fractional occupation numbers are indicated and the same data is also presented in Table .Figure: Orbital energy levels with occupation for Z 3 Z_3-Z 2 Z_2 at varying temperatures with orbital energy shifted by +5+5.", "Fractional occupation numbers are indicated and the same data is also presented in Table .Figure: Orbital energy levels with occupation for Z 4 Z_4-Z 3 Z_3 at varying temperatures with orbital energy shifted by +10+10.", "Fractional occupation numbers are indicated and the same data is also presented in Table .The change in occupation numbers with rising temperature is graphically presented in Figures REF , REF , and REF .", "The same data is repeated in Tables REF , REF , and REF .", "At $T=0$ the lowest electronic orbitals are fully occupied for $Z_4$ -$Z_3$ , while one electron is split between two degenerate orbitals for $Z_2$ .", "Even though there is a small gap ($1.69\\times 10^{-2}$ ) between the fifth and sixth electron orbitals for $Z_3$ -$Z_2$ the fifth electron is split (0.55 vs 0.45) between these orbitals.", "We successfully replicated this split with a 3000 round sequential SD orbital occupation number optimization.", "Furthermore, restarting the SD iteration with a five orbital initial guess based on the split orbital reference solution results in convergence to a higher energy state.", "Figure: Energy convergence for Z 2 Z_2 at varying temperatures.Figure: Energy convergence for Z 3 Z_3-Z 2 Z_2 at varying temperatures.Figure: Energy convergence for Z 4 Z_4-Z 3 Z_3 at varying temperatures.Figures REF , REF , and REF illustrate energy convergence for the different methods.", "The simultaneous methods generally perform better than the sequential methods, and the simultaneous NLCG method is more robust than the simultaneous QN approach.", "In the energy convergence for the sequential optimization routines the switch between orbital and occupation optimization is readily seen in the steplike energy convergence.", "Furthermore, the performance of the sequential QN and NLCG methods is nearly identical for all models.", "This might be due to the limited number of step available for orbital optimization before occupation optimization is enabled.", "Figure: Energy convergence of the simultaneous QN method with restarts for Z 4 Z_4-Z 3 Z_3 at varying temperatures.The simultaneous NLCG method outperforms the QN method for the $Z_4$ -$Z_3$ system shown in Figure REF .", "Increasing history generally improves the convergence rate of the QN method, but this did not significantly change the rate of convergence for this model.", "Frequent restarts limit history length and provide at least a partial explanation for this effect.", "Figure REF presents $Z_4$ -$Z_3$ restarts for the simultaneous QN method.", "Restarts are frequent for this model at all temperatures compared to $Z_2$ and $Z_3$ -$Z_2$ .", "However, for $T > 0$ there is generally sufficiently many steps between restarts for the history to grow to full length, and the rate of convergence does improve somewhat.", "Figure: Energy convergence for Z 4 Z_4-Z 3 Z_3 at varying temperatures T<1T < 1.For the $Z_4$ -$Z_3$ model the energy difference between the highest occupied and lowest unoccupied orbital is $1.69\\times 10^{-2}$ , see Figure REF and Table REF .", "This difference is comparatively small and could explain the poor performance of the QN method, particularly for $T=0$ .", "In Figure REF the convergence rate of the optimization procedures for $Z_4$ -$Z_3$ for $T = 0.3,0.5,0.7$ , and the convergence rate for $T=0$ is included for reference.", "At $T = 0.3$ the rate of convergence for the QN method is considerably improved and the convergence rate remains superior to $T=0$ for $T=0.5$ and $T=0.7$ .", "The elevated temperature broadens the Fermi surface and this could explain the improved convergence at $T=0.3$ , while the convergence of higher energy orbitals makes the problem more challenging at higher temperatures.", "This would also explain the decreasing performance of NLCG for higher temperatures.", "Table: Orbital energy levels with occupation for Z 2 Z_2 at varying temperatures.", "The same data is graphically presented in Figure .Table: Orbital energy levels with occupation for Z 3 Z_3-Z 2 Z_2 at varying temperatures with orbital energies shifted by +5+5.", "The same data is graphically presented in Figure .Table: Orbital energy levels with occupation for Z 4 Z_4-Z 3 Z_3 at varying temperatures with orbital energies shifted by +10+10.", "The same data is graphically presented in Figure ." ], [ "Conclusion", "We have presented two schemes for energy optimization of ensemble DFT computations.", "The updates take the problem constraints into account and permits us to use information obtained from previous evaluations of the target functional and gradients to improve rate of convergence.", "We have further demonstrated the methods numerically on a model problem inspired by the electronic structure theory and compared simultaneous and sequential schemes based on the QN and NLCG methods.", "The ensemble model successfully concentrates occupation to low energy orbitals at low temperatures, and gradually increases occupation of higher energy orbitals at increasing temperature to increase the entropy of the system.", "Optimization of the occupation numbers also enables ensemble DFT calculations to automatically handle degenerate and near degenerate orbitals at $T=0$ , which are challenging for methods that construct the electron density by the Aufbau principle.", "Furthermore, is seems possible to broaden the Fermi surface by increasing temperature to accelerate convergence of small gap systems.", "Simultaneous optimization schemes provide improved convergence compared to sequential approaches for both the NLCG and QN methods.", "While the NLCG and QN methods are often comparable in performance, the NLCG method is overall more robust.", "In contrast, Reference [3] found that QN method is more robust than the NLCG method.", "It is possible that the quadratic approximate line search gives a better result for the model problem.", "As the NLCG method depends heavily on a high quality line search this might provide a possible explanation.", "In the present case, the QN method performs poorly for problems with frequent restarts and while this effect does not fully explain the lack of convergence it can be used as a problem indicator." ] ]

[ [ "Dark Matter collisions with the Human Body" ], [ "Abstract We investigate the interactions of Weakly Interacting Massive Particles (WIMPs) with nuclei in the human body.", "We are motivated by the fact that WIMPs are excellent candidates for the dark matter in the Universe.", "Our estimates use a 70 kg human and a variety of WIMP masses and cross-sections.", "The contributions from individual elements in the body are presented and it is found that the dominant contribution is from scattering off of oxygen (hydrogen) nuclei for the spin-independent (spin-dependent) interactions.", "For the case of 60 GeV WIMPs, we find that, of the billions of WIMPs passing through a human body per second, roughly ~10 WIMPs hit one of the nuclei in the human body in an average year, if the scattering is at the maximum consistent with current bounds on WIMP interactions.", "We also study the 10-20 GeV WIMPs with much larger cross-sections that best fit the DAMA, COGENT, and CRESST data sets and find much higher rates: in this case as many as $10^5$ WIMPs hit a nucleus in the human body in an average year, corresponding to almost one a minute.", "Though WIMP interactions are a source of radiation in the body, the annual exposure is negligible compared to that from other natural sources (including radon and cosmic rays), and the WIMP collisions are harmless to humans." ], [ "=1 Dark Matter collisions with the Human Body Katherine Freese [][email protected] Michigan Center for Theoretical Physics, Department of Physics, University of Michigan, Ann Arbor, MI 48109 Christopher Savage [][email protected] The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden We investigate the interactions of Weakly Interacting Massive Particles (WIMPs) with nuclei in the human body.", "We are motivated by the fact that WIMPs are excellent candidates for the dark matter in the Universe.", "Our estimates use a 70 kg human and a variety of WIMP masses and cross-sections.", "The contributions from individual elements in the body are presented and it is found that the dominant contribution is from scattering off of oxygen (hydrogen) nuclei for the spin-independent (spin-dependent) interactions.", "For the case of 60 GeV WIMPs, we find that, of the billions of WIMPs passing through a human body per second, roughly $\\sim 10$ WIMPs hit one of the nuclei in the human body in an average year, if the scattering is at the maximum consistent with current bounds on WIMP interactions.", "We also study the 10–20 GeV WIMPs with much larger cross-sections that best fit the DAMA, COGENT, and CRESST data sets and find much higher rates: in this case as many as $10^5$ WIMPs hit a nucleus in the human body in an average year, corresponding to almost one a minute.", "Though WIMP interactions are a source of radiation in the body, the annual exposure is negligible compared to that from other natural sources (including radon and cosmic rays), and the WIMP collisions are harmless to humans.", "A variety of astrophysical observations has shown conclusively that the majority of the matter in the Universe consists of an unknown nonluminous, nonbaryonic component.", "Understanding the nature of this dark matter is one of the major outstanding problems of astrophysics and particle physics.", "Most cosmologists believe that the solution to this puzzle lies in the discovery of a new type of fundamental particle.", "Leading candidates for the dark matter are Weakly Interacting Massive Particles (WIMPs), a generic class of particles that are electrically neutral and do not participate in strong interactions, yet have weak interactions with ordinary matter.", "Possible WIMP candidates include supersymmetric particles and Kaluza-Klein particles motivated by theories with extra dimensions.", "These particles are thought to have masses in the range 1 GeV–10 TeV, consistent with their being part of an electroweak theory.", "Searches for WIMPs [1], [2], [3] include direct detection laboratory experiments, which look for the elastic scattering of WIMPs in the Galaxy as they pass through terrestrial detectors situated in deep underground sites.", "These efforts are ongoing worldwide.", "Currently there are intriguing hints of discovery with the DAMA [4], CoGeNT [5], [6], and CRESST [7] experiments although no consensus has been reached in the community.", "The null results of a host of other experiments, including CDMS [8] and XENON [9], [10] have been used to place bounds on the scattering rates of WIMPs as a function of WIMP mass.", "In the standard framework used in this work, there is a strong tension between the results of the first three experiments and the null results of the latter two.", "Many efforts in both the experimental and theoretical directions are ongoing to understand these discrepancies; in this paper we will simply use the currently published results of these experiments.", "In this paper we consider this same elastic scattering of WIMPs with nuclei in the human body.", "Billions of WIMPs pass through our bodies every second, yet most of them pass through unimpeded.", "Only rarely does WIMP actually hit one of our nuclei.", "To perform our analysis we will assume a human of 70 kg and consider a variety of WIMP masses in the GeV–TeV range.", "First we will study 60 GeV WIMPs with the maximum scattering cross-section allowed by the null results of the XENON and CDMS experiments.", "Then we will turn to the lower mass WIMPs (10–20 GeV) that provide the best fits to the hints of discovery in DAMA, CRESST, and COGENT as well as TeV benchmark cases again compatible with the null result experiments.", "Finally, we examine the radiation exposure these interactions represent and how it compares to other natural radiation sources.", "The scattering rate of WIMPs with an element (indexed by $k$ ) in a human body of mass $M_{\\mathrm {body}}$ is given byThe rate here is a pure rate, not a rate per unit target mass as is commonly used in the dark matter direct detection literature.", "$R_k = N_k n_{\\chi }\\langle v \\sigma _k \\rangle = \\left( \\frac{f_k M_{\\mathrm {body}}}{m_k} \\right)\\left( \\frac{\\rho _{\\chi }}{m_{\\chi }} \\right)\\int d^3v \\, v f(\\mathbf {v}) \\sigma _k(v) \\, ,$ where $N_k = \\frac{f_k M_{\\mathrm {body}}}{m_k}$ is the number of nuclei of that element in the body, with $m_k$ the nuclear mass and $f_k$ the mass fraction of that element; $n_{\\chi }= \\frac{\\rho _{\\chi }}{m_{\\chi }}$ is the number density of WIMPs, with $m_{\\chi }$ the WIMP mass and $\\rho _{\\chi }$ the local dark matter mass density; $f(\\mathbf {v})$ is the WIMP velocity distribution; and $\\sigma _k(v)$ is the (velocity-dependent) WIMP-nucleus scattering cross-section.", "To a reasonable first approximation, the dark matter halo can be treated as a non-rotating, isothermal sphere (the Standard Halo Model) [2], [3].", "For the resulting Maxwellian velocity distribution, a 3D velocity dispersion of 270 km/s is assumed.", "The velocity distribution is truncated at 550 km/s to account for the fact that high velocity particles would escape the galaxy [11], though the results of this paper are fairly insensitive to this cutoff as such high velocity particles would otherwise make only a small contribution to the total scattering rateThe same cannot always be said for the rates in direct detection experiments as these experiments are sensitive to events that produce energies above a threshold, not the total number of events.", "In some cases, only high velocity WIMPs produce scattering events above threshold, so the choice of cutoff becomes important..", "The local density of the dark matter halo is taken to be 0.4 GeV/cm$^3$ .", "While the smooth halo component is likely to be supplemented by a variety of substructures such as streams, clumps, or debris flow, their contributions are unlikely to be large enough to substantially modify the results of this paper.", "Dropping the isotope index $k$ , the scattering cross-section is given by $\\sigma (v) = \\int _0^{q_{\\mathrm {max}}^2} dq^2 \\frac{d\\sigma }{dq^2}(q^2,v) \\, ,$ where $q$ is the momentum transferred in a scatter, $q_{\\mathrm {max}}= 2 \\mu v$ is the maximum momentum transfer in a scatter at a relative velocity $v$ , $\\mu $ is the WIMP-nucleus reduced mass, and $\\frac{d\\sigma }{dq^2}(q^2,v) = \\frac{\\sigma _{0}}{4 \\mu ^2 v^2}F^2(q) \\, \\Theta (q_{\\mathrm {max}}-q) \\, ,$ with $\\Theta $ the step function and $\\sigma _0$ the scattering cross-section in the zero-momentum-transfer limit.", "Here, $F^2(q)$ is a form factor to account for the finite size of the nucleus.", "For small momentum transfers, the WIMP coherently scatters off the entire nucleus; the nucleus is essentially a point particle in this case, with $F^2(q) \\rightarrow 1$ .", "For sufficiently small $v$ , such that the possible momentum transfer remains small, $\\sigma (v) \\rightarrow \\sigma _0$ .", "As the de Broglie wavelength of the momentum transfer becomes comparable to the size of the nucleus, the interaction becomes sensitive to the spatial structure of the nucleus and $F^2(q) < 1$ , with $F^2(q) \\ll 1$ at higher momentum transfers.", "For velocities at which this form factor becomes relevant, $\\sigma (v) < \\sigma _0$ (with $\\sigma (v) \\ll \\sigma _0$ at very high velocities).", "The velocity at which this form factor causes the cross-section $\\sigma (v)$ to start to significantly deviate from the zero-momentum-transfer limit $\\sigma _0$ is dependent on the nuclei in question for two reasons: (1) the size of the nucleus grows as the nucleus gets heavier and (2) the momentum transferred becomes larger as the nucleus gets heavier, assuming the WIMP is heavier than the nuclei in question.", "For the typical WIMP velocities in the halo, the form factor suppression is negligible for nuclei much lighter than iron ($\\sigma (v) \\approx \\sigma _0$ ), while it is significant for nuclei much heavierFor the same reasons as given in the previous footnote (the application of a threshold), the form factor is more important for direct detection and it can significantly suppress the direct detection rates above threshold even when the total rate does is not significantly affected..", "There are two types of interactions commonly considered for WIMP scattering: spin-independent (SI) and spin-dependent (SD).", "Each coupling has its own form factor and Eqn.", "(REF ) must be summed over these two contributions.", "In the SI case, the WIMP essentially couples to the mass in the nucleus, with a zero-momentum-transfer limit cross-section $ \\sigma _{0,\\mathrm {SI}}= \\frac{4 \\mu ^2}{\\pi } \\left[ Z f_{\\mathrm {p}}+ (A-Z) f_{\\mathrm {n}}\\right]^2 \\, ,$ where $f_{\\mathrm {p}}$ and $f_{\\mathrm {n}}$ are the couplings to the proton and neutron, respectively, $Z$ is the number of protons in the nucleus, and $A-Z$ is the number of neutrons.", "For many WIMP candidates, $f_{\\mathrm {p}}\\approx f_{\\mathrm {n}}$ and the cross-section scales as $ \\sigma _{0,\\mathrm {SI}}= \\frac{\\mu ^2}{\\mu _{\\mathrm {p}}^2} A^2 \\, \\sigma _{\\mathrm {p,SI}}\\, ,$ where $\\mu _{\\mathrm {p}}$ is the WIMP-proton reduced mass and $\\sigma _{\\mathrm {p,SI}}$ is the SI WIMP-proton scattering cross-section.", "We will assume $f_{\\mathrm {p}}= f_{\\mathrm {n}}$ below, though the results are only very mildly sensitive to the ratio of these two couplings except in the case $\\frac{f_{\\mathrm {p}}}{f_{\\mathrm {n}}} \\approx -\\frac{A-Z}{Z}$ where the terms in Eqn.", "(REF ) cancel.", "In the SD case, as the name implies, the WIMP couples to the spin of the nucleus, with $ \\sigma _{0,\\mathrm {SD}}= \\frac{32 \\mu ^2}{\\pi } G_{F}^{2} J(J+1) \\Lambda ^2 \\, ,$ where $J$ is the spin of the nucleus, $ \\Lambda \\equiv \\frac{1}{J} \\left( a_{\\mathrm {p}}\\langle S_{\\mathrm {p}} \\rangle + a_{\\mathrm {n}}\\langle S_{\\mathrm {n}} \\rangle \\right) \\, ,$ $a_{\\mathrm {p}}$ and $a_{\\mathrm {n}}$ are the couplings to the proton and neutron, respectively, and $\\langle S_{\\mathrm {p}} \\rangle $ and $\\langle S_{\\mathrm {n}} \\rangle $ are the spin contributions from the proton and neutron groups, respectively.", "In our analysis, we shall assume identical couplings to the proton and neutron ($a_{\\mathrm {p}}= a_{\\mathrm {n}}$ ), so that $ \\sigma _{0,\\mathrm {SD}}= \\frac{\\mu ^2}{\\mu _{\\mathrm {p}}^2}\\frac{J(J+1)}{\\frac{1}{2}(\\frac{1}{2}+1)}\\left( \\frac{\\langle S_{\\mathrm {p}} \\rangle +\\langle S_{\\mathrm {n}} \\rangle }{J} \\right)^2\\, \\sigma _{\\mathrm {p,SD}}\\, .$ Whereas the couplings to neutrons and protons are roughly identical for SI scattering for many WIMP candidates, in the case of SD scattering they may differ.", "Typically, however, the two SD couplings are found to be within a factor of 2–3 of each other.", "Our results, using identical couplings, will thus be order of magnitude estimates of the general case.", "More detailed discussions of dark matter scattering kinematics, cross-sections, and form factors can be found in Refs.", "[12], [13], [14], [15]; other reviews can be found in Refs.", "[16], [17].", "Table: Interactions of 60 GeV WIMPs on various nuclei in the human body.The mass fraction of the most significant elements in the humanbody, taken from Ref.", "(which in turn refers toRefs.", ", ), is shown.Also shown are the number of WIMP scatters per year for eachelement at the largest spin-independent (SI) andspin-dependent (SD) scattering cross-sections not currentlyexcluded by XENON100 , which areσ p, SI =10 -8 \\sigma _{\\mathrm {p,SI}}= 10^{-8} pb and σ p, SD =2×10 -3 \\sigma _{\\mathrm {p,SD}}= 2 \\times 10^{-3} pb,respectively.We assume a human mass of 70 kg and identical couplings to theproton and neutron.", "(†\\dag ) The SD rate for nitrogen-14 has not been calculated but maybe non-negligible and perhaps as large as 𝒪(10)\\mathcal {O}(10);see the text.Table REF shows the mass fractions of the most significant elements in the human body as well as the scattering rates for each element for a 70 kg body and a 60 GeV WIMP.", "Rates are shown for both SI and SD scattering, assuming scattering cross-sections of $\\sigma _{\\mathrm {p,SI}}= 10^{-8}$  pb and $\\sigma _{\\mathrm {p,SD}}= 2 \\times 10^{-3}$  pb, respectively, the largest cross-sections not excluded by XENON at that WIMP mass.", "Oxygen and carbon are the largest components in the human body by mass and also contribute the most to the SI scattering rate, with oxygen accounting for 65% of the SI scatters at this WIMP mass.", "However, hydrogen, the largest component by number of atoms (representing about 60% of the atoms in the human body), has a much smaller SI scattering rate than many other elements with significantly smaller mass fractions (as well as number of atoms).", "For example, iron, while accounting for less than 1/1000 the mass of the hydrogen, nevertheless has an SI scattering rate $\\sim $ 20 times larger.", "The reason for this lies in the scaling of the SI cross-section shown in Eqn.", "(REF ).", "In addition to the explicit $A^2$ factor, the $\\frac{\\mu ^2}{\\mu _{\\mathrm {p}}^2}$ factor also scales as $A^2$ (for nuclei much lighter than the WIMP), so that the cross-section scales as $A^4$ .", "For a given mass fraction, the number of nuclei is proportional to $1/A$ , so the interaction rate scales as $A^3$ .", "With this scaling and the mass fractions shown in the table, the relative oxygen-to-hydrogen SI scattering rate should then approximately be $\\frac{0.61}{0.10} \\left(\\frac{16}{1} \\right)^3\\approx 25,000$ , in reasonable agreement with the actual value of $\\frac{3.49}{0.00023} \\approx 15,000$ ; the overestimate in the first case is due to the fact that $\\frac{\\mu ^2}{\\mu _{\\mathrm {p}}^2} \\rightarrow A^2$ applies in the limit that the WIMP is much heavier than the nucleus, a limit that has not been fully reached here.", "As the nuclei become heavier, the form factor becomes more and more significant, so the $A^3$ scaling in the interaction rate for a given mass fraction no longer holds, though the rate still grows rapidly.", "On the other hand, scattering with hydrogen is the dominant contribution in the SD case.", "The primary difference is that, unlike the SI case, there is no explicit $A^2$ scaling in the scattering cross-section: the spin factors in Eqn.", "(REF ) are of $\\mathcal {O}(1)$ for all nuclei.", "With the $\\mu ^2$ factor, the SD cross-section scales as $\\sim A^2$ .", "After accounting for the $1/A$ scaling of the number of nuclei for a given mass fraction, the total scattering rate scales as $\\sim A$ (neglecting form factors).", "However, isotopes with zero nuclear spin ($J=0$ ) have $\\sigma _{0,\\mathrm {SD}}= 0$ , so they do not contribute at all to the SD scattering rate.", "Many of the elements listed in Table REF , including oxygen and carbon, are mainly composed of spinless isotopes, with non-zero spin isotopes representing only a small fraction of that element's natural composition.", "The SD scattering rate is thus suppressed in these cases.", "Hydrogen, on the other hand, is mainly composed of spin-1/2 $^1$ H; even spin-1 deuterium contributes to SD scattering.", "Because of the $A$ scaling of the scattering rate for a given mass fraction and the relative isotopic compositions between spinless and non-zero spin nuclei, hydrogen dominates the SD capture rate.", "In our analysis, we have neglected the SD contribution of spin-1 $^{14}$ N. As this is the dominant isotope of nitrogen, nitrogen is expected to have a significant SD scattering rate.", "However, this nucleus belongs to a small group of proton-odd, neutron-odd isotopes with non-zero spin that are not well characterized in the scattering literature and we are unaware of existing estimates for $\\langle S_{\\mathrm {p}} \\rangle $ and $\\langle S_{\\mathrm {n}} \\rangle $ .", "Taking $|\\langle S_{\\mathrm {p}} \\rangle | \\sim |\\langle S_{\\mathrm {n}} \\rangle | \\sim 0.1$ , similar to nearby nuclei (except one of these two quantities is nearly zero in these other nuclei), we can expect $\\mathcal {O}(10)$ SD scattering events per year with nitrogen in the human body.", "This would make nitrogen one of the larger contributors to the total SD rate, though hydrogen still remains the dominant source of SD interactions.", "The overall scattering rates of $\\mathcal {O}(10)$ should not be unexpected for the benchmark WIMP mass and cross-sections here.", "These benchmarks would produce a few events/year in the $\\sim $ 100 kg of liquid xenon that is the target mass in the XENON experiment, the currently measured event rate in the detector (though the measured rate is also consistent with backgrounds alone).", "With a similar mass between the human body and the XENON detector, the rates should be of similar orders of magnitude, though detection efficiencies, thresholds, and different target elements mean the rates are not simply proportional to the target mass.", "Since xenon ($A \\approx 130$ ) is much heavier than oxygen ($A \\approx 16$ ), one might expect a much higher rate in XENON than the human body for SI scattering due to the $\\sim A^4$ cross-section scaling ($\\sigma _{0,\\mathrm {Xe}}$ is $\\mathcal {O}(10^3)$ larger than $\\sigma _{0,\\mathrm {O}}$ ).", "However, due to a threshold and a $<\\!\\!100\\%$ detection efficiency, the few events/year rate measured in XENON is not the total rate in the detector, which is somewhat higher (by an order of magnitude or more).", "In addition, xenon scattering will be form factor suppressed, so that the total scattering rate for xenon is not as high as would be expected from the $A^4$ scaling alone.", "For the SD case, the $\\mathcal {O}(10)$ higher scattering rate in the human body versus the XENON experiment can be attributed to the much larger number of non-zero spin nuclei in the former case (mainly hydrogen).", "Table: The total number of scatters within a human body per year forthe given WIMP masses and WIMP-proton scattering cross-sections.The CoGeNT, CRESST, and DAMA benchmarks are those that best fitthe data for the respective experiments (CRESST has two maximumlikelihood points);these points are all strongly disfavored by the null results ofCDMS and XENON in the standard framework used in this analysis.The XENON benchmarks are compatible with the null results ofCDMS and XENON.We assume a human mass of 70 kg and identical couplings to theproton and neutron.In Table REF , we show scattering rates in the body for several WIMP benchmarks.", "The benchmarks are chosen to correspond to the approximate best-fit WIMP mass and scattering cross-section for the CoGeNT [5], [6], CRESST [7], and DAMA [4] experiments.", "Two CRESST benchmark points are included, corresponding to the two sets of parameters that maximize their likelihood function, M1 (the global maximum) and M2 (a local maximum).", "While DAMA likewise has two best-fit points, we have included only the lower mass one as the higher mass point is in strong conflict with the null results of XENON [9], [10] and CDMS [8].", "We note that, in fact, all of the CoGeNT, CRESSST, and DAMA benchmark points are incompatible with XENON and CDMS under the analysis framework we are using here.", "Many researchers are trying to understand the origin of these differences; in this paper we simply follow the published results in choosing our benchmark points.", "Two additional benchmark points are included, corresponding to the maximum cross-section consistent with the null results of XENON (and CDMS, which has a slightly weaker constraint) for WIMP masses of 60 GeV and 1 TeV; the former case is the benchmark used in Table REF .", "All benchmarks are included for the SI case, while only the DAMA best-fit and XENON-allowed benchmarks are included in the SD case.", "The scattering rates for the CoGeNT, CRESST, and DAMA benchmark points are all significantly larger than the rates for the XENON-allowed benchmarks, as the former are all at cross-sections higher than those that would produce the allowed few events/year observed in XENON.", "The rates for these positive-signal benchmarks vary from $\\sim $ 4 per day (CRESST M1) to $\\sim $ 20 per hour (DAMA, SI case).", "For the XENON-allowed cases, the rates are several per year in the SI case, but a moderately larger $\\sim $ 2 per month in the SD case.", "At WIMP masses below 60 GeV, XENON begins to lose sensitivity: the rate above threshold becomes a smaller and smaller portion of the total rate.", "For low masses, one can thus choose cross-sections resulting in very large total rates (in both the human body and XENON detector), that produce only a few events above threshold and are thus not excluded by XENON.", "WIMP interactions represent a source of radiation in the human body, so a question arises: are WIMP collisions dangerous to humans?", "Here we compare the radiation due to WIMPs with that from natural sources, namely radioactivity here on Earth (including radon) as well as cosmic rays coming down through the atmosphere.", "The natural radiation background varies by location, with a typical annual exposure of 0.4–4 mSv (see Refs.", "[21], [22] for a review; here the unit of radiation exposure is Sieverts, or Sv).", "The cosmic-ray contribution is 0.3 mSv/yr at sea level and increases at higher elevations.", "Cosmic-ray muons deposit far more energy in the human body than do WIMPs.", "These muons pass through the human body at a rate of a few per second, depositing $\\sim $ 10–100 MeV of energy each, far larger than the $\\sim $ 10 keV deposited by a WIMP.", "For comparison, for the XENON-allowed benchmarks we have considered, the dose-equivalent exposure due to WIMP interactions is $\\mathcal {O}(10^{-11})$  mSv/yr, a negligible exposure compared to other natural radiation sources.", "Indeed we find that the radiation dose from cosmic-rays received each second exceeds the lifetime WIMP dose.", "Even for the higher WIMP interaction rates for the masses and cross-sections that can reproduce the CoGeNT, CRESST, and DAMA results, the WIMP radiation dose is negligible compared to other radiation sources.", "Thus WIMPs are harmless to the human body.", "In conclusion, we have studied the interactions of WIMPs with nuclei in a human body of mass 70 kg.", "We examined the contributions from a variety of elements in the body and found that the dominant contribution is from scattering off of oxygen nuclei for spin-independent (SI) interactions and hydrogen nuclei for spin-dependent (SD) interactions.", "For a canonical case of 60 GeV WIMP mass and the maximum elastic scattering cross-sections compatible with the experimental bounds from XENON and CDMS ($\\sigma _{\\mathrm {p,SI}}= 10^{-8}$  pb = $10^{-44}$  cm$^2$ and $\\sigma _{\\mathrm {p,SD}}= 2 \\times 10^{-3}$  pb), we found that on average five WIMPs hit one of the nuclei in the human body in a year via SI scattering and 30 via SD scattering.", "We also studied the 10–20 GeV WIMPs with much larger cross-sections that best fit the DAMA, COGENT, and CRESST data sets, and found much higher rates: in this case as many as $10^5$ WIMPs hit a nucleus in the human body in an average year, corresponding to almost one a minute.", "Finally, we have determined that, while these WIMP interactions represent a source of radiation in the body, the exposure rate is negligible compared to that from other natural sources of radiation and WIMP collisions are harmless to humans.", "K.F.", "acknowledges the support of the DOE and the Michigan Center for Theoretical Physics via the University of Michigan.", "K.F.", "thanks the Caltech Physics Dept for hospitality during her visit.", "C.S.", "is grateful for financial support from the Swedish Research Council (VR) through the Oskar Klein Centre." ] ]

[ [ "Holographic Storage of Biphoton Entanglement" ], [ "Abstract Coherent and reversible storage of multi-photon entanglement with a multimode quantum memory is essential for scalable all-optical quantum information processing.", "Although single photon has been successfully stored in different quantum systems, storage of multi-photon entanglement remains challenging because of the critical requirement for coherent control of photonic entanglement source, multimode quantum memory, and quantum interface between them.", "Here we demonstrate a coherent and reversible storage of biphoton Bell-type entanglement with a holographic multimode atomic-ensemble-based quantum memory.", "The retrieved biphoton entanglement violates Bell's inequality for 1 microsecond storage time and a memory-process fidelity of 98% is demonstrated by quantum state tomography." ], [ "Holographic Storage of Biphoton Entanglement Han-Ning Dai* Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China Han Zhang* Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China Sheng-Jun Yang Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China Tian-Ming Zhao Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China Jun Rui Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China You-Jin Deng Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China Li Li Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China Nai-Le Liu Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China Shuai Chen Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China Xiao-Hui Bao Physikalisches Institut, Reprecht-Karls-Universitat Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China Xian-Min Jin Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China Bo Zhao Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria Jian-Wei Pan Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics, University of Science and Technology of China,Hefei,Anhui 230026, China 03.67.Bg, 42.50.Ex Coherent and reversible storage of multi-photon entanglement with a multimode quantum memory is essential for scalable all-optical quantum information processing.", "Although single photon has been successfully stored in different quantum systems, storage of multi-photon entanglement remains challenging because of the critical requirement for coherent control of photonic entanglement source, multimode quantum memory, and quantum interface between them.", "Here we demonstrate a coherent and reversible storage of biphoton Bell-type entanglement with a holographic multimode atomic-ensemble-based quantum memory.", "The retrieved biphoton entanglement violates Bell's inequality for 1 $\\mu $ s storage time and a memory-process fidelity of 98% is demonstrated by quantum state tomography.", "[1]These authors contributed equally to this work.", "Faithfully mapping multi-photon entanglement into and out of quantum memory is of crucial importance for scalable linear-optical quantum computation [1] and long-distance quantum communication [2].", "Recently, storage of nonclassical light [3], [4] and single photons [5], [6], [7], [8], [9], [10], [11], [12] has been demonstrated in various quantum systems, such as atomic ensemble, solid system, and single atom.", "Among these, the atomic-ensemble-based quantum memory holds the promise to implement multimode quantum memory for multi-photon entanglement.", "A natural method for this purpose is to select spatially separated sub-ensembles of a large atomic ensemble as different quantum registers [13], [8], for which the number of stored modes is limited by the spatial dimension of the atomic ensemble.", "More powerful methods, such as exploring large optical depth of an atomic ensemble [14], [15], or utilizing photon echoes [16] or atomic frequency combs techniques [17], have been employed to demonstrate atomic-ensemble-based multi-mode memories.", "Figure: (a) Schematic view of the quad-mode holographic quantummemory.", "The control field is shined into the atomic cloud horizontally, and the signalmodes are incident from different directions in the same plane, with angles (θ 1 ,θ 2 ,θ 3 ,θ 4 )=(-1 ∘ ,-0.6 ∘ ,0.6 ∘ ,1 ∘ )(\\protect \\theta _{1},\\protect \\theta _{2},\\protect \\theta _{3},\\protect \\theta _{4})=(-1^{\\circ },-0.6^{\\circ },0.6^{\\circ },1^{\\circ }) relative to the the control field.", "(b) The typical Λ\\Lambda -type energy levels, with two ground states |g〉|g\\rangle and |s〉|s\\rangle and an excited state |e〉|e\\rangle .", "The |e〉-|g〉|e\\rangle - |g\\rangle and the |e〉-|s〉|e\\rangle -|s\\rangle transition are coupled to the signal and control fields, respectively.", "(c) Illustration of the wave vectors ofthe spin waves.", "The input signal field with wave vector of 𝐤 s,i \\mathbf {k}_{s,i} (i=1,2,3,4i=1,2,3,4) is mapped to a spin wave withwave vector 𝐪 i =𝐤 s,i -𝐤 c \\mathbf {q}_{i}=\\mathbf {k}_{s,i}-\\mathbf {k}_{c}, with 𝐤 c \\mathbf {k}_{c} the wave vector of thecontrol field.An alternative and elegant method is to implement the atomic ensemble as a holographic multimode quantum memory [18], [19], [20], using spatially overlapped but orthogonal spin waves as different quantum registers.", "For clarity, we illustrate this in the example of storing a single-photon state in an atomic ensemble of $N$ atoms that have two long-lived ground states $|g\\rangle $ and $|s\\rangle $  [21], [22], [23], [24].", "Initially, a “vacuum” state $|\\mathrm {vac}\\rangle =|g_{1}...g_{N}\\rangle $ is prepared such that all the atoms are at the $|g\\rangle $ state.", "The single-photon state is then mapped into the ensemble as a collective state $|1,\\mathbf {q}\\rangle =S_{\\mathbf {q} }^{\\dagger }|$ vac$\\rangle =(1/\\sqrt{N})\\sum _{j}e^{i\\mathbf {q}\\cdot \\mathbf {x}_{j}}|g_{1}\\cdots s_{j}\\cdots g_{N}\\rangle $ , where $\\mathbf {x}_{j}$ is the position of the $j$ th atom, $S_{\\mathbf {q}}^{\\dagger }=(1/\\sqrt{N})\\sum _{j}e^{i\\mathbf {q}\\cdot \\mathbf {x}_{j}}|s\\rangle _{j}\\langle g|$ is the collective creation operator of a spin wave with wave vector $\\mathbf {q}$ .", "For $N \\gg 1$ , one has $[S_{\\mathbf {q}_{1}},S_{\\mathbf {q}_{2}}^{\\dagger }]\\approx \\delta _{\\mathbf {q}_{1}\\mathbf {q}_{2}}$ , namely, the collective states satisfy the orthogonality relation $\\langle 1,\\mathbf {q}_{1}|1,\\mathbf {q}_{2}\\rangle \\approx \\delta _{\\mathbf {q}_{1}\\mathbf {q}_{2}}.$ Therefore, one can encode different qubits by different phase patterns and employ a single atomic ensemble as a holographic multimode quantum memory.", "Since the information is stored globally throughout the medium, one can achieve high-capacity data storage.", "Recently, holographic storage of classical light and microwave pulses have been demonstrated [25], [26], [27].", "Figure: Illustration of the experimental setup.", "The twopolarization-entangled photons produced by a narrowband SPDC source aredirected to the memory lab through 20-m fibers.", "Different polarizationcomponents are separated by the polarization beam splitter (PBS)1, and are coupled to the quad-mode holographic quantum memory.", "All thelight fields are turned into right-hand circular polarization (σ + \\protect \\sigma ^{+}) by wave plates.", "The biphoton entanglement are stored into the four quantum registers by adiabaticallyswitching off the control light.", "After a controllable delay, the photons areretrieved out, transferred back to their original polarization states by wave plates and combined on PBS 2 _{2}.", "Then the retrieved entangled photons are guided into filters, containing a Fabry-Perot cavity and a hot atomic cell tofilter out the leakages from the control light, and detected by single photondetectors.", "The path length difference between PBS 1 _{1} and PBS 2 _{2} is activelystabilized by an additional phase beam (dashed line).", "Inset: The experimental laser lights and atomic levels.Here we report an experimental demonstration of holographic storage of biphoton Bell-type entanglement with a single atomic ensemble, in which four orthogonal spin waves with different wave vectors are used as a quad-mode quantum memory.", "The posterior biphoton entanglement is mapped into and out of the quad-register holographic quantum memory, via a technique based on electromagnetically induced transparency (EIT).", "Violation of Bell's inequality is observed for storage time up to 1 $\\mu $ s and a memory-process fidelity of 98%, calculated by quantum state tomography, is achieved.", "The experimental scheme and setup are shown in Fig.", "REF and REF , respectively.", "In the memory lab, we prepare, within 14 ms, a cold $^{87}$ Rb atomic ensemble consisting of about $10^{8}$ atoms in a dark Magnetic-Optical-Trap (MOT).", "The temperature of the atomic cloud is about 140 $\\mu $ K, and the optical depth (OD) is about 10.", "The typical $\\Lambda $ -type energy-level configuration is shown in Fig.", "REF b, where $|g \\rangle $ , $|e\\rangle $ , and $|s\\rangle $ correspond to the $^{87}$ Rb hyper-fine states $|5S_{1/2},F=1\\rangle $ , $|5P_{1/2},F=2\\rangle $ , and $|5S_{1/2},F=2\\rangle $ , respectively.", "All the atoms are initially prepared at $|g\\rangle $ .", "A strong classical control field couples $|e\\rangle $ -$|s\\rangle $ transition with wave vector $\\mathbf {k}_{c}$ and beam waist diameter $w_{c}\\approx 850$ $\\mu $ m, while the to-be-stored quantum field, which has four components (see below), couples $|g\\rangle $ -$|e\\rangle $ transition with beam waist diameter $w_{s}\\approx 450$ $\\mu $ m. The control field is focused at the ensemble center, and the four components of the signal field are guided through the atomic cloud along four different directions, which are in the same plane but with different angles $\\theta _{i} \\, (i=1-4)$ relative to the control-light direction [28].", "We set $ (\\theta _{1},\\theta _{2},\\theta _{3},\\theta _{4})=(-1^{\\circ },-0.6^{\\circ },0.6^{\\circ },1^{\\circ })$ , as illustrated in Fig.", "REF a.", "By carefully adjusting the directions of the control and signal beams, we make all the light modes overlap in the center of the atomic ensemble.", "The atomic ensemble has a length $L\\approx 2$ mm, and the signal fields propagate within the control field during storage.", "Each component $i$ of the signal field is associated with a wave vector $\\mathbf {k}_{s,i}$ , and is to be stored in a spin wave with $\\mathbf {q}_{i}=\\mathbf {k}_{s,i}-\\mathbf {k}_{c}$ .", "By careful alignment, a holographic quad-mode quantum memory, with approximately equivalent optical depth and similar performance, is established.", "We measure the EIT transmission spectrum and perform slow-light experiment.", "For a control light with a Rabi frequency of about 7 MHz, we observe an EIT window of $2.2$ MHz and a delay time of about 160 ns for all the four modes.", "Note that such a holographic quantum memory is different from the scheme in Ref.", "[8], [13], where each signal mode requests a spatially separated atomic sub-ensemble.", "The biphoton entanglement comes from a narrowband cavity-enhanced spontaneous parametric down conversion (SPDC) entanglement source as in previous work [29], [9].", "The source cavity contains three main parts, i.e., a nonlinear crystal, a tuning crystal and an output coupler.", "The nonlinear crystal is a 25-mm type-II a periodically poled KTiOPO$_{4}$ (PPKTP) crystal, whose operational wavelength $\\lambda \\approx 795$ nm is designed to match the D1 transition line of $^{87}$ Rb.", "The cavity is locked intermittently to a Ti: Sapphire laser using the Pound-Drever-Hall method.", "The linewidth and finesse of the cavity are measured to be 5 MHz and 170, respectively.", "Polarization-perpendicular photon pairs are created by applying a ultraviolet (UV) pumping light, which is up converted from the Ti: Sapphire laser.", "Single-mode output is achieved by using a filter cavity (made of a single piece of fused silica of about 6.35 mm) with a finesse of 30, which removes the background modes.", "Polarization-entangled photon pairs are post-selected by interfering the twin photons at polarization beam splitters (PBSs).", "The ideal outcome state corresponds to a Bell state $|\\phi ^{+}\\rangle _{p}=(|H\\rangle _{1}|H\\rangle _{2}+|V\\rangle _{1}|V\\rangle _{2})/\\sqrt{2}$ with H(V) represents the horizontal(vertical) polarization of the photons.", "Under a continuous wave (CW) pump with a pump power of 4 mW, the spectrum brightness of the polarization-entangled pairs after the filter cavity is about 50 s$^{-1}$ mW$^{-1}$ MHz$^{-1}$ .", "In the storage experiment, the entangled signal photons are created by a 200 ns pump pulse, which is cut from a 28 mW CW pump laser.", "The production rate is about 33 s$^{-1}$ .", "The measured ratio of counts under $|HH\\rangle /|HV\\rangle $ and $|++\\rangle /|+-\\rangle $ bases are 14.3:1 and 23.1:1, respectively, with $|\\pm \\rangle =(|H\\rangle \\pm |V\\rangle )/\\sqrt{2}$ .", "Figure: Average visibility (red, left axis) and overall storage efficiency (blue, right axis) of the retrieved biphoton state versus storage time.", "An exponential fitting (blue solid line) of the storage efficiency yields a lifetime of τ=2.8±0.2\\protect \\tau =2.8\\pm 0.2 μ\\protect \\mu s. The retrieved biphoton state is measured under |H/V〉|H/V\\rangle , |±〉|\\pm \\rangle , and |R/L〉|R/L\\rangle bases with |±〉=(|H〉±|V〉)/2|\\pm \\rangle =(|H\\rangle \\pm |V\\rangle )/\\sqrt{2}, and |R/L〉=(|H〉±i|V〉)/2|R/L\\rangle =(|H\\rangle \\pm i|V\\rangle )/\\sqrt{2}.The average visibility is fitted using V=1/(a+be 2t/τ )V=1/(a+b e^{2t/\\tau }) (red solid line) with aa and bb the fitting parameters.", "The result shows that within about 1.6 μ\\protect \\mu s the visibility is above the threshold of 0.71 to violatethe CHSH-Bell's inequality.", "Error bars represent ±\\pm standard deviationThe signal photon pair is directed to the memory lab with 20-meter single-mode fibers.", "The different polarization components are spatially separated by PBS$_{1}$ , then transferred to right-hand circular polarized ($\\sigma ^{+}$ ) by wave plates, and then guided to the four quantum registers by lens (see Fig.", "REF ).", "More precisely, the $|H\\rangle _{1}$ , $|H\\rangle _{2}$ , $|V\\rangle _{2}$ , and $|V\\rangle _{1}$ polarization components are coupled to mode 1-4, respectively.", "After these components entering the atomic ensemble, we adiabatically switch off the control light, and the photonic entanglement is mapped into the atomic ensemble.", "This yields an entanglement among the four quantum registers $|\\psi \\rangle _{a}=(S_{\\mathbf {q}_{1}}^{\\dagger }S_{\\mathbf {q}_{2}}^{\\dagger }+S_{\\mathbf {q}_{3}}^{\\dagger }S_{\\mathbf {q}_{4}}^{\\dagger })|\\text{vac}\\rangle /\\sqrt{2}$ After a controllable delay, we adiabatically switch on the control light and convert the atomic entanglement back into photonic entanglement.", "The polarization states of the output photons are transferred back linearly polarized by wave plates and combined by PBS$_{2}$ to reconstruct the biphoton entanglement.", "The two retrieved entangled photons are respectively guided into a filter consisting of a Fabry-Perot cavity (transmission window 600 MHz) and a pure $^{87}$ Rb vapor cell with atoms prepared in $|5S_{1/2},F=2\\rangle $ , and then detected by single-photon detectors.", "The measured overall average storage efficiency is shown in Fig.", "REF , which yields a $1/e$ lifetime of $2.8\\pm 0.2$ $\\mu $ s. The measured coincidence rate without storage and after 1 $\\mu $ s storage time is 1.3 s$^{-1}$ and 0.03 s$^{-1}$ , respectively.", "The propagating phase between PBS$_{1}$ and PBS$_{2}$ is actively stabilized within $\\lambda _{l} /30$ by an additional phase lock beam with $\\lambda _{l}\\approx 780$ nm [28], [9].", "To verify that the biphoton Bell-type entanglement is faithfully mapped into and out of the four holographic quantum registers, we first measure the retrieved biphoton state in $|H/V\\rangle $ , $|\\pm \\rangle $ , and $|R/L\\rangle =(|H\\rangle \\pm i|V\\rangle )/\\sqrt{2} $ bases at different storage time.", "The average visibility is shown in Fig.", "REF , which for storage time less than 1.6 $\\mu $ s, exceeds the threshold 0.71 to violate CHSH-Bell's inequality.", "Note that the reduction of the visibility with storage time is mainly due to the background coincidences caused by the dark counts and the leakage from the control field.", "We further measure the correlation function $E(\\phi _{1},\\phi _{2})$ , with $\\phi _{1}(\\phi _{2})$ the polarization angle for signal photon 1(2), and calculate quantity $S=|-E(\\phi _{1},\\phi _{2})+E(\\phi _{1},\\phi _{2}^{\\prime })+E(\\phi _{1}^{\\prime },\\phi _{2})+E(\\phi _{1}^{\\prime },\\phi _{2}^{\\prime })|,$ where $(\\phi _{1},\\phi _{1}^{\\prime },\\phi _{2},\\phi _{2}^{\\prime })=(0^{\\circ },45^{\\circ },22.5^{\\circ },67.5^{\\circ }).$ We obtain $S=2.54\\pm 0.03$ for the input state, and $S=2.25\\pm 0.08$ for the retrieved state after 1 $\\mu $ s storage.", "The violation of the CHSH-Bell's inequality ($S>2$  )[30] confirms the entanglement has been coherently and reversibly stored in the quad-mode holographic quantum memory.", "Figure: Density matrix of the input state (a, b) and of the output state after 1 μ\\mu s storage (c, d), obtained from quantum state tomography.", "a and c are for the real parts, and b andd are for the imaginary parts.To quantitatively assess the fidelity of the storage process, we perform the quantum state tomography [31], [32] to construct the density matrix $\\rho _{\\rm {in}}$ of the input and $\\rho _{\\rm {out}}$ of the output state after 1 $\\mu $ s storage, in which the polarization state of each photon is measured with two single-photon detectors under different detection settings.", "The results are illustrated in Fig.", "REF , from which the fidelity of the measured state $\\rho $ on the ideal Bell state $\\rho _{\\phi ^{+}}$ is calculated as $F(\\rho _{\\phi ^+}, \\rho )=(\\mathrm {Tr}\\left(\\sqrt{\\sqrt{\\rho _{\\phi ^+} }\\rho \\sqrt{\\rho _{\\phi ^+}}}\\right))^{2} $ .", "A Monte Carlo simulation technique [32] is applied to calculate the uncertainties of the fidelity.", "Briefly, an ensemble of 100 random sets of data are generated according to Possionian distribution and then the density matrices are obtained by means of the maximum likelihood method.", "This yields a distribution of fidelities, from which the mean value and uncertainties of the fidelity are calculated.", "We obtain $F(\\rho _{\\phi ^{+}},\\rho _{\\mathrm {in}})=(87.9\\pm 0.5)\\%$ for the input state $\\rho _{\\mathrm {in}}$ and $F(\\rho _{\\phi ^{+}},\\rho _{\\mathrm {out}})=(81\\pm 2)\\%$ , beyond the threshold [33] 78% for Werner states to violate Bell's inequality.", "The fidelity of the memory process is given by $F(\\rho _{\\text{in}},\\rho _{\\text{out}}) =(98.2\\pm 0.9)\\%$ .", "In summary, we have experimentally demonstrated the coherent mapping of a biphoton Bell-type entanglement, created from a narrowband SPDC source, into and out of a four-register holographic quantum memory, with a high memory-process fidelity of 98% for 1 $\\mu $ s storage time.", "The narrowband photonic entanglement source inherits the advantage of conventional broadband SPDC source, and can be used to generate multi-photon entanglement beyond biphoton entanglement.", "A novel feature of the holographic quantum memory is that one can use more modes by simply choosing the directions of the signal and control fields.", "The memory capacity $N_m$ for a coplanar configuration may be estimated by the geometric mean of the Fresnel numbers of the illuminated regions as $N_m\\sim w_{c}w_{s}/(\\lambda L)$  [20], which is $N_m\\sim 240$ for our experimental parameters.", "Increasing the beam waist diameters or extending to a three-dimensional geometry would allow much more modes.", "Individual control of each quantum register may be achieved by using an optical cavity and employing the stimulated Raman adiabatic passage technique [18] or employing the phase match method [20].", "To extend our work to storage of multi-photon entanglement, we have to improve the brightness of the entanglement source, and increase the retrieval efficiency and lifetime of the quantum memory.", "The storage efficiency is about 15%, which can be improved by further increasing the optical depth and reducing the linewidth of the narrowband entanglement source.", "The storage time is about 1 $\\mu $ s, which is limited by inhomogeneous broadening induced by residual magnetic field, and can be improved to be of order of millisecond by trapping the atoms in optical lattice and using the magnetic-insensitive state [34], [35].", "Our work opens up the possibility of scalable preparation and high-capacity storage of multi-photon entanglement, and also sheds light on the emerging field of holographic quantum information processing.", "This work was supported by the National Natural Science Foundation of China, the National Fundamental Research Program of China (grant no.", "2011CB921300), the Chinese Academy of Sciences, the Austrian Science Fund, the European Commission through the European Research Council Grant and the Specific Targeted Research Projects of Hybrid Information Processing." ] ]

[ [ "Absolute linear instability in laminar and turbulent gas/liquid\n two-layer channel flow" ], [ "Abstract We study two-phase stratified flow where the bottom layer is a thin laminar liquid and the upper layer is a fully-developed gas flow.", "The gas flow can be laminar or turbulent.", "To determine the boundary between convective and absolute instability, we use Orr--Sommerfeld stability theory, and a combination of linear modal analysis and ray analysis.", "For turbulent gas flow, and for the density ratio r=1000, we find large regions of parameter space that produce absolute instability.", "These parameter regimes involve viscosity ratios of direct relevance to oil/gas flows.", "If, instead, the gas layer is laminar, absolute instability persists for the density ratio r=1000, although the convective/absolute stability boundary occurs at a viscosity ratio that is an order of magnitude smaller than in the turbulent case.", "Two further unstable temporal modes exist in both the laminar and the turbulent cases, one of which can exclude absolute instability.", "We compare our results with an experimentally-determined flow-regime map, and discuss the potential application of the present method to non-linear analyses." ], [ "Introduction", "We investigate linear absolute and convective instability for a liquid film sheared by laminar and turbulent gas streams in a channel.", "In the oil/gas industries, this approach serves as a model that can be used to predict the onset of droplet entrainment [15].", "The motivation for this work is twofold: previous work on the turbulent case focussed uniquely on temporal stability analysis [23], [6], [5], [26], while previous work on the laminar case [34] omitted large regions of parameter space that are relevant to the oil-and-gas industries, and which are found herein to be absolutely unstable.", "The current work aims to fill in these two gaps in the literature and introduces modifications and extensions of existing methodologies (developed previously for single-phase flows or for temporal stability analysis only) that are potentially of interest in other areas.", "The route to droplet entrainment from a liquid layer into a gas stream in pipe and channel flows is still unclear.", "The idealized system of fully-developed flow with a flat gas/liquid interface is linearly unstable to infinitesimally small perturbations.", "For a laminar base state, a stratification in dynamic viscosity produces instability; for a turbulent base state, the mechanism of [24] may also dominate for deep liquid layers at large Froude numbers [33].", "Other mechanisms, such as a Tollmien–Schlichting mode in the liquid [6] may also be important in certain parameter regimes.", "Although linear (temporal) instability is arguably a necessary condition for droplet entrainment in the gas, it does not provide much insight regarding whether a localized disturbance grows whilst it is merely convected downstream or whether instead the disturbance destabilizes the entire system.", "To answer these questions a spatio-temporal analysis is required.", "A recent spatio-temporal analysis by [34] has revealed a region in parameter space wherein the laminar base state is absolutely unstable, indicating that the system does not merely act as an amplifier but also as a generator of disturbances.", "This was found to include a large range of practically useful viscosity ratios, but was limited to a density ratio of $O(1)$ .", "The laminar density-matched problem studied by [34] has only limited applicability in oil/gas transport, where the density ratio is large, and where the operating conditions produce turbulence in the gas layer, or in both layers (see, e.g., the visualisation of droplet entrainment events by [20]).", "The present study therefore investigates the corresponding problem for a turbulent base state, but the laminar case is also revisited.", "Although replacement of the base state by a turbulent one [30], [31] in a linear modal spatio-temporal analysis may seem trivial, the results turn out to be difficult to interpret, due to the presence of multiple unstable modes.", "Therefore, in this study, we have developed a twin-track approach, in which modal analysis and ray analysis are combined to locate and characterize absolute instability.", "The ray analysis used herein extends the work of [12] for single-phase flows.", "This approach yields surprising results.", "In particular, it has revealed significant regions in parameter space where the turbulent base state is absolutely unstable for large density ratios.", "This also holds for the laminar base state, thereby contradicting [34], who only found absolute instability for density ratios of $O(1)$ .", "We have therefore revisited the spatio-temporal work of [34], and have established using both ray and modal analyses that the laminar system is indeed absolutely unstable at large density ratios for a substantial range of viscosity ratios and liquid-film depths.", "Reasons for the oversight in the previous work are given.", "The paper is organized as follows.", "The turbulence model and the linear stability analysis are formulated in Section .", "We discuss some theoretical and numerical aspects of the linear stability analysis in Section , paying close attention to the development of a ray analysis for two-phase flows.", "We apply this technique to the turbulent base state in Section , while the laminar case is revisited in Section .", "In Section  we argue for the importance of using the ray analysis and the modal analysis simultaneously, for complete and accurate results.", "We also compare the flow-regime boundaries identified herein with those found in experiments, and discuss the generalisation of our work to non-linear and non-parallel flows." ], [ "Linear stability analysis", "In this section we review a model of turbulent channel flow used elsewhere by the authors [30], [31].", "This is a Reynolds-averaged model describing co-current flow in a stably-stratified system where the upper layer is a turbulent gas and the lower layer a laminar liquid film.", "We also recall the Orr–Sommerfeld Table: Table of parameters in the two-phase problem and their typical values.", "The range of values of the liquid-layer dynamic viscosity μ L \\mu _L, the liquid-layer density ρ L \\rho _L, and the liquid-film thickness d L d_L can be backed out from the gas-layer analogues and the ratios mm, rr, and ϵ\\epsilon , respectively.technique used to determine the stability of the interface in this two-layer system.", "For reference, typical values of the problem parameters are given in Table REF , where the subscripts $G,L$ indicate the gas and liquid, respectively." ], [ "The base state", "We consider a flat-interface base state in two-layer stratified flow (Figure REF ).", "The bottom layer is a thin, laminar, liquid layer, and the top layer is gaseous, turbulent and fully-developed.", "A pressure gradient is applied along the channel.", "The base-state profile of the system is a uni-directional flow in the horizontal, $x$ - direction as a function of the cross-flow coordinate $z$ .", "In the bottom layer, the profile is determined by balancing the viscous and the pressure forces; in the top layer, the viscosity in the balance law must be supplemented by the turbulent eddy viscosity: $ \\mu _G\\frac{\\partial U_0}{\\partial z}+\\tau _{0}=\\tau _{\\mathrm {i}}+\\frac{dP}{dL}z,$ where $U_0(z)$ is the base-state velocity in the gas, $\\tau _\\mathrm {i}$ is the interfacial shear stress, and $dP/dL$ is the applied pressure gradient.", "Moreover, $\\tau _{0}=-\\rho _G\\langle u^{\\prime } w^{\\prime }\\rangle $ is the turbulent shear stress due to the averaged effect of the turbulent fluctuating velocities, $u^{^{\\prime }}$ and $w^{^{\\prime }}$ .", "In channel flows, it is appropriate to model this term using an eddy-viscosity model [25].", "In mixing-length theory, the eddy viscosity depends on the local rate of strain [7], which means that the turbulent shear stress depends on the square of the rate of strain.", "Instead, as in the work of [31] and [4], we use an interpolation function for the eddy viscosity.", "This mimics the ordinary mixing-length theory near the interface and near the wall, and transitions smoothly from the near-wall and near-interfacial regions to the zone surrounding the gas centreline.", "Thus, the turbulent shear stress is linear in the rate of strain, and $\\tau _{0}=\\mu _T\\frac{\\partial U_0}{\\partial z},\\qquad \\mu _T=\\kappa \\rho _G d_GU_{*\\mathrm {w}}\\mathcal {F}\\left(s\\right)\\psi _{\\mathrm {i}}\\left(s\\right)\\psi _{\\mathrm {w}}\\left(1-s\\right),\\qquad s=z/d_G,$ where $\\mu _T$ is the eddy viscosity and $\\kappa $ is the von Kármán constant, taken as 0.4.", "Additionally, $U_{*\\mathrm {w}}$ is the friction velocity at the upper wall.", "The corresponding stress is $\\tau _\\mathrm {w}$ , with $U_{*\\mathrm {w}}=\\sqrt{|\\tau _\\mathrm {w}|/\\rho _G}$ .", "Similarly, the interfacial friction velocity is defined as $U_*=\\sqrt{\\tau _\\mathrm {i}/\\rho _G}$ .", "The function $\\mathcal {F}$ is the interpolation function described in [30] and [31]: $\\mathcal {F}\\left(s\\right)=s\\left(1-s\\right)\\left[\\frac{s^3+|R|^{5/2}\\left(1-s\\right)^3}{R^2\\left(1-s\\right)^2+Rs\\left(1-s\\right)+s^2}\\right],\\qquad R=\\tau _{\\mathrm {i}}/\\tau _{\\mathrm {w}}.\\\\$ Finally, $\\psi _{\\mathrm {i}}$ and $\\psi _{\\mathrm {w}}$ are interface and wall functions respectively, which damp the effects of turbulence to zero rapidly near the interface and the wall.", "These are given below.", "Figure: A schematic diagram of the base flow.", "The liquid layer is laminar,while the gas layer exhibits fully-developed turbulence, described here bya Reynolds-averaged velocity profile.", "A pressure gradient in the xx-directiondrives the flow.We non-dimensionalize the problem on the gas-layer depth $d_G$ , the gas-layer density $\\rho _G$ , the gas-layer viscosity $\\mu _G$ , and the velocity scale $U_p$ , where $\\rho _G U_p^2=d_G|dP/dL|.$ Then, integration of Equation (REF ) yields the non-dimensional base state: $U_0\\left(z\\right)={\\left\\lbrace \\begin{array}{ll}\\frac{\\mu _G}{\\mu _L}\\left[-\\tfrac{1}{2}Re\\left(z^2-\\epsilon ^2\\right)+\\frac{Re_*^2}{Re}\\left(z+\\epsilon \\right)\\right],&-\\epsilon \\le z\\le 0,\\\\\\frac{\\mu _G}{\\mu _L}\\left(\\tfrac{1}{2}\\epsilon ^2 Re+\\epsilon \\frac{Re_*^2}{Re}\\right)+\\frac{Re_*^2}{Re}\\int _0^{z}\\frac{\\left(1-\\frac{Re^2}{Re_*^2}s\\right)ds}{1+\\frac{\\kappa Re_{*}}{\\sqrt{|R|}}g\\left(s\\right)\\psi _\\mathrm {i}\\left(s\\right)\\psi _\\mathrm {w}\\left(1-s\\right)},&0\\le z\\le 1,\\end{array}\\right.", "}$ where $\\epsilon =d_L/d_G$ , $Re=\\rho _G U_p d_G/\\mu _G$ , and where $Re_*=(U_*/U_p)Re$ .", "Knowledge of $Re_*$ amounts to knowledge of the interfacial shear stress.", "This is not known a priori as a function of the externally-imposed parameters.", "However, it is available within the model, and the root-finding procedure $U_0\\left(1;Re_*\\right)=0$ yields $Re_*$ as a function of the parameters $\\left(Re,\\epsilon ,\\mu _L/\\mu _G\\right)$ .", "For completeness, we also list the interfacial and wall functions: $\\psi _\\mathrm {i}\\left(s\\right)&=&1-e^{-C_A Re_*^2s^2},\\\\\\psi _\\mathrm {w}\\left(s\\right)&=&1-e^{-C_A Re_*^2s^2/R^2},$ where $C_A$ is a constant fixed such that the interfacial and wall viscous sublayers are five wall-units in extent [27].", "The functional forms for $\\mathcal {F}$ and the wall functions are confirmed by the excellent agreement between the model predictions of the base state and experiments and numerical simulation [31].", "Having constituted the base state, we now introduce the theory necessary to determine its stability." ], [ "The perturbation equations", "We base the dynamical equations for the interfacial motion on the Reynolds-averaged Navier–Stokes (RANS) equations.", "The turbulent velocity is decomposed into averaged and fluctuating parts.", "We make the quasi-laminar approximation, which means that the fluctuations are only considered in the base state, where they are modelled using the eddy viscosity (Section REF ).", "Before deriving equations for perturbations induced by small waves at the interface, we discuss the dynamics of these perturbations with respect to their interactions with the turbulent eddies in the flow.", "In a realisation of the three-dimensional turbulent two-layer flow with small-amplitude waves, a Fourier mode decomposition can be made of the interface elevation and field variables.", "Here, ensemble-averaged Fourier modes are assumed to be predominantly two-dimensional.", "The correspondingly averaged equations of motion are linearized in terms of wave amplitude.", "The linearized problem contains wave-induced Reynolds stress terms (WIRSs), but these have been found recently not to be significant in two-layer flows such as those studied here [31], [33].", "This can also be seen from an order-of-magnitude estimate of the WIRSs terms compared to inertial terms in the perturbation momentum equation.", "Further theoretical justification exists for the case of viscosity-contrast instability, where the instability is dominated by conditions close to the interface, a zone where the perturbation turbulent stresses are damped rapidly to zero by the existence of viscous sublayers.", "In practical terms, the quasi-laminar approximation, wherein the WIRSs are ignored and the effect of turbulence is assumed to be entirely through the base-state velocity profile, while brutal in its simplicity, yields similar results to other turbulence models that explicitly include the WIRSs.", "It also predicts critical Reynolds numbers for the onset of wavy flow that agree with the laboratory experiments of [9] and [10].", "The reader is referred to the papers by [31], [33] for further details.", "Here, we study linear spatio-temporal instability.", "Although the above-mentioned prior findings are limited to temporal Fourier modes, there is an analytical connection between spatio-temporal and temporal modes [32] (see also Appendix ), such that the properties of the temporal study are inherited by the spatio-temporal one.", "Therefore, we make here also the quasi-laminar approximation.", "Thus, a small disturbance $z=\\eta \\left(x,t\\right)$ centred around the flat interface $z=0$ gives rise to disturbances in the velocity and pressure fields, which satisfy the following linearized equations of motion in the $j^{\\text{th}}$ phase ($j=L,G$ ): $r_j\\left[\\frac{\\partial }{\\partial t}\\delta u+U_0\\frac{\\partial }{\\partial x} \\delta u+\\frac{\\mathrm {d}U_0}{\\mathrm {d}{z}}\\delta w\\right]&=&-\\frac{\\partial }{\\partial x}\\delta p+\\frac{m_j}{Re}\\left(\\frac{\\partial ^2}{\\partial x^2}+\\frac{\\partial ^2}{\\partial z^2}\\right)\\delta u,\\\\r_j\\left[\\frac{\\partial }{\\partial t}\\delta w+U_0\\frac{\\partial }{\\partial x}\\delta w\\right]&=&-\\frac{\\partial }{\\partial z}\\delta p+\\frac{m_j}{Re}\\left(\\frac{\\partial ^2}{\\partial x^2}+\\frac{\\partial ^2}{\\partial z^2}\\right)\\delta w,\\\\\\frac{\\partial }{\\partial x}\\delta u+\\frac{\\partial }{\\partial z}\\delta w&=&0,$ where $\\left(r_L,r_G\\right)=\\left(r,1\\right)$ and $\\left(m_L,m_G\\right)=\\left(m,1\\right)$ .", "Using the incompressibility condition, this system of equations reduces to a single equation in the streamfunction $\\phi $ .", "Further simplification occurs when the streamfunction is written as a sum of normal modes: $\\phi \\left(x,z,t\\right)=\\frac{1}{2\\pi }\\int _{C_\\alpha =\\mathbb {R}}\\mathrm {d}\\alpha \\, \\mathrm {e}^{\\mathrm {i}\\alpha x}\\phi _\\alpha \\left(z,t\\right),$ which in turn can be written in Laplace-transform notation: $\\phi \\left(x,z,t\\right)=\\frac{1}{4\\pi ^2}\\int _{C_\\alpha }\\mathrm {d}\\alpha \\int _{C_\\omega }\\mathrm {d}\\omega \\, \\mathrm {e}^{\\mathrm {i}\\left(\\alpha x-\\omega t\\right)}\\phi _{\\alpha \\,\\omega }\\left(z\\right),$ where $C_\\omega $ is the Bromwich contour [2].", "If the $\\omega $ -singularities in the function $\\phi _{\\alpha \\,\\omega }\\left(z\\right)$ lie below the real axis in the complex-$\\omega $ plane, then the integral (REF ) is an ordinary double Fourier integral.", "Using the Fourier and Laplace decompositions, Equations (REF ) reduce to the Orr–Sommerfeld equation: $\\mathrm {i}\\alpha r_j\\left[\\left(U_0-\\frac{\\omega }{\\alpha }\\right)\\left(\\mathrm {D}^2-\\alpha ^2\\right)\\phi _{\\alpha \\,\\omega }-\\frac{\\mathrm {d}^2U_0}{\\mathrm {d}z^2}\\phi _{\\alpha \\,\\omega }\\right]=\\frac{m_j}{Re}\\left(\\mathrm {D}^2-\\alpha ^2\\right)^2\\phi _{\\alpha \\,\\omega },$ where $\\mathrm {D}=\\mathrm {d}/\\mathrm {d}z$ .", "Equation (REF ) only holds in the interior parts of the domain, $z\\in \\left(-\\epsilon ,0^-\\right)\\cup \\left(0^+,1\\right)$ .", "To close the Equation (REF ), no-slip and no-penetration conditions are applied at $z=-\\epsilon $ and $z=1$ : $\\phi _{\\alpha \\,\\omega }\\left(-\\epsilon \\right)=\\mathrm {D}\\phi _{\\alpha \\,\\omega }\\left(-\\epsilon \\right)=\\phi _{\\alpha \\,\\omega }\\left(1\\right)=\\mathrm {D}\\phi _{\\alpha \\,\\omega }\\left(1\\right)=0,$ and the streamfunction is matched across the interface $z=0$ , where the following conditions hold (we use the notation $c=\\omega /\\alpha $ ): $\\phi _{L}&=\\phi _{G},\\\\\\mathrm {D}\\phi _{L}&=\\mathrm {D}\\phi _{G}+\\frac{\\phi _G}{c-U_0}\\left(\\frac{\\mathrm {d}U_{0}}{\\mathrm {d}{z}}\\bigg |_{0^+}-\\frac{\\mathrm {d}U_{0}}{\\mathrm {d}{z}}\\bigg |_{0^-}\\right),\\\\m\\left(\\mathrm {D}^2+\\alpha ^2\\right)\\phi _L&=\\left(\\mathrm {D}^2+\\alpha ^2\\right)\\phi _{G},$ $m\\left(\\mathrm {D}^3\\phi _L-3\\alpha ^2\\mathrm {D}\\phi _L\\right)+\\mathrm {i}\\alpha rRe\\left(c-U_0\\right)\\mathrm {D}\\phi _L+\\mathrm {i}\\alpha rRe \\frac{\\mathrm {d}U_0}{\\mathrm {d}z}\\bigg |_{0^-}\\phi _L-\\frac{\\mathrm {i}\\alpha r Re}{c-U_0}\\left(F+\\alpha ^2 S\\right)\\phi _L\\\\=\\left(\\mathrm {D}^3\\phi _{G}-3\\alpha ^2\\mathrm {D}\\phi _{G}\\right)+\\mathrm {i}\\alpha Re\\left(c-U_0\\right)\\mathrm {D}\\phi _{G}+\\mathrm {i}\\alpha Re\\frac{\\mathrm {d}U_{0}}{\\mathrm {d}z}\\bigg |_{0^+}\\phi _{G}.$ Here $F$ and $S$ denote parameters that encode the effects of gravity and surface tension, respectively; they are defined here for the first time as $F=\\frac{g d_G}{\\left(\\mu _G/\\rho _G d_G\\right)^2}\\frac{r-1}{Re^2}&:=&F_0(r-1)/Re^2,\\\\S=\\frac{\\gamma }{\\mu _G^2/\\rho _G d_G}\\frac{1}{Re^2}&:=&S_0/Re^2,$ where $g$ is acceleration due to gravity and $\\gamma $ is surface tension.", "The appropriate range of values for $F_0$ and $S_0$ is discussed in Sec. .", "We abbreviate the Orr-Sommerfeld (or OS) equation (REF ) and the matching conditions (REF )–(REF ) using operator notation, $\\mathcal {L}_{\\alpha \\,\\omega }\\phi _{\\alpha \\,\\omega }=\\mathrm {i}\\omega \\mathcal {M}_{\\alpha \\,\\omega }\\phi _{\\alpha \\,\\omega }.$ This equation amounts to an eigenvalue equation, which we solve numerically by introducing a trial solution: $\\phi _{\\alpha \\,\\omega }\\left(z\\right)&\\approx &\\sum _{n=0}^{N_1}a_n T_n\\left(\\frac{2z}{\\epsilon }+1\\right),\\qquad -\\epsilon \\le z\\le 0,\\\\\\phi _{\\alpha \\,\\omega }\\left(z\\right)&\\approx &\\sum _{n=0}^{N_2}b_n T_n\\left(2z-1\\right),\\qquad 0 \\le z\\le 1,$ where $T_n(\\cdot )$ is the $n^{\\mathrm {th}}$ Chebyshev polynomial.", "We substitute Equation (REF ) into Equation (REF ) and evaluate the result at $N_1+N_2-6$ interior points.", "The ansatz (REF ) is also substituted into the eight boundary and interfacial conditions.", "This yields $N_1+N_2+2$ linear equations in as many unknowns.", "In matrix terms, we have to solve $L_{\\alpha \\,\\omega }\\mathbf {v}=\\mathrm {i}\\omega M_{\\alpha \\,\\omega }\\mathbf {v},$ where $\\mathbf {v}=\\left(a_0,..,a_{N_1},b_0,..,b_{N_2}\\right)^T$ .", "Such an equation is readily solved using linear-algebra packages.", "This method is described in more detail and its implementation is tested against benchmarks in another paper by the present authors [34].", "The number of collocation points $(N_1+1,N_2+1)$ is adjusted until convergence is achieved.", "The application of this numerical method will be the subject of the following sections." ], [ "Further numerical methods and postprocessing", "In this section we revisit the basic definition of absolute instability, namely that the streamfunction response to a localized disturbance should grow exponentially in time at the origin of the disturbance.", "Solving the associated Cauchy problem gives a quick and clear method to characterize the instability.", "This approach also enables us to pinpoint the source of the instability through an energy-budget analysis.", "We also review herein an equivalent method to determine absolute instability, namely modal analysis." ], [ "Modal analysis", "A purely temporal analysis involves the solution of the eigenvalue problem (REF ), where we write $\\alpha =\\alpha _{\\mathrm {r}}+\\mathrm {i}\\alpha _{\\mathrm {i}}$ , $\\omega =\\omega _{\\mathrm {r}}+\\mathrm {i}\\omega _{\\mathrm {i}}$ , for $\\alpha =\\alpha _{\\mathrm {r}}$ only.", "This gives a dispersion relation $\\left(\\omega _{\\mathrm {i}}^{\\mathrm {temp}}(\\alpha _{\\mathrm {r}}),\\omega _{\\mathrm {i}}^{\\mathrm {temp}}(\\alpha _{\\mathrm {r}})\\right)=\\left(\\omega _{\\mathrm {r}}(\\alpha _{\\mathrm {r}},\\alpha _{\\mathrm {i}}=0),\\omega _{\\mathrm {i}}(\\alpha _{\\mathrm {r}},\\alpha _{\\mathrm {i}}=0)\\right),$ with associated group velocity $c_{\\mathrm {g}}=d\\omega _{\\mathrm {i}}^{\\mathrm {temp}}/d\\alpha _{\\mathrm {r}}$ .", "The pair $(\\alpha _{\\mathrm {r}},\\omega _{\\mathrm {i}}^{\\mathrm {temp}}(\\alpha _{\\mathrm {r}}))$ that maximizes $\\omega _{\\mathrm {i}}$ is called the most dangerous mode.", "The flow is linearly unstable if $\\omega _{\\mathrm {i}}^{\\mathrm {temp}}>0$ for the most dangerous mode.", "Unstable parallel flows are further classified as convectively unstable if initially localized pulses are amplified in at least one moving frame of reference but are damped in the laboratory frame, and absolutely unstable if such pulses lead to growing disturbances in the entire domain in the laboratory frame.", "To describe such an instability, we use the description of [19].", "An unstable parallel flow is absolutely unstable if the following criteria have all been met: (i) $\\omega _{\\mathrm {i}0}:=\\omega _{\\mathrm {i}}(\\alpha _0)>0$ , where $\\alpha _0$ is the wave number at which the complex derivative $d\\omega /d\\alpha $ is zero, (ii) the corresponding saddle point $\\alpha _0$ in the complex $\\alpha $ -plane is the result of the coalescence of spatial branches that originate from opposite half-planes at a larger and positive value of $\\omega _{\\mathrm {i}}$ and (iii) the saddle point pinches at $\\omega _{\\mathrm {i}0}$ ; this is verified by locating a cusp at $\\omega _{\\mathrm {i}0}$ in the complex $\\omega $ plane [21], [34] and ensuring that the complex wave number corresponding to the pinching point coincides with $\\alpha _0$ ." ], [ "Ray analysis", "The linear stability equations (REF ) in streamfunction form (Equation (REF )), together with the boundary and initial conditions (REF )–(REF ) can be neatly encoded in linear-operator form: $\\mathcal {L}\\overline{\\phi }=\\mathcal {M}\\partial _t\\overline{\\phi },$ where we study $\\overline{\\phi }(x,z,t)$ , the filtered streamfunction containing only positive real wave numbers [19]; here $\\mathcal {L}$ and $\\mathcal {M}$ are linear operators (cf.", "Equation (REF )).", "When an impulsive, localized force is applied to the streamfunction, Equation (REF ) is modified: $\\mathcal {L}\\overline{\\phi }+\\delta (x)\\delta (z)\\delta (t)=\\mathcal {M}\\partial _t\\overline{\\phi },$ in terms of the Dirac delta function $\\delta (.", ")$ .", "The solution of Equation (REF ) (determined here in a domain with periodic boundaries at $x=\\pm L_x/2$ ) can be used to characterize the instability that develops from the impulse.", "We use the following algorithm developed by [11] and [12], here applied to two-phase flows: Compute the complex-valued filtered streamfunction $\\overline{\\phi }\\left(x,z,t\\right)$ ; Form the $L^2$ -norm $A\\left(x,t\\right)=\\sqrt{\\int _{-\\epsilon }^1\\mathrm {d}{z}\\,|\\overline{\\phi }\\left(x,z,t\\right)|^2};$ Examine the norm along rays, $A(v,t)=A(x=vt,t)$ .", "If $A(0,t)$ is a decreasing function of time, the instability is convective.", "It suffices to consider positive and zero ray velocities only, since this enables a classification of the instability as either absolute or convective and, furthermore, in the convective case, gives information about the leading- and trailing-edge velocities of the downstream-propagating disturbance.", "Additional information can be extracted from the evolution of the norm, provided the contributions to the growth of the streamfunction are dominated by a single eigenmode.", "This caveat does not appear in the single-phase work of [11] and [12]: those problems contain a simple means of projecting the streamfunction on to the eigenmode of interest; the spatial symmetries that produce this projection do not exist in the current problem, and this approach is therefore not applicable.", "We therefore assume that the evolution is dominated by a single eigenmode, and justify this assumption a posteriori.", "Thus, along rays $x=vt$ , we assume that $A\\left(x=vt,t\\right)\\sim t^{-1/2}e^{\\sigma \\left(v\\right)t}$ , where $\\sigma $ is the spatiotemporal growth rate of the dominant eigenmode.", "Therefore, we extract the finite-time estimate of the spatiotemporal growth rate as follows: $\\sigma \\left(v\\right)=\\frac{\\ln A\\left(vt_2,t_2\\right)-\\ln A\\left(vt_1,t_1\\right)}{t_2-t_1}+\\tfrac{1}{2}\\frac{\\ln t_2-\\ln t_1}{t_2-t_1},$ where $t_1$ and $t_2$ are large but finite times and $t_2>t_1$ .", "The complex wave number and frequency along the ray $x=vt$ also follow from this analysis [11], [12]: $\\alpha _i(v)&=&-\\frac{\\mathrm {d}\\sigma }{\\mathrm {d}v},\\\\\\omega _{\\mathrm {i}}(v)&=& \\sigma (v)+\\alpha _{\\mathrm {i}}v,\\\\\\alpha _{\\mathrm {r}}(v)&=&\\Re \\left(\\frac{-\\mathrm {i}}{\\overline{\\phi }}\\frac{\\partial \\overline{\\phi }}{\\partial x}\\right)_{z=0,x=vt},$ where Equation () holds because the right-hand side is independent of time as $t\\rightarrow \\infty $ ." ], [ "Transient direct numerical simulations", "Transient direct numerical simulation (DNS) of Equation (REF ) is complicated by the fact that the operator $\\mathcal {M}$ is non-invertible.", "To solve this equation in an optimal way, we have developed our own numerical method, which we outline here.", "As in Section , we write $\\overline{\\phi }\\left(z,t\\right)$ as a finite sum of Chebyshev polynomials: $\\overline{\\phi }\\left(x,z,t\\right)=\\sum _{\\alpha >0} e^{\\mathrm {i}\\alpha x}{\\left\\lbrace \\begin{array}{ll}\\sum _n a_{\\alpha \\,n}T_n\\left(\\frac{2z}{\\epsilon }+1\\right),&-\\epsilon \\le z\\le 0,\\\\\\sum _n b_{\\alpha \\,n}T_n\\left(2z-1\\right),&0\\le z\\le 1,\\end{array}\\right.", "}$ or more compactly, $\\overline{\\phi }\\left(x,z,t\\right)=\\sum _{\\alpha >0} \\sum _n e^{\\mathrm {i}\\alpha x} v_{\\alpha \\,n} T_n\\left(\\eta _j\\right),\\qquad j=L,G.$ We substitute Equation (REF ) into Equation (REF ).", "This yields the following equation for the normal mode $\\alpha $ : $M_\\alpha \\frac{\\mathrm {d}\\mathbf {v}_\\alpha }{\\mathrm {d}t}=L_\\alpha \\mathbf {v}_\\alpha ,\\qquad t>0,$ where $M_\\alpha $ and $L_\\alpha $ are the Orr–Sommerfeld matrices described in Equation (REF ).", "The matrix $M_\\alpha $ is not invertible: it has rows of zeros corresponding to the no-slip boundary conditions, the continuity of the streamfunction at the interface, and the continuity of the tangential stress at the interface.", "Equation (REF ) is therefore a differential algebraic equation (DAE).", "There are several standard methods for solving DAEs with computational software packages [29].", "For a singular matrix $M$ , the DAE $M(t,y)y^{\\prime }=f(t,y)$ has a solution only when the initial condition $y_0$ is consistent, that is, if there is an initial slope $y_{p0}$ such that $M(t_0,y_0)y_{p0}=f(t_0,y_0)$ .", "In general, computational packages for solving DAEs demand not only that the initial data be consistent, but also that the slope be prescribed as an input to the numerical solver [29].", "We develop herein a numerical method for linear DAEs that removes the necessity to specify the slope.", "Moreover, long-time integrations of DAEs using computational packages can be costly, especially for the modal decomposition (REF ), which contains a large number of wave numbers.", "Thus, we resort to a semi-analytical solution method that holds for constant-coefficient DAEs such as Equation (REF ).", "We re-write Equation (REF ) as $\\frac{\\mathrm {d}}{\\mathrm {d}t}M_\\alpha \\mathbf {v}_\\alpha =L_\\alpha \\mathbf {v}_\\alpha ,$ and integrate it over a numerical time step $\\Delta t$ : $M_\\alpha \\mathbf {v}_\\alpha \\left(t+\\Delta t\\right)-M_\\alpha \\mathbf {v}_\\alpha \\left(t\\right)=\\int _{t}^{t+\\Delta t}L_\\alpha \\mathbf {v}_\\alpha \\left(s\\right)\\mathrm {d}s.$ For a sufficiently small timestep, the integral on the right-hand side may be approximated by the trapezoidal rule.", "Thus, the equation is approximated by $M_\\alpha \\mathbf {v}_\\alpha \\left(t+\\Delta t\\right)-M_\\alpha \\mathbf {v}_\\alpha \\left(t\\right)=\\tfrac{1}{2}\\Delta t\\left[L_\\alpha \\mathbf {v}_\\alpha \\left(t+\\Delta t\\right)+L_\\alpha \\mathbf {v}_\\alpha \\left(t\\right)\\right].$ Re-arrangement gives $\\left[M_\\alpha -\\tfrac{1}{2}\\Delta t L_\\alpha \\right]\\mathbf {v}_\\alpha ^{p+1}=\\left[M_\\alpha +\\tfrac{1}{2}\\Delta t L_\\alpha \\right]\\mathbf {v}_\\alpha ^{p},$ where $\\mathbf {v}_\\alpha ^{p+1}=\\mathbf {v}_\\alpha \\left(t+\\Delta t\\right)$ and $\\mathbf {v}_\\alpha ^p=\\mathbf {v}_\\alpha \\left(t\\right)$ .", "The solution at an arbitrary time $p$ is therefore available from the initial condition using only three matrix operations: $\\mathbf {v}_\\alpha ^{p}=X_\\alpha ^p\\mathbf {v}_{0\\alpha },\\qquad X_\\alpha =\\left[M_\\alpha -\\tfrac{1}{2}\\Delta t L_\\alpha \\right]^{-1}\\left[M_\\alpha +\\tfrac{1}{2}\\Delta t L_\\alpha \\right].$ Equation (REF ) becomes an exact solution to the DAE (REF ) in the limit when $\\Delta t\\rightarrow 0$ : $\\mathbf {v}_\\alpha \\left(t\\right)=\\tilde{X}_{\\alpha }\\left(t\\right)\\mathbf {v}_{0\\alpha },\\qquad \\tilde{X}_{\\alpha }\\left(t\\right)=\\lim _{\\Delta t\\rightarrow 0\\atop t=p\\Delta t}X_{\\alpha }^p.$ Finally, to approximate a delta-function impulse, we fix the initial condition $\\mathbf {v}_{0\\alpha }$ as follows: $\\overline{\\phi }\\left(x,z,t=0\\right)=\\sum _{\\alpha >0} \\mathrm {e}^{\\mathrm {i}\\alpha x}\\mathrm {e}^{-\\alpha ^2 w_x^2}{\\left\\lbrace \\begin{array}{ll}\\sum _n\\frac{N_n}{\\sqrt{\\pi w_z^2}}T_n(\\eta _L)\\int _{-1}^1 \\mathrm {d}s\\,\\mathrm {e}^{-\\left[\\epsilon \\left(s-1\\right)/2\\right]^2/w_z^2}\\frac{T_n\\left(s\\right)}{\\sqrt{1-s^2}},&z\\le 0,\\\\\\sum _n\\frac{N_n}{\\sqrt{\\pi w_z^2}}T_n(\\eta _G)\\int _{-1}^1\\mathrm {d}s\\,\\mathrm {e}^{-\\left[\\left(s+1\\right)/2\\right]^2/w_z^2}\\frac{T_n\\left(s\\right)}{\\sqrt{1-s^2}},&z\\ge 0,\\\\\\end{array}\\right.", "}$ where $N_n=1/\\pi $ if $n=0$ and $2/\\pi $ otherwise.", "The coefficients $w_x$ and $w_z$ are the widths of the approximate delta functions in the $x$ - and $z$ -directions respectively.", "This solution method is appropriate for the kind of long-time simulations performed herein because the time required on a computer to implement Equation (REF ) depends only very weakly on $p$ : the computation time required to raise an arbitrary square matrix to the $p^\\mathrm {th}$ power depends (in a worst-case-scenario) only logarithmically on $p$  [1].", "Thus a DNS of 1000 time units takes (at most) only three times longer than a DNS of 10 time units.", "We have validated the solution method by computing the growth rate for delocalized monochromatic initial conditions, and checking that the dispersion curve agrees with standard (temporal) eigenvalue analysis: the results are identical in each case and are not reported further here.", "We have further validated the spatio-temporal transient DNS by using Equations (REF ) and computing the growth rate and wave number at the most dangerous spatio-temporal mode.", "This is necessarily the most dangerous temporal mode [19].", "Thus, if our transient DNS is correct, then the maximum growth rate computed in this manner must agree with standard temporal eigenvalue analysis.", "The results for this comparison are shown in Table REF , and Table: Comparison between modal and spatio-temporal analyses at various values of mm, with r=1000r=1000, ϵ=0.1\\epsilon =0.1, and Re=2500Re=2500.The parameters SS and FF are given in Equation ().", "The DNS parameters are N 1 =21N_1=21, N 2 =60N_2=60, L x =150L_x=150, Δx=0.01\\Delta x=0.01, and Δt=0.01\\Delta t=0.01.confirm the correctness of our transient DNS.", "We now apply this method to classify completely the instability in turbulent two-phase stratified flow." ], [ "Modal analysis and flow-regime maps", "In this section we examine the spatio-temporal instability wherein the upper layer is turbulent.", "We base the parameter values on an upper layer of air of depth $d_G=5\\,\\mathrm {cm}$ , together with the values of surface tension and gravitational acceleration given in Table REF (these parameter values were used in [31]).", "Substituting these values into Equation (REF ), we obtain $F_0= 3.7809\\times 10^6,\\qquad S_0= 1.1420\\times 10^7.$ Starting with the base state described in Section , we carry out a modal analysis for the turbulent case.", "The results are shown in Figure REF .", "Here, we have fixed $Re=2350$ and $\\epsilon =0.1$ , and have Figure: Re=5000,m=1100\\,\\,Re=5000,m=1100chosen the value of $m$ to give a convective/absolute (C/A) transition, such that the main saddle point in the contour plot for the largest value of $\\omega _{\\mathrm {i}}$ has just become positive.", "The other conditions described in Section REF that are required for absolute instability also apply.", "However, from Figure REF , it is not certain that the system is indeed absolutely unstable, given the odd features near the imaginary axis caused by different eigenmodes being dominant in different parts of the complex wave number plane.", "Therefore, we turn to the ray analysis, representative results from which are shown in Figure REF .", "The two cases are chosen for clarity, since they lie away from (and at opposite sides of) the C/A transition such that the respective convective and absolute behaviour is clearly visible.", "In this figure, the amplification of a pulse is very large.", "This poses a practical problem when trying to infer a C/A transition: when starting from an absolutely unstable case and repeating the analysis for a slightly different value of the parameter $m$ , a convective instability is found when the left tail of the signal decreases as time goes by.", "But the tails of the pulse are close to the part of the $x$ -domain that is affected by numerical error.", "Moreover, for large $m$ , the strong amplification of all parts of the pulse makes a comparison of signal tails difficult.", "The accuracy with which a C/A transition can be determined with the ray analysis is therefore limited.", "That being as it may be, close inspection of the ray analysis results near the C/A transitions inferred from a modal analysis confirms the latter.", "Figure: The norm A(x,t)A(x,t) for (a) m=400m=400 (convective); (b) m=1300m=1300 (absolute), at t=30t=30, 40, 50, 60, 70 (from bottom to top).", "The other parameters are r=1000r=1000, Re=2500Re=2500, ϵ=0.1\\epsilon =0.1, and SS and FF are given in Equation ().", "The DNS parameters are N 1 =21N_1=21, N 2 =60N_2=60, L x =150L_x=150, Δx=0.01\\Delta x=0.01, and Δt=0.01\\Delta t=0.01.Only a fraction of the xx domain is shown.Carrying on from these studies, we have constructed a flow-regime map using the modal analysis.", "We have carefully followed the dominant saddle points such as those in Figure REF , and have performed the necessary checks required for confirming absolute instability (Section REF ).", "Our modal studies have furthermore been confirmed by the ray analysis discussed above.", "In Figure REF (a), we see that if $Re$ is increased for a fixed value of $m$ , the system generally goes from a stable state, through a convectively unstable to an absolutely unstable state.", "Figure REF (b) shows that the convectively unstable regime can disappear entirely, when large values of $m$ are used.", "The same figure also shows that the neutral curve `pushes' the C/A transition at large values of $m$ back to higher $Re$ -values; this is discussed further below.", "We note in Figure REF (b) the presence of two Figure: Flow-regime map for the turbulent base state.", "Here, r=1000r=1000 and SS and FF, are given in Equation ().", "Mode M1 can be stable (S, neutral curve marked with circles), convectively unstable (C), or absolutely unstable (A, C/A transition curve marked with squares).", "Further modes, called M2 and M3, can be convectively unstable, with neutral curves (`NC') given by dashed lines and labelled by triangles and diamonds respectively.", "(a) Variations in ϵ\\epsilon at fixed m=1000m=1000; (b) Variations in mm at fixed ϵ=0.1\\epsilon =0.1.additional modes of instability whose neutral curves are marked `NC:M2' and `NC:M3'; hence, three modes are convectively unstable.", "A standard temporal energy-budget analysis conducted at $Re = 5000$ , $m = 600$ , confirms that M2 and M3 both derive most of the destabilizing energy from the interfacial region, and a small contribution from the liquid layer.", "The profile of the wave-induced Reynolds stress has also confirmed that they are both conventional `internal modes' in the liquid [6], [31].", "However, these modes differ in one respect: M2 has a speed comparable to, but less than, the interfacial velocity whereas M3 is much slower, especially at large Reynolds numbers.", "In Figure REF , we examine the complex wave number $(\\alpha _{\\mathrm {r}},\\alpha _{\\mathrm {i}})$ at the C/A transition.", "We have plotted $\\alpha _{\\mathrm {r}}$ as a function of $Re$ for various $(\\epsilon ,m)$ combinations; we have also plotted $\\alpha _{\\mathrm {i}}(Re_\\mathrm {int})^{1/2}$ in the same manner.", "Here $Re_\\mathrm {int}=rU_\\mathrm {int}\\epsilon /m$ is the liquid Reynolds number based on the base-state velocity at the interface, $U_\\mathrm {int}$ (the reason for this rescaling will become apparent in what follows).", "Figure REF (a,b) demonstrates that $\\alpha _{\\mathrm {r}}$ at the transition is governed primarily by $Re$ .", "The real and imaginary parts of the wave number both increase significantly with $Re$ .", "By rescaling the wavenumbers with respect to the lower-layer depth (i.e.", "letting $\\alpha _{\\mathrm {r}}\\rightarrow \\epsilon \\alpha _{\\mathrm {r}}$ ), this variation is given some context: for $Re=1000$ the wavelength is comparable to the liquid-layer depth, while for $Re=5000$ it is approximately 10 times smaller than the liquid-layer depth.", "Figure: The complex wave number along the C/A transition in Figure  as a function of ReRe (squares, solid lines).In (a), the wavenumber of the most dangerous temporal mode is also shown (circles, dashed line).Filled squares: variation in ϵ\\epsilon at fixed mm; empty squares: variation in mm at fixed ϵ\\epsilon .", "Panel (c) is a rescaled version of the imaginary wave number in (b).Figure REF (a) also indicates that $\\alpha _{\\mathrm {r}}$ at the saddle point along the neutral C/A curve closely follows $\\alpha _{\\mathrm {r}}$ for the most-dangerous temporal mode.", "This confirms the observation in Figure REF that the saddle point lies almost directly below the most dangerous temporal mode.", "This is explained as follows: for all the cases considered here, the group velocity $d\\omega _{\\mathrm {r}}/d\\alpha _{\\mathrm {r}}$ calculated for temporal modes depends only weakly on $\\alpha _{\\mathrm {r}}$ , and a straightforward application of the Cauchy–Riemann conditions to the analytic function $\\omega =\\omega _{\\mathrm {r}}(\\alpha _{\\mathrm {r}},\\alpha _{\\mathrm {i}})+\\mathrm {i}\\omega _{\\mathrm {i}}(\\alpha _{\\mathrm {r}},\\alpha _{\\mathrm {i}})$ shows that for such group velocities, the $\\alpha _{\\mathrm {r}}$ -values for the most-dangerous temporal mode and the saddle-point mode coincide.", "Most of the results in Figure REF (a,b) collapse into one plot, namely Figure REF (a).", "Almost all corresponding boundaries obtained for Figure REF (a,b) collapse.", "Especially of interest is the large-$Re$ /large-$Re_\\mathrm {int}$ behaviour of the C/A transition (branch I), which corresponds to a critical value of $Re_\\mathrm {int}$ that is virtually independent of $Re$ .", "Figure: Reduced flow-regime map for the turbulent base state.Filled symbols: variation in ϵ\\epsilon at fixed mm; empty symbols: variation in mm at fixed ϵ\\epsilon .It is possible to explain this scaling behaviour using the theory developed by [32] (see also Appendix ).", "There, the spatio-temporal growth rate $\\omega _{\\mathrm {i}}(\\alpha _{\\mathrm {r}},\\alpha _{\\mathrm {i}})$ is prescribed in terms of a Taylor series in $\\alpha _{\\mathrm {i}}$ , where the coefficients in the Taylor series are derived from the purely temporal linear stability analysis, and depend on $\\alpha _{\\mathrm {r}}$ .", "We have verified that the C/A transition curves in Figure REF are described accurately by a quadratic truncation of this Taylor series (Appendix ).", "In this `quadratic approximation', the saddle point occurs at $\\alpha _{\\mathrm {r}}=\\alpha _{\\mathrm {r}}^*$ , where $\\alpha _{\\mathrm {r}}^*$ solves $\\frac{d\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}}\\frac{d^2\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^2}\\Big |_{\\alpha _{\\mathrm {r}}^*}=-c_{\\mathrm {g}}(\\alpha _{\\mathrm {r}}^*)\\frac{dc_{\\mathrm {g}}}{d\\alpha _{\\mathrm {r}}}\\Big |_{\\alpha _{\\mathrm {r}}^*}.$ In the current application, the group velocity is approximately constant, hence $\\alpha _{\\mathrm {r}}^*\\approx \\alpha _{\\mathrm {r,max}}$ , the location of the most-dangerous temporal mode.", "The same quadratic approximation gives the following condition for the onset of absolute instability: $-\\frac{d^2\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^2}\\bigg |_{\\alpha _{\\mathrm {r}}^*}\\omega _{\\mathrm {i}}^{\\mathrm {temp}}(\\alpha _{\\mathrm {r}}^*)=\\tfrac{1}{2}c_{\\mathrm {g}}^2(\\alpha _{\\mathrm {r}}^*).$ We now approximate each term in Equation (REF ).", "Consider first of all the right-hand side.", "Previous work [31] demonstrates that the group velocity $c_{\\mathrm {g}}$ is only slightly in excess of the interfacial velocity $U_\\mathrm {int}$ (this is also consistent with experiments, e.g. [9]).", "Thus, we approximate the group velocity as $c_{\\mathrm {g}}\\approx U_\\mathrm {int}$ .", "Moreover, since the liquid layer is thin and viscous, the base-state velocity in the liquid is close to linear shear flow.", "In addition, for thin films, $Re_*=\\theta Re$ , where $\\theta $ is a geometric factor independent of the flow parameters [31].", "Thus, we have the following string of equalities: $\\frac{U_\\mathrm {int}}{U_0}\\approx \\frac{\\tau _{\\mathrm {i}}d_L}{\\mu _L U_0}=\\frac{\\epsilon }{m}\\frac{Re_*^2}{Re}=\\frac{\\epsilon }{m}\\theta ^2 Re.$ But $Re_\\mathrm {int}= \\frac{\\rho _L U_\\mathrm {int}d_L}{\\mu _L}=\\frac{r}{\\theta ^2} \\left(\\frac{U_\\mathrm {int}}{U_0}\\right)^2,$ hence $\\frac{U_\\mathrm {int}}{U_0}=\\left(\\theta /r^{1/2}\\right)Re_\\mathrm {int}^{1/2},$ and $c_{\\mathrm {g}}^2\\approx \\left(\\theta ^2/r\\right)Re_\\mathrm {int}.$ Next, we consider the left-hand side.", "We use the quadratic approximation $\\omega _{\\mathrm {i}}^{\\mathrm {temp}}=A\\alpha -\\tfrac{1}{2}B\\alpha ^2,\\qquad A,B>0.$ Note that this is not a long-wave approximation, but is instead a fit to the data around the most-dangerous temporal mode, where the fitting parameters $A$ and $B$ are selected with respect to the actual, computed values of the maximum growth rate, expressed as $\\max (\\omega _{\\mathrm {i}}^{\\mathrm {temp}})=A^2/(2B)$ , and the cutoff wave number, written as $\\alpha _0=2A/B$ .", "We now consider three parameter regimes and investigate the functional dependence of $A$ and $B$ on the Reynolds number $Re$ .", "Functional dependence for fixed $\\epsilon \\approx 0.1$ and varying $m$ : At large Reynolds numbers, both the growth rate and the cutoff wave number increase linearly with $Re$ .", "We write $\\max (\\omega _{\\mathrm {i}}^{\\mathrm {temp}})=k_1 Re,\\qquad \\alpha _0=k_2 Re,$ where $k_1$ and $k_2$ are constants of proportionality; these are measured to be $m$ -independent (Figure REF ).", "In other words, $\\tfrac{1}{2}(A^2/B)=k_1Re, \\qquad 2A/B=k_2 Re.$ Hence, $A^2=2k_1Re(2A/Re k_2)$ , or $A=4k_1/k_2$ .", "At maximum growth, $-\\omega _{\\mathrm {i}}^{\\mathrm {temp}}\\frac{d^2\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^2}\\bigg |_{\\alpha _{\\mathrm {r},\\mathrm {max}}}=\\tfrac{1}{2}A^2=8k_1^2/k_2^2,$ and the LHS of the C/A transition criterion is independent of $Re$ for large values of $Re$ .", "Figure: Dependence of the temporal problem on the system parameters.", "(a) The cutoff wavenumber α 0 \\alpha _0, for which ω i temp (α 0 )=0\\omega _{\\mathrm {i}}^{\\mathrm {temp}}(\\alpha _0)=0; (b) the maximum growth rate max(ω i temp )\\max (\\omega _{\\mathrm {i}}^{\\mathrm {temp}}).", "The parameters to be varied are the pair (m,ϵ)(m,\\epsilon ).", "Circles: (4000,0.1), Squares: (2000,0.1), Crosses: (1000,0.1); Diamonds: (500,0.1); Dashes: (1000,0.005); Stars: (1000,0.05).", "Trendlines have been added to the insets.Combining Equations (REF ), (REF ), and (REF ), we have the following criterion for the onset of absolute instability: $Re_\\mathrm {int}=C_1,\\qquad C_1=\\frac{16 rk_1^2}{k_2^2\\theta ^2},$ where $C_1$ is a parameter-independent constant.", "Hence, at large $Re$ , there is a critical Reynolds number $Re_\\mathrm {int}$ for the onset of absolute instability.", "This is consistent with figure  REF (branch I).", "On the other hand, for smaller values of $Re$ , Figure REF gives $\\max (\\omega _{\\mathrm {i}}^{\\mathrm {temp}})\\propto |Re-Re_\\mathrm {c}|^p,\\qquad p>1,$ where $Re_\\mathrm {c}$ is approximately independent of $m$ .", "We have also measured the cutoff wave number as $\\alpha _{0}=\\alpha _{0\\mathrm {c}}+k_3 |Re-Re_\\mathrm {c}|^{1/q},\\qquad q>1,$ where $\\alpha _{0\\mathrm {c}}$ and $k_3$ are approximately independent of $m$ .", "(We have found $p\\approx 5/2$ , $q\\approx 2$ ; see Figure REF .)", "Putting these facts together, we get $-\\omega _{\\mathrm {i}}^{\\mathrm {temp}}\\frac{d^2\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^2}\\bigg |_{\\mathrm {max}}\\propto \\frac{1}{\\alpha _{0\\mathrm {c}}^2}|Re-Re_\\mathrm {c}|^{2p},\\qquad Re\\rightarrow Re_\\mathrm {c},$ and the criterion for the onset of absolute instability is therefore $Re_\\mathrm {int}=C_2 |Re-Re_\\mathrm {c}|^{2p},$ where $C_2$ is independent of $m$ .", "This relation is consistent with figure  REF (branch II).", "Functional dependence for fixed $m$ and varying $\\epsilon $ : At large Reynolds numbers and relatively large values $Re_\\mathrm {int}$ such that $\\epsilon \\approx 0.1$ , both the growth rate and the cutoff wave number increase linearly with $Re$ (Figure REF ).", "Thus, the scaling arguments employed for fixed $\\epsilon $ and varying $m$ pertain here also, and the criterion for absolute instability is again $Re_\\mathrm {int}=C_1$ , where $C_1$ is a parameter-free constant.", "This relation is again consistent with Figure REF (branch I).", "For smaller values of $Re$ , but for still relatively large values of $Re_\\mathrm {int}$ , Figure REF gives $\\max (\\omega _{\\mathrm {i}}^{\\mathrm {temp}})\\propto |Re-Re_\\mathrm {c}(\\epsilon )|^p,\\qquad p>1,$ where $Re_\\mathrm {c}(\\epsilon )$ is approximately independent of $m$ but depends on $\\epsilon $ .", "We have also measured the cutoff wave number as $\\alpha _{0}=\\alpha _{0\\mathrm {c}}(\\epsilon )+k_3 |Re-Re_\\mathrm {c}(\\epsilon )|^{1/q},\\qquad q>1,$ where $\\alpha _{0\\mathrm {c}}(\\epsilon )$ is approximately independent of $m$ but dependent on $\\epsilon $ , and where $k_3$ is independent of $m$ and $\\epsilon $ .", "Putting these facts together, we get $-\\omega _{\\mathrm {i}}^{\\mathrm {temp}}\\frac{d^2\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^2}\\bigg |_{\\alpha _{\\mathrm {r},\\mathrm {max}}}\\propto \\frac{1}{\\alpha _{0\\mathrm {c}}^2(\\epsilon )}|Re-Re_\\mathrm {c}(\\epsilon )|^{2p},\\qquad Re\\rightarrow Re_\\mathrm {c},$ and the criterion for the onset of absolute instability is therefore $Re_\\mathrm {int}=\\frac{C_2}{\\alpha _{0\\mathrm {c}}(\\epsilon )} |Re-Re_\\mathrm {c}(\\epsilon )|^{2p},$ where $C_1$ is independent of $m$ .", "This relation is consistent with Figure REF (branch III, large $Re_\\mathrm {int}$ ), in the sense that the stability boundary depends on $Re_\\mathrm {int}$ , $Re$ , and $\\epsilon $ .", "As $\\epsilon $ is reduced, the Reynolds number $Re_\\mathrm {int}$ is also reduced.", "In such a regime, and for large values of $Re$ , we have $\\max (\\omega _{\\mathrm {i}}^{\\mathrm {temp}})=k_4 \\epsilon Re^2/m,\\qquad \\alpha _0=k_5 Re,$ where $k_4$ and $k_5$ are constants of proportionality.", "In other words, $\\tfrac{1}{2}(A^2/B)=k_4\\epsilon Re^2/m, \\qquad 2A/B=k_5 Re,$ Hence, $A=4k_4\\epsilon Re/(k_5 m)$ , and $B=8k_4\\epsilon /(k_5 m)$ .", "Thus, at maximum growth, $-\\omega _{\\mathrm {i}}^{\\mathrm {temp}}\\frac{d^2\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^2}\\bigg |_{\\mathrm {max}}=\\tfrac{1}{2}a^2=\\frac{8k_4^2\\epsilon ^2 Re^2}{k_5^2 m^2}\\propto Re_\\mathrm {int}^2.$ Therefore, in this case, the LHS of the C/A transition criterion is proportional to $Re_{\\rm int}^2$ , while the RHS is proportional to $Re_\\mathrm {int}$ .", "Therefore, there is a minimum value of $Re_\\mathrm {int}$ (rather than a maximum value) for absolute instability.", "This corresponds precisely to the lower-branch C/A transition curve in Figure REF (branch III, small $Re_\\mathrm {int}$ ).", "These results have been presented for fixed values of $F_0$ and $S_0$ .", "We briefly sketch the effect of varying these parameters.", "Increasing $F_0$ is stabilizing, and raises the critical Reynolds number $Re_c$ for the onset of temporal instability.", "This suggests that branches II and III should shift to the right under such an increase.", "Similarly, increasing $S_0$ makes the temporal disturbances more stable, the main effect of which is (counter-intuitively) to increase $-\\omega _{\\mathrm {i}}^{\\mathrm {temp}}d^2\\omega _{\\mathrm {i}}^{\\mathrm {temp}}/d\\alpha _{\\mathrm {r}}^2$ at $\\alpha _{\\mathrm {r}}=\\alpha _{\\mathrm {r},\\mathrm {max}}$ , which implies that the critical Reynolds number $Re_\\mathrm {int}$ for the onset of absolute instability at large $Re$ and $Re_\\mathrm {int}$ should be raised.", "We have performed some detailed calculations for the onset of absolute instability with the `full' dispersion relation, the results of which agree with this description provided by the quadratic approximation.", "We also comment on the scaling behaviour for the imaginary part of the wave number at the C/A transition.", "The value of $\\alpha _{\\mathrm {i}}$ at transition in the quadratic approximation is $\\alpha _{\\mathrm {i}}=-2\\max (\\omega _{\\mathrm {i}}^{\\mathrm {temp}})/c_{\\mathrm {g}}.$ Approximating $c_{\\mathrm {g}}\\propto \\sqrt{Re_\\mathrm {int}}$ , we have $\\alpha _{\\mathrm {i}}\\sqrt{Re_\\mathrm {int}}\\propto -\\max (\\omega _{\\mathrm {i}}^{\\mathrm {temp}})$ .", "For large values of the Reynolds numbers $Re$ and $Re_\\mathrm {int}$ , we have $\\max (\\omega _{\\mathrm {i}}^{\\mathrm {temp}})\\propto Re$ , where the constant of proportionality is parameter-free.", "This gives $-\\alpha _{\\mathrm {i}}\\sqrt{Re_\\mathrm {int}}\\propto Re$ , in agreement with Figure REF .", "One further result concerns the alignment of branch I (main C/A transition, Figure REF ) with the neutral temporal stability curve of M2.", "It is as if the absolute instability is quenched when M2 becomes unstable.", "This is due to mode competition, which occurs in a convectively unstable regime close to the C/A boundary, in which both M1 and M2 are unstable, with a wave length comparable to the liquid-layer depth.", "In Figure REF , the contours of $\\omega _{\\mathrm {r}}$ are shown for the least stable mode at each complex $\\alpha $ .", "This case involves the parameter values $Re = 5000$ , $\\epsilon = 0.1$ , and $m = 600$ .", "The spatial curve $\\omega _{\\mathrm {i}}= 0$ of the most dangerous mode is identified with M1 by a Gaster-type analysis (see [13] and Appendix ).", "The Figure: m=900\\,\\,m=900contours $\\omega _{\\mathrm {r}}= \\mathrm {Const.", "}$ that connect orthogonally to the spatial curve are identified also with M1.", "The orthogonality of contours of the real and imaginary parts of an analytic function is a straightforward consequence of the Cauchy–Riemann conditions.", "In this way, we have established in Figure REF (a) that the saddle point (a necessary precursor for absolute instability) does not originate from the M1-eigenmode.", "However, upon increasing $m$ (Figs.", "REF (b–d)), the region of $\\alpha $ -space in which M1 dominates fills out, and the spatial curve of M1 connects to a saddle: it is as if the dominant saddle point `switches' from non-M1 to M1.", "Going to higher values of $m$ , the sign of $\\omega _{\\mathrm {i}}$ at the saddle point becomes positive, and absolute instability ensues.", "We have investigated in more depth the identity of the rival saddle points by using the quadratic approximation, which provides a direct connection between the spatio-temporal and temporal modes.", "By reconstructing Figure REF from the quadratic approximation (not shown), we have confirmed the spatio-temporal mode competition.", "Care has been taken to ensure correct identification of the dispersion curves by double-checking the continuity in $\\alpha _{\\mathrm {r}}$ of the corresponding $\\omega _{\\rm r}$ curves.", "In a reconstruction of the $\\omega _{\\mathrm {i}}$ -landscape of M1, a saddle point exists at $\\alpha _{\\mathrm {r}}\\approx \\alpha _{\\mathrm {r},\\mathrm {max}}$ .", "In a further reconstruction, two additional saddle points occur, and are connected with M2: one lies near $\\alpha _{\\mathrm {r},\\mathrm {max}}$ , while a further saddle point resides at large $\\alpha _{\\mathrm {r}}$ .", "Finally, we have verified that as $m$ increases, the values of $\\omega _{\\mathrm {i}}$ associated with M1 increase (especially near the saddle point), eventually leading to absolute instability; the other saddles are simultaneously overwhelmed by this increase and no longer produce mode competition." ], [ "Energy budget", "The purpose of applying an energy-budget analysis is to infer the mechanism for instability generation.", "In view of the connection between spatio-temporal and temporal modes exemplified in the quadratic approximation considered in Section REF , a spatio-temporal energy-budget analysis necessarily inherits the properties of the corresponding temporal analysis.", "Nevertheless, we consider the energy-budget analysis in this section, and investigate how such a study applies to the growth of a pulse (as opposed to a normal mode).", "The ray analysis provides a means of examining such pulses.", "We examine the mechanism by which the pulse grows, noting that this does not have to be the same throughout the pulse.", "We introduce the kinetic energy density $K\\left(x,t\\right)=\\tfrac{1}{2}\\int _{-\\epsilon }^0\\mathrm {d}{z}\\,r_L|\\delta \\mathbf {u}|^2+\\tfrac{1}{2}\\int _{0}^1\\mathrm {d}{z}\\,r_G|\\delta \\mathbf {u}|^2,$ where $\\delta \\mathbf {u}=(\\delta u,\\delta w)$ is the perturbation velocity.", "We differentiate this expression with respect to time at a fixed location $x$ , and apply the equations of motion (REF ) and Gauss's divergence theorem to obtain the following flux-conservation equation: $\\frac{\\partial K}{\\partial t}+\\frac{\\partial F_K}{\\partial x}=s_B\\left(x,t\\right)+s_I\\left(x,t\\right),$ where the source/sink terms $s_B$ and $s_I$ , and the flux $F_K$ are described in what follows.", "First, we introduce the following notation for the perturbative contribution to the viscous stress tensor $\\delta T_{ij}$ : $\\delta T_{ij}=-\\delta _{ij}\\delta p+\\frac{m_j}{Re}\\left(\\frac{\\partial }{\\partial x_i}\\delta u_j+\\frac{\\partial }{\\partial x_j}\\delta u_i\\right);$ we also denote the separate fluid domains by $\\left(a_L,b_L\\right)=\\left(-\\epsilon ,0\\right)$ and $\\left(a_G,b_G\\right)=\\left(0,d_G\\right)$ .", "The flux $F_K$ can then be written as $F_K\\left(x,t\\right)=\\int _{a_L}^{b_L}\\mathrm {d}{z}\\left[\\tfrac{1}{2}r_L U_0\\left(z\\right)|\\delta \\mathbf {u}|^2-\\delta u \\,\\delta T_{xx}-\\delta w\\,\\delta T_{xz}\\right]\\\\+\\int _{a_G}^{b_G}\\mathrm {d}{z}\\left[\\tfrac{1}{2}r_G U_0\\left(z\\right)|\\delta \\mathbf {u}|^2-\\delta u\\,\\delta T_{xx}-\\delta w\\,\\delta T_{xz}\\right].$ In a similar manner, the bulk source/sink term takes the form $s_B\\left(x,t\\right)&=&\\sum _{j=L,G}\\left[REY_j\\left(x,t\\right)+DISS_j\\left(x,t\\right)\\right],\\\\REY_j\\left(x,t\\right)&=&-r_j\\int _{a_j}^{b_j}\\mathrm {d}{z}\\,\\delta u\\,\\delta w\\frac{\\mathrm {d}U_0}{\\mathrm {d}z},\\\\DISS_j\\left(x,t\\right)&=&-\\frac{m_j}{Re}\\int _{a_j}^{b_j}\\mathrm {d}{z}\\left[2\\left(\\delta u_x\\right)^2+2\\left(\\delta w_z\\right)^2+\\left(\\delta u_z+\\delta w_x\\right)^2\\right],$ and finally, the interfacial source/sink $s_I\\left(x,t\\right)$ is given by $s_I\\left(x,t\\right)&=&TAN\\left(x,t\\right)+NOR\\left(x,t\\right),\\\\TAN(x,t)&=&\\left(\\delta T^L_{zx}\\,\\delta u^L-\\delta T^G_{zx}\\,\\delta u^G\\right)_{z=0},\\\\NOR(x,t)&=&\\left(\\delta T^L_{zz}\\,\\delta w^L-\\delta T^G_{zz}\\,\\delta w^G\\right)_{z=0}.$ There are no contributions to the energy balance from the perturbation turbulent stresses because these are neglected in the quasi-laminar approximation (see Section REF ).", "Figure: Energy budget for (a,b) Re=2000Re=2000, m=2000m=2000 (convective); (c) Re=4000Re=4000, m=300m=300 (convective); (d-e), Re=4000Re=4000, m=2000m=2000.", "Solid line, TANTAN; Solid line with circles, REY G REY_G; dotted line segments represent absolute value of the same variable as along the rest of the curve.", "All are at t=50t=50 for r=1000r=1000, ϵ=0.1\\epsilon =0.1.As in previous studies of the purely temporal instability, the term $TAN$ is identified with the viscosity-contrast instability (`Yih mode') [35], [6].", "A positive value of this term indicates work done by the perturbations on the interface due to the viscosity jump across the interface.", "Again, in analogy with the purely temporal case, the term $REY_L$ is due to an instability of Tollmien–Schlichting type in the bulk liquid flow, while positive values of $REY_G$ correspond to an instability of the Miles type [24] near the critical layer in the gas.", "Equally, the terms $REY_L$ and $REY_G$ can be thought of as giving the rate at which energy is transferred from the mean flow to the disturbance via the wave-induced Reynolds stresses.", "We examine these terms in detail now (the sinks $DISS_{L,G}$ are not of interest, since they are necessarily stabilising).", "To this end, we have selected three states $(Re,m)$ from Figure REF (b) that intersect the near-horizontal and near-vertical parts of the C/A transition.", "For these states, we plot the sources and sinks as a function of $x$ at a fixed point in time (the $t$ -value is chosen such that all transience has been eliminated from the pulse).", "Naturally, the curves exhibit oscillatory behaviour (the distribution of the phase of even a single temporal mode shows oscillations).", "The energy budget for the purely temporal study is averaged over a single wave length, but spatial averaging is not conducted here, since the spatial distribution is the focus of the study.", "Therefore, we examine in Figure REF (a) a snapshot of the spatial distribution of the largest terms in the budget for $Re=2000$ , $m=2000$ , $\\epsilon =0.1$ .", "This parameter set is seen in Figure REF (b) to be convectively unstable and to lie to the left of the C/A critical curve.", "The term $TAN$ dominates throughout the pulse, followed by $REY_G$ (other, smaller terms are not shown).", "For $Re=4000$ , $m=300$ , $\\epsilon =0.1$ , this is also the case but, although $TAN$ reaches locally the largest values over a wave length, its sign changes, whereas $REY_G$ is positive virtually throughout the pulse (a zoomed view is shown in Figure REF (c)).", "Finally, crossing the C/A boundary (Figure REF (d-e)) causes the distribution of $REY_G$ to become asymmetric, while the features regarding the signs of $TAN$ and $REY_G$ previously observed for $m=300$ still hold.", "Having characterized the turbulent base state in detail, we revisit the laminar problem and investigate whether large-$r$ absolute instability is possible there." ], [ "Revisiting the laminar problem", "In this section we review the problem of interfacial instability where the upper layer is laminar.", "Although the main conclusions for the turbulent case carry over here, some differences arise.", "The base-state flow $U_0(z)$ is determined in a standard fashion by solving the momentum balance $\\mu _j\\frac{d^2 U_0}{dz^2}-\\frac{dP}{dL}=0,\\qquad j=L,G,$ and is subject to continuity of velocity and shear stress across the interface at $z=0$ .", "To facilitate comparison with previous work on laminar flows, for this section only, we adopt the non-dimensionalization scheme of [34].", "We set $\\epsilon =d_L/(d_L+d_G)$ and $Re=\\rho _G V (d_L+d_G)/\\mu _G$ , where the characteristic velocity $V$ is chosen to be the superficial velocity $(d_L+d_G)^{-1}\\int _{-d_L}^{d_G} U_0(z)\\mathrm {d}z$ .", "The gravity and surface tension are parameterized as $\\mathcal {G}:= (\\rho _L-\\rho _G)g(d_L+d_G)^2/(\\mu _G V)$ and $\\mathcal {S}:= \\gamma /(\\mu _G V)$ respectively; the value of these parameters is varied in the following parameter studies.", "Because the model for the turbulent case relies on the quasi-laminar theory for the Orr–Sommerfeld perturbation equations, the streamfunction equations and the transient-DNS numerical method carry over directly to the laminar case, once the necessary rescaling and non-dimensionalization have been performed (e.g., using $\\mathcal {G}$ and $\\mathcal {S}$ in the normal stress condition)." ], [ "Parametric study (1): $\\mathcal {G}$ and {{formula:0c0a0a68-d035-4d80-ad94-73f21472d961}} taken as constant", "We set $\\mathcal {G}=0$ and take $\\mathcal {S}=0.01$ , corresponding exactly to the paper of [34]; this choice enables us to relate the current investigation to the results of [34] for liquid/liquid systems.", "For these parameter values, the ray analysis has revealed that the laminar base state is absolutely unstable for a significant portion of parameter space, even for a density ratio of $r=1000$ .", "Figure: ϵ=0.18\\,\\,\\epsilon =0.18This is seen in Figure REF , where the $L^2$ -norm $A(x,t)$ associated with the broadband disturbance is shown at several times.", "These curves have been truncated at the points where they decrease below a specified fraction of their maximum value, as the finite working precision makes the results too uncertain far away from the pulse.", "The existence of absolute instability at large density ratios was not found in the modal analysis of [34], who only ascertained that density-matched fluids can become absolutely unstable.", "The cause for this oversight is that the magnitude of the wave number at the saddle point is very large, beyond the range searched by [34].", "Figure: Flow-regime map showing laminar base state in parametric study (1).", "Here, r=1000r=1000, 𝒮=0.01\\mathcal {S}=0.01, and 𝒢=0\\mathcal {G}=0.", "The system is always either convectively (C) unstable or absolutely (A) unstable.", "A second temporal mode is also unstable below the neutral curve (dashed curve labelled `M2', with triangles).", "(a) Variations in ϵ\\epsilon at fixed m=150m=150; (b) Variations in mm at fixed thickness ϵ=0.1\\epsilon =0.1.Figure: Flow-regime map of parametric study (1) in terms of the liquid Reynolds number Re int Re_\\mathrm {int}.", "Filled symbols: variations in ϵ\\epsilon at constant m=150m=150; open symbols: variations in mm at fixed ϵ=0.1\\epsilon =0.1.", "As described in the text, a further mode (convectively unstable) exists for Re8000Re8000, whose neutral curve is unaffected by changes in Re int Re_\\mathrm {int}, mm, and ϵ\\epsilon .Motivated by the plots in Figure REF showing absolute instability, we revisit the otherwise more accurate modal analysis and perform a large scan through the complex $\\alpha $ -plane to obtain the C/A boundaries.", "These boundaries are consistent with the results of the ray analysis shown in Figure REF .", "Also, the results are similar to those in [34] for $r=1$ , suggesting that the transition found by [34] extends to density ratios of at least $r=1000$ .", "We have verified that, upon lowering the density ratio at a point in the absolute regime of Figure REF , also identified as such in [34], the system remains absolutely unstable at intermediate values of $r$ .", "In contrast to the $r=1$ case in [34], for $r=1000$ , absolute instability occurs only at large $m$ -values.", "As in the turbulent case, we have examined the flow-regime boundaries in the ($Re,Re_\\mathrm {int}$ ) plane, where $Re_\\mathrm {int}:= r\\epsilon Re U_{\\mathrm {int}}/m$ is a Reynolds number based on the liquid-film properties and the interfacial velocity $U_{\\mathrm {int}}$ of the base state.", "In contrast to the turbulent case, the results of Figure REF do not collapse, although the overall trends otherwise bear some resemblance to the turbulent case (e.g.", "Figure REF ): branch III corresponds to a critical value of $Re_\\mathrm {int}$ (consistent with [34] for $r=1$ ), while branch II corresponds to a critical value of $Re$ .", "These two branches shift when changing the value of $m$ , respectively $\\epsilon $ , thereby contracting or expanding the absolutely unstable regime.", "Instead, we plot the regime boundaries in the ($Re,Re_\\mathrm {int}/\\epsilon $ ) plane in Figure REF .", "Although the two branches labelled `II' virtually coincide in this plane (this is discussed further below), the functional form $Re_\\mathrm {int}=\\epsilon f(Re)$ of the neutral curve differs substantially from that already encountered in the turbulent study.", "Figure: The real (a) and imaginary (b) component of the wave number at the saddle point along all C/A transitions in Figure (a) (squares) and Figure (b) (triangles), as functions of Re int Re_\\mathrm {int}.", "The open symbols represent secondary saddle points that do not correspond to absolute instability.We also investigate variations in the wave number at the saddle point (taken at the onset of absolute instability) in Figure REF , where $\\epsilon (\\alpha _{\\mathrm {r}},\\alpha _{\\mathrm {i}})$ is plotted as a function of $Re_\\mathrm {int}$ .", "The contrast between these results and the earlier turbulent results is remarkable.", "The wave number is $O(1/\\epsilon )$ .", "While the flow-regime map does not collapse in $(Re,Re_\\mathrm {int})$ -space, the wave numbers $\\epsilon (\\alpha _{\\mathrm {r}},\\alpha _{\\mathrm {i}})$ depend mainly on $Re_\\mathrm {int}$ , and not separately on $Re$ .", "The wave number $\\alpha _0$ at the saddle point follows from $(d\\omega /d\\alpha )_{\\alpha _0} =0$ , which therefore only involves a relatively simple scaling, whereas the flow regime is determined by $\\omega _{\\mathrm {i}}(\\alpha _0) >0$ , which involves more parameters.", "Nevertheless, the dependency of $\\epsilon \\alpha $ on $Re_\\mathrm {int}$ is complex.", "Above $Re_\\mathrm {int}\\approx 5$ (corresponding to branch I) the curves nearly collapse; a small difference remains, as is also seen in the corresponding conditions C/A transition curves in Figure REF .", "Upon decreasing the value of $Re_\\mathrm {int}$ , two saddle points emerge.", "However, only one of the saddle points produces absolute instability.", "Most of the results in Figure REF lie on the branch that corresponds to somewhat longer waves (note, the data represented by the filled triangles in Figure REF (a) jump from the upper to the lower branch when $Re_\\mathrm {int}$ is decreased).", "These substantial differences between the turbulent and the laminar cases are now addressed in a further parametric study." ], [ "Parametric study (2): $\\mathcal {G}$ and {{formula:f31d5799-8f44-4a56-b994-93bcc25d48c9}} vary inversely with Reynolds number", "We extend parametric study (1) to allow for effects of gravity and surface tension corresponding to systems with a large density ratio, and we set $\\mathcal {G}=\\mathcal {G}_0 (r-1)/Re$ and $\\mathcal {S}=\\mathcal {S}_0/Re$ .", "This extension is more reflective of real systems because it corresponds to a fixed value of the dimensional surface tension $\\gamma $ , while allowing for variations in $\\mathcal {S}$ through changes in the Reynolds number.", "Here, the values of $\\mathcal {G}_0$ and $\\mathcal {S}_0$ are the same as $F_0$ and $S_0$ (Section ) but such that these are for channels that are ten times smaller: $\\mathcal {G}_0=F_0/10^3$ and $\\mathcal {S}_0=S_0/10$ unless indicated otherwise (the effect of changing these values is discussed below).", "The results of this parameter modification are shown in Figure REF .", "A striking feature of this study is the blurring of the previously sharp distinction between the turbulent case (Figure REF ), and the laminar study (Figure REF ).", "Moreover, increasing $\\mathcal {G}$ and $\\mathcal {S}$ leads to a substantial modification of the absolute region in parameter space.", "Figure: Flow-regime map for parametric study (2).", "Filled symbols: variations in ϵ\\epsilon at constant m=100m=100; open symbols: variations in mm at fixed ϵ=0.1\\epsilon =0.1.The role played by $\\mathcal {G}$ and $\\mathcal {S}$ in this modification is further highlighted in Tables REF –REF .", "Table REF corresponds to branch I, and demonstrates that an increase in $\\mathcal {G}$ or $\\mathcal {S}$ at fixed $m$ and $\\epsilon $ calls for an increase in $Re$ to sustain absolute instability.", "Similarly, Table REF corresponds to branch II, and demonstrates that an increase in $\\mathcal {G}$ at fixed $Re$ and $\\epsilon $ requires a corresponding, destabilizing increase in $m$ to sustain absolute instability.", "This produces a proportional decrease in $Re_\\mathrm {int}$ .", "The situation concerning increases in $\\mathcal {S}$ is somewhat counter-intuitive (increases in $\\mathcal {S}$ produce decreases in the critical value of $m$ , which produce increases in the critical value of $Re_\\mathrm {int}$ ).", "However, this agrees with the turbulent case, which has already been explained using the quadratic approximation.", "Table: Dependence of the critical parameters on surface tension and gravity at low ReRe (branch II, Re=10 4 Re=10^4, ϵ=0.1\\epsilon =0.1).", "The wavenumber at the saddle point is also stated.The uncertainty in mm is ±2.5\\pm 2.5, which also introduces an uncertainty in the values of Re int Re_\\mathrm {int} and α\\alpha .The scaling of the transition curves in Figure REF can again be elucidated with the quadratic approximation.", "For example, for large $Re$ and $Re_\\mathrm {int}$ , we have measured $\\max (\\omega _{\\mathrm {i}}^{\\mathrm {temp}})\\propto (\\epsilon /m^{1/2})Re^{1/4},\\qquad \\alpha _0-\\alpha _{0\\mathrm {c}}\\propto (\\epsilon ^{1/2}/m^{1/2})Re^{3/4},\\qquad c_{\\mathrm {g}}\\approx U_{\\mathrm {int}}\\propto \\epsilon /m.$ Plugging these scaling rules into the quadratic approximation gives $\\epsilon Re/m^2=\\mathrm {const.", "}$ , or $Re_\\mathrm {int}/\\epsilon =\\mathrm {const.", "}$ at large $Re$ and $Re_\\mathrm {int}$ , which implies the existence of a critical value of $Re_\\mathrm {int}/\\epsilon $ along branch I.", "The dramatic distinction between these scaling rules and those encountered in the turbulent case owes not to some deep distinction between the respective base states, but rather arises from the distinct non-dimensionalization schemes chosen in each case.", "Indeed, as a final test, we have verified that a laminar temporal study, based on the non-dimensionalization scheme previously applied in the turbulent case, yields the scaling rules described in Section REF (not shown).", "In Figure REF we examine again the wave number $\\epsilon (\\alpha _{\\mathrm {r}},\\alpha _{\\mathrm {i}})$ at the saddle point, at the inception of absolute instability.", "The contrast between this figure and parametric study (1) is remarkable.", "In Figure REF , the results resemble those for turbulence, with one significant difference: the saddle-point mode corresponds to a much longer wave.", "The reason for the contrast between parametric studies (1) and (2) is due to the fact that the wave length is controlled mostly by surface tension (as demonstrated by Table REF ); the surface tension in study (2) greatly exceeds that in study (1), consequently, the most unstable wave length is longer.", "Figure: The real (a) and imaginary (b) component of the wave number at the saddle point along all C/A transitions in Figure (a) (squares) and Figure (b) (triangles), as functions of Re int Re_\\mathrm {int}.", "The open symbols represent secondary saddle points that do not correspond to absolute instability.Referring to parametric study (1) and figure REF (b), branch II (varying $m$ ) of the C/A transition `follows' closely the neutral stability curve for a second temporal mode, M2.", "The same neutral curve is also present in parametric study (2), but is further removed from the C/A transition curve, and occurs at $Re_\\mathrm {int}/\\epsilon \\approx 1.5\\times 10^4$ .", "As in the turbulent case, it is as if the existence of the M2 mode quenches the M1 absolute instability.", "Again, as in the turbulent case, there is clear evidence of competition between spatio-temporal modes, in addition to temporal mode competition.", "In Figure REF , $\\omega _{\\mathrm {r}}$ is shown for the least stable mode at each complex $\\alpha $ .", "Results are presented at different values of $m$ for $Re=20000$ , $\\epsilon =0.1$ , and $r=1000$ .", "The value $m=85$ describes a situation where M2 has just become stable while M1 is convectively unstable (the point of neutral stability for M2 is at $m=75$ ).", "The C/A transition for M1 occurs at $m=170$ .", "The spatial curve $\\omega _{\\mathrm {i}}= 0$ of the most dangerous mode is identified with M1 by a Gaster-type analysis [13].", "The Figure: m=130\\,\\,m=130contours $\\omega _{\\mathrm {r}}= \\mathrm {Const}.$ that connect orthogonally to the spatial curve are identified also with M1.", "In this way, we have established that the saddle point in figure REF (a) does not correspond to M1.", "Thus mode competition interferes most dramatically with the saddle point near the point of neutral temporal stability of M2 (near $m=75$ ), while an approach to the C/A transition from within the convective regime (i.e., an increase in $m$ from a low value) causes the mode competition to disappear gradually, thus producing a conventional single-mode saddle point.", "This strongly suggests that the proximity of the M2 neutral curve and the M1 C/A transition in the flow diagram is no coincidence, and is in agreement with the earlier results for the turbulent case.", "The same phenomenon persists for lower Reynolds numbers (e.g.", "$Re=5000$ ).", "However, we have presented the results for $Re=20000$ because this regime exhibits the mode competition most clearly.", "At such high Reynolds numbers, a third unstable mode comes into existence (visible in Fig.", "REF in the neighbourhood $\\alpha _{\\mathrm {i}}=0$ and $\\alpha _{\\mathrm {r}}=2$ ).", "This mode has a critical Reynolds number $Re\\approx 8000$ and its neutral curve is almost independent of $Re_\\mathrm {int}$ , $m$ , and $\\epsilon $ .", "However, this third mode does not play any role in the spatio-temporal mode competition and is not discussed any further.", "Finally, we have applied the spatio-temporal energy budget to the laminar flow for the cases corresponding to Figure REF (a), (c) and (e).", "The results indicate that over most of the pulse width, $TAN$ dominates: this represents a source of energy for the instability that is associated with the viscosity-contrast mechanism, which dominates in temporal instability.", "However, $REY_G$ also plays a significant role.", "In all three cases, $REY_G$ is negative on the downstream side of the pulse, but positive on the upstream side.", "This is most visible in the absolutely unstable case.", "In contrast, for the turbulent base state, it is only in the absolutely unstable case that $REY_G$ develops this asymmetry, where it even becomes the dominant term on the upstream side in the energy-budget analysis." ], [ "Ray analysis revisited", "Some further information about the absolute instability of M1 is obtained from a ray analysis (Figure REF ).", "The parameters are taken from the cases studied in Figure REF .", "The branches of the C/A boundary in Figure REF (a) can be mapped on to families of $\\sigma (v)$ -curves in Figure REF : decreasing $\\epsilon $ from $\\epsilon \\approx 0.18$ in the latter gives a family of $\\sigma (v)$ -curves that is associated with the upper branch I of the C/A transition in Figure REF (a) ($m$ constant, $\\epsilon $ varying).", "As $\\epsilon $ is decreased, the $\\sigma (v)$ -curves shift downwards in their entirety, until the growth rate $\\sigma (v=0)$ eventually becomes positive, indicating a switchover to absolute instability.", "Conversely, increasing $\\epsilon $ from $\\epsilon \\approx 0.025$ gives a family of $\\sigma (v)$ -curves that is associated with the lower branch III in Figure REF (a).", "These curves are steep for small $\\epsilon $ -values, such that $\\sigma (v)$ is negative.", "As $\\epsilon $ increases, the slope diminishes, until $\\sigma (v=0)$ is positive, and absolute instability is attained from below.", "Furthermore, in a convectively unstable regime at large $\\epsilon $ -values, $\\sigma (v)$ is linear in $v$ for small $v$ , and the corresponding $\\alpha _{\\mathrm {r}}$ - and $\\omega _{\\mathrm {i}}$ -values approximate negative constants, consistent with Equations (REF ).", "Again with reference to Equations (REF ), the upper branch in Figure REF (a) corresponds to a shift in the value $v$ for which $\\sigma =\\sigma _{\\rm max}$ (hence $\\alpha _{\\mathrm {i}}=0$ ), whereas the transition at the lower branch is associated with an increase in temporal relative to spatial growth.", "Figure: Ray analysis: (a) spatio-temporal growth rates, (b) spatial growth rates along rays; (c) temporal growth rates along rays, for laminar flow at various values of ϵ\\epsilon , at r=1000r=1000, m=150m=150, Re=1500Re=1500.", "The rest of the parameters are the same as in Figure .", "Trend lines have been added in (a) merely to guide the eye.Finally, for large values of $\\epsilon $ , the $\\sigma (v)$ -curve is very wide, indicating a rapid spreading of an initial pulse, whereas for low values the pulse remains more localized.", "For the cases considered herein, we have verified that the wavenumber of the spatio-temporally most dangerous mode coincides with that of the temporally most dangerous mode (Table REF ).", "Table: Comparison between modal and spatio-temporal analyses at various values of ϵ\\epsilon , at r=1000r=1000, m=150m=150, and Re=1500Re=1500.", "The rest of the parameters are the same as in Figure .It was not possible to compute this wavenumber for all of the $\\epsilon $ -values, and our finite-difference discretization of Equation () is not always accurate.", "It is for this reason that we did not include $\\alpha _r$ -$v$ plots in Figure REF .", "Nevertheless, Figure REF still contains important information: a previous set of purely spatial and purely temporal analyses (see [34] and Appendix ) found extremely large purely spatial growth rates (exceeding the least-negative purely spatial growth rate by an order of magnitude) that were not excited in a DNS.", "From the spatial growth rates along rays in Figure REF (b), we see that such extremely large spatial growth rates are not selected, instead, the spatial growth rate comes from the same eigenmode as the temporally most dangerous mode." ], [ "Flow-regime maps", "It is of interest to compare our regime boundaries (e.g., Figure REF ) with flow-regime maps determined from experiments.", "Our theory is of course applicable to water/air systems, for which channel-flow experimental studies are available.", "However, the results in Sec.", "prove that water/air systems are usually convectively unstable, while the main focus of the present paper is on absolutely unstable systems.", "Consequently, we have searched the literature for experiments concerning air and oil, for which the viscosity contrast is much higher, and for which absolute instability is expected.", "We have found that such experiments were conducted on pipes (with the exception of the work by [14] on Hele-Shaw cells).", "Nevertheless, a comparison between our channel-flow model and a pipe-flow experiment is justified, since a 2D analysis may be representative of the mid-plane in pipe flow, especially for thin films, linear transverse waves (in the third dimension) are usually overwhelmed by 2D waves, and in any case it is necessary to demonstrate that the boundaries determined in our study are not restricted to an obscure limit.", "The results of Figure REF (a) are therefore presented in Figure REF in terms of superficial velocities $U_G$ and $U_L$ , where $c_g=U_G/(U_G+U_L)$ and $U_m=U_G+U_L$ is the mixture velocity for parameter values approximately corresponding to the flow of air and spindle oil through a pipe of 91mm diameter, and flow-regime boundaries determined experimentally by [18] are superposed (our data are for $m=1000$ , $r=1000$ and the choice of $S$ and $F$ mentioned at the start of Section , which are also representative of the experimental conditions stated in [18]).", "The more widely used flow-regime map by [22] cannot be used here, as it is for water/air systems, but they observed good agreement with water/air data also published by [18].", "We first note from Figure REF that the neutral stability curve from the temporal analysis agrees reasonably well with the experimental data on the transition from stratified flow (i.e., no waves) to wavy stratified flow.", "This is expected based on the comparisons with experiments in our previous work on temporal instability, [31].", "We note here that the superficial velocities in a 2D system are expected to be larger than for a pipe flow at the same maximal velocities in a fluid, which may have reduced the difference between the results somewhat.", "On the other hand, the fact that the liquid layer at the centre of the pipe is expected to be smaller due to gas pushing the liquid to the side of the pipe is expected to be of less significance, as the liquid height is the parameter along the theoretical data in the figure.", "The main C/A transition is predicted theoretically here at a total superficial velocity somewhat below the experimentally determined transition from wavy-stratified flow to mist flow (waves with droplets).", "A relation with the onset of atomization for very viscous liquids cannot be ruled out.", "In fact the experimental mist flow regime lies almost entirely in the absolutely unstable regime, since the critical value of the parameter $c_g$ for this regime is close to the high-$Re$ C/A transition from the theory (and even more so, the M2 neutral curve).", "In the experiments, slug flow is observed at $c_g$ below this critical value at large $U_m$ .", "Hence slug flow is observed experimentally when according to the 2D theory the flow is convectively unstable." ], [ "Potential use in global and non-linear analyses", "Although we have identified and classified the nature of the linear instability for our stated two-phase stratified problem, we briefly outline extensions that follow naturally from the present work.", "The stability analysis considered throughout this paper is a `local analysis' in the sense that the base flow varies on a length scale which is long relative to the wave length of the instability waves (this being true whether the analysis is linear or not).", "A more general approach is global stability analysis, wherein the whole of the physical domain is considered [19], [8].", "Under such a technique, spatial streamwise variation is accounted for in both the base flow and the perturbation terms, permitting the study of nonparallel flows, i.e., no restrictions are placed on spatial scalings.", "Of course, such studies require appropriate computational power and therefore have only been pursued recently.", "In the context of the present work, such an approach permits changes in the thickness of the liquid layer which is important for many of the stated applications, but it means that ray analysis in the form used herein can no longer be employed, as periodic boundary conditions no longer apply.", "The theoretical analysis required to elucidate non-linear effects discussed herein is elaborate.", "Thus, an important additional step might involve a numerical (e.g., DNS) study and experiments, with the aim of identifying which key non-linear effects dominate.", "Specifically, whether a transition to absolute instability in the nonlinear regime may precede the linear C/A transition (this has been observed by [14] in Hele-Shaw cells).", "Another question that a full numerical simulation might address concerns the location of the dominant non-linear interactions, i.e., at the phase interface or in the bulk of the liquid layer.", "A full numerical simulation would also permit a comparison between periodic boundary conditions (as used in the present work), and inlet conditions (as in a true channel flow).", "In this paper, the ray analysis is conducted in Fourier space, and required periodic boundary conditions.", "However, the pulse-type initial conditions considered herein involve extremely long domains, and comparisons with even larger domains are self-consistent.", "Nevertheless, full numerical simulations would further validate these observations.", "Such a study and the subsequent non-linear analysis will form the basis of future work." ], [ "Summary", "We have studied the linear stability of a liquid layer sheared by laminar or turbulent gas flow in a two-dimensional Cartesian geometry.", "A Poiseuille (laminar) or Reynolds-averaged (turbulent) model is used to describe the stratified two-phase base flow.", "Results from a normal-mode analysis and a ray analysis show that the base flow is absolutely unstable to linear perturbations for large regions of parameter space.", "In particular, for density ratios of $O(1000)$ , clear evidence is given of oil/gas flows being absolutely unstable for a sufficiently large dynamic viscosity ratio (at least $O(10^2)$ for the laminar and $O(10^3)$ for the turbulent base state), provided the Reynolds number of the flow is sufficiently high.", "Since the dynamic viscosity ratio for air/water systems is only $m\\approx 55$ at $20^{o}$ C, the present results for the turbulent case lead us to conclude that laboratory experiments carried out for water/air may not be representative for oil/gas systems.", "In both turbulent and laminar base states, the flow-regime map collapses in the ($Re$ , $Re_\\mathrm {int}$ ) plane, where $Re_\\mathrm {int}$ is the liquid-based Reynolds number (with a small subset of exceptional cases).", "A recently developed theory for C/A transitions [32] has enabled us to formulate a simple criterion for the onset of absolute instability, based on a competition between instability growth and convection by the group velocity.", "A single number, based on the temporal growth properties, encodes the tendency for instability growth.", "This number is governed by $\\epsilon $ , $m$ , and $Re$ , but becomes parameter-free in certain limits, while the behaviour of the group velocity is governed simply by $Re_\\mathrm {int}$ .", "This criterion therefore explains the importance of $Re_\\mathrm {int}$ in the C/A transition curves, and, moreover, when examined in detail, explains the collapse of the transition curves into universal forms.", "Independent confirmation of our results from the modal analysis has been obtained using a ray analysis.", "The ray analysis benefits from our development of an efficient method of solution for linear differential-algebraic equations, which allows us to extend drastically the simulation time over which the evolution of an initially pulsed disturbance can be traced, thereby eliminating the problems encountered with DNS by [34].", "This method all but guarantees the detection of absolute instability, if it exists.", "Additionally, the ray analysis leads naturally to an energy-budget method that determines the origin of the spatio-temporal instability.", "Results from this method demonstrate the importance of the viscosity-contrast mechanism acting at the interface, as well as a transfer of energy from the bulk gas flow to the waves, as the sources of the instability.", "Nevertheless, there exist several internal modes (which also derive energy from the interface) that are at least as significant in other cases.", "A further application of the ray analysis is envisaged for flows wherein the complex dispersion relation contains singularities (e.g.", "if $\\omega (\\alpha )$ has a pole [16], [17] or root-type behaviour [3] at some point $\\alpha _0$ ).", "In such problems, the application of the saddle-point criterion, with its associated construction of the steepest-descent path enclosing all the dynamically-relevant singularities, is a non-obvious task [28].", "For these cases, a straightforward ray analysis could be used to verify the correctness of one's implementation of steepest-descent method.", "The ray analysis suffers from the amplification of numerical error when the spatial growth rates are large, meaning that only a small region around the impulse centre can be used to extract meaningful information.", "Thus, both methods are necessary to obtain a complete picture of the stability properties.", "The ray analysis and the quadratic approximation are quite general, and we anticipate that a wide range of multiphase flow scenarios can be tackled using these techniques." ], [ "Validity of the quadratic approximation as applied to the C/A transition of two-phase stratified flow", "We review the so-called `quadratic approximation' developed by [32] and demonstrate that it approximates well the precise criteria for the onset of absolute instability.", "The quadratic approximation is based on the following identity for the analytic continuation of the growth rate $\\omega _{\\mathrm {i}}$ into the complex plane: $\\omega _{\\mathrm {i}}(\\alpha _{\\mathrm {r}},\\alpha _{\\mathrm {i}})=\\omega _{\\mathrm {i}}^{\\mathrm {temp}}(\\alpha _{\\mathrm {r}})+\\sum _{n=0}^\\infty \\frac{(-1)^n}{(2n+1)!", "}\\frac{d^{2n}c_{\\mathrm {g}}}{d\\alpha _{\\mathrm {r}}^{2n}}\\alpha _{\\mathrm {i}}^{2n+1}+\\sum _{n=0}^\\infty \\frac{(-1)^{n+1}}{(2n+2)!", "}\\frac{d^{2n+2}\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^{2n+2}}\\alpha _{\\mathrm {i}}^{2n+2},$ where $c_{\\mathrm {g}}$ is the group velocity $d\\omega _{\\mathrm {r}}/d\\alpha _{\\mathrm {r}}$ in a purely temporal analysis.", "This is a consequence of the Cauchy–Riemann conditions on $\\omega (\\alpha )$ viewed as a holomorphic function on an appropriate open subset in the complex plane, and has been derived elsewhere by [32].", "We truncate this series at quadratic order in $\\alpha _{\\mathrm {i}}$ and apply the conditions $\\partial \\omega _{\\mathrm {i}}/\\partial \\alpha _{\\mathrm {r}}=\\partial \\omega _{\\mathrm {i}}/\\partial \\alpha _{\\mathrm {i}}=0$ for a saddle point.", "This yields the conditions $\\frac{d\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}}+\\frac{dc_{\\mathrm {g}}}{d\\alpha _{\\mathrm {r}}}\\alpha _{\\mathrm {i}}-\\tfrac{1}{2}\\frac{d^3\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^3}\\alpha _{\\mathrm {i}}^2=0,\\qquad c_{\\mathrm {g}}(\\alpha _{\\mathrm {r}})-\\frac{d^2\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^2}\\alpha _{\\mathrm {i}}=0.$ Simultaneous solution of these equations yields the following $\\alpha _{\\mathrm {r}}$ -value for the location of the saddle point: $c_{\\mathrm {g}}(\\alpha _{\\mathrm {r}})\\frac{dc_{\\mathrm {g}}}{d\\alpha _{\\mathrm {r}}}=-\\frac{d\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}}\\frac{d^2\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^2}+\\tfrac{1}{2}c_{\\mathrm {g}}^2(\\alpha _{\\mathrm {r}})\\left(\\frac{d^3\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^3}\\bigg \\slash \\frac{d^2\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^2}\\right).$ (Note that for a strictly quadratic approximation, as in the main part of the paper, the third derivative should be neglected here.)", "At the onset of absolute instability, $\\omega _{\\mathrm {i}}=0$ at the saddle point.", "Substitution of this condition into the quadratic approximation to Equation (REF ) yields the following criterion for the onset of absolute instability: $-\\frac{d^2\\omega _{\\mathrm {i}}^{\\mathrm {temp}}}{d\\alpha _{\\mathrm {r}}^2}\\bigg |_{\\alpha _{\\mathrm {r}}^*}\\omega _{\\mathrm {i}}^{\\mathrm {temp}}(\\alpha _{\\mathrm {r}}^*)=\\tfrac{1}{2}c_{\\mathrm {g}}^2(\\alpha _{\\mathrm {r}}^*).$ where $\\alpha _{\\mathrm {r}}^*$ is the root of Equation (REF ).", "In Figure REF , the results of the quadratic approximation are compared with the full modal results that have been taken from Figures REF and REF for the turbulent and laminar base states, respectively.", "All Figure: Comparison of flow-regime maps resulting from the quadratic approximation (open symbols, keeping mm constant; filled symbols, keeping ϵ\\epsilon constant) and the full modal analysis (solid lines) for the turbulent (a) and laminar (b) base state.branches are qualitatively correctly predicted, and quantitatively accurate results are even obtained at some, especially all results at large $Re$ in the turbulent case, and those for low $Re$ and a large viscosity ratio in both laminar and turbulent cases." ], [ "Analysis of purely spatial modes", "The focus throughout the paper has been on spatio-temporal analysis.", "In particular, our modal analysis has involved the extraction of saddle points from the solution of an eigenvalue problem in a complex wave number space.", "In this section, we review the Gaster theory for the extraction of purely spatial growth rates.", "The focus herein is on the turbulent base state.", "The most basic description of linear instability is a temporal analysis, which involves the solution of the eigenvalue problem (REF ) for $\\alpha =\\alpha _{\\mathrm {r}}$ only.", "This gives a dispersion relation $(\\omega _{\\mathrm {r}}(\\alpha _{\\mathrm {r}}),\\omega _{\\mathrm {i}}(\\alpha _{\\mathrm {r}}))$ .", "The pair $(\\omega _{\\mathrm {r}}(\\alpha _{\\mathrm {r}}),\\omega _{\\mathrm {i}}(\\alpha _{\\mathrm {r}}))$ that maximizes $\\omega _{\\mathrm {i}}$ is called the most dangerous temporal mode.", "The maximum is taken over the complete spectrum of modes in the linear Orr–Sommerfeld (OS) equations.", "The system is unstable if $\\omega _{\\mathrm {i}}>0$ at the most dangerous temporal mode.", "This case has been investigated exhaustively by the present authors [30], [31], [33].", "We therefore turn to the spatial case, wherein we are concerned with the dispersion relation $\\alpha _{\\mathrm {i}}=\\alpha _{\\mathrm {i}}\\left(\\alpha _{\\mathrm {r}},\\omega _{\\mathrm {i}}=0\\right)$ , which can be obtained by treating the OS problem (REF ) as a non-linear eigenvalue problem in $\\alpha $ .", "In practice, we do not solve this complicated problem.", "Instead, we solve Equation (REF ) in the complex $\\alpha $ -plane, $\\alpha :=\\alpha _{\\mathrm {r}}+\\mathrm {i}\\alpha _{\\mathrm {i}}$ .", "The result is a discrete spectrum of frequencies $\\lbrace \\omega _n(\\alpha )\\rbrace _n$ , which we order according to the magnitude of $\\Im \\left[\\omega _n(\\alpha )\\right]$ : the most dangerous mode at complex wave number $\\alpha $ is denoted by $\\omega _C(\\alpha )$ , and is defined such that $\\Im \\left[\\omega _C(\\alpha )\\right]=\\max _{n,\\text{ all modes}} \\Big \\lbrace \\Im \\left[\\omega _{n}\\left(\\alpha \\right)\\right]\\Big \\rbrace _{\\alpha \\in \\mathbb {C}}.$ We plot the zero contour of $\\Im \\left[\\omega _C(\\alpha )\\right]=0$ , which gives the desired relation $\\alpha _{\\mathrm {i}}=\\alpha _{\\mathrm {i}}\\left(\\alpha _{\\mathrm {r}},\\omega _{\\mathrm {i}}=0\\right)$ .", "We repeat these steps for the less dangerous modes.", "This results in multiple spatial curves.", "These are displayed in Figure REF (a) for the parameter values $m=55$ , $r=1000$ , $\\epsilon =0.05$ , with $F$ and $S$ given by Equation (REF ).", "However, in this figure, we plot only the spatial curves produced by the most dangerous mode; the less dangerous modes do not produce the same large spatial amplification (spatial amplification corresponds to $\\alpha _{\\mathrm {i}}<0$ ), and are not shown.", "The large spatial growth rates in the figure would appear to dominate in an evolving flow.", "In practice, however, they are not observed, neither in turbulence nor in laminar flows.", "Because the large spatial growth rates are not accessed, we focus our attention on the spatial curves near $\\alpha _\\mathrm {i}=0$ , for which an analytical theory is available.", "Figure: (a) Spatial stability study; (b) Comparison with Gaster analysis.", "Here `MDF' refers to the `most-dangerous frequency', or, equivalently, the most-dangerous mode.", "[13] has shown that small spatial growth rates are related to the temporal ones through the formula $-\\alpha _\\mathrm {i}=\\omega _\\mathrm {i} \\Big / \\frac{\\partial \\omega _\\mathrm {r}}{\\partial \\alpha _\\mathrm {r}},$ where the quantities on the right-hand side are derived from the purely temporal analysis.", "We test the applicability of Equation (REF ) in Figure REF (b).", "We plot the Gaster curves associated with the temporally most dangerous mode and the second most dangerous mode in the figure, and compare the result with the contour in Figure REF (a).", "There is good agreement between the two theories.", "However, the curve generated from the contour analysis exhibits `kinks' where the most dangerous mode switches from one eigenmode to another.", "To obtain perfect overlap between the two curve sets, it is necessary to pick out all the eigenmodes from the contour analysis and to cut and re-connect the contours so as to obtain smooth curves.", "This exercise is difficult but yields no new information, and we do not pursue it here.", "Instead, we focus on the Gaster study, where two spatial curves overlap with the contour-generated ones.", "The Gaster curves are obtained from the first and second temporally most dangerous modes.", "Thus, the character of a spatial instability is identified with the character of the temporal instability through the Gaster formula (REF ).", "The Gaster curve suggests that the second most dangerous temporal mode contributes most strongly to the spatial instability.", "However, it is not obvious from this simple analysis which eigenmode is excited in an impulse-response scenario: this question is studied in detail in Section , where it is demonstrated that the spatial curve associated with the temporally most dangerous mode is the one that is selected." ], [ "Acknowledgements", "The authors would also like to thank J.-C. Loiseau for carrying out preliminary numerical investigations for this project." ] ]

[ [ "Self inductance of a wire loop as a curve integral" ], [ "Abstract It is shown that the self inductance of a wire loop may be written as a curve integral akin to the Neumann formula for the mutual inductance of two wire loops.", "The only difference is that contributions where the two integration variables get too close to each other must be excluded from the curve integral and evaluated in detail.", "The contributions of these excluded segments depend on the distribution of the current in the cross section of the wire.", "They add to a simple constant proportional to the wire length.", "The error of the new expression is of first order in the wire radius if there are sharp corners and of second order in the wire radius for smooth wire loops." ], [ "Introduction", "Electrical inductance plays a crucial role in power plants, transformers and electronic devices.", "The coefficients of self and mutual inductance required to quantitatively describe inductance belong to the field of magnetostatics.", "Calculating inductance coefficients with analytic techniques is, however, impossible except in simple cases.", "The mathematical reason for the difficulty is that the Laplace equation allows analytic solutions only for some symmetric constellations.", "There thus are only a few closed-form expressions for these coefficients.", "In practice one often is forced to use approximations, finite element methods or other numerical techniques.", "The situation simplifies when the current flows in thin wires.", "This situation is analogous to an electrostatic system of point charges, where electric field and electrostatic energy directly follow from the given charge distribution, while in a generic system charge and current distributions also are unknown at the outset.", "The purpose of this article is to derive a new expression for the self inductance of a wire loop, giving self inductance as a curve integral similar to the Neumann formula for mutual inductance.", "The starting point is the expression $W=\\frac{\\mu _{0}}{8\\pi }\\int \\frac{\\mathbf {j}\\left(\\mathbf {x}\\right)\\cdot \\mathbf {j}\\left(\\mathbf {x}^{\\prime }\\right)}{|\\mathbf {x-x}^{\\prime }|}d^{3}xd^{3}x^{\\prime }$ for the magnetic field energy of a system with current density $\\mathbf {j}\\left(\\mathbf {x}\\right)$ , where $\\mu _{0}$ is the magnetic constant.", "[1] This expression essentially was already given by Neumann in 1845.", "[2] It resembles the expression for gravitational or electrostatic potential energy, the only new ingredient is the scalar product between the current elements.", "For a current density $\\mathbf {j}\\left(\\mathbf {x}\\right)=\\sum I_{m}\\mathbf {j}_{m}\\left(\\mathbf {x}\\right)$ corresponding to $N$ separate current loops with currents $I_{m}$ and normalized current densities $\\mathbf {j}_{m}$ it follows $W=\\frac{\\mu _{0}}{8\\pi }\\sum _{m,n=1}^{N}I_{m}I_{n}\\int \\frac{\\mathbf {j}_{m}\\left(\\mathbf {x}\\right)\\cdot \\mathbf {j}_{n}\\left(\\mathbf {x}^{\\prime }\\right)}{|\\mathbf {x-x}^{\\prime }|}d^{3}xd^{3}x^{\\prime }\\overset{!", "}{=}\\frac{1}{2}\\sum _{m,n=1}^{N}L_{m,n}I_{m}I_{n}.$ If the currents flow in thin wires, then the integrals become curve integrals, and one immediately reads off the Neumann expression for mutual inductance of two (filamentary) current loops[2] $L_{1,2}=\\frac{\\mu _{0}}{4\\pi }\\oint \\frac{d\\mathbf {x}_{1}\\cdot d\\mathbf {x}_{2}}{\\left|\\mathbf {x}_{1}\\mathbf {-x}_{2}\\right|}.$ It is plausible that there exists a similar expression for the self inductance of a wire loop, but we were not able to find any hint in the literature.", "Formally one might read off from equation (REF ) an expression similar to equation (REF ), where the two closed curves coincide.", "But this cannot be correct, because $|\\mathbf {x-x}^{\\prime }|$ now vanishes and the integral isn't defined.", "Instead we will prove $L=\\frac{\\mu _{0}}{4\\pi }\\left(\\oint \\frac{d\\mathbf {x}\\cdot d\\mathbf {x}^{\\prime }}{\\left|\\mathbf {x-x}^{\\prime }\\right|}\\right)_{\\left|s-s^{\\prime }\\right|>a/2}+\\frac{\\mu _{0}}{4\\pi }lY+...$ where $a$ denotes the wire radius and $l$ the length of the wire.", "The variable $s$ measures the length along the wire axis.", "The constant $Y$ depends on the distribution of the current in the cross section of the wire: $Y=0$ if the current flows in the wire surface, $Y=1/2$ when the current is homogeneous across the wire.", "The ellipses represents terms like $O\\left(\\mu _{0}a\\right)$ and $O\\left(\\mu _{0}a^{2}/l\\right)$ , which are negligible for $l\\gg a$ .", "In hindsight it is completely natural to use a cutoff of order $a$ in the curve integral.", "In fact, the exact value of this cutoff is arbitrary, because the contribution proportional to $lY$ also depends on this cutoff.", "The simplest way to determine $Y$ would be to compare the expression with the self inductance of a long rectangle." ], [ "Simple derivation", "Consider equation (REF ) with $N=1$ for a thin wire with circular cross section, radius $a$ and length $l$ .", "Let $s$ denote the length along the axis of the wire.", "The planes perpendicular to the wire axis then define a projection from the bulk of the wire onto the axis, $\\mathbf {x}\\rightarrow s\\left(\\mathbf {x}\\right)$ .", "Selecting a length scale $b$ satisfying $a\\ll b\\ll l$ allows to write $L=\\left(\\mu _{0}/4\\pi \\right)\\left(\\overline{L}+\\widehat{L}\\right)$ with $\\overline{L} & = & \\left(\\int \\frac{\\mathbf {j}\\left(\\mathbf {x}\\right)\\mathbf {j}\\left(\\mathbf {x}^{\\prime }\\right)}{|\\mathbf {x-x}^{\\prime }|}d^{3}xd^{3}x^{\\prime }\\right)_{\\left|s-s^{\\prime }\\right|>b},\\\\\\widehat{L} & = & \\left(\\int \\frac{\\mathbf {j}\\left(\\mathbf {x}\\right)\\mathbf {j}\\left(\\mathbf {x}^{\\prime }\\right)}{|\\mathbf {x-x}^{\\prime }|}d^{3}xd^{3}x^{\\prime }\\right)_{\\left|s-s^{\\prime }\\right|<b}.$ The second part contains contributions from all point pairs $\\left\\lbrace \\mathbf {x},\\mathbf {x}^{\\prime }\\right\\rbrace $ with a distance along the axis smaller than $b$ , the first the complement of this set ($s$ is a cyclic quantity).", "For given $\\mathbf {x}$ the planes at $s\\left(\\mathbf {x}\\right)\\pm b$ delimitate the points $\\mathbf {x}^{\\prime }$ contributing to the first or second integral, see figure (REF ).", "$\\overline{L}$ now approximately becomes a curve integral and $\\widehat{L}$ essentially consists of cylinders of length $2b$ .", "Figure: A section of a wire with radius aa, with a segment of length 2b2b,and a plane perpendicular to the wire axis at the center of the segment.The strategy then is to replace $\\overline{L}$ with the curve integral and to explicitly evaluate the contribution of the cylinders in $\\widehat{L}$ .", "The cylinders are long in comparison to the radius because of $a\\ll b$ and straight (at least most of them) because of $b\\ll l$ .", "Actually the only requirement for the lengths is $a\\ll l$ , the length $b=\\sqrt{al}$ then satisfies $a\\ll b\\ll l$ .", "The approximation thus is exact in the limit $a\\ll l$ except in special cases.", "Inserting $\\widehat{L}_{0}$ for a straight segment from equation (REF ) in the appendix thus leads to $L=\\frac{\\mu _{0}}{4\\pi }\\left(\\oint \\frac{d\\mathbf {x}\\cdot d\\mathbf {x}^{\\prime }}{|\\mathbf {x-x}^{\\prime }|}\\right)_{\\left|s-s^{\\prime }\\right|>b}+\\frac{\\mu _{0}l}{2\\pi }\\left(\\ln \\left(\\frac{2b}{a}\\right)+\\frac{Y}{2}\\right)+...$ This expression cannot depend on the (more or less) arbitrary length scale $b$ .", "The curve integral thus isThis also can easily be verified with an explicit calculation.", "$\\overline{L}\\left(b\\right)=const-\\frac{\\mu _{0}l}{2\\pi }\\ln \\left(2b/b_{0}\\right)$ .", "But $b$ is the only “short” length scale in the curve integral, and $\\overline{L}\\left(b\\right)$ thus also is valid for $b=a/2$ .", "The expression (REF ) therefore doesn't change if one formally sets $b=a/2$ .", "Equation (REF ) now agrees with equation (REF ), but some questions remain.", "First of all, how accurate is formula (REF )?", "The curve integral is a purely geometric quantity with dimension “length” and order of magnitude $l$ .", "Plausible expressions for the order of magnitude of the relative error are $a/l$ , $\\left(a/l\\right)^{2}$ and $\\left(a/R\\right)^{2}$ , with $R$ a typical curvature radius of the wire loop.", "Errors of this order normally are negligible (and also occur in the Neumann formula for mutual inductance).", "But the derivation of formula (REF ) is not as straightforward, and so what are the actual limits or exceptions?" ], [ "Examples and comparison with exact self inductance", "To get an impression of the accuracy we have compared self-inductances calculated with the curve integral with the result of a numeric evaluation of the nominally 6-dimensional integral in equation (REF ).", "This integral becomes 4-dimensional if the currents flow in the wire surface (skin effect, $Y=0$ ), and the results below correspond to this skin effect case.", "The order of magnitudes of the error terms identified here are corroborated below in a more detailed derivation of formula (REF )." ], [ "Straight segment", "The first example is a straight segment with length $c$ and complete skin effect.", "This of course isn't a closed circuit, but it might be an edge of a rectangle.", "The orthogonal edges of the rectangle don't interact with the segment because of the scalar product $\\mathbf {j}\\cdot \\mathbf {j}^{\\prime }$ .", "What is missing for a rectangle are the interaction terms with the opposite edges (and the small contributions from the corners).", "In this case the volume integral (REF ) may even be evaluated analytically with the result $L=\\frac{\\mu _{0}}{4\\pi }\\left\\lbrace 2c\\left[\\ln \\left(\\frac{2c}{a}\\right)-1\\right]+8a/\\pi -a^{2}/c+...\\right\\rbrace ,$ while the curve integral (REF ) leads to $L_{c}\\left(c\\right)=\\frac{\\mu _{0}}{4\\pi }\\left\\lbrace 2c\\left[\\ln \\left(\\frac{2c}{a}\\right)-1\\right]+a\\right\\rbrace .$ The difference is of order $O\\left(\\mu _{0}a\\right)$ , much smaller than $\\mu _{0}c$ for $c\\gg a$ .", "The next example is a ring with radius $R$ .", "The curve integral (REF ) gives $L_{c}=\\mu _{0}R\\left(\\ln \\left(8R/a\\right)-2+Y/2\\right)+\\mu _{0}O\\left(a^{2}/R\\right).$ This expression also may be found in the literature, derived with the help of elliptic functions and some approximations in a much more complicated way.", "The table displays the ratio of the exact inductance and $L_{c}$ for some values of $R/a$ , Table: NO_CAPTIONThe expression $L_{c}$ is more accurate than one might expect.", "It gives a reasonable approximation already for $R=3a$ , and the error roughly decays like $O\\left(a^{2}/R^{2}\\right)$ .", "This case is more complicated because in principle also the shape of the corners comes into play (curvature radius?).", "But the simplest thing to do is to evaluate the curve integral (REF ) for a rectangle with edges of length $c$ and $d$ .", "Orthogonal edges decouple because of the scalar product $\\mathbf {j}\\cdot \\mathbf {j}^{\\prime }$ , and the first contribution are the terms (REF ) for the four edges by themselves.", "The second contribution are the parts of the curve integral (REF ) with $\\mathbf {x}$ on one edge and $\\mathbf {x}^{\\prime }$ on the opposite one.", "The condition $\\left|s-s^{\\prime }\\right|>a/2$ is irrelevant for these cross terms, and one easily obtains for parallel edges of length $c$ and distance $d$ $L_{c}\\left(c,d\\right)=\\frac{\\mu _{0}}{4\\pi }\\left(4\\sqrt{c^{2}+d^{2}}-4d-4c\\operatorname{asinh}\\left(c/d\\right)\\right),$ and the sum together with the $Y$ -term of equation (REF ) is $L_{c} & = & \\frac{\\mu _{0}}{\\pi }\\lbrace \\; c\\ln \\frac{2c}{a}+d\\ln \\frac{2d}{a}-\\left(c+d\\right)\\left(2-Y/2\\right)\\\\& & +2\\sqrt{c^{2}+d^{2}}-c\\operatorname{asinh}\\left(c/d\\right)-d\\operatorname{asinh}\\left(d/c\\right)+a\\;\\rbrace .$ This expression also may be found in the literature, with sometimes a factor 2 at the $a$ -term.", "[3] The table displays the ratio of the numerically evaluated self-inductance $L$ of a square with border length $c$ and corners with curvature radius $a$ and the curve integral $L_{c}$ for different border length $c$ , Table: NO_CAPTIONThe curvature radius $a$ is minimal in that the centre of curvature lies on the inner border of the wire.", "It is remarkable that the absolute error nearly remains constant.", "The last row of the table displays the ratio of the exact self inductance and the exact curve integral (REF ) (with round corners), also evaluated numerically.", "This expression is a better approximation for small $c/a$ , where the square with round corners degenerates to a ring.", "The curve integral (REF ) for an equilateral triangle with edge length $c$ consists of three times the expression (REF ) for the edges by themselves and three times the interaction energy $L_{c}\\left(c,c,120\\right)$ of adjacent edges (with $s$ on one edge and $s^{\\prime }$ on the other, see appendix .", "There is no such interaction for rectangles because of the scalar product), $L_{c}=\\frac{\\mu _{0}}{2\\pi }3c\\left\\lbrace \\ln \\left(\\frac{c}{a}\\right)-1-\\ln \\frac{3}{2}\\right\\rbrace .$ The table displays the ratio of the exact self-inductance $L$ of an equilateral triangle with border length $c$ and corners with curvature radius $a$ and the curve integral $L_{c}$ for different border length $c$ , Table: NO_CAPTIONThe absolute error $\\left(L-L_{c}\\right)/\\mu _{0}$ is nearly constant also here.", "The last row again is the ratio of the exact self inductance and the curve integral (REF ) with round corners, evaluated numerically.", "For a loop consisting of infinitely long parallel wires the condition $a\\ll l$ is perfectly met and the curve integral gives the exact self inductance even for minimal distance $d=2a,$ $L_{c}=\\frac{\\mu _{0}l}{\\pi }\\left(\\ln \\frac{d}{a}+Y/2\\right).$ This expression is the limiting case of the expression (REF ) for a long rectangle.", "The point is, that the contribution of the corners becomes negligible for a long rectangle, and that the replacement of the (circular symmetric) current by a line current doesn't change the magnetic field according to Ampere's law.", "Of course, the assumption of a circular symmetric current distribution gets wrong in the skin effect case if the wires are close to each other because of additional screening currents.", "To summarize, formula (REF ) is rather accurate even for circuits with a linear extension as small as 20 times the wire radius, even if the circuit contains sharp corners." ], [ "Error estimation", "According to equation (REF ) the self inductance may be written as $L=\\left(\\mu _{0}/4\\pi \\right)\\left(\\overline{L}+\\widehat{L}\\right)$ , where $\\widehat{L}$ contains the short segments and $\\overline{L}$ the complement.", "The self inductance $L$ of course doesn't depend on the arbitrary segment length $b$ .", "Let us now introduce some notation.", "We use a coordinate system $\\left\\lbrace s,r,\\phi \\right\\rbrace $ in the wire where the length $s$ along the wire axis is cyclic with period $l$ , and the coordinates $\\left\\lbrace r,\\phi \\right\\rbrace $ describe planes perpendicular to the wire axis.", "The intersections of the planes and the wire are assumed to be circular, $0\\leqq r\\leqq a$ and $0\\leqq \\phi \\leqq 2\\pi $ .", "The volume element reads $dV=\\left(1+r\\cos \\phi /R\\left(s\\right)\\right)rdrd\\phi ds$ , where $R\\left(s\\right)$ denotes the curvature radius of the wire, and $\\phi =0$ at the outer border of the wire (this is possible at least locally).", "The volume element also may be written as $dV=dsd\\widetilde{A}$ with $d\\widetilde{A}=\\left(1+r\\cos \\phi /R\\left(s\\right)\\right)dA$ and area element $dA=rdrd\\phi $ .", "The coordinates become cylindrical coordinates for straight wire segments, i.e.", "for $R=\\infty $ .", "The current density $\\mathbf {j}$ is normalized, that is $\\int dA\\left|\\mathbf {j}\\right|\\mathbf {=}\\int d\\widetilde{A}\\left|\\mathbf {j}\\right|=1$ .", "We will also need the radial moments $a_{n}=\\left\\langle r^{n}\\right\\rangle =\\int d\\widetilde{A}r^{n}\\left|\\mathbf {j}\\right|$ of the current distribution.", "In the skin effect case of course $a_{n}=a^{n}$ .", "One quantity of interest then is $\\overline{L}\\left(s\\right)=\\int ds^{\\prime }\\theta \\left(\\left|s^{\\prime }-s\\right|-b\\right)\\int d\\widetilde{A}d\\widetilde{A}^{\\prime }\\frac{\\mathbf {j}\\left(s,r,\\phi \\right)\\cdot \\mathbf {j}\\left(\\mathbf {x}^{\\prime }\\right)}{|\\mathbf {x}\\left(s,r,\\phi \\right)\\mathbf {-x}\\left(s^{\\prime },r^{\\prime },\\phi ^{\\prime }\\right)|},$ the energy of the current in the plane at $s=0$ with respect to the current at $\\left|s^{\\prime }-s\\right|>b$ , see figure (REF ).", "The symbol $\\theta $ denotes the step function.", "To obtain $\\overline{L}$ from $\\overline{L}\\left(s\\right)$ requires to integrate over $s$ .", "The curve integral $\\overline{L}_{\\gamma }\\left(s\\right)=\\oint ds^{\\prime }\\theta \\left(\\left|s^{\\prime }-s\\right|-b\\right)\\frac{\\cos \\left(\\mathbf {j}\\left(s\\right),\\mathbf {j}\\left(s^{\\prime }\\right)\\right)}{|\\mathbf {x}\\left(s,0,0\\right)\\mathbf {-x}\\left(s^{\\prime },0,0\\right)|},$ is an approximation for $\\overline{L}\\left(s\\right)$ .", "Similarly we write the short segment around the plane at $s=0$ as $\\widehat{L}\\left(s\\right)=\\int ds^{\\prime }\\theta \\left(b-\\left|s^{\\prime }-s\\right|\\right)\\int d\\widetilde{A}d\\widetilde{A}^{\\prime }\\frac{\\mathbf {j}\\left(s,r,\\phi \\right)\\cdot \\mathbf {j}\\left(s^{\\prime },r^{\\prime },\\phi ^{\\prime }\\right)}{|\\mathbf {x}\\left(s,r,\\phi \\right)\\mathbf {-x}\\left(s^{\\prime },r^{\\prime },\\phi ^{\\prime }\\right)|},$ These definitions allow to write $\\frac{4\\pi }{\\mu _{0}}L\\left(s\\right) & = & \\overline{L}\\left(s\\right)+\\widehat{L}\\left(s\\right)=\\\\& = & \\left\\lbrace \\left(\\overline{L}_{\\gamma }+\\widehat{L}_{\\gamma }\\right)+\\left(\\overline{L}-\\overline{L}_{\\gamma }+\\hat{P}_{0}\\right)+\\left(\\widehat{L}-\\widehat{L}_{0}\\right)\\right\\rbrace _{s},$ where we have added and subtracted the curve integral $\\overline{L}_{\\gamma }\\left(s\\right)$ , the segment integral $\\widehat{L}_{0}\\left(s\\right)$ for a straight segment from equation (REF ) and the approximation $\\widehat{L}_{\\gamma }\\left(s\\right)=2\\left(\\ln \\left(2b/a\\right)+Y/2\\right)$ for $\\widehat{L}_{0}\\left(s\\right)$ .", "The first bracket in equation (REF ) now is formula (REF ), and the two other brackets thus represent the error.", "The second bracket contains the difference of the volume and the curve integral (for $\\left|s^{\\prime }-s\\right|>b$ ) plus the power series $\\hat{P}_{0}=\\widehat{L}_{0}-\\widehat{L}_{\\gamma }$ from equation (REF ), the third bracket is the difference of the actual segment integral (REF ) and the segment integral for a straight segment.", "It is evident that the error becomes small in suitable limits, and we now want to determine the order of magnitude of the error." ], [ "Smooth current loops", "We first consider smooth current loops, that is current loops with a minimal curvature radius $R$ comparable with the system size.", "We also assume that the loop returns immediately and doesn't touch itself anywhere in between.", "Such complications are considered below.", "The coordinates $\\mathbf {x}$ and $\\mathbf {x}^{\\prime }$ in the integral (REF ) above may be expanded like $\\mathbf {x}=\\mathbf {x}\\left(s,0,0\\right)+\\mathbf {x}_{1}\\left(s,r,\\phi \\right)$ , and thus $\\mathbf {x}-\\mathbf {x}^{\\prime } & = & \\mathbf {x}_{s,s^{\\prime }}+\\mathbf {x}_{1}-\\mathbf {x}_{1}^{\\prime },\\\\\\left(\\mathbf {x}-\\mathbf {x}^{\\prime }\\right)^{2} & = & x_{s,s^{\\prime }}^{2}+2\\mu x_{s,s^{\\prime }}+\\nu ^{2}$ with $\\mathbf {x}_{s,s^{\\prime }}=\\mathbf {x}\\left(s,0,0\\right)-\\mathbf {x}\\left(s^{\\prime },0,0\\right)$ the distance of the projections onto the axis and $\\left|\\mathbf {x}_{1}\\right|$ and $\\left|\\mathbf {x}_{1}^{\\prime }\\right|$ of order $O\\left(a\\right)$ .", "The abbreviations are $\\mu & = & \\widehat{\\mathbf {x}}_{ss^{\\prime }}\\cdot \\mathbf {x}_{1}+\\widehat{\\mathbf {x}}_{s^{\\prime }s}\\cdot \\mathbf {x}_{1}^{\\prime },\\\\\\nu ^{2} & = & \\left(\\mathbf {x}_{1}-\\mathbf {x}_{1}^{\\prime }\\right)^{2}.$ The procedure now is to use the multipole expansion $\\frac{1}{\\left|\\mathbf {x}-\\mathbf {x}^{\\prime }\\right|} & = & \\frac{1}{x_{s,s^{\\prime }}}-\\frac{\\mu }{x_{s,s^{\\prime }}^{2}}+\\frac{1}{2x_{s,s^{\\prime }}^{3}}\\left(3\\mu ^{2}-\\nu ^{2}\\right)-\\frac{1}{2x_{s,s^{\\prime }}^{4}}\\left(5\\mu ^{3}-3\\mu \\nu ^{2}\\right)\\\\& & +\\frac{1}{8x_{s,s^{\\prime }}^{5}}\\left(35\\mu ^{4}-30\\mu ^{2}\\nu ^{2}+3\\nu ^{4}\\right)+O\\left(a^{5}x_{s,s^{\\prime }}^{-6}\\right).$ This expansion converges for $x_{s,s^{\\prime }}\\geqq \\left|\\mathbf {x}_{1}^{\\prime }\\mathbf {-x}_{1}\\right|$ , the coefficients are the coefficients of the Legendre polynomials.", "Inserting the leading monopole term $1/x_{s,s^{\\prime }}$ of equation (REF ) into equation (REF ) reproduces the curve integral $\\overline{L}_{\\gamma }\\left(s\\right)$ .", "The higher multipole terms describe the difference between the volume integral $\\overline{L}\\left(s\\right)$ and the curve integral.", "The $s^{\\prime }$ -integral in the multipole terms converges and the difference thus is mainly a local quantity.", "The nominal order of magnitude of the multipole terms $\\overline{L}_{n}$ is $a^{n}/b^{n}$ , $n\\geqq 1$ .", "With a length $b=\\sqrt{aR}$ the order of magnitude becomes $\\left(a/R\\right)^{n/2}$ , and the expansion up to the hexadecupole ($n=4$ ) is needed to get the $\\left(a/R\\right)^{2}$ approximation.", "The volume element is $\\left(1+r\\cos \\phi /R\\left(s\\right)\\right)rdrd\\phi ds=\\left(1+\\mathbf {x}_{1}\\cdot \\mathbf {R}\\left(s\\right)/R^{2}\\left(s\\right)\\right)rdrd\\phi ds,$ where $\\mathbf {R}\\left(s\\right)$ is the local curvature radius vector.", "Integrals over $\\phi $ and $\\phi ^{\\prime }$ then may be evaluated with the help of $\\left\\langle \\mathbf {x}_{1}\\right\\rangle & = & 0,\\\\\\left\\langle \\left(\\mathbf {x}_{1}\\right)_{m}\\left(\\mathbf {x}_{1}\\right)_{n}\\right\\rangle & = & r^{2}P_{m,n}/2,$ where $P_{m,n}$ is the projection operator projecting onto the plane perpendicular to the wire axis.", "This implies $\\mathbf {P\\cdot R}=\\mathbf {R}$ .", "Inserting now the multipole expansion (REF ) into the volume integral (REF ) generates an expansion of the difference of the volume and the curve integral in the region $\\left|s^{\\prime }\\right|>b$ ." ], [ "Dipole", "The voltage drops by the same amount along inner and outer border along curved parts of the loop.", "Electric field and current density thus are larger at the inner border, and the current density thus depends on $r$ and $\\varphi $ .", "This $\\varphi $ -dependence of the current density compensates the $\\varphi $ -dependence $1+r\\cos \\left(\\varphi \\right)/R$ from the volume element.", "The simple result then is: there is no dipole contribution.", "The factors $\\cos \\varphi $ and $\\cos \\varphi ^{\\prime }$ from $\\mu $ give 0 after integration over the angles.", "There are in fact short transition regions between straight and curved parts of the loop where the current distribution changes from uniform to non-uniform, but this only leads to corrections of higher order." ], [ "Quadrupole", "After integration over the angles $\\phi $ and the radial coordinates $r$ there remains $\\overline{L}_{2}\\left(s\\right)=a_{2}\\oint ds^{\\prime }\\theta \\left(\\left|s^{\\prime }-s\\right|-b\\right)\\left(\\frac{3}{4}\\left(2-\\left(\\widehat{\\mathbf {n}}\\cdot \\widehat{\\mathbf {x}}_{s,s^{\\prime }}\\right)^{2}-\\left(\\widehat{\\mathbf {n}}^{\\prime }\\cdot \\widehat{\\mathbf {x}}_{s,s^{\\prime }}\\right)^{2}\\right)-1\\right)\\frac{\\cos \\alpha }{x_{s,s^{\\prime }}^{3}},$ where $\\widehat{\\mathbf {n}}$ denotes a unit vector in the direction of the wire axis and the expression $\\mathbf {P}=1-\\widehat{\\mathbf {n}}\\widehat{\\mathbf {n}}$ for the projection operator was used.", "In $\\overline{L}_{2}\\left(s\\right)$ one may recognize $1-\\left(\\widehat{\\mathbf {n}}\\cdot \\widehat{\\mathbf {x}}_{s,s^{\\prime }}\\right)^{2}=\\sin ^{2}\\psi $ and $1-\\left(\\widehat{\\mathbf {n}}^{\\prime }\\cdot \\widehat{\\mathbf {x}}_{s,s^{\\prime }}\\right)^{2}=\\sin ^{2}\\psi ^{\\prime }$ , where $\\psi $ denotes the angle between the distance vector $\\mathbf {x}_{s,s^{\\prime }}$ and the wire axis.", "With $\\psi ^{\\prime }=\\psi =2\\alpha =\\left(s-s\\right)^{\\prime }/2R$ for a smooth current loop there remains an error $\\left(a^{2}/R^{2}\\right)\\ln \\left(R/b\\right)$ .", "The $-a_{2}/b^{2}$ term (from the $-1$ ) is peculiar.", "With the choice $b=\\sqrt{aR}$ it would be of order $a/R$ , but it gets cancelled against the leading term of the power series $\\hat{P}_{0}$ of equation (REF ).", "This of course is the reason for combining $\\hat{P}_{0}$ with the multipole expansion in equation (REF ): for a straight segment the formula is exact, and the third bracket in equation (REF ) vanishes.", "The multipole expansion together with $\\hat{P}_{0}$ thus also vanishes for $R=\\infty $ ." ], [ "Oktupole", "The oktupole contributes at most a term of order $a^{3}/b^{3}$ .", "But the $\\cos \\phi $ and $\\sin \\psi $ from the odd power of $\\mu $ make the actual contribution smaller than the expected $\\sim a^{2}/R^{2}$ ." ], [ "Hexadecupole", "The hexadecupole term is of order $a^{4}/b^{4}\\sim a^{2}/R^{2}$ and its leading part must be kept.", "The $\\mu $ factors contain a factor $\\sin \\psi \\sim b/R$ and may be dropped.", "The $\\nu ^{4}$ leads to $\\overline{L}_{4}\\left(s\\right)=\\frac{3}{8}\\oint ds^{\\prime }\\theta \\left(\\left|s^{\\prime }-s\\right|-b\\right)\\left[2a_{4}+a_{2}^{2}\\left(3+\\cos ^{2}\\alpha \\right)\\right]\\frac{\\cos \\alpha }{x_{s,s^{\\prime }}^{5}}.$ The factors $\\cos \\alpha $ may be replaced with 1 because of $\\alpha =\\left(s^{\\prime }-s\\right)/R$ is small.", "The integral converges and contributes an error like $a^{2}/R^{2}$ .", "But this term gets cancelled by the second term of the power series $\\hat{P}_{0}$ from equation (REF ).", "For smooth current loops it finally follows $\\frac{4\\pi }{\\mu _{0}}L\\left(s\\right)=\\left(\\overline{L}_{\\gamma }+\\widehat{L}_{\\gamma }\\right)+O\\left(\\left(a^{2}/R\\right)\\ln \\frac{R}{b}\\right)+\\left(\\widehat{L}-\\widehat{L}_{0}\\right),$ where the first bracket on the r.h.s is formula (REF ), evaluated with $b=\\sqrt{aR}$ .", "The remaining segment integral $\\widehat{L}$ for a segment with curvature radius $R$ is evaluated in appendix .", "The result (REF ) contributes another logarithmic error $\\left(a^{2}/R^{2}\\right)\\ln \\left(R/b\\right)$ , and a larger error of order $b^{2}/R^{2}\\sim a/R$ .", "We now drop all errors of order $O\\left(\\left(a^{2}/R\\right)\\ln \\left(R/b\\right)\\right)$ and $O\\left(a^{2}/R\\right)$ from the multipole expansion and the curved segment .", "As in the simple derivation of the formula above one may notice that there is no small lenght scale in the formula $\\bar{L}_{\\gamma }+\\widehat{L}_{\\gamma }$ and its $b$ -dependence is under control for all $b<\\sqrt{aR}$ and given by (REF ).", "It contains a “large” $b^{2}/R^{2}\\sim a/R$ term.", "The essential point now is that this $b$ -dependence of the formula exactly absorbs the remaining large error from $\\widehat{L}$ (the cancellation comes about, because $b$ is the lower limit in the curve integral and the upper limit in the volume integral).", "Nothing therefore changes if we now set $b=a/2$ , except that there only remains formula (REF ) evaluated for $b=a/2$ and small terms of order $O\\left(\\left(a^{2}/R\\right)\\ln \\left(R/a\\right)\\right)$ and $O\\left(a^{2}/R\\right)$ .", "For a circular loop the leading error terms can even be evaluated in closed form.", "The results for the self-inductances in the skin effect and volume current case are $L_{S} & = & \\mu _{0}R\\left\\lbrace \\left(1+\\frac{3}{4}\\frac{a^{2}}{R^{2}}\\right)\\ln \\left(\\frac{8R}{a}\\right)-2-\\frac{3}{2}\\frac{a^{2}}{R^{2}}\\right\\rbrace ,\\\\L_{V} & = & \\mu _{0}R\\left\\lbrace \\left(1+\\frac{3}{8}\\frac{a^{2}}{R^{2}}\\right)\\ln \\left(\\frac{8R}{a}\\right)-1.75-\\frac{2}{3}\\frac{a^{2}}{R^{2}}\\right\\rbrace .$ The corrections perfectly agree with a numeric evaluation of the 6- or 4-dimensional integrals, but strongly disagree with expressions in the literature.", "[4] The reason appears to be that these calculations assume a constant current distribution in the surface or cross section of the wire, wrong just in the curved parts of the loop." ], [ "Current loops with sharp corners", "The errors originate from the curved parts of the current loop, and for current loops with sharp corners one may expect larger errors.", "A simple way to construct such loops is to insert straight segments into a circular loop.", "It was shown above that the absolute error for a circular loop with radius $R$ is of order $O\\left(\\mu _{0}a{}^{2}/R\\right)$ .", "This estimation is valid even for minimal curvature radius $R=a$ (the inductance has dimension $length\\times \\mu _{0}$ and $a$ is the only available length for such loops).", "Since the loop with inserted straight segments is better approximated by the curve integral the absolute error can only be smaller.", "The formal reason is that a filamentary straight segment generates the same magnetic field as the actual axially symmetric current distribution.", "The inductance of a loop of extension $l\\gg a$ generally is of order $O\\left(\\mu _{0}l\\cdot \\ln \\left(l/a\\right)\\right)$ .", "The ratio with the absolute error leads to a generic estimation of the relative error of formula (REF ) $\\Delta L/L=\\sum _{n}O\\left(\\frac{a^{2}}{lR_{n}}\\right),$ where the index $n$ enumerates the corners and the logarithmic factor is neglected.", "We have allowed here corners with different curvature radius $R_{n}$ .", "The special case with curvature radii of order $O\\left(l\\right)$ leads back to the estimation of the error for smooth current loops above.", "Sharp corners with curvature radius $R=O\\left(a\\right)$ contribute a relative error of order $O\\left(a/l\\right)$ .", "Corners with a small angle of course should get a smaller weight in the sum.", "This estimation could be made more rigorous, but the procedure only is circumstantial and of no interest here.", "The error estimations above fail if the current loop comes close to itself before it actually closes, that is for tight spirals, coils or something like that (the estimation $x_{s,s^{\\prime }}\\sim \\left|s^{\\prime }-s\\right|$ gets invalid).", "In this case the replacement of the actual current with a filamentary current generates additional errors at these positions.", "But this isn't specific for equation (REF ), exactly the same (small) errors are contained in the Neumann formula for mutual inductance.", "The error may easily be estimated, because the condition $\\left|s^{\\prime }-s\\right|>b$ is irrelevant if $s^{\\prime }$ is on one winding and $s$ on another.", "A simple possibility is to consider as a worst case scenario two current loops of radius $R$ a distance $d>2a$ on top of each other.", "The error comes from the multipole expansion (REF ) valid everywhere now, and it is a simple matter to estimate the integrals.", "The order of magnitude of the relative error is $O\\left(a^{2}/Rd\\right)$ , small except for small curvature radius and small distance (both of order $O\\left(a\\right)$ )." ], [ "Conclusions", "The curve integral (REF ) for the self inductance of a wire loop is only a little bit more complicated than the Neumann formula for the mutual inductance of two wire loops.", "The exact expression for self inductance is a 6-dimensional integral with a logarithmic divergence and several length scales.", "Nevertheless clear statements follow for the accuracy of formula (REF ), for loops consisting of straight segments as well as for smooth loops.", "The error originates from the curved parts of the loop, and is of order $\\mu _{0}a$ or $\\mu _{0}a^{2}/l$ , negligible for most practical purposes.", "The leading error presumably even might be given as a sum over the corners for loops consisting of straight segments or as additional curve integrals for smooth current loops (see the quadrupole contribution above).", "The techniques used for error estimation also may be used for the Neumann formula.", "The self inductance curve integral can be evaluated analytically in many cases, for instance for current loops consisting of coplanar straight segments (not a new result; see also appendix and ).", "But equation (REF ) is valid for abitrary curves, and the numerical evaluation of two-dimensionals integrals with a computer program is a breeze with appropriate numerical libraries.", "The information for the self inductance is contained in the curve spanned by the current loop, and any self inductance calculation at least requires a double integral along the curve.", "Simpler methods or estimations based on “partial inductance” or only the magnetic flux miss this point and only may work in special cases.", "The fact that two coinciding points cause problems in self inductance calculations is well known and has been circumvented in several ways, for instance by distributing the current onto two filamentary loops.", "But no systematic approximations can be obtained in this way.", "Current distributions which are not circular symmetric also lead to formula (REF ), with a cutoff and a constant $Y$ depending on the current distribution.", "An example are circuits consisting of coplaner flat strips of width $w$ .", "The self inductance of such circuits is given in the accuracy described above by formula (REF ) with $a=w$ and $Y=3$ ." ], [ "Contribution from straight segments of length $2b$", "The contribution $\\widehat{L}$ to the self inductance in equation (REF ) is due to the interaction of the current in the plane $s$ with the current in all planes $s^{\\prime }$ with $\\left|s^{\\prime }-s\\right|<b$ .", "This value depends on the current distribution in the wire and on the wire geometry within the segment $\\left[s-b\\text{,}s+b\\right]$ , but may be evaluated if the segment is straight or slightly curved.", "This $s$ -dependent value of course still is to be integrated over all $s$ .", "To get an approximation for $\\widehat{L}$ in the straight wire case use cylinder coordinates with a length $s$ along the axis and area element $dA=rdrd\\phi $ (see figure (REF )).", "This leads to $\\widehat{L}_{0} & = & \\oint ds\\widehat{L}\\left(s\\right),\\\\\\widehat{L}_{0}\\left(s\\right) & = & \\left(\\int \\frac{\\mathbf {j}\\left(r\\right)\\mathbf {j}\\left(r^{\\prime }\\right)}{|\\mathbf {x}\\left(s,r,\\phi \\right)\\mathbf {-x}^{\\prime }|}ds^{\\prime }dA^{\\prime }dA\\right)_{\\left|s\\left(\\mathbf {x}^{\\prime }\\right)-s\\right|<b}.$ In the latter integral $\\mathbf {x}$ extends over the plane through the centre of a cylindrical segment, $\\mathbf {x}^{\\prime }$ extends over the complete segment.", "The integral $\\widehat{L}_{0}\\left(s\\right)$ of course is independent of $s$ .", "The integral over $s^{\\prime }$ (from $-b$ to $b$ ) may be performed using $|\\mathbf {x-x}^{\\prime }|^{2}=N^{2}+s^{\\prime 2}$ , $N^{2}=r^{2}+r^{\\prime 2}-2rr^{\\prime }\\cos \\left(\\phi -\\phi ^{\\prime }\\right)$ , $\\widehat{L}_{0}\\left(0\\right) & \\cong & 2\\int dAdA^{\\prime }\\mathbf {j}\\left(r\\right)\\mathbf {j}\\left(r^{\\prime }\\right)\\operatorname{asinh}\\left(b/N\\right)\\\\& = & 2\\int dAdA^{\\prime }\\mathbf {j}\\left(r\\right)\\mathbf {j}\\left(r^{\\prime }\\right)\\left\\lbrace \\ln \\left(2b/a\\right)-\\ln \\left(N/a\\right)+A_{1}\\left(N/b\\right)+...\\right\\rbrace $ In the second line the expansion $\\operatorname{asinh}\\left(x\\right) & = & \\ln \\left(2x\\right)+A_{1}\\left(1/x\\right)\\\\A_{1}\\left(x\\right) & = & \\sum \\nolimits _{n=1}^{\\infty }\\frac{1\\cdot 3...\\left(2n-1\\right)}{2\\cdot 4...2n}\\frac{\\left(-1\\right)^{n+1}}{2n}x^{2n}=x^{2}/4-3x^{4}/32+...$ was used.", "The expansion converges because of $N=O\\left(a\\right)\\ll b$ .", "It doesn't matter which $\\phi ^{\\prime }$ occurs in the $\\phi $ -integral and thus we now set $\\phi ^{\\prime }=0$ .", "Because of $\\int dA\\left|\\mathbf {j}\\right|=1$ the leading term simply becomes $2\\ln \\left(2b/a\\right)$ .", "The second term follows from $\\ln \\left(N/a\\right)=\\frac{1}{2}\\ln \\left(\\rho ^{2}+\\rho ^{\\prime 2}-2\\rho \\rho ^{\\prime }\\cos \\phi \\right)$ with $\\rho =r/a$ and $\\rho ^{\\prime }=r^{\\prime }/a$ and $\\frac{1}{2\\pi }\\int _{0}^{2\\pi }\\ln \\left(\\rho ^{2}+\\rho ^{\\prime 2}-2\\rho \\rho ^{\\prime }\\cos \\phi \\right)d\\phi =2\\ln \\left(\\rho _{>}\\right),$ where $\\ln \\left(\\rho _{>}\\right)=\\theta \\left(\\rho -\\rho ^{\\prime }\\right)\\ln \\rho +\\theta \\left(\\rho ^{\\prime }-\\rho \\right)\\ln \\rho ^{\\prime }$ .", "This term thus vanishes in the skin effect case where the current differs from 0 only for $\\rho =\\rho ^{\\prime }=1$ .", "The current density in the constant current case is $j\\left(r\\right)=1/\\left(\\pi a^{2}\\right)$ and the second term becomes $-\\frac{\\left(2\\pi \\right)^{2}}{\\pi ^{2}}2\\int _{0}^{1}d\\rho \\rho \\int _{0}^{1}d\\rho ^{\\prime }\\rho ^{\\prime }\\ln \\left(\\rho _{>}\\right)=1/2.$ A rapidly convergent expansion approximation for $b\\gg a$ thus is $\\widehat{L}_{0}\\left(0\\right) & = & \\widehat{L}_{\\gamma }\\left(0\\right)+\\widehat{P}_{0}\\left(0\\right),\\\\\\widehat{L}_{\\gamma }\\left(0\\right) & = & 2\\ln \\left(2b/a\\right)+Y,\\\\\\widehat{P}_{0}\\left(0\\right) & =\\left\\langle 2A_{1}\\left(\\frac{N}{b}\\right)\\right\\rangle = & \\frac{a_{2}}{b^{2}}-\\frac{3}{8b^{4}}\\left(a_{4}+2a_{2}^{2}\\right)+O\\left(\\frac{a^{6}}{b^{6}}\\right)$ with $Y=1/2$ for a constant current distribution and $Y=0$ in the skin effect case." ], [ "Contribution from a curved segment", "The goal is to evaluate the integral $\\widehat{L}\\left(0\\right)$ from equation (REF ) for a segment of length $2b$ and constant curvature radius $R$ , $\\widehat{L}_{R}\\left(0\\right)=\\int \\frac{ds^{\\prime }d\\widetilde{A}d\\widetilde{A}^{\\prime }}{|\\mathbf {x}\\left(0,r,\\phi \\right)\\mathbf {-x}^{\\prime }|}\\theta \\left(b-|s^{\\prime }|\\right)j\\left(r\\right)j\\left(r^{\\prime }\\right)\\cos \\left(\\frac{s^{\\prime }}{R}\\right).$ The distance up to order $O\\left(R^{-2}\\right)$ follows from $\\left(\\mathbf {x-x}^{\\prime }\\right)^{2} & \\cong & s^{\\prime 2}+N^{2}+q,\\\\N^{2} & = & r^{2}+r^{\\prime 2}-2rr^{\\prime }\\cos \\left(\\phi -\\phi ^{\\prime }\\right),\\\\q & = & s^{\\prime 2}\\left[\\left(r^{\\prime }\\cos \\phi ^{\\prime }+r\\cos \\phi \\right)R^{-1}+rr^{\\prime }R^{-2}\\cos \\phi \\cos \\phi ^{\\prime }\\right]-s^{\\prime 4}/\\left(12R^{2}\\right)+s^{\\prime 6}/\\left(360R^{4}\\right).$ Expanding in $q$ gives $\\frac{1}{|\\mathbf {x-x}^{\\prime }|}=\\frac{1}{\\left(s^{\\prime 2}+N^{2}\\right)^{1/2}}-\\frac{q}{2\\left(s^{\\prime 2}+N^{2}\\right)^{3/2}}+\\frac{3q^{2}}{8\\left(s^{\\prime 2}+N^{2}\\right)^{5/2}}+...$ The $N$ in the denominators is negligible for $b\\gg a$ and the elementary integrals lead to $\\widehat{L}_{R}\\left(0\\right)-\\widehat{L}_{0}\\left(0\\right)=-\\frac{11}{24}\\frac{b^{2}}{R^{2}}+O\\left(\\frac{a_{2}}{R^{2}}\\ln \\frac{b}{a}\\right).$ The term of order $O\\left(a_{2}/R^{2}\\right)$ contributes to the error of the self inductance formula with the expected order of magnitude.", "It is essential however, that the $O\\left(b^{2}/R^{2}\\right)$ term, which is of order $a/R$ because of $b=\\sqrt{aR}$ , cancels against a contribution from the curve integral $\\overline{L}_{\\gamma }\\left(0\\right)$ from equation (REF ).", "To verify this cancellation start with $\\frac{d}{db}\\overline{L}_{\\gamma }\\left(0\\right)=\\frac{-\\cos \\left(\\mathbf {j}\\left(0\\right)\\mathbf {,j}\\left(b\\right)\\right)}{|\\mathbf {x}\\left(0\\right)\\mathbf {-x}\\left(b\\right)|}-\\frac{-\\cos \\left(\\mathbf {j}\\left(0\\right)\\mathbf {,j}\\left(-b\\right)\\right)}{|\\mathbf {x}\\left(0\\right)\\mathbf {-x}\\left(-b\\right)|}.$ Inserting $\\left|x-x^{\\prime }\\right|=2R\\sin b/2R$ and $\\cos \\left(\\mathbf {j}\\left(0\\right)\\mathbf {,j}\\left(b\\right)\\right)=\\cos b/r$ for a section with curvature radius $R$ gives $\\frac{d}{db}\\overline{L}_{\\gamma }\\left(0\\right)=\\frac{-2}{b}\\left(1-\\frac{11}{24}\\left(\\frac{b}{R}\\right)^{2}+...\\right),$ with integral $\\overline{L}_{\\gamma }\\left(0\\right)=const-2\\ln \\frac{2b}{a}+\\frac{11}{24}\\frac{b^{2}}{R^{2}}+...$" ], [ "Curve integral for adjacent straight segments", "For completeness we display here the curve integral contribution to the self inductance from adjacent straight segments of length $c$ and $d$ with an angle $\\alpha $ between the currents, $L_{\\gamma }\\left(c,d,\\alpha \\right) & = & \\frac{\\mu _{0}}{2\\pi }\\cos \\left(\\alpha \\right)\\lbrace c\\operatorname{asinh}\\frac{d+c\\cos \\alpha }{c\\sin \\alpha }+d\\operatorname{asinh}\\frac{c+d\\cos \\alpha }{d\\sin \\alpha }\\\\& & -\\left(c+d\\right)\\operatorname{asinh}\\frac{\\cos \\alpha }{\\sin \\alpha }-\\frac{2b}{\\sqrt{1\\left(1-\\cos \\alpha \\right)}}\\operatorname{asinh}\\frac{1-\\cos \\alpha }{\\sin \\alpha }\\rbrace .$ For each corner such a term is to be added to the contribution (REF ) of the (straight) segments by themselves.", "The $b$ -term is of order $O\\left(a\\right)$ for $b=a/2$ and normally may be neglected." ], [ "Curve integral for non-adjacent coplanar straight segments", "What then is missing for calculating the self inductance of a loop consisting of arbitrary coplanar straight segments is the mutual contribution from non-adjacent straight segments, see figure (REF ).", "$L\\left(m,m^{\\prime },n,n^{\\prime },\\alpha \\right) & = & \\frac{\\mu _{0}}{2\\pi }\\left\\lbrace A\\left(m^{\\prime },n^{\\prime },n,\\alpha \\right)+A\\left(n^{\\prime },m^{\\prime },m,\\alpha \\right)+A\\left(m,n,n^{\\prime },\\alpha \\right)+A\\left(n,m,m^{\\prime },\\alpha \\right)\\right\\rbrace ,\\\\A\\left(w,u,v,\\alpha \\right) & = & \\left[w\\operatorname{asinh}\\left(\\frac{u+w\\cos \\alpha }{w\\sin \\alpha }\\right)-w\\operatorname{asinh}\\left(\\frac{v+w\\cos \\alpha }{w\\sin \\alpha }\\right)\\right]\\cos \\alpha .$ This leads to an unwieldy expression already for a hexagon, but the calculation of the self inductance of such loops is a matter of algebra and geometry (not a new result)." ] ]

[ [ "A pseudo-matched filter for chaos" ], [ "Abstract A matched filter maximizes the signal-to-noise ratio of a signal.", "In the recent work of Corron et al.", "[Chaos 20, 023123 (2010)], a matched filter is derived for the chaotic waveforms produced by a piecewise-linear system.", "Motivated by these results, we describe a pseudo-matched filter, which removes noise from the same chaotic signal.", "It consists of a notch filter followed by a first-order, low-pass filter.", "We compare quantitatively the matched filter's performance to that of our pseudo-matched filter using correlation functions in a simulated radar application.", "On average, the pseudo-matched filter performs with a correlation signal-to-noise ratio that is 2.0 dB below that of the matched filter.", "Our pseudo-matched filter, though somewhat inferior in comparison to the matched filter, is easily realizable at high speed (> 1 GHz) for potential radar applications." ], [ "Introduction", "A conventional radar system measures the distances of targets in the field of view using a signal source, a transmitter, and a receiver.", "In Fig.", "REF , a radar transmitter broadcasts a signal $u(t)$ from the source toward an intended target, and the receiver detects a version of the transmitted signal that is reflected off of the target.", "Prior to transmission, a copy of the radar's signal is digitally sampled and stored as $s_n$ .", "The received signal $v(t)$ , which picks up environmental noise, is filtered and correlated with $s_n$ .", "Typical radar signals are non-repeating in order to avoid multiple points of correlation and, therefore, the correlation will peak only when the transmitted and received signals are aligned.", "Using the time of the output peak in the correlation, the measured range from the transmitter to the target is determined.", "Figure: Example chaos radar.", "A signal u(t)u(t) is transmitted to measure target distances and stored digitally as s n s_n.", "The received signal v(t)v(t) is filtered and correlated with s n s_n.", "The correlation output χ(t)\\chi (t) peaks at time t=2τ pd +τ f t = 2\\tau _{\\text{pd}} + \\tau _{\\text{f}}, where τ pd \\tau _{\\text{pd}} is the propagation delay from the radar to the target and τ f \\tau _{\\text{f}} is the delay through the filter.The performance of a radar is determined by its ability to identify the correlation time between the transmitted and received signal.", "In the correlation function, the width of the peak scales inversely with the bandwidth of the transmitted signal and sets the spatial resolution of the radar.", "In addition, the height of the correlation peak above the noise floor, also known as signal-to-noise ratio ($SNR$ ), is determined by the length of the transmitted and sampled waveforms as well as the noise from the environment.", "Thus, the digital storage capacity of the radar sets the maximum $SNR$ in the correlation measurement.", "State-of-the-art radar systems use high-frequency, broadband signals, where the digital sampling and storage of the signals can be costly.", "These radars must balance the bandwidth and cost of the system’s design while maintaining its performance.", "A simple example of an inexpensive, broadband signal source is amplified electrical noise.", "In the past, electrical noise has been used by radar systems to perform ranging measurements.", "[1], [2] The high bandwidth of these noise generators yields high-resolution ranging information, but requires fast sampling and large data storage capacities.", "Some recent techniques have been proposed to minimize the necessary sampling capacity of noise radars using analog delay lines for signal storage.", "[3] But, these methods limit the ranging capabilities of the radar.", "Further research has shown that more efficient radars benefit from a structured, rather than a stochastic, signal.", "For instance, various deterministic signal sources have been studied in efforts to minimize the necessary data storage rate and capacity of a radar.", "As one example, pseudo-random binary sequences (PRBS's) are often used as a radar signal sources.", "To be implemented as a radar signal source, a PRBS is up-converted to a suitable frequency band before transmitting and then down-converted at the receiver before correlation.", "[4], [5] The main advantage of a PRBS is the ability to use one-bit digital samplings of the binary sequences, thereby requiring low amounts of data-storage capacity.", "This allows for longer sequences of the transmitted waveform to be stored, thus enhancing the radar's $SNR$ without increasing the cost of the system.", "The main disadvantage of a PRBS is that it requires computational power to generate and its sequence eventually repeats, which ultimately limits its performance.", "Many other radar concepts like this one exist, each with advantages and disadvantages, and today the radar community continues to develop broadband signal sources.", "[6], [7] One novel approach to a radar is to use chaotic waveform generators, which are believed to have several properties that make them ideally suited as signal sources for radar applications.", "[8] One defining feature of a chaotic system is that it can generate a signal that does not repeat in time.", "Chaotic signals are also often inherently broadband.", "High-speed chaos has been observed in optical and electronic systems with frequency bandwidths that stretch across several gigahertz.", "[9], [10], [11] Such broadband chaos has been studied for its applications in high-resolution ranging and in imaging.", "[12], [13], [14], [15] These applications use the non-repeating aspects of the high-speed chaos.", "However, conventional chaos radars do not take full advantage of the chaotic signal source.", "In addition to producing broadband, non-repeating signals, chaotic systems are deterministic and extremely sensitive to small perturbations.", "By not using these properties, chaos radars add no benefit over noise radars, requiring high-sampling to perform correlations.", "Thus, to take full advantage of chaotic systems in a radar application, a chaos radar needs to benefit from the determinism or sensitivity of the chaos in addition to its noise-like properties.", "Recently, Corron et al.", "proposed a novel chaos radar concept that uses the dynamics produced by a piecewise-linear harmonic oscillator.", "[16], [17], [18] It produces simultaneously a chaotic signal and a binary switching state that completely characterizes its chaotic dynamics.", "In the proposed radar, the chaotic signal serves as the signal source (see Fig.", "REF ) and is transmitted, while a copy of a switching state is stored using a one-bit digital sampling.", "For a radar receiver, Corron et al.", "derived the analytical form of a filter that is matched to the chaos produced by the system.", "A matched filter is a linear operation that optimizes the $SNR$ of a signal in the presence of additive white Gaussian noise ($AWGN$ ).", "Their matched filter recovers the switching states from the received signal, which are then passed to the correlation operation of Fig.", "REF .", "This technique uses deterministic aspects of the chaos, making it an improvement over a noise radar.", "With reduced data storage and an optimal $SNR$ , their architecture could effectively reduce the cost of a radar system.", "Corron et al.", "implement their piecewise-linear design using an LRC (inductance-resistance-capacitance) oscillator that operates in the kHz frequency range.", "[17] It is difficult to a realize high-frequency version ($>$ 1 GHz) of this system because of parasitic capacitances, inductances, and the inherent time delays in the propagation of signals in LRC circuits.", "As it stands, there is no high-frequency realization of the piecewise-linear system from Ref.", "[Corron2010] or the associated matched filter.", "In order to have resolutions that are comparable to state-of-the art radar systems, the waveforms and switching states produced by this chaotic system must be scaled to higher-frequencies and to broader bandwidths.", "Thus, techniques for increasing the speed of this system are of interest.", "As a first step toward a high-frequency implementation of Corron et al.", "'s system, we compare the performance of the matched filter presented in Ref.", "[Corron2010] to a set of standard filters (first order low-pass filter and notch filter).", "These standard filters operate at high-frequencies, are inexpensive, well characterized, and readily available.", "Cascading these standard filters allows us to realize a pseudo-matched filter for the chaos produced by the piecewise-linear harmonic oscillator.", "We define a pseudo-matched filter as a sub-optimal linear operation (when compared to the matched filter) that removes $AWGN$ from the corresponding (matched) waveform and performs comparably to the matched filter for the system.", "We are interested in a chaos-based radar system such as that described by Ref.", "[Blakely2010] that could benefit from integrating readily available filters for high-speed applications." ], [ "Matched Filter Review", "To motivate our analysis, we briefly review the characteristics of the chaotic system and matched filter presented in Ref.", "[Corron2010] within the context of a radar application.", "Consider a harmonic oscillator with negative damping $-\\beta $ and with a piece-wise constant driving term $s(t)$ whose behavior is governed by the differential equation $\\ddot{u}(t) - 2 \\beta \\dot{u}(t)+ (\\omega _{\\text{o}}^2+\\beta ^2)(u(t)-s(t)) = 0,$ together with a guard condition on the output variable $u(t)$ that switches the sign of $s(t)$ $s(t) = \\left\\lbrace \\begin{array}{lr}\\;\\;\\,1, & \\text{if } u(t^*) \\ge 0 \\\\-1, & \\text{if } u(t^*) < 0\\end{array}\\right..$ Figure REF shows a time series of the variables $u(t)$ and $s(t)$ with the parameters $\\omega _{\\text{o}} = 2\\pi $ and $\\beta = ln(2)$ .", "We integrate Eqs.", "(REF )-(REF ) using MATLAB's ODE45, where the switching condition is monitored using the integrator's event-detection algorithm.", "The attractor for this system is low dimensional and is plotted in Fig.", "REF b.", "The dynamics can also be viewed as a chaotic shift map,[17] as seen in Fig.", "REF c. For this system, $u(t)$ oscillates with a growing amplitude and fixed oscillation frequency $f_{\\text{o}} = \\omega _{\\text{o}} /2\\pi $ about a piece-wise constant line $s(t)$ .", "The switching times of $s(t)$ depend on the local maxima and minima of $u(t)$ , and the times between the local maxima and minima of $u(t)$ are fixed by the fundamental frequency $f_{\\text{o}}$ .", "Thus, the maximum rate of switching in $s(t)$ is limited to $1/f_{\\text{o}}$ .", "Using a one-bit digital sampling of $s(t)$ at a sampling frequency that is greater than or equal to $f_{\\text{o}}$ , we are able to store a record of all switching state values $s_n$ .", "Similar to the case for a PSRB, information about the transmitted waveform can be stored with minimal sampling and memory, enhancing the potential $SNR$ of a radar correlation measurement.", "Figure: Chaos from a piecewise-linear harmonic oscillator with negative damping.", "(a) Time series of the analog variable u(t)u(t) (green) and the nonlinear switching state s(t)s(t) (blue dashed line).", "(b) Chaotic attractor in phase space.", "(c) Chaotic shift map created by sampling u(t)u(t) using the times t * t^* from Eq.", "(2), where v n =u(t * )v_n = u(t^*) if |u(t * )-1|<0|u(t^*) - 1| < 0.In addition to its data storage capabilities, chaos from this system can be further exploited using a matched filter.", "Corron et al.", "demonstrate that the switching information $s(t)$ can be recovered using a matched filter at the radar receiver.", "[18] The match filter is described by $\\dot{y}(t) = v(t-2\\pi /\\omega _{\\text{o}})-v(t),$ $\\ddot{\\xi _m}(t)+2 \\beta \\dot{\\xi _m}(t)+ (\\omega _{\\text{o}}^2+\\beta ^2)\\xi _m(t) = (\\omega _{\\text{o}}^2+\\beta ^2)y(t),$ where $v(t)$ is the input signal and $\\xi _{\\text{m}}(t)$ is the analog output of the matched filter.", "In Fig.", "REF , we examine the output of the matched filter when it is driven by $v(t) = u(t)$ + $AWGN$ .", "The original signals $u(t)$ and $s(t)$ are plotted in Figs.", "REF a-b.", "The switching state $s(t)$ is plotted with a one-bit digital sampling $s_n$ at a sampling frequency $f_{\\text{o}}$ .", "In Fig.", "REF c, the time series $v(t)$ has a $SNR$ of $-5.9$ dB, where $SNR = 10\\text{log}_{10}(SNR_{\\text{input}})$ , $SNR_{\\text{input}} = \\sigma _{u}^2/\\sigma _{AWGN}^2$ , and where $\\sigma _{u}^2$ and $\\sigma _{AWGN}^2$ are the input signal $u(t)$ and additive noise powers, respectively (see the Appendix for details on the additive noise).", "The matched filter is optimized to remove the noise from $v(t)$ and recover the binary switching information $s_n$ for the correlation operation.", "We compare $s_n$ to the solution for Eqs.", "(REF ) - (REF ), $\\xi _{\\text{m}}(t)$ , shown in Fig.", "REF d. The solution $\\xi _{\\text{m}}(t)$ is the output of the matched filter and is a nearly noise-free waveform that transitions approximately between two states, one positive and negative (defined by a dotted black line at $\\xi _{\\text{m}}(t) = 0$ ).", "In between these transitions, $\\xi _{\\text{m}}(t)$ oscillates with a small, irregular amplitude at a frequency approximately equal to $f_{\\text{o}}$ .", "Upon further examination, the matched filter output signal $\\xi _{\\text{m}}(t)$ follows the same transitions from the original switching state $s(t)$ .", "Using a correlation between $s(t)$ and $\\xi _{\\text{m}}(t)$ , we determine the time delay through the matched filter $\\tau _{\\text{m}}$ to be approximately $1/(2f_{\\text{o}})$ .", "We compensate for the delay, and sample $\\xi _{\\text{m}}(t)$ at $f_{\\text{o}}$ , assigning binary values using the relation: $-1$ if $\\xi _{\\text{m}}(t_n) \\le 0$ and $+1$ if $\\xi _{\\text{m}}(t_n) > 0$ , where $t_n$ is the $n^{\\text{th}}$ sampling time.", "These binary values, shown in Fig.", "REF d, match $s_n$ , demonstrating that the matched filter has recovered the switching information from $v(t)$ .", "Figure: Temporal evolution of (a) u(t)u(t), (b) s(t)s(t), (c) v(t)v(t) with SNR=-5.9SNR = -5.9 dB, and (d) ξ m (t)\\xi _{\\text{m}}(t).", "The horizontal axes in (b) and (c) are shifted by τ m \\tau _{\\text{m}} and (2τ pd +τ m )(2\\tau _{\\text{pd}} + \\tau _{\\text{m}}), respectively, where τ pd \\tau _{\\text{pd}} is the propagation distance to an intended target and τ m \\tau _{\\text{m}} is the time delay through the matched filter.", "Above the signals ξ m (t)\\xi _{\\text{m}}(t) and s(t)s(t) (blue), a single-bit discrete sampling of the waveforms (red dots) is shown.In a simulated radar application, Corron et al.", "use a tapped delay line to perform a time-domain correlation operation between the digitally stored $s_n$ and the analog output of the matched filter $\\xi _{\\text{m}}(t)$ .", "The tapped delay line is described by $\\chi _{\\text{m}}(t) = \\sum \\limits _{n=1}^N s_n \\xi _{\\text{m}}(t-\\tau _n),$ where $N$ is the length of the stored sequence and $\\chi _{\\text{m}}(t)$ is the output of the correlator (we have used the linearity of Eqs.", "(REF ) - (REF ) to rearrange the operations in Ref.", "[Blakely2010]).", "The tapped delay line is a convenient technique for calculating in real time the correlation between a binary and an analog signal.", "To better understand the operations performed by a tapped delay line, we use a pictorial representation of Eq.", "(REF ).", "In Fig.", "REF , the output of the matched filter $\\xi _{\\text{m}}(t)$ splits into $N$ copies, each of which are successively delayed by $\\tau _n f_{\\text{o}} = n$ , where $n$ is an integer.", "The resulting copies are multiplied by the corresponding stored $s_n$ and summed continuously in time.", "The output $\\chi _{\\text{m}}(t)$ peaks $2\\tau _{\\text{pd}} + \\tau _{\\text{m}}$ , where $\\tau _{\\text{pd}}$ and $\\tau _{\\text{m}}$ are the propagation delays of the signal to the target and through the matched filter, respectively.", "In the context of Fig.", "REF , the correlation operation of this chaos radar can be performed between a transmitted and received signal using the tapped delay line in Eq.", "(REF ).", "Thus, using the chaotic waveform generator from Ref.", "[Corron2010], combined with the matched filter and tapped delay line, we arrive at a chaos radar.", "[18] Figure: Schematic representation of the tapped delay line in Eq.", "().", "The switching state s(t)s(t) is sampled and s n s_n is stored for n=1n = 1 to n=Nn = N, where, N=100N=100 in this case.", "The output of the matched filter ξ m (t)\\xi _{\\text{m}}(t) drives the tapped delay line and the output sum χ m (t)\\chi _{\\text{m}}(t) peaks at time t=2τ pd +τ m t = 2\\tau _{\\text{pd}}+\\tau _{\\text{m}}.This particular chaos radar benefits in two ways from the deterministic characteristics of the system's chaos.", "The first benefit is the link between the non-repeating waveform $u(t)$ and the switching state $s(t)$ : A binary sampling of $s(t)$ can completely characterize the dynamics in $u(t)$ .", "A second benefit is the ability to derive a matched filter that optimizes the output $SNR$ of the receiver.", "The matched filter for chaos, combined with a tapped delay line, provides an architecture for relatively quick and inexpensive correlations between a binary switching signal and a recovered analog signal, both generated from a single chaotic system.", "These benefits present a platform for a high-performance radar with improved $SNR$ over conventional chaos radars." ], [ "Matched Filter Analysis", "With this background and motivation, we now examine Corron et al.", "'s matched filter for chaos in the frequency domain.", "Using the transfer functions of Eqs.", "(REF ), (REF ), and (REF ), we examine the spectral properties of the matched filter.", "For the purposes of our analysis, we focus on the magnitudes of transfer functions.", "Because the phase of the matched filter is approximately linear with frequency $f$ when $f < f_{\\text{o}}$ , it preserves the transition information in $s_n$ .", "We will derive empirically a pseudo-matched filter using a combination of standard filters and that also preserves the transition information in $s_n$ .", "Lastly, we compare the pseudo-matched filter performance to the true matched filter for chaos in a simulated radar application.", "First, we analyze the spectral properties of the chaotic dynamics from $u(t)$ and $s(t)$ as well as the driving signal $v(t)$ .", "In Fig.", "REF , we plot the spectral amplitudes for $u(t)$ , $s(t)$ , and $v(t)$ , where $v(t) = u(t)$ + $AWGN$ ($SNR = -5.9$ dB).", "One should take note that, in Fig.", "REF a, there is no maxima in the frequency spectrum at $f_{\\text{o}}$ , the fundamental frequency of oscillation, because the phase of $u(t)$ switches by $\\pi $ each time $s(t)$ switches states.", "This demonstrates that, if the spectrum of $u(t)$ is scaled-up in high-frequency system ($> 1$ GHz), the bandwidth in $u(t)$ would stretch over several gigahertz.", "In addition to its radar properties, a high-frequency broadband spectrum in $u(t)$ would provide a useful carrier signal for low-profile or ultra-wideband technologies.", "[19], [20] In Fig.", "REF c, the frequency spectrum of the $AWGN$ in $v(t)$ covers information about $u(t)$ .", "The matched filter is engineered to process the spectrum of $v(t)$ to recover $s_n$ .", "Figure: Frequency spectra for the chaotic time series of (a) u(t)u(t), (b) s(t)s(t), and (c) v(t)v(t).", "The frequency spectra are calculated using u ˜\\tilde{u}, s ˜\\tilde{s}, and v ˜\\tilde{v}, the discrete Fourier transforms of u(t)u(t), s(t)s(t), and v(t)v(t), respectively.", "In these plots, the frequency axes have a spectral resolution of 10 -4 10^{-4}, and the spectral amplitudes have been averaged over a window of 10 -2 10^{-2}.Next, we examine the transfer function of the matched filter.", "We take the Fourier transform of Eqs.", "(REF ) - (REF ) and obtain the transfer functions $H_{\\text{input}}$ and $H_{\\text{o}}$ .", "We combine these transfer functions to obtain the transfer function for the matched filter $H_{\\text{m}}$ .", "They read $H_{\\text{input}}(\\nu ) = \\frac{\\tilde{y}}{\\tilde{v}} = \\frac{e^{2\\pi i \\nu }-1}{2 \\pi i \\nu },$ $H_{\\text{o}}(\\nu )= \\frac{\\tilde{\\xi }_{\\text{m}}}{\\tilde{y}} = \\frac{4 \\pi ^2 + \\beta ^2}{4\\pi ^2(1-\\nu ^2)+\\beta (4\\pi i \\nu + \\beta )},$ $H_{\\text{m}}(\\nu ) = \\frac{\\tilde{\\xi }_{\\text{m}}}{\\tilde{v}} = H_{\\text{input}} H_{\\text{o}},$ where $\\nu = f/f_{\\text{o}}$ and $\\tilde{\\xi }_{\\text{m}}$ , $\\tilde{y}$ , and $\\tilde{v}$ are the Fourier transforms of $\\xi _{\\text{m}}(t)$ , $y(t)$ , and $v(t)$ , respectively.", "We plot the magnitudes of $H_{\\text{input}}$ , $H_{\\text{o}}$ , and $H_{\\text{m}}$ as a function of frequency $\\nu $ in Fig.", "REF .", "In Fig.", "REF c, we also plot the phase of $H_{\\text{m}}$ , where the phase is approximately linear with frequency for $\\nu <1$ and thus preserves timing information in $s_n$ .", "Figure: Magnitudes the transfer functions (a) H input H_{\\text{input}}, (b) H o H_{\\text{o}}, and (c) H m H_{\\text{m}}.", "In (d), the phase of H m H_{\\text{m}} is given as a function frequency.We analyze $H_{\\text{input}}$ and $H_{\\text{o}}$ individually to better understand the matched filter's transfer function.", "We factorize $H_{\\text{input}}$ into two linear operations ($H_{\\text{input}} = H_{\\text{notch}} H_{\\text{integrator}}$ ), a notch filter and an integrator $H_{\\text{notch}}(\\nu )= \\frac{\\tilde{y}}{\\tilde{q}}=e^{2\\pi i \\nu }-1,$ $H_{\\text{integrator}}(\\nu )= \\frac{\\tilde{q}}{\\tilde{v}} = \\frac{1}{2 \\pi i \\nu },$ where $\\tilde{q}$ is the Fourier transform of an intermediate input-output variable and $H_{\\text{notch}}$ and $H_{\\text{integrator}}$ are the transfer functions of a notch filter and integrator, respectively.", "We plot the magnitudes of $H_{\\text{notch}}$ and $H_{\\text{integrator}}$ in Fig.", "REF .", "In Fig.", "REF a, the magnitude of the notch filter's transfer function goes to minus infinity at integer multiples of the fundamental frequency $f_{\\text{o}}$ .", "Similar types of filters have been used previously in chaotic systems to stabilize periodic orbits using continuous-time control methods.", "[21], [22] The transfer function in Eq.", "(REF ) is a standard operation for integration.", "In Fig.", "REF b, the function $|H_{\\text{integrator}}|$ diverges to infinity at $\\nu = 0$ and falls off at a rate of $1/\\nu $ with increasing $\\nu $ .", "When cascaded, $H_{\\text{notch}}$ and $H_{\\text{integrator}}$ complement one another to form a filter that preserves low frequencies, eliminates integer multiples of $f_{\\text{o}}$ , and cuts out high-frequencies (see Fig.", "REF a).", "Figure: Magnitudes of the transfer functions (a) H notch H_{\\text{notch}} and (b) H integrator H_{\\text{integrator}}.Upon inspection of $H_{\\text{o}}$ in Eq.", "(REF ), we see that it takes on a functional form that is equivalent to the Fourier transform of the dynamical system in Eq.", "(REF ), given by $\\frac{\\tilde{u}}{\\tilde{s}} = \\frac{4 \\pi ^2 + \\beta ^2}{4\\pi ^2(1-\\nu ^2)+\\beta (4\\pi i \\nu + \\beta )},$ where $\\tilde{u}$ and $\\tilde{s}$ are the Fourier transforms of $u(t)$ and $s(t)$ , respectively.", "Thus, Eq.", "(REF ) contains specific spectral information about the system's dynamics.", "When the transfer function $H_{\\text{o}}$ is applied with $H_{\\text{input}}$ , it reshapes the spectrum of the transfer function for the matched filter $H_{\\text{m}}$ near $\\nu = 1$ , as seen best in Fig.", "REF c. Thus, the operations of the matched filter for chaos can be separated into four criteria: (i) a notch filter that eliminates the fundamental oscillation frequency $f_{\\text{o}}$ , (ii) an integrator that preserves the low frequencies and (iii) cuts off high frequencies falling off as $1/\\nu $ , and (iv) a dynamical filter that reshapes the transfer function near the fundamental oscillation frequency $f_{\\text{o}}$ .", "These four criteria are the foundation for our derivation of the pseudo-matched filter for chaos." ], [ "Pseudo-matched Filter", "Our strategy for designing a pseudo-matched filter for chaos is to simplify the four criteria of the matched filter using transfer functions from components that are readily available at high-speed.", "In the upcoming section, we satisfy criteria (i) and (ii) using a single transfer function.", "We also show that criterion (iii) is more easily accomplished without an integrator.", "Lastly, we demonstrate that criterion (iv) is not necessary for our applications.", "To begin constructing our pseudo-matched filter, we select a different notch filter that is shifted in frequency but still blocks the fundamental frequency $f_{\\text{o}}$ .", "Most notch filters block integer multiples ($n = 0, 1, 2, 3, ...$ ) of a single frequency.", "Instead, we choose a notch filter that is shifted to block odd integer multiples ($2n+1 = 1, 3, 5, ...$ ) of a single frequency.", "Since the matched filter attenuates frequencies above $\\nu = 1$ , we conjecture that the only important spectral notch is at $f_o$ , and all high-order even notches are not included in our pseudo-matched filter.", "The transfer function of our shifted-notch filter is $H_{\\text{shifted\\text{-}notch}}(\\nu ) = \\frac{\\tilde{v}_{out}}{\\tilde{v}_{in}} = \\frac{1}{2}(1+e^{\\pi i \\nu }),$ where $\\tilde{v}_{in}$ and $\\tilde{v}_{out}$ are the Fourier transforms of the input signal $v_{\\text{in}}$ and output signal $v_{\\text{out}}$ , respectively.", "We plot the magnitude of $H_{\\text{shifted\\text{-}notch}}$ in Fig.", "REF a (compare to $H_{\\text{notch}}$ from Fig.", "REF a).", "In both plots, the fundamental frequency $f_{\\text{o}}$ is blocked.", "However, in Fig.", "REF a, the lower frequencies ($\\nu < 0.5$ ) are not cut.", "Thus, the shifted-notch filter performs two of the four operations from the matched filter; (i) it eliminates the fundamental oscillation frequency $f_o$ and (ii) preserves low frequencies.", "Figure: Magnitudes of the transfer functions (a) H bad hbox H_{\\text{shifted\\text{-}notch}}, (b) H bad hbox H_{\\text{low\\text{-}pass}}, (c) H p H_{\\text{p}}.", "In (d), the phase of H p H_{\\text{p}} is given as a function frequency.In the time domain, the shifted-notch filter of Eq.", "(REF ) is expressed by $v_{\\text{out}}(t) = \\frac{1}{2}(v_{\\text{in}}(t) + v_{\\text{in}}(t-\\pi /\\omega _{\\text{o}})),$ We compare Eq.", "(REF ) to Eq.", "(REF ) and note that the output is no longer related to the input through a derivative.", "Also, the time-shift on the input signal is halved ($\\pi /\\omega _{\\text{o}}$ instead of $2\\pi /\\omega _{\\text{o}}$ ) and the shifted input $v_{\\text{in}}(t - \\pi /\\omega _{\\text{o}})$ is summed with the present state $v_{\\text{in}}(t)$ .", "In an experimental setting using high-speed electronics, where $v_{\\text{in}}$ and $v_{\\text{out}}$ are voltages, this shifted-notch filter can be realized using a voltage divider, time-delays (realized, for example, by coaxial cables), and an isolating hybrid junction, as illustrated in Fig.", "REF .", "The lengths of the two cables used in this realization of the filter are chosen such that the difference in propagation times for electromagnetic waves to propagate through them is: $\\tau _{\\text{B}}-\\tau _{\\text{A}} = \\pi /\\omega _{\\text{o}}$ .", "The isolating hybrid junction sums the outputs: $v(t-\\tau _{\\text{A}}) + v(t-\\tau _{\\text{B}})$ .", "We shift time $t \\rightarrow t + \\tau _{\\text{A}}$ to arrive at the output signal $v_{\\text{out}}$ in Eq.", "(14).", "This realization of the shifted-notch filter can scale to high-speed voltages ($> 1$ GHz).", "Figure: Pictorial realization for a high-speed (>1> 1 GHz) shifted-notch filter for voltages v in v_{\\text{in}} and v out v_{\\text{out}}.", "An example of a broadband, high-frequency power splitter is the Mini-Circuits ZFRSC-42-S, and an example of a broadband, high-frequency hybrid-junction is the M/A-COM H-9.Continuing the construction of the pseudo-matched filter, we use a first-order low-pass filter to attenuate high frequencies, rather than an integrator.", "We avoid the need for an integrator because the shifted-notch does not cut off low frequencies.", "The transfer function of the low-pass filter is $H_{\\text{low\\text{-}pass}}(\\nu ) = \\frac{\\tilde{x}_{\\text{out}}}{\\tilde{x}_{\\text{in}}} = \\frac{1}{1+2 \\pi i \\nu /\\nu _{\\text{L}}},$ where $x_{\\text{in}}$ and $x_{\\text{out}}$ are the Fourier transforms of the input and output signals, respectively, and the low-pass cutoff frequency is $\\nu _{\\text{L}} = f_{\\text{L}}/f_{\\text{o}}$ .", "We plot the magnitude of $H_{\\text{low\\text{-}pass}}$ in Fig.", "REF b.", "In the figure, $H_{\\text{low\\text{-}pass}}$ leaves the spectral amplitude of frequencies below $\\nu _L$ unchanged, while suppressing frequencies above $\\nu _{\\text{L}}$ .", "Beyond $\\nu = \\nu _{\\text{L}}$ , the rate of the spectral “roll-off” of $|H_{\\text{low-pass}}|$ is not $\\sim \\nu ^{-1}$ , but rather $\\sim (1+\\nu )^{-1}$ .", "A first-order low-pass filter is a standard electronic component for filtering an input voltage $x_{\\text{in}}$ to obtain an output voltage $x_{\\text{out}}$ and satisfies approximately the third component of the matched filter criteria (iii).", "We note that higher-order low-pass filters (Butterworth, Chebyshev, etc.)", "are also available at high-speed.", "When constructing our pseudo-matched filter, we neglect the dynamical filter that reshapes the spectrum (iv).", "We show that using just the shifted-notch and low-pass filters allows us to achieve comparable performance to the true matched filter in a simulated radar application.", "Thus, we cascade the shifted-notch and low-pass filters ($H_{\\text{shifted\\text{-}notch}} H_{\\text{low\\text{-}pass}}$ ) to arrive at the transfer function of our pseudo-matched filter $H_{\\text{p}}(\\nu )= \\frac{\\tilde{v}_{out}}{\\tilde{v}_{in}} = \\frac{1+e^{\\pi i \\nu }}{2+4\\pi i \\nu /\\nu _L},$ where $\\tilde{v}_{in}$ and $\\tilde{v}_{out}$ are the Fourier transforms of the input and output signals of the filter, respectively.", "We plot the magnitude and phase of $H_{\\text{p}}$ in Fig.", "REF c. In the figure, the phase of $H_{\\text{p}}$ is approximately linear and thus preserves timing information from $v(t)$ .", "For comparison to the matched filter, see Fig.", "REF c. Qualitatively, the two filters follow similar trends in both magnitude and phase.", "The phase in the pseudo-matched filter has a lower slope in its frequency dependence; a lower slope just constitutes a shorter time delay through the filter.", "However, it is clear by comparison of the magnitudes that the pseudo-matched filter is not performing the same operations as the matched filter.", "We now apply the pseudo-matched filter to the chaotic waveform generated by Eqs.", "(REF )-(REF ) and examine its output in the time-domain.", "We drive the pseudo-matched filter with $v(t) = u(t) + AWGN$ , where $v(t)$ has a $SNR$ of $-5.9$ dB (see Fig.", "REF c).", "In Fig.", "REF a, we plot the pseudo-matched filter's output $\\xi _{\\text{p}}(t)$ .", "From the figure, we see that pseudo-matched filter has effectively removed the main oscillation frequency $f_{\\text{o}}$ and what remains is a digital-like signal $\\xi _{\\text{p}}(t)$ , where the noise has also been reduced.", "We note that a considerable amount of noise is still present in comparison to Fig.", "REF d. Using a correlation between the original $s(t)$ and $\\xi _{\\text{p}}(t)$ , we determine the time delay through the pseudo-matched filter $\\tau _{\\text{p}}$ to be approximately $0.14/f_{\\text{o}}$ .", "We compensate for the delay, and sample $\\xi _{\\text{p}}(t)$ at $f_{\\text{o}}$ , assigning binary values using the relation: $-1$ if $\\xi _{\\text{p}} (t_n) \\le 0$ and $+1$ if $\\xi _{\\text{p}} (t_n) > 0$ , where $t_n$ is the $n^{\\text{th}}$ sampling time.", "In Fig.", "REF a, we see that, with this particular $SNR$ , the discrete sampling of $\\xi _{\\text{p}}(t)$ is equivalent to $s_n$ from Fig.", "REF b.", "Figure: Output of pseudo-matched filter.", "(a) Time series of the output of the matched filter ξ p (t)\\xi _{\\text{p}}(t) (blue) while driven by v(t)=u(t)+AWGNv(t) = u(t) + AWGN (SNR=-5.9SNR = -5.9 dB).", "The signal ξ p (t)\\xi _{\\text{p}}(t) is sampled with uniform spacing (red dots) at a clock frequency f o =ω o /2πf_{\\text{o}} = \\omega _{\\text{o}}/2\\pi .", "Above the waveform, a single-bit discrete sampling of the waveform is shown.", "From the figure, we see that all of the relevant information from s n s_n is encoded in ξ p (t)\\xi _{\\text{p}}(t).", "(b) The switching state s(t)s(t) is sampled and s n s_n is stored for N=100N=100.", "The output of the pseudo-matched filter ξ p (t)\\xi _{\\text{p}}(t) drives Eq.", "() and the output χ p (t)\\chi _{\\text{p}}(t) peaks at time t=2τ pd +τ p t = 2\\tau _{\\text{pd}}+\\tau _{\\text{p}}.Next, we parallel the construction of the time delay tap from Eq.", "(REF ) for the output of the pseudo-matched filter.", "In this case, the time delay tap is $\\chi _{\\text{p}}(t) = \\sum \\limits _{n=1}^N s_n \\xi _{\\text{p}}(t-\\tau _n),$ where $\\chi _{\\text{p}}(t)$ is the output of the time-domain correlation.", "We plot an example of $\\chi _{\\text{p}}(t)$ in Fig.", "REF b using the the same chaos and $s_n$ that were used to calculate $\\chi _{\\text{m}}(t)$ in Fig.", "REF b.", "By visually comparing $\\chi _{\\text{p}}(t)$ to $\\chi _{\\text{m}}(t)$ , we see that the output correlation peaks are qualitatively similar, but $\\chi _{\\text{p}}(t)$ has more noise.", "In the remaining section, we establish criteria for quantitatively comparing these correlation waveforms and use these criteria to weigh each filter's performance." ], [ "Matched vs. Pseudo-matched", "In radar applications, the ability to correctly identify the location of the correlation peak in the correlation operation is the useful measure.", "Therefore, in order to compare quantitatively the output from the matched filter and pseudo-matched filter, we weigh each filter's performance based on the peak width and output $SNR$ of $\\chi _{\\text{m,p}}(t)$ .", "We also present an approximate analytical form for each correlation's output $SNR$ .", "The peak widths of the output-correlation functions give the resolutions of each radar system.", "We measure $\\Delta _{\\text{m}}$ and $\\Delta _{\\text{p}}$ , the full-width at half maximum (FWHM) time of the correlation output peaks using the matched and pseudo-matched filters, respectively.", "For the most ideal measure of each filter's correlation peak width, we measure $\\Delta _{\\text{m,p}}$ in cases where no noise is present in the received waveform $v(t) = u(t)$ .", "We note that these widths are independent of $N$ , the number of stored data points in the correlation calculation of Eqs.", "(REF ) and (REF ).", "Using a Gaussian fit to the peak of $\\chi _{\\text{m,p}}(t)$ , we obtain peak widths $\\Delta _{\\text{m}} f_{\\text{o}} = 0.55$ and $\\Delta _{\\text{p}} f_{\\text{o}} = 0.73$ (see Appendix).", "Using these values of $\\Delta _{\\text{m,p}}$ and scaling $f_{\\text{o}}$ to 1 GHz, we calculate the theoretical resolutions of the matched and pseudo-matched filters to be 0.17 m and 0.22 m, respectively.", "In this example, the ranging resolutions differ by 5 cm.", "Thus, this is not a critical difference for radar applications that localize targets like planes or cars, and the pseudo-matched filter has an acceptable ranging resolution in comparison to the matched filter.", "Next, we measure the output $SNR$ 's of the matched and pseudo-matched filter correlations using the correlation peak heights $a_{\\text{m,p}}$ and the surrounding correlation noise floors.", "The output $SNR$ in $\\chi _{\\text{m,p}}(t)$ is $SNR_{\\text{m,p}} = \\frac{a_{\\text{m,p}}^2}{\\sigma _{\\text{N}|\\text{m,p}}^2},$ where $a_{\\text{m,p}}$ is the peak height of $\\xi _{\\text{m,p}}(t)$ from a Gaussian fit (see Appendix) and $\\sigma _{\\text{N}|\\text{m,p}}^2$ is the output variance of the correlation noise floor (note that the mean of the noise floor $\\sim 0$ ) for the matched and pseudo-matched filters, respectively.", "We present a summary of these quantities in the block diagram shown in Fig.", "REF a.", "In the diagram, we also review the waveforms and processes used for generating $\\xi _{\\text{m,p}}(t)$ and $\\chi _{\\text{m,p}}(t)$ and highlight the two relevant quantities, $SNR_{\\text{input}}$ and $SNR_{\\text{m,p}}$ .", "We calculate $SNR_{\\text{m,p}}$ as a function of the input $SNR_{\\text{input}}$ .", "The results of these calculations are given in Fig.", "REF b.", "In addition, we use the distributions from $\\xi _{\\text{m}}(t)$ and $\\xi _{\\text{p}}(t)$ from the two different cases $v(t) = AWGN$ and $v(t) = u(t)$ to derive a an analytical prediction for $SNR_{\\text{m,p}}$ (see Appendix for derivations).", "These theoretical predictions are plotted with dotted lines in Fig.", "REF b.", "These plots represent the performances of the matched and pseudo-matched filters in a simulated radar.", "Figure: (a) Block diagram for testing the matched and pseudo-matched filters in a simulated radar application.", "(b) Output-correlation SNRSNR's of the matched (blue □\\square ) and pseudo-matched (red ◯\\bigcirc ) filters scaled by NN on a logarithmic scale as a function of 1/SNR input 1/SNR_{\\text{input}}.", "For each value of SNR input SNR_{\\text{input}}, 100 calculations of SNR m,p SNR_{\\text{m,p}} were performed using sequence of s n s_{n} for n=n o n = n_{\\text{o}} to n=n o +Nn = n_{\\text{o}} + N, where n o n_{\\text{o}} is a random positive integer and N=50N = 50.", "The mean value of the calculated SNR m,p SNR_{\\text{m,p}} is plotted with the respective standard deviations.", "The bue and red dotted lines give the theoretical predictions of the SNR m,p SNR_{\\text{m,p}} as references for the matched and pseudo-matched filters, respectively.", "Cases for larger NN were verified to have quantitatively similar results.From Fig.", "REF b, it is clear that the matched filter outperforms the pseudo-matched filter in the output $SNR$ of a radar correlation.", "Without noise in the system, the matched and pseudo-matched filters perform with output correlation $SNR$ 's of $2.6+10\\text{log}_{10}(N)$ dB and $1.3+10\\text{log}_{10}(N)$ dB, respectively.", "For $SNR_{\\text{input}}$ = 1/100, the output $SNR$ 's decrease to $-2.0+10\\text{log}_{10}(N)$ dB and $-4.4+10\\text{log}_{10}(N)$ dB, respectively.", "In Fig.", "REF b, the average difference between $SNR_{\\text{m}}$ and $SNR_{\\text{p}}$ is 2.0 dB.", "We note that this difference is independent of $N$ and therefore fully characterizes the filter performances.", "Thus, where small loss is acceptable in the performance of the radar, the pseudo-matched filter is a simpler alternative to the system's analytically matched filter for chaos.", "As a final example, we use the theoretical $SNR_{\\text{m,p}}$ to predict when the matched and pseudo-matched filters will fail in a radar application.", "Failure occurs when $SNR_{\\text{m,p}}$ falls below a certain threshold.", "For a radar system that is capable of storing $N$ = 50 data points and has a desired output correlation $SNR$ of 33 dB, the matched and pseudo-matched filters will fail at a $1/SNR_{\\text{input}}$ of approximately 25 and 70, respectively.", "If, in this application, the input $SNR$ is such that $1/SNR_{\\text{intput}} < 10$ , then a radar with either the matched or pseudo-matched filter will be able to range, on average, without failure.", "The choice between the matched and pseudo-matched filter is therefore an application-dependent problem, and, as the bandwidth of this system scales higher, one must also begin to weigh each filter's high-frequency capabilities as well as its baseline performance." ], [ "CONCLUSIONS", "In conclusion, for the chaotic system presented in Ref.", "[Corron2010], we derive empirically a sub-optimal filter for removing noise from the waveforms generated by this dynamical system.", "This sub-optimal filter performs approximately three out of four of the linear operations from the matched filter: (i) eliminates the fundamental oscillation frequency $f_{\\text{o}}$ , (ii) preserves the low frequencies, and (iii) cuts off high frequencies.", "Our filter, deemed a pseudo-matched filter, is composed of a shifted-notch filter and a first-order low-pass filter.", "In the context of a radar concept that uses a time delay tap as a correlation measure, we have shown that, depending on the application, the pseudo-matched filter may be an acceptable substitute for the matched filter.", "In addition, we acknowledge that the psuedo-matched filter can be further improved using higher order low-pass filters and additional shifted-notch filters.", "We present this current version of the pseudo-matched filter to illustrate our method for its derivation and emphasize its simplicity.", "Lastly, we note that our analysis highlights the flexibility of Corron et al.", "'s findings.", "The chaos from the dynamical system in Eqs.", "(REF ) - (REF ) can be processed by a linear filter to recover an underlying digital waveform.", "We capitalize on this system's elegance to create a pseudo-matched filter.", "Although it is less optimal when compared to the matched filter, the pseudo-matched filter shows that the advantages of this chaotic system can be adapted for applied settings that use commercially available, high-speed filters." ], [ "ACKNOWLEDGEMENTS", "We gratefully acknowledge Damien Rontani for useful discussions, G. Martin Hall with help in radar concepts, and the financial support of Propagation Research Associates (PRA) Grant No.", "W31P4Q-11-C-0279." ], [ "Additive Noise", "Because the output from MATLAB's ODE45 uses a variable timestep, we resample $u(t)$ using a linear interpolation with time steps $\\delta _t$ , where $\\delta _t f_{\\text{o}} = 10^{-2}$ .", "To simulate environmental noise, we add noise to the waveform $u(t)$ using random numbers spaced by time units $\\delta _t$ .", "The random numbers are calculated from a Gaussian distribution with zero mean.", "For different points along the $1/SNR_{\\text{input}}$ axis of Fig.", "REF b, the variance of the $AWGN$ is varied accordingly." ], [ "Gaussian Fits", "We measure the correlation peak width and height using a Gaussian fit $f(t) = a_{\\text{m,p}}e^{(t-(2\\tau _{\\text{pd}}+\\tau _{\\text{m,p}}))^2/2c_{\\text{m,p}}^2},$ where $a_{\\text{m,p}}$ and $c_{\\text{m,p}}$ are free parameters that are fit to the correlation peak heights and widths.", "Using $f(t)$ to fit $\\chi _{\\text{m,p}}(t)$ , we obtain a FWHM peak width $\\Delta _{\\text{m,p}} f_{\\text{o}} = 2\\sqrt{2ln(2)}c_{\\text{m,p}}$ and peak height $a_{\\text{m,p}}$ ." ], [ "Analytical SNR's", "We derive analytical forms for the output-correlation $SNR$ of the matched and pseudo-matched filters.", "To do so, we approximate Eq.", "(REF ) as $SNR_{\\text{m,p}} = \\frac{a_{\\text{m,p}}^2}{\\sigma _{\\text{N}|\\text{m,p}}^2} \\sim \\frac{(A_{\\text{m,p}} N)^2}{\\sigma _{1|\\text{m,p}}^2 N +\\sigma _{2|\\text{m,p}}^2 N},$ where $A_{\\text{m,p}}$ is a constant that characterizes the growth rate of the correlation peak height with $N$ , $\\sigma _{1|\\text{m,p}}^2$ is a constant determined in the noise-free case where $v(t) = u(t)$ , and $\\sigma _{2|\\text{m,p}}^2$ is a function of $SNR_{\\text{input}}$ in the case where $v(t) = AWGN$ .", "Recall that the numerators and denominators of Eq.", "(REF ) represent the power of peak heights of the correlation and the surrounding noise floor, respectively.", "We derive each of the three terms $A_{\\text{m,p}}$ , $\\sigma _{1|\\text{m,p}}^2$ , and $\\sigma _{2|\\text{m,p}}^2$ in the following sections.", "The correlation peak heights for the matched and pseudo-matched filters grow at different rates.", "In the correlation operations of Eqs.", "(REF ) and (REF ), the peaks occur at times $t_{\\text{m,p}}^* = 2\\tau _{\\text{pd}}+\\tau _{\\text{m,p}}$ for the matched and pseudo-matched filters, respectively.", "At time $t^*_{\\text{m,p}}$ , $\\xi _{\\text{m,p}}(t)$ is aligned with $s_n$ and the output correlation is $\\chi _{\\text{m,p}}(t^*_{\\text{m,p}}) \\sim \\sum \\limits _{n=1}^N |\\xi _{\\text{m,p}}(t^*_{\\text{m,p}}-\\tau _n)| \\sim A_{\\text{m,p}}N.$ We approximate $A_{\\text{m,p}}$ from the local maxima of $|\\xi _{\\text{m,p}}(t)|$ .", "To do so, we examine the noise-free case where $v(t) = u(t)$ and collect a subset of points $|\\xi _{\\text{m,p}}(t_{\\text{m,p}}^{(r)})|$ where $t_{\\text{m,p}}^{(r)}$ are the times of local maxima in $|\\xi _{\\text{m,p}}(t)|$ .", "We average $|\\xi _{\\text{m,p}}(t_{\\text{m,p}}^{(r)})|$ to obtain $A_{\\text{m}} = 0.67$ and $A_{\\text{p}} = 0.51$ using a time-length $t f_{\\text{o}} \\sim 10^4$ .", "To approximate the value of $\\sigma _{1|\\text{m,p}}^2$ , we also examine $\\xi _{\\text{m,p}}(t)$ in the noise-free case where $v(t) = u(t)$ .", "The deterministic noise floor in a correlation measurement is also known as its side-lobes; the side-lobes result from non-zero contributing terms in the correlation $\\chi _{\\text{m,p}}(t)$ when $t \\ne t_{\\text{m,p}}^*$ .", "Using the central limit theorem, we approximate the variance of these nonzero terms as $\\sigma _{1|\\text{m,p}}^2N$ , where $\\sigma _{1|\\text{m,p}}^2$ is the variance of the signal $\\xi _{\\text{m,p}}(t)$ .", "In this approximation, we find that $\\sigma _{1|\\text{m}}^2 = 0.25$ and $\\sigma _{1|\\text{p}}^2 = 0.20$ .", "It remains to calculate the contributions to the noise floor of the correlation from additive noise.", "To do so, we examine $\\xi _{\\text{m,p}}(t)$ in the case where $v(t) = AWGN$ .", "Similar to the case for the side-lobes, we use the central limit theorem to approximate the contribution of the $AWGN$ to the noise floor of the correlation as $\\sigma _{2|\\text{m,p}}^2N$ , where $\\sigma _{2|\\text{m,p}}^2$ is the variance of the signal $\\xi _{\\text{m,p}}(t)$ .", "However, $\\sigma _{2|\\text{m,p}}^2$ depends on the variance of the input $AWGN$ $\\sigma _{2|\\text{m,p}}^2 = \\alpha _{\\text{m,p}}\\sigma _{AWGN}^2,$ where $\\alpha _{\\text{m,p}}$ is the noise attenuation factor of the matched and pseudo-matched filters, respectively.", "We measure the values $\\alpha _{\\text{m}} = 1/289$ and $\\alpha _{\\text{p}} = 1/255$ .", "Lastly, we use that $\\sigma _{AWGN}^2 = \\sigma _{u}^2/(SNR_{\\text{input}})$ to rewrite Eq.", "(REF ) as $SNR_{\\text{m,p}} \\sim N\\frac{A_{\\text{m,p}}^2}{\\sigma _{1|\\text{m,p}}^2 +\\alpha _{\\text{m,p}}\\frac{\\sigma _{u}^2}{SNR_{\\text{input}}}},$ where $\\sigma _{u}^2 = 1.34$ is the power of the chaotic signal $u(t)$ .", "We plot Eq.", "(REF ) as a function of $1/SNR_{\\text{input}}$ for the matched and pseudo-matched filters in Fig.", "REF b." ] ]

[ [ "A single-exponential FPT algorithm for the $K_4$-minor cover problem" ], [ "Abstract Given an input graph G and an integer k, the parameterized K_4-minor cover problem asks whether there is a set S of at most k vertices whose deletion results in a K_4-minor-free graph, or equivalently in a graph of treewidth at most 2.", "This problem is inspired by two well-studied parameterized vertex deletion problems, Vertex Cover and Feedback Vertex Set, which can also be expressed as Treewidth-t Vertex Deletion problems: t=0 for Vertex Cover and t=1 for Feedback Vertex Set.", "While a single-exponential FPT algorithm has been known for a long time for \\textsc{Vertex Cover}, such an algorithm for Feedback Vertex Set was devised comparatively recently.", "While it is known to be unlikely that Treewidth-t Vertex Deletion can be solved in time c^{o(k)}.n^{O(1)}, it was open whether the K_4-minor cover problem could be solved in single-exponential FPT time, i.e.", "in c^k.n^{O(1)} time.", "This paper answers this question in the affirmative." ], [ "Introduction", " Given a set $\\mathcal {F}$ of graphs, the parameterized $\\mathcal {F}$ -minor cover problem is to identify a set $S$ of at most $k$ vertices — if it exists — in an input graph $G$ such that the deletion of $S$ results in a graph which does not have any graph from $\\mathcal {F}$ as a minor; the parameter is $k$ .", "Such a set $S$ is called an $\\mathcal {F}$ -minor cover (or an $\\mathcal {F}$ -hitting set) of $G$ .", "A number of fundamental graph problems can be viewed as $\\mathcal {F}$ -minor cover problems.", "Well-known examples include Vertex Cover ($\\mathcal {F}=\\lbrace K_2\\rbrace $ ), Feedback Vertex Set ($\\mathcal {F}=\\lbrace K_3\\rbrace $ ), and more generally the Treewidth-$t$ Vertex Deletion for any constant $t$ , which asks whether an input graph can be converted to one with treewidth at most $t$ by deleting at most $k$ vertices.", "Observe that for $t=0$ and 1, Treewidth-$t$ Vertex Deletion is equivalent to Vertex Cover and Feedback Vertex Set, respectively.", "The importance of Treewidth-$t$ Vertex Deletion is not only theoretical.", "For example, even for small values of $t$ , efficient algorithms for this problem would improve algorithms for inference in Bayesian Networks as a subroutine of the cutset conditioning method [1].", "This method is practical only with small value $t$ and efficient algorithms for small treewidth $t$ , though not for any fixed $t$ , are desirable.", "In this paper we consider the parameterized $\\mathcal {F}$ -minor cover problem for $\\mathcal {F}=\\lbrace K_4\\rbrace $ , which is equivalent to the Treewidth-2 Vertex Deletion.", "The NP-hardness of this problem is due to [24].", "Fixed-parameter tractability (i.e.", "can be solved in time $f(k)\\cdot n^{O(1)}$ for some computable function $f$ ) follows from two celebrated meta-results: the Graph Minor Theorem of Robertson and Seymour [27] and Courcelle's theorem [8].", "Unfortunately, the resulting algorithms involve huge exponential functions in $k$ and are impractical even for small values of $k$ .", "In recent years, single-exponential time parameterized algorithms — those which run in $c^k\\cdot n^{O(1)}$ time for some constant $c$ — and also sub-exponential time parameterized algorithms have been developed for a wide variety of problems.", "Of special interest is the bidimensionality theory introduced by Demaine et al.", "[11] as a tool to obtain sub-exponential parameterized algorithms for the so-called bidimensional problems on $H$ -minor-free graphs.", "It is also known to be unlikely that every fixed parameter tractable problem can be solved in sub-exponential time [6].", "For problems which probably do not allow sub-exponential time algorithms, ensuring a single exponential upper bound on the time complexity is highly desirable.", "For example, Bodleander et al.", "[4] proved that all problems that have finite integer index and satisfy some compactness conditions admit a linear kernel on graphs of bounded genus [4], implying single-exponential running times for such problems.", "More recently Cygan et al.", "developed the “cut-and-count” technique to derive (randomized) single-exponential parameterized algorithms for many connectivity problems parameterized by treewidth [9].", "In contrast, some problems are unlikely to have single-exponential algorithms [23].", "For treewidth-$t$ vertex deletion, single-exponential parameterized algorithms are known only for $t=0$ and $t=1$ .", "Indeed, for $t=0$ (Vertex Cover), the $O(2^k\\cdot n)$ -time bounded search tree algorithm is an oft-quoted first example for a parameterized algorithm [13], [25], [15].", "For $t=1$ (Feedback Vertex Set), no single-exponential algorithm was known for many years until Guo et al.", "[19] and Dehne et al.", "[10] independently discovered such algorithms.", "The fastest known deterministic algorithm for this problem runs in time $O(3.83^k\\cdot n^2)$  [5].", "The fastest known randomized algorithm, developed by Cygan et al., runs in $O(3^k\\cdot n^{O(1)})$ time [9].", "Very recently, Fomin et al.", "[18] presented $2^{O(k\\log k)}\\cdot n^{O(1)}$ -time algorithms for treewidth-$t$ vertex deletion.", "In this paper we prove the following result for $t=2$ : Theorem 1 The $K_4$ -minor cover problem can be solved in $2^{O(k)}\\cdot n^{O(1)}$ time.", "Our single-exponential parameterized algorithm for $K_4$ -minor cover is based on iterative compression.", "This allows us, with a single-exponential time overhead, to focus on the disjoint version of the $K_4$ -minor cover problem: given a solution $S$ , find a smaller solution disjoint from $S$ .", "We employ a search tree method to solve the disjoint problem.", "Although our algorithm shares the spirit of Chen et al.", "'s search tree algorithm for Feedback Vertex Set [7], that we want to cover $K_4$ -minor instead of $K_3$ requires a nontrivial effort.", "In order to bound the branching degree by a constant, three key ingredients are exploited.", "First, we employ protrusion replacement, a technique developed to establish a meta theorem for polynomial-size kernels [4], [16], [17].", "We need to modify the existing notions so as to use the protrusion technique in the context of iterative compression.", "Second, we introduce a notion called the extended SP-decomposition, which makes it easier to explore the structure of treewidth-two graphs.", "Finally, the technical running time analysis depends on the property of the extended SP-decomposition and a measure which keeps track of the biconnectivity." ], [ "Notation and preliminaries", " We follow standard graph terminology as found in, e.g., Diestel's textbook [12].", "Any graph considered in this paper is undirected, loopless and may contain parallel edges.", "A cut vertex (resp.", "cut edge) is a vertex (resp.", "an edge) whose deletion strictly increases the number of connected components in the graph.", "A connected graph without a cut vertex is biconnected.", "A subgraph of $G$ is called a block if it is a maximal biconnected subgraph.", "A biconnected graph is itself a block.", "In particular, an edge which is not a part of any cycle is a block as well.", "For a vertex set $X$ in a graph $G=(V,E)$ , the boundary $\\partial _G(X)$ of $X$ is the set $N(V\\setminus X)$ , i.e.", "the set of vertices in $X$ which are adjacent with at least one vertex in $V\\setminus X$ .", "We may omit the subscript when it is clear from the context.", "Minors.", "The contraction of an edge $e=(u,v)$ in a graph $G$ results in a graph denoted $G/e$ where vertices $u$ and $v$ have been replaced by a single vertex $uv$ which is adjacent to all the former neighbors of $u$ and $v$ .", "A subdivision of an edge $e$ is the operation of deleting $e$ and introducing a new vertex $x_e$ which is adjacent to both the end vertices of $e$ .", "A subdivision of a graph $H$ is a graph obtained from $H$ by a series of edge subdivisions.", "A graph $H$ is a minor of graph $G$ if it can be obtained from a subgraph of $G$ by contracting edges.", "A graph $H$ is a topological minor of $G$ if a subdivision of $H$ is isomorphic to a subgraph $G^{\\prime }$ of $G$ .", "In these cases we say that $G$ contains $H$ as a (topological) minor and that $G^{\\prime }$ is an $H$ -subdivision in $G$ .", "In an $H$ -subdivision $G^{\\prime }$ of $G$ , the vertices which correspond to the original vertices of $H$ are called branching nodes; the other vertices of $G^{\\prime }$ are called subdividing nodes.", "It is well known that if the maximum degree of $H$ is at most three, then $G$ contains $H$ as a minor if and only if it contains $H$ as a topological minor [12].", "A $\\theta _3$ -subdivision is a graph which consists of three vertex disjoint paths between two branching vertices.", "Series-parallel graphs and treewidth-two graphs.", "A two-terminal graph is a triple $(G,s,t)$ where $G$ is a graph and the terminals $s$ , $t$ .", "The series composition of $(G_1,s_1,t_1)$ and $(G_2,s_2,t_2)$ is obtained by taking the disjoint union of $G_1$ and $G_2$ and identifying $t_1$ with $s_2$ .", "The resulting graph has $s_1$ and $t_2$ as terminals.", "The parallel composition of $(G_1,s_1,t_1)$ and $(G_2,s_2,t_2)$ is obtained by taking the disjoint union of $G_1$ and $G_2$ and identifying $s_1$ with $s_2$ and $t_1$ with $t_2$ .", "Series and parallel compositions generalize to any number of two-terminal graphs.", "Two-terminal series-parallel graphs are formed from the single edge and successive series or parallel compositions.", "A graph $G$ is a series-parallel graph (SP-graph) if $(G,s,t)$ is a two-terminal series-parallel graph for some $s,t\\in V(G)$ .", "The recursive construction of a series-parallel graph $G$ defines an SP-tree $(T,\\mathcal {X}=\\lbrace X_{\\alpha }:\\alpha \\in V(T)\\rbrace )$ , where $T$ is a tree whose leaves correspond to the edges of $G$ .", "Every internal node $\\alpha $ is either an S-node or a P-node and represents the subgraph $G_{\\alpha }$ resulting from the series composition or the parallel composition, respectively, of the graphs associated with its children.", "Every node $\\alpha $ of $T$ is labelled by the set $X_{\\alpha }$ of the terminals of $G_{\\alpha }$ .", "Interested readers are referred to Valdes et al.", "'s seminal paper on the subject [28].", "We may assume that an SP-tree satisfies additional conditions.", "We use, for example, canonicalFull definition, proofs of lemmas, theorems ...marked by $\\star $ are also deferred to the appendix SP-trees for the purpose of analysis, whose definition will not be given in the extended abstract.", "We remark that any SP-graph can be represented as a canonical SP-tree [3] and it can be computed in linear time.", "We refer to Diestel's textbook [12] for the definition of the treewidth of a graph $G$ which we denote $tw(G)$ .", "It is well known that a graph has treewidth at most two graphs if and only if it is $K_4$ -minor-free.", "We also make use of the following alternative characterization: $tw(G)\\leqslant 2$ if and only if every block of $G$ is a series-parallel graph [2], [3].", "Extended SP-decompositon.", "A connected graph $G$ can be decomposed into blocks which are joined by the cut vertices of $G$ in a tree-like manner.", "To be precise, we can associate a block tree $\\mathcal {B}_G$ to $G$ , in which the node set consists of all blocks and cut vertices of $G$ , and a block $B$ and a cut vertex $c$ are adjacent in $\\mathcal {B}_G$ if and only if $B$ contains $c$ .", "To explore the structure of a treewidth-two graph $G$ efficiently, we combine its block tree $\\mathcal {B}_G$ with (canonical) SP-trees of its blocks into an extended SP-decomposition as described below.", "We assume that $G$ is connected: in general, an extended SP-decomposition of $G$ is a collection of extended SP-decompositions of its connected components.", "Let $\\mathcal {B}_G$ be the block tree of a treewidth-two graph $G$ .", "We fix an arbitrary cut node $c_{root}$ of $\\mathcal {B}_G$ if one exists.", "The oriented block tree $\\vec{\\mathcal {B}}_G$ is obtained by orienting the edges of $\\mathcal {B}_G$ outward from $c_{root}$ .", "If $\\mathcal {B}_G$ consists of a single node, it is regarded as an oriented block tree itself.", "We construct an extended SP-decomposition of a connected graph $G$ by replacing the nodes of $\\vec{\\mathcal {B}}_G$ by the corresponding SP-trees and connecting distinct SP-trees to comply the orientations of edges in $\\vec{\\mathcal {B}}_G$ .", "To be precise, an extended SP-decomposition is a pair $(T, \\mathcal {X}=\\lbrace X_{\\alpha }:\\alpha \\in V(T)\\rbrace )$ , where $T$ is a rooted tree whose vertices are called nodes and $\\mathcal {X}=\\lbrace X_{\\alpha }:\\ \\alpha \\in V(T)\\rbrace $ is a collection of subsets of $V(G)$ , one for each node in $T$ .", "We say that $X_{\\alpha }$ is the label of node $\\alpha $ .", "For each block $B$ of $G$ , let $(T^B,\\mathcal {X}^B)$ be a (canonical) SP-tree of $G[B]$ such that $c(B)$ is one of the terminal associated to the root node of $T^B$ .", "A leaf node of $T^B$ is called an edge node.", "For each cut vertex $c$ of $G$ , add to $(T,\\mathcal {X})$ a cut node $\\alpha $ with $X_{\\alpha }=\\lbrace c\\rbrace $ .", "For each block $B$ of $G$ , let the root node of $(T^B,\\mathcal {X}^B)$ be a child of the unique cut node $\\alpha $ in $T$ which satisfies $X_{\\alpha }=\\lbrace c(B)\\rbrace $ .", "For a cut vertex $c$ of $G$ , let $B=B(c)$ be the unique block such that $(B,c)\\in E(\\vec{\\mathcal {B}}_G)$ .", "Let $\\beta $ be an arbitrary leaf node of the (canonical) SP-tree $(T^B,\\mathcal {X}^B)$ such that $c\\in X_{\\beta }$ (note that such a node always exists).", "Make the cut node $\\alpha $ of $(T,\\mathcal {X})$ labeled by $\\lbrace c\\rbrace $ a child of the leaf node $\\beta $ .", "Let $\\alpha $ be a node of $T$ .", "Then $T_{\\alpha }$ is the subtree of $T$ rooted at node $\\alpha $ ; $E_{\\alpha }$ is the set of edges $(u,v)\\in E(G)$ such that there exists an edge node $\\alpha ^{\\prime } \\in V(T_{\\alpha })$ with $X_{\\alpha ^{\\prime }}=\\lbrace u,v\\rbrace $ ; and $G_{\\alpha }$ is the — not necessarily induced — subgraph of $G$ with the vertex set $V_{\\alpha }:=\\bigcup _{\\alpha ^{\\prime } \\in V(T_{\\alpha })} X_{\\alpha ^{\\prime }}$ and the edge set $E_{\\alpha }$ .", "Recall that $X_{\\alpha }$ is the set of vertices which form the label of the node $\\alpha $ , and that $|X_{\\alpha }|\\in \\lbrace 1,2\\rbrace $ .", "We define $Y_{\\alpha }:=V_{\\alpha }\\setminus X_{\\alpha }$ .", "Observe that in the construction above, every node $\\alpha $ of $(T,\\mathcal {X})$ is either a cut node or corresponds to a node from the SP-tree $(T^B,\\mathcal {X}^B)$ of some block $B$ of $G$ .", "We say that a node $\\alpha $ which is not a cut node is inherited from $(T^B,\\mathcal {X}^B)$ , where $B$ is the block to which $\\alpha $ belongs.", "Let $\\alpha $ be inherited from $(T^B,\\mathcal {X}^B)$ .", "We use $T^{B}_{\\alpha }$ to denote the SP-tree naturally associated with the subtree of $T^{B}$ rooted at $\\alpha $ .", "By $G^B_{\\alpha }$ we denote the SP-graph represented by the SP-tree $T^{B}_{\\alpha }$ , where $(T^B,\\mathcal {X}^B)$ inherits $\\alpha $ .", "The vertex set of $G^B_{\\alpha }$ is denoted $V^B_{\\alpha }$ .", "We observe that for every node $\\alpha $ , $G_{\\alpha }$ is connected and that $\\partial _G(V_{\\alpha })\\subseteq X_{\\alpha }$ .", "It is well-known that one can decide whether $tw(G)\\leqslant 2$ in linear time [28].", "It is not difficult to see that in linear time we can also construct an extended SP-decomposition of $G$ ." ], [ "The algorithm", " Our algorithm for $K_4$ -minor cover uses various techniques from parameterized complexity.", "First, an iterative compression [26] step reduces $K_4$ -minor cover to the so-called disjoint $K_{4}$ -minor cover problem, where in addition to the input graph we are given a solution set to be improved.", "Then a Branch-or-reduce process develops a bounded search tree.", "We start with a definition of the compression problem for $K_4$ -minor cover.", "Iterative compression.", "Given a subset $S$ of vertices, a $K_4$ -minor cover $W$ of $G$ is $S$ -disjoint if $W\\cap S=\\emptyset $ .", "We omit the mention of $S$ when it is clear from the context.", "If $\\vert W\\vert \\le k-1$ , then we say that $W$ is small.", "disjoint $K_{4}$ -minor cover problem Input:           [t]14cmA graph $G$ and a $K_4$ -minor cover $S$ of $G$ Parameter: The integer $k=|S|$ Output: A small $S$ -disjoint $K_4$ -minor cover $W$ of $G$ , if one exists.", "Otherwise return NO.", "An FPT algorithm for the disjoint $K_{4}$ -minor cover problem can be used as a subroutine to solve the $K_4$ -minor cover problem.", "Such a procedure has now become a standard in the context of iterative compression problems [7], [22], [20].", "Lemma 1 ($\\star $ ) If disjoint $K_{4}$ -minor cover can be solved in $c^k\\cdot n^{O(1)}$ time, then $K_4$ -minor cover can be solved in $(c+1)^k\\cdot n^{O(1)}$ time.", "Observe that both $G[V\\setminus S]$ and $G[S]$ is $K_4$ -minor-free.", "Indeed if $G[S]$ is not $K_4$ -minor-free, then the answer to disjoint $K_{4}$ -minor cover is trivially NO.", "Protrusion rule.", "A subset $X$ of the vertex set of a graph $G$ is a $t$ -protrusion of $G$ if $tw(G[X])\\leqslant t$ and $|\\partial (X)|\\leqslant t$ .", "Our algorithm deeply relies on protrusion reduction technique, which made a huge success lately in discovering meta theorems for kernelization [16], [4].", "However, we need to adapt the notions developed for protrusion technique so that we can apply the technique to our “disjoint” problem, which arises in the iterative compression-based algorithm.", "In essence, our (adapted) protrusion lemma for disjoint parameterized problems says that a 'large' protrusion which is disjoint from the forbidden set $S$ can be replaced by a 'small' protrusion which is again disjoint from $S$ .", "Due to its generality, this result may be of independent interest.", "Reduction Rule 1 ($\\star $ ) (Generic disjoint protrusion rule) Let $(G,S,k)$ be an instance of disjoint $K_{4}$ -minor cover and $X$ be a $t$ -protrusion such that $X\\cap S=\\emptyset $ .", "Then there exists a computable function $\\gamma (.", ")$ and an algorithm which computes an equivalent instance in time $O(|X|)$ such that $G[S]$ and $G^{\\prime }[S]$ are isomorphic, $G^{\\prime }-S$ is $K_4$ -minor-free, $|V(G^{\\prime })|<|V(G)|$ and $k^{\\prime }\\leqslant k$ , provided $|X|>\\gamma (2t+1)$ .", "We remark that some of the reduction rules we shall present in the next subsection are instantiations the generic disjoint protrusion rule.", "However, to ease the algorithm analysis, the generic rule above is used only on $t$ -protrusion whose boundary size is 3 or 4.", "For protrusions with boundary size 1 or 2, we shall instead apply the following explicit reduction rules." ], [ "(Explicit) Reduction rules", " We say that a reduction rule is safe if, given an instance $(G,S,k)$ , the rule returns an equivalent instance $(G^{\\prime },S^{\\prime },k^{\\prime })$ ; that is, $(G,S,k)$ is a YES-instance if and only if $(G^{\\prime },S^{\\prime },k^{\\prime })$ is.", "Let $F$ denote the subset $V(G)\\setminus S$ of vertices.", "For a vertex $v\\in F$ , let $N_S(v)$ denote the neighbors of $v$ which belong to $S$ .", "By $N_i\\subseteq F$ we refer to the set of vertices $v$ in $F$ with $|N_S(v)|=i$ .", "The next three rules are simple rule that can be applied in polynomial time.", "In each of them, $S$ and $k$ are unchanged ($S^{\\prime }=S$ , $k^{\\prime }=k$ ).", "Observe that reduction rule REF (b) can be seen as a disjoint 1-protrusion rule.", "Reduction Rule 2 ($\\star $ ) (1-boundary rule) Let $X$ be a subset of $F$ .", "(a) If $G[X]$ is a connected component of $G$ or of $G\\setminus e$ for some cut edge $e$ , then delete $X$ .", "(b) If $|\\partial _G(X)|=1$ , then delete $X\\setminus \\partial _G(X)$ .", "Reduction Rule 3 ($\\star $ ) (Bypassing rule) Bypass every vertex $v$ of degree two in $G$ with neighbors $u_1\\in V$ , $u_2\\in F$ .", "That is, delete $v$ and its incident edges, and add the new edge $(u_1,u_2)$ .", "Reduction Rule 4 ($\\star $ ) (Parallel rule) If there is more than one edge between $u\\in V$ and $v\\in F$ , then delete all these edges except for one.", "The next two reduction rules are somewhat more technical, and their proofs of correctness require a careful analysis of the structure of the $K_4$ -subdivisions in a graph.", "Reduction Rule 5 ($\\star $ ) (Chandelier rule) Let $X=\\lbrace u_1,\\ldots , u_{\\ell }\\rbrace $ be a subset of $F$ , and let $x$ be a vertex in $S$ such that $G[X]$ contains the path $u_1,\\ldots , u_{\\ell }$ , $N_S(u_i)=\\lbrace x\\rbrace $ for every $i=1,\\ldots ,\\ell $ , and vertices $u_{2},\\ldots ,u_{\\ell -1}$ have degree exactly 3 in $G$ .", "If $\\ell \\ge 4$ , contract the edge $e=(u_2,u_3)$ (and apply Rule REF to remove the parallel edges created).", "The intuition behind the correctness of Chandelier rule REF is that such a set $X$ cannot host all four branching nodes of a $K_4$ -subdivision.", "Our last reduction rule is an explicit 2-protrusion rule.", "In the particular case when the boundary size is exactly two, the candidate protrusions for replacement are either a single edge or a $\\theta _3$ (see Figure REF ).", "Reduction Rule 6 ($\\star $ ) (2-boundary rule) Let $X\\subseteq F$ be such that $G[X]$ is connected, $\\partial (X)=\\lbrace s,t\\rbrace $ (and thus, $X\\setminus \\lbrace s,t\\rbrace \\subseteq N_0$ ).", "Then we do the following.", "(1) Delete $X\\setminus \\lbrace s,t\\rbrace $ .", "(2) If $G[X]+(s,t)$ is a series parallel graph and $|X|>2$ , then add the edge $(s,t)$ (if it is not present).", "Else if $G[X]+(s,t)$ is not a series parallel graph and $|X|>4$ , add two new vertices $a,b$ and the edges $\\lbrace (a,b),(a,t),(a,s),(b,t),(b,s)\\rbrace $ (see Figure REF ).", "Figure: If G[X]+(s,t)G[X]+(s,t) is an SP-graph, we can safely replace G[X]G[X] by the edge (s,t)(s,t).", "Otherwise G[X]G[X] can be replaced by a subdivision of θ 3 \\theta _3 with poles aa and bb in which ss and tt are subdividing nodes.An instance of disjoint $K_{4}$ -minor cover is reduced if none of the Reduction rules REF - REF applies." ], [ "Branching rules", " A branching rule is an algorithm which, given an instance $(G,S,k)$ , outputs a set of $d$ instances $(G_1,S_1,k_1)\\dots (G_d,S_d,k_d)$ for some constant $d>1$ ($d$ is the branching degree).", "A branching rule is safe if $(G,S,k)$ is a YES-instance if and only if there exists $i$ , $1\\leqslant i\\leqslant d$ such that $(G_i,S_i,k_i)$ is a YES instance.", "We now present three generic branching rules, with potentially unbounded branching degrees.", "Later we describe how to apply these rules so as to bound the branching degree by a constant.", "Given a vertex $s\\in S$ , we denote by $cc_S(s)$ the connected component of $G[S]$ which contains $s$ .", "Likewise, $bc_S(s)$ denotes the biconnected component of $G[S]$ containing $s$ .", "It is easy to see that three branching rules below are safe.", "Branching Rule 1 Let $(G,S,k)$ be an instance of disjoint $K_{4}$ -minor cover and let $X$ be a subset of $F$ such that $G[S\\cup X]$ contains a $K_4$ -subdivision.", "Then branch into the instances $(G-\\lbrace x\\rbrace ,S,k-1)$ for every $x\\in X$ .", "Branching Rule 2 Let $(G,S,k)$ be an instance of disjoint $K_{4}$ -minor cover and let $X$ be be a connected subset of $F$ .", "If $S$ contains two vertices $s_1$ and $s_2$ each having a neighbor in $X$ and such that $cc_S(s_1)\\ne cc_S(s_2)$ , then branch into the instances $(G-\\lbrace x\\rbrace ,S,k-1)$ for every $x\\in X$ $(G,S\\cup X, k)$ Branching Rule 3 Let $(G,S,k)$ be an instance of disjoint $K_{4}$ -minor cover and let $X$ be a connected subset of $F$ .", "If $S$ contains two vertices $s_1$ and $s_2$ each having a neighbor in $X$ such that $cc_S(s_1)=cc_S(s_2)$ and $bc_S(s_1)\\ne bc_S(s_2)$ , then branch into the instances $(G-\\lbrace x\\rbrace ,S,k-1)$ for every $x\\in X$ $(G,S\\cup X, k)$ We shall apply branching rule REF under three different situations: (i) $X$ is a singleton $\\lbrace x\\rbrace $ for every $x\\in F$ , (ii) $X$ is connected, and (iii) $X$ consists of a pair of non-adjacent vertices of $F$ .", "Let us discuss these three settings in further details.", "An instance $(G,S,k)$ is said to be a simplified instance if it is a reduced instance and if none of the branching rules REF  - REF applies on singleton sets $X=\\lbrace v\\rbrace $ , for any $v\\in F$ .", "A simplified instance, in which branching rule REF cannot be applied under (i), has a useful property.", "Lemma 2 ($\\star $ ) If $(G,S,k)$ is a simplified instance of disjoint $K_{4}$ -minor cover, then $F=N_0\\cup N_1\\cup N_2$ .", "An instance $(G,S,k)$ of disjoint $K_{4}$ -minor cover is independent if (a) $F$ is an independent set; (b) every vertex of $F$ belongs to $N_2$ ; (c) the two neighbors of every vertex of $F$ belong to the same biconnected component of $G[S]$ and (d) $G[S\\cup \\lbrace x\\rbrace ]$ is $K_4$ -minor-free for every $x\\in F$ .", "In essence, next lemma shows that the instance is independent once branching rule REF has been exhaustively applied under (ii).", "Theorem 2 ($\\star $ ) Let $(G,S,k)$ be an instance of disjoint $K_{4}$ -minor cover.", "If none of the reduction rules applies nor branching rules on connected subsets $X\\subseteq F$ applies, then $(G,S,k)$ is an independent instance.", "Next lemma shows that in an independent instance, it is enough to cover the $K_4$ -subdivisions containing exactly two vertices of $F$ .", "To see this, we construct an auxiliary graph $G^*(S)$ as follows: its vertex set is $F$ ; $(u,v)$ is an edge in $G^*(S)$ if and only if $G[S\\cup \\lbrace u,v\\rbrace ]$ contains $K_4$ as a minor.", "Then the following theorem holds, which essentially states that we obtain a solution for disjoint $K_{4}$ -minor cover by applying branching rule REF exhaustively under (iii).", "Theorem 3 ($\\star $ ) Let $(G,S,k)$ be an independent instance of disjoint $K_{4}$ -minor cover.", "Then $W\\subseteq F$ is a disjoint $K_4$ -minor cover of $G$ if and only if it is a vertex cover of $G^*(S)$ .", "Observe that we do not need to build $G^*(S)$ to solve the disjoint $K_{4}$ -minor cover problem on an independent instanceA more careful analysis shows that $G^*(S)$ is a circle graph.", "As Vertex Cover is polynomial time solvable on circle graphs, so is disjoint $K_{4}$ -minor cover problem on an independent instance..", "Indeed, for every pair of vertices $u,v\\in F$ , it is enough to test whether $G[S\\cup \\lbrace u,v\\rbrace ]$ contains $K_4$ as a minor (this can be done in linear time [28]) and if so we apply branching rule REF on the set $X=\\lbrace u,v\\rbrace $ ." ], [ "Algorithm and complexity analysis ", " Let us present the whole search tree algorithm.", "At each node of the computation tree associated with a given instance $(G,S,k)$ , one of the followings operations is performed.", "As each operation either returns a solution (as in (a),(e)) or generates a set of instances (as in (b)-(d)), the overall application of the operations can be depicted as a search tree.", "if ($k<0$ ) or ($k\\le 0$ , $tw(G)> 2$ ) or ($tw(G[S]) > 2$ ), then return no; if the instance is not reduced, apply one of Reduction rules REF –REF (note that we apply Reduction rules REF –REF first whenever possible, and Reduction rule REF is applied when none of the rules REF –REF can be applied); if the instance is not simplified, apply one of Branching rules REF –REF on the singleton sets $\\lbrace x\\rbrace $ for each $x\\in F$ ; if the instance is simplified, apply the procedure Branch-or-reduce; if the application of Branch-or-reduce marks every node of $(T,\\mathcal {X})$ , the instance is an independent instance; solve it in $2^k\\cdot n^{O(1)}$ using branching rule REF on pairs of vertices of $F$ .", "We now describe the procedure Branch-or-reduce as a systematic way of applying the branching and reduction rules.", "It works in a bottom-up manner on an extended SP-decomposition $(T,\\mathcal {X})$ of $G[F]$ .", "Initially the nodes of $(T,\\mathcal {X})$ are unmarked.", "Starting from a lowest node, Branch-or-reduce recursively tests if we can apply one of the branching rules on a subgraph associated with a lowest unmarked node.", "If the branching rules do not apply, it may be due to a large protrusion.", "In that case, we detect the protrusion (see Lemma REF ) and reduce the instance using the protrusion rule REF .", "Once either a branching rule or the protrusion rule has been applied, the procedure Branch-or-reduce terminates.", "The output is a set of instances of disjoint $K_{4}$ -minor cover, possibly a singleton.", "[h] A simplified instance $(G,S,k)$ of disjoint $K_{4}$ -minor cover, together with an extended SP-decomposition $(T,\\mathcal {X})$ of $G[F]$ .", "A set of instances of disjoint $K_{4}$ -minor cover.", "$T$ contains unmarked nodes l:alpha Let $\\alpha $ be an unmarked node at the farthest distance from the root of $T$ l:br2-test $S$ contains two vertices $x_u\\in N_S(u)$ and $x_v\\in N_S(v)$ with $u,v\\in V_{\\alpha }$ and $cc_S(x_u)\\ne cc_S(x_v)$ l:Xbr2 Let $X$ be a path in $G_{\\alpha }$ between two such vertices $u$ and $v$ such that $X\\setminus \\lbrace u,v\\rbrace \\subseteq N_0$ l:br2 Apply Branching rule REF to $X$ ; terminate l:br3-test $S$ contains two vertices $x_u\\in N_S(u)$ and $x_v\\in N_S(v)$ with $u,v\\in V_{\\alpha }$ and $bc_S(x_u)\\ne bc_S(x_v)$ l:Xbr3 Let $X$ be a path in $G_{\\alpha }$ between two such vertices $u$ and $v$ such that $X\\setminus \\lbrace u,v\\rbrace \\subseteq N_0$ l:br3 Apply Branching rule REF to $X$ ; terminate l:br1-test $G[S\\cup V_{\\alpha }]$ contains a $K_4$ -subdivision l:Xbr1 Let $X \\subseteq V_{\\alpha }$ be a connected set such that $G[S+X]$ contains a $K_4$ -subdivision l:br1 Apply Branching rule REF to $X$ ; terminate l:protrusion-test $\\alpha $ is a P-node and $|V^B_{\\alpha }|\\geqslant \\gamma (9)$ l:Xprot $X = V^B_{\\alpha }$ is a 4-protrusion (see Lemma REF ) l:prot Apply the protrusion Reduction rule REF with $X$ ; terminate l:mark Mark the node $\\alpha $ Branch-or-reduce The complexity analysis relies on a series of technical lemmas such as Lemma REF .", "We say that a path $P$ avoids a set $X$ if no internal vertex of $P$ belongs to $X$ .", "To simplify the notation, we use $G_{\\alpha }$ instead of $G[F]_{\\alpha }$ for a node $\\alpha $ of $T$ .", "Similarly, we use the names $V_{\\alpha }$ , $Y_{\\alpha }=V_{\\alpha }\\setminus X_{\\alpha }$ and $V^B_{\\alpha }$ to denote the various named subsets of $V(G[F]_{\\alpha })$ .", "Lemma 3 ($\\star $ ) Let $W$ and $Z$ be disjoint vertex subsets of a graph $G$ such that $G[W]$ is biconnected, $G[Z]$ is connected and $|N_W(Z)|\\ge 3$ .", "Then $G[W\\cup Z]$ contains a $K_4$ -subdivision.", "Lemma 4 Let $(G,S,k)$ be a simplified instance and let $\\alpha $ be a lowest node of the extended SP-decomposition $(T,\\mathcal {X})$ of $G[F]$ which is considered at line  of Algorithm REF .", "If $\\alpha $ is a P-node inherited from the SP-tree of block $B$ , then $|\\partial _G(V^B_{\\alpha })\\setminus X_{\\alpha }|\\le 2$ and $V^B_{\\alpha }$ is a 4-protrusion.", "As $\\alpha $ is a P-node, $G^B_{\\alpha }$ is biconnected.", "We argue $|\\partial _G(V^B_{\\alpha })\\setminus X_{\\alpha }|\\le 2$ and the second statement easily follows.", "Suppose $\\partial _G(V^B_{\\alpha })\\setminus X_{\\alpha }$ contains three distinct vertices, say, $x$ , $y$ and $z$ .", "We claim that there exist three internally vertex-disjoint paths $P_x$ , $P_y$ and $P_z$ from $S$ to each of $x$ , $y$ and $z$ avoiding $V^B_{\\alpha }$ .", "Without loss of generality, we show that $G[S\\cup V_{\\alpha }]$ contains a path $P_x$ between $S$ and $x$ avoiding $V^B_{\\alpha }$ and the claim follows as a corollary.", "If $x\\in N_1\\cup N_2$ , then it is trivial.", "Suppose $x \\notin N_1\\cup N_2$ and thus $x$ is a cut vertex of $G[F]$ .", "Then $(T,\\mathcal {X})$ contains a cut node $\\beta $ with $X_{\\beta }=\\lbrace x\\rbrace $ such that $\\beta $ is a descendent of $\\alpha $ .", "It can be shownLemma REF in the appendix that $Y_{\\beta }\\cap (N_1\\cup N_2)\\ne \\emptyset $ .", "Since $G_{\\beta }$ is connected, $G[S\\cup V_{\\beta }]$ contains a path $P_x$ between $S$ and $x$ and $P_x$ is a path avoiding $V^B_{\\alpha }$ .", "As $\\alpha $ fails the test of line , the vertices of $N_{S}(V_{\\alpha })$ belong to the same connected component, say $C$ , of $G[S]$ .", "Now Lemma REF applies to the biconnected graph $G^B_{\\alpha }$ and $(C\\cup P_x\\cup P_y\\cup P_z)\\setminus \\lbrace x,y,z\\rbrace $ , showing that $G[V^B_{\\alpha }\\cup P_x\\cup P_y \\cup P_z \\cup S]$ contains a $K_4$ -subdivision: a contradiction to the fact that Branching rule REF does not apply.", "Therefore, $\\partial _G(V^B_{\\alpha })\\setminus X_{\\alpha }$ contains at most two vertices.", "The next two lemmas show that applying Branch-or-reduce in a bottom-up manner enables us to bound the branching degree of the Branch-or-reduce procedure.", "Lemma REF states that for every marked node $\\alpha $ , the graph $G_{\\alpha }$ is of constant-size.", "Lemma 5 ($\\star $ ) Let $(G,S,k)$ be a simplified instance of disjoint $K_{4}$ -minor cover and let $\\alpha $ be a marked node of the extended SP-decomposition $(\\mathcal {X},T)$ of $G[F]$ .", "Then $|V_{\\alpha }|\\leqslant c_1:=12(\\gamma (8)+2c_0)$ .", "Lemma 6 ($\\star $ ) Let $(G,S,k)$ be a simplified instance of disjoint $K_{4}$ -minor cover and let $\\alpha $ be a lowest unmarked node of $(T,\\mathcal {X})$ of $G[F]$ .", "In polynomial time, one can find a path $X$ of size at most $2c_1$ satisfying the conditions of line  (resp.", "line ) if the test at line  (resp. )", "succeeds; a subset $X\\subseteq V_{\\alpha }$ of size bounded by $2c_1$ satisfying the condition of line  if the test at line  succeeds; For running time analysis of our algorithm, we introduce the following measure $\\mu := (2c_1+2)k +(2c_1+2)\\# cc(G[S]) + \\#bc(G[S])$ where $\\#cc(G[S])$ and $\\#bc(G[S])$ respectively denote the number of connected and biconnected components of $G[S]$ .", "Reminder of Theorem REF The $K_4$ -minor cover problem can be solved in $2^{O(k)}\\cdot n^{O(1)}$ time.", "Due to Lemma REF , it is sufficient to show that one can solve disjoint $K_{4}$ -minor cover in time $2^{O(k)}\\cdot n^{O(1)}$ .", "The recursive application of operations (a)-(e) at the beginning of the section to a given instance $(G,S,k)$ produces a search tree $\\Upsilon $ .", "It is not difficult to see that $(G,S,k)$ is a YES-instance if and only if at least one of the leaf nodes in $\\Upsilon $ corresponds to a YES-instance.", "This follows from the fact that reduction and branching rules are safe.", "Let us see the running time to apply the operations (a)-(e) at each node of $\\Upsilon $ .", "Every instance corresponding to a leaf node either is a trivial instance or is an independent instance (see Theorem REF ) which can be solved in $2^k\\cdot n^{O(1)}$ using branching rule REF on pairs of vertices of $F$ (see Theorem REF ).", "Clearly, the operations (a)–(c) can be applied in polynomial time.", "Consider the operation (d).", "The while-loop in the algorithm Branch-or-reduce iterates $O(n)$ times.", "At each iteration, we are in one of the three situations: we detect in polynomial time (Lemma REF ) a connected subset $X$ on which to apply one of Branching rules, or apply the protrusion rule in polynomial time (Reduction rule REF ), or none of these two cases occur and the node under consideration is marked.", "Observe that the branching degree of the search tree is at most $2c_1+1$ by Lemma REF .", "To bound the size of $\\Upsilon $ , we need the following claim.", "Claim 1 In any application of Branching rules REF –REF , the measure $\\mu $ strictly decreases.", "Proof of claim.", "The statement holds for Branching rule REF since $k$ reduces by one and $G[S]$ is unchanged.", "Recall that Branching rules REF and REF put a vertex in the potential solution or add a path $X\\subseteq F$ to $S$ .", "In the first case, $\\mu $ strictly decreases because $k$ decreases and $\\#cc(G[S])$ and $\\#bc(G[S])$ remain unchanged.", "Let us see that $\\mu $ strictly decreases also when we add a path $X$ to $S$ .", "If Branching rule REF is applied, the number of biconnected components may increase by at most $2c_1+1$ .", "This happens if every edge on the path $X$ together with the two edges connecting the two end vertices of $X$ to $S$ add to the biconnected components of $G[S\\cup X]$ .", "Hence we have that the new value of $\\mu $ is $\\mu ^{\\prime }= (2c_1+2)k+(2c_1+2)\\#cc(G[S\\cup X])+\\#bc(G[S\\cup X])\\leqslant (2c_1+2)k+(2c_1+2)(\\#cc(G[S])-1)+(\\#bc(G[S])+2c_1+1)\\leqslant \\mu -1$ .", "It remains to observe that an application Branching rule REF strictly decreases the number of biconnected components while does not increase the number of connected components.", "Thereby $\\mu ^{\\prime }\\leqslant \\mu -1$ .", "$\\Diamond $ By Claim REF , at every root-leaf computation path in $\\Upsilon $ we have at most $\\mu = (2c_1+2)k +(2c_1+2)\\#cc(G[S]) + \\#bc(G[S]) \\le (4c_1+5)k$ nodes at which a branching rule is applied.", "Since we branch into at most $(2c_1+1)$ ways, the number of leaves is bounded by $(2c_1+1)^{(4c_1+5)k}$ .", "Also note that any root-leaf computation path contains $O(n)$ nodes at which a reduction rule is applied since any reduction rule strictly decreases the size of the instance and does not affect $G[S]$ .", "It follows that the running time is bounded by $((4c_1+5)k+ O(n))\\cdot (2c_1+1)^{(4c_1+3)k} \\cdot poly(n)=2^{O(k)}\\cdot n^{O(1)}$ ." ], [ "Conclusion and open problems", " Due to the use of the generic protrusion rule (on $t$ -protrusion for $t=3$ or 4), the result in this paper is existential.", "A tedious case by case analysis would eventually leads to an explicit $c^k\\cdot n^{O(1)}$ exponential FPT algorithm for some constant value $c$ .", "It is an intriguing challenge to reduce the basis to a small $c$ and/or get a simple proof of such an explicit algorithm.", "More generally, it would be interesting to investigate the systematic instantiation of protrusion rules.", "We strongly believe that our method will apply to similar problems.", "The first concrete example is the parameterized Outerplanar Vertex Deletion, or equivalently the $\\lbrace K_{2,3},K_4\\rbrace $ -minor cover problem.", "For that problem, we need to adapt the reduction and branching rules in order to preserve (respectively, eliminate) the existence of a $K_{2,3}$ as well.", "For example, the by-passing rule (Reduction rule REF ) may destroy a $K_{2,3}$ unless we only bypass a degree-two vertices when it is adjacent to another degree-two vertex.", "Similarly in Reduction Rule REF , we cannot afford to replace the set $X$ by an edge.", "It would be safe with respect to $\\lbrace K_{2,3},K_4\\rbrace $ -minor if instead $X$ is replaced by a length-two path or by two parallel paths of length two (depending on the structure of $X$ ).", "So we conjecture that for Outerplanar Vertex Deletion our reduction and branching rules can be adapted to design a single exponential FPT algorithm.", "A more challenging problem would be to get a single exponential FPT algorithm for the treewidth-$t$ vertex deletion for any value of $t$ .", "Up to now and to the best of our knowledge, the fastest algorithm runs in $2^{O(k\\log k)}\\cdot n^{O(1)}$  [18].", "Acknowledgements.", "We would like to thank Saket Saurabh for his insightful comments on an early draft and Stefan Szeider for pointing out the application of our problem in Bayesian Networks." ], [ "Minors and tree-width", "Observation 1 A $K_4$ -subdivision is biconnected; equivalently, it is connected and does not contain a cut vertex.", "Since there are three distinct paths between any two branching nodes in a $K_4$ -subdivision, we need at least three vertices in order to separate any two of them.", "Hence we have: Observation 2 Let $\\lbrace s,t\\rbrace $ be a separator of graph $G$ , and let $H$ be a $K_4$ -subdivision in $G$ .", "Then there exists a connected component $X_0$ of $G-\\lbrace s,t\\rbrace $ such that all four branching nodes of $H$ belong to $X_0\\cup \\lbrace s,t\\rbrace $ .", "A tree decomposition of $G$ is a pair $(T,\\mathcal {X})$ , where $T$ is a tree whose vertices we will call nodes and $\\mathcal {X}=\\lbrace X_{i}:i\\in V(T)\\rbrace $ is a collection of subsets of $V(G)$ (called bags) with the following properties: $\\bigcup _{i \\in V(T)} X_{i} = V(G)$ , for each edge $(v,w) \\in E(G)$ , there is an $i\\in V(T)$ such that $v,w\\in X_{i}$ , and for each $v\\in V(G)$ the set of nodes $\\lbrace i :\\ v \\in X_{i}\\rbrace $ form a subtree of $T$ .", "The width of a tree decomposition $(T,\\lbrace X_{i}:\\ i \\in V(T)\\rbrace )$ equals $\\max _{i \\in V(T)} \\lbrace |X_{i}| - 1\\rbrace $ .", "The treewidth of a graph $G$ is the minimum width over all tree decompositions of $G$ .", "We use the notation $tw(G)$ to denote the treewidth of a graph $G$ ." ], [ "Block, canonical SP-tree and extended SP-decomposition", "Without loss of generality, we may assume [3] that an SP-tree satisfies the following conditions: (1) an S-node does not have another S-node as a child; each child of an S-node is either a P-node or a leaf; and (2) a P-node has exactly two children — see Figure REF .", "Figure: A canonical SP-tree.", "P-nodes are coloured grey andS-nodes are coloured white.", "Observe that as P-nodes are binaryand may have a P-node as a child, while S-nodes do not haveany S-node as a child, conditions (1) and (2) are satisfied.By Lemma REF , we may further assume that for a biconnected series-parallel graph $G$ and any fixed vertex $s\\in V(G)$ , (3) $G$ has an SP-tree whose root is a P-node with $s$ as one of its two terminals.", "We say that an SP-tree is canonical if it satisfies the conditions (1) and (2), and also (3) when $G$ is biconnected.", "Lemma 7 [14] Let $G$ be a series-parallel graph, and let $s,t$ be two vertices in $G$ .", "Then $G$ is an SP-graph with terminals $s$ and $t$ if and only if $G+(s,t)$ is an SP-graph.", "Moreover, if $G$ is biconnected, then the last operation is a parallel join.", "The following is a well-known characterization relating forbidden minors, treewidth, and series-parallel graphs [2], [3].", "Lemma 8 Given a graph $G$ , the followings are equivalent.", "$G$ does not contain $K_4$ as a minor (That is, $G$ is $K_4$ -minor-free.).", "The treewidth of $G$ is at most two.", "Every block of $G$ is a series-parallel graph.", "It is well-known that one can decide whether $tw(G)\\leqslant 2$ in linear time [28].", "It is not difficult to see that in linear time we can also construct an extended SP-decomposition of $G$ .", "Though the next lemma is straightforward, we sketch the proof for completeness.", "Lemma 9 Given a graph $G$ , one can decide whether $tw(G)\\leqslant 2$ (or equivalently, whether $G$ is $K_4$ -minor-free) in linear time.", "Further, we can construct an extended SP-decomposition of $G$ in linear time if $tw(G)\\leqslant 2$ .", "The classical algorithm due to Hopcroft and Tarjan [21] identifies the blocks and cut vertices of $G$ in linear time.", "Due to Lemma REF , testing $tw(G)\\le 2$ reduces to testing whether each block of $G$ is a series-parallel graph.", "It is known [28] that the recognition of a series-parallel graph and the construction of an SP-decomposition can be done in linear time.", "Further, an SP-decomposition can be transformed into a canonical SP-decomposition in linear time.", "Given an oriented block tree $\\vec{\\mathcal {B}}_G$ and a canonical SP-decomposition for every block, we can construct the extended SP-decomposition in linear time, and the statement follows.", "Figure: A K 4 K_4-minor-free graph GG and its block tree ℬ G \\mathcal {B}_G." ], [ "Proof of Generic disjoint protrusion rule", "Definition 1 ($t$ -Boundaried Graphs) A $t$ -boundaried graph is a graph $G=(V,E)$ with $t$ distinguished vertices, uniquely labeled from 1 to $t$ .", "The set $\\partial (G)\\subseteq V$ of labeled vertices is called the boundary of $G$ .", "The vertices in $\\partial (G)$ are referred to as boundary vertices or terminals.", "Definition 2 (Gluing by $\\oplus $ ) Let $G_1$ and $G_2$ be two $t$ -boundaried graphs.", "We denote by $G_1\\oplus G_2$ the $t$ -boundaried graph such that: its vertex set is obtained by taking the disjoint union of $V(G_1)$ and $V(G_2)$ , and identifying each vertex of $\\partial (G_1)$ with the vertex of $\\partial (G_2)$ having the same label; and its edge set is the union of $E(G_1)$ and $E(G_2)$ .", "(That is, we glue $G_1$ and $G_2$ together on their boundaries.)", "Many graph optimization problems can be rephrased as a task of finding an optimal number of vertices or edges satisfying a property expressible in Monadic Second Order logic (MSO).", "A parameterized graph problem $\\Pi \\subseteq \\Sigma ^* \\times \\mathbb {N}$ is given with a graph $G$ and an integer $k$ as an input.", "When the goal is to decide whether there exists a subset $W$ of at most $k$ vertices for which an MSO-expressible property $P_{\\Pi }(G,W)$ holds, we say that $\\Pi $ is a $p$ -min-MSO graph problem.", "When $P_{\\Pi }(G,\\emptyset )$ holds, we write that $P_{\\Pi }(G)$ holds (or that $G$ satisfies $P_{\\Pi }$ ).", "In the (parameterized) disjoint version $\\Pi ^d$ of a $p$ -min-MSO problem $\\Pi $ , we are given a triple $(G,S,k)$ , where $G$ is a graph, $S$ a subset of $V(G)$ and $k$ the parameter, and we seek for a solution set $W$ which is disjoint from $S$ , and whose size is at most $k$ .", "The fact that a set $W$ is such a solution is expressed by the MSO-property $P_{\\Pi ^d}(G,S,W):P_{\\Pi }(G,W) \\wedge (S\\cap W = \\emptyset )$ .", "Definition 3 For a disjoint parameterized problem $\\Pi ^d$ and two $t$ -boundaried graphs We use this notation since later in this section, $G_p$ plays the role of a (large) protrusion and $G_r$ , its replacement.", "$G_p$ and $G_r$ , we say that $G_p\\equiv _{\\Pi ^d} G_r$ if there exists a constant $c$ such that for all $t$ -boundaried graphs $G$ , for every vertex set $S\\subseteq V(G)\\setminus \\partial (G)$ , and for every integer $k$ , $(G_p\\oplus G, S,k)\\in \\Pi ^d$ if and only if $(G_r\\oplus G,S,k+c)\\in \\Pi ^d$ Definition 4 (Disjoint Finite integer index) For a disjoint parameterized graph problem $\\Pi ^d$ , we say that $\\Pi ^d$ has disjoint finite integer index if the following property is satisfied: for every $t$ , there exists a finite set $\\mathcal {R}$ of $t$ -boundaried graphs such that for every $t$ -boundaried graph $G_p$ there exists $G_r\\in \\mathcal {R}$ with $G_p\\equiv _{\\Pi ^d} G_r$ .", "Such a set $\\mathcal {R}$ is called a set of representatives for $(\\Pi ^d,t)$ .", "It is often convenient to pair up a $t$ -boundaried graph $G$ with a set $W\\subseteq V(G)$ of vertices.", "We define $\\mathcal {H}_t$ to be the set of pairs $(G,W)$ , where $G$ is a $t$ -boundaried graph and $W\\subseteq V(G)$ .", "For an $p$ -min-MSO problem $\\Pi $ and a $t$ -boundaried graph $G$ , we define the signature function $\\zeta _G:\\mathcal {H}_t\\rightarrow \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ as follows.", "$\\zeta _G((G^{\\prime },W^{\\prime }))=\\left\\lbrace \\begin{array}{ll}\\infty & \\text{ if } \\nexists W \\subseteq V(G) ~s.t.~ P_{\\Pi }(G\\oplus G^{\\prime },W\\cup W^{\\prime }) \\\\\\min _{W\\subseteq V(G)} \\lbrace |W| : P_{\\Pi }(G\\oplus G^{\\prime },W\\cup W^{\\prime })\\rbrace & \\text{ otherwise}\\end{array} \\right.$ To ease the notation, we write $\\zeta _G(G^{\\prime },W^{\\prime })$ to denote $\\zeta _G((G^{\\prime },W^{\\prime }))$ .", "Definition 5 (Strong monotonicity) A $p$ -min-MSO problem $\\Pi $ is said to be strongly monotone if there exists a function $f: \\mathbb {N} \\rightarrow \\mathbb {N}$ satisfying the following condition: for every $t$ -boundaried graph $G$ , there exists a set $W_G \\subseteq V(G)$ such that for every $(G^{\\prime },W^{\\prime })\\in \\mathcal {H}_t$ with finite value $\\zeta _G(G^{\\prime },W^{\\prime })$ , $P_{\\Pi }(G\\oplus G^{\\prime },W_G\\cup W^{\\prime })$ holds and $|W_G|\\leqslant \\zeta _G(G^{\\prime },W^{\\prime })+f(t)$ .", "Bodlaender et al.", "show [4] that if $\\mathcal {F}$ is a finite set of connected planar graphs, then $\\mathcal {F}$ -minor cover problem is strongly monotone.", "The following lemma is a corollary of this fact.", "We give the proof for completeness.", "Lemma 10 The $K_4$ -minor cover problem is strongly monotone.", "Let $G$ be a $t$ -boundaried graph and $\\partial (G)$ be its boundary.", "Let $W\\subseteq V(G)$ be a minimum size vertex subset such that $G[V\\setminus W]$ is $K_4$ -minor-free.", "Define $W_G=W\\cup \\partial (G)$ .", "Then for every pair $(G^{\\prime },W^{\\prime })\\in \\mathcal {H}_t$ such that $\\zeta _G(G^{\\prime },W^{\\prime })$ is finite, $W_G\\cup W^{\\prime }$ is a $K_4$ -minor cover of $G\\oplus G^{\\prime }$ and moreover by construction $|W_G|\\leqslant \\zeta _G(G^{\\prime },W^{\\prime })+t$ .", "Lemma 11 Let $\\Pi $ be a strongly monotone $p$ -min-MSO problem.", "Then its disjoint version $\\Pi ^d$ has disjoint finite integer index.", "We consider the following equivalence relation $\\sim _{\\Pi }$ on $\\mathcal {H}_t$ : $(G,W)\\sim _{\\Pi } (G^{\\prime },W^{\\prime })$ if and only if for every $(G_p,W_p)\\in \\mathcal {H}_t$ we have $P_{\\Pi } (G_p\\oplus G, W_p \\cup W) \\Leftrightarrow P_{\\Pi }(G_p\\oplus G^{\\prime }, W_p\\cup W^{\\prime })$ Since $P_{\\Pi }$ is an MSO-property, it has a finite state property of $t$ -boundaried graphs [8].", "[disable,inline]What is an extended MSO property, and why is this implication true?", "We should give a reference.", "Is [8] a valid reference for this?", "– PhilipThat is, there exists a finite set $\\mathcal {S}\\subseteq \\mathcal {H}_t$ with the property that for every pair $(G,W)\\in \\mathcal {H}_t$ , there exists a pair $(G^{\\prime },W^{\\prime })\\in \\mathcal {S}$ with $(G,W)\\sim _{\\Pi } (G^{\\prime },W^{\\prime })$ .", "Let $G_p$ be a $t$ -boundaried graph.", "By the definition of strong monotonicity, there exists $W_{G_p}\\subseteq V(G_p)$ such that for every $(G,W)\\in \\mathcal {H}_t$ with finite value $\\zeta _{G_p}(G,W)$ , $P_{\\Pi }(G_p\\oplus G,W_{G_p}\\cup W)$ holds, and $|W_{G_p}|\\leqslant \\zeta _{G_p}(G,W)+f(t)$ .", "Observe also that by definition of the function $\\zeta _{G_p}$ , $\\zeta _{G_p}(G,W)\\leqslant |W_{G_p}|$ .", "It follows that $|W_{G_p}|-f(t) \\leqslant \\zeta _{G_p}(G,W)\\leqslant |W_{G_p}|$ We define the equivalence relation $\\sim _{\\mathcal {R}}$ on $t$ -boundaried graphs as follows: $G_p\\sim _{\\mathcal {R}} G_r$ if and only if there exist sets $W_{G_p}\\subseteq V(G_{p})$ and $W_{G_r}\\subseteq V(G_{r}$ meeting the condition of strong monotonicity such that for every $(G,W)\\in \\mathcal {S}$ we have $ |W_{G_p}|-\\zeta _{G_p}(G,W)=|W_{G_r}|-\\zeta _{G_r}(G,W)$ .", "By (REF ) and the finiteness of $\\mathcal {S}$ , there exists a set $\\mathcal {R}$ of at most $(f(t)+2)^{|\\mathcal {S}|}$ $t$ -boundaried graphs such that for every $t$ -boundaried graph $G_p$ , there exists $G_r\\in \\mathcal {R}$ with $G_p\\sim _{\\mathcal {R}} G_r$ .", "Let $G_p$ and $G_{r}$ be $t$ -boundaried graphs such that $G_p\\sim _{\\mathcal {R}} G_r$ .", "As a consequence of (REF ), there is a constant $c_r=|W_{G_{p}}|-|W_{G_{r}}|$ (which depends only on $G_p$ and $G_r$ ) such that $\\zeta _{G_p}(G,W)=\\zeta _{G_r}(G,W)+c_r$ for every $(G,W)\\in \\mathcal {S}$ .", "The rest of the proof is devoted to the following claim: Claim 2 For two $t$ -boundaried graphs $G_p$ and $G_r$ , if $G_p\\sim _{\\mathcal {R}} G_r$ then $G_p\\equiv _{\\Pi ^d}G_r$ .", "Specifically, for every $t$ -boundaried graph $G$ and $S\\in V(G)\\setminus \\partial (G)$ , we have $(G_p\\oplus G, S,k)\\in \\Pi ^d \\mbox{ if and only if~} (G_r\\oplus G,S,k-c_r)\\in \\Pi ^d$ Proof of claim.", "We only prove the forward direction, the reverse follows with symmetric arguments.", "[disable,inline]This symmetry is not obvious to me, since the above relation, and the relation $\\equiv _{\\Pi ^d}$ are not symmetric.", "– Philip Suppose that $(G_p\\oplus G, S,k)\\in \\Pi ^d$ .", "Consider $Z\\subseteq V(G_p\\oplus G)$ such that $Z\\cap S=\\emptyset $ , $P_{\\Pi }(G_p\\oplus G,Z)$ is satisfied and $Z$ has the minimum size.", "We denote $W=Z\\cap V(G)$ and $W_p=Z\\setminus W$ .", "Observe that since $P_{\\Pi }(G_p\\oplus G,Z)$ holds, $P_{\\Pi }(G_p\\oplus G,W_p\\cup W)$ also holds.", "Let us consider $(G^{\\prime },W^{\\prime })\\in \\mathcal {S}$ such that $(G,W)\\sim _{\\Pi }(G^{\\prime },W^{\\prime })$ .", "We first prove that $|W_p|=\\zeta _{G_p}(G^{\\prime },W^{\\prime })$ .", "Since $P_{\\Pi }(G_p\\oplus G,W_p\\cup W)$ holds and $(G,W)\\sim _{\\Pi }(G^{\\prime },W^{\\prime })$ , we have that $P_{\\Pi }(G_p\\oplus G^{\\prime },W_p\\cup W^{\\prime })$ holds.", "Hence $|W_{p}|\\ge \\zeta _{G_p}(G^{\\prime },W^{\\prime })$ .", "For the sake of contradiction, assume that there exists $W^{\\prime }_p\\subseteq V(G_{p})$ such that $|W^{\\prime }_p|<|W_p|$ and $P_{\\Pi }(G_p\\oplus G^{\\prime },W^{\\prime }_p\\cup W^{\\prime })$ holds.", "Since $(G,W)\\sim _{\\Pi } (G^{\\prime },W^{\\prime })$ , $P_{\\Pi }(G_p\\oplus G,W^{\\prime }_p\\cup W)$ is satisfied.", "As $W\\cap W_p=\\emptyset $ , we have $|W^{\\prime }_p\\cup W|<|Z|$ ; this contradicts the choice of $Z$ .", "Since $G_p\\sim _R G_r$ and $(G^{\\prime },W^{\\prime })\\in \\mathcal {S}$ , there exists $W_r\\subseteq V(G_r)$ such that $P_{\\Pi }(G_r\\oplus G^{\\prime },W_r\\cup W^{\\prime })$ holds and $|W_r|=|W_p|-c_r$ .", "And finally, $(G,W)\\sim _{\\Pi }(G^{\\prime },W^{\\prime })$ implies that $P_{\\Pi }(G_r\\oplus G,W_r\\cup W)$ .", "To conclude the proof observe first that $S\\subseteq V(G)\\setminus \\partial (G)$ implies that $(W_r\\cup W)\\cap S=\\emptyset $ .", "Moreover we have $|W_r\\cup W|\\leqslant |W_r|+|W|=|W_p|-c_r+|W|= |Z|-c_r\\leqslant k-c_r$ It follows that $(G_r\\oplus G,S,k-c_r)\\in \\Pi ^d$ .", "$\\Diamond $ By Claim REF , we conclude that $\\mathcal {R}$ is a set of representatives for $(\\Pi ^d, t)$ and thus the disjoint version $\\Pi ^d$ of a strongly monotone $p$ -min-MSO problem $\\Pi $ has disjoint finite integer index.", "[disable,inline]BLABLABLA introducing the idea of protrusion reduction rules, add a figure of a protrusion Definition 6 A subset $X$ of the vertex set of a graph $G$ is a $t$ -protrusion of $G$ if $tw(G[X])\\leqslant t$ and $|\\partial (X)|\\leqslant t$ .", "Lemma 12 Let $\\Pi ^d$ be the disjoint version of a strongly monotone p-min-MSO problem $\\Pi $ .", "There exists a computable function $\\gamma :\\mathbb {N}\\rightarrow \\mathbb {N}$ and an algorithm that given: an instance $(G,S,k)$ of $\\Pi ^d$ such that $P_{\\Pi }(G,S)$ holds a $t$ -protrusion $X$ of $G$ such that $|X|>\\gamma (2t+1)$ and $X\\cap S=\\emptyset $ in time $O(|X|)$ outputs an instance $(G^{\\prime },S,k^{\\prime })$ such that $|V(G^{\\prime })|<|V(G)|$ , $k^{\\prime }\\le k$ , $(G^{\\prime },S,k^{\\prime })\\in \\Pi ^d$ if and only if $(G,S,k)\\in \\Pi ^d$ , and $P_{\\Pi }(G^{\\prime },S)$ holds.", "Let $\\sim _{\\mathcal {R}}$ be the equivalence relation on $(2t+1)$ -boundaried graphs defined in the proof of Lemma REF .We refine the equivalence relation $\\sim _{\\mathcal {R}}$ into $\\sim _{\\mathcal {R}^*}$ according to whether a $(2t+1)$ -boundaried graph satisfies $P_{\\Pi }$ .", "be precise, we have $G_p\\sim _{\\mathcal {R}^*} G_r$ if and only if (a) $G_p\\sim _{\\mathcal {R}} G_r$ and (b) for every $(2t+1)$ -boundaried graph $H$ : $P_{\\Pi }(G_p\\oplus H)$ if and only $P_{\\Pi }(G_r\\oplus H)$ We know that $\\sim _{\\mathcal {R}}$ has finite index.", "As $P_{\\Pi }$ is an MSO-expressible graph property, the equivalence relation (b) has finite index [8].", "Therefore $\\sim _{\\mathcal {R}^*}$ also defines finitely many equivalence classes.", "We select a set $\\mathcal {R}^*$ of representatives for $\\sim _{\\mathcal {R}^*}$ with one further restriction: Claim REF is satisfied for some nonnegative constant $c_r$ .", "Such a set of representatives $\\mathcal {R}^*$ can be constituted by picking up a representative $G_r$ for each equivalence class so that the constant $\\zeta _{G_p}(G,W)-\\zeta _{G_r}(G,W)$ , following the condition (a), is nonnegative for every $G_p\\sim _{\\mathcal {R}^*} G_r$ .", "Here $\\zeta $ is the signature function for $\\Pi $ .", "Define $\\gamma (2t+1)$ to be the size of the vertex set of the largest graph in $\\mathcal {R}^*$ .", "Let $\\phi $ and $\\rho $ be mappings from the set of $(2t+1)$ -boundaried graphs of size at most $2\\gamma (2t+1)$ to $\\mathcal {R}^*$ and $\\mathbb {N}$ respectively such that for every $(2t+1)$ -boundaried graph $G$ and $S\\subseteq V(G)\\setminus \\partial (G)$ , we have $(G_p\\oplus G,S,k)\\in \\Pi ^d$ if and only if $(\\phi (G_p)\\oplus G,S,k-\\rho (G_p))\\in \\Pi ^d$ .", "Such mappings exist: we take $\\phi (G_p):=G_r \\in \\mathcal {R}^*$ such that $G_p\\sim _{\\mathcal {R}^*} G_r$ , and $\\rho (G_p):=\\zeta _{G_p}(G,W)-\\zeta _{\\phi (G_p)}(G,W)$ which is a constant by the definition of $\\sim _{\\mathcal {R}}$ (and thus of $\\sim _{\\mathcal {R}^*}$ ) and nonnegative by the way we constitute $\\mathcal {R}^*$ as explained in the previous paragraph.", "Suppose that $|X|>\\gamma (2t+1)$ .", "We build a nice tree-decomposition of $G[X]$ of width $t$ in $O(|X|)$ time and identify a bag $b$ of the tree-decomposition farthest from its root such that the subgraph $G_b$ induced by the vertices appearing in bag $b$ or below contains at least $\\gamma (2t+1)$ and at most $2\\gamma (2t+1)$ vertices.", "The existence of such a bag is guaranteed by the properties of a nice tree decomposition.", "Note that for any $X^{\\prime }\\subset X$ , we have $X^{\\prime }\\cap S=\\emptyset $ .", "Let $X^{\\prime }=V(G_{v})$ , so that that $|X^{\\prime }|\\le 2\\gamma (2t+1)$ .", "We replace $G[X]$ by $\\phi (G[X^{\\prime }])$ [disable,inline]The function $\\phi ()$ is defined only for $2t$ -boundaried graphs.", "It is not clear why $G[X^{\\prime }]$ is a $2t$ -boundaried graph.", "In particular, nothing seems to rule out the possibility that $|\\partial (X^{\\prime })|=2t+1$ .", "– Philip We have fixed this by replacing 2t with 2t+1 everywhere.", "– Philip to obtain $G^{\\prime }$ , and decrease $k$ by $\\rho (X^{\\prime })$ .", "It follows that $(G,S,k)\\in \\Pi ^d$ if and only if $(G^{\\prime },S,k^{\\prime })\\in \\Pi ^d$ .", "Observe that $k^{\\prime }=k-\\rho (X^{\\prime })\\le k$ and $|V(G^{\\prime })|<|V(G)|$ as $|\\phi (G[X])|\\le \\gamma (2t+1)<|X|$ .", "Finally, observe that the condition (b) of $\\sim _{\\mathcal {R}^*}$ ensures that $G^{\\prime }-S$ is $K_4$ -minor-free.", "This completes the proof.", "As a corollary, since the $K_4$ -minor cover is strongly monotone, the following reduction rule for $\\textsc {disjoint $ K4$-minor cover}{}$ is safe.", "We state the rule for an arbitrary value of $t$ , but in practice, our reduction rule will only be based on $t$ -protrusions for $t\\leqslant 4$ .", "Reduction Rule 1 (Generic disjoint protrusion rule) Let $(G,S,k)$ be an instance of disjoint $K_{4}$ -minor cover and $X$ be a $t$ -protrusion such that $X\\cap S=\\emptyset $ .", "Then there exists a computable function $\\gamma (.", ")$ and an algorithm which computes an equivalent instance in time $O(|X|)$ such that $G[S]$ and $G^{\\prime }[S]$ are isomorphic, $G^{\\prime }-S$ is $K_4$ -minor-free, $|V(G^{\\prime })|<|V(G)|$ and $k^{\\prime }\\leqslant k$ , provided $|X|>\\gamma (2t+1)$ .", "We remark on Reduction rule REF that $|\\partial (X^{\\prime })|$ may be strictly smaller than $2t+1$ .", "In that case, we can identify some vertices of $X^{\\prime }\\setminus \\partial (X^{\\prime })$ as boundary vertices and construe $X^{\\prime }$ as $(2t+1)$ -boundaried graph.", "This is always possible for $|X^{\\prime }|>\\gamma (2t+1)\\ge 2t$ .", "[disable,inline]Again, we do not seem to consider the case when $|\\partial (X^{\\prime })|$ is strictly larger than $2t$ .", "The change from 2t to 2t+1 has fixed this problem also.", "– Philip" ], [ "Deferred proof of Lemma ", "Reminder of Lemma REF If disjoint $K_{4}$ -minor cover can be solved in $c^k\\cdot n^{O(1)}$ time, then $K_4$ -minor cover can be solved in $(c+1)^k\\cdot n^{O(1)}$ time.", "Let $\\mathcal {A}$ be an FPT algorithm which solves the disjoint $K_{4}$ -minor cover problem in $c^k\\cdot n^{O(1)}$ time.", "Let $(G,k)$ be the input graph for the $K_4$ -minor cover problem and let $v_1,\\ldots ,v_n$ be any enumeration of the vertices of $G$ .", "Let $V_i$ and $G_i$ respectively denote the subset $\\lbrace v_1\\dots v_i\\rbrace $ of vertices and the induced subgraph $G[V_i]$ .", "We iterate over $i=1,\\ldots ,n$ in the following manner.", "At the $i$ -th iteration, suppose we have a $K_4$ -minor cover $S_i \\subseteq V_i$ of $G_{i}$ of size at most $k$ .", "At the next iteration, we set $S_{i+1}:=S_i\\cup \\lbrace v_{i+1}\\rbrace $ (notice that $S_{i+1}$ is a $K_4$ -minor cover for $G_{i+1}$ of size at most $k+1$ ).", "If $|S_{i+1}|\\le k$ , we can safely move on to the $i+2$ -th iteration.", "If $|S_{i+1}|=k+1$ , we look at every subset $S \\subseteq S_{i+1}$ and check whether there is a $K_4$ -minor cover $W$ of size at most $k$ such that $W\\cap S_{i+1}=S_{i+1}\\setminus S$ .", "To do this, we use the FPT algorithm $\\mathcal {A}$ for $\\textsc {disjoint $ K4$-minor cover}{}$ on the instance $(H,S)$ with $H=G_{i+1}-(S_{i+1}\\setminus S)$ .", "If $\\mathcal {A}$ returns a $K_4$ -minor cover $W$ of $H$ with $|W|< |S|$ , then observe that the vertex set $(S_{i+1}\\setminus S) \\cup W$ is a $K_4$ -minor cover of $G$ whose size is strictly smaller than $S_{i+1}$ .", "If there is a $K_4$ -minor cover of $G_{i+1}$ of size strictly smaller than $S_{i+1}$ , then for some $S\\subseteq S_{i+1}$ , there is a small $S$ -disjoint $K_4$ -minor cover in $G_{i+1}-(S_{i+1}\\setminus S)$ and $\\mathcal {A}$ correctly returns a solution.", "The time required to execute $\\mathcal {A}$ for every subset $S$ at the $i$ -th iteration is $\\sum _{i=0}^{k+1} \\binom{k+1}{i} \\cdot c^i \\cdot n^{O(1)}=(c+1)^{k+1}\\cdot n^{O(1)}$ .", "Thus we have an algorithm for $K_4$ -minor cover which runs in time $(c+1)^k \\cdot n^{O(1)}$ ." ], [ "Deferred proofs for (explicit) reduction rules", "Lemma 13   Reduction rules REF , REF and REF are safe and can be applied in polynomial time.", "It is not difficult to see that each of these rules can be applied in polynomial time.", "We now prove that each of them is safe.", "Reduction rule REF .", "Let $W$ be a small $S$ -disjoint $K_{4}$ -minor cover of $G$ .", "Observe that $G^{\\prime }-(W\\setminus X)$ is a subgraph of $G-W$ .", "It follows that $(W\\setminus X)$ is a small $S$ -disjoint $K_{4}$ -minor cover of $G- X$ .", "By the same reasoning, $(W\\setminus (X\\setminus \\partial (X)))$ is a small $S$ -disjoint $K_{4}$ -minor cover of $G-(X\\setminus \\partial (X))$ .", "For the opposite direction, let $W^{\\prime }$ be a small $S$ -disjoint $K_{4}$ -minor cover of $G^{\\prime }:=(G-X)$ .", "Then $G^{\\prime }-W^{\\prime }$ is $K_4$ -minor-free.", "Since $G- W^{\\prime }$ is a disjoint union of $G^{\\prime }\\setminus W^{\\prime }$ and $G[X]$ and any $K_4$ -subdivision is biconnected, $G-W^{\\prime }$ is $K_4$ -minor-free as well.", "Thus $W^{\\prime }$ is a small $S$ -disjoint $K_{4}$ -minor cover of $G$ .", "The same argument goes through when $G^{\\prime }=(G\\setminus (X\\setminus \\partial (X)))$ , as well.", "Reduction rule REF .", "Let $W$ be a small $S$ -disjoint $K_{4}$ -minor cover of $G$ .", "Without loss of generality, assume that the vertex $v$ is not in $W$ .", "Indeed, any $K_4$ -subdivision containing $v$ also contains $u_2$ and thus, we can take $(W\\setminus \\lbrace v\\rbrace )\\cup \\lbrace u_2\\rbrace $ to hit such a $K_4$ -subdivision.", "Let $G^{\\prime }$ be the graph obtained from $G$ by applying the rule.", "Observe that $G_{2}=G^{\\prime }\\setminus W$ is a minor of $G_{1}=G\\setminus W$ , that is: If $W\\cap \\lbrace u_{1},u_{2}\\rbrace =\\emptyset $ , then $G_{2}$ can be obtained from $G_{1}$ by contracting the edge $(v,u_{1})$ .", "If $W\\cap \\lbrace u_{1},u_{2}\\rbrace \\ne \\emptyset $ , then $G_{2}$ can be obtained from $G_{1}$ by deleting $v$ .", "It follows that $W$ is a small $S$ -disjoint $K_{4}$ -minor cover of $G^{\\prime }$ as well.", "For the opposite direction, let $W^{\\prime }$ be a small $S$ -disjoint $K_{4}$ -minor cover of $G^{\\prime }$ .", "Observe that $G_{1}^{\\prime }=G\\setminus W^{\\prime }$ can be obtained from the $K_{4}$ -minor-free graph $G_{2}^{\\prime }=G^{\\prime }\\setminus W^{\\prime }$ in the following ways: If $W^{\\prime }\\cap \\lbrace u_{1},u_{2}\\rbrace =\\lbrace u_{1},u_{2}\\rbrace $ , then $G_{1}^{\\prime }$ can be obtained from $G_{2}^{\\prime }$ by adding an isolated vertex $v$ .", "If $W^{\\prime }\\cap \\lbrace u_{1},u_{2}\\rbrace =\\lbrace u_{2}\\rbrace $ , then $G_{1}^{\\prime }$ can be obtained from $G_{2}^{\\prime }$ by attaching a vertex $v$ to $u_{1}$ .", "If $W^{\\prime }\\cap \\lbrace u_{1},u_{2}\\rbrace =\\emptyset $ , then $G_{1}^{\\prime }$ can be obtained from $G_{2}^{\\prime }$ by subdividing the edge $(u_{1},u_{2})$ .", "In the first two cases, note that any $K_4$ -subdivision is biconnected and thus $v$ is never contained in a $K_4$ -subdivision.", "By the assumption that $G^{\\prime }_2$ is $K_4$ -minor-free, $G^{\\prime }_1$ is also $K_4$ -minor-free.", "In the third case, $G^{\\prime }_1$ is also $K_4$ -minor-free since subdividing an edge in a $K_{4}$ -minor-free graph does not introduce a $K_{4}$ minor.", "It follows that $W^{\\prime }$ is a small $S$ -disjoint $K_{4}$ -minor cover of $G$ as well.", "Reduction rule REF .", "In the forward direction, observe that the graph obtained by applying the rule is a subgraph of the original graph.", "In the reverse direction, observe that increasing the multiplicity (number of parallel edges) of any edge in a $K_{4}$ -minor-free graph does not introduce a $K_{4}$ -minor in the graph.", "Figure: Contraction of the edge e=u 2 u 3 e=u_2u_3 into u e u_e (the grey vertex) when Reduction rule  applies.Lemma 14 Reduction Rule REF is safe and can be applied in polynomial time.", "Let $u_e$ be the vertex obtained by contracting $e$ , and let $W$ be a small disjoint $K_4$ -minor cover of $G$ .", "If $W\\cap \\lbrace u_{2},u_{3}\\rbrace =\\emptyset $ , then let $W^{\\prime }\\leftarrow W$ ; otherwise let $W^{\\prime }\\leftarrow (W\\setminus \\lbrace u_2,u_3\\rbrace )\\cup \\lbrace u_e\\rbrace $ .", "In either case $\\vert W^{\\prime }\\vert \\leqslant \\vert W\\vert \\leqslant k$ , and $(G/e)\\setminus W^{\\prime }$ is a minor of $G\\setminus W$ .", "Since $G\\setminus W$ is $K_4$ -minor-free, so is $(G/e)\\setminus W^{\\prime }$ , and so $W^{\\prime }$ is a small disjoint $K_4$ -minor cover of $G/e$ .", "Conversely, let $W^{\\prime }$ be a small disjoint $K_4$ -minor cover of $G/e$ .", "We first consider the case $u_e\\in W^{\\prime }$ .", "Then let $W\\leftarrow (W^{\\prime }\\setminus \\lbrace u_e\\rbrace )\\cup \\lbrace u_2\\rbrace $ .", "We claim that $W$ is a small disjoint $K_{4}$ -minor cover of $G$ .", "It is not difficult to see that $W$ is both small and $S$ -disjoint; we now show that it is a $K_{4}$ -minor cover of $G$ .", "Assume to the contrary that $G- W$ contains a $K_{4}$ -subdivision $H$ .", "Observe that $G-(W\\cup \\lbrace u_{3}\\rbrace )$ is isomorphic to $(G/e)- W^{\\prime }$ which is $K_4$ -minor-free, and so $u_3\\in V(H)$ .", "Now $u_3$ is a degree 2 vertex in $G-W$ and so is a subdividing node of $H$ , implying that $u_4$ and $x$ (the neighbors of $u_3$ ) belongs to $V(H)$ .", "As $x$ and $u_4$ are adjacent, $G-W$ contains a $K_4$ -subdivision $H^{\\prime }$ with $V(H^{\\prime })=V(H)\\setminus \\lbrace u_3\\rbrace $ .", "Thus $G-(W\\cup \\lbrace u_{3}\\rbrace )$ contains a $K_{4}$ -subdivision, a contradiction.", "Suppose now that $u_e\\notin W^{\\prime }$ .", "We claim that $W^{\\prime }$ is a $K_4$ -minor cover of $G$ as well.", "Assume to the contrary that $H$ is a $K_4$ -subdivision in $G- W^{\\prime }$ .", "We claim that every $K_4$ -subdivision $H$ in $G-W^{\\prime }$ contains $u_2$ and $u_3$ as branching nodes.", "Assume that $u_2\\notin V(H)$ .", "Then since $G-(W^{\\prime }\\cup \\lbrace u_2\\rbrace )$ is a (non-induced) subgraph of $G/e-W^{\\prime }$ , $H$ is also a $K_4$ -subdivision in $G/e-W^{\\prime }$ : a contradiction.", "So every $K_4$ -subdivision in $G-W^{\\prime }$ contains $u_2$ .", "By a symmetric argument, $u_{3}\\in V(H)$ as well.", "[disable,color=lightgray,inline]I have replaced the earlier argument by an appeal to symmetry.", "I cannot now see why this is wrong, but since we had a different, more complicated argument before, it is possible that this new argument is wrong.", "This is the previous argument : “If $u_3\\notin V(H)$ , then $u_2$ has to be a subdividing node of $H$ with neighbors $x$ and $x_1$ .", "Observe that since $u_1$ and $x$ are adjacent, $G-W^{\\prime }$ also contains a $K_4$ -subdivision $H^{\\prime }$ avoiding $u_2$ : contradiction, so $u_2,u_3\\in V(H)$ .” — Philip [disable,inline]Christophe says: “I had some doubt about validity of the use of symmetry (Eunjung also used symmetry).", "My concern was the folllowing: we need to argue that both $u_2$ and $u_3$ belong to $V(H)$ .", "I feel (but maybe wrong) that the symmetry only guarantee that $u_2$ or $u_3$ belong to $V(H)$ .", "So I argued that assuming $u_2$ belong to $V(H)$ , $u_3$ also has to be in.", "What do you think ?” Since our argument which says $u_{2}\\in V(H)$ holds irrespective of whether $u_{3}\\in V(H)$ , I feel the symmetry argument is valid.", "We can always put the other one back if we feel otherwise.", "— Philip Now a simple case by case analysis (see Figure REF ) shows that if $u_2$ or $u_3$ is a subdividing node, then $G/e-W^{\\prime }$ also contains a $K_4$ -subdivision $H^{\\prime }$ with $V(H^{\\prime })=(V(H)\\setminus \\lbrace u_2,u_3\\rbrace )\\cup \\lbrace u_e\\rbrace $ : a contradiction.", "Figure: The different possible intersections of HH withG[{u 1 ,u 2 ,u 3 ,u 4 ,x}]G[\\lbrace u_1,u_2,u_3,u_4,x\\rbrace ].", "The bold lines denote those edges inHH which are incident on u 2 u_{2} or u 3 u_{3}.", "In cases(1), (2) and (3) we can argue that there exists aK 4 K_4-subdivision in G-W ' G-W^{\\prime } avoiding either u 2 u_2 or u 3 u_3: acontradiction.", "In cases (4), (5) and (6), we observe theexistence of a K 4 K_4-subdivision in G/eG/e: a contradiction.", "[disable,color=lightgray,inline]Do you think we need to detail the arguments for the above mentioned case by case analysis ?", "I think these pictures are good enough, and that no more detail is required.", "— Philip It follows that $u_{2},u_{3}$ are both present as branching nodes in $H$ (see case (7) in Figure REF ).", "As these vertices both have degree 3 in $G$ , every edge incident to $u_2$ or $u_3$ is used in $H$ .", "Therefore the common neighbor $x$ of $u_2$ and $u_3$ also appears in $H$ as a branching node.", "So at most one vertex in $\\lbrace u_{1},u_{4}\\rbrace $ is a branching node; assume without loss of generality that $u_{4}$ is a subdividing node.", "It lies on the path between $u_3$ and a branching node $y\\notin \\lbrace u_2,u_3,x\\rbrace $ , and we can make $u_{4}$ a branching node instead of $u_{3}$ to obtain a new $K_{4}$ -subdivision $H^{\\prime }$ by replacing in $H$ the edge $(x,u_3)$ by the edge $(x,u_4)$ .", "But then $H^{\\prime }$ is a $K_4$ -subdivision in $G\\setminus W^{\\prime }$ which does not contain $u_{3}$ as a branching node, a contradiction.", "It follows that $W^{\\prime }$ is a small disjoint $K_4$ -minor cover of $G$ .", "It is not difficult to see that the rule can be applied in polynomial time.", "Lemma 15 Let $(G,S,k)$ be an instance reduced with respect to Reduction Rules REF , REF and REF .", "Then Reduction Rule REF is safe and can be applied in polynomial time.", "Since $(G,S,k)$ is reduced with respect to Rule REF , $G[F]$ does not contain any cut vertex.", "Let $(G^{\\prime },S,k)$ be the instance obtained by applying Reduction Rule REF to $(G,S,k)$ .", "Let $X^{\\prime }$ be the set of vertices with which the rule replaced $X$ and let $X_0:=X\\setminus \\lbrace s,t\\rbrace $ , $X_0^{\\prime }:=X^{\\prime }\\setminus \\lbrace s,t\\rbrace $ .", "We can assume that $X_0\\ne \\emptyset $ since otherwise the reduction rule is useless.", "To prove that $(G,S,k)$ has a small disjoint $K_4$ -minor cover of $G$ if and only if $(G^{\\prime },S,k)$ does, we need the following claim.", "Claim 1 $G[X]+(s,t)$ is an SP-graph if and only if $G[X]+(s,t)$ is $K_4$ -minor-free.", "Proof of claim.", "The forward direction follows directly from Lemma REF .", "Assume now that $G[X]+(s,t)$ is $K_4$ -minor-free.", "As $(G,S,k)$ is reduced with respect to Reduction Rule REF , the block tree of $G[X]$ is a path and moreover $s$ and $t$ belong to the two leaf blocks, respectively (these blocks may also coincide).", "This implies that the addition of the edge $(s,t)$ to $G[X]$ yields a biconnected graph.", "This concludes the proof since by Lemma REF a biconnected $K_4$ -minor free graph is an SP-graph.", "$\\Diamond $ [disable,color=lightgray,inline]What happens in the forward direction is the following: In both the cases, the new graph $G^{\\prime }$ is a minor of the graph $G$ , and so any $K_{4}$ -minor cover of $G$ is also a $K_{4}$ -minor cover of $G^{\\prime }$ .", "The fact that $G^{\\prime }$ is a minor of $G$ is easy to show in the first case, but for the second case I cannot clinch the argument.", "What we need in the second case is the following lemma: “Let $G$ be a $K_{4}$ -minor-free graph, and let $s,t$ be two vertices in $G$ such that adding the edge $(s,t)$ to $G$ yields a graph which contains a $K_{4}$ -minor.", "Then $G$ contains the bottom right graph as a minor, where $s$ and $t$ are mapped to the two vertices of degree three.” — Philip We now resume the proof of the lemma.", "Let $W$ be a small disjoint $K_4$ -minor cover of $G$ .", "If $W\\cap X_0 \\ne \\emptyset $ , set $W^*:=(W\\setminus X_0)\\cup \\lbrace t\\rbrace $ .", "Since $\\lbrace s\\rbrace $ is a cut vertex in $G-W^*$ isolating $X_0$ , no $K_4$ -subdivision in $G-W^*$ uses any vertex from $X_0$ .", "Also $|W^*|\\le k$ , and so $W^{*}\\subseteq V(F)\\setminus X_{0}$ is a small disjoint $K_4$ -minor cover of $G$ .", "So we can assume without loss of generality that $W\\cap X_0= \\emptyset $ .", "Let us prove that $W$ is a $K_4$ -minor cover of $G^{\\prime }$ .", "For the sake of contradiction, let $H^{\\prime }$ be a $K_4$ -subdivision in $G^{\\prime }-W$ .", "There are two cases to consider: Reduction Rule REF replaces $G[X]$ by the edge $(s,t)$ : Observe that all the branching nodes of $H^{\\prime }$ belong to $V(G)\\setminus (W\\cup X_0)$ .", "Suppose $H^{\\prime }$ uses the edge $(s,t)$ for a path between two branching nodes, say $u$ and $v$ .", "As $W\\cap X_0= \\emptyset $ , using an arbitrary $s,t$ -path $P$ in $G[X]$ instead of the edge $(s,t)$ witnesses the existence of a $u,v$ -path $G-W$ .", "This implies that $G-W$ contains a $K_4$ -subdivision $H$ such that $V(H)=V(H^{\\prime })\\cup V(P)$ , a contradiction.", "Reduction Rule REF replaces $G[X]$ by a $\\theta _3$ on vertex set $X^{\\prime }=\\lbrace a,b,s,t\\rbrace $ : this occurs when $G[X]+(s,t)$ is not an SP-graph and so by Claim REF contains a $K_4$ -subdivision.", "By Observation REF , the branching nodes of $V(H^{\\prime })$ belong either to $X^{\\prime }$ or to $V(G)\\setminus \\lbrace a,b\\rbrace $ .", "In the latter case, vertex $a$ or $b$ may be used by $H^{\\prime }$ as a subdividing node to create a path through $s$ and $t$ between two branching nodes of $H^{\\prime }$ .", "The same argument as above then yields a contradiction.", "In the former case, observe that every vertex of $X^{\\prime }$ is a branching node of $H^{\\prime }$ and some vertices out of $X^{\\prime }$ may be used by $H^{\\prime }$ as subdividing nodes to create the missing path $P$ between $s$ and $t$ in $G^{\\prime }-W$ .", "As $G[X]+(s,t)$ also contains a $K_4$ -subdivision, say $H$ , we can construct a $K_4$ -subdivision in $G-W$ on vertex set $V(H)\\cup V(P)$ , a contradiction.", "For the reverse direction, let $W^{\\prime }$ be a small disjoint $K_4$ -minor cover of $G^{\\prime }$ .", "Again we can assume that $W^{\\prime }\\cap X_0^{\\prime }=\\emptyset $ .", "Indeed, if $W^{\\prime }\\cap X_0^{\\prime }\\ne \\emptyset $ , it is easy to see that $(W^{\\prime }\\setminus X_0^{\\prime })\\cup \\lbrace t\\rbrace $ is also a small disjoint $K_4$ -minor cover of $G^{\\prime }$ .", "Let us prove that $W^{\\prime }$ is also a $K_4$ -minor cover of $G$ (the arguments are basically the same as above).", "For the sake of contradiction, assume $H$ is a $K_4$ -subdivision of $G-W^{\\prime }$ .", "By Observation REF , since $\\lbrace s,t\\rbrace $ is a separator of size two, the branching nodes of $V(H)$ belong either to $X$ or to $V(G)\\setminus X_0$ .", "In the former case, $G[X]+(s,t)$ is not an SP-graph, and thus $X$ as been replaced by a $\\theta _3$ on $\\lbrace a,b,s,t\\rbrace $ .", "Let $P$ be the $s,t$ -path of $G-(X_0\\cup W^{\\prime })$ used by $H$ .", "As $W^{\\prime }\\cap X_0^{\\prime }=\\emptyset $ , $\\lbrace a,b,s,t\\rbrace \\cup V(P)$ induces a $K_4$ -subdivision in $G^{\\prime }-W^{\\prime }$ , a contradiction.", "In the latter case, if $H$ uses a path between $s$ and $t$ in $G[X]-W^{\\prime }$ , then such a path also exists in $G^{\\prime }-W^{\\prime }$ witnessing a $K_4$ -subdivision in $G^{\\prime }-W^{\\prime }$ , a contradiction." ], [ "Deferred proofs of Lemmas ", "Reminder of Lemma REF Let $W$ and $Z$ be disjoint vertex subsets of a graph $G$ such that $G[W]$ is biconnected, $G[Z]$ is connected and $|N_W(Z)|\\ge 3$ .", "Then $G[W\\cup Z]$ contains a $K_4$ -subdivision.", "Let $x$ , $y$ and $z$ be three vertices of $N_W(Z)$ .", "Since $G[Z]$ is connected and since contracting edges does not introduce a new $K_{4}$ -subdivision, we may assume without loss of generality that there is a single vertex, say $u$ , in $Z$ such that $\\lbrace x,y,z\\rbrace \\subseteq N(u)$ .", "Since $G[W]$ is biconnected, it follows from Menger's Theorem that there are at least two distinct paths in $G[W]$ between any two vertices in $W$ .", "Therefore, every pair of vertices in $W$ belong to at least one cycle of $G[W]$ .", "Let $C$ be a cycle in $G[W]$ to which $x$ and $y$ belong.", "If $z$ also belongs to $C$ , then the subgraph $G[C\\cup \\lbrace u\\rbrace ]$ contains a $K_{4}$ -subdivision with $x, y, z,u$ as the branching nodes, and we are done.", "So let $z$ not belong to the cycle $C$ .", "Since $G[W]$ is biconnected, $|N_{W}(z)|\\ge 2$ .", "From Menger's Theorem applied to $C$ and $N_{W}(z)$ , we get that there are at least two paths from $z$ to $C$ which intersect only at $z$ .", "These paths together with the cycle $C$ constitute a $\\theta _{3}$ -subdivision in which $x$ and $y$ are branching nodes and $z$ is a subdividing node.", "Together with the vertex $u$ , this $\\theta _{3}$ -subdivision forms a $K_{4}$ in $G[W\\cup Z]$ .", "Reminder of Lemma REF If $(G,S,k)$ is a simplified instance of disjoint $K_{4}$ -minor cover, then $F=N_0\\cup N_1\\cup N_2$ .", "As $(G,S,k)$ is a simplified instance, $G[S\\cup \\lbrace x\\rbrace ]$ is $K_4$ -minor-free for every $x\\in F$ (by Branching rule REF ) and there exists a biconnected component $B$ of $G[S]$ containing $N_S(x)$ (otherwise we could apply Branching rule REF or REF ).", "It directly follows from Lemma REF , that for every vertex $x\\in F$ , $|N_S(x)|\\leqslant 2$ ." ], [ "Deferred proofs of Theorem ", "Reminder of Theorem REF Let $(G,S,k)$ be an instance of disjoint $K_{4}$ -minor cover.", "If none of the reduction rules nor branching rules applies, then $(G,S,k)$ is an independent instance.", "Once we show that $F$ is an independent set, condition (b) follows from Corollary REF and the fact that $(G,S,k)$ is reduced with respect to Reduction rule REF .", "Conditions (c) and (d) are also satisfied in this case since $(G,S,k)$ is simplified, specifically since Branching rules REF , REF and REF do not apply on singleton sets $X$ .", "We now prove that $F$ is an independent set.", "Suppose $G[F]$ contains a connected component $X$ with at least two vertices.", "Since $(G,S,k)$ is a simplified instance, $G[X\\cup S]$ does not contain $K_{4}$ as a minor.", "Hence from Lemma REF , we have $|N_S(X)|\\le 2$ .", "We consider two cases, whether $G[X]$ is a tree or not.", "Let us assume that $X$ is a tree.", "Observe that every leaf of $X$ belongs to $N_2$ , for otherwise Rule REF or Rule REF would apply.", "So $X$ contains two leaves, say $u$ and $v$ , having the same two neighbors in $S$ , say $x$ and $y$ .", "But then observe that $x$ and $y$ belong to the same connected component of $G[S]$ (otherwise Branching Rule REF would apply).", "It clearly follows that $x$ , $y$ , $u$ and $v$ are the four branching nodes of a $K_4$ -subdivision in $G[S\\cup X]$ , which contradicts the assumption that Branching Rule REF cannot apply to $(G,S,k)$ .", "We now consider the case where $X$ is not a tree.", "Before we proceed further we observe the following.", "A nontrivial block is a block which is more than just an edge.", "Claim 3 Let $B$ be a nontrivial block of $G[F]$ .", "Let $F_{B}$ be the graph obtained from $G[F]$ by removing $B\\setminus \\partial _G(B)$ and all the edges in $G[\\partial _G(B)]$ .", "Then every connected component of $F_{B}$ contains a vertex of $N_1\\cup N_2$ .", "Proof of claim.", "Observe that any connected component of $F_{B}$ shares at most one vertex with $B$ .", "Thus if a connected component of $G[F\\setminus (B\\setminus \\partial _G(B))]$ is entirely contained in $N_0$ , then we can apply Reduction rule REF .", "$\\Diamond $ As $X$ is not a tree, it contains a non-trivial block $B$ .", "Since $(G,S,k)$ is reduced with respect to Reduction Rule REF , $|\\partial _G(B)|\\geqslant 2$ .", "We first assume that $|\\partial _G(B)|=2$ with $\\partial (B)=\\lbrace s,t\\rbrace $ .", "Observe that $G[B]+(s,t)$ is not a series-parallel graph since otherwise $B$ would be a single edge $(s,t)$ due to Reduction rule REF .", "As $(G,S,k)$ is reduced with respect to Reduction rule REF , $B$ is a $\\theta _3$ with $s$ and $t$ as subdividing nodes.", "Due to Branching rule REF , $N_S(X)$ is contained in a single connected component of $S$ .", "Together with the observation of Claim REF , this implies that there exists an $s,t$ -path $P$ in $G[S\\cup X]$ in which no internal vertex lies in $B$ .", "However, $G[B\\cup P]$ is a $K_4$ -subdivision and Branching rule REF would apply, a contradiction.", "So we have that $|\\partial _G(B)|\\ge 3$ and let $\\lbrace x,y,z\\rbrace \\subseteq \\partial (B)$ .", "By Claim REF , there exist three internally vertex-disjoint paths $P_x$ , $P_y$ and $P_z$ from $x$ , $y$ and $z$ respectively to a connected component $G[S]$ such that no internal vertex of them lies in $B$ .", "Since $B$ is biconnected, Lemma REF applies by taking $B$ and $(S\\cup P_x\\cup P_y\\cup P_z)\\setminus \\lbrace x,y,z\\rbrace $ showing that $G[B\\cup P_x \\cup P_y \\cup P_z \\cup S]$ contains a $K_4$ -subdivision: a contradiction of the fact that Branching rule REF does not apply.", "Reminder of Theorem REF Let $(G,S,k)$ be an independent instance of disjoint $K_{4}$ -minor cover.", "Then $W\\subseteq F$ is a disjoint $K_4$ -minor cover of $G$ if and only if it is a vertex cover of $G^*(S)$ .", "If $W\\subseteq F$ is a $K_4$ -minor cover of $G$ , then by construction $G^*(S)-W$ is an independent set and thus, $W$ is a vertex cover of $G^*(S)$ .", "To show the converse, we can assume that $G[S]$ is biconnected.", "Indeed, for every $v\\in F$ , its two neighbors $x_v,y_v\\in S$ belong to the same biconnected component and thus any cut vertex of $G[S]$ remains a cut vertex of $G-W$ .", "Since $K_4$ -subdivision is biconnected, any such subdivision in $G-W$ must not contain $u,v\\in F\\setminus W$ such that $N_S(u)$ and $N_S(v)$ belong to distinct biconnected components of $G[S]$ .", "An SP-tree is minimal if any S-node (resp.", "P-node) does not have S-nodes (resp.", "P-nodes) as a child [3].", "Furthermore, any SP-tree obtained will be converted into a minimal one via standard operations on the given SP-tree: if there is an S-node (resp.", "P-node) with another S-node (resp.", "P-node) as a child, contract along the edge and if an S-node or P-node has exactly one child, delete it and connect its child and its parent by an edge.", "Throughout the proof, we fix a minimal SP-tree $\\mathcal {T}_S$ of $G[S]$ .", "Furthermore, we take the root as follows: (a) $G[S]$ is a cycle, we let two adjacent vertices be the terminals of the root.", "(2) otherwise, the last parallel operation has at least three children.", "For a node $\\alpha $ of the SP-tree $\\mathcal {T}_S$ , let $Z_\\alpha $ be the set of terminals of its children $\\alpha _1\\dots \\alpha _c$ , that is, $Z_{\\alpha }=\\bigcup _{1\\leqslant i\\leqslant c} X_{\\alpha _i}$ .", "Claim 4 For every $u\\in F$ , either $X_{\\alpha }=\\lbrace x_u,y_u\\rbrace $ for some node $\\alpha $ of $\\mathcal {T}_S$ or there is a unique $S$ -node $\\alpha $ such that $\\lbrace x_u,y_u\\rbrace \\subseteq Z_{\\alpha }$ .", "Proof of claim.", "Let us suppose that for $u\\in F$ , there no $\\alpha $ in $\\mathcal {T}_S$ such that $X_{\\alpha }=\\lbrace x_u,y_u\\rbrace $ .", "We argue that for such $u$ , there exists an $S$ -node $\\alpha $ such that $\\lbrace x_u,y_u\\rbrace \\subseteq Z_{\\alpha }$ .", "To this end, take a lowest node $\\alpha $ such that $x_u,u_y\\in V_{\\alpha }$ and let $X_{\\alpha }=\\lbrace s,t\\rbrace $ .", "Then $\\alpha $ should be an S-node.", "Suppose $\\alpha $ is a P-node.", "As we choose $\\alpha $ to be lowest, there are two children $\\beta _x$ and $\\beta _y$ of $\\alpha $ such that $x_u\\in Y_{\\beta _x}$ and $y_u\\in Y_{\\beta _y}$ .", "This implies $G[S]$ is not a cycle as we fix the terminals of the root to be adjacent vertices in this case.", "Note that $X_{\\alpha }=X_{\\beta _x}=X_{\\beta _y}$ and $X_\\alpha $ separates $x_u$ and $y_u$ .", "Since $G[V_{\\beta _x}]$ is an SP-graph, there is a path $P_x$ from $s$ to $t$ visiting $x_u$ .", "Likewise, $G[V_{\\beta _y}]$ contains a path $P_y$ from $s$ to $t$ visiting $y_u$ .", "On the other hand, since $G[S]$ is not a simple cycle, there is a P-node $\\alpha ^{\\prime }$ such that either (a) $\\alpha ^{\\prime }=\\alpha $ and $\\alpha ^{\\prime }$ has a child $\\beta \\ne \\lbrace \\beta _x,\\beta _y\\rbrace $ , or (b) $\\alpha ^{\\prime }$ is an ancestor of $\\alpha $ and it has a child $\\beta $ which is not an ancestor of $\\alpha $ .", "In both cases, the subgraph $G[S\\setminus (Y_{\\beta _x}\\cup Y_{\\beta _y})]$ is connected and contains a path $P$ connecting $s$ and $t$ .", "The three paths $P_x$ , $P_y$ , $P$ and the length-two path between $x_u$ and $y_u$ via $u$ form a $K_4$ -subdivision with $\\lbrace v_x,v_y,s,t\\rbrace $ branching nodes.", "Now we argue the uniqueness of such an S-node.", "For some $u\\in F$ , suppose that there are two distinct S-nodes $\\alpha $ and $\\alpha ^{\\prime }$ such that $\\lbrace x_u,y_u\\rbrace \\subseteq Z_{\\alpha }$ and $\\lbrace x_u,y_u\\rbrace \\subseteq Z_{\\alpha ^{\\prime }}$ .", "Since $X_{\\alpha }$ is a separator of $G[S]$ , the only possibility is to have $X_{\\alpha }=X_{\\alpha ^{\\prime }}=\\lbrace x_u,y_u\\rbrace $ .", "This contradicts to our assumption that there is no vertex $u$ such that $\\lbrace x_u,y_u\\rbrace $ labels a node of $\\mathcal {T}_S$ .", "$\\Diamond $ Let $F_0$ and $F_1$ form a partition of $F$ : $u\\in F_0$ if $X_{\\alpha }=\\lbrace x_u,y_u\\rbrace $ for some node $\\alpha $ of $\\mathcal {T}_S$ , otherwise $u$ belongs to $F_1$ .", "For $u\\in F_1$ , we denote as $\\alpha (u)$ the unique S-node of $\\mathcal {T}_S$ with $\\lbrace x_u,y_u\\rbrace \\subseteq Z_{\\alpha }$ .", "Suppose $W\\subseteq F$ is a vertex cover of $G^*(S)$ .", "We shall then incrementally extend $\\mathcal {T}_S$ to an SP-tree of $G[S]+(F\\setminus W)$ .", "For $u\\in F$ , let $\\mathcal {T}_u$ be the minimal SP-tree with $\\lbrace x_u,y_u\\rbrace $ as terminals of the length-two path $x_uuy_u$ .", "It is not difficult to increment $\\mathcal {T}_S$ to an SP-tree $\\mathcal {T}_{S+F_0}$ of $G[S\\cup F_0]$ .", "Let $u\\in F_0$ and $\\alpha $ be the node labeled by $\\lbrace x_u,y_u\\rbrace $ .", "If $\\alpha $ is an S-node, there is a P-node labeled by the same terminals.", "Hence we assume that $\\alpha $ is either a leaf node or a P-node.", "We do the following: (1) if $\\alpha $ is a P-node, make $\\mathcal {T}_u$ to be a child of $\\alpha $ , (2) if $\\alpha $ is an edge node, convert $\\alpha $ into a P-node and make $\\mathcal {T}_u$ to be a child of $\\alpha $ .", "The resulting SP-tree is again minimal, via standard manipulation if necessary.", "It is worth noting that none of S-nodes are affected during the entire manipulation and thus $\\alpha (u)$ remains unaffected for $u\\in F_1$ .", "We wish to show that $\\mathcal {T}_{S+F_0}$ can be extended to contain all $F_1\\setminus W$ as well.", "When $\\alpha $ is an S-node, $Z_{\\alpha }$ can be construed as an interval on the terminals of its children: the the ordering of series compositions imposes an ordering on the elements of $Z_{\\alpha }$ .", "The crucial observation is that if $\\alpha (u)=\\alpha (v)$ for $u,v\\in F_1\\setminus W$ , then the intervals $[x_u,y_u]$ and $[x_v,y_v]$ in $\\alpha (u)$ do not overlap.", "Suppose they overlap.", "We can take a cycle $C$ containing all the vertices of $Z_{\\alpha }$ .", "Then $C$ together with the two paths $P_u=x_uuy_u$ and $P_v=x_vvy_v$ form a $K_4$ -subdivision in $G[C\\cup \\lbrace u,v\\rbrace ]$ .", "Therefore, we have an edge $(u,v)$ in $G^*(S)$ , a contradiction.", "Starting from $\\mathcal {T}_{S+F_0}$ , now we increment the SP-tree by attaching $\\mathcal {T}_u$ for every $u\\in F_1\\setminus W$ .", "Given $u\\in F_1\\setminus W$ , add a P-node $\\alpha ^{\\prime }$ with $X_{\\alpha ^{\\prime }}=\\lbrace x_u,y_u\\rbrace $ as a child of $\\alpha (u)$ and make $\\alpha ^{\\prime }$ to become the father of every former child $\\alpha _i$ of $\\alpha $ for which $X_{\\alpha _i}$ is contained in the interval $[x_u,y_u]$ .", "Note that no S-node other than $\\alpha (u)$ is affected by this manipulation.", "Moreover, $\\alpha (u)$ remains as an S-node.", "Indeed, if we need to change $\\alpha (u)$ , it is only because $\\alpha (u)$ has a unique child after the operation.", "This implies $x_u,y_u$ are in fact the terminals of $X_{\\alpha (u)}$ .", "However, the parent of $\\alpha (u)$ , which is a P-node due to minimality of the SP-tree, is labeled by $\\lbrace x_u,y_u\\rbrace $ , a contradiction.", "Finally due to the crucial observation from the previous paragraph, this incremental extension can be performed for all vertices of $F_1\\setminus W$ .", "Implying $G-W$ is an SP-graph, this complete the proof.", "[disable, inline]The proof above has to be completed: the missing references refer to the circle graph lemma, which we decided to cancel in this version" ], [ "Deferred proof of Lemma ", "Lemma 16 Let $(G,S,k)$ be a reduced instance.", "If $\\alpha $ is a non-leaf node of an extended SP-decomposition $(T,\\mathcal {X})$ of $G[F]$ , then $(V_{\\alpha }\\setminus Y_{\\alpha })\\setminus N_0 \\ne \\emptyset $ .", "Observe that for every non-leaf node $\\alpha $ of $(T,\\mathcal {X})$ , the set $Y_{\\alpha }=V_{\\alpha }\\setminus X_{\\alpha }$ is nonempty.", "This can be easily verified when $\\alpha $ is a cut node, an edge node which is not a leaf (this happens only when the edge node is the parent of a cut node in the extended decomposition), or an S-node.", "When $\\alpha $ is a P-node, the fact that $(G,S,k)$ is reduced with respect to Reduction Rule REF ensures $Y_{\\alpha }\\ne \\emptyset $ .", "For the sake of contradiction, suppose that $Y_{\\alpha }\\subseteq N_0$ .", "Observe that no vertex in $Y_{\\alpha }$ has a neighbor in $F\\setminus V_{\\alpha }$ .", "By assumption, no vertex in $Y_{\\alpha }$ has a neighbor in $S$ .", "Hence $\\partial (V_{\\alpha })\\subseteq X_{\\alpha }$ and thus $Y_{\\alpha }\\subseteq V_{\\alpha }\\setminus \\partial (V_{\\alpha })$ .", "If $|\\partial (V_{\\alpha })|=1$ then Reduction Rule REF applies, a contradiction.", "Thus $|\\partial (V_{\\alpha })|=2$ , and so $\\partial (V_{\\alpha })=X_{\\alpha }$ .", "Furthermore, no descendant of $\\alpha $ is a cut node in $G[F]$ (otherwise Reduction Rule REF applies), which implies that $V_{\\alpha }$ is contained in a leaf block of $G[F]$ .", "$G_{\\alpha }$ is thus a series-parallel graph having $X_{\\alpha }=\\lbrace s,t\\rbrace $ as terminals and thus by Lemma REF $G_{\\alpha }+(s,t)$ is an SP-graph.", "Since $\\alpha $ is a non-leaf node and $(G,S,k)$ is reduced with respect to Reduction rule REF , we have $|V_{\\alpha }|>2$ .", "Thus $G_{\\alpha }$ is not isomorphic to any of the two excluded graphs of Reduction Rule REF .", "So Reduction Rule REF applies deleting the nonempty set $Y_{\\alpha }$ , a contradiction.", "Lemma 17 Let $(G,S,k)$ be a simplified instance of disjoint $K_{4}$ -minor cover and $\\alpha $ be a marked node of the extended SP-decomposition $(T,\\mathcal {X})$ of $G[F]$ .", "Then every block $B$ in $G_{\\alpha }$ satisfies $|B|< \\gamma (9)$ .", "[disable,inline]Since the graph $G_{\\alpha }$ is not necessarily an induced subgraph of $G[F]$ , it seems possible that the block structure of $G_{\\alpha }$ is not necessarily the same as that of $G[V_{\\alpha }]$ .", "It is therefore not clear how the previous assertion holds.", "– Philip From the discussion with EJK: This is clearly true for all blocks of $G_{\\alpha }$ which are also blocks in $G$ .", "The block structure of $G_{\\alpha }$ is different from that of $G[V_{\\alpha }]$ only on the subgraph defined by the vertices in the block of $G[F]$ to which $\\alpha $ belongs.", "If $\\alpha $ is a parallel node, then Lemma REF applies.", "Else if it is a serial node, we can apply the proof of Lemma REF to each child of $\\alpha $ .", "– Philip Recall that the root of the SP-tree of $B$ is a P-node $\\beta $ inherited from $(T,\\mathcal {X})$ .", "As a descendent of $\\alpha $ , $\\beta $ is a marked node.", "By Lemma REF , $V^B_{\\beta }$ is a 4-protrusion.", "As $\\beta $ is marked, $V^B_{\\beta }$ is reduced under protrusion rule (Reduction Rule REF ) and so $|B|\\leqslant |V^B_{\\beta }|< \\gamma (9)$ .", "Lemma 18 Let $(G,S,k)$ be a simplified instance of disjoint $K_{4}$ -minor cover and let $\\alpha $ be a marked cut node of the extended SP-decomposition $(T,\\mathcal {X})$ of $G[F]$ with $X_{\\alpha }=\\lbrace c\\rbrace $ .", "Then $|V_{\\alpha }|\\leqslant c_0=\\gamma (9)+7$ .", "Moreover, the block tree of $G_{\\alpha }$ is a path.", "[disable,inline]I had some formal problems with this proof, as we are dealing here with $\\vec{\\mathcal {B}}_{G_{\\alpha }}$ which is a totally different object than $(T, \\mathcal {X})$ .", "So the Lemmas we have so far did not strictly apply: ${\\mathcal {X},T}$ is obtained from $\\vec{\\mathcal {B}}_G$ .", "Isn't this enough?", "Let $\\vec{\\mathcal {B}}_{F_{\\alpha }}$ be the oriented block tree of $G_{\\alpha }$ rooted at $B_c$ , the block containing $c$ .", "Let $B_1$ be a leaf block in $\\vec{\\mathcal {B}}_{F_{\\alpha }}$ and $c_1$ be the cut vertex such that $(c_1,B_1)\\in E(\\vec{\\mathcal {B}}_F)$ .", "Observe that $(T,\\mathcal {X})$ contains a cut node $\\beta _1$ such that $X_{\\beta _1}=\\lbrace c_1\\rbrace $ and by the construction of $(T,\\mathcal {X})$ , the node $\\beta _1$ is a descendant of $\\alpha $ .", "By Lemma REF , $B_1$ contains a vertex of $N_1\\cup N_2$ , say $x_1\\in B_1$ such that $x_1\\ne c_1$ .", "We consider two cases.", "(a) $B_1$ is a nontrivial block.", "Consider the remaining part of $G_{\\alpha }$ , i.e.", "$C_1:=(V_{\\alpha }\\setminus B_1) \\cup \\lbrace c_1\\rbrace $ .", "We shall show that $C_1\\subseteq N_0$ , i.e.", "no vertex of $C_1$ has a neighbor in $S$ .", "Suppose the contrary and observe that $G[C_1\\cup S]$ contains a path $P_1$ between $c_1$ and $S$ avoiding $B_1$ .", "If there is a vertex $y_1 \\in B_1$ s.t.", "$y_1\\notin \\lbrace c_1,x_1\\rbrace $ and $y_1 \\subseteq N_1\\cup N_2$ , then by Lemma REF , $G[V_{\\alpha }\\cup S]$ contains a $K_4$ -subdivision, a contradiction.", "If no such vertex $y_1$ exists, observe that $\\lbrace x_1,c_1\\rbrace $ forms a boundary of $B_1$ .", "Due to the assumption that $\\alpha $ is marked, the subgraph $G[V_{\\alpha }\\cup S]$ is $K_4$ -minor-free.", "In particular, the subgraph $G[B_1\\cup P]$ is $K_4$ -minor-free, where $P$ is a path between $x_1$ and $c_1$ in $G[V_{\\alpha }\\cup S]$ avoiding $B_1$ .", "The existence of such $P$ is ensured due to the existence of $P_1$ , that $x_1\\in N_1\\cup N_2$ and the fact that $N_S(V_{\\alpha })$ belong to the same connected component of $G[S]$ .", "Now that $G[B_1]+(x_1,c_1)$ is a biconnected $K_4$ -minor-free graph, hence an SP-graph.", "It follows that Reduction rule REF applies to $B_1$ and reduces it to a single edge: a contradiction to the fact that the instance is simplified.", "It follows $C_1\\subseteq N_0$ .", "As a corollary we know that $\\vec{\\mathcal {B}}_{F_{\\alpha }}$ contains no other leaf block and thus it is a path.", "It remains to bound the size of $V_{\\alpha }$ .", "Since $C_1\\subseteq N_0$ and $\\lbrace c_1,c\\rbrace $ forms a boundary of $C_1$ , whenever $|C_1|>4$ Reduction rule REF applies, contradiction.", "Hence $|V_{\\alpha }|=|B_1|+|C_1\\setminus \\lbrace c_1\\rbrace |$ and combining the bound given by Lemma REF , we obtain the upper bound $\\gamma (9) + 3$ .", "(b) $B_1$ is a trivial block (i.e.", "an edge) W.l.o.g.", "$\\vec{\\mathcal {B}}_{F_{\\alpha }}$ does not contain a nontrivial leaf block.", "Consider the remaining part of $G_{\\alpha }$ , i.e.", "$C_1:=V_{\\alpha }\\setminus \\lbrace x_1\\rbrace $ .", "Here we claim that $|N_S(C_1)|\\le 1$ .", "Suppose the contrary.", "By Lemma REF , we have $|N_S(V_{\\alpha })|\\leqslant 2$ .", "Hence considering the case when $N_S(x_1)=N_S(C_1)=\\lbrace u,v\\rbrace $ is sufficient.", "It remains to see that $u$ and $v$ belong to the same connected component of $G[S]$ , and $G[V_{\\alpha }\\cup S]$ contains a $K_4$ -subdivision with $x_1,C_1,u,v$ as branching nodes, a contradiction.", "As a corollary we know that $\\vec{\\mathcal {B}}_{F_{\\alpha }}$ contains no other leaf block and thus it is a path.", "It remains to bound the size of $V_{\\alpha }$ .", "Consider the case when every block of $\\vec{\\mathcal {B}}_{F_{\\alpha }}$ trivial, i.e.", "$G_{\\alpha }$ is a path.", "From the argument of the previous paragraph, we know that $|N_S(C_1)|\\le 1$ and $N_S(C_1)\\subseteq N_S(x_1)$ .", "Since the instance is reduced with respect to 1-Boundary rule REF and Chandelier rule REF , we can conclude that $|V_{\\alpha }|\\leqslant 4$ .", "Now consider the case $\\vec{\\mathcal {B}}_{F_{\\alpha }}$ contains a nontrivial block and let $B_2$ be the nontrivial block which is farthest from $c$ .", "Since $\\vec{\\mathcal {B}}_{F_{\\alpha }}$ is a path, it can be partitioned into two subpaths: the one starting from the cut node $c$ to the block $B_2$ and the remaining part.", "Let $G_0$ and $G_1$ be the associated subgraphs of $G_{\\alpha }$ , i.e.", "containing the vertices which appear in each subpath as part of a block or as a cut node.", "As every block of $G_1$ is trivial, the bound in the previous paragraph applies and $|G_1|\\le 4$ .", "Observe that the bound obtained in (a) applies to $G_0$ : to be precise, applies to the graph obtained from $G_{\\alpha }$ by contracting $G_1$ into a single vertex.", "Hence we get the desired bound $|V_{\\alpha }|\\le |G_1|+|G_2| =\\gamma (9)+ 7$ .", "Reminder of Lemma REF Let $(G,S,k)$ be a simplified instance of disjoint $K_{4}$ -minor cover and let $\\alpha $ be a marked node of the extended SP-decomposition $(T,\\mathcal {X})$ of $G[F]$ , then $|V_{\\alpha }|\\leqslant c_1=12(\\gamma (9)+2c_0)$ .", "We consider each possible type of node separately.", "Recall that since $\\alpha $ is marked, the neighbourhood $N_S(V_{\\alpha })$ belongs to a single biconnected component and $G[S\\cup V_{\\alpha }]$ is $K_4$ -minor-free.", "When $\\alpha $ is a cut node, Lemma REF directly provides the bound.", "We now consider the remaining cases.", "(1) $\\alpha $ is an edge node: By the construction of an extended SP-decomposition $(T,\\mathcal {X})$ , any child of $\\alpha $ is a cut node.", "Since $\\alpha $ can have at most two children, Lemma REF implies $|V_{\\alpha }|\\leqslant 2c_0$ .", "(2) $\\alpha $ is a P-node: Recall that we have $|V^B_{\\alpha }|<\\gamma (9) $ by Lemma REF and $\\alpha $ has at most two attachment vertices by Lemma REF .", "Each attachment vertex of $\\alpha $ either belongs to $N_1\\cup N_2$ or is a cut vertex.", "Hence we can apply the bound on cut node size given by Lemma REF .", "It follows that $|V_{\\alpha }|\\leqslant \\gamma (9) + 2c_0$ .", "(3) $\\alpha $ is an S-node: Let $\\beta _1,\\ldots , \\beta _q$ be the children of $\\alpha $ and denote bye $x_1\\dots x_{q+1}$ the vertices such that for $1\\leqslant j\\leqslant q$ , $X_{\\beta _j}=\\lbrace x_j,x_{j+1}\\rbrace $ .", "Since every child of an S-node is either a P-node or an edge node, from case 1 and 2 we have $|V_{\\beta _j}|\\le \\gamma (9) + 2c_0$ .", "We now prove that if $q\\geqslant 13$ , then either the instance is not simplified or $G[S\\cup V_{\\alpha }]$ contains $K_4$ as a minor.", "Since the lemma holds trivially if every $V_{\\beta _{j}}$ has at most four vertices, in the rest of the proof we assume without loss of generality that for each P-node $\\beta _{j}$ which we consider, $|V_{\\beta _{j}}|>4$ .", "Claim 5 For $1\\leqslant j\\leqslant q-1$ , let $Z_j:= V_{\\beta _{j}} \\cup V_{\\beta _{j+1}}$ .", "Then $Z_j\\setminus \\partial _{F}(Z_j)$ contains at least one vertex in $N_1\\cup N_2$ .", "Proof of claim.", "Suppose one of $\\beta _j$ and $\\beta _{j+1}$ , say $\\beta _j$ , is a P-node.", "By Lemma REF , $Y_{\\beta _j}= V_{\\beta _j}\\setminus X_{\\beta _j}$ contains a vertex of $N_1\\cup N_2$ .", "If both of $\\beta _j$ and $\\beta _{j+1}$ are edge nodes, then $x_{j+1}\\in N_1\\cup N_2$ , since otherwise its degree in $G$ is two and we can apply Reduction Rule REF , a contradiction.", "$\\Diamond $ Suppose that $q\\geqslant 13$ .", "First, suppose there exists $j$ , $3\\leqslant j \\leqslant q-2$ , such that $\\beta _j$ is a P-node.", "By Lemma REF , we have $Y_{\\beta _j}\\cap (N_1\\cup N_2)\\ne \\emptyset $ .", "On the other hand, Claim REF says that the subsets $Z_{j-2}$ and $Z_{j+1}$ both contain at least one vertex in $N_1\\cup N_2$ each.", "Since $G[V^B_{\\beta _j}]$ is biconnected and $G[(S\\cup Z_{j-2}\\cup Z_{j+1})\\setminus X_{\\beta _j}]$ is connected, Lemma REF applies to these two graphs and there is a $K_4$ -subdivision in $G[S\\cup V_{\\alpha }]$ , a contradiction.", "Therefore, we can assume that for every $j$ , $3\\leqslant j \\le q-2$ , $\\beta _j$ is an edge node.", "It follows that $G[X^{\\prime }]$ , with $X^{\\prime }=\\lbrace x_j:3\\leqslant j \\leqslant q-2\\rbrace $ , is a chordless path.", "Claim REF implies that every internal vertex of $X^{\\prime }$ is an attachment vertex, that is, either it belongs to $N_1\\cup N_2$ or it is a cut vertex belonging to some $A(\\beta _j)$ .", "We consider the two sets $X_1:=\\bigcup _{3\\leqslant j\\leqslant 6}V_{\\beta _j}$ and $X_2:=\\bigcup _{8\\leqslant j\\leqslant 11}V_{\\beta _j}$ .", "Claim 6 $|N_S(X_1)|\\geqslant 2$ and $|N_S(X_2)|\\geqslant 2$ .", "Proof of claim.", "Consider $X^{\\prime }_1=\\lbrace x_4,x_5,x_6,x_7\\rbrace $ .", "Suppose that every vertex on $X^{\\prime }_1$ belongs $N_1\\cup N_2$ .", "As the instance is reduced with respect to Rule REF and $|X^{\\prime }_1|=4$ , clearly we have $|N_S(X_1)|\\geqslant 2$ .", "Hence we may assume there exists a cut vertex $x\\in X^{\\prime }_1$ and let $\\alpha _x$ be the cut node of $(T,\\mathcal {X})$ with $X_{\\alpha _x}=\\lbrace x\\rbrace $ .", "By Lemma REF , there is only one leaf block $B_x$ in $G_{\\alpha _x}$ .", "If $B_x$ is a single edge, $B_x$ contains a pendant vertex $y$ .", "Observe that $N_S(y)=2$ and the claim holds.", "Consider the case $B_x$ is a nontrivial block.", "By Lemma REF , $B_x$ contains a vertex $y\\ne c$ in $N_1\\cup N_2$ , where $c$ is the unique cut vertex contained in $B_x$ .", "In fact, $B_x$ does not contain $z\\ne y$ such that $z\\in N_1\\cup N_2$ , since otherwise $|\\partial _G(B_x)|\\ge 3$ and applying Lemma REF on $Y:=B_x$ , $W:=C\\cup (V_{\\alpha }\\setminus B_x)$ (with $C$ the connected component of $N_S(V_{\\alpha })$ ) witnesses $K_4$ -subdivision in $G[S\\cup V_{\\alpha }]$ , a contradiction.", "So we have $\\partial _G(B_x)=\\lbrace c,y\\rbrace $ .", "As we assume that the instance is reduced, in particular with respect to Reduction rule REF , and $B_x$ is a nontrivial block, we conclude that $B_x$ is a $\\theta _3$ with $c$ and $y$ as subdividing nodes.", "On the other hand, it is not difficult to see that $G[S\\cup V_{\\alpha }]$ contains a $c,y$ -path $P$ avoiding $B_x$ .", "It remains to observe that $G[B_x\\cup P]$ is a $K_4$ -model, a contradiction.", "$\\Diamond $ If $|N_S(V_{\\alpha })|\\geqslant 3$   then Lemma REF applies to the biconnected component of $N_S(V_{\\alpha })$ and $V_{\\alpha }$ , thus we obtain a $K_4$ -subdivision, a contradiction.", "If $|N_S(V_{\\alpha })|=2$ , then $N_S(X_1)=N_S(X_2)$ and $G[S\\cup V_{\\alpha }]$ contains a $K_4$ -model with branching nodes being the following four connected subsets, a contradiction: $X_1$ , $X_2$ , each of the two vertices of $N_S(X)$ .", "That is, we have a $K_4$ -model in $G[S\\cup V_{\\alpha }]$ whenever $q\\geqslant 13$ .", "Therefore, we have $q\\leqslant 12$ if $\\alpha $ is marked." ], [ "Deferred proof of Lemma ", "Reminder of Lemma REF Let $(G,S,k)$ be a simplified instance of disjoint $K_{4}$ -minor cover and let $\\alpha $ be a lowest unmarked node of $(T,\\mathcal {X})$ of $G[F]$ .", "In polynomial time, one can find a path $X$ of size at most $2c_1$ satisfying the conditions of line  (resp.", "line ) if the test at line  (resp. )", "succeeds; a subset $X\\subseteq V_{\\alpha }$ of size bounded by $2c_1$ satisfying the condition of line  if the test at line  succeeds; Suppose that $\\alpha $ is a cut node.", "If the test at line or at line succeeds, then there are two children $\\beta _1$ and $\\beta _2$ of $\\alpha $ such that $X:=V_{\\beta _1}\\cup V_{\\beta _2}$ satisfies the conditions of line  or line , respectively.", "In case of (b), the proof of Lemma REF shows that if $\\alpha $ has two children $\\beta _1$ and $\\beta _2$ , then the subgraph $G[X\\cup S]$ contains $K_4$ as a minor, where $X:=V_{\\beta _1}\\cup V_{\\beta _2}$ .", "With the bound provided by Lemma REF , now it suffices to argue that $X$ is a connected set.", "We claim that $c\\in X_{\\beta _1}\\cap X_{\\beta _2}$ .", "Indeed, $\\beta _i$ is either a P-node or an edge node.", "Obviously, $c\\in X_{\\beta _i}$ if $\\beta _i$ is an edge node.", "If $\\beta _i$ is a P-node, recall that this is the root node of the canonical SP-tree $(T^B, \\mathcal {X}^B)$ from which $\\beta _i$ is inherited.", "Since $(c,G^B_{\\beta _i})\\in E(\\vec{\\mathcal {B}}_G)$ , the construction of $(T^B, \\mathcal {X}^B)$ requires that $c\\in X_{\\beta _i}$ .", "As a result, $c\\in X_{\\beta _1}\\cap X_{\\beta _2}$ and the subgraph $G[V_{\\beta _1}\\cup V_{\\beta _2}]$ is connected.", "If $\\alpha $ is an edge node, $\\alpha $ can have at most two children, all of which are cut nodes.", "Take $X=V_{\\alpha }$ .", "Since every child of $\\alpha $ is marked already, the bound of Lemma REF holds and $|X|\\leqslant 2c_0$ .", "In $G[X]$ , one can identify a path or a subset satisfying the condition (a) or (b).", "If $\\alpha $ is a P-node, let $\\beta _1$ and $\\beta _2$ be its two children.", "By Lemma REF , we know that $|V_{\\beta _1}|, |V_{\\beta _2}|\\leqslant c_1$ .", "Take $X=V_{\\alpha }$ .", "In $G[X]$ , one can identify a path or a subset satisfying the condition (a) or (b) if this is the case.", "Let us consider the case when $\\alpha $ is an S-node with $\\beta _1,\\ldots , \\beta _q$ as its children.", "Suppose that there are $u,v \\in V_{\\alpha }\\cap (N_1\\cup N_2)$ which have neighbors in distinct connected components of $G[S]$ .", "Then there exist $1\\leqslant k < k^{\\prime } \\leqslant q$ such that $u\\in V_{\\beta _k}$ and $v\\in V_{\\beta _{k^{\\prime }}}$ .", "Choose $k$ and $k^{\\prime }$ such that $k^{\\prime }-k$ is minimized.", "We claim that $k^{\\prime }-k\\leqslant 2$ .", "Suppose not.", "Then we can find an alternative vertex $w \\in Z_{k+1}\\cap (N_1\\cup N_2)$ due to Claim REF in the proof of Lemma REF and decrease $k^{\\prime }-k$ , a contradiction.", "Therefore, there exists $k$ such that $X:=V_{\\beta _k}\\cup V_{\\beta _{k+1}} \\cup V_{\\beta _{k+2}}$ contains $u,v$ .", "It remains to observe that $|X|\\leqslant 3\\times (\\gamma (9)+2c_0)$ and we can find a path $P$ between $u$ and $v$ within $X$ , satisfying (a).", "The proof remains the same when there are $u,v \\in V_{\\alpha }\\cap (N_1\\cup N_2)$ with $bc_S(u)\\ne bc_S(v)$ .", "On the other hand if the test at line  succeeds, the proof of Case (3) in Lemma REF shows one can find a bounded-size subset $X$ .", "Indeed, if $q\\leqslant 12$ , one can take $X:=V_{\\alpha }$ and observe that $|X|\\leqslant 12(\\gamma (9) + 2c_0)\\leqslant 2c_1$ .", "If $q\\geqslant 13$ , take $X:=\\bigcup _{j=1}^{13} V_{\\beta _j}$ and observe that $|X|\\leqslant 13(\\gamma (9) +2c_0)\\leqslant 2c_1$ ." ] ]

[ [ "Evolution of pi^0 suppression in Au+Au collisions from sqrt(s_NN) = 39\n to 200 GeV" ], [ "Abstract Neutral-pion, pi^0, spectra were measured at midrapidity (|y|<0.35) in Au+Au collisions at sqrt(s_NN) = 39 and 62.4 GeV and compared to earlier measurements at 200 GeV in the 1<p_T<10 GeV/c transverse-momentum (p_T) range.", "The high-p_T tail is well described by a power law in all cases and the powers decrease significantly with decreasing center-of-mass energy.", "The change of powers is very similar to that observed in the corresponding p+p-collision spectra.", "The nuclear-modification factors (R_AA) show significant suppression and a distinct energy dependence at moderate p_T in central collisions.", "At high p_T, R_AA is similar for 62.4 and 200 GeV at all centralities.", "Perturbative-quantum-chromodynamics calculations that describe R_AA well at 200 GeV, fail to describe the 39 GeV data, raising the possibility that the relative importance of initial-state effects and soft processes increases at lower energies.", "A conclusion that the region where hard processes are dominant is reached only at higher p_T, is also supported by the x_T dependence of the x_T-scaling power-law exponent." ], [ "Evolution of $\\pi ^{0}$ suppression in Au$+$ Au collisions from $\\sqrt{s_{\\rm _{NN}}}$ = 39 to 200 GeV Abilene Christian University, Abilene, Texas 79699, USA Department of Physics, Banaras Hindu University, Varanasi 221005, India Bhabha Atomic Research Centre, Bombay 400 085, India Baruch College, City University of New York, New York, New York, 10010 USA Collider-Accelerator Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA University of California - Riverside, Riverside, California 92521, USA Charles University, Ovocný trh 5, Praha 1, 116 36, Prague, Czech Republic Chonbuk National University, Jeonju, 561-756, Korea Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan University of Colorado, Boulder, Colorado 80309, USA Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA Czech Technical University, Zikova 4, 166 36 Prague 6, Czech Republic Dapnia, CEA Saclay, F-91191, Gif-sur-Yvette, France Debrecen University, H-4010 Debrecen, Egyetem tér 1, Hungary ELTE, Eötvös Loránd University, H - 1117 Budapest, Pázmány P. s. 1/A, Hungary Ewha Womans University, Seoul 120-750, Korea Florida State University, Tallahassee, Florida 32306, USA Georgia State University, Atlanta, Georgia 30303, USA Hanyang University, Seoul 133-792, Korea Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Prague 8, Czech Republic Iowa State University, Ames, Iowa 50011, USA Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia Helsinki Institute of Physics and University of Jyväskylä, P.O.Box 35, FI-40014 Jyväskylä, Finland KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan Korea University, Seoul, 136-701, Korea Russian Research Center “Kurchatov Institute\", Moscow, 123098 Russia Kyoto University, Kyoto 606-8502, Japan Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France Lawrence Livermore National Laboratory, Livermore, California 94550, USA Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA LPC, Université Blaise Pascal, CNRS-IN2P3, Clermont-Fd, 63177 Aubiere Cedex, France Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden University of Maryland, College Park, Maryland 20742, USA Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003-9337, USA Muhlenberg College, Allentown, Pennsylvania 18104-5586, USA Myongji University, Yongin, Kyonggido 449-728, Korea Nagasaki Institute of Applied Science, Nagasaki-shi, Nagasaki 851-0193, Japan University of New Mexico, Albuquerque, New Mexico 87131, USA New Mexico State University, Las Cruces, New Mexico 88003, USA Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA IPN-Orsay, Universite Paris Sud, CNRS-IN2P3, BP1, F-91406, Orsay, France PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Physics Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan Saint Petersburg State Polytechnic University, St. Petersburg, 195251 Russia Universidade de São Paulo, Instituto de Física, Caixa Postal 66318, São Paulo CEP05315-970, Brazil Department of Physics and Astronomy, Seoul National University, Seoul, Korea Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA University of Tennessee, Knoxville, Tennessee 37996, USA Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Vanderbilt University, Nashville, Tennessee 37235, USA Weizmann Institute, Rehovot 76100, Israel Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences (Wigner RCP, RMKI) H-1525 Budapest 114, POBox 49, Budapest, Hungary Yonsei University, IPAP, Seoul 120-749, Korea A. Adare University of Colorado, Boulder, Colorado 80309, USA S. Afanasiev Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia C. Aidala Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA N.N.", "Ajitanand Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA Y. Akiba RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA R. Akimoto Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan H. Al-Ta'ani New Mexico State University, Las Cruces, New Mexico 88003, USA J. Alexander Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA A. Angerami Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA K. Aoki RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan N. Apadula Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA Y. Aramaki Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan H. Asano Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan E.C.", "Aschenauer Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA E.T.", "Atomssa Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA T.C.", "Awes Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA B. Azmoun Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA V. Babintsev IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia M. Bai Collider-Accelerator Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA B. Bannier Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA K.N.", "Barish University of California - Riverside, Riverside, California 92521, USA B. Bassalleck University of New Mexico, Albuquerque, New Mexico 87131, USA S. Bathe Baruch College, City University of New York, New York, New York, 10010 USA RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA V. Baublis PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia S. Baumgart RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan A. Bazilevsky Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA R. Belmont Vanderbilt University, Nashville, Tennessee 37235, USA A. Berdnikov Saint Petersburg State Polytechnic University, St. Petersburg, 195251 Russia Y. Berdnikov Saint Petersburg State Polytechnic University, St. Petersburg, 195251 Russia X. Bing Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA D.S.", "Blau Russian Research Center “Kurchatov Institute\", Moscow, 123098 Russia K. Boyle RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA M.L.", "Brooks Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA H. Buesching Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA V. Bumazhnov IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia S. Butsyk University of New Mexico, Albuquerque, New Mexico 87131, USA S. Campbell Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA P. Castera Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA C.-H. Chen Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA C.Y.", "Chi Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA M. Chiu Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA I.J.", "Choi University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA J.B. Choi Chonbuk National University, Jeonju, 561-756, Korea S. Choi Department of Physics and Astronomy, Seoul National University, Seoul, Korea R.K. Choudhury Bhabha Atomic Research Centre, Bombay 400 085, India P. Christiansen Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden T. Chujo Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan O. Chvala University of California - Riverside, Riverside, California 92521, USA V. Cianciolo Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Z. Citron Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA B.A.", "Cole Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA M. Connors Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA M. Csanád ELTE, Eötvös Loránd University, H - 1117 Budapest, Pázmány P. s. 1/A, Hungary T. Csörgő Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences (Wigner RCP, RMKI) H-1525 Budapest 114, POBox 49, Budapest, Hungary S. Dairaku Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan A. Datta Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003-9337, USA M.S.", "Daugherity Abilene Christian University, Abilene, Texas 79699, USA G. David Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA A. Denisov IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia A. Deshpande RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA E.J.", "Desmond Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA K.V.", "Dharmawardane New Mexico State University, Las Cruces, New Mexico 88003, USA O. Dietzsch Universidade de São Paulo, Instituto de Física, Caixa Postal 66318, São Paulo CEP05315-970, Brazil L. Ding Iowa State University, Ames, Iowa 50011, USA A. Dion Iowa State University, Ames, Iowa 50011, USA M. Donadelli Universidade de São Paulo, Instituto de Física, Caixa Postal 66318, São Paulo CEP05315-970, Brazil O. Drapier Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France A. Drees Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA K.A.", "Drees Collider-Accelerator Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA J.M.", "Durham Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA A. Durum IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia L. D'Orazio University of Maryland, College Park, Maryland 20742, USA S. Edwards Collider-Accelerator Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Y.V.", "Efremenko Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA T. Engelmore Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA A. Enokizono Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA S. Esumi Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan K.O.", "Eyser University of California - Riverside, Riverside, California 92521, USA B. Fadem Muhlenberg College, Allentown, Pennsylvania 18104-5586, USA D.E.", "Fields University of New Mexico, Albuquerque, New Mexico 87131, USA M. Finger Charles University, Ovocný trh 5, Praha 1, 116 36, Prague, Czech Republic M. Finger, Jr. Charles University, Ovocný trh 5, Praha 1, 116 36, Prague, Czech Republic F. Fleuret Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France S.L.", "Fokin Russian Research Center “Kurchatov Institute\", Moscow, 123098 Russia J.E.", "Frantz Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA A. Franz Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA A.D. Frawley Florida State University, Tallahassee, Florida 32306, USA Y. Fukao RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan T. Fusayasu Nagasaki Institute of Applied Science, Nagasaki-shi, Nagasaki 851-0193, Japan K. Gainey Abilene Christian University, Abilene, Texas 79699, USA C. Gal Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA A. Garishvili University of Tennessee, Knoxville, Tennessee 37996, USA I. Garishvili Lawrence Livermore National Laboratory, Livermore, California 94550, USA A. Glenn Lawrence Livermore National Laboratory, Livermore, California 94550, USA X. Gong Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA M. Gonin Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France Y. Goto RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA R. Granier de Cassagnac Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France N. Grau Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA S.V.", "Greene Vanderbilt University, Nashville, Tennessee 37235, USA M. Grosse Perdekamp University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA T. Gunji Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan L. Guo Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA H.-Å.", "Gustafsson Deceased Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden T. Hachiya RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan J.S.", "Haggerty Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA K.I.", "Hahn Ewha Womans University, Seoul 120-750, Korea H. Hamagaki Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan J. Hanks Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA K. Hashimoto RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan Physics Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan E. Haslum Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden R. Hayano Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan X.", "He Georgia State University, Atlanta, Georgia 30303, USA T.K.", "Hemmick Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA T. Hester University of California - Riverside, Riverside, California 92521, USA J.C. Hill Iowa State University, Ames, Iowa 50011, USA R.S.", "Hollis University of California - Riverside, Riverside, California 92521, USA K. Homma Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan B. Hong Korea University, Seoul, 136-701, Korea T. Horaguchi Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Y. Hori Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan S. Huang Vanderbilt University, Nashville, Tennessee 37235, USA T. Ichihara RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA H. Iinuma KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan Y. Ikeda RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan J. Imrek Debrecen University, H-4010 Debrecen, Egyetem tér 1, Hungary M. Inaba Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan A. Iordanova University of California - Riverside, Riverside, California 92521, USA D. Isenhower Abilene Christian University, Abilene, Texas 79699, USA M. Issah Vanderbilt University, Nashville, Tennessee 37235, USA A. Isupov Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia D. Ivanischev PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia B.V. Jacak[PHENIX Spokesperson: ][email protected] Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA M. Javani Georgia State University, Atlanta, Georgia 30303, USA J. Jia Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA X. Jiang Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA B.M.", "Johnson Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA K.S.", "Joo Myongji University, Yongin, Kyonggido 449-728, Korea D. Jouan IPN-Orsay, Universite Paris Sud, CNRS-IN2P3, BP1, F-91406, Orsay, France J. Kamin Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA S. Kaneti Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA B.H.", "Kang Hanyang University, Seoul 133-792, Korea J.H.", "Kang Yonsei University, IPAP, Seoul 120-749, Korea J.S.", "Kang Hanyang University, Seoul 133-792, Korea J. Kapustinsky Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA K. Karatsu Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan M. Kasai RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan Physics Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan D. Kawall Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003-9337, USA RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA A.V.", "Kazantsev Russian Research Center “Kurchatov Institute\", Moscow, 123098 Russia T. Kempel Iowa State University, Ames, Iowa 50011, USA A. Khanzadeev PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia K.M.", "Kijima Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan B.I.", "Kim Korea University, Seoul, 136-701, Korea C. Kim Korea University, Seoul, 136-701, Korea D.J.", "Kim Helsinki Institute of Physics and University of Jyväskylä, P.O.Box 35, FI-40014 Jyväskylä, Finland E.J.", "Kim Chonbuk National University, Jeonju, 561-756, Korea H.J.", "Kim Yonsei University, IPAP, Seoul 120-749, Korea K.-B.", "Kim chonbuk Y.-J.", "Kim University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA Y.K.", "Kim Hanyang University, Seoul 133-792, Korea E. Kinney University of Colorado, Boulder, Colorado 80309, USA Á.", "Kiss ELTE, Eötvös Loránd University, H - 1117 Budapest, Pázmány P. s. 1/A, Hungary E. Kistenev Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA J. Klatsky Florida State University, Tallahassee, Florida 32306, USA D. Kleinjan University of California - Riverside, Riverside, California 92521, USA P. Kline Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA Y. Komatsu Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan B. Komkov PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia J. Koster University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA D. Kotchetkov Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA D. Kotov Saint Petersburg State Polytechnic University, St. Petersburg, 195251 Russia A. Král Czech Technical University, Zikova 4, 166 36 Prague 6, Czech Republic F. Krizek Helsinki Institute of Physics and University of Jyväskylä, P.O.Box 35, FI-40014 Jyväskylä, Finland G.J.", "Kunde Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA K. Kurita RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan Physics Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan M. Kurosawa RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan Y. Kwon Yonsei University, IPAP, Seoul 120-749, Korea G.S.", "Kyle New Mexico State University, Las Cruces, New Mexico 88003, USA R. Lacey Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA Y.S.", "Lai Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA J.G.", "Lajoie Iowa State University, Ames, Iowa 50011, USA A. Lebedev Iowa State University, Ames, Iowa 50011, USA B. Lee Hanyang University, Seoul 133-792, Korea D.M.", "Lee Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA J. Lee Ewha Womans University, Seoul 120-750, Korea K.B.", "Lee Korea University, Seoul, 136-701, Korea K.S.", "Lee Korea University, Seoul, 136-701, Korea S.H.", "Lee Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA S.R.", "Lee Chonbuk National University, Jeonju, 561-756, Korea M.J. Leitch Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA M.A.L.", "Leite Universidade de São Paulo, Instituto de Física, Caixa Postal 66318, São Paulo CEP05315-970, Brazil M. Leitgab University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA B. Lewis Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA S.H.", "Lim Yonsei University, IPAP, Seoul 120-749, Korea L.A. Linden Levy University of Colorado, Boulder, Colorado 80309, USA A. Litvinenko Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia M.X.", "Liu Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA B.", "Love Vanderbilt University, Nashville, Tennessee 37235, USA C.F.", "Maguire Vanderbilt University, Nashville, Tennessee 37235, USA Y.I.", "Makdisi Collider-Accelerator Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA M. Makek Weizmann Institute, Rehovot 76100, Israel A. Malakhov Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia A. Manion Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA V.I.", "Manko Russian Research Center “Kurchatov Institute\", Moscow, 123098 Russia E. Mannel Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA S. Masumoto Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan M. McCumber University of Colorado, Boulder, Colorado 80309, USA P.L.", "McGaughey Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA D. McGlinchey Florida State University, Tallahassee, Florida 32306, USA C. McKinney University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA M. Mendoza University of California - Riverside, Riverside, California 92521, USA B. Meredith University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA Y. Miake Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan T. Mibe KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan A.C. Mignerey University of Maryland, College Park, Maryland 20742, USA A. Milov Weizmann Institute, Rehovot 76100, Israel D.K.", "Mishra Bhabha Atomic Research Centre, Bombay 400 085, India J.T.", "Mitchell Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Y. Miyachi RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan S. Miyasaka RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan A.K.", "Mohanty Bhabha Atomic Research Centre, Bombay 400 085, India H.J.", "Moon Myongji University, Yongin, Kyonggido 449-728, Korea D.P.", "Morrison Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA S. Motschwiller Muhlenberg College, Allentown, Pennsylvania 18104-5586, USA T.V.", "Moukhanova Russian Research Center “Kurchatov Institute\", Moscow, 123098 Russia T. Murakami Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan J. Murata RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan Physics Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan T. Nagae Kyoto University, Kyoto 606-8502, Japan S. Nagamiya KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan J.L.", "Nagle University of Colorado, Boulder, Colorado 80309, USA M.I.", "Nagy Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences (Wigner RCP, RMKI) H-1525 Budapest 114, POBox 49, Budapest, Hungary I. Nakagawa RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Y. Nakamiya Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan K.R.", "Nakamura Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan T. Nakamura RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan K. Nakano RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan C. Nattrass University of Tennessee, Knoxville, Tennessee 37996, USA A. Nederlof Muhlenberg College, Allentown, Pennsylvania 18104-5586, USA M. Nihashi Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan R. Nouicer Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA N. Novitzky Helsinki Institute of Physics and University of Jyväskylä, P.O.Box 35, FI-40014 Jyväskylä, Finland A.S. Nyanin Russian Research Center “Kurchatov Institute\", Moscow, 123098 Russia E. O'Brien Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA C.A.", "Ogilvie Iowa State University, Ames, Iowa 50011, USA K. Okada RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA A. Oskarsson Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden M. Ouchida Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan K. Ozawa Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan R. Pak Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA V. Papavassiliou New Mexico State University, Las Cruces, New Mexico 88003, USA B.H.", "Park Hanyang University, Seoul 133-792, Korea I.H.", "Park Ewha Womans University, Seoul 120-750, Korea S.K.", "Park Korea University, Seoul, 136-701, Korea S.F.", "Pate New Mexico State University, Las Cruces, New Mexico 88003, USA L. Patel Georgia State University, Atlanta, Georgia 30303, USA H. Pei Iowa State University, Ames, Iowa 50011, USA J.-C. Peng University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA H. Pereira Dapnia, CEA Saclay, F-91191, Gif-sur-Yvette, France V. Peresedov Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia D.Yu.", "Peressounko Russian Research Center “Kurchatov Institute\", Moscow, 123098 Russia R. Petti Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA C. Pinkenburg Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA R.P.", "Pisani Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA M. Proissl Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA M.L.", "Purschke Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA H. Qu Abilene Christian University, Abilene, Texas 79699, USA J. Rak Helsinki Institute of Physics and University of Jyväskylä, P.O.Box 35, FI-40014 Jyväskylä, Finland I. Ravinovich Weizmann Institute, Rehovot 76100, Israel K.F.", "Read Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA University of Tennessee, Knoxville, Tennessee 37996, USA R. Reynolds Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA V. Riabov PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia Y. Riabov PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia E. Richardson University of Maryland, College Park, Maryland 20742, USA D. Roach Vanderbilt University, Nashville, Tennessee 37235, USA G. Roche LPC, Université Blaise Pascal, CNRS-IN2P3, Clermont-Fd, 63177 Aubiere Cedex, France S.D.", "Rolnick University of California - Riverside, Riverside, California 92521, USA M. Rosati Iowa State University, Ames, Iowa 50011, USA P. Rukoyatkin Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia B. Sahlmueller Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA N. Saito KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan T. Sakaguchi Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA V. Samsonov PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia M. Sano Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan M. Sarsour Georgia State University, Atlanta, Georgia 30303, USA S. Sawada KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan K. Sedgwick University of California - Riverside, Riverside, California 92521, USA R. Seidl RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA A. Sen Georgia State University, Atlanta, Georgia 30303, USA R. Seto University of California - Riverside, Riverside, California 92521, USA D. Sharma Weizmann Institute, Rehovot 76100, Israel I. Shein IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia T.-A.", "Shibata RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan K. Shigaki Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan M. Shimomura Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan K. Shoji Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan P. Shukla Bhabha Atomic Research Centre, Bombay 400 085, India A. Sickles Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA C.L.", "Silva Iowa State University, Ames, Iowa 50011, USA D. Silvermyr Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA K.S.", "Sim Korea University, Seoul, 136-701, Korea B.K.", "Singh Department of Physics, Banaras Hindu University, Varanasi 221005, India C.P.", "Singh Department of Physics, Banaras Hindu University, Varanasi 221005, India V. Singh Department of Physics, Banaras Hindu University, Varanasi 221005, India M. Slunečka Charles University, Ovocný trh 5, Praha 1, 116 36, Prague, Czech Republic R.A. Soltz Lawrence Livermore National Laboratory, Livermore, California 94550, USA W.E.", "Sondheim Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA S.P.", "Sorensen University of Tennessee, Knoxville, Tennessee 37996, USA M. Soumya Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA I.V.", "Sourikova Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA P.W.", "Stankus Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA E. Stenlund Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden M. Stepanov Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003-9337, USA A. Ster Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences (Wigner RCP, RMKI) H-1525 Budapest 114, POBox 49, Budapest, Hungary S.P.", "Stoll Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA T. Sugitate Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan A. Sukhanov Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA J.", "Sun Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA J. Sziklai Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences (Wigner RCP, RMKI) H-1525 Budapest 114, POBox 49, Budapest, Hungary E.M. Takagui Universidade de São Paulo, Instituto de Física, Caixa Postal 66318, São Paulo CEP05315-970, Brazil A. Takahara Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan A. Taketani RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Y. Tanaka Nagasaki Institute of Applied Science, Nagasaki-shi, Nagasaki 851-0193, Japan S. Taneja Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA K. Tanida RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Department of Physics and Astronomy, Seoul National University, Seoul, Korea M.J. Tannenbaum Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA S. Tarafdar Department of Physics, Banaras Hindu University, Varanasi 221005, India A. Taranenko Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA E. Tennant New Mexico State University, Las Cruces, New Mexico 88003, USA H. Themann Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA T. Todoroki RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan L. Tomášek Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Prague 8, Czech Republic M. Tomášek Czech Technical University, Zikova 4, 166 36 Prague 6, Czech Republic Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Prague 8, Czech Republic H. Torii Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan R.S.", "Towell Abilene Christian University, Abilene, Texas 79699, USA I. Tserruya Weizmann Institute, Rehovot 76100, Israel Y. Tsuchimoto Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan T. Tsuji Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan C. Vale Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA H.W.", "van Hecke Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA M. Vargyas ELTE, Eötvös Loránd University, H - 1117 Budapest, Pázmány P. s. 1/A, Hungary E. Vazquez-Zambrano Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA A. Veicht Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA J. Velkovska Vanderbilt University, Nashville, Tennessee 37235, USA R. Vértesi Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences (Wigner RCP, RMKI) H-1525 Budapest 114, POBox 49, Budapest, Hungary M. Virius Czech Technical University, Zikova 4, 166 36 Prague 6, Czech Republic A. Vossen University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA V. Vrba Czech Technical University, Zikova 4, 166 36 Prague 6, Czech Republic Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Prague 8, Czech Republic E. Vznuzdaev PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia X.R.", "Wang New Mexico State University, Las Cruces, New Mexico 88003, USA D. Watanabe Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan K. Watanabe Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Y. Watanabe RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Y.S.", "Watanabe Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan F. Wei Iowa State University, Ames, Iowa 50011, USA R. Wei Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400, USA S.N.", "White Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA D. Winter Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA S. Wolin University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA C.L.", "Woody Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA M. Wysocki University of Colorado, Boulder, Colorado 80309, USA Y.L.", "Yamaguchi Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan R. Yang University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA A. Yanovich IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia J. Ying Georgia State University, Atlanta, Georgia 30303, USA S. Yokkaichi RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Z.", "You Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA I. Younus University of New Mexico, Albuquerque, New Mexico 87131, USA I.E.", "Yushmanov Russian Research Center “Kurchatov Institute\", Moscow, 123098 Russia W.A.", "Zajc Columbia University, New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533, USA A. Zelenski Collider-Accelerator Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA L. Zolin Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia PHENIX Collaboration Neutral-pion, $\\pi ^{0}$ , spectra were measured at midrapidity ($|y|<0.35$ ) in Au$+$ Au collisions at $\\sqrt{s_{\\rm _{NN}}}$ = 39 and 62.4 GeV and compared to earlier measurements at 200 GeV in the $1<p_T<10$  GeV/$c$ transverse-momentum ($p_T$ ) range.", "The high-$p_T$ tail is well described by a power law in all cases and the powers decrease significantly with decreasing center-of-mass energy.", "The change of powers is very similar to that observed in the corresponding $p$ +$p$ -collision spectra.", "The nuclear-modification factors ($R_{\\rm AA}$ ) show significant suppression and a distinct energy dependence at moderate $p_T$ in central collisions.", "At high $p_T$ , $R_{\\rm AA}$ is similar for 62.4 and 200 GeV at all centralities.", "Perturbative-quantum-chromodynamics calculations that describe $R_{\\rm AA}$ well at 200 GeV, fail to describe the 39 GeV data, raising the possibility that the relative importance of initial-state effects and soft processes increases at lower energies.", "A conclusion that the region where hard processes are dominant is reached only at higher $p_T$ , is also supported by the $x_T$ dependence of the $x_T$ -scaling power-law exponent.", "25.75.Dw Large transverse-momentum ($p_T$) particles produced in high-energy nucleus-nucleus (AB) collisions play a crucial role in studying the properties of the medium created in relativistic heavy-ion collisions.", "Most hadrons at sufficiently high $p_T$ are fragmentation products of hard-scattered partons and their production rate in vacuum, as measured in $p$ +$p$ collisions, is well described by perturbative quantum chromodynamics (pQCD) [1].", "In the absence of any nuclear effects the production rate in relativistic heavy-ion collisions in the pQCD regime, i.e.", "at sufficiently high $p_T$, would scale with the increased probability that a hard scattering occurs, due to the large number of nucleons.", "This probability is characterized by the nuclear thickness function $T_{\\rm AB}$  [2].", "However, such scaling has been violated to various degrees depending both on collision energy, $\\sqrt{s_{\\rm _{NN}}}$, and hadron $p_T$.", "At lower collision energies, the hadron yield is enhanced above the expected scaling.", "This was first observed in $p$ +A and this enhancement is generally attributed to multiple soft scattering (“Cronin effect”[3]), and is presumed to occur in ion-ion collisions as well.", "Initial parton distribution functions in nuclei (nPDF) are different from those in protons [4].", "Finally, if a dense, colored medium is formed in the AB collision, the hard-scattered parton may traverse some of it, losing energy in the process.", "Therefore, the observed yield at a given (high) $p_T$ will be lower than that expected from $T_{\\rm AB}$ scaling, exhibiting “suppression” or “jet quenching,” described in terms of the nuclear-modification factor, $R_{\\rm AA}$ (see Eq.", "(1)).", "Alternatively, other studies divide the yields for heavy-ion collisions at one energy with those for the same colliding species at a lower energy Au+Au , rather than scaled $p$ +$p$ reference data, to study energy and centrality scaling [5].", "One of the first discoveries at the Relativistic Heavy-Ion Collider (RHIC) was a very large hadron suppression at high $p_T$ (above $\\approx $ 3 GeV/$c$) in $\\sqrt{s_{\\rm _{NN}}}$ = 130 and 200 GeV Au+Au collisions [6], [7], [8], [9].", "This suppression was attributed to the dominance of parton energy loss in the medium, i.e.", "to final state effects.", "To test this hypothesis, the same measurements were performed in $d$ +Au collisions [10], where the formation of the hot, dense partonic medium is not expected, and initial-state effects (if any) prevail.", "No suppression in $d$ +Au data was observed leaving little (if any) room for the initial-state effects as the origin of the large jet quenching observed in ${\\rm Au+Au}$.", "Studies with the lighter Cu+Cu system at three energies ($\\sqrt{s_{\\rm _{NN}}}$ = 22.4, 62.4 and 200 GeV [11]) have revealed that at $\\sqrt{s_{\\rm _{NN}}}$ = 22.4 GeV mechanisms that enhance $R_{\\rm AA}$ ($>1$ ) dominate at all centralities.", "Note, however, that this data set had very limited $p_T$ range ($p_T$ $<$ 4 GeV/$c$).", "At 62.4 GeV, jet quenching overwhelmes any enhancement and leads to a suppression ($R_{\\rm AA}$ $<1$ ) in more central collisions.", "The low-energy scan at RHIC provides an opportunity to study the transition from enhancement ($R_{\\rm AA}$ $>1$ ) to suppression ($R_{\\rm AA}$ $<1$ ) and the evolution of $R_{\\rm AA}$ with collision energy, centrality and $p_T$.", "The results put constraints on energy-loss models (see [12] and references therein).", "Here, we present new measurements by the PHENIX experiment at RHIC of $\\pi ^{0}$ invariant yields and $R_{\\rm AA}$ in ${\\rm Au+Au}$ collisions at $\\sqrt{s_{\\rm _{NN}}}$ = 39 and 62.4 GeV.", "The data were taken during the 2010 run and the $p_T$ limits (statistics) were 8 GeV/$c$ and 10 GeV/$c$, respectively.", "Reference $p$ +$p$-collision data for $\\sqrt{s_{\\rm _{NN}}}$ = 62.4 GeV were taken in the same experiment in the 2006 run [1], while for $\\sqrt{s_{\\rm _{NN}}}$ = 39 GeV, data measured in the FERMILAB experiment E706 were used [13].", "Neutral pions were measured on a statistical basis via their $\\pi ^0 \\rightarrow \\gamma \\gamma $ decay branch with the electromagnetic calorimeter (EMCal) [14].", "The EMCal comprises two calorimeter types: 6 sectors of lead scintillator sampling calorimeter (PbSc) and 2 sectors of lead glass Čerenkov calorimeter (PbGl).", "Each sector is located $\\approx 5$  m from the beamline and subtends $|\\eta | < 0.35$ in pseudorapidity and $\\Delta \\phi $ = 22.5$^\\circ $ in azimuth.", "This Letter presents results obtained with the PbSc sectors only.", "The segmentation of the PbSc ($\\Delta \\eta \\times \\Delta \\phi = 0.01\\times 0.01$ ) ensures that the two photons from the $\\pi ^0 \\rightarrow \\gamma \\gamma $ decays are very well resolved up to $p_T$ $<$ 12 GeV/$c$, i.e.", "across the entire $p_T$ range of this measurement.", "The results are based on data sets of $\\,3.5\\cdot 10^{8}\\,$ and $\\,7.0\\cdot 10^{8}\\,$ minimum bias ${\\rm Au+Au}$ events at 39 and 62.4 GeV, respectively.", "The minimum bias (MB) trigger for both $\\sqrt{s_{\\rm _{NN}}}$ = 39 and 62.4 GeV was provided by the Beam-Beam-Counters (BBC) [15], located close to the beam axis in both directions and covering $3.0\\le |\\eta | \\le 3.9$ .", "In order to reduce background at least two hits were required in both BBC's.", "This condition selects $\\sim 86$ % of the total inelastic cross section.", "The centrality selection in ${\\rm Au+Au}$ collisions at both energies was based on the charged signal sum of the BBC's, which is proportional to the charged particle multiplicity.", "For each centrality the average number of binary collisions ($\\left<N_{\\rm coll}\\right>$ ) and the number of participants ($\\left<N_{\\rm part}\\right>$ ) were calculated using a Glauber model [2] based Monte Carlo code.", "Table: Sources of systematic uncertainties and their relativeeffect (in %) on the invariant yieldsThe PHENIX analysis of neutral pions is described in detail elsewhere [9].", "Table REF lists the sources of systematic uncertainties on the extracted-$\\pi ^{0}$ invariant yields in this analysis.", "They can be divided into three different categories: (1) Type-A, $p_T$-uncorrelated; (2) Type-B, $p_T$-correlated, where the correlation may be an arbitrary smooth function; (3) Type-C, $p_T$-correlated, where all points move by the same fraction up or down.", "The main sources of systematic uncertainties in the $\\pi ^{0}$ measurement are the energy scale, yield extraction and particle-identification (PID) efficiency correction.", "Figure: (Color online)Invariant yields of π 0 \\pi ^{0} in Au + Au {\\rm Au+Au} at s NN \\sqrt{s_{\\rm _{NN}}} = 39 GeV (a)and 62.4 GeV (b) in all centralities and minimumbias.", "Only statistical uncertainties are shown.Figure REF shows the invariant yields of the $\\pi ^{0}$ s for all centralities and also in minimum bias collisions.", "From fitting the $\\sqrt{s_{\\rm _{NN}}}$ = 39 and 62.4 GeV minimum bias spectra with a power law function ($\\propto p_T^{n}$ ) for $p_T$ $>$ 4 GeV/$c$, we obtained powers $n_{39} = -13.04~\\pm ~0.08$ and $n_{62.4} = -10.60~\\pm ~0.03 $ , respectively, significantly steeper than at $\\sqrt{s_{\\rm _{NN}}}$ =200 GeV, where $n_{200} = -8.06~\\pm ~0.01$ for MB collisions [9].", "The slopes of the corresponding $p$ +$p$-collision spectra are somewhat different, but comparable, $n_{39}^{pp}=-13.59~\\pm ~0.21$ , $n_{62.4}^{pp}=-9.82 \\pm 0.18$ and $n_{200}^{pp}=-8.22 \\pm 0.09$ , respectively.", "Nuclear effects on the $\\pi ^{0}$ production are quantified using the nuclear modification factor $R_{\\rm AA} (p_T) = \\frac{(1/N_{\\rm AA}^{\\rm evt}) {\\rm d}^2N_{\\rm AA}^{\\pi ^0}/{\\rm d}p_T{\\rm dy}}{\\left<T_{\\rm AB}\\right>\\times {\\rm d}^2\\sigma _{\\rm pp}^{\\pi ^0}/{\\rm d}p_T{\\rm dy}},$ where $\\sigma _{pp}^{\\pi ^0}$ is the production cross section of $\\pi ^{0}$ in $p$ +$p$ collisions, and $\\left<T_{\\rm AB}\\right> = \\left<N_{\\rm coll}\\right>/\\sigma _{pp}^{\\rm inel}$ is the nuclear thickness function averaged over the range of impact parameters contributing to the given centrality class according to the Glauber model.", "Thus $R_{\\rm AA}$ compares the yield observed in $A+A$ collisions to the yield expected from the superposition of $N_{\\rm coll}$ independent $p$ +$p$ interactions.", "In the absence of nuclear effects, $R_{\\rm AA}$ should be equal to unity.", "However, $R_{\\rm AA}$ $\\approx 1$ does not necessarily imply the absence of suppression, it may also indicate a balance between enhancing and depleting mechanisms.", "In order to calculate $R_{\\rm AA}$, a reference $p_T$ distribution in $p$ +$p$ collisions is needed.", "Preferably this is measured with the same detector, in which case many systematic uncertainties cancel in the ratio.", "The PHENIX experiment has measured the $\\pi ^{0}$ cross section in $p$ +$p$ collisions at $\\sqrt{s_{\\rm _{NN}}}$ = 62.4  GeV [1] but only up to $p_T$ = 7 GeV/$c$ while the current Au+Au measurement reaches up to 10 GeV/$c$.", "Hence the $p$ +$p$ data were fitted with a power law function between $4.5 < p_T < 7$  GeV/$c$ and then extrapolated.", "The systematic uncertainty resulting from this extrapolation reaches 20% at 10 GeV/$c$, estimated from a series of fits, where each time one or more randomly selected points are omitted and the remaining points are re-fitted.", "So far PHENIX has not measured the $p$ +$p$ spectrum of $\\pi ^{0}$ at $\\sqrt{s_{\\rm _{NN}}}$ = 39 GeV.", "Therefore, data from the Fermilab experiment E706 [13] were used.", "However, the E706 acceptance ($-1.", "< |\\eta | < 0.5$ ) is different from that of PHENIX ($|\\eta |<0.35$ ), and since $dN/d\\eta $ is not flat, a $p_T$-dependent correction was applied to the E706 data.", "This correction factor was determined from a pythia simulation by means of the ratio of yields (normalized per unit rapidity) when calculated from the observed yield in the PHENIX and E706 acceptance windows.", "The systematic uncertainty of the correction is 1–2% at 3 GeV/$c$ but reaches 20% at 8 GeV/$c$.", "Figure: (Color online)Nuclear modification factor (R AA R_{\\rm AA}) of π 0 \\pi ^{0} in Au + Au {\\rm Au+Au} collisions in most central 0–10% (a) and mid-peripheral40–60% (b).", "Error bars are the quadratic sum of statisticaland p T p_T-correlated systematic uncertainties (including systematicuncertainties from the pp+pp-collision reference).Boxes around 1 are the quadratic sum of the C-typeuncertainties combined with the N coll N_{\\rm coll} uncertainties.", "These arefully correlated between different energies.Also shown for central collisions are pQCDcalculations with regular Cronin-effect (solid lines) and with theCronin-effect reduced by a factor of two for all three energies (bands).Figure REF shows the nuclear modification factor of $\\pi ^{0}$ s measured in ${\\rm Au+Au}$ collisions at $\\sqrt{s_{\\rm _{NN}}}$ = 39, 62.4 and 200 GeV (data from [9]) as a function of $p_T$ for most central collisions (a) and 40–60% centrality (b).", "In the most central collisions (0–10%) there is a significant suppression for all three energies, while in mid-peripheral collisions (40–60%) at $\\sqrt{s_{\\rm _{NN}}}$ =39 GeV, $R_{\\rm AA}$ is consistent with unity above $p_T$ $>$ 3 GeV/$c$.", "For 0–10% pQCD calculations [17], [16] are also shown.", "The solid curves are calculated with the same parametrization that was successful for 200 GeV ${\\rm Au+Au}$ data (and also 200 GeV ${\\rm Cu+Cu}$  [11]).", "Neither the 62.4, nor the 39 GeV data are consistent with the predictions.", "The only qualitative agreement is that the turnover point of the $R_{\\rm AA}$ curves moves to higher $p_T$ with lower collision energy as observed in the data.", "The bands are calculated within the same framework but with the Cronin-effect reduced and the energy loss varied by $\\pm $ 10%.", "The 200 GeV data are still well described, the 62.4 GeV data are consistent within uncertainties, but the 39 GeV $R_{\\rm AA}$, particularly the shape, is inconsistent with the corresponding band.", "Coupled with the observations that the slopes at high $p_T$ become much steeper, but the bulk properties (like elliptic flow, energy density, apparent temperature) change only slowly in the collision energy range in question, it is quite concievable that hard scattering as a source of particles at a given $p_T$ becomes completely dominant only at higher transverse momentum, i.e.", "jet quenching will be “masked\" up to higher $p_T$.", "Note that while the shapes at lower $p_T$ are different, at $p_T$ $>\\approx $ 7 GeV/$c$ $R_{\\rm AA}$ is essentially the same for the 62.4 and 200 GeV data, irrespective of centrality (see also Fig.", "REF ).", "While in the 39 GeV data $R_{\\rm AA}$ also shows a decreasing trend at higher $p_T$, unfortunately the $p_T$ reach of the current data sample precludes any conclusion as to what would happen to their $R_{\\rm AA}$ at even higher $p_T$.", "Figure: (Color online)Nuclear modification factor averagedfor p T >6p_T>6 GeV/cc.", "Uncertainties are shown as error bars(statictical), boxes (sum of p T p_T-uncorrelated and N coll N_{\\rm coll}), boxesaround one (Type B and C and uncertainties from the pp+pp-collision reference).Figrue REF shows $p_T$-averaged $R_{\\rm AA}$ as a function of the number of participants.", "The averaging was done above $p_T>$ 6 GeV/$c$.", "Our first observation is that $R_{\\rm AA}$ decreases with increasing centrality even for the lowest-energy system.", "Similarly, as already discussed in the context of Fig.", "REF , at high enough $p_T$ the suppression is the same at 62.4 and 200 GeV, at all centralities.", "This is remarkable because the power $n$ of the fit to the spectra changes approximately by two units from 200 to 62.4 GeV, so the average momentum loss of the partons also has to be different in order to compensate the effect of the changing slope.", "The average momentum loss is usually defined by the fractional momentum shift $\\delta p_T / p_T$ between the corresponding ${\\rm Au+Au}$ and $T_{\\rm AA}$ -scaled $p$ +$p$ spectra as follows.", "Since the power law tails of the $p$ +$p$ and ${\\rm Au+Au}$ spectra are similar, they can be fitted simultaneously with the same function and same power $n$ $f(p_T) = \\frac{A}{(p_T (1 + \\delta p_T / p_T))^n}$ with $\\delta p_T$ being the horizontal shift between the scaled $p$ +$p$ and the ${\\rm Au+Au}$ spectra.", "In panel (a) of Fig.", "REF , the observed fractional momentum shifts are shown for central collisions, as a function of the ${\\rm Au+Au}$ $p_T$.", "Figure: (Color online)(a) Fractional momentum shift δp T /p T \\delta p_T / p_T between Au + Au {\\rm Au+Au} and T AA T_{\\rm AA}-scaled pp+pp data as a function of the Au + Au {\\rm Au+Au} p T p_T.", "(b) Power n eff n_{\\rm eff} of x T x_T-scaling for pp+pp and Au + Au {\\rm Au+Au} (minimum bias)at various collision energies.Inclusive single-particle spectra at sufficiently high $p_T$ and collision energy were predicted to exhibit scaling with the variable $x_T=2p_T/\\sqrt{s}$ such that the production cross section can be written in a form [18], [19] $E\\frac{d^3\\sigma }{dp^3} =\\frac{1}{\\sqrt{s}^{n(x_T,\\sqrt{s})}} G(x_T)$ where $G(x_T)$ is a universal function and $n(x_T,\\sqrt{s})$ characterizes the specific process [19].", "The scaling power $n_{\\rm eff}(x_T)$ between any pair of $\\sqrt{s_{\\rm _{NN}}}$ energies is then calculated as $n_{\\rm eff}(x_T) = \\frac{log(Yield(x_T,\\sqrt{s_1})/Yield(x_T,\\sqrt{s_2}))}{log(\\sqrt{s_2}/\\sqrt{s_1})}$ In panel (b) of Fig.", "REF , $n_{\\rm eff}(x_T)$ is shown when comparing invariant-$\\pi ^{0}$ yields in $p$ +$p$ and ${\\rm Au+Au}$ collisions at different energies.", "Both the shape and the magnitude of $n_{\\rm eff}(x_T)$ is similar for the 62.4/200 GeV $p$ +$p$ and ${\\rm Au+Au}$ as well as for the 39/200 GeV $p$ +$p$ data.", "The rise of $n_{\\rm eff}(x_T)$ at lower $x_T$ can be attributed to the dominance of soft processes [20], while at higher $x_T$ they deviate strongly from leading-twist scaling predictions [21], [19].", "However, the shape of $n_{\\rm eff}(x_T)$ in the 39 and 200 GeV ${\\rm Au+Au}$ comparison is very different from all others.", "It may not even reach its maximum in the measured $x_T$ range, and its constant rise is similar to the rise observed in the low-$x_T$ (soft) region of the other data shown.", "One possible explanation could be that while present, hard scattering is still not the overwhelming source of high-$p_T$ $\\pi ^{0}$ s in the currently available $p_T$ range in 39 GeV ${\\rm Au+Au}$ collisions.", "In summary, the $\\pi ^{0}$ $p_T$ spectra were measured in ${\\rm Au+Au}$ collisions at two different energies, $\\sqrt{s_{\\rm _{NN}}}$ = 39 and 62.4 GeV, and compared to the earlier result for $\\sqrt{s_{\\rm _{NN}}}$ = 200 GeV.", "In all cases the high $p_T$ part of the invariant yields can be well described with a single power law function.", "The powers decrease considerably at lower $\\sqrt{s_{\\rm _{NN}}}$, and since the soft processes change only slowly with collision energy, jet quenching might be “masked” up to higher transverse momenta.", "The high-$p_T$ $\\pi ^{0}$ yields in ${\\rm Au+Au}$ at 62.4 GeV are suppressed, and above $p_T$  $>$  6 GeV/$c$ the data points are comparable with the 200 GeV results at all centralities.", "The $\\pi ^{0}$ yields in ${\\rm Au+Au}$ at 39 GeV are suppressed in the most central collisions, but no suppression is apparent in more peripheral collisions.", "At lower energies, a decreasing momentum shift compensates for the steeper slopes at high $p_T$, making the $R_{\\rm AA}$'s comparable, in fact, identical in the case of 62.4 and 200 GeV.", "When related to 200 GeV, $n_{\\rm eff}(x_T)$ is similar for 62.4 and 39 GeV $p$ +$p$ and 62.4 GeV ${\\rm Au+Au}$, but very different for the 39 GeV ${\\rm Au+Au}$ data.", "The new data provided in a wide energy range of ${\\rm Au+Au}$ collisions will help to constrain energy-loss models.", "We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions.", "We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (P. R. China), Ministry of Education, Youth and Sports (Czech Republic), Jyväskylä University (Finland), Centre National de la Recherche Scientifique, Commissariat à l'Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Ministry of Industry, Science and Tekhnologies, Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), Hungarian National Science Fund, OTKA (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), National Research Foundation and WCU program of the Ministry Education Science and Technology (Korea), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and Wallenberg Foundation (Sweden), the U.S.", "Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the Hungarian American Enterprise Scholarship Fund, and the US-Israel Binational Science Foundation." ] ]

[ [ "On the structure and stability of magnetic tower jets" ], [ "Abstract Modern theoretical models of astrophysical jets combine accretion, rotation, and magnetic fields to launch and collimate supersonic flows from a central source.", "Near the source, magnetic field strengths must be large enough to collimate the jet requiring that the Poynting flux exceeds the kinetic-energy flux.", "The extent to which the Poynting flux dominates kinetic energy flux at large distances from the engine distinguishes two classes of models.", "In magneto-centrifugal launch (MCL) models, magnetic fields dominate only at scales $\\lesssim 100$ engine radii, after which the jets become hydrodynamically dominated (HD).", "By contrast, in Poynting flux dominated (PFD) magnetic tower models, the field dominates even out to much larger scales.", "To compare the large distance propagation differences of these two paradigms, we perform 3-D ideal MHD AMR simulations of both HD and PFD stellar jets formed via the same energy flux.", "We also compare how thermal energy losses and rotation of the jet base affects the stability in these jets.", "For the conditions described, we show that PFD and HD exhibit observationally distinguishable features: PFD jets are lighter, slower, and less stable than HD jets.", "Unlike HD jets, PFD jets develop current-driven instabilities that are exacerbated as cooling and rotation increase, resulting in jets that are clumpier than those in the HD limit.", "Our PFD jet simulations also resemble the magnetic towers that have been recently created in laboratory astrophysical jet experiments." ], [ "Introduction", "Non-relativistic jets are observed in the vicinities of many Protostellar Objects, Young Stellar Objects (YSOs) and post-AGB stars.", "Plausible models suggest that jets are launched and collimated by a symbiosis of accretion, rotation and magnetic mechanisms, which occur at the jet “central engine” (see [42], for a review).", "The jet material must be accelerated to velocities beyond the escape speed and magnetically mediated launch models are often favoured because they can provide the needed directed momentum (see Livio 2004, Pudritz 2004 for reviews).", "Astrophysical jets are expected to be Poynting flux dominated (PFD) close to their engine.", "It is however unclear how far from the launch region the Poynting flux continues to dominate over kinetic energy flux, or whether the jets eventually become essentially hydrodynamic [4].", "The difference between these two possibilities is a difference between two magnetically launched outflow classes: (1) magnetocentrifugal jets ([5]; [40]; [3]; [34]), in which magnetic fields only dominate out to the Alfvén radius, or (2) Poynting flux dominated magnetic tower jets ([45]; [30]; [50]; [29]; [37]) in which magnetic fields dominate the jet structure, acting as a magnetic piston over very large distances from the engine.", "Indeed, magnetic fields with initially poloidal (radial and vertical) dominant geometries anchored to accretion discs have been shown to form tall, highly wound and helical magnetic structures, or magnetic towers, that expand vertically when laterally supported in pressure equilibrium with the ambient gas ([45], [30], [31]).", "PFD jets carry large electric currents which generate strong, tightly wound helical magnetic fields around the jet axis.", "Simulations of such jets have found that magnetic fields play a role in the formation of current-driven kink instabilities and the stabilization of Kelvin-Helmholtz (KH) modes (e.g.", "see [37]).", "However, while the correlation between the mechanical power of astrophysical jets and their main observable features (e.g.", "length, velocity, cocoon geometry, etc.)", "has been widely studied for kinetic-energy dominated jets, this is not the case for PFD magnetic tower jets.", "Recently, magnetized jets have been produced in laboratory experiments.", "These flows appear to exhibit key aspects of magnetic tower evolution ([25], [47]).", "In these experiments, performed on Pulsed Power current generators, the local injection of purely toroidal magnetic energy produced high Mach number ($\\sim \\,$ 20), fully radiative and fully magnetized jets.", "These magnetic towers exhibit Poynting flux dominated cavities with $\\beta <1$ (where $\\beta $ is the ratio of thermal to magnetic pressures) which expand supersonically into an unmagnetized ambient medium.", "Within the cavity, a central jet forms via hoop stresses.", "While the body of these jets has $\\beta < 1$ , their core is a high $\\beta $ , kinetic energy dominated flow.", "The central jet evolution of these experiments also showed the growth of current-driven, non-linear instabilities, in particular the kink, $m=\\,$ 1, mode.", "As a result, the laboratory jets are eventually corrugated and become a collimated chain of magnetized “clumps” [25].", "These experiments were then modeled via resistive MHD simulations specifically developed for laboratory studies [10], where the details of the flow, including current distributions, were followed.", "The break-up of the jet into a sequence of collimated clumps has been suggested as an explanation for clumpy flows observed in YSO outflow systems (Hartigan & Morse 2007; Yirak et al.", "2010; Hartigan et al.", "2011).", "We note that the effect of plasma cooling via optically thin radiation has not been followed before in simulations of magnetic tower jets.", "Studies of weakly magnetized, kinetic energy-dominated jets show that this type of cooling can make the flow more susceptible to instabilities, such as KH modes ([17], and references therein).", "Recently, [38] studied magnetocentrifugally launched jets with 2D radiation-MHD simulations.", "Ohsuga et al.", "found that the strength of radiative cooling, which they control by changing the plasma density, affects the structure and evolution of both accretion disks and their associate jets.", "Although we follow thermal energy losses in the present study, we do not compute radiate transfer.", "In this paper we study the effects that thermal energy losses and rotation, independently of one another, have on the stability of PFD magnetic towers.", "For comparison we also run simulations of collimated asymptotically HD jets.", "Such HD jets could represent the asymptotic propagation regimes of magneto-centrifugally launched flows, which are distinct from PFD ones in that the latter remain dominated by magnetic flux out to much larger distances.", "Our comparison allows us to articulate how PDF flows differ from their hydrodynamic counterparts.", "This paper is organized as follows: in section  we describe the methodology and numerical code that we use for this study as well as our implementation of the gas, the velocity and the magnetic field.", "The results of our simulations are presented in section , where we follow the evolution, structure and stability of our model jets.", "In section  we discuss the implications of our simulations and how they compare with specific laboratory experiments and astronomical observations.", "Finally, we conclude in section ." ], [ "Model and Initial Set-up", "We model PFD and HD jets numerically by solving the equations of ideal (i.e.", "no explicit microphysical diffusivities) magnetohydrodynamics (MHD) in three-dimensions.", "In non-dimensional conservative form these are given by $\\frac{\\partial \\rho }{\\partial t} + \\nabla \\cdot (\\rho \\mbox{\\bf V}) &=&\\dot{\\rho }_\\mathrm {inj}\\\\\\frac{\\partial (\\rho {\\bf V})}{\\partial t} +\\nabla \\cdot \\left( \\rho {\\bf V V } + p \\, {\\bf \\hat{I}} + (B^2/2) {\\bf \\hat{I}} -{\\bf B B}\\right) &= &\\dot{\\bf P}_\\mathrm {inj}\\\\\\frac{\\partial E}{\\partial t} + \\nabla \\cdot \\left[\\left(E+p + B^2/2 \\right){\\bf V}-{\\bf B}({\\bf V}\\cdot {\\bf B})\\right] &=&\\dot{E}_\\mathrm {inj} - \\Delta (T)\\\\\\frac{\\partial {\\bf B}}{\\partial t} - \\nabla \\times ( {\\bf V}\\times {\\bf B}) &= &\\dot{{\\bf B}}_\\mathrm {inj},$ where $\\rho $ , $p$ , V, B and Î are the gas density, thermal pressure, flow velocity, magnetic field and the unitary tensor, respectively.", "In (3), $E=p/(\\gamma -1)+\\rho V^2/2+B^2/2$ and represents the total energy density whereas $\\gamma $ is the ratio of specific heats.", "We have implemented source terms in the right hand side of equations (1)–(4) to account for the injection of mass, momentum, total energy, and magnetic flux.", "Since the cross sectional area of the jet base is fixed, these injections are respectively accomplished by injecting a mass density per unit time $\\dot{\\rho }_\\mathrm {inj}$ , a momentum flux, $\\dot{\\bf P}_\\mathrm {inj}$ , total energy flux minus radiation loss $\\dot{E}_\\mathrm {inj}-\\Delta (T)$ and magnetic field per unit time $\\dot{{\\bf B}}_\\mathrm {inj}$ .", "We solve these equations using the adaptive mesh refinement (AMR) numerical code AstroBEAR2.0https://clover.pas.rochester.edu/trac/astrobear/wiki which uses the single step, second-order accurate, shock capturing CTU+CT [16] scheme [12], [9].", "While AstroBEAR2.0 is able to compute several microphysical processes, such as gas self-gravity and heat conduction, we do not consider these in the present study.", "Our computational domain is defined within $|x|,|y|\\le \\,$ 160 AU and 0$\\le z \\le $ 400 AU, where 20 AU is equivalent to one computational length unit.", "We use a coarse grid of 64$\\times $ 64$\\times $ 80 cells plus two levels of AMR refinement which attain an effective resolution of 1.25 AU.", "Outflow boundary conditions were set at the left and right domain faces of both $x$ and $y$ , as well as in the upper $z$ face.", "At the lower $z$ face we combine two boundary conditions: reflective, in those cells located at $\\sqrt{x^2+y^2} \\ge r_e$ , and magnetic/jet source term values in those cells located at smaller radii.", "$r_e=\\,$ 31.4 AU.", "The latter represents the characteristic radius of the energy injection region, equal to the jets' radius, which is resolved by 24 cells.", "We use BlueGene$/$ Phttps://www.rochester.edu/its/web/wiki/crc/index.php/Blue_Gene/P, an IBM massively parallel processing supercomputer of the Center for Integrated Research Computing of the University of Rochester, to run simulations for about 1 day using 512 processors." ], [ "Initial conditions", "We initialize our simulations with a static gas which has a uniform density of 200 particles per cm$^{-3}$ and a temperature of 10000 K. Gas is modelled with an ideal gas equation of state and a ratio of specific heats of $\\gamma =\\,$ 5$/$ 3.", "Magnetic fields are placed in a central cylinder of equal radius and height $r_e$ .", "In cylindrical coordinates the magnetic vector potential is given by ${\\bf A}(r,z) = \\left\\lbrace \\begin{array}{c l}\\frac{r}{4} [cos(2\\,r) + 1][ cos(2\\,z) + 1 ] \\hat{\\phi } +\\frac{\\alpha }{8} [cos(2\\,r) + 1][ cos(2\\,z) + 1 ] \\hat{k},& \\mbox{for}~r,z < r_e; \\\\0, & \\mbox{for}~r,z \\ge r_e,\\end{array} \\right.$ where the parameter $\\alpha $ has units of length and determines the ratio of toroidal to poloidal magnetic fluxes.", "We use $\\alpha =\\,$ 40 (computational length units) which is an arbitrary choice, yet consistent with the wound helical magnetic configurations expected from accretion discs [5], [30], [26] and produced in high energy density laboratory experiments of magnetic towers [25], [10].", "Our choice of ${\\bf A}$ is in part motivated by the work of [26].", "However, in our model, ${\\bf A}$ is strictly localized to the central part of the grid.", "We obtain the initial magnetic field, ${\\bf B}^{\\mathrm {init}}$ , by taking the curl of ${\\bf A}$ : $\\begin{array}{c l l}B^{\\mathrm {init}}_r &=& - \\frac{\\partial }{\\partial z} (A_{\\phi })= 2 r cos^2(r) cos(z) sin(z), \\\\B^{\\mathrm {init}}_{\\phi } &=& - \\frac{\\partial }{\\partial r} (A_z)= \\alpha cos^2(z) cos(r) sin(r), \\\\B^{\\mathrm {init}}_z &=& \\frac{1}{r} \\frac{\\partial }{\\partial r} (r A_{\\phi })= 2 cos^2(z) [ cos^2(r) - r cos(r) sin(r) ].\\end{array}$ The magnetic field is normalized so that the initial thermal to magnetic pressure ratio $\\beta $ is less than unity for $r<r_e$ , and unity at $r=r_e$ .", "In Figure 1 we show profiles of the initial magnetic field components (top and middle rows) and $\\beta $ (bottom panel) as a function the distance form the origin.", "Figure: Magnetic field initial conditions.", "The field has ahelical structure which is dominated by the toroidalcomponent." ], [ "Energy injection", "We model jets by continually injecting energy into the central region of the grid, where $r,z \\le r_e$ .", "Because one of the key goals of this work is to compare the observable propagation signatures of PFD jets versus HD ones, (e.g.", "their length, velocity, density distribution), we inject either pure magnetic energy for the PFD case, hence the name magnetic towers, or pure kinetic energy for the HD case.", "We will now give details about the implementation of the jets." ], [ "Magnetic towers", "For these simulations we inject magnetic flux by adding the initial magnetic field configuration (REF ) to the instantaneous central magnetic fields, ${\\bf B}^n$ .", "i.e.", "${\\bf B}^{n+1}={\\bf B}^n+\\dot{\\bf B}_{\\mathrm {inj}} \\,dt, \\\\$ where ${\\bf B}^{n+1}$ represents the central magnetic fields ($r,z \\le r_e$ ) corresponding to the next computational timestep, $dt$ is the current timestep, $\\dot{\\bf B}_{\\mathrm {inj}}={\\bf B}^{\\mathrm {init}} \\, B_c$ and $B_c$ is the magnetic flux injection rate (see below).", "For numerical stabilization we also continually add static gas to the grid within $r,z<r_e$ .", "This is accomplished using the expression $\\rho ^{n+1}(r,z)=\\rho ^n(r,z)+ \\rho _c \\, |{\\bf B}(r,z)|^2 \\, dt, \\\\$ where $\\rho ^{n+1}(r,z)$ and $\\rho ^n(r,z)$ represent the gas densities corresponding to the next and the current timesteps, respectively.", "We set the constant factor $\\rho _c$ (which has units of kg m$^{-3}$  s$^{-1}$  T$^{-2}$ ) to be 0.01 computational units.", "Hence the average amount of injected gas is of order 0.001 $\\rho _{\\mathrm {amb}}^0$ per unit time, where $\\rho _{\\mathrm {amb}}^0$ is the initial ($t=\\,$ 0 yr) grid density of 200 particles per cm$^3$ ; very dilute.", "Because of the factor $|B^2|$ , the distribution of gas provided by (REF ) matches the gradients of magnetic pressure, thus we inject more gas at regions where the jet and magnetic cavity densities tend to be lower (section REF )." ], [ "Hydrodynamical jets", "For these simulations we continuously inject kinetic energy and gas to the cells located at $r<r_e$ and $z<$ 0, i.e.", "within the bottom $z$ boundary.", "This region is equivalent to the base of the magnetic towers (discussed above).", "We impose constant boundary conditions in this region, based on the following three assumptions.", "(1) The collimation of the HD jet is presumed to have occurred at sub-resolution scales.", "(2) the HD jet is taken to have the same time averaged, maximum propagation speed as the PFD magnetic tower, that is $v_j = v_z \\approx |{\\bf B_{\\mathrm {max}}}| (4 \\, \\pi \\, \\rho _{\\mathrm {amb}}^0)^{-1/2}.$ (3) The injected energy fluxes of the HD and PFD magnetic tower jets are taken to be equal, i.e.", "$0.5 \\rho _j v_z^3 \\, a = ( |{\\bf B}|^2 / 8 \\pi ) \\,( |{\\bf B}| ( 4 \\pi \\rho _{\\mathrm {amb}}^0)^{-1/2} ) \\, a,$ where $\\rho _j$ is the jet's density and $a$ ($=\\pi r_e^2$ ) is the area of the energy injection region.", "Hence, $\\rho _j = |{\\bf B}_{\\mathrm {max}}|^2 ( 4 \\pi v_j^2 )^{-1}.$ To ensure the condition (REF ) at all times, we set $B_c=\\,$ 10$/$ (1 time computational unit) in equation (REF ).", "We note that for the HD run, ${\\bf B}=\\,$ 0 everywhere and at all times, and that the values of ${\\bf B_{\\mathrm {max}}}$ and ${\\bf B}$ in equations (REF )–(REF ) are taken from the magnetic tower simulation (above)." ], [ "Simulations", "We carry out six simulations: three magnetic tower runs and their corresponding hydrodynamical versions.", "The adiabatic tower.", "This is a magnetic tower model which we have implemented as described in section REF .", "The cooling tower.", "This is a magnetic tower model which is identically to the adiabatic tower except for the addition of optically thin cooling which we have implemented using the tables of Dalgarno & McCray (1972) via the source term $\\Delta (T)$ in equation (3).", "The rotating tower.", "This is a magnetic tower model which is identically to the adiabatic tower except for the addition of a rotation profile at the jet base.", "This is accomplished by continually driving an azimuthal velocity to the central gas and frozen in magnetic fields of the tower.", "We use a velocity equal to the Keplerian speed corresponding to a two solar mass star.", "Specifically we impose $v_{\\phi } = \\left\\lbrace \\begin{array}{c l}\\sqrt{G 2 M_{\\odot }/r}, &\\mbox{for}~r,z < r_e; \\\\0, & \\mbox{for}~r,z \\ge r_e.\\end{array} \\right.$ Our choice of two solar masses is arbitrary but within the expected values for protostellar and young stellar object (YSO) jet engines [20].", "We note that the gas in our simulations is unaffected by gravitational forces, hence the centrifugal expansion produced by (REF ) is only balanced by magnetic pressure gradients.", "We do not expect significant dynamical differences with respect to a case in which gas was affected by gravity because we simulate jets far from the central star [32].", "Also, in our magnetic towers the magnetic fields are quite strong and the magnetic cavities contain very light gas (see below).", "The HD jet.", "This is an adiabatic hydrodynamical jet model which we have implemented as described in section REF .", "The cooling HD jet.", "This is a hydrodynamical jet model which is identical to the HD jet except for the addition of the same thermal cooling source term that we use for the cooling tower run (above).", "The rotating HD jet.", "This is an adiabatic hydrodynamical jet model which is identical to the HD jet except for the addition of the base ($r<r_e$ and $z$ within the bottom boundary of the computational domain) rotation profile described by equation (REF ).", "The structure and evolution of the HD, the cooling HD and the rotating HD jet simulations are similar in terms of their global propagation characteristics (see section REF ).", "Hence, without loss of generality, in what follows we will only discuss about the adiabatic HD jet." ], [ "Plasma structure and evolution", "In Figure REF we show the evolution of the plasma with logarithmic false color particle density maps.", "From left to right, columns in the Figure show the adiabatic, the rotating and the cooling magnetic towers, and then the HD jet.", "Time increases downward by row.", "We denote the structures in the simulation as follows, based on the left-most panel of row 2 (Figure REF ): the jet core (white plasma within $r 0.4$ ); the jet beam (lightest-orange plasma within $r 1.6$ ); the magnetic cavity (dark-orange plasma within $1.6 r 4$ , outside the jet); the contact discontinuity (CD, thin surface between the magnetic cavity and the swept up external medium).", "Beyond the CD we see the (light-orange) shocked ambient plasma.", "The simulations show that the initial helically wound magnetic field launches PFD jets via magnetic pressure gradients: The low $\\beta $ , low density cavities expand via the z-gradient of the toroidal magnetic pressure between the tower and ambient medium.", "Inside the cavity, a central jet beam forms collimated by hoop stresses of the toroidal field (section REF ).", "The field in the cavity is in turn, radially collimated by the pressure of the external high $\\beta $ plasma.", "The jets and their corresponding magnetic cavities expand and accelerate, especially along the $z$ -axis.", "This drives bow shocks on the external unmagnetized media.", "This magnetic tower evolution is consistent with the analytical model of [30], [31], as well as with previous simulations of PFD jets and magnetic towers (see e.g.", "[45], [37], [26]).", "Comparison of PFD magnetic towers with the HD jet reveals the following characteristics (Figure REF ).", "The towers propagate with very similar vertical velocities but decelerate, by about 20%, relative to the HD jet.", "This results because although the towers and the hydro jet have the same injected energy flux, the towers produce not only axial but radial expansion.", "The pre-collimated HD jet can only expand radially via a much lower thermal pressure.", "Thus all of the energy flux in the hydro-case for our set up is more efficiently directed to axial mechanical power.", "Moreover, the towers and the hydro jet show different structures: towers have a thin central jet which is susceptible to instabilities, whereas the HD jet's beam is thicker, smoother and stable.", "We consistently see lower densities in the PFD tower cavities than in that of the HD case.", "The laboratory experiment magnetic towers of [25] and [47] also show a magnetic cavity mostly void of plasma.", "The gas distribution inside the cavities shows more complex and smaller scale structures in the magnetic tower cases than in the HD one.", "We see that the magnetic towers are affected by either cooling or rotation after their early expansion phase.", "Instabilities develop in their jet beam after $\\sim \\,$ 70 yr (section REF ).", "The cocoon geometry of the cooling case (third column from left to right in Figure REF ) is the fattest.", "We find that the volume of the ambient region which is affected by the towers is smaller in the cooling case, as expected [14], [18].", "The above findings imply very different emission distributions for PFD and HD dominated jets.", "Future studies should address the creation of synthetic observations to assess these differences.", "Figure: Evolution of the plasma gas density.", "These false colorlogarithmic maps show the magnetic tower structures in the adiabatic(1st column), the rotating (2nd column) and the cooling (3rd column)cases, and the HD jet structure (4th column).", "From top to bottomtime is equal to 42, 84 and 118 yr.The evolution of the magnetic towers' gas density is consistent with that of their compressive MHD and hydrodynamic waves and shocks.", "In Figure REF we show profiles of the relevant velocities of the towers ($v_x, v_y, v_z$ , the sound speed and the Alfvén speed) along the jet axis, $r=\\,0$ , as a function of cooling, base rotation, and time.", "During their early stable propagation phase, the jet cores are mostly sub-Alfvénic and trans-sonic, independent of cooling or rotation.", "Fast-forward compressive MHD (FF) and hydrodynamic bow shocks are evident in the ambient medium, ahead of the jet heads.", "Some evolutionary features are worth noting.", "From figure 3 we see the FF shocks steepen in time (compare top to middle and middle to bottom rows).", "The swept of shells of unmagnetized ambient medium become relatively thin when radiative cooling is included (right column: compare top to middle and middle to bottom rows).", "The adiabatic and rotating cases show regions within the lower half of the jet where the flow speed is super-Alfvénic.", "Such regions are bounded by the reverse and the forward slow-modes of compressive MHD waves, and characterized by high thermal to magnetic pressure ratios (section REF ).", "At $t \\,$ 90 yr the distribution of waves and shocks of both the rotating and cooling cases (bottom row, middle and right columns) is significantly affected.", "This is due to the growth of non-linear current-driven instabilities (section REF ).", "Possibly, pressure driven modes coexist with the current driven ones in regions of high $\\beta $ .", "We see fast, though mostly sub-Alfvénic, azimuthal velocities, in the central parts of these jet cores.", "Figure: Evolution of the plasma velocities along the jets' axis.", "These are the magnetic towers in the adiabatic (left), the rotating (middle) andthe cooling (right) cases.", "From top to bottom time is equal to 42, 84 and118 yr.Each computational velocity unit is equivalent to 9.1 km s -1 ^{-1}.To clarify the effect of cooling on our magnetic towers we present temperature maps in Figure REF below.", "We model radiation losses using (3), where $\\Delta T \\propto \\rho ^2 \\Lambda (T)$ and $\\Lambda (T)$ is taken from the tables of [13].", "Figures 2 and 4 help to form a complete picture of the cooling strength.", "In the non-cooling cases we see shocked ambient medium at temperatures of $T \\sim \\,$ 10$^5$  K. This material forms an extended shell surrounding the magnetic cavity formed by inflowing Poynting flux.", "In the cooling case this shocked shell of ambient gas has cooled significantly to temperatures of $T \\le \\,$ 10$^4$  K. The cooling has decreased the pressure in this region on the shell now becomes both thin and dense.", "Note we also see low temperature via cooling occur in the jet beam and the knots that form once the beam becomes unstable.", "Figure: Evolution of the towers' temperature.", "These logarithmic color maps show the magnetic towers in the adiabatic (left), the rotating (middle) andthe cooling (right) cases.", "From top to bottom time is equal to 42, 84 and118 yr." ], [ "Magnetic fields and current density", "In Figure REF we show the distribution of the towers' magnetic fields on the plane that contains the jets' axis.", "These are linear color maps of the absolute value of the toroidal to poloidal field component ratio.", "From left to right we show the adiabatic, the rotating and the cooling cases, respectively, and time increases downwards with row.", "We see that the magnetic flux changes sign along the radial direction.", "In general there are four main nested surfaces or layers of magnetic field lines (e.g.", "see middle row, left column panel): at the very core we see predominately poloidal (vertical $B_z$ ) fields surrounded by a surface of primarily toroidal (azimuthal $B_\\phi $ ) flux.", "These field components represent the central core of the jet plasma column.", "They are collimated by two outer magnetic surfaces.", "The smaller of these two is dominated by poloidal lines, whereas the larger one is dominated by toroidal lines.", "These outer field lines are collimated by the thermal pressure and inertia of the external media.", "As expected, the geometry of the towers' magnetic fields changes in time.", "Initially, the field lines have a highly wound helical configuration (section REF ).", "The magnetic pressure is very high and unbalanced in the vertical direction.", "The toroidal field lines thus move away from each other and the magnetic towers rise.", "The injection of magnetic flux sustains a non-force-free configuration at the base of the tower.", "“New” field lines push the “old” ones upwards then.", "The latter stretch and expand radially, making way for, and collimating, the jets' new field lines.", "After the towers early expansion phase ($t \\,$ 90 yr), we find, in agreement with the results of the previous section, that the jets of both the cooling and the rotating cases are affected by instabilities (section REF ).", "The final magnetic structure of the towers is clearest in the field line maps of Figure REF .", "These are the lines in the central part ($r\\,$ 1.2 $r_e$ ) of the adiabatic (left), the rotating (middle) and the cooling (right) towers at $t=\\,$ 118 yr.", "The top and bottom panels shows the towers edge-on and pole-on, respectively.", "The adiabatic case shows quite ordered helical field lines and the strongest jet fields (red color) of all the towers.", "We also see that toroidal field lines tend to pile up at the tower's tip.", "Such a concentration of lines causes acceleration of the plasma the tip of the adiabatic jet to supersonic speeds (see bottom, left panel in Figure REF , $z\\approx \\,$ 15).", "In contrast, the cooling tower (right panel) shows the weakest and most disordered field lines.", "The middle and right panels show clear differences between the cooling and the rotating cases.", "The instabilities that develop in these towers are clearer in the rotating case (middle column; setion REF ).", "Figure: Evolution of the towers' magnetic fields.", "This is the ratio of the toroidal componentover the poloidal one.", "B y =B phi B_y = B_{phi} and is perpendicular to the maps.These linear colour maps show the magnetic towers in the adiabatic (left), the rotating (middle) andthe cooling (right) cases.", "From top to bottom time is equal to 42, 84 and118 yr.Figure: Central (rr \\, 1.2 r e r_e) magnetic field lines att=t=\\,118 yr. From left to rightthese are the adiabatic, the rotating and the cooling magnetictowers, respectively.", "Bottom panels show an upper view, pole-on.Open field lines are a visualization effect.The magnetic fields are ultimately sustained by electric currents.", "In Figure REF we show the evolution of the axial current density, $J_z$ (panels in this Figure are arranged as in Figure REF ).", "As expected we see a clear correlation between the distributions of the axial current density and the magnetic field.", "The jets carry a high axial current (red region) which is contained within a current-free region (white one) at larger radii.", "The main part of the return current (blue region) moves along the contact surface of the towers' cavity.", "This forms a closed circulation current system which is consistent with previous simulations of PFD jets (see e.g.", "[27], [24], [37], [10]) and the magnetic tower laboratory experiments of [25] and [47].", "We note however that both the magnetic field and the current density are strictly localized in our model, i.e.", "no components of the current in the external medium.", "This is a characteristic feature of magnetic towers.", "We see that the current $J_z$ is also affected by the instabilities that develop in the rotating and cooling towers after their early expansion phase (bottom row, middle and right columns).", "The effect of instabilities is most pronounced in the jet beam.", "As the jet breaks up into clumps the current becomes more localized.", "Numerical reconnection allows some of the sections of tangled fields to become isolated however the overall flow of axial current density continues as does the outer sheath of return current.", "Figure: Evolution of the axial current density.J z J_z has been normalized to the absolute value of its maximum value, |max(J z )||max(J_z)|,for display purposes.", "These linear color maps show the magnetic towers in the adiabatic (left), the rotating (middle) andthe cooling (right) cases.", "From top to bottom time is equal to 42, 84 and118 yr." ], [ "Energy Flux", "To study the relative magnetic vs. kinetic energy content of our magnetic towers we compute the Poynting flux, $f_P$ , and the kinetic flux, $f_k$ , defined respectively as $\\begin{array}{c l l}f_P = & \\int \\limits _s \\, [{\\bf B \\times (V \\times B ) }]_z \\, dS, \\\\f_k = & \\int \\limits _s \\, \\frac{1}{2} \\, \\rho \\, |{\\bf V}|^2 \\, V_z \\, dS.\\end{array}$ The above integrals are taken over the area of the jet beams.", "In Figure REF we show logarithmic color maps of the distribution of the jet Poynting to kinetic flux ratio, $\\log \\left| Q({\\bf x},t) \\right|$ , where $Q({\\bf x},t)= f_P/f_k$ , as a function of colling, jet base rotation and time.", "The maps show that only the core of the jets is dominated by kinetic energy flux ($Q < \\, $ 1, blue region) while the bulk of the beam is PFD ($Q > \\, $ 1, red region) for all the cases (i.e.", "adiabatic, rotating and cooling).", "This distribution is consistent with that found in the laboratory jets of [25].", "We confirm that our magnetic towers are indeed PFD.", "We note that the dark red stripes in Figure REF correspond to regions where the toroidal field components are particularly strong (see Figure REF ).", "To stress and clarify this point we also show logarithmic maps of $f_k$ (left panels) and $f_P$ (right panels) in Figure REF .", "Figure: Distribution and evolution of the jet beam Poynting to kinetic flux ratio.These logarithmic maps show the jets of the magnetic towers in the adiabatic (left), the rotating (middle) andthe cooling (right) cases.", "From top to bottom time is equal to 42, 84 and118 yr.Figure: Distribution and evolution of the towers' kinetic energy(left) and Poynting (right) polar fluxes in computational units.These logarithmic maps show the magnetic towers in the adiabatic (left), the rotating (middle) andthe cooling (right) cases.", "From top to bottom time is equal to 42, 84 and118 yr.We find that the time average mean $Q$ of our magnetic tower beams – averaged over the adiabatic, cooling and rotating cases – is $\\sim \\,$ 6.", "This is about 2$/$ 3 of the time average mean $Q$ in the magnetic towers simulated by [19].", "We note that the spatial distribution of $Q$ in both our and their simulations is not isotropic and time-dependent.", "Early in the evolution of our towers $Q$ is axisymmetric however the growth of the kink instabilities eventually leads to the development to far more spatial variability in $Q(x,y,z)$ .", "Our simulations show the ratio of Poyting flux to kinetic energy flux is always greater than unity for the magnetic tower ($Q >\\,$ 1).", "This should be compared with the models of jets created by magneto-centrifugal (MCL) processes.", "While MCL jets begin with $Q>\\,$ 1 on scales less than the Alfvén radius, in the asymptotic limit the kinetic energy flux comes to dominate the flux electromagnetic energy leading.", "Simulations of MCL launching in which the flow is cold and gas pressure can be ignored show typical values of $Q \\sim \\,$ 0.7 at observationally-resolved distances from the engine (Krasnopolsky et al.", "1999, 2003).", "We leave a detailed comparison of PDF and MCL jets for the future." ], [ "Forces", "Magnetic towers expand due to a combination of magnetic, thermal and inertial forces.", "In Figure REF we show the thermal to magnetic pressure ratio, $\\beta $ , using logarithmic grey scale maps (arranged as in Figure REF ).", "We find $\\beta $ for the magnetic towers is generally and consistently well below unity.", "The adiabatic and rotating cases (left and middle columns, respectively) do show regions where $\\beta > \\,$ 1 close to $(r,z)=($ 0,6).", "Such regions are located between the reverse an forward slow-mode compressive MHD waves (Figure REF , left and middle columns), and filled with subsonic, weakly magnetized plasma.", "This high-$\\beta $ region is strongly affected by cooling (right column) which reduces the thermal energy (see also Figure REF ).", "Hence the total pressure of the surrounding plasma becomes further dominated by the magnetic component, and it collapses yielding higher compression ratios than the adiabatic case.", "The field in the cooling case also takes on a configuration amenable to instability.", "Thus the plasma in the high-$\\beta $ jet-core region plays a critical role on the overall stably of PFD outflows.", "Figure: Evolution of the thermal to magnetic pressure ratio.", "These logarithmicgrey scale maps show the magnetic towers in the adiabatic (left), the rotating (middle) andthe cooling (right) cases.", "From top to bottom time is equal to 42, 84 and118 yr.In Figure REF we show the radial component of the forces in the magnetic towers during their intermediate evolutionary phase.", "Form small to large radii these linear color maps show the jet core (dark colored regions), the jet beam edge (light colored regions), the cavities' central force-free region (white region), the CD (light colors), and finally the swept-up ambient gas (outer-most light features, bottom row) of the towers.", "In general these figures show that the inward Lorentz force (top panels) is slightly stronger than both the inertial (or specific centrifugal $v_{\\perp }^2/x$ ), and the thermal pressure, $({\\bf \\nabla } P)_x$ , forces which push plasma outward.", "This fact is consistent with the results of Takeuchi et al.", "(2010) and Ohsuga & Mineshige (2011).", "Figures REF and REF show the character of the force density distribution responsible for confining the jets and their cavities.", "The jets are self-confined in the current-free region located at a few jet radii from the core (i.e.", "hoop stress).", "At larger radii, near the towers' contact surface, which is also the return current surface (blue outer region in Figure REF ), the magnetic pressure is weak and thus it only requires a mild ambient pressure (light-grey outer region, Figure REF ) to confine the outer part of the towers.", "Figure: Radial forces at the intermediate evolutionary phase (t=t=\\,84 yr) of the towers.Forces are normalized to the maximum value, F max F_{max}.", "The horizontal axis is x=rx=r,the vertical axis is zz and v ⊥ v_{\\perp } is perpendicular to the maps.In section REF we saw that the magnetic towers decelerate with respect to the HD jet.", "This can be understood with the bottom panel in Figure REF where we see that the magnetic flux that is injected into the towers (within model $r 1.5$ ; section REF ) causes not only axial ($z$ ) expansion, but also radial expansion via magnetic pressure.", "In contrast, the kinetic-energy flux in the HD jet (not shown) is overwhelmingly axial." ], [ "Stability", "The structure and expansion of our PFD magnetic tower jets are affected by current-driven instabilities.", "We see evidence of the pinch, $m=\\,$ 0, the kink, $m=\\,$ 1, and the $m=\\,$ 2 normal mode perturbations.", "These are expected in expanding magnetized plasma columns and consistent with the models of [37] and Ciardi et al.", "(2007) – but see also [46] – and also with the laboratory experiments of [25] and [47].", "We find that the kink perturbations grow and lead to instabilities in the cooling tower, firstly, and later also in the rotating tower.", "Perturbations with modes $m=\\,$ 0 and 2 develop in the adiabatic jet after expanding for $\\sim \\,$ 80 yr ($\\sim $ 70% of the total simulation running time).", "These are caused by radial gradients in the magnetic fields located within the jet beam, at the boundary of the current-free, force-free region.", "The thermal and magnetic components of the total pressure balance each others' perturbations locally.", "As a result, the core of the adiabatic jet becomes a helical column with an elliptical cross-section.", "The growth rate of these m=0 and m=2 perturbations is $>$ 120 yr, which is longer than the simulation final time.", "Thus we are seeing only the linear develop of the modes.", "Finally we note resemblances (Figure REF , bottom left panel) to structures seen in the S$_{\\rm II}$ emission distribution of the jet in HH 34 [43].", "The central part of the towers' jet beams are high beta plasma columns where $|B_{\\phi } / B_z | \\ll 1$ (Figure REF , $B_{\\phi }=B_y$ ).", "To understand their development we can appeal to standard Kruskal-Shafranov criterion for the kink instability, namely [7] $\\left| \\frac{B_{\\phi }}{B_z} \\right| > | (\\beta _z - 1)k r_{jet} |,$ where $\\beta _z=2 \\mu _0 P / B_z^2$ and $k^{-1}$ is the characteristic wavelength of the current-driven perturbations.", "In Figure REF (right column) we see that the cooling jet's core shows $\\beta _z \\sim \\,$ 1 ($z \\sim \\,$ 4–5 at time$=\\,$ 84 yr, and $z \\sim \\,$ 6–11 at time$=\\,$ 118 yr).", "This means that the cooling tower does not have sufficient thermal energy, in comparison with the adiabatic and rotating cases, to balance the magnetic pressure kink perturbations.", "This is consistent with what we see in the towers' density and temperature maps, Figures 2 and  4.", "In addition, we see that the jet core radius of the cooling tower is about 20% smaller than that of the adiabatic tower.", "This is consistent with what is found in laboratory experiments of magnetized supersonic jets, in which outflows with different cooling rates are compared (Ciardi et al.", "2012, in prep).", "Both of these effects (thermal energy losses and core radial compression) reduce the right hand side of (REF ), making the system more susceptible to the growth of kink instabilities in the cooling tower.", "For the rotating case, we find that rotation at the base of the jet beam, equation (REF ), causes a progressive, slow amplification of the toroidal magnetic field component of the jet.", "This process is evident in the four panels at the bottom left of Figure REF , where we see that in general the Alfvén speed is higher in the rotating case (middle column) than in the adiabatic one (left column).", "This growth is likely sufficient to amplify the left hand side of equation (REF ) pushing the jet into the unstable regime.", "Since the field grows linearly with the differential rotation, the growth rate is likely proportional to the imposed amount of rotation.", "We have not tested this as we have used only one vale of the differential rotation.", "Note that the towers are not completely destroyed even when unstable, and the amplitude of the kink perturbations in the jet are about twice its radius (Figure REF ), in agreement with the Kruskal-Shafranov criterion [23], [44]." ], [ "The HD cooling and HD rotating cases", "In addition to the four simulations presented above, we have carried out two variations of the HD jet run: one with cooling, and one which has a base rotation profile which follows equation (REF ), just as in the rotating magnetic tower run (section REF ).", "We found that the results of the cooling HD jet simulation were consistent, as expected, with those found in previous similar studies, i.e.", "thin jet-produced shocks with high compression factors [14], [17].", "For the regimes we have studied, the propagation and structure of the HD jets is affected by both cooling and rotation in ways which have been studied before and which do not alter the global propagation properties of the flow, i.e.", "no instabilities are introduced as in the magnetic tower case." ], [ "Implications for Jet Observations, Experiments and Future Work", "The results of the simulations help guide our understanding of the evolution of PFD magnetic towers.", "In particular the simulations show new details of the cavity-jet connection, the evolution of the tower given different assumptions (cooling, rotation, etc.)", "as well as providing some insight into the stability properties of the central jets which form in the flow.", "For non-relativistic collimated flows magnetic towers have been proposed as mechanisms for launching some classes of YSO.", "While the flows downstream at observable distances ($>10^3$  AU) seem to be kinetic energy dominated, at smaller scales a PFD region may be expected.", "As Hartigan et al.", "(2007) have shown, what few measurements of magnetic fields exist in YSO jets indicate there must be a region of sub-Alvénic, PFD dominated flow on scales of order 100 AU or less.", "In addition, these simulations demonstrate (and as laboratory experiments have shown) the long term evolution of magnetic towers may yield a series of collimated clumps whose magnetization properties vary over time.", "In this way PFD flows may evolve into kinetic energy dominated jets at large distances from the central engine.", "Planetary Nebula (PN) offer another potential application of non-relativistic PFD dominated flows.", "Magnetic fields are already expected to play an important role in launching pre-Planetary Nebulae (PPN) based on an observed mismatch between momentum in the PPN flow and momentum available through radiation.", "A number of papers have discussed how strong magnetic fields might create PPN or PN collimated flows (Blackman et al.", "2001a,b, Frank & Blackman 2004, Matt, Frank & Blackman 2006).", "Observations of PPN and PN offer morphological similarities to the kinds of features seen in our simulations, such as hollow lobes and axial clumps.", "Future work might address these connections.", "Of particular importance is the connection between the models presented in this paper and recent “laboratory astrophysics” experiments.", "These studies utilized Pulsed Power technologies and were successful in creating high Mach number, fully radiative, magnetized outflows (Lebedev et al.", "2005, Ciardi et al.", "2007, Ampleford et al.", "2008, Ciardi et al.", "2009, Frank et al.", "2009).", "The outflows were created when TW electrical pulses (1 MA, 250 ns) are applied to a radial array of fine metallic wires.", "Lorentz forces ablated plasma from the wires creating an ambient plasma above the array.", "After the complete ablation of wires near the central electrode, the current switches to the plasma and creates a magnetic cavity with a central jet (i.e.", "a magnetic tower).", "The central part of the jet is confined and accelerated by the pressure of the toroidal field.", "Return current flows along the walls of the magnetic cavity, which is in turn confined by the thermal pressure and by inertia of the ambient plasma.", "As the magnetic cavity expands, the jet becomes detached and propagates away from the source at $\\sim $ 200 km s$^{-1}$ .", "Instabilities which resemble the kink mode ($m =\\,$ 1) develop within the body of these jets fragmenting them into well collimated structures with characteristic axial non-uniformities.", "Thus the evolution of magnetic towers in the laboratory show a range of features that are strikingly similar to what is seen in our simulations.", "This concordance is all the more noteworthy in that our initial conditions were in no way tuned to the experiments and are, in fact, a modified version of what can be found in a number of purely astrophysical studies [26].", "Thus it appears that the laboratory experiments and the simulations support each other, as well as the conclusion that both are revealing generic properties of PFD outflows.", "While we did not study relativistic outflows, some aspects of the comparative behavior between HD and PFD jets revealed by our models might still apply.", "The fragmentation of the PFD magnetic tower core, for example, implies that rather than continuous jet beams we would expect high resolution observations to reveal essentially “clumpy” jets with a distribution of velocities, densities and magnetization.", "In this way our models, except the cooling ones, can be considered to articulate classes of flow features in AGN radio jets [49], X-ray binaries [33] and, perhaps, GRBs [36].", "But both relativistic and different radiative cooling generalizations are needed to confirm or refute the implications of our present calculations for such regimes.", "There is opportunity for future work to focus more closely on the links with the laboratory experiments.", "In particular, issues related to the development of kink mode instabilities, their non-linear resolution and the evolution of clumpy magnetized jets should be explored more fully and in more detail.", "Regarding the effect of rotation at the base of the jets on their stability, we note that [35] have carried out 3-D simulations of magnetocentrifugally driven, conical jets, and found that kink instabilities are stronger when a rigid rotation profile is imposed, in comparison to a Keplerian rotation profile.", "Rigid rotation seems to induce a shearless magnetic field [35].", "A direct comparison with our calculations must be made carefully though; our initial magnetic configuration has a dominant toroidal component and no radial component, while the initial field setup of [35] is purely radial.", "Also, our rotating magnetic tower is continually affected by injection of magnetic flux, which is not the case of the conical jets of [35]." ], [ "CONCLUSIONS", "We have carried out 3-D ideal MHD simulations of PFD and HD dominated jets to compare their structure and evolution subject to the same injected energy flux, and to study the effects of cooling and jet rotation on the jet stability.", "We note that our HD cases can, in principle, emulate asymptotic propagation regimes of magneto-centrifugally launched jets if those jets become kinetic energy dominated at large distances.", "Magnetic towers will however remain PFD at large radii.", "Our simulations read us to the following conclusions.", "Helical localized magnetic fields injected into a region of low pressure will launch PFD, magnetic towers via magnetic pressure gradients.", "Towers consist of a low density low beta plasma, the radial collimation of which is caused by the pressure of the external plasma.", "Within the towers a higher density, higher beta jet forms collimated by the magnetic field lines located within the cavity.", "We found that PFD jets create structures that are more susceptible to instabilities relative to purely hydrodynamical jets given the same injected energy flux.", "Unstable modes in the magnetic towers differ according to conditions within the flow.", "The adiabatic PFD jet is unstable to $m=\\,$ 0 and 2 mode perturbations, and its core adopts a elliptical cross-section.", "On the other hand, the PFD jet with a Keplerian rotating base exhibits an $m=\\,$ 1 kink mode instability.", "The beam is not completely destroyed but adopts a chaotic clumpy structure.", "Base rotation causes a slow amplification of the toroidal field exacerbating a pressure unbalance in the jet's core that leads to instability.", "The cooling PFD jet also shows a $m=\\,$ 1 kink mode instability.", "Cooling reduces the thermal energy of the jet's core, making the thermal pressure insufficient to damp the magnetic pressure kink perturbations.", "The cooling PFD beam shows the fastest growth rate of the kink instability.", "Our magnetic tower (PDF jets) simulations are in good agreement with the laboratory experiments of [25].", "In both our simulations and the experiments: (1) jets carry axial currents which return along the contact discontinuities, (2) the jet cores have a high $\\beta $ , (3) jet beams and cavities are PFD, (4) jets are eventually corrugated by current driven instabilities becoming a collimated chain of magnetized “clumps” or “bullets”.", "The similarity between our models and the experiments is particularly noteworthy because our implementation was not tuned to represent the laboratory results.", "This strengthens the case for the usefulness of laboratory experiments in articulating new features of astrophysical MHD flows in cases where similarity conditions can be obtained.", "We found that PFD jets decelerate by about 20% relative to the HD ones given the same injected energy flux.", "This is because PFD jets produce not only axial but radial expansion due to magnetic pressure.", "All of the pre-collimated energy flux of the HD case is more efficiently directed to axial mechanical power.", "Also, the long term evolution of PFD jets yield a series of collimated clumps, the magnetization properties of which may vary over time.", "PFD flows may thus eventually evolve into HD jets at large distances from the central engine.", "Our work shows that outflows launched as magnetic towers show a different behavior compared with those launched by magneto-centrifugal (MCL) mechanisms when the MCL flows become asymptotically kinetic energy flux dominated.", "As it was shown by Hartigan et al.", "2007, in YSO flows some mechanism may be needed to reduce the magnetization of plasma close to the jet source.", "If these flows begin as magnetic towers then the disruption of the central jets via kink modes may provide a means to produce collimated high beta clumps of material as is observed in HH flows.", "Thus our work may help to lead methods for distinguishing between different launch mechanisms by providing descriptions of asymptotic flow characteristics where observations might be possible.", "Financial support for this project was provided by the Space Telescope Science Institute grants HST-AR-11251.01-A and HST-AR-12128.01-A; by the National Science Foundation under award AST-0807363; by the Department of Energy under award DE-SC0001063; and by Cornell University grant 41843-7012.", "SL acknowledges support from EPSRC Grant No. EP/G001324/1.", "We thank David Meier and Neil Turner for helpful discussions." ] ]

[ [ "A New Approach to Online Scheduling: Approximating the Optimal\n Competitive Ratio" ], [ "Abstract We propose a new approach to competitive analysis in online scheduling by introducing the novel concept of competitive-ratio approximation schemes.", "Such a scheme algorithmically constructs an online algorithm with a competitive ratio arbitrarily close to the best possible competitive ratio for any online algorithm.", "We study the problem of scheduling jobs online to minimize the weighted sum of completion times on parallel, related, and unrelated machines, and we derive both deterministic and randomized algorithms which are almost best possible among all online algorithms of the respective settings.", "We also generalize our techniques to arbitrary monomial cost functions and apply them to the makespan objective.", "Our method relies on an abstract characterization of online algorithms combined with various simplifications and transformations.", "We also contribute algorithmic means to compute the actual value of the best possi- ble competitive ratio up to an arbitrary accuracy.", "This strongly contrasts all previous manually obtained competitiveness results for algorithms and, most importantly, it reduces the search for the optimal com- petitive ratio to a question that a computer can answer.", "We believe that our concept can also be applied to many other problems and yields a new perspective on online algorithms in general." ], [ "Introduction", "Competitive analysis [44], [25] is the most popular method for studying the performance of online algorithms.", "It provides an effective framework to analyze and classify algorithms based on their worst-case behavior compared to an optimal offline algorithm over an infinite set of input instances.", "For some problem types, e.g., online paging, competitive analysis may not be adequate to evaluate the performance of algorithms, but for a vast majority of online problems it is practical, natural, and yields meaningful results.", "A classical such problem is online scheduling to minimize the weighted average completion time.", "It has received a lot of attention in the past two decades.", "For different machine environments, a long sequence of papers emerged introducing new techniques and algorithms, improving upper and lower bounds on the competitive ratio of particular algorithms as well as on the best possible competitive ratio that any online algorithm can achieve.", "Still, unsatisfactory gaps remain.", "As for most online problems, a provably optimal online algorithm, w.r.t.", "competitive analysis, among all online algorithms is only known for very special cases.", "In this work we close these gaps and present nearly optimal online scheduling algorithms.", "We provide competitive-ratio approximation schemes that compute algorithms with a competitive ratio that is at most a factor $1+\\varepsilon $ larger than the optimal ratio for any $\\varepsilon >0$ .", "To that end, we introduce a new way of designing online algorithms.", "Apart from structuring and simplifying input instances, we find an abstract description of online scheduling algorithms, which allows us to reduce the infinite-size set of all online algorithms to a relevant set of finite size.", "This is the key for eventually allowing an enumeration scheme that finds an online algorithm with a competitive ratio arbitrarily close to the optimal one.", "Besides improving on previous algorithms, our method also provides an algorithm to compute the competitive ratio of the designed algorithm, and even the best possible competitive ratio, up to any desired accuracy.", "This is clearly in strong contrast to all previously given (lower) bounds that stem from manually designed input instances.", "We are aware of only very few online problems for which a competitive ratio, or even the optimal competitive ratio, are known to be computable by some algorithm (for a not inherently finite problem).", "Our result is surprising, as there are typically no means of enumerating all possible input instances and all possible online algorithms.", "Even for only one given algorithm, usually one cannot compute its competitive ratio simply due to difficulties like the halting problem.", "We overcome these issues and pave the way for computer-assisted design of online algorithms.", "We believe that our concept of abstraction for online algorithms can be applied successfully to other problems.", "We show this for other scheduling problems with jobs arriving online over time.", "We hope that our new approach to competitive analysis contributes to a better understanding of online algorithms and may lead to a new line of research in online optimization." ], [ "Problem Definition and Previous Results", "Competitive analysis.", "Given a minimization problem, a deterministic online algorithm A is called $\\rho $ -competitive if, for any problem instance $I$ , it achieves a solution of value $\\textsf {\\textsc {A}}(I) \\le \\rho \\cdot \\textsf {\\textsc {Opt}}(I)$ , where $\\textsf {\\textsc {Opt}}(I)$ denotes the value of an optimal offline solution for the same instance $I$ .", "A randomized online algorithm is called $\\rho $ -competitive, if it achieves in expectation a solution of value $\\mathbb {E}\\left[\\,\\textsf {\\textsc {A}}(I)\\,\\right] \\le \\rho \\cdot \\textsf {\\textsc {Opt}}(I)$ for any instance $I$ .", "The competitive ratio $\\rho _{\\textsf {\\textsc {A}}}$ of $\\textsf {\\textsc {A}}$ is the infimum over all $\\rho $ such that $\\textsf {\\textsc {A}}$ is $\\rho $ -competitive.", "The minimum competitive ratio $\\rho ^*$ achievable by any online algorithm is called optimal.", "Note that there are no requirements on the computational complexity of competitive algorithms.", "Indeed, the competitive ratio measures the best possible performance under the lack of information given unbounded computational resources.", "We define a competitive-ratio approximation scheme as a procedure that computes a nearly optimal online algorithm and at the same time provides a nearly exact estimate of the optimal competitive ratio.", "Definition 1.1 A competitive-ratio approximation scheme computes for a given $\\varepsilon >0$ an online algorithm $\\textsf {\\textsc {A}}$ with a competitive ratio $\\rho _{\\textsf {\\textsc {A}}}\\le (1+\\varepsilon )\\rho ^*$ .", "Moreover, it determines a value $\\rho ^{\\prime }$ such that $\\rho ^{\\prime }\\le \\rho ^* \\le (1+\\varepsilon ) \\rho ^{\\prime }$ .", "Online scheduling.", "A scheduling instance consists of a fixed set of $m$ machines and a set of jobs $J$ , where each job $j\\in J$ has processing time $p_j\\in \\mathbb {Q}^+$ , weight $w_j\\in \\mathbb {Q}^+$ , and release date $r_j\\in \\mathbb {Q}^+$ .", "The jobs arrive online over time, i.e., each job becomes known to the scheduling algorithm only at its release date.", "We consider three different machine environments: identical parallel machines (denoted by P), related machines (Q) where each machine $i$ has associated a speed $s_i$ and processing a job $j$ on machine $i$ takes $p_j/s_i$ time, and unrelated machines (R) where the processing time of a job $j$ on each machine $i$ is explicitly given as a value $p_{ij}$ .", "The main problem considered in this paper is to schedule the jobs on the given set of machines so as to minimize $\\sum _{j\\in J}w_jC_j$ , where $C_j$ denotes the completion time of job $j$ .", "We consider the problem with and without preemption.", "Using standard scheduling notation [19], we denote the non-preemptive (preemptive) problems that we consider in this paper by $\\textup {Pm}|\\,r_j, (pmtn)\\,|\\sum w_jC_j$ , $\\textup {Qm}|\\,r_j,(pmtn)\\,|\\sum w_jC_j$ , and $\\textup {Rm}|\\,r_j,pmtn\\,|\\sum w_jC_j$ .", "We also briefly consider more general min-sum objectives $\\sum _{j\\in J}w_jf(C_j)$ , where $f$ is an arbitrary monomial function $f(x)=k\\cdot x^{\\alpha }$ , with constant $\\alpha \\ge 1, k >0$ , and the classical makespan $C_{\\max }:=\\max _{j\\in J}C_j$.", "Previous results.", "The offline variants of nearly all problems under consideration are NP-hard [16], [27], [28], but in most cases polynomial-time approximation schemes have been developed [1], [29], [21], [22].", "The corresponding online settings have been a highly active field of research in the past fifteen years.", "A whole sequence of papers appeared introducing new algorithms, new relaxations and analytical techniques that decreased the gaps between lower and upper bounds on the optimal competitive ratio [18], [39], [20], [43], [42], [2], [6], [24], [10], [17], [35], [34], [9], [40], [30], [31], [46], [36], [5], [41], [14], [7].", "Interestingly, despite the considerable effort, optimal competitive ratios are known only for $\\textup {1}|\\,r_j, pmtn\\,|\\sum C_j$  [38] and for non-preemptive single-machine scheduling [2], [46], [24], [6].", "In all other scheduling settings remain unsatisfactory, even quite significant gaps.", "See Appendix  for a detailed description of the state of the art for each individual problem.", "Very recently, our new concept of competitive-ratio approximation schemes was applied also to job shop scheduling $\\textup {Jm}|\\,r_j,op\\le \\mu \\,|C_{\\max }$ and non-preemptive scheduling on unrelated machines $\\textup {Rm}|\\,rj\\,|C_{\\max }$  [26].", "To the best of our knowledge, there are only very few problems in online optimization for which an optimal competitive ratio can be determined, bounded, or approximated by computational means.", "Lund and Reinhold [32] present a framework for upper-bounding the optimal competitive ratio of randomized algorithms by a linear program.", "For certain cases, e.g., the 2-server problem in a space of three points, this yields a provably optimal competitive ratio.", "Ebenlendr et al.", "[11], [12] study various online and semi-online variants of scheduling preemptive jobs on uniformly related machines to minimize the makespan.", "In contrast to our model, they assume the jobs to be given one by one (rather than over time).", "They prove that the optimal competitive ratio can be computed by a linear program for any given set of speeds.", "In terms of approximating the best possible performance guarantee, the work by Augustine, Irani, and Swamy [3] is closest to ours.", "They show how to compute a nearly optimal power-down strategy for a processor with a finite number of power states." ], [ "New Results and Methodology", " In this paper, we introduce the concept of competitive-ratio approximation schemes and present such schemes for various scheduling problems with jobs arriving online over time.", "We present our technique focussing on the problems $\\textup {Pm}|\\,r_j, (pmtn)\\,|\\sum w_jC_j$ , $\\textup {Qm}|\\,r_j,(pmtn)\\,|\\sum w_jC_j$  (assuming a constant range of machine speeds without preemption), and $\\textup {Rm}|\\,r_j,pmtn\\,|\\sum w_jC_j$ , and we comment on how it applies to other cost functions such as the makespan, $C_{\\max }$ , and $\\sum _{j\\in J}w_jf(C_j)$ , where $f$ is an arbitrary monomial function with fixed exponent.", "For any $\\varepsilon >0$ , we show that the competitive ratios of our new algorithms are by at most a factor $1+\\varepsilon $ larger than the respective optimal competitive ratios.", "We obtain such nearly optimal online algorithms for the deterministic as well as the randomized setting, for any number of machines $m$ .Moreover, we give an algorithm which estimates the optimal competitive ratio for these problems to any desired accuracy.", "Thus, we reduce algorithmically the performance gaps for all considered problems to an arbitrarily small value.", "These results reduce the long-time ongoing search for the best possible competitive ratio for the considered problems to a question that can be answered by a finite algorithm.", "To achieve our results, we introduce a new and unusual way of designing online scheduling algorithms.", "We present an abstraction in which online algorithms are formalized as algorithm maps.", "Such a map receives as input a set of unfinished jobs together with the schedule computed so far.", "Based on this information, it returns a schedule for the next time instant.", "This view captures exactly how online algorithms operate under limited information.", "The total number of algorithm maps is unbounded.", "However, we show that there is a finite subset which approximates the entire set.", "More precisely, for any algorithm map there is a map in our subset whose competitive ratio is at most by a factor $1+\\varepsilon $ larger.", "To achieve this reduction, we first apply several standard techniques, such as geometric rounding, time-stretch, and weight-shift, to transform and simplify the input problem without increasing the objective value too much; see, e.g., [1].", "The key, however, is the insight that it suffices for an online algorithm to base its decisions on the currently unfinished jobs and a very limited part of the so far computed schedule—rather than the entire history.", "This allows for an enumeration of all relevant algorithm maps (see also [33] for an enumeration routine for online algorithms for a fixed task system with finitely many states).", "For randomized algorithms we even show that we can restrict to instances with only constantly many jobs.", "As all our structural insights also apply to offline algorithms for the same problems, they might turn out to be useful for other settings as well.", "Our algorithmic scheme contributes more than an improved competitive ratio.", "It also outputs (up to a factor $1+\\varepsilon $ ) the exact value of the competitive ratio of the derived algorithm, which implies a $(1+\\varepsilon )$ -estimate for the optimal competitive ratio.", "This contrasts strongly all earlier results where (matching) upper and lower bounds on the competitive ratio of a particular and of all online algorithm had to be derived manually, instead of executing an algorithm using, e.g., a computer.", "In general, there are no computational means to determine the competitive ratio of an algorithm—even when it is a constant.", "It is simply not possible to enumerate all possible input instances.", "Even more, there are no general means of enumerating all possible online algorithms to determine the optimal competitive ratio.", "However, for the scheduling problems studied in this paper our extensive simplification of input instances and our abstract view on online algorithms allow us to overcome these obstacles, losing only a factor of $1+\\varepsilon $ in the objective.", "Although the enumeration scheme for identifying the (nearly) optimal online algorithm heavily exploits unbounded computational resources, the resulting algorithm itself has polynomial running time.", "As a consequence, there are efficient online algorithms for the considered problems with almost optimal competitive ratios.", "Hence, the granted additional, even unbounded, computational power of online algorithms does not yield any significant benefit here.", "Outline of the paper.", "In Section  we introduce several general transformations and observations that simplify the structural complexity of online scheduling in the setting of $\\textup {Pm}|\\,r_j, pmtn\\,|\\sum w_jC_j$ .", "Based on this, we present our abstraction of online algorithms and develop a competitive-ratio approximation scheme in Section .", "Next, we sketch in Section  how to extend these techniques to the non-preemptive setting and more general machine environments such that the approximation scheme (Sec. )", "remains applicable.", "In Section , we present competitive-ratio approximation schemes for the randomized setting.", "Finally, in Section , we extend our results to other objective functions." ], [ "General Simplifications and Techniques", "In this section, we discuss several transformations that simplify the input and reduce the structural complexity of online schedules for $\\textup {Pm}|\\,r_j, pmtn\\,|\\sum w_jC_j$ .", "Later, we outline how to adapt these for more general settings.", "Our construction combines several transformation techniques known for offline PTASs (see [1] and the references therein) and a new technique to subdivide an instance online into parts which can be handled separately.", "We will use the terminology that at $1+O(\\varepsilon )$ loss we can restrict to instances or schedules with certain properties.", "This means that we lose at most a factor $1+O(\\varepsilon )$ , as $\\varepsilon \\rightarrow 0$ , by limiting our attention to those.", "We bound several relevant parameters by constants.", "If not stated differently, any mentioned constant depends only on $\\varepsilon $ and $m$ .", "Lemma 2.1 ([1]) At $1+O(\\varepsilon )$ loss we can restrict to instances where all processing times, release dates, and weights are powers of $1+\\varepsilon $ , no job is released before time $t=1$ , and $r_{j}\\ge \\varepsilon \\cdot p_{j}$ for all jobs $j$ .", "This standard geometric rounding procedure used in the lemma above allows us to see intervals of the form $I_{x}:=[R_{x},R_{x+1})$ with $R_{x}:=(1+\\varepsilon )^{x}$ as atomic entities.", "An online algorithm can define the corresponding schedule at the beginning of an interval since no further jobs are released until the next interval.", "Moreover, we assume at $1+\\varepsilon $ loss that all jobs that finish within $I_{x}$ have completion time $R_{x+1}$ ." ], [ "Simplification within intervals.", "Our goal is to reduce the number of situations that can arise at the beginning of an interval.", "To this end, we partition the set of jobs released at time $R_x$ into the set of large jobs $L_x$ , with processing times at least $\\varepsilon ^3R_x$ , and the set of small jobs $S_{\\!x}$ with all remaining jobs.", "Running Smith's Rule [45] on small jobs allows us to group very small jobs to job packs, which we treat as single jobs.", "Together with Lemma REF we obtain bounds on the lengths of jobs of each release date.", "Lemma 2.2 At $1+O(\\varepsilon )$ loss, we can assume that for each interval $I_x$ there are lower and upper bounds for the lengths of the jobs $S_x \\cup L_x$ that are within a constant factor of $R_x$ and the constants are independent of $x$ .", "Also, the number of distinct processing times of jobs in each interval is upper-bounded by a constant.", "We look for jobs in $S_{\\!x}$ and $L_x$ which can be excluded from processing within $I_x$ at a loss of not more than $1+O(\\varepsilon )$ .", "This allows us to bound the number of released jobs per interval.", "Lemma 2.3 At $1+O(\\varepsilon )$ loss, we can restrict to instances where for each $x$ , the number of jobs released at time $R_x$ is bounded by a constant $\\Delta $ .", "To prove the above lemmas, we use the technique of time-stretching, see [1].", "In an online interpretation of this method, we shift the work assigned to any interval $I_x$ to the interval $I_{x+1}$ .", "This can be done at a loss of $1+\\varepsilon $ and we obtain free space of size $\\varepsilon \\cdot I_{x^{\\prime }-1}$ in each interval $I_{x^{\\prime }}$ .", "Again using time-stretching, we can show that no job needs to be completed later than constantly many intervals after its release interval.", "Lemma 2.4 There is a constant $s$ such that at $1+O(\\varepsilon )$ loss we can restrict to schedules such that for each interval $I_{x}$ there is a subinterval of $I_{x+s-1}$ which is large enough to process all jobs released at $R_x$ and during which only those jobs are executed.", "We call this subinterval the safety net of interval $I_{x}$ .", "We can assume that each job released at $R_x$ finishes before time $R_{x+s}$ .", "We can also simplify the complexity of the computed schedules by limiting the way jobs are preempted.", "We say that two large jobs are of the same type if they have the same processing time and the same release date.", "A job is partially processed if it has been processed, but not yet completed.", "Lemma 2.5 There is a constant $\\mu >0$ such that at $1+O(\\varepsilon )$ loss we can restrict to schedules such that at the end of each interval, there are at most $m$ large jobs of each type which are partially processed and each of them is processed to an extent which is a multiple of $p_{j}\\cdot \\mu $ and each small job finishes without preemption in the same interval where it started.", "Irrelevant history.", "The schedule that an online algorithm computes for an interval may depend on the set of currently unfinished jobs and possibly the entire schedule used so far.", "In the remainder of this section we show why we can assume that an online algorithm only takes a finite amount of history into account in its decision making, namely, the jobs with relatively large weight released in the last constantly many intervals.", "Our strategy is to partition the time horizon into periods.", "For each integer $k\\ge 0$ , we define a period $Q_{k}$ which consists of the $s$ consecutive intervals $I_{k\\cdot s},...,I_{(k+1)\\cdot s-1}$ .", "For ease of notation, we will treat a period $Q$ as the set of jobs released in that period.", "For a set of jobs $J$ we denote by $rw(J):=\\sum _{j\\in J}r_{j}w_{j}$ their release weight.", "Note that $rw(J)$ forms a lower bound on the quantity that these jobs must contribute to the objective in any schedule.", "Due to Lemma REF , we also obtain an upper bound of $\\left(1+\\varepsilon \\right)^{s}\\cdot rw(J)$ for the latter quantity.", "Lemma 2.6 Let $Q_{k},...,Q_{k+p}$ be consecutive periods such that period $Q_{k+p}$ is the first of this series with $rw(Q_{k+p})\\le \\frac{\\varepsilon }{\\left(1+\\varepsilon \\right)^{s}}\\cdot \\sum _{i=0}^{p-1}rw(Q_{k+i}).$ Then at $1+\\varepsilon $ loss we can move all jobs in $Q_{k+p}$ to their safety nets.", "The above observation defines a natural partition of a given instance $I$ into parts by the insignificant periods.", "Formally, let $a_{1},...,a_{\\ell }$ be all ordered indices such that $Q_{a_{i}}$ is insignificant compared to the preceding periods according to Lemma REF  ($a_{0}:=0$ ).", "Let $a_{\\ell +1}$ be the index of the last period.", "For each $i\\in \\lbrace 0,...,\\ell \\rbrace $ we define a part $P_{\\!i}$ consisting of all periods $Q_{a_{i}+1},...,Q_{a_{i+1}}$ .", "Again, identify with $P_{\\!i}$ all jobs released in this part.", "We treat now each part $P_{\\!i}$ as a separate instance that we present to a given online algorithm.", "For the final output, we concatenate the computed schedules for the different parts.", "It then suffices to bound $\\textsf {\\textsc {A}}(P_{\\!i})/\\textsf {\\textsc {Opt}}(P_{\\!i})$ for each part $P_{\\!i}$ since $\\textsf {\\textsc {A}}(I)/\\textsf {\\textsc {Opt}}(I)\\le \\max _{i}\\lbrace \\textsf {\\textsc {A}}(P_{\\!i})/\\textsf {\\textsc {Opt}}(P_{\\!i})\\rbrace \\cdot (1+O(\\varepsilon )).$ Lemma 2.7 At $1+O(\\varepsilon )$ loss we can restrict to instances which consist of only one part.", "Each but the last period of one part fulfills the opposite condition of the one from Lemma REF .", "This implies exponential growth for the series of partial sums of release weights (albeit with a small growth factor).", "From this observation, we get: Lemma 2.8 There is a constant $K$ such that the following holds: Let $Q_{1},Q_{2},...,Q_{p}$ be consecutive periods such that $rw(Q_{i+1})>\\frac{\\varepsilon }{\\left(1+\\varepsilon \\right)^{s}}\\cdot \\sum _{\\ell =1}^{i}rw(Q_{\\ell })$ for all $i$ .", "Then in any schedule in which each job $j$ finishes no later than by time $r_{j}\\cdot (1+\\varepsilon )^{s}$ (e.g., using the safety net) it holds that $\\sum _{i=1}^{p-K-1}\\sum _{j\\in Q_{i}}w_{j}C_{j}\\le \\varepsilon \\cdot \\sum _{i=p-K}^{p}\\sum _{j\\in Q_{i}}w_{j}C_{j}.$ The objective value of one part is therefore dominated by the contribution of the last $K$ periods of this part.", "We will need this later to show that at $1+\\varepsilon $ loss we can assume that an online algorithm bases its decisions only on a constant amount of information.", "Denote the corresponding number of important intervals by $\\Gamma := Ks$ .", "This enables us to partition the jobs into relevant and irrelevant jobs.", "Intuitively, a job is irrelevant if it is released very early (cf.", "Lemma REF ) or its weight is very small in comparison to some other job.", "The subsequent lemma states that the irrelevant jobs can almost be ignored for the objective value of a schedule.", "Definition 2.9 A job $j$ is irrelevant at time $R_{x}$ if it was irrelevant at time $R_{x-1}$ , or $r_{j}<R_{x-\\Gamma }$ , or it is dominated at time $R_{x}$.", "This is the case if there is a job $j^{\\prime }$ , either released at time $R_{x}$ or already relevant at time $R_{x-1}$ with release date at least $R_{x-\\Gamma }$ , such that $w_{j}<\\frac{\\varepsilon }{\\Delta \\cdot \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}$ .", "Otherwise, a job released until $R_x$ is relevant at time $R_{x}$ .", "Denote the respective subsets of some job set $J$ by $\\mathrm {Rel}_{x}(J)$ and $\\mathrm {Ir}_{x}(J)$ .", "Lemma 2.10 Consider a schedule of one part in which no job $j$ finishes later than at time $r_{j}\\cdot (1+\\varepsilon )^{s}$ (e.g., using the safety net) and let $x$ be an interval index in this part.", "Then $\\sum _{j\\in \\mathrm {Ir}_{x}(J)}w_{j}C_{j}\\le O(\\varepsilon )\\cdot \\sum _{j\\in \\mathrm {Rel}_{x}(J)}w_{j}C_{j}$ .", "The above lemma implies that at $1+O(\\varepsilon )$ loss we can restrict to online algorithms which schedule the remaining part of a job in its safety net, once it has become irrelevant.", "Abstraction of Online Algorithms In this section we show how to construct a competitive-ratio approximation scheme based on the observations of Section .", "To do so, we restrict ourselves to such simplified instances and schedules.", "The key idea is to characterize the behavior of an online algorithm by a map: For each interval, the map gets as input the schedule computed so far and all information about the currently unfinished jobs.", "Based on this information, the map outputs how to schedule the available jobs within this interval.", "More precisely, we define the input by a configuration and the output by an interval-schedule.", "Definition 3.1 An interval-schedule $S$ for an interval $I_{x}$ is defined by the index $x$ of the interval, a set of jobs $J(S)$ available for processing in $I_x$ together with the properties $r_{j}, p_{j}, w_{j}$ of each job $j\\in J(S)$ and its already finished part $f_{j}<p_{j}$ up to $R_{x}$ , for each job $j\\in J(S)$ the information whether $j$ is relevant at time $R_{x}$ , and for each job $j\\in J(S)$ and each machine $i$ a value $q_{ij}$ specifying for how long $j$ is processed by $S$ on machine $i$ during $I_{x}$ .", "An interval-schedule is called feasible if there is a feasible schedule in which the jobs of $J(S)$ are processed corresponding to the $q_{j}$ values within the interval $I_{x}$ .", "Denote the set of feasible interval-schedules as $\\mathcal {S}$ .", "Definition 3.2 A configuration $C$ for an interval $I_{x}$ consists of the index $x$ of the interval, a set of jobs $J(C)$ released up to time $R_{x}$ together with the properties $r_{j}, p_{j}, w_{j}, f_{j}$ of each job $j\\in J(C)$ , an interval-schedule for each interval $I_{x^{\\prime }}$ with $x^{\\prime }<x$ .", "The set of all configurations is denoted by $.", "An \\emph {end-configuration} is a configuration~$ C$ for an interval~$ Ix$ such that at time~$ Rx$,and not earlier, no jobs are left unprocessed.$ We say that an interval-schedule $S$ is feasible for a configuration $C$ if the set of jobs in $J(C)$ which are unfinished at time $R_{x}$ matches the set $J(S)$ with respect to release dates, total and remaining processing time, weight and relevance of the jobs.", "Instead of online algorithms we work from now on with algorithm maps, which are defined as functions $f:\\mathcal {S}$ .", "An algorithm map determines a schedule $f(I)$ for a given scheduling instance $I$ by iteratively applying $f$ to the corresponding configurations.", "W.l.o.g.", "we consider only algorithm maps $f$ such that $f(C)$ is feasible for each configuration $C$ and $f(I)$ is feasible for each instance $I$ .", "Like for online algorithms, we define the competitive ratio $\\rho _{f}$ of an algorithm map $f$ by $\\rho _{f}:=\\max _{I}f(I)/\\textsf {\\textsc {Opt}}(I)$ .", "Due to the following observation, algorithm maps are a natural generalization of online algorithms.", "Proposition 3.3 For each online algorithm $\\textsf {\\textsc {A}}$ there is an algorithm map $f_{\\textsf {\\textsc {A}}}$ such that when $\\textsf {\\textsc {A}}$ is in configuration $C\\in atthe beginning of an interval~$ Ix$, algorithm~$A$ schedules thejobs according to~$ fA(C)$.$ Recall, that we restrict our attention to algorithm maps describing online algorithms which obey the simplifications introduced in Section .", "The essence of such online algorithms are the decisions for the relevant jobs.", "To this end, we define equivalence classes for configurations and for interval-schedules.", "Intuitively, two interval-schedules (configurations) are equivalent if we can obtain one from the other by scalar multiplication with the same value, while ignoring the irrelevant jobs.", "Definition 3.4 Let $S,S^{\\prime }$ be two feasible interval-schedules for two intervals $I_{x},I_{x^{\\prime }}$ .", "Denote by $J_{\\mathrm {Rel}}(S)\\subseteq J(S)$ and $J_{\\mathrm {Rel}}(S^{\\prime })\\subseteq J(S^{\\prime })$ the relevant jobs in $J(S)$ and $J(S^{\\prime })$ .", "Let further $\\sigma :J_{\\mathrm {Rel}}(S)\\rightarrow J_{\\mathrm {Rel}}(S^{\\prime })$ be a bijection and $y$ an integer.", "The interval-schedules $S,S^{\\prime }$ are $(\\sigma ,y)$ -equivalent if $r_{\\sigma (j)}=r_{j}(1+\\varepsilon )^{x^{\\prime }-x}, p_{\\sigma (j)}=p_{j}(1+\\varepsilon )^{x^{\\prime }-x},f_{\\sigma (j)}=f_{j}(1+\\varepsilon )^{x^{\\prime }-x}, q_{\\sigma (j)}=q_{j}(1+\\varepsilon )^{x^{\\prime }-x}$ and $w_{\\sigma (j)}=w_{j}\\left(1+\\varepsilon \\right)^{y}$ for all $j\\in J_{\\mathrm {Rel}}(S)$ .", "The interval-schedules $S,S^{\\prime }$ are equivalent (denoted by $S\\sim S^{\\prime }$ ) if a map $\\sigma $ and an integer $y$ exist such that they are $(\\sigma ,y)$ -equivalent.", "Definition 3.5 Let $C,C^{\\prime }$ be two configurations for two intervals $I_{x},I_{x^{\\prime }}$ .", "Denote by $J_{\\mathrm {Rel}}(C),J_{\\mathrm {Rel}}(C^{\\prime })$ the jobs which are relevant at times $R_{x},R_{x^{\\prime }}$ in $C,C^{\\prime }$ , respectively.", "Configurations $C,C^{\\prime }$ are equivalent (denoted by $C\\sim C^{\\prime }$ ) if there is a bijection $\\sigma :J_{\\mathrm {Rel}}(C)\\rightarrow J_{\\mathrm {Rel}}(C^{\\prime })$ and an integer $y$ such that $r{}_{\\sigma (j)}=r_{j}(1+\\varepsilon )^{ x^{\\prime }-x },p{}_{\\sigma (j)}=p_{j}(1+\\varepsilon )^{x^{\\prime }-x},f{}_{\\sigma (j)}=f_{j}(1+\\varepsilon )^{x^{\\prime }-x}$ and $w_{\\sigma (j)}=w_{j}\\left(1+\\varepsilon \\right)^{y}$ for all $j\\in J_{\\mathrm {Rel}}(C)$ , and the interval-schedules of $I_{x-k}$ and $I_{x^{\\prime }-k}$ are $(\\sigma ,y)$ -equivalent when restricted to the jobs in $J_{\\mathrm {Rel}}(C)$ and $J_{\\mathrm {Rel}}(C^{\\prime })$ for each $k\\in \\mathbb {N}$ .", "A configuration $C$ is realistic for an algorithm map $f$ if there is an instance $I$ such that if $f$ processes $I$ then at time $R_x$ it is in configuration $C$ .", "The following lemma shows that we can restrict the set of algorithm maps under consideration to those which treat equivalent configurations equivalently.", "We call algorithm maps obeying this condition in addition to the restrictions of Section  simplified algorithm maps.", "Lemma 3.6 At $1+O(\\varepsilon )$ loss we can restrict to algorithm maps $f$ such that $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations $C,C^{\\prime }$ .", "Let $f$ be an algorithm map.", "For each equivalence class $\\subseteq of the set of configurations we pick arepresentative $ C$ which is realistic for~$ f$.", "For each configuration $ C'$, we define a newalgorithm map $ f$ by setting $ f(C')$ to be theinterval-schedule for $ C'$ which is equivalent to $ f(C)$.One can show by induction that $ f$ is always in aconfiguration such that an equivalent configuration is realistic for$ f$.", "Hence, equivalence classes without realistic configurationsfor~$ f$ are not relevant.", "We claim that$ f(1+O())f$.$ Consider an instance $\\bar{I}$ .", "We show that there is an instance $I$ such that $\\bar{f}(\\bar{I})/\\textsf {\\textsc {Opt}}(\\bar{I})\\le (1+O(\\varepsilon ))f(I)/\\textsf {\\textsc {Opt}}(I)$ which implies the claim.", "Consider an interval $I_{\\bar{x}}$ .", "Let $\\bar{C}$ be the end-configuration obtained when $\\bar{f}$ is applied iteratively on $\\bar{I}$ .", "Let $C$ be the representative of the equivalence class of $\\bar{C}$ , which was chosen above and which is realistic for $f$ ($C$ is also an end-configuration).", "Therefore, there is an instance $I$ such that $C$ is reached at time $R_{x}$ when $f$ is applied on $I$ .", "Hence, $I$ is the required instance since the relevant jobs dominate the objective value (see Lemma REF ) and $C \\sim \\bar{C}$ .", "Lemma 3.7 There are only constantly many simplified algorithm maps.", "Each simplified algorithm map can be described using finite information.", "From the simplifications introduced in Section  follows that the domain of the algorithm maps under consideration contains only constantly many equivalence classes of configurations.", "Also, the target space contains only constantly many equivalence classes of interval-schedules.", "For an algorithm map $f$ which obeys the restrictions of Section , the interval-schedule $f(C)$ is fully specified when knowing only $C$ and the equivalence class which contains $f(C)$ (since the irrelevant jobs are moved to their safety net anyway).", "Since $f(C)\\sim f(C^{\\prime })$ for a simplified algorithm map $f$ if $C\\sim C^{\\prime }$ , we conclude that there are only constantly many simplified algorithm maps.", "Finally, each equivalence class of configurations and interval-schedules can be characterized using only finite information, and hence the same holds for each simplified algorithm map.", "The next lemma shows that up to a factor $1+\\varepsilon $ worst case instances of simplified algorithm maps span only constantly many intervals.", "Using this property, we will show in the subsequent lemmas that the competitive ratio of a simplified algorithm map can be determined algorithmically up to a $1+\\varepsilon $ factor.", "Lemma 3.8 There is a constant $E$ such that for any instance $I$ and any simplified algorithm map $f$ there is a realistic end-configuration $\\tilde{C}$ for an interval $I_{\\tilde{x}}$ with $\\tilde{x}\\le E$ which is equivalent to the corresponding end-configuration when $f$ is applied to $I$ .", "Consider a simplified algorithm map $f$ .", "For each interval $I_{x}$ , denote by ${x}^{f}$ the set of realistic equivalence classes for $I_{x}$ , i.e., the equivalence classes which have a realistic representative for $I_{x}$ .", "Since there are constantly many equivalence classes and thus constantly many sets of equivalence classes, there must be a constant $E$ independent of $f$ such that ${\\bar{x}}^{f}={\\bar{x}^{\\prime }}^{f}$ for some $\\bar{x}<\\bar{x}^{\\prime }\\le E$ .", "Since $f$ is simplified it can be shown by induction that ${\\bar{x}+k}^{f}={\\bar{x}^{\\prime }+k}^{f}$ for any $k\\in \\mathbb {N}$ , i.e., $f$ cycles with period length $\\bar{x}^{\\prime }-\\bar{x}$ .", "Consider now some instance $I$ and let $C$ with interval $I_{x}$ be the corresponding end-configuration when $f$ is applied to $I$ .", "If $x\\le E$ we are done.", "Otherwise there must be some $k\\le \\bar{x}^{\\prime }-\\bar{x}$ such that ${\\bar{x}+k}^{f}={x}^{f}$ since $f$ cycles with this period length.", "Hence, by definition of ${\\bar{x}+k}^{f}$ there must be a realistic end-configuration $\\tilde{C}$ which is equivalent to $C$ for the interval $I_{\\tilde{x}}$ with $\\tilde{x}:=\\bar{x}+k\\le E$ .", "Lemma 3.9 Let $f$ be a simplified algorithm map.", "There is an algorithm which approximates $\\rho _{f}$ up to a factor $1+\\varepsilon $ , i.e., it computes a value $\\rho ^{\\prime }$ with $\\rho ^{\\prime }\\le \\rho _{f}\\le (1+O(\\varepsilon ))\\rho ^{\\prime }$ .", "[Proof sketch.]", "By Lemma REF , the relevant jobs in a configuration dominate the entire objective value.", "In particular, we do not need to know the irrelevant jobs of a configuration if we only want to approximate its objective value up to a factor of $1+O(\\varepsilon )$ .", "For an end-configuration $C$ denote by $val_{C}(J_{\\mathrm {Rel}}(C))$ the objective value of the jobs in $J_{\\mathrm {Rel}}(C)$ in the history of $C$ .", "We define $r(C):=val_{C}(J_{\\mathrm {Rel}}(C))/\\textsf {\\textsc {Opt}}(J_{\\mathrm {Rel}}(C))$ to be the achieved competitive ratio of $C$ when restricted to the relevant jobs.", "According to Lemma REF , it suffices to construct the sets ${0}^{f},...,{E}^{f}$ in order to approximate the competitive ratio of all end-configurations in these sets.", "We start with ${0}^{f}$ and determine $f(C)$ for one representant $C$ of each equivalence class ${0}^{f}$ .", "Based on this we determine the set ${1}^{f}$ .", "We continue inductively to construct all sets ${x}^{f}$ with $x\\le E$ .", "We define $r_{\\max }$ to be the maximum ratio $r(C)$ for an end-configuration $C\\in \\cup _{0\\le x\\le E}{x}^{f}$ .", "Due to Lemma REF and Lemma REF the value $r_{\\max }$ implies the required $\\rho ^{\\prime }$ fulfilling the properties claimed in this lemma.", "Our main algorithm works as follows.", "We first enumerate all simplified algorithm maps.", "For each simplified algorithm map $f$ we approximate $\\rho _{f}$ using Lemma REF .", "We output the map $f$ with the minimum (approximated) competitive ratio.", "Note that the resulting online algorithm has polynomial running time: All simplifications of a given instance can be done efficiently and for a given configuration, the equivalence class of the schedule for the next interval can be found in a look-up table of constant size.", "Theorem 3.10 For any $m\\in \\mathbb {N}$ we obtain a competitive-ratio approximation scheme for $\\textup {Pm}|\\,r_{j},pmtn\\,|\\sum w_{j}C_{j}$ .", "Extensions to Other Settings Non-preemptive Scheduling.", "When preemption is not allowed, the definition of the safety net (Lemma REF ) needs to be adjusted since we cannot ensure that at the end of each interval $I_{x+s}$ there is a machine idle.", "However, we can guarantee that there is a reserved space somewhere in $[R_{x},R_{x+s})$ to process the small and big jobs in $S_{x} \\cup L_{x}$ .", "Furthermore, we cannot enforce that a big job $j$ is processed for exactly a certain multiple of $p_{j}\\mu $ in each interval (Lemma REF ).", "To solve this, we pretend that we could preempt $j$ and ensure that after $j$ has been preempted its machine stays idle until $j$ continues.", "Next, we can no longer assume that each part can be treated independently (Lemma REF ).", "Since some of the remaining jobs at the end of a part may have already started processing, we cannot simply move them to their safety net.", "Here we use the following simplification.", "Lemma 4.1 Let $\\textnormal {first}(i)$ denote the job that is released first in part $P_{\\!i}$ .", "At $1+\\varepsilon $ loss, we can restrict to instances such that $\\sum _{\\ell =1}^{i-1} rw(P_{\\!\\ell }) \\le \\frac{\\varepsilon }{\\left(1+\\varepsilon \\right)^{s}}\\cdot rw(\\textnormal {first}(i))$ , i.e., $\\textnormal {first}(i)$ dominates all previous parts.", "Therefore, at $1+\\varepsilon $ loss it is enough to consider only the currently running jobs from the previous part and the last $\\Gamma $ intervals from the current part when taking decisions.", "Finally, we add some minor modifications to handle the case that a currently running job is dominated by some other job.", "With these adjustments, we have only constantly many equivalence classes for interval-schedules and configurations, which allows us to construct a competitive-ratio approximation scheme as in Section .", "Theorem 4.2 For any $m\\in \\mathbb {N}$ we obtain a competitive-ratio approximation scheme for $\\textup {Pm}|\\,r_j\\,|\\sum w_jC_j$  .", "Scheduling on Related Machines.", "In this setting, each machine $i$ has associated a speed $s_i$ , such that processing job $j$ on machine $i$ takes $p_{j}/s_{i}$ time units.", "W.l.o.g.", "the slowest machine has unit speed.", "Let $s_{\\max }$ denote the maximum speed in an instance.", "An adjusted version of Lemma REF ensures that at $1+\\varepsilon $ loss $r_{j} \\ge \\varepsilon \\, p_{j}/s_{\\max }$ for all jobs $j$  (rather than $r_{j} \\ge \\varepsilon p_{j}$ ).", "Furthermore, we can bound the number of distinct processing times and the number of released jobs of each interval, using similar arguments as in the unit-speed case.", "Lemma 4.3 At $1+O(\\varepsilon )$ loss we can restrict to instances where for each release date the number of released jobs and the number of distinct processing times is bounded by a constant depending only on $\\varepsilon $ , $m$ , and $s_{\\max }$ .", "We establish the safety net for the jobs of each release date $R_{x}$ only on the fastest machine and thereby ensure the condition of Lemma REF in the related machine setting.", "For the non-preemptive setting we incorporate the adjustments introduced in Section REF .", "Since at $1+\\varepsilon $ loss we can round the speeds of the machines to powers of $1+\\varepsilon $ we obtain the following result.", "Theorem 4.4 For any $m\\in \\mathbb {N}$ we obtain competitive-ratio approximation schemes for $\\textup {Qm}|\\,r_j, pmtn\\,|\\sum w_jC_j$ and $\\textup {Qm}|\\,r_j\\,|\\sum w_jC_j$ , assuming that the speeds of any two machines differ by at most a constant factor.", "In the preemptive setting we can strengthen the result and give a competitive-ratio approximation scheme for the case that machine speeds are part of the input, that is, we obtain a nearly optimal competitive ratio for any speed vector.", "The key is to bound the variety of different speeds.", "To that end, we show that at $1+\\varepsilon $ loss a very fast machine can simulate $m-1$ very slow machines.", "Lemma 4.5 For $\\textup {Qm}|\\,r_j, pmtn\\,|\\sum w_jC_j$ , at $1+O(\\varepsilon )$ loss, we can restrict to instances in which $s_{\\max }$ is bounded by $m/\\varepsilon $ .", "As speeds are geometrically rounded, we have for each value $m$ only finitely many speed vectors.", "Thus, our enumeration scheme finds a nearly optimal online algorithm with a particular routine for each speed vector.", "Theorem 4.6 For any $m\\in \\mathbb {N}$ we obtain a competitive-ratio approximation scheme for $\\textup {Qm}|\\,r_j,pmtn\\,|\\sum w_jC_j$  .", "Preemptive Scheduling on Unrelated Machines.", "When each job $j$ has its individual processing time $p_{ij}$ on machine $i$ , the problem complexity increases significantly.", "We restrict to preemptive scheduling and show how to decrease the complexity to apply our approximation scheme.", "The key is to bound the range of the finite processing times for each job (which is unfortunately not possible in the non-preemptive case, see [1] for a counterexample).", "Lemma 4.7 At $1+\\varepsilon $ loss we can restrict to instances in which for each job $j$ the ratio of any two of its finite processing times is bounded by $m/\\varepsilon $ .", "The above lemma allows us to introduce the notion of job classes.", "Two jobs $j,j^{\\prime }$ are of the same class if they have finite processing times on exactly the same machines and  $p_{ij}/p_{ij^{\\prime }}=p_{i^{\\prime }j}/p_{i^{\\prime }j^{\\prime }}$ for any two such machines $i$ and $i^{\\prime }$ .", "For fixed $m$ , the number of different job classes is bounded by a constant $W$ .", "For each job class, we define large and small tasks similar to the identical machine case: for each job $j$ we define a value $\\tilde{p}_{j}:=\\max _{i}\\lbrace p_{ij}|p_{ij}<\\infty \\rbrace $ and say a job is large if  $\\tilde{p}_{j}\\ge \\varepsilon ^{2}r_{j}/W$ and small otherwise.", "For each job class separately, we perform the adjustments of Section .", "This yields the following lemma.", "Lemma 4.8 At $1+O(\\varepsilon )$ loss we can restrict to instances and schedules such that for each job class, the number of distinct values $\\tilde{p}_j$ of jobs $j$ with the same release date is bounded by a constant, for each job class, the number of jobs with the same release date is bounded by a constant $\\tilde{\\Delta }$ , a large job $j$ is only preempted at integer multiples of $\\tilde{p}_{j} \\cdot \\tilde{\\mu }$ for some constant $\\tilde{\\mu }$ and small jobs are never preempted and finish in the same interval where they start.", "The above lemmas imply that both, the number of equivalence classes of configurations and the number of equivalence classes for interval-schedules are bounded by constants.", "Thus, we can apply the enumeration scheme from Section .", "Theorem 4.9 For any $m\\in \\mathbb {N}$ we obtain a competitive-ratio approximation scheme for $\\textup {Rm}|\\,r_j,pmtn\\,|\\sum w_jC_j$  .", "Randomized algorithms When algorithms are allowed to make random choices and we consider expected values of schedules, we can restrict to instances which span only constantly many periods.", "Assuming the simplifications of Section , this allows a restriction to instances with a constant number of jobs.", "Lemma 5.1 For randomized algorithms, at $1+\\varepsilon $ loss we can restrict to instances in which all jobs are released in at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ consecutive periods.", "[Proof idea.]", "Beginning at a randomly chosen period $Q_i$ with $i\\in [0,M)$ , with $M:=\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\rceil $ , we move all jobs released in $Q_{i+kM}$ , $k=0,1,\\ldots $ , to their safety net.", "At $1+\\varepsilon $ loss, this gives us a partition into parts, at the end of which no job remains, and we can treat each part independently.", "A randomized online algorithm can be viewed as a function that maps every possible configuration $C$ to a probability distribution of interval-schedules which are feasible for $C$ .", "To apply our algorithmic framework from the deterministic setting that enumerates all algorithm maps, we discretize the probability space and define discretized algorithm maps.", "To this end, let $\\bar{\\Gamma }$ denote the maximum number of intervals in instances with at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $  periods.", "Definition 5.2 (Discretized algorithm maps) Let $\\bar{ be the set of configurations for intervals~I_{x}with~x\\le \\bar{\\Gamma }, let~\\bar{\\mathcal {S}} be the set ofinterval-schedules for intervals~I_{x} with~x\\le \\bar{\\Gamma },and let~\\delta >0.", "A \\emph {\\delta -discretized algorithm map }is afunction~f:\\bar{\\times \\bar{\\mathcal {S}}\\rightarrow [0,1] suchthatf(C,S)=k\\cdot \\delta with some~k\\in \\mathbb {N}_{0} for all~C\\in \\bar{ and~S\\in \\bar{\\mathcal {S}}, and\\sum _{S\\in \\bar{\\mathcal {S}}}f(C,S)=1 for all ~C\\in \\bar{.", "}By restricting to \\delta -discretizedalgorithm maps we do not lose too much in the competitive ratio.", "}\\begin{lem}There is a value~\\delta >0 such that for any (randomized) algorithmmap~f there is a~\\delta -discretized randomized algorithm map~gwith~\\rho _{g}\\le \\rho _{f}\\left(1+\\varepsilon \\right).\\end{lem}\\begin{proof}[Proof idea.", "]Let~f be a randomized algorithm map andlet~\\delta >0 such that 1/\\delta \\in \\mathbb {N}.", "We define anew~\\delta -discretized algorithm map~g.", "For eachconfiguration~C we define the values~g(C,S) suchthat~\\left\\lfloor f(C,S)/\\delta \\right\\rfloor \\cdot \\delta \\le g(C,S)\\le \\left\\lceil f(C,S)/\\delta \\right\\rceil \\cdot \\delta and~\\sum _{S\\in \\mathcal {S}}g(C,S)=1.", "To seethat~\\rho _{g}\\le \\left(1+\\varepsilon \\right)\\rho _{f}, consider an instance~I and apossible schedule~S(I) for I.", "There is a probability~pthat~f outputs~S(I).", "We show that the schedules which have largeprobability~p dominate~\\mathbb {E}\\left[\\,f(I)\\,\\right]/\\textsf {\\textsc {Opt}}(I).", "We show furtherthat if p is sufficiently large, the probability that~gproduces~S(I) is in~[p/(1+\\varepsilon ),p(1+\\varepsilon )), which implies theLemma.\\end{proof}}Like in the deterministic case, we can now show that at 1+\\varepsilon lossit suffices to restrict to \\emph {simplified \\delta -discretizedalgorithm maps} which treat equivalent configurations equivalently,similar to Lemma~\\ref {lem:equal-confs} (replacing~\\Gamma by~\\bar{\\Gamma } in Definition~\\ref {def:irrelevant_jobs} of theirrelevant jobs).As there are only constantly many of these maps, we enumerate all ofthem, test each map for its competitive ratio, and select the best ofthem.", "}\\begin{thm} We obtain randomized competitive-ratio approximation schemes for \\textup {Pm}|\\,r_j,(pmtn)\\,|\\sum w_jC_j,\\textup {Qm}|\\,r_j, pmtn\\,|\\sum w_jC_j, and\\textup {Rm}|\\,r_j, pmtn\\,|\\sum w_jC_j\\, and for \\textup {Qm}|\\,r_j\\,|\\sum w_jC_j with a bounded range of speeds for any fixed m\\in \\mathbb {N}.\\end{thm}$ General Min-Sum Objectives and Makespan In this section we briefly argue how the techniques presented above for minimizing $\\sum _{j}w_{j}C_{j}$ can be used for constructing online algorithm schemes for other scheduling problems with jobs arriving online over time, namely for minimizing $\\sum _{j\\in J}w_jf(C_{j})$ , with $f(x)=k\\cdot x^{\\alpha }$ and constant $\\alpha \\ge 1, k>0$ , and the makespan.", "Since monomial functions $f$ have the property that $f((1+\\varepsilon ) C_j)\\le (1+O(\\varepsilon )) f(C_j)$ , the arguments in previous sections apply almost directly to the generalized min-sum objective.", "In each step of simplification and abstraction, we have an increased loss in the performance guarantee, but it is covered by the $O(\\varepsilon )$ -term.", "Consider now the makespan objective.", "The simplifications within intervals of Section  are based on arguing on completion times of individual jobs, and clearly hold also for the last job.", "Thus, they directly apply to makespan minimization.", "We simplify the definition of irrelevant history by omitting the partition of the instance into parts and we define a job $j$ to be irrelevant at time $R_{x}$ if $r_{j}\\le R_{x-s}$ .", "Based on this definition, we define equivalence classes of configurations (ignoring weights and previous interval-schedules) and again restrict to algorithm maps $f$ with $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations (Lemma 3.6).", "Lemmas 3.7–3.9 then hold accordingly and yield a competitive-ratio approximation scheme.", "Finally, the adjustments of Sections  and  can be made accordingly.", "Without the partition of the instance into parts, this even becomes easier in the non-preemptive setting.", "Thus, we can state the following result.", "Theorem 6.1 For any $m \\in \\mathbb {N}$ there are deterministic and randomized competitive-ratio approximation schemes for preemptive and non-preemptive scheduling, on $m$ identical, related (with bounded speed ratio when non-preemptive), and unrelated machines (only preemptive) for the objectives of minimizing $C_{\\max }$ and minimizing $\\sum _{j\\in J}w_j f(C_j)$ , with $f(x)=k\\cdot x^{\\alpha }$ and constant $\\alpha \\ge 1,k>0$ .", "Conclusions We introduce the concept of competitive-ratio approximation schemes that compute online algorithms with a competitive ratio arbitrarily close to the best possible competitive ratio.", "We provide such schemes for various problem variants of scheduling jobs online to minimize the weighted sum of completion times, arbitrary monomial cost functions, and the makespan.", "The techniques derived in this paper provide a new and interesting view on the behavior of online algorithms.", "We believe that they contribute to the understanding of such algorithms and possibly open a new line of research in which they yield even further insights.", "In particular, it seems promising that our methods could also be applied to other online problems than scheduling jobs arriving online over time.", "Related work Sum of weighted completion times.", "The offline variants of nearly all problems under consideration are NP-hard.", "This is true already for the special case of a single machine [27], [28].", "Two restricted variants can be solved optimally in polynomial time.", "Smith's Rule solves the problem $\\textup {1}|\\,\\,|\\sum w_jC_j$ to optimality by scheduling jobs in non-increasing order of weight-to-processing-time ratios [45].", "Furthermore, scheduling by shortest remaining processing times yields an optimal schedule for $\\textup {1}|\\,r_j,pmtn\\,|\\sum w_jC_j$  [38].", "However, for the other settings polynomial-time approximation schemes have been developed [1], even when the number of machines is part of the input.", "The online setting has been a highly active field of research in the past fifteen years.", "A whole sequence of papers appeared introducing new algorithms, new relaxations and analytical techniques that decreased the gaps between lower and upper bounds on the optimal competitive ratio [18], [39], [20], [43], [42], [2], [6], [24], [10], [17], [35], [34], [9], [40], [30], [31], [46], [36], [5], [41], [14].", "We do not intend to give a detailed history of developments; instead, we refer the reader to overviews, e.g., in [34], [10].", "Table REF summarizes the current state-of-the-art on best known lower and upper bounds on the optimal competitive ratios.", "Interestingly, despite the considerable effort, optimal competitive ratios are known only for $\\textup {1}|\\,r_j, pmtn\\,|\\sum C_j$  [38] and for non-preemptive single-machine scheduling [2], [46], [24], [6].", "In all other scheduling settings remain unsatisfactory, even quite significant gaps.", "Table: Lower and upper bounds on the competitive ratio for deterministic and randomized online algorithms.", "[1]For $m=1,2,3,4,5, \\dots 100$ the lower bound is $LB=2, 1.520, 1.414, 1.373, 1.364, \\dots 1.312$ .", "More general min-sum (completion time) objectives.", "Recently, there has been an increasing interest in studying generalized cost functions.", "So far, this research has focussed on offline problems.", "The most general case is when each job may have its individual non-decreasing cost function $f_j$ .", "For scheduling on a single machine with release dates and preemption, $1|r_j,pmtn|\\sum f_j$ , Bansal and Pruhs [4] gave a randomized $\\mathcal {O}(\\log \\log (nP))$ -approximation, where $P=\\max _{j\\in J}p_j$ .", "In the case that all jobs have identical release dates, the approximation factor reduces to 16.", "Cheung and Shmoys [8] improved this latter result and gave a deterministic $(2+\\varepsilon )$ -approximation.", "This result applies also on a machine of varying speed.", "The more restricted problem with a global cost function $1|r_j,pmtn|\\sum w_jf(C_j)$ has been studied by Epstein et al.", "[13] in the context of universal solutions.", "They gave an algorithm that produces for any job instance one scheduling solution that is a $(4+\\varepsilon )$ -approximation for any cost function and even under unreliable machine behavior.", "Höhn and Jacobs [23] studied the same problem without release dates.", "They analyzed the performance of Smith's Rule [45] and gave tight approximation guarantees for all convex and all concave functions $f$ .", "Makespan.", "The online makespan minimization problem has been extensively studied in a different online paradigm where jobs arrive one by one (see [15], [37] and references therein).", "Our model, in which jobs arrive online over time, is much less studied.", "In the identical parallel machine environment, Chen and Vestjens [7] give nearly tight bounds on the optimal competitive ratio, $1.347 \\le \\rho ^* \\le 3/2$ , using a natural online variant of the well-known largest processing time first algorithm.", "In the offline setting, polynomial time approximation schemes are known for identical [21] and uniform machines [22].", "For unrelated machines, the problem is NP-hard to approximate with a better ratio than $3/2$ and a 2-approximation is known [29].", "If the number of machines is bounded by a constant there is a PTAS [29].", "Proofs of Section  First, we will show that the number of distinct processing times of large jobs in each interval can be upper-bounded by a constant.", "To achieve this, we partition the jobs of an instance into large and small jobs.", "With respect to a release date $R_{x}$ we say that a job $j$ with $r_{j}=R_{x}$ is large if $p_{j}\\ge \\varepsilon ^{2}I_{x}=\\varepsilon ^{3}R_{x}$ and small otherwise.", "Abusing notation, we refer to $|I_{x}|$ also by $I_{x}$ .", "Note that $I_x=\\varepsilon \\cdot (1+\\varepsilon )^x$ .", "Lemma B.1 The number of distinct processing times of jobs in each set $L_x$ is bounded by $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ .", "For any $j\\in L_{x}$ the processing time $p_{j}$ is a power of $1+\\varepsilon $ , say $p_{j}=(1+\\varepsilon )^{y}$ .", "Hence, we have that $\\varepsilon ^{3}\\left(1+\\varepsilon \\right)^{x}<\\left(1+\\varepsilon \\right)^{y}\\le \\frac{1}{\\varepsilon }\\left(1+\\varepsilon \\right)^{x}$ .", "The number of integers $y$ which satisfy the above inequalities is bounded by $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ , which yields the constant claimed in the lemma.", "Furthermore, we can bound the number of large jobs of each job size which are released at the same time.", "Lemma B.2 Without loss, we can restrict to instances with $|L_{x}|\\le (m/\\varepsilon ^{2}+m)4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ for each set $L_{x}$ .", "Let $L_{x,p}\\subseteq L_{x}$ denote the set of jobs in $L_{x}$ with processing time $p$ .", "By an exchange argument, one can restrict to schedules such that at each point in time at most $m$ jobs in $L_{x,p}$ are partially (i.e., to some extent but not completely) processed.", "Since $p_{j}\\ge \\varepsilon ^{2}I_{x}$ for each job $j\\in L_{x}$ , at most $m/\\varepsilon ^{2}+m$ jobs in $L_{x,p}$ are touched within $I_{x}$ .", "By an exchange argument we can assume that they are the $m/\\varepsilon ^{2}+m$ jobs with the largest weight in $L_{x,p}$ .", "Hence, the release date of all other jobs in $L_{x,p}$ can be moved to $R_{x+1}$ without any cost.", "Since due to Lemma REF there are at most $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ distinct processing times $p$ of large jobs in $L_{x}$ , the claim follows.", "We now just need to take care of the small jobs.", "Denote by $w_{j}/p_{j}$ the Smith's ratio of a job $j$ .", "An ordering where the jobs are ordered non-increasingly by their Smith's ratios is an ordering according to Smith's rule.", "The next lemma shows that scheduling the small jobs according to Smith's Rule is almost optimal and small jobs do not even need to be preempted or to cross intervals.", "For a set of jobs $J$ we define $p(J):=\\sum _{j\\in J}p_{j}$ .", "Lemma B.3 At $1+\\varepsilon $ loss we can restrict to schedules such that for each interval $I_{x}$ the small jobs scheduled within this interval are chosen by Smith's Rule from the set $\\bigcup _{x^{\\prime }\\le x}S_{x^{\\prime }}$ , no small job is preempted, any small job finishes in the same interval where it started and $p(S_{\\!x})\\le m\\cdot I_{x}$ for each interval $I_{x}$ .", "By an exchange argument one can show that it is optimal to schedule the small jobs by Smith's Rule if they can be arbitrarily divided into smaller jobs (where the weight is divided proportional to the processing time of the smaller jobs).", "Start with such a schedule and stretch time once.", "The gained free space is enough to finish all small jobs which are partially scheduled in each interval.", "For the last claim of the lemma, note that the total processing time in each interval $I_{x}$ is $mI_{x}$ .", "Order the small jobs non-increasingly by their Smith's Ratios and pick them until the total processing time of picked jobs just does not exceed $mI_{x}$ .", "The release date of all other jobs in $S_{x}$ can be safely moved to $R_{x+1}$ since due to our modifications we would not schedule them in $I_{x}$ anyway.", "Lemma REF (restated) There is a constant $s$ such that at $1+O(\\varepsilon )$ loss we can restrict to schedules such that for each interval $I_{x}$ there is a subinterval of $I_{x+s-1}$ which is large enough to process all jobs released at $R_x$ and during which only jobs in $R_x$ are executed.", "We call this subinterval the safety net of interval $I_{x}$ .", "We can assume that each job released at $R_x$ finishes before time $R_{x+s}$ .", "By Lemmas REF and $\\ref {lem:number-large-jobs}$ we bound $p(S_{x})+p(L_{x})$ by $p(S_{x})+p(L_{x}) & \\le & m\\cdot I_{x}+(m/\\varepsilon ^{2}+m) \\cdot \\left(4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }\\right) \\cdot \\frac{1}{\\varepsilon }\\left(1+\\varepsilon \\right)^{x}\\\\& \\le & m\\cdot \\left(1+\\varepsilon \\right)^{x}\\left(\\varepsilon +\\frac{8}{\\varepsilon ^{3}}\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }\\right)\\\\& = & \\varepsilon \\cdot I_{x+s-1}$ for a suitable constant $s$ , depending on $\\varepsilon $ and $m$ .", "Stretching time once, we gain enough free space at the end of each interval $I_{x+s-1}$ to establish the safety net for each job set $p(S_{x})+p(L_{x})$ .", "Lemma B.4 There is a constant $d$ such that we can at $1+O(\\varepsilon )$ loss restrict to instances such that $p_{j}>\\frac{\\varepsilon }{2d}\\cdot I_{x}$ for each job $j \\in S_x \\cup L_x$ .", "We call a job $j$ tiny if $p_j \\le \\frac{\\varepsilon }{2d}\\cdot I_{x}$ .", "Let $T_{x}=\\lbrace j_{1},j_{2},...,j_{|T_{x}|}\\rbrace $ denote all tiny jobs released at $R_{x}$ .", "W.l.o.g.", "assume that they are ordered non-increasingly by their Smith's Ratios $w_{j}/p_{j}$ .", "Let $\\ell $ be the largest integer such that $\\sum _{i=1}^{\\ell }p_{i}\\le \\frac{\\varepsilon }{d}\\cdot I_{x}$ .", "We define the pack $P_{x}^{1}:=\\lbrace j_{1},...,j_{\\ell }\\rbrace $ .", "We denote by $\\sum _{i=1}^{\\ell }p_{i}$ the processing time of pack $P_{x}^{1}$ and by $\\sum _{i=1}^{\\ell }w_{i}$ its weight.", "We continue iteratively until we assigned all tiny jobs to packs.", "By definition of the processing time of tiny jobs, the processing time of all but possibly the last pack released at time $R_{x}$ is in the interval $[\\frac{\\varepsilon }{2d}\\cdot I_{x},\\frac{\\varepsilon }{d}\\cdot I_{x}]$ .", "Using timestretching, we can show that at $1+O(\\varepsilon )$ loss all tiny jobs of the same pack are scheduled in the same interval on the same machine.", "Here we use that in any schedule obeying Smith's Rule and using the safety net (see Lemma REF ) in each interval there is at most one partially but unfinished pack from each of at most $s$ previous release dates.", "Hence, we can treat the packs as single jobs whose processing time and weight matches the respective values of the packs.", "Also, at $1+\\varepsilon $ loss we can ensure that also the very last pack has a processing time in $[\\frac{\\varepsilon }{2d}\\cdot I_{x},\\frac{\\varepsilon }{d}\\cdot I_{x}]$ .", "Finally, at $1+O(\\varepsilon )$ loss we can ensure that the processing times and weights of the new jobs (which replace the packs) are powers of $1+\\varepsilon $ .", "Lemma B.5 Assume that there is a constant $d$ such that $p_{j}>\\frac{\\varepsilon }{2d}\\cdot I_{x}$ for each job $j \\in S_x$ .", "Then at $1+O(\\varepsilon )$ loss, the number of distinct processing times of jobs each set $S_x$ is upper-bounded by $(\\log _{1+\\varepsilon }\\varepsilon \\cdot 2d)$ .", "From the previous lemmas, we have $\\frac{e^2}{2d}\\cdot (1+\\varepsilon )^x<(1+\\varepsilon )^y<\\varepsilon ^3(1+\\varepsilon )^x.$ The number of integers $y$ satisfying these inequalities is upper-bounded by the claimed constant.", "Lemma REF now follows from the lemmas REF and REF .", "Lemma REF follows from lemmas REF , REF and REF .", "Next, we prove Lemma REF : [Proof of Lemma REF ] The claim about the number of partially processed jobs of each type can be assumed without any loss.", "For the extent of processing, note that due to Lemmas REF , REF , and REF there is a constant $c$ such that at each time $R_{x}$ the total processing time of unfinished large jobs is bounded by $c\\cdot R_{x}$ .", "We stretch time once.", "The gained space is sufficient to schedule $p_{j}\\cdot \\mu $ processing units of each unfinished large job $j$ (for an appropriately chosen universal constant $\\mu $ ).", "This allows us to enforce the claim.", "The claim about the non-preemptive behavior of small jobs follows from Lemma REF .", "[Proof of Lemma REF ] In any schedule the jobs in $\\cup _{i=0}^{p-1}Q_{k+i}$ contribute at least $\\sum _{i=0}^{p-1}rw(Q_{k+i})$ towards the objective.", "If we move all jobs in $Q_{k+p}$ to their safety nets, they contribute at most $\\sum _{j\\in Q_{k+p}}r_{j}\\left(1+\\varepsilon \\right)^{s}\\cdot w_{j} & = & \\left(1+\\varepsilon \\right)^{s}\\cdot rw(Q_{k+p})\\\\& \\le & \\varepsilon \\cdot \\sum _{i=0}^{p-1}rw(Q_{k+i})\\\\& \\le & \\varepsilon \\cdot OPT$ to the objective.", "[Proof of Lemma REF ] We modify a given online algorithm such that each part is treated as a separate instance.", "To bound the cost in the competitive ratio, we show that $\\frac{\\textsf {\\textsc {A}}(I)}{\\textsf {\\textsc {Opt}}(I)}\\le \\max _{i}\\lbrace \\frac{\\textsf {\\textsc {A}}(P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\rbrace \\cdot (1+O(\\varepsilon )).$ By the above lemmas, there is a $(1+O(\\varepsilon ))$ -approximative (offline) solution in which at the end of each part $P_{\\!i}$ each job has either completed or has been moved to its safety net.", "Denote this solution by $\\textsf {\\textsc {Opt}}^{\\prime }(I)$ and by $\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})$ its respective part for each part $P_{\\!i}$ .", "Note that $\\textsf {\\textsc {Opt}}^{\\prime }(I)=\\sum _{i}\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})$ .", "Then, $\\frac{\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i}))}{\\textsf {\\textsc {Opt}}(I)} \\le \\frac{\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i}))}{\\sum _{i=1}^{k}\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})}\\cdot (1+O(\\varepsilon )) \\le \\max _{i=1,...,k}\\lbrace \\frac{\\textsf {\\textsc {A}}(P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\rbrace \\cdot (1+O(\\varepsilon )).$ [Proof of Lemma REF ] We show that $\\left(1+\\varepsilon \\right)^{s}\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\varepsilon \\cdot \\sum _{i=p-K}^{p}rw(Q_{i})$ for a sufficiently large value $K$ .", "This will then be the claimed constant.", "Let $\\delta ^{\\prime }:=\\frac{\\varepsilon }{\\left(1+\\varepsilon \\right)^{s}}$ .", "By assumption, we have that $rw(Q_{i+1})>\\delta ^{\\prime }\\cdot \\sum _{\\ell =1}^{i}rw(Q_{\\ell })$ for each $i$ .", "This implies that $\\frac{rw(Q_{i+1})}{\\sum _{\\ell =1}^{i+1}rw(Q_{\\ell })}>\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}$ for each $i$ .", "Hence, $\\frac{\\sum _{\\ell =1}^{i}rw(Q_{\\ell })}{rw(Q_{i+1})+\\sum _{\\ell =1}^{i}rw(Q_{\\ell })}\\le 1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}<1$ for each $i$ and hence, $\\sum _{\\ell =1}^{i}rw(Q_{\\ell })\\le (1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }})\\sum _{\\ell =1}^{i+1}rw(Q_{\\ell }).$ In other words, if we remove $Q_{i+1}$ from $\\cup _{\\ell =1}^{i+1}Q_{\\ell }$ then the total release weight of the set decreases by a factor of at least $1-\\delta ^{\\prime }/(1+\\delta ^{\\prime })<1$ .", "For any $K$ this implies that $\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}\\sum _{\\ell =1}^{p}rw(Q_{\\ell })$ and hence $\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\frac{1}{1-\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}}\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}\\sum _{i=p-K}^{p}rw(Q_{i}).$ By choosing $K$ sufficiently large, the claim follows.", "[Proof of Lemma REF ] We partition $\\mathrm {Ir}_{x}(J)$ into two groups: $\\mathrm {Ir}_{x}^{\\mathrm {old}}(J):=\\lbrace j\\in \\mathrm {Ir}_{x}(J)|r_{j}<R_{x-\\Gamma }\\rbrace $ and $\\mathrm {Ir}_{x}^{\\mathrm {new}}(J):=\\lbrace j\\in \\mathrm {Ir}_{x}(J)|r_{j}\\ge R_{x-\\Gamma }\\rbrace $ .", "Lemma REF implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))\\le \\varepsilon \\cdot rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)\\cup \\mathrm {Rel}_{x}(J))$ (recall that the former value is an upper bound on the total weighted completion time of the jobs in $\\mathrm {Ir}_{x}^{\\mathrm {old}}(J)$ ).", "For every job $j\\in \\mathrm {Ir}_{x}^{\\mathrm {new}}(J)$ there must be a job $j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)$ such that $w_{j}<\\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}$ .", "We say that such a job $j^{\\prime }$ dominates $j$ .", "At most $\\Delta $ jobs are released at the beginning of each interval and hence $|\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)|\\le \\Delta \\Gamma $ .", "In particular, if $\\mathrm {dom}(j^{\\prime })$ denotes all jobs in $\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)$ which are dominated by $j^{\\prime }$ then $\\sum _{j\\in \\mathrm {dom}(j^{\\prime })}w_{j}r_{j}\\le \\Delta \\Gamma \\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}r_{j^{\\prime }}\\cdot (1+\\varepsilon )^{\\Gamma }$ This implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)) & \\le & (1+\\varepsilon )^{s}\\sum _{j\\in \\mathrm {Ir}_{x}^{\\mathrm {new}}(J)}w_{j}r_{j}\\\\& \\le & \\sum _{j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)}\\Delta \\Gamma \\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}r_{j^{\\prime }}\\cdot (1+\\varepsilon )^{\\Gamma +s}\\\\& \\le & \\varepsilon \\cdot \\sum _{j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)}w_{j^{\\prime }}r_{j^{\\prime }}\\\\& = & \\varepsilon \\cdot (rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))+rw(\\mathrm {Rel}_{x}(J)))$ Together with Inequality REF this implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}(J)) & = & (1+\\varepsilon )^{s}(rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)+rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J)))\\\\& \\le & \\left(\\varepsilon \\cdot (rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))+rw(\\mathrm {Rel}_{x}(J))\\right)+\\left(\\varepsilon \\cdot rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)\\cup \\mathrm {Rel}_{x}(J))\\right)\\\\& \\le & 2\\varepsilon \\cdot rw(\\mathrm {Rel}_{x}(J))+\\varepsilon (rw(\\mathrm {Ir}_{x}(J))$ and the latter inequality implies that $\\sum _{j\\in \\mathrm {Ir}_{x}(J)}C_{j}w_{j} & \\le & (1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}(J))\\\\& \\le & 2\\varepsilon \\frac{(1+\\varepsilon )^{s}}{(1+\\varepsilon )^{s}-\\varepsilon }rw(\\mathrm {Rel}_{x}(J))\\\\& \\le & 3\\varepsilon \\cdot rw(\\mathrm {Rel}_{x}(J))$ Proofs of Section  Lemma C.1 In the non-preemptive setting, at $1+O(\\varepsilon )$ loss we can ensure that at the end of each interval $I_{x}$ , there are at most $m$ large jobs from each type which are partially (i.e., neither fully nor not at all) processed, and for each partially but not completely processed large job $j$ there is a value $k_{x,j}$ such that $j$ is processed for at least $k_{x,j}\\cdot p_{j}\\cdot \\mu $ time units in $I_{x}$ , we calculate the objective with adjusted completion times $\\bar{C}_{j}=R_{c(j)}$ for some value $c(j)$ for each job $j$ such that $\\sum _{x<c(j)}k_{x,j}\\cdot p_{j}\\cdot \\mu \\ge p_{j}$ .", "Note that the first property holds for any non-preemptive schedule and is listed here only for the sake of clarity.", "The other two properties can be shown similiarly as in the proof of Lemma REF .", "[Proof of Lemma REF ] Assume that we have an online algorithm $\\textsf {\\textsc {A}}$ with competitive factor $\\rho _{\\textsf {\\textsc {A}}}$ on instances in which for every $i$ the first job $\\textnormal {first}(i)$ released in part $P_{\\!i}$ satisfies $\\sum _{\\ell =1}^{i-1} rw(P_{\\!\\ell }) \\le w_{\\!\\textnormal {first}(i)}\\varepsilon /\\left(1+\\varepsilon \\right)^{s}$  (i.e., $\\textnormal {first}(i)$ dominates all previously released parts).", "Based on $\\textsf {\\textsc {A}}$ we construct a new algorithm $\\textsf {\\textsc {A}}^{\\prime }$ for arbitrary instances with competitive ratio at most $\\left(1+\\varepsilon \\right)\\rho _{\\textsf {\\textsc {A}}}$ : When a new part $P_{\\!i}$ begins, we scale the weights of all jobs in $P_{\\!i}$ such that $\\sum _{\\ell =1}^{i-1}rw(P_{\\!\\ell }) \\le w^{\\prime }_{\\!\\textnormal {first}(i)}\\varepsilon /\\left(1+\\varepsilon \\right)^{s}$ , where the values $w^{\\prime }_{\\!j}$ denote the adjusted weights.", "Denote by $\\bar{I}(i)$ the resulting instance up to (and including) part $P_{\\!i}$ .", "We schedule the resulting instance using $\\textsf {\\textsc {A}}$ .", "We take the computed schedule for each part $P_{\\!i}$ and use it for the jobs with their original weight, obtaining a new algorithm $\\textsf {\\textsc {A}}^{\\prime }$ .", "The following calculations shows that this procedure costs only a factor $1+\\varepsilon $ .", "To this end, we proof that for any instance $I$ it holds that $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I)}{\\textsf {\\textsc {Opt}}(I)} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}(\\bar{I}(i))}{\\textsf {\\textsc {Opt}}(\\bar{I}(i))}\\cdot (1+O(\\varepsilon ))\\le (1+O(\\varepsilon ))\\rho _{\\textsf {\\textsc {A}}}.$ For each $P_{\\!i}$ we define $\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})$ to be the amount that the jobs in $P_{\\!i}$ contribute in $\\textsf {\\textsc {A}}^{\\prime }(I)$ .", "Similarly, we define $\\textsf {\\textsc {Opt}}(I|P_{\\!i})$ .", "We have that $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I)}{\\textsf {\\textsc {Opt}}(I)} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(I|P_{\\!i})} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}.$ We claim that for each $i$ holds $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\le (1+O(\\varepsilon ))\\cdot \\frac{\\textsf {\\textsc {A}}(\\bar{I}(i))}{\\textsf {\\textsc {Opt}}(\\bar{I}(i))}$ .", "For each part $P_{\\!i}$ let $v_{i}$ denote the scale factor of the weight of each job in $\\bar{I}(i)$ in comparison to its original weight.", "The optimum for the instance $\\bar{I}(i)$ can be bounded by $\\textsf {\\textsc {Opt}}(\\bar{I}(i)) & \\le & \\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i} +\\left(1+\\varepsilon \\right)^{s}\\sum _{\\ell =1}^{i-1}rw(P_{\\!\\ell }) \\le \\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i}+\\varepsilon \\cdot r_{\\textnormal {first}(i)} \\cdot w^{\\prime }_{\\textnormal {first}(i)}\\le \\left(1+\\varepsilon \\right)\\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i}.$ Furthermore holds by construction $\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})\\cdot v_{i}\\le \\textsf {\\textsc {A}}(\\bar{I}^{i})$ .", "Thus, $\\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}(\\bar{I}^{i})}{\\textsf {\\textsc {Opt}}(\\bar{I}^{i})}\\cdot (1+O(\\varepsilon ))$ .", "[Proof of Lemma REF ] Given a schedule on related machines with speed values $s_{1},...,s_{\\max }$ , we stretch time twice.", "Thus, we gain in each interval $I_{x}$ free space of size $\\varepsilon I_{x}$ on the fastest machine.", "For each machine whose speed is at most $\\frac{\\varepsilon }{m}s_{\\max }$ , we take its schedule of the interval $I_{x}$ and simulate it on the fastest machine.", "Thus, those slow machines are not needed and can be removed.", "The remaining machines have speeds in $[\\frac{\\varepsilon }{m}\\,s_{\\max },s_{\\max }]$ .", "Assuming the slowest machines has unit speed gives the desired bound.", "[Proof of Lemma REF ] Consider a schedule for an instance which does not satisfy the property.", "We stretch time twice and thus we gain a free space of $\\varepsilon I_{x}$ in each interval $I_{x}$ .", "Consider some $I_{x}$ and a job $j$ which is scheduled in $I_{x}$ .", "Let $i$ be a fastest machine for $j$ .", "We remove the processing volume of $j$ scheduled in $I_x$ on slow machines $i^{\\prime }$ with $p_{i^{\\prime }j} >\\frac{m}{\\varepsilon }\\, p_{ij}$ and schedule it on $i$ in the gained free space.", "This way, we obtain a feasible schedule even if a job never runs on a machine where it is slow.", "Thus, we can set $p_{i^{\\prime }j}=\\infty $ if there is a fast machine $i$ such that $p_{ij} \\le \\frac{\\varepsilon }{m}p_{i^{\\prime }j}$ .", "Proofs of Section  [Proof of Lemma REF ] Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances $I$ such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )$ .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "In instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We need to bound the increase in the total expected cost caused by moving all jobs in periods in $\\mathcal {Q}$ to their safety nets.", "This increase is bounded by $\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}\\sum _{j\\in Q}\\left(1+\\varepsilon \\right)^{s}r_{j}\\cdot w_{j}\\right] & \\le &\\left(1+\\varepsilon \\right)^{s}\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}rw(Q)\\right]\\\\& \\le & \\left(1+\\varepsilon \\right)^{s}\\frac{1}{M}\\sum _{Q\\in I}rw(Q)\\\\& \\le & \\varepsilon \\cdot rw(I)\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)\\,.$ Thus, the total expected cost of the computed schedule is $\\mathbb {E}\\left[\\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i})\\right] & \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(P_{\\!i})\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & (\\rho _{\\textsf {\\textsc {A}}}+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I).$ Thus, at $1+\\varepsilon $ loss in the competitive ratio we can restrict to parts $I_i$ which span a constant number of periods.", "[Proof of Lemma ] Consider an instance $I$ .", "Let $\\delta >0$ and $k\\in \\mathbb {N}$ be values to be determined later with the property that $1/\\delta \\in \\mathbb {N}$ .", "For each configuration $C$ and each interval-schedule $S$ we define a value $g(C,S)$ such that $\\left\\lfloor \\frac{f(C,S)}{\\delta }\\right\\rfloor \\cdot \\delta \\le g(C,S)\\le \\left\\lceil \\frac{f(C,S)}{\\delta }\\right\\rceil \\cdot \\delta $ and $\\sum _{S\\in \\mathcal {S}}g(C,S)=1$ .", "Now we want to bound $\\rho _{g}$ .", "The idea is that for determining the ratio $\\mathbb {E}\\left[\\,g(I)\\,\\right]/\\textsf {\\textsc {Opt}}(I)$ it suffices to consider schedules $S(I)$ which are computed with sufficiently large probability.", "We show that also $f$ computes them with almost the same probability.", "Let $S(I)$ denote a schedule for the entire instance $I$ .", "We denote by $P_{f}(S(I))$ and $P_{g}(S(I))$ the probability that $f$ and $g$ compute the schedule $S(I)$ when given the instance $I$ .", "Assume that $P_{f}(S(I))\\ge k\\cdot \\delta $ .", "Denote by $C_{0},...,C_{\\bar{\\Gamma }-1}$ the configurations that algorithms are faced with when computing $S(I)$ , i.e., each configuration $C_{x}$ contains the jobs which are released but unfinished at the beginning of interval $I_{x}$ in $S(I)$ and as history the schedule $S(I)$ restricted to the intervals $I_{0},...,I_{x-1}$ .", "Denote by $S_{\\!x}$ the schedule of $S(I)$ in the interval $I_{x}$ .", "Hence, $P_{f}(S(I))=\\prod _{x=0}^{\\bar{\\Gamma }-1}f(C_{x},S_{\\!x})$ .", "Note that from $P_{f}(S(I))\\ge k\\cdot \\delta $ follows that $f(C_{x},S_{\\!x})\\ge k\\cdot \\delta $ for all $x$ .", "For these schedules, $P_{g}(S(I))$ is not much larger since $P_{g}(S(I)) & = & \\prod _{x=0}^{\\bar{\\Gamma }-1}g(C_{x},S_{\\!x})\\le \\prod _{x=0}^{\\bar{\\Gamma }-1}\\frac{k+1}{k}f(C_{x},S_{\\!x})\\le \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}P_{f}(S(I)).$ Let $\\mathcal {S}(I)$ denote the set of all schedules for $I$ .", "We partition $\\mathcal {S}(I)$ into schedule sets $\\mathcal {S}_{H}^{g}(I):=\\lbrace S(I)|P_{g}(S(I))\\ge k\\cdot \\delta \\rbrace $ and $\\mathcal {S}_{L}^{g}(I):=\\mathcal {S}(I)\\setminus \\mathcal {S}_{H}(I)$ .", "We estimate the expected value of a schedule computed by algorithm map $g$ on $I$ by $\\mathbb {E}\\left[\\,g(I)\\,\\right] & = & \\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{g}(S(I))\\cdot S(I) + \\sum _{S(I)\\in \\mathcal {S}_{L}^{g}(I)}P_{g}(S(I)) \\cdot S(I)\\\\& \\le &\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\cdot S(I) + |\\mathcal {S}(I)| \\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s} \\cdot rw(I)\\\\& \\le &\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot S(I) + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I)\\\\& \\le & \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\mathbb {E}\\left[\\,f(I)\\,\\right] + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I).$ We choose $k$ such that $\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\le 1+\\varepsilon /2$ and $\\delta $ such that $|\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\le \\varepsilon /2$ for all instances $I$  (note here that $|\\mathcal {S}(I)|$ can be upper bounded by a value independent of $I$ since our instances contain only constantly many jobs).", "This yields $\\frac{\\mathbb {E}\\left[\\,g(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon /2)\\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} + \\varepsilon /2\\cdot \\frac{rw(I)}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon ) \\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)}\\,,$ and we conclude that $\\rho _{g}\\le \\left(1+\\varepsilon \\right)\\rho _{f}$ .", "Competitive-Ratio Approximation Schemes for Minimizing $C_{\\max }$ (cf.", "Section ) Consider the objective of minimizing the makespan.", "The simplifications within intervals of Section  are based on arguing on completion times of individual jobs, and clearly hold also for the last job.", "Thus, they directly apply to minimizing the makespan.", "Furthermore, we simplify the definition of irrelevant history in Section  by omitting the partition of the instance into parts.", "We observe that when then last job is released at time $R_{x^{*}}$ then all jobs $j$ with $r_{j}\\le R_{x^{*}-s}$ are irrelevant for the objective: such a job $j$ finishes at time $R_{x^{*}}$ the latest in any schedule (due to the safety net) and $OPT\\ge R_{x^{*}}$ .", "Therefore, we define a job $j$ to be irrelevant at time $R_{x}$ if $r_{j}\\le R_{x-s}$ .", "We keep Definition REF for the equivalence relation of schedules as it is except for the notion of job weights which are not important for the makespan.", "Based on the above definition for relevant jobs we define equivalence classes of configurations.", "With this definition, we can still restrict to algorithm maps $f$ with $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations $C,C^{\\prime }$ (Lemma REF ).", "Lemmas REF to REF then hold accordingly.", "Finally, note that since we do not split the instance into parts, we do not need (an adjusted version of) Lemma REF in the non-preemptive case.", "Theorem E.1 For any $m\\in \\mathbb {N}$ we obtain competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds.", "For constructing randomized online algorithm schemes for minimizing the makespan, similarly to Lemma REF we can show that we can restrict our attention to instances which span only a constant number of periods.", "Lemma E.2 For randomized algorithms for minimizing the makespan, at $1+O(\\varepsilon )$ loss we can restrict to instances in which all jobs are released in at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ consecutive periods.", "Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm for minimizing the makespan over time with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+O(\\varepsilon ))$ .", "Our reasoning is similar to the proof of Lemma REF .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "Given an instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We present each part separately to $\\textsf {\\textsc {A}}$ .", "We bound the competitive ratio $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ of the resulting algorithm.", "Let $R_{x^{*}}:=\\max _{j \\in I}r_{j}$ .", "Moving the jobs from periods $\\mathcal {Q}$ has an effect on the optimal makespan only if $o$ is chosen such that at least one job $j$ with $r_{j}>R_{x^{*}-s}$ is moved.", "There are at most two offsets $o$ such that this happens.", "In that case, the algorithm still achieves a competitive ratio of at most $\\left(1+\\varepsilon \\right)^{s}$ .", "In all other cases, $\\textsf {\\textsc {A}}^{\\prime }$ achieves a competitive ratio of at most $\\rho _{\\textsf {\\textsc {A}}}$ .", "Thus, we can bound $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ by $\\rho _{\\textsf {\\textsc {A}}^{\\prime }} & \\le & \\frac{2}{M}\\left(1+\\varepsilon \\right)^{s}+\\frac{M-2}{M}\\rho _{\\textsf {\\textsc {A}}}\\le 2\\varepsilon +\\rho _{\\textsf {\\textsc {A}}}\\le (1+O(\\varepsilon ))\\rho _{\\textsf {\\textsc {A}}}.$ We can prove similarly as in Lemma  that any randomized algorithm map $f$ can be well approximated by a discretized randomized algorithm map $g$ .", "Hence, we obtain the following theorem.", "Theorem E.3 For any $m\\in \\mathbb {N}$ we obtain randomized competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds." ], [ "Abstraction of Online Algorithms", "In this section we show how to construct a competitive-ratio approximation scheme based on the observations of Section .", "To do so, we restrict ourselves to such simplified instances and schedules.", "The key idea is to characterize the behavior of an online algorithm by a map: For each interval, the map gets as input the schedule computed so far and all information about the currently unfinished jobs.", "Based on this information, the map outputs how to schedule the available jobs within this interval.", "More precisely, we define the input by a configuration and the output by an interval-schedule.", "Definition 3.1 An interval-schedule $S$ for an interval $I_{x}$ is defined by the index $x$ of the interval, a set of jobs $J(S)$ available for processing in $I_x$ together with the properties $r_{j}, p_{j}, w_{j}$ of each job $j\\in J(S)$ and its already finished part $f_{j}<p_{j}$ up to $R_{x}$ , for each job $j\\in J(S)$ the information whether $j$ is relevant at time $R_{x}$ , and for each job $j\\in J(S)$ and each machine $i$ a value $q_{ij}$ specifying for how long $j$ is processed by $S$ on machine $i$ during $I_{x}$ .", "An interval-schedule is called feasible if there is a feasible schedule in which the jobs of $J(S)$ are processed corresponding to the $q_{j}$ values within the interval $I_{x}$ .", "Denote the set of feasible interval-schedules as $\\mathcal {S}$ .", "Definition 3.2 A configuration $C$ for an interval $I_{x}$ consists of the index $x$ of the interval, a set of jobs $J(C)$ released up to time $R_{x}$ together with the properties $r_{j}, p_{j}, w_{j}, f_{j}$ of each job $j\\in J(C)$ , an interval-schedule for each interval $I_{x^{\\prime }}$ with $x^{\\prime }<x$ .", "The set of all configurations is denoted by $.", "An \\emph {end-configuration} is a configuration~$ C$ for an interval~$ Ix$ such that at time~$ Rx$,and not earlier, no jobs are left unprocessed.$ We say that an interval-schedule $S$ is feasible for a configuration $C$ if the set of jobs in $J(C)$ which are unfinished at time $R_{x}$ matches the set $J(S)$ with respect to release dates, total and remaining processing time, weight and relevance of the jobs.", "Instead of online algorithms we work from now on with algorithm maps, which are defined as functions $f:\\mathcal {S}$ .", "An algorithm map determines a schedule $f(I)$ for a given scheduling instance $I$ by iteratively applying $f$ to the corresponding configurations.", "W.l.o.g.", "we consider only algorithm maps $f$ such that $f(C)$ is feasible for each configuration $C$ and $f(I)$ is feasible for each instance $I$ .", "Like for online algorithms, we define the competitive ratio $\\rho _{f}$ of an algorithm map $f$ by $\\rho _{f}:=\\max _{I}f(I)/\\textsf {\\textsc {Opt}}(I)$ .", "Due to the following observation, algorithm maps are a natural generalization of online algorithms.", "Proposition 3.3 For each online algorithm $\\textsf {\\textsc {A}}$ there is an algorithm map $f_{\\textsf {\\textsc {A}}}$ such that when $\\textsf {\\textsc {A}}$ is in configuration $C\\in atthe beginning of an interval~$ Ix$, algorithm~$A$ schedules thejobs according to~$ fA(C)$.$ Recall, that we restrict our attention to algorithm maps describing online algorithms which obey the simplifications introduced in Section .", "The essence of such online algorithms are the decisions for the relevant jobs.", "To this end, we define equivalence classes for configurations and for interval-schedules.", "Intuitively, two interval-schedules (configurations) are equivalent if we can obtain one from the other by scalar multiplication with the same value, while ignoring the irrelevant jobs.", "Definition 3.4 Let $S,S^{\\prime }$ be two feasible interval-schedules for two intervals $I_{x},I_{x^{\\prime }}$ .", "Denote by $J_{\\mathrm {Rel}}(S)\\subseteq J(S)$ and $J_{\\mathrm {Rel}}(S^{\\prime })\\subseteq J(S^{\\prime })$ the relevant jobs in $J(S)$ and $J(S^{\\prime })$ .", "Let further $\\sigma :J_{\\mathrm {Rel}}(S)\\rightarrow J_{\\mathrm {Rel}}(S^{\\prime })$ be a bijection and $y$ an integer.", "The interval-schedules $S,S^{\\prime }$ are $(\\sigma ,y)$ -equivalent if $r_{\\sigma (j)}=r_{j}(1+\\varepsilon )^{x^{\\prime }-x}, p_{\\sigma (j)}=p_{j}(1+\\varepsilon )^{x^{\\prime }-x},f_{\\sigma (j)}=f_{j}(1+\\varepsilon )^{x^{\\prime }-x}, q_{\\sigma (j)}=q_{j}(1+\\varepsilon )^{x^{\\prime }-x}$ and $w_{\\sigma (j)}=w_{j}\\left(1+\\varepsilon \\right)^{y}$ for all $j\\in J_{\\mathrm {Rel}}(S)$ .", "The interval-schedules $S,S^{\\prime }$ are equivalent (denoted by $S\\sim S^{\\prime }$ ) if a map $\\sigma $ and an integer $y$ exist such that they are $(\\sigma ,y)$ -equivalent.", "Definition 3.5 Let $C,C^{\\prime }$ be two configurations for two intervals $I_{x},I_{x^{\\prime }}$ .", "Denote by $J_{\\mathrm {Rel}}(C),J_{\\mathrm {Rel}}(C^{\\prime })$ the jobs which are relevant at times $R_{x},R_{x^{\\prime }}$ in $C,C^{\\prime }$ , respectively.", "Configurations $C,C^{\\prime }$ are equivalent (denoted by $C\\sim C^{\\prime }$ ) if there is a bijection $\\sigma :J_{\\mathrm {Rel}}(C)\\rightarrow J_{\\mathrm {Rel}}(C^{\\prime })$ and an integer $y$ such that $r{}_{\\sigma (j)}=r_{j}(1+\\varepsilon )^{ x^{\\prime }-x },p{}_{\\sigma (j)}=p_{j}(1+\\varepsilon )^{x^{\\prime }-x},f{}_{\\sigma (j)}=f_{j}(1+\\varepsilon )^{x^{\\prime }-x}$ and $w_{\\sigma (j)}=w_{j}\\left(1+\\varepsilon \\right)^{y}$ for all $j\\in J_{\\mathrm {Rel}}(C)$ , and the interval-schedules of $I_{x-k}$ and $I_{x^{\\prime }-k}$ are $(\\sigma ,y)$ -equivalent when restricted to the jobs in $J_{\\mathrm {Rel}}(C)$ and $J_{\\mathrm {Rel}}(C^{\\prime })$ for each $k\\in \\mathbb {N}$ .", "A configuration $C$ is realistic for an algorithm map $f$ if there is an instance $I$ such that if $f$ processes $I$ then at time $R_x$ it is in configuration $C$ .", "The following lemma shows that we can restrict the set of algorithm maps under consideration to those which treat equivalent configurations equivalently.", "We call algorithm maps obeying this condition in addition to the restrictions of Section  simplified algorithm maps.", "Lemma 3.6 At $1+O(\\varepsilon )$ loss we can restrict to algorithm maps $f$ such that $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations $C,C^{\\prime }$ .", "Let $f$ be an algorithm map.", "For each equivalence class $\\subseteq of the set of configurations we pick arepresentative $ C$ which is realistic for~$ f$.", "For each configuration $ C'$, we define a newalgorithm map $ f$ by setting $ f(C')$ to be theinterval-schedule for $ C'$ which is equivalent to $ f(C)$.One can show by induction that $ f$ is always in aconfiguration such that an equivalent configuration is realistic for$ f$.", "Hence, equivalence classes without realistic configurationsfor~$ f$ are not relevant.", "We claim that$ f(1+O())f$.$ Consider an instance $\\bar{I}$ .", "We show that there is an instance $I$ such that $\\bar{f}(\\bar{I})/\\textsf {\\textsc {Opt}}(\\bar{I})\\le (1+O(\\varepsilon ))f(I)/\\textsf {\\textsc {Opt}}(I)$ which implies the claim.", "Consider an interval $I_{\\bar{x}}$ .", "Let $\\bar{C}$ be the end-configuration obtained when $\\bar{f}$ is applied iteratively on $\\bar{I}$ .", "Let $C$ be the representative of the equivalence class of $\\bar{C}$ , which was chosen above and which is realistic for $f$ ($C$ is also an end-configuration).", "Therefore, there is an instance $I$ such that $C$ is reached at time $R_{x}$ when $f$ is applied on $I$ .", "Hence, $I$ is the required instance since the relevant jobs dominate the objective value (see Lemma REF ) and $C \\sim \\bar{C}$ .", "Lemma 3.7 There are only constantly many simplified algorithm maps.", "Each simplified algorithm map can be described using finite information.", "From the simplifications introduced in Section  follows that the domain of the algorithm maps under consideration contains only constantly many equivalence classes of configurations.", "Also, the target space contains only constantly many equivalence classes of interval-schedules.", "For an algorithm map $f$ which obeys the restrictions of Section , the interval-schedule $f(C)$ is fully specified when knowing only $C$ and the equivalence class which contains $f(C)$ (since the irrelevant jobs are moved to their safety net anyway).", "Since $f(C)\\sim f(C^{\\prime })$ for a simplified algorithm map $f$ if $C\\sim C^{\\prime }$ , we conclude that there are only constantly many simplified algorithm maps.", "Finally, each equivalence class of configurations and interval-schedules can be characterized using only finite information, and hence the same holds for each simplified algorithm map.", "The next lemma shows that up to a factor $1+\\varepsilon $ worst case instances of simplified algorithm maps span only constantly many intervals.", "Using this property, we will show in the subsequent lemmas that the competitive ratio of a simplified algorithm map can be determined algorithmically up to a $1+\\varepsilon $ factor.", "Lemma 3.8 There is a constant $E$ such that for any instance $I$ and any simplified algorithm map $f$ there is a realistic end-configuration $\\tilde{C}$ for an interval $I_{\\tilde{x}}$ with $\\tilde{x}\\le E$ which is equivalent to the corresponding end-configuration when $f$ is applied to $I$ .", "Consider a simplified algorithm map $f$ .", "For each interval $I_{x}$ , denote by ${x}^{f}$ the set of realistic equivalence classes for $I_{x}$ , i.e., the equivalence classes which have a realistic representative for $I_{x}$ .", "Since there are constantly many equivalence classes and thus constantly many sets of equivalence classes, there must be a constant $E$ independent of $f$ such that ${\\bar{x}}^{f}={\\bar{x}^{\\prime }}^{f}$ for some $\\bar{x}<\\bar{x}^{\\prime }\\le E$ .", "Since $f$ is simplified it can be shown by induction that ${\\bar{x}+k}^{f}={\\bar{x}^{\\prime }+k}^{f}$ for any $k\\in \\mathbb {N}$ , i.e., $f$ cycles with period length $\\bar{x}^{\\prime }-\\bar{x}$ .", "Consider now some instance $I$ and let $C$ with interval $I_{x}$ be the corresponding end-configuration when $f$ is applied to $I$ .", "If $x\\le E$ we are done.", "Otherwise there must be some $k\\le \\bar{x}^{\\prime }-\\bar{x}$ such that ${\\bar{x}+k}^{f}={x}^{f}$ since $f$ cycles with this period length.", "Hence, by definition of ${\\bar{x}+k}^{f}$ there must be a realistic end-configuration $\\tilde{C}$ which is equivalent to $C$ for the interval $I_{\\tilde{x}}$ with $\\tilde{x}:=\\bar{x}+k\\le E$ .", "Lemma 3.9 Let $f$ be a simplified algorithm map.", "There is an algorithm which approximates $\\rho _{f}$ up to a factor $1+\\varepsilon $ , i.e., it computes a value $\\rho ^{\\prime }$ with $\\rho ^{\\prime }\\le \\rho _{f}\\le (1+O(\\varepsilon ))\\rho ^{\\prime }$ .", "[Proof sketch.]", "By Lemma REF , the relevant jobs in a configuration dominate the entire objective value.", "In particular, we do not need to know the irrelevant jobs of a configuration if we only want to approximate its objective value up to a factor of $1+O(\\varepsilon )$ .", "For an end-configuration $C$ denote by $val_{C}(J_{\\mathrm {Rel}}(C))$ the objective value of the jobs in $J_{\\mathrm {Rel}}(C)$ in the history of $C$ .", "We define $r(C):=val_{C}(J_{\\mathrm {Rel}}(C))/\\textsf {\\textsc {Opt}}(J_{\\mathrm {Rel}}(C))$ to be the achieved competitive ratio of $C$ when restricted to the relevant jobs.", "According to Lemma REF , it suffices to construct the sets ${0}^{f},...,{E}^{f}$ in order to approximate the competitive ratio of all end-configurations in these sets.", "We start with ${0}^{f}$ and determine $f(C)$ for one representant $C$ of each equivalence class ${0}^{f}$ .", "Based on this we determine the set ${1}^{f}$ .", "We continue inductively to construct all sets ${x}^{f}$ with $x\\le E$ .", "We define $r_{\\max }$ to be the maximum ratio $r(C)$ for an end-configuration $C\\in \\cup _{0\\le x\\le E}{x}^{f}$ .", "Due to Lemma REF and Lemma REF the value $r_{\\max }$ implies the required $\\rho ^{\\prime }$ fulfilling the properties claimed in this lemma.", "Our main algorithm works as follows.", "We first enumerate all simplified algorithm maps.", "For each simplified algorithm map $f$ we approximate $\\rho _{f}$ using Lemma REF .", "We output the map $f$ with the minimum (approximated) competitive ratio.", "Note that the resulting online algorithm has polynomial running time: All simplifications of a given instance can be done efficiently and for a given configuration, the equivalence class of the schedule for the next interval can be found in a look-up table of constant size.", "Theorem 3.10 For any $m\\in \\mathbb {N}$ we obtain a competitive-ratio approximation scheme for $\\textup {Pm}|\\,r_{j},pmtn\\,|\\sum w_{j}C_{j}$ .", "Extensions to Other Settings Non-preemptive Scheduling.", "When preemption is not allowed, the definition of the safety net (Lemma REF ) needs to be adjusted since we cannot ensure that at the end of each interval $I_{x+s}$ there is a machine idle.", "However, we can guarantee that there is a reserved space somewhere in $[R_{x},R_{x+s})$ to process the small and big jobs in $S_{x} \\cup L_{x}$ .", "Furthermore, we cannot enforce that a big job $j$ is processed for exactly a certain multiple of $p_{j}\\mu $ in each interval (Lemma REF ).", "To solve this, we pretend that we could preempt $j$ and ensure that after $j$ has been preempted its machine stays idle until $j$ continues.", "Next, we can no longer assume that each part can be treated independently (Lemma REF ).", "Since some of the remaining jobs at the end of a part may have already started processing, we cannot simply move them to their safety net.", "Here we use the following simplification.", "Lemma 4.1 Let $\\textnormal {first}(i)$ denote the job that is released first in part $P_{\\!i}$ .", "At $1+\\varepsilon $ loss, we can restrict to instances such that $\\sum _{\\ell =1}^{i-1} rw(P_{\\!\\ell }) \\le \\frac{\\varepsilon }{\\left(1+\\varepsilon \\right)^{s}}\\cdot rw(\\textnormal {first}(i))$ , i.e., $\\textnormal {first}(i)$ dominates all previous parts.", "Therefore, at $1+\\varepsilon $ loss it is enough to consider only the currently running jobs from the previous part and the last $\\Gamma $ intervals from the current part when taking decisions.", "Finally, we add some minor modifications to handle the case that a currently running job is dominated by some other job.", "With these adjustments, we have only constantly many equivalence classes for interval-schedules and configurations, which allows us to construct a competitive-ratio approximation scheme as in Section .", "Theorem 4.2 For any $m\\in \\mathbb {N}$ we obtain a competitive-ratio approximation scheme for $\\textup {Pm}|\\,r_j\\,|\\sum w_jC_j$  .", "Scheduling on Related Machines.", "In this setting, each machine $i$ has associated a speed $s_i$ , such that processing job $j$ on machine $i$ takes $p_{j}/s_{i}$ time units.", "W.l.o.g.", "the slowest machine has unit speed.", "Let $s_{\\max }$ denote the maximum speed in an instance.", "An adjusted version of Lemma REF ensures that at $1+\\varepsilon $ loss $r_{j} \\ge \\varepsilon \\, p_{j}/s_{\\max }$ for all jobs $j$  (rather than $r_{j} \\ge \\varepsilon p_{j}$ ).", "Furthermore, we can bound the number of distinct processing times and the number of released jobs of each interval, using similar arguments as in the unit-speed case.", "Lemma 4.3 At $1+O(\\varepsilon )$ loss we can restrict to instances where for each release date the number of released jobs and the number of distinct processing times is bounded by a constant depending only on $\\varepsilon $ , $m$ , and $s_{\\max }$ .", "We establish the safety net for the jobs of each release date $R_{x}$ only on the fastest machine and thereby ensure the condition of Lemma REF in the related machine setting.", "For the non-preemptive setting we incorporate the adjustments introduced in Section REF .", "Since at $1+\\varepsilon $ loss we can round the speeds of the machines to powers of $1+\\varepsilon $ we obtain the following result.", "Theorem 4.4 For any $m\\in \\mathbb {N}$ we obtain competitive-ratio approximation schemes for $\\textup {Qm}|\\,r_j, pmtn\\,|\\sum w_jC_j$ and $\\textup {Qm}|\\,r_j\\,|\\sum w_jC_j$ , assuming that the speeds of any two machines differ by at most a constant factor.", "In the preemptive setting we can strengthen the result and give a competitive-ratio approximation scheme for the case that machine speeds are part of the input, that is, we obtain a nearly optimal competitive ratio for any speed vector.", "The key is to bound the variety of different speeds.", "To that end, we show that at $1+\\varepsilon $ loss a very fast machine can simulate $m-1$ very slow machines.", "Lemma 4.5 For $\\textup {Qm}|\\,r_j, pmtn\\,|\\sum w_jC_j$ , at $1+O(\\varepsilon )$ loss, we can restrict to instances in which $s_{\\max }$ is bounded by $m/\\varepsilon $ .", "As speeds are geometrically rounded, we have for each value $m$ only finitely many speed vectors.", "Thus, our enumeration scheme finds a nearly optimal online algorithm with a particular routine for each speed vector.", "Theorem 4.6 For any $m\\in \\mathbb {N}$ we obtain a competitive-ratio approximation scheme for $\\textup {Qm}|\\,r_j,pmtn\\,|\\sum w_jC_j$  .", "Preemptive Scheduling on Unrelated Machines.", "When each job $j$ has its individual processing time $p_{ij}$ on machine $i$ , the problem complexity increases significantly.", "We restrict to preemptive scheduling and show how to decrease the complexity to apply our approximation scheme.", "The key is to bound the range of the finite processing times for each job (which is unfortunately not possible in the non-preemptive case, see [1] for a counterexample).", "Lemma 4.7 At $1+\\varepsilon $ loss we can restrict to instances in which for each job $j$ the ratio of any two of its finite processing times is bounded by $m/\\varepsilon $ .", "The above lemma allows us to introduce the notion of job classes.", "Two jobs $j,j^{\\prime }$ are of the same class if they have finite processing times on exactly the same machines and  $p_{ij}/p_{ij^{\\prime }}=p_{i^{\\prime }j}/p_{i^{\\prime }j^{\\prime }}$ for any two such machines $i$ and $i^{\\prime }$ .", "For fixed $m$ , the number of different job classes is bounded by a constant $W$ .", "For each job class, we define large and small tasks similar to the identical machine case: for each job $j$ we define a value $\\tilde{p}_{j}:=\\max _{i}\\lbrace p_{ij}|p_{ij}<\\infty \\rbrace $ and say a job is large if  $\\tilde{p}_{j}\\ge \\varepsilon ^{2}r_{j}/W$ and small otherwise.", "For each job class separately, we perform the adjustments of Section .", "This yields the following lemma.", "Lemma 4.8 At $1+O(\\varepsilon )$ loss we can restrict to instances and schedules such that for each job class, the number of distinct values $\\tilde{p}_j$ of jobs $j$ with the same release date is bounded by a constant, for each job class, the number of jobs with the same release date is bounded by a constant $\\tilde{\\Delta }$ , a large job $j$ is only preempted at integer multiples of $\\tilde{p}_{j} \\cdot \\tilde{\\mu }$ for some constant $\\tilde{\\mu }$ and small jobs are never preempted and finish in the same interval where they start.", "The above lemmas imply that both, the number of equivalence classes of configurations and the number of equivalence classes for interval-schedules are bounded by constants.", "Thus, we can apply the enumeration scheme from Section .", "Theorem 4.9 For any $m\\in \\mathbb {N}$ we obtain a competitive-ratio approximation scheme for $\\textup {Rm}|\\,r_j,pmtn\\,|\\sum w_jC_j$  .", "Randomized algorithms When algorithms are allowed to make random choices and we consider expected values of schedules, we can restrict to instances which span only constantly many periods.", "Assuming the simplifications of Section , this allows a restriction to instances with a constant number of jobs.", "Lemma 5.1 For randomized algorithms, at $1+\\varepsilon $ loss we can restrict to instances in which all jobs are released in at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ consecutive periods.", "[Proof idea.]", "Beginning at a randomly chosen period $Q_i$ with $i\\in [0,M)$ , with $M:=\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\rceil $ , we move all jobs released in $Q_{i+kM}$ , $k=0,1,\\ldots $ , to their safety net.", "At $1+\\varepsilon $ loss, this gives us a partition into parts, at the end of which no job remains, and we can treat each part independently.", "A randomized online algorithm can be viewed as a function that maps every possible configuration $C$ to a probability distribution of interval-schedules which are feasible for $C$ .", "To apply our algorithmic framework from the deterministic setting that enumerates all algorithm maps, we discretize the probability space and define discretized algorithm maps.", "To this end, let $\\bar{\\Gamma }$ denote the maximum number of intervals in instances with at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $  periods.", "Definition 5.2 (Discretized algorithm maps) Let $\\bar{ be the set of configurations for intervals~I_{x}with~x\\le \\bar{\\Gamma }, let~\\bar{\\mathcal {S}} be the set ofinterval-schedules for intervals~I_{x} with~x\\le \\bar{\\Gamma },and let~\\delta >0.", "A \\emph {\\delta -discretized algorithm map }is afunction~f:\\bar{\\times \\bar{\\mathcal {S}}\\rightarrow [0,1] suchthatf(C,S)=k\\cdot \\delta with some~k\\in \\mathbb {N}_{0} for all~C\\in \\bar{ and~S\\in \\bar{\\mathcal {S}}, and\\sum _{S\\in \\bar{\\mathcal {S}}}f(C,S)=1 for all ~C\\in \\bar{.", "}By restricting to \\delta -discretizedalgorithm maps we do not lose too much in the competitive ratio.", "}\\begin{lem}There is a value~\\delta >0 such that for any (randomized) algorithmmap~f there is a~\\delta -discretized randomized algorithm map~gwith~\\rho _{g}\\le \\rho _{f}\\left(1+\\varepsilon \\right).\\end{lem}\\begin{proof}[Proof idea.", "]Let~f be a randomized algorithm map andlet~\\delta >0 such that 1/\\delta \\in \\mathbb {N}.", "We define anew~\\delta -discretized algorithm map~g.", "For eachconfiguration~C we define the values~g(C,S) suchthat~\\left\\lfloor f(C,S)/\\delta \\right\\rfloor \\cdot \\delta \\le g(C,S)\\le \\left\\lceil f(C,S)/\\delta \\right\\rceil \\cdot \\delta and~\\sum _{S\\in \\mathcal {S}}g(C,S)=1.", "To seethat~\\rho _{g}\\le \\left(1+\\varepsilon \\right)\\rho _{f}, consider an instance~I and apossible schedule~S(I) for I.", "There is a probability~pthat~f outputs~S(I).", "We show that the schedules which have largeprobability~p dominate~\\mathbb {E}\\left[\\,f(I)\\,\\right]/\\textsf {\\textsc {Opt}}(I).", "We show furtherthat if p is sufficiently large, the probability that~gproduces~S(I) is in~[p/(1+\\varepsilon ),p(1+\\varepsilon )), which implies theLemma.\\end{proof}}Like in the deterministic case, we can now show that at 1+\\varepsilon lossit suffices to restrict to \\emph {simplified \\delta -discretizedalgorithm maps} which treat equivalent configurations equivalently,similar to Lemma~\\ref {lem:equal-confs} (replacing~\\Gamma by~\\bar{\\Gamma } in Definition~\\ref {def:irrelevant_jobs} of theirrelevant jobs).As there are only constantly many of these maps, we enumerate all ofthem, test each map for its competitive ratio, and select the best ofthem.", "}\\begin{thm} We obtain randomized competitive-ratio approximation schemes for \\textup {Pm}|\\,r_j,(pmtn)\\,|\\sum w_jC_j,\\textup {Qm}|\\,r_j, pmtn\\,|\\sum w_jC_j, and\\textup {Rm}|\\,r_j, pmtn\\,|\\sum w_jC_j\\, and for \\textup {Qm}|\\,r_j\\,|\\sum w_jC_j with a bounded range of speeds for any fixed m\\in \\mathbb {N}.\\end{thm}$ General Min-Sum Objectives and Makespan In this section we briefly argue how the techniques presented above for minimizing $\\sum _{j}w_{j}C_{j}$ can be used for constructing online algorithm schemes for other scheduling problems with jobs arriving online over time, namely for minimizing $\\sum _{j\\in J}w_jf(C_{j})$ , with $f(x)=k\\cdot x^{\\alpha }$ and constant $\\alpha \\ge 1, k>0$ , and the makespan.", "Since monomial functions $f$ have the property that $f((1+\\varepsilon ) C_j)\\le (1+O(\\varepsilon )) f(C_j)$ , the arguments in previous sections apply almost directly to the generalized min-sum objective.", "In each step of simplification and abstraction, we have an increased loss in the performance guarantee, but it is covered by the $O(\\varepsilon )$ -term.", "Consider now the makespan objective.", "The simplifications within intervals of Section  are based on arguing on completion times of individual jobs, and clearly hold also for the last job.", "Thus, they directly apply to makespan minimization.", "We simplify the definition of irrelevant history by omitting the partition of the instance into parts and we define a job $j$ to be irrelevant at time $R_{x}$ if $r_{j}\\le R_{x-s}$ .", "Based on this definition, we define equivalence classes of configurations (ignoring weights and previous interval-schedules) and again restrict to algorithm maps $f$ with $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations (Lemma 3.6).", "Lemmas 3.7–3.9 then hold accordingly and yield a competitive-ratio approximation scheme.", "Finally, the adjustments of Sections  and  can be made accordingly.", "Without the partition of the instance into parts, this even becomes easier in the non-preemptive setting.", "Thus, we can state the following result.", "Theorem 6.1 For any $m \\in \\mathbb {N}$ there are deterministic and randomized competitive-ratio approximation schemes for preemptive and non-preemptive scheduling, on $m$ identical, related (with bounded speed ratio when non-preemptive), and unrelated machines (only preemptive) for the objectives of minimizing $C_{\\max }$ and minimizing $\\sum _{j\\in J}w_j f(C_j)$ , with $f(x)=k\\cdot x^{\\alpha }$ and constant $\\alpha \\ge 1,k>0$ .", "Conclusions We introduce the concept of competitive-ratio approximation schemes that compute online algorithms with a competitive ratio arbitrarily close to the best possible competitive ratio.", "We provide such schemes for various problem variants of scheduling jobs online to minimize the weighted sum of completion times, arbitrary monomial cost functions, and the makespan.", "The techniques derived in this paper provide a new and interesting view on the behavior of online algorithms.", "We believe that they contribute to the understanding of such algorithms and possibly open a new line of research in which they yield even further insights.", "In particular, it seems promising that our methods could also be applied to other online problems than scheduling jobs arriving online over time.", "Related work Sum of weighted completion times.", "The offline variants of nearly all problems under consideration are NP-hard.", "This is true already for the special case of a single machine [27], [28].", "Two restricted variants can be solved optimally in polynomial time.", "Smith's Rule solves the problem $\\textup {1}|\\,\\,|\\sum w_jC_j$ to optimality by scheduling jobs in non-increasing order of weight-to-processing-time ratios [45].", "Furthermore, scheduling by shortest remaining processing times yields an optimal schedule for $\\textup {1}|\\,r_j,pmtn\\,|\\sum w_jC_j$  [38].", "However, for the other settings polynomial-time approximation schemes have been developed [1], even when the number of machines is part of the input.", "The online setting has been a highly active field of research in the past fifteen years.", "A whole sequence of papers appeared introducing new algorithms, new relaxations and analytical techniques that decreased the gaps between lower and upper bounds on the optimal competitive ratio [18], [39], [20], [43], [42], [2], [6], [24], [10], [17], [35], [34], [9], [40], [30], [31], [46], [36], [5], [41], [14].", "We do not intend to give a detailed history of developments; instead, we refer the reader to overviews, e.g., in [34], [10].", "Table REF summarizes the current state-of-the-art on best known lower and upper bounds on the optimal competitive ratios.", "Interestingly, despite the considerable effort, optimal competitive ratios are known only for $\\textup {1}|\\,r_j, pmtn\\,|\\sum C_j$  [38] and for non-preemptive single-machine scheduling [2], [46], [24], [6].", "In all other scheduling settings remain unsatisfactory, even quite significant gaps.", "Table: Lower and upper bounds on the competitive ratio for deterministic and randomized online algorithms.", "[1]For $m=1,2,3,4,5, \\dots 100$ the lower bound is $LB=2, 1.520, 1.414, 1.373, 1.364, \\dots 1.312$ .", "More general min-sum (completion time) objectives.", "Recently, there has been an increasing interest in studying generalized cost functions.", "So far, this research has focussed on offline problems.", "The most general case is when each job may have its individual non-decreasing cost function $f_j$ .", "For scheduling on a single machine with release dates and preemption, $1|r_j,pmtn|\\sum f_j$ , Bansal and Pruhs [4] gave a randomized $\\mathcal {O}(\\log \\log (nP))$ -approximation, where $P=\\max _{j\\in J}p_j$ .", "In the case that all jobs have identical release dates, the approximation factor reduces to 16.", "Cheung and Shmoys [8] improved this latter result and gave a deterministic $(2+\\varepsilon )$ -approximation.", "This result applies also on a machine of varying speed.", "The more restricted problem with a global cost function $1|r_j,pmtn|\\sum w_jf(C_j)$ has been studied by Epstein et al.", "[13] in the context of universal solutions.", "They gave an algorithm that produces for any job instance one scheduling solution that is a $(4+\\varepsilon )$ -approximation for any cost function and even under unreliable machine behavior.", "Höhn and Jacobs [23] studied the same problem without release dates.", "They analyzed the performance of Smith's Rule [45] and gave tight approximation guarantees for all convex and all concave functions $f$ .", "Makespan.", "The online makespan minimization problem has been extensively studied in a different online paradigm where jobs arrive one by one (see [15], [37] and references therein).", "Our model, in which jobs arrive online over time, is much less studied.", "In the identical parallel machine environment, Chen and Vestjens [7] give nearly tight bounds on the optimal competitive ratio, $1.347 \\le \\rho ^* \\le 3/2$ , using a natural online variant of the well-known largest processing time first algorithm.", "In the offline setting, polynomial time approximation schemes are known for identical [21] and uniform machines [22].", "For unrelated machines, the problem is NP-hard to approximate with a better ratio than $3/2$ and a 2-approximation is known [29].", "If the number of machines is bounded by a constant there is a PTAS [29].", "Proofs of Section  First, we will show that the number of distinct processing times of large jobs in each interval can be upper-bounded by a constant.", "To achieve this, we partition the jobs of an instance into large and small jobs.", "With respect to a release date $R_{x}$ we say that a job $j$ with $r_{j}=R_{x}$ is large if $p_{j}\\ge \\varepsilon ^{2}I_{x}=\\varepsilon ^{3}R_{x}$ and small otherwise.", "Abusing notation, we refer to $|I_{x}|$ also by $I_{x}$ .", "Note that $I_x=\\varepsilon \\cdot (1+\\varepsilon )^x$ .", "Lemma B.1 The number of distinct processing times of jobs in each set $L_x$ is bounded by $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ .", "For any $j\\in L_{x}$ the processing time $p_{j}$ is a power of $1+\\varepsilon $ , say $p_{j}=(1+\\varepsilon )^{y}$ .", "Hence, we have that $\\varepsilon ^{3}\\left(1+\\varepsilon \\right)^{x}<\\left(1+\\varepsilon \\right)^{y}\\le \\frac{1}{\\varepsilon }\\left(1+\\varepsilon \\right)^{x}$ .", "The number of integers $y$ which satisfy the above inequalities is bounded by $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ , which yields the constant claimed in the lemma.", "Furthermore, we can bound the number of large jobs of each job size which are released at the same time.", "Lemma B.2 Without loss, we can restrict to instances with $|L_{x}|\\le (m/\\varepsilon ^{2}+m)4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ for each set $L_{x}$ .", "Let $L_{x,p}\\subseteq L_{x}$ denote the set of jobs in $L_{x}$ with processing time $p$ .", "By an exchange argument, one can restrict to schedules such that at each point in time at most $m$ jobs in $L_{x,p}$ are partially (i.e., to some extent but not completely) processed.", "Since $p_{j}\\ge \\varepsilon ^{2}I_{x}$ for each job $j\\in L_{x}$ , at most $m/\\varepsilon ^{2}+m$ jobs in $L_{x,p}$ are touched within $I_{x}$ .", "By an exchange argument we can assume that they are the $m/\\varepsilon ^{2}+m$ jobs with the largest weight in $L_{x,p}$ .", "Hence, the release date of all other jobs in $L_{x,p}$ can be moved to $R_{x+1}$ without any cost.", "Since due to Lemma REF there are at most $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ distinct processing times $p$ of large jobs in $L_{x}$ , the claim follows.", "We now just need to take care of the small jobs.", "Denote by $w_{j}/p_{j}$ the Smith's ratio of a job $j$ .", "An ordering where the jobs are ordered non-increasingly by their Smith's ratios is an ordering according to Smith's rule.", "The next lemma shows that scheduling the small jobs according to Smith's Rule is almost optimal and small jobs do not even need to be preempted or to cross intervals.", "For a set of jobs $J$ we define $p(J):=\\sum _{j\\in J}p_{j}$ .", "Lemma B.3 At $1+\\varepsilon $ loss we can restrict to schedules such that for each interval $I_{x}$ the small jobs scheduled within this interval are chosen by Smith's Rule from the set $\\bigcup _{x^{\\prime }\\le x}S_{x^{\\prime }}$ , no small job is preempted, any small job finishes in the same interval where it started and $p(S_{\\!x})\\le m\\cdot I_{x}$ for each interval $I_{x}$ .", "By an exchange argument one can show that it is optimal to schedule the small jobs by Smith's Rule if they can be arbitrarily divided into smaller jobs (where the weight is divided proportional to the processing time of the smaller jobs).", "Start with such a schedule and stretch time once.", "The gained free space is enough to finish all small jobs which are partially scheduled in each interval.", "For the last claim of the lemma, note that the total processing time in each interval $I_{x}$ is $mI_{x}$ .", "Order the small jobs non-increasingly by their Smith's Ratios and pick them until the total processing time of picked jobs just does not exceed $mI_{x}$ .", "The release date of all other jobs in $S_{x}$ can be safely moved to $R_{x+1}$ since due to our modifications we would not schedule them in $I_{x}$ anyway.", "Lemma REF (restated) There is a constant $s$ such that at $1+O(\\varepsilon )$ loss we can restrict to schedules such that for each interval $I_{x}$ there is a subinterval of $I_{x+s-1}$ which is large enough to process all jobs released at $R_x$ and during which only jobs in $R_x$ are executed.", "We call this subinterval the safety net of interval $I_{x}$ .", "We can assume that each job released at $R_x$ finishes before time $R_{x+s}$ .", "By Lemmas REF and $\\ref {lem:number-large-jobs}$ we bound $p(S_{x})+p(L_{x})$ by $p(S_{x})+p(L_{x}) & \\le & m\\cdot I_{x}+(m/\\varepsilon ^{2}+m) \\cdot \\left(4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }\\right) \\cdot \\frac{1}{\\varepsilon }\\left(1+\\varepsilon \\right)^{x}\\\\& \\le & m\\cdot \\left(1+\\varepsilon \\right)^{x}\\left(\\varepsilon +\\frac{8}{\\varepsilon ^{3}}\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }\\right)\\\\& = & \\varepsilon \\cdot I_{x+s-1}$ for a suitable constant $s$ , depending on $\\varepsilon $ and $m$ .", "Stretching time once, we gain enough free space at the end of each interval $I_{x+s-1}$ to establish the safety net for each job set $p(S_{x})+p(L_{x})$ .", "Lemma B.4 There is a constant $d$ such that we can at $1+O(\\varepsilon )$ loss restrict to instances such that $p_{j}>\\frac{\\varepsilon }{2d}\\cdot I_{x}$ for each job $j \\in S_x \\cup L_x$ .", "We call a job $j$ tiny if $p_j \\le \\frac{\\varepsilon }{2d}\\cdot I_{x}$ .", "Let $T_{x}=\\lbrace j_{1},j_{2},...,j_{|T_{x}|}\\rbrace $ denote all tiny jobs released at $R_{x}$ .", "W.l.o.g.", "assume that they are ordered non-increasingly by their Smith's Ratios $w_{j}/p_{j}$ .", "Let $\\ell $ be the largest integer such that $\\sum _{i=1}^{\\ell }p_{i}\\le \\frac{\\varepsilon }{d}\\cdot I_{x}$ .", "We define the pack $P_{x}^{1}:=\\lbrace j_{1},...,j_{\\ell }\\rbrace $ .", "We denote by $\\sum _{i=1}^{\\ell }p_{i}$ the processing time of pack $P_{x}^{1}$ and by $\\sum _{i=1}^{\\ell }w_{i}$ its weight.", "We continue iteratively until we assigned all tiny jobs to packs.", "By definition of the processing time of tiny jobs, the processing time of all but possibly the last pack released at time $R_{x}$ is in the interval $[\\frac{\\varepsilon }{2d}\\cdot I_{x},\\frac{\\varepsilon }{d}\\cdot I_{x}]$ .", "Using timestretching, we can show that at $1+O(\\varepsilon )$ loss all tiny jobs of the same pack are scheduled in the same interval on the same machine.", "Here we use that in any schedule obeying Smith's Rule and using the safety net (see Lemma REF ) in each interval there is at most one partially but unfinished pack from each of at most $s$ previous release dates.", "Hence, we can treat the packs as single jobs whose processing time and weight matches the respective values of the packs.", "Also, at $1+\\varepsilon $ loss we can ensure that also the very last pack has a processing time in $[\\frac{\\varepsilon }{2d}\\cdot I_{x},\\frac{\\varepsilon }{d}\\cdot I_{x}]$ .", "Finally, at $1+O(\\varepsilon )$ loss we can ensure that the processing times and weights of the new jobs (which replace the packs) are powers of $1+\\varepsilon $ .", "Lemma B.5 Assume that there is a constant $d$ such that $p_{j}>\\frac{\\varepsilon }{2d}\\cdot I_{x}$ for each job $j \\in S_x$ .", "Then at $1+O(\\varepsilon )$ loss, the number of distinct processing times of jobs each set $S_x$ is upper-bounded by $(\\log _{1+\\varepsilon }\\varepsilon \\cdot 2d)$ .", "From the previous lemmas, we have $\\frac{e^2}{2d}\\cdot (1+\\varepsilon )^x<(1+\\varepsilon )^y<\\varepsilon ^3(1+\\varepsilon )^x.$ The number of integers $y$ satisfying these inequalities is upper-bounded by the claimed constant.", "Lemma REF now follows from the lemmas REF and REF .", "Lemma REF follows from lemmas REF , REF and REF .", "Next, we prove Lemma REF : [Proof of Lemma REF ] The claim about the number of partially processed jobs of each type can be assumed without any loss.", "For the extent of processing, note that due to Lemmas REF , REF , and REF there is a constant $c$ such that at each time $R_{x}$ the total processing time of unfinished large jobs is bounded by $c\\cdot R_{x}$ .", "We stretch time once.", "The gained space is sufficient to schedule $p_{j}\\cdot \\mu $ processing units of each unfinished large job $j$ (for an appropriately chosen universal constant $\\mu $ ).", "This allows us to enforce the claim.", "The claim about the non-preemptive behavior of small jobs follows from Lemma REF .", "[Proof of Lemma REF ] In any schedule the jobs in $\\cup _{i=0}^{p-1}Q_{k+i}$ contribute at least $\\sum _{i=0}^{p-1}rw(Q_{k+i})$ towards the objective.", "If we move all jobs in $Q_{k+p}$ to their safety nets, they contribute at most $\\sum _{j\\in Q_{k+p}}r_{j}\\left(1+\\varepsilon \\right)^{s}\\cdot w_{j} & = & \\left(1+\\varepsilon \\right)^{s}\\cdot rw(Q_{k+p})\\\\& \\le & \\varepsilon \\cdot \\sum _{i=0}^{p-1}rw(Q_{k+i})\\\\& \\le & \\varepsilon \\cdot OPT$ to the objective.", "[Proof of Lemma REF ] We modify a given online algorithm such that each part is treated as a separate instance.", "To bound the cost in the competitive ratio, we show that $\\frac{\\textsf {\\textsc {A}}(I)}{\\textsf {\\textsc {Opt}}(I)}\\le \\max _{i}\\lbrace \\frac{\\textsf {\\textsc {A}}(P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\rbrace \\cdot (1+O(\\varepsilon )).$ By the above lemmas, there is a $(1+O(\\varepsilon ))$ -approximative (offline) solution in which at the end of each part $P_{\\!i}$ each job has either completed or has been moved to its safety net.", "Denote this solution by $\\textsf {\\textsc {Opt}}^{\\prime }(I)$ and by $\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})$ its respective part for each part $P_{\\!i}$ .", "Note that $\\textsf {\\textsc {Opt}}^{\\prime }(I)=\\sum _{i}\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})$ .", "Then, $\\frac{\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i}))}{\\textsf {\\textsc {Opt}}(I)} \\le \\frac{\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i}))}{\\sum _{i=1}^{k}\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})}\\cdot (1+O(\\varepsilon )) \\le \\max _{i=1,...,k}\\lbrace \\frac{\\textsf {\\textsc {A}}(P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\rbrace \\cdot (1+O(\\varepsilon )).$ [Proof of Lemma REF ] We show that $\\left(1+\\varepsilon \\right)^{s}\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\varepsilon \\cdot \\sum _{i=p-K}^{p}rw(Q_{i})$ for a sufficiently large value $K$ .", "This will then be the claimed constant.", "Let $\\delta ^{\\prime }:=\\frac{\\varepsilon }{\\left(1+\\varepsilon \\right)^{s}}$ .", "By assumption, we have that $rw(Q_{i+1})>\\delta ^{\\prime }\\cdot \\sum _{\\ell =1}^{i}rw(Q_{\\ell })$ for each $i$ .", "This implies that $\\frac{rw(Q_{i+1})}{\\sum _{\\ell =1}^{i+1}rw(Q_{\\ell })}>\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}$ for each $i$ .", "Hence, $\\frac{\\sum _{\\ell =1}^{i}rw(Q_{\\ell })}{rw(Q_{i+1})+\\sum _{\\ell =1}^{i}rw(Q_{\\ell })}\\le 1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}<1$ for each $i$ and hence, $\\sum _{\\ell =1}^{i}rw(Q_{\\ell })\\le (1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }})\\sum _{\\ell =1}^{i+1}rw(Q_{\\ell }).$ In other words, if we remove $Q_{i+1}$ from $\\cup _{\\ell =1}^{i+1}Q_{\\ell }$ then the total release weight of the set decreases by a factor of at least $1-\\delta ^{\\prime }/(1+\\delta ^{\\prime })<1$ .", "For any $K$ this implies that $\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}\\sum _{\\ell =1}^{p}rw(Q_{\\ell })$ and hence $\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\frac{1}{1-\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}}\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}\\sum _{i=p-K}^{p}rw(Q_{i}).$ By choosing $K$ sufficiently large, the claim follows.", "[Proof of Lemma REF ] We partition $\\mathrm {Ir}_{x}(J)$ into two groups: $\\mathrm {Ir}_{x}^{\\mathrm {old}}(J):=\\lbrace j\\in \\mathrm {Ir}_{x}(J)|r_{j}<R_{x-\\Gamma }\\rbrace $ and $\\mathrm {Ir}_{x}^{\\mathrm {new}}(J):=\\lbrace j\\in \\mathrm {Ir}_{x}(J)|r_{j}\\ge R_{x-\\Gamma }\\rbrace $ .", "Lemma REF implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))\\le \\varepsilon \\cdot rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)\\cup \\mathrm {Rel}_{x}(J))$ (recall that the former value is an upper bound on the total weighted completion time of the jobs in $\\mathrm {Ir}_{x}^{\\mathrm {old}}(J)$ ).", "For every job $j\\in \\mathrm {Ir}_{x}^{\\mathrm {new}}(J)$ there must be a job $j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)$ such that $w_{j}<\\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}$ .", "We say that such a job $j^{\\prime }$ dominates $j$ .", "At most $\\Delta $ jobs are released at the beginning of each interval and hence $|\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)|\\le \\Delta \\Gamma $ .", "In particular, if $\\mathrm {dom}(j^{\\prime })$ denotes all jobs in $\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)$ which are dominated by $j^{\\prime }$ then $\\sum _{j\\in \\mathrm {dom}(j^{\\prime })}w_{j}r_{j}\\le \\Delta \\Gamma \\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}r_{j^{\\prime }}\\cdot (1+\\varepsilon )^{\\Gamma }$ This implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)) & \\le & (1+\\varepsilon )^{s}\\sum _{j\\in \\mathrm {Ir}_{x}^{\\mathrm {new}}(J)}w_{j}r_{j}\\\\& \\le & \\sum _{j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)}\\Delta \\Gamma \\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}r_{j^{\\prime }}\\cdot (1+\\varepsilon )^{\\Gamma +s}\\\\& \\le & \\varepsilon \\cdot \\sum _{j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)}w_{j^{\\prime }}r_{j^{\\prime }}\\\\& = & \\varepsilon \\cdot (rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))+rw(\\mathrm {Rel}_{x}(J)))$ Together with Inequality REF this implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}(J)) & = & (1+\\varepsilon )^{s}(rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)+rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J)))\\\\& \\le & \\left(\\varepsilon \\cdot (rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))+rw(\\mathrm {Rel}_{x}(J))\\right)+\\left(\\varepsilon \\cdot rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)\\cup \\mathrm {Rel}_{x}(J))\\right)\\\\& \\le & 2\\varepsilon \\cdot rw(\\mathrm {Rel}_{x}(J))+\\varepsilon (rw(\\mathrm {Ir}_{x}(J))$ and the latter inequality implies that $\\sum _{j\\in \\mathrm {Ir}_{x}(J)}C_{j}w_{j} & \\le & (1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}(J))\\\\& \\le & 2\\varepsilon \\frac{(1+\\varepsilon )^{s}}{(1+\\varepsilon )^{s}-\\varepsilon }rw(\\mathrm {Rel}_{x}(J))\\\\& \\le & 3\\varepsilon \\cdot rw(\\mathrm {Rel}_{x}(J))$ Proofs of Section  Lemma C.1 In the non-preemptive setting, at $1+O(\\varepsilon )$ loss we can ensure that at the end of each interval $I_{x}$ , there are at most $m$ large jobs from each type which are partially (i.e., neither fully nor not at all) processed, and for each partially but not completely processed large job $j$ there is a value $k_{x,j}$ such that $j$ is processed for at least $k_{x,j}\\cdot p_{j}\\cdot \\mu $ time units in $I_{x}$ , we calculate the objective with adjusted completion times $\\bar{C}_{j}=R_{c(j)}$ for some value $c(j)$ for each job $j$ such that $\\sum _{x<c(j)}k_{x,j}\\cdot p_{j}\\cdot \\mu \\ge p_{j}$ .", "Note that the first property holds for any non-preemptive schedule and is listed here only for the sake of clarity.", "The other two properties can be shown similiarly as in the proof of Lemma REF .", "[Proof of Lemma REF ] Assume that we have an online algorithm $\\textsf {\\textsc {A}}$ with competitive factor $\\rho _{\\textsf {\\textsc {A}}}$ on instances in which for every $i$ the first job $\\textnormal {first}(i)$ released in part $P_{\\!i}$ satisfies $\\sum _{\\ell =1}^{i-1} rw(P_{\\!\\ell }) \\le w_{\\!\\textnormal {first}(i)}\\varepsilon /\\left(1+\\varepsilon \\right)^{s}$  (i.e., $\\textnormal {first}(i)$ dominates all previously released parts).", "Based on $\\textsf {\\textsc {A}}$ we construct a new algorithm $\\textsf {\\textsc {A}}^{\\prime }$ for arbitrary instances with competitive ratio at most $\\left(1+\\varepsilon \\right)\\rho _{\\textsf {\\textsc {A}}}$ : When a new part $P_{\\!i}$ begins, we scale the weights of all jobs in $P_{\\!i}$ such that $\\sum _{\\ell =1}^{i-1}rw(P_{\\!\\ell }) \\le w^{\\prime }_{\\!\\textnormal {first}(i)}\\varepsilon /\\left(1+\\varepsilon \\right)^{s}$ , where the values $w^{\\prime }_{\\!j}$ denote the adjusted weights.", "Denote by $\\bar{I}(i)$ the resulting instance up to (and including) part $P_{\\!i}$ .", "We schedule the resulting instance using $\\textsf {\\textsc {A}}$ .", "We take the computed schedule for each part $P_{\\!i}$ and use it for the jobs with their original weight, obtaining a new algorithm $\\textsf {\\textsc {A}}^{\\prime }$ .", "The following calculations shows that this procedure costs only a factor $1+\\varepsilon $ .", "To this end, we proof that for any instance $I$ it holds that $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I)}{\\textsf {\\textsc {Opt}}(I)} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}(\\bar{I}(i))}{\\textsf {\\textsc {Opt}}(\\bar{I}(i))}\\cdot (1+O(\\varepsilon ))\\le (1+O(\\varepsilon ))\\rho _{\\textsf {\\textsc {A}}}.$ For each $P_{\\!i}$ we define $\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})$ to be the amount that the jobs in $P_{\\!i}$ contribute in $\\textsf {\\textsc {A}}^{\\prime }(I)$ .", "Similarly, we define $\\textsf {\\textsc {Opt}}(I|P_{\\!i})$ .", "We have that $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I)}{\\textsf {\\textsc {Opt}}(I)} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(I|P_{\\!i})} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}.$ We claim that for each $i$ holds $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\le (1+O(\\varepsilon ))\\cdot \\frac{\\textsf {\\textsc {A}}(\\bar{I}(i))}{\\textsf {\\textsc {Opt}}(\\bar{I}(i))}$ .", "For each part $P_{\\!i}$ let $v_{i}$ denote the scale factor of the weight of each job in $\\bar{I}(i)$ in comparison to its original weight.", "The optimum for the instance $\\bar{I}(i)$ can be bounded by $\\textsf {\\textsc {Opt}}(\\bar{I}(i)) & \\le & \\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i} +\\left(1+\\varepsilon \\right)^{s}\\sum _{\\ell =1}^{i-1}rw(P_{\\!\\ell }) \\le \\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i}+\\varepsilon \\cdot r_{\\textnormal {first}(i)} \\cdot w^{\\prime }_{\\textnormal {first}(i)}\\le \\left(1+\\varepsilon \\right)\\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i}.$ Furthermore holds by construction $\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})\\cdot v_{i}\\le \\textsf {\\textsc {A}}(\\bar{I}^{i})$ .", "Thus, $\\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}(\\bar{I}^{i})}{\\textsf {\\textsc {Opt}}(\\bar{I}^{i})}\\cdot (1+O(\\varepsilon ))$ .", "[Proof of Lemma REF ] Given a schedule on related machines with speed values $s_{1},...,s_{\\max }$ , we stretch time twice.", "Thus, we gain in each interval $I_{x}$ free space of size $\\varepsilon I_{x}$ on the fastest machine.", "For each machine whose speed is at most $\\frac{\\varepsilon }{m}s_{\\max }$ , we take its schedule of the interval $I_{x}$ and simulate it on the fastest machine.", "Thus, those slow machines are not needed and can be removed.", "The remaining machines have speeds in $[\\frac{\\varepsilon }{m}\\,s_{\\max },s_{\\max }]$ .", "Assuming the slowest machines has unit speed gives the desired bound.", "[Proof of Lemma REF ] Consider a schedule for an instance which does not satisfy the property.", "We stretch time twice and thus we gain a free space of $\\varepsilon I_{x}$ in each interval $I_{x}$ .", "Consider some $I_{x}$ and a job $j$ which is scheduled in $I_{x}$ .", "Let $i$ be a fastest machine for $j$ .", "We remove the processing volume of $j$ scheduled in $I_x$ on slow machines $i^{\\prime }$ with $p_{i^{\\prime }j} >\\frac{m}{\\varepsilon }\\, p_{ij}$ and schedule it on $i$ in the gained free space.", "This way, we obtain a feasible schedule even if a job never runs on a machine where it is slow.", "Thus, we can set $p_{i^{\\prime }j}=\\infty $ if there is a fast machine $i$ such that $p_{ij} \\le \\frac{\\varepsilon }{m}p_{i^{\\prime }j}$ .", "Proofs of Section  [Proof of Lemma REF ] Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances $I$ such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )$ .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "In instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We need to bound the increase in the total expected cost caused by moving all jobs in periods in $\\mathcal {Q}$ to their safety nets.", "This increase is bounded by $\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}\\sum _{j\\in Q}\\left(1+\\varepsilon \\right)^{s}r_{j}\\cdot w_{j}\\right] & \\le &\\left(1+\\varepsilon \\right)^{s}\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}rw(Q)\\right]\\\\& \\le & \\left(1+\\varepsilon \\right)^{s}\\frac{1}{M}\\sum _{Q\\in I}rw(Q)\\\\& \\le & \\varepsilon \\cdot rw(I)\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)\\,.$ Thus, the total expected cost of the computed schedule is $\\mathbb {E}\\left[\\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i})\\right] & \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(P_{\\!i})\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & (\\rho _{\\textsf {\\textsc {A}}}+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I).$ Thus, at $1+\\varepsilon $ loss in the competitive ratio we can restrict to parts $I_i$ which span a constant number of periods.", "[Proof of Lemma ] Consider an instance $I$ .", "Let $\\delta >0$ and $k\\in \\mathbb {N}$ be values to be determined later with the property that $1/\\delta \\in \\mathbb {N}$ .", "For each configuration $C$ and each interval-schedule $S$ we define a value $g(C,S)$ such that $\\left\\lfloor \\frac{f(C,S)}{\\delta }\\right\\rfloor \\cdot \\delta \\le g(C,S)\\le \\left\\lceil \\frac{f(C,S)}{\\delta }\\right\\rceil \\cdot \\delta $ and $\\sum _{S\\in \\mathcal {S}}g(C,S)=1$ .", "Now we want to bound $\\rho _{g}$ .", "The idea is that for determining the ratio $\\mathbb {E}\\left[\\,g(I)\\,\\right]/\\textsf {\\textsc {Opt}}(I)$ it suffices to consider schedules $S(I)$ which are computed with sufficiently large probability.", "We show that also $f$ computes them with almost the same probability.", "Let $S(I)$ denote a schedule for the entire instance $I$ .", "We denote by $P_{f}(S(I))$ and $P_{g}(S(I))$ the probability that $f$ and $g$ compute the schedule $S(I)$ when given the instance $I$ .", "Assume that $P_{f}(S(I))\\ge k\\cdot \\delta $ .", "Denote by $C_{0},...,C_{\\bar{\\Gamma }-1}$ the configurations that algorithms are faced with when computing $S(I)$ , i.e., each configuration $C_{x}$ contains the jobs which are released but unfinished at the beginning of interval $I_{x}$ in $S(I)$ and as history the schedule $S(I)$ restricted to the intervals $I_{0},...,I_{x-1}$ .", "Denote by $S_{\\!x}$ the schedule of $S(I)$ in the interval $I_{x}$ .", "Hence, $P_{f}(S(I))=\\prod _{x=0}^{\\bar{\\Gamma }-1}f(C_{x},S_{\\!x})$ .", "Note that from $P_{f}(S(I))\\ge k\\cdot \\delta $ follows that $f(C_{x},S_{\\!x})\\ge k\\cdot \\delta $ for all $x$ .", "For these schedules, $P_{g}(S(I))$ is not much larger since $P_{g}(S(I)) & = & \\prod _{x=0}^{\\bar{\\Gamma }-1}g(C_{x},S_{\\!x})\\le \\prod _{x=0}^{\\bar{\\Gamma }-1}\\frac{k+1}{k}f(C_{x},S_{\\!x})\\le \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}P_{f}(S(I)).$ Let $\\mathcal {S}(I)$ denote the set of all schedules for $I$ .", "We partition $\\mathcal {S}(I)$ into schedule sets $\\mathcal {S}_{H}^{g}(I):=\\lbrace S(I)|P_{g}(S(I))\\ge k\\cdot \\delta \\rbrace $ and $\\mathcal {S}_{L}^{g}(I):=\\mathcal {S}(I)\\setminus \\mathcal {S}_{H}(I)$ .", "We estimate the expected value of a schedule computed by algorithm map $g$ on $I$ by $\\mathbb {E}\\left[\\,g(I)\\,\\right] & = & \\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{g}(S(I))\\cdot S(I) + \\sum _{S(I)\\in \\mathcal {S}_{L}^{g}(I)}P_{g}(S(I)) \\cdot S(I)\\\\& \\le &\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\cdot S(I) + |\\mathcal {S}(I)| \\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s} \\cdot rw(I)\\\\& \\le &\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot S(I) + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I)\\\\& \\le & \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\mathbb {E}\\left[\\,f(I)\\,\\right] + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I).$ We choose $k$ such that $\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\le 1+\\varepsilon /2$ and $\\delta $ such that $|\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\le \\varepsilon /2$ for all instances $I$  (note here that $|\\mathcal {S}(I)|$ can be upper bounded by a value independent of $I$ since our instances contain only constantly many jobs).", "This yields $\\frac{\\mathbb {E}\\left[\\,g(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon /2)\\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} + \\varepsilon /2\\cdot \\frac{rw(I)}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon ) \\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)}\\,,$ and we conclude that $\\rho _{g}\\le \\left(1+\\varepsilon \\right)\\rho _{f}$ .", "Competitive-Ratio Approximation Schemes for Minimizing $C_{\\max }$ (cf.", "Section ) Consider the objective of minimizing the makespan.", "The simplifications within intervals of Section  are based on arguing on completion times of individual jobs, and clearly hold also for the last job.", "Thus, they directly apply to minimizing the makespan.", "Furthermore, we simplify the definition of irrelevant history in Section  by omitting the partition of the instance into parts.", "We observe that when then last job is released at time $R_{x^{*}}$ then all jobs $j$ with $r_{j}\\le R_{x^{*}-s}$ are irrelevant for the objective: such a job $j$ finishes at time $R_{x^{*}}$ the latest in any schedule (due to the safety net) and $OPT\\ge R_{x^{*}}$ .", "Therefore, we define a job $j$ to be irrelevant at time $R_{x}$ if $r_{j}\\le R_{x-s}$ .", "We keep Definition REF for the equivalence relation of schedules as it is except for the notion of job weights which are not important for the makespan.", "Based on the above definition for relevant jobs we define equivalence classes of configurations.", "With this definition, we can still restrict to algorithm maps $f$ with $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations $C,C^{\\prime }$ (Lemma REF ).", "Lemmas REF to REF then hold accordingly.", "Finally, note that since we do not split the instance into parts, we do not need (an adjusted version of) Lemma REF in the non-preemptive case.", "Theorem E.1 For any $m\\in \\mathbb {N}$ we obtain competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds.", "For constructing randomized online algorithm schemes for minimizing the makespan, similarly to Lemma REF we can show that we can restrict our attention to instances which span only a constant number of periods.", "Lemma E.2 For randomized algorithms for minimizing the makespan, at $1+O(\\varepsilon )$ loss we can restrict to instances in which all jobs are released in at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ consecutive periods.", "Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm for minimizing the makespan over time with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+O(\\varepsilon ))$ .", "Our reasoning is similar to the proof of Lemma REF .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "Given an instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We present each part separately to $\\textsf {\\textsc {A}}$ .", "We bound the competitive ratio $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ of the resulting algorithm.", "Let $R_{x^{*}}:=\\max _{j \\in I}r_{j}$ .", "Moving the jobs from periods $\\mathcal {Q}$ has an effect on the optimal makespan only if $o$ is chosen such that at least one job $j$ with $r_{j}>R_{x^{*}-s}$ is moved.", "There are at most two offsets $o$ such that this happens.", "In that case, the algorithm still achieves a competitive ratio of at most $\\left(1+\\varepsilon \\right)^{s}$ .", "In all other cases, $\\textsf {\\textsc {A}}^{\\prime }$ achieves a competitive ratio of at most $\\rho _{\\textsf {\\textsc {A}}}$ .", "Thus, we can bound $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ by $\\rho _{\\textsf {\\textsc {A}}^{\\prime }} & \\le & \\frac{2}{M}\\left(1+\\varepsilon \\right)^{s}+\\frac{M-2}{M}\\rho _{\\textsf {\\textsc {A}}}\\le 2\\varepsilon +\\rho _{\\textsf {\\textsc {A}}}\\le (1+O(\\varepsilon ))\\rho _{\\textsf {\\textsc {A}}}.$ We can prove similarly as in Lemma  that any randomized algorithm map $f$ can be well approximated by a discretized randomized algorithm map $g$ .", "Hence, we obtain the following theorem.", "Theorem E.3 For any $m\\in \\mathbb {N}$ we obtain randomized competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds." ], [ "Non-preemptive Scheduling.", "When preemption is not allowed, the definition of the safety net (Lemma REF ) needs to be adjusted since we cannot ensure that at the end of each interval $I_{x+s}$ there is a machine idle.", "However, we can guarantee that there is a reserved space somewhere in $[R_{x},R_{x+s})$ to process the small and big jobs in $S_{x} \\cup L_{x}$ .", "Furthermore, we cannot enforce that a big job $j$ is processed for exactly a certain multiple of $p_{j}\\mu $ in each interval (Lemma REF ).", "To solve this, we pretend that we could preempt $j$ and ensure that after $j$ has been preempted its machine stays idle until $j$ continues.", "Next, we can no longer assume that each part can be treated independently (Lemma REF ).", "Since some of the remaining jobs at the end of a part may have already started processing, we cannot simply move them to their safety net.", "Here we use the following simplification.", "Lemma 4.1 Let $\\textnormal {first}(i)$ denote the job that is released first in part $P_{\\!i}$ .", "At $1+\\varepsilon $ loss, we can restrict to instances such that $\\sum _{\\ell =1}^{i-1} rw(P_{\\!\\ell }) \\le \\frac{\\varepsilon }{\\left(1+\\varepsilon \\right)^{s}}\\cdot rw(\\textnormal {first}(i))$ , i.e., $\\textnormal {first}(i)$ dominates all previous parts.", "Therefore, at $1+\\varepsilon $ loss it is enough to consider only the currently running jobs from the previous part and the last $\\Gamma $ intervals from the current part when taking decisions.", "Finally, we add some minor modifications to handle the case that a currently running job is dominated by some other job.", "With these adjustments, we have only constantly many equivalence classes for interval-schedules and configurations, which allows us to construct a competitive-ratio approximation scheme as in Section .", "Theorem 4.2 For any $m\\in \\mathbb {N}$ we obtain a competitive-ratio approximation scheme for $\\textup {Pm}|\\,r_j\\,|\\sum w_jC_j$  ." ], [ "Scheduling on Related Machines.", "In this setting, each machine $i$ has associated a speed $s_i$ , such that processing job $j$ on machine $i$ takes $p_{j}/s_{i}$ time units.", "W.l.o.g.", "the slowest machine has unit speed.", "Let $s_{\\max }$ denote the maximum speed in an instance.", "An adjusted version of Lemma REF ensures that at $1+\\varepsilon $ loss $r_{j} \\ge \\varepsilon \\, p_{j}/s_{\\max }$ for all jobs $j$  (rather than $r_{j} \\ge \\varepsilon p_{j}$ ).", "Furthermore, we can bound the number of distinct processing times and the number of released jobs of each interval, using similar arguments as in the unit-speed case.", "Lemma 4.3 At $1+O(\\varepsilon )$ loss we can restrict to instances where for each release date the number of released jobs and the number of distinct processing times is bounded by a constant depending only on $\\varepsilon $ , $m$ , and $s_{\\max }$ .", "We establish the safety net for the jobs of each release date $R_{x}$ only on the fastest machine and thereby ensure the condition of Lemma REF in the related machine setting.", "For the non-preemptive setting we incorporate the adjustments introduced in Section REF .", "Since at $1+\\varepsilon $ loss we can round the speeds of the machines to powers of $1+\\varepsilon $ we obtain the following result.", "Theorem 4.4 For any $m\\in \\mathbb {N}$ we obtain competitive-ratio approximation schemes for $\\textup {Qm}|\\,r_j, pmtn\\,|\\sum w_jC_j$ and $\\textup {Qm}|\\,r_j\\,|\\sum w_jC_j$ , assuming that the speeds of any two machines differ by at most a constant factor.", "In the preemptive setting we can strengthen the result and give a competitive-ratio approximation scheme for the case that machine speeds are part of the input, that is, we obtain a nearly optimal competitive ratio for any speed vector.", "The key is to bound the variety of different speeds.", "To that end, we show that at $1+\\varepsilon $ loss a very fast machine can simulate $m-1$ very slow machines.", "Lemma 4.5 For $\\textup {Qm}|\\,r_j, pmtn\\,|\\sum w_jC_j$ , at $1+O(\\varepsilon )$ loss, we can restrict to instances in which $s_{\\max }$ is bounded by $m/\\varepsilon $ .", "As speeds are geometrically rounded, we have for each value $m$ only finitely many speed vectors.", "Thus, our enumeration scheme finds a nearly optimal online algorithm with a particular routine for each speed vector.", "Theorem 4.6 For any $m\\in \\mathbb {N}$ we obtain a competitive-ratio approximation scheme for $\\textup {Qm}|\\,r_j,pmtn\\,|\\sum w_jC_j$  ." ], [ "Preemptive Scheduling on Unrelated Machines.", "When each job $j$ has its individual processing time $p_{ij}$ on machine $i$ , the problem complexity increases significantly.", "We restrict to preemptive scheduling and show how to decrease the complexity to apply our approximation scheme.", "The key is to bound the range of the finite processing times for each job (which is unfortunately not possible in the non-preemptive case, see [1] for a counterexample).", "Lemma 4.7 At $1+\\varepsilon $ loss we can restrict to instances in which for each job $j$ the ratio of any two of its finite processing times is bounded by $m/\\varepsilon $ .", "The above lemma allows us to introduce the notion of job classes.", "Two jobs $j,j^{\\prime }$ are of the same class if they have finite processing times on exactly the same machines and  $p_{ij}/p_{ij^{\\prime }}=p_{i^{\\prime }j}/p_{i^{\\prime }j^{\\prime }}$ for any two such machines $i$ and $i^{\\prime }$ .", "For fixed $m$ , the number of different job classes is bounded by a constant $W$ .", "For each job class, we define large and small tasks similar to the identical machine case: for each job $j$ we define a value $\\tilde{p}_{j}:=\\max _{i}\\lbrace p_{ij}|p_{ij}<\\infty \\rbrace $ and say a job is large if  $\\tilde{p}_{j}\\ge \\varepsilon ^{2}r_{j}/W$ and small otherwise.", "For each job class separately, we perform the adjustments of Section .", "This yields the following lemma.", "Lemma 4.8 At $1+O(\\varepsilon )$ loss we can restrict to instances and schedules such that for each job class, the number of distinct values $\\tilde{p}_j$ of jobs $j$ with the same release date is bounded by a constant, for each job class, the number of jobs with the same release date is bounded by a constant $\\tilde{\\Delta }$ , a large job $j$ is only preempted at integer multiples of $\\tilde{p}_{j} \\cdot \\tilde{\\mu }$ for some constant $\\tilde{\\mu }$ and small jobs are never preempted and finish in the same interval where they start.", "The above lemmas imply that both, the number of equivalence classes of configurations and the number of equivalence classes for interval-schedules are bounded by constants.", "Thus, we can apply the enumeration scheme from Section .", "Theorem 4.9 For any $m\\in \\mathbb {N}$ we obtain a competitive-ratio approximation scheme for $\\textup {Rm}|\\,r_j,pmtn\\,|\\sum w_jC_j$  .", "Randomized algorithms When algorithms are allowed to make random choices and we consider expected values of schedules, we can restrict to instances which span only constantly many periods.", "Assuming the simplifications of Section , this allows a restriction to instances with a constant number of jobs.", "Lemma 5.1 For randomized algorithms, at $1+\\varepsilon $ loss we can restrict to instances in which all jobs are released in at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ consecutive periods.", "[Proof idea.]", "Beginning at a randomly chosen period $Q_i$ with $i\\in [0,M)$ , with $M:=\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\rceil $ , we move all jobs released in $Q_{i+kM}$ , $k=0,1,\\ldots $ , to their safety net.", "At $1+\\varepsilon $ loss, this gives us a partition into parts, at the end of which no job remains, and we can treat each part independently.", "A randomized online algorithm can be viewed as a function that maps every possible configuration $C$ to a probability distribution of interval-schedules which are feasible for $C$ .", "To apply our algorithmic framework from the deterministic setting that enumerates all algorithm maps, we discretize the probability space and define discretized algorithm maps.", "To this end, let $\\bar{\\Gamma }$ denote the maximum number of intervals in instances with at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $  periods.", "Definition 5.2 (Discretized algorithm maps) Let $\\bar{ be the set of configurations for intervals~I_{x}with~x\\le \\bar{\\Gamma }, let~\\bar{\\mathcal {S}} be the set ofinterval-schedules for intervals~I_{x} with~x\\le \\bar{\\Gamma },and let~\\delta >0.", "A \\emph {\\delta -discretized algorithm map }is afunction~f:\\bar{\\times \\bar{\\mathcal {S}}\\rightarrow [0,1] suchthatf(C,S)=k\\cdot \\delta with some~k\\in \\mathbb {N}_{0} for all~C\\in \\bar{ and~S\\in \\bar{\\mathcal {S}}, and\\sum _{S\\in \\bar{\\mathcal {S}}}f(C,S)=1 for all ~C\\in \\bar{.", "}By restricting to \\delta -discretizedalgorithm maps we do not lose too much in the competitive ratio.", "}\\begin{lem}There is a value~\\delta >0 such that for any (randomized) algorithmmap~f there is a~\\delta -discretized randomized algorithm map~gwith~\\rho _{g}\\le \\rho _{f}\\left(1+\\varepsilon \\right).\\end{lem}\\begin{proof}[Proof idea.", "]Let~f be a randomized algorithm map andlet~\\delta >0 such that 1/\\delta \\in \\mathbb {N}.", "We define anew~\\delta -discretized algorithm map~g.", "For eachconfiguration~C we define the values~g(C,S) suchthat~\\left\\lfloor f(C,S)/\\delta \\right\\rfloor \\cdot \\delta \\le g(C,S)\\le \\left\\lceil f(C,S)/\\delta \\right\\rceil \\cdot \\delta and~\\sum _{S\\in \\mathcal {S}}g(C,S)=1.", "To seethat~\\rho _{g}\\le \\left(1+\\varepsilon \\right)\\rho _{f}, consider an instance~I and apossible schedule~S(I) for I.", "There is a probability~pthat~f outputs~S(I).", "We show that the schedules which have largeprobability~p dominate~\\mathbb {E}\\left[\\,f(I)\\,\\right]/\\textsf {\\textsc {Opt}}(I).", "We show furtherthat if p is sufficiently large, the probability that~gproduces~S(I) is in~[p/(1+\\varepsilon ),p(1+\\varepsilon )), which implies theLemma.\\end{proof}}Like in the deterministic case, we can now show that at 1+\\varepsilon lossit suffices to restrict to \\emph {simplified \\delta -discretizedalgorithm maps} which treat equivalent configurations equivalently,similar to Lemma~\\ref {lem:equal-confs} (replacing~\\Gamma by~\\bar{\\Gamma } in Definition~\\ref {def:irrelevant_jobs} of theirrelevant jobs).As there are only constantly many of these maps, we enumerate all ofthem, test each map for its competitive ratio, and select the best ofthem.", "}\\begin{thm} We obtain randomized competitive-ratio approximation schemes for \\textup {Pm}|\\,r_j,(pmtn)\\,|\\sum w_jC_j,\\textup {Qm}|\\,r_j, pmtn\\,|\\sum w_jC_j, and\\textup {Rm}|\\,r_j, pmtn\\,|\\sum w_jC_j\\, and for \\textup {Qm}|\\,r_j\\,|\\sum w_jC_j with a bounded range of speeds for any fixed m\\in \\mathbb {N}.\\end{thm}$ General Min-Sum Objectives and Makespan In this section we briefly argue how the techniques presented above for minimizing $\\sum _{j}w_{j}C_{j}$ can be used for constructing online algorithm schemes for other scheduling problems with jobs arriving online over time, namely for minimizing $\\sum _{j\\in J}w_jf(C_{j})$ , with $f(x)=k\\cdot x^{\\alpha }$ and constant $\\alpha \\ge 1, k>0$ , and the makespan.", "Since monomial functions $f$ have the property that $f((1+\\varepsilon ) C_j)\\le (1+O(\\varepsilon )) f(C_j)$ , the arguments in previous sections apply almost directly to the generalized min-sum objective.", "In each step of simplification and abstraction, we have an increased loss in the performance guarantee, but it is covered by the $O(\\varepsilon )$ -term.", "Consider now the makespan objective.", "The simplifications within intervals of Section  are based on arguing on completion times of individual jobs, and clearly hold also for the last job.", "Thus, they directly apply to makespan minimization.", "We simplify the definition of irrelevant history by omitting the partition of the instance into parts and we define a job $j$ to be irrelevant at time $R_{x}$ if $r_{j}\\le R_{x-s}$ .", "Based on this definition, we define equivalence classes of configurations (ignoring weights and previous interval-schedules) and again restrict to algorithm maps $f$ with $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations (Lemma 3.6).", "Lemmas 3.7–3.9 then hold accordingly and yield a competitive-ratio approximation scheme.", "Finally, the adjustments of Sections  and  can be made accordingly.", "Without the partition of the instance into parts, this even becomes easier in the non-preemptive setting.", "Thus, we can state the following result.", "Theorem 6.1 For any $m \\in \\mathbb {N}$ there are deterministic and randomized competitive-ratio approximation schemes for preemptive and non-preemptive scheduling, on $m$ identical, related (with bounded speed ratio when non-preemptive), and unrelated machines (only preemptive) for the objectives of minimizing $C_{\\max }$ and minimizing $\\sum _{j\\in J}w_j f(C_j)$ , with $f(x)=k\\cdot x^{\\alpha }$ and constant $\\alpha \\ge 1,k>0$ .", "Conclusions We introduce the concept of competitive-ratio approximation schemes that compute online algorithms with a competitive ratio arbitrarily close to the best possible competitive ratio.", "We provide such schemes for various problem variants of scheduling jobs online to minimize the weighted sum of completion times, arbitrary monomial cost functions, and the makespan.", "The techniques derived in this paper provide a new and interesting view on the behavior of online algorithms.", "We believe that they contribute to the understanding of such algorithms and possibly open a new line of research in which they yield even further insights.", "In particular, it seems promising that our methods could also be applied to other online problems than scheduling jobs arriving online over time.", "Related work Sum of weighted completion times.", "The offline variants of nearly all problems under consideration are NP-hard.", "This is true already for the special case of a single machine [27], [28].", "Two restricted variants can be solved optimally in polynomial time.", "Smith's Rule solves the problem $\\textup {1}|\\,\\,|\\sum w_jC_j$ to optimality by scheduling jobs in non-increasing order of weight-to-processing-time ratios [45].", "Furthermore, scheduling by shortest remaining processing times yields an optimal schedule for $\\textup {1}|\\,r_j,pmtn\\,|\\sum w_jC_j$  [38].", "However, for the other settings polynomial-time approximation schemes have been developed [1], even when the number of machines is part of the input.", "The online setting has been a highly active field of research in the past fifteen years.", "A whole sequence of papers appeared introducing new algorithms, new relaxations and analytical techniques that decreased the gaps between lower and upper bounds on the optimal competitive ratio [18], [39], [20], [43], [42], [2], [6], [24], [10], [17], [35], [34], [9], [40], [30], [31], [46], [36], [5], [41], [14].", "We do not intend to give a detailed history of developments; instead, we refer the reader to overviews, e.g., in [34], [10].", "Table REF summarizes the current state-of-the-art on best known lower and upper bounds on the optimal competitive ratios.", "Interestingly, despite the considerable effort, optimal competitive ratios are known only for $\\textup {1}|\\,r_j, pmtn\\,|\\sum C_j$  [38] and for non-preemptive single-machine scheduling [2], [46], [24], [6].", "In all other scheduling settings remain unsatisfactory, even quite significant gaps.", "Table: Lower and upper bounds on the competitive ratio for deterministic and randomized online algorithms.", "[1]For $m=1,2,3,4,5, \\dots 100$ the lower bound is $LB=2, 1.520, 1.414, 1.373, 1.364, \\dots 1.312$ .", "More general min-sum (completion time) objectives.", "Recently, there has been an increasing interest in studying generalized cost functions.", "So far, this research has focussed on offline problems.", "The most general case is when each job may have its individual non-decreasing cost function $f_j$ .", "For scheduling on a single machine with release dates and preemption, $1|r_j,pmtn|\\sum f_j$ , Bansal and Pruhs [4] gave a randomized $\\mathcal {O}(\\log \\log (nP))$ -approximation, where $P=\\max _{j\\in J}p_j$ .", "In the case that all jobs have identical release dates, the approximation factor reduces to 16.", "Cheung and Shmoys [8] improved this latter result and gave a deterministic $(2+\\varepsilon )$ -approximation.", "This result applies also on a machine of varying speed.", "The more restricted problem with a global cost function $1|r_j,pmtn|\\sum w_jf(C_j)$ has been studied by Epstein et al.", "[13] in the context of universal solutions.", "They gave an algorithm that produces for any job instance one scheduling solution that is a $(4+\\varepsilon )$ -approximation for any cost function and even under unreliable machine behavior.", "Höhn and Jacobs [23] studied the same problem without release dates.", "They analyzed the performance of Smith's Rule [45] and gave tight approximation guarantees for all convex and all concave functions $f$ .", "Makespan.", "The online makespan minimization problem has been extensively studied in a different online paradigm where jobs arrive one by one (see [15], [37] and references therein).", "Our model, in which jobs arrive online over time, is much less studied.", "In the identical parallel machine environment, Chen and Vestjens [7] give nearly tight bounds on the optimal competitive ratio, $1.347 \\le \\rho ^* \\le 3/2$ , using a natural online variant of the well-known largest processing time first algorithm.", "In the offline setting, polynomial time approximation schemes are known for identical [21] and uniform machines [22].", "For unrelated machines, the problem is NP-hard to approximate with a better ratio than $3/2$ and a 2-approximation is known [29].", "If the number of machines is bounded by a constant there is a PTAS [29].", "Proofs of Section  First, we will show that the number of distinct processing times of large jobs in each interval can be upper-bounded by a constant.", "To achieve this, we partition the jobs of an instance into large and small jobs.", "With respect to a release date $R_{x}$ we say that a job $j$ with $r_{j}=R_{x}$ is large if $p_{j}\\ge \\varepsilon ^{2}I_{x}=\\varepsilon ^{3}R_{x}$ and small otherwise.", "Abusing notation, we refer to $|I_{x}|$ also by $I_{x}$ .", "Note that $I_x=\\varepsilon \\cdot (1+\\varepsilon )^x$ .", "Lemma B.1 The number of distinct processing times of jobs in each set $L_x$ is bounded by $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ .", "For any $j\\in L_{x}$ the processing time $p_{j}$ is a power of $1+\\varepsilon $ , say $p_{j}=(1+\\varepsilon )^{y}$ .", "Hence, we have that $\\varepsilon ^{3}\\left(1+\\varepsilon \\right)^{x}<\\left(1+\\varepsilon \\right)^{y}\\le \\frac{1}{\\varepsilon }\\left(1+\\varepsilon \\right)^{x}$ .", "The number of integers $y$ which satisfy the above inequalities is bounded by $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ , which yields the constant claimed in the lemma.", "Furthermore, we can bound the number of large jobs of each job size which are released at the same time.", "Lemma B.2 Without loss, we can restrict to instances with $|L_{x}|\\le (m/\\varepsilon ^{2}+m)4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ for each set $L_{x}$ .", "Let $L_{x,p}\\subseteq L_{x}$ denote the set of jobs in $L_{x}$ with processing time $p$ .", "By an exchange argument, one can restrict to schedules such that at each point in time at most $m$ jobs in $L_{x,p}$ are partially (i.e., to some extent but not completely) processed.", "Since $p_{j}\\ge \\varepsilon ^{2}I_{x}$ for each job $j\\in L_{x}$ , at most $m/\\varepsilon ^{2}+m$ jobs in $L_{x,p}$ are touched within $I_{x}$ .", "By an exchange argument we can assume that they are the $m/\\varepsilon ^{2}+m$ jobs with the largest weight in $L_{x,p}$ .", "Hence, the release date of all other jobs in $L_{x,p}$ can be moved to $R_{x+1}$ without any cost.", "Since due to Lemma REF there are at most $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ distinct processing times $p$ of large jobs in $L_{x}$ , the claim follows.", "We now just need to take care of the small jobs.", "Denote by $w_{j}/p_{j}$ the Smith's ratio of a job $j$ .", "An ordering where the jobs are ordered non-increasingly by their Smith's ratios is an ordering according to Smith's rule.", "The next lemma shows that scheduling the small jobs according to Smith's Rule is almost optimal and small jobs do not even need to be preempted or to cross intervals.", "For a set of jobs $J$ we define $p(J):=\\sum _{j\\in J}p_{j}$ .", "Lemma B.3 At $1+\\varepsilon $ loss we can restrict to schedules such that for each interval $I_{x}$ the small jobs scheduled within this interval are chosen by Smith's Rule from the set $\\bigcup _{x^{\\prime }\\le x}S_{x^{\\prime }}$ , no small job is preempted, any small job finishes in the same interval where it started and $p(S_{\\!x})\\le m\\cdot I_{x}$ for each interval $I_{x}$ .", "By an exchange argument one can show that it is optimal to schedule the small jobs by Smith's Rule if they can be arbitrarily divided into smaller jobs (where the weight is divided proportional to the processing time of the smaller jobs).", "Start with such a schedule and stretch time once.", "The gained free space is enough to finish all small jobs which are partially scheduled in each interval.", "For the last claim of the lemma, note that the total processing time in each interval $I_{x}$ is $mI_{x}$ .", "Order the small jobs non-increasingly by their Smith's Ratios and pick them until the total processing time of picked jobs just does not exceed $mI_{x}$ .", "The release date of all other jobs in $S_{x}$ can be safely moved to $R_{x+1}$ since due to our modifications we would not schedule them in $I_{x}$ anyway.", "Lemma REF (restated) There is a constant $s$ such that at $1+O(\\varepsilon )$ loss we can restrict to schedules such that for each interval $I_{x}$ there is a subinterval of $I_{x+s-1}$ which is large enough to process all jobs released at $R_x$ and during which only jobs in $R_x$ are executed.", "We call this subinterval the safety net of interval $I_{x}$ .", "We can assume that each job released at $R_x$ finishes before time $R_{x+s}$ .", "By Lemmas REF and $\\ref {lem:number-large-jobs}$ we bound $p(S_{x})+p(L_{x})$ by $p(S_{x})+p(L_{x}) & \\le & m\\cdot I_{x}+(m/\\varepsilon ^{2}+m) \\cdot \\left(4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }\\right) \\cdot \\frac{1}{\\varepsilon }\\left(1+\\varepsilon \\right)^{x}\\\\& \\le & m\\cdot \\left(1+\\varepsilon \\right)^{x}\\left(\\varepsilon +\\frac{8}{\\varepsilon ^{3}}\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }\\right)\\\\& = & \\varepsilon \\cdot I_{x+s-1}$ for a suitable constant $s$ , depending on $\\varepsilon $ and $m$ .", "Stretching time once, we gain enough free space at the end of each interval $I_{x+s-1}$ to establish the safety net for each job set $p(S_{x})+p(L_{x})$ .", "Lemma B.4 There is a constant $d$ such that we can at $1+O(\\varepsilon )$ loss restrict to instances such that $p_{j}>\\frac{\\varepsilon }{2d}\\cdot I_{x}$ for each job $j \\in S_x \\cup L_x$ .", "We call a job $j$ tiny if $p_j \\le \\frac{\\varepsilon }{2d}\\cdot I_{x}$ .", "Let $T_{x}=\\lbrace j_{1},j_{2},...,j_{|T_{x}|}\\rbrace $ denote all tiny jobs released at $R_{x}$ .", "W.l.o.g.", "assume that they are ordered non-increasingly by their Smith's Ratios $w_{j}/p_{j}$ .", "Let $\\ell $ be the largest integer such that $\\sum _{i=1}^{\\ell }p_{i}\\le \\frac{\\varepsilon }{d}\\cdot I_{x}$ .", "We define the pack $P_{x}^{1}:=\\lbrace j_{1},...,j_{\\ell }\\rbrace $ .", "We denote by $\\sum _{i=1}^{\\ell }p_{i}$ the processing time of pack $P_{x}^{1}$ and by $\\sum _{i=1}^{\\ell }w_{i}$ its weight.", "We continue iteratively until we assigned all tiny jobs to packs.", "By definition of the processing time of tiny jobs, the processing time of all but possibly the last pack released at time $R_{x}$ is in the interval $[\\frac{\\varepsilon }{2d}\\cdot I_{x},\\frac{\\varepsilon }{d}\\cdot I_{x}]$ .", "Using timestretching, we can show that at $1+O(\\varepsilon )$ loss all tiny jobs of the same pack are scheduled in the same interval on the same machine.", "Here we use that in any schedule obeying Smith's Rule and using the safety net (see Lemma REF ) in each interval there is at most one partially but unfinished pack from each of at most $s$ previous release dates.", "Hence, we can treat the packs as single jobs whose processing time and weight matches the respective values of the packs.", "Also, at $1+\\varepsilon $ loss we can ensure that also the very last pack has a processing time in $[\\frac{\\varepsilon }{2d}\\cdot I_{x},\\frac{\\varepsilon }{d}\\cdot I_{x}]$ .", "Finally, at $1+O(\\varepsilon )$ loss we can ensure that the processing times and weights of the new jobs (which replace the packs) are powers of $1+\\varepsilon $ .", "Lemma B.5 Assume that there is a constant $d$ such that $p_{j}>\\frac{\\varepsilon }{2d}\\cdot I_{x}$ for each job $j \\in S_x$ .", "Then at $1+O(\\varepsilon )$ loss, the number of distinct processing times of jobs each set $S_x$ is upper-bounded by $(\\log _{1+\\varepsilon }\\varepsilon \\cdot 2d)$ .", "From the previous lemmas, we have $\\frac{e^2}{2d}\\cdot (1+\\varepsilon )^x<(1+\\varepsilon )^y<\\varepsilon ^3(1+\\varepsilon )^x.$ The number of integers $y$ satisfying these inequalities is upper-bounded by the claimed constant.", "Lemma REF now follows from the lemmas REF and REF .", "Lemma REF follows from lemmas REF , REF and REF .", "Next, we prove Lemma REF : [Proof of Lemma REF ] The claim about the number of partially processed jobs of each type can be assumed without any loss.", "For the extent of processing, note that due to Lemmas REF , REF , and REF there is a constant $c$ such that at each time $R_{x}$ the total processing time of unfinished large jobs is bounded by $c\\cdot R_{x}$ .", "We stretch time once.", "The gained space is sufficient to schedule $p_{j}\\cdot \\mu $ processing units of each unfinished large job $j$ (for an appropriately chosen universal constant $\\mu $ ).", "This allows us to enforce the claim.", "The claim about the non-preemptive behavior of small jobs follows from Lemma REF .", "[Proof of Lemma REF ] In any schedule the jobs in $\\cup _{i=0}^{p-1}Q_{k+i}$ contribute at least $\\sum _{i=0}^{p-1}rw(Q_{k+i})$ towards the objective.", "If we move all jobs in $Q_{k+p}$ to their safety nets, they contribute at most $\\sum _{j\\in Q_{k+p}}r_{j}\\left(1+\\varepsilon \\right)^{s}\\cdot w_{j} & = & \\left(1+\\varepsilon \\right)^{s}\\cdot rw(Q_{k+p})\\\\& \\le & \\varepsilon \\cdot \\sum _{i=0}^{p-1}rw(Q_{k+i})\\\\& \\le & \\varepsilon \\cdot OPT$ to the objective.", "[Proof of Lemma REF ] We modify a given online algorithm such that each part is treated as a separate instance.", "To bound the cost in the competitive ratio, we show that $\\frac{\\textsf {\\textsc {A}}(I)}{\\textsf {\\textsc {Opt}}(I)}\\le \\max _{i}\\lbrace \\frac{\\textsf {\\textsc {A}}(P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\rbrace \\cdot (1+O(\\varepsilon )).$ By the above lemmas, there is a $(1+O(\\varepsilon ))$ -approximative (offline) solution in which at the end of each part $P_{\\!i}$ each job has either completed or has been moved to its safety net.", "Denote this solution by $\\textsf {\\textsc {Opt}}^{\\prime }(I)$ and by $\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})$ its respective part for each part $P_{\\!i}$ .", "Note that $\\textsf {\\textsc {Opt}}^{\\prime }(I)=\\sum _{i}\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})$ .", "Then, $\\frac{\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i}))}{\\textsf {\\textsc {Opt}}(I)} \\le \\frac{\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i}))}{\\sum _{i=1}^{k}\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})}\\cdot (1+O(\\varepsilon )) \\le \\max _{i=1,...,k}\\lbrace \\frac{\\textsf {\\textsc {A}}(P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\rbrace \\cdot (1+O(\\varepsilon )).$ [Proof of Lemma REF ] We show that $\\left(1+\\varepsilon \\right)^{s}\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\varepsilon \\cdot \\sum _{i=p-K}^{p}rw(Q_{i})$ for a sufficiently large value $K$ .", "This will then be the claimed constant.", "Let $\\delta ^{\\prime }:=\\frac{\\varepsilon }{\\left(1+\\varepsilon \\right)^{s}}$ .", "By assumption, we have that $rw(Q_{i+1})>\\delta ^{\\prime }\\cdot \\sum _{\\ell =1}^{i}rw(Q_{\\ell })$ for each $i$ .", "This implies that $\\frac{rw(Q_{i+1})}{\\sum _{\\ell =1}^{i+1}rw(Q_{\\ell })}>\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}$ for each $i$ .", "Hence, $\\frac{\\sum _{\\ell =1}^{i}rw(Q_{\\ell })}{rw(Q_{i+1})+\\sum _{\\ell =1}^{i}rw(Q_{\\ell })}\\le 1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}<1$ for each $i$ and hence, $\\sum _{\\ell =1}^{i}rw(Q_{\\ell })\\le (1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }})\\sum _{\\ell =1}^{i+1}rw(Q_{\\ell }).$ In other words, if we remove $Q_{i+1}$ from $\\cup _{\\ell =1}^{i+1}Q_{\\ell }$ then the total release weight of the set decreases by a factor of at least $1-\\delta ^{\\prime }/(1+\\delta ^{\\prime })<1$ .", "For any $K$ this implies that $\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}\\sum _{\\ell =1}^{p}rw(Q_{\\ell })$ and hence $\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\frac{1}{1-\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}}\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}\\sum _{i=p-K}^{p}rw(Q_{i}).$ By choosing $K$ sufficiently large, the claim follows.", "[Proof of Lemma REF ] We partition $\\mathrm {Ir}_{x}(J)$ into two groups: $\\mathrm {Ir}_{x}^{\\mathrm {old}}(J):=\\lbrace j\\in \\mathrm {Ir}_{x}(J)|r_{j}<R_{x-\\Gamma }\\rbrace $ and $\\mathrm {Ir}_{x}^{\\mathrm {new}}(J):=\\lbrace j\\in \\mathrm {Ir}_{x}(J)|r_{j}\\ge R_{x-\\Gamma }\\rbrace $ .", "Lemma REF implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))\\le \\varepsilon \\cdot rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)\\cup \\mathrm {Rel}_{x}(J))$ (recall that the former value is an upper bound on the total weighted completion time of the jobs in $\\mathrm {Ir}_{x}^{\\mathrm {old}}(J)$ ).", "For every job $j\\in \\mathrm {Ir}_{x}^{\\mathrm {new}}(J)$ there must be a job $j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)$ such that $w_{j}<\\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}$ .", "We say that such a job $j^{\\prime }$ dominates $j$ .", "At most $\\Delta $ jobs are released at the beginning of each interval and hence $|\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)|\\le \\Delta \\Gamma $ .", "In particular, if $\\mathrm {dom}(j^{\\prime })$ denotes all jobs in $\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)$ which are dominated by $j^{\\prime }$ then $\\sum _{j\\in \\mathrm {dom}(j^{\\prime })}w_{j}r_{j}\\le \\Delta \\Gamma \\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}r_{j^{\\prime }}\\cdot (1+\\varepsilon )^{\\Gamma }$ This implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)) & \\le & (1+\\varepsilon )^{s}\\sum _{j\\in \\mathrm {Ir}_{x}^{\\mathrm {new}}(J)}w_{j}r_{j}\\\\& \\le & \\sum _{j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)}\\Delta \\Gamma \\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}r_{j^{\\prime }}\\cdot (1+\\varepsilon )^{\\Gamma +s}\\\\& \\le & \\varepsilon \\cdot \\sum _{j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)}w_{j^{\\prime }}r_{j^{\\prime }}\\\\& = & \\varepsilon \\cdot (rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))+rw(\\mathrm {Rel}_{x}(J)))$ Together with Inequality REF this implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}(J)) & = & (1+\\varepsilon )^{s}(rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)+rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J)))\\\\& \\le & \\left(\\varepsilon \\cdot (rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))+rw(\\mathrm {Rel}_{x}(J))\\right)+\\left(\\varepsilon \\cdot rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)\\cup \\mathrm {Rel}_{x}(J))\\right)\\\\& \\le & 2\\varepsilon \\cdot rw(\\mathrm {Rel}_{x}(J))+\\varepsilon (rw(\\mathrm {Ir}_{x}(J))$ and the latter inequality implies that $\\sum _{j\\in \\mathrm {Ir}_{x}(J)}C_{j}w_{j} & \\le & (1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}(J))\\\\& \\le & 2\\varepsilon \\frac{(1+\\varepsilon )^{s}}{(1+\\varepsilon )^{s}-\\varepsilon }rw(\\mathrm {Rel}_{x}(J))\\\\& \\le & 3\\varepsilon \\cdot rw(\\mathrm {Rel}_{x}(J))$ Proofs of Section  Lemma C.1 In the non-preemptive setting, at $1+O(\\varepsilon )$ loss we can ensure that at the end of each interval $I_{x}$ , there are at most $m$ large jobs from each type which are partially (i.e., neither fully nor not at all) processed, and for each partially but not completely processed large job $j$ there is a value $k_{x,j}$ such that $j$ is processed for at least $k_{x,j}\\cdot p_{j}\\cdot \\mu $ time units in $I_{x}$ , we calculate the objective with adjusted completion times $\\bar{C}_{j}=R_{c(j)}$ for some value $c(j)$ for each job $j$ such that $\\sum _{x<c(j)}k_{x,j}\\cdot p_{j}\\cdot \\mu \\ge p_{j}$ .", "Note that the first property holds for any non-preemptive schedule and is listed here only for the sake of clarity.", "The other two properties can be shown similiarly as in the proof of Lemma REF .", "[Proof of Lemma REF ] Assume that we have an online algorithm $\\textsf {\\textsc {A}}$ with competitive factor $\\rho _{\\textsf {\\textsc {A}}}$ on instances in which for every $i$ the first job $\\textnormal {first}(i)$ released in part $P_{\\!i}$ satisfies $\\sum _{\\ell =1}^{i-1} rw(P_{\\!\\ell }) \\le w_{\\!\\textnormal {first}(i)}\\varepsilon /\\left(1+\\varepsilon \\right)^{s}$  (i.e., $\\textnormal {first}(i)$ dominates all previously released parts).", "Based on $\\textsf {\\textsc {A}}$ we construct a new algorithm $\\textsf {\\textsc {A}}^{\\prime }$ for arbitrary instances with competitive ratio at most $\\left(1+\\varepsilon \\right)\\rho _{\\textsf {\\textsc {A}}}$ : When a new part $P_{\\!i}$ begins, we scale the weights of all jobs in $P_{\\!i}$ such that $\\sum _{\\ell =1}^{i-1}rw(P_{\\!\\ell }) \\le w^{\\prime }_{\\!\\textnormal {first}(i)}\\varepsilon /\\left(1+\\varepsilon \\right)^{s}$ , where the values $w^{\\prime }_{\\!j}$ denote the adjusted weights.", "Denote by $\\bar{I}(i)$ the resulting instance up to (and including) part $P_{\\!i}$ .", "We schedule the resulting instance using $\\textsf {\\textsc {A}}$ .", "We take the computed schedule for each part $P_{\\!i}$ and use it for the jobs with their original weight, obtaining a new algorithm $\\textsf {\\textsc {A}}^{\\prime }$ .", "The following calculations shows that this procedure costs only a factor $1+\\varepsilon $ .", "To this end, we proof that for any instance $I$ it holds that $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I)}{\\textsf {\\textsc {Opt}}(I)} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}(\\bar{I}(i))}{\\textsf {\\textsc {Opt}}(\\bar{I}(i))}\\cdot (1+O(\\varepsilon ))\\le (1+O(\\varepsilon ))\\rho _{\\textsf {\\textsc {A}}}.$ For each $P_{\\!i}$ we define $\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})$ to be the amount that the jobs in $P_{\\!i}$ contribute in $\\textsf {\\textsc {A}}^{\\prime }(I)$ .", "Similarly, we define $\\textsf {\\textsc {Opt}}(I|P_{\\!i})$ .", "We have that $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I)}{\\textsf {\\textsc {Opt}}(I)} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(I|P_{\\!i})} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}.$ We claim that for each $i$ holds $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\le (1+O(\\varepsilon ))\\cdot \\frac{\\textsf {\\textsc {A}}(\\bar{I}(i))}{\\textsf {\\textsc {Opt}}(\\bar{I}(i))}$ .", "For each part $P_{\\!i}$ let $v_{i}$ denote the scale factor of the weight of each job in $\\bar{I}(i)$ in comparison to its original weight.", "The optimum for the instance $\\bar{I}(i)$ can be bounded by $\\textsf {\\textsc {Opt}}(\\bar{I}(i)) & \\le & \\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i} +\\left(1+\\varepsilon \\right)^{s}\\sum _{\\ell =1}^{i-1}rw(P_{\\!\\ell }) \\le \\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i}+\\varepsilon \\cdot r_{\\textnormal {first}(i)} \\cdot w^{\\prime }_{\\textnormal {first}(i)}\\le \\left(1+\\varepsilon \\right)\\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i}.$ Furthermore holds by construction $\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})\\cdot v_{i}\\le \\textsf {\\textsc {A}}(\\bar{I}^{i})$ .", "Thus, $\\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}(\\bar{I}^{i})}{\\textsf {\\textsc {Opt}}(\\bar{I}^{i})}\\cdot (1+O(\\varepsilon ))$ .", "[Proof of Lemma REF ] Given a schedule on related machines with speed values $s_{1},...,s_{\\max }$ , we stretch time twice.", "Thus, we gain in each interval $I_{x}$ free space of size $\\varepsilon I_{x}$ on the fastest machine.", "For each machine whose speed is at most $\\frac{\\varepsilon }{m}s_{\\max }$ , we take its schedule of the interval $I_{x}$ and simulate it on the fastest machine.", "Thus, those slow machines are not needed and can be removed.", "The remaining machines have speeds in $[\\frac{\\varepsilon }{m}\\,s_{\\max },s_{\\max }]$ .", "Assuming the slowest machines has unit speed gives the desired bound.", "[Proof of Lemma REF ] Consider a schedule for an instance which does not satisfy the property.", "We stretch time twice and thus we gain a free space of $\\varepsilon I_{x}$ in each interval $I_{x}$ .", "Consider some $I_{x}$ and a job $j$ which is scheduled in $I_{x}$ .", "Let $i$ be a fastest machine for $j$ .", "We remove the processing volume of $j$ scheduled in $I_x$ on slow machines $i^{\\prime }$ with $p_{i^{\\prime }j} >\\frac{m}{\\varepsilon }\\, p_{ij}$ and schedule it on $i$ in the gained free space.", "This way, we obtain a feasible schedule even if a job never runs on a machine where it is slow.", "Thus, we can set $p_{i^{\\prime }j}=\\infty $ if there is a fast machine $i$ such that $p_{ij} \\le \\frac{\\varepsilon }{m}p_{i^{\\prime }j}$ .", "Proofs of Section  [Proof of Lemma REF ] Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances $I$ such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )$ .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "In instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We need to bound the increase in the total expected cost caused by moving all jobs in periods in $\\mathcal {Q}$ to their safety nets.", "This increase is bounded by $\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}\\sum _{j\\in Q}\\left(1+\\varepsilon \\right)^{s}r_{j}\\cdot w_{j}\\right] & \\le &\\left(1+\\varepsilon \\right)^{s}\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}rw(Q)\\right]\\\\& \\le & \\left(1+\\varepsilon \\right)^{s}\\frac{1}{M}\\sum _{Q\\in I}rw(Q)\\\\& \\le & \\varepsilon \\cdot rw(I)\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)\\,.$ Thus, the total expected cost of the computed schedule is $\\mathbb {E}\\left[\\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i})\\right] & \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(P_{\\!i})\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & (\\rho _{\\textsf {\\textsc {A}}}+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I).$ Thus, at $1+\\varepsilon $ loss in the competitive ratio we can restrict to parts $I_i$ which span a constant number of periods.", "[Proof of Lemma ] Consider an instance $I$ .", "Let $\\delta >0$ and $k\\in \\mathbb {N}$ be values to be determined later with the property that $1/\\delta \\in \\mathbb {N}$ .", "For each configuration $C$ and each interval-schedule $S$ we define a value $g(C,S)$ such that $\\left\\lfloor \\frac{f(C,S)}{\\delta }\\right\\rfloor \\cdot \\delta \\le g(C,S)\\le \\left\\lceil \\frac{f(C,S)}{\\delta }\\right\\rceil \\cdot \\delta $ and $\\sum _{S\\in \\mathcal {S}}g(C,S)=1$ .", "Now we want to bound $\\rho _{g}$ .", "The idea is that for determining the ratio $\\mathbb {E}\\left[\\,g(I)\\,\\right]/\\textsf {\\textsc {Opt}}(I)$ it suffices to consider schedules $S(I)$ which are computed with sufficiently large probability.", "We show that also $f$ computes them with almost the same probability.", "Let $S(I)$ denote a schedule for the entire instance $I$ .", "We denote by $P_{f}(S(I))$ and $P_{g}(S(I))$ the probability that $f$ and $g$ compute the schedule $S(I)$ when given the instance $I$ .", "Assume that $P_{f}(S(I))\\ge k\\cdot \\delta $ .", "Denote by $C_{0},...,C_{\\bar{\\Gamma }-1}$ the configurations that algorithms are faced with when computing $S(I)$ , i.e., each configuration $C_{x}$ contains the jobs which are released but unfinished at the beginning of interval $I_{x}$ in $S(I)$ and as history the schedule $S(I)$ restricted to the intervals $I_{0},...,I_{x-1}$ .", "Denote by $S_{\\!x}$ the schedule of $S(I)$ in the interval $I_{x}$ .", "Hence, $P_{f}(S(I))=\\prod _{x=0}^{\\bar{\\Gamma }-1}f(C_{x},S_{\\!x})$ .", "Note that from $P_{f}(S(I))\\ge k\\cdot \\delta $ follows that $f(C_{x},S_{\\!x})\\ge k\\cdot \\delta $ for all $x$ .", "For these schedules, $P_{g}(S(I))$ is not much larger since $P_{g}(S(I)) & = & \\prod _{x=0}^{\\bar{\\Gamma }-1}g(C_{x},S_{\\!x})\\le \\prod _{x=0}^{\\bar{\\Gamma }-1}\\frac{k+1}{k}f(C_{x},S_{\\!x})\\le \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}P_{f}(S(I)).$ Let $\\mathcal {S}(I)$ denote the set of all schedules for $I$ .", "We partition $\\mathcal {S}(I)$ into schedule sets $\\mathcal {S}_{H}^{g}(I):=\\lbrace S(I)|P_{g}(S(I))\\ge k\\cdot \\delta \\rbrace $ and $\\mathcal {S}_{L}^{g}(I):=\\mathcal {S}(I)\\setminus \\mathcal {S}_{H}(I)$ .", "We estimate the expected value of a schedule computed by algorithm map $g$ on $I$ by $\\mathbb {E}\\left[\\,g(I)\\,\\right] & = & \\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{g}(S(I))\\cdot S(I) + \\sum _{S(I)\\in \\mathcal {S}_{L}^{g}(I)}P_{g}(S(I)) \\cdot S(I)\\\\& \\le &\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\cdot S(I) + |\\mathcal {S}(I)| \\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s} \\cdot rw(I)\\\\& \\le &\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot S(I) + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I)\\\\& \\le & \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\mathbb {E}\\left[\\,f(I)\\,\\right] + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I).$ We choose $k$ such that $\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\le 1+\\varepsilon /2$ and $\\delta $ such that $|\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\le \\varepsilon /2$ for all instances $I$  (note here that $|\\mathcal {S}(I)|$ can be upper bounded by a value independent of $I$ since our instances contain only constantly many jobs).", "This yields $\\frac{\\mathbb {E}\\left[\\,g(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon /2)\\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} + \\varepsilon /2\\cdot \\frac{rw(I)}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon ) \\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)}\\,,$ and we conclude that $\\rho _{g}\\le \\left(1+\\varepsilon \\right)\\rho _{f}$ .", "Competitive-Ratio Approximation Schemes for Minimizing $C_{\\max }$ (cf.", "Section ) Consider the objective of minimizing the makespan.", "The simplifications within intervals of Section  are based on arguing on completion times of individual jobs, and clearly hold also for the last job.", "Thus, they directly apply to minimizing the makespan.", "Furthermore, we simplify the definition of irrelevant history in Section  by omitting the partition of the instance into parts.", "We observe that when then last job is released at time $R_{x^{*}}$ then all jobs $j$ with $r_{j}\\le R_{x^{*}-s}$ are irrelevant for the objective: such a job $j$ finishes at time $R_{x^{*}}$ the latest in any schedule (due to the safety net) and $OPT\\ge R_{x^{*}}$ .", "Therefore, we define a job $j$ to be irrelevant at time $R_{x}$ if $r_{j}\\le R_{x-s}$ .", "We keep Definition REF for the equivalence relation of schedules as it is except for the notion of job weights which are not important for the makespan.", "Based on the above definition for relevant jobs we define equivalence classes of configurations.", "With this definition, we can still restrict to algorithm maps $f$ with $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations $C,C^{\\prime }$ (Lemma REF ).", "Lemmas REF to REF then hold accordingly.", "Finally, note that since we do not split the instance into parts, we do not need (an adjusted version of) Lemma REF in the non-preemptive case.", "Theorem E.1 For any $m\\in \\mathbb {N}$ we obtain competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds.", "For constructing randomized online algorithm schemes for minimizing the makespan, similarly to Lemma REF we can show that we can restrict our attention to instances which span only a constant number of periods.", "Lemma E.2 For randomized algorithms for minimizing the makespan, at $1+O(\\varepsilon )$ loss we can restrict to instances in which all jobs are released in at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ consecutive periods.", "Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm for minimizing the makespan over time with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+O(\\varepsilon ))$ .", "Our reasoning is similar to the proof of Lemma REF .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "Given an instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We present each part separately to $\\textsf {\\textsc {A}}$ .", "We bound the competitive ratio $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ of the resulting algorithm.", "Let $R_{x^{*}}:=\\max _{j \\in I}r_{j}$ .", "Moving the jobs from periods $\\mathcal {Q}$ has an effect on the optimal makespan only if $o$ is chosen such that at least one job $j$ with $r_{j}>R_{x^{*}-s}$ is moved.", "There are at most two offsets $o$ such that this happens.", "In that case, the algorithm still achieves a competitive ratio of at most $\\left(1+\\varepsilon \\right)^{s}$ .", "In all other cases, $\\textsf {\\textsc {A}}^{\\prime }$ achieves a competitive ratio of at most $\\rho _{\\textsf {\\textsc {A}}}$ .", "Thus, we can bound $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ by $\\rho _{\\textsf {\\textsc {A}}^{\\prime }} & \\le & \\frac{2}{M}\\left(1+\\varepsilon \\right)^{s}+\\frac{M-2}{M}\\rho _{\\textsf {\\textsc {A}}}\\le 2\\varepsilon +\\rho _{\\textsf {\\textsc {A}}}\\le (1+O(\\varepsilon ))\\rho _{\\textsf {\\textsc {A}}}.$ We can prove similarly as in Lemma  that any randomized algorithm map $f$ can be well approximated by a discretized randomized algorithm map $g$ .", "Hence, we obtain the following theorem.", "Theorem E.3 For any $m\\in \\mathbb {N}$ we obtain randomized competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds." ], [ "Randomized algorithms", "When algorithms are allowed to make random choices and we consider expected values of schedules, we can restrict to instances which span only constantly many periods.", "Assuming the simplifications of Section , this allows a restriction to instances with a constant number of jobs.", "Lemma 5.1 For randomized algorithms, at $1+\\varepsilon $ loss we can restrict to instances in which all jobs are released in at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ consecutive periods.", "[Proof idea.]", "Beginning at a randomly chosen period $Q_i$ with $i\\in [0,M)$ , with $M:=\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\rceil $ , we move all jobs released in $Q_{i+kM}$ , $k=0,1,\\ldots $ , to their safety net.", "At $1+\\varepsilon $ loss, this gives us a partition into parts, at the end of which no job remains, and we can treat each part independently.", "A randomized online algorithm can be viewed as a function that maps every possible configuration $C$ to a probability distribution of interval-schedules which are feasible for $C$ .", "To apply our algorithmic framework from the deterministic setting that enumerates all algorithm maps, we discretize the probability space and define discretized algorithm maps.", "To this end, let $\\bar{\\Gamma }$ denote the maximum number of intervals in instances with at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $  periods.", "Definition 5.2 (Discretized algorithm maps) Let $\\bar{ be the set of configurations for intervals~I_{x}with~x\\le \\bar{\\Gamma }, let~\\bar{\\mathcal {S}} be the set ofinterval-schedules for intervals~I_{x} with~x\\le \\bar{\\Gamma },and let~\\delta >0.", "A \\emph {\\delta -discretized algorithm map }is afunction~f:\\bar{\\times \\bar{\\mathcal {S}}\\rightarrow [0,1] suchthatf(C,S)=k\\cdot \\delta with some~k\\in \\mathbb {N}_{0} for all~C\\in \\bar{ and~S\\in \\bar{\\mathcal {S}}, and\\sum _{S\\in \\bar{\\mathcal {S}}}f(C,S)=1 for all ~C\\in \\bar{.", "}By restricting to \\delta -discretizedalgorithm maps we do not lose too much in the competitive ratio.", "}\\begin{lem}There is a value~\\delta >0 such that for any (randomized) algorithmmap~f there is a~\\delta -discretized randomized algorithm map~gwith~\\rho _{g}\\le \\rho _{f}\\left(1+\\varepsilon \\right).\\end{lem}\\begin{proof}[Proof idea.", "]Let~f be a randomized algorithm map andlet~\\delta >0 such that 1/\\delta \\in \\mathbb {N}.", "We define anew~\\delta -discretized algorithm map~g.", "For eachconfiguration~C we define the values~g(C,S) suchthat~\\left\\lfloor f(C,S)/\\delta \\right\\rfloor \\cdot \\delta \\le g(C,S)\\le \\left\\lceil f(C,S)/\\delta \\right\\rceil \\cdot \\delta and~\\sum _{S\\in \\mathcal {S}}g(C,S)=1.", "To seethat~\\rho _{g}\\le \\left(1+\\varepsilon \\right)\\rho _{f}, consider an instance~I and apossible schedule~S(I) for I.", "There is a probability~pthat~f outputs~S(I).", "We show that the schedules which have largeprobability~p dominate~\\mathbb {E}\\left[\\,f(I)\\,\\right]/\\textsf {\\textsc {Opt}}(I).", "We show furtherthat if p is sufficiently large, the probability that~gproduces~S(I) is in~[p/(1+\\varepsilon ),p(1+\\varepsilon )), which implies theLemma.\\end{proof}}Like in the deterministic case, we can now show that at 1+\\varepsilon lossit suffices to restrict to \\emph {simplified \\delta -discretizedalgorithm maps} which treat equivalent configurations equivalently,similar to Lemma~\\ref {lem:equal-confs} (replacing~\\Gamma by~\\bar{\\Gamma } in Definition~\\ref {def:irrelevant_jobs} of theirrelevant jobs).As there are only constantly many of these maps, we enumerate all ofthem, test each map for its competitive ratio, and select the best ofthem.", "}\\begin{thm} We obtain randomized competitive-ratio approximation schemes for \\textup {Pm}|\\,r_j,(pmtn)\\,|\\sum w_jC_j,\\textup {Qm}|\\,r_j, pmtn\\,|\\sum w_jC_j, and\\textup {Rm}|\\,r_j, pmtn\\,|\\sum w_jC_j\\, and for \\textup {Qm}|\\,r_j\\,|\\sum w_jC_j with a bounded range of speeds for any fixed m\\in \\mathbb {N}.\\end{thm}$" ], [ "General Min-Sum Objectives and Makespan", "In this section we briefly argue how the techniques presented above for minimizing $\\sum _{j}w_{j}C_{j}$ can be used for constructing online algorithm schemes for other scheduling problems with jobs arriving online over time, namely for minimizing $\\sum _{j\\in J}w_jf(C_{j})$ , with $f(x)=k\\cdot x^{\\alpha }$ and constant $\\alpha \\ge 1, k>0$ , and the makespan.", "Since monomial functions $f$ have the property that $f((1+\\varepsilon ) C_j)\\le (1+O(\\varepsilon )) f(C_j)$ , the arguments in previous sections apply almost directly to the generalized min-sum objective.", "In each step of simplification and abstraction, we have an increased loss in the performance guarantee, but it is covered by the $O(\\varepsilon )$ -term.", "Consider now the makespan objective.", "The simplifications within intervals of Section  are based on arguing on completion times of individual jobs, and clearly hold also for the last job.", "Thus, they directly apply to makespan minimization.", "We simplify the definition of irrelevant history by omitting the partition of the instance into parts and we define a job $j$ to be irrelevant at time $R_{x}$ if $r_{j}\\le R_{x-s}$ .", "Based on this definition, we define equivalence classes of configurations (ignoring weights and previous interval-schedules) and again restrict to algorithm maps $f$ with $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations (Lemma 3.6).", "Lemmas 3.7–3.9 then hold accordingly and yield a competitive-ratio approximation scheme.", "Finally, the adjustments of Sections  and  can be made accordingly.", "Without the partition of the instance into parts, this even becomes easier in the non-preemptive setting.", "Thus, we can state the following result.", "Theorem 6.1 For any $m \\in \\mathbb {N}$ there are deterministic and randomized competitive-ratio approximation schemes for preemptive and non-preemptive scheduling, on $m$ identical, related (with bounded speed ratio when non-preemptive), and unrelated machines (only preemptive) for the objectives of minimizing $C_{\\max }$ and minimizing $\\sum _{j\\in J}w_j f(C_j)$ , with $f(x)=k\\cdot x^{\\alpha }$ and constant $\\alpha \\ge 1,k>0$ ." ], [ "Conclusions", "We introduce the concept of competitive-ratio approximation schemes that compute online algorithms with a competitive ratio arbitrarily close to the best possible competitive ratio.", "We provide such schemes for various problem variants of scheduling jobs online to minimize the weighted sum of completion times, arbitrary monomial cost functions, and the makespan.", "The techniques derived in this paper provide a new and interesting view on the behavior of online algorithms.", "We believe that they contribute to the understanding of such algorithms and possibly open a new line of research in which they yield even further insights.", "In particular, it seems promising that our methods could also be applied to other online problems than scheduling jobs arriving online over time." ], [ "Sum of weighted completion times.", "The offline variants of nearly all problems under consideration are NP-hard.", "This is true already for the special case of a single machine [27], [28].", "Two restricted variants can be solved optimally in polynomial time.", "Smith's Rule solves the problem $\\textup {1}|\\,\\,|\\sum w_jC_j$ to optimality by scheduling jobs in non-increasing order of weight-to-processing-time ratios [45].", "Furthermore, scheduling by shortest remaining processing times yields an optimal schedule for $\\textup {1}|\\,r_j,pmtn\\,|\\sum w_jC_j$  [38].", "However, for the other settings polynomial-time approximation schemes have been developed [1], even when the number of machines is part of the input.", "The online setting has been a highly active field of research in the past fifteen years.", "A whole sequence of papers appeared introducing new algorithms, new relaxations and analytical techniques that decreased the gaps between lower and upper bounds on the optimal competitive ratio [18], [39], [20], [43], [42], [2], [6], [24], [10], [17], [35], [34], [9], [40], [30], [31], [46], [36], [5], [41], [14].", "We do not intend to give a detailed history of developments; instead, we refer the reader to overviews, e.g., in [34], [10].", "Table REF summarizes the current state-of-the-art on best known lower and upper bounds on the optimal competitive ratios.", "Interestingly, despite the considerable effort, optimal competitive ratios are known only for $\\textup {1}|\\,r_j, pmtn\\,|\\sum C_j$  [38] and for non-preemptive single-machine scheduling [2], [46], [24], [6].", "In all other scheduling settings remain unsatisfactory, even quite significant gaps.", "Table: Lower and upper bounds on the competitive ratio for deterministic and randomized online algorithms.", "[1]For $m=1,2,3,4,5, \\dots 100$ the lower bound is $LB=2, 1.520, 1.414, 1.373, 1.364, \\dots 1.312$ ." ], [ "More general min-sum (completion time) objectives.", "Recently, there has been an increasing interest in studying generalized cost functions.", "So far, this research has focussed on offline problems.", "The most general case is when each job may have its individual non-decreasing cost function $f_j$ .", "For scheduling on a single machine with release dates and preemption, $1|r_j,pmtn|\\sum f_j$ , Bansal and Pruhs [4] gave a randomized $\\mathcal {O}(\\log \\log (nP))$ -approximation, where $P=\\max _{j\\in J}p_j$ .", "In the case that all jobs have identical release dates, the approximation factor reduces to 16.", "Cheung and Shmoys [8] improved this latter result and gave a deterministic $(2+\\varepsilon )$ -approximation.", "This result applies also on a machine of varying speed.", "The more restricted problem with a global cost function $1|r_j,pmtn|\\sum w_jf(C_j)$ has been studied by Epstein et al.", "[13] in the context of universal solutions.", "They gave an algorithm that produces for any job instance one scheduling solution that is a $(4+\\varepsilon )$ -approximation for any cost function and even under unreliable machine behavior.", "Höhn and Jacobs [23] studied the same problem without release dates.", "They analyzed the performance of Smith's Rule [45] and gave tight approximation guarantees for all convex and all concave functions $f$ ." ], [ "Makespan.", "The online makespan minimization problem has been extensively studied in a different online paradigm where jobs arrive one by one (see [15], [37] and references therein).", "Our model, in which jobs arrive online over time, is much less studied.", "In the identical parallel machine environment, Chen and Vestjens [7] give nearly tight bounds on the optimal competitive ratio, $1.347 \\le \\rho ^* \\le 3/2$ , using a natural online variant of the well-known largest processing time first algorithm.", "In the offline setting, polynomial time approximation schemes are known for identical [21] and uniform machines [22].", "For unrelated machines, the problem is NP-hard to approximate with a better ratio than $3/2$ and a 2-approximation is known [29].", "If the number of machines is bounded by a constant there is a PTAS [29]." ], [ "Proofs of Section ", "First, we will show that the number of distinct processing times of large jobs in each interval can be upper-bounded by a constant.", "To achieve this, we partition the jobs of an instance into large and small jobs.", "With respect to a release date $R_{x}$ we say that a job $j$ with $r_{j}=R_{x}$ is large if $p_{j}\\ge \\varepsilon ^{2}I_{x}=\\varepsilon ^{3}R_{x}$ and small otherwise.", "Abusing notation, we refer to $|I_{x}|$ also by $I_{x}$ .", "Note that $I_x=\\varepsilon \\cdot (1+\\varepsilon )^x$ .", "Lemma B.1 The number of distinct processing times of jobs in each set $L_x$ is bounded by $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ .", "For any $j\\in L_{x}$ the processing time $p_{j}$ is a power of $1+\\varepsilon $ , say $p_{j}=(1+\\varepsilon )^{y}$ .", "Hence, we have that $\\varepsilon ^{3}\\left(1+\\varepsilon \\right)^{x}<\\left(1+\\varepsilon \\right)^{y}\\le \\frac{1}{\\varepsilon }\\left(1+\\varepsilon \\right)^{x}$ .", "The number of integers $y$ which satisfy the above inequalities is bounded by $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ , which yields the constant claimed in the lemma.", "Furthermore, we can bound the number of large jobs of each job size which are released at the same time.", "Lemma B.2 Without loss, we can restrict to instances with $|L_{x}|\\le (m/\\varepsilon ^{2}+m)4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ for each set $L_{x}$ .", "Let $L_{x,p}\\subseteq L_{x}$ denote the set of jobs in $L_{x}$ with processing time $p$ .", "By an exchange argument, one can restrict to schedules such that at each point in time at most $m$ jobs in $L_{x,p}$ are partially (i.e., to some extent but not completely) processed.", "Since $p_{j}\\ge \\varepsilon ^{2}I_{x}$ for each job $j\\in L_{x}$ , at most $m/\\varepsilon ^{2}+m$ jobs in $L_{x,p}$ are touched within $I_{x}$ .", "By an exchange argument we can assume that they are the $m/\\varepsilon ^{2}+m$ jobs with the largest weight in $L_{x,p}$ .", "Hence, the release date of all other jobs in $L_{x,p}$ can be moved to $R_{x+1}$ without any cost.", "Since due to Lemma REF there are at most $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ distinct processing times $p$ of large jobs in $L_{x}$ , the claim follows.", "We now just need to take care of the small jobs.", "Denote by $w_{j}/p_{j}$ the Smith's ratio of a job $j$ .", "An ordering where the jobs are ordered non-increasingly by their Smith's ratios is an ordering according to Smith's rule.", "The next lemma shows that scheduling the small jobs according to Smith's Rule is almost optimal and small jobs do not even need to be preempted or to cross intervals.", "For a set of jobs $J$ we define $p(J):=\\sum _{j\\in J}p_{j}$ .", "Lemma B.3 At $1+\\varepsilon $ loss we can restrict to schedules such that for each interval $I_{x}$ the small jobs scheduled within this interval are chosen by Smith's Rule from the set $\\bigcup _{x^{\\prime }\\le x}S_{x^{\\prime }}$ , no small job is preempted, any small job finishes in the same interval where it started and $p(S_{\\!x})\\le m\\cdot I_{x}$ for each interval $I_{x}$ .", "By an exchange argument one can show that it is optimal to schedule the small jobs by Smith's Rule if they can be arbitrarily divided into smaller jobs (where the weight is divided proportional to the processing time of the smaller jobs).", "Start with such a schedule and stretch time once.", "The gained free space is enough to finish all small jobs which are partially scheduled in each interval.", "For the last claim of the lemma, note that the total processing time in each interval $I_{x}$ is $mI_{x}$ .", "Order the small jobs non-increasingly by their Smith's Ratios and pick them until the total processing time of picked jobs just does not exceed $mI_{x}$ .", "The release date of all other jobs in $S_{x}$ can be safely moved to $R_{x+1}$ since due to our modifications we would not schedule them in $I_{x}$ anyway.", "Lemma REF (restated) There is a constant $s$ such that at $1+O(\\varepsilon )$ loss we can restrict to schedules such that for each interval $I_{x}$ there is a subinterval of $I_{x+s-1}$ which is large enough to process all jobs released at $R_x$ and during which only jobs in $R_x$ are executed.", "We call this subinterval the safety net of interval $I_{x}$ .", "We can assume that each job released at $R_x$ finishes before time $R_{x+s}$ .", "By Lemmas REF and $\\ref {lem:number-large-jobs}$ we bound $p(S_{x})+p(L_{x})$ by $p(S_{x})+p(L_{x}) & \\le & m\\cdot I_{x}+(m/\\varepsilon ^{2}+m) \\cdot \\left(4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }\\right) \\cdot \\frac{1}{\\varepsilon }\\left(1+\\varepsilon \\right)^{x}\\\\& \\le & m\\cdot \\left(1+\\varepsilon \\right)^{x}\\left(\\varepsilon +\\frac{8}{\\varepsilon ^{3}}\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }\\right)\\\\& = & \\varepsilon \\cdot I_{x+s-1}$ for a suitable constant $s$ , depending on $\\varepsilon $ and $m$ .", "Stretching time once, we gain enough free space at the end of each interval $I_{x+s-1}$ to establish the safety net for each job set $p(S_{x})+p(L_{x})$ .", "Lemma B.4 There is a constant $d$ such that we can at $1+O(\\varepsilon )$ loss restrict to instances such that $p_{j}>\\frac{\\varepsilon }{2d}\\cdot I_{x}$ for each job $j \\in S_x \\cup L_x$ .", "We call a job $j$ tiny if $p_j \\le \\frac{\\varepsilon }{2d}\\cdot I_{x}$ .", "Let $T_{x}=\\lbrace j_{1},j_{2},...,j_{|T_{x}|}\\rbrace $ denote all tiny jobs released at $R_{x}$ .", "W.l.o.g.", "assume that they are ordered non-increasingly by their Smith's Ratios $w_{j}/p_{j}$ .", "Let $\\ell $ be the largest integer such that $\\sum _{i=1}^{\\ell }p_{i}\\le \\frac{\\varepsilon }{d}\\cdot I_{x}$ .", "We define the pack $P_{x}^{1}:=\\lbrace j_{1},...,j_{\\ell }\\rbrace $ .", "We denote by $\\sum _{i=1}^{\\ell }p_{i}$ the processing time of pack $P_{x}^{1}$ and by $\\sum _{i=1}^{\\ell }w_{i}$ its weight.", "We continue iteratively until we assigned all tiny jobs to packs.", "By definition of the processing time of tiny jobs, the processing time of all but possibly the last pack released at time $R_{x}$ is in the interval $[\\frac{\\varepsilon }{2d}\\cdot I_{x},\\frac{\\varepsilon }{d}\\cdot I_{x}]$ .", "Using timestretching, we can show that at $1+O(\\varepsilon )$ loss all tiny jobs of the same pack are scheduled in the same interval on the same machine.", "Here we use that in any schedule obeying Smith's Rule and using the safety net (see Lemma REF ) in each interval there is at most one partially but unfinished pack from each of at most $s$ previous release dates.", "Hence, we can treat the packs as single jobs whose processing time and weight matches the respective values of the packs.", "Also, at $1+\\varepsilon $ loss we can ensure that also the very last pack has a processing time in $[\\frac{\\varepsilon }{2d}\\cdot I_{x},\\frac{\\varepsilon }{d}\\cdot I_{x}]$ .", "Finally, at $1+O(\\varepsilon )$ loss we can ensure that the processing times and weights of the new jobs (which replace the packs) are powers of $1+\\varepsilon $ .", "Lemma B.5 Assume that there is a constant $d$ such that $p_{j}>\\frac{\\varepsilon }{2d}\\cdot I_{x}$ for each job $j \\in S_x$ .", "Then at $1+O(\\varepsilon )$ loss, the number of distinct processing times of jobs each set $S_x$ is upper-bounded by $(\\log _{1+\\varepsilon }\\varepsilon \\cdot 2d)$ .", "From the previous lemmas, we have $\\frac{e^2}{2d}\\cdot (1+\\varepsilon )^x<(1+\\varepsilon )^y<\\varepsilon ^3(1+\\varepsilon )^x.$ The number of integers $y$ satisfying these inequalities is upper-bounded by the claimed constant.", "Lemma REF now follows from the lemmas REF and REF .", "Lemma REF follows from lemmas REF , REF and REF .", "Next, we prove Lemma REF : [Proof of Lemma REF ] The claim about the number of partially processed jobs of each type can be assumed without any loss.", "For the extent of processing, note that due to Lemmas REF , REF , and REF there is a constant $c$ such that at each time $R_{x}$ the total processing time of unfinished large jobs is bounded by $c\\cdot R_{x}$ .", "We stretch time once.", "The gained space is sufficient to schedule $p_{j}\\cdot \\mu $ processing units of each unfinished large job $j$ (for an appropriately chosen universal constant $\\mu $ ).", "This allows us to enforce the claim.", "The claim about the non-preemptive behavior of small jobs follows from Lemma REF .", "[Proof of Lemma REF ] In any schedule the jobs in $\\cup _{i=0}^{p-1}Q_{k+i}$ contribute at least $\\sum _{i=0}^{p-1}rw(Q_{k+i})$ towards the objective.", "If we move all jobs in $Q_{k+p}$ to their safety nets, they contribute at most $\\sum _{j\\in Q_{k+p}}r_{j}\\left(1+\\varepsilon \\right)^{s}\\cdot w_{j} & = & \\left(1+\\varepsilon \\right)^{s}\\cdot rw(Q_{k+p})\\\\& \\le & \\varepsilon \\cdot \\sum _{i=0}^{p-1}rw(Q_{k+i})\\\\& \\le & \\varepsilon \\cdot OPT$ to the objective.", "[Proof of Lemma REF ] We modify a given online algorithm such that each part is treated as a separate instance.", "To bound the cost in the competitive ratio, we show that $\\frac{\\textsf {\\textsc {A}}(I)}{\\textsf {\\textsc {Opt}}(I)}\\le \\max _{i}\\lbrace \\frac{\\textsf {\\textsc {A}}(P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\rbrace \\cdot (1+O(\\varepsilon )).$ By the above lemmas, there is a $(1+O(\\varepsilon ))$ -approximative (offline) solution in which at the end of each part $P_{\\!i}$ each job has either completed or has been moved to its safety net.", "Denote this solution by $\\textsf {\\textsc {Opt}}^{\\prime }(I)$ and by $\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})$ its respective part for each part $P_{\\!i}$ .", "Note that $\\textsf {\\textsc {Opt}}^{\\prime }(I)=\\sum _{i}\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})$ .", "Then, $\\frac{\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i}))}{\\textsf {\\textsc {Opt}}(I)} \\le \\frac{\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i}))}{\\sum _{i=1}^{k}\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})}\\cdot (1+O(\\varepsilon )) \\le \\max _{i=1,...,k}\\lbrace \\frac{\\textsf {\\textsc {A}}(P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\rbrace \\cdot (1+O(\\varepsilon )).$ [Proof of Lemma REF ] We show that $\\left(1+\\varepsilon \\right)^{s}\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\varepsilon \\cdot \\sum _{i=p-K}^{p}rw(Q_{i})$ for a sufficiently large value $K$ .", "This will then be the claimed constant.", "Let $\\delta ^{\\prime }:=\\frac{\\varepsilon }{\\left(1+\\varepsilon \\right)^{s}}$ .", "By assumption, we have that $rw(Q_{i+1})>\\delta ^{\\prime }\\cdot \\sum _{\\ell =1}^{i}rw(Q_{\\ell })$ for each $i$ .", "This implies that $\\frac{rw(Q_{i+1})}{\\sum _{\\ell =1}^{i+1}rw(Q_{\\ell })}>\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}$ for each $i$ .", "Hence, $\\frac{\\sum _{\\ell =1}^{i}rw(Q_{\\ell })}{rw(Q_{i+1})+\\sum _{\\ell =1}^{i}rw(Q_{\\ell })}\\le 1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}<1$ for each $i$ and hence, $\\sum _{\\ell =1}^{i}rw(Q_{\\ell })\\le (1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }})\\sum _{\\ell =1}^{i+1}rw(Q_{\\ell }).$ In other words, if we remove $Q_{i+1}$ from $\\cup _{\\ell =1}^{i+1}Q_{\\ell }$ then the total release weight of the set decreases by a factor of at least $1-\\delta ^{\\prime }/(1+\\delta ^{\\prime })<1$ .", "For any $K$ this implies that $\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}\\sum _{\\ell =1}^{p}rw(Q_{\\ell })$ and hence $\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\frac{1}{1-\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}}\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}\\sum _{i=p-K}^{p}rw(Q_{i}).$ By choosing $K$ sufficiently large, the claim follows.", "[Proof of Lemma REF ] We partition $\\mathrm {Ir}_{x}(J)$ into two groups: $\\mathrm {Ir}_{x}^{\\mathrm {old}}(J):=\\lbrace j\\in \\mathrm {Ir}_{x}(J)|r_{j}<R_{x-\\Gamma }\\rbrace $ and $\\mathrm {Ir}_{x}^{\\mathrm {new}}(J):=\\lbrace j\\in \\mathrm {Ir}_{x}(J)|r_{j}\\ge R_{x-\\Gamma }\\rbrace $ .", "Lemma REF implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))\\le \\varepsilon \\cdot rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)\\cup \\mathrm {Rel}_{x}(J))$ (recall that the former value is an upper bound on the total weighted completion time of the jobs in $\\mathrm {Ir}_{x}^{\\mathrm {old}}(J)$ ).", "For every job $j\\in \\mathrm {Ir}_{x}^{\\mathrm {new}}(J)$ there must be a job $j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)$ such that $w_{j}<\\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}$ .", "We say that such a job $j^{\\prime }$ dominates $j$ .", "At most $\\Delta $ jobs are released at the beginning of each interval and hence $|\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)|\\le \\Delta \\Gamma $ .", "In particular, if $\\mathrm {dom}(j^{\\prime })$ denotes all jobs in $\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)$ which are dominated by $j^{\\prime }$ then $\\sum _{j\\in \\mathrm {dom}(j^{\\prime })}w_{j}r_{j}\\le \\Delta \\Gamma \\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}r_{j^{\\prime }}\\cdot (1+\\varepsilon )^{\\Gamma }$ This implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)) & \\le & (1+\\varepsilon )^{s}\\sum _{j\\in \\mathrm {Ir}_{x}^{\\mathrm {new}}(J)}w_{j}r_{j}\\\\& \\le & \\sum _{j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)}\\Delta \\Gamma \\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}r_{j^{\\prime }}\\cdot (1+\\varepsilon )^{\\Gamma +s}\\\\& \\le & \\varepsilon \\cdot \\sum _{j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)}w_{j^{\\prime }}r_{j^{\\prime }}\\\\& = & \\varepsilon \\cdot (rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))+rw(\\mathrm {Rel}_{x}(J)))$ Together with Inequality REF this implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}(J)) & = & (1+\\varepsilon )^{s}(rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)+rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J)))\\\\& \\le & \\left(\\varepsilon \\cdot (rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))+rw(\\mathrm {Rel}_{x}(J))\\right)+\\left(\\varepsilon \\cdot rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)\\cup \\mathrm {Rel}_{x}(J))\\right)\\\\& \\le & 2\\varepsilon \\cdot rw(\\mathrm {Rel}_{x}(J))+\\varepsilon (rw(\\mathrm {Ir}_{x}(J))$ and the latter inequality implies that $\\sum _{j\\in \\mathrm {Ir}_{x}(J)}C_{j}w_{j} & \\le & (1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}(J))\\\\& \\le & 2\\varepsilon \\frac{(1+\\varepsilon )^{s}}{(1+\\varepsilon )^{s}-\\varepsilon }rw(\\mathrm {Rel}_{x}(J))\\\\& \\le & 3\\varepsilon \\cdot rw(\\mathrm {Rel}_{x}(J))$" ], [ "Proofs of Section ", "Lemma C.1 In the non-preemptive setting, at $1+O(\\varepsilon )$ loss we can ensure that at the end of each interval $I_{x}$ , there are at most $m$ large jobs from each type which are partially (i.e., neither fully nor not at all) processed, and for each partially but not completely processed large job $j$ there is a value $k_{x,j}$ such that $j$ is processed for at least $k_{x,j}\\cdot p_{j}\\cdot \\mu $ time units in $I_{x}$ , we calculate the objective with adjusted completion times $\\bar{C}_{j}=R_{c(j)}$ for some value $c(j)$ for each job $j$ such that $\\sum _{x<c(j)}k_{x,j}\\cdot p_{j}\\cdot \\mu \\ge p_{j}$ .", "Note that the first property holds for any non-preemptive schedule and is listed here only for the sake of clarity.", "The other two properties can be shown similiarly as in the proof of Lemma REF .", "[Proof of Lemma REF ] Assume that we have an online algorithm $\\textsf {\\textsc {A}}$ with competitive factor $\\rho _{\\textsf {\\textsc {A}}}$ on instances in which for every $i$ the first job $\\textnormal {first}(i)$ released in part $P_{\\!i}$ satisfies $\\sum _{\\ell =1}^{i-1} rw(P_{\\!\\ell }) \\le w_{\\!\\textnormal {first}(i)}\\varepsilon /\\left(1+\\varepsilon \\right)^{s}$  (i.e., $\\textnormal {first}(i)$ dominates all previously released parts).", "Based on $\\textsf {\\textsc {A}}$ we construct a new algorithm $\\textsf {\\textsc {A}}^{\\prime }$ for arbitrary instances with competitive ratio at most $\\left(1+\\varepsilon \\right)\\rho _{\\textsf {\\textsc {A}}}$ : When a new part $P_{\\!i}$ begins, we scale the weights of all jobs in $P_{\\!i}$ such that $\\sum _{\\ell =1}^{i-1}rw(P_{\\!\\ell }) \\le w^{\\prime }_{\\!\\textnormal {first}(i)}\\varepsilon /\\left(1+\\varepsilon \\right)^{s}$ , where the values $w^{\\prime }_{\\!j}$ denote the adjusted weights.", "Denote by $\\bar{I}(i)$ the resulting instance up to (and including) part $P_{\\!i}$ .", "We schedule the resulting instance using $\\textsf {\\textsc {A}}$ .", "We take the computed schedule for each part $P_{\\!i}$ and use it for the jobs with their original weight, obtaining a new algorithm $\\textsf {\\textsc {A}}^{\\prime }$ .", "The following calculations shows that this procedure costs only a factor $1+\\varepsilon $ .", "To this end, we proof that for any instance $I$ it holds that $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I)}{\\textsf {\\textsc {Opt}}(I)} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}(\\bar{I}(i))}{\\textsf {\\textsc {Opt}}(\\bar{I}(i))}\\cdot (1+O(\\varepsilon ))\\le (1+O(\\varepsilon ))\\rho _{\\textsf {\\textsc {A}}}.$ For each $P_{\\!i}$ we define $\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})$ to be the amount that the jobs in $P_{\\!i}$ contribute in $\\textsf {\\textsc {A}}^{\\prime }(I)$ .", "Similarly, we define $\\textsf {\\textsc {Opt}}(I|P_{\\!i})$ .", "We have that $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I)}{\\textsf {\\textsc {Opt}}(I)} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(I|P_{\\!i})} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}.$ We claim that for each $i$ holds $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\le (1+O(\\varepsilon ))\\cdot \\frac{\\textsf {\\textsc {A}}(\\bar{I}(i))}{\\textsf {\\textsc {Opt}}(\\bar{I}(i))}$ .", "For each part $P_{\\!i}$ let $v_{i}$ denote the scale factor of the weight of each job in $\\bar{I}(i)$ in comparison to its original weight.", "The optimum for the instance $\\bar{I}(i)$ can be bounded by $\\textsf {\\textsc {Opt}}(\\bar{I}(i)) & \\le & \\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i} +\\left(1+\\varepsilon \\right)^{s}\\sum _{\\ell =1}^{i-1}rw(P_{\\!\\ell }) \\le \\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i}+\\varepsilon \\cdot r_{\\textnormal {first}(i)} \\cdot w^{\\prime }_{\\textnormal {first}(i)}\\le \\left(1+\\varepsilon \\right)\\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i}.$ Furthermore holds by construction $\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})\\cdot v_{i}\\le \\textsf {\\textsc {A}}(\\bar{I}^{i})$ .", "Thus, $\\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}(\\bar{I}^{i})}{\\textsf {\\textsc {Opt}}(\\bar{I}^{i})}\\cdot (1+O(\\varepsilon ))$ .", "[Proof of Lemma REF ] Given a schedule on related machines with speed values $s_{1},...,s_{\\max }$ , we stretch time twice.", "Thus, we gain in each interval $I_{x}$ free space of size $\\varepsilon I_{x}$ on the fastest machine.", "For each machine whose speed is at most $\\frac{\\varepsilon }{m}s_{\\max }$ , we take its schedule of the interval $I_{x}$ and simulate it on the fastest machine.", "Thus, those slow machines are not needed and can be removed.", "The remaining machines have speeds in $[\\frac{\\varepsilon }{m}\\,s_{\\max },s_{\\max }]$ .", "Assuming the slowest machines has unit speed gives the desired bound.", "[Proof of Lemma REF ] Consider a schedule for an instance which does not satisfy the property.", "We stretch time twice and thus we gain a free space of $\\varepsilon I_{x}$ in each interval $I_{x}$ .", "Consider some $I_{x}$ and a job $j$ which is scheduled in $I_{x}$ .", "Let $i$ be a fastest machine for $j$ .", "We remove the processing volume of $j$ scheduled in $I_x$ on slow machines $i^{\\prime }$ with $p_{i^{\\prime }j} >\\frac{m}{\\varepsilon }\\, p_{ij}$ and schedule it on $i$ in the gained free space.", "This way, we obtain a feasible schedule even if a job never runs on a machine where it is slow.", "Thus, we can set $p_{i^{\\prime }j}=\\infty $ if there is a fast machine $i$ such that $p_{ij} \\le \\frac{\\varepsilon }{m}p_{i^{\\prime }j}$ .", "Proofs of Section  [Proof of Lemma REF ] Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances $I$ such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )$ .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "In instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We need to bound the increase in the total expected cost caused by moving all jobs in periods in $\\mathcal {Q}$ to their safety nets.", "This increase is bounded by $\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}\\sum _{j\\in Q}\\left(1+\\varepsilon \\right)^{s}r_{j}\\cdot w_{j}\\right] & \\le &\\left(1+\\varepsilon \\right)^{s}\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}rw(Q)\\right]\\\\& \\le & \\left(1+\\varepsilon \\right)^{s}\\frac{1}{M}\\sum _{Q\\in I}rw(Q)\\\\& \\le & \\varepsilon \\cdot rw(I)\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)\\,.$ Thus, the total expected cost of the computed schedule is $\\mathbb {E}\\left[\\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i})\\right] & \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(P_{\\!i})\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & (\\rho _{\\textsf {\\textsc {A}}}+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I).$ Thus, at $1+\\varepsilon $ loss in the competitive ratio we can restrict to parts $I_i$ which span a constant number of periods.", "[Proof of Lemma ] Consider an instance $I$ .", "Let $\\delta >0$ and $k\\in \\mathbb {N}$ be values to be determined later with the property that $1/\\delta \\in \\mathbb {N}$ .", "For each configuration $C$ and each interval-schedule $S$ we define a value $g(C,S)$ such that $\\left\\lfloor \\frac{f(C,S)}{\\delta }\\right\\rfloor \\cdot \\delta \\le g(C,S)\\le \\left\\lceil \\frac{f(C,S)}{\\delta }\\right\\rceil \\cdot \\delta $ and $\\sum _{S\\in \\mathcal {S}}g(C,S)=1$ .", "Now we want to bound $\\rho _{g}$ .", "The idea is that for determining the ratio $\\mathbb {E}\\left[\\,g(I)\\,\\right]/\\textsf {\\textsc {Opt}}(I)$ it suffices to consider schedules $S(I)$ which are computed with sufficiently large probability.", "We show that also $f$ computes them with almost the same probability.", "Let $S(I)$ denote a schedule for the entire instance $I$ .", "We denote by $P_{f}(S(I))$ and $P_{g}(S(I))$ the probability that $f$ and $g$ compute the schedule $S(I)$ when given the instance $I$ .", "Assume that $P_{f}(S(I))\\ge k\\cdot \\delta $ .", "Denote by $C_{0},...,C_{\\bar{\\Gamma }-1}$ the configurations that algorithms are faced with when computing $S(I)$ , i.e., each configuration $C_{x}$ contains the jobs which are released but unfinished at the beginning of interval $I_{x}$ in $S(I)$ and as history the schedule $S(I)$ restricted to the intervals $I_{0},...,I_{x-1}$ .", "Denote by $S_{\\!x}$ the schedule of $S(I)$ in the interval $I_{x}$ .", "Hence, $P_{f}(S(I))=\\prod _{x=0}^{\\bar{\\Gamma }-1}f(C_{x},S_{\\!x})$ .", "Note that from $P_{f}(S(I))\\ge k\\cdot \\delta $ follows that $f(C_{x},S_{\\!x})\\ge k\\cdot \\delta $ for all $x$ .", "For these schedules, $P_{g}(S(I))$ is not much larger since $P_{g}(S(I)) & = & \\prod _{x=0}^{\\bar{\\Gamma }-1}g(C_{x},S_{\\!x})\\le \\prod _{x=0}^{\\bar{\\Gamma }-1}\\frac{k+1}{k}f(C_{x},S_{\\!x})\\le \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}P_{f}(S(I)).$ Let $\\mathcal {S}(I)$ denote the set of all schedules for $I$ .", "We partition $\\mathcal {S}(I)$ into schedule sets $\\mathcal {S}_{H}^{g}(I):=\\lbrace S(I)|P_{g}(S(I))\\ge k\\cdot \\delta \\rbrace $ and $\\mathcal {S}_{L}^{g}(I):=\\mathcal {S}(I)\\setminus \\mathcal {S}_{H}(I)$ .", "We estimate the expected value of a schedule computed by algorithm map $g$ on $I$ by $\\mathbb {E}\\left[\\,g(I)\\,\\right] & = & \\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{g}(S(I))\\cdot S(I) + \\sum _{S(I)\\in \\mathcal {S}_{L}^{g}(I)}P_{g}(S(I)) \\cdot S(I)\\\\& \\le &\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\cdot S(I) + |\\mathcal {S}(I)| \\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s} \\cdot rw(I)\\\\& \\le &\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot S(I) + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I)\\\\& \\le & \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\mathbb {E}\\left[\\,f(I)\\,\\right] + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I).$ We choose $k$ such that $\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\le 1+\\varepsilon /2$ and $\\delta $ such that $|\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\le \\varepsilon /2$ for all instances $I$  (note here that $|\\mathcal {S}(I)|$ can be upper bounded by a value independent of $I$ since our instances contain only constantly many jobs).", "This yields $\\frac{\\mathbb {E}\\left[\\,g(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon /2)\\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} + \\varepsilon /2\\cdot \\frac{rw(I)}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon ) \\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)}\\,,$ and we conclude that $\\rho _{g}\\le \\left(1+\\varepsilon \\right)\\rho _{f}$ .", "Competitive-Ratio Approximation Schemes for Minimizing $C_{\\max }$ (cf.", "Section ) Consider the objective of minimizing the makespan.", "The simplifications within intervals of Section  are based on arguing on completion times of individual jobs, and clearly hold also for the last job.", "Thus, they directly apply to minimizing the makespan.", "Furthermore, we simplify the definition of irrelevant history in Section  by omitting the partition of the instance into parts.", "We observe that when then last job is released at time $R_{x^{*}}$ then all jobs $j$ with $r_{j}\\le R_{x^{*}-s}$ are irrelevant for the objective: such a job $j$ finishes at time $R_{x^{*}}$ the latest in any schedule (due to the safety net) and $OPT\\ge R_{x^{*}}$ .", "Therefore, we define a job $j$ to be irrelevant at time $R_{x}$ if $r_{j}\\le R_{x-s}$ .", "We keep Definition REF for the equivalence relation of schedules as it is except for the notion of job weights which are not important for the makespan.", "Based on the above definition for relevant jobs we define equivalence classes of configurations.", "With this definition, we can still restrict to algorithm maps $f$ with $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations $C,C^{\\prime }$ (Lemma REF ).", "Lemmas REF to REF then hold accordingly.", "Finally, note that since we do not split the instance into parts, we do not need (an adjusted version of) Lemma REF in the non-preemptive case.", "Theorem E.1 For any $m\\in \\mathbb {N}$ we obtain competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds.", "For constructing randomized online algorithm schemes for minimizing the makespan, similarly to Lemma REF we can show that we can restrict our attention to instances which span only a constant number of periods.", "Lemma E.2 For randomized algorithms for minimizing the makespan, at $1+O(\\varepsilon )$ loss we can restrict to instances in which all jobs are released in at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ consecutive periods.", "Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm for minimizing the makespan over time with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+O(\\varepsilon ))$ .", "Our reasoning is similar to the proof of Lemma REF .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "Given an instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We present each part separately to $\\textsf {\\textsc {A}}$ .", "We bound the competitive ratio $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ of the resulting algorithm.", "Let $R_{x^{*}}:=\\max _{j \\in I}r_{j}$ .", "Moving the jobs from periods $\\mathcal {Q}$ has an effect on the optimal makespan only if $o$ is chosen such that at least one job $j$ with $r_{j}>R_{x^{*}-s}$ is moved.", "There are at most two offsets $o$ such that this happens.", "In that case, the algorithm still achieves a competitive ratio of at most $\\left(1+\\varepsilon \\right)^{s}$ .", "In all other cases, $\\textsf {\\textsc {A}}^{\\prime }$ achieves a competitive ratio of at most $\\rho _{\\textsf {\\textsc {A}}}$ .", "Thus, we can bound $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ by $\\rho _{\\textsf {\\textsc {A}}^{\\prime }} & \\le & \\frac{2}{M}\\left(1+\\varepsilon \\right)^{s}+\\frac{M-2}{M}\\rho _{\\textsf {\\textsc {A}}}\\le 2\\varepsilon +\\rho _{\\textsf {\\textsc {A}}}\\le (1+O(\\varepsilon ))\\rho _{\\textsf {\\textsc {A}}}.$ We can prove similarly as in Lemma  that any randomized algorithm map $f$ can be well approximated by a discretized randomized algorithm map $g$ .", "Hence, we obtain the following theorem.", "Theorem E.3 For any $m\\in \\mathbb {N}$ we obtain randomized competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds." ], [ "General Min-Sum Objectives and Makespan", "In this section we briefly argue how the techniques presented above for minimizing $\\sum _{j}w_{j}C_{j}$ can be used for constructing online algorithm schemes for other scheduling problems with jobs arriving online over time, namely for minimizing $\\sum _{j\\in J}w_jf(C_{j})$ , with $f(x)=k\\cdot x^{\\alpha }$ and constant $\\alpha \\ge 1, k>0$ , and the makespan.", "Since monomial functions $f$ have the property that $f((1+\\varepsilon ) C_j)\\le (1+O(\\varepsilon )) f(C_j)$ , the arguments in previous sections apply almost directly to the generalized min-sum objective.", "In each step of simplification and abstraction, we have an increased loss in the performance guarantee, but it is covered by the $O(\\varepsilon )$ -term.", "Consider now the makespan objective.", "The simplifications within intervals of Section  are based on arguing on completion times of individual jobs, and clearly hold also for the last job.", "Thus, they directly apply to makespan minimization.", "We simplify the definition of irrelevant history by omitting the partition of the instance into parts and we define a job $j$ to be irrelevant at time $R_{x}$ if $r_{j}\\le R_{x-s}$ .", "Based on this definition, we define equivalence classes of configurations (ignoring weights and previous interval-schedules) and again restrict to algorithm maps $f$ with $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations (Lemma 3.6).", "Lemmas 3.7–3.9 then hold accordingly and yield a competitive-ratio approximation scheme.", "Finally, the adjustments of Sections  and  can be made accordingly.", "Without the partition of the instance into parts, this even becomes easier in the non-preemptive setting.", "Thus, we can state the following result.", "Theorem 6.1 For any $m \\in \\mathbb {N}$ there are deterministic and randomized competitive-ratio approximation schemes for preemptive and non-preemptive scheduling, on $m$ identical, related (with bounded speed ratio when non-preemptive), and unrelated machines (only preemptive) for the objectives of minimizing $C_{\\max }$ and minimizing $\\sum _{j\\in J}w_j f(C_j)$ , with $f(x)=k\\cdot x^{\\alpha }$ and constant $\\alpha \\ge 1,k>0$ ." ], [ "Conclusions", "We introduce the concept of competitive-ratio approximation schemes that compute online algorithms with a competitive ratio arbitrarily close to the best possible competitive ratio.", "We provide such schemes for various problem variants of scheduling jobs online to minimize the weighted sum of completion times, arbitrary monomial cost functions, and the makespan.", "The techniques derived in this paper provide a new and interesting view on the behavior of online algorithms.", "We believe that they contribute to the understanding of such algorithms and possibly open a new line of research in which they yield even further insights.", "In particular, it seems promising that our methods could also be applied to other online problems than scheduling jobs arriving online over time." ], [ "Sum of weighted completion times.", "The offline variants of nearly all problems under consideration are NP-hard.", "This is true already for the special case of a single machine [27], [28].", "Two restricted variants can be solved optimally in polynomial time.", "Smith's Rule solves the problem $\\textup {1}|\\,\\,|\\sum w_jC_j$ to optimality by scheduling jobs in non-increasing order of weight-to-processing-time ratios [45].", "Furthermore, scheduling by shortest remaining processing times yields an optimal schedule for $\\textup {1}|\\,r_j,pmtn\\,|\\sum w_jC_j$  [38].", "However, for the other settings polynomial-time approximation schemes have been developed [1], even when the number of machines is part of the input.", "The online setting has been a highly active field of research in the past fifteen years.", "A whole sequence of papers appeared introducing new algorithms, new relaxations and analytical techniques that decreased the gaps between lower and upper bounds on the optimal competitive ratio [18], [39], [20], [43], [42], [2], [6], [24], [10], [17], [35], [34], [9], [40], [30], [31], [46], [36], [5], [41], [14].", "We do not intend to give a detailed history of developments; instead, we refer the reader to overviews, e.g., in [34], [10].", "Table REF summarizes the current state-of-the-art on best known lower and upper bounds on the optimal competitive ratios.", "Interestingly, despite the considerable effort, optimal competitive ratios are known only for $\\textup {1}|\\,r_j, pmtn\\,|\\sum C_j$  [38] and for non-preemptive single-machine scheduling [2], [46], [24], [6].", "In all other scheduling settings remain unsatisfactory, even quite significant gaps.", "Table: Lower and upper bounds on the competitive ratio for deterministic and randomized online algorithms.", "[1]For $m=1,2,3,4,5, \\dots 100$ the lower bound is $LB=2, 1.520, 1.414, 1.373, 1.364, \\dots 1.312$ ." ], [ "More general min-sum (completion time) objectives.", "Recently, there has been an increasing interest in studying generalized cost functions.", "So far, this research has focussed on offline problems.", "The most general case is when each job may have its individual non-decreasing cost function $f_j$ .", "For scheduling on a single machine with release dates and preemption, $1|r_j,pmtn|\\sum f_j$ , Bansal and Pruhs [4] gave a randomized $\\mathcal {O}(\\log \\log (nP))$ -approximation, where $P=\\max _{j\\in J}p_j$ .", "In the case that all jobs have identical release dates, the approximation factor reduces to 16.", "Cheung and Shmoys [8] improved this latter result and gave a deterministic $(2+\\varepsilon )$ -approximation.", "This result applies also on a machine of varying speed.", "The more restricted problem with a global cost function $1|r_j,pmtn|\\sum w_jf(C_j)$ has been studied by Epstein et al.", "[13] in the context of universal solutions.", "They gave an algorithm that produces for any job instance one scheduling solution that is a $(4+\\varepsilon )$ -approximation for any cost function and even under unreliable machine behavior.", "Höhn and Jacobs [23] studied the same problem without release dates.", "They analyzed the performance of Smith's Rule [45] and gave tight approximation guarantees for all convex and all concave functions $f$ ." ], [ "Makespan.", "The online makespan minimization problem has been extensively studied in a different online paradigm where jobs arrive one by one (see [15], [37] and references therein).", "Our model, in which jobs arrive online over time, is much less studied.", "In the identical parallel machine environment, Chen and Vestjens [7] give nearly tight bounds on the optimal competitive ratio, $1.347 \\le \\rho ^* \\le 3/2$ , using a natural online variant of the well-known largest processing time first algorithm.", "In the offline setting, polynomial time approximation schemes are known for identical [21] and uniform machines [22].", "For unrelated machines, the problem is NP-hard to approximate with a better ratio than $3/2$ and a 2-approximation is known [29].", "If the number of machines is bounded by a constant there is a PTAS [29]." ], [ "Proofs of Section ", "First, we will show that the number of distinct processing times of large jobs in each interval can be upper-bounded by a constant.", "To achieve this, we partition the jobs of an instance into large and small jobs.", "With respect to a release date $R_{x}$ we say that a job $j$ with $r_{j}=R_{x}$ is large if $p_{j}\\ge \\varepsilon ^{2}I_{x}=\\varepsilon ^{3}R_{x}$ and small otherwise.", "Abusing notation, we refer to $|I_{x}|$ also by $I_{x}$ .", "Note that $I_x=\\varepsilon \\cdot (1+\\varepsilon )^x$ .", "Lemma B.1 The number of distinct processing times of jobs in each set $L_x$ is bounded by $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ .", "For any $j\\in L_{x}$ the processing time $p_{j}$ is a power of $1+\\varepsilon $ , say $p_{j}=(1+\\varepsilon )^{y}$ .", "Hence, we have that $\\varepsilon ^{3}\\left(1+\\varepsilon \\right)^{x}<\\left(1+\\varepsilon \\right)^{y}\\le \\frac{1}{\\varepsilon }\\left(1+\\varepsilon \\right)^{x}$ .", "The number of integers $y$ which satisfy the above inequalities is bounded by $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ , which yields the constant claimed in the lemma.", "Furthermore, we can bound the number of large jobs of each job size which are released at the same time.", "Lemma B.2 Without loss, we can restrict to instances with $|L_{x}|\\le (m/\\varepsilon ^{2}+m)4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ for each set $L_{x}$ .", "Let $L_{x,p}\\subseteq L_{x}$ denote the set of jobs in $L_{x}$ with processing time $p$ .", "By an exchange argument, one can restrict to schedules such that at each point in time at most $m$ jobs in $L_{x,p}$ are partially (i.e., to some extent but not completely) processed.", "Since $p_{j}\\ge \\varepsilon ^{2}I_{x}$ for each job $j\\in L_{x}$ , at most $m/\\varepsilon ^{2}+m$ jobs in $L_{x,p}$ are touched within $I_{x}$ .", "By an exchange argument we can assume that they are the $m/\\varepsilon ^{2}+m$ jobs with the largest weight in $L_{x,p}$ .", "Hence, the release date of all other jobs in $L_{x,p}$ can be moved to $R_{x+1}$ without any cost.", "Since due to Lemma REF there are at most $4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }$ distinct processing times $p$ of large jobs in $L_{x}$ , the claim follows.", "We now just need to take care of the small jobs.", "Denote by $w_{j}/p_{j}$ the Smith's ratio of a job $j$ .", "An ordering where the jobs are ordered non-increasingly by their Smith's ratios is an ordering according to Smith's rule.", "The next lemma shows that scheduling the small jobs according to Smith's Rule is almost optimal and small jobs do not even need to be preempted or to cross intervals.", "For a set of jobs $J$ we define $p(J):=\\sum _{j\\in J}p_{j}$ .", "Lemma B.3 At $1+\\varepsilon $ loss we can restrict to schedules such that for each interval $I_{x}$ the small jobs scheduled within this interval are chosen by Smith's Rule from the set $\\bigcup _{x^{\\prime }\\le x}S_{x^{\\prime }}$ , no small job is preempted, any small job finishes in the same interval where it started and $p(S_{\\!x})\\le m\\cdot I_{x}$ for each interval $I_{x}$ .", "By an exchange argument one can show that it is optimal to schedule the small jobs by Smith's Rule if they can be arbitrarily divided into smaller jobs (where the weight is divided proportional to the processing time of the smaller jobs).", "Start with such a schedule and stretch time once.", "The gained free space is enough to finish all small jobs which are partially scheduled in each interval.", "For the last claim of the lemma, note that the total processing time in each interval $I_{x}$ is $mI_{x}$ .", "Order the small jobs non-increasingly by their Smith's Ratios and pick them until the total processing time of picked jobs just does not exceed $mI_{x}$ .", "The release date of all other jobs in $S_{x}$ can be safely moved to $R_{x+1}$ since due to our modifications we would not schedule them in $I_{x}$ anyway.", "Lemma REF (restated) There is a constant $s$ such that at $1+O(\\varepsilon )$ loss we can restrict to schedules such that for each interval $I_{x}$ there is a subinterval of $I_{x+s-1}$ which is large enough to process all jobs released at $R_x$ and during which only jobs in $R_x$ are executed.", "We call this subinterval the safety net of interval $I_{x}$ .", "We can assume that each job released at $R_x$ finishes before time $R_{x+s}$ .", "By Lemmas REF and $\\ref {lem:number-large-jobs}$ we bound $p(S_{x})+p(L_{x})$ by $p(S_{x})+p(L_{x}) & \\le & m\\cdot I_{x}+(m/\\varepsilon ^{2}+m) \\cdot \\left(4\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }\\right) \\cdot \\frac{1}{\\varepsilon }\\left(1+\\varepsilon \\right)^{x}\\\\& \\le & m\\cdot \\left(1+\\varepsilon \\right)^{x}\\left(\\varepsilon +\\frac{8}{\\varepsilon ^{3}}\\log _{1+\\varepsilon }\\frac{1}{\\varepsilon }\\right)\\\\& = & \\varepsilon \\cdot I_{x+s-1}$ for a suitable constant $s$ , depending on $\\varepsilon $ and $m$ .", "Stretching time once, we gain enough free space at the end of each interval $I_{x+s-1}$ to establish the safety net for each job set $p(S_{x})+p(L_{x})$ .", "Lemma B.4 There is a constant $d$ such that we can at $1+O(\\varepsilon )$ loss restrict to instances such that $p_{j}>\\frac{\\varepsilon }{2d}\\cdot I_{x}$ for each job $j \\in S_x \\cup L_x$ .", "We call a job $j$ tiny if $p_j \\le \\frac{\\varepsilon }{2d}\\cdot I_{x}$ .", "Let $T_{x}=\\lbrace j_{1},j_{2},...,j_{|T_{x}|}\\rbrace $ denote all tiny jobs released at $R_{x}$ .", "W.l.o.g.", "assume that they are ordered non-increasingly by their Smith's Ratios $w_{j}/p_{j}$ .", "Let $\\ell $ be the largest integer such that $\\sum _{i=1}^{\\ell }p_{i}\\le \\frac{\\varepsilon }{d}\\cdot I_{x}$ .", "We define the pack $P_{x}^{1}:=\\lbrace j_{1},...,j_{\\ell }\\rbrace $ .", "We denote by $\\sum _{i=1}^{\\ell }p_{i}$ the processing time of pack $P_{x}^{1}$ and by $\\sum _{i=1}^{\\ell }w_{i}$ its weight.", "We continue iteratively until we assigned all tiny jobs to packs.", "By definition of the processing time of tiny jobs, the processing time of all but possibly the last pack released at time $R_{x}$ is in the interval $[\\frac{\\varepsilon }{2d}\\cdot I_{x},\\frac{\\varepsilon }{d}\\cdot I_{x}]$ .", "Using timestretching, we can show that at $1+O(\\varepsilon )$ loss all tiny jobs of the same pack are scheduled in the same interval on the same machine.", "Here we use that in any schedule obeying Smith's Rule and using the safety net (see Lemma REF ) in each interval there is at most one partially but unfinished pack from each of at most $s$ previous release dates.", "Hence, we can treat the packs as single jobs whose processing time and weight matches the respective values of the packs.", "Also, at $1+\\varepsilon $ loss we can ensure that also the very last pack has a processing time in $[\\frac{\\varepsilon }{2d}\\cdot I_{x},\\frac{\\varepsilon }{d}\\cdot I_{x}]$ .", "Finally, at $1+O(\\varepsilon )$ loss we can ensure that the processing times and weights of the new jobs (which replace the packs) are powers of $1+\\varepsilon $ .", "Lemma B.5 Assume that there is a constant $d$ such that $p_{j}>\\frac{\\varepsilon }{2d}\\cdot I_{x}$ for each job $j \\in S_x$ .", "Then at $1+O(\\varepsilon )$ loss, the number of distinct processing times of jobs each set $S_x$ is upper-bounded by $(\\log _{1+\\varepsilon }\\varepsilon \\cdot 2d)$ .", "From the previous lemmas, we have $\\frac{e^2}{2d}\\cdot (1+\\varepsilon )^x<(1+\\varepsilon )^y<\\varepsilon ^3(1+\\varepsilon )^x.$ The number of integers $y$ satisfying these inequalities is upper-bounded by the claimed constant.", "Lemma REF now follows from the lemmas REF and REF .", "Lemma REF follows from lemmas REF , REF and REF .", "Next, we prove Lemma REF : [Proof of Lemma REF ] The claim about the number of partially processed jobs of each type can be assumed without any loss.", "For the extent of processing, note that due to Lemmas REF , REF , and REF there is a constant $c$ such that at each time $R_{x}$ the total processing time of unfinished large jobs is bounded by $c\\cdot R_{x}$ .", "We stretch time once.", "The gained space is sufficient to schedule $p_{j}\\cdot \\mu $ processing units of each unfinished large job $j$ (for an appropriately chosen universal constant $\\mu $ ).", "This allows us to enforce the claim.", "The claim about the non-preemptive behavior of small jobs follows from Lemma REF .", "[Proof of Lemma REF ] In any schedule the jobs in $\\cup _{i=0}^{p-1}Q_{k+i}$ contribute at least $\\sum _{i=0}^{p-1}rw(Q_{k+i})$ towards the objective.", "If we move all jobs in $Q_{k+p}$ to their safety nets, they contribute at most $\\sum _{j\\in Q_{k+p}}r_{j}\\left(1+\\varepsilon \\right)^{s}\\cdot w_{j} & = & \\left(1+\\varepsilon \\right)^{s}\\cdot rw(Q_{k+p})\\\\& \\le & \\varepsilon \\cdot \\sum _{i=0}^{p-1}rw(Q_{k+i})\\\\& \\le & \\varepsilon \\cdot OPT$ to the objective.", "[Proof of Lemma REF ] We modify a given online algorithm such that each part is treated as a separate instance.", "To bound the cost in the competitive ratio, we show that $\\frac{\\textsf {\\textsc {A}}(I)}{\\textsf {\\textsc {Opt}}(I)}\\le \\max _{i}\\lbrace \\frac{\\textsf {\\textsc {A}}(P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\rbrace \\cdot (1+O(\\varepsilon )).$ By the above lemmas, there is a $(1+O(\\varepsilon ))$ -approximative (offline) solution in which at the end of each part $P_{\\!i}$ each job has either completed or has been moved to its safety net.", "Denote this solution by $\\textsf {\\textsc {Opt}}^{\\prime }(I)$ and by $\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})$ its respective part for each part $P_{\\!i}$ .", "Note that $\\textsf {\\textsc {Opt}}^{\\prime }(I)=\\sum _{i}\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})$ .", "Then, $\\frac{\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i}))}{\\textsf {\\textsc {Opt}}(I)} \\le \\frac{\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i}))}{\\sum _{i=1}^{k}\\textsf {\\textsc {Opt}}^{\\prime }(P_{\\!i})}\\cdot (1+O(\\varepsilon )) \\le \\max _{i=1,...,k}\\lbrace \\frac{\\textsf {\\textsc {A}}(P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\rbrace \\cdot (1+O(\\varepsilon )).$ [Proof of Lemma REF ] We show that $\\left(1+\\varepsilon \\right)^{s}\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\varepsilon \\cdot \\sum _{i=p-K}^{p}rw(Q_{i})$ for a sufficiently large value $K$ .", "This will then be the claimed constant.", "Let $\\delta ^{\\prime }:=\\frac{\\varepsilon }{\\left(1+\\varepsilon \\right)^{s}}$ .", "By assumption, we have that $rw(Q_{i+1})>\\delta ^{\\prime }\\cdot \\sum _{\\ell =1}^{i}rw(Q_{\\ell })$ for each $i$ .", "This implies that $\\frac{rw(Q_{i+1})}{\\sum _{\\ell =1}^{i+1}rw(Q_{\\ell })}>\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}$ for each $i$ .", "Hence, $\\frac{\\sum _{\\ell =1}^{i}rw(Q_{\\ell })}{rw(Q_{i+1})+\\sum _{\\ell =1}^{i}rw(Q_{\\ell })}\\le 1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}<1$ for each $i$ and hence, $\\sum _{\\ell =1}^{i}rw(Q_{\\ell })\\le (1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }})\\sum _{\\ell =1}^{i+1}rw(Q_{\\ell }).$ In other words, if we remove $Q_{i+1}$ from $\\cup _{\\ell =1}^{i+1}Q_{\\ell }$ then the total release weight of the set decreases by a factor of at least $1-\\delta ^{\\prime }/(1+\\delta ^{\\prime })<1$ .", "For any $K$ this implies that $\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}\\sum _{\\ell =1}^{p}rw(Q_{\\ell })$ and hence $\\sum _{i=1}^{p-K-1}rw(Q_{i})<\\frac{1}{1-\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}}\\left(1-\\frac{\\delta ^{\\prime }}{1+\\delta ^{\\prime }}\\right)^{K}\\sum _{i=p-K}^{p}rw(Q_{i}).$ By choosing $K$ sufficiently large, the claim follows.", "[Proof of Lemma REF ] We partition $\\mathrm {Ir}_{x}(J)$ into two groups: $\\mathrm {Ir}_{x}^{\\mathrm {old}}(J):=\\lbrace j\\in \\mathrm {Ir}_{x}(J)|r_{j}<R_{x-\\Gamma }\\rbrace $ and $\\mathrm {Ir}_{x}^{\\mathrm {new}}(J):=\\lbrace j\\in \\mathrm {Ir}_{x}(J)|r_{j}\\ge R_{x-\\Gamma }\\rbrace $ .", "Lemma REF implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))\\le \\varepsilon \\cdot rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)\\cup \\mathrm {Rel}_{x}(J))$ (recall that the former value is an upper bound on the total weighted completion time of the jobs in $\\mathrm {Ir}_{x}^{\\mathrm {old}}(J)$ ).", "For every job $j\\in \\mathrm {Ir}_{x}^{\\mathrm {new}}(J)$ there must be a job $j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)$ such that $w_{j}<\\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}$ .", "We say that such a job $j^{\\prime }$ dominates $j$ .", "At most $\\Delta $ jobs are released at the beginning of each interval and hence $|\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)|\\le \\Delta \\Gamma $ .", "In particular, if $\\mathrm {dom}(j^{\\prime })$ denotes all jobs in $\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)$ which are dominated by $j^{\\prime }$ then $\\sum _{j\\in \\mathrm {dom}(j^{\\prime })}w_{j}r_{j}\\le \\Delta \\Gamma \\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}r_{j^{\\prime }}\\cdot (1+\\varepsilon )^{\\Gamma }$ This implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)) & \\le & (1+\\varepsilon )^{s}\\sum _{j\\in \\mathrm {Ir}_{x}^{\\mathrm {new}}(J)}w_{j}r_{j}\\\\& \\le & \\sum _{j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)}\\Delta \\Gamma \\frac{\\varepsilon }{\\Delta \\Gamma \\cdot (1+\\varepsilon )^{\\Gamma +s}}w_{j^{\\prime }}r_{j^{\\prime }}\\cdot (1+\\varepsilon )^{\\Gamma +s}\\\\& \\le & \\varepsilon \\cdot \\sum _{j^{\\prime }\\in \\mathrm {Ir}_{x}^{\\mathrm {old}}(J)\\cup \\mathrm {Rel}_{x}(J)}w_{j^{\\prime }}r_{j^{\\prime }}\\\\& = & \\varepsilon \\cdot (rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))+rw(\\mathrm {Rel}_{x}(J)))$ Together with Inequality REF this implies that $(1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}(J)) & = & (1+\\varepsilon )^{s}(rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)+rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J)))\\\\& \\le & \\left(\\varepsilon \\cdot (rw(\\mathrm {Ir}_{x}^{\\mathrm {old}}(J))+rw(\\mathrm {Rel}_{x}(J))\\right)+\\left(\\varepsilon \\cdot rw(\\mathrm {Ir}_{x}^{\\mathrm {new}}(J)\\cup \\mathrm {Rel}_{x}(J))\\right)\\\\& \\le & 2\\varepsilon \\cdot rw(\\mathrm {Rel}_{x}(J))+\\varepsilon (rw(\\mathrm {Ir}_{x}(J))$ and the latter inequality implies that $\\sum _{j\\in \\mathrm {Ir}_{x}(J)}C_{j}w_{j} & \\le & (1+\\varepsilon )^{s}rw(\\mathrm {Ir}_{x}(J))\\\\& \\le & 2\\varepsilon \\frac{(1+\\varepsilon )^{s}}{(1+\\varepsilon )^{s}-\\varepsilon }rw(\\mathrm {Rel}_{x}(J))\\\\& \\le & 3\\varepsilon \\cdot rw(\\mathrm {Rel}_{x}(J))$" ], [ "Proofs of Section ", "Lemma C.1 In the non-preemptive setting, at $1+O(\\varepsilon )$ loss we can ensure that at the end of each interval $I_{x}$ , there are at most $m$ large jobs from each type which are partially (i.e., neither fully nor not at all) processed, and for each partially but not completely processed large job $j$ there is a value $k_{x,j}$ such that $j$ is processed for at least $k_{x,j}\\cdot p_{j}\\cdot \\mu $ time units in $I_{x}$ , we calculate the objective with adjusted completion times $\\bar{C}_{j}=R_{c(j)}$ for some value $c(j)$ for each job $j$ such that $\\sum _{x<c(j)}k_{x,j}\\cdot p_{j}\\cdot \\mu \\ge p_{j}$ .", "Note that the first property holds for any non-preemptive schedule and is listed here only for the sake of clarity.", "The other two properties can be shown similiarly as in the proof of Lemma REF .", "[Proof of Lemma REF ] Assume that we have an online algorithm $\\textsf {\\textsc {A}}$ with competitive factor $\\rho _{\\textsf {\\textsc {A}}}$ on instances in which for every $i$ the first job $\\textnormal {first}(i)$ released in part $P_{\\!i}$ satisfies $\\sum _{\\ell =1}^{i-1} rw(P_{\\!\\ell }) \\le w_{\\!\\textnormal {first}(i)}\\varepsilon /\\left(1+\\varepsilon \\right)^{s}$  (i.e., $\\textnormal {first}(i)$ dominates all previously released parts).", "Based on $\\textsf {\\textsc {A}}$ we construct a new algorithm $\\textsf {\\textsc {A}}^{\\prime }$ for arbitrary instances with competitive ratio at most $\\left(1+\\varepsilon \\right)\\rho _{\\textsf {\\textsc {A}}}$ : When a new part $P_{\\!i}$ begins, we scale the weights of all jobs in $P_{\\!i}$ such that $\\sum _{\\ell =1}^{i-1}rw(P_{\\!\\ell }) \\le w^{\\prime }_{\\!\\textnormal {first}(i)}\\varepsilon /\\left(1+\\varepsilon \\right)^{s}$ , where the values $w^{\\prime }_{\\!j}$ denote the adjusted weights.", "Denote by $\\bar{I}(i)$ the resulting instance up to (and including) part $P_{\\!i}$ .", "We schedule the resulting instance using $\\textsf {\\textsc {A}}$ .", "We take the computed schedule for each part $P_{\\!i}$ and use it for the jobs with their original weight, obtaining a new algorithm $\\textsf {\\textsc {A}}^{\\prime }$ .", "The following calculations shows that this procedure costs only a factor $1+\\varepsilon $ .", "To this end, we proof that for any instance $I$ it holds that $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I)}{\\textsf {\\textsc {Opt}}(I)} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}(\\bar{I}(i))}{\\textsf {\\textsc {Opt}}(\\bar{I}(i))}\\cdot (1+O(\\varepsilon ))\\le (1+O(\\varepsilon ))\\rho _{\\textsf {\\textsc {A}}}.$ For each $P_{\\!i}$ we define $\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})$ to be the amount that the jobs in $P_{\\!i}$ contribute in $\\textsf {\\textsc {A}}^{\\prime }(I)$ .", "Similarly, we define $\\textsf {\\textsc {Opt}}(I|P_{\\!i})$ .", "We have that $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I)}{\\textsf {\\textsc {Opt}}(I)} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(I|P_{\\!i})} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}.$ We claim that for each $i$ holds $\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})}\\le (1+O(\\varepsilon ))\\cdot \\frac{\\textsf {\\textsc {A}}(\\bar{I}(i))}{\\textsf {\\textsc {Opt}}(\\bar{I}(i))}$ .", "For each part $P_{\\!i}$ let $v_{i}$ denote the scale factor of the weight of each job in $\\bar{I}(i)$ in comparison to its original weight.", "The optimum for the instance $\\bar{I}(i)$ can be bounded by $\\textsf {\\textsc {Opt}}(\\bar{I}(i)) & \\le & \\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i} +\\left(1+\\varepsilon \\right)^{s}\\sum _{\\ell =1}^{i-1}rw(P_{\\!\\ell }) \\le \\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i}+\\varepsilon \\cdot r_{\\textnormal {first}(i)} \\cdot w^{\\prime }_{\\textnormal {first}(i)}\\le \\left(1+\\varepsilon \\right)\\textsf {\\textsc {Opt}}(P_{\\!i})\\cdot v_{i}.$ Furthermore holds by construction $\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})\\cdot v_{i}\\le \\textsf {\\textsc {A}}(\\bar{I}^{i})$ .", "Thus, $\\max _{i}\\frac{\\textsf {\\textsc {A}}^{\\prime }(I|P_{\\!i})}{\\textsf {\\textsc {Opt}}(P_{\\!i})} \\le \\max _{i}\\frac{\\textsf {\\textsc {A}}(\\bar{I}^{i})}{\\textsf {\\textsc {Opt}}(\\bar{I}^{i})}\\cdot (1+O(\\varepsilon ))$ .", "[Proof of Lemma REF ] Given a schedule on related machines with speed values $s_{1},...,s_{\\max }$ , we stretch time twice.", "Thus, we gain in each interval $I_{x}$ free space of size $\\varepsilon I_{x}$ on the fastest machine.", "For each machine whose speed is at most $\\frac{\\varepsilon }{m}s_{\\max }$ , we take its schedule of the interval $I_{x}$ and simulate it on the fastest machine.", "Thus, those slow machines are not needed and can be removed.", "The remaining machines have speeds in $[\\frac{\\varepsilon }{m}\\,s_{\\max },s_{\\max }]$ .", "Assuming the slowest machines has unit speed gives the desired bound.", "[Proof of Lemma REF ] Consider a schedule for an instance which does not satisfy the property.", "We stretch time twice and thus we gain a free space of $\\varepsilon I_{x}$ in each interval $I_{x}$ .", "Consider some $I_{x}$ and a job $j$ which is scheduled in $I_{x}$ .", "Let $i$ be a fastest machine for $j$ .", "We remove the processing volume of $j$ scheduled in $I_x$ on slow machines $i^{\\prime }$ with $p_{i^{\\prime }j} >\\frac{m}{\\varepsilon }\\, p_{ij}$ and schedule it on $i$ in the gained free space.", "This way, we obtain a feasible schedule even if a job never runs on a machine where it is slow.", "Thus, we can set $p_{i^{\\prime }j}=\\infty $ if there is a fast machine $i$ such that $p_{ij} \\le \\frac{\\varepsilon }{m}p_{i^{\\prime }j}$ .", "Proofs of Section  [Proof of Lemma REF ] Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances $I$ such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )$ .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "In instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We need to bound the increase in the total expected cost caused by moving all jobs in periods in $\\mathcal {Q}$ to their safety nets.", "This increase is bounded by $\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}\\sum _{j\\in Q}\\left(1+\\varepsilon \\right)^{s}r_{j}\\cdot w_{j}\\right] & \\le &\\left(1+\\varepsilon \\right)^{s}\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}rw(Q)\\right]\\\\& \\le & \\left(1+\\varepsilon \\right)^{s}\\frac{1}{M}\\sum _{Q\\in I}rw(Q)\\\\& \\le & \\varepsilon \\cdot rw(I)\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)\\,.$ Thus, the total expected cost of the computed schedule is $\\mathbb {E}\\left[\\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i})\\right] & \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(P_{\\!i})\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & (\\rho _{\\textsf {\\textsc {A}}}+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I).$ Thus, at $1+\\varepsilon $ loss in the competitive ratio we can restrict to parts $I_i$ which span a constant number of periods.", "[Proof of Lemma ] Consider an instance $I$ .", "Let $\\delta >0$ and $k\\in \\mathbb {N}$ be values to be determined later with the property that $1/\\delta \\in \\mathbb {N}$ .", "For each configuration $C$ and each interval-schedule $S$ we define a value $g(C,S)$ such that $\\left\\lfloor \\frac{f(C,S)}{\\delta }\\right\\rfloor \\cdot \\delta \\le g(C,S)\\le \\left\\lceil \\frac{f(C,S)}{\\delta }\\right\\rceil \\cdot \\delta $ and $\\sum _{S\\in \\mathcal {S}}g(C,S)=1$ .", "Now we want to bound $\\rho _{g}$ .", "The idea is that for determining the ratio $\\mathbb {E}\\left[\\,g(I)\\,\\right]/\\textsf {\\textsc {Opt}}(I)$ it suffices to consider schedules $S(I)$ which are computed with sufficiently large probability.", "We show that also $f$ computes them with almost the same probability.", "Let $S(I)$ denote a schedule for the entire instance $I$ .", "We denote by $P_{f}(S(I))$ and $P_{g}(S(I))$ the probability that $f$ and $g$ compute the schedule $S(I)$ when given the instance $I$ .", "Assume that $P_{f}(S(I))\\ge k\\cdot \\delta $ .", "Denote by $C_{0},...,C_{\\bar{\\Gamma }-1}$ the configurations that algorithms are faced with when computing $S(I)$ , i.e., each configuration $C_{x}$ contains the jobs which are released but unfinished at the beginning of interval $I_{x}$ in $S(I)$ and as history the schedule $S(I)$ restricted to the intervals $I_{0},...,I_{x-1}$ .", "Denote by $S_{\\!x}$ the schedule of $S(I)$ in the interval $I_{x}$ .", "Hence, $P_{f}(S(I))=\\prod _{x=0}^{\\bar{\\Gamma }-1}f(C_{x},S_{\\!x})$ .", "Note that from $P_{f}(S(I))\\ge k\\cdot \\delta $ follows that $f(C_{x},S_{\\!x})\\ge k\\cdot \\delta $ for all $x$ .", "For these schedules, $P_{g}(S(I))$ is not much larger since $P_{g}(S(I)) & = & \\prod _{x=0}^{\\bar{\\Gamma }-1}g(C_{x},S_{\\!x})\\le \\prod _{x=0}^{\\bar{\\Gamma }-1}\\frac{k+1}{k}f(C_{x},S_{\\!x})\\le \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}P_{f}(S(I)).$ Let $\\mathcal {S}(I)$ denote the set of all schedules for $I$ .", "We partition $\\mathcal {S}(I)$ into schedule sets $\\mathcal {S}_{H}^{g}(I):=\\lbrace S(I)|P_{g}(S(I))\\ge k\\cdot \\delta \\rbrace $ and $\\mathcal {S}_{L}^{g}(I):=\\mathcal {S}(I)\\setminus \\mathcal {S}_{H}(I)$ .", "We estimate the expected value of a schedule computed by algorithm map $g$ on $I$ by $\\mathbb {E}\\left[\\,g(I)\\,\\right] & = & \\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{g}(S(I))\\cdot S(I) + \\sum _{S(I)\\in \\mathcal {S}_{L}^{g}(I)}P_{g}(S(I)) \\cdot S(I)\\\\& \\le &\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\cdot S(I) + |\\mathcal {S}(I)| \\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s} \\cdot rw(I)\\\\& \\le &\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot S(I) + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I)\\\\& \\le & \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\mathbb {E}\\left[\\,f(I)\\,\\right] + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I).$ We choose $k$ such that $\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\le 1+\\varepsilon /2$ and $\\delta $ such that $|\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\le \\varepsilon /2$ for all instances $I$  (note here that $|\\mathcal {S}(I)|$ can be upper bounded by a value independent of $I$ since our instances contain only constantly many jobs).", "This yields $\\frac{\\mathbb {E}\\left[\\,g(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon /2)\\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} + \\varepsilon /2\\cdot \\frac{rw(I)}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon ) \\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)}\\,,$ and we conclude that $\\rho _{g}\\le \\left(1+\\varepsilon \\right)\\rho _{f}$ .", "Competitive-Ratio Approximation Schemes for Minimizing $C_{\\max }$ (cf.", "Section ) Consider the objective of minimizing the makespan.", "The simplifications within intervals of Section  are based on arguing on completion times of individual jobs, and clearly hold also for the last job.", "Thus, they directly apply to minimizing the makespan.", "Furthermore, we simplify the definition of irrelevant history in Section  by omitting the partition of the instance into parts.", "We observe that when then last job is released at time $R_{x^{*}}$ then all jobs $j$ with $r_{j}\\le R_{x^{*}-s}$ are irrelevant for the objective: such a job $j$ finishes at time $R_{x^{*}}$ the latest in any schedule (due to the safety net) and $OPT\\ge R_{x^{*}}$ .", "Therefore, we define a job $j$ to be irrelevant at time $R_{x}$ if $r_{j}\\le R_{x-s}$ .", "We keep Definition REF for the equivalence relation of schedules as it is except for the notion of job weights which are not important for the makespan.", "Based on the above definition for relevant jobs we define equivalence classes of configurations.", "With this definition, we can still restrict to algorithm maps $f$ with $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations $C,C^{\\prime }$ (Lemma REF ).", "Lemmas REF to REF then hold accordingly.", "Finally, note that since we do not split the instance into parts, we do not need (an adjusted version of) Lemma REF in the non-preemptive case.", "Theorem E.1 For any $m\\in \\mathbb {N}$ we obtain competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds.", "For constructing randomized online algorithm schemes for minimizing the makespan, similarly to Lemma REF we can show that we can restrict our attention to instances which span only a constant number of periods.", "Lemma E.2 For randomized algorithms for minimizing the makespan, at $1+O(\\varepsilon )$ loss we can restrict to instances in which all jobs are released in at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ consecutive periods.", "Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm for minimizing the makespan over time with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+O(\\varepsilon ))$ .", "Our reasoning is similar to the proof of Lemma REF .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "Given an instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We present each part separately to $\\textsf {\\textsc {A}}$ .", "We bound the competitive ratio $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ of the resulting algorithm.", "Let $R_{x^{*}}:=\\max _{j \\in I}r_{j}$ .", "Moving the jobs from periods $\\mathcal {Q}$ has an effect on the optimal makespan only if $o$ is chosen such that at least one job $j$ with $r_{j}>R_{x^{*}-s}$ is moved.", "There are at most two offsets $o$ such that this happens.", "In that case, the algorithm still achieves a competitive ratio of at most $\\left(1+\\varepsilon \\right)^{s}$ .", "In all other cases, $\\textsf {\\textsc {A}}^{\\prime }$ achieves a competitive ratio of at most $\\rho _{\\textsf {\\textsc {A}}}$ .", "Thus, we can bound $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ by $\\rho _{\\textsf {\\textsc {A}}^{\\prime }} & \\le & \\frac{2}{M}\\left(1+\\varepsilon \\right)^{s}+\\frac{M-2}{M}\\rho _{\\textsf {\\textsc {A}}}\\le 2\\varepsilon +\\rho _{\\textsf {\\textsc {A}}}\\le (1+O(\\varepsilon ))\\rho _{\\textsf {\\textsc {A}}}.$ We can prove similarly as in Lemma  that any randomized algorithm map $f$ can be well approximated by a discretized randomized algorithm map $g$ .", "Hence, we obtain the following theorem.", "Theorem E.3 For any $m\\in \\mathbb {N}$ we obtain randomized competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds." ], [ "Proofs of Section ", "[Proof of Lemma REF ] Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances $I$ such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )$ .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "In instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We need to bound the increase in the total expected cost caused by moving all jobs in periods in $\\mathcal {Q}$ to their safety nets.", "This increase is bounded by $\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}\\sum _{j\\in Q}\\left(1+\\varepsilon \\right)^{s}r_{j}\\cdot w_{j}\\right] & \\le &\\left(1+\\varepsilon \\right)^{s}\\mathbb {E}\\left[\\sum _{Q\\in \\mathcal {Q}}rw(Q)\\right]\\\\& \\le & \\left(1+\\varepsilon \\right)^{s}\\frac{1}{M}\\sum _{Q\\in I}rw(Q)\\\\& \\le & \\varepsilon \\cdot rw(I)\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)\\,.$ Thus, the total expected cost of the computed schedule is $\\mathbb {E}\\left[\\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\textsf {\\textsc {A}}(P_{\\!i})\\right] & \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\sum _{i=1}^{k}\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(P_{\\!i})\\\\& \\le & \\varepsilon \\cdot \\textsf {\\textsc {Opt}}(I)+\\rho _{\\textsf {\\textsc {A}}}\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & (\\rho _{\\textsf {\\textsc {A}}}+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I)\\\\& \\le & \\rho _{\\textsf {\\textsc {A}}}(1+\\varepsilon )\\cdot \\textsf {\\textsc {Opt}}(I).$ Thus, at $1+\\varepsilon $ loss in the competitive ratio we can restrict to parts $I_i$ which span a constant number of periods.", "[Proof of Lemma ] Consider an instance $I$ .", "Let $\\delta >0$ and $k\\in \\mathbb {N}$ be values to be determined later with the property that $1/\\delta \\in \\mathbb {N}$ .", "For each configuration $C$ and each interval-schedule $S$ we define a value $g(C,S)$ such that $\\left\\lfloor \\frac{f(C,S)}{\\delta }\\right\\rfloor \\cdot \\delta \\le g(C,S)\\le \\left\\lceil \\frac{f(C,S)}{\\delta }\\right\\rceil \\cdot \\delta $ and $\\sum _{S\\in \\mathcal {S}}g(C,S)=1$ .", "Now we want to bound $\\rho _{g}$ .", "The idea is that for determining the ratio $\\mathbb {E}\\left[\\,g(I)\\,\\right]/\\textsf {\\textsc {Opt}}(I)$ it suffices to consider schedules $S(I)$ which are computed with sufficiently large probability.", "We show that also $f$ computes them with almost the same probability.", "Let $S(I)$ denote a schedule for the entire instance $I$ .", "We denote by $P_{f}(S(I))$ and $P_{g}(S(I))$ the probability that $f$ and $g$ compute the schedule $S(I)$ when given the instance $I$ .", "Assume that $P_{f}(S(I))\\ge k\\cdot \\delta $ .", "Denote by $C_{0},...,C_{\\bar{\\Gamma }-1}$ the configurations that algorithms are faced with when computing $S(I)$ , i.e., each configuration $C_{x}$ contains the jobs which are released but unfinished at the beginning of interval $I_{x}$ in $S(I)$ and as history the schedule $S(I)$ restricted to the intervals $I_{0},...,I_{x-1}$ .", "Denote by $S_{\\!x}$ the schedule of $S(I)$ in the interval $I_{x}$ .", "Hence, $P_{f}(S(I))=\\prod _{x=0}^{\\bar{\\Gamma }-1}f(C_{x},S_{\\!x})$ .", "Note that from $P_{f}(S(I))\\ge k\\cdot \\delta $ follows that $f(C_{x},S_{\\!x})\\ge k\\cdot \\delta $ for all $x$ .", "For these schedules, $P_{g}(S(I))$ is not much larger since $P_{g}(S(I)) & = & \\prod _{x=0}^{\\bar{\\Gamma }-1}g(C_{x},S_{\\!x})\\le \\prod _{x=0}^{\\bar{\\Gamma }-1}\\frac{k+1}{k}f(C_{x},S_{\\!x})\\le \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}P_{f}(S(I)).$ Let $\\mathcal {S}(I)$ denote the set of all schedules for $I$ .", "We partition $\\mathcal {S}(I)$ into schedule sets $\\mathcal {S}_{H}^{g}(I):=\\lbrace S(I)|P_{g}(S(I))\\ge k\\cdot \\delta \\rbrace $ and $\\mathcal {S}_{L}^{g}(I):=\\mathcal {S}(I)\\setminus \\mathcal {S}_{H}(I)$ .", "We estimate the expected value of a schedule computed by algorithm map $g$ on $I$ by $\\mathbb {E}\\left[\\,g(I)\\,\\right] & = & \\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{g}(S(I))\\cdot S(I) + \\sum _{S(I)\\in \\mathcal {S}_{L}^{g}(I)}P_{g}(S(I)) \\cdot S(I)\\\\& \\le &\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\cdot S(I) + |\\mathcal {S}(I)| \\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s} \\cdot rw(I)\\\\& \\le &\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\sum _{S(I)\\in \\mathcal {S}_{H}^{g}(I)}P_{f}(S(I))\\cdot S(I) + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I)\\\\& \\le & \\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }} \\mathbb {E}\\left[\\,f(I)\\,\\right] + |\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\cdot rw(I).$ We choose $k$ such that $\\left(\\frac{k+1}{k}\\right)^{\\bar{\\Gamma }}\\le 1+\\varepsilon /2$ and $\\delta $ such that $|\\mathcal {S}(I)|\\cdot k\\cdot \\delta \\cdot \\left(1+\\varepsilon \\right)^{s}\\le \\varepsilon /2$ for all instances $I$  (note here that $|\\mathcal {S}(I)|$ can be upper bounded by a value independent of $I$ since our instances contain only constantly many jobs).", "This yields $\\frac{\\mathbb {E}\\left[\\,g(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon /2)\\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)} + \\varepsilon /2\\cdot \\frac{rw(I)}{\\textsf {\\textsc {Opt}}(I)} \\le (1+\\varepsilon ) \\cdot \\frac{\\mathbb {E}\\left[\\,f(I)\\,\\right]}{\\textsf {\\textsc {Opt}}(I)}\\,,$ and we conclude that $\\rho _{g}\\le \\left(1+\\varepsilon \\right)\\rho _{f}$ ." ], [ "Competitive-Ratio Approximation Schemes for Minimizing $C_{\\max }$ (cf. Section ", "Consider the objective of minimizing the makespan.", "The simplifications within intervals of Section  are based on arguing on completion times of individual jobs, and clearly hold also for the last job.", "Thus, they directly apply to minimizing the makespan.", "Furthermore, we simplify the definition of irrelevant history in Section  by omitting the partition of the instance into parts.", "We observe that when then last job is released at time $R_{x^{*}}$ then all jobs $j$ with $r_{j}\\le R_{x^{*}-s}$ are irrelevant for the objective: such a job $j$ finishes at time $R_{x^{*}}$ the latest in any schedule (due to the safety net) and $OPT\\ge R_{x^{*}}$ .", "Therefore, we define a job $j$ to be irrelevant at time $R_{x}$ if $r_{j}\\le R_{x-s}$ .", "We keep Definition REF for the equivalence relation of schedules as it is except for the notion of job weights which are not important for the makespan.", "Based on the above definition for relevant jobs we define equivalence classes of configurations.", "With this definition, we can still restrict to algorithm maps $f$ with $f(C)\\sim f(C^{\\prime })$ for any two equivalent configurations $C,C^{\\prime }$ (Lemma REF ).", "Lemmas REF to REF then hold accordingly.", "Finally, note that since we do not split the instance into parts, we do not need (an adjusted version of) Lemma REF in the non-preemptive case.", "Theorem E.1 For any $m\\in \\mathbb {N}$ we obtain competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds.", "For constructing randomized online algorithm schemes for minimizing the makespan, similarly to Lemma REF we can show that we can restrict our attention to instances which span only a constant number of periods.", "Lemma E.2 For randomized algorithms for minimizing the makespan, at $1+O(\\varepsilon )$ loss we can restrict to instances in which all jobs are released in at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ consecutive periods.", "Let $\\textsf {\\textsc {A}}$ be a randomized online algorithm for minimizing the makespan over time with a competitive ratio of $\\rho _{\\textsf {\\textsc {A}}}$ on instances which span at most $\\left(1+\\varepsilon \\right)^{s}/\\varepsilon $ periods.", "We construct a new randomized online algorithm $\\textsf {\\textsc {A}}^{\\prime }$ which works on arbitrary instances such that $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}\\le \\rho _{\\textsf {\\textsc {A}}}(1+O(\\varepsilon ))$ .", "Our reasoning is similar to the proof of Lemma REF .", "At the beginning of $\\textsf {\\textsc {A}}^{\\prime }$ , we choose an offset $o\\in \\lbrace 0,...,M-1\\rbrace $ uniformly at random with $M:=\\left\\lceil \\left(1+\\varepsilon \\right)^{s}/\\varepsilon \\right\\rceil $ .", "Given an instance $I$ , we move all jobs to their safety net which are released in periods $Q\\in \\mathcal {Q}:=\\lbrace Q_{i}|i\\equiv o\\bmod M\\rbrace $ .", "This splits the instance into parts $P_{\\!0},...,P_{\\!k}$ where each part $P_{\\!\\ell }$ consists of the periods $Q_{o+(\\ell -1)\\cdot M},...,Q_{o+\\ell \\cdot M-1}$ .", "Note that at the end of each part no job remains.", "We present each part separately to $\\textsf {\\textsc {A}}$ .", "We bound the competitive ratio $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ of the resulting algorithm.", "Let $R_{x^{*}}:=\\max _{j \\in I}r_{j}$ .", "Moving the jobs from periods $\\mathcal {Q}$ has an effect on the optimal makespan only if $o$ is chosen such that at least one job $j$ with $r_{j}>R_{x^{*}-s}$ is moved.", "There are at most two offsets $o$ such that this happens.", "In that case, the algorithm still achieves a competitive ratio of at most $\\left(1+\\varepsilon \\right)^{s}$ .", "In all other cases, $\\textsf {\\textsc {A}}^{\\prime }$ achieves a competitive ratio of at most $\\rho _{\\textsf {\\textsc {A}}}$ .", "Thus, we can bound $\\rho _{\\textsf {\\textsc {A}}^{\\prime }}$ by $\\rho _{\\textsf {\\textsc {A}}^{\\prime }} & \\le & \\frac{2}{M}\\left(1+\\varepsilon \\right)^{s}+\\frac{M-2}{M}\\rho _{\\textsf {\\textsc {A}}}\\le 2\\varepsilon +\\rho _{\\textsf {\\textsc {A}}}\\le (1+O(\\varepsilon ))\\rho _{\\textsf {\\textsc {A}}}.$ We can prove similarly as in Lemma  that any randomized algorithm map $f$ can be well approximated by a discretized randomized algorithm map $g$ .", "Hence, we obtain the following theorem.", "Theorem E.3 For any $m\\in \\mathbb {N}$ we obtain randomized competitive-ratio approximation schemes for $\\textup {Pm}|\\,r_{j},(pmtn)\\,|C_{\\max }$ , $\\textup {Qm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and $\\textup {Rm}|\\,r_{j},pmtn\\,|C_{\\max }$ , and for $\\textup {Qm}|\\,r_{j}\\,|C_{\\max }$ assuming a constant range of machine speeds." ] ]

[ [ "Computationally efficient methods for modelling laser wakefield\n acceleration in the blowout regime" ], [ "Abstract Electron self-injection and acceleration until dephasing in the blowout regime is studied for a set of initial conditions typical of recent experiments with 100 terawatt-class lasers.", "Two different approaches to computationally efficient, fully explicit, three-dimensional particle-in-cell modelling are examined.", "First, the Cartesian code VORPAL using a perfect-dispersion electromagnetic solver precisely describes the laser pulse and bubble dynamics, taking advantage of coarser resolution in the propagation direction, with a proportionally larger time step.", "Using third-order splines for macroparticles helps suppress the sampling noise while keeping the usage of computational resources modest.", "The second way to reduce the simulation load is using reduced-geometry codes.", "In our case, the quasi-cylindrical code CALDER-CIRC uses decomposition of fields and currents into a set of poloidal modes, while the macroparticles move in the Cartesian 3D space.", "Cylindrical symmetry of the interaction allows using just two modes, reducing the computational load to roughly that of a planar Cartesian simulation while preserving the 3D nature of the interaction.", "This significant economy of resources allows using fine resolution in the direction of propagation and a small time step, making numerical dispersion vanishingly small, together with a large number of particles per cell, enabling good particle statistics.", "Quantitative agreement of the two simulations indicates that they are free of numerical artefacts.", "Both approaches thus retrieve physically correct evolution of the plasma bubble, recovering the intrinsic connection of electron self-injection to the nonlinear optical evolution of the driver." ], [ "Introduction", "Relativistic Langmuir waves driven by short, intense laser pulses in rarefied plasmas maintain accelerating gradients several orders of magnitude higher than those accessible in conventional metallic structures [71], [18], [9].", "The technical simplicity and compactness of these laser-plasma accelerators (LPAs) is attractive for a broad range of applications, such as nuclear activation and on-site isotope production [38], [61], long-distance probing of defects in shielded structures [60], and testing radiation resistivity of electronic components [24].", "Realisation of compact, inexpensive, bright x- and gamma-ray sources using electron beams from LPAs [64], [63], [69], [34], [3] holds the promise to enable a much wider user community than can be served by existing large-scale facilities.", "These applications are not especially demanding as regards electron beam quality, and in fact sometimes draw benefits from poor beam collimation and a broad energy spectrum [24].", "However, there are also important applications with much tighter beam requirements.", "Such applications include generating coherent x-rays using an external magnetic undulator [20], [66], [65], [14], producing x-rays for the phase contrast imaging [12], [33], and high-brightness, quasi-monochromatic gamma-ray Compton sources [36], [23]; these require electron beams with a multi-kA current, low phase space volume, and energy in the few-gigaelectronvolt (GeV) range.", "Achieving this high level of accelerator performance is a major near-term goal of the LPA community.", "Modern laser systems capable of concentrating up to 10 Joules of energy in a sub-50 femtosecond pulse [81], [13], [35], [12] make it possible to achieve the so-called blowout (or “bubble”) regime, which is desirable due to its technical simplicity and scalability [19], [44].", "In this regime, motion of the electrons in the focus of the laser pulse is highly relativistic.", "The laser ponderomotive force expels plasma electrons from the region of the pulse, while the fully stripped ions remain essentially immobile, creating a column of positive charge in the laser wake.", "The charge separation force attracts bulk plasma electrons to the axis, creating a closed bubble devoid of electrons.", "This co-propagating electron density bubble [62], [50], [58], [19], [43] guides the laser pulse over many Rayleigh lengths [50], [44].", "The bubble readily traps initially quiescent background electrons, accelerating them to hundreds of megaelectronvolts (MeV) over a few mm, creating a collimated electron bunch [58], [27].", "It is in this regime that the first quasi-monoenergetic electrons were produced from laser plasmas in the laboratory [15], [47], [10], and the GeV energy range was approached [37], [32], [21], [13], [35], [4], [42], [40], [55].", "Multi-dimensional particle-in-cell (PIC) simulations have played a key role in understanding the physics of the fully kinetic, strongly relativistic blowout regime.", "The PIC method [25], [2] self-consistently models both electromagnetic fields and charged particles, representing field quantities on a grid and particles in a continuous phase space.", "Given sufficient computing power, electromagnetic PIC codes can simulate the plasma electrons (and ions, if necessary), the laser pulse driving the plasma wake, and the dynamics of electrons injected into the accelerating potential.", "In particular, two- and three-dimensional PIC simulations have been essential in understanding the dynamical nature of the electron self-injection process [80], [53], [79], [85], [26], [30], [27], [28], [29].", "However, to capture precisely the correlation between driver dynamics, electron self-injection, and GeV-scale acceleration in the bubble regime, a simulation must meet a number of challenging requirements.", "Optimisation of a GeV-scale LPA performance, even with the use of massively parallel computation, is a challenging task especially because of the necessary cm-scale laser-plasma interaction length.", "The laser energy is used most effectively if electrons are accelerated until they outrun the bubble and exit the accelerating phase, at which point they will have gained the maximum possible energy in an LPA stage, $E_d\\approx [2.7\\gamma _g^{4/3}P_{\\text{TW}}^{1/3}]{MeV}.$ Acceleration to this dephasing limit occurs over the distance [44] $L_d\\approx 0.6\\lambda _0\\gamma _g^{8/3}P_{\\text{TW}}^{1/6}.$ Here, $P_{\\text{TW}}$ is the laser power in terawatts ($[1]{TW} = [10^{12}]{W}$ ), $\\gamma _g = \\omega _0/\\omega _{\\text{pe}}\\gg 1$ is the Lorentz factor associated with the linear group velocity of the pulse in plasma, $\\omega _0$ is the laser frequency, $\\lambda _0 = 2\\pi c/\\omega _0$ is the laser wavelength, $\\omega _{\\text{pe}} = (n_0e^2/m_e\\epsilon _0)^{1/2}$ is the electron Langmuir frequency, $m_e$ is the electron rest mass, $n_0$ is the background electron density, $e$ is the electron charge, and $\\epsilon _0$ is the permittivity of free space.", "The scalings (REF ) and (REF ) imply that the pulse remains self-guided, namely, that it remains longer than $c\\, \\omega _{\\text{pe}}^{-1}$ [67], [17], and that its power exceeds the critical power for relativistic self-focusing, $[P_{\\text{cr}}=16.2\\gamma _g^2]{GW}$ [68].", "Increasing the electron energy therefore requires reduction of the electron plasma density, increasing both the bubble velocity and size, $L_{\\text{acc}}\\approx 0.9\\lambda _0\\gamma _g^{2/3}P_{\\text{TW}}^{1/6},$ where $L_{\\text{acc}}$ is the length of the accelerating phase of the wakefield (roughly equal to the bubble radius).", "Electron dephasing scales as $L_d\\sim n_0^{-4/3}$ and thus the final energy gain scales as $E_d\\sim n_0^{-2/3}$ .", "For instance, reaching [1]GeV energy with a [200]TW pulse and a wavelength of $\\lambda _0 = [0.8]{m}$ may be achieved in a [0.47]cm length plasma with density $n_0 = [3.5\\times 10^{18}]{cm^{-3}}$ , and doubling that energy would require nearly four times the plasma length and three times lower density, also increasing the bubble size by $\\mathbin {\\sim }$ 40%.", "Simulations of LPA commonly use a moving-widow, where the simulation box propagates with the speed of light colinearly with the laser pulse.", "This optimisation notwithstanding, even the experiments with currently operating [100]TW systems bring forth the task of modelling the pulse propagation in cm-length plasmas, with the size of 3D simulation box on the order of hundred(s) of microns longitudinally and transversely.", "The greatest challenge arises from the great disparity of physical scales between the laser wavelength and plasma length, which is the hallmark of high-energy laser-plasma acceleration.", "The need to resolve the laser wavelength, $\\lambda _0\\sim [1]{m}$ , fixes the grid resolution, and, due to stability conditions [6], also limits the time step to a small fraction of $\\omega _0^{-1}$ .", "Furthermore, the strong localisation of the injection process imposes even stricter limit on grid resolution; the vast majority of injection candidates are concentrated in the inner lining of the bubble (the sheath), and penetrate into the bubble near its rear, where the sheath is longitudinally compressed to a few tens of nanometres [79], [27].", "Resolving this structure, together with ensuring sufficient particle statistics in the sheath, is necessary to avoid excessive sampling noise and eliminate unphysical effects.", "In this situation, extending the plasma length to centimeters and increasing the size of the simulation window to hundreds of microns, while at the same time maintaining sufficient macroparticle statistics, would require solving Maxwell's equations on meshes amounting to billions of grid points, and advancing 1–10 billion macroparticles over millions of time steps.", "Performing such simulations with standard electromagnetic solvers and particle movers requires a national-scale supercomputing facility.", "As a result, an attempt to reproduce the long time scale evolution of the laser and the bubble together with fine details of the electron self-injection dynamics is usually a compromise between affordable simulation load and unavoidable coarseness of the results.", "However, the high precision of modern LPA experiments and high beam quality requirements of the applications are rather unforgiving to these compromises and do not tolerate numerical artefacts [5].", "These considerations make it clear that PIC algorithms must be modified in order to reduce the required computational resources without compromising precision.", "One of the main directions is development of electromagnetic solvers that minimize numerical error while using the lowest possible grid resolution.", "One particular limitation of PIC that requires high longitudinal resolution is that of numerical dispersion.", "In PIC, electromagnetic fields are typically updated using the finite-difference time-domain (FDTD) method on a staggered Yee grid [82], [70].", "This method is second-order accurate, and since it is explicit and local, it parallelizes efficiently enabling large-scale simulations.", "However, it is known that this algorithm experiences numerical dispersion error for waves propagating along the axis, which leads to errors in the group velocity of the laser pulse.", "This artificial slowdown of the driver and the bubble leads to incorrect dephasing of accelerated electrons and also permits synchronisation of the sheath electrons with the bubble, leading to their unphysical injection.", "Mitigating this effect by using higher resolution increases the computation time quadratically.", "Because of the deleterious effects of numerical dispersion in FDTD schemes, efforts have been made to develop perfect dispersion algorithms, which exhibit no numerical dispersion for waves propagating along a grid axis.", "For accelerator applications, several modifications to FDTD have been described that correct for numerical dispersion using implicit methods [83], [84].", "Because LPA simulations tend to be quite large-scale (using thousands of processor cores), an explicit algorithm is desirable for reasons of computational efficiency.", "Such an algorithm has been described in 2D [59] and in 3D for cubic cells [31].", "These algorithms have also been explored for LPA as a means of reducing noise in boosted-frame simulations [76].", "In this paper, we use two complementary simulation codes (with different numerical approaches and physics content) to explore physical phenomena involved in self-injection and acceleration of electrons until dephasing under typical conditions of recent experiments with [100]TW-class lasers.", "We use a newly-developed perfect-dispersion algorithm [7] implemented in the fully explicit 3D Cartesian vorpal simulation framework [52], subsequently referred to as vorpal-pd.", "The algorithm, briefly described in Sec.", ", eliminates numerical dispersion in the direction of pulse propagation.", "Thus, even with a relatively large longitudinal grid spacing ($\\mathbin {\\sim }$ 15 grid points per $\\lambda _0$ ), the correct group velocity of a broad-bandwidth laser pulse is obtained.", "The other code used here, calder-circ, uses cylindrical geometry.", "This code uses poloidal mode decomposition of fields and currents defined on a radial grid, while macroparticles retain their full 3D dynamics in Cartesian coordinates [39].", "Well-preserved cylindrical symmetry of the laser-plasma interaction enables using just a few lower-order modes.", "Neglecting higher-order, non-axisymmetric contributions to the wakefields and currents makes it possible to approach the performance of a 2D code.", "calder-circ thus allows for fast, extra-high resolution runs with excellent macroparticle statistics [30], [27].", "The paper is organized as follows.", "In section we outline the main features of the recently implemented perfect dispersion algorithm in the vorpal-pd code.", "Section is dedicated to the benchmarking of vorpal-pd against calder-circ.", "Sec.", "summarizes the results and indicates the directions of future work." ], [ "The perfect dispersion method", "In this section we give a brief overview of the perfect dispersion method we use; a more complete description together with detailed benchmarks will be presented in [7].", "Our method is based on that in [59], [31], in which the FDTD algorithm is modified by smoothing the fields in the curl operator in one of Maxwell's equations.", "We choose to smooth the electric fields for the magnetic field update; our update equations are then $D_tB = -\\nabla ^{\\prime }\\times E, \\quad D_tE = c^2\\nabla \\times B - \\frac{J}{\\epsilon _0},$ where $J$ is the electric current deposited from particle motion.", "Here $D_t$ is the finite difference time derivative, $\\nabla \\times $ is the standard finite difference curl operator, and $\\nabla ^{\\prime }\\times $ is the modified curl operator.", "Our modification to the curl operator involves applying smoothing transverse to the coordinate axis along which the derivative is taken.", "For instance, when computing $\\partial E_y/\\partial x$ , $E_y$ is smoothed in the $y$ and $z$ directions.", "This is equivalent to applying a smoothing operator before the numerical derivative operator.", "The electric field is smoothed only for the update of the magnetic field; the smoothed fields are not stored for the next time step.", "The smoothed curl operator $\\nabla ^{\\prime }\\times $ is formed by modifying the finite difference operation.", "If $D_i$ is the numerical derivative operator in the $i$ -th direction, then for the modified curl we use $D_iS_i$ in place of $D_i$ , where $S_i$ is the smoothing operator for the derivative.", "The smoothing operator $S_x$ is defined by the stencil in the $y$ and $z$ directions $\\begin{bmatrix}\\gamma _{yz} & \\beta _z & \\gamma _{yz} \\\\\\beta _y & \\alpha _x & \\beta _y \\\\\\gamma _{yz} & \\beta _z & \\gamma _{yz}\\end{bmatrix},$ and similar relations hold for cyclic permutations of the coordinate indices.", "The coefficients $\\alpha _i$ , $\\beta _i$ , and $\\gamma _{ij}$ are chosen to guarantee that waves propagating along the $x$ axis (the laser propagation direction in our simulations) in vacuum experience no numerical dispersion, as described in [7].", "The only constraint is that the longitudinal grid spacing $\\Delta x$ must satisfy $\\Delta x \\le \\Delta y, \\Delta z$ for the transverse grid spacings $\\Delta y$ and $\\Delta z$ ." ], [ "Benchmarking", "While a technological path to high-quality GeV beams exists, experimental progress is impeded by an incomplete understanding of the intrinsic relation between electron self-injection and nonlinear optical evolution of the driver, and hence by the lack of suitable criteria for selection of the optimal regimes that produce beams with the smallest possible phase-space volume.", "Control and optimisation of the fully kinetic, intrinsically 3D process of electron self-injection is a daunting task.", "It involves a systematic study of the links among the dynamics of self-injection and the nonlinear optical processes involving the laser pulse and the bubble.", "Due to the extended acceleration length, the interaction of the laser pulse with the plasma is rich in nonlinear phenomena.", "Even a Gaussian beam which is perfectly matched to the electron density gradient in which it propagates is not immune to nonlinear optical processes.", "Oscillations of the pulse spot-size due to non-linear refraction [53], [85], [30], self-phase modulation leading to the formation of a relativistically intense optical piston [75], [41], [11], [54], [77], [27], [28], and relativistic filamentation [1], [73], [72] are processes which result in pulse deformations.", "Electron self-injection appears to be extremely sensitive to such changes in pulse shape, which lead to contamination of the electron beam with polychromatic, poorly collimated background [28].", "Such contamination is readily seen even in simulations with idealised initial conditions [35], [13], [49], [30], [27].", "The complicated modal structure of the incident pulse further aggravates the situation, leading to continuous off-axis injection, collective betatron oscillations [16], [46], [8], and electron beam steering [56].", "In practice, these phenomena currently preclude operation reliable enough to enable high-precision user experiments; reported islands of stability for self-injection in laser and plasma parameter space remain relatively narrow [32], [48], [73], [21], [45], [78].", "Numerical codes used in predictive modelling of LPAs must be able to reproduce these phenomena with high precision in order not to confuse the instability of acceleration caused by physical processes with unphysical artefacts caused by intrinsic deficiencies of numerical algorithms, such as numerical dispersion, high sampling noise, and grid heating." ], [ "Simulation parameters", "The simulations presented here extend the earlier case study by [27] and use the same set of initial conditions.", "A transform-limited Gaussian laser pulse with full width at half-maximum (FWHM) in intensity $\\tau _L = [30]{fs}$ , wavelength $\\lambda _0 = [0.805]{m}$ , and [70]TW power is focussed at the plasma border ($x = 0$ ) into a spot size $r_0 = [13.6]{m}$ , and propagates in the positive $x$ direction.", "The laser pulse is polarised in the $y$ direction.", "The peak intensity at the focus is $[2.3\\times 10^{19}]{W/cm^2}$ , giving a normalised vector potential of $a_0 = 3.27$ .", "The plasma density has a [0.5]mm linear entrance ramp followed by a [2]mm plateau and a [0.5]mm linear exit ramp.", "The density in the plateau region, $n_0 = [6.5\\times 10^{18}]{cm^{-3}}$ , corresponds to $\\gamma _g\\approx P/P_{\\text{cr}}\\approx 16.3$ and dephasing length $L_d\\approx [1.7]{mm}$ .", "The simulations carried out with vorpal-pd use grid spacings of $\\Delta x = 0.06\\lambda _0 = [48.3]{nm}$ longitudinally and $\\Delta y = \\Delta z = 0.5\\lambda = [403]{nm}$ transversely, with four macroparticles per cell.", "Use of third-order splines for the macroparticle shapes reduces the sampling noise, mitigating the adverse effect of the coarse grid.", "The domain in the vorpal-pd simulation is $[72]{m}$ long and $[91]{m}$ wide, and is surrounded transversely by a 16-layer perfectly-matched layer absorbing boundary.", "The code is fully parallelised, and was run using 6 144 cores on the Hopper supercomputer at the National Energy Research Scientific Computing Center (NERSC).", "Completion of a typical run took $\\mathbin {\\sim }3\\times 10^{5}$ CPU hours.", "The calder-circ simulation uses 45 macroparticles per cylindrical cell, formed by the revolution of the grid cell around the propagation axis.", "The longitudinal grid spacing is $\\Delta x =0.125c/\\omega _0\\approx [16]{nm}$ .", "The aspect ratio $\\Delta r/\\Delta x = 15.6$ (where $r=\\sqrt{y^2 + z^2}$ ), and the time step $\\Delta t = 0.1244\\omega _0^{-1}$ .", "With these grid parameters, numerical dispersion is negligible, and sampling noise is significantly reduced.", "This high resolution simulation does not indicate any new physical effects compared to the vorpal-pd simulation, and does not exhibit significant differences in the quantitative results.", "Well preserved cylindrical symmetry during the interaction (confirmed in the vorpal-pd simulation) enables us to approximate fields and currents using just the two lowest-order poloidal modes, thus reducing the 3D problem to an essentially 2D one.", "These results confirm the earlier established fact [39] that, in the case of a linearly polarised laser, higher-order modes contribute only weakly to the electric field.", "Comparison with the results of the vorpal-pd runs shows that our restriction to only two modes is sufficiently precise to reproduce all relevant physical effects, and to simulate the propagation through a [3]mm plasma in 2 625 CPU hours on 250 cores." ], [ "Formation of quasi-monoenergetic bunches and physical origin of dark current", "Upon entering the plasma, the strongly overcritical pulse rapidly self-focuses, reaching its highest intensity at $x\\approx [0.8]{mm}$ , soon after entering the density plateau.", "Full blowout is maintained over the entire propagation distance.", "In both simulations, electrons are accelerated until dephasing in two distinct stages, each characterised by completely different laser pulse dynamics.", "Transverse evolution of the laser pulse is the hallmark of Stage I.", "The pulse spot size oscillates, first causing expansion and then contraction of the bubble.", "The bubble expansion produces self-injection of electrons from the sheath; stabilisation and contraction of the bubble extinguish injection, limiting the beam charge to a fraction of a nC.", "Phase space rotation creates a well-collimated quasi-monoenergetic bunch long before dephasing.", "Further acceleration (Stage II) is dominated by longitudinal (temporal) self-compression of the pulse, leading to gradual elongation of the bubble and continuous injection, producing a polychromatic, poorly collimated energy tail with a few nC charge.", "This two-stage evolution has been noticed in earlier simulations [13], [35], and explained in detail in [27].", "The correlation between the plasma bubble evolution and the self-injection process is quantified in figure REF .", "Panel (a) shows the length of the accelerating phase on axis, viz.", "the length of the region inside the bubble where the longitudinal electric field is negative.", "Panel (b) shows the longitudinal “collection phase space”, viz.", "momenta of macroparticles reaching the dephasing point, $p_{x}(x = x_{\\text{deph}})$ , vs. their initial position in plasma.", "Panel (c) shows the collection volume: the initial positions of electrons reaching the dephasing point.", "Comparison of these three panels shows that electrons are injected only during the periods of bubble expansion.", "Figure: (a) Length of the accelerating phase vs. propagation distance in calder-circ (black) and vorpal-pd simulations (red/grey).", "Expansion and contraction of the bubble due to nonlinear focusing ofthe driver (Stage I) is followedby continuous expansion caused by pulse self-compression (Stage II).", "(b) Longitudinal momentum of electrons reaching the dephasing point, x deph ≈[2.4]mmx_{\\text{deph}}\\approx [2.4]{mm}, vs. their initiallongitudinal positions.", "Black dots are the calder-circ macroparticles; the colourmap represents the normalised number density of vorpal-pd macroparticles.", "Electrons are injected only during periods ofbubbleexpansion.", "A quasi-monoenergetic bunch forms during Stage I and maintains its low energy spread until dephasing, indicated by the group of early-injected particles with E≈[500]MeVE\\approx [500]{MeV}.Groupsof electrons encompassed by the ellipses were injected into the second and third buckets, to be further captured and accelerated by the expanding first bucket.", "Continuous injection during Stage IIcreatesa polychromatic energy tail.", "(c) Collection volume: initial radial offsets of electrons reaching dephasing limit R in =y in 2 +z in 2 R_{\\mathrm {in}}=\\sqrt{y_{\\mathrm {in}}^2+z_{\\mathrm {in}}^2}, vs. their initiallongitudinal positions x in x_{\\mathrm {in}}.", "Black (red/grey) dots are calder-circ (vorpal-pd) macroparticles.", "This collection volume indicates that the vast majority of electrons are collected from ahollowconical cylinder with a radius slightly smaller than the local bubble size.During Stage I, radial oscillation of the laser pulse tail inside the bubble causes alternating expansion and contraction of the first bucket, clearly seen in the progression from $x = 0.6$ to [1.24]mm in figure REF (a).", "The bubble size oscillates around the average value predicted by the estimate (REF ), $L_{\\text{acc}}\\approx [9.5]{m}$ .", "Electron self-injection into the oscillating bubble leads to the formation of a quasi-monoenergetic component in the energy spectrum.", "At the end of Stage I, at $z\\approx [1.24]{mm}$ , the bubble contracts to the same size in both runs, truncating the tail of injected bunch and expelling electrons injected between $x = 0.825$ and $x\\approx [0.95]{mm}$ .", "These electrons do not reach dephasing and thus are missing in figures REF (b) and REF (c).", "Electrons injected between $x = 0.65$ and [0.825]mm, remain in the bubble and are further accelerated.", "This well-separated group of particles is clearly seen in figure REF (b).", "In both the vorpal-pd and calder-circ simulations, these electrons reach dephasing first, preserving low energy spread, and are accelerated to the highest energy, $E\\approx [500]{MeV}$ .", "The bubble expands more rapidly and stabilises sooner in the vorpal-pd simulation, causing stronger reduction of the phase velocity in the subsequent buckets (second and third).", "Hence, in contrast to the calder-circ run, vorpal-pd gives a noticeable amount of charge trapped and preaccelerated in these buckets.", "These electrons, indicated by the red ellipse in figure REF (b), are swallowed by the expanding first bucket during Stage II and are further accelerated, contributing to the dark current.", "This contribution, however, appears to be fairly minimal in comparison to the amount of continuously injected charge during Stage II.", "The leading edge of the laser pulse constantly experiences a negative gradient of the nonlinear index of refraction.", "As a result, by the end of Stage I, it accumulates considerable redshift.", "During Stage II, plasma-induced group velocity dispersion slows the red-shifted spectral components relative to the unshifted components, leading to the front etching and pulse self-compressing into a relativistically intense, few-cycle long optical piston [75], [41], [11], [27].", "As the pulse transforms into a piston, the bubble constantly elongates, resulting in copious trapping and creating a poorly collimated, polychromatic tail, clearly seen in figure REF (b).", "At the dephasing point, $x_{\\text{deph}}\\approx [2.4]{mm}$ , the bubble size becomes nearly twice the estimate $L_{\\text{acc}}\\approx [9.5]{m}$ based on the scaling law (REF ).", "Even though figure REF (a) shows a larger bubble expansion in the vorpal-pd run, the sections of collection phase space corresponding to Stage II look nearly identical for both codes in figure REF (b).", "The collection volume depicted in figure REF (c) indicates that the electrons are collected from a conical shell with a radius slightly smaller than the bubble radius.", "This structure of the collection volume indicates that the vast majority of trapped and accelerated electrons have impact parameters of sheath electrons [74], [79], [57], [30], [27].", "Collection volumes in the vorpal-pd and calder-circ runs are almost identical during Stage I, whereas the radius of the cone is larger for vorpal-pd during Stage II, on account of the greater expansion due to pulse diffraction.", "Snapshots of electron density, longitudinal phase space, and energy spectra at the points of maximal expansion and contraction of the bubble are presented in figures REF , REF , and REF .", "Data for panels (a), (b), and (c) in these figures are from the calder-circ simulation, and for panels (d), (e), and (f) from the vorpal-pd simulation.", "Figure: Electron density (in cm -3 {cm}^{-3}) in the plane of laser polarisation in calder-circ (top row) and vorpal-pd simulations (bottom).", "Panels (a) and (d) show the fully expanded bubble in themiddle of Stage I,(b) and (e) the fully contracted bubble at the end of Stage I, and (c) and (f) the bubble in the vicinity of electron dephasing point at the end of Stage II.", "x=ctx = ct is the trajectory of the laserpulse maximum in vacuum; (a) and (d) correspond to the distance x=ct≈[930]mx = ct \\approx [930]{m} from the plasma edge, (b) and (e) to x=ct≈[1210]mx = ct\\approx [1210]{m}, and (c) and (f) to x=ct≈[2364]mx= ct\\approx [2364]{m}.", "Before the dephasing point, the bubble, elongated and deformed due to the laser pulse self-compression, traps considerable charge.", "Beam loading, however, is yetunable to terminate self-injection (cf.", "panels (c) and (f)).Figure: Electron density (arbitrary units) in longitudinal phase space in calder-circ (top row) and vorpal-pd simulations (bottom).", "Each panel corresponds to the same panel offigure .", "Full expansion of the bubble saturates injection and initiates phase space rotation (panels (a) and (d)).", "Contraction of the bubble terminates injection, clipping therear of injected bunch, eliminating low-energy tail.", "Phase space rotation makes the bunch quasi-monoenergetic (panels (b) and (e)).", "Elongation and deformation of the bubble due to the laser pulseself-compression causes continuous injection, producing an electron beam with a continuous spectrum of longitudinal momenta (panels (c) and (f)).Figure: Electron energy spectra in calder-circ (black) and vorpal-pd simulations (red/grey).", "Panels (a), (b), and (c) correspond to the phase space snapshots (a) and (d), (b) and (e), and (c) and (f)of figure , respectively.", "(a) At the point of full expansion, the electron energy spectrum is broad.", "(b) Full contraction of the bubble suppresses the low-energy tail and reducesthe energy spread.", "Electrons from the second bucket contribute to the background, seen in the diffuse peaks around [150]MeV.", "(c) Continuous injection caused by the bubble expansion anddeformation produces a massive polychromatic tail.", "The leading bunch, at E≈[500]MeVE\\approx [500]{MeV}, reaches dephasing, but is still distinct from the tail.The fully expanded bubble in the middle of Stage I is shown in figures REF (a) and REF (d).", "As soon as the bubble fully expands, injection terminates.", "Uninterrupted injection of sheath electrons before this point produces a large spread of longitudinal momentum and energy, shown in figures REF (a), REF (d), and REF (a).", "The slight contraction of the bubble between $x = 0.95$ and [1.24]mm truncates the bunch.", "Electrons injected at the very end of the expansion interval are expelled, while particles remaining in the bucket are further accelerated.", "The transverse self-fields of the bunch are unable to prevent the bucket contraction.", "Snapshots of the contracted bubble are presented in figures REF (b) and REF (e).", "During the contraction interval, the tail of electron bunch, exposed to the highest accelerating gradient, equalises in energy with earlier injected electrons, thus producing a characteristic `U' shape in the longitudinal phase space.", "This feature (also observed in the similar situation by [44]) is clearly seen in figures REF (b) and REF (e).", "As a result of this evolution, quasi-monoenergetic bunches form in both vorpal-pd and calder-circ simulations at the end of Stage I.", "These quasi-monoenergetic spikes with $<10\\%$ energy spread can be seen in figure REF (b).", "In addition to the quasi-monoenergetic spikes, these energy spectra also reveal diffuse features near [150]MeV, corresponding to the electrons trapped in the second bucket; these particles can be seen in the snapshots of electron density shown in figures REF (b) and REF (e).", "These electrons, however, never equalise in energy with the leading high-energy bunch.", "Table: Parameters of the quasi-monoenergetic bunch (E>[200]MeVE > [200]{MeV}) at the end of Stage I (cf.", "the spectra in figure (b)).", "Q mono Q_{\\text{mono}} is the charge in pC;E mono E_{\\text{mono}} is the energy corresponding to the spectral peak (in MeV); ΔE mono \\Delta E_{\\text{mono}} is the absolute energy spread (FWHM) in MeV; ε N,y \\varepsilon _{N,y} and ε N,z \\varepsilon _{N,z} arethe normalised transverse emittance (in mm mrad) in and out of the laser polarisation plane, respectively.Parameters of the bunches, summarised in table REF , appear to be very similar.", "Normalised transverse emittances presented in this table are calculated according to the usual definition $\\varepsilon _{N,i}=(m_ec)^{-1}[ (\\langle p_i^2\\rangle - \\langle p_i\\rangle ^2)(\\langle r_i^2\\rangle - \\langle r_i\\rangle ^2) - (\\langle p_i r_i \\rangle - \\langle r_i \\rangle \\langle p_i\\rangle )^2]^{1/2}$ , where $i=y$ and $z$ correspond to the emittance in and out of polarisation plane.", "The beam asymmetry is more pronounced in the vorpal-pd simulation, presumably on account of the inclusion of the complete electromagnetic field, in contrast to just two poloidal modes in calder-circ.", "Agreement between the two codes worsens during Stage II.", "As has already been noted, the bubble expansion is larger in the vorpal-pd simulation.", "As a result, the amount of continuously injected charge at the dephasing point (2.5 nC) is about 60% higher and the divergence of the continuously injected beam (80 mrad) is about twice that in the calder-circ simulation.", "The difference in charge can be easily inferred from figure REF (c).", "On the other hand, parameters of the leading bunches are in reasonable agreement, with the central energy $[485\\pm 20]{MeV}$ in vorpal-pd against $[515\\pm 25]{MeV}$ in calder-circ run.", "In both simulations, the emittance of the quasi-monoenergic component increases by $\\mathbin {\\sim }30\\%$ over its value at the end of Stage I.", "The lower energy of the leading bunch in the vorpal-pd run can be explained by its earlier dephasing due to more rapid expansion of the bubble.", "Both codes agree that the bubble not only elongates during Stage II, but becomes more and more asymmetric in the laser polarisation plane.", "The “pennant-like” bubble shape is responsible for massive off-axis injection, leading to the noticeable beam centroid oscillations in the laser polarisation plane seen in figures REF (c) and REF (f).", "Such phenomenon has been observed in similar situations by others [16].", "This violation of symmetry is a manifestation of carrier-envelope phase effects in the interaction of a relativistically intense, linearly polarised, few-cycle piston with the plasma [51].", "Conversely, in the plane orthogonal to the laser polarisation, both the bucket and the beam remain perfectly symmetric (not shown).", "Surprisingly, the two poloidal radiation modes of calder-circ still capture the field evolution well.", "Inclusion of higher order modes should improve the situation.", "On the other hand, figures REF (c) and REF (f) indicate that electromagnetic solvers of both codes agree on the group velocity of the laser pulse even in the situation where the pulse shrinks down to less than two cycles and remains strongly relativistic.", "This means that (a) poloidal mode decomposition does not damage dispersion in the axial direction, and (b) the coarse grid and dispersion properties of vorpal-pd are sufficient to describe well the extreme case of pulse spectral broadening to $\\Delta \\omega \\sim \\omega _0$ and compression to nearly a single cycle.", "Examination of the bubble evolution and collection volumes (cf.", "figure REF ), together with individual snapshots of electron density in coordinate and longitudinal phase space, indicate that, in spite of the great difference in the algorithms, vorpal-pd and calder-circ reproduce the same correlation between the evolution of the bubble and the self-injection of sheath electrons, and agree quantitatively on the parameters of quasi-monoenergetic beams produced by the oscillating bubble.", "Self-injection begins, terminates, and resumes at exactly the same positions along the propagation axis in both runs, and electrons are collected from the same plasma volume.", "Despite differences in minor details, both codes consistently reproduce physical details of the self-injection process over the entire dephasing length.", "This level of agreement between very different numerical models indicates that the results are largely free of numerical artefacts.", "Importantly, the discrepancies emerge when the interaction develops noticeable non-cylindrically symmetric features, and hence the reduced field description of calder-circ loses precision.", "We believe that the agreement between the models may be improved in a straightforward fashion (viz.", "using a larger number of poloidal modes) without significantly reducing computational efficiency." ], [ "Effects of numerical dispersion control", "As described above, simulating upcoming experiments will require economising on computational cost as much as possible without sacrificing physical accuracy.", "One means of reducing longitudinal resolution requirements, and hence allowing a larger time step, is to minimise numerical dispersion through a modified electromagnetic update.", "Here we show how numerical dispersion quantitatively affects the injected electron bunch.", "The immediate effect of numerical dispersion is an unphysically low group velocity of the laser pulse.", "While this effect is difficult to observe directly in the laser pulse because of more significant changes in the pulse shape, it can be seen in the electron phase space, which is of experimental importance.", "We examine the initially-injected electrons at the point where they have rotated in phase space such that the beam has achieved minimal energy spread.", "The minimal energy spread condition is characterised by the phase space of the bunch being roughly longitudinally symmetric and in the shape of a `U'.", "We find from the perfect dispersion simulation that this occurs after the laser has propagated approximately [1.8]mm into the plasma.", "We show longitudinal momentum spectra and phase space at this point for both perfect dispersion and normal dispersion in figure REF .", "We find that with the normal dispersion algorithm, the beam achieves lower energy and exhibits higher energy spread than with the perfect dispersion algorithm.", "We also find that phase space rotation has occurred more quickly.", "Figure: Top: Electron longitudinal momentum spectra after [1.8]mm propagation for perfect and normal dispersion.", "Bottom: Longitudinal phase space for perfect dispersion (left)and normal dispersion (right).Figure: Longitudinal phase space for perfect dispersion (left) and normal dispersion (right) at the minimal energy spread point for both algorithms.", "For perfect dispersion, this is after[1.8]mm of propagation, and for normal dispersion after [1.6]mm.Figure: Comparisons of longitudinal momentum and phase space.", "Momentum spectra for perfect and normal dispersion (left), the perfect-dispersion phase space (center),and the normal-dispersion phase space (right).", "The top row shows the electron distribution at [960]m[960]{m} of propagation, the middle row at [1.24]mm, and the bottom rowat [2.4]mm.We also compare the two dispersion algorithms at points of minimal energy spread.", "Without dispersion control, the more rapid phase space rotation causes the minimal energy spread to occur after just [1.6]mm of propagation rather than [1.8]mm.", "We show the two phase space plots in figure REF .", "This comparison is relevant since for applications, as one would want to design the system such that the injected beam exits the plasma at this point of minimum energy spread [22].", "It is clear from these plots that with the normal dispersion algorithm, the beam has reached lower mean energy ([390]MeV) at the minimum energy spread point than with perfect dispersion, where the beam has mean energy of [460]MeV.", "In addition, the normal dispersion case exhibits slightly higher energy spread and total charge in the bunch.", "We also compare the longitudinal momentum spectra and phase space at the points compared with calder-circ simulations in the previous section, namely $[960]{m}$ , [1.24]mm, and [2.4]mm.", "These comparisons are shown in figure REF .", "We can see, especially in the later two plots, that the injected bunch in the normal dispersion simulation shows both lower mean energy and greater phase space rotation than in the perfect dispersion run, which agrees better with the calder-circ simulations as seen in the previous section.", "While these discrepancies are small, they are noticeable and consistent with numerical group velocity error.", "As LPA system designs are refined, and diagnostics and control over the laser pulse and plasma improve, it will be important to control numerical effects on this level to optimise parameters through simulation.", "The perfect dispersion algorithm allows us to do so while still using low longitudinal resolution for computational efficiency." ], [ "Conclusions", "In this paper we have demonstrated the utility of using computationally efficient, fully explicit 3D PIC codes to describe and explain the physical phenomena accompanying electron acceleration until dephasing in a self-guided LPA in the blowout regime.", "Electron self-injection and its relation to nonlinear dynamical processes involving the laser pulse and bubble were explored.", "Two approaches to reducing the computational cost of the simulations were considered.", "First, using the Cartesian code vorpal with a newly developed perfect dispersion algorithm [7], vorpal-pd, made it possible to use large grid spacings ($\\mathbin {\\sim }15$ grid points per wavelength in the direction of propagation) and proportionally larger time steps.", "This approach reproduces the correct group velocity of a broad-bandwidth laser pulse.", "The red-shift, self-compression, and depletion of the laser pulse were thus described correctly, with proper resolution of all important physical scales.", "Second, the well-preserved axial symmetry of the problem allowed us to use a reduced geometry description, with poloidal-mode decomposition of currents and fields.", "This approach was realised in the code calder-circ [39].", "By using only two modes, we approached the performance of a 2D code, at the same time preserving the correct cylindrical geometry of the interaction.", "The high computational efficiency of calder-circ allowed us to use a very high longitudinal resolution ($\\mathbin {\\sim }50$ grid points per laser wavelength in the direction of propagation) and a large number of macroparticles ($\\mathbin {\\sim }50$ per cylindrical cell), eliminating numerical dispersion and strongly reducing the sampling noise.", "This high resolution simulation did not indicate any new physical effects relative to the vorpal-pd runs, and did not exhibit significant differences in the quantitative results.", "Even with strong violation of cylindrical symmetry (such as near the dephasing limit, when the pulse was transformed into a two-cycle relativistic piston), the calder-circ results remained qualitatively correct.", "Both codes described precisely the self-focusing of the laser pulse, the oscillations of its spot size, and related oscillations of the bubble; electron self-injection into the oscillating bubble and formation of a quasi-monoenergetic bunch; laser pulse frequency broadening and self-compression into the relativistic piston; constant elongation of the bubble during the piston formation; and uninterrupted electron injection eventually overloading the bubble.", "The codes showed excellent agreement on the locations of initiation and extinction of injection, on the collection volume, and on parameters of the quasi-monoenergetic component in the electron spectrum, indicating that the results are free of numerical artefacts.", "It is especially interesting that the calder-circ simulation with just two poloidal modes did not lose accuracy and preserved the correct group velocity (agreeing with the vorpal-pd run) even when the laser pulse was compressed down two cycles.", "We thus conclude that (1) using perfect dispersion, taking a coarser grid and larger time steps, and using higher-order splines for macroparticle shapes to suppress the sampling noise, or (2) neglecting high-order non-axisymmetric field and current components, thus reducing the dimensionality of problem are both effective and promising means to increase the computational efficiency without sacrificing fidelity.", "Both of these methods are applicable to the design of upcoming experiments on GeV-scale acceleration of electrons with [100]TW-scale lasers.", "The work of B. M. C. and D. L. B. was partially supported by U. S. DOE Contract No.", "DE-SC0006245 (SBIR).", "The work of S. Y. K. and B. A. S.", "was partially supported by U. S. DOE Contract No.", "DE-FG02-08ER55000 and by NSF Grant PHY-1104683.", "X. D. and A. F. L. thank Victor Malka for his support during the development of calder-circ.", "A. B.", "and E. L. acknowledge the support of LASERLAB-EUROPE/LAPTECH through EC FP7 contract no.", "228334.", "The calder-circ simulations in this work were performed using high-performance computing resources of GENCI-CCRT (grant 2010-x2010056304).", "vorpal simulations used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U. S. DOE under Contract No.", "DE-AC02-05CH11231." ] ]

[ [ "Diffusion-less recrystallization at high uniaxial deformation" ], [ "Abstract It was shown by computer simulation that coarsened grains with lower content of defects are formed at uniaxial deformation of a four-grain infinite nano-wire.", "The structure similar to crystal filaments was formed in the case of tension.", "The case of compression demonstrated formation of two grains (from four initial one) disoriented at an angle of four degrees." ], [ "Introduction", "It is well known that the process of megaplastic (severe) plastic deformation (SPD) is accompanied by fragmentation (size reduction) of grains due to multiplication of dislocations, grain boundaries, and other defects.", "It is difficult to imagine that the same force factors allow occurrence of opposite diffusion-free processes directed toward grain coarsening, which can be considered as a specific kind of crystallization.", "Recently, a conception was developed that the processes of metal grain fragmentation and recrystallization periodically alternate in time [1].", "At that, new relaxation mechanisms are activated such as low-temperature dynamical recrystallization [1], [2].", "It is assumed that diffusion accelerated by external stresses plays an important role in these processes [3].", "However it is known that diffusion processes are slow enough.", "At recrystallization anneal, time of treatment is about one hour, and the time of homogenization anneal is ten hours.", "So long time periods are required for realization of structural transformations.", "Even if we suggest diffusion accelerated by an order, it is impossible to explain high rate of these processes.", "At the same time, samples of tens millimeters in size are processed with SPD methods during several fractions of a second.", "It was shown earlier by one of the authors that methods of molecular dynamics predict another diffusion-free recrystallization based on the same principle as martensitic transition [4].", "The experiment was carried out at a two-grain infinite nano-wire with free external boundaries at uniaxial tension, i.e.", "at conditions far from a real SPD.", "The experiment was organized in such a way that the boundaries of the external blocks had lugs, so the correlation with the real situation was complicated.", "Here the results of simulation of four-grain nano-wire are presented, with the wire constructed without lugs of the external free faces (Fig.", "REF a).", "Figure: Structure changes at the atomic level at uniaxial tensile deformation:а) the initial state ; b, c) the deformed state; steps = 423, 3259, 20755 correspondingly.Both tension and compression tests were carried out.", "It was noticed by Tott that the initial texture of the sample exerts great influence on deformation processes [5].", "He suggested a model of grain fragmentation in metal at severe deformations, which was based on a combination of deformation-strengthening model and Taylor’s model with a new element added, i.e.", "possible evolution of grain population by means of fragmentation.", "Structure evolution of fcc metals in Brigeman anvils demonstrated that grains with ideal orientation relative to the external stress (or deformation) were much less fragmented compared to the grains of other orientations.", "To a certain extent, we have checked this feature in the course of computer simulation.", "In our case, we form the initial texture of double type with a half of grains oriented at an angle of 90 degree about the other half of grains, i.e.", "50 percent of «crystallites» are ideally oriented." ], [ "Calculation technique", "Numerical methods of molecular dynamics are often used for studying of peculiarities of the kinetics of defects at atomic layer [6], [7].", "Here two aspects of the problem are of special importance, that is generation and motion of structure defects and the related heat production (entropy production).", "We suggest that the particles interact with the aid of Lennard-Jones pair potential (REF ): $U_{ijklmn}=E_{b}((\\dfrac{r_{0}}{r_{ijklmn}})^{12}-2(\\dfrac{r_{0}}{r_{ijklmn}})^{6}), \\ $ where $r_{ijklmn}$ is the distance between the particles with the numbers of $i$ , $j$ , $k$ and $l$ , $m$ , $n$ and with Cartesian coordinates $X_{ijk}$ , $Y_{ijk}$ , $Z_{ijk}$ and $X_{lmn}$ , $Y_{lmn}$ , $Z_{lmn}$ .", "The accepted short numeration of the particles is convenient for determination of the initial state of $3D$ lattice and the succeeding control of its evolution.", "Indexes $i$ and $l$ , $j$ and $m$ , $k$ and $n$ numerate atoms of the lattice along $X$ , $Y$ and $Z$ direction, correspondingly.", "$E_{b}$ , $r_{0}$ are the binding energy and equilibrium distance between the particles in a dimer, respectively.", "Evaluations can be applied to copper, so we assume $E_{b}=0.83·10^{-19} J, r_{0}=0.3615·10^{-9} m$ , and $M=1.054·10^{-25} kg$ .", "Time step was selected to be $Δt = 10^{-14}s$ , the period of linear oscillations of a diatomic molecule was $T=3·10^{-13}$ s. The calculations are conveniently done with using reduced units: $1m_{r} = 0.362 nm$ ; $1kg_{r} = 1.054•10^{-23} kg; 1s_{r} = 0.263•10^{-13}s.$ In this case, the parameter values used in the calculations will correspond to copper.", "The binding energy measured in reduced units is (REF ) $E_{b}=\\dfrac{A}{12(r_{0})^{6}}=0.417*10^{-4}J_{r}, \\ $ At $XY$ cross-section, the model consists of 4 rectangles.", "Along one side of each rectangle, 14 atoms are located at the distance of $a_{0}$ , and 10 atoms are located along the other side at the distance of $a_{0}\\sqrt{2}$ in such a way, so the rectangle sides appear approximately equal.", "This scheme allows us to arrange them, in order the maximal sides of the neighbor squares are mutually orthogonal (Fig.REF a) Four more squares just like these are located along $Z$ axis, with their atoms to be placed above the centers of the rectangles of the preceding layer at a height of $a_{0}/2$ .", "Being continued periodically along $Z$ axis, such a construction results in creation of a stable fcc structure.", "To form a stable structure, two such layers are enough.", "Thus, every two-layer square will contain 280 atoms and the sum over the calculation cell will be 1120 atoms.", "In addition to periodical conditions along $Z$ axis, periodical boundary conditions are imposed along the vertical $Y$ axis.", "Summarized over two squares, the period of boundary conditions along $Y$ axis will be $28a_{0}$ or 24 atomic layers, whereas the period along $Z$ axis consists of two layers.", "The principle of the calculation is that the nearest neighbors over several coordination spheres are determined and the simulation is carried out over them.", "In terms of texture, the initial state of the system is approximated by infinite texture of aches-board type.", "The rate of deformation both at tension and compression was $10^{9}s^{-1}$ ." ], [ "Recrystallization at uniaxial stretching strain", "At the initial stage of every type of deformation, the total elastic energy of the system accumulated in the form of potential energy increases with deformation rise as the second degree according to the law of elastic behavior (curve 2, Fig.", "REF ).", "Figure: Time dependence of the kinetic energy (1), the total energy (2) and the potential energy (3).Dislocation appearance at the i-th time step associated with the time moment of $t = Δt•i$ results in a leap reduction of the potential energy.", "The leap lasts till the time of $t+dt$ .", "The beginning of the leap is related to the appearance of a dislocation, and the duration is determined by dislocation motion.", "Thus, the deformation of the sample of atomic size is divided into two stages: the elastic stage of energy accumulation and the stage of plastic motion.", "Further, these stages may alternate.", "The reduction of potential (elastic) energy is spent partially on dislocation generation (energy of inner stresses around dislocations), partially on the increase of the energy of heat motion at the moment of dislocation nucleation and during dislocation motion.", "This fact is rendered by dispersion of potential energy values at the curve.", "When the plastic stage is accomplished, the tage of elastic deformation starts again and the potential energy increases according to a slower square relationship.", "At this stage, the leap is of smaller magnitude and lasts for a longer time period.", "The recurring leap takes place at lower level of the accumulated potential energy that is related to activating role of thermal fluctuations increased after the first leap.", "Stage character of atomic-size sample deformation can be traced also by the measurements of other characteristics of the system, i.e.", "the total inner energy of the system (curve 1, REF ) and the kinetic energy (curve 3).", "We should mention that the inner energy of the system is rising all the time that is related to the constant filling at the rate of the work done over the system.", "However the rate of the filling (incline of the plot) is not constant, being dependent on the stage of the process and the character of the occurring processes.", "After every leap the rate of increase of the inner energy is reduced.", "The plot shows also that thermal fluctuations of the kinetic and potential energy are strictly in antiphase That is expressed as absence of fluctuations of the very inner energy been evidences of the executed energy conservation law.", "The first elastic stage of deformation begins at zero initial velocity of particles corresponding to the absolute zero temperature.", "During the first leap, the kinetic energy increases and then it remains stable at the succeeding stage of elastic deformation (curve 1 in REF ).", "This is an evidence of the absent dissipative processes in the form of dislocation motion at this stage.", "After the leap, the temperature increases slowly, being associated with both elastic and plastic deformation occurring simultaneously.", "Under tension, a preferred direction is present in the material that is why the rest of crystallites tend to rearrangement along this direction.", "Crystallites are aimed to $<100>$ orientation under tension, and to $<0-11>$ under compressing.", "Such reorientation becomes possible due to different mechanisms: grain boundary slip, twinning, martensitic transition.", "In our case, a transformation involving the shear mechanism (by twinning) takes place, with the illustration presented in Fig.", "REF .", "Figure: Illustration of shear mechanism of structure transformation.The plot presents the grid connecting atoms been neighbors at the initial time moment (atoms located at neighbor sites of the calculated array).", "The figure demonstrates also some lattice distortions determined by diffusion of separate atoms.", "We should note that as a result of evolution, that defect ensemble is aimed to reduction of the excess of potential energy that is seen in the distribution of potential energy presented in Fig.REF .", "It was mentioned above that deformation process consists of several stages, i.e.", "the regions of the elastic deformation are followed by the regions of the plastic one (Fig.", "REF ).", "Figure: Distribution of the potential energy.", "a) is the initial state, б) is the deformed one." ], [ "Recrystallization at compression deformation", "An analogous situation is observed at compression deformation with some distinctions.", "Thus, uniaxial compressing forms a low-angle boundary at the final stage (Fig.", "REF c), Figure: Structure changes at atomic level under compression deformation: а-в – time steps =404,6205,15204,16703= 404, 6205, 15204,16703 correspondingly.", "The initial stage is presented in Fig.а.which was not registered in the case of tension.", "Besides, after compression deformation, a texture with prevailing $<0-11>$ orientation is formed that agrees with experimental data [8].", "Analogous to tension, ensemble of defects tends to minimize the excess energy, i.e.", "the system trends to perfection.", "Some divergences are present on the plot of potential energy, that is significant increase in the kinetic energy with simultaneous drop of the potential energy (in global sense) at the final stage of deformation.", "This fact is I probably related to the start of amorphysation (melting) of the material.", "Because the plot of the energy change is similar to that in the case of tension, we do not reproduce it.", "However this effect requires a more detailed study." ], [ "Effect of deformation rate on the structural transformations of the material", "In the present work, we have also considered the cases of deceleration of the deformation.", "For instance, if the deformation rate is reduced tenfold, no distinctions in kind are observed.", "However, if the rate drops by two orders, another evolution of the system is registered.", "The main distinction is that contradicting to the above theory, the potential energy does not change at first glance (Fig.", "REF .", ").", "Figure: STime dependence of the kinetic energy (1), the total energy (2),and the potential energy (3) (deformation rate is reduced by a factor of 100).But actually there are no contradictions because in the case of such slow deformation rate, the system is «shallowly unloaded».", "When speaking about «shallowly unload» we suggest the system accommodation occurring at lower level of the accumulated potential energy that is related to activating role of thermal fluctuations.", "Their level has increased after the preceding structural transformation.", "Besides, global leaps of the total energy are observed, in contrast to high deformation rates.", "This effect may be determined by the work performed by the system.", "Undoubtedly, this phenomenon is worth of more detailed consideration.", "As for the structure transformation, all the stages of structure reconstruction observed at the relatively high rate are present here.", "For example, the system passes the stage of twin formation, with twins formed in grains without prevailing orientation.", "Further, the formation of phase synchronism occurs, that propagate along the whole studied subject both in vertical and horizontal direction.", "Then the system is deformed as a single whole." ], [ "SHORT CONCLUSION", "It was shown that in both cases, a martensitic-type transition results in formation of coarsened grains with lower content of defects of smaller dimensionality as compared to grain boundaries.", "At tension, a structure of a crystal filament type was formed.", "Structure transformation at uniaxial compression was finished with formation of two grains disoriented at an angle of 4 degree (from four initial ones).", "We have shown conceptual possibility of diffusion-free structure transformation using shear mechanism (the way of twinning) at the deformation of a nan-subject under uniaxial loading .", "The effect of the deformation rate on the behavior of the system of deformed nano-grains is also presented.", "At lower deformation rates, system accommodation takes place at smaller level of the accumulated potential energy.", "The authors express their gratitude to Dr. of Sci.", "Pashinskaya E.G.", "and Dr. of Sci.", "Konstantinova T.E.", "for effective discussion and valuable advices and comments." ] ]

[ [ "Distribution-Dependent Sample Complexity of Large Margin Learning" ], [ "Abstract We obtain a tight distribution-specific characterization of the sample complexity of large-margin classification with L2 regularization: We introduce the margin-adapted dimension, which is a simple function of the second order statistics of the data distribution, and show distribution-specific upper and lower bounds on the sample complexity, both governed by the margin-adapted dimension of the data distribution.", "The upper bounds are universal, and the lower bounds hold for the rich family of sub-Gaussian distributions with independent features.", "We conclude that this new quantity tightly characterizes the true sample complexity of large-margin classification.", "To prove the lower bound, we develop several new tools of independent interest.", "These include new connections between shattering and hardness of learning, new properties of shattering with linear classifiers, and a new lower bound on the smallest eigenvalue of a random Gram matrix generated by sub-Gaussian variables.", "Our results can be used to quantitatively compare large margin learning to other learning rules, and to improve the effectiveness of methods that use sample complexity bounds, such as active learning." ], [ "Introduction", "In this paper we pursue a tight characterization of the sample complexity of learning a classifier, under a particular data distribution, and using a particular learning rule.", "Most learning theory work focuses on providing sample-complexity upper bounds which hold for a large class of distributions.", "For instance, standard distribution-free VC-dimension analysis shows that if one uses the Empirical Risk Minimization (ERM) learning rule, then the sample complexity of learning a classifier from a hypothesis class with VC-dimension $d$ is at most $O\\left(\\frac{d}{\\epsilon ^2}\\right)$ , where $\\epsilon $ is the maximal excess classification error [32], [1].", "Such upper bounds can be useful for understanding the positive aspects of a learning rule.", "However, it is difficult to understand the deficiencies of a learning rule, or to compare between different rules, based on upper bounds alone.", "This is because it is possible, and is often the case, that the actual number of samples required to get a low error, for a given data distribution using a given learning rule, is much lower than the sample-complexity upper bound.", "As a simple example, suppose that the support of a given distribution is restricted to a subset of the domain.", "If the VC-dimension of the hypothesis class, when restricted to this subset, is smaller than $d$ , then learning with respect to this distribution will require less examples than the upper bound predicts.", "Of course, some sample complexity upper bounds are known to be tight or to have an almost-matching lower bound.", "For instance, the VC-dimension upper bound is tight [33].", "This means that there exists some data distribution in the class covered by the upper bound, for which this bound cannot be improved.", "Such a tightness result shows that there cannot be a better upper bound that holds for this entire class of distributions.", "But it does not imply that the upper bound characterizes the true sample complexity for every specific distribution in the class.", "The goal of this paper is to identify a simple quantity, which is a function of the distribution, that does precisely characterize the sample complexity of learning this distribution under a specific learning rule.", "We focus on the important hypothesis class of linear classifiers, and on the popular rule of margin-error-minimization (MEM).", "Under this learning rule, a learner must always select a linear classifier that minimizes the margin-error on the input sample.", "The VC-dimension of the class of homogeneous linear classifiers in $\\mathbb {R}^d$ is $d$ [14].", "This implies a sample complexity upper bound of $O\\left(\\frac{d}{\\epsilon ^2}\\right)$ using any MEM algorithm, where $\\epsilon $ is the excess error relative to the optimal margin error.This upper bound can be derived analogously to the result for ERM algorithms with $\\epsilon $ being the excess classification error.", "It can also be concluded from our analysis in Theorem  below.", "We also have that the sample complexity of any MEM algorithm is at most $O\\big (\\frac{B^2}{\\gamma ^2\\epsilon ^2}\\big )$ , where $B^2$ is the average squared norm of the data and $\\gamma $ is the size of the margin [6].", "Both of these upper bounds are tight.", "For instance, there exists a distribution with an average squared norm of $B^2$ that requires as many as $C\\cdot \\frac{B^2}{\\gamma ^2\\epsilon ^2}$ examples to learn, for some universal constant $C$ [1].", "However, the VC-dimension upper bound indicates, for instance, that if a distribution induces a large average norm but is supported by a low-dimensional sub-space, then the true number of examples required to reach a low error is much smaller.", "Thus, neither of these upper bounds fully describes the sample complexity of MEM for a specific distribution.", "We obtain a tight distribution-specific characterization of the sample complexity of large-margin learning for a rich class of distributions.", "We present a new quantity, termed the margin-adapted dimension, and use it to provide a tighter distribution-dependent upper bound, and a matching distribution-dependent lower bound for MEM.", "The upper bound is universal, and the lower bound holds for a rich class of distributions with independent features.", "The margin-adapted dimension refines both the dimension and the average norm of the data distribution, and can be easily calculated from the covariance matrix and the mean of the distribution.", "We denote this quantity, for a margin of $\\gamma $ , by $k_\\gamma $ .", "Our sample-complexity upper bound shows that $\\tilde{O}(\\frac{k_\\gamma }{\\epsilon ^2})$ examples suffice in order to learn any distribution with a margin-adapted dimension of $k_\\gamma $ using a MEM algorithm with margin $\\gamma $ .", "We further show that for every distribution in a rich family of `light tailed' distributions—specifically, product distributions of sub-Gaussian random variables—the number of samples required for learning by minimizing the margin error is at least $\\Omega (k_{\\gamma })$ .", "Denote by $m(\\epsilon ,\\gamma ,D)$ the number of examples required to achieve an excess error of no more than $\\epsilon $ relative to the best possible $\\gamma $ -margin error for a specific distribution $D$ , using a MEM algorithm.", "Our main result shows the following matching distribution-specific upper and lower bounds on the sample complexity of MEM: $\\Omega (k_\\gamma (D)) \\le m(\\epsilon ,\\gamma ,D) \\le \\tilde{O}\\left(\\frac{k_{\\gamma }(D)}{\\epsilon ^2}\\right).$ Our tight characterization, and in particular the distribution-specific lower bound on the sample complexity that we establish, can be used to compare large-margin ($L_2$ regularized) learning to other learning rules.", "We provide two such examples: we use our lower bound to rigorously establish a sample complexity gap between $L_1$ and $L_2$ regularization previously studied in [22], and to show a large gap between discriminative and generative learning on a Gaussian-mixture distribution.", "The tight bounds can also be used for active learning algorithms in which sample-complexity bounds are used to decide on the next label to query.", "In this paper we focus only on large margin classification.", "But in order to obtain the distribution-specific lower bound, we develop new tools that we believe can be useful for obtaining lower bounds also for other learning rules.", "We provide several new results which we use to derive our main results.", "These include: Linking the fat-shattering of a sample with non-negligible probability to a difficulty of learning using MEM.", "Showing that for a convex hypothesis class, fat-shattering is equivalent to shattering with exact margins.", "Linking the fat-shattering of a set of vectors with the eigenvalues of the Gram matrix of the vectors.", "Providing a new lower bound for the smallest eigenvalue of a random Gram matrix generated by sub-Gaussian variables.", "This bound extends previous results in analysis of random matrices." ], [ "Paper Structure", "We discuss related work on sample-complexity upper bounds in Section .", "We present the problem setting and notation in Section , and provide some necessary preliminaries in Section .", "We then introduce the margin-adapted dimension in Section .", "The sample-complexity upper bound is proved in Section .", "We prove the lower bound in Section .", "In Section  we show that any non-trivial sample-complexity lower bound for more general distributions must employ properties other than the covariance matrix of the distribution.", "We summarize and discuss implication in Section .", "Proofs omitted from the text are provided in Appendix" ], [ "Related Work", "As mentioned above, most work on “sample complexity lower bounds” is directed at proving that under some set of assumptions, there exists a data distribution for which one needs at least a certain number of examples to learn with required error and confidence [2], [15], [16].", "This type of a lower bound does not, however, indicate much on the sample complexity of other distributions under the same set of assumptions.", "For distribution-specific lower bounds, the classical analysis of [31] provides not only sufficient but also necessary conditions for the learnability of a hypothesis class with respect to a specific distribution.", "The essential condition is that the metric entropy of the hypothesis class with respect to the distribution be sub-linear in the limit of an infinite sample size.", "In some sense, this criterion can be seen as providing a “lower bound” on learnability for a specific distribution.", "However, we are interested in finite-sample convergence rates, and would like those to depend on simple properties of the distribution.", "The asymptotic arguments involved in Vapnik's general learnability claim do not lend themselves easily to such analysis.", "[9] show that if the distribution is known to the learner, a specific hypothesis class is learnable if and only if there is a finite $\\epsilon $ -cover of this hypothesis class with respect to the distribution.", "[8] consider a similar setting, and prove sample complexity lower bounds for learning with any data distribution, for some binary hypothesis classes on the real line.", "[34] provide distribution-specific sample complexity upper bounds for hypothesis classes with a limited VC-dimension, as a function of how balanced the hypotheses are with respect to the considered distributions.", "These bounds are not tight for all distributions, thus they also do not fully characterize the distribution-specific sample complexity.", "As can be seen in Equation (REF ), we do not tightly characterize the dependence of the sample complexity on the desired error [29], thus our bounds are not tight for asymptotically small error levels.", "Our results are most significant if the desired error level is a constant well below chance but bounded away from zero.", "This is in contrast to classical statistical asymptotics that are also typically tight, but are valid only for very small $\\epsilon $ .", "As was recently shown by [19], the sample complexity for very small $\\epsilon $ (in the classical statistical asymptotic regime) depends on quantities that can be very different from those that control the sample complexity for moderate error rates, which are more relevant for machine learning." ], [ "Problem Setting and Definitions", "Consider a domain $\\mathcal {X}$ , and let $D$ be a distribution over $\\mathcal {X}\\times \\lbrace \\pm 1\\rbrace $ .", "We denote by $D_X$ the marginal distribution of $D$ on $\\mathcal {X}$ .", "The misclassification error of a classifier $h:\\mathcal {X}\\rightarrow \\mathbb {R}$ on a distribution $D$ is $\\ell _0(h,D) \\triangleq \\mathbb {P}_{(X,Y)\\sim D}[Y\\cdot h(X) \\le 0].$ The margin error of a classifier $w$ with respect to a margin $\\gamma > 0$ on $D$ is $\\ell _\\gamma (h,D) \\triangleq \\mathbb {P}_{(X,Y)\\sim D}[Y\\cdot h(X) \\le \\gamma ].$ For a given hypothesis class $\\mathcal {H}\\subseteq \\lbrace \\pm 1\\rbrace ^\\mathcal {X}$ , the best achievable margin error on $D$ is $\\ell ^*_\\gamma (\\mathcal {H}, D) \\triangleq \\inf _{h \\in \\mathcal {H}}\\ell _\\gamma (h,D).$ We usually write simply $\\ell ^*_\\gamma (D)$ since $\\mathcal {H}$ is clear from context.", "A labeled sample is a (multi-)set $S = \\lbrace (x_i,y_i)\\rbrace _{i=1}^m \\subseteq \\mathcal {X}\\times \\lbrace \\pm 1\\rbrace $ .", "Given $S$ , we denote the set of its examples without their labels by $S_X \\triangleq \\lbrace x_1,\\ldots ,x_m\\rbrace $ .", "We use $S$ also to refer to the uniform distribution over the elements in $S$ .", "Thus the misclassification error of $h:\\mathcal {X}\\rightarrow \\lbrace \\pm 1\\rbrace $ on $S$ is $\\ell (h,S) \\triangleq \\frac{1}{m}|\\lbrace i \\mid y_i \\cdot h(x_i) \\le 0\\rbrace |,$ and the $\\gamma $ -margin error on $S$ is $\\ell _\\gamma (h,S) \\triangleq \\frac{1}{m}|\\lbrace i \\mid y_i \\cdot h(x_i) \\le \\gamma \\rbrace |.$ A learning algorithm is a function $\\mathcal {A}:\\cup _{m=1}^\\infty (\\mathcal {X}\\times \\lbrace \\pm 1\\rbrace )^m \\rightarrow \\mathbb {R}^\\mathcal {X}$ , that receives a training set as input, and returns a function for classifying objects in $\\mathcal {X}$ into real values.", "The high-probability loss of an algorithm $\\mathcal {A}$ with respect to samples of size $m$ , a distribution $D$ and a confidence parameter $\\delta \\in (0,1)$ is $\\ell (\\mathcal {A},D,m,\\delta ) = \\inf \\lbrace \\epsilon \\ge 0 \\mid \\mathbb {P}_{S \\sim D^m}[\\ell (\\mathcal {A}(S), D) \\ge \\epsilon ] \\le \\delta \\rbrace .$ In this work we investigate the sample complexity of learning using margin-error minimization (MEM).", "The relevant class of algorithms is defined as follows.", "An margin-error minimization (MEM) algorithm $\\mathcal {A}$ maps a margin parameter $\\gamma > 0$ to a learning algorithm $\\mathcal {A}_\\gamma $ , such that $\\forall S \\subseteq \\mathcal {X}\\times \\lbrace \\pm 1\\rbrace , \\quad \\mathcal {A}_\\gamma (S) \\in \\operatornamewithlimits{argmin}_{h \\in \\mathcal {H}} \\ell _\\gamma (h,S).$ The distribution-specific sample complexity for MEM algorithms is the sample size required to guarantee low excess error for the given distribution.", "Formally, we have the following definition.", "[Distribution-specific sample complexity] Fix a hypothesis class $\\mathcal {H}\\subseteq \\lbrace \\pm 1\\rbrace ^\\mathcal {X}$ .", "For $\\gamma >0$ , $\\epsilon ,\\delta \\in [0,1]$ , and a distribution $D$ , the distribution-specific sample complexity, denoted by $m(\\epsilon ,\\gamma ,D,\\delta )$ , is the minimal sample size such that for any MEM algorithm $\\mathcal {A}$ , and for any $m \\ge m(\\epsilon ,\\gamma ,D,\\delta )$ , $\\ell _0(\\mathcal {A}_\\gamma ,D,m,\\delta ) - \\ell ^*_\\gamma (D) \\le \\epsilon .$ Note that we require that all possible MEM algorithms do well on the given distribution.", "This is because we are interested in the MEM strategy in general, and thus we study the guarantees that can be provided regardless of any specific MEM implementation.", "We sometimes omit $\\delta $ and write simply $m(\\epsilon , \\gamma , D)$ , to indicate that $\\delta $ is assumed to be some fixed small constant.", "In this work we focus on linear classifiers.", "For simplicity of notation, we assume a Euclidean space $\\mathbb {R}^d$ for some integer $d$ , although the results can be easily extended to any separable Hilbert space.", "For a real vector $x$ , $\\Vert x\\Vert $ stands for the Euclidean norm.", "For a real matrix $\\mathbb {X}$ , $\\Vert \\mathbb {X}\\Vert $ stands for the Euclidean operator norm.", "Denote the unit ball in $\\mathbb {R}^d$ by $\\mathbb {B}^d_1\\triangleq \\lbrace w\\in \\mathbb {R}^d \\mid \\Vert w\\Vert \\le 1\\rbrace $ .", "We consider the hypothesis class of homogeneous linear separators, $\\mathcal {W}= \\lbrace x \\mapsto \\langle x,w \\rangle \\mid w \\in \\mathbb {B}^d_1\\rbrace $ .", "We often slightly abuse notation by using $w$ to denote the mapping $x \\mapsto \\langle x,w \\rangle $ .", "We often represent sets of vectors in $\\mathbb {R}^d$ using matrices.", "We say that $\\mathbb {X} \\in \\mathbb {R}^{m\\times d}$ is the matrix of a set $\\lbrace x_1,\\ldots ,x_m\\rbrace \\subseteq \\mathbb {R}^d$ if the rows in the matrix are exactly the vectors in the set.", "For uniqueness, one may assume that the rows of $\\mathbb {X}$ are sorted according to an arbitrary fixed full order on vectors in $\\mathbb {R}^d$ .", "For a PSD matrix $\\mathbb {X}$ denote the largest eigenvalue of $\\mathbb {X}$ by $\\lambda _{\\max }(\\mathbb {X})$ and the smallest eigenvalue by $\\lambda _{\\min }(\\mathbb {X})$ .", "We use the $O$ -notation as follows: $O(f(z))$ stands for $C_1+C_2f(z)$ for some constants $C_1,C_2\\ge 0$ .", "$\\Omega (f(z))$ stands for $C_2f(z)-C_1$ for some constants $C_1,C_2\\ge 0$ .", "$\\widetilde{O}(f(z))$ stands for $f(z)p(\\ln (z)) + C$ for some polynomial $p(\\cdot )$ and some constant $C > 0$ ." ], [ "Preliminaries", "As mentioned above, for the hypothesis class of linear classifiers $\\mathcal {W}$ , one can derive a sample-complexity upper bound of the form $O(B^2/\\gamma ^2 \\epsilon ^2)$ , where $B^2 = \\mathbb {E}_{X \\sim D}[\\Vert X\\Vert ^2]$ and $\\epsilon $ is the excess error relative to the $\\gamma $ -margin loss.", "This can be achieved as follows [6].", "Let $\\mathcal {Z}$ be some domain.", "The empirical Rademacher complexity of a class of functions $\\mathcal {F}\\subseteq \\mathbb {R}^\\mathcal {Z}$ with respect to a set $S = \\lbrace z_i\\rbrace _{i\\in [m]} \\subseteq \\mathcal {Z}$ is $\\mathcal {R}(\\mathcal {F}, S) = \\frac{1}{m}\\mathbb {E}_\\sigma [|\\sup _{f \\in \\mathcal {F}} \\sum _{i\\in [m]}\\sigma _i f(z_i)|],$ where $\\sigma = (\\sigma _1,\\ldots ,\\sigma _m)$ are $m$ independent uniform $\\lbrace \\pm 1\\rbrace $ -valued variables.", "The average Rademacher complexity of $\\mathcal {F}$ with respect to a distribution $D$ over $\\mathcal {Z}$ and a sample size $m$ is $\\mathcal {R}_m(\\mathcal {F}, D) = \\mathbb {E}_{S \\sim D^m}[\\mathcal {R}(\\mathcal {F}, S)].$ Assume a hypothesis class $\\mathcal {H}\\subseteq \\mathbb {R}^\\mathcal {X}$ and a loss function $\\ell :\\lbrace \\pm 1\\rbrace \\times \\mathbb {R}\\rightarrow \\mathbb {R}$ .", "For a hypothesis $h \\in \\mathcal {H}$ , we introduce the function $h_\\ell :\\mathcal {X}\\times \\lbrace \\pm 1\\rbrace \\rightarrow \\mathbb {R}$ , defined by $h_\\ell (x,y) = \\ell (y,h(x))$ .", "We further define the function class $\\mathcal {H}_\\ell = \\lbrace h_\\ell \\mid h \\in \\mathcal {H}\\rbrace \\subseteq \\mathbb {R}^{\\mathcal {X}\\times \\lbrace \\pm 1\\rbrace }$ .", "Assume that the range of $\\mathcal {H}_\\ell $ is in $[0,1]$ .", "For any $\\delta \\in (0,1)$ , with probability of $1-\\delta $ over the draw of samples $S \\subseteq \\mathcal {X}\\times \\lbrace \\pm 1\\rbrace $ of size $m$ according to $D$ , every $h \\in \\mathcal {H}$ satisfies [6] $\\ell (h, D) \\le \\ell (h, S) + 2\\mathcal {R}_m(\\mathcal {H}_\\ell , D) + \\sqrt{\\frac{8\\ln (2/\\delta )}{m}}.$ To get the desired upper bound for linear classifiers we use the ramp loss, which is defined as follows.", "For a number $r$ , denote $\\llbracket {r} \\rrbracket \\triangleq \\min (\\max (r,0),1)$ .", "The $\\gamma $ -ramp-loss of a labeled example $(x,y) \\in \\mathbb {R}^d \\times \\lbrace \\pm 1\\rbrace $ with respect to a linear classifier $w \\in \\mathbb {B}^d_1$ is $\\mathrm {ramp}_\\gamma (w,x,y) = \\llbracket {1-y\\langle w,x \\rangle /\\gamma } \\rrbracket $ .", "Let $\\mathrm {ramp}_\\gamma (w,D) = \\mathbb {E}_{(X,Y)\\sim D}[\\mathrm {ramp}_\\gamma (w,X,Y)]$ , and denote the class of ramp-loss functions by $\\textsc {ramp}_\\gamma = \\lbrace (x,y) \\mapsto \\mathrm {ramp}_\\gamma (w,x,y) \\mid w \\in \\mathbb {B}^d_1\\rbrace .$ The ramp-loss is upper-bounded by the margin loss and lower-bounded by the misclassification error.", "Therefore, the following result can be shown.", "For any MEM algorithm $\\mathcal {A}$ , we have $\\ell _0(\\mathcal {A}_\\gamma , D, m, \\delta ) \\le \\ell ^*_\\gamma (\\mathcal {H},D) + 2\\mathcal {R}_m(\\textsc {ramp}_\\gamma , D) + \\sqrt{\\frac{14\\ln (2/\\delta )}{m}}.$ We give the proof in Appendix REF for completeness.", "Since the $\\gamma $ -ramp loss is $1/\\gamma $ Lipschitz, it follows from [6] that $\\mathcal {R}_m(\\textsc {ramp}_\\gamma , D) \\le \\sqrt{\\frac{B^2}{\\gamma ^2 m}}.$ Combining this with Proposition  we can conclude a sample complexity upper bound of $O(B^2/\\gamma ^2 \\epsilon ^2)$ .", "In addition to the Rademacher complexity, we will also use the classic notions of fat-shattering [17] and pseudo-shattering [24], defined as follows.", "Let $\\mathcal {F}$ be a set of functions $f:\\mathcal {X}\\rightarrow \\mathbb {R}$ , and let $\\gamma >0$ .", "The set $\\lbrace x_1,\\ldots ,x_m\\rbrace \\subseteq \\mathcal {X}$ is $\\gamma $ -shattered by $\\mathcal {F}$ with the witness $r \\in \\mathbb {R}^m$ if for all $y \\in \\lbrace \\pm 1\\rbrace ^m$ there is an $f \\in \\mathcal {F}$ such that $\\forall i\\in [m],\\:y[i](f(x_i)-r[i]) \\ge \\gamma $ .", "The $\\gamma $ -shattering dimension of a hypothesis class is the size of the largest set that is $\\gamma $ -shattered by this class.", "We say that a set is $\\gamma $ -shattered at the origin if it is $\\gamma $ -shattered with the zero vector as a witness.", "Let $\\mathcal {F}$ be a set of functions $f:\\mathcal {X}\\rightarrow \\mathbb {R}$ , and let $\\gamma >0$ .", "The set $\\lbrace x_1,\\ldots ,x_m\\rbrace \\subseteq \\mathcal {X}$ is pseudo-shattered by $\\mathcal {F}$ with the witness $r \\in \\mathbb {R}^m$ if for all $y \\in \\lbrace \\pm 1\\rbrace ^m$ there is an $f \\in \\mathcal {F}$ such that $\\forall i\\in [m],\\:y[i](f(x_i)-r[i]) > 0$ .", "The pseudo-dimension of a hypothesis class is the size of the largest set that is pseudo-shattered by this class." ], [ "The Margin-Adapted Dimension", "When considering learning of linear classifiers using MEM, the dimension-based upper bound and the norm-based upper bound are both tight in the worst-case sense, that is, they are the best bounds that rely only on the dimensionality or only on the norm respectively.", "Nonetheless, neither is tight in a distribution-specific sense: If the average norm is unbounded while the dimension is small, then there can be an arbitrarily large gap between the true distribution-dependent sample complexity and the bound that depends on the average norm.", "If the converse holds, that is, the dimension is arbitrarily large while the average-norm is bounded, then the dimensionality bound is loose.", "Seeking a tight distribution-specific analysis, one simple approach to tighten these bounds is to consider their minimum, which is proportional to $\\min (d,B^2/\\gamma ^2)$ .", "Trivially, this is an upper bound on the sample complexity as well.", "However, this simple combination is also not tight: Consider a distribution in which there are a few directions with very high variance, but the combined variance in all other directions is small (see Figure REF ).", "We will show that in such situations the sample complexity is characterized not by the minimum of dimension and norm, but by the sum of the number of high-variance dimensions and the average squared norm in the other directions.", "This behavior is captured by the margin-adapted dimension which we presently define, using the following auxiliary definition.", "Figure: Illustrating covariance matrix ellipsoids.", "left: norm bound is tight; middle: dimension bound is tight; right: neither bound is tight.Let $b>0$ and let $k$ be a positive integer.", "A distribution $D_X$ over $\\mathbb {R}^d$ is $(b,k)$ -limited if there exists a sub-space $V \\subseteq \\mathbb {R}^d$ of dimension $d-k$ such that $\\mathbb {E}_{X\\sim D_X}[\\Vert \\mathbb {O}_V\\cdot X\\Vert ^2] \\le b,$ where $\\mathbb {O}_V$ is an orthogonal projection onto $V$ .", "[margin-adapted dimension]The margin-adapted dimension of a distribution $D_X$ , denoted by $k_\\gamma (D_X)$ , is the minimum $k$ such that the distribution is $(\\gamma ^2k,k)$ -limited.", "We sometimes drop the argument of $k_\\gamma $ when it is clear from context.", "It is easy to see that for any distribution $D_X$ over $\\mathbb {R}^d$ , $k_{\\gamma }(D_X) \\le \\min (d,\\mathbb {E}[\\Vert X\\Vert ^2]/\\gamma ^2)$ .", "Moreover, $k_\\gamma $ can be much smaller than this minimum.", "For example, consider a random vector $X \\in \\mathbb {R}^{1001}$ with mean zero and statistically independent coordinates, such that the variance of the first coordinate is 1000, and the variance in each remaining coordinate is $0.001$ .", "We have $k_1=1$ but $d = \\mathbb {E}[\\Vert X\\Vert ^2] =1001$ .", "$k_\\gamma (D_X)$ can be calculated from the uncentered covariance matrix $\\mathbb {E}_{X\\sim D_X}[XX^T]$ as follows: Let $\\lambda _1 \\ge \\lambda _2 \\ge \\cdots \\lambda _d\\ge 0$ be the eigenvalues of this matrix.", "Then $k_{\\gamma } = \\min \\lbrace k \\mid \\sum _{i=k+1}^d \\lambda _i \\le \\gamma ^2 k\\rbrace .$ A quantity similar to this definition of $k_\\gamma $ was studied previously in [12].", "The eigenvalues of the empirical covariance matrix were used to provide sample complexity bounds, for instance in [28].", "However, $k_\\gamma $ generates a different type of bound, since it is defined based on the eigenvalues of the distribution and not of the sample.", "We will see that for small finite samples, the latter can be quite different from the former.", "Finally, note that while we define the margin-adapted dimension for a finite-dimensional space for ease of notation, the same definition carries over to an infinite-dimensional Hilbert space.", "Moreover, $k_\\gamma $ can be finite even if some of the eigenvalues $\\lambda _i$ are infinite, implying a distribution with unbounded covariance." ], [ "A Distribution-Dependent Upper Bound", "In this section we prove an upper bound on the sample complexity of learning with MEM, using the margin-adapted dimension.", "We do this by providing a tighter upper bound for the Rademacher complexity of $\\textsc {ramp}_\\gamma $ .", "We bound $\\mathcal {R}_m(\\textsc {ramp}_\\gamma ,D)$ for any $(B^2,k)$ -limited distribution $D_X$ , using $L_2$ covering numbers, defined as follows.", "Let $(\\mathcal {X}, \\Vert \\cdot \\Vert _\\circ )$ be a normed space.", "An $\\eta $ -covering of a set $\\mathcal {F}\\subseteq \\mathcal {X}$ with respect to the norm $\\Vert \\cdot \\Vert _\\circ $ is a set $\\mathcal {C}\\subseteq \\mathcal {X}$ such that for any $f\\in \\mathcal {F}$ there exists a $g \\in \\mathcal {C}$ such that $\\Vert f -g\\Vert _\\circ \\le \\eta .$ The covering-number for given $\\eta > 0$ , $\\mathcal {F}$ and $\\circ $ is the size of the smallest such $\\eta $ -covering, and is denoted by $\\mathcal {N}(\\eta , \\mathcal {F}, \\circ )$ .", "Let $S = \\lbrace x_1,\\ldots ,x_m\\rbrace \\subseteq \\mathbb {R}^d$ .", "For a function $f:\\mathbb {R}^d \\rightarrow \\mathbb {R}$ , the $L_2(S)$ norm of $f$ is $\\Vert f\\Vert _{L_2(S)} = \\sqrt{\\mathbb {E}_{X \\sim S}[f(X)^2]}$ .", "Thus, we consider covering-numbers of the form $\\mathcal {N}(\\eta , \\textsc {ramp}_\\gamma , L_2(S))$ .", "The empirical Rademacher complexity of a function class can be bounded by the $L_2$ covering numbers of the same function class as follows [20]: Let $\\epsilon _i = 2^{-i}$ .", "Then $\\sqrt{m}\\mathcal {R}(\\textsc {ramp}_\\gamma ,S) \\le C\\sum _{i\\in [N]}\\epsilon _{i-1}\\sqrt{\\ln \\mathcal {N}(\\epsilon _{i}, \\textsc {ramp}_\\gamma , L_2(S))} + 2\\epsilon _{N}\\sqrt{m}.$ To bound the covering number of $\\textsc {ramp}_\\gamma $ , we will restate the functions in $\\textsc {ramp}_\\gamma $ as sums of two functions, each selected from a function class with bounded complexity.", "The first function class will be bounded because of the norm bound on the subspace $V$ used in Definition , and the second function class will have a bounded pseudo-dimension.", "However, the second function class will depend on the choice of the first function in the sum.", "Therefore, we require the following lemma, which provides an upper bound on such sums of functions.", "We use the notion of a Hausdorff distance between two sets $\\mathcal {G}_1,\\mathcal {G}_2\\subseteq \\mathcal {X}$ , defined as $\\Delta _H(\\mathcal {G}_1,\\mathcal {G}_2) = \\sup _{g_1 \\in \\mathcal {G}_1} \\inf _{g_2 \\in \\mathcal {G}_2} \\Vert g_1 - g_2\\Vert _\\circ $ .", "Let $(\\mathcal {X}, \\Vert \\cdot \\Vert _\\circ )$ be a normed space.", "Let $\\mathcal {F}\\subseteq \\mathcal {X}$ be a set, and let $\\mathcal {G}:\\mathcal {X}\\rightarrow 2^{\\mathcal {X}}$ be a mapping from objects in $\\mathcal {X}$ to sets of objects in $\\mathcal {X}$ .", "Assume that $\\mathcal {G}$ is $c$ -Lipschitz with respect to the Hausdorff distance on sets, that is assume that $\\forall f_1,f_2\\in \\mathcal {X}, \\Delta _H(\\mathcal {G}(f_1),\\mathcal {G}(f_2)) \\le c\\Vert f_1 - f_2\\Vert _\\circ .$ Let $\\mathcal {F}_\\mathcal {G}= \\lbrace f+g \\mid f \\in \\mathcal {F}, g \\in \\mathcal {G}(f)\\rbrace $ .", "Then $\\mathcal {N}(\\eta , \\mathcal {F}_\\mathcal {G}, \\circ ) \\le \\mathcal {N}(\\eta /(2+c), \\mathcal {F}, \\circ )\\cdot \\sup _{f\\in \\mathcal {F}}\\mathcal {N}(\\eta /(2+c), \\mathcal {G}(f),\\circ ).$ For any set $A \\subseteq \\mathcal {X}$ , denote by $\\mathcal {C}_A$ a minimal $\\eta $ -covering for $A$ with respect to $\\Vert \\cdot \\Vert _\\circ $ , so that $|\\mathcal {C}_A| = \\mathcal {N}(\\eta , A, \\circ )$ .", "Let $f+g \\in \\mathcal {F}_\\mathcal {G}$ such that $f \\in \\mathcal {F}, g \\in \\mathcal {G}(f)$ .", "There is a $\\hat{f} \\in \\mathcal {C}_\\mathcal {F}$ such that $\\Vert f - \\hat{f}\\Vert _\\circ \\le \\eta $ .", "In addition, by the Lipschitz assumption there is a $\\tilde{g} \\in \\mathcal {G}(\\hat{f})$ such that $\\Vert g - \\tilde{g}\\Vert _\\circ \\le c\\Vert f - \\hat{f}\\Vert _\\circ \\le c\\eta $ .", "Lastly, there is a $\\hat{g} \\in \\mathcal {C}_{\\mathcal {G}(\\hat{f})}$ such that $\\Vert \\tilde{g} - \\hat{g}\\Vert _\\circ \\le \\eta $ .", "Therefore $\\Vert f + g - (\\hat{f} + \\hat{g})\\Vert _\\circ \\le \\Vert f - \\hat{f}\\Vert _\\circ + \\Vert g- \\tilde{g}\\Vert _\\circ + \\Vert \\tilde{g} - \\hat{g}\\Vert _\\circ \\le (2+c)\\eta .$ Thus the set $\\lbrace f+g \\mid f \\in \\mathcal {C}_\\mathcal {F}, g \\in \\mathcal {C}_{\\mathcal {G}(f)}\\rbrace $ is a $(2+c)\\eta $ cover of $\\mathcal {F}_\\mathcal {G}$ .", "The size of this cover is at most $|\\mathcal {C}_\\mathcal {F}|\\cdot \\sup _{f \\in \\mathcal {F}}|\\mathcal {C}_{\\mathcal {G}(f)}| \\le \\mathcal {N}(\\eta , \\mathcal {F}, \\circ )\\cdot \\sup _{f\\in \\mathcal {F}}\\mathcal {N}(\\eta , \\mathcal {G}(f),\\circ )$ .", "The following lemma provides us with a useful class of mappings which are 1-Lipschitz with respect to the Hausdorff distance, as required in Lemma .", "The proof is provided in Appendix REF .", "Let $f:\\mathcal {X}\\rightarrow \\mathbb {R}$ be a function and let $Z \\subseteq \\mathbb {R}^\\mathcal {X}$ be a function class over some domain $\\mathcal {X}$ .", "Let $\\mathcal {G}:\\mathbb {R}^\\mathcal {X}\\rightarrow 2^{\\mathbb {R}^\\mathcal {X}}$ be the mapping defined by $\\mathcal {G}(f) \\triangleq \\lbrace x \\mapsto \\llbracket {f(x) + z(x)} \\rrbracket - f(x) \\mid z \\in Z\\rbrace .$ Then $\\mathcal {G}$ is 1-Lipschitz with respect to the Hausdorff distance.", "The function class induced by the mapping above preserves the pseudo-dimension of the original function class, as the following lemma shows.", "The proof is provided in Appendix REF .", "Let $f:\\mathcal {X}\\rightarrow \\mathbb {R}$ be a function and let $Z \\subseteq \\mathbb {R}^\\mathcal {X}$ be a function class over some domain $\\mathcal {X}$ .", "Let $\\mathcal {G}(f)$ be defined as in Equation (REF ).", "Then the pseudo-dimension of $\\mathcal {G}(f)$ is at most the pseudo-dimension of $Z$ .", "Equipped with these lemmas, we can now provide the new bound on the Rademacher complexity of $\\textsc {ramp}_\\gamma $ in the following theorem.", "The subsequent corollary states the resulting sample-complexity upper bound for MEM, which depends on $k_\\gamma $ .", "Let $D$ be a distribution over $\\mathbb {R}^d \\times \\lbrace \\pm 1\\rbrace $ , and assume $D_X$ is $(B^2,k)$ -limited.", "Then $\\mathcal {R}(\\textsc {ramp}_\\gamma , D) \\le \\sqrt{\\frac{O(k + B^2/\\gamma ^2)\\ln (m)}{m}}.$ In this proof all absolute constants are assumed to be positive and are denoted by $C$ or $C_i$ for some integer $i$ .", "Their values may change from line to line or even within the same line.", "Consider the distribution $\\tilde{D}$ which results from drawing $(X,Y) \\sim D$ and emitting $(Y\\cdot X, 1)$.", "It too is $(B^2,k)$ -limited, and $\\mathcal {R}(\\textsc {ramp}_\\gamma ,D) = \\mathcal {R}(\\textsc {ramp}_\\gamma ,\\tilde{D})$ .", "Therefore, we assume without loss of generality that for all $(X,Y)$ drawn from $D$ , $Y = 1$ .", "Accordingly, we henceforth omit the $y$ argument from $\\mathrm {ramp}_\\gamma (w,x,y)$ and write simply $\\mathrm {ramp}_\\gamma (w,x) \\triangleq \\mathrm {ramp}_\\gamma (w,x,1)$ .", "Following Definition , Let $\\mathbb {O}_V$ be an orthogonal projection onto a sub-space $V$ of dimension $d-k$ such that $\\mathbb {E}_{X\\sim D_X}[\\Vert \\mathbb {O}_V\\cdot X\\Vert ^2] \\le B^2$ .", "Let $\\bar{V}$ be the complementary sub-space to $V$ .", "For a set $S = \\lbrace x_1,\\ldots ,x_m\\rbrace \\subseteq \\mathbb {R}^d$ , denote $B(S) = \\sqrt{\\frac{1}{m}\\sum _{i\\in [m]}\\Vert \\mathbb {O}_V\\cdot X\\Vert ^2}$ .", "We would like to use Equation (REF ) to bound the Rademacher complexity of $\\textsc {ramp}_\\gamma $ .", "Therefore, we will bound $\\mathcal {N}(\\eta ,\\textsc {ramp}_\\gamma , L_2(S))$ for $\\eta > 0$ .", "Note that $\\mathrm {ramp}_\\gamma (w,x) = \\llbracket {1-\\langle w,x \\rangle /\\gamma } \\rrbracket = 1 - \\llbracket {\\langle w,x \\rangle /\\gamma } \\rrbracket .$ Shifting by a constant and negating do not change the covering number of a function class.", "Therefore, $\\mathcal {N}(\\eta ,\\textsc {ramp}_\\gamma , L_2(S))$ is equal to the covering number of $\\lbrace x \\mapsto \\llbracket {\\langle w,x \\rangle /\\gamma } \\rrbracket \\mid w \\in \\mathbb {B}^d_1\\rbrace $ .", "Moreover, let $\\textsc {ramp}_\\gamma ^{\\prime } = \\lbrace x \\mapsto \\llbracket {\\langle w_a + w_b,x \\rangle /\\gamma } \\rrbracket \\mid w_a \\in \\mathbb {B}^d_1\\cap V,\\: w_b \\in \\bar{V}\\rbrace .$ Then $\\lbrace x \\mapsto \\llbracket {\\langle w,x \\rangle /\\gamma } \\rrbracket \\mid w \\in \\mathbb {B}^d_1\\rbrace \\subseteq \\textsc {ramp}_\\gamma ^{\\prime }$ , thus it suffices to bound $\\mathcal {N}(\\eta ,\\textsc {ramp}^{\\prime }_\\gamma , L_2(S))$ .", "To do that, we show that $\\textsc {ramp}_\\gamma ^{\\prime }$ satisfies the assumptions of Lemma  for the normed space $(\\mathbb {R}^{\\mathbb {R}^d},\\Vert \\cdot \\Vert _{L_2(S)})$ .", "Define $\\mathcal {F}= \\lbrace x \\mapsto \\langle w_a,x \\rangle /\\gamma \\mid w_a \\in \\mathbb {B}^d_1\\cap V\\rbrace .$ Let $\\mathcal {G}:\\mathbb {R}^{\\mathbb {R}^d} \\rightarrow 2^{\\mathbb {R}^{\\mathbb {R}^d}}$ be the mapping defined by $\\mathcal {G}(f) \\triangleq \\lbrace x \\mapsto \\llbracket {f(x) + \\langle w_b,x \\rangle /\\gamma } \\rrbracket - f(x) \\mid w_b \\in \\bar{V}\\rbrace .$ Clearly, $\\mathcal {F}_\\mathcal {G}= \\lbrace f + g \\mid f \\in \\mathcal {F}, g \\in \\mathcal {G}(f)\\rbrace = \\textsc {ramp}_\\gamma ^{\\prime }$ .", "Furthermore, by Lemma , $\\mathcal {G}$ is 1-Lipschitz with respect to the Hausdorff distance.", "Thus, by Lemma  $\\mathcal {N}(\\eta , \\textsc {ramp}_\\gamma ^{\\prime }, L_2(S)) \\le \\mathcal {N}(\\eta /3, \\mathcal {F}, L_2(S))\\cdot \\sup _{f\\in \\mathcal {F}}\\mathcal {N}(\\eta /3, \\mathcal {G}(f),L_2(S)).$ We now proceed to bound the two covering numbers on the right hand side.", "First, consider $\\mathcal {N}(\\eta /3, \\mathcal {G}(f),L_2(S))$ .", "By Lemma , the pseudo-dimension of $\\mathcal {G}(f)$ is the same as the pseudo-dimension of $\\lbrace x \\mapsto \\langle w,x \\rangle /\\gamma \\mid w \\in \\bar{V}\\rbrace $ , which is exactly $k$ , the dimension of $\\bar{V}$ .", "The $L_2$ covering number of $\\mathcal {G}(f)$ can be bounded by the pseudo-dimension of $\\mathcal {G}(f)$ as follows [5]: $\\mathcal {N}(\\eta /3,\\mathcal {G}(f),L_2(S)) \\le C_1\\left(\\frac{C_2}{\\eta ^2}\\right)^k.$ Second, consider $\\mathcal {N}(\\eta /3, \\mathcal {F}, L_2(S))$ .", "Sudakov's minoration theorem ([30], and see also [18], Theorem 3.18) states that for any $\\eta > 0$ $\\ln \\mathcal {N}(\\eta , \\mathcal {F}, L_2(S)) \\le \\frac{C}{m\\eta ^2}\\mathbb {E}_{s}^2[\\sup _{f \\in \\mathcal {F}} \\sum _{i\\in [m]}s_i f(x_i)],$ where $s = (s_1,\\ldots ,s_m)$ are independent standard normal variables.", "The right-hand side can be bounded as follows: $&\\gamma \\mathbb {E}_s[\\sup _{f \\in \\mathcal {F}}|\\sum _{i=1}^m s_i f(x_i)|]= \\mathbb {E}_s[\\sup _{w \\in \\mathbb {B}^d_1\\cap V}|\\langle w,\\sum _{i=1}^m s_i x_i \\rangle |]\\\\&\\quad \\le \\mathbb {E}_s[\\Vert \\sum _{i=1}^m s_i\\mathbb {O}_V x_i\\Vert ]\\le \\sqrt{\\mathbb {E}_s[\\Vert \\sum _{i=1}^m s_i\\mathbb {O}_V x_i\\Vert ^2]}= \\sqrt{\\sum _{i\\in [m]}\\Vert \\mathbb {O}_V x_i\\Vert ^2} = \\sqrt{m}B(S).$ Therefore $\\ln \\mathcal {N}(\\eta , \\mathcal {F}, L_2(S)) \\le C\\frac{B^2(S)}{\\gamma ^2\\eta ^2}.$ Substituting this and Equation (REF ) for the right-hand side in Equation (REF ), and adjusting constants, we get $\\ln \\mathcal {N}(\\eta , \\textsc {ramp}_\\gamma , L_2(S)) \\le \\ln \\mathcal {N}(\\eta , \\textsc {ramp}_\\gamma ^{\\prime }, L_2(S)) \\le C_1(1 + k\\ln (\\frac{C_2}{\\eta })+\\frac{B^2(S)}{\\gamma ^2\\eta ^2}),$ To finalize the proof, we plug this inequality into Equation (REF ) to get $&\\sqrt{m}\\mathcal {R}(\\textsc {ramp}_\\gamma , S) \\le C_1\\sum _{i\\in [N]}\\epsilon _{i-1}\\sqrt{1 + k\\ln (C_2/\\epsilon _i)+\\frac{B^2(S)}{\\gamma ^2\\epsilon _i^2}} + 2\\epsilon _{N}\\sqrt{m} \\\\&\\le C_1\\left(\\sum _{i\\in [N]}\\epsilon _{i-1}\\left(1 + \\sqrt{k\\ln (C_2/\\epsilon _i)}+\\sqrt{\\frac{B^2(S)}{\\gamma ^2\\epsilon _i^2}}\\right)\\right) + 2\\epsilon _{N}\\sqrt{m}\\\\&= C_1\\left(\\sum _{i\\in [N]}2^{-i+1} + \\sqrt{k}\\sum _{i\\in [N]}2^{-i+1}\\ln (C_2/2^{-i}) + \\sum _{i\\in [N]}\\frac{B(S)}{\\gamma }\\right) +2^{-N+1}\\sqrt{m} \\\\&\\le C\\left(1 + \\sqrt{k} + \\frac{B(S)\\cdot N}{\\gamma }\\right) + 2^{-N+1}\\sqrt{m}.$ In the last inequality we used the fact that $\\sum _{i}i2^{-i+1} \\le 4$ .", "Setting $N = \\ln (2m)$ we get $&\\mathcal {R}(\\textsc {ramp}_\\gamma , S) \\le \\frac{C}{\\sqrt{m}}\\left(1 + \\sqrt{k} + \\frac{B(S)\\ln (2m)}{\\gamma }\\right).$ Taking expectation over both sides, and noting that $\\mathbb {E}[B(S)]\\le \\sqrt{\\mathbb {E}[B^2(S)]} \\le B$ , we get $\\mathcal {R}(\\textsc {ramp}_\\gamma , S) \\le \\frac{C}{\\sqrt{m}}(1 + \\sqrt{k} + \\frac{B\\ln (2m)}{\\gamma })\\le \\sqrt{\\frac{O(k + B^2\\ln ^2(2m)/\\gamma ^2)}{m}}.$ Corollary (Sample complexity upper bound) Let $D$ be a distribution over $\\mathbb {R}^d\\times \\lbrace \\pm 1\\rbrace $ .", "Then $m(\\epsilon ,\\gamma ,D) \\le \\tilde{O}\\left(\\frac{k_\\gamma (D_X)}{\\epsilon ^2}\\right).$ By Proposition , we have $\\ell _0(\\mathcal {A}_\\gamma , D, m, \\delta ) \\le \\ell ^*_\\gamma (\\mathcal {W},D) + 2\\mathcal {R}_m(\\textsc {ramp}_\\gamma , D) + \\sqrt{\\frac{14\\ln (2/\\delta )}{m}}.$ By definition of $k_\\gamma (D_X)$ , $D_X$ is $(\\gamma ^2 k_\\gamma , k_\\gamma )$ -limited.", "Therefore, by Theorem , $\\mathcal {R}_m(\\textsc {ramp}_\\gamma , D) \\le \\sqrt{\\frac{O(k_\\gamma (D_X))\\ln (m)}{m}}.$ We conclude that $\\ell _0(\\mathcal {A}_\\gamma , D, m, \\delta ) \\le \\ell ^*_\\gamma (\\mathcal {W},D) + \\sqrt{\\frac{O(k_\\gamma (D_X)\\ln (m) + \\ln (1/\\delta ))}{m}}.$ Bounding the second right-hand term by $\\epsilon $ , we conclude that $m(\\epsilon ,\\gamma ,D) \\le \\tilde{O}(k_\\gamma /\\epsilon ^2)$ .", "One should note that a similar upper bound can be obtained much more easily under a uniform upper bound on the eigenvalues of the uncentered covariance matrix.This has been pointed out to us by an anonymous reviewer of this manuscript.", "An upper bound under sub-Gaussianity assumptions can be found in [27].", "However, such an upper bound would not capture the fact that a finite dimension implies a finite sample complexity, regardless of the size of the covariance.", "If one wants to estimate the sample complexity, then large covariance matrix eigenvalues imply that more examples are required to estimate the covariance matrix from a sample.", "However, these examples need not be labeled.", "Moreover, estimating the covariance matrix is not necessary to achieve the sample complexity, since the upper bound holds for any margin-error minimization algorithm." ], [ "A Distribution-Dependent Lower Bound", "The new upper bound presented in Corollary REF can be tighter than both the norm-only and the dimension-only upper bounds.", "But does the margin-adapted dimension characterize the true sample complexity of the distribution, or is it just another upper bound?", "To answer this question, we first need tools for deriving sample complexity lower bounds.", "Section REF relates fat-shattering with a lower bound on sample complexity.", "In Section REF we use this result to relate the smallest eigenvalue of a Gram-matrix to a lower bound on sample complexity.", "In Section REF the family of sub-Gaussian product distributions is presented.", "We prove a sample-complexity lower bound for this family in Section REF ." ], [ "A Sample Complexity Lower Bound Based on Fat-Shattering", "The ability to learn is closely related to the probability of a sample to be shattered, as evident in Vapnik's formulations of learnability as a function of the $\\epsilon $ -entropy [31].", "It is well known that the maximal size of a shattered set dictates a sample-complexity upper bound.", "In the theorem below, we show that for some hypothesis classes it also implies a lower bound.", "The theorem states that if a sample drawn from a data distribution is fat-shattered with a non-negligible probability, then MEM can fail to learn a good classifier for this distribution.In contrast, the average Rademacher complexity cannot be used to derive general lower bounds for MEM algorithms, since it is related to the rate of uniform convergence of the entire hypothesis class, while MEM algorithms choose low-error hypotheses [7].", "This holds not only for linear classifiers, but more generally for all symmetric hypothesis classes.", "Given a domain $\\mathcal {X}$ , we say that a hypothesis class $\\mathcal {H}\\subseteq \\mathbb {R}^\\mathcal {X}$ is symmetric if for all $h \\in \\mathcal {H}$ , we have $-h \\in \\mathcal {H}$ as well.", "This clearly holds for the class of linear classifiers $\\mathcal {W}$ .", "Let $\\mathcal {X}$ be some domain, and assume that $\\mathcal {H}\\subseteq \\mathbb {R}^\\mathcal {X}$ is a symmetric hypothesis class.", "Let $D$ be a distribution over $\\mathcal {X}\\times \\lbrace \\pm 1\\rbrace $ .", "If the probability of a sample of size $m$ drawn from $D_X^{m}$ to be $\\gamma $ -shattered at the origin by $\\mathcal {W}$ is at least $\\eta $ , then $m(\\epsilon ,\\gamma ,D,\\eta /2) \\ge \\lfloor m/2\\rfloor $ for all $\\epsilon < 1/2 - \\ell ^*_\\gamma (D)$ .", "Let $\\epsilon \\le \\frac{1}{2} - \\ell ^*_\\gamma (D)$ .", "We show a MEM algorithm $\\mathcal {A}$ such that $\\ell _0(\\mathcal {A}_\\gamma ,D,\\lfloor m/2\\rfloor ,\\eta /2) \\ge \\frac{1}{2} > \\ell ^*_\\gamma (D) + \\epsilon ,$ thus proving the desired lower bound on $m(\\epsilon ,\\gamma ,D,\\eta /2)$ .", "Assume for simplicity that $m$ is even (otherwise replace $m$ with $m-1$ ).", "Consider two sets $S,\\tilde{S} \\subseteq \\mathcal {X}\\times \\lbrace \\pm 1\\rbrace $ , each of size $m/2$ , such that $S_X\\cup \\tilde{S}_X$ is $\\gamma $ -shattered at the origin by $\\mathcal {W}$ .", "Then there exists a hypothesis $h_1 \\in \\mathcal {H}$ such that the following holds: For all $x \\in S_X \\cup \\tilde{S}_X$ , $|h_1(x)| \\ge \\gamma $ .", "For all $(x,y) \\in S$ , $\\textrm {sign}(h_1(x)) = y$ .", "For all $(x,y) \\in \\tilde{S}$ , $\\textrm {sign}(h_1(x)) = -y$ .", "It follows that $\\ell _\\gamma (h_1,S) = 0$ .", "In addition, let $h_2 = -h_1$ .", "Then $\\ell _\\gamma (h_2,\\tilde{S}) = 0$ .", "Moreover, we have $h_2 \\in \\mathcal {H}$ due to the symmetry of $\\mathcal {H}$ .", "On each point in $\\mathcal {X}$ , at least one of $h_1$ and $h_2$ predict the wrong sign.", "Thus $\\ell _0(h_1,D) + \\ell _0(h_2,D) \\ge 1$ .", "It follows that for at least one of $i \\in \\lbrace 1,2\\rbrace $ , we have $\\ell _0(h_i, D) \\ge {\\frac{1}{2}}$.", "Denote the set of hypotheses with a high misclassification error by $\\mathcal {H}_{\\otimes }= \\lbrace h\\in \\mathcal {H}\\mid \\ell _0(h,D) \\ge {\\frac{1}{2}}\\rbrace .$ We have just shown that if $S_X\\cup \\tilde{S}_X$ is $\\gamma $ -shattered by $\\mathcal {W}$ then at least one of the following holds: (1) $h_1 \\in \\mathcal {H}_{\\otimes }\\cap \\operatornamewithlimits{argmin}_{h\\in \\mathcal {H}} \\ell _\\gamma (h,S)$ or (2) $h_2 \\in \\mathcal {H}_{\\otimes }\\cap \\operatornamewithlimits{argmin}_{h\\in \\mathcal {H}} \\ell _\\gamma (h,\\tilde{S})$ .", "Now, consider a MEM algorithm $\\mathcal {A}$ such that whenever possible, it returns a hypothesis from $\\mathcal {H}_{\\otimes }$ .", "Formally, given the input sample $S$ , if $\\mathcal {H}_{\\otimes }\\cap \\operatornamewithlimits{argmin}_{h\\in \\mathcal {H}} \\ell _\\gamma (h,S) \\ne \\emptyset $ , then $\\mathcal {A}(S) \\in \\mathcal {H}_{\\otimes }\\cap \\operatornamewithlimits{argmin}_{h\\in \\mathcal {H}} \\ell _\\gamma (h,S)$ .", "It follows that $&\\mathbb {P}_{S \\sim D^{m/2}}[\\ell _0(\\mathcal {A}(S),D) \\ge \\tfrac{1}{2}] \\ge \\mathbb {P}_{S \\sim D^{m/2}}[\\mathcal {H}_{\\otimes }\\cap \\operatornamewithlimits{argmin}_{h\\in \\mathcal {H}} \\ell _\\gamma (h, S) \\ne \\emptyset ]\\\\&\\quad = {\\frac{1}{2}}(\\mathbb {P}_{S \\sim D^{m/2}}[\\mathcal {H}_{\\otimes }\\cap \\operatornamewithlimits{argmin}_{h\\in \\mathcal {H}} \\ell _\\gamma (h, S)\\ne \\emptyset ] + \\mathbb {P}_{\\tilde{S}\\sim D^{m/2}}[\\mathcal {H}_{\\otimes }\\cap \\operatornamewithlimits{argmin}_{h\\in \\mathcal {H}} \\ell _\\gamma (h, \\tilde{S}) \\ne \\emptyset ]) \\\\&\\quad \\ge {\\frac{1}{2}}(\\mathbb {P}_{S,\\tilde{S} \\sim D^{m/2}}[\\mathcal {H}_{\\otimes }\\cap \\operatornamewithlimits{argmin}_{h\\in \\mathcal {H}} \\ell _\\gamma (h, S)\\ne \\emptyset \\:\\text{ OR }\\: \\mathcal {H}_{\\otimes }\\cap \\operatornamewithlimits{argmin}_{h\\in \\mathcal {H}} \\ell _\\gamma (h, \\tilde{S}) \\ne \\emptyset ])\\\\&\\quad \\ge {\\frac{1}{2}}\\mathbb {P}_{S,\\tilde{S} \\sim D^{m/2}}[S_X\\cup \\tilde{S}_X \\text{ is $\\gamma $-shattered at the origin }].$ The last inequality follows from the argument above regarding $h_1$ and $h_2$ .", "The last expression is simply half the probability that a sample of size $m$ from $D_X$ is shattered.", "By assumption, this probability is at least $\\eta $ .", "Thus we conclude that $\\mathbb {P}_{S \\sim D^{m/2}}[\\ell _0(\\mathcal {A}(S),D) \\ge {\\frac{1}{2}}] \\ge \\eta /2.$ It follows that $\\ell _0(\\mathcal {A}_\\gamma ,D,m/2,\\eta /2) \\ge {\\frac{1}{2}}$ .", "As a side note, it is interesting to observe that Theorem REF does not hold in general for non-symmetric hypothesis classes.", "For example, assume that the domain is $\\mathcal {X}= [0,1]$ , and the hypothesis class is the set of all functions that label a finite number of points in $[0,1]$ by $+1$ and the rest by $-1$ .", "Consider learning using MEM, when the distribution is uniform over $[0,1]$ , and all the labels are $-1$ .", "For any $m > 0$ and $\\gamma \\in (0,1)$ , a sample of size $m$ is $\\gamma $ -shattered at the origin with probability 1.", "However, any learning algorithm that returns a hypothesis from the hypothesis class will incur zero error on this distribution.", "Thus, shattering alone does not suffice to ensure that learning is hard." ], [ "A Sample Complexity Lower Bound with Gram-Matrix Eigenvalues", "We now return to the case of homogeneous linear classifiers, and link high-probability fat-shattering to properties of the distribution.", "First, we present an equivalent and simpler characterization of fat-shattering for linear classifiers.", "We then use it to provide a sufficient condition for the fat-shattering of a sample, based on the smallest eigenvalue of its Gram matrix.", "Let $\\mathbb {X} \\in \\mathbb {R}^{m\\times d}$ be the matrix of a set of size $m$ in $\\mathbb {R}^d$ .", "The set is $\\gamma $ -shattered at the origin by $\\mathcal {W}$ if and only if $\\mathbb {X}\\mathbb {X}^T$ is invertible and for all $y \\in \\lbrace \\pm 1\\rbrace ^m$ , $y^T (\\mathbb {X}\\mathbb {X}^T)^{-1} y \\le \\gamma ^{-2}$ .", "To prove Theorem REF we require two auxiliary lemmas.", "The first lemma, stated below, shows that for convex function classes, $\\gamma $ -shattering can be substituted with shattering with exact $\\gamma $ -margins.", "Let $\\mathcal {F}\\subseteq \\mathbb {R}^\\mathcal {X}$ be a class of functions, and assume that $\\mathcal {F}$ is convex, that is $\\forall f_1,f_2\\in \\mathcal {F}, \\forall \\lambda \\in [0,1],\\quad \\lambda f_1 + (1-\\lambda )f_2\\in \\mathcal {F}.$ If $S = \\lbrace x_1,\\ldots ,x_m\\rbrace \\subseteq \\mathcal {X}$ is $\\gamma $ -shattered by $\\mathcal {F}$ with witness $r \\in \\mathbb {R}^m$ , then for every $y \\in \\lbrace \\pm 1\\rbrace ^m$ there is an $f \\in \\mathcal {F}$ such that for all $i \\in [m],\\:y[i] (f(x_i) - r[i]) = \\gamma $ .", "The proof of this lemma is provided in Appendix REF .", "The second lemma that we use allows converting the representation of the Gram-matrix to a different feature space, while keeping the separation properties intact.", "For a matrix $\\mathbb {M}$ , denote its pseudo-inverse by $\\mathbb {M}^+$ .", "Let $\\mathbb {X} \\in \\mathbb {R}^{m\\times d}$ be a matrix such that $\\mathbb {X} \\mathbb {X}^T$ is invertible, and let $\\mathbb {Y}\\in \\mathbb {R}^{m\\times k}$ such that $\\mathbb {X}\\mathbb {X}^T = \\mathbb {Y} \\mathbb {Y}^T$ .", "Let $r \\in \\mathbb {R}^m$ be some real vector.", "If there exists a vector $\\widetilde{w} \\in \\mathbb {R}^k$ such that $\\mathbb {Y} \\widetilde{w} = r$ , then there exists a vector $w \\in \\mathbb {R}^d$ such that $\\mathbb {X} w = r \\text{ and } \\Vert w\\Vert = \\Vert \\mathbb {Y}^T (\\mathbb {Y}^T)^+ \\widetilde{w}\\Vert \\le \\Vert \\tilde{w}\\Vert $ .", "Denote $\\mathbb {K} = \\mathbb {X}\\mathbb {X}^T = \\mathbb {Y}\\mathbb {Y}^T$ .", "Let $\\mathbb {S} = \\mathbb {Y}^T \\mathbb {K}^{-1} \\mathbb {X}$ and let $w = \\mathbb {S}^T \\widetilde{w}$ .", "We have $\\mathbb {X} w = \\mathbb {X} \\mathbb {S}^T \\widetilde{w} = \\mathbb {X} \\mathbb {X}^T \\mathbb {K}^{-1} \\mathbb {Y}\\widetilde{w} = \\mathbb {Y} \\widetilde{w} = r.$ In addition, $\\Vert w \\Vert ^2 = w^T w = \\widetilde{w}^T \\mathbb {S} \\mathbb {S}^T \\widetilde{w}.$ By definition of $\\mathbb {S}$ , $\\mathbb {S} \\mathbb {S}^T = \\mathbb {Y}^T \\mathbb {K}^{-1} \\mathbb {X} \\mathbb {X}^T \\mathbb {K}^{-1} \\mathbb {Y} = \\mathbb {Y}^T \\mathbb {K}^{-1} \\mathbb {Y} = \\mathbb {Y}^T (\\mathbb {Y} \\mathbb {Y}^T)^{-1} \\mathbb {Y} = \\mathbb {Y}^T (\\mathbb {Y}^T)^{+}.$ Denote $\\mathbb {O} = \\mathbb {Y}^T (\\mathbb {Y}^T)^{+}$ .", "$\\mathbb {O}$ is an orthogonal projection matrix: by the properties of the pseudo-inverse, $\\mathbb {O} = \\mathbb {O}^T$ and $\\mathbb {O}^2 = \\mathbb {O}$ .", "Therefore $\\Vert w\\Vert ^2 = \\widetilde{w}^T \\mathbb {S} \\mathbb {S}^T \\widetilde{w} = \\widetilde{w}^T \\mathbb {O} \\widetilde{w} =\\widetilde{w}^T \\mathbb {O} \\mathbb {O}^T \\widetilde{w} = \\Vert \\mathbb {O} \\widetilde{w} \\Vert ^2 \\le \\Vert \\widetilde{w}\\Vert ^2.$ [of Theorem REF ] We prove the theorem for 1-shattering.", "The case of $\\gamma $ -shattering follows by rescaling $X$ appropriately.", "Let $\\mathbb {X}\\mathbb {X}^T = \\mathbb {U} \\Lambda \\mathbb {U}^T$ be the SVD of $\\mathbb {X}\\mathbb {X}^T$ , where $\\mathbb {U}$ is an orthogonal matrix and $\\Lambda $ is a diagonal matrix.", "Let $\\mathbb {Y} = \\mathbb {U} \\Lambda ^{\\frac{1}{2}}$ .", "We have $\\mathbb {X}\\mathbb {X}^T = \\mathbb {Y} \\mathbb {Y}^T$ .", "We show that the specified conditions are sufficient and necessary for the shattering of the set: Sufficient: If $\\mathbb {X}\\mathbb {X}^T$ is invertible, then $\\Lambda $ is invertible, thus so is $\\mathbb {Y}$ .", "For any $y\\in \\lbrace \\pm 1\\rbrace ^m$ , Let $w_y = \\mathbb {Y}^{-1} y$ .", "Then $\\mathbb {Y} w_y = y$ .", "By Lemma REF , there exists a separator $w$ such that $\\mathbb {X}w = y$ and $\\Vert w\\Vert \\le \\Vert w_y\\Vert = \\sqrt{y^T (\\mathbb {Y}\\mathbb {Y}^T)^{-1}y} = \\sqrt{y^T (\\mathbb {X}\\mathbb {X}^T)^{-1}y} \\le 1$ .", "Necessary: If $\\mathbb {X}\\mathbb {X}^T$ is not invertible then the vectors in $S$ are linearly dependent, thus $S$ cannot be shattered using linear separators [31].", "The first condition is therefore necessary.", "Assume $S$ is 1-shattered at the origin and show that the second condition necessarily holds.", "By Lemma REF , for all $y \\in \\lbrace \\pm 1\\rbrace ^m$ there exists a $w_y \\in \\mathbb {B}^d_1$ such that $\\mathbb {X}w_y = y$ .", "Thus by Lemma REF there exists a $\\widetilde{w}_y$ such that $\\mathbb {Y} \\widetilde{w}_y = y$ and $\\Vert \\widetilde{w}_y\\Vert \\le \\Vert w_y\\Vert \\le 1$ .", "$\\mathbb {X}\\mathbb {X}^T$ is invertible, thus so is $\\mathbb {Y}$ .", "Therefore $\\widetilde{w}_y = \\mathbb {Y}^{-1}y$ .", "Thus $y^T(\\mathbb {X}\\mathbb {X}^T)^{-1}y = y^T(\\mathbb {Y}\\mathbb {Y}^T)^{-1}y = \\Vert \\widetilde{w}_y\\Vert \\le 1$ .", "We are now ready to provide a sufficient condition for fat-shattering based on the smallest eigenvalue of the Gram matrix.", "Corollary Let $\\mathbb {X} \\in \\mathbb {R}^{m\\times d}$ be the matrix of a set of size $m$ in $\\mathbb {R}^d$ .", "If $\\lambda _{\\min }(\\mathbb {X}\\mathbb {X}^T) \\ge m\\gamma ^2$ then the set is $\\gamma $ -shattered at the origin by $\\mathcal {W}$ .", "If $\\lambda _{\\min }(\\mathbb {X}\\mathbb {X}^T) \\ge m\\gamma ^2$ then $\\mathbb {X}\\mathbb {X}^T$ is invertible and $\\lambda _{\\max }((\\mathbb {X}\\mathbb {X}^T)^{-1})\\le (m\\gamma ^2)^{-1}$ .", "For any $y \\in \\lbrace \\pm 1\\rbrace ^m$ we have $\\Vert y\\Vert =\\sqrt{m}$ and $y^T (\\mathbb {X}\\mathbb {X}^T)^{-1} y \\le \\Vert y\\Vert ^2\\lambda _{\\max }((\\mathbb {X}\\mathbb {X}^T)^{-1}) \\le m(m\\gamma ^2)^{-1} = \\gamma ^{-2}.$ By Theorem REF the sample is $\\gamma $ -shattered at the origin.", "Corollary REF generalizes the requirement of linear independence for shattering with no margin: A set of vectors is shattered with no margin if the vectors are linearly independent, that is if $\\lambda _{\\min }>0$ .", "The corollary shows that for $\\gamma $ -fat-shattering, we can require instead $\\lambda _{\\min }\\ge m\\gamma ^2$ .", "We can now conclude that if it is highly probable that the smallest eigenvalue of the sample Gram matrix is large, then MEM might fail to learn a good classifier for the given distribution.", "This is formulated in the following theorem.", "Let $D$ be a distribution over $\\mathbb {R}^d\\times \\lbrace \\pm 1\\rbrace $ .", "Let $m > 0$ and let $\\mathbb {X}$ be the matrix of a sample drawn from $D^{m}_X$ .", "Let $\\eta = \\mathbb {P}[\\lambda _{\\min }(\\mathbb {X} \\mathbb {X}^T) \\ge m \\gamma ^2]$ .", "Then for all $\\epsilon < 1/2 - \\ell ^*_\\gamma (D)$ , $m(\\epsilon ,\\gamma ,D,\\eta /2) \\ge \\lfloor m/2\\rfloor $ .", "The proof of the theorem is immediate by combining Theorem REF and Corollary REF .", "Theorem REF generalizes the case of learning a linear separator without a margin: If a sample of size $m$ is linearly independent with high probability, then there is no hope of using $m/2$ points to predict the label of the other points.", "The theorem extends this observation to the case of learning with a margin, by requiring a stronger condition than just linear independence of the points in the sample.", "Recall that our upper-bound on the sample complexity from Section  is $\\tilde{O}(k_{\\gamma })$ .", "We now define the family of sub-Gaussian product distributions, and show that for this family, the lower bound that can be deduced from Theorem REF is also linear in $k_{\\gamma }$ ." ], [ "Sub-Gaussian Distributions", "In order to derive a lower bound on distributionspecific sample complexity in terms of the covariance of $X \\sim D_X$ , we must assume that $X$ is not too heavy-tailed.", "This is because for any data distribution there exists another distribution which is almost identical and has the same sample complexity, but has arbitrarily large covariance values.", "This can be achieved by mixing the original distribution with a tiny probability for drawing a vector with a huge norm.", "We thus restrict the discussion to multidimensional sub-Gaussian distributions.", "This ensures light tails of the distribution in all directions, while still allowing a rich family of distributions, as we presently see.", "Sub-Gaussianity is defined for scalar random variables as follows [13].", "[Sub-Gaussian random variables] A random variable $X \\in \\mathbb {R}$ is sub-Gaussian with moment $B$, for $B \\ge 0$ , if $\\forall t \\in \\mathbb {R}, \\quad \\mathbb {E}[\\exp (tX)]\\le \\exp (t^2B^2 /2).$ In this work we further say that $X$ is sub-Gaussian with relative moment $\\rho > 0$ if $X$ is sub-Gaussian with moment $\\rho \\sqrt{\\mathbb {E}[X^2]}$ , that is, $\\forall t \\in \\mathbb {R}, \\quad \\mathbb {E}[\\exp (tX)]\\le \\exp (t^2\\rho ^2\\mathbb {E}[X^2] /2).$ Note that a sub-Gaussian variable with moment $B$ and relative moment $\\rho $ is also sub-Gaussian with moment $B^{\\prime }$ and relative moment $\\rho ^{\\prime }$ for any $B^{\\prime } \\ge B$ and $\\rho ^{\\prime } \\ge \\rho $ .", "The family of sub-Gaussian distributions is quite extensive: For instance, it includes any bounded, Gaussian, or Gaussian-mixture random variable with mean zero.", "Specifically, if $X$ is a mean-zero Gaussian random variable, $X \\sim N(0, \\sigma ^2)$ , then $X$ is sub-Gaussian with relative moment 1 and the inequalities in the definition above hold with equality.", "As another example, if $X$ is a uniform random variable over $\\lbrace \\pm b\\rbrace $ for some $b \\ge 0$ , then $X$ is sub-Gaussian with relative moment 1, since $\\mathbb {E}[\\exp (tX)] = {\\frac{1}{2}}(\\exp (tb) + \\exp (-tb)) \\le \\exp (t^2b^2/2) = \\exp (t^2\\mathbb {E}[X^2]/2).$ Let $\\mathbb {B} \\in \\mathbb {R}^{d\\times d}$ be a symmetric PSD matrix.", "A random vector $X \\in \\mathbb {R}^d$ is a sub-Gaussian random vector with moment matrix $\\mathbb {B}$ if for all $u \\in \\mathbb {R}^d$ , $\\mathbb {E}[\\exp (\\langle u,X \\rangle )] \\le \\exp (\\langle \\mathbb {B}u,u \\rangle /2)$ .", "The following lemma provides a useful connection between the trace of the sub-Gaussian moment matrix and the moment-generating function of the squared norm of the random vector.", "The proof is given in Appendix REF .", "Let $X \\in \\mathbb {R}^d$ be a sub-Gaussian random vector with moment matrix $\\mathbb {B}$ .", "Then for all $t \\in (0,\\frac{1}{4\\lambda _{\\max }(\\mathbb {B})}]$ , $\\mathbb {E}[\\exp (t \\Vert X\\Vert ^2)] \\le \\exp (2t \\cdot \\operatorname{trace}(\\mathbb {B})).$ Our lower bound holds for the family of sub-Gaussian product distributions, defined as follows.", "[Sub-Gaussian product distributions] A distribution $D_X$ over $\\mathbb {R}^d$ is a sub-Gaussian product distribution with moment $B$ and relative moment $\\rho $ if there exists some orthonormal basis $a_1,\\ldots ,a_d \\in \\mathbb {R}^d$ , such that for $X \\sim D_X$ , $\\langle a_i, X \\rangle $ are independent sub-Gaussian random variables, each with moment $B$ and relative moment $\\rho $ .", "Note that a sub-Gaussian product distribution has mean zero, thus its covariance matrix is equal to its uncentered covariance matrix.", "For any fixed $\\rho \\ge 0$ , we denote by $\\mathcal {D}^\\textrm {sg}_\\rho $ the family of all sub-Gaussian product distributions with relative moment $\\rho $ , in arbitrary dimension.", "For instance, all multivariate Gaussian distributions and all uniform distributions on the corners of a centered hyper-rectangle are in $\\mathcal {D}^\\textrm {sg}_1$ .", "All uniform distributions over a full centered hyper-rectangle are in $\\mathcal {D}^\\textrm {sg}_{3/2}$ .", "Note that if $\\rho _1 \\le \\rho _2$ , $\\mathcal {D}^\\textrm {sg}_{\\rho _1}\\subseteq \\mathcal {D}^\\textrm {sg}_{\\rho _2}$ .", "We will provide a lower bound for all distributions in $\\mathcal {D}^\\textrm {sg}_\\rho $ .", "This lower bound is linear in the margin-adapted dimension of the distribution, thus it matches the upper bound provided in Corollary REF .", "The constants in the lower bound depend only on the value of $\\rho $ , which we regard as a constant." ], [ "A Sample-Complexity Lower Bound for Sub-Gaussian Product Distributions", "As shown in Section REF , to obtain a sample complexity lower bound it suffices to have a lower bound on the value of the smallest eigenvalue of a random Gram matrix.", "The distribution of the smallest eigenvalue of a random Gram matrix has been investigated under various assumptions.", "The cleanest results are in the asymptotic case where the sample size and the dimension approach infinity, the ratio between them approaches a constant, and the coordinates of each example are identically distributed.", "[[3]] Let $\\lbrace \\mathbb {X}_i\\rbrace _{i=1}^\\infty $ be a series of matrices of sizes $m_i \\times d_i$ , whose entries are i.i.d.", "random variables with mean zero, variance $\\sigma ^2$ and finite fourth moments.", "If $\\lim _{i\\rightarrow \\infty }\\frac{m_i}{d_i} = \\beta < 1$ , then $\\lim _{i\\rightarrow \\infty } \\lambda _{\\min }(\\frac{1}{d_i}\\mathbb {X}_i\\mathbb {X}_i^T) = \\sigma ^2(1-\\sqrt{\\beta })^2.$ This asymptotic limit can be used to approximate an asymptotic lower bound on $m(\\epsilon ,\\gamma ,D)$ , if $D_X$ is a product distribution of i.i.d.", "random variables with mean zero, variance $\\sigma ^2$ , and finite fourth moment.", "Let $\\mathbb {X} \\in \\mathbb {R}^{m\\times d}$ be the matrix of a sample of size $m$ drawn from $D_X$ .", "We can find $m = m_\\circ $ such that $\\lambda _{m_\\circ }(\\mathbb {X}\\mathbb {X}^T) \\approx \\gamma ^2m_\\circ $ , and use Theorem REF to conclude that $m(\\epsilon ,\\gamma ,D) \\ge m_\\circ /2$ .", "If $d$ and $m$ are large enough, we have by Theorem REF that for $\\mathbb {X}$ drawn from $D_X^m$ : $\\lambda _{\\min }(\\mathbb {X}\\mathbb {X}^T) \\approx d \\sigma ^2 (1-\\sqrt{m/d})^2 = \\sigma ^2(\\sqrt{d}-\\sqrt{m})^2.$ Solving the equality $\\sigma ^2(\\sqrt{d}-\\sqrt{m_\\circ })^2=m_\\circ \\gamma ^2$ we get $m_\\circ = d/(1+\\gamma /\\sigma )^2$ .", "The margin-adapted dimension for $D_X$ is $k_\\gamma \\approx d/(1+\\gamma ^2/\\sigma ^2)$ , thus $\\tfrac{1}{2}k_\\gamma \\le m_\\circ \\le k_\\gamma $ .", "In this case, then, the sample complexity lower bound is indeed the same order as $k_\\gamma $ , which controls also the upper bound in Corollary REF .", "However, this is an asymptotic analysis, which holds for a highly limited set of distributions.", "Moreover, since Theorem REF holds asymptotically for each distribution separately, we cannot use it to deduce a uniform finite-sample lower bound for families of distributions.", "For our analysis we require finite-sample bounds for the smallest eigenvalue of a random Gram-matrix.", "[26], [25] provide such finite-sample lower bounds for distributions which are products of identically distributed sub-Gaussians.", "In Theorem REF below we provide a new and more general result, which holds for any sub-Gaussian product distribution.", "The proof of Theorem REF is provided in Appendix REF .", "Combining Theorem REF with Theorem REF above we prove the lower bound, stated in Theorem REF below.", "For any $\\rho > 0$ and $\\delta \\in (0,1)$ there are $\\beta > 0$ and $C > 0$ such that the following holds.", "For any $D_X \\in \\mathcal {D}^\\textrm {sg}_\\rho $ with covariance matrix $\\Sigma \\le I$ , and for any $m \\le \\beta \\cdot \\operatorname{trace}(\\Sigma ) - C$ , if $\\mathbb {X}$ is the $m\\times d$ matrix of a sample drawn from $D_X^m$ , then $\\mathbb {P}[\\lambda _{\\min }(\\mathbb {X} \\mathbb {X}^T) \\ge m] \\ge \\delta .$ [Sample complexity lower bound for distributions in $\\mathcal {D}^\\textrm {sg}_\\rho $ ] For any $\\rho >0$ there are constants $\\beta > 0,C\\ge 0$ such that for any $D$ with $D_X \\in \\mathcal {D}^\\textrm {sg}_\\rho $ , for any $\\gamma > 0$ and for any $\\epsilon < \\frac{1}{2} - \\ell ^*_\\gamma (D)$ , $m(\\epsilon ,\\gamma ,D,1/4) \\ge \\beta k_\\gamma (D_X)-C.$ Assume w.l.o.g.", "that the orthonormal basis $a_1,\\ldots ,a_d$ of independent sub-Gaussian directions of $D_X$ , defined in Definition REF , is the natural basis $e_1,\\ldots ,e_d$ .", "Define $\\lambda _i = \\mathbb {E}_{X\\sim D_X}[X[i]^2]$ , and assume w.l.o.g.", "$\\lambda _1 \\ge \\ldots \\ge \\lambda _d > 0$ .", "Let $\\mathbb {X}$ be the $m\\times d$ matrix of a sample drawn from $D_X^m$ .", "Fix $\\delta \\in (0,1)$ , and let $\\beta $ and $C$ be the constants for $\\rho $ and $\\delta $ in Theorem REF .", "Throughout this proof we abbreviate $k_\\gamma \\triangleq k_\\gamma (D_X)$ .", "Let $m \\le \\beta (k_\\gamma -1) - C$ .", "We would like to use Theorem REF to bound $\\lambda _{\\min }(\\mathbb {X}\\mathbb {X}^T)$ with high probability, so that Theorem REF can be applied to get the desired lower bound.", "However, Theorem REF holds only if $\\Sigma \\le I$ .", "Thus we split to two cases—one in which the dimensionality controls the lower bound, and one in which the norm controls it.", "The split is based on the value of $\\lambda _{k_\\gamma }$ .", "Case I: Assume $\\lambda _{k_\\gamma } \\ge \\gamma ^2$ .", "Then $\\forall i\\in [k_\\gamma ],\\lambda _i \\ge \\gamma ^2$ .", "By our assumptions on $D_X$ , for all $i\\in [d]$ the random variable $X[i]$ is sub-Gaussian with relative moment $\\rho $ .", "Consider the random variables $Z[i] = X[i]/\\sqrt{\\lambda _i}$ for $i \\in [k_\\gamma ]$ .", "$Z[i]$ is also sub-Gaussian with relative moment $\\rho $ , and $\\mathbb {E}[Z[i]^2] = 1$ .", "Consider the product distribution of $Z[1],\\ldots ,Z[k_\\gamma ]$ , and let $\\Sigma ^{\\prime }$ be its covariance matrix.", "We have $\\Sigma ^{\\prime } = I_{k_\\gamma }$ , and $\\operatorname{trace}(\\Sigma ^{\\prime }) = k_\\gamma $ .", "Let $\\mathbb {Z}$ be the matrix of a sample of size $m$ drawn from this distribution.", "By Theorem REF , $\\mathbb {P}[\\lambda _{\\min }(\\mathbb {Z} \\mathbb {Z}^T)\\ge m] \\ge \\delta $ , which is equivalent to $\\mathbb {P}[\\lambda _{\\min }(\\mathbb {X} \\cdot \\mathrm {diag}(1/\\lambda _1,\\ldots ,1/\\lambda _{k_\\gamma },0,\\ldots ,0) \\cdot \\mathbb {X}^T)\\ge m] \\ge \\delta .$ Since $\\forall i\\in [k_\\gamma ],\\lambda _i \\ge \\gamma ^2$ , we have $\\mathbb {P}[\\lambda _{\\min }(\\mathbb {X} \\mathbb {X}^T)\\ge m\\gamma ^2] \\ge \\delta $ .", "Case II: Assume $\\lambda _{k_\\gamma } < \\gamma ^2$ .", "Then $\\lambda _i < \\gamma ^2$ for all $i \\in \\lbrace k_\\gamma ,\\ldots ,d\\rbrace $ .", "Consider the random variables $Z[i] = X[i]/\\gamma $ for $i \\in \\lbrace k_\\gamma ,\\ldots ,d\\rbrace $ .", "$Z[i]$ is sub-Gaussian with relative moment $\\rho $ and $\\mathbb {E}[Z[i]^2] \\le 1$ .", "Consider the product distribution of $Z[k_\\gamma ],\\ldots ,Z[d]$ , and let $\\Sigma ^{\\prime }$ be its covariance matrix.", "We have $\\Sigma ^{\\prime } < I_{d-k_\\gamma +1}$ .", "By the minimality in Equation (REF ) we also have $\\operatorname{trace}(\\Sigma ^{\\prime }) = \\frac{1}{\\gamma ^2}\\sum _{i=k_\\gamma }^d \\lambda _i \\ge k_\\gamma -1$ .", "Let $\\mathbb {Z}$ be the matrix of a sample of size $m$ drawn from this product distribution.", "By Theorem REF , $\\mathbb {P}[\\lambda _{\\min }(\\mathbb {Z} \\mathbb {Z}^T)\\ge m] \\ge \\delta $ .", "Equivalently, $\\mathbb {P}[\\lambda _{\\min }(\\mathbb {X} \\cdot \\mathrm {diag}(0,\\ldots ,0,1/\\gamma ^2,\\ldots ,1/\\gamma ^2) \\cdot \\mathbb {X}^T)\\ge m] \\ge \\delta ,$ therefore $\\mathbb {P}[\\lambda _{\\min }(\\mathbb {X} \\mathbb {X}^T)\\ge m\\gamma ^2] \\ge \\delta $ .", "In both cases $\\mathbb {P}[\\lambda _{\\min }(\\mathbb {X}\\mathbb {X}^T)\\ge m\\gamma ^2] \\ge \\delta $ .", "This holds for any $m \\le \\beta (k_\\gamma -1) -C$ , thus by Theorem REF $m(\\epsilon , \\gamma ,D, \\delta /2) \\ge \\lfloor (\\beta (k_\\gamma -1)-C)/2\\rfloor $ for $\\epsilon < 1/2-\\ell _\\gamma ^*(D)$ .", "We finalize the proof by setting $\\delta = \\frac{1}{2}$ and adjusting $\\beta $ and $C$ ." ], [ "On the Limitations of the Covariance Matrix", "We have shown matching upper and lower bounds for the sample complexity of learning with MEM, for any sub-Gaussian product distribution with a bounded relative moment.", "This shows that the margin-adapted dimension fully characterizes the sample complexity of learning with MEM for such distributions.", "What properties of a distribution play a role in determining the sample complexity for general distributions?", "In the following theorem we show that these properties must include more than the covariance matrix of the distribution, even when assuming sub-Gaussian tails and bounded relative moments.", "For any integer $d > 1$ , there exist two distributions $D$ and $P$ over $\\mathbb {R}^d \\times \\lbrace \\pm 1\\rbrace $ with identical covariance matrices, such that for any $\\epsilon ,\\delta \\in (0,\\frac{1}{4})$ , $m(\\epsilon , 1, P, \\delta ) \\ge \\Omega (d)$ while $m(\\epsilon , 1, D,\\delta ) \\le {\\lceil \\log _2(1/\\delta ) \\rceil }$ .", "Both $D_X$ and $P_X$ are sub-Gaussian random vectors, with a relative moment of $\\sqrt{2}$ in all directions.", "Let $D_a$ and $D_b$ be distributions over $\\mathbb {R}^d$ such that $D_a$ is uniform over $\\lbrace \\pm 1\\rbrace ^d$ and $D_b$ is uniform over $\\lbrace \\pm 1\\rbrace \\times \\lbrace 0\\rbrace ^{d-1}$ .", "Let $D_X$ be a balanced mixture of $D_a$ and $D_b$ .", "Let $P_X$ be uniform over $\\lbrace \\pm 1\\rbrace \\times \\lbrace \\frac{1}{\\sqrt{2}}\\rbrace ^{d-1}$ .", "For both $D$ and $P$ , let $\\mathbb {P}[Y = \\langle e_1, X \\rangle ] = 1$ .", "The covariance matrix of $D_X$ and $P_X$ is $\\mathrm {diag}(1,{\\frac{1}{2}}, \\ldots , {\\frac{1}{2}})$ , thus $k_1(D_X) = k_1(P_X) \\ge \\Omega (d)$ .", "By Equation (REF ), $P_X,D_a$ and $D_b$ are all sub-Gaussian product distribution with relative moment 1, thus also with moment $\\sqrt{2} > 1$ .", "The projection of $D_X$ along any direction $u \\in \\mathbb {R}^d$ is sub-Gaussian with relative moment $\\sqrt{2}$ as well, since $&\\mathbb {E}_{X \\sim D_X}[\\exp (\\langle u,X \\rangle )] = {\\frac{1}{2}}(\\mathbb {E}_{X \\sim D^a}[\\exp (\\langle u,X \\rangle )] + \\mathbb {E}_{X \\sim D^b}[\\exp (\\langle u,X \\rangle )]) \\\\&={\\frac{1}{2}}(\\prod _{i\\in [d]}(\\exp (u_i)+\\exp (-u_i))/2 + (\\exp (u_1)+\\exp (-u_1))/2) \\\\&\\le {\\frac{1}{2}}(\\prod _{i\\in [d]}\\exp (u_i^2/2) + \\exp (u_1^2/2))\\le \\exp (\\Vert u\\Vert ^2/2) \\le \\exp ((\\Vert u\\Vert ^2+u_1^2)/2)\\\\&=\\exp (\\mathbb {E}_{X\\sim D_X} [\\langle u, X \\rangle ^2]).$ For $P$ we have by Theorem REF that for any $\\epsilon \\le \\frac{1}{4}$ , $m(\\epsilon , 1, P,\\frac{1}{4}) \\ge \\Omega (k_1(P_X)) \\ge \\Omega (d)$ .", "In contrast, any MEM algorithm $\\mathcal {A}_1$ will output the correct separator for $D$ whenever the sample has at least one point drawn from $D_b$ .", "This is because the separator $e_1$ is the only $w\\in \\mathbb {B}^d_1$ that classifies this point with zero 1-margin errors.", "Such a point exists in a sample of size $m$ with probability $1-2^{-m}$ .", "Therefore $\\ell _0(\\mathcal {A}_1,D,m,1/2^m) = 0$ .", "It follows that for all $\\epsilon > 0$ , $m(\\epsilon ,1,D,\\delta ) \\le {\\lceil \\log _2(1/\\delta ) \\rceil }$ ." ], [ "Conclusions", "Corollary REF and Theorem REF together provide a tight characterization of the sample complexity of any sub-Gaussian product distribution with a bounded relative moment.", "Formally, fix $\\rho > 0$ .", "For any $D$ such that $D_X \\in \\mathcal {D}^\\textrm {sg}_\\rho $ , and for any $\\gamma > 0$ and $\\epsilon \\in (0,\\frac{1}{2} - \\ell ^*_\\gamma (D))$ $\\Omega (k_\\gamma (D_X)) \\le m(\\epsilon ,\\gamma ,D) \\le \\tilde{O}\\left(\\frac{k_{\\gamma }(D_X)}{\\epsilon ^2}\\right).$ The upper bound holds uniformly for all distributions, and the constants in the lower bound depend only on $\\rho $ .", "This result shows that the true sample complexity of learning each of these distributions with MEM is characterized by the margin-adapted dimension.", "An interesting conclusion can be drawn as to the influence of the conditional distribution of labels $D_{Y|X}$ : Since Equation (REF ) holds for any $D_{Y|X}$ , the effect of the direction of the best separator on the sample complexity is bounded, even for highly non-spherical distributions.", "We note that the upper bound that we have proved involves logarithmic factors which might not be necessary.", "There are upper bounds that depend on the margin alone and on the dimension alone without logarithmic factors.", "On the other hand, in our bound, which combines the two quantities, there is a logarithmic dependence which stems from the margin component of the bound.", "It might be possible to tighten the bound and remove the logarithmic dependence.", "Equation (REF ) can be used to easily characterize the sample complexity behavior for interesting distributions, to compare $L_2$ margin minimization to other learning methods, and to improve certain active learning strategies.", "We elaborate on each of these applications in the following examples.", "[Gaps between $L_1$ and $L_2$ regularization in the presence of irrelevant features]       [22] considers learning a single relevant feature in the presence of many irrelevant features, and compares using $L_1$ regularization and $L_2$ regularization.", "When $\\Vert X\\Vert _{\\infty } \\le 1$ , upper bounds on learning with $L_1$ regularization guarantee a sample complexity of $O(\\ln (d))$ for an $L_1$ -based learning rule [35].", "In order to compare this with the sample complexity of $L_2$ regularized learning and establish a gap, one must use a lower bound on the $L_2$ sample complexity.", "The argument provided by Ng actually assumes scale-invariance of the learning rule, and is therefore valid only for unregularized linear learning.", "In contrast, using our results we can easily establish a lower bound of $\\Omega (d)$ for many specific distributions with a bounded $\\Vert X\\Vert _{\\infty }$ and $Y=\\textrm {sign}(X[i])$ for some $i$ .", "For instance, if each coordinate is a bounded independent sub-Gaussian random variable with a bounded relative moment, we have $k_1 = {\\lceil d/2 \\rceil }$ and Theorem REF implies a lower bound of $\\Omega (d)$ on the $L_2$ sample complexity.", "[Gaps between generative and discriminative learning for a Gaussian mixture] Let there be two classes, each drawn from a unit-variance spherical Gaussian in $\\mathbb {R}^d$ with a large distance $2v>> 1$ between the class means, such that $d >> v^4$ .", "Then $\\mathbb {P}_D[X|Y=y]= \\mathcal {N}(y v\\cdot e_1,I_d)$ , where $e_1$ is a unit vector in $\\mathbb {R}^d$ .", "For any $v$ and $d$ , we have $D_X \\in \\mathcal {D}^\\textrm {sg}_1$ .", "For large values of $v$ , we have extremely low margin error at $\\gamma =v/2$ , and so we can hope to learn the classes by looking for a large-margin separator.", "Indeed, we can calculate $k_\\gamma ={\\lceil d/(1+\\frac{v^2}{4}) \\rceil }$ , and conclude that the required sample complexity is $\\tilde{\\Theta }(d/v^2)$ .", "Now consider a generative approach: fitting a spherical Gaussian model for each class.", "This amounts to estimating each class center as the empirical average of the points in the class, and classifying based on the nearest estimated class center.", "It is possible to show that for any constant $\\epsilon >0$ , and for large enough $v$ and $d$ , $O(d/v^4)$ samples are enough in order to ensure an error of $\\epsilon $ .", "This establishes a rather large gap of $\\Omega (v^2)$ between the sample complexity of the discriminative approach and that of the generative one.", "[Active learning] In active learning, there is an abundance of unlabeled examples, but labels are costly, and the active learning algorithm needs to decide which labels to query based on the labels seen so far.", "A popular approach to active learning involves estimating the current set of possible classifiers using sample complexity upper bounds [4], [11].", "Without any distribution-specific information, only general distribution-free upper bounds can be used.", "However, since there is an abundance of unlabeled examples, the active learner can use these to estimate tighter distribution-specific upper bounds.", "In the case of linear classifiers, the margin-adapted dimension can be calculated from the uncentered covariance matrix of the distribution, which can be easily estimated from unlabeled data.", "Thus, our sample complexity upper bounds can be used to improve the active learner's label complexity.", "Moreover, the lower bound suggests that any further improvement of such active learning strategies would require more information other than the distribution's covariance matrix.", "To summarize, we have shown that the true sample complexity of large-margin learning of each of a rich family of distributions is characterized by the margin-adapted dimension.", "Characterizing the true sample complexity allows a better comparison between this learning approach and other algorithms, and has many potential applications.", "The challenge of characterizing the true sample complexity extends to any distribution and any learning approach.", "Theorem  shows that other properties but the covariance matrix must be taken into account for general distributions.", "We believe that obtaining answers to these questions is of great importance, both to learning theory and to learning applications.", "The authors thank Boaz Nadler for many insightful discussions.", "During part of this research, Sivan Sabato was supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.", "This work is partly supported by the Gatsby Charitable Foundation, The DARPA MSEE project, the Intel ICRI-CI center, and the Israel Science Foundation center of excellence grant." ], [ "Proofs Omitted from the Text", "In this appendix we give detailed proofs which were omitted from the text." ], [ "Proof of Proposition ", "Let $w^* \\in \\operatornamewithlimits{argmin}_{w\\in \\mathbb {B}^d_1} \\ell _\\gamma (w,D)$ .", "By Equation (REF ), with probability $1-\\delta /2$ $\\mathrm {ramp}_\\gamma (\\mathcal {A}_\\gamma (S), D) \\le \\mathrm {ramp}_\\gamma (\\mathcal {A}_\\gamma (S), S) + 2\\mathcal {R}_m(\\textsc {ramp}_\\gamma , D) + \\sqrt{\\frac{8\\ln (2/\\delta )}{m}}.$ Set $h^* \\in \\mathcal {H}$ such that $\\ell _\\gamma (h^*, D) = \\ell _\\gamma ^*(\\mathcal {H},D)$ .", "We have $\\mathrm {ramp}_\\gamma (\\mathcal {A}_\\gamma (S), S) \\le \\ell _\\gamma (\\mathcal {A}_\\gamma (S),S) \\le \\ell _\\gamma (h^*,S).$ The first inequality follows since the ramp loss is upper bounded by the margin loss.", "The second inequality follows since $\\mathcal {A}$ is a MEM algorithm.", "Now, by Hoeffding's inequality, since the range of $\\mathrm {ramp}_\\gamma $ is in $[0,1]$ , with probability at least $1-\\delta /2$ $\\ell _\\gamma (h^*,S) \\le \\ell _\\gamma (h^*, D) + \\sqrt{\\frac{\\ln (2/\\delta )}{2m}}.$ It follows that with probability $1-\\delta $ $\\mathrm {ramp}_\\gamma (\\mathcal {A}_\\gamma (S), D) \\le \\ell _\\gamma ^*(\\mathcal {H},D) + 2\\mathcal {R}_m(\\textsc {ramp}_\\gamma , D) + \\sqrt{\\frac{14\\ln (2/\\delta )}{m}}.$ We have $\\ell _0 \\le \\mathrm {ramp}_\\gamma $ .", "Combining this with Equation (REF ) we conclude Equation (REF )." ], [ "Proof of Lemma ", "[of Lemma ] For a function $f:\\mathcal {X}\\rightarrow \\mathbb {R}$ and a $z \\in Z$ , define the function $G[f,z]$ by $\\forall x \\in \\mathcal {X},\\quad G[f,z](x) = \\llbracket {f(x) + z(x)} \\rrbracket - f(x).$ Let $f_1,f_2 \\in \\mathbb {R}^\\mathcal {X}$ be two functions, and let $g_1 = G[f_1,z] \\in \\mathcal {G}(f_1)$ for some $w_b \\in \\bar{V}$ .", "Then, since $G[f_2,z] \\in \\mathcal {G}(f_2)$ , we have $\\inf _{g_2 \\in \\mathcal {G}(f_2)}\\Vert g_1 -g_2\\Vert _{L_2(S)} \\le \\Vert G[f_1,z] - G[f_2,z]\\Vert .$ Now, for all $x\\in \\mathbb {R}$ , $|G[f_1,z](x) - G[f_2,z](x)| &=|\\llbracket {f_1(x) + z(x)} \\rrbracket - f_1(x)- \\llbracket {f_2(x) + z(x)} \\rrbracket + f_2(x)| \\\\&\\le |f_1(x) - f_2(x)|.$ Thus, for any $S \\subseteq \\mathcal {X}$ , $\\Vert G[f_1,z] - G[f_2,z]\\Vert ^2_{L_2(S)} &= \\mathbb {E}_{X \\sim S}(G[f_1,z](X) - G[f_2,z](X))^2\\\\&\\le \\mathbb {E}_{X \\sim S}(f_1(X) - f_2(X))^2 = \\Vert f_1 -f_2\\Vert ^2_{L_2(S)}.$ It follows that $\\inf _{g_2 \\in \\mathcal {G}(f_2)}\\Vert g_1 -g_2\\Vert _{L_2(S)} \\le \\Vert f_1 -f_2\\Vert _{L_2(S)}$ .", "This holds for any $g_1 \\in \\mathcal {G}(f_1)$ , thus $\\Delta _H(\\mathcal {G}(f_1),\\mathcal {G}(f_2))\\le \\Vert f_1 -f_2\\Vert _{L_2(S)}$ ." ], [ "Proof of Lemma ", "[of Lemma ] Let $k$ be the pseudo-dimension of $\\mathcal {G}(f)$ , and let $\\lbrace x_1,\\ldots ,x_k\\rbrace \\subseteq \\mathcal {X}$ be a set which is pseudo-shattered by $\\mathcal {G}(f)$ .", "We show that the same set is pseudo-shattered by $Z$ as well, thus proving the lemma.", "Since $\\mathcal {G}(f)$ is pseudo-shattered, there exists a vector $r \\in \\mathbb {R}^k$ such that for all $y \\in \\lbrace \\pm 1\\rbrace ^k$ there exists a $g_y \\in \\mathcal {G}(f)$ such that $\\forall i\\in [m], \\textrm {sign}(g_y(x_i)-r[i]) = y[i]$ .", "Therefore for all $y\\in \\lbrace \\pm 1\\rbrace ^k$ there exists a $z_y \\in Z$ such that $\\forall i \\in [k], \\textrm {sign}(\\llbracket {f(x_i) + z_y(x_i)} \\rrbracket -f(x_i) - r[i]) = y[i].$ By considering the case $y[i] = 1$ , we have $0 < \\llbracket {f(x_i) + z_y(x_i)} \\rrbracket -f(x_i) - r[i] \\le 1 -f(x_i) - r[i].$ By considering the case $y[i] = -1$ , we have $0 > \\llbracket {f(x_i) + z_y(x_i)} \\rrbracket -f(x_i) - r[i] \\ge -f(x_i) - r[i].$ Therefore $0 < f(x_i) + r[i] < 1$ .", "Now, let $y \\in \\lbrace \\pm 1\\rbrace ^k$ and consider any $i \\in [k]$ .", "If $y[i] = 1$ then $\\llbracket {f(x_i) + z_y(x_i)} \\rrbracket -f(x_i) - r[i] > 0$ It follows that $\\llbracket {f(x_i) + z_y(x_i)} \\rrbracket > f(x_i) + r[i] > 0,$ thus $f(x_i) + z_y(x_i) > f(x_i) + r[i].$ In other words, $\\textrm {sign}(z_y(x_i)-r[i]) = 1 = y[i]$ .", "If $y[i] = -1$ then $\\llbracket {f(x_i) + z_y(x_i)} \\rrbracket -f(x_i) - r[i] < 0.$ It follows that $\\llbracket {f(x_i) + z_y(x_i)} \\rrbracket < f(x_i) + r[i] < 1,$ thus $f(x_i) + z_y(x_i) < f(x_i) + r[i].$ in other words, $\\textrm {sign}(z_y(x_i)-r[i]) = -1 = y[i]$ .", "We conclude that $Z$ shatters $\\lbrace x_1,\\ldots ,x_k\\rbrace $ as well, using the same vector $r \\in \\mathbb {R}^k$ .", "Thus the pseudo-dimension of $Z$ is at least $k$ ." ], [ "Proof of Lemma ", "To prove Lemma REF , we first prove the following lemma.", "Denote by $\\mathrm {conv}(A)$ the convex hull of a set $A$ .", "Let $\\gamma > 0$ .", "For each $y \\in \\lbrace \\pm 1\\rbrace ^m$ , select $r_y \\in \\mathbb {R}^m$ such that for all $i \\in [m]$ , $r_y[i]y[i] \\ge \\gamma $ .", "Let $R = \\lbrace r_y \\in \\mathbb {R}^m \\mid y \\in \\lbrace \\pm 1\\rbrace ^m\\rbrace $ .", "Then $\\lbrace \\pm \\gamma \\rbrace ^m \\subseteq \\mathrm {conv}(R)$ .", "We will prove the claim by induction on the dimension $m$ .", "For the base case, if $m=1$ , we have $R = \\lbrace a,b\\rbrace \\subseteq \\mathbb {R}$ where $a \\le -\\gamma $ and $b \\ge \\gamma $ .", "Clearly, $\\mathrm {conv}(R) = [a,b]$ , and $\\pm \\gamma \\in [a,b]$ .", "For the inductive step, assume the lemma holds for $m-1$ .", "For a vector $t \\in \\mathbb {R}^m$ , denote by $\\bar{t}$ its projection $(t[1],\\ldots , t[m-1])$ on $\\mathbb {R}^{m-1}$ .", "Similarly, for a set of vectors $S \\subseteq \\mathbb {R}^m$ , let $\\bar{S} = \\lbrace \\bar{s} \\mid s \\in S\\rbrace \\subseteq \\mathbb {R}^{m-1}$.", "Define $Y_+ = \\lbrace \\pm 1\\rbrace ^{m-1} \\times \\lbrace +1\\rbrace $ and $Y_- = \\lbrace \\pm 1\\rbrace ^{m-1} \\times \\lbrace -1\\rbrace $ .", "Let $R_+ = \\lbrace r_y \\mid y\\in Y_+ \\rbrace $ , and similarly for $R_-$ .", "Then the induction hypothesis holds for $\\bar{R}_+$ and $\\bar{R}_-$ with dimension $m-1$ .", "Let $z \\in \\lbrace \\pm \\gamma \\rbrace ^m$ .", "We wish to prove $z \\in \\mathrm {conv}(R)$ .", "From the induction hypothesis we have $\\bar{z} \\in \\mathrm {conv}(\\bar{R}_+)$ and $\\bar{z} \\in \\mathrm {conv}(\\bar{R}_-)$ .", "Thus, for all $y\\in \\lbrace \\pm 1\\rbrace $ there exist $\\alpha _y,\\beta _y \\ge 0$ such that $\\sum _{y\\in Y_+} \\alpha _y = \\sum _{y\\in Y_-} \\beta _y =1$ , and $\\bar{z} = \\sum _{y\\in Y_+} \\alpha _y \\bar{r}_y = \\sum _{y\\in Y_-} \\beta _y \\bar{r}_y.$ Let $z_a = \\sum _{y\\in Y_+} \\alpha _y r_y$ and $z_b = \\sum _{y\\in Y_-} \\beta _y r_y$ We have that $\\forall y \\in Y_+, r_y[m] \\ge \\gamma $ , and $\\forall y \\in Y_-,r_y[m] \\le -\\gamma $ .", "Therefore, $z_b[m] \\le -\\gamma \\le z[m] \\le \\gamma \\le z_a[m].$ In addition, $\\bar{z}_a = \\bar{z}_b = \\bar{z}$ .", "Select $\\lambda \\in [0,1]$ such that $z[m] = \\lambda z_a[m] + (1-\\lambda ) z_b[m]$ , then $z = \\lambda z_a + (1-\\lambda ) z_b$ .", "Since $z_a,z_b \\in \\mathrm {conv}(R)$ , we have $z \\in \\mathrm {conv}(R)$ .", "[of Lemma REF ] Denote by $f(S)$ the vector $(f(x_1), \\ldots , f(x_m))$ .", "Recall that $r \\in \\mathbb {R}^m$ is the witness for the shattering of $S$ , and let $L = \\lbrace f(S) - r \\mid f \\in \\mathcal {F}\\rbrace \\subseteq \\mathbb {R}^m.$ Since $S$ is shattered, for any $y \\in \\lbrace \\pm 1\\rbrace ^m$ there is an $r_y \\in L$ such that $\\forall i\\in [m], r_y[i]y[i] \\ge \\gamma $ .", "By Lemma REF , $\\lbrace \\pm \\gamma \\rbrace ^m \\subseteq \\mathrm {conv}(L)$ .", "Since $\\mathcal {F}$ is convex, $L$ is also convex.", "Therefore $\\lbrace \\pm \\gamma \\rbrace ^m \\subseteq L$ ." ], [ "Proof of Lemma ", "[of Lemma REF ] It suffices to consider diagonal moment matrices: If $\\mathbb {B}$ is not diagonal, let $\\mathbb {V} \\in \\mathbb {R}^{d\\times d}$ be an orthogonal matrix such that $\\mathbb {V} \\mathbb {B} \\mathbb {V}^T$ is diagonal, and let $Y = \\mathbb {V}X$ .", "We have $\\mathbb {E}[\\exp (t \\Vert Y\\Vert ^2)] = \\mathbb {E}[\\exp (t \\Vert X\\Vert ^2)]$ and $\\operatorname{trace}(\\mathbb {V}\\mathbb {B}\\mathbb {V}^T) = \\operatorname{trace}(\\mathbb {B})$ .", "In addition, for all $u \\in \\mathbb {R}^d$ , $\\mathbb {E}[\\exp (\\langle u,Y \\rangle )] &= \\mathbb {E}[\\exp (\\langle \\mathbb {V}^T u, X \\rangle )]\\le \\exp ({\\frac{1}{2}}\\langle \\mathbb {B} \\mathbb {V}^T u, \\mathbb {V}^T u \\rangle ) = \\exp ({\\frac{1}{2}}\\langle \\mathbb {V} \\mathbb {B} \\mathbb {V}^T u, u \\rangle ).$ Therefore $Y$ is sub-Gaussian with the diagonal moment matrix $\\mathbb {V}\\mathbb {B}\\mathbb {V}^T$ .", "Thus assume w.l.o.g.", "that $\\mathbb {B} = \\mathrm {diag}(\\lambda _1,\\ldots ,\\lambda _d)$ where $\\lambda _1 \\ge \\ldots \\ge \\lambda _d \\ge 0$ .", "We have $\\exp (t\\Vert X\\Vert ^2) = \\prod _{i\\in [d]}\\exp (tX[i]^2)$ .", "In addition, for any $t > 0$ and $x\\in \\mathbb {R}$ , $2\\sqrt{\\Pi t}\\cdot \\exp (t x^2) = \\int _{-\\infty }^\\infty \\exp (s x-\\frac{s^2}{4 t})ds.$ Therefore, for any $u\\in \\mathbb {R}^d$ , $(2\\sqrt{\\Pi t})^d \\cdot \\mathbb {E}[\\exp (t\\Vert X\\Vert ^2)] &= \\mathbb {E}\\left[\\prod _{i\\in [d]} \\int _{-\\infty }^\\infty \\exp (u[i] X[i]-\\frac{u[i]^2}{4 t})du[i]\\right]\\\\&= \\mathbb {E}\\left[\\int _{-\\infty }^{\\infty } \\ldots \\int _{-\\infty }^{\\infty } \\prod _{i\\in [d]} \\exp (u[i] X[i]-\\frac{u[i]^2}{4 t})du[i]\\right]\\\\&= \\mathbb {E}\\left[\\int _{-\\infty }^{\\infty } \\ldots \\int _{-\\infty }^{\\infty } \\exp (\\langle u,X \\rangle -\\frac{\\Vert u\\Vert ^2}{4 t})\\prod _{i\\in [d]} du[i]\\right]\\\\&= \\int _{-\\infty }^{\\infty } \\ldots \\int _{-\\infty }^{\\infty } \\mathbb {E}[\\exp (\\langle u, X \\rangle )]\\exp (-\\frac{\\Vert u\\Vert ^2}{4 t})\\prod _{i\\in [d]} du[i]$ By the sub-Gaussianity of $X$ , the last expression is bounded by $&\\le \\int _{-\\infty }^{\\infty } \\ldots \\int _{-\\infty }^{\\infty } \\exp ({\\frac{1}{2}}\\langle \\mathbf {\\mathbb {B}}u,u \\rangle -\\frac{\\Vert u\\Vert ^2}{4 t})\\prod _{i\\in [d]} du[i]\\\\&= \\int _{-\\infty }^{\\infty } \\ldots \\int _{-\\infty }^{\\infty } \\prod _{i\\in [d]} \\exp ( \\frac{\\lambda _iu[i]^2}{2} - \\frac{u[i]^2}{4t}) du[i]\\\\&= \\prod _{i\\in [d]} \\int _{-\\infty }^{\\infty } \\exp (u[i]^2(\\frac{\\lambda _i}{2}-\\frac{1}{4t}))du[i] = \\Pi ^{d/2}\\big (\\prod _{i\\in [d]}(\\frac{1}{4t}-\\frac{\\lambda _i}{2})\\big )^{-{\\frac{1}{2}}}.$ The last equality follows from the fact that for any $a > 0$ , $\\int _{-\\infty }^{\\infty } \\exp (-a \\cdot s^2)ds= \\sqrt{\\Pi /a}$ , and from the assumption $t \\le \\frac{1}{4\\lambda _1}$ .", "We conclude that $\\mathbb {E}[\\exp (t \\Vert X\\Vert ^2)] \\le (\\prod _{i\\in [d]}(1-2\\lambda _i t))^{-{\\frac{1}{2}}} \\le \\exp (2t \\cdot \\sum _{i=1}^d \\lambda _i) = \\exp (2t\\cdot \\operatorname{trace}(\\mathbb {B})),$ where the second inequality holds since $\\forall x \\in [0,1]$ , $(1-x/2)^{-1}\\le \\exp (x)$ ." ], [ "Proof of Theorem ", "In the proof of Theorem REF we use the fact $\\lambda _{\\min }(\\mathbb {X}\\mathbb {X}^T) = \\inf _{\\Vert x\\Vert _2=1}\\Vert \\mathbb {X}^Tx\\Vert ^2$ and bound the right-hand side via an $\\epsilon $ -net of the unit sphere in $\\mathbb {R}^m$ , denoted by $S^{m-1} \\triangleq \\lbrace x \\in \\mathbb {R}^m \\mid \\Vert x\\Vert _2 = 1\\rbrace $ .", "An $\\epsilon $ -net of the unit sphere is a set $C \\subseteq S^{m-1}$ such that $\\forall x \\in S^{m-1}, \\exists x^{\\prime } \\in C, \\Vert x-x^{\\prime }\\Vert \\le \\epsilon $ .", "Denote the minimal size of an $\\epsilon $ -net for $S^{m-1}$ by $\\mathcal {N}_m(\\epsilon )$ , and by $\\mathcal {C}_m(\\epsilon )$ a minimal $\\epsilon $ -net of $S^{m-1}$ , so that $\\mathcal {C}_m(\\epsilon ) \\subseteq S^{m-1}$ and $|\\mathcal {C}_m(\\epsilon )| = \\mathcal {N}_m(\\epsilon )$ .", "The proof of Theorem REF requires several lemmas.", "First we prove a concentration result for the norm of a matrix defined by sub-Gaussian variables.", "Then we bound the probability that the squared norm of a vector is small.", "Let $\\mathbb {Y}$ be a $d\\times m$ matrix with $m \\le d$ , such that $\\mathbb {Y}_{ij}$ are independent sub-Gaussian variables with moment $B$ .", "Let $\\Sigma $ be a diagonal $d\\times d$ PSD matrix such that $\\Sigma \\le I$ .", "Then for all $t \\ge 0$ and $\\epsilon \\in (0,1)$ , $\\mathbb {P}[\\Vert \\sqrt{\\Sigma }\\mathbb {Y}\\Vert \\ge t] \\le \\mathcal {N}_m(\\epsilon )\\exp (\\frac{\\operatorname{trace}(\\Sigma )}{2}-\\frac{t^2(1-\\epsilon )^2}{4B^2}).$ We have $\\Vert \\sqrt{\\Sigma }\\mathbb {Y}\\Vert \\le \\max _{x \\in \\mathcal {C}_m(\\epsilon )}\\Vert \\sqrt{\\Sigma }\\mathbb {Y}x\\Vert /(1-\\epsilon )$ , see for instance in [10].", "Therefore, $\\mathbb {P}[\\Vert \\sqrt{\\Sigma }\\mathbb {Y}\\Vert \\ge t] \\le \\sum _{x \\in \\mathcal {C}_m(\\epsilon )}\\mathbb {P}[\\Vert \\sqrt{\\Sigma }\\mathbb {Y}x\\Vert \\ge (1-\\epsilon )t].$ Fix $x \\in \\mathcal {C}_m(\\epsilon )$ .", "Let $V = \\sqrt{\\Sigma }\\mathbb {Y}x$ , and assume $\\Sigma = \\mathrm {diag}(\\lambda _1,\\ldots ,\\lambda _d)$ .", "For $u \\in \\mathbb {R}^d$ , $&\\mathbb {E}[\\exp (\\langle u, V \\rangle )] = \\mathbb {E}[\\exp (\\sum _{i\\in [d]}u_i\\sqrt{\\lambda }_i \\sum _{j\\in [m]}\\mathbb {Y}_{ij}x_j)]= \\prod _{j,i}\\mathbb {E}[\\exp (u_i\\sqrt{\\lambda }_i \\mathbb {Y}_{ij}x_j)]\\\\&\\quad \\le \\prod _{j,i}\\exp (u_i^2\\lambda _i B^2 x_j^2/2) = \\exp (\\frac{B^2}{2}\\sum _{i\\in [d]}u_i^2\\lambda _i \\sum _{j\\in [m]}x_j^2 )\\\\&\\quad = \\exp (\\frac{B^2}{2} \\sum _{i\\in [d]}u_i^2\\lambda _i) = \\exp (\\langle B^2\\Sigma u,u \\rangle /2).$ Thus $V$ is a sub-Gaussian vector with moment matrix $B^2\\Sigma $ .", "Let $s = 1/(4B^2)$ .", "Since $\\Sigma \\le I$ , we have $s \\le 1/(4B^2\\max _{i\\in [d]}\\lambda _i)$ .", "Therefore, by Lemma REF , $\\mathbb {E}[\\exp (s\\Vert V\\Vert ^2)] \\le \\exp (2sB^2\\operatorname{trace}(\\Sigma )).$ By Chernoff's method, $\\mathbb {P}[\\Vert V\\Vert ^2 \\ge z^2] \\le \\mathbb {E}[\\exp (s\\Vert V\\Vert ^2)]/\\exp (sz^2)$ .", "Thus $\\mathbb {P}[\\Vert V\\Vert ^2 \\ge z^2] \\le \\exp (2sB^2\\operatorname{trace}(\\Sigma ) - sz^2) = \\exp (\\frac{\\operatorname{trace}(\\Sigma )}{2}-\\frac{z^2}{4B^2}).$ Set $z = t(1-\\epsilon )$ .", "Then for all $x \\in S^{m-1}$ $\\mathbb {P}[\\Vert \\sqrt{\\Sigma }\\mathbb {Y}x\\Vert \\ge t(1-\\epsilon )] = \\mathbb {P}[\\Vert V\\Vert \\ge t(1-\\epsilon )] \\le \\exp (\\frac{\\operatorname{trace}(\\Sigma )}{2}-\\frac{t^2(1-\\epsilon )^2}{4B^2}).$ Therefore, by Equation (REF ), $\\mathbb {P}[\\Vert \\sqrt{\\Sigma }\\mathbb {Y}\\Vert \\ge t] \\le \\mathcal {N}_m(\\epsilon )\\exp (\\frac{\\operatorname{trace}(\\Sigma )}{2}-\\frac{t^2(1-\\epsilon )^2}{4B^2}).$ Let $\\mathbb {Y}$ be a $d\\times m$ matrix with $m \\le d$ , such that $\\mathbb {Y}_{ij}$ are independent centered random variables with variance 1 and fourth moments at most $B$ .", "Let $\\Sigma $ be a diagonal $d\\times d$ PSD matrix such that $\\Sigma \\le I$ .", "There exist $\\alpha > 0$ and $\\eta \\in (0,1)$ that depend only on $B$ such that for any $x \\in S^{m-1}$ $\\mathbb {P}[\\Vert \\sqrt{\\Sigma }\\mathbb {Y}x\\Vert ^2\\le \\alpha \\cdot (\\operatorname{trace}(\\Sigma )-1)] \\le \\eta ^{\\operatorname{trace}(\\Sigma )}.$ To prove Lemma REF we require Lemma REF [25] and Lemma REF , which extends Lemma 2.6 in the same work.", "Let $T_1,\\ldots ,T_n$ be independent non-negative random variables.", "Assume that there are $\\theta > 0$ and $\\mu \\in (0,1)$ such that for any $i$ , $\\mathbb {P}[T_i \\le \\theta ] \\le \\mu $ .", "There are $\\alpha > 0$ and $\\eta \\in (0,1)$ that depend only on $\\theta $ and $\\mu $ such that $\\mathbb {P}[\\sum _{i=1}^n T_i < \\alpha n] \\le \\eta ^n.$ Let $\\mathbb {Y}$ be a $d\\times m$ matrix with $m \\le d$ , such that the columns of $\\mathbb {Y}$ are i.i.d.", "random vectors.", "Assume further that $\\mathbb {Y}_{ij}$ are centered, and have a variance of 1 and a fourth moment at most $B$ .", "Let $\\Sigma $ be a diagonal $d \\times d$ PSD matrix.", "Then for all $x \\in S^{m-1}$ , $\\mathbb {P}[\\Vert \\sqrt{\\Sigma } \\mathbb {Y} x\\Vert \\le \\sqrt{\\operatorname{trace}(\\Sigma )/2}] \\le 1-1/(196B).$ Let $x\\in S^{m-1}$ , and $T_i = (\\sum _{j=1}^m \\mathbb {Y}_{ij} x_j)^2$ .", "Let $\\lambda _1,\\ldots ,\\lambda _d$ be the values on the diagonal of $\\Sigma $ , and let $T_\\Sigma = \\Vert \\sqrt{\\Sigma } \\mathbb {Y} x\\Vert ^2 = \\sum _{i=1}^d \\lambda _i T_i$ .", "First, since $\\mathbb {E}[\\mathbb {Y}_{ij}] = 0$ and $\\mathbb {E}[\\mathbb {Y}_{ij}] = 1$ for all $i,j$ , we have $\\mathbb {E}[T_i] = \\sum _{i\\in [m]} x^2_j\\mathbb {E}[\\mathbb {Y}_{ij}^2] = \\Vert x\\Vert ^2 = 1.$ Therefore $\\mathbb {E}[T_\\Sigma ] = \\operatorname{trace}(\\Sigma )$ .", "Second, since $\\mathbb {Y}_{i1},\\ldots ,\\mathbb {Y}_{im}$ are independent and centered, we have [18] $\\mathbb {E}[T_i^2] = \\mathbb {E}[(\\sum _{j\\in [m]} \\mathbb {Y}_{ij} x_j)^4] \\le 16\\mathbb {E}_\\sigma [(\\sum _{j\\in [m]} \\sigma _j\\mathbb {Y}_{ij} x_j)^4],$ where $\\sigma _1,\\ldots ,\\sigma _m$ are independent uniform $\\lbrace \\pm 1\\rbrace $ variables.", "Now, by Khinchine's inequality [21], $\\mathbb {E}_\\sigma [(\\sum _{j\\in [m]} \\sigma _j\\mathbb {Y}_{ij} x_j)^4] \\le 3\\mathbb {E}[(\\sum _{j\\in [m]} \\mathbb {Y}^2_{ij} x^2_j)^2]= 3\\sum _{j,k\\in [m]} x^2_j x^2_k\\mathbb {E}[\\mathbb {Y}^2_{ij}]\\mathbb {E}[\\mathbb {Y}^2_{ik}].$ Now $\\mathbb {E}[\\mathbb {Y}^2_{ij}]\\mathbb {E}[\\mathbb {Y}^2_{ik}] \\le \\sqrt{\\mathbb {E}[\\mathbb {Y}^4_{ij}]\\mathbb {E}[\\mathbb {Y}^4_{ik}]} \\le B$ .", "Thus $\\mathbb {E}[T_i^2] \\le 48B\\sum _{j,k\\in [m]} x^2_j x^2_k = 48B\\Vert x\\Vert ^4 = 48B.$ Thus, $\\mathbb {E}[T_\\Sigma ^2] &= \\mathbb {E}[(\\sum _{i=1}^d \\lambda _i T_i)^2] = \\sum _{i,j=1}^d \\lambda _i \\lambda _j \\mathbb {E}[ T_i T_j] \\\\&\\le \\sum _{i,j=1}^d \\lambda _i \\lambda _j \\sqrt{\\mathbb {E}[T_i^2] \\mathbb {E}[T_j^2]} \\le 48B (\\sum _{i=1}^d \\lambda _i)^2 = 48B\\cdot \\operatorname{trace}(\\Sigma )^2.$ By the Paley-Zigmund inequality [23], for $\\theta \\in [0,1]$ $\\mathbb {P}[T_\\Sigma \\ge \\theta \\mathbb {E}[T_\\Sigma ]] \\ge (1-\\theta )^2 \\frac{\\mathbb {E}[T_\\Sigma ]^2}{\\mathbb {E}[T_\\Sigma ^2]} \\ge \\frac{(1-\\theta )^2}{48B}.$ Therefore, setting $\\theta = 1/2$ , we get $\\mathbb {P}[T_\\Sigma \\le \\operatorname{trace}(\\Sigma )/2] \\le 1 - 1/(196B).$ [of Lemma REF ] Let $\\lambda _1,\\ldots ,\\lambda _d \\in [0,1]$ be the values on the diagonal of $\\Sigma $ .", "Consider a partition $Z_1,\\ldots ,Z_k$ of $[d]$ , and denote $L_j =\\sum _{i\\in Z_j} \\lambda _i$ .", "There exists such a partition such that for all $j\\in [k]$ , $L_j \\le 1$ , and for all $j \\in [k-1]$ , $L_j > {\\frac{1}{2}}$ .", "Let $\\Sigma [j]$ be the sub-matrix of $\\Sigma $ that includes the rows and columns whose indexes are in $Z_j$ .", "Let $\\mathbb {Y}[j]$ be the sub-matrix of $\\mathbb {Y}$ that includes the rows in $Z_j$ .", "Denote $T_j = \\Vert \\sqrt{\\Sigma [j]} \\mathbb {Y}[j] x\\Vert ^2$ .", "Then $\\Vert \\sqrt{\\Sigma }\\mathbb {Y}x\\Vert ^2 =\\sum _{j\\in [k]} \\sum _{i\\in Z_j} \\lambda _i (\\sum _{j=1}^m \\mathbb {Y}_{ij} x_j)^2 =\\sum _{j\\in [k]} T_j.$ We have $\\operatorname{trace}(\\Sigma ) = \\sum _{i=1}^d \\lambda _i \\ge \\sum _{j\\in [k-1]} L_j \\ge {\\frac{1}{2}}(k-1)$ .", "In addition, $L_j \\le 1$ for all $j \\in [k]$ .", "Thus $\\operatorname{trace}(\\Sigma ) \\le k \\le 2\\operatorname{trace}(\\Sigma )+1.$ For all $j \\in [k-1]$ , $L_j \\ge {\\frac{1}{2}}$ , thus by Lemma REF , $\\mathbb {P}[T_j \\le 1/4] \\le 1 - 1/(196B)$ .", "Therefore, by Lemma REF there are $\\alpha > 0$ and $\\eta \\in (0,1)$ that depend only on $B$ such that $&\\mathbb {P}[\\Vert \\sqrt{\\Sigma }\\mathbb {Y}x\\Vert ^2 < \\alpha \\cdot (\\operatorname{trace}(\\Sigma )-1)] \\le \\mathbb {P}[\\Vert \\sqrt{\\Sigma }\\mathbb {Y}x\\Vert ^2 < \\alpha (k-1)] \\\\&\\quad = \\mathbb {P}[\\sum _{j\\in [k]} T_j < \\alpha (k-1)] \\le \\mathbb {P}[\\sum _{j\\in [k-1]} T_j < \\alpha (k-1)] \\le \\eta ^{k-1} \\le \\eta ^{2\\operatorname{trace}(\\Sigma )}.$ The lemma follows by substituting $\\eta $ for $\\eta ^2$ .", "[of Theorem REF ] We have $&\\sqrt{\\lambda _{\\min }(\\mathbb {X}\\mathbb {X}^T)} = \\inf _{x\\in S^{m-1}}\\Vert \\mathbb {X}^Tx\\Vert \\ge \\min _{x \\in \\mathcal {C}_m(\\epsilon )}\\Vert \\mathbb {X}^Tx\\Vert -\\epsilon \\Vert \\mathbb {X}^T\\Vert .$ For brevity, denote $L = \\operatorname{trace}(\\Sigma )$ .", "Assume $L \\ge 2$ .", "Let $m\\le L\\cdot \\min (1,(c-K\\epsilon )^2)$ where $c, K, \\epsilon $ are constants that will be set later such that $c- K\\epsilon > 0$ .", "By Equation (REF ) $&\\mathbb {P}[\\lambda _{\\min }(\\mathbb {X}\\mathbb {X}^T) \\le m] \\le \\mathbb {P}[\\lambda _{\\min }(\\mathbb {X}\\mathbb {X}^T) \\le (c-K\\epsilon )^2L] \\\\&\\quad \\le \\mathbb {P}[\\min _{x\\in \\mathcal {C}_m(\\epsilon )}\\Vert \\mathbb {X}^Tx\\Vert -\\epsilon \\Vert \\mathbb {X}^T\\Vert \\le (c-K\\epsilon )\\sqrt{L}] \\\\ &\\quad \\le \\mathbb {P}[\\Vert \\mathbb {X}^T\\Vert \\ge K\\sqrt{L}] + \\mathbb {P}[\\min _{x\\in \\mathcal {C}_m(\\epsilon )}\\Vert \\mathbb {X}^Tx\\Vert \\le c\\sqrt{L}].$ The last inequality holds since the inequality in line (REF ) implies at least one of the inequalities in line ().", "We will now upper-bound each of the terms in line ().", "We assume w.l.o.g.", "that $\\Sigma $ is not singular (since zero rows and columns can be removed from $\\mathbb {X}$ without changing $\\lambda _{\\min }(\\mathbb {X}\\mathbb {X}^T)$ ).", "Define $\\mathbb {Y} \\triangleq \\sqrt{\\Sigma ^{-1}}\\mathbb {X}^T $ .", "Note that $\\mathbb {Y}_{ij}$ are independent sub-Gaussian variables with (absolute) moment $\\rho $ .", "To bound the first term in line (), note that by Lemma REF , for any $K > 0$ , $\\mathbb {P}[\\Vert \\mathbb {X}^T\\Vert \\ge K\\sqrt{L}] = \\mathbb {P}[\\Vert \\sqrt{\\Sigma }\\mathbb {Y}\\Vert \\ge K\\sqrt{L}] \\le \\mathcal {N}_m({\\frac{1}{2}})\\exp (L({\\frac{1}{2}}-\\frac{K^2}{16\\rho ^2})).$ By [26], Proposition 2.1, for all $\\epsilon \\in [0,1]$ , $\\mathcal {N}_m(\\epsilon ) \\le 2m(1+\\frac{2}{\\epsilon })^{m-1}.$ Therefore $\\mathbb {P}[\\Vert \\mathbb {X}^T\\Vert \\ge K\\sqrt{L}] \\le 2m5^{m-1}\\exp (L({\\frac{1}{2}}-\\frac{K^2}{16\\rho ^2})).$ Let $K^2 = 16\\rho ^2(\\frac{3}{2}+\\ln (5)+\\ln (2/\\delta ))$ .", "Recall that by assumption $m \\le L$ , and $L \\ge 2$ .", "Therefore $&\\mathbb {P}[\\Vert \\mathbb {X}^T\\Vert \\ge K\\sqrt{L}] \\le 2m5^{m-1}\\exp (-L(1+\\ln (5)+\\ln (2/\\delta )))\\\\&\\quad \\le 2L5^{L-1}\\exp (-L(1+\\ln (5)+\\ln (2/\\delta ))).$ Since $L \\ge 2$ , we have $2L\\exp (-L) \\le 1$ .", "Therefore $\\mathbb {P}[\\Vert \\mathbb {X}^T\\Vert \\ge K\\sqrt{L}]\\le 2L\\exp (-L-\\ln (2/\\delta ))\\le \\exp (-\\ln (2/\\delta )) = \\frac{\\delta }{2}.$ To bound the second term in line (), since $\\mathbb {Y}_{ij}$ are sub-Gaussian with moment $\\rho $ , $\\mathbb {E}[\\mathbb {Y}_{ij}^4] \\le 5\\rho ^4$ [13].", "Thus, by Lemma REF , there are $\\alpha >0$ and $\\eta \\in (0,1)$ that depend only on $\\rho $ such that for all $x\\in S^{m-1}$ , $\\mathbb {P}[\\Vert \\sqrt{\\Sigma }\\mathbb {Y}x\\Vert ^2\\le \\alpha (L-1)] \\le \\eta ^{L}$ .", "Set $c = \\sqrt{\\alpha /2}$ .", "Since $L\\ge 2$ , we have $c\\sqrt{L} \\le \\sqrt{\\alpha (L-1)}$ .", "Thus $&\\mathbb {P}[\\min _{x\\in \\mathcal {C}_m(\\epsilon )}\\Vert \\mathbb {X}^Tx\\Vert \\le c\\sqrt{L}] \\le \\sum _{x\\in \\mathcal {C}_m(\\epsilon )}\\mathbb {P}[\\Vert \\mathbb {X}^Tx\\Vert \\le c \\sqrt{L}] \\\\&\\quad \\le \\sum _{x\\in \\mathcal {C}_m(\\epsilon )}\\mathbb {P}[\\Vert \\sqrt{\\Sigma }\\mathbb {Y}x\\Vert \\le \\sqrt{\\alpha (L-1)}]\\le \\mathcal {N}_m(\\epsilon )\\eta ^{L}.$ Let $\\epsilon = c/(2K)$ , so that $c - K\\epsilon > 0$ .", "Let $\\theta = \\min ({\\frac{1}{2}},\\frac{\\ln (1/\\eta )}{2\\ln (1+2/\\epsilon )})$ .", "Set $L_\\circ $ such that $\\forall L \\ge L_\\circ $ , $L \\ge \\frac{2\\ln (2/\\delta )+2\\ln (L)}{\\ln (1/\\eta )}$ .", "For $L \\ge L_\\circ $ and $m \\le \\theta L \\le L/2$ , $\\mathcal {N}_m(\\epsilon )\\eta ^{L} &\\le 2m(1+2/\\epsilon )^{m-1}\\eta ^{L} \\\\&\\le L\\exp (L(\\theta \\ln (1+2/\\epsilon )-\\ln (1/\\eta )))\\\\&= \\exp (\\ln (L) + L(\\theta \\ln (1+2/\\epsilon )-\\ln (1/\\eta )/2) - L\\ln (1/\\eta )/2) \\\\&\\le \\exp (L(\\theta \\ln (1+2/\\epsilon )-\\ln (1/\\eta )/2)+\\ln (\\delta /2))\\\\&\\le \\exp (\\ln (\\delta /2)) = \\frac{\\delta }{2}.$ Line (REF ) follows from $L \\ge L_\\circ $ , and line () follows from $\\theta \\ln (1+2/\\epsilon )-\\ln (1/\\eta )/2 \\le 0$ .", "Set $\\beta = \\min \\lbrace (c-K\\epsilon )^2,1,\\theta \\rbrace $ .", "Combining Equation (), Equation (REF ) and Equation () we have that if $L \\ge \\bar{L} \\triangleq \\max (L_\\circ ,2)$ , then $\\mathbb {P}[\\lambda _{\\min }(\\mathbb {X}\\mathbb {X}^T) \\le m] \\le \\delta $ for all $m \\le \\beta L$ .", "Specifically, this holds for all $L \\ge 0$ and for all $m \\le \\beta (L-\\bar{L})$ .", "Letting $C = \\beta \\bar{L}$ and substituting $\\delta $ for $1-\\delta $ we get the statement of the theorem." ] ]

[ [ "Higher Derivative Corrections to Non-Abelian Vortex Effective Theory" ], [ "Abstract We give a systematic method to calculate higher derivative corrections to low-energy effective theories of solitons, which are in general non-linear sigma models on the moduli spaces of the solitons.", "By applying it to the effective theory of a single BPS non-Abelian vortex in U(N) gauge theory with N fundamental Higgs fields, we obtain four derivative corrections to the effective sigma model on the moduli space C \\times CP^{N-1}.", "We compare them with the Nambu-Goto action and the Faddeev-Skyrme model.", "We also show that Yang-Mills instantons/monopoles trapped inside a non-Abelian vortex membrane/string are not modified in the presence of higher derivative terms." ], [ "Introduction", "Solitons are smooth localized solutions of nonlinear partial differential equations, and they are ubiquitous in diverse fields in physics and mathematics.", "Topological solitons are smooth localized solutions of field equations in quantum field theories, and their stability is ensured by topological charges.", "A particularly interesting class of topological solitons is Bogomol'nyi-Prasad-Sommerfield (BPS) solitons [1], which saturate minimum energy bounds called the Bogomol'nyi bounds.", "They are the most stable solutions with given topological numbers and satisfy first order partial differential equations called BPS equations.", "Prominent examples of such BPS solitons are instantons in Euclidean Yang-Mills theory [2], `t Hooft-Polyakov monopoles [3] in the Bogomol'nyi limit [1], and Abrikosov-Nielsen-Olesen (ANO) vortices [4] at the critical coupling (a standard reference is [5]).", "Recent examples of BPS topological solitons in gauge theories are BPS kinks (domain walls) in Abelian [6] and non-Abelian [7] gauge theories.", "While they preserve half of supersymmetry and are quantum mechanically stable in supersymmetric gauge theories [8] on one hand, one of the most important features of BPS solitons is that there exist no static forces among them and consequently a continuous family of configurations with degenerate energy is allowed.", "As a result, generic solutions of BPS solitons contain collective coordinates, called moduli parameters as integration constants.", "The moduli parameters of BPS solitons parametrize the space of whole solutions of BPS equations, called the moduli space.", "Although there is no static force among BPS solitons, they non-trivially scatter each other when they are moving.", "While such dynamics was studied by computer simulations, it is very difficult to study it by means of a fully analytic approach.", "The seminal idea of Manton [9] is that when solitons move slowly, their dynamics can be described as geodesics on their moduli space.", "This low-energy approximation is now called the moduli, geodesic, or Manton approximation.", "After his work, the moduli spaces are recognized as the most important objects associated with BPS solitons [5].", "The Atiyah-Hitchin metric on two BPS monopoles is one of the most prominent examples [10].", "By examining geodesics on it, one can study the scattering of two BPS monopoles and find some interesting phenomena such as the right angle scattering in head on collisions.", "The moduli space of BPS monopoles was further studied in [11], [12].", "The moduli space and soliton scattering were studied for ANO vortices in the BPS limit [13], [14], [15], [16], [17], [18], [19] and for BPS domain walls [20].", "More recently, the moduli space dynamics has been successfully applied even to 1/4 BPS composite solitons such as domain wall networks [21] and strings stretched between parallel domain walls [22].", "Although there has been a lot of remarkable progress in study of the moduli space of BPS solitons, the analysis of their dynamics has been restricted to the leading order of the low-energy (small-velocity) limit, except for a few examples [23], [24].", "Non-Abelian vortices were found recently in ${\\cal N}=2$ supersymmetric $U(N)$ QCD [25], [26].", "Although there has been a lot of progress in the study of non-Abelian vortices [27], [28], here we concentrate on their moduli space.", "A single non-Abelian vortex has moduli space ${\\mathbb {C}} \\times {\\mathbb {C}}P^{N-1}$ where the former denotes the position and the latter, called orientational moduli, comes from the $SU(N)$ vacuum symmetry broken in the presence of the vortex [25], [26].", "One of the most important facts about the orientational moduli is that confined monopoles and trapped Yang-Mills instantons in the Higgs phase are realized, respectively, as kinks [29] and lumps [30], [31] in the vortex world-sheet ${\\mathbb {C}}P^{N-1}$ sigma model.", "This world-sheet effective theory provides a physical explanation for the relationship between BPS spectra in two-dimensional $\\mathcal {N} = (2,2)$ sigma models and four-dimensional $\\mathcal {N} = 2$ supersymmetric QCD [32], [33].", "The moduli space of multiple vortices with full moduli parameters was completely determined (without metric) in [34], [28], [35]; The moduli space for $k$ separated vortices is a $k$ -symmetric product $({\\mathbb {C}} \\times {\\mathbb {C}}P^{N-1})^k/{\\mathcal {S}}_k$ of the single vortex moduli space [34] while the whole space is regular.", "General formula for the moduli space metric and its Kähler potential were given in [36].", "The metric of the moduli space on the coincident vortices was found [37], [38] and was applied to low-energy dynamics of two non-Abelian vortices in head-on collision, reconnections of two non-Abelian cosmic strings [37] and flux matching of vortex-monopole composite [39].", "Recently the moduli space metric of multiple non-Abelian vortices has been finally obtained for well separated vortices [40], and their low-energy dynamics has been examined [41].", "The purpose of this paper is to propose a systematic method to study dynamics of BPS solitons moving with higher velocities beyond the Manton approximation; we give a general scheme to calculate derivative corrections and apply it to obtain four-derivative corrections to the low-energy effective theory on a single non-Abelian vortex.", "In the spirit of the low-energy effective action, the leading order terms can be obtained by integrating out massive modes.", "The lowest order terms take the form of nonlinear sigma models with two derivative terms for light or massless fields, typically Nambu-Goldstone modes.", "A famous example is the chiral Lagrangian or chiral perturbation theory for pions which are the Nambu-Goldstone bosons of chiral symmetry breaking.", "The next-leading terms consist of four or higher derivative corrections, which are the focus of our study.", "In order to obtain four derivative terms, we solve equations for massive modes and eliminate them order by order.", "As a concrete example, we consider a single non-Abelian vortex.", "We obtain four-derivative corrections for the translational moduli ${\\mathbb {C}}$ , the orientational moduli (the ${\\mathbb {C}}P^{N-1}$ model) and their mixing terms.", "In the literature a different expansion is known: an expansion from the Nambu-Goto action [42] in powers of (the inverse of) the width of strings.", "The effective action of a string in the thin limit can be described by the Nambu-Goto action [43].", "The finite-width correction in general takes the form of extrinsic curvature squared, which is called stiffness or rigidity of string [44].", "Dynamics of a string with the correction term was studied [45].", "The explicit calculation of the rigidity term was performed in the Abelian Higgs model by many authors in the context of cosmic strings and QCD strings [46].", "This expansion is in powers of $\\# (\\partial ) - \\# (X)$ , where we denote the number of derivatives and fields as $\\# (\\partial )$ and $\\# (X)$ , respectively.", "The leading term, the Nambu-Goto action, is the zeroth order term in the series, and thus contains the same number of derivatives and fields.", "The next-leading term, the stiffness term, contains two more derivatives than the number of fields starting from ${\\cal O}(\\partial ^6 X^4)$ .", "One advantage of this width expansion is that even the leading term, the Nambu-Goto action, contains infinite number of derivatives so that it can describe large fluctuations or bends of a string.", "However the expansion in powers of $\\# (\\partial ) - \\# (X)$ is not applicable for dynamics of moduli in general; for instance multiple solitons or even a single non-Abelian soliton with internal degrees of freedom.", "— On the other hand, our expansion is purely a derivative expansion commonly used in the literature of field theories.", "It is valid at low energy and even the leading term, the nonlinear sigma model, can describe dynamics of multiple solitons.", "In the present paper, we will discuss the higher derivative corrections to the low-energy effective theory on a non-Abelian vortex world-volume.", "By a symmetry argument, one can easily write down the generic form of the higher derivative terms and find that the corrections cause instability for the lumps (sigma model instantons) which are identified with the Yang-Mills instantons trapped inside a vortex.", "According to the Derrick's scaling argument, the size of a lump expands in the presence of the generic higher derivative terms.", "Since the lumps are responsible for non-perturbative effects in the vortex world-sheet effective theory, one may think that the correspondence between the BPS spectra in 2d and 4d [32], [33] would be modified by the higher derivative corrections.", "However, as we will see in Sec.", "REF , the lump solutions are still stable if the higher derivative terms have a specific form.", "In this paper, we will calculate the explicit form of the higher derivative terms and find that they do not modify the lump solutions.", "This paper is organized as follows.", "We first review non-Abelian vortices and their moduli space in Sec. .", "After illustrating our method for the derivative expansion in a simple example of classical mechanics in Sec.", ", we develop a systematic method to obtain derivative corrections to the effective action for non-Abelian vortices in Sec. .", "In Sec.", "REF we compare our result for the effective action of a single vortex with previously known models, the Nambu-Goto action and the Faddeev-Skyrme model [47].", "In Sec.", "REF we discuss that the four derivative terms do not modify the Yang-Mills instanton solutions trapped inside a non-Abelian vortex [30], [31].", "In Sec.", ", we discuss higher order corrections to the vortex effective action in a mass deformed model and show that the classical BPS spectrum is not modified by the higher order corrections.", "Sec.", "is devoted to summary and discussion." ], [ "The non-Abelian vortices", "In this section, we briefly review the non-Abelian vortices and summarize the basic tools to describe the moduli space of BPS configurations.", "Let us consider the $U(N)$ gauge theory in $(d+1)$ -dimensional spacetime with gauge field $W_\\mu $ and $N_{\\rm F}=N$ Higgs fields $H$ ($N$ -by-$N$ matrix) in the fundamental representation.", "The Lagrangian of our model takes the form $\\mathcal {L} &=& {\\rm Tr}\\left[ - \\frac{1}{2g^2} F_{\\mu \\nu } F^{\\mu \\nu }+ \\mu H (\\mu H)^\\dagger - \\frac{g^2}{4} (H H^\\dagger - v^2 \\mathbf {1}_N)^2 \\right], $ where $g$ is the gauge coupling constant and $v^2$ is the Fayet-Iliopoulos parameter.", "We use the almost minus metric $\\eta _{\\mu \\nu } = (+1,-1,-1,\\cdots ,-1)$ .", "Although we can choose different gauge coupling constants for the $U(1)$ and $SU(N)$ parts of the gauge group, we set them equal for notational simplicity.", "Our notation for the covariant derivative and the field strength is $\\mu H = (\\partial _\\mu + i W_\\mu ) H, \\hspace{28.45274pt}F_{\\mu \\nu } = \\partial _\\mu W_\\nu - \\partial _\\nu W_\\mu + i [ W_\\mu , W_\\nu ].$ As is well known, the Lagrangian can be embedded into a supersymmetric theory with eight supercharges.", "The vacuum condition is solved by $H = v \\, \\mathbf {1}_N.$ This vacuum expectation value (VEV) of the Higgs field completely breaks the gauge symmetry, whereas the following $SU(N)_{\\rm C + \\rm F}$ diagonal symmetry remains unbroken $H ~\\rightarrow ~ U_{\\rm C} H U_{\\rm F}, \\hspace{28.45274pt} U_{\\rm C}^\\dagger = U_{\\rm F} \\in SU(N)_{\\rm C +F}.$ Let us consider the non-Abelian vortices in this color-flavor locked Higgs vacuum.", "We assume that the vortices are localized in the $z = x_{d-1} + i x_d$ plane.", "The topological sectors of the field configurations are classified by the vorticity, i.e.", "the magnetic flux of the overall $U(1)$ gauge group $k ~\\equiv ~ - \\frac{1}{2\\pi } \\int dz \\wedge d \\bar{z} \\ {\\rm Tr}\\, F_{z \\bar{z}}, \\hspace{28.45274pt} k \\in \\mathbb {Z}.$ For a given vorticity $k$ , the tension (the energy per unit volume) of static configurations is bounded from below as $T \\hspace{-5.69054pt} &=& \\hspace{-5.69054pt} \\int d^2 x \\, {\\rm Tr}\\Bigg [ 4 {\\bar{z}} H ({\\bar{z}} H)^\\dagger + \\frac{4}{g^2} \\left| i F_{z \\bar{z}} - \\frac{g^2}{4} ( v^2 \\mathbf {1}_N - H H^\\dagger ) \\right|^2 - 4 {[z} ( {\\bar{z}]} H H^\\dagger ) + 2 v^2 i F_{z \\bar{z}} \\Bigg ] \\\\&\\ge & 2 \\pi v^2 k,$ where we have assumed that $i H \\rightarrow 0$ at the infinity $|z| \\rightarrow \\infty $ so that the third term does not contribute to the energy.", "This Bogomol'nyi bound is saturated if the following equations are satisfied ${\\bar{z}} H = 0, \\hspace{28.45274pt}i F_{z \\bar{z}} = \\frac{g^2}{4} ( v^2 \\mathbf {1}_N - H H^\\dagger ).$ These equations are the BPS equations for non-Abelian vortices and describe the configurations of static BPS vortices which are minimal energy configurations in a given topological sector.", "Since the same equations can be derived by imposing the condition that one half of supersymmetry is preserved, Eq.", "(REF ) is also called 1/2 BPS equations.", "We can show that any of the BPS configurations is a solution of the following full equations of motion of the system $0 &=& \\mu \\mu H + \\frac{g^2}{2} ( H H^\\dagger - v^2 \\mathbf {1}_N ) H, \\\\0 &=& \\frac{2}{g^2} \\mu F_{\\mu \\nu } + i \\left[ H ( \\nu H )^\\dagger - (\\nu H) H^\\dagger \\right].", "$ In order to describe the moduli space of the BPS configurations, it is convenient to write the BPS solution as $H = v \\, S^{-1} H_0, \\hspace{28.45274pt}W_{\\bar{z}} = - i S^{-1} \\bar{\\partial }S,$ where $H_0$ and $S$ are $N$ -by-$N$ matrices.", "Then, the first BPS equation in Eq.", "(REF ) becomes $\\partial _{\\bar{z}} H_0 = 0.$ Therefore all the entries of the $N$ -by-$N$ matrix $H_0$ are arbitrary holomorphic polynomials.", "The matrix function $S \\in GL(N,$ is determined from the second equation in Eq.", "(REF ), which can be rewritten into the following equation for $\\Omega \\equiv SS^\\dagger $ : [48], [34] $\\frac{4}{g^2 v^2} \\partial _{\\bar{z}} ( \\Omega \\partial _z \\Omega ^{-1} )= H_0 H_0^\\dagger \\Omega ^{-1} - \\mathbf {1}_{N}.$ For a given $H_0$ , this equation can be viewed as a non-linear differential equation for $\\Omega $ .", "The boundary condition for $\\Omega $ should be chosen so that the energy density vanishes at the spatial infinity $|z| \\rightarrow \\infty $ .", "In general, this condition is solved by $\\Omega ~\\underset{|z| \\rightarrow \\infty }{\\longrightarrow }~ H_0 H_0^\\dagger .$ The matrix $S$ can be determined from the solution $\\Omega $ uniquely up to the unphysical $U(N)$ gauge transformation $S \\rightarrow S u(x)$ (with $u(x) \\in U(N)$ ).", "Solving Eq.", "(REF ) for an arbitrarily chosen $H_0(z)$ , we can obtain a BPS vortex solutions through Eq.", "(REF ).", "In other words, the matrix $H_0(z)$ classifies all the BPS configurations.", "Hence the holomorphic matrix $H_0(z)$ is called “the moduli matrix\" and Eq.", "(REF ) is called “the master equation\" for vortices.", "The matrices $H_0(z)$ and $\\Omega $ are invariant under the original $U(N)$ gauge transformation.", "On the other hand, the original fields $H$ and $W_{\\bar{z}}$ are invariant under “the $V$ -transformations\" defined by $H_0(z) \\rightarrow V(z) H_0(z), \\hspace{19.91692pt}\\Omega \\rightarrow V(z) \\, \\Omega \\, V(z)^\\dagger , \\hspace{28.45274pt}V(z) \\in GL(N,.$ This is the gauge symmetry of the master equation Eq.", "(REF ) which does not change the physical quantities.", "Therefore, the solutions of the BPS equations are completely classified by the equivalence classes of the moduli matrix $H_0(z) ~\\sim ~ V(z) H_0(z).$ This implies that the parameters contained in an appropriately gauge fixed moduli matrix $H_0$ can be interpreted as the moduli parameters of the BPS configurations.", "Let us next discuss how the moduli matrix $H_0$ determines the topological sector of the corresponding solution.", "From Eq.", "(REF ), we can see that the magnetic flux of the overall $U(1)$ is given in terms of $\\Omega $ by ${\\rm Tr}\\, F_{z \\bar{z}} = - i \\partial _{\\bar{z}} \\partial _z \\log \\det \\Omega .$ By using the Stokes' theorem and the boundary condition Eq.", "(REF ), we find that the vorticity Eq.", "(REF ) is given by $k ~=~ \\frac{1}{4 \\pi i} \\oint _{S^1_\\infty } \\,(d z \\partial _z - d \\bar{z} \\partial _{\\bar{z}}) \\log \\det (H_0 H_0^\\dagger ),$ where $S^1_{\\infty }$ denotes the clock-wise circle at infinity $|z| \\rightarrow \\infty $ .", "This equation shows that the order of the polynomial $\\det H_0(z)$ corresponds to the vorticity.", "As an example, let us consider the case of single vortex configurations.", "The simplest single vortex solution can be obtained by embedding the single Abelian vortex solution in the upper-left corner of the $N$ -by-$N$ matrices $H$ and $W_{\\bar{z}}$ $H ~=~ \\left( \\begin{array}{cc} H^{\\rm ANO} & 0 \\\\ 0 & v \\, \\mathbf {1}_{N-1} \\end{array} \\right), \\hspace{28.45274pt}W_{\\bar{z}} ~=~ \\left( \\begin{array}{cc} W_{\\bar{z}}^{\\rm ANO} & 0 \\\\ 0 & \\mathbf {0}_{N-1} \\end{array} \\right).$ The other single vortex solutions can be obtained by acting on the embedded solution with the $SU(N)_{\\rm C+F}$ symmetry $H \\rightarrow U^\\dagger H U, \\hspace{28.45274pt} W_{\\bar{z}} \\rightarrow U^\\dagger W_{\\bar{z}} U, \\hspace{28.45274pt} U \\in SU(N)_{\\rm C + F}.$ Since the vortex embedded into the upper-left corner breaks $SU(N)_{\\rm C+F} \\rightarrow SU(N-1) \\times U(1)$ , the internal vortex moduli space is the complex projective space $P̏^{N-1} \\cong \\frac{SU(N)}{SU(N-1) \\times U(1)}.$ By modding out the unbroken $SU(N-1) \\times U(1)$ , we can fix the unitary matrix $U$ as $U &=& \\left( \\begin{array}{cc} 1 & - \\vec{b}^\\dagger \\\\ 0 & \\mathbf {1}_{N-1} \\end{array} \\right)\\left( \\begin{array}{cc} X^{\\frac{1}{2}} & 0 \\\\ 0 & Y^{-\\frac{1}{2}} \\end{array} \\right)\\left( \\begin{array}{cc} 1 & 0 \\\\ \\vec{b} & \\mathbf {1}_{N-1} \\end{array} \\right),$ where the parameters $\\vec{b} = (b^1, \\cdots , b^{N-1})$ can be interpreted as the inhomogeneous coordinates of $P̏^{N-1}$ and $X$ and $Y$ are given byThe square root of the matrix $Y$ is given by $Y^{\\pm \\frac{1}{2}} = (1-P) + (1+|\\vec{b}|^2)^{\\pm \\frac{1}{2}} P$ with $P = |\\vec{b}|^{-2} \\vec{b} \\otimes \\vec{b}^\\dagger $ .", "$X \\equiv 1 + \\vec{b}^\\dagger \\cdot \\vec{b}, \\hspace{28.45274pt}Y \\equiv \\mathbf {1}_{N-1} + \\vec{b} \\otimes \\vec{b}^\\dagger .$ The physical meaning of the orientational moduli can be seen from the magnetic flux $F_{z \\bar{z}} ~=~ F_{z \\bar{z}}^{\\rm ANO} ~\\times ~ \\frac{1}{1+|\\vec{b}|^2} \\left( \\begin{array}{cc} \\ \\, 1 & - \\vec{b}^\\dagger \\\\ -\\vec{b} & \\ \\vec{b} \\otimes \\vec{b}^\\dagger \\end{array} \\right).$ Thus the orientational moduli $\\vec{b}$ determines the $U(1)$ generator in which the vortex magnetic flux is embedded.", "The moduli matrix $H_0$ and the solution of the master equation $\\Omega $ corresponding to the single vortex configurations are given by $H_0(z) ~=~V \\left( \\begin{array}{cc} z - Z & 0 \\\\ 0 & \\mathbf {1}_{N-1} \\end{array} \\right)U, \\hspace{28.45274pt}\\Omega ~=~ V \\left( \\begin{array}{cc} e^{\\psi } & 0 \\\\ 0 & \\mathbf {1}_{N-1} \\end{array} \\right)V^\\dagger ,$ where $\\psi $ is a real profile function satisfying $\\frac{4}{g^2 v^2} \\partial _{\\bar{z}} \\partial _z \\psi = 1 - |z-Z|^2 e^{-\\psi },$ with the boundary condition $\\psi \\rightarrow \\log |z-Z|^2$ .", "The complex parameter $Z$ in the moduli matrix $H_0$ can be interpreted as the position moduli of the vortex.", "Although the holomorphic matrix $V(z)$ can be an arbitrary element of $GL(N,$ , it will be convenient to use the matrix of the form $V(z) &=&\\left( \\begin{array}{cc} X^{-\\frac{1}{2}} & 0 \\\\ 0 & Y^{\\frac{1}{2}} \\end{array} \\right)\\left( \\begin{array}{cc} 1 & (z-Z) \\vec{b}^\\dagger \\\\ 0 & \\mathbf {1}_{N-1} \\end{array} \\right).$ This $V$ -transformation is chosen so that the moduli matrix $H_0(z)$ takes the form $H_0(z) ~=~ \\left( \\begin{array}{cc} z - Z & 0 \\\\ \\vec{b} & \\mathbf {1}_{N-1} \\end{array} \\right).$ The important point is that the $V$ -transformation is completely fixed so that the moduli matrix is holomorphic not only in $z$ but also in the moduli parameters $Z$ and $\\vec{b}$ .", "In general, there exists such a fixed form of the moduli matrix in each coordinate patch of the moduli space.", "The $V$ -transformation between a pair of the fixed moduli matrices induces a coordinate transformation from one patch to another.", "For instance, in the case of $N=2$ , there are two fixed forms $H_0 = \\left( \\begin{array}{cc} z - Z & 0 \\\\ b & 1 \\end{array} \\right), \\hspace{28.45274pt} H_0^{\\prime } = \\left( \\begin{array}{cc} 1 & b^{\\prime } \\\\ 0 & z - Z \\end{array} \\right).$ These two matrices are related by $H_0^{\\prime }= V H_0 , \\hspace{28.45274pt} V = \\left( \\begin{array}{cc} 0 & b^{\\prime }\\\\ -b & z-Z \\end{array} \\right), \\hspace{14.22636pt}b^{\\prime }=\\frac{1}{b}.$ Thus, the induced coordinate transformation is the standard transition map between the inhomogeneous coordinates of $P̏^1$ .", "As this example shows, the coordinate fransformations are holomorphic, so that the moduli space is a complex manifold.", "For a general winding number $k$ , we can fix generic moduli matrices to the following form [34] $H_0(z) ={{\\left( \\begin{array}{cc} (z-Z_1) (z-Z_2) \\cdots (z-Z_k) & 0 \\\\ \\vec{b}_1 \\, e_1(z) + \\vec{b}_2 \\, e_2(z) \\cdots + \\vec{b}_k \\, e_k(z) & \\mathbf {1}_{N-1} \\end{array} \\right)}}, \\hspace{19.91692pt}e_I(z) = \\prod _{J \\ne I} \\frac{z-Z_J}{Z_I-Z_J}.$ The parameters $Z_I$ and $\\vec{b}_I~(I=1,\\cdots ,k)$ are position and orientational moduli of $I$ -th vortex which cover a local coordinate patch of the moduli space of the vortices $\\mathcal {M}_k$ .", "As in the case of the single vortex, the moduli matrix $H_0(z)$ is holomorphic with respect to the moduli parameters.", "This fact will be important when we derive general formulas for the second and fourth order effective Lagrangian in section ." ], [ "Preliminary: a particle in $\\mathbb {R}^n$", "Before studying higher derivative corrections to the vortex effective action in field theories, we first explain our basic strategy in a simple system of a particle in $\\mathbb {R}^n$ with the following Lagrangian $L ~=~ \\frac{m}{2} \\dot{\\bf x} \\cdot \\dot{\\bf x} - V({\\bf x}), \\hspace{28.45274pt}{\\bf x}(t) = (x^1,x^2,\\cdots ,x^n).$ In a minimum energy configuration, the particle stays at the bottom of the potential $V$ where the gradient of $V$ vanishes ${\\rm grad} \\, V = 0.$ Now let us assume that the potential $V$ has flat directions which are parameterized by $\\phi ^i$ .", "Then we can define “the moduli space” of the minimum energy configurations by ${\\cal M} = \\left\\lbrace \\ {\\bf x}^{(0)}(\\phi ^i) \\in \\mathbb {R}^n \\ |\\ {\\rm grad} \\, V = 0\\ \\right\\rbrace ,$ and $\\phi ^i$ can be interpreted as “the moduli parameters”.", "Since one can shift the particle to any points on the bottom of the potential without loss of energy, we can assume that the particle slowly moves along the moduli space $\\mathcal {M}$ for sufficiently small excitation energy.", "This motion of the particle can be represented by the moduli parameters $\\phi ^i(t)$ which weakly depend on the time $t$ , that is ${\\bf x}(t) = {\\bf x}^{(0)}(\\phi ^i(t)).$ Then, the low energy effective Lagrangian of the particle can be obtained by substituting Eq.", "(REF ) into the original Lagrangian Eq.", "(REF ) $L_{\\rm eff} ~=~ L_{\\rm eff}^{(0)} + L_{\\rm eff}^{(2)} ~= - V({\\bf x}^{(0)}) + \\frac{m}{2} \\dot{\\bf x}^{(0)} \\cdot \\dot{\\bf x}^{(0)},$ where $L_{\\rm eff}^{(0)} = - V({\\bf x}^{(0)})$ is the constant value of the potential at the bottom.", "The second order effective Lagrangian $L_{\\rm eff}^{(2)}$ can be rewritten by using the metric $g_{ij}$ on the moduli space $\\mathcal {M}$ as $L_{\\rm eff}^{(2)} ~=~ \\frac{m}{2} g_{ij} \\dot{\\phi }^i\\dot{\\phi }^j.$ The metric is given by the inner products of the basis $\\lbrace \\Phi _i \\rbrace $ of “the zero modes\" $g_{ij} \\equiv \\Phi _i \\cdot \\Phi _j, \\hspace{28.45274pt}\\Phi _i ~\\equiv ~ \\frac{\\partial }{\\partial \\phi ^i} {\\bf x}^{(0)}.$ The effective equations of motion for the moduli parameters take the form of the geodesic equation $\\ddot{\\phi }^i + \\Gamma ^i_{jk} \\dot{\\phi }^j \\dot{\\phi }^k ~=~ 0,$ where the connection is given by $\\Gamma ^i_{jk} ~=~ \\frac{1}{2} g^{il} \\left(g_{lj,k}+g_{lk,j}-g_{jk,l}\\right) ~=~g^{il} \\, \\Phi _l \\cdot \\frac{\\partial }{\\partial \\phi ^j} \\Phi _k.$ This approximation is valid if the velocity of the particle is sufficiently small so that the time derivative is much less than the typical mass scale of the massive modes determined from the Hessian matrix $\\mathbf {H}$ of the potential $V$ defined by $\\big [ \\mathbf {H} ({\\bf x}^{(0)}) \\big ]_{ab} = \\frac{\\partial ^2}{\\partial x^a \\partial x^b} V \\bigg |_{{\\bf x}={\\bf x}^{(0)}}.$ Next, let us consider higher derivative corrections to the effective Lagrangian by taking the massive modes into account.", "To this end, we first introduce a correction to (REF ) by adding small fluctuations to “the slowly moving background\" ${\\bf x}^{(0)}(\\phi ^i(t))$ as ${\\bf x}(t) ~=~ {\\bf x}^{(0)}(\\phi ^i(t)) + \\delta {\\bf x}(t).$ Since the motion of the particle along the flat direction is already represented by the moduli parameters $\\phi ^i(t)$ , we impose the following condition to avoid the double-counting of the degrees of freedom of the zero modes: $\\delta {\\bf x} \\cdot \\Phi _i = 0.$ This means that the fluctuations $\\delta {\\bf x}$ is orthogonal to the tangent space of ${\\cal M}$ , namely the fluctuation $\\delta {\\bf x}$ contains only massive modes.", "Then, the original Lagrangian Eq.", "(REF ) can be rewritten as $L &=& L_{\\rm eff}^{(0)} + L_{\\rm eff}^{(2)} + \\delta L + \\lambda ^i \\left( \\delta {\\bf x} \\cdot \\Phi _i \\right) , \\phantom{\\bigg [}\\\\\\delta L &=&m \\, \\dot{\\bf x}^{(0)} \\cdot \\delta \\dot{\\bf x}+ \\frac{m}{2} \\delta \\dot{\\bf x} \\cdot \\delta \\dot{\\bf x}- \\frac{1}{2} \\delta {\\bf x} \\, \\mathbf {H} \\, \\delta {\\bf x} + \\cdots , \\phantom{\\bigg [}$ where we have introduced the Lagrange multipliers $\\lambda ^i$ to impose the constraint Eq.", "(REF ).", "Note that there is no linear term in the Taylor expansion of the potential since the background satisfies ${\\rm grad} \\, V( {\\bf x}^{(0)} ) = 0$ .", "Now let us consider the expansion of $\\delta {\\bf x}$ with respect to the time derivative $\\partial _t$ $\\delta {\\bf x} = {\\bf x}^{(2)} + {\\bf x}^{(4)} + \\cdots , \\hspace{28.45274pt}{\\bf x}^{(n)} \\sim \\mathcal {O}(\\partial _t^n).$ Note that $\\delta {\\bf x}$ can have only terms with even numbers of the time derivatives due to the reflection symmetry $t \\rightarrow - t$ .", "There is no zeroth order term since the fluctuation $\\delta {\\bf x}$ vanishes for the static configurations.", "Correspondingly, the Lagrangian $\\delta L$ can also be expanded as $\\delta L = L^{(4)} + L^{(6)} + \\cdots .$ The fourth order Lagrangian contains the background ${\\bf x}^{(0)}$ and the second order fluctuation ${\\bf x}^{(2)}$ $L^{(4)} = m \\, \\dot{\\bf x}^{(0)} \\cdot \\dot{\\bf x}^{(2)}- \\frac{1}{2} {\\bf x}^{(2)} \\, \\mathbf {H} \\, {\\bf x}^{(2)}.$ Note that $\\mathbf {x}^{(4)}$ does not contribute to the fourth order Lagrangian since ${\\rm grad} \\, V( {\\bf x}^{(0)} ) = 0$ .", "The massive modes ${\\bf x}^{(2)}$ can be eliminated by solving their equation of motion $\\mathbf {H} \\, {\\bf x}^{(2)} + m \\, \\ddot{\\bf x}^{(0)} = \\lambda ^i \\Phi _i.$ To determine the Lagrange multiplier $\\lambda ^i$ , let us take the inner products of the both hand sides and the zero modes $\\Phi _j$ $m \\, \\Phi _j \\cdot \\ddot{\\bf x}^{(0)} &=& g_{ij} \\lambda ^i,$ where we have used the fact that the zero modes $\\left\\lbrace \\Phi _i \\right\\rbrace $ satisfy $\\Phi _i \\, \\mathbf {H} ~=~ \\frac{\\partial }{\\partial \\phi ^i} \\big [ \\, {\\rm grad} \\, V({\\bf x}^{(0)}) \\, \\big ] ~=~ 0.$ We can show that the Lagrange multiplier is proportional to the second order equation of motion Eq.", "(REF ) $\\lambda ^i ~=~ m \\, g^{ij} \\, \\Phi _j \\cdot \\ddot{\\bf x}^{(0)} ~=~ m \\left( \\ddot{\\phi }^i + \\Gamma ^i_{kl} \\dot{\\phi }^k \\dot{\\phi }^l \\right).$ Substituting back $\\lambda ^i$ into Eq.", "(REF ), we obtain the following equations of motion for the massive modes $\\mathbf {H} \\, {\\bf x}^{(2)} = - m \\, \\mathbf {P} \\ddot{\\bf x}^{(0)},$ where $\\mathbf {P}$ is the projection matrix which project out the zero modes $\\mathbf {P} \\ddot{\\bf x}^{(0)} ~\\equiv ~ \\ddot{\\bf x}^{(0)} - g^{ij} (\\ddot{\\bf x}^{(0)} \\cdot \\Phi _j ) \\Phi _i,$ Since the right hand side of Eq.", "(REF ) does not contain the zero mode directions, the matrix $\\mathbf {H}$ can be “inverted\" as ${\\bf x}^{(2)} = - m \\, \\mathbf {G} \\ddot{\\bf x}^{(0)},$ where $\\mathbf {G}$ is the matrix satisfying $\\mathbf {H} \\mathbf {G} = \\mathbf {P}, \\hspace{28.45274pt}\\mathbf {G} \\mathbf {P} = \\mathbf {P} \\mathbf {G} = \\mathbf {G}, \\hspace{28.45274pt}\\mathbf {G} \\Phi _i = 0.$ Substituting the solution for the massive modes Eq.", "(REF ) into Eq.", "(REF ), we obtain the following fourth order effective Lagrangian $L_{\\rm eff}^{(4)} ~=~ \\frac{1}{2} {\\bf x}^{(2)} \\mathbf {H} \\, {\\bf x}^{(2)}~=~ \\frac{m^2}{2} \\dot{\\phi }^i \\dot{\\phi }^j \\dot{\\phi }^k \\dot{\\phi }^l \\left( \\frac{\\partial \\Phi _j}{\\partial \\phi ^i} \\mathbf {G} \\frac{\\partial \\Phi _l}{\\partial \\phi ^k} \\right).$ Let us see a simple example of a particle in $\\mathbb {R}^2$ .", "We assume that the potential is rotationally symmetric $V = V(|{\\bf x}|)$ and has a minimum at $|{\\bf x}| = r_0$ , namely $V^{\\prime }(r_0)=0$ and $V^{\\prime \\prime }(r_0) > 0$ .", "The static configurations are parameterized by the moduli parameter $\\theta $ as ${\\bf x}^{(0)} ~= \\left( \\begin{array}{cc} r_0 \\cos \\theta \\\\ r_0 \\sin \\theta \\end{array} \\right).$ The second order effective Lagrangian is given by $L_{\\rm eff}^{(2)} ~=~ \\frac{m}{2} \\dot{\\bf x}^{(0)} \\cdot \\dot{\\bf x}^{(0)} ~=~ \\frac{m}{2} r_0^2 \\, \\dot{\\theta }^2.$ The corresponding equation of motion for the moduli parameter is $\\ddot{\\theta }= 0$ and describes the particle rotating around the circle at $|{\\bf x}| = r_0$ .", "The matrices $\\mathbf {H}$ and $\\mathbf {G}$ are respectively given by $\\mathbf {H} ~=~ V^{\\prime \\prime }(r_0) \\ \\mathbf {P}, \\hspace{28.45274pt}\\mathbf {G} ~=~ V^{\\prime \\prime }(r_0)^{-1} \\mathbf {P},$ where $\\mathbf {P}$ is the projection operator $\\mathbf {P} ~= \\left( \\begin{array}{cc} \\cos ^2 \\theta & \\sin \\theta \\cos \\theta \\\\ \\sin \\theta \\cos \\theta & \\sin ^2 \\theta \\end{array} \\right).$ From Eq.", "(REF ), we obtain the following fourth order Lagrangian $L_{\\rm eff}^{(4)} ~=~ \\frac{(m r_0)^2}{2} V^{\\prime \\prime }(r_0)^{-1} \\, \\dot{\\theta }^4.$ Even if we take into account this fourth order Lagrangian, the equation of motion is not modified $\\ddot{\\theta }=0$ and solved by $\\theta = \\omega t + \\theta _0$ .", "On the other hand, the relation between the angular velocity $\\omega $ and the angular momentum $l$ is modified as $l ~=~ m r_0^2 \\left[ 1 + 2 m \\omega ^2 V^{\\prime \\prime }(r_0)^{-1} \\right] \\omega .$ This is just because the rotation radius is increased by the centrifugal force as $r_0 ~~\\longrightarrow ~~ r_0 + m r_0 \\, \\dot{\\theta }^2 \\, V^{\\prime \\prime }(r_0)^{-1}.$ The shift of the rotation radius can be also seen in the solution for the massive mode ${\\bf x}^{(0)} + \\delta {\\bf x} ~\\approx ~ {\\bf x}^{(0)} - m \\, \\mathbf {G} \\ddot{\\bf x}^{(0)} ~=~ \\left[ r_0 + m r_0 \\dot{\\theta }^2 \\, V^{\\prime \\prime }(r_0)^{-1} \\right] \\left( \\begin{array}{cc} \\cos \\theta \\\\ \\sin \\theta \\end{array} \\right).$ Therefore, the higher derivative term gives the correction from the massive mode which is slightly shifted by the motion of the zero mode (see Fig.", "REF ).", "Figure: The shift of the rotation radius." ], [ "Derivative expansion", "In this section, we discuss the effective Lagrangian for non-Abelian vortices by generalizing the method of the derivative expansion discussed in the previous section.", "To deal with the vortex positions and orientations $Z_I,\\,\\vec{b}_I~(I=1,\\cdots ,k)$ on an equal footing, we combine them into a set of complex moduli parameters $\\phi ^i~(i=1,\\cdots ,k N = {\\rm dim}_ \\mathcal {M}_k)$ , and assume that the moduli matrix $H_0(z)$ is always holomorphic in $\\phi ^i$ like Eq.", "(REF ).", "The zero mode fluctuations along the vortex world-volume are described by the vortex moduli $\\phi ^i$ promoted to fields which weakly depend on the world-volume coordinates $\\phi ^i \\rightarrow \\phi ^i(x^\\alpha ), \\hspace{28.45274pt} (\\alpha = 0,1,\\cdots ,d-2).$ These moduli fields induce fluctuations of massive modes around the weakly fluctuating vortex background $H (x^\\mu )~ &=& H^{(0)} \\, (z,\\bar{z},\\phi (x^\\alpha )) ~+~ \\delta H(x^\\mu ), \\\\W_{\\bar{z}} (x^\\mu ) &=& W_{\\bar{z}}^{(0)}(z,\\bar{z},\\phi (x^\\alpha )) ~+~ \\delta W_{\\bar{z}}(x^\\mu ), \\\\W_\\alpha (x^\\mu ) &=& \\hspace{34.1433pt} 0 \\hspace{45.5244pt} ~+~ \\delta W_\\alpha (x^\\mu ),$ where $H^{(0)}$ and $W_{\\bar{z}}^{(0)}$ are the BPS vortex background Eq.", "(REF ) depending on the world-volume coordinates $x^\\alpha $ through the moduli fields $\\phi (x^\\alpha )$ .", "We assume that the excitation energy of the fluctuations are much less than the typical mass scale of the massive modes.", "Then we can expand the induced fields with respect to the derivative by assuming that $\\partial _\\alpha ~\\ll ~ g v.$ Note that $g v$ is the unique mass scale controlling the mass of the bulk fields, inverse width of the vortex and the mass scale of the massive modes localized on the vortex world-volume.", "The induced fluctuations are expanded with respect to the derivative $\\partial _\\alpha $ as $\\delta H~ &=& H^{(2)} + H^{(4)} + \\cdots , \\\\\\delta W_{\\bar{z}} &=& W_{\\bar{z}}^{(2)} + W_{\\bar{z}}^{(4)} + \\cdots , \\\\\\delta W_\\alpha &=& W_\\alpha ^{(1)} + W_\\alpha ^{(3)} + \\cdots .$ These fluctuations can be determined by solving the equations of motion (REF ) and () order-by-order.", "Note that odd (even) order equations of motion for $\\delta H$ and $\\delta W_{\\bar{z}}$ ($\\delta W_\\alpha $ ) are trivial due to the reflection symmetry $x^\\alpha \\rightarrow - x^\\alpha $ .", "Then, the effective Lagrangian can be obtained by eliminating the induced fluctuations from original Lagrangian (REF ) expanded with respect to the derivative $\\partial _\\alpha $ $\\mathcal {L} ~=~ \\mathcal {L}^{(0)} + \\mathcal {L}^{(2)} + \\mathcal {L}^{(4)} + \\cdots .$ As we will see, the Lagrangian is quadratic in the induced fluctuations up to the fourth order in the derivative $\\partial _\\alpha $ , so that we can use the linearized equations of motion to eliminate the induced fluctuations.", "In order to determine the fluctuations, we have to specify the boundary conditions for them.", "In the singular gauge, the vortex background takes the following form at the infinity $|z| \\rightarrow \\infty $ $H^{(0)} ~=~ v \\mathbf {1}_N + \\mathcal {O}(e^{-g v |z|}),\\hspace{28.45274pt}W_{\\bar{z}}^{(0)} ~=~ \\mathcal {O} ( \\bar{z}^{-1} ).$ Since this vortex background is independent of the zero mode fluctuations $\\phi (x^\\alpha )$ at infinity, the minimal excitations induced by the zero modes should vanish at infinity.", "Therefore, we impose the following boundary conditions for the induced fluctuations $\\delta H \\underset{|z| \\rightarrow \\infty }{\\longrightarrow } 0 , \\hspace{28.45274pt}\\delta W_\\mu \\underset{|z| \\rightarrow \\infty }{\\longrightarrow } 0, \\hspace{28.45274pt} (\\mbox{in the singular gauge}).$" ], [ "The zeroth order effective Lagrangian", "The zeroth order Lagrangian contains only the background fields $\\mathcal {L}^{(0)} = {\\rm Tr}\\left[ - \\frac{4}{g^2} |F_{z \\bar{z}}^{(0)}|^2- 2 z H^{(0)} (z H^{(0)})^\\dagger - \\frac{g^2}{4} (H^{(0)}H^{(0)\\dagger } - v^2 \\mathbf {1}_N)^2 \\right].$ Substituting the solution (REF ) and using the boundary condition Eq.", "(REF ), we find that the zeroth order term $\\mathcal {L}_{\\rm eff}^{(0)}$ of the low-energy effective Lagrangian is given by the sum of the tension of the individual vortices $\\mathcal {L}_{\\rm eff}^{(0)}~\\equiv ~ \\int d^2 x \\, \\mathcal {L}^{(0)}= - k T, \\hspace{28.45274pt} T ~\\equiv ~ 2 \\pi v^2.$" ], [ "The second order effective Lagrangian", "The second order Lagrangian takes the form [25], [26] (see also [33], [48], [30], [34]).", "$\\mathcal {L}^{(2)} &=& {\\rm Tr}\\left[\\frac{4}{g^2}F_{\\alpha \\bar{z}}^{(1)} {F^\\alpha }_{z}^{(1)}+ \\alpha H^{(0)} (\\alpha H^{(0)})^\\dagger \\right].$ where $F_{\\alpha \\bar{z}}^{(1)}$ and $\\alpha H^{(0)}$ are given by $F_{\\alpha \\bar{z}}^{(1)} ~=~ \\partial _\\alpha W_{\\bar{z}}^{(0)} - \\partial _{\\bar{z}} W_\\alpha ^{(1)} + i [W_\\alpha ^{(1)}, W_{\\bar{z}}^{(0)}], \\hspace{14.22636pt}\\alpha H^{(0)} ~=~ \\partial _\\alpha H^{(0)} + i W_\\alpha ^{(1)} H^{(0)}.$ Note that the terms containing the second order fields $H^{(2)}$ and $W_{\\bar{z}}^{(2)}$ do not contribute to the second order Lagrangian since these terms are proportional to the background equations of motion $0 ~=~ {\\rm Tr}\\left[ H^{(2)} \\left( \\frac{\\delta }{\\delta H} \\int d^{d+1} x \\, \\mathcal {L}^{(0)} \\right)_{\\rm BPS \\ background} \\right], ~~\\cdots .$ The dynamical degrees of freedom in the second order Lagrangian are not only the zero modes $\\phi ^i(x^\\alpha )$ contained in the background fields ($H^{(0)}$ , $W_{\\bar{z}}^{(0)}$ ) but also the first order fluctuations $W_\\alpha ^{(1)}$ .", "We can eliminate $W_\\alpha ^{(1)}$ by using the equations of motionNote that the orthogonality condition between zero modes and fluctuations is trivial for $W_\\alpha ^{(1)}$ since $W_\\alpha = 0$ in the vortex background.", "$\\frac{4}{g^2} \\left( z F_{\\bar{z} \\alpha }^{(1)} + {\\bar{z}} F_{z \\alpha }^{(1)} \\right) ~=~ i \\Big [ H^{(0)} (\\alpha H^{(0)})^\\dagger - \\alpha H^{(0)} H^{(0)\\dagger } \\Big ].$ By using the fact that the moduli matrix $H_0(z)$ is holomorphic in the moduli parameters $\\phi ^i$ and the matrix $\\Omega $ is the solution of the master equation Eq.", "(REF ), we can check that the solution is given by $W_\\alpha ^{(1)}~=~ i (\\delta _\\alpha S^\\dagger S^{\\dagger -1} - S^{-1} \\delta _\\alpha ^\\dagger S),$ where we have defined the differential operators $\\delta _\\alpha $ and $\\delta _\\alpha ^\\dagger $ by $\\delta _\\alpha \\equiv \\partial _\\alpha \\phi ^i \\frac{\\partial }{\\partial \\phi ^i}, \\hspace{28.45274pt}\\delta _\\alpha ^\\dagger \\equiv \\partial _\\alpha \\bar{\\phi }^i \\frac{\\partial }{\\partial \\bar{\\phi }^i}.$ The solution Eq.", "(REF ) satisfies the boundary condition Eq.", "(REF ) since the asymptotic form of the matrix $S$ in the singular gauge is $S \\rightarrow H_0(z,\\phi ^i)$ and the moduli matrix $H_0(z,\\phi ^i)$ is holomorphic with respect to $\\phi ^i$ .", "Substituting the solution (REF ) into the second order Lagrangian (REF ), we obtain the following formal expression of the second order effective Lagrangian $\\mathcal {L}_{\\rm eff}^{(2)} ~=~ v^2\\int d^2 x \\ \\delta _\\alpha ^\\dagger {\\rm Tr}\\left[ \\delta _\\alpha H_0H_0^\\dagger \\Omega ^{-1} \\right] ~=~ g_{i \\bar{j}} \\partial _\\alpha \\phi ^i\\partial ^\\alpha \\bar{\\phi }^j.", "$ This effective Lagrangian gives a natural Kähler metric on the moduli space of vortices $g_{i \\bar{j}} ~\\equiv ~ v^2 \\int d^2 x \\, \\frac{\\partial }{\\partial \\bar{\\phi }^j} {\\rm Tr}\\left[ \\frac{\\partial H_0}{\\partial \\phi ^i}H_0^\\dagger \\Omega ^{-1} \\right].$ Note that the above formulas for the first order solution (REF ) and the moduli space metric are not invariant under the generic $V$ -transformation Eq.", "(REF ).", "However they are invariant under the $V$ -transformation respecting the holomorphy of $H_0(z)$ as we pointed out at the end of Sec. .", "In order to obtain the explicit form of the moduli space metric, one needs to know the solution of the master equation $\\Omega $ .", "However, no analytic solution of $\\Omega $ has been known even for the minimal winding vortex, and hence it is quite difficult to obtain the explicit form of the effective Lagrangian in general.", "Nevertheless, we can obtain an exact form of the second order effective Lagrangian for the single vortex since the degrees of freedom $Z,\\,\\vec{b}$ in this case are nothing but Nambu-Goldstone zero modes [30].", "By substituting the solution Eq.", "(REF ) and using the boundary condition for the profile function $\\psi \\rightarrow \\log |z-Z|^2$ , we can show that $\\mathcal {L}^{(2)}_{\\rm eff} ~=~ \\frac{T}{2} \\partial _\\alpha Z \\partial ^\\alpha \\bar{Z} + \\frac{4\\pi }{g^2} g_{i \\bar{j}}^{\\rm FS} \\partial ^\\alpha b^i \\partial _\\alpha \\bar{b}^j ,$ where $g_{i \\bar{j}}^{\\rm FS}$ is the Fubini-Study metric on $P̏^{N-1}$ $g_{i \\bar{j}}^{\\rm FS} ~\\equiv ~ \\frac{\\partial }{\\partial b^i} \\frac{\\partial }{\\partial \\bar{b}^j} \\log (1 + |\\vec{b}|^2).$ Another example of analytic moduli space metric for higher winding vortices has been obtained for well-separated vortices [40].", "Note the moduli space metric can be rewritten as inner products of zero modes as in the case of the example discussed in the previous section (see Eq.", "(REF )).", "Before closing this subsection, let us review the zero modes and the derivative operator defining them [25].", "The physical zero modes in a BPS background are defined as the solutions of the linearized BPS equations for the fluctuations around the background.", "With a suitable gauge fixing condition, the linearized BPS equations can be written as $\\Delta \\Phi ~\\equiv ~{\\left( \\begin{array}{cc}i {\\bar{z}}^f & - \\frac{g}{2} H_r^{(0)} \\\\\\frac{g}{2} H_r^{(0)\\dagger } & i z^a \\end{array} \\right)\\left( \\begin{array}{cc} \\delta H \\\\ \\frac{2}{g} \\delta W_{\\bar{z}} \\end{array} \\right)} ~=~ 0,$ where the subscript $r$ denotes the fact that $H^{(0)}$ acts as right multiplication and ${\\bar{z}}^f$ and $z^a$ are covariant derivatives with the zeroth order gauge field which act on the fundamental and adjoint fields, respectively.", "There exists one zero mode $\\Phi _i$ for each moduli parameter $\\phi ^i$ and the set $\\lbrace \\Phi _i\\rbrace ~(i=1,\\cdots ,{\\rm dim}_ \\mathcal {M})$ forms a basis of the zero modesThe linearized equations of motion in a static BPS background can be summarized into the form of $\\Delta ^\\dagger \\Delta \\Phi = 0$ .", "Here, the Hermitian conjugate of the operator $\\Delta $ $\\Delta ^\\dagger ~=~\\left( \\begin{array}{cc}i { z}^f & \\frac{g}{2} H_r^{(0)} \\\\-\\frac{g}{2} H_r^{(0)\\dagger } & i {\\bar{z}}^a\\end{array} \\right), $ turns out to have no zero mode [25]; $\\Delta ^\\dagger \\Phi = 0 \\, \\Rightarrow \\, \\Phi =0$ , so that $\\lbrace \\Phi _i\\rbrace $ gives a full set of the zero modes.", ".", "In terms of $\\Omega , S$ and $H_0$ , the basis $\\Phi _i$ can be written as $\\Phi _i~=~{ \\left( \\begin{array}{c}vS^{-1} \\Omega \\, \\displaystyle \\frac{\\partial }{\\partial \\phi ^i}\\left[ \\Omega ^{-1} H_0 \\right] \\\\\\displaystyle \\frac{2i}{g}S^{-1} \\bar{\\partial }\\left[ \\Omega \\displaystyle \\frac{\\partial }{\\partial \\phi ^i} \\Omega ^{-1} \\right] S \\end{array} \\right)}.$ By using the solution for $W_\\alpha ^{(1)}$ given in Eq.", "(REF ), these zero modes can be summarized as $\\partial _\\alpha \\phi ^i \\, \\Phi _i ~=~ \\left( \\begin{array}{c} \\alpha H^{(0)} \\\\\\frac{2}{g} F_{\\alpha \\bar{z}}^{(1)} \\end{array} \\right).$ Then, the moduli space metric Eq.", "(REF ) can be rewritten as inner products of the physical zero modes $\\Phi _i$ $g_{i \\bar{j}} \\equiv \\big < \\Phi _j \\,,\\, \\Phi _i \\big >.$ where we have defined the hermitian inner product for pairs of fields in the fundamental and adjoint representations $\\big < \\Phi , \\Phi ^{\\prime } \\big > ~\\equiv ~ \\int d^2 x \\, {\\rm Tr}\\left[ f^{\\prime } f^\\dagger + a^\\dagger a^{\\prime } \\right], \\hspace{28.45274pt} \\Phi = \\left( \\begin{array}{c} f \\\\ a \\end{array} \\right), ~~ \\Phi ^{\\prime } = \\left( \\begin{array}{c} f^{\\prime } \\\\ a^{\\prime } \\end{array} \\right).$" ], [ "The fourth order effective Lagrangian", "Let us now calculate the fourth order effective Lagrangian by solving the linearized equations of motion and eliminating the massive modes.", "The fourth order Lagrangian takes the form $\\mathcal {L}^{(4)} \\hspace{-5.69054pt} &=& \\hspace{-5.69054pt} {\\rm Tr}\\Bigg [ - \\frac{1}{2g^2} (F_{\\alpha \\beta }^{(2)})^2 + \\left( \\frac{4}{g^2} \\alpha W_{\\bar{z}}^{(2)} F_{\\alpha z}^{(1)} + \\alpha H^{(2)} ( \\alpha H^{(0)} )^\\dagger + (h.c.) \\right) \\\\&{}& ~~~ \\, - \\frac{4}{g^2} \\left| i z W_{\\bar{z}}^{(2)} + \\frac{g^2}{4} H^{(2)} H^{(0)\\dagger } + (h.c.) \\right|^2 - 4 \\left| {\\bar{z}} H^{(2)} + i W_{\\bar{z}}^{(2)} H^{(0)} \\right|^2 \\Bigg ].$ Here, terms proportional to $W_\\alpha ^{(3)},H^{(4)}$ and $,W_z^{(4)}$ automatically vanish due to the same mechanism as Eq.", "(REF ).", "It will be convenient to combine $H^{(2)}$ and $W_{\\bar{z}}^{(2)}$ into a column vector $\\Phi ^{(2)} ~\\equiv ~ \\left( \\begin{array}{c} H^{(2)} \\\\{\\frac{2}{g}} W_{\\bar{z}}^{(2)} \\end{array} \\right).$ Note that the fluctuations transform under the gauge transformations as $H^{(2)} \\rightarrow H^{(2)} + i \\Lambda H^{(0)}, \\hspace{28.45274pt}W_{\\bar{z}}^{(2)} \\rightarrow W_{\\bar{z}}^{(2)} - {\\bar{z}} \\Lambda ,$ where $\\Lambda $ is an arbitrary hermitian matrix of order $\\partial _\\alpha ^2$ .", "In order to eliminate this unphysical degrees of freedom, let us impose the following gauge fixing conditions for the the fluctuations $i z W_{\\bar{z}}^{(2)} + \\frac{g^2}{4} H^{(2)} H^{(0) \\dagger } &=& (h.c.).$ This constraint is equivalent to the condition that the second order fluctuation $\\Phi ^{(2)}$ is orthogonal to the unphysical gauge zero modes $\\Phi _\\Lambda $ $0 ~=~ \\Big < \\Phi _\\Lambda \\,, \\, \\Phi ^{(2)} \\Big > + (h.c.),\\hspace{28.45274pt}\\Phi _\\Lambda ~=~ \\left( \\begin{array}{c} i \\Lambda H^{(0)} \\\\ - \\frac{2}{g} {\\bar{z}} \\Lambda \\end{array} \\right).$ Then, the terms containing $(H^{(2)},\\,W_{\\bar{z}}^{(2)})$ in the fourth order effective Lagrangian ${\\cal L}_{\\rm eff}^{(4)}$ are summarized as $- 4 \\Big < \\, \\Delta \\Phi ^{(2)} \\,,\\, \\Delta \\Phi ^{(2)} \\, \\Big >+\\bigg [ \\Big < \\, \\alpha \\Phi ^{(2)} \\,,\\, \\partial ^\\alpha \\phi ^i \\Phi _i \\ \\Big >+ \\lambda ^i \\Big < \\, \\Phi ^{(2)} \\,,\\, \\Phi _i \\, \\Big > + (c.c.)", "\\bigg ],$ where we have introduced the Lagrange multiplier $\\lambda ^i$ to impose the condition that the second order fluctuation $\\Phi ^{(2)}$ is orthogonal to the physical zero modes $\\Phi _i$ , as we have done in Eq.", "(REF ) in order to separate the massive modes from the zero modes.", "The linearized equations of motion for $H^{(2)}$ and $W_{\\bar{z}}^{(2)}$ can be written as (cf.", "${\\bf H} \\rightarrow \\Delta ^\\dag \\Delta $ in Eq.", "(REF )) $4 \\Delta ^\\dagger \\Delta \\Phi ^{(2)} + \\alpha ( \\partial ^\\alpha \\phi ^i \\Phi _i ) ~=~ \\lambda ^i \\Phi _i.$ Let us first determine the Lagrange multipliers $\\lambda ^i $ .", "By taking the inner products of the zero modes $\\Phi _i^\\dagger $ and the both hand sides of the linearized equation Eq.", "(REF ), we find that $\\partial _\\alpha \\partial ^\\alpha \\phi ^i + \\Gamma ^i_{jk} \\partial _\\alpha \\phi ^j \\partial ^\\alpha \\phi ^k ~=~ \\lambda ^i,$ where we have used $\\left< \\Phi _j \\,,\\, \\alpha \\Phi _i \\right>~=~ \\delta _\\alpha g_{i \\bar{j}}~=~ g_{l \\bar{j}} \\Gamma ^l_{i k} \\partial _\\alpha \\phi ^k, \\hspace{22.76219pt}\\Big < \\Phi _j\\,,\\, \\Delta ^\\dagger \\Delta \\Phi ^{(2)} \\Big >= \\Big <\\Delta \\Phi _j\\,,\\, \\Delta \\Phi ^{(2)} \\Big > = 0.$ We find that $\\lambda ^i=0$ is nothing but the equation of motion for $\\phi ^i$ with the second order Lagrangian Eq.", "(REF ).", "Then, the linearized equation Eq.", "(REF ) becomes $\\Delta ^\\dagger \\Delta \\Phi ^{(2)} + \\frac{1}{4} \\mathbf {P} \\Big [ \\alpha (\\partial ^\\alpha \\phi ^i \\Phi _i) \\Big ] = 0,$ where $\\mathbf {P}$ the following projection operator which projects out the zero modes $\\mathbf {P} \\Phi ~\\equiv ~ \\Phi - \\big < \\Phi _j\\,,\\, \\Phi \\big > g^{\\bar{j} i} \\Phi _i.$ It seems that in order to solve Eq.", "(REF ) in terms of $\\Phi ^{(2)}$ , we need to know the explicit form of the background BPS solution .", "However, we can find the following formal expression for $\\Delta \\Phi ^{(2)}$ without solving the BPS equations $\\Delta \\Phi ^{(2)} ~&=&~ {\\frac{i}{2g}\\left( \\begin{array}{c} \\displaystyle \\frac{2}{g v} \\partial _\\alpha \\phi ^i \\partial ^\\alpha \\phi ^j ~ S^\\dagger \\ \\left[ \\nabla _i \\frac{\\partial }{\\partial \\phi ^j} \\left( \\bar{\\partial }\\Omega ^{-1} \\Omega \\right) \\right] H_0^{\\dagger -1}\\\\ \\displaystyle i \\partial _\\alpha \\phi ^i \\partial ^\\alpha \\bar{\\phi }^j \\, S^{-1} \\left[ \\frac{\\partial }{\\partial \\bar{\\phi }^j} \\left( \\Omega \\frac{\\partial }{\\partial \\phi ^i} \\Omega ^{-1} \\right) \\right] S \\end{array} \\right)},$ where $\\nabla _i$ is the covariant derivative on the moduli space, which acts on $\\frac{\\partial }{\\partial \\phi ^j} \\left( \\bar{\\partial }\\Omega ^{-1} \\Omega \\right)$ as $\\nabla _i \\frac{\\partial }{\\partial \\phi ^j} \\left( \\bar{\\partial }\\Omega ^{-1} \\Omega \\right) ~ \\equiv ~ \\left( \\frac{\\partial }{\\partial \\phi ^i} \\frac{\\partial }{\\partial \\phi ^j} - \\Gamma ^k_{ij} \\frac{\\partial }{\\partial \\phi ^k} \\right) \\left( \\bar{\\partial }\\Omega ^{-1} \\Omega \\right).$ We can confirm that Eq.", "(REF ) satisfies the linearized equation Eq.", "(REF ) by checking that the following equivalent equation is satisfied $(\\Delta \\Delta ^\\dagger ) \\Delta \\Phi ^{(2)} ~=~ - \\frac{1}{4} \\Delta \\Big [ \\alpha ( \\partial ^\\alpha \\phi ^i \\Phi _i) \\Big ],$ where the operator $\\Delta \\Delta ^\\dagger $ takes the form $\\Delta \\Delta ^\\dagger &=&\\left(\\begin{array}{cc}-{f}_{\\bar{z}}f_z+\\frac{g^2}{4} (H^{(0)\\dagger }H^{(0)})_r & {\\bf 0} \\\\{\\bf 0} & -{a}_{z}a_{\\bar{z}}+\\frac{g^2}{4} (H^{(0)}H^{(0)\\dagger })_r\\end{array} \\right).$ The solution $\\Delta \\Phi ^{(2)}$ is unique since the operator $\\Delta \\Delta ^\\dagger $ obviously has no zero mode and hence it is invertible.", "Although it has apparent singularities due to the factor $H_0^{\\dagger -1}$ , we can show that the solution Eq.", "(REF ) is smooth everywhere (see Appendix for the proof).", "It is in general not easy to solve Eq.", "(REF ) in terms of $\\Phi ^{(2)}$ .", "However, the explicit form of $\\Delta \\Phi ^{(2)}$ is sufficient for the purpose of determining the fourth order Lagrangian since it can be rewritten as $\\mathcal {L}_{\\rm eff}^{(4)}= \\int d^2 x \\, {\\rm Tr}\\left[ - \\frac{1}{2g^2}(F_{\\alpha \\beta }^{(2)})^2 \\right]+ 4 \\Big < \\, \\Delta \\Phi ^{(2)} \\,,\\, \\Delta \\Phi ^{(2)} \\, \\Big >.$ By using the first order solution (REF ), the first term in the fourth order Lagrangian (REF ) can be calculated as ${\\rm Tr}\\left[ - \\frac{1}{2g^2} (F_{\\alpha \\beta }^{(2)})^2 \\right] &=& \\ \\frac{2}{g^2} {\\rm Tr}\\left[\\delta _{[\\alpha }^\\dagger (\\Omega \\delta _{\\beta ]} \\Omega ^{-1}) \\delta ^{\\dagger [\\alpha } ( \\Omega \\delta ^{\\beta ]} \\Omega ^{-1} ) \\right].$ Substituting (REF ) and (REF ) into (REF ), we obtain the following form of the fourth order effective Lagrangian for the moduli fields $\\mathcal {L}_{\\rm eff}^{(4)} ~=~ (A_{i j \\bar{k} \\bar{l}} + B_{i j \\bar{k} \\bar{l}}) (\\partial _\\alpha \\phi ^i \\partial ^\\alpha \\phi ^j) \\overline{(\\partial _\\beta \\phi ^k \\partial ^\\beta \\phi ^l)} + 2 B_{i j \\bar{k} \\bar{l}} (\\partial _{[\\alpha } \\phi ^i \\partial _{\\beta ]} \\phi ^j) \\overline{(\\partial ^{[\\alpha } \\phi ^k \\partial ^{\\beta ]} \\phi ^l)},$ where the tensors $A_{i j \\bar{k} \\bar{l}}$ and $B_{i j \\bar{k} \\bar{l}}$ are given by $A_{i j \\bar{k} \\bar{l}} &=& \\frac{1}{g^2} \\int d^2 x \\, {\\rm Tr}\\Bigg [ \\frac{4}{g^2 v^2} \\left( \\nabla _i \\frac{\\partial }{\\partial \\phi ^j} (\\partial _{\\bar{z}} \\Omega ^{-1} \\Omega ) \\right) (H_0 H_0^\\dagger )^{-1} \\left( \\bar{\\nabla }_k \\frac{\\partial }{\\partial \\bar{\\phi }^l} (\\Omega \\partial _z \\Omega ^{-1} ) \\right) \\Omega \\Bigg ], \\\\B_{i j \\bar{k} \\bar{l}} &=& \\frac{1}{g^2} \\int d^2 x \\, {\\rm Tr}\\Bigg [ \\frac{\\partial }{\\partial \\bar{\\phi }^k} \\left( \\Omega \\frac{\\partial }{\\partial \\phi ^i} \\Omega ^{-1} \\right) \\frac{\\partial }{\\partial \\bar{\\phi }^l} \\left( \\Omega \\frac{\\partial }{\\partial \\phi ^j} \\Omega ^{-1} \\right) \\Bigg ].$" ], [ "Correction to the single vortex effective Lagrangian", "Applying the formula Eq.", "(REF ) to the single vortex solution Eq.", "(REF ), we obtain the following fourth order effective Lagrangian for a single non-Abelian vortex $\\mathcal {L}^{(4)}_{\\rm eff} = \\frac{T}{8} |\\partial _\\alpha Z \\partial ^\\alpha Z|^2 + \\frac{4\\pi }{g^2} ( g^{\\rm FS}_{i \\bar{j}} \\partial _\\alpha b^i \\partial _{\\beta } \\bar{b}^j ) \\left[ \\partial ^{\\lbrace \\alpha } Z \\, \\partial ^{\\beta \\rbrace } \\bar{Z} - \\frac{1}{2} \\eta ^{\\alpha \\beta } (\\partial _\\gamma Z \\partial ^\\gamma \\bar{Z}) + c ( g^{\\rm FS}_{k \\bar{l}}\\partial ^\\alpha b^k \\partial ^\\beta \\bar{b}^l ) \\right].$ where $T = 2 \\pi v^2$ is the vortex tension and $c$ is a constant given by $c = \\frac{1}{2\\pi } \\int d^2 x \\, \\left( 1 - |z-Z|^2 e^{-\\psi } \\right)^2\\sim 0.830707 \\times \\frac{1}{g^2 v^2} .$ In summary, we have obtained the following derivative expansion of the effective Lagrangian for a single non-Abelian vortex $\\mathcal {L}_{\\rm eff} \\hspace{-5.69054pt} &=& - \\ T \\ \\left[ 1 - \\frac{1}{2} \\partial _\\alpha Z \\partial ^\\alpha \\bar{Z} - \\frac{1}{8} |\\partial _\\alpha Z \\partial ^\\alpha Z|^2 \\right] \\phantom{\\Bigg [} \\\\&& + \\frac{4\\pi }{g^2} \\left[ \\eta ^{\\alpha \\beta } \\Big ( 1 - \\frac{1}{2} \\partial _\\beta Z \\, \\partial ^\\beta \\bar{Z} \\Big ) + \\partial ^{\\lbrace \\alpha } Z \\, \\partial ^{\\beta \\rbrace } \\bar{Z} \\right] \\left( g^{\\rm FS}_{i \\bar{j}} \\partial _\\alpha b^i \\partial _{\\beta } \\bar{b}^j \\right) \\phantom{\\Bigg [} \\\\&& + \\frac{4\\pi c}{g^2} \\left( g^{\\rm FS}_{i \\bar{j}}\\partial _\\alpha b^i \\partial _\\beta \\bar{b}^j \\right) \\left( g^{\\rm FS}_{k \\bar{l}}\\partial ^\\alpha b^k \\partial ^\\beta \\bar{b}^l \\right) + \\mathcal {O}(\\partial _\\alpha ^6).", "\\phantom{\\Bigg [} $" ], [ "Comparison with other models", "Let us compare our result, Eq.", "(REF ), of the four derivative terms in the effective action for a single non-Abelian vortex with those in different context.", "First we compare the translation modes $Z$ with the Nambu-Goto action, and second we compare the orientational modes $b^i$ with the Faddeev-Skyrme model." ], [ "Translational zero modes and mixed terms ", "First, by setting the orientational moduli $b^i$ to zero in Eq.", "(REF ), we have the effective action for the translational modes $Z$ as $S_{Z} = - 2 \\pi v^2 \\int d^{d-1} x\\left( 1 - {1\\over 2} \\partial _\\alpha Z \\partial ^\\alpha \\bar{Z}- {1\\over 8} |\\partial _\\alpha Z \\partial ^\\alpha Z|^2 \\right),$ which is precisely the one of an Abelian (ANO) vortex.", "At this order, this effective Lagrangian coincides with the Nambu-Goto action [42] given by $S_{\\rm NG} = - T \\int d^{d-1} x \\, \\sqrt{- \\det (- \\gamma _{\\alpha \\beta })},$ where $\\gamma _{\\alpha \\beta }$ is the induced metric on the world-volume given by $\\gamma _{\\alpha \\beta } ~=~ \\partial _{\\alpha } X^\\mu \\partial _{\\beta } X^\\nu \\eta _{\\mu \\nu }~=~ \\eta _{\\alpha \\beta } - \\frac{1}{2} \\left( \\partial _\\alpha Z \\partial _\\beta \\bar{Z} + \\partial _\\alpha \\bar{Z} \\partial _\\beta Z \\right).$ In Eq.", "(REF ), there are the four derivative terms containing both the translational zero modes $Z$ and the orientational zero modes $b^i$ .", "These are precisely the terms appearing in the derivative expansion of the following action describing the $P̏^{N-1}$ sigma model on the vortex world-volume $S_{\\rm eff} = \\int d^{d-1} x \\, \\sqrt{- \\det (- \\gamma _{\\alpha \\beta })} \\left[ - T + \\frac{4\\pi }{g^2} \\gamma ^{\\alpha \\beta } (g^{\\rm FS}_{i \\bar{j}} \\partial _\\alpha b^i \\partial _\\beta \\bar{b}^j) + \\mathcal {O}(\\partial _a^4) \\right].$" ], [ "Orientational zero modes ", "Next, we consider the orientational zero modes in the internal space.", "For simplicity, let us restrict ourselves to the case of $N=2$ in which the internal moduli space is ${\\mathbb {C}}P^{1}$ .", "By setting $Z=0$ in Eq.", "(REF ), we obtain the following effective Lagrangian of the orientational zero modes with the higher derivative correction $\\mathcal {L}_{{\\mathbb {C}}P^1} = \\frac{4\\pi }{g^2} \\left[ \\frac{\\partial _\\alpha b \\, \\partial ^\\alpha \\bar{b}}{(1+|b|^2)^2}+ c \\frac{|\\partial _\\alpha b \\, \\partial ^\\alpha b|^2}{(1+|b|^2)^4} \\right].$ Now let us compare this with the Skyrme-Faddeev model which is a ${\\mathbb {C}}P^{1}$ model with a four-derivative term [47].", "To this end, we formulate the ${\\mathbb {C}}P^1$ model (at the leading order) by two complex fields $h =(h^1,h^2)$ charged under $U(1)$ gauge symmetry.", "By introducing auxiliary gauge field $a_{\\alpha }$ and scalar field $\\lambda $ as Lagrange multipliers, it can be written as ${\\cal L} ~=~ (\\partial _{\\alpha } + i a_{\\alpha }) h \\, (\\partial _{\\alpha } - i a_{\\alpha }) h^\\dagger - \\lambda \\left(h h^\\dagger - {4 \\pi \\over g^2} \\right)$ The variation of $\\lambda $ gives the constraint $hh^\\dagger = {4 \\pi \\over g^2}$ , which can be solved by $h = \\sqrt{{(g^2/4\\pi ) \\over {1+|b|^2}}}(1,b)$ .", "Then the variation of $a_{\\alpha }$ gives $&& a_{\\alpha }= \\frac{i}{2}{\\bar{b} \\partial _{\\alpha } b - b \\partial _{\\alpha } \\bar{b}\\over (1 + |b|^2)} .$ Substituting these back into the original Lagrangian (REF ), the ${\\mathbb {C}}P^1$ model (at the leading order) is recovered.", "The field strength of the gauge field (REF ) is $f_{\\alpha \\beta } ~=~ \\partial _{\\alpha } a_{\\beta } - \\partial _{\\beta } a_{\\alpha } ~=~-i{\\partial _{\\alpha } b \\, \\partial _{\\beta } \\bar{b} - \\partial _{\\beta } b \\, \\partial _{\\alpha } \\bar{b} \\over (1+|b|^2)^2 }.", "$ Then the Faddeev-Skyrme term [47] can be written as the field strength squared asThis term is also called the baby Skyrme term in $d=2+1$ [50].", "$f_{\\alpha \\beta } f^{\\alpha \\beta }~=~ 2 {(\\partial _{\\alpha } \\bar{b} \\, \\partial ^{\\alpha } b)^2 - |\\partial _{\\alpha } b \\, \\partial ^{\\alpha } b|^2 \\over (1+|b|^2)^4} .$ This term is quadratic in the time derivative and does not coincide with Eq.", "(REF ).", "The other term containing four time-derivatives appears in an ${\\cal N}=1$ supersymmetric extension of the Skyrme-Faddeev term [51]: This term also arises when one constructs the Faddeev-Skyrme-like model as the low-energy effective theory of pure $SU(2)$ Yang-Mills theory [52], and hence is called the Gies term in that context.", "${(\\partial _{\\alpha } b \\, \\partial ^{\\alpha } \\bar{b})^2 \\over (1+|b|^2)^4}.$ As shown in Appendix , this higher derivative term can be obtained by adding a higher derivative term to the original Lagrangian (REF ).", "In general the fourth order terms are summarized as ${\\cal L}^{(4)}_{\\rm general} ~=~ c_1 f_{\\alpha \\beta } f^{\\alpha \\beta } + c_2 {(\\partial _{\\alpha } b \\, \\partial ^{\\alpha } \\bar{b})^2 \\over (1+|b|^2)^4}~=~ { (2c_1+c_2)(\\partial _{\\alpha } \\bar{b} \\, \\partial ^{\\alpha } b)^2- 2 c_1 |\\partial _{\\alpha } b \\, \\partial ^{\\alpha } b|^2 \\over (1+|b|^2)^4} .$ This reduces to the four derivative term in Eq.", "(REF ) when $c_2 = - 2 c_1 = c.$ The condition $c_2 = -2c_1$ is precisely the condition for ${\\cal N}=1$ supersymmetry [51].", "This must be the case because there remains ${\\cal N}=1$ supersymmetry (four supercharges) in the vortex effective theory, because vortices are 1/2 BPS states in supersymmetric theories with eight supercharges." ], [ "Instantons trapped inside a non-Abelian vortex", "In this section, we consider $d=4+1$ dimensions where vortices are membranes with 2+1 dimensional world-volume.", "Besides the vortices, there also exist Yang-Mills instantons which are particle-like solitons in $d=4+1$ dimensions.", "But they cannot exist stably in the Higgs phase which we are considering.", "Instead, they can stably exist inside the world-volume of a non-Abelian vortex [30], [31].", "Instanton-vortex composite configurations are 1/4 BPS states in supersymmetric gauge theories with 8 supercharges.", "In the supersymmetric ${\\mathbb {C}}P^{N-1}$ model with 4 supercharges, at the leading order of the 1/2 BPS vortex effective theory, instantons can be regarded as 1/2 BPS sigma model lumps [30] which are point-like solitons in $d=2+1$ dimensions.", "Here we show that instantons (= lumps) are not modified even if we include derivative corrections found in the previous sections.", "For simplicity, we set $Z=0$ and consider the case of $N=2$ (the $P̏^1$ sigma model) with the Lagrangian (REF ) for the orientational modes.", "In $d=4+1$ dimensions, the vortex is a membrane which has coordinates $(x^0=t,x^1,x^2)$ .", "We parametrize the two spatial coordinates by complex variables $w=x^1 + ix^2$ and $\\bar{w} = x^1 - i x^2$ .", "We discuss only static configurations in this model.", "First we recall the lump solution in the model without the four derivative correction.", "The energy density can be written as $E_{{\\mathbb {C}}P^1}^{(2)}&=& \\frac{4\\pi }{g^2} \\int d^2 x \\sum _{\\alpha =1,2} \\frac{\\partial _\\alpha b \\, \\partial ^\\alpha \\bar{b}}{(1+|b|^2)^2}~=~ \\frac{8\\pi }{g^2} \\int d^2 x \\frac{ 2 |\\partial _{\\bar{w}} b|^2 + (|\\partial _w b|^2 - |\\partial _{\\bar{w}} b|^2)}{(1+|b|^2)^2}.$ The first term is positive semi-definite $|\\partial _{\\bar{w}} b|^2 \\ge 0$ and the second term can be rewritten as $I ~=~ \\frac{4\\pi i}{g^2} \\int \\frac{\\partial _w b \\, \\partial _{\\bar{w}} \\bar{b} - \\partial _{\\bar{w}} b \\partial _w \\bar{b}}{(1+|b|^2)^2} \\, dw \\wedge d\\bar{w}~=~ \\frac{4\\pi i}{g^2} \\int \\frac{d b \\wedge d \\bar{b}}{(1+|b|^2)^2} ~=~ \\frac{8\\pi ^2}{g^2} k,$ where the integer $k$ is the degree of the map from the vortex world-volume to the target space $P̏^1$ .", "This is the topological charge for lumps: $\\pi _2({\\mathbb {C}}P^1) = {\\mathbb {Z}}$ .", "The energy (REF ) is bound from below by this topological charge $I$ , and the Bogomol'nyi bound is saturated by the BPS equation for lumps, given by $\\partial _{\\bar{w}} b = 0.$ The BPS solutions $b(w)$ are holomorphic in $w$ and satisfy the static equation of motion.", "If we fix the boundary condition as $b \\rightarrow \\infty ~(w \\rightarrow \\infty )$ , the solutions with $I = \\frac{8\\pi ^2}{g^2} k$ are given by $k$ -th order rational maps $b(w) ~=~ \\frac{a_0 w^k + a_1 w^{k-1} + \\cdots + a_k}{\\tilde{a}_0 w^{k-1} + \\tilde{a}_1 w^{k-2} + \\cdots + \\tilde{a}_{k-1} }.$ This is the general solutions for the 1/2 BPS sigma model lumps.", "Next let us consider the effect of the four derivative term $E_{{\\mathbb {C}}P^1}^{(4)} &=&-\\frac{64\\pi c}{g^2} \\int d^2 x \\, \\frac{|\\partial _w b \\, \\partial _{\\bar{w}} b|^2}{(1+|b|^2)^4}.$ The total energy can be rewritten as $E_{P̏^1}^{(2)+(4)} ~=~ \\frac{16 \\pi }{g^2} \\int d^2 x \\left[ 1 - 4 c \\frac{|\\partial _w b|^2}{(1+|b|^2)^2} \\right] \\frac{|\\partial _{\\bar{w}} b|^2}{(1+|b|^2)^2} ~+~ \\frac{8 \\pi ^2}{g^2} k.$ Again, the first term is positive semi-definite as long as the correction term is sufficiently small and vanishes if the BPS equation (REF ) is satisfied.", "We thus have found that there is no contribution from the four derivative term to the BPS configurations and consequently the lump solutions are not modified.", "The energy of lumps $\\frac{8\\pi ^2}{g^2} k$ agrees with that of instantons in the Higgs phase, which can be found by rewriting the energy of the original bulk theory to the Bogomol'nyi form for 1/4 BPS configurations.", "Therefore, it is natural to conjecture that all higher order terms vanish for 1/2 BPS lump configurations.", "By considering 1/2 BPS lumps on the ${\\mathbb {C}}P^{N-1}$ model, we have obtained 1/4 BPS composite states of instantons inside a vortex.", "On the other hand, there exist another 1/4 BPS composite states of intersecting vortex-membranes [30], [31].", "Vortices of one kind have codimensions in $z$ -plane and extend to $w$ -plane, and those of another kind have codimensions in $w$ -plane and extend to $z$ -plane.", "This intersecting vortices can be constructed by considering holomorphic maps of the translational moduli $Z(w)$ , instead of the orientational moduli $b^i(w)$ considered in this section.", "Now let us show that, in general, the fourth derivative terms do not modify the 1/2 BPS states.", "The BPS equation can be found by rewriting the second order energy as $E_{\\rm eff}^{(2)} = \\int d^2 x \\, 4 g_{i \\bar{j}} \\partial _{\\bar{w}} \\phi ^i \\partial _w \\bar{\\phi }^j + \\int i g_{i \\bar{j}} d \\phi ^i \\wedge d \\bar{\\phi }^j.$ The second term gives the area of a two cycle of the moduli space on which the map $\\phi ^i$ wraps.", "The first term is positive semi-definite and vanishes for any holomorphic maps satisfying $\\partial _{\\bar{w}} \\phi ^i ~=~ 0.$ On the other hand, it follows from the general form of the higher derivative corrections Eq.", "(REF ) that the fourth order energy vanishes for the holomorphic maps $E_{\\rm eff}^{(4)} ~\\propto ~\\partial _{\\bar{w}} \\phi ^i \\partial _w \\phi ^j \\partial _w \\bar{\\phi }^k \\partial _{\\bar{w}} \\bar{\\phi }^l ~=~ 0.$ Therefore, the higher derivative corrections do not modify the 1/2 BPS configurations.", "There is a possibility that the holomorphic maps are unstable if the fourth order energy is negative.", "However, this would be an artifact of the truncation at the fourth order.", "For example, the translational part of the energy can be rewritten as $E_{\\rm eff} ~=~ T \\int d^2 x \\, 2 (1 - |\\partial _w Z|^2) |\\partial _{\\bar{w}} Z|^2 + T \\int \\frac{i}{2} \\left( d w \\wedge d \\bar{w} + d Z \\wedge d \\bar{Z} \\right).$ The second term is a topological term and the first term vanishes for a holomorphic map $Z(w)$ but it is not positive semi-definite.", "Although a holomorphic map $Z(w)$ appears to be an unstable solution if $|\\partial _w Z|^2 > 1$ , such a case is beyond the validity of our approximation.", "We can show that all the holomorphic maps are stable BPS solutions if we consider the full order action, i.e.", "the Nambu-Goto action, whose energy can be rewritten as $E_{\\rm NG} &=& T \\int d^2 x \\left[\\sqrt{(1 + |\\partial _{\\bar{w}} Z|^2 - |\\partial _w Z|^2)^2 + 4 |\\partial _{\\bar{w}} Z|^2} - (1 + |\\partial _{\\bar{w}} Z|^2 - |\\partial _w Z|^2) \\right] \\\\&+& T \\int \\frac{i}{2} \\left( d w \\wedge d \\bar{w} + d Z \\wedge d \\bar{Z} \\right).$ The first term is positive semi-definite and can be expanded as $T \\int d^2 x \\, 2 \\Big [ 1 - |\\partial _w Z|^2 + |\\partial _w Z|^2 (|\\partial _w Z|^2 + |\\partial _{\\bar{w}} Z|^2) + \\cdots \\Big ] |\\partial _{\\bar{w}} Z|^2.$ As this example shows, we should take into account the full order corrections to prove the stability of the holomorphic maps." ], [ "Higher derivative terms and massive modes", "To see the physical meaning of the higher derivative corrections for the internal orientation, let us consider the equation of motion with higher order corrections in the $N=2$ case.", "Assuming that the orientational zero mode $b$ is independent of the spatial world-volume coordinates, we can show that the following configuration satisfies the equation of motion even in the presence of the higher derivative terms : $b ~=~ \\exp \\left( i \\omega t \\right).$ This solution corresponds to an excited state of the non-Abelian vortex whose orientation is rotating along the equator of $P̏^1$ (Fig.", "REF ).", "Figure: The rotation of the orientationThe rotation of the orientation induces one of three components of the $SU(2)_{\\rm C+F}$ conserved charges, which is given by $\\rho _3 ~=~ \\frac{i}{2} \\left( \\frac{\\partial \\mathcal {L}}{\\partial (\\partial _t b)} b - \\frac{\\partial \\mathcal {L}}{\\partial (\\partial _t \\bar{b})} \\bar{b}\\right) ~=~ \\frac{\\pi }{g^2} \\left( 1 + \\frac{1}{2} c \\, \\omega ^2 \\right) \\omega .$ The relation between the angular velocity $\\omega $ and the conserved charge $\\rho _3$ (shown in Fig.", "REF -(a)) is modified by the higher derivative term, as in the case of the angular velocity and the angular momentum of the particle (see Eq.", "(REF )).", "By numerically solving the full equations of motion Eqs.", "(REF ) and (), we can see that the size of the excited vortex is slightly larger than that of the static configuration (see Fig.", "REF -(b)).", "This is analogous to the case of the particle discussed in Sec.", ": the higher derivative term corresponds to the correction from the massive mode (vortex size) which is slightly shifted by the conserved charge.", "Figure: (a) The relation between the angular velocity ω\\omega and the conserved charge ρ 3 \\rho _3 for g=v=1g=v=1.Compared to the second order relationρ 3 =π g 2 ω\\rho _3 = \\frac{\\pi }{g^2} \\omega (dashed line),the fourth order relationρ 3 =π g 2 (1+c 2ω 2 )ω\\rho _3=\\frac{\\pi }{g^2}(1 + \\frac{c}{2} \\omega ^2) \\omega (solid line)shows the better agreement with the numerical result (dots)obtained by solving the full equations of motionEqs.", "() and ().", "(b) The energy density distributions in the transverse plane(zz-plane) for g=v=1g=v=1.The size of the excited vortex with ω=0.5\\omega = 0.5 (solid line)is slightly larger than that of the static vortex with ω=0\\omega = 0(dashed line)." ], [ "Higher order corrections from mass terms", "In this section, we consider higher order corrections from supersymmetric mass deformations of the original theory.", "Without breaking the supersymmetry, we can deform the original model by adding the following terms to the Lagrangian Eq.", "(REF ) $\\mathcal {L}_{{\\rm adjoint} + {\\rm mass}}~=~ {\\rm Tr}\\left[ \\frac{1}{g^2} \\mu \\Sigma _I \\mu \\Sigma _I+ \\frac{1}{2g^2} [ \\Sigma _I , \\Sigma _J ]^2 - |\\Sigma _I H - H M_I |^2 \\right],$ where $\\Sigma _I$ ($I=1,\\cdots ,n$ ) are the real adjoint scalar fields in the vector multiplets and $M_I$ are mass matrices in the Cartan subalgebra of $SU(N)_{\\rm F}$ $M_I ~\\equiv ~ \\mathbf {m}_I \\cdot \\mathbf {H}~=~ {\\rm diag} (m_{I,0},\\,m_{I,1},\\,\\cdots ,\\,m_{I,N-1}).$ If all the masses are non-degenerate, the $SU(N)_{\\rm F}$ flavor symmetry is maximally broken to the Cartan subalgebra of $SU(N)_{\\rm F}$ .", "These mass deformations can be obtained by dimensional reductions from the six dimensional model on $\\mathbb {R}^{6-n} \\times (S^1)^n$ with the following twisted boundary conditions around periodic dimensions $H(x^\\mu , \\theta _I + 2 \\pi R_I) = H (x^\\mu , \\theta _I) e^{2 \\pi i R_I M_I}.$ Ignoring the infinite tower of the Kaluza-Klein modes, we obtain the mass deformation Eq.", "(REF ) with the following identification $\\Sigma _I(x^\\mu ) = - W_{\\theta _I}(x^\\mu ), \\hspace{28.45274pt}H(x^\\mu ) e^{ i \\theta _I M_I } = H(x^\\mu , \\theta _I).$ The mass terms do not change the VEV of $H$ (given in Eq.", "(REF )), while they induce those of the adjoint scalars $\\langle \\Sigma _I \\rangle = M_I.$ In this mass deformed model, the color-flavor global symmetry $SU(N)_{\\rm C+F}$ is explicitly broken to the Cartan subgroup $U(1)^{N-1}$ .", "Hence the orientational moduli, which were the Nambu-Goldstone zero modes of $SU(N)_{\\rm C+F}$ , are lifted by a potential induced by the mass terms.", "The potential on the moduli space can be calculated by finding the minimum energy configuration and evaluating the energy for each values of the moduli parameters.", "This can be done perturbatively with respect to the masses $m_{I,A}$ .", "As a zeroth order configuration, we consider the vortex solution Eq.", "(REF ) satisfying the BPS equations without the mass deformations.", "Then we can determine the corrections to $\\Sigma _I$ by solving their equations of motion $\\frac{2}{g^2} \\Big ( \\mu \\mu \\Sigma _I + \\big [ \\Sigma _J \\big [ \\Sigma _J, \\Sigma _I \\big ] \\big ] \\Big ) = (\\Sigma _I H - H M_I ) H^\\dagger + H (H^\\dagger \\Sigma _I - M_I H^\\dagger ).$ Due to the fact that $\\Sigma _I$ originate from the higher dimensional gauge fields $W_{\\theta _I}$ , we can solve the equations of motion in the similar way to the gauge fields $W_\\alpha $ $\\Sigma _I^{(1)} = M + i ( \\delta _I S^\\dagger S^{\\dagger -1} - S^{-1} \\delta _I^\\dagger S ) + \\mathcal {O}(m^3, m \\partial _\\alpha ^2) ,$ where we have assumed that the derivatives $\\partial _\\alpha $ and the masses $m$ are of the same order $m ~\\sim ~ \\partial _\\alpha .$ The differential operators $\\delta _I$ and $\\delta _I^\\dagger $ are defined by $\\delta _I = k_I^i \\frac{\\partial }{\\partial \\phi ^i}, \\hspace{28.45274pt}\\delta _I^\\dagger = \\bar{k}_I^i \\frac{\\partial }{\\partial \\bar{\\phi }^i},$ where $k_I^i$ are holomorphic Killing vectors on the moduli space which are the following linear combinations of the Killing vectors of the unbroken symmetry $U(1)^{N-1}$ $k_I^i ~\\equiv ~ \\mathbf {m}_I \\cdot \\xi ^i.$ For example, the holomorphic Killing vectors on the moduli space of the single vortex solutions are given by $k_I^i ~=~ i ( m_{I,i} - m_{I,0} ) b^i~~(\\mbox{no sum over $i$}).$ Inserting the solution $\\Sigma _I^{(1)}$ into the deformation terms Eq.", "(REF ), we obtain the effective potential of the form $V_{\\rm eff}^{(2)} = g_{i \\bar{j}} \\, k_I^i \\bar{k}_I^j.$ This is the sum of the squared norm of the Killing vectors $k_I^i$ evaluated with respect to the moduli space metric $g_{i \\bar{j}}$ given in Eq.", "(REF ).", "The potential of this form can also be obtained by dimensional reductions from the four dimensional vortex world-volume action (the effective theory of the vortex in six spacetime dimensions) $S_{\\rm eff}^{4d} ~=~ \\int d^4 x \\, g_{i \\bar{j}}\\Big [ \\partial _\\alpha \\phi ^i \\partial ^\\alpha \\bar{\\phi }^j- \\partial _{\\theta _I} \\phi ^i \\partial _{\\theta _I} \\bar{\\phi }^j \\Big ].$ As in the case of the bulk theory, we impose the twisted boundary condition for the moduli fields $\\phi ^i(x^\\alpha , \\theta _I + 2 \\pi R_I) = \\phi ^i(x^\\alpha , \\theta _I)e^{2 \\pi R_I k_I^i}.$ Keeping only the lowest modes and evaluating the kinetic terms in the effective action, we obtain the effective action with the potential Eq.", "(REF ) $S_{\\rm eff}^{(4-n)d} ~=~ \\int d^{4-n} x \\, g_{i \\bar{j}} \\Big [ \\partial _\\alpha \\phi ^i \\partial ^\\alpha \\bar{\\phi }^j - k_I^i \\bar{k}_I^j \\Big ].$ Therefore, the procedure of the dimensional reductions and the calculation of the effective action are commutative.", "The higher order corrections to $H$ and $W_{\\bar{z}}$ can also be determined from the equations of motion with the mass deformations.", "They can also be solved in a similar way to the case without mass deformation as $\\Delta \\Phi ^{(2)} ~=~ \\frac{i}{2g}{\\left( \\begin{array}{c} \\displaystyle \\frac{2}{g v} \\, \\left[ \\partial _\\alpha \\phi ^j \\partial ^\\alpha \\phi ^k - k_I^j k_I^k \\right] ~ \\, S^\\dagger \\ \\left[\\nabla _j \\frac{\\partial }{\\partial \\phi ^k} \\left( \\bar{\\partial }\\Omega ^{-1} \\Omega \\right) \\right] H_0^{\\dagger -1}\\\\ \\displaystyle \\ i \\, \\left[ \\partial _\\alpha \\phi ^j \\partial ^\\alpha \\bar{\\phi }^k - k_I^j \\bar{k}_I^k \\right] \\, S^{-1} \\left[ \\frac{\\partial }{\\partial \\bar{\\phi }^k} \\left( \\Omega \\frac{\\partial }{\\partial \\phi ^j} \\Omega ^{-1} \\right) \\right] S \\end{array} \\right)}.$ Then the higher order corrections are obtained by substituting the solution into $\\mathcal {L}^{(4)}_{\\rm eff} = \\int d^2 x \\, {\\rm Tr}\\left[ - \\frac{1}{2g^2} (F_{\\alpha \\beta }^{(2)})^2 + \\frac{1}{g^2} (\\alpha \\Sigma _I^{(1)})^2 + \\frac{1}{2g^2} [\\Sigma _I^{(1)}, \\Sigma _J^{(1)}]^2 \\right] + 4 \\Big < \\Delta \\Phi ^{(2)} \\,,\\, \\Delta \\Phi ^{(2)} \\Big >.$ In the case of the effective Lagrangian of a single vortex, the induced terms in take the form $\\mathcal {L}_{\\rm eff \\, mass}^{(2) + (4)} &=& - \\frac{4\\pi }{g^2} g_{i \\bar{j}}^{\\rm FS} k_I^i \\bar{k}_I^j \\Big ( 1 - \\frac{1}{2} \\partial _\\alpha Z \\partial ^\\alpha \\bar{Z} \\Big )\\\\&& + \\frac{4\\pi c}{g^2} \\Big [ (g^{\\rm FS}_{i \\bar{j}} k_I^i \\partial _\\alpha \\bar{b}^j)\\, (g^{\\rm FS}_{k \\bar{l}} k_I^k \\partial ^\\alpha \\bar{b}^l) +{\\rm c.c.}", "\\Big ] \\\\&& + \\frac{4\\pi c}{g^2} (g^{\\rm FS}_{i \\bar{j}} k_I^i \\bar{k}_J^j)(g^{\\rm FS}_{k \\bar{l}} k_I^k \\bar{k}_J^l) .$ These corrections terms can also be obtained by using the dimensional reductions from the four-dimensional world-volume with the higher derivative corrections." ], [ "Kink monopoles", "In this section we consider a single 1/2 BPS non-Abelian vortex-string in four spacetime dimensions and its low-energy effective theory in two spacetime dimensions.", "Besides the vortex, an another important topological soliton, 1/2 BPS monopole, arises in four dimensions.", "However, since the gauge symmetry is completely broken in the Higgs phase, the monopole cannot stay alone and must be accompanied with the vortex-string.", "If the vortex-strings are attached to the monopoles from the both left and right hand sides, the composite vortex-monopole state is indeed a stable BPS state preserving 1/4 supersymmetry (two supercharges) [29].", "Such confined monopoles can also be viewed as kinks in the two-dimensional vortex effective theory.", "For example, let us consider $N=2$ case with only one mass term $M ~=~ \\frac{1}{2}(m,-m).$ The orientational part of the second order effective Lagrangian of a single vortex-string in the mass deformed $U(2)$ theory takes the form $\\mathcal {L}_{\\rm eff}^{(2)} ~=~ \\frac{4\\pi }{g^2}\\frac{\\partial _\\alpha b \\, \\partial ^\\alpha \\bar{b} - m^2 |b|^2}{(1+|b|^2)^2},\\qquad (\\alpha = 0,1).$ In this sigma model, the orientational moduli are lifted by the potential and there are discrete vacua at $b=0,\\infty $ .", "In the $(1+1)$ -dimensional vortex world-sheet, we can consider a BPS kink interpolating between the discrete vacua $E_{\\rm eff}^{(2)} = \\int dx \\left[ \\frac{4\\pi }{g^2}\\frac{|\\partial _x b - m b|^2}{(1+|b|^2)^2} + \\partial _x \\sigma \\right],$ where $\\sigma $ is the moment map of the unbroken $U(1)$ symmetry defined by $\\sigma = - \\frac{2\\pi m}{g^2} \\frac{1-|b|^2}{1+|b|^2}.$ The BPS equation can be easily solved as $\\partial _x b = m b, ~~~~~\\longrightarrow ~~~~~b(x) = \\exp \\left[ m(x-x_0)+i\\theta \\right], $ where $x_0$ and $\\theta $ are kink position and phase moduli, respectively.", "The mass of the kink is $M_{\\rm BPS} ~=~ \\sigma (x \\rightarrow \\infty )- \\sigma (x \\rightarrow - \\infty ) ~=~ \\frac{4\\pi m}{g^2}.$ As expected, this BPS mass of the kink coincides with that of the monopole in $(3+1)$ dimensions.", "To see the effect of the higher order corrections to the BPS kinks, let us take into account the forth order corrections in the effective theory $\\mathcal {L}_{\\rm eff}^{(2+4)} = \\frac{4\\pi }{g^2}\\left[ \\frac{\\partial _\\alpha b \\, \\partial ^\\alpha \\bar{b} - m^2 |b|^2}{(1+|b|^2)^2}+ c \\frac{|\\partial _\\alpha b \\, \\partial ^\\alpha b + m^2 b^2 |^2}{(1+|b|^2)^4} \\right].$ The energy of static configurations can be rewritten as $E_{\\rm eff}^{(2+4)} = \\int dx \\left[ \\frac{4\\pi }{g^2}\\frac{|\\partial _x b - m b|^2}{(1+|b|^2)^2}\\left( 1 - c \\frac{|\\partial _x b + m b|^2}{(1+|b|^2)^2} \\right)+ \\partial _x \\sigma \\right].$ Therefore, the BPS kink solution and its mass Eqs.", "(REF ) and (REF ) are not modified by the higher order corrections.", "Thus, we again encountered the case where the topological soliton in the vortex effective theory is exact at the second order.", "This is consistent with the observation that the kink on the vortex corresponds to the monopoles attached by the vortices in the four dimensional full theory." ], [ "Q-solitons", "Let us again consider the effective theory of the single non-Abelian vortex in $4+1$ dimensions.", "In the case of $N=2$ , the effective theory is ($2+1$ )-dimensional massive $\\mathbb {C}P^1$ sigma model.", "The effective Lagrangian at the second order is the same as Eq.", "(REF ) with $\\alpha = 0,1,2$ .", "As is well known, from the Derrick's theorem, there are no stable lump solitons: the potential makes them collapse.", "However, one can still construct stable solitons by adding Noether charges to the lumps.", "It is the so-called Q-lumps [54], [55].", "The Q-lump solutions have the same form as the $\\mathbb {C}P^1$ but their phases are time-dependent.", "To see this, let us see the energy $E^{(2)} &=& \\frac{4\\pi }{g^2}\\int d^2x\\ \\frac{ |\\dot{b}|^2 + |\\partial _i b|^2 - m^2 |b|^2}{(1+|b|^2)^2} \\nonumber \\\\&=& \\frac{4\\pi }{g^2}\\int d^2x\\ \\frac{ |\\dot{b} \\mp i m b|^2 + |\\partial _{\\bar{w}} b|^2/2}{(1+|b|^2)^2} + I + Q,$ with $I$ and $Q$ being the topological charge and the Noether charge associated with the $U(1)$ global symmetry $I &=& \\frac{8\\pi i}{g^2}\\int \\frac{db \\wedge d\\bar{b}}{(1+|b|^2)^2} = \\frac{8\\pi ^2k}{g^2},\\\\Q &=& \\pm \\frac{4\\pi m}{g^2}\\int d^2x\\ \\frac{i ( b \\dot{b}^* - \\dot{b} b^*)}{(1+|b|^2)^2}.$ The BPS equations are of the form $\\dot{b} = \\pm i m b,\\quad \\partial _{\\bar{w}} b = 0.$ Thus the solution is given by $b(t,x^1,x^2) = e^{\\pm i m t}\\, \\tilde{b}(w),$ where $\\tilde{b}(w)$ can be any holomorphic function of $w$ as in the case of the static configuration Eq.", "(REF ).", "Note that the Q-lump solution with the minimal winding number ($\\tilde{b} = \\frac{\\lambda }{w}$ ) has infinite energy because its Noether charge density has an asymptotic tail $\\sim 1/|\\lambda ^2w|$  [55].", "The minimal configuration with finite energy, for example, is given by $\\tilde{b} = \\frac{\\lambda }{w-w_1} - \\frac{\\lambda }{w - w_2}$ .", "Let us next see the effects from the higher derivative corrections: Do the Q-lumps receive corrections from the higher order terms?", "Indeed, as one can easily check, the solution given in Eq.", "(REF ) solves the full equations of motion including the fourth order corrections.", "In the presence of the higher order corrections, the energy density can be rewritten as follows $E^{(4)} &=& \\frac{4\\pi c}{g^2}\\int d^2x\\ \\frac{2\\left(|\\dot{b}|^4 - |(\\partial _i b)^2-m^2 b^2|^2\\right) + |\\dot{b}^2-(\\partial _ib)^2 + m^2 b^2|^2}{(1+|b|^2)^4}\\nonumber \\\\&=& \\frac{32\\pi c}{g^2}\\int d^2x\\ \\frac{m^2\\left(\\bar{\\tilde{b}}^2 \\partial _w \\tilde{b} \\partial _{\\bar{w}}\\tilde{b} + \\tilde{b}^2 \\partial _w \\bar{\\tilde{b}} \\partial _{\\bar{w}} \\bar{\\tilde{b}} \\right)- 2 | \\partial _w \\tilde{b} \\partial _{\\bar{w}} \\tilde{b} |^2}{(1+|\\tilde{b}|^2)^4}.$ Since $\\tilde{b}$ is a holomorphic function in $w$ , we can immediately conclude that $E^{(4)} = 0$ for the Q-lump solutions $b = e^{imt}\\tilde{b}(w)$ .", "Therefore, the BPS Q-lump solutions and their masses are not modified by the higher order corrections." ], [ "Summary and Discussion", "We have proposed a systematic method to obtain higher derivative terms in the low-energy effective theories on solitons.", "We have applied our method to a single non-Abelian vortex and have obtained four-derivative terms in the ${\\mathbb {C}}\\times {\\mathbb {C}}P^{N-1}$ model on the vortex world-volume.", "We have compared our four-derivative terms with the Nambu-Goto action and the Faddeev-Skyrme model.", "The action for the translational moduli $Z$ coincides with the Nambu-Goto action and the terms for the orientational moduli $b^i$ coincide with those for the supersymmetric extension of the Faddeev-Skyrme term.", "We have also shown that the contribution from the four-derivative terms disappears for 1/4 BPS states of instantons trapped inside a non-Abelian vortex and consequently the solutions are not modified in the presence of the four derivative terms.", "In this paper, we have derived the four derivative terms.", "In principle we can go on to any order in our formalism.", "The sixth order is considered to be prominently important.", "We have confirmed that the effective action of the translational zero modes $Z$ at the fourth order is consistent with the Nambu-Goto action.", "On the other hand, in the width expansion from the Nambu-Goto action, the first correction term written as the extrinsic curvature squared starts from the sixth derivatives in the derivative expansion [44].", "In field theory calculation, it seems that there is no agreement on the signature of that term [46].", "Therefore, we can in principle determine that term for the BPS case.", "In this paper we have studied local non-Abelian vortices which exist in the theory with the number $N_{\\rm F}$ of flavors equals to the number $N_{\\rm C}$ of color.", "When the theory has more flavors, $N_{\\rm F}>N_{\\rm C}$ , vortices are called semi-local [56].", "It is known in this case that the orientational zero modes of a single vortex is non-normalizable, i.e., the integration over the codimensions diverges [57], unless the size modulus is zero and the vortex shrinks to a local vortex [58].", "For two vortices, the relative orientational zero modes are normalizable even with a non-zero size moduli, while the overall orientational zero modes are non-normalizable [58].", "Although we can formally extend our method to semi-local vortices, we should check if there exists a divergence in four derivative terms even for normalizable moduli.", "Remember that in the derivative expansion, derivatives are assumed to be less than the lowest mass $m$ of the mass spectrum in the vacuum.", "The existence of the vacuum moduli for $N_{\\rm F}>N_{\\rm C}$ implies that $m=0$ and hence the convergence radius of the derivative expansion seems to be zero.", "Therefore, it is interesting to see if there exists a special mechanism which justifies the derivative expansion for semi-local vortices.", "Non-Abelian vortices were extended to arbitrary gauge groups $G$ in the form of ${(U(1) \\times G)/ C(G)}$ with the center $C(G)$ of $G$ [59].", "Especially the cases of $G=SO(N),\\,USp(2N)$ have been studied in detail [60], [61].", "We can straightforwardly extend our analysis to the cases of arbitrary gauge groups but we should be careful to the normalizability since they are semi-local vortices in general.", "Although we have studied BPS vortices in supersymmetric gauge theories, our method to obtain higher derivative corrections by solving equations of motion for massive fields is robust and can be extended to non-supersymmetric theories.", "For instance, we can apply it to non-Abelian vortices in non-supersymmetric theory [62].", "In reality, non-Abelian vortices exist in high-density QCD which may be realized in the core of neutron stars [63].", "In this case, the low-energy world-sheet theory on the vortex is described by the bosonic ${\\mathbb {C}}P^2$ model at the leading order [64].", "The four-derivative correction to it should be important especially for the fate of confined monopoles which have been recently shown to exist as kinks on the vortex [65].", "This is because they are non-BPS and higher derivative terms do not vanish automatically, unlike BPS instantons discussed in Sec.", "REF .", "Finally our method is general so that we can apply it to other BPS solitons such as domain walls, monopoles and instantons, or non-BPS solitons such as Skyrmions.", "In the same spirit, four derivative term in the form of the Skyrme term was obtained in the effective theory of non-Abelian domain walls [23], and four derivative terms for collective coordinates of a rotating Skyrmion were calculated [24]." ], [ "Acknowledgments", "We would like to thank Kaneyasu Asakuma for a discussion in the early stage of this work.", "The work of M. E. , M. N. and N.S.", "are supported in part by Grant-in Aid for Scientific Research No.", "23740226 (M.E.", "), No.", "23740226 (M.N.", "), the “Topological Quantum Phenomena”Grant-in Aid for Scientific Research on Innovative Areas (No.", "23103515) (M.N.", "), No.", "21540279 (N.S.)", "and No.", "21244036 (N.S.)", "from the Ministry of Education, Culture, Sports, Science and Technology(MEXT) of Japan, and by Japan Society for the Promotion of Science (JSPS) and Academy of Sciences of the Czech Republic (ASCR) under the Japan - Czech Republic Research Cooperative Program (M.E.", "and N.S.", ")." ], [ "O(3) model", "From the isomorphism ${\\mathbb {C}}P^1 \\simeq S^2 \\simeq O(3)/O(2)$ , the ${\\mathbb {C}}P^1$ model is equivalent to the $O(3)$ model.", "In order to see this equivalence, let us introduce a three vector ${\\bf n} = (n_1,n_2,n_3)$ by ${\\bf n} &=& \\left( \\begin{array}{cc} \\frac{1}{\\sqrt{1+|\\beta |^2}}, & \\frac{\\bar{\\beta }}{\\sqrt{1+|\\beta |^2}} \\end{array} \\right)\\vec{\\sigma }\\left( \\begin{array}{c} \\frac{1}{\\sqrt{1+|\\beta |^2}} \\\\ \\frac{\\beta }{\\sqrt{1+|\\beta |^2}} \\end{array} \\right)\\nonumber \\\\&=& \\left( {\\beta + \\bar{\\beta }\\over 1+|\\beta |^2},-i{\\beta - \\bar{\\beta }\\over 1+|\\beta |^2},{1 -|\\beta |^2 \\over 1+|\\beta |^2}\\right) $ which satisfies the constraint ${\\bf n}^2 = 1.$ Conversely, $\\beta $ is the stereographic coordinate, given by $\\beta = {n_1 + i n_2 \\over 1 + n_3}= {1 - n_3 \\over n_1 - i n_2}.$ The kinetic term becomes ${\\partial _\\mu \\beta \\partial ^{\\mu } \\bar{\\beta }\\over (1+|\\beta |^2)^2}= {1\\over 2} \\partial _{\\mu } {\\bf n} \\cdot \\partial ^{\\mu } {\\bf n} ,$ and the field strength can be rewritten as $f_{\\mu \\nu }= {\\bf n} \\cdot (\\partial _{\\mu } {\\bf n} \\times \\partial _{\\nu }{\\bf n}).$ Therefore the Skyrme-Faddeev term and the other four derivative term become $f_{\\mu \\nu } f^{\\mu \\nu } ~~&=& (\\partial _{\\mu } {\\bf n} \\times \\partial _{\\nu } {\\bf n})^2 , \\phantom{{(\\partial _{\\mu } \\beta \\partial ^{\\mu } \\bar{\\beta })^2 \\over (1+|\\beta |^2)^4}} \\\\{(\\partial _{\\mu } \\beta \\partial ^{\\mu } \\bar{\\beta })^2 \\over (1+|\\beta |^2)^4}&=& {1\\over 4} (\\partial _{\\mu } {\\bf n} \\cdot \\partial ^{\\mu } {\\bf n})^2 ,$ respectively.", "The total four derivative terms in Eq.", "(REF ) can be rewritten as ${\\cal L}_4= c_1 (\\partial _{\\mu } {\\bf n} \\times \\partial _{\\nu } {\\bf n})^2+ {c_2 \\over 4} (\\partial _{\\mu } {\\bf n} \\cdot \\partial ^{\\mu } {\\bf n})^2.$" ], [ "Gauged linear sigma model and higher derivative corrections", "In this section, we consider a gauged linear sigma model with a higher derivative term which reproduces the orientational part of the higher derivative terms in Eq.", "(REF ).", "The model is described by $N$ charged scalar fields $h=(h_1,\\cdots ,h_N)$ coupled to a $U(1)$ auxiliary gauge field $a_\\alpha $ .", "The Lagrangian is given by $\\mathcal {L} ~=~ \\alpha h (\\alpha h)^\\dagger + d \\, \\alpha h (\\beta h)^\\dagger \\, \\alpha h (\\beta h)^\\dagger + \\lambda \\left( h h^\\dagger - v^2 \\right),$ where $d$ is a constant and $\\lambda $ is a Lagrange multiplier for the constraint $h h^\\dagger = v^2$ , which is solved by $h = \\frac{v}{\\sqrt{1+|b_i|^2}} (b_1,\\cdots ,b_{N-1},1).$ The covariant derivative is defined by $\\alpha h = (\\partial _\\alpha + i a_\\alpha ) h$ .", "The equation of motion for the auxiliary gauge field $a_\\alpha $ can be solved as $a_\\alpha = -\\frac{i}{2} \\frac{b_i \\partial _\\alpha \\bar{b}_i - \\partial _\\alpha b_i \\bar{b}_i}{1 + |b_i|^2}.$ This solution is independent of the parameter $d$ .", "In other words, the higher derivative term in Eq.", "(REF ) does not change the solution for the auxiliary gauge field.", "We can easily show that Eq.", "(REF ) is the solution by using the following relations $\\alpha h h^\\dagger = h (\\alpha h)^\\dagger = 0.$ Substituting Eq.", "(REF ) back into the Lagrangian Eq.", "(REF ), we obtain $\\mathcal {L} = v^2 g_{i \\bar{j}}^{\\rm FS} \\partial _\\alpha b^i \\partial ^\\alpha \\bar{b}^j + v^4 d (g_{i \\bar{j}}^{\\rm FS} \\partial _\\alpha b^i \\partial _\\beta \\bar{b}^j) (g_{k \\bar{l}}^{\\rm FS} \\partial ^\\alpha b^k \\partial ^\\beta \\bar{b}^l).$ Therefore, if $v^2 = \\frac{4\\pi }{g^2}$ and $d= \\frac{g^2}{4\\pi } c$ , this Lagrangian coincides with the orientational part of the vortex effective action Eq.", "(REF )." ], [ "Regularity of the second order solution", "In this section, we show that there is no singularity in the first component of the solution $\\Delta \\Phi ^{(2)}$ given in Eq.", "(REF ).", "Here we assume that $Z_I \\ne Z_J \\, ( I \\ne J ) $ , so that the constant matrix $\\Psi ^I \\equiv \\left[ (z-Z_I) H_0^{-1} \\right]_{z=Z_I}$ is well-defined.", "Then the singular behavior around the $I$ -th vortex position $z=Z_I$ can be written as $\\frac{1}{\\bar{z} - \\bar{Z}_I} \\left[ \\Psi _I \\bar{\\nabla }_i \\frac{\\partial }{\\partial \\bar{\\phi }^j} \\left(\\Omega \\partial _z \\Omega ^{-1} \\right) \\right]_{z=Z_I}^\\dagger + \\mathcal {O}(1).$ We can show that $\\left[ \\bar{\\nabla }_i \\frac{\\partial }{\\partial \\bar{\\phi }^j} \\left(\\Omega \\partial _z \\Omega ^{-1} \\right) \\right]_{z=Z_I} &=& \\bar{\\nabla }_i \\frac{\\partial }{\\partial \\bar{\\phi }^j} \\left[ \\, \\Omega \\partial _z \\Omega ^{-1} \\right]_{z=Z_I} - \\bar{\\nabla }_i \\left[ \\frac{\\partial Z_I}{\\partial \\bar{\\phi }^j} \\, \\partial _{\\bar{z}} (\\Omega \\partial _z \\Omega ^{-1} ) \\right]_{z=Z_I} \\\\&-& \\frac{\\partial \\bar{Z}_I}{\\partial \\bar{\\phi }^i} \\frac{\\partial }{\\partial \\bar{\\phi }^j} \\left[ \\partial _{\\bar{z}} ( \\Omega \\partial _z \\Omega ^{-1} ) \\right]_{z=Z_I} + \\frac{\\partial \\bar{Z}_I}{\\partial \\bar{\\phi }^i} \\frac{\\partial Z_I}{\\partial \\bar{\\phi }^j} \\left[ \\partial _{\\bar{z}}^2 (\\Omega \\partial _z \\Omega ^{-1}) \\right]_{z=Z_I}, $ where we have used the following identity for any function of the form $f(z, \\bar{z}, \\phi ^i,\\bar{\\phi }^i)$ $\\left[ \\frac{\\partial }{\\partial \\bar{\\phi }^i} f \\right]_{z=Z_I} = \\frac{\\partial }{\\partial \\bar{\\phi }^i} \\left[ f \\right]_{z=Z_I} - \\frac{\\partial \\bar{Z}_I}{\\partial \\bar{\\phi }^i} \\left[ \\partial _{\\bar{z}} f \\right]_{z=Z_I}.$ Then, the singular part can be rewritten into the following form $- \\frac{g^2v^2}{4} \\frac{1}{\\bar{z} - \\bar{Z}_I} \\left[ \\Psi _I \\bar{\\nabla }_i\\frac{\\partial }{\\partial \\bar{\\phi }^j} \\left( B_I - \\bar{Z}_I {\\bf 1}_N \\right) \\right]^\\dagger ,\\quad {\\rm with} \\quad B_I \\equiv -\\frac{4}{g^2v^2} \\left[ \\Omega \\partial _z \\Omega ^{-1} \\right]_{z=Z_I},$ where we used the following relations which can be derived from the master equation Eq.", "(REF ): $\\Psi _I\\partial _{\\bar{z}} \\left[ \\Omega \\partial _z \\Omega ^{-1} \\right]_{z=Z_I} =-\\frac{g^2v^2}{4} \\Psi _I, \\qquad \\Psi _I\\partial _{\\bar{z}}^{2} \\left[ \\Omega \\partial _z \\Omega ^{-1}\\right]_{z=Z_I}=0.$ Now let us use the explicit form of the generic moduli matrix Eq.", "(REF ).", "Then, $\\Psi _I$ is given by $\\Psi _I ~=~ \\prod _{I \\ne J} \\frac{1}{Z_I - Z_J} \\left( \\begin{array}{cc} \\ 1 & \\ 0 \\\\ -\\vec{b} & \\ \\mathbf {0} \\end{array} \\right).$ By assuming that the matrix $B_I$ takes the form $B_I \\equiv \\left( \\begin{array}{cc} p_I & (\\vec{q}_I)^{\\rm T} \\\\ \\vec{r}_I & s_I \\end{array} \\right),$ we can rewrite the singular part as $- \\frac{g^2 v^2}{4} \\frac{1}{\\bar{z} - \\bar{Z}_I} \\left[ \\prod _{J \\ne I} \\frac{1}{Z_I - Z_J} \\left( \\begin{array}{cc} \\ 1 & \\ 0 \\\\ -\\vec{b} & \\ \\mathbf {0} \\end{array} \\right)\\bar{\\nabla }_i \\frac{\\partial }{\\partial \\bar{\\phi }^j} \\left( \\begin{array}{cc} p_I - \\bar{Z}_I & (\\vec{q}_I)^{\\rm T} \\\\ \\vec{r}_I & s_I - Z_I \\end{array} \\right)\\right]^\\dagger .$ We can show that $\\bar{\\nabla }_i \\frac{\\partial }{\\partial \\bar{\\phi }^j} (\\bar{Z}_I - p_I) = \\bar{\\nabla }_i \\frac{\\partial }{\\partial \\bar{\\phi }^j} \\vec{q}_I = 0$ as follows.", "It has been shown that the matrix $B_I$ is related to the moduli space metric as [40] $g_{i\\bar{j}} &=& \\pi v^2 \\sum _{I=1}^{k}\\left( \\frac{\\partial Z_I}{\\partial \\phi ^i} \\frac{\\partial \\bar{Z}_I}{\\partial \\bar{\\phi }^j}+{\\rm Tr} \\left[ \\frac{\\partial H_0}{\\partial \\phi ^i} \\Psi _I \\frac{\\partial B^I}{\\partial \\bar{\\phi }^j}\\right]_{z=Z_I} \\right) \\\\&=& \\pi v^2 \\sum _{I=1}^{k} \\left( \\frac{\\partial Z_I}{\\partial \\phi ^i} \\frac{\\partial }{\\partial \\bar{\\phi }^j} (\\bar{Z}_I - p_I) + \\prod _{J \\ne I} \\frac{1}{Z_I - Z_J} \\left[ \\frac{\\partial \\vec{b}_I}{\\partial \\phi ^i} - \\sum _{J=1}^k \\frac{\\partial Z_J}{\\partial \\phi ^i} D_J{}^I \\vec{b}_J \\right] \\cdot \\frac{\\partial \\vec{q}_I}{\\partial \\bar{\\phi }^j} \\right), $ where $D_J{}^I \\equiv \\left[ \\partial _z e_J(z) \\right]_{z=Z_I}$ .", "Since the vortex moduli space is a Kähler manifold, there exist a Kähler potential $\\mathcal {K}$ such that $g_{i \\bar{j}} ~=~ \\frac{\\partial \\mathcal {K}}{\\partial \\phi ^i \\partial \\bar{\\phi }^j}.$ By comparing this equation with Eq.", "(REF ), we can read $p_I$ and $\\vec{q}_I$ as $\\frac{\\partial }{\\partial \\bar{\\phi }^j} (\\bar{Z}_I - p_I) &=& \\ \\ \\frac{1}{\\pi v^2} \\left[ \\frac{\\partial {\\cal K}}{\\partial \\bar{\\phi }^j \\partial Z_I} - \\sum _{J=1}^k D_I{}^J \\vec{b}_I \\cdot \\frac{\\partial {\\cal K}}{\\partial \\bar{\\phi }^j \\partial \\vec{b}_J} \\right], \\\\\\frac{\\partial \\vec{q}_I}{\\partial \\bar{\\phi }^j} \\hspace{19.91692pt} &=& \\ \\ \\frac{1}{\\pi v^2} \\prod _{J \\ne I} (Z_I - Z_J) \\frac{\\partial {\\cal K}}{\\partial \\bar{\\phi }^j \\partial \\vec{b}_I},$ Since the moduli space metric is covariantly constant, it follows that $\\bar{\\nabla }_i \\frac{\\partial }{\\partial \\bar{\\phi }^j} (\\bar{Z}_I - p_I) ~=~ \\bar{\\nabla }_i \\frac{\\partial \\vec{q}_I}{\\partial \\bar{\\phi }^j} ~=~ 0.$ This shows that there is no singularity in the solution Eq.", "(REF )." ] ]

[ [ "Influence of realistic parameters on state-of-the-art LWFA experiments" ], [ "Abstract We examine the influence of non-ideal plasma-density and non-Gaussian transverse laser-intensity profiles in the laser wakefield accelerator analytically and numerically.", "We find that the characteristic amplitude and scale length of longitudinal density fluctuations impacts on the final energies achieved by electron bunches.", "Conditions that minimize the role of the longitudinal plasma density fluctuations are found.", "The influence of higher order Laguerre-Gaussian laser pulses is also investigated.", "We find that higher order laser modes typically lead to lower energy gains.", "Certain combinations of higher order modes may, however, lead to higher electron energy gains." ], [ "Introduction", "Laser wakefield accelerators (LWFAs) [1] have the potential to play an important role in a wide range of applications, including high-energy physics experiments (by providing more compact accelerators [2], [3], [4], [5], [6]), nuclear physics experiments (by providing electron beams which can be used for the photonuclear activation of nuclei [7], [8]), and medical applications [9] (by providing, for instance, compact x-rays sources [10], [11]).", "For these applications, the recent developments of LWFAs, which currently deliver low divergence ($\\sim 10~\\mathrm {mrad}$ ), high-energy ($\\sim 1~\\mathrm {GeV}$ ), high charge ($\\sim 10~\\mathrm {pC}$ ) electron beams, with $\\lesssim 10~\\%$ energy spread, in externally- [5] or self-guided regimes [6], are placing strong emphasis on the shot-to-shot stability of the accelerators.", "It is therefore useful to understand the influence of shot-to-shot fluctuations on key outputs of LWFAs.", "Shot-to-shot fluctuations of the plasma density and laser intensity profiles impact on the reproducibility of LWFAs.", "In particular, changes in the laser and plasma profiles may affect beam divergence, emittance, pointing stability, energy spread, and maximum energy.", "In this paper we will focus on their influence in energy gain.", "Previous research indicated that fluctuations in experimental plasma density profiles may prevent self-injection, thus constraining the applicability of LWFAs [12].", "However, plasma density ramps [13], typically present in experiments with gas jets, can be used to control the phase-velocity of the plasma wave, and facilitate the self-injection of plasma electrons.", "Fluctuations of the laser intensity profile, which frequently exhibits higher-order modes, can affect the quality of the electron beams [14].", "Nevertheless, properly tailored higher-order laser modes can be used to lower the self-trapping thresholds [15] or increase the amplitude of the self-injected beam betatron oscillations [16], [17].", "Thus, either to obtain low shot-to-shot fluctuations on the outputs of LWFAs, or to obtain highly optimized acceleration regimes, an adequate control of the plasma density [12], and laser intensity [18] is required.", "In order to investigate the influence of non standard plasma density and laser intensity profiles we use analytical modeling, and particle-in-cell simulations.", "Standard particle-in-cell algorithms are very computationally expensive.", "Reduced codes such as QuickPIC [19], or advanced PIC algorithms in boosted frames [20], [21] are ideal to perform systematic parameter scans of the LWFA.", "In the boosted frame technique no physical approximations are involved.", "The associated computational gains result from bringing closer the disparate scales of LWFAs, i.e.", "the laser and the plasma wavelengths.", "The analysis of the relevant laser and plasma dynamics, however, is not straightforward, since simultaneous events in the boosted frame occur at different instants of time in the laboratory frame [22].", "The use of reduced models can circumvent these difficulties.", "The Quasi-Static Approximation (QSA), for instance, is widely used to model LWFAs in reduced PIC codes [23], [24].", "It precludes the self-injection physics, but the dynamics of the laser, and acceleration of externally injected electron beams can be investigated systematically.", "Typically, the plasma response is determined through a Quasi-Static [23], [24] plasma field solver, and the laser is advanced in the Ponderomotive Guiding Center Approximation (PGCA)  [24].", "Simulations under the QSA and PGCA require lower spatial resolutions in comparison to standard full PIC codes.", "In addition, larger time steps can also be used [25], [26].", "Thus, reduced codes employing the QSA and PGCA can be much faster than standard full PIC codes.", "Here we employ the quasi-static, massively parallel, fully relativistic, PIC code QuickPIC which can be more than two orders of magnitude faster than standard full PIC codes [19].", "In this paper, we investigate the influence of non-standard plasma and laser configurations in the outputs of LWFAs for a wide range of parameters relevant for state-of-the-art LWFAs.", "In Section , a reference simulation using state-of-the-art laser and plasma parameters is presented and discussed.", "Then, in Sec.", ", the influence of inhomogeneous longitudinal plasma density perturbations is considered.", "It is shown that the impact of the density fluctuations can be minimized when the plasma-density variations are small in comparison to the background density, and when the typical wavelengths of the perturbations are much shorter than the acceleration distance.", "Section  analyses the influence of non-Gaussian transverse laser intensity profiles in the acceleration.", "Analytical estimates for the energy gain are derived and confirmed with QuickPIC simulations.", "A parameter region where higher order modes can lead to higher energy gains was also identified.", "Finally the conclusions are stated." ], [ "Self-guiding of 10 J laser pulses", "In this Section, the propagation of a 10 J laser pulse, with parameters close to Ref.", "[6] is investigated using QuickPIC [19].", "In the baseline simulation conditions that we use in this paper, a laser pulse with central wavelength $\\lambda _0=0.8~\\mu \\mathrm {m}$ , normalized peak vector potential $a_0=3.85$ , spot-size $W_0=19~\\mu \\mathrm {m}$ ($1/e$ ), and duration $\\tau _0=55~\\mathrm {fs}$ ($1/e$ ) is initialized at the entrance the pre-formed, transversely uniform plasma.", "The plasma density rises linearly to $n_0=5.7\\times 10^{18}~\\mathrm {cm}^{-3}$ for $0.065~\\mathrm {cm}$ .", "Uniform density is then used in the following $0.73~\\mathrm {cm}$ , falling linearly until the end of the plasma, during $0.12~\\mathrm {cm}$ .", "The simulation uses a window which moves at the speed of the light in vacuum (c), with $250~\\mu \\mathrm {m}\\times 250~\\mu \\mathrm {m}\\times 80~\\mu \\mathrm {m}$ , divided into $512\\times 512\\times 256$ cells, with 4 particles per cell.", "An electron beam with $Q=0.16~\\mathrm {pC}$ is at the back of the first plasma wave, in a region where the accelerating fields are maximum.", "The charge of the beam is much lower than the beam loading charge [27], [28], in order to provide a quasi test particle acceleration regime.", "The electron beam density profile is given by: $N \\propto \\exp \\left(-\\frac{(\\xi -\\xi _0)^2}{2 \\sigma _z^2}\\right) \\exp \\left(-\\frac{r^2}{2 \\sigma _r^2}\\right),$ where $\\xi =z-c t$ , $z$ is the propagation distance, $t$ the time, $\\xi _0$ the center of the electron beam, $r=\\sqrt{x^2+y^2}$ is the transverse coordinate, $\\sigma _z=1.2~\\mathrm {\\mu m}$ is the length of the electron beam and $\\sigma _r=1.2~\\mathrm {\\mu m}$ its width.", "The electron beam was initialized with a relativistic factor $\\gamma =200$ , with corresponding longitudinal velocity above the phase velocity of the plasma wave.", "Furthermore, the simulations included radiation damping, which results in a decelerating-like force, due to the betatron motion of the beam electrons betatron oscillations [29].", "For these parameters, we have found that radiation damping is negligible [30].", "The laser pulse peak $a_0$ , spot-size $W_0$ , blowout radius $r_b$ , and electron beam peak energy, are depicted in Fig.", "REF .", "For $z\\le 0.1~\\mathrm {cm}$ , strong laser spacial and temporal compression (Fig.", "REF a) occurrs, where the $a_0$ increases from $a_0=3.85$ to $a_0= 13$  [31].", "Simultaneously, the $W_0$ decreases by the same factor, from $W_0=19~\\mathrm {\\mu m}$ to $W_0=7.5~\\mathrm {\\mu m}$ (Fig.", "REF b).", "As a result, the ponderomotive force $F_p\\propto a_0/W_0$ increases nearly by one order of magnitude, leading to complete electron blowout after $z=0.1~\\mathrm {cm}$ .", "This behavior, also present in [32], may be due to the mismatch between the initial laser spot-size and blowout radius $r_b$ .", "For $0.1~\\mathrm {cm}<z<0.8~\\mathrm {cm}$ , both the $a_0$ and $W_0$ variations are smoother (Fig.", "REF ) indicating that matched conditions established self consistently.", "In average, $a_0$ increases from $a_0=8$ to $a_0=14$ until $z=0.8~\\mathrm {cm}$ , rapidly decreasing from then on.", "The increase of $a_0$ is accompanied by spot-size oscillations between $W_0=8~\\mathrm {\\mu m}$ and $W_0=16~\\mathrm {\\mu m}$ , indicating that the body of the laser pulse is self-guided.", "Figures REF a and REF b also show that, for $0.1~\\mathrm {cm}<z <0.8~\\mathrm {cm}$ , both the $a_0$ and $W_0$ vary by more than $40\\%$ in each oscillation.", "The corresponding variations of $r_b$ , however, are lower than $10~\\%$ , thus guaranteeing a stable acceleration regime (in the blowout regime, the accelerating gradients are solely determined by $r_b$  [33]).", "A closer investigation of Figs.", "REF a and REF b reveals that $W_0$ and $a_0$ oscillate out of phase, i.e.", "$a_0$ is minimum for maximum $W_0$ , and vice-versa, which lead to large variations of the ponderomotive force.", "However, the ponderomotive force at $r\\simeq r_b$ is always similar, thus resulting in lower variations for $r_b$ .", "The stable accelerating fields lead to beam peak energies of almost 1 GeV at $z\\sim 0.7~\\mathrm {cm}$ (in agreement with [6]), which saturated from then on.", "The stabilization of the final electron beam peak energy was a result of the single beam electrons dynamics (not from the phase-space rotation due to the dephasing of the electron beam as in [34]): a fraction of the beam electrons are in regions of accelerating fields and gain energy.", "The remaining fraction are in a region of decelerating fields, and loose energy.", "As a result, the energy spectrum broadened with constant peak energy.", "These results establish the baseline conditions to analyze possible perturbations/deviations from the idealized parameters.", "Figure: Baseline QuickPIC simulation results.", "(a) Evolution of the normalized peak laser pulse vector potential a 0 a_0 associated with an initially transversely Gaussian laser pulse.", "(b) Evolution of the laser pulse spot-size (solid line) and blowout radius (dashed line).", "The blowout radius corresponds to the location of the maximum plasma density for a fixed transverse slice, located roughly at the center of the laser pulse.", "(c) Evolution of the electron beam peak energy." ], [ "Electron acceleration in inhomogeneous plasma density profiles", "Nondeterministic plasma density fluctuations present in experiments impact on the acceleration gradients and final energies of electron bunches.", "Understanding the role of density non-uniformities in the LWFA is thus important to design stable accelerators.", "To this end we estimate analytically the single particle energy gain in a LWFA with small density perturbations.", "The expression for the energy gain of a trapped electron in the frame that moves with the laser group velocity $v_{g0}$ , and in a reference plasma frequency $\\omega _{p0}$ , is given by: $\\Delta E = q \\int _{-r_b}^{\\xi _c} \\frac{E_{\\mathrm {accel}}(\\xi _p)}{1-\\beta _{\\phi 0}} \\mathrm {d} \\xi _p,$ where $E_{\\mathrm {accel}}$ is the accelerating field, $\\xi _p = z_p - v_{g0}t = z_p - v_{\\phi 0} t \\simeq z_p (1-\\beta _{\\phi 0})$ is the distance of the externally injected electron to a point moving with the phase velocity of the plasma wave $v_{\\phi 0}=\\beta _{\\phi 0} c \\equiv v_{g0}$ , and $z_p=c t$ is the electron longitudinal position that moves at the speed of light.", "We consider that the electron, initially placed at $\\xi _p=-r_b$ , i.e.", "at the back of the bubble, accelerates in the positive $\\xi _p$ direction until the center of the bubble $\\xi _c$ is reached.", "At the center of the bubble, $E_{\\mathrm {accel}}=0$ , and the acceleration stops.", "In order to determine an expression for $E_{\\mathrm {accel}}$ we assume that the center of the bubble $\\xi _c$ travels at the wake phase velocity in the blowout regime $v_{\\phi }=c(1-\\alpha \\omega _{p}^2/\\omega _0^2)$ , where $\\alpha = 3/2$ , $\\omega _0$ is the laser central frequency, $\\omega _{p}=\\omega _{p}(\\xi _p)$ is the position dependent plasma wave frequency, and $\\omega _0/\\omega _p \\gg 1$ .", "Since in the blowout regime the accelerating fields are linear, we may write $E_{\\mathrm {accel}}= (m_e \\omega _p^2 / q) (\\xi _p-\\xi _c)/2$ .", "In order to derive an expression for $\\xi _c$ we assume that $\\xi _c=0$ when $\\xi _p = -r_b$ .", "Hence the trajectory of the center of the bubble is $\\xi _c = \\int _{0}^{t_{\\mathrm {accel}}}(v_{\\phi }-v_{\\phi 0})\\mathrm {d}t$ , where $t_{\\mathrm {accel}}$ is the acceleration time.", "Since a trapped electron moves at $c$ , $\\xi _p=(c-v_{\\phi 0}) t$ , and hence $\\xi _c(\\xi _p)=\\int _{-r_b}^{\\xi _p} \\left[\\omega _{p}(\\xi )^2/\\omega _{p0}^2 -1\\right] d\\xi $ .", "In an ideal scenario where the plasma is uniform ($\\omega _p(\\xi _p)$ is constant) the acceleration stops at $\\xi _c=0$ .", "In general, if $\\omega _p(\\xi _p)$ is not constant $\\xi _c$ may be different than zero.", "As a result, Eq.", "(REF ) can be re-written as: $\\Delta E = - \\frac{3}{2} m_e \\int _{-r_b}^{\\xi _c} \\frac{\\omega _0^2 \\omega _p^2(\\xi _p) \\left[\\xi _p-\\xi _c(\\xi _p)\\right]}{2 \\omega _{p0}^2} \\mathrm {d} \\xi _p,$ To further evaluate Eq.", "(REF ) we assume small plasma density perturbations such that $\\omega _p^2(\\xi _p) = \\omega _{p0}^2 [1+ \\delta \\omega _p(\\xi _p)]$ where $\\delta \\omega _p(z_p)\\ll 1$ is a small perturbation.", "For the sake of simplicity we also assume that the plasma density varies sinusoidally, and that the average plasma density (frequency) is $n_0$ ($\\omega _{p0}$ ).", "Hence, $\\delta \\omega _p(\\xi _p) = \\delta \\cos (k_1 \\xi _p + \\phi )$ where $\\delta \\ll 1$ is the amplitude of the sinusoidal plasma density perturbation.", "The initial phase $\\phi =-\\pi /2 - k_1 \\xi _{p0}$ guarantees that $\\delta \\omega _p(-r_b) = 0$ , and that the plasma density rises for $\\xi _p \\gtrsim -r_b$ if $\\delta > 0$ ($\\xi _{p0}=-r_b$ is the initial electron bunch position).", "The wavenumber $k_1$ is the typical wave number of the non-uniformity in the frame that travels with $v_{\\phi 0}$ .", "In the laboratory frame coordinates ($z_p(\\xi _p),t$ ), $k_1 =(2/3) k_1^{\\mathrm {lab}} \\omega _0^2/\\omega _{p0}^2$ .", "For the sake of simplicity, we consider that the upper limit of the integral in Eq.", "(REF ) is $\\xi _c=0$ .", "Retaining the leading order terms on the order of $\\mathcal {O}(\\delta )$ leads to: $\\Delta E = m_e c^2 \\left[\\frac{3}{8}\\frac{\\omega _0^2}{\\omega _{p0}^2}\\frac{r_b^2 \\omega _{p0}^2}{c^2} - \\frac{3 \\delta }{2} \\frac{k_{p0}}{k_1}\\frac{\\omega _0^2}{\\omega _{p0}^2}\\left(\\frac{r_b\\omega _{p0}}{c} - \\frac{k_{p0}}{k_1} \\sin \\left(r_b k_1\\right)\\right)\\right],$ The first term on the right-hand-side of Eq.", "(REF ), $(\\Delta E)_{\\mathrm {uni}} = m_e c^2 (3/8) (r_b\\omega _p/c)^2 \\omega _0^2/\\omega _{p0}^2$ , corresponds to the maximum energy gain in an uniform plasma.", "Making the substitution $r_b =2 \\sqrt{a_0} c/\\omega _p$ yields $(\\Delta E)_{\\mathrm {uni}} = m_e c^2 (3/2) a_0 \\omega _0^2/\\omega _{p0}^2$ which is close to the result from Ref. [30].", "The differences in the numerical factors are due to the fact that Ref.", "[30] considers a constant average $E_{\\mathrm {accel}}$ .", "Instead, Eq.", "(REF ) uses linear accelerating fields.", "The second term, $(\\Delta E)_{\\mathrm {sin}} = - m_e c^2(3 \\delta \\omega _0^2 k_{p0}/2 \\omega _{p0}^2 k_1) \\left[r_b \\omega _{p0}/c - (k_{p0}/k_1)\\sin \\left(r_b k_1\\right)\\right]$ are corrections associated with the density non-uniformities.", "These corrections are due equally to the variations on the accelerating gradients, and to the variations of the laser group velocity through fluctuations of $\\xi _c$ .", "According to Eq.", "(REF ), the ratio $(\\Delta E)_{\\mathrm {sin}}/(\\Delta E)_{\\mathrm {uni}}$ is: $\\frac{(\\Delta E)_{\\mathrm {sin}}}{(\\Delta E)_{\\mathrm {uni}}} = \\frac{4 \\delta [r_b k_1 -\\sin (r_b k_1)]}{r_b^2 k_1^2}.$ Equation (REF ) is plotted in Fig.", "REF as a function of $r_b k_1$ .", "The ratio $(\\Delta E)_{\\mathrm {sin}}/(\\Delta E)_{\\mathrm {uni}}$ is directly proportional to $\\delta $ .", "Larger final energy fluctuations are hence expected in plasmas with larger density fluctuations.", "In addition, the sign of $\\delta $ determines the sign of $(\\Delta E)_{\\mathrm {sin}}$ .", "A density increase (reduction) at the start of the acceleration may then lead to larger (lower) energy gains in comparison to uniform plasma densities.", "This is associated with the fact that the acceleration gradients decrease as the acceleration progresses.", "Hence, the density fluctuations at the start of the acceleration will be more important than those occurring later.", "The initial density fluctuations then give the dominant contribution for $(\\Delta E)_{\\mathrm {sin}}$ .", "Equation (REF ) also shows that $(\\Delta E)_{\\mathrm {sin}}$ depends on the typical wavenumber of the density non-uniformities.", "For $r_b k_1 \\ll \\pi $ the density fluctuations can be neglected because their amplitude is arbitrary small in this limit.", "For $r_b k_1 = \\pi $ , $(\\Delta E)_{\\mathrm {sin}}/(\\Delta E)_{\\mathrm {uni}} = 4 \\delta /\\pi \\simeq 1.27\\delta $ is maximum.", "In the laboratory frame coordinates ($z,t$ ) this corresponds to a wavenumber $k_1^{\\mathrm {lab}}=(3 \\pi /2) \\omega _{p0}^2/(r_b \\omega _0^2)$ , and to a wavelength $\\lambda _1^{\\mathrm {lab}} = 2 \\pi / k_1^{\\mathrm {lab}} = (4/3) r_b \\omega _0^2 / \\omega _p^2$ .", "The total acceleration length in the matched blowout regime is $L_{\\mathrm {accel}} \\simeq (2/3)\\omega _0^2 r_b/\\omega _{p0}^2$  [30].", "Thus $\\lambda _1^{\\mathrm {lab}} \\simeq 2 L_{\\mathrm {accel}}$ .", "In the limit $r_b k_1 \\gg \\pi $ , or equivalently $\\lambda _1^{\\mathrm {lab}} \\ll 2 L_{\\mathrm {accel}}$ , $(\\Delta E)_{\\mathrm {sin}}\\rightarrow 0$ because the effects associated with the density non-uniformities in the acceleration will average to zero.", "Figure: Normalized ratio [(ΔE) sin /(ΔE) uni ]/δ[(\\Delta E)_{\\mathrm {sin}}/(\\Delta E)_{\\mathrm {uni}}]/\\delta as a function of k 1 r b k_1 r_b.", "Maximum [(ΔE) sin /(ΔE) uni ]/δ[(\\Delta E)_{\\mathrm {sin}}/(\\Delta E)_{\\mathrm {uni}}]/\\delta occurs for r b k 1 =πr_b k_1=\\pi , where [(ΔE) sin /(ΔE) uni ]/δ=4/π≃1.27[(\\Delta E)_{\\mathrm {sin}}/(\\Delta E)_{\\mathrm {uni}}]/\\delta = 4/\\pi \\simeq 1.27.Equation (REF ) assumes that the density variations are sinusoidal, that the wakefield amplitude is constant, and that the acceleration distance does not change, and neglects pump-depletion and beam loading effects.", "In order to verify that Eq.", "(REF ) holds even when these approximations are not verified, we performed particle-in-cell simulations in QuickPIC in conditions relevant for experiments.", "The baseline simulation parameters for the laser and plasma are identical to those of Sec. .", "We first investigate the laser dynamics in the experimental density profile shown in Fig.", "REF a, which was obtained from interferometric measurements of the plasma channel profile during the experiment reported in [6].", "Figures REF b, and REF c show that the global evolution of $a_0$ and $W_0$ in the inhomogeneous density profile is identical to that described in Sec.", ": the propagation is dominated by strong initial self-focusing that leads to the complete blowout of plasma electrons, followed by a self-guided propagation stage.", "In spite of the similar laser dynamics associated with the profiles represented in Fig.", "REF a, two differences can still be mentioned: first of all, the variation of the laser $a_0$ ($W_0$ ) in Sec.", "is systematically higher (lower) than the corresponding variations associated with the experimental profile of Fig.", "REF a.", "Moreover, the frequency of the $a_0$ ($W_0$ ) oscillations is also higher in Sec.", "than by using the experimental profile.", "These effects are both due to the higher plasma density used in Sec. .", "On one hand, the use of higher densities leads to stronger laser pulse ponderomotive forces, and self-focusing, and thus to higher peak $a_0$ 's.", "Consequently, the frequencies of oscillation of both $W_0$ and $a_0$ are higher.", "Furthermore, the frequency shifts are stronger and faster for high densities, which further increases the peak $a_0$ 's obtained using the experimental profile from Fig.", "REF .", "Figure: QuickPIC simulation results comparison between an experimental (solid-black line) and idealized (dash-black line) plasma density profiles (a) Realistic density profile, with small (≲30%\\lesssim 30\\%) fluctuations from the idealized profile, characterized by two ramps at the edges of a uniform plasma density region.", "(b) comparison between the peak normalized vector potentials, and (c) comparison of the evolution of the laser pulse spot-size.Simulation results from a systematic parameter scan with different amplitudes of the plasma density non-uniformities is shown in Fig.", "REF .", "The simulated density profiles are variations of that of Fig.", "REF a with different $|\\delta |$ 's.", "The value of $\\delta $ corresponds to the standard deviation of the density variations along the plasma multiplied by $\\sqrt{2}$ such that it is consistent with the amplitude of the sine function used in the analytical model.", "In addition, the average plasma density is identical for each simulated profile, and the test electron bunches were injected at the same distance from the laser in each simulation.", "Figure REF b shows the energy gain of external electron bunches as a function of the propagation distance for each of the profiles of Fig.", "REF a.", "Figure REF b confirms that larger plasma density fluctuations lead to larger energy gain variations in comparison to uniform plasma densities.", "In addition, larger $|\\delta |$ 's lead to lower energy gains since the first part of the propagation occurs in regions where the plasma density is lower.", "These findings agree qualitatively with Eq.", "(REF ).", "In order to compare the model and simulations quantitatively we plot in the inset of Fig.", "REF b the theoretical value for the maximum $(\\Delta E)_{\\mathrm {sin}}/(\\Delta E)_{\\mathrm {uni}}=4 \\delta /\\pi $ , and the maximum energy gain for each simulation as a function of $\\delta $ .", "The simulation results follow the theoretical results for maximum $(\\Delta E)_{\\mathrm {sin}}/(\\Delta E)_{\\mathrm {uni}}$ .", "We also analyzed the role of the plasma density ramps in the acceleration gradients.", "Simulations results with ramp lengths identical to those of Figs.", "REF a and REF a but with different plasma lengths are illustrated in Fig.", "REF c. These simulations indicate that the energy gain does not depend on the presence of the plasma ramps.", "Naturally, larger plasma ramps will affect final energy gains.", "However, our results illustrate that in the typical experimental considered conditions, electron energy gains are insensitive to the presence of the plasma ramps.", "All simulations showed large energy spreads on the order of 100% at the end of the plasma.", "This is due to the fact that initially the wake is not beam loaded.", "We use low beam charges, such that the acceleration occurs in a test-particle-regime.", "Consequently the acceleration differed for particles at different longitudinal positions.", "The results from these simulations then did not show a clear influence of $\\delta $ on the particle energy spread.", "However, it is expected that plasma density non-uniformities will influence the energy spread when the plasma wave is beam loaded.", "Figure: Energy gain by externally injected electrons propagating in non-uniform plasma profiles with different amplitudes of the density variations and plasma lengths.", "(a) Simulated longitudinal plasma density profiles as a function of the propagation distance.", "(b) Corresponding energy gain by externally injected electrons.", "The inset in (b) shows a comparison between the simulated energy gain variation after 0.8 cm of propagation with the theoretical estimates.", "(c) Energy gain by externally injected electrons using different plasma lengths with ramps identical to those shown in (a)." ], [ "Role of higher-order Laguerre-Gaussian laser pulses in the LWFA", "The presence of higher order modes in the transverse laser intensity profile is ubiquitous in LWFAs.", "Hence, understanding the role of higher order modes in the laser propagation and in the final electron bunch energy is important to plan and interpret experiments.", "In this section we will examine LWFAs excited by radially symmetric laser pulses with higher order Laguerre-Gaussian modes [35].", "In vacuum, the slowly varying envelope of the normalized vector potential associated with an arbitrary laser beam profile can be written as a sum of Laguerre-Gaussian modes as: $\\mathbf {a} & = & f(z)\\sum _{l,p} \\mathbf {a}^{(l,p)} \\left(\\frac{r \\sqrt{2}}{W_0(z)}\\right)^{|l|}\\exp \\left(-\\frac{r^2}{W_0^2(z)}\\right) \\mathrm {L}_p^{|l|}\\left(\\frac{2 r^2}{W_0^2(z)}\\right) \\times \\nonumber \\\\&\\times &\\exp \\left(i \\frac{k_0 r^2}{2 R(z)}\\right) \\exp \\left(i l \\phi \\right),$ where $f(z)$ is the longitudinal laser profile, $\\phi $ is the azimuthal coordinate, $W_0(z)=W_0 \\sqrt{1+z^2/z_r^2}$ is the vacuum laser spot size as a function $z$ , $z_r = k_0 W_0^2/2$ is the Rayleigh length, $R(z) = z\\left(1+z_r^2/z^2\\right)$ is the radius of curvature of the laser wavefronts, and $L_p^{|l|}$ are generalized Laguerre polynomials of degree $p$ .", "Since we analyze the propagation of radially symmetric laser beams we consider $l=0$ .", "Eq.", "(REF ) then becomes: $\\mathbf {a}=f(z)\\sum _{p} \\mathbf {a}^{(p)} \\exp \\left(-\\frac{r^2}{W_0^2(z)}\\right) \\mathrm {L}_p\\left(\\frac{2 r^2}{W_0^2(z)}\\right) \\exp \\left(i \\frac{k_0 r^2}{2 R(z)}\\right),$ where $\\mathrm {L}_p\\equiv \\mathrm {L}_p^{(0)}$ .", "Equation (REF ) shows how the different combinations of higher order Laguerre-Gaussian modes lead to different radial laser intensity, energy and power distributions.", "Since the laser power $P$ is conserved for each transverse laser slice, we assume that initial laser power distribution determines the electron energy gain.", "The energy gain $\\Delta E$ in the blowout regime as a function $P$ is [30]: $\\Delta E [GeV] = 1.7 \\left(\\frac{P[\\mathrm {TW}]}{100}\\right)^{1/3}\\left(\\frac{10^{18}}{n_0[\\mathrm {cm}^{-3}]}\\right)^{2/3}\\left(\\frac{0.8}{\\lambda _0 [\\mu \\mathrm {m}]}\\right)^{4/3},$ where $\\lambda _0 = 2 \\pi c/k_0$ is the central laser wavelength.", "The fraction of the laser power which is far from the blowout region may diffract, not contributing to wake excitation and particle acceleration.", "Thus, we consider that the only laser power useful for particle acceleration is that contained within $r\\lesssim r_b$ .", "According to Fig.", "REF this corresponds to the laser power contained within $r\\lesssim W_0$ .", "The power at the focal plane contained within a laser spot size for lasers with $p=0$ (Gaussian beam) and $p=1$ is: $P & \\propto & \\int _{0}^{W_0} r \\left[a^{(0)}+ a^{(1)} \\exp \\left(-\\frac{r^2}{W_0^2}\\right) \\mathrm {L}_p\\left(\\frac{2 r^2}{W_0^2}\\right)\\right]^2 \\mathrm {d}r \\nonumber \\\\& \\propto & \\frac{W_0^2 \\left(a^{(0)2}+a^{(1)2}\\right)}{4} -\\frac{W_0^2 \\left(a^{(0)2} - 4 a^{(0)} a^{(1)} + 5 a^{(1)2}\\right)}{4 ^2},$ To make quantitative predictions we assume that the laser pulse energy $E^{\\mathrm {laser}}$ is constant for any combination of $a^{(0)}$ and $a^{(1)}$ .", "According to Eq.", "(REF ), the laser energy is proportional to $E^{\\mathrm {laser}}\\propto (a^{(0)2}+a^{(1)2})\\equiv a_0^2$ , where $a_0$ is the normalized vector potential of a purely Gaussian laser beam with $W_0$ and $f(z)$ identical to the higher order mode.", "Thus, $a^{(0)}=\\sqrt{a_0^2-a^{(1)2}}$ .", "Making use of the latter when inserting Eq.", "(REF ) into Eq.", "(REF ), we can calculate the ratio $\\Delta E^{(1)}/\\Delta E^{(0)}$ given by $\\frac{\\Delta E^{(1)}}{\\Delta E^{(0)}} = \\left(1-\\frac{4 a^{(1)2}-4 a^{(1)} \\sqrt{a_0^2-a^{(1)2}}}{a_0^2 \\left[^2-1\\right]}\\right)^{1/3},$ where $\\Delta E^{(1)}$ ($\\Delta E^{(0)}$ ) is the energy gain of the laser pulse with higher order (purely Gaussian) modes.", "Note that when the higher order laser pulse power inside $r<W_0$ is the same as a Gaussian pulse then $\\Delta E^{(1)}/\\Delta E^{(0)}=1$ .", "Equation (REF ) can be used to examine the impact of the first $p=0$ and $p=1$ Laguerre-Gaussian modes in the acceleration of electrons, and can be readily generalized including the presence of modes with $p>1$ .", "It shows that the presence of higher order Laguerre-Gaussian modes generally leads to lower energy gains in comparison to Gaussian laser pulses.", "This is because the laser power is typically distributed around wider radial regions for lasers with $a^{(1)}\\ne 0$ .", "Minimum energy gains occur when $a^{(1)}= -(a_0/2)\\sqrt{2+\\sqrt{2}}\\simeq 0.92 a_0$ for which $\\Delta E^{(1)}/\\Delta E^{(0)}=[(^2-3-2\\sqrt{2})/(^2-1)]^{1/3}\\approx 0.625$ .", "Interestingly, Eq.", "(REF ) also predicts that the energy gain in the range $0<a^{(1)}<a_0/\\sqrt{2}$ is higher than that of a Gaussian pulse.", "Maximum energy gain $\\Delta E^{(1)}/\\Delta E^{(0)}$ is obtained when $a^{(1)}= (a_0/2)\\sqrt{2-\\sqrt{2}}\\simeq 0.38 a_0$ for which $\\Delta E^{(1)}/\\Delta E^{(0)}=[(^2+2\\sqrt{2}-3)/(^2-1)]^{1/3}\\approx 1.04$ .", "In this case more laser power is contained within $r<W_0$ in comparison to a purely Gaussian laser leading to $\\Delta E^{(1)}/\\Delta E^{(0)}>1$ .", "In order to confirm these findings we performed two sets of QuickPIC simulations using lasers with higher order Laguerre-Gaussian modes.", "The baseline parameters (laser energy, spot-size, duration, and plasma density) are similar to those presented in Sec. .", "In the first set of simulations we analyzed electron energy gain by lasers with $p=0$ and $p=1$ .", "This permitted to make direct comparisons with Eq.", "(REF ).", "Results are summarized in Fig.", "(REF ).", "Very good agreement was found between the analytical model (cf.", "Eq.", "(REF )) and simulation results.", "In particular, simulations confirmed that higher order Laguerre-Gaussian laser beams can lead to higher electron energies.", "Simulations also revealed that the theory underestimates the energy gain for $(a^{(0)},a^{(1)})=(0,\\pm 3.85)$ .", "Simulations in this scenario show that a fraction of the laser initially at $r\\gtrsim W_0$ self-focuses and enters the blowout region during the propagation.", "This mechanism increases the total laser power inside $r<W_0$ that can be used in particle acceleration, but is not accounted for in Eq.", "(REF ) Figure: Influence of higher order Laguerre-Gaussian laser beam modes in the energy gain in the LWFA.", "(a) Initial central laser lineouts for each simulated scenario.", "(b) Corresponding energy gain as a function of the propagation distance.", "(c) Comparison between the analytical model Eq.", "() and the simulation results.In a second set of QuickPIC simulations we test the model using “realistic” laser pulse profiles.", "The simulated laser parameters are a numerical fit of measurements of the focal spot laser intensity distribution from [6].", "The simulation uses $a^{(0)}=1.9$ , $a^{(1)}=-0.44$ , $a^{(2)}=1.14$ , $a^{(3)}=0.05$ , $a^{(4)}=0.46$ , $a^{(5)}=0.13$ , $a^{(6)}=0.33$ , $a^{(0)}=-0.39$ , and $W_0=30~\\mu \\mathrm {m}$ .", "The plasma density, laser duration and laser energy are similar to the baseline simulation parameters of Sec. .", "Figure REF shows the main simulation results.", "For comparison, Fig.", "REF a plots the Gaussian laser profile associated with the baseline simulation parameters.", "The corresponding plasma density profile is also illustrated in Fig.", "REF b.", "These figures reveal that higher order modes form during the propagation even if the initial laser profile is Gaussian.", "This is because higher order modes appear naturally when Gaussian lasers propagate in non-parabolic plasma density profiles such as those associated with the blowout regime.", "Figure REF c-d show simulation results using the fit to the experimental laser profile including higher order modes.", "Since the initial laser profile already contains higher order Laguerre-Gaussian modes, the laser filaments at $z=0.3~\\mathrm {cm}$ are much more pronounced than in Fig.", "REF a-b.", "As the laser propagates it self-focuses leading to the generation of non-linear plasma waves in the blowout regime.", "A sharp electron interface that separates the bubble from the surrounding plasma is then created.", "This thin plasma sheet at $r=r_b$ bends the laser pulse wavefronts outwards for $r\\gtrsim W_0\\sim r_b$ , and inwards for $r\\lesssim W_0 \\sim r_b$ , which thus confines the central region of the laser pulse inside the blowout region, and prevents the entrance of the side filaments inside it.", "The blowout region thus acts as a spatial filter of the laser pulse.", "Most of the laser light outside the blowout region is then lost via diffraction and pump depletion.", "This process confirms that only a fraction of the initial laser pulse power or energy is used to create the blowout and accelerate electrons.", "As a result, the final electron beam energies are lower in comparison with the results from Sec. .", "The electron energy gain as a function of the propagation distance is shown in the inset of Fig.", "REF d. Despite the theory does not account for the details of the wake excitation by lasers with higher order modes, and for the laser evolution, it agrees very well with simulations.", "Figure: QuickPIC simulation results comparing a scenario using a purely Gaussian laser beam (a)-(b), and a laser with higher order Laguerre-Gaussian modes (c)-(d).", "(a) is a 2D slice at y=0 illustrating the laser vector potential after 0.3 cm.", "The arrow indicates the laser propagation direction.", "The dashed line indicates the position where a transverse line-out was taken in (b).", "(b) Lineout of the plasma density, and laser vector potential after 0.3 cm.", "The dashed curve is the initial laser profile.", "(c) is a 2D slice at y=0 illustrating the profile of a fit to an experimental laser vector potential after 0.3 cm.", "(d) lineouts of the plasma and laser profiles.", "The laser profile (dashed-line) is characterized by a central filament, with several filaments in the side.", "The inset in (d) shows the energy gain of externally injected electrons as a function of the propagation distance.", "The theoretical prediction is shown by the dark square, the gray curve represents the simulation results with higher order laser modes (H.O.L.", "), and the black curve represents the simulation results with a purely Gaussian Laser (G.L.", ").The final energy achieved by LWFAs driven by lasers with higher order modes is also connected with the peak laser pulse vector potential $a_{0L}=\\sum _p a^{(p)}$ .", "To illustrate this relation, consider two lasers with identical energy.", "One of them is purely Gaussian with parameters matched to the blowout regime, i.e.", "$W_{0G}=2\\sqrt{a_{0G}}$ , and $P_G \\propto (W_{0G} a_{0G})^2 = 4 a_{0G}^3$ .", "Equation (REF ) then gives $\\Delta E_{G}\\propto a_{0G}$ .", "In the second laser, the intensity profile is redistributed into well defined filaments sufficiently far from each other such that they cannot interact through the plasma.", "Consider in addition that the central laser filament spot-size is matched to the blowout radius, i.e.", "$W_{0L}=2\\sqrt{a_{0L}}<W_{0G}$ , and $P_{L}\\propto (W_{0L} a_{0L})^2 = 4 a_{0L}^3$ (this condition may be satisfied after some propagation in the plasma).", "The corresponding energy gain is then $\\Delta E_{L}\\propto a_{0L}$ .", "As a result, $\\Delta E_L/\\Delta E_{G} \\propto a_{0L}/a_{0G}<1$ .", "This reasoning establishes a direct relation between the maximum energy gain in LWFAs with the peak $a_0$ of lasers with higher order modes.", "Note however, that this is only a special case of Eq.", "(REF ).", "Still, for the case of Fig.", "REF , $a_L = 3.18$ , $a_G=3.85$ , and then $\\Delta E_L/\\Delta E_{G} =0.82$ which is very close to simulation results, and to the prediction of the refined analytical model (generalization of Eq.", "(REF ) up to modes with $p=7$ ).", "In general, however, the full driver laser intensity distribution needs to be considered to predict the energy gain more acuratelly.", "Similarly to Sec.", ", simulations results in this section showed little correlation between the energy spread and the considered laser transverse intensity profiles.", "This is also due to the fact that initially the wake is not beam loaded.", "It is expected that in beam loaded scenarios where the energy spreads are lower, higher order modes will also affect the energy spreads.", "Moreover the wake excitation in the presence of higher order modes may contrast significantly with that associated with a Gaussian laser.", "The wake wavelength, and amplitude may then vary using non-Gaussian lasers.", "This may impact in the beam loading charge, in the acceleration distance, and hence in the final energy spread of the accelerated particles." ], [ "Conclusions", "The influence of non-ideal plasma and laser parameters has been examined through analytical and numerical modeling.", "A reference simulation in a self-guided propagation regime with state-of-the-art parameters served as a baseline to address the most standard deviations to idealized parameters.", "An analytical model was developed to predict the energy gain in non-homogeneous plasma density profiles.", "The analysis showed that the effects of the longitudinal density fluctuations are minimized when the amplitude of the fluctuations is low in comparison to the background plasma density.", "In addition, the density fluctuations can also be neglected when their characteristic wavelength is much shorter than the total acceleration distance.", "The analytical model can be generalized for arbitrary density fluctuations by writing the Fourier series of the plasma density, and performing the required integrations.", "However, based on simulation results using realistic plasma profiles, we expect that this will not change our conclusions.", "The theory also ignored the laser pulse dynamics (self-focusing, self-steepening, pump-depletion, self-compression) except for the laser pulse group velocity.", "Still, it captured the relevant physics and is in very good agreement with PIC simulations in QuickPIC.", "The acceleration of electrons in the wake of drivers with higher order Laguerre-Gaussian modes was also investigated.", "An analytical model was derived to determine the electron energy gain in the wake of a higher order laser pulse.", "The model assumes that only the laser power inside the blowout region can effectively accelerate electrons, but neglects the details of wakefield excitation.", "The theory showed that higher order Laguerre-Gaussian modes generally lead to lower electron energy gains.", "This occurs because the laser pulse energy is typically distributed along wider radial regions in comparison to Gaussian beams.", "These results then emphasize that better quality focal spots lead to optimized acceleration regimes, also in agreement with previous works on self-injection [36].", "However, the parameters of higher order Laguerre-Gaussian laser beams leading to higher energy gains were identified analytically and confirmed in QuickPIC simulations.", "Simulations using transverse laser profiles with larger deviations from Gaussian showed that some laser power initially at $r>W_0$ self-focused, joining the blowout region.", "In these situations the analytical model underestimated electron energy gain.", "Typically, however, the analytical and numerical modeling are in very good agreement.", "It is important to remark that higher-order Gaussian beams may lead to optimized conditions for wake excitation.", "For instance, the ponderomotive force which is associated with the laser-pulse side filaments modifies the properties (thickness and height) of thin electron sheet which surrounds the blowout region [33].", "Since the accelerating gradients are determined by the details of this thin sheet, the use of higher-order Laguerre-Gaussian modes may enhance the accelerating fields of the plasma wave.", "In addition, the use of higher-order Laguerre-Gaussian laser pulses may also lead to more effective self-guiding.", "For instance lasers with transverse super-Gaussian profiles may be more efficiently guided in the blowout." ], [ "Acknowledgments", "The authors acknowledge fruitful discussions with Dr. N. Lopes.", "Work partially supported by FCT (Portugal) through grants SFRH/BD/22059/2005, PTDC/FIS/111720/2009, and CERN/FP/116388/2010, EC FP7 through LaserLab-Europe/Laptech; UC Lab Fees Research Award No.", "09-LR-05-118764-DOUW, the US DOE under DE-FC02-07ER41500 and DE-FG02-92ER40727, and the NSF under NSF PHY-0904039 and PHY-0936266.", "Simulations were done on the IST Cluster at IST, on the Jugene supercomputer under a ECFP7 and a DEISA Award, and on Jaguar computer under an INCITE Award." ] ]

[ [ "The Blanco Cosmology Survey: Data Acquisition, Processing, Calibration,\n Quality Diagnostics and Data Release" ], [ "Abstract The Blanco Cosmology Survey (BCS) is a 60 night imaging survey of $\\sim$80 deg$^2$ of the southern sky located in two fields: ($\\alpha$,$\\delta$)= (5 hr, $-55^{\\circ}$) and (23 hr, $-55^{\\circ}$).", "The survey was carried out between 2005 and 2008 in $griz$ bands with the Mosaic2 imager on the Blanco 4m telescope.", "The primary aim of the BCS survey is to provide the data required to optically confirm and measure photometric redshifts for Sunyaev-Zel'dovich effect selected galaxy clusters from the South Pole Telescope and the Atacama Cosmology Telescope.", "We process and calibrate the BCS data, carrying out PSF corrected model fitting photometry for all detected objects.", "The median 10$\\sigma$ galaxy (point source) depths over the survey in $griz$ are approximately 23.3 (23.9), 23.4 (24.0), 23.0 (23.6) and 21.3 (22.1), respectively.", "The astrometric accuracy relative to the USNO-B survey is $\\sim45$ milli-arcsec.", "We calibrate our absolute photometry using the stellar locus in $grizJ$ bands, and thus our absolute photometric scale derives from 2MASS which has $\\sim2$% accuracy.", "The scatter of stars about the stellar locus indicates a systematics floor in the relative stellar photometric scatter in $griz$ that is $\\sim$1.9%, $\\sim$2.2%, $\\sim$2.7% and$\\sim$2.7%, respectively.", "A simple cut in the AstrOmatic star-galaxy classifier {\\tt spread\\_model} produces a star sample with good spatial uniformity.", "We use the resulting photometric catalogs to calibrate photometric redshifts for the survey and demonstrate scatter $\\delta z/(1+z)=0.054$ with an outlier fraction $\\eta<5$% to $z\\sim1$.", "We highlight some selected science results to date and provide a full description of the released data products." ], [ "Introduction", "Since the discovery of cosmic acceleration at the end of the last millenium [70], [63], understanding the underlying causes has remained as one of the key mysteries in modern astrophysics.", "As the most massive collapsed structures in the universe, galaxy cluster populations and their evolution with redshift provide a powerful probe of, for example, the dark energy equation of state parameter as well as alternate gravity theories which mimic cosmic acceleration [81], [33], [37].", "Evolution of the cluster abundance depends on a combination of the angular-diameter distance vs. redshift relation and the growth rate of density perturbations.", "This sensitivity enables one to constrain a range of cosmological parameters, including the matter density, the sum of the neutrino masses [38], the present day amplitude of density fluctuations, and the presence of primordial non-Gaussianity in the initial density fluctuations [23], [22].", "In addition, galaxy clusters provide an ideal laboratory to study galaxy evolution [26].", "Interesting studies of the galaxy properties and their evolution within clusters include studies of the blue fraction and the halo occupation distribution [14], [48], [49], [47], [34], [84].", "The first large scale attempt to identify and catalog galaxy clusters was by Abell in 1958.", "He discovered galaxy clusters by looking for over-densities of galaxies in Palomar Observatory photographic plates within a radius of about 2.1 Mpc around a given cluster position [2].", "Abell's catalogs contained about 4700 clusters [3].", "However Abell's catalog suffered from incompleteness and contamination from projection effects as well as human bias [10].", "With the advent of CCD cameras one could apply objective automated algorithms to look for galaxy clusters, and this has led to significant progress in cosmological as well astrophysical studies using galaxy clusters.", "In the last decade, many optical photometric surveys such as SDSS, CFHTLS, RCS covering contiguous regions of the sky have discovered several new galaxy clusters spanning a broad range of masses and redshifts.", "The CFHTLS-W [4] has observed about 171 deg$^2$ in $griz$ bands with 80 % completeness up to $i$ band magnitude of 23.", "The RCS-2 [30] survey has covered approximately 1000 deg$^2$ in $grz$ bands with 10$\\sigma $ magnitude depths of around 23.55 in $r$ -band.", "The SDSS MaxBCG catalog  [42] has covered about 7,500 deg$^2$ in $ugriz$ bands and with 10 $\\sigma $ $r$ band magnitude limits of about 22.35.", "The largest optical galaxy cluster survey in terms of area is the Northern Sky Optical Cluster Survey III which has imaged about 11,400 deg$^2$ up to a redshift of about 0.25 [29].", "The deepest optical cluster survey to date is the CFHTLS-D survey [4], which reaches 80 % completeness for $i$ band magnitudes of 26 and has detected clusters up to redshift of 1.5.", "Two upcoming photometric galaxy cluster surveys which will start around October 2012 include DES which will cover about 5,000 deg$^2$ in $grizY$ bands with 10 $\\sigma $ $r$ -band limiting magnitudes of 24.8, and KIDS [25] which will cover 1,500 deg$^2$ in $ugri$ bands with 10$\\sigma $ $r$ -band limiting magnitude of 24.45.", "One can use such surveys for cosmological studies using galaxy clusters.", "For example,  [31] showed that a large optical galaxy cluster survey could constrain cosmological parameters using the self-calibration method [50], [51].", "First cosmological constraints using SDSS optical catalogs are described in  [68].", "Over the last decade there have been several mm-wave cluster studies in the Southern Hemisphere, including ACBAR [65], ACT [28], APEX [32] and SPT [69].", "All these projects have attempted to carry out galaxy cluster surveys using the Sunyaev-Zel'dovich effect (SZE).", "The SZE is the distortion of cosmic microwave background spectrum due to inverse Compton scattering of CMB photons by hot electrons in galaxy clusters [76], and it provides a promising way to discover galaxy clusters.", "Because the surface brightness of the SZE signature of a particular cluster is independent of redshift, SZE survey cluster samples can in principle have sensitivity over a broad range of redshifts [9], [15].", "However, to make use of SZE selected galaxy cluster samples, one needs a well-understood selection of galaxy clusters (sample contamination and completeness), cluster redshift estimates and a link between SZE signature and the cluster halo masses.", "It is important to note that redshift estimates cannot be obtained using SZ experiments alone, and so one needs dedicated optical surveys to follow up these galaxy clusters detected by SZ surveys.", "The Blanco Cosmology Survey (BCS) is an optical photometric survey which was designed for this purpose and positioned to overlap the ACBAR, ACT, APEX and SPT surveys in the southern hemisphere.", "The goal of BCS is to enable cluster cosmology by providing the data to confirm galaxy clusters from the above surveys and to measure their photometric redshifts.", "This was done by surveying two patches totalling $\\sim $ 80 deg$^2$ positioned so that they could be observed with good efficiency over the full night during the period October – December from Chile.", "The BCS observing strategy was chosen to obtain depths roughly two magnitudes deeper than SDSS, so that one could estimate photometric redshifts for $L \\ge L_{*}$ galaxies out to a redshift $z=1$ .", "The outline of this paper is as follows: Sect.", "describes the Blanco Cosmology Survey, including the camera, observing strategy and site characteristics.", "In Sect.", "we describe in detail the processing and calibration of the dataset using the Dark Energy Survey Data Management System.", "In Sect.", "REF , we describe the photometric characteristics of the BCS dataset and present single galaxy photometric redshifts that are tuned using fields containing large numbers of spectroscopic redshifts.", "In this paper all magnitudes refer to AB magnitudes.", "Figure: BCS survey footprint of coadded tiles in the 5 hr and 23 hr fields.", "There are 104 tiles covering ∼\\sim 35 deg 2 ^2 in the 23 hr field and 138 tiles covering ∼\\sim 45 deg 2 ^2 in the 5 hr field for a total coverage of ∼\\sim 80 deg 2 ^2.", "The black vertically hatched boxes represent tiles which have passed our quality checks.", "The red horizontally hatched boxes represent tiles with some data quality problems that we have not corrected." ], [ "BCS survey", "Blanco Cosmology Survey was a NOAO Large Survey project (2005B-0043, PI: Joseph Mohr) which was awarded 60 nights between 2005 (starting from semester 2005B) and 2008 on the Cerro Tololo Inter American Observatory (CTIO) Blanco 4 m telescope using the Mosaic2 imager with $griz$ bands.", "Because of shared nights with other programs, the data acquisition included 69 nights, and the final processed dataset only consists of 66 nights, because two nights were entirely clouded out and the pointing solution for one night (20071105) was wrong due to observer error.", "We now describe the Mosaic2 imager on the Blanco telescope and then discuss the BCS observing strategy." ], [ "Mosaic2 Imager", "The Mosaic2 imager is a prime focus camera on the Blanco 4m telescope that contains eight $2048\\times 4096$ CCD detectors.", "The 8 SITe $2\\mathrm {K}\\times 4\\mathrm {K}$ CCDs are read out in dual-amplifier mode, where different halves of each CCD are read out in parallel through separate amplifiers.", "The CCDs are read out through a single amplifier per chip simultaneously to 8 controller inputs.", "Read noise is about $6-8$ electrons and readout time is about 110 seconds.", "The dark current rate is less than 1 electron/pixel/hour at 90 K. The resulting mosaic array is a square of about 5 inches on an edge.", "The gaps between CCDs are kept to about 0.7 mm in the row direction and 0.5 mm in the column direction.", "Given the fast optics at the prime focus on the Blanco, the pixels subtend $0.27^{\\prime \\prime }$ on the sky.", "Total field of view is 36.8 arc-minute on a side for a total solid angle per exposure of $\\sim $ 0.4 deg$^2$ .", "More details on the Mosaic2 imager can be found in the online CTIO documentationhttp://www.ctio.noao.edu/mosaic/manual/index.html.." ], [ "Field selection and multi-wavelength coverage", "The survey was divided into two fields to allow efficient use of the allocated nights between October and December.", "Both fields lie near $\\delta = -55^{\\circ }$ which allows for overlap with the SPT and other mm-wave surveys.", "One field is centered near $\\alpha = 23.5$  hr and the other is at $\\alpha $ = 5.5 hr.", "The 5 hr 30 min $-52^{\\circ }$ patch consists of a $12\\times 11$ array of Blanco pointings and the 23 hr $-55^{\\circ }$ patch is a $10\\times 10$ array of pointings.", "The 5 hr field lies within the Boomerang field where the ACBAR experiment took data.", "The 23 hr field has been observed by the APEX, ACT and SPT experiments.", "In addition to the large science fields, BCS also covers nine small fields that overlap large spectroscopic surveys so that photometric redshifts using BCS data can be trained and tested using a sample of over 5,000 galaxies with spectroscopic redshifts.", "BCS also surveyed standard star fields for photometric calibration.", "The coverage of BCS in 5 hr and 23 hr fields is shown in Figs REF .", "For convenience of data processing and building catalogs, we divide the survey region into $36^{\\prime } \\times 36^{\\prime }$ square regions called tiles.", "Each tile is a $8192\\times 8192$ pixel portion of a tangent plane projection.", "These tiles are set on a grid of point separated by $34^{\\prime }$ , allowing for approximately $1^{\\prime }$ overlaps of sky between neighboring tiles.", "The black vertical hatches in Fig.", "REF indicate locations of tiles which passed various quality checks.", "The red horizontal hatches indicate locations of tiles which were observed and processed, but failed data quality checks.", "We also secured other multi-wavelength observations overlapping parts of the BCS fields.", "About 14 deg$^2$ of the 23 hr BCS field was surveyed using XMM-Newton (known as XMM-BCS survey) and results from those observations are reported elsewhere [78].", "An $\\sim $ 12 deg$^2$ region of the same field was also targeted in a Spitzer survey (S-BCS).", "More recently, the XMM-Newton survey has been expanded to 25 deg$^2$ , and the Spitzer survey has been expanded to 100 deg$^2$ .", "Most of the BCS region has been observed in the near-infrared as part of the the ESO VISTA survey program [16].", "Figure: Redshift evolution of a passively evolving L * L_{*} galaxy along with target 10σ\\sigma photometric BCS depths in each band.", "The exposure times in each band were tuned so that photometric depth meets or exceeds L * L_{*} out to the redshift where the 4000Åbreak shifts out of that band, but also limited to z=1z=1 due to the low sensitivity of the Mosaic2 camera in the zz-band." ], [ "Observing Strategy", "The BCS observing strategy was designed to allow us to accurately measure cluster photometric redshifts out to redshift $z=1$ .", "Because the 4000Å break is redshifting out to 8000Å by $z=1$ , obtaining reliable photo-z's for $0< z<1$ requires all four photometric bands $g$ , $r$ , $i$ , $z$ (i.e.", "one loses all clusters at $z<0.4$ if you drop $g$ -band and with $z$ -band we can actually push beyond $z=1$ ).", "The redshift at which the 4000Å break redshifts beyond a particular band sets, crudely speaking, the maximum redshift for which that band is useful for cluster photo-z's; for $griz$ this is $z=0.35$ , 0.7, 1.0 and 1.4, respectively.", "Because the central wavelength of the g-band is about 4800Å  with a FWHM of 1537Å  we start losing sensitivity to very low redshift clusters, because it is not possible to straddle the 4000Å break.", "Although detailed studies of the sensitivity of optical cluster detection at low redshifts have not been done, our ability to estimate unbiased red sequence redshifts for clusters is reduced below redshifts $z\\sim 0.1$ .", "We calculate our photometric limits in each band by requiring that the depth allows us to probe at least to $L_*$ at that maximum redshift with 10$\\sigma $ photometry.", "We use a Bruzual and Charlot $z=3$ single burst model with passive evolution [12] to calculate the evolution of $L_*$ in the four bands (see Fig.", "REF ).", "We select our $z$ depth to probe to $L_*$ at $z=1$ rather than at $z=1.4$ , because of the low sensitivity of the Mosaic2 detectors in the $z$ -band.", "The survey was designed to reach 10$\\sigma $ photometric limits within a 2.2 arcsec aperture of $g=24.0$ , $r=23.9$ , $i=23.6$ and $z=22.3$ .", "These limits assume an airmass of 1.3 and $0.9^{\\prime \\prime }$ median seeing for all bands.", "Assuming bright time for $z$ and $i$ and dark time for $g$ and $r$ these limits require exposures of 250 s, 600 s, 1400 s and 700 s in $griz$ , respectively.", "In all, we observed about 288 tiles spanning our survey fields.", "For each field we typically took 2 exposures in $g$ of 125 secs each, 2 exposures in $r$ of 300s each, 3 exposures in $i$ of 450s each and three exposures in $z$ of 235 seconds each.", "A limitation of the Mosaic2 detector is a very low saturation of around 25,000 ADU for most of the detectors, and this forced us to take short exposures even through the readout time for each was quite high.", "Neighboring pointings have small overlaps, but multiple exposures were offset by approximately half the width of an amplifier to help us tie the survey together photometrically.", "Having two shifted exposures allows us to largely overcome the gaps in our survey left by spaces between neighboring chips.", "In addition to this primary survey tiling, we also constructed another layer of tilings which was designed to sit at the vertices of unique groups of four adjacent primary pointings.", "These tiles were observed using shorter exposures during poor seeing conditions on photometric nights.", "The 110 s readout of the Mosaic2 camera makes the efficiency of short exposures low, and so in each band we have chosen the minimum number of exposures allowable given the sky brightness.", "The total exposure per tile is 3000 s and after including the readout time, the total observation time per science field is about 4200 s, giving us an overall efficiency of about 70%.", "The dome flats and bias frames were taken in the afternoon, and we did not take any twilight flats.", "Over the course of the survey we acquired just over 3000 science exposures and an additional 455 photometric overlap exposures.", "In addition to science exposures, on photometric nights we also observed photometric calibration fields as well as fields for calibrating our photo-z algorithms.", "These fields were CNOC2, DEEP, CFRS, CDFS, SSA22 and VVDS fields.", "For the photometric calibration fields we typically observed two or three fields during evening and morning twilight and a single field during the transition from the 23 hr field to the 5 hr 30 min field.", "We observed in all four bands during these calibration exposures.", "The spectroscopic standard fields were observed to full science depth using the same strategy as for the full survey.", "Figure: Sky brightness distributions for all four bands averaged on a per exposure basis during the BCS survey.", "Typically we observed in gg and rr during dark time and ii and zz during bright time.", "The brightness valuesare peaked at around 22.5, 21.5, 20.5, and 18.75 mag/arcsec 2 ^2 in grizgriz bands, respectively." ], [ "Site Characteristics", "The BCS survey provides a sampling of the CTIO site characteristics over a 69 night period in the October to December timeframe over four observing seasons.", "Because this is the same timeframe planned for DES observations this provides an interesting glimpse into the expected site characteristics for DES.", "Given that the entire Mosaic2 camera and wide field corrector are being replaced by DECam and the new DECam wide field corrector [73], the seeing distribution for the DES data could be significantly improved relative to the BCS seeing distribution.", "The seeing distribution is shown in top panel of Fig.", "REF .", "The seeing was obtained by running PSFEX software on all single-epoch images and using the FWHM_MEAN parameter.", "The FWHM_MEAN is derived from elliptical Moffat fits to the non-parametric PSF models.", "These FWHMs include the pixel footprint.", "The modal seeing values integrated over the survey are $\\simeq 1^{\\prime }$ , $0.95^{\\prime }$ , $0.8^{\\prime }$ , $0.95^{\\prime }$ for $griz$ bands, respectively.", "The median seeing values are 1.07, 0.99, 0.95, and 0.95 arcseconds, while the upper and lower quartile seeing values are [0.96, 1.26], [0.89, 1.16], [0.84, 1.13], [0.83, 1.11] arcseconds respectively.", "The sky brightness is shown in Fig.", "REF .", "The sky brightness is calculated using $ZP- 2.5\\log {B}$ , where $ZP$ is the calculated zeropoint for that image and B is the sky brightness in ADU/arcsec$^{2}$ .", "The sky brightness distributions in the $griz$ bands have modal values of approximately 22.5, 21.5, 20.5 and 18.75 mag/arcsec$^2$ , respectively.", "Moreover, almost all $i$ and $z$ band data were taken with the moon up, while almost all $g$ and $r$ band data were taken with the moon set.", "The median values are 22.3, 21.3, 20.3 and 18.7 mag/arcsec$^2$ , respectively.", "Given the division of the survey into a 23 hr and a 5 hr field, it was possible to obtain most of the data at relatively low airmass.", "Fig.", "REF shows the airmass distributions for each band during primary survey observations.", "We often obtained photometric calibration field observations over a broader range of airmasses, but we tried to restrict our primary survey observations to air masses of $<$ 1.6.", "The median air mass in bands $griz$ are 1.144, 1.147, 1.138 and 1.141 respectively.", "Figure: The airmass distributions for BCS exposures, color coded by band and normalized by total number of exposures.", "The peak airmass values in grizgriz bands are 1.144, 1.147, 1.138 and 1.141 respectively." ], [ "Data Processing and Calibration", "The processing of BCS data is carried out using the automated Dark Energy Survey data management (DESDM) system which has been under development since Fall 2005 at University of Illinois [60], [57].", "DESDM will be used to process, calibrate and store data from the Dark Energy Survey once it begins operations in October 2012.", "Since 2005, the DESDM system has been validated through a series of data challenges with simulated DECam data, which enabled us to improve various steps of the pipeline.", "The same automated pipeline was used to analyze BCS data.", "The only addition/change to the DESDM pipeline to analyze BCS data was in the crosstalk correction code, for which the routine had to be customized for the Mosaic2 camera.", "Processing of the BCS data presented here has been carried out on National Teragrid resources at NCSA and LONI supercomputers together with dedicated workstations needed for orchestrating the jobs and hosting the database.", "The middleware for the data reduction pipeline is designed using Condor batch processing system.", "Each night takes about 300 CPU hours for processing.", "We have processed BCS data multiple times in a process of discovery where we found problems with the data that required changes to our system.", "Scientific results from earlier rounds of processing of BCS data have already appeared, including the optical confirmation of the first ever SZE selected galaxy clusters [75] and the discovery of strong gravitational lensing arc using data from the firstå round of processing in Spring 2008 [13].", "Additional galaxy cluster science arising from subsequent rounds of BCS processing have also been published [35], [84], [78].", "Currently our latest processing is being used for additional SZE cluster science within SPT, continued studies of the X-BCS region, and for the followup of the broader XMM-XXL survey over the 23 hr field.", "The BCS data were made public one year after their acquisition, as is standard policy at NOAO.", "This has enabled multiple independent teams to access the data and use it for their own scientific aims.", "The first three seasons of BCS data were processed using an independent pipeline developed at Rutgers University [55], [54].", "All four seasons of BCS data have also been processed independently using NOAO pipeline as part of the current automated processing program, and with the PHOTPIPE analysis pipeline [67]." ], [ "Detrending", "In this section we describe in detail the key steps involved in the DESDM pipeline used for reduction of Mosaic2 data to convert raw data products to science ready catalogs and images.", "Data from every night are processed through a nightly processing or detrending pipeline.", "Then data from different nights in the same part of the sky are combined using the co-addition pipeline.", "The detrending pipeline briefly consists of crosstalk corrections, overscan, flatfield, bias and illumination correction, astrometric calibration and cataloging.", "We now describe in detail each step of the detrending pipeline.", "A common feature of multi-CCD cameras, such as Mosaic2 imager, is crosstalk among the signals from otherwise independent amplifiers or CCDs.", "This leads to a CCD image containing not only the flux distribution that it collected from the sky, but also a low amplitude version of the sky flux distributions that appear in other CCDs.", "The crosstalk correction equation is described by: $I_{i} = \\sum _{j=1}^N \\alpha _{ij} I_\\mathrm {raw}^{j} ;$ where $I_i$ denotes the crosstalk corrected image pixel value in $i^\\mathrm {th}$ CCD, $\\alpha _{ij}$ denote the cross-talk coefficients and $I_\\mathrm {raw}^{j}$ is the raw image pixel value.", "We used cross-talk coefficients provided by NOAO through the survey.", "As part of the crosstalk-correction stage, the raw image (which contains 16 extensions) is split into one single-extension file per CCD.", "The processing and calibration of CCD mosaics can proceed independently for each CCD after the crosstalk correction, and therefore we split the images to enable efficient staging of the data to the compute resources." ], [ "Image Detrending", "Detrending is the process that removes the instrumental signatures from the images.", "Detrending, in this context, includes overscan correction, bias subtraction, flat fielding, pixel-scale correction, fringe and illumination correction.", "Both the overscan correction and bias correction are required to remove the bias level present in the CCD and any residual, recurrent structure in the DC bias.", "Overscan correction is done for all raw science and calibration images.", "We subtract the median pixel value in the overscan region in each row for both the amplifiers in each CCD from the raw image pixel values after the crosstalk correction stage.", "The median bias frame is created using nightly bias frames taken during the late afternoon, and subtracted from the nightly data.", "The flat field correction is typically derived from dome flats taken for each observing band.", "The input dome flat images are overscan corrected, bias corrected, and then scaled to a common mode and then median combined.", "The resulting flat field correction is scaled by the inverse of the image mode, creating a correction with mean value of about unity.", "For the bias correction and the flat correction the variation among the input images is used to create an inverse variance weight map that is stored as a second extension in the correction images.", "The creation of correction images also requires a bad pixel map, which is an image where pixels with poor response or with high dark current are masked and excluded from the images.", "These bad pixel maps are created initially using bias correction and flat field correction images to identify the troublesome pixels.", "The bias and flat field corrections are then applied to the science images to remove pixel to pixel sensitivity variations.", "These corrections are only applied to those science pixels that are not masked.", "In this process, each science image receives an associated inverse variance weight map that encodes the Poisson noise levels and Gaussian propagated noise from each correction step on a per pixel basis.", "In addition, each science image has an associated bad pixel map (short integer) where a bit is assigned to each type of masking (i.e.", "pixels masked from the original bad pixel map, or masked due to saturation, cosmic ray, etc).", "In our data model the science image has three extensions: image, weight map, bad pixel map.", "Each measured flux at the pixel level comes along with its statistical weight and a history of any masking that has been done on that pixel.", "For the Mosaic2 imager which has significant focal plane distortion, the pixel scale varies significantly over the field, leading to a significant trend in delivered pixel brightness as a function of position even with a flat input sky.", "For such detectors flattening the sky introduces a photometric non-flatness to the focal plane.", "Typically this pixel scale variation is corrected during the process of remapping to a portion of a tangent plane, but in our case we prefer to do the single epoch cataloging on images that do not suffer from correlated noise.", "Therefore, we apply a pixel scale correction to account for variation of pixel response as a function of $x$ and $y$ position for each CCD.", "We first created master template images to determine photometric flatness corrections using astrometrically refined images from the Mosaic2 camera that we use to calculate the solid angle of each pixel.", "The correction image is then normalized by the median value, providing a flat field like correction image that can be used to bring all pixels to a uniform flux sensitivity.", "To avoid reintroducing trends in the sky with this correction, we apply this correction only to the values of each pixel after subtraction of the modal sky value.", "Effectively, this correction scales only source flux while maintaining a flat sky.", "Illumination and fringe corrections are derived from fully processed science observations in a particular band.", "These can be from a single night or shared across nights.", "Usually if there was only one exposure from a given band in a night we use science observations from neighbouring nights to create the illumination and fringe correction images.", "Illumination corrections are done for all images, but fringe corrections on the Mosaic2 camera are needed only for $i$ and $z$ bands.", "To create these correction images, we first create sky flat templates.", "This requires a process of stacking all the detrended images in a band-CCD combination after first flagging all pixels contaminated by source flux.", "Source contaminated pixels are determined by applying a simple threshold above background with a variable grow radius so that all neighboring pixels of a pixel determined to contain source flux are also masked.", "Modal sky values are then calculated for each image using pixels that are not flagged for any reason (object pixels, hot column, saturated, interpolated, etc).", "The reduced images are then scaled to a common modal sky value, median combined and then rescaled to a unit modal value.", "This science sky image then contains a combination of any illumination and fringe signatures that are common to the input images.", "To create the illumination correction we adaptively smooth the science sky images with a kernel that is large in the center and grows smaller near the edges.", "This effectively averages out the effects of any fringing, leaving an illumination correction image behind.", "The fringe correction is then produced by first differencing the science sky image and the illumination correction image, leaving behind an image of the small scale structure (i.e.", "fringe signature) that is common to all the science images.", "This fringe correction image is then scaled by the model value of the science flat image to produce a fractional fringe correction image.", "The illumination correction image is applied like a flat field correction to all previously corrected images, thereby removing any trends that are introduced by the differences in illumination of the dome flats and the flat sky.", "The fringe correction is applied by first scaling the correction image by the modal value of the sky in the science image and then subtracting it.", "The results of these two corrections are visually very impressive.", "The fringe effects in $i$ and $z$ band are nicely removed in almost all cases.", "We have found some problem images where the fringe correction leaves clearly visible fringe signatures, and these are cases where only a few frames in $i$ or $z$ were taken on a particular night, and the use of images from neighboring nights to create the corrections was not adequate.", "We expect that the residual scatter we measure could be further reduced using a star flat technique to better characterize the non-uniformities in the pupil ghost.", "Nevertheless, the delivered data quality from our current flattening prescription produces data that meet our data quality requirements.", "We note that the same prescription has been used previously to meet the data quality requirements of the SuperMACHO experiment in the processing of Mosaic2 data.", "At the end of this series of image detrending steps which includes overscan, bias, flat-field, pixel-scale, illumination and fringe corrections, the pipeline creates eight images (one for each CCD) for every science exposure.", "These single epoch image FITS files are called red images, and they contain 3 extensions: the main image, a bad pixel mask (BPM) and an inverse variance weight image.", "The BPM contains a short integer image where any unusable pixels have non-zero values (coded according to the source of the problem).", "The weight-map is an inverse variance image map that tracks the noise on the pixel scale and where the weight is set to zero for all masked pixels.", "Figure: Distortion map produced by SCAMP for one Mosaic2 exposure consisting of 8 images.", "TPV distortion model was used.", "The Mosaic2 distortions were modelled for each CCD by expressing distortions along the RA and DEC direction each with a third order polynomial in CCD xx and yy." ], [ "Astrometric Calibration", "Besides pointing errors, wide-field imagers exhibit instrumental distortions that generally deviate significantly from those of a pure tangential projection.", "In addition, the vertical gradient of atmospheric refractivity creates a small image flattening of the order of a few hundredths of a percent (corresponding to a few pixels on a large mosaic), with direction and amplitude depending on the direction of the pointing.", "These three contributions are modeled in the SCAMP [6] package that we use for astrometric calibration.", "SCAMP uses the TPV distortion modelcurrently under review for inclusion in the registry of FITS conventions; see http://fits.gsfc.nasa.gov/registry/tpvwcs.html., which maps detector coordinates to true tangent plane coordinates using a polynomial expansion.", "SCAMP is normally meant to be run on a large set of SExtractor catalogs extracted from overlapping exposures together with a reference catalog, in order to derive a global solution.", "However, since our pipeline operates on an image-by-image basis, we proceed in two steps: we first run SCAMP once on a small subset of catalogs extracted from BCS mosaic images to derive an accurate polynomial model of the distortions where the distortions in RA/DEC tangential plane are expressed as a third degree polynomial function of the CCD $x$ /$y$ position.", "This Mosaic2 distortion map modeled using a third order polynomial per CCD for a BCS exposure is shown in Fig.", "REF .", "The astrometric solution computed in this first step of calibration is based on a set of overlapping catalogs from dithered exposures which provides tighter constraints on non-linear distortion terms (than catalogs taken individually).", "Using this model, we create a distortion catalog that encodes the fixed distortion pattern of the detector.", "We then run SCAMP on catalogs from each individual exposure (i.e.", "the union of the catalogs from each of the eight single epoch detrended images), allowing only linear terms (two for small position offsets and four for the linear distortion matrix) describing the whole focal plane to vary from exposure to exposure.", "The solutions for the World Coordinate System (WCS) including the TPV model parameters are then inserted back into image headers.", "This approach capitalizes on the expected constancy of the instrumental distortions over time.", "Figure: Median value of the difference in RA and DEC for objects in BCS coadd catalogs vs USNOB catalogfor every tile in arcsecs.", "The matching is done in a 2 '' 2^{\\prime \\prime } window.", "The histograms are peaked at ∼0.0104 '' \\sim 0.0104^{\\prime \\prime } and 0.0084 '' 0.0084^{\\prime \\prime } in Δ\\Delta -RA and Δ\\Delta -DEC respectively.", "The rms of the histograms in RA and DEC is about 0.047 and 0.045 arcsecs.", "Note that the intrinsic accuracy of the USNOB catalog is about 0.2 arcsecs .We use the USNO-B1 [58] catalog as the astrometric reference.", "For astrometric refinement, the cataloging is done using SExtractor, and using WINdowed barycenters to estimate the positions of sources.", "The astrometric accuracy is quite good, as can be demonstrated with the BCS coadds.", "First, the accuracy is at the level of a fraction of a PSF or else significant PSF distortions would appear in the coadds, and this is not the case.", "Second, we can measure the absolute accuracy relative to the calibrating catalog USNOB by probing for systematic offsets in RA or DEC between our object catalogs and those from the calibration source.", "Fig.", "REF shows the distribution of median offsets within all the coadd tiles for both RA and DEC.", "The mean of the histograms is $0.0104^{\\prime \\prime }$ in RA and $0.0084^{\\prime \\prime }$ in DEC, and the corresponding rms scatter is 47 milli-arcsec and 45 milli-arcsec, respectively.", "The USNOB catalog itself has an absolute accuracy with characteristic uncertainty of 200 milli-arcseconds [58], which then clearly dominates the astrometric uncertainty of our final catalogs." ], [ "Single-Epoch Cataloging", "To catalog all objects from single-epoch images we run SExtractor using PSF modeling and model-fitting photometry.", "A PSF model is derived for each CCD image using the PSFEx package [7].", "PSF variations within the each CCD are modeled as a $N^\\mathrm {th}$ degree polynomial expansion in CCD coordinates.", "For our application we adopt a $26\\times 26$ pixel kernel and follow variations to 3$^\\mathrm {rd}$ order.", "An example of variation of the Full-Width at Half Maximum (FWHM) of the PSF model across a single-epoch image is shown in Fig.", "REF .", "The FWHM varies at the 10% level across this CCD due to both instrumental and integrated atmospheric effects.", "lr Sextractor Detection Parameters 0pt Parameter Values DETECT TYPE CCD DETECT MINAREA 5 DETECT THRESH 1.5 ANALYSIS THRESH 1.5 FILTER Y FILTER NAME gauss 3.0 3x3.conv DEBLEND NTHRESH 32 DEBLEND MINCONT 0.005 CLEAN Y CLEAN PARAM 1.0 BACKPHOTO THICK 24.0 A new version of SExtractor (version 2.14.2) uses this PSF model to carry out PSF corrected model fitting photometry over each image.", "The code proceeds by fitting a PSF model and a galaxy model to every source in the image.", "The two-dimensional modeling uses a weighted $\\chi ^2$ that captures the goodness of fit between the observed flux distribution and the model and iterates to a minimum.", "The resulting model parameters are stored and “asymptotic” magnitude estimates are extracted by integrating over these models.", "This code has been extensively tested within the DESDM program on simulated images, but the BCS data provide the first large scale real world test.", "For the BCS application we adopt a Sérsic profile galaxy model that has an ellipticity and orientation.", "This model fitting is computationally intensive and slows the “lightning-fast” SExtractor down to a rate on the order of 10 objects/s on a single core.", "The SExtractor config file detection parameters are shown in Table REF .", "Figure: Variation of the PSF model FWHM for gg-band across a single-epoch image from the BCS night 20061030.", "Variations across the roughly 10 ' ×20 ' 10^{\\prime }\\times 20^{\\prime } image are at the 10% level.The advantages of model fitting photometry on single epoch images that have not been remapped are manifold.", "First, pixel to pixel noise correlations are not present in the data and do not have to be corrected for in estimating measurement uncertainties.", "Second, unbiased PSF and galaxy model fitting photometry is available across the image, allowing one to go beyond an approximate aperture correction to aperture magnitudes often used to extract galaxy and stellar photometry.", "Third, there are morphological parameters that can be extracted after directly accounting for the local PSF, which allows for improvements in star-galaxy classification and the extraction of PSF corrected galaxy shear.", "A more detailed description of these new SExtractor capabilities along with the results from an extensive testing program within DESDM will appear elsewhere (Bertin et al, in preparation)." ], [ "Remapping", "From the WCS parameters which are computed for every reduced image, one can approximate the footprint of the CCD on the sky using frame boundaries in Right Ascension and Declination.", "For the BCS survey we have a predefined grid of $36^{\\prime }\\times 36^{\\prime }$ tangent plane tiles covering the observed fields.", "Based on this, for every red image which is astrometrically calibrated, we determine which tiles it overlaps.", "We then use SWarp [8] to produce background-subtracted remapped images that conform to sections of these tangent plane tiles.", "A particular red image can be remapped to up to four different remap images in this process.", "Pixels are resampled using Lanczós-3 interpolation.", "Remapping also produces a pixel weightmap and we also remap the bad pixel map (using nearest neighbor remapping).", "In this process of remapping, zero weight pixels in the reduced images generically impact multiple pixels in the remap image given the size of the interpolation kernel.", "These remaps are then stored for later photometric calibration and coaddition.", "This on-the-fly remapping need not be done, because at a later stage of coaddition one could in principle return to the red images, but given the PSF homogenization we do prior to coaddition we have found it convenient to do the remapping as we are processing the nightly datasets." ], [ "Nightly Photometric Calibration", "Our initial strategy for photometric calibration involved traditional photometric calibration using the standard fields observed on photometric nights along with the image overlaps to create a common zeropoint across all our tiles.", "In fact, within DESDM we have developed a so-called Photometric Standards Module [77] (PSM) that we use to fit for nightly photometric solutions, and then we apply those solutions to all science images and associated catalogs from that night.", "For BCS this involves determining the zeropoints of all images on photometric nights through calibration to identified non-variable Standard stars from SDSS Stripe 82 field [72].", "This procedure was used for processing and calibration of the BCS data processed in Spring 2008.", "But closer analysis of these data showed that we were not able to control photometric zeropoints to the required level to allow for cluster photometric redshifts over the full survey area.", "We therefore abandoned this method for BCS in favor of relative photometric calibration using common stars in overlapping red images followed by absolute calibration using the stellar locus (described in more detail in Sec.", "REF ).", "One problem we faced is that so-called photometric nights exhibited non-photometric behavior in the standard field observations.", "There was no reliable photometric monitor camera at CTIO during our survey, and so observers simply used the time honored tradition of watching for clouds to make the call on a night being photometric.", "Because of our strategy for standard star observations (beginning, middle, end of night), even those nights that exhibit consistent photometric solutions need not have been photometric over the full nights.", "Therefore, we felt it safer to assume that no night was truly photometric and to calibrate the data using an entirely different approach.", "The results from the PSM module for those nights exhibiting good photometric solutions are still useful.", "They have allowed us to monitor changes in the detectors and to measure the color terms in transforming our photometry onto the SDSS system.", "We provide a brief description of this procedure, although no science results in this paper are based on PSM related direct photometric calibration.", "We expect to apply this method for absolute photometric calibration of DES data where we will indeed have an IR photometric monitoring camera on the mountain.", "The PSM solves for the following equation : $m_\\mathrm {inst} - m_\\mathrm {std} = a_n + b_n \\times (stdColor - stdcolor_0) + kX$ where $a_n$ is the photometric zeropoint for all 8 CCDs, $b_n$ is the color term, $stdColor$ is the fiducial color around which we define our standard solutions, which is $g-r$ for $g$ and $r$ bands, and $r-i$ for $i$ and $z$ bands, $stdcolor_0$ is a constant equal to $g-r=0.53$ for $g$ and $r$ bands and $r-i=0.09$ for $i$ and $z$ bands, $k$ is the first-order extinction coefficient and $X$ is the airmass.", "The PSM module solves for $a_n$ , $b_n$ and $k$ for each photometric night.", "Using these values for the PSM $a$ , $b$ and $k$ , one can also estimate the expected zeropoint for every exposure.", "We calculate it as follows $ZP = -a + 2.5 \\log \\mathrm {(exptime)} -kX$ We applied the PSM on about 30 BCS nights which were classified as photometric.", "We also checked for trends in variation of color terms as a function of CCD number.", "Only the $i$ -band color term shows some variation, and this approximate constancy of color terms greatly simplifies the coaddition of the data, because we don't have to track which CCDs have contributed to each pixel on the sky.", "The color terms we have used for photometric calibration are -0.1221, -0.0123, -0.1907, and 0.0226 in $griz$ respectively.", "We also examine the band dependent extinction coefficient ($k$ ) calculated using data from the photometric nights.", "For the ensemble of about 30 photometric solutions in each band, we find the median $griz$ extinction coefficients at CTIO over the life of the survey to be 0.181, 0.104, 0.087 and 0.067 mag./airmass respectively.", "This completes the description of all the steps of the nightly processing or single-epoch processing that we do for BCS." ], [ "Coaddition", "Once we have data processed for all of the BCS nights, we then combine data within common locations on the sky to build deeper images that we call coadds.", "This process is called co-addition and is complicated because it involves combining data taken in widely separated times and under very different observing conditions.", "Co-addition processing is done on a tile by tile basis.", "We describe our approach below." ], [ "Relative Photometric Calibration", "During single-epoch processing we extract instrumental magnitudes.", "To produce science ready catalogs, we must calculate the zeropoint for every image and re-calibrate the magnitudes.", "The photometric calibration is done in two steps.", "The first step is a relative zeropoint calibration that uses the same object in overlapping exposures, and the second is an absolute calibration using the stellar locus.", "The relative calibration is done tile by tile rather than simultaneously across the full survey.", "We use two different pieces of information to calculate the relative zeropoints.", "The primary constraint comes from the average magnitude differences from pairs of red images with overlapping stars.", "The stars are selected based on the SExtractor flags and spread_model (discussed later in Sec.", "REF ) values.", "In cases where there aren't enough overlapping stars, we use the average CCD to CCD zeropoint differences derived from photometric nights.", "In previous versions of the reduction we also used direct zeropoints derived from photometric nights (see Sec.", "REF ) and relative sky brightnesses on pairs of CCDs.", "As previously mentioned, the direct photometric zeropoint information is contaminated at some level.", "The sky brightness constraints also seem to be problematic for BCS, because only $g$ and $r$ band data were taken on dark nights with no moon present which can introduce a gradient across the camera.", "To avoid a degradation of the calibration we used neither the sky brightness constraints nor the direct photometric zeropoints.", "We determine the zeropoints for all images in a tile by doing a least squares solution using the inputs described above.", "For this least squares solution there are $N$ input images, each with an unknown zeropoint in the vector $z$ .", "We arbitrarily fix the zeropoint for one image and calibrate the remaining images relative to it.", "We have $M$ different constraints in the constraint vector $c$ .", "The matrix $A$ is $N \\times M$ and denotes the images involved for each constraint.", "The resulting system of equations is described by $Az = c$ where we use singular value decomposition to solve for the vector $z$ .", "This gives the relative zeropoints needed to coadd the data for a particular tile." ], [ "PSF Homogenization", "Combining images with variable seeing generically leads to a PSF that varies discontinuously over the coadded image.", "This affects star galaxy separation and contributes to variation across the image in the completeness at a given photometric depth.", "The PSF accuracy could be quite poor in regions where there are abrupt changes to the PSF which would translate into biases in the photometry that would be difficult to track.", "The main steps involved in the process of PSF homogenization include: (1) modeling the PSF using PSFEx for all remap images contributing to a coadd tile, (2) choosing the parameters of the target PSF, (3) using PSFEx to generate the homogenization kernel, and (4) carrying out the convolution to homogenize all the remap images to a common PSF.", "To reduce PSF variation we processed our images to bring them to a common PSF within an image and from image to image within a coadd tile.", "To do this we apply position dependent convolution kernels that are determined using power spectrum weighting functions that adjust the relative contributions of large scale and small scale power within an image in such a way as to bring the PSFs within and among the image samples into agreement.", "The target PSF is defined to be a circular Moffat function with the FWHM set to be the same as the median value of all input PSFs.", "$\\chi ^2 = \\vert \\Psi - \\sum _{l}Y_l(x_i)\\kappa _l\\ast \\Psi _\\mathrm {median} \\vert ^2$ where $Y_l$ are the elements of a polynomial basis in $x-y$ .", "The target PSF is defined to be a circular Moffat function with the FWHM set to be the median FWHM of the input images.", "We imposed a cut on input image PSF $\\mathrm {FWHM}<1.6$  arcsec.", "This selects only images with relatively good seeing.", "Images from each band are homogenized separately.", "The FWHM of the target PSF for all BCS tiles is shown in bottom panel of Fig.", "REF .", "Figure: FWHM of single epoch images using PSFex (top panel) along with the target PSF FWHM used for homogenizingthe coadd images for the full BCS survey (bottom panel).", "The peak values of target PSFs are about 1 '' 1^{\\prime \\prime } for gg and rr bands, 0.9 '' 0.9^{\\prime \\prime } for ii-band and 0.8 '' 0.8^{\\prime \\prime } for zz-band respectively.Another price of homogenization is that noise is correlated on the scale of the PSF.", "While the noise is already correlated to some degree through the remapping interpolation kernel, PSF homogenization characteristically affects larger angular scales than does the remapping kernel.", "This leads to biases in photometric and morphological uncertainties, and can also affect initial object detection process in SExtractor.", "To address this within DES we account for the noise correlations on two critical scales by producing two different weight maps.", "The first weight map is used to track the pixel scale noise, and the second weight map is used to correct for the correlated noise on the scale of the PSF.", "The pixel scale weight map is used by SExtractor in determining photometric and morphological uncertainties.", "The PSF scale weight map is used by SExtractor in the detection process.", "Extensive tests within DESDM have shown this approach to be adequate to produce unbiased photometric and morphological uncertainties and to enable unbiased detection of objects within coadds built from homogenized images.", "These results will be presented in detail elsewhere.", "For the BCS processing we used only a single pixel-scale weight map, tuned to return the correct measurement uncertainties within SExtractor.", "Figure: Variation of the PSF model FWHM for a gg-band coadd image for the coadd tile BCS0516-5441.", "Because of the homogenization process the variations are at the level of 1% across the 36 ' 36^{\\prime } image." ], [ "Stacking Single Epoch Images", "We use SWarp to combine the PSF-homogenized images to build the coadd tile.", "Inputs include the relative flux scales derived from the calibration described in Sec.", "REF ).", "We combine the homogenized remap images using the associated weight maps and bpm for each image.", "The values of the flux-scaled, resampled pixels for each image are then median combined to create the output image.", "This allows us to be more robust to transient features such as cosmic rays in the $i$ and $z$ bands where there are three overlapping images.", "Also, objects with saturated pixels in all single epoch images will contain pixels that are marked as saturated in the coadd images as well.", "This ensures accurate flagging of objects with untrustworthy photometry during the coadd cataloging stage.", "The resulting output coadd image's size is $8192\\times 8192$ pixels or approximately $0.6 \\times 0.6$ degrees.", "Fig.", "REF shows a map of the FWHM as a function of position over one homogenized coadd image.", "Variations are at the level of $\\sim $ 1% over the coadd, as compared to the $\\sim $ 10% variations that are typical for Mosaic2 across a single CCD (see Fig.", "REF ).", "The constancy of PSF as a function of a position ensures that the PSF model can be modeled accurately and that the PSF corrected model fitting photometry is unbiased.", "The PSF homogenization process also circularizes the PSF.", "Fig REF shows the distribution of ellipticities for the Mosaic2 single epoch images and coadded images (color coded by band).", "The single epoch ellipticity varies up to 0.1 with a modal value around 0.02.", "By contrast, the ellipticity distribution of the BCS coadds is peaked at a fraction of a percent with a median value of 0.001.", "Figure: Mean Ellipticity calculated by PSFEx for single-epoch images (top panel) and for PSF homogenized coadds (bottom panel), color coded by band.", "Ellipticity is defined as (a-b)/(a+b){(a-b)}/{(a+b)} where aa and bb refer to semi-major and semi-minor axis respectively.", "For single-epoch images the median ellipticty for grizgriz bands is 0.0342, 0.0326, 0.0374 and 0.04 respectively.", "For coadds, typical values are around 0.004, 0.0024, 0.0026, 0.0033." ], [ "Cataloging of Coadded images", "To catalog the objects from coadded images, we run SExtractor in dual-image mode with a common detection image across all bands.", "For BCS we use the $i$ -band image as the detection image, because it has three overlapping images so the cosmic ray removal is good, and it is by design the deepest of the bands.", "We then run SExtractor using model-fitting photometry using this detection image and coadded image in each band.", "This ensures that a common set of objects are cataloged across all bands.", "In both single epoch and coaddition cataloging, the detection criterion was that a minimum of 5 adjacent pixels had to have flux levels about 1.5$\\sigma $ above background noise.", "The full SExtractor detection parameters used for both coadded and single-epoch images are shown in Tab.", "REF .", "In all we catalog about 800 columns across four bands.", "However for the public data release, we have released 60 columns from SExtractor per object.", "This full list in can be found in Tab. .", "Most of the parameters are described in the SExtractor manual online.", "There are a few additional parameters which are not yet released in the public version of SExtractor.", "These include model magnitudes and a new star-galaxy classifier called spread_model, which is a normalized simplified linear discriminant between the best fitting local PSF model ($\\phi $ ) and a slightly more extended model ($G$ ) made from the same PSF convolved with a circular exponential disk model with $\\mathrm {scale-length} = \\mathrm {FWHM}/16$ (where $FWHM$ is the full-width-half maximum of the PSF model).", "It can be defined by the following equation ${\\tt spread\\_model} = \\frac{{\\bf \\phi ^T x}}{{\\bf \\phi ^T\\phi }} - \\frac{{\\bf G^T x}}{{\\bf G^T\\phi }} ;$ where $x$ is the image vector centered on the source.", "The distribution of spread_model for BCS catalogs is discussed in  REF .", "More details of spread_model will be described elsewhere (Bertin et al in preparation).", "Figure: The stellar locus in three different color-color spaces for the BCS tile BCS0510-5043.", "The blue line shows the expected distribution derived from studies of a large ensemble of stars within the SDSS and 2MASS surveys.", "Red points show model magnitudes of stars from the BCS catalogs of this tile.", "The stellar locus distributions allow us to calibrate the absolute photometry and to assess the quality of the photometry for each tile." ], [ "Absolute Photometric Calibration", "Once all objects from the coadd are cataloged (in instrumental magnitudes), we proceed to obtain the absolute photometric calibration using Stellar Locus Regression [36].", "The principle behind this is that the regularity of the stellar main sequence leads to a pre-determined line in color-color space called the stellar locus.", "This stellar locus is observed to be invariant over the sky, at least for fields that lie outside the galactic plane.", "The constancy of the stellar locus has been used as a cross check of the photometric calibration within the SDSS survey [40].", "Absolute photometric calibration is done after the end of coaddition.", "We select star-like objects using a cut on the SExtractor spread_model parameter and magnitude error.", "We then match the observed stars to 2MASS stars from the NOMAD catalog, which is a combination of USNOB and 2MASS datasets, and which have $JHK$ magnitudes [71].", "Color offsets are varied until the observed locus matches the known locus.", "Because the 2MASS magnitudes are calibrated with a zeropoint accuracy at the $\\sim $ 2% level, one can bootstrap the calibration to the other bands.", "The known locus is derived using the high quality “superclean” SDSS-2MASS matched catalog from [20].", "It consists of $\\sim $ 300,000 high quality stars with data in $ugrizJHK$ .", "A median locus is calculated for each possible color combination in bins of $g-i$ .", "The fit is done in two stages.", "First, a three paramater fit is done to the $g-r$ , $r-i$ , $i-z$ colors.", "Another fit is done using $g-r$ and $r-J$ where the shift in the $g-r$ color is fixed from the first fit.", "The first fit provides an accurate calibration of the colors and the second fit fixes the absolute scale.", "The fit is done this way because only a fraction of the stars that overlap with 2MASS are saturated in all bands.", "We perform the stellar locus calibration for model and 3 arcsec aperture magnitudes, separately.", "The model magnitude calibration is then used to calibrate the other magnitudes in the catalog (except for the 3 arcsec magnitudes).", "The calibration of the 3 arcsec aperture magnitudes determines a PSF dependent aperture correction for the mag_aper_3 magnitudes only.", "We have found these small aperture magnitudes to provide higher signal to noise colors for faint galaxies in comparison to mag_model and mag_auto.", "An example of the stellar locus fits for one coadd tile is shown in Fig.", "REF .", "Red points are the observed colors of the stars, and the blue line is the median SDSS-2MASS locus.", "The orthogonal scatter about the stellar locus for all 3 color combinations for BCS tiles is shown in Fig  REF .", "The rms orthogonal scatter about the stellar locus in $(g-r, r-i)$ , $(r-i, i-z)$ and $(g-r, r-j)$ is 0.059, 0.061 and 0.075 respectively.", "Given the scatter and the number of stars available for calibration we can determine the zeropoints in our bands with sub-percent accuracy.", "Figure: Stellar locus scatter (above) for three color combinations for all tiles in BCS survey (top panel) and the same for SDSS (bottom panel).", "Typical BCS scatter is in the 5% to 8% range, and offsets after calibration are characteristically 1% or less.", "Typical scatter and offsets in the SDSS dataset are smaller than in the BCS survey, reflecting the tighter requirements on photometric quality in SDSS." ], [ "Testing Stellar Locus Calibration in SDSS", "To validate our photometric calibration algorithms, we applied exactly the same procedure to the full SDSS-2MASS catalog in [20].", "This catalog includes noiser objects than the catalog we used to derive the median stellar locus.", "We selected 4 areas (between RA of $120^{\\circ }$ and $350^{\\circ }$ ) and divided each into $1^{\\circ } \\times 1^{\\circ }$ patches.", "We match the objects to obtain 2MASS magnitudes and then apply the same calibration procedure as we did for the BCS catalogs.", "The rms scatter distributions for all the three color combinations can be found in Fig.", "REF (bottom panel).", "The corresponding scatter for SDSS in ($g-r,r-i$ ), ($r-i,i-z$ ), and ($g-r,r-j$ ) is about 0.041, 0.035 and 0.05 respectively and is about 1.5 times smaller than for the BCS catalogs.", "This is clear evidence for higher scatter in our stellar photometry as compared to the SDSS photometry.", "Assuming this additional source of scatter adds in quadrature with the SDSS observed scatter, we estimate the extra noise in BCS color combinations compared to SDSS is 0.039, 0.054 and 0.048 in ($g-r,r-i$ ), ($r-i,i-z$ ), and ($g-r,r-j$ ) respectively.", "Because these noise sources are getting contributions from each color, we can estimate that the noise floors are $\\delta (g-r)\\sim 0.027$ , $\\delta (r-i)\\sim 0.038$ , $\\delta (i-z)\\sim 0.038$ .", "These then imply noise floors in the stellar photometry within $griz$ bands of approximately 1.9%, 2.3%, 2.7% and 2.7%, respectively.", "This is then in good agreement with the typical repeatability scatter seen in these bands (see Fig.", "REF ) when one considers that $g$ and $r$ bands each have two overlapping exposures and $i$ and $z$ band each have three." ], [ "Star-Galaxy Classification", "Our current catalogs contain two star-galaxy classification parameters provided by SExtractor : class_star which has been extensively studied and spread_model, which has been newly developed as part of the DESDM development program.", "In order to test their performance and range of magnitudes up to which these measures can be reliably used, we plot the behavior of these two classifiers in $i$ -band as a function of $mag\\_model$ in Fig.", "REF .", "class_star lies in the range from 0 to 1.", "At bright magnitudes one can see two sequences in class_star for galaxies and stars near 0 and 1 respectively.", "The two sequences begin merging as bright as $i=20$ and are significantly merged beyond $i=22$ .", "As described in Sec.", "REF spread_model uses the local PSF model to quantify the differences between PSF-like objects and resolved objects.", "In the spread_model panel it's clear that there is a strong stellar sequence around the value 0.0, and that galaxies exhibit more positive values.", "The narrow stellar sequence and the broad galaxy sequence begin merging at $i=22$ in the BCS, but there is significant separation in the two distributions of points down to $i=23$ .", "Along with spread_model comes a measurement uncertainty, and so it is, for example, possible to define a sample of objects that lie off the stellar sequence in a statically significant way.", "For the BCS data, a good cut to separate stars would be ${\\tt spread\\_model}<0.003$ .", "Detailed studies of this new classification tool have been carried out within the DESDM project and will be carried out elsewhere.", "Figure: Plots of spread_model (top panel) and class_star (bottom panel) as a function of ii-band magnitudes for the full BCS catalog.", "Note that both measurements exhibit separate sequences for stars and galaxies, and that as one moves to fainter magnitudes these sequences merge.", "This is simply due to low signal to noise objects not containing enough morphological information for a reliable classification.", "However, note also that the new spread_model retains good capability of separating galaxies from stars to fainter magnitudes than class_star." ], [ "Quality Control and Science Ready Catalogs", "During the processing within the DESDM system a variety of quality checks are carried out.", "These include, for example, thresholding checks on the fraction of flagged pixels within an image and the $\\chi ^2$ and number of stars used in the astrometric fit of each exposure.", "In addition, the system is set up to report on the similarity between correction images (bias, flat, illum, and fringe) against stored templates that have been fully vetted.", "During the BCS processing this last facility was not used.", "Our experience has been that problems at any level of processing are most likely to show up in the stages of relative and absolute photometric calibration.", "Therefore, for the BCS processing done here we capture a range of photometric quality tests including the number of stars used in the stellar locus calibration and the rms scatter about the true stellar locus of the calibrated data (see Fig.", "REF ).", "In addition, we examine the photometric repeatability for common objects within overlapping images contributing to each tile.", "In Fig.", "REF we show an example for the $g$ -band in tile BCS0549-5043.", "This shows the magnitude difference between pairs of overlapping objects versus the average magnitude.", "The scatter here includes both statistical and systematic contributions, and the envelope of scatter grows toward faint magnitudes, as expected.", "Outlier rejection is done on the point distribution, and all 3$\\sigma $ outliers are filtered out and colored red.", "In the top panel we plot the mean and rms as well as the outlier fraction of these repeatability distributions as a function of magnitude.", "The mean and rms numbers are listed in milli-mags.", "Also, the statistical uncertainties of the model magnitudes are used to estimate the systematic magnitude error contribution to the rms.", "On the bright end where the statistical noise is very small, the systematic contribution to the rms is close to the total, which is 10 millimags in this case.", "As one moves toward the faint end the statistical contribution increases and the estimated systematic contribution plays only a small role in explaining the scatter.", "This is just how we expect the photometry to behave.", "Figure: Repeatability plots for single-epoch images for BCS tile BCS0549-5043 in gg-band.", "The repeatability is used to test the quality of the photometry in each band and tile.", "The top panel shows the mean magnitude difference between different single-epoch images which cover same region of sky, binned as a function of magnitude along with statistical and systematic errors.", "The bottom panel shows an un-binned representation of the same.", "Characteristic scatter on the bright end (i.e.", "the systematic floor) is 2% to 3% for gg and rr and 3% to 4% for ii and zz.Repeatability plots indicate systematic contributions to the photometric errors at the 10 to 20 milli-mag levels for typical $g$ and $r$ band tiles.", "For $i$ and $z$ band tiles the systematic noise is closer to 30 to 40 milli-mags.", "For all the BCS tiles we have examined these repeatability and stellar locus plots to probe whether the scatter is in acceptable ranges.", "In cases where tiles didn't meet these quality control tests we worked on the relative and absolute photometric calibration to improve the data.", "In addition to these photometry tests we examined the sky distribution of cataloged objects within each tile.", "In cases where large numbers of faint “junk” objects were found we attempted to remove them by adjusting the cataloging.", "At present all our tiles meet these quality tests except for a handful of tiles that are marked in red in Fig.", "REF .", "This includes 4 tiles in the 5 hr field and 6 tiles in the 23 hr field, corresponding to $\\sim $ 4% of the 80 deg$^2$ region.", "Ideally, we would reimage these regions to obtain better data.", "For every BCS night, the detrending pipeline creates three main types of science image files which we denote as raw, red and remap.", "The coadd pipeline produces four coadd images per tile for each of the four bands.", "Once we have calibrated coadd catalogs for all the processed tiles we run a post-processing program to remove duplicate objects near edges of the tiles.", "This is necessary because there is a 2 arcmin overlap between neighboring tiles.", "The program selects sources that appear in neighboring tiles that lie within in $0.9$  arcsec radius and for each pair it keeps the object that lies farthest from the edge of its tile.", "In this way a single, science ready catalog is prepared for each field.", "The 23 hr field catalog contains 1,877,088 objects, and the 5 hr field contains 2,952,282 objects with $i$ model magnitude $< 23.5$ .", "In the next section we review additional tests of the data quality.", "Figure: Histogram of 10σ\\sigma magnitude limits for all BCS tiles using mag_automag\\_auto errors in all four bands.", "The median depth values for all BCS tiles are 23.3, 23.4, 23 and 21.3 in grizgriz, respectively.", "The corresponding 10σ\\sigma point source depths are 23.9, 24.0, 23.6 and 22.1." ], [ "Survey Depth", "We estimate the 10$\\sigma $ photometric depths for galaxies using SExtractor $mag\\_auto$ errors.", "This is obtained by doing a linear fit to the relationship between the magnitude and the log of the inverse magnitude error.", "As a cross-check we also estimated the depths using information in the weight maps, and the results were comparable.", "The distributions of depth for each band over the full survey is shown in Fig.", "REF .", "The median magnitude depths for $griz$ bands are 23.3, 23.4, 23.0 and 21.3 respectively.", "These numbers are shallower than the depths we estimated using the NOAO exposure time calculator during the survey planning; those depths were 24.0, 23.9, 23.6, and 22.3.", "Our originally proposed depths assume a 2.2 arcsec diameter aperture, whereas galaxies near the 10$\\sigma $ detection threshold are typically larger in our images.", "We examine the depths of $2^{\\prime \\prime }$ aperture photometry and find that the median depths are 24.1, 24.1, 23.5 and 22.2 in $griz$ , respectively.", "These are within 0.2 mag of our naive estimates, explaining the bulk of the difference.", "In addition, we know that during our survey often the conditions were not photometric, and this could introduce another 0.1 to 0.2 mag offset.", "Another reason for the difference is that the calibrated observed magnitudes also include a correction for galactic extinction and reddening, whereas the estimated depths did not have extinction corrections included.", "Corresponding $10\\sigma $ point source depths are extracted using model fitting $mag\\_psf$ uncertainties.", "The results in bands $griz$ are 23.9, 24.0, 23.6 and 22.1, respectively.", "These are in better agreement with the small aperture photometry depths we used to estimate the exposure times for the survey.", "Another way of probing the depth of the survey is to look at the number counts of sources as a function of magnitude.", "Fig.", "REF contains the logN-logS from the combined 5 hr and 23 hr fields using $mag\\_auto$ .", "No star-galaxy separation is carried out, because near the detection limit there is not enough morphological information to reliably classify.", "The magnitudes of the turnover in the counts corresponds to 24.15, 23.55, 23.25 and 22.35.", "These turnover magnitudes mark the onset of significant incompleteness in the catalogs.", "Estimates of the depth of the 50% and 90% completenesss limit for a subset of the tiles appear in [84], but we do not apply that analysis to the whole survey.", "Figure: Number counts of BCS objects for all four bands in the BCS field using mag_automag\\_auto.", "The turnover magnitudes are 24.15, 23.55, 23.25 and 22.35 in grizgriz, respectively.", "The corresponding median mag_automag\\_auto 10σ\\sigma depths in grizgriz are 23.0, 23.4, 23.0 and 21.3 respectively.Finally, we probe for spatial variations in photometry by examining the distribution of sources above certain flux cuts over the two survey regions.", "The distributions of all sources at $i<22.5$ in both the 5 hr and the 23 hr fields are shown in Fig.", "REF .", "Objects are excluded for tiles that did not pass our quality tests, and this produces 4 black squares in the 5 hr field, and 6 black squares in the 23 hr field.", "The general uniformity of this object density distribution is an indication that the absolute photometric calibration is reasonably consistent across the fields.", "In the 5 hr field it is clear that one tile in the lower left does not reach the depth $i=22.5$ reached by the other tiles.", "This defect disappears if we examine the density distribution roughly 1 magnitude brighter, indicating this is a depth issue and not a photometric calibration problem.", "For the 23 hr field there is a small black rectangular notch in the upper right of the field with an associated dark path.", "Within this tile we have verified that too few of our $i$ -band exposures met the seeing requirements, and that has led to an uncovered region (the notch) as well as the shadow of lower object density to the right.", "Again, this is a depth issue rather than a photometric issue.", "There is another shadowed tile visible in the lower right part of the 23 hr field, and this is also a depth issue.", "Figure: Distribution of sources in 5 hr (top) and 23 hr fields (bottom) from the combined catalogsafter a mag_automag\\_auto magnitude cut (i<22.5i<22.5).", "The gaps show tiles which were not included in the releasedue to data quality problems.", "Some other tiles have only partial coverage or do not push to the depth of the magnitude cut with good completeness.", "A logarithmic scale (zscale option in ds9) is used.", "The uniformity of the source distribution is a demonstration of the photometric uniformity across the survey.In Fig.", "REF we show similar object density plots for stars and galaxies for the 5 hr field.", "The stars and galaxies are chosen based on spread_model cut at 0.003, where all objects with values greater than this threshold were considered galaxies and the rest were considered stars.", "A catalog depth cut at $i<22.5$ was imposed.", "This is shown in Fig.", "REF .", "The stellar distribution is quite uniform across this field, indicating that spread_model performance is quite robust to variations of PSF across a survey.", "Note that the shallow tile in the lower left portion of the survey exhibits edge effects, which we believe are associated with the reduced depth of this tile relative to the others.", "In the lower panel is the galaxy distribution.", "The same shallow tile shows up in the lower left portion.", "In addition, it is clear that the galaxy density is varying as a function of position as expected for the large scale structure of the Universe.", "We are quite happy with this performance.", "We have explored the same plots in the 23 hr field, and the results are similar.", "Moreover, we have explored these plots created using star_class as the classifier.", "The spatial distribution is highly inhomogeneous, indicating that class_star cannot be used to reliably separate stars and galaxies in a uniform manner across a large survey.", "Figure: Distribution of stars (top) and galaxies (bottom) in BCS 5 hr field with mag_automag\\_auto i<22.5i<22.5 based on spread_model cut of 0.003.", "The stars look uniformly distributed and you can see traces of large-scalestructure in galaxy density plot.", "A logarithmic scale (zscale option in ds9) is used.", "We have explored similar plots with class_star, and these contain very large inhomogeneities in the stellar and galaxy distribution, indicating that spread_model offers significant advantages over class_star in the classification of objects in large surveys." ], [ "BCS Data release", "We are publicly releasing the BCS catalogs, images and the photo-z training fields to the astrophysical community.", "All public BCS data products can be downloaded from http://www.usm.uni-muenchen.de/BCS.", "The BCS catalogs are divided into ascii files for the 5 hour and 23 hour fields.", "Separate catalogs are available for the tiles that passed our quality analysis and for the tiles that did not.", "Each catalog contains 63 columns which are described in Table .", "We are also making available the coadded images for the BCS survey at the same site.", "These images are available in a PSF homogenized form (used for the cataloging) and in the non-homogenized form.", "As in the case of the catalogs, we split the tar files by field and by whether the tiles passed our quality tests or did not.", "These tarballs contain FITS tile compressed images, which reduces the volume by a factor of $\\sim $ 5 relative to the uncompressed coadds.", "Initial tests of data quality are undertaken by obtaining photometric redshifts for BCS objects using an artificial neural network.", "Neural networks have been used to determine accurate photometric redshifts in past optical surveys [17], [62].", "We use annz, a feed-forward multi-layer perceptron network designed for finding photometric redshifts [18].", "The network is composed of a series of inputs, several layers of nodes, and one or more outputs.", "Each node is made of a function that takes its input as a weighted output of each of the previous layer's nodes.", "The weights are tuned by training the network on a representative dataset with known outputs.", "The optimal set of weights are those that minimize a cost function, which reflects the difference between a known output value and the network's predicted value.", "The training process can result in a set of weights that are over-fit to a particular training set.", "Furthermore, a given training process can converge to a local minimum of the cost function instead of the true minimum.", "In annz, the first issue is overcome by finding the set of weights that minimizes the cost function on a separate validation set rather than on the training set itself.", "The second is avoided by training a committee of several networks with randomized initial weights.", "The mean weights from each committee are used in the final network.", "We train our neural network on 5,820 objects with known redshifts.", "It is run with eight input parameters: four magnitudes $griz$ ; three colors $g - r$ , $r -i$ , and $i - z$ ; and a concentration index.", "$Mag\\_auto$ magnitudes are used for individual filters, $mag\\_aper\\_3$ magnitudes are used to determine colors, and the $i$ -band spread_model is used for the concentration index.", "Following the guidelines of [27] and [18], we use a minimally sufficient network architecture and committee size in hopes of achieving the highest quality results.", "We find this to be a committee of 8 neural networks that each have an architecture of 8:16:16:1 (eight inputs, two hidden layers of 16 nodes each, and one output).", "We denote photometric and spectroscopic redshifts as $z_{\\mathrm {phot}}$ and $z_{\\mathrm {spec}}$ , respectively and have $\\Delta z$ represent $z_{\\mathrm {phot}}- z_{\\mathrm {spec}}$ .", "Figure: Comparison between calibrated model magnitudes for stars from four BCS standardtiles (after stacking them together) and SDSS magnitudes after color and extinction corrections for gg-band.", "The stars are chosen by requiring that 𝚌𝚕𝚊𝚜𝚜_𝚜𝚝𝚊𝚛>0.8{\\tt class\\_star} > 0.8 in all four bands and alsoSExtractor flag << 5.", "The histograms are normalized to unity.", "The peak offset between BCS model magnitudes and SDSS is -0.06-0.06 in gg, rr, iiand 0.02 in zz bands." ], [ "Photometry Crosschecks with SDSS", "We compare our photometry with SDSS data by looking at spectroscopic calibration tiles which overlap with SDSS data and which contain significant numbers of spectroscopic redshifts.", "As explained in Sec.", "REF , these spectroscopic redshifts are then used for training our neural networks to obtained photometric redshifts.", "To do a comparison with SDSS catalogs, we applied color and extinction corrections to SDSS catalogs from these tiles.", "The fields which we consider for this purposes are from CNOC and DEEP fields centered at RA, DEC values of (02 hr 25 min, $7^{\\circ }$ ), (02 hr 29 min, $35^{\\circ }$ ), (23 hr 27 min, $8^{\\circ }$ ), and (23 hr 29 min, $12^{\\circ }$ ).", "We then do an object by object comparison of colors and magnitudes of all stars from SDSS versus those from BCS catalogs in these tiles.", "The SDSS magnitudes for objects which overlap BCS tiles go up to 23.4 in $g$ and 21.6 in $r$ , $i$ , and $z$ .", "We consider an object to be matched if it spatially overlaps to within $2^{\\prime \\prime }$ .", "Since the number of objects in each tile which overlap with SDSS is small, we combine results from all tiles into one plot for each magnitude or color as necessary.", "The magnitude comparison for all four bands (using $mag\\_model$ ) is shown in Fig.", "REF .", "The peak offset between BCS model magnitudes and SDSS magnitudes is approximately $-0.06$ in $g$ ,$r$ and $i$ bands and about 0.02 in $z$ bands, while the median offset is $-0.0562$ in $g$ , $r$ and $i$ and 0.0087 in $z$ .", "We also do a color comparison using the same cuts for these tiles between BCS and SDSS colors using $mag\\_aper3$ (magnitude within a 3 arcsec aperture), because colors are determined using this magnitude in photo-z estimation (Fig.", "REF ).", "The peak offset in colors in $g-r$ ,$r-i$ , and $i-z$ is about $-0.01$ , $-0.03$ , and $-0.02$ magnitudes respectively.", "The median offset is about $-0.01$ for $g-r$ and $i-z$ and about $-0.05$ for $r-i$ .", "The rms scatter about the median is 0.052, 0.061 and 0.081 for $g-r$ , $r-i$ , and $i-z$ , respectively.", "Figure: Difference in (g-r)(g-r), (r-i)(r-i), and (i-z)(i-z) colors between stars from BCS tiles and SDSSusing mag_aper3mag\\_aper3.", "All cuts are same as in Fig.", "and histograms are normalized to unity." ], [ "Photometric Redshift Calibration", "We obtain our training dataset by dedicating nine of the survey pointings to fields overlapping spectroscopic surveys: CDFS, CFRS, two CNOC2 fields, SSA 22, three DEEP2 fields, and VVDS.", "Objects from these fields share their photometric depth and reduction pipeline with the BCS data as well as have known spectroscopic redshifts.", "Although this training set is not representative of the survey in sky position, [1] show that limiting a neural network training set to small patches of sky does not result in biased redshifts for large surveys.", "The key issue is having uniformity of photometry between the training and application fields.", "lccr Photo-z Training Fields 0pt Survey RA Dec Redshifts ACESa 03:32 -27:48 2846 CFRSb 22:17 00:91 65 CNOC2c 02:25 00:07 318 CNOC2c 02:26 00:43 164 SSA 22d 01:40 00:01 818 DEEP2e 02:29 00:35 226 DEEP2e 23:27 00:08 414 DEEP2e 23:29 00:12 600 VVDSf 14:00 05:00 329 a[19] b[46], [45] c[83] d[21] e[24], [59] f[43], [44] Only objects that have reliable redshifts and photometric parameters are used to train the neural network.", "Objects with an $i$ -band magnitude $>22.5$ or an $i$ -band error $>0.1$ are removed from the training set.", "Objects that are unresolved in one or more bands or that have a SExtractor flag greater than 2 are removed as well.", "Similar cuts are made based on spectroscopic redshift errors, however the nature of the cut varies by catalog.", "The DEEP2, CNOC2, and CFRS catalogs provide redshift errors for each measurement.", "Objects from these fields are removed if their spectroscopic redshift errors are greater than 0.01.", "The ACES catalog (providing coverage of the CDFS field) and the VVDS catalog assign a confidence parameter for each object.", "In this case we only include objects with a confidence of 3 or 4 (see respective surveys for definitions).", "Both primary and secondary targets from the VVDS survey are included.", "Additional cuts were experimented with but found to produce more outliers, a larger sigma, or to reduce the size of the training set too much.", "The final training set contains 5,820 objects.", "Table REF breaks down the number of training objects that pass the filter criteria from each pointing.", "Figure REF further breaks down these objects by redshift bin.", "The pointings combine to provide a consistent distribution of redshifts from $0<z\\le 1.1$ .", "We have released the matched catalogs of spectroscopic redshifts along with information from BCS catalogs for these fields.", "This would enable others to develop their own photometric redshift estimates using these data.", "Figure: Redshift distribution of 5,820 objects from the calibration fields used to train annz.", "The redshift distribution is color coded by source.We evaluate the performance of annz on our data by randomly selecting half of the objects from the training set to train annz while the other half remains for testing.", "One sixth of the objects from the training half are removed to form the validation set (see above).", "The result provides 2,910 objects with both photometric and spectroscopic redshifts.", "Figure: Top panel: Two-dimensional histogram of z phot z_{\\mathrm {phot}} vs. z spec z_{\\mathrm {spec}} fortraining set objects that have z phot z_{\\mathrm {phot}} error <0.13< 0.13.", "Bin sizes are 0.015z×0.015z0.015z\\times 0.015z.", "Red bins count catastrophic outliers as defined by Equation.", "Blue bins count all other objects.", "5 objectswith z spec z_{\\mathrm {spec}} or z phot z_{\\mathrm {phot}} >1.5> 1.5 are not displayed.", "Bottom panel: Thesame training set data are shown with each point representing a bin of 50objects.We measure the photometric redshift performance using three metrics.", "The first, following [39], is the normalized median absolute deviation $\\sigma _{\\Delta z/(1+z)} = 1.48 \\times {\\rm median}\\left(\\frac{|\\Delta z|}{1+z_{\\rm spec}}\\right).", "\\nonumber $ This metric is better suited for our data than the standard deviation as it is less affected by catastrophic outliers.", "The second is the fraction of catastrophic outliers $\\eta $ defined as the percentage of objects that satisfy $\\frac{|\\Delta z|}{1+z_{\\rm spec}} > 0.15.$ The third metric is the net bias in redshift, averaged over all $N$ objects and defined as $z_\\mathrm {bias} = \\frac{1}{N}\\sum _{i=1}^N \\Delta z_i.", "\\nonumber $ Our training set performs as $\\sigma _{\\Delta z/(1+z)} = 0.061$ with $\\eta =7.49\\%$ .", "Over the entire range of redshifts there is little net bias: $z_\\mathrm {bias} = 0.0005$ .", "These statistics, particularly the fraction of catastrophic outliers, can be improved by culling objects based on their photometric redshift error.", "annz provides redshift errors that are derived from the errors of the input parameters, however there are several other methods of determining photometric redshift errors.", "[61] evaluate how well various methods improve $z_{\\mathrm {phot}}$ statistics.", "They show that culling objects based on redshift errors derived from magnitude errors are competitive with other methods at reducing the redshift scatter and catastrophic outlier fraction.", "We analyze our photometric performance after culling our data of objects with $z_{\\mathrm {phot}}$ error $\\ge 0.13$ based on errors provided by annz.", "The performance of the culled data improves to $\\sigma _{\\Delta z/(1+z)} = 0.054$ and $\\eta =4.93\\%$ .", "However, the $z_\\mathrm {bias}$ increases slightly to 0.0022.", "While the bias increases, it is still negligible.", "Figure REF demonstrates the performance of annz in determining redshifts.", "For objects within the range $0.3 \\lesssim z_{\\mathrm {spec}}\\lesssim 0.9$ , our photometric and spectroscopic redshifts match with little bias.", "For objects with redshifts below 0.3, there is a positive bias and for objects with redshifts beyond $z_{\\mathrm {spec}}\\sim 0.9$ there is a negative bias." ], [ "Application to the full BCS Catalog", "The 5,820 objects from the training set are used to train a committee of 8 annz networks, each with an architecture 8:16:16:1.", "This committee is used to determine redshifts and errors for every object in the BCS catalog.", "These are included in columns 61 and 62 of the data release.", "Because we found a negligible net bias when testing our calibration set, we do not perform a bias correction to redshifts of the BCS catalog.", "Many of the objects of the BCS catalog lie outside of the parameter space of data used to train annz.", "While [18] have demonstrated success using annz to determine redshifts of galaxies outside the parameter space used to train the network, this was done using a set of galaxies with a very uniform distribution of spectral types.", "For the generic distribution of galaxies provided in the BCS catalog, neural networks are unreliable in predicting redshifts outside the trained parameter space.", "Therefore we indicate whether an object lies inside or outside of the parameter space of the training set with a flag (column 63).", "A value of 1 means the object is within the parameter space of the training set and the redshift is reliable.", "A value of 0 means the object lays outside the parameter space and the redshift is unreliable.", "The flag is based only on the magnitude and magnitude error cuts that were made on the training set (i.e.", "$i < 22.5$ , $i$ -error $<0.1$ , resolved in all bands).", "It is not based on the SExtractor flag, the star-galaxy separation criteria, or on photometric redshift errors.", "We were able to obtain photo-z's for about 1,955,400 objects from the BCS catalog with $i <22$ .", "From these, there are $\\sim $ 204,600 objects in the catalog that pass the star-galaxy separation criteria in all bands and lie within the training set parameter space.", "The redshift distribution of these BCS objects in different magnitude ranges is shown in Figure REF .", "The peak redshift is around $z_{\\mathrm {phot}}=0.4$ for $20<i<22$ .", "Out of these, there are about 200 objects with $z_{\\mathrm {phot}}>1.0$ .", "Many of the objects that do not pass star-galaxy separation are stars.", "Since annz was trained with only galaxies, stellar objects lie outside the parameter space and therefore do necessarily not get assigned a correct redshift of $z_{\\mathrm {phot}}= 0$ .", "In fact, only a handful of objects in the entire catalog are assigned a $z_{\\mathrm {phot}}$ close to zero.", "We investigate the performance of the photo-z's when training a network with both stars and galaxies.", "Using the same inputs and network architecture as above but including $\\sim $ 1,000 stars in the training set, annz was successful in assigning stars a redshift below 0.1 only 70% of the time.", "However, redshift assignment of galaxies was not adversely affected.", "Only 4 out of approximately 3,400 galaxies were assigned a redshift less than 0.015.", "The fraction of catastrophic outliers as well as $\\sigma _{\\Delta z/(1+z)}$ were not significantly affected either, so long as stars are not included in the statistics.", "While the results of training annz with stars are not sufficient to use for the entire BCS catalog, these preliminary results show some promise.", "Furthermore, [17] have shown better results when training an annz network specifically for star-galaxy separation.", "Figure: Distribution of photometric redshifts of the galaxies that lie within the annz training set parameter space and have z phot z_{\\mathrm {phot}} error <0.05(1+z phot )< 0.05 (1+z_{\\mathrm {phot}}) and pass the star-galaxy separation test and in different ii-band model magnitude ranges." ], [ "Conclusions and Discussion", "In this paper we present an overview of the Blanco Cosmology Survey (BCS), an $\\sim $ 80 deg$^2$ optical photometric survey in $griz$ bands carried out with the Mosaic2 imager on Blanco 4m telescope between $2005-2008$ .", "We discuss the observing strategy within the context of our scientific goals, and we present basic observing characteristics at CTIO such as the sky brightness and delivered image quality.", "We provide a detailed description of the data processing, calibration and quality control, which we have carried out using a development version of the Dark Energy Survey Data Management System.", "The processing steps in going from raw exposures to science ready catalogs include image detrending and astrometric calibration; this processing is run independently on every night of observations.", "This is followed by image co-addition, which combines data from the same region of the sky into deeper coadd images.", "The processing of real data from the Blanco telescope provides a real world stress test of the DESDM system.", "Many novel algorithmic features, which will be used to process upcoming DES data, were tested on BCS data.", "These include PSF homogenization, cataloging using PSF corrected, model-fitting photometry, object classification using the new spread_model, absolute photometric calibration using the stellar locus and a variety of quality control tests.", "We present the characteristics of the dataset, including the median estimated $10\\sigma $ galaxy photometry depth in the coadds for bands $griz$ , which are 23.3, 23.4, 23.0, and 21.3, respectively.", "The corresponding point source $10\\sigma $ depths in $griz$ are 23.9, 24.0, 23.6, and 22.1, respectively.", "We measure the systematic noise floor in our photometry using photometric repeatability in single epoch images and comparisons of the stellar locus scatter from BCS and SDSS.", "Both results indicate a noise floor at the $\\sim 1.9$ % level in $g$ , $\\sim 2.2$ % in $r$ , and $\\sim 2.7$ % in $i$ and $z$ bands.", "This noise floor does not impact the core galaxy cluster science for which the BCS was designed.", "We expect that with an improved characterization of the illumination correction using the star flat technique demonstrated in the CFHT Legacy Survey [64] it would be possible to reduce this noise floor further, but given that the current floor is adequate for our science needs we have not included these corrections in our BCS processing.", "Our absolute photometric calibration is obtained using the stellar locus and including the 2MASS $J$ band photometry.", "We can calibrate our zeropoints at better than $\\sim $ 1% (statistical) to the stellar locus, and so our overall photometric uniformity is driven by the $\\sim $ 2% accuracy of the 2MASS survey [71].", "We show that our photometric zeropoint calibration is quite uniform across the survey by showing star and galaxy counts across the survey.", "We also demonstrate that with spread_model it is possible to carry out uniform star-galaxy separation even across a large extragalactic survey.", "As an additional data quality test, we present photometric redshifts derived from a neural network trained on a sample of objects with spectroscopic redshifts that we targeted during the BCS survey.", "The performance of our four band $griz$ photometric redshifts are evaluated based on analysis of a calibration set of over 5,000 galaxies with measured spectroscopic redshifts.", "We find good performance with a characteristic scatter of $\\sigma _{\\Delta z/(1+z)}=0.054$ and an outlier fraction of $\\eta =4.93$ %.", "Finally, we provided a summary of the output data products from our co-added images and catalogs along with information on how to download them.", "Finally, the BCS data have been used for a range of scientific pursuits, which we briefly summarize and reference here to allow the reader to seek additional information as needed.", "Within the SPT survey, the first four SZE selected clusters were optically confirmed with redshift estimates using BCS data [75] and detailed studies of galaxy populations using these clusters were reported in  [84].", "The total number of SPT cluster candidates with signal-to-noise ratio $> 4.5$ in BCS footprint is 15 [66] and among these 10 have been confirmed with the BCS data and the remaining 5 have redshift lower limits between 1 and 1.5 [74].", "These clusters and their BCS derived redshifts have figured prominently in SPT publications to date [75], [80], [35], [5], [82], [66], [74].", "The BCS data enabled the serendipitous discovery of a strong lensing arc of a galaxy at $z = 0.9057$ by a massive galaxy cluster at a redshift of $z=0.3838$  [13].", "Additional automated searches for strong lensing arcs have also been carried out, and further analysis of BCS data for weak lensing is in progress.", "A sample of about 105 galaxy clusters was found using first three seasons of BCS data using an independent processing [56], [55], and the BCS data were also used for optical confirmation of ACT clusters [53].", "Other studies include estimates of weak lensing cluster masses [52] and a search for QSO candidates using $r$ -band data [41].", "We used the BCS data to measure photometric redshifts of about 46 X-ray selected clusters in the XMM-BCS survey [78].", "This X-ray selected sample is currently being used in combination with SPT data to explore the low mass cluster population and its SZE properties (Liu et al., in prep).", "In addition, these BCS data are also being used in the analysis of the larger XMM-XXL survey in the 23 hr field (Pierre, private communication).", "The BCS data continue to provide an important dataset for SPT.", "Recently, the data were used to trace the galaxy populations and were correlated against the SPT CMB-lensing maps [79], demonstrating correlations significant at the 4$\\sigma $ to 5$\\sigma $ level in both BCS fields [11].", "The BCS data will provide a valuable optical dataset for combination with a 100 deg$^2$ Spitzer survey over the same region (Stanford, private communication), a 100 deg$^2$ Herschel survey (Carlstrom, private communication), and they will overlap one of the deep mm-wave fields being targeted by SPT-pol (Carlstrom, private communication) until the Dark Energy Survey data are available.", "We would like to acknowledge Len Cowie for providing us spectroscopic redshifts for objects from the SSA 22 field.", "The Munich group acknowledges the support of the Excellence Cluster Universe and from the program TR33: The Dark Universe, both of which are funded by the Deutsche Forschungs Gemeinschaft.", "We acknowledge support from the National Science Foundation (NSF) through grants NSF AST 05-07688, NSF AST 07-08539, NSF AST 07-15036, and NSF AST 08-13534.", "We acknowledge the support of the University of Illinois where this project was begun.", "This paper includes data gathered with the Blanco 4-meter telescope, located at the Cerro Tololo Inter-American Observatory in Chile, which is part of the U.S. National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA), under contract with the NSF.", "Facilities: Blanco (MOSAIC)" ], [ "BCS Catalog Description", "We created an ASCII catalog files which is obtained from the catalogs of each individual tile and after removing duplicates.", "The description of each column in the BCS catalog is provided in Tab. .", "lccc Details of BCS Catalogs 0pt Column Parameter Units Definition 1 tilename - Name of tile 2 objectid - ID from DESDM database table coadd_objects 3 RA degree Right Ascension 4 DEC degree Declination 5 mag_model_g AB mag Model magnitude ($g$ ) 6 magerr_model_g AB mag Error in model magnitude ($g$ ) 7 mag_auto_g AB mag Kron magnitude ($g$ ) 8 magerr_auto_g AB mag Error in Kron magnitude ($g$ ) 9 mag_psf_g AB mag PSF magnitude ($g$ ) 10 magerr_psf_g AB mag Error in PSF magnitude ($g$ ) 11 mag_petro_g AB mag Petrosian magnitude ($g$ ) 12 magerr_petro_g AB mag Error in Petrosian magnitude ($g$ ) 13. mag_aper3_g AB mag Magnitude in 3-arcsec aperture ($g$ ) 14 magerr_aper3_g AB mag Magnitude error in 3-arcsec aperture ($g$ ) 15 flags_g - SExtractor flag ($g$ ) 16 class_star_g - SExtractor star/galaxy separator 17 spread_model_g - Difference in PSF and Sérsic magnitude ($g$ ) 18 spread_modelerr_g - Error in spread_model ($g$ ) 19 mag_model_r AB mag Model magnitude ($r$ ) 20 magerr_model_r AB mag Error in model magnitude ($r$ ) 21 mag_auto_r AB mag Kron magnitude ($r$ ) 22 magerr_auto_r AB mag Error in Kron magnitude ($r$ ) 23 mag_psf_r AB mag PSF magnitude ($r$ ) 24 magerr_psf_r AB mag Error in PSF magnitude ($r$ ) 25 mag_petro_r AB mag Petrosian magnitude ($r$ ) 26 magerr_petro_r AB mag Error in Petrosian magnitude ($r$ ) 27 mag_aper3_r AB mag Magnitude in 3-arcsec aperture ($r$ ) 28 magerr_aper3_r AB mag Magnitude error in 3-arcsec aperture ($r$ ) 29 flags_r - SExtractor flag ($r$ ) 30 class_star_r - SExtractor star/galaxy separator 31 spread_model_r - Difference in PSF and Sérsic magnitude ($r$ ) 32 spread_modelerr_r - Error in spread_model ($r$ ) 33 mag_model_i AB mag Model magnitude ($i$ ) 34 magerr_model_i AB mag Error in model magnitude ($i$ ) 35 mag_auto_i AB mag Kron magnitude ($i$ ) 36 magerr_auto_i AB mag Error in Kron magnitude ($i$ ) 37 mag_psf_i AB mag PSF magnitude ($i$ ) 38 magerr_psf_i AB mag Error in PSF magnitude ($i$ ) 39 mag_petro_i AB mag Petrosian magnitude ($i$ ) 40 magerr_petro_i AB mag Error in Petrosian magnitude ($i$ ) 41 mag_aper3_i AB mag Magnitude in 3-arcsec aperture ($i$ ) 42 magerr_aper3_i AB mag Magnitude error in 3-arcsec aperture ($i$ ) 43 flags_i - SExtractor flag ($i$ ) 44 class_star_i - SExtractor star/galaxy separator 45 spread_model_i - Difference in PSF and Sérsic magnitude ($i$ ) 46 spread_modelerr_i - Error in spread_model ($i$ ) 47 mag_model_z AB mag Model magnitude ($z$ ) 48 magerr_model_z AB mag Error in model magnitude ($z$ ) 49 mag_auto_z AB mag Kron magnitude ($z$ ) 50 magerr_auto_z AB mag Error in Kron magnitude ($z$ ) 51 mag_psf_z AB mag PSF magnitude ($z$ ) 52 magerr_psf_z AB mag Error in PSF magnitude ($z$ ) 53 mag_petro_z AB mag Petrosian magnitude ($z$ ) 54 magerr_petro_z AB mag Error in Petrosian magnitude ($z$ ) 55 mag_aper3_z AB mag Magnitude in 3-arcsec aperture ($z$ ) 56 magerr_aper3_z AB mag Magnitude error in 3-arcsec aperture ($z$ ) 57 flags_z - SExtractor flag ($z$ ) 58 class_star_z - SExtractor star/galaxy separator 59 spread_model_z - Difference in PSF and Sérsic magnitude ($z$ ) 60 spread_modelerr_z - Error in spread_model ($z$ ) 61 z_phot - Photometric redshift 62 z_phot_err - Photometric redshift error 63 z_phot_flag - Within annz training set parameter space Explanation and contents of catalogs in the BCS survey release.", "More details on some of the parameters can be found in the SExtractor manual.", "The magnitudes are corrected for galactic extinction." ] ]

[ [ "Enhanced charge order in a photoexcited one-dimensional strongly\n correlated system" ], [ "Abstract We present a compelling response of a low-dimensional strongly correlated system to an external perturbation.", "Using the time-dependent Lanczos method we investigate a nonequilibrium evolution of the half-filled one-dimensional extended Hubbard model, driven by a transient laser pulse.", "When the system is close to the phase boundary, by tuning the laser frequency and strength, a sustainable charge order enhancement is found that is absent in the Mott insulating phase.", "We analyze the conditions and investigate possible mechanisms of emerging charge order enhancement.", "Feasible experimental realizations are proposed." ], [ "Enhanced Charge Order in a Photoexcited One-Dimensional Strongly Correlated System Hantao Lu Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan Shigetoshi Sota Computational Materials Science Research Team, RIKEN AICS, Kobe, Hyogo 650-0047, Japan Hiroaki Matsueda Sendai National College of Technology, Sendai, 989-3128, Japan Janez Bonča Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia J. Stefan Institute, SI-1000 Ljubljana, Slovenia Takami Tohyama Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan We present a compelling response of a low-dimensional strongly correlated system to an external perturbation.", "Using the time-dependent Lanczos method we investigate a nonequilibrium evolution of the half-filled one-dimensional extended Hubbard model, driven by a transient laser pulse.", "When the system is close to the phase boundary, by tuning the laser frequency and strength, a sustainable charge order enhancement is found that is absent in the Mott insulating phase.", "We analyze the conditions and investigate possible mechanisms of emerging charge order enhancement.", "Feasible experimental realizations are proposed.", "71.10.Fd, 74.40.Gh, 78.47.J-, 78.20.Bh Nonequilibrium processes in strongly correlated electron systems can provide new insights into the dynamical properties of these systems, which, in many aspects, can be qualitatively different from their weakly interacting counterparts.", "One such example is nonequilibrium induced phase transition [1], [2], [3].", "As the system is driven away from the equilibrium, under certain conditions, a “crossover” from one state to another (metastable) state may occur.", "A well-known example is the insulator-to-metal transition induced either by strong electric field or transient laser pulse, as a result of photodoping [4], [5], [6], [7], [8].", "In the case of Mott insulators, photocarriers are doublons and holons.", "In one dimension, just after the doping, generally, the system is in a metallic state due to the existence of itinerant carriers and the benefit of the spin-charge separation [8], [9].", "Experimentally, the dielectric breakdown in one-dimensional (1D) Mott insulators has been observed long ago in organic materials [10], [11] and oxides [12].", "Photoinduced metallic state by transient light has also been investigated [13], [8].", "More recently, the nonequilibrium and nonlinear phenomena have stimulated significant interest within the cold atom community, where a novel realization of the fermionic Mott insulating state has been achieved [14].", "In this Letter we address the question of whether it is possible to reach other characteristically different states besides the metallic one after the photodoping.", "One possibility is the enhancement of charge-order when the attractive interaction between doublons and holons is incorporated [15].", "We seek such enhancement in the extended half-filled 1D Hubbard model with an additional nearest-neighbor interaction.", "At half filling, its phase is well-known and understood [16], [17], [18], [19].", "In large on-site Coulomb interaction, the model possesses two phases: spin-density-wave (SDW) and charge-density-wave (CDW), connected by a first order quantum phase transition, with algebraic decay of spin correlations and long-range charge order, respectively.", "In general, as a correlated system approaches the phase boundary, the response to an external perturbation becomes more elaborate.", "In this Letter we show that in the latter case a substantial change in the electronic structure can be triggered by the optical pulse.", "In particular, when the system is originally in the Mott-insulating phase but close to the transition to CDW, a sustainable enhancement of charge order parameter is achieved by exposing the system to an external optical pulse with a rather carefully tuned frequency and amplitude.", "We consider the 1D extended Hubbard model at half filling.", "The laser pump is incorporated by means of the Peierls substitution in the Hamiltonian: $H(t)&=&-t_h\\sum _{i,\\sigma }\\left(e^{iA(t)}c_{i,\\sigma }^{\\dagger }c_{i+1,\\sigma }+\\text{H.c.}\\right) \\nonumber \\\\&+&U\\sum _{i}\\left(n_{i,\\uparrow }-\\frac{1}{2}\\right)\\left(n_{i,\\downarrow }-\\frac{1}{2}\\right) \\nonumber \\\\&+&V\\sum _i\\left(n_i-1\\right)\\left(n_{i+1}-1\\right),$ where $c_{i,\\sigma }^\\dagger $ ($c_{i,\\sigma }$ ) creates (annihilates) electrons with spin $\\sigma $ at site $i$ , $n_{i,\\sigma }=c_{i,\\sigma }^\\dagger c_{i,\\sigma }$ , $n_i=n_{i,\\uparrow }+n_{i,\\downarrow }$ , $t_h$ is the hopping constant while $U$ and $V$ are the on-site and nearest neighbor interaction strengths, respectively.", "We model the external laser pulse in the temporal gauge via the time-dependent vector potential $A(t)$  [20] $A(t)=A_0e^{-\\left(t-t_0\\right)^2/2t_d^2}\\cos \\left[\\omega _{\\text{pump}}\\left(t-t_0\\right)\\right],$ where $A_0$ controls the laser amplitude, which reaches its full strength at $t=t_0$ ; $t_d$ characterizes the duration time of light action.", "Notice that due to finite $t_d$ , the incoming photon frequency is broadened into a Gaussian-like distribution, with the variance of $1/t_d^2$ around the central value $\\omega _{\\text{pump}}$ .", "We set $t_h$ and $t_h^{-1}$ as energy and time units.", "We now give a short overview of the equilibrium properties (with $A=0$ ) of the model, related to our work.", "We set $U=10$ and vary $V$ .", "With increasing $V$ the system undergoes around $U\\approx 2V$ a first-order phase transition directly between SDW and CDW phases.", "Choosing rather large $U$ we avoid the intermediate bond-order-wave phase that separates SDW and CDW at smaller $U$  [17], [18], [19].", "In both phases charge excitations are gapped.", "In the Mott insulating, i.e., SDW phase, gapless spin excitations exist and the system displays no charge order.", "In contrast, a charge-density wave is characteristic for the CDW phase while spin excitations are gapped.", "In order to solve the time-dependent Hamiltonian $H(t)$ , starting from the Schrödinger equation $i\\partial \\psi (t)/\\partial t=H(t)\\psi (t)$ , we employ the time-dependent Lanczos method, which is originally described in Ref.", "[21] and later applied and analyzed in more detail in Ref.", "[22], followed by its applications in nonequilibrium dynamics of strongly correlated systems in Ref. [23].", "The basic idea is that we approximate the time evolution of ${\\vert \\psi (t)\\rangle }$ by a step-vise change of time $t$ in small increments $\\delta t$ .", "At each step, the Lanczos basis with dimension $M$ is generated resulting in the time evolution ${\\vert \\psi (t+\\delta t)\\rangle }\\simeq e^{-iH(t)\\delta t}{\\vert \\psi (t)\\rangle }\\simeq \\sum _{l=1}^M e^{-i\\epsilon _l\\delta t}{\\vert \\phi _l\\rangle }\\langle \\phi _l\\vert \\psi (t)\\rangle ,$ where $\\epsilon _l$ and ${\\vert \\phi _l\\rangle }$ , respectively, are eigenvalues and eigenvectors of the tridiagonal matrix generated in Lanczos iteration.", "We have confirmed that in our calculation, generally $M=30$ provides adequate accuracy when $\\delta t\\lesssim 0.1$ .", "For even smaller $\\delta t$ , $M$ can be smaller.", "In the succeeding numerical calculations, we employ periodic boundary conditions and set $t_d=5$ and $t_0=12.5$ .", "To investigate the time evolution of the charge order, we define the charge-charge correlation as $C(j;t)=\\frac{(-1)^j}{L}\\sum _{i=0}^{L-1}\\langle \\psi (t)\\vert (n_{i+j}-1)(n_i-1)\\vert \\psi (t)\\rangle ,$ where $L$ is the lattice size.", "Figure: (color online) Time dependence of the charge-chargecorrelations as functions of distance (labeled by jj) for 14-sitelattice.", "The laser pulse with Gaussian magnitude modulation reachesits full strength at t=12.5t=12.5, as indicated by solid lines.", "Withoutexception, the pumping frequencies are set to match the resonancepeaks of the optical absorption spectrum.", "Parameters: U=10U=10,δt=0.02\\delta t=0.02, M=100M=100.", "(a) V=1.0V=1.0, ω pump =7.1\\omega _{\\text{pump}}=7.1,A 0 =0.10A_0=0.10; (b) V=3.0V=3.0, ω pump =6.1\\omega _{\\text{pump}}=6.1, A 0 =0.30A_0=0.30; (c)V=4.5V=4.5, ω pump =4.0\\omega _{\\text{pump}}=4.0, A 0 =0.07A_0=0.07; (d) V=5.5V=5.5,ω pump =4.1\\omega _{\\text{pump}}=4.1, A 0 =0.60A_0=0.60.Figure REF shows the results of charge-charge correlation functions $C(j;t)$ for the system with 14 lattice sites, where the largest distance between two sites is 7.", "In order that the photon energy can be effectively transmitted to the electrons, we match the pumping frequency $\\omega _{\\text{pump}}$ with the first optical absorption peak.", "Figure REF (a) to REF (c) show the results of the system approaching the phase boundary from the SDW side, as the nearest-neighbor interaction $V$ increases from $1.0$ to $4.5$ (while keeping fixed $U=10$ ).", "For $V=1.0$ and $3.0$ the response of the system shows only a slight increase of mostly short-distance correlations during and after the pulse.", "In contrast, at $V=4.5$ the system displays clear tendency towards a sustainable charge order enhancement that emerges as the external pulse reaches its maximum and remains nearly constant after the pulse.", "By performing a time-dependent density-matrix renormalization group method under open boundary conditions on larger system size, up to 30, similar behavior can be observed (not shown here).", "We note that the length scale with which prominent charge order enhancement can be identified in the case of $V=4.5$ is confined within distance less than ten, in the units of lattice constant.", "The opposite effect is found when starting from the CDW side of the phase diagram, at $V=5.5$ [Fig.", "REF (d)], where after the pulse, the charge order is substantially diminished, along with a limited recovery of SDW order.", "For deeper understanding of the photoinduced charge order enhancement we note that the quantum phase transition between SDW-CDW phases is driven by the competition between the energy cost for doublon generation and the energy gain due to the attraction between doublon-holon pairs.", "The transition point is roughly $V\\approx U/2$ .", "In the SDW phase, we expect charge-order favorite states to proliferate in the low energy regime as the transition point is approached, which makes them likely candidates to be picked up by the laser pulse.", "This is in contrast to the case when the system is far from the transition point where CDW states represent high-energy states.", "For a more quantitative analysis, we compare the spectra of systems with different values of $V$ in the time-independent Hamiltonian, i.e., Eq.", "(REF ) but $A(t)=0$ .", "In order to describe the CDW order with finite spatial extension, we define $\\mathcal {O}_{\\text{CDW}}=\\frac{1}{LL_c}\\sum _{i=0}^{L-1}\\sum _{j=1}^{L_c}(-1)^j\\left(n_{i+j}-1\\right)\\left(n_i-1\\right).$ Here, $L_c$ is introduced as a cut off parameter for the correlation length.", "The expectation values of the order parameter $\\langle \\mathcal {O}_{\\text{CDW}}\\rangle $ for eigenstates of a smaller 10-site system with $L_c=5$ are produced in Fig.", "REF .", "Comparison between different excited states shows that we can distinguish states with large values of the CDW order.", "Note that despite the external laser pulse, the total momentum $P_{\\text{tot}}$ and the total spin $S_{\\text{tot}}$ remain good quantum values throughout the time evolution.", "For the ground state of a 10-site lattice at half filling, we have $P_{\\text{tot}}= S_{\\text{tot}}=0$ .", "Accordingly, only results of the eigenstates with the same values of $P_{\\text{tot}}$ and $S_{\\text{tot}}$ are displayed.", "Figure: (color online) The expectations of CDW order parameter ofeigenstates for 10-site lattice with V=1.0V=1.0, 3.03.0, 4.54.5, and5.55.5.", "The energy EE is measured from the ground state.", "Only thedata of states with P tot =S tot =0P_{\\text{tot}}=S_{\\text{tot}}=0 are shown, upto E=60E=60.", "Note that the largest value of〈𝒪 CDW 〉\\langle \\mathcal {O}_{\\text{CDW}}\\rangle in (a) through (d) is marked by across.In Fig.", "REF , we find that in the case of small $V=1.0$ , the eigenstates with predominant CDW features are positioned in the high energy regime, located around $E\\sim 45$ above the ground state energy.", "With the increase of $V$ the states with large values of $\\langle \\mathcal {O}_{\\text{CDW}}\\rangle $ move towards the lower part of the energy spectra.", "At $V=5.5$ the eigenstate with $\\langle \\mathcal {O}_{\\text{CDW}}\\rangle \\approx 0.65$ turns to be the ground state [Fig.", "REF (d)].", "Keeping this picture in mind, it becomes more plausible that a well-tuned laser pulse may trigger the enhancement of the charge order on the SDW side of the phase diagram.", "Our numerical calculations suggest that the necessary precondition for such enhancement is the proximity of the system to the phase boundary as well as matching conserved quantum numbers between the SDW and CDW phases.", "We further elaborate on the condition for the emergence of the CDW order enhancement induced by the laser pulse.", "To this effect we perform parameter-sweeping calculations on the 10-site lattice.", "We sweep the pumping frequency $\\omega _{\\text{pump}}$ and laser intensity $A_0$ and carry out the time evolutions up to $t=52.5$ .", "For a given pair of $\\omega _{\\text{pump}}$ and $A_0$ , we then calculate the expectation of CDW order $\\langle \\mathcal {O}_{\\text{CDW}}\\rangle _{\\text{av}}$ , and the energy increase measured from the ground state $\\Delta E$ , by averaging on the last 50 time steps (corresponding to time length $\\Delta t=5$ ).", "Contour plots of $\\langle \\mathcal {O}_{\\text{CDW}}\\rangle _{\\text{av}}$ and $\\Delta E$ are shown in Fig.", "REF .", "To facilitate the analysis we present in Fig.", "REF the corresponding optical spectra, obtained from the imaginary part of the dynamical current-current correlation function: $\\text{Im}\\chi _j(\\omega )=\\frac{1}{L}\\sum _n\\left| \\langle n\\vert \\hat{j}\\vert 0\\rangle \\right|^2\\delta \\left(\\omega -E_n+E_0\\right),$ where ${\\vert 0\\rangle }$ and ${\\vert n\\rangle }$ represent the ground state and excited states with energy $E_0$ and $E_n$ , respectively.", "The current operator $\\hat{j}$ defined as $\\hat{j}=-i\\sum _{i,\\sigma }\\left(c_{i,\\sigma }^{\\dagger }c_{i+1,\\sigma }-\\text{H.c.}\\right)$ .", "Figure: (color online) Contour plots of final time-evolution resultsof CDW order 〈𝒪 CDW 〉 av \\langle \\mathcal {O}_{\\text{CDW}}\\rangle _{\\text{av}} (firstcolumn) and energy increase ΔE\\Delta E (second column) as functionsof ω pump \\omega _{\\text{pump}} and A 0 A_0, obtained by averaging on thelast 50 time steps (time length Δt=5\\Delta t=5), for 10-sitelattice.", "Here, we take δt=0.1\\delta t=0.1, M=30M=30.", "(a), (b) V=1.0V=1.0;(c), (d) V=3.0V=3.0; (e), (f) V=4.5V=4.5; and (g), (h) V=5.5V=5.5.Figure: (color online) The imaginary part of the dynamicalcurrent-current correlation function for 10-site lattice withV=1.0V=1.0, 3.03.0, 4.54.5, and 5.55.5, obtained by Lanczos method.", "Theδ\\delta function is approached by a Lorentzian with a width of0.10.1.In Fig.", "REF , we notice the emergence of similar patterns between $\\langle \\mathcal {O}_{\\text{CDW}}\\rangle _{\\text{av}}$ (left) and $\\Delta E$ (right) in each row panel.", "When starting from the SDW side [Figs.", "REF (a)-REF (f)], in the case of moderate values of $A_0$ , the contour plots representing the CDW enhancement and energy increase roughly match.", "On the other hand, starting from the CDW ground state, when $V=5.5$ [Figs.", "REF (g) and REF (h)], the effect is just the opposite: the energy increase occurs along the destruction of charge order.", "Furthermore, the positions of the enhanced energy stripes largely coincide with the resonance-peak positions of optical absorption, as presented in Fig.", "REF , except for some cases in the large $A_0$ region, which will be discussed later.", "From these observations, we can conclude that at moderate values of $A_0$ , in order to obtain enhanced $\\langle \\mathcal {O}_{\\text{CDW}}\\rangle _{\\text{av}}$ , the incoming photon frequency should be tuned close to absorption windows, that match resonance peaks of the optical spectrum.", "Let us investigate the case of $V=4.5$ in more detail.", "The first optical peak is located at $\\omega \\simeq 4.3$ [Fig.", "REF (c)].", "States with enhanced CDW order are located around the same energy (the highest one with energy $3.9$ ), as shown in Fig.", "REF (c).", "Not surprisingly, a sustainable enhancement of $\\langle \\mathcal {O}_{\\text{CDW}}\\rangle _{\\text{av}}$ can be found with $\\omega _{\\text{pump}}\\simeq 4.3$ when $A_0\\sim 0.08$ [Fig.", "REF (e)].", "On the other hand, systems with $V=1.0$ , and $3.0$ [Figs.", "REF (a) and REF (c)] display no, or at most a very slight, increase of $\\langle \\mathcal {O}_{\\text{CDW}}\\rangle _{\\text{av}}$ around the resonance frequencies, which match the optical frequency peaks, i.e., $\\omega \\simeq 7.5$ and $6.3$ , respectively [Figs.", "REF (a) and REF (b)].", "We thus propose two conditions for the observation of the enhancement of $\\langle \\mathcal {O}_{\\text{CDW}}\\rangle _{\\text{av}} $ after the pulse: 1) the current matrix element $\\langle n\\vert \\hat{j}\\vert 0\\rangle $ in Eq.", "(REF ) should be sizable and 2) $\\langle n\\vert \\mathcal {O}_{\\text{CDW}}\\vert n\\rangle $ should be large as well.", "Further detailed analysis on the $V=4.5$ case shows that at $\\omega _{\\text{pump}}\\approx 4.3$ , with $A_0$ growing beyond $0.1$ , $\\Delta E$ temporally drops, reaches a local minimum and then keeps increasing [Fig.", "REF (f)].", "The CDW signal up to $A_0\\sim 0.25$ follows the same trend and then decreases [Fig.", "REF (e)].", "This suggests a third condition to obtain enhancement of $\\langle \\mathcal {O}_{\\text{CDW}}\\rangle _{\\text{av}}$ : the pulse should deliver the optimal energy increase which can be controlled either by $A_0$ , or another parameter $t_d$ .", "In some simulations we have noticed a temporary increase of CDW order that lasts only up to the midpoint of the pulse action and then diminishes.", "The reason is that the system has passed the CDW enhancement region during the process – it has gained excessive energy.", "Such a case can be found in Fig.", "REF (b) ($V=3.0$ , $A_0=0.30$ ), where we observe that two successive enhancements of the CDW signal appear during the pulse and then die out.", "The fact that the double-peak structure also emerges in the corresponding energy profile is consistent with the above argument.", "Moreover, during the irradiation, photons can be absorbed or emitted at different stages following the time evolution.", "We also notice the enhancements of the CDW signal with the accompanying energy increase when $\\omega _{\\text{pump}}$ is less than the resonance frequency, such as for $V=4.5$ , when $\\omega _{\\text{pump}}\\approx 2$ [Fig.", "REF (e)].", "We speculate that this effect is generated by the multiphoton, or more precisely, two-photon process, partly supported by the fact that it appears at large $A_0$ and the energy increase is close to 4, the same as what happens when the $\\omega _{\\text{pump}}$ is around 4.", "The analysis on this possible multiphoton process in the off-resonant regime is out of reach of the present Letter [24].", "In conclusion, we address the issues of the thermodynamic limit and feasible experimental realizations of the proposed CDW enhancement.", "It is well known that for the 1D extended Hubbard model, excitonic bound states can be found at the edge of the optical spectrum when $V\\ge 2t_h$  [25].", "The existence of well-formed excitons is a preliminary condition of the enhancement of the CDW order parameter after a laser pumping.", "With further increase of $V$ , we can expect that at the edge of the absorption spectrum, charge-order states multiply and gradually dominate.", "They can thus be captured by a laser pulse with well tuned frequency and strength, as proposed by our numerical simulations.", "Although it is quite difficult to find a quasi-1D Mott insulator with proper $U$ and $V$ values that position the system near the phase boundary, there are other mechanisms which can take up the role of $V$ .", "When additional on-site Holstein phonons are taken into account, it has been shown that the system can be driven from SDW to CDW phase by electron-phonon interactions [26], [27].", "Representative materials near the phase boundary with substantial electron-phonon interaction can be found in tetracyanoquinodimethane (TCNQ) series [28].", "Another way to approach the phase boundary can be chemical substitutions, as in halogen-bridged transition metal compounds [29].", "Moreover, a transient phase transition from CDW to Mott or metallic phases has already been observed [30].", "These might prove to be useful in the search of proper candidates to observe the proposed effect.", "As an alternative to electronic systems, CDW enhancement can possibly be realized in cold atoms [31].", "Finally, we would like to emphasize that, while our work has concentrated on a particular CDW order enhancement induced by a laser pulse, the proposed mechanism may be effective in other cases when observing the response of correlated systems to an external perturbation near a phase boundary such as enhanced antiferromagnetic correlations, stripes, or possibly even superconducting fluctuations.", "This work was also supported by the Strategic Programs for Innovative Research (SPIRE), the Computational Materials Science Initiative (CMSI), the global COE program \"Next Generation Physics, Spun from Universality and Emergence\" from MEXT, the Yukawa International Program for Quark-Hadron Sciences at YITP, Kyoto University, and SLO-Japan collaboration project from ARRS and JSPS.", "T.T.", "acknowledges support by the Grant-in-Aid for Scientific Research (Grant No.", "22340097) from MEXT.", "J.B. acknowledges support by the P1-0044 of ARRS, and CINT user program, Los Alamos National Laboratory, NM USA.", "A part of numerical calculations was performed in the supercomputing facilities in YITP and ACCMS, Kyoto University, and ISSP in the University of Tokyo." ] ]

[ [ "The impact of the supersonic baryon-dark matter velocity difference on\n the z~20 21cm background" ], [ "Abstract Recently, Tseliakhovich and Hirata (2010) showed that during the cosmic Dark Ages the baryons were typically moving supersonically with respect to the dark matter with a spatially variable Mach number.", "Such supersonic motion may source shocks that heat the Universe.", "This motion may also suppress star formation in the first halos.", "Even a small amount of coupling of the 21cm signal to this motion has the potential to vastly enhance the 21cm brightness temperature fluctuations at 15<z<40 as well as to imprint acoustic oscillations in this signal.", "We present estimates for the size of this coupling, which we calibrate with a suite of cosmological simulations.", "Our simulations, discussed in detail in a companion paper, are initialized to self-consistently account for gas pressure and the dark matter-baryon relative velocity, v_bc (in contrast to prior simulations).", "We find that the supersonic velocity difference dramatically suppresses structure formation at 10-100 comoving kpc scales, it sources shocks throughout the Universe, and it impacts the accretion of gas onto the first star-forming minihalos (even for halo masses as large as ~10^7 Msun).", "However, we find that the v_bc-sourced temperature fluctuations can contribute only as much as ~10% of the fluctuations in the 21cm signal.", "We do find that v_bc could source an O(1) component in the power spectrum of the 21cm signal via the X-ray (but not ultraviolet) backgrounds produced once the first stars formed.", "In a scenario in which ~10^6 Msun minihalos reheated the Universe via their X-ray backgrounds, we find that the pre-reionization 21cm signal would be larger than previously anticipated and exhibit significant acoustic features.", "We show that structure formation shocks are unable to heat the Universe sufficiently to erase a strong 21cm absorption trough at z ~ 20 that is found in most models of the sky-averaged 21cm intensity." ], [ "Introduction", "Sometime between the redshifts of 30 and 10, it is thought that the Universe transitioned from a pristine landscape, where the distribution of matter was calculable from the cosmological initial conditions alone, to a vastly more complex system in which stars abound.", "Once these stars formed, within just a few hundred million years they had reheated and reionized the entire Universe.", "These cosmic times can be observed directly via redshifted 21cm radiation from neutral hydrogen.", "In fact, predictions are that the 21cm line is visible in absorption against the Cosmic Microwave Background (CMB) from $15\\lesssim z \\lesssim 40$ , with a larger amplitude for its mean brightness temperature than from any other cosmological epoch (e.g., [57], [24]).", "Motivated by these predictions, the radio instruments LEDA, DARE, and LOFAR are being designed to observe 21cm emission from the era prior to reionization, the so-called cosmic Dark Ages [13], [6], [31].www.cfa.harvard.edu/LEDA; http://lunar.colorado.edu/dare/; http://www.lofar.org/ These efforts are in addition to those aiming to detect the 21cm signal from the Reionization Epoch (targeting $z\\sim 6-12$ ), which includes EDGES, PAPER, MWA, LOFAR (with their high band antennae), and GMRT [11], [55], [31], [53].", "EDGES, LEDA, and DARE are targeting the sky-averaged 21cm signal, whereas the other instruments are attempting to measure fluctuations in the 21cm intensity on $\\sim 10~$ comoving Mpc scales.", "Plans for the next generation of instruments are also in development.https://www.cfa.harvard.edu/$\\sim $ lincoln/astro2010.hera.pdf; http://www.skatelescope.org/ Yet, despite the prospects for observations of 21cm emission from $10\\lesssim z \\lesssim 40$ , key theoretical questions regarding these times remain unanswered – even questions which are independent of the large uncertainties inherent to modeling the astrophysical processes that impact the 21cm signal.", "For example, it is unclear whether weak structure formation shocks would have significantly heated the cosmic gas [25], [23].", "In fact, [25] found that the reheating from structure formation shocks significantly impacted their simulated 21cm absorption signal.", "The sky-averaged 21cm brightness temperature, which LEDA and DARE aim to observe, is likely to be inversely proportional to the gas temperature at the time of emission, so that colder temperatures would result in a larger signal.", "Here we show that structure formation shocks are not likely to suppress this absorption signal, in contrast to [25].", "In addition, recently [68] argued that a previously unconsidered effect could impress large spatial fluctuations in the 21cm signal: At the time of recombination, the cosmic gas was moving with respect to the dark matter at a root mean square velocity of $30~$ km s$^{-1}$ and in a coherent manner on $\\lesssim 10$  comoving Mpc separations.", "These initial velocity differences translate into the dark matter moving through the gas with a typical Mach number of ${\\cal M}\\approx 2$ from $z = 150$ until the time the gas was reheated.", "Semi-analytic models predict that astrophysical backgrounds reheated the Universe somewhere between $z=10$ and 20 [22], [58].", "The baryon-dark matter supersonic motion could suppress the formation of the first stars [68], [42], [65], [47].", "It could also introduce spatially variable kinetic temperatures by the entropy generated in shocks that these supersonic flows sourced.", "Both of these effects would result in fluctuations in the 21cm signal.", "In fact, the 21cm correlation function may have a significantly larger amplitude on the $10-100~$ comoving Mpc scales that observations are most sensitive to if this differential velocity impacts the 21cm background [18].", "Already, many 21cm models that do not account for the [68] effect find that the fluctuations in the 21cm background from $z=20$ are likely to be no more difficult to detect than those from $z=10$ [57].", "Existing interferometric efforts to detect highly redshifted 21cm emission are primarily focused on $z= 6-10$ .", "If the signal from $z=20$ were further boosted by the dark matter–baryon velocity differential (a scenario the results of this paper support under specific circumstances), then more effort should be channeled towards detecting 21cm fluctuations from this earlier epoch.", "Furthermore, the power spectrum of the differential velocity features significant acoustic features similar to those in the CMB.", "These features could provide a more distinctive signature that would aid the separation of the 21cm signal from the foregrounds.", "The aim of this paper is to estimate the coupling strength of the 21cm signal to the dark matter–baryon velocity differential.", "To do so, this study presents semi-analytic estimates for the size of this coupling, which are calibrated with numerical simulations of the Dark Ages.", "Our simulations, described in detail in a companion paper [49], employ more physical cosmological initial conditions than the simulations used in prior studies.", "Unlike previous studies, our simulations are initialized with a transfer function that consistently incorporates the dark matter–baryon differential velocity.", "(On the scale of our simulations, $\\le 1~{\\rm Mpc}/h$ , this differential velocity is a uniform wind to an excellent approximation.)", "In Paper I, we show that these improvements provide a much better match to the linear evolution of cosmological perturbations.", "In addition, we run both the Enzo [50] and GADGET [64] cosmological codes, with many different box sizes and particle numbers/grid sizes, to explore the robustness of our results.", "This paper is organized as follows: Section provides a short introduction into the 21cm signal, its detectability, and introduces a parameterization for how this signal could couple to the dark matter–baryon velocity differential.", "We then provide estimates for how the 21cm signal is impacted by this velocity differential under various assumptions, using a suite of simulations to calibrate these assumptions (Section ).", "This study assumes a flat $\\Lambda $ CDM cosmological model with $\\Omega _m=0.27$ , $\\Omega _\\Lambda =0.73$ , $h=0.71$ , $\\sigma _8= 0.8$ , $n_s=0.96$ , $Y_{\\rm He} = 0.24$ , and $\\Omega _b = 0.046$ , consistent with recent measurements [38].", "We will subsequently abbreviate proper Mpc as pMpc and Mpc and kpc will be reserved for comoving lengths.", "Some of our calculations use the Sheth-Tormen mass function, for which we adopt the parameters $p = 0.3$ , $a=0.75$ , $A=0.322$ [62].", "While this study was nearing completion, [70] published a related study using semi-numeric methods.", "[70] investigated one model for how the baryon–dark matter velocity difference could impact the 21cm signal.", "We provide here a more complete census of the different scenarios where the relative velocity could contribute to the 21cm signal." ], [ "The Pre-reionization 21cm Signal", "This section provides a brief introduction to the temporal evolution of the redshifted 21cm signal.", "This introduction differs slightly from prior expositions (e.g., [22], [24]) in that it characterizes the transition in terms of the star formation rate density, $\\dot{\\rho }_{\\rm SFR}$ , a quantity that is also constrained by optical/infrared observations of the $z\\sim 10$ Universe.", "One goal of §REF is to motivate why it is likely for the IGM to appear with ${T_{\\rm K}}< {T_{\\rm CMB}}$ in 21cm absorption soon after the first stars formed (the situation this paper primarily investigates).", "A second goal is to understand whether minihalos could source the pre-reionization radiation background, which is the scenario that leads to the differential velocity having its largest impact on the 21cm signal.", "In addition, minihalo scenarios have largely (and we argue unjustly) been ignored in previous research on the extremely-redshifted 21cm background.", "Next, §REF provides an introduction to the fluctuating 21cm signal and also motivates why the velocity difference between the baryons and dark matter could lead to larger fluctuations than in previous models that had neglected this effect.", "Finally, §REF discusses the detectability of 21cm signals." ], [ "the temporal evolution of this signal", "The 21cm line of atomic hydrogen offers a brightness temperature contrast with respect to the CMB brightness temperature of $T_b^{21{\\rm cm}} &=& 41.5 \\, x_{\\rm H} \\,(1+\\delta _b) \\left(1 - \\frac{T_{\\rm CMB}(z)}{{T_{\\rm S}}} \\right) Z_{20}^{1/2}{\\rm ~ mK} ,\\\\&=& -216 \\, x_{\\rm H} \\,(1+\\delta _b) \\left( \\frac{1 - \\frac{T_{\\rm CMB}}{{T_{\\rm S}}}}{1 - \\frac{T_{\\rm CMB}}{T_{\\rm K}^{\\rm ad}}}\\right) Z_{20}^{1/2} {\\rm ~~ mK}, $ where $T_{\\rm CMB}(z)$ is the CMB temperature at redshift $z$ , $Z_{20} \\equiv (1+z)/21$ , ${T_{\\rm S}}$ is the spin temperature of the 21cm line [21], $x_{\\rm H}$ is the neutral hydrogen fraction, and $\\delta _b$ is the overdensity in baryons.", "Note that for ensuing expressions we will sometimes drop the superscript “$21{\\rm cm}$ ” in $T_b^{21{\\rm cm}}$ .", "Equation () was evaluated for ${T_{\\rm S}}$ equal to the kinetic temperature of the gas prior to reheating and after thermal decoupling or $T_{\\rm K}^{\\rm ad} = 0.021 \\,(1+z)^2~$ K. This equation demonstrates that the absolute brightness temperature of the 21cm signal is likely to peak at times when the gas was kinetically cold.", "The spin temperature of the 21cm transition interpolates between the gas temperature, ${T_{\\rm K}}$ , and ${T_{\\rm CMB}}(z)$ as [21] ${T_{\\rm S}}^{-1} = \\frac{{T_{\\rm CMB}}^{-1} + T_{\\rm K}^{-1} (x_\\alpha + x_c)}{1+ x_\\alpha + x_c},$ where $x_c$ describes how well particle collisions couple ${T_{\\rm S}}$ to ${T_{\\rm K}}$ .", "This coefficient is effectively zero in the low-density IGM at $z<40$ (the redshifts we focus on), and $x_\\alpha \\approx 1.8 \\times 10^{11} \\, (1+z)^{-1} J_\\alpha $ parametrizes the efficacy of Ly$\\alpha $ scattering at pumping the 21cm transition.", "This scattering couples ${T_{\\rm S}}$ to ${T_{\\rm K}}$ via the Wouthuysen-Field mechanism [21].", "Equation (REF ) assumes c.g.s.", "units for $J_\\alpha $ , where $J_\\alpha $ is the average photon specific intensity at the frequency of Ly$\\alpha $ and is given by (e.g., [56]) $J_\\alpha &=& \\sum _{n=2}^{\\infty } f_{\\rm rec}(n) \\int _z^{z_{\\rm max}(n)} dz^{\\prime } \\frac{(1+z)^2}{4\\pi } \\frac{c}{H(z)} \\epsilon (\\nu _n^{\\prime }, z^{\\prime }),~~~~\\\\ &\\approx & \\frac{5}{27}\\frac{c \\,(1+z)^{3}}{4 \\pi H(z)} \\; \\epsilon _*(z).$ Here, $H(z)$ is the Hubble expansion rate, $\\epsilon $ is the spatially-averaged comoving specific emissivity, $f_{\\rm rec}(n)$ is the probability that an absorption into the $n^{\\rm th}$ Rydberg level of atomic hydrogen – and the resulting radiative cascade – produces a Ly$\\alpha $ photon [56], $z_{\\rm max}$ is the maximum redshift at which a photon could have been produced that was absorbed into the $n^{\\rm th}$ level at $z$ , and $\\nu _n^{\\prime } = \\nu _n (1+z^{\\prime })/(1+z)$ .", "The approximation that yields equation () includes only the $n=2$ contribution to the summation: Photons that redshift into the Ly$\\beta $ resonance contribute negligibly to the pumping of $T_{\\rm S}$ (these photons are destroyed after a few scatterings) and higher Lyman resonances than Ly$\\alpha $ are pumped by photons in a more restricted spectral range.", "Equation () also makes the further simplification that the specific emissivity is frequency-independent between 1 and $3~$ Ry with value $\\epsilon _*$ .", "The evolution of the 21cm brightness temperature is thought to be determined by the production of ultraviolet photons that pump the 21cm line (affecting $x_\\alpha $ ), X-ray heating of the intergalactic gas ($T_{\\rm K}$ ), and the reionization of hydrogen ($x_{\\rm H}$ ).", "In what follows, we discuss the order in which these different processes likely impacted the 21cm line: Ly$\\alpha $ pumping: Given equations (REF ), (REF ), and the number of $0.75-1~$ Ry photons produced per baryon incorporated in stars, $N_{\\alpha }$ , we can solve for the comoving star formation rate density required to satisfy $x_\\alpha = 1$ and thereby couple ${T_{\\rm S}}$ to ${T_{\\rm K}}$ : $[\\dot{\\rho }_{\\rm SFR}]_{\\alpha } = 1.7 \\times 10^{-3} \\, Z_{20}^{-1/2} \\left(\\frac{N_{\\alpha }}{10^4}\\right)^{-1}~{\\rm {M_{\\odot }}~yr^{-1} ~{\\rm Mpc}^{-3}}.$ Stellar population synthesis calculations find $N_{\\alpha } \\approx 10^4$ for Pop II stars with a standard initial mass function (IMF; [39]) and $N_{\\alpha } \\approx 5000$ for a top-heavy Pop III stellar population [12].", "Let us estimate the epoch at which $[\\dot{\\rho }_{\\rm SFR}]_{\\alpha }$ is surpassed, assuming that the fraction $f_\\star $ of the baryons in halos above mass $m_h$ are incorporated into stars such that $\\dot{\\rho }_{\\rm SFR} = f_\\star \\, \\bar{\\rho }_b \\, df_{\\rm coll}(m_h)/dt $ .", "Here, $f_{\\rm coll}(m_h)$ is the fraction of matter that has collapsed in halos more massive than $m_h$ , for which we use the Sheth-Tormen halo mass function.", "For $f_\\star = \\lbrace 0.1\\%, 1\\%, 10\\%\\rbrace $ , the characteristic star formation rate, $[\\dot{\\rho }_{\\rm SFR}]_{\\alpha }$ , would have been encountered at $\\lbrace 6,~ 15,~ 19\\rbrace $ if star formation traced the mass in halos that could cool atomically.", "These estimates assumed that such halos have virial temperatures of $T_{\\rm vir} > 10^4~$ K, yielding halo masses of $m_H > 3\\times 10^7 \\, Z_{20}^{-3/2}$ .", "Most previous discussions of 21cm radiation have assumed that halos that cool atomically are the main driver of the Ly$\\alpha $ pumping [25], [22].", "However, stars in smaller “minihalos” (halos that can only cool by molecular hydrogen transitions) may contribute to the production of ultraviolet photons.", "Section will show that $\\sim 10^6~{M_{\\odot }}$ halos must generate much of the $0.75-1~$ Ry photons for the differential dark matter–baryon differential velocity to modulate the local star formation rate in a detectable manner.", "Minihalos with circular velocities of at least $3.7~$ km s$^{-1}$ are massive enough for the gas to be able to cool by molecular hydrogen [67], [1], [40], and they correspond to halo masses of $> 3.5\\times 10^5\\, Z_{20}^{-3/2}~{M_{\\odot }}$ .", "Such halos would have achieved $x_\\alpha = 1$ at $z = \\lbrace 13,~ 22,~ 28\\rbrace $ for $f_\\star = \\lbrace 0.1\\%, 1\\%, 10\\%\\rbrace $ .", "If instead the bulk of the star formation occurred in more robust $1.5\\times 10^6\\, Z_{20}^{-3/2}~{M_{\\odot }}$ , $6~$ km s$^{-1}$ halos, $x_\\alpha = 1$ would have occurred at somewhat lower redshifts of $z = \\lbrace 11, 20, 25\\rbrace $ .", "The Lyman-Werner band background (e.g., the spectral band $11.2-13.6$  eV) acts to destroy the molecular hydrogen and sterilize star formation, especially in the lowest mass minihalos [30], [29].", "The numerical calculations of [40], which were confirmed in [71], found that $4\\%$ of the gas is able to cool (and collapse to much higher densities) in halos with mass $m_{\\rm crit} = 2.5 \\times 10^{5} + 8.7 \\times 10^{5} \\, F_{LW, 21}^{0.47}~~~{M_{\\odot }},$ where $F_{LW, 21}$ is the Lyman-Werner intensity integrated over solid angle in units of $10^{-21} ~{\\rm erg~s^{-1}~Hz^{-1}~sr^{-1}}$ .", "[52] found a similar relation, but with a $\\sim 3$ times higher normalization.", "The critical star formation rate to raise $m_{\\rm crit}$ turns out to be much less than $[\\dot{\\rho }_{\\rm SFR}]_{\\alpha } $ : $x_\\alpha = 1$ requires a radiation background in this band with intensity of $F_{\\alpha , 21} = 4 \\pi h \\nu _{\\rm {\\rm Ly} \\alpha }\\,(1+z)/ (1.8 \\times 10^{11})\\times 10^{21}$ in the same units as $F_{LW, 21}$ (e.g., eqn.", "REF ).", "Plugging in the numbers, $x_\\alpha = 1$ yields a Lyman-Werner intensity of $F_{LW, 21} \\approx 20 \\,Z_{20} \\, \\exp [-\\tau _{LW}],$ noting that $F_{LW, 21} \\sim F_{\\alpha , 21}$ and $\\tau _{LW}$ is the typical intergalactic opacity for Lyman-Werner photons which can can be $1-2$ in the absence of dissociations [60] and can be larger once the first HII regions have formed [35].", "Once $x_\\alpha = 1$ and assuming equation (REF ), the Lyman-Werner background is sufficient to suppress cooling in halos with masses $<4 \\times 10^6 \\exp [-0.47 \\tau _{LW}]~{M_{\\odot }}$ .In addition, if an appreciable number of solar mass stars form, these produce infrared radiation that dissociates H$^-$ , preventing the formation of molecular hydrogen.", "[73] found that such an infrared background becomes more effective at preventing star formation than the Lyman-Werner background if $\\sim $ solar mass stars comprise $>90\\%$ of the stellar mass.", "This would require the high-redshift IMF to be more bottom heavy than the $z=0$ IMF.", "[72] studied the star formation rate density in cosmological simulations that followed Pop III star formation in minihalos, including self-consistently Lyman-Werner radiation backgrounds.", "They found that star formation rate densities sufficient for $x_\\alpha =1$ occurred by $z= 22-25$ in their simulations.", "X-ray heating: Another critical juncture in the evolution of the 21cm signal occurred when ${\\dot{\\rho }_{\\rm SFR}}$ was sufficient for X-rays to have heated the gas above the CMB temperature.", "Penetrating X-rays are likely the most efficient mechanism for reheating the IGM [16].", "However, relating X-ray production to the star formation rate (SFR) is more uncertain than relating ultraviolet emission to SFR since X-ray production depends on the abundances of X-ray binaries and supernovae.", "To do so, we follow the methodology taken in [22], using relations calibrated on low-redshift galaxies between X-ray luminosity and the SFR.", "In particular, the critical SFR density to heat the IGM with X-rays by an amount of $T_{\\rm CMB}(z)$ is $[\\dot{\\rho }_{\\rm SFR}]_{\\rm X} &=& 4.0\\times 10^{-2} \\, Z_{20}^{5/2}\\,\\left( \\frac{t_{\\rm SFR}}{0.1 \\,t_H} \\right)^{-1} \\, \\left(\\frac{f_X}{0.2} \\, \\right)^{-1} \\\\& \\times & \\left(\\frac{L_X/{\\rm SFR}}{10^{40} {\\rm ~erg~ s^{-1}~ {M_{\\odot }}^{-1} ~yr}} \\right)^{-1} ~{\\rm {M_{\\odot }}~yr^{-1} ~Mpc^{-3}}, \\nonumber $ where $f_X$ is the fraction of energy that heats the IGM [63], $L_X/{\\rm SFR}$ is the $\\sim 0.1-2~$ keV luminosity per unit SFR, $t_{\\rm SFR}$ is the timescale over which the emitting population had been active, and $t_H = H(z)^{-1}$ .", "Equation (REF ) evaluated $L_X/{\\rm SFR}$ at $10^{40} {\\rm ~erg~ s^{-1}~ {M_{\\odot }}^{-1} ~yr}$ , which is a factor of $\\sim 5$ higher than low redshift measurements for the same relationship except between $2-10~$ keV [28], [45].The spectral index of the X-ray emission is uncertain, but empirical determinations at low-redshifts are consistent with having equal energy per log in frequency [59], [66].", "Low-redshift X-ray emission that traces star formation is dominated by high-mass X-ray binaries.", "Interestingly, there is no evidence for evolution in the $L_X/{\\rm SFR}$ , even to $z\\approx 6$ [17].", "Using the Sheth-Tormen mass function, the $e$ -folding time, $[d \\log f_{\\rm coll}(m_h)/dt]^{-1}$ , is $0.11$ and $0.06$ of a Hubble time at $z=20$ for $m_h =10^6~{M_{\\odot }}$ and $m_h =10^8~{M_{\\odot }}$ , respectively.", "Is it possible for minihalos to have also dominated the X-ray reheating of the Universe?", "For our fiducial parameters, $F_{LW, 21}$ would have been even larger by a factor of 20 when $[\\dot{\\rho }_{\\rm SFR}]_{\\rm X}$ was satisfied than when $[\\dot{\\rho }_{\\rm SFR}]_{\\alpha }$ was.", "Larger $f_X$ will reduce the resulting $F_{LW, 21}$ , making it more difficult for the Lyman-Werner background to sterilize minihalos.", "However, X-rays can also catalyze the formation of $H_2$ , combatting its destruction via the Lyman-Werner background [30].", "We find that at the epoch when $[\\dot{\\rho }_{\\rm SFR}]_{\\rm X}$ is satisfied the electron fraction is increased by a factor of 10 over the relic electron fraction at the cosmic mean density.", "A 10 times higher electron fraction means that 10 times more molecular hydrogen will form, such that a 10 times larger Lyman-Werner background is required to yield the same $H_2$ fraction.", "However, this estimate represents an upper bound on how much the critical $F_{LW, 21}$ of a halo would be increased by $X$ -rays, as the recombination time is $\\sim 1$ Hubble time for $\\delta _b = 200$ gas with the relic electron fraction at $z=20$ .", "Thus, it is difficult to significantly boost the electron fraction in dense, star forming regions with X-rays.", "In fact, [41] found in simulations that an X-ray background only mildly promotes the formation of molecular hydrogen in halos.", "Ionizations: The final effect that stars have on the 21cm signal is via their ionizations.", "If a stellar population produces $N_{\\rm ion}$ hydrogen ionizing photons per stellar baryon, the critical SFR density required to reionize the Universe to an ionized fraction of $x_i$ is $[\\dot{\\rho }_{\\rm SFR}]_{\\rm ion} &= & 4.4 \\times 10^{-1} \\, \\bar{x}_i \\, Z_{20}^{3/2} \\,\\left( \\frac{t_{\\rm SFR}}{0.1 \\,t_H} \\right)^{-1} \\nonumber \\\\& & \\times \\left(\\frac{f_{\\rm esc}}{0.1} \\frac{N_{\\rm ion}}{4000}\\right)^{-1} ~{\\rm {M_{\\odot }}~yr^{-1} ~Mpc^{-3}},$ where $f_{\\rm esc}$ is the fraction of ionizing photons that escape from their sites of production into the IGM.", "The factor $f_{\\rm esc}$ is highly uncertain (e.g., [37]) and likely to be $\\ll 1$ .", "For Pop II stars with a Scalo initial mass function (IMF), $N_{\\rm ion} \\approx 4000$ for $Z = 0.05 \\, Z_$ [3].", "This number varies at the factor of 2-level when changing assumptions regarding the metallicity and the IMF, at least for empirically-determined IMFs.", "However, Pop III stars with a top-heavy IMF are much more efficient producers of ionizing photons, with $N_{\\rm ion} \\approx 40, 000$ [12].", "Figure: Model history of the gas temperature and spin temperature (top panel) and of the mean 21cm brightness temperature (bottom panel) for the parameters N α =10 4 N_\\alpha = 10^4, f X =1f_X = 1, f * =0.02f_* = 0.02, N ion =4000N_{\\rm ion} = 4000, and f esc =0.1f_{\\rm esc} = 0.1, assuming that star formation traces the mass in atomic cooling halos.", "The shaded regions qualitatively delineate the phases where different radiation backgrounds drive the signal: first radiative pumping by far ultraviolet photons, then heating by penetrating X-ray ionizations, and lastly reionization by extreme ultraviolet photons.", "The LEDA and DARE instruments aim to constrain this signal between 10≲z≲3010 \\lesssim z \\lesssim 30.Thus, we find $[\\dot{\\rho }_{\\rm SFR}]_{\\alpha } \\ll [\\dot{\\rho }_{\\rm SFR}]_{\\rm X} \\ll [\\dot{\\rho }_{\\rm SFR}]_{\\rm ion}$ in agreement with the progression found in [22].", "With this ordering, radiation from star formation first coupled the spin temperature to the gas temperature such that the 21cm signal would appear in absorption.", "Next, radiation associated with star formation reheated the Universe, and, lastly, ultraviolet radiation from stars reionized the cosmic gas.", "In fact, [58] demonstrated that one could change both $N_\\alpha $ and $f_X$ by orders of magnitude and still have a phase in which the 21cm signal was in absorption owing to Ly$\\alpha $ coupling ${T_{\\rm S}}$ to ${T_{\\rm K}}$ at times prior to X-ray reheating above ${T_{\\rm CMB}}$ .", "The amplitude of the 21cm signal is larger during this phase compared to its amplitude at other times (eqn. ).", "The discussion in this paper focuses on this phase.", "Figure REF shows the history evolution of ${T_{\\rm K}}$ , ${T_{\\rm S}}$ and the 21cm brightness temperature for the parameters $f_X = 1$ , $f_* = 0.02$ , and $f_{\\rm esc} = 0.1$ (defined in eqn.s REF , REF , and REF ), and assuming that star formation traces atomic cooling halos.", "The shaded regions qualitatively delineate the phases where different processes drive the signal.", "In this model, the 21cm signal appears in absorption until $z\\approx 11.5$ (bottom panel, Fig.", "REF ).", "The temperature history for a model in which minihalos dominate the backgrounds can be similar, but the minihalo case is more likely to be important as redshift increases.", "Our discussion has ignored shock heating as a contribution to the gas temperature.", "The Universe could heat owing to structure formation shocks.", "Such heating would also suppress the absorption signal that occurs at $z \\approx 12-22$ in Figure REF .", "Shock heating would be unfortunate as this dip (the sharpest and strongest feature that is expected in the sky-averaged 21cm signal) is being targeted by the 21cm instruments LEDA and DARE.", "Its presence is also important for the effects studied in this paper.", "In the only previous numerical study of this dip's existence, [25] found that it was significantly impacted by shock heating.", "Fortunately, our simulations (which are better suited for this calculation than those in [25]) show that shock heating does not significantly impact the temperature of the intergalactic gas at these redshifts (see Appendix ).", "Relating the critical ${\\dot{\\rho }_{\\rm SFR}}$ derived in this section to the ${\\dot{\\rho }_{\\rm SFR}}$ measured in rest-frame ultraviolet observations is also helpful for gauging when these different ${\\dot{\\rho }_{\\rm SFR}}$ thresholds are satisfied.", "In particular, [9] found ${\\dot{\\rho }_{\\rm SFR}}\\approx 10^{-2}~{M_{\\odot }}~{\\rm Mpc}^{-3}~{\\rm yr}^{-1}$ at $z=8$ as well as that ${\\dot{\\rho }_{\\rm SFR}}$ was decreasing quickly with increasing redshift.", "A star formation rate density of $10^{-2}~{M_{\\odot }}~{\\rm Mpc}^{-3}~{\\rm yr}^{-1}$ is only just sufficient to reionize the Universe in one Hubble time for $f_{\\rm esc} =0.5$ – a value that is higher than anticipated –, but it is more than sufficient to satisfy $[\\dot{\\rho }_{\\rm SFR}]_{\\alpha }$ .", "At face value, the rest-frame ultraviolet determinations of ${\\dot{\\rho }_{\\rm SFR}}$ also suggest that the ${\\dot{\\rho }_{\\rm SFR}}$ thresholds outlined in this section are satisfied at relatively low redshifts.", "However, it is likely that faint, dwarf galaxies that are missed by the [9] observations contribute significantly to the true value of ${\\dot{\\rho }_{\\rm SFR}}$ at $z\\gtrsim 8$ [10], [37]." ], [ "Fluctuations", "Interferometric efforts targeting redshifted 21cm radiation are not directly sensitive to the mean 21cm signal, but instead to spatial fluctuations in the 21cm brightness temperature.", "In the limit of small fluctuations (appropriate for the models we consider), the observed brightness temperature contrast with respect to ${T_{\\rm CMB}}$ can be approximated as (e.g., [24]) $\\delta T_b^{21{\\rm cm}} \\approx ~~\\bar{T}_b^{21{\\rm cm}} \\, &\\bigg (& 1+\\delta _b + \\frac{1}{1+\\bar{x}_\\alpha }\\delta _\\alpha \\nonumber \\\\& +& \\frac{T_{\\rm CMB}}{\\bar{T}_{\\rm K} - T_{\\rm CMB}}\\delta _T - \\delta _{{\\mathbf {\\nabla }}v} \\bigg ),$ where $\\bar{T}_b^{21{\\rm cm}}$ is the spatial average of equation (REF ) and bars over other quantities also denote a spatial average.", "Equation (REF ) is valid at $z \\lesssim 40$ , once collisions can no longer pump the 21cm line, and prior to reionization.", "Respectively, $\\delta _b$ , $\\delta _\\alpha $ , $\\delta _T$ , and $\\delta _{{\\mathbf {\\nabla }}v}$ are the overdensities in baryons, Ly$\\alpha $ radiation, temperature, and the line-of-sight derivative of the line-of-sight proper velocity (arising from redshift-space distortions).", "At lowest order, the overdensities in the temperature and in the intensity at the Ly$\\alpha $ resonance can be related to the overdensity in baryons and the square of the baryon–dark matter velocity difference, $v_{\\rm bc}^2$ , as $\\delta _\\alpha &=& \\left(b_{\\delta , \\alpha }\\, \\delta _b + b_{v^2, \\alpha } \\, \\delta _{v^2} \\right) \\star W_\\alpha , \\\\\\delta _T &=& \\left(b_{\\delta , T}\\, \\delta _b + b_{v^2, T} \\, \\delta _{v^2} \\right),$ where $\\delta _{v^2} \\equiv \\left[v_{\\rm bc}^2({x})/\\sigma _{\\rm bc}^2 - 1\\right].$ [18] demonstrated that the expansion to first order in $\\delta _{v^2}$ is a good approximation if $\\delta _\\alpha $ and $\\delta _T$ are local functions of $|{v}_{\\rm bc}|$ , at least on scales where fluctuations in $v_{\\rm bc}^2$ are much less than unity.", "In addition, $b_{\\delta , \\alpha }$ is the bias of star forming regions, $b_{\\delta , T}$ is the bias of density-tracing temperature fluctuations (equal to $2/3$ for adiabatic evolution)Both temperature “biases” formally enter in convolution at times when X-ray heating was important, reflecting the propagation of X-ray photons.", "and $b_{v^2, \\alpha }$ [$b_{v^2, \\alpha }$ ] is the bias with which star formation [heating] traces $\\delta _{v^2}$ .", "Lastly, $\\sigma _{\\rm bc}^2 \\equiv \\langle v_{\\rm bc}^2 \\rangle _v$ , where $\\langle ... \\rangle _v$ denotes an ensemble average over the Maxwell-Boltzmann probability distribution of $v_{\\rm bc}$ .", "In addition, $W_\\alpha (r)$ describes how Ly$\\alpha $ photons travel from a point source to a distance $r$ away and is normalized to have unit norm, entering equation (REF ) in convolution.", "It is given by a similar expression to equation () [5].", "Interferometric 21cm fluctuation efforts aim to measure the power spectrum of the 21cm brightness temperature fluctuations, $P_{21}({k}) = \\langle |\\delta \\tilde{T}_b({k})|^2 \\rangle $ , where tildes signify the Fourier dual.Any significant contribution to the power spectrum from $v_{\\rm bc}$ also means the signal is highly non-Gaussian such that higher-order statistics such as the trispectrum will be easier to detect than previously anticipated [74].", "However, we concentrate on the power spectrum here.", "Combining equations (REF ), (REF ), and () yields $& &\\bar{T}_b^{-2} P_{21}({k}) \\approx \\nonumber \\\\&& ~~~\\left(1+ \\, \\frac{b_{\\delta , \\alpha }\\, \\tilde{W}_\\alpha (k)}{1+\\bar{x}_\\alpha } + \\frac{T_{\\rm CMB} \\, b_{\\delta , T}}{\\bar{T}_{\\rm K} - T_{\\rm CMB}} + \\mu ^2 \\right)^2 P_\\delta (k) \\nonumber \\\\& &~~~ + \\left(\\frac{b_{v^2, \\alpha }\\, \\tilde{W}_\\alpha (k)}{1+\\bar{x}_\\alpha } + \\frac{T_{\\rm CMB} \\,b_{v^2, T}}{\\bar{T}_{\\rm K} - T_{\\rm CMB}} \\right)^2 P_{v^2}(k), $ where $P_\\delta \\equiv \\langle |\\tilde{\\delta }(k)|^2 \\rangle $ and $P_{v^2}\\equiv \\langle |\\tilde{\\delta }_{v^2}(k)|^2 \\rangle $ .", "We are working in the limit in which the baryons trace the matter such that $\\delta _b = \\delta $ , and we have dropped the much smaller terms that are $\\propto \\langle \\tilde{\\delta }(k)^2 \\tilde{\\delta }_{v^2}(k) \\rangle $ , even though they formally are the same order as $P_{v^2}$ .", "The terms with $\\mu ^2$ owe to redshift-space distortions [36], where $\\mu = {\\hat{n}}\\cdot {k}/k$ and ${\\hat{n}}$ is a unit vector that points along the line-of-sight.", "Previous analyses aside from [18] had not included the $P_{v^2}$ contribution to $P_{21}$ .", "This new term's amplitude peaks at smaller $k$ than $P_\\delta $ .", "In fact, even if the coefficient that couples it to $P_{21}$ is $10^4$ times smaller than the analogous coefficient for $P_\\delta $ , it still would contribute comparable power to this other term at $k = 0.1~$ Mpc$^{-1}$ – roughly the scale at which 21cm experiments are most sensitive [44], [54].", "The Fourier transform of $P_{v^2}$ is equal to $\\left\\langle \\delta _{v^2}({x}) \\, \\delta _{v^2} ({x}+{r}) \\right\\rangle = \\frac{4}{9} \\, \\psi _1(r)^2 + \\frac{2}{9}\\, \\left[\\psi _1(r) + \\psi _2(r) \\right]^2 ,$ where $\\psi _1(r) &\\equiv & \\frac{3}{\\sigma _{\\rm bc}^2}\\int _0^\\infty \\frac{k^2 dk}{2 \\pi ^2} P_{v}(k) \\frac{j_1(k r)}{k r},\\\\\\psi _2(r) &\\equiv & \\frac{3}{\\sigma _{\\rm bc}^2}\\int _0^\\infty \\frac{k^2 dk}{2 \\pi ^2} P_{v}(k) \\ j_2(k r),\\\\\\sigma _{\\rm bc}^2 &=& \\int _0^\\infty \\frac{k^2 dk}{2 \\pi ^2} P_{v}(k) ,$ $P_{v} \\equiv \\langle |\\tilde{v}_{\\rm bc}({k})|^2\\rangle $ , and $\\tilde{v}_{\\rm bc}({k}) = - i \\frac{a \\, {k}}{k^2} \\left[\\dot{T_c}(k, a) - \\dot{T_b}(k, a)\\right] \\tilde{\\delta }_{\\rm pri}.$ A dot over a variable signifies differentiation with respect to time.", "$T_c$ and $T_b$ are the baryonic and dark matter transfer functions that map from the primordial overdensity, $\\delta _{\\rm pri }$ , to the overdensity in these components.", "Equation (REF ) can be derived from equation (REF ) using Wick's theorem and the correlation function between the cartesian components of the velocity field: $\\left\\langle v_i({x}) v_j({x}+{r}) \\right\\rangle _v = \\frac{\\sigma _{\\rm bc}^2}{3} \\, \\left( \\psi _1(r) \\, \\delta _{ij}^{\\rm K} + \\psi _2(r) \\, \\frac{r_i r_j}{r^2} \\right),$ where $\\delta _{ij}^{\\rm K}$ is the Kronecker delta.", "One of the primary objectives of this paper is to calculate the contribution of the term proportional to $P_{v^2}$ to $P_{21}$ , namely to calculate $b_{v^2,\\alpha }$ and $b_{v^2, T}$ in equation (REF ).", "An estimate for $b_{v^2,\\alpha }$ : If star formation traces the amount of matter that collapses into halos above some mass cutoff, $f_{\\rm coll}$ , [18] found that $b_{v^2, \\alpha } \\approx \\frac{3}{2} \\left(\\frac{\\langle v_{\\rm bc}^2 f_{\\rm coll} \\rangle _v}{ \\sigma _{\\rm bc}^2 \\langle f_{\\rm coll} \\rangle _v} -1 \\right),$ provided a good fit to the large-scale bias found in a full calculation for the source clustering in the model they considered.", "An alternative bias coefficient comes from equating the variance in $f_{\\rm coll}$ as a function of $v_{\\rm bc}$ to $\\langle \\delta _{v^2}^2 \\rangle _v$ (which equals $2/3$ ), and is given by $b_{v^2, \\alpha } \\approx \\sqrt{\\frac{3}{2}}\\, \\left( \\frac{\\langle f_{\\rm coll}^2 \\rangle _v}{\\langle f_{\\rm coll} \\rangle _v^2} -1 \\right)^{1/2}.$ This alternative bias should hold to the extent that the $v_{\\rm bc}$ –driven fluctuations are linear on all scales.", "Such a condition does not hold for the density field at $z\\lesssim 20$ , but is more valid for $\\delta _{v^2}$ .", "The exercise of plugging $f_{\\rm coll}(v_{\\rm bc}) \\rightarrow \\langle f_{\\rm coll} \\rangle _v \\, (1+ b_{v^2, \\alpha } \\, \\delta _{v^2})$ into the right-hand-side of equations (REF ) and (REF ) reveals that both expressions for the bias are exact in the limit that higher than linear order terms are subdominant.", "We find that either choice of bias agrees within $\\sim 10\\%$ among the models we consider.", "However, because the calculations in [18] find that the bias given by equation (REF ) fares excellently even in the somewhat nonlinear cases they considered, our subsequent calculations will use equation (REF ) for $b_{v^2, \\alpha }$ and for $b_{v^2, T}$ when radiation backgrounds source the fluctuations.", "An estimate for $b_{v^2,T}$ : We derive $b_{v^2, T}$ with a similar logic to how equation (REF ) was derived: If temperature-overdensity fluctuations were coupled to $v_{\\rm bc}^2$ with root mean square (RMS) fluctuation amplitude of $\\sigma _T$ when averaging over regions, then $b_{v^2, T} = \\sqrt{\\frac{3}{2}} \\, \\sigma _T,$ in order for the variance of the field $b_{v^2, T} \\, \\delta _{v^2}$ to equal $\\sigma _T$ .", "As an example of heating, if the fraction $\\beta $ of $v_{\\rm bc}^2$ were locally thermalized in shocks (such that the region was heated by the factor $1+ 5\\beta {\\cal M}^2/9$ ), $b_{v^2, T} = \\frac{5 \\beta }{9} \\, \\left\\langle {\\cal M}_{\\rm bc}^2 \\right\\rangle _v / \\left(1+\\frac{5\\beta }{9} \\left\\langle {\\cal M}_{\\rm bc}^2 \\right\\rangle _v \\right).$" ], [ "detectability", "It is not necessarily more difficult to detect the fluctuating 21cm signal from the era of focus here, $z\\sim 20$ , than from $z \\approx 8$ – the epoch most efforts are targeting: The absolute brightness temperature of the signal is likely to be much larger at $z\\sim 20$ than at $z\\sim 8$ , which compensates for the increased brightness temperature of the foregrounds [57].", "For subsequent estimates of the 21cm fluctuating signal, we quantify its potential to be detected by considering an interferometric configuration similar to the Murchison Widefield array (MWA) but that is optimized to the higher redshifts in question (by dilating the sizes of the dipoles and baselines).", "We also consider an interferometer with 10 times the collecting area that is a scaled up replica of the MWA.", "See Appendix for additional details regarding these hypothetical instruments.", "The statistical sensitivities of these instruments (excluding sample variance) to $P_{21}$ measured in spherical shells of width $\\Delta k = 0.2 \\,k$ are shown in the ensuing 21cm power spectrum figures.", "These calculations assume an observing time of $1000~$ hr and a bandwidth of $5~$ MHz.", "They also assume that the foregrounds can be removed from all wavevectors that do not utilize the zero mode along the line-of-sight such that the sensitivity at these wavevectors is limited by thermal noise [44].", "For $B = 5~$ MHz, an instrument can at best observe $k>0.05~$ Mpc$^{-1}$ , corresponding to the fundamental line-of-sight wavenumber for the chosen bandwidth.", "Larger bandwidths (which allow a measurement at smaller $k$ ) are desirable to probe the $k$ where $P_{v^2}$ has its largest impact on the $P_{21}(k)$ .", "However, the bandwidth (redshift) interval must also be chosen so that there is not significant temporal evolution in the signal across the band.", "The amount of collapsed mass evolves on a timescale of $\\Delta z/(1+z) \\sim 0.1$ (Section REF ), which is comparable at $z=20$ to the $\\Delta z$ covered by a bandwidth of $5~$ MHz.", "Thus, we expect the amount of star formation evolves substantially across the redshifts covered in this instrumental band.", "This evolution would result in spurious large-scale power at the $[T_b^{\\rm 21cm}]^2$ –level, which dwarfs the signal we are discussing.", "Thus, extracting the $v_{\\rm bc}$ –induced signal from the lowest $k$ to which a survey is sensitive will be challenging (which happen to be where the effects of $v_{\\rm bc}$ are most prominent).", "This section provides analytic estimates for the imprint of $v_{\\rm bc}$ on the 21cm signal.", "These estimates are calibrated with the cosmological simulations discussed in Paper I.", "They also require a calculation for how $v_{\\rm bc}$ alters the linear growth of density fluctuations.", "To do so, we solve the linear system of equations given in Paper I, and originally in [68], for the growth of modes in the presence of a nonzero $v_{\\rm bc}$ .", "This calculation is initialized with $z=1000$ transfer functions from the CAMB code.http://camb.info/ It assumes an instantaneous kinetic decoupling of the baryons from the CMB at this redshift (as motivated in § 5.3.1 in [33] and [19]) to solve for the growth of structure in two gravitationally coupled fluids, the dark matter and the baryons.", "This approximation for the growth of modes avoids the full Boltzmann code calculation.", "We find that solutions with this approximation excellently reproduce the growth of modes in CAMB for the case CAMB solves, $v_{\\rm bc} = 0$ .", "Figure REF features the results of this calculation for $z=20$ , showing the small-scale power spectrum of the total matter (top panel) and the baryons (bottom panel).", "A nonzero $v_{\\rm bc}$ acts as an effective pressure in the baryons, decreasing the $k$ above which pressure smooths fluctuations and in an anisotropic fashion.", "The linear variance in the baryonic density contrast is altered at the $\\sim 20 \\%$ level at $z=20$ by $v_{\\rm bc}$ .", "Nonlinear evolution further amplifies these differences." ], [ "temperature fluctuations from structure formation and shocking", "It is possible that shocking or structure formation changed the temperature of the Universe between regions with different $v_{\\rm bc}$ sufficiently to result in an enhancement of the 21cm signal.", "The period when such heating is most likely to source a significant component in the 21cm signal is after Ly$\\alpha $ pumping started to couple the spin temperature to the temperature of the gas but before the gas was reheated above the temperature of the CMB by X-rays.", "Figure REF illustrates how the 21cm power spectrum, $\\Delta T_b(k)^2 \\equiv k^3 P_{21}/ (2 \\pi ^2)$ , would be affected if $0.33$ , 1, and $3.3\\%$ of the energy in $v_{\\rm bc}$ were thermalized using equation (REF ) and assuming $x_\\alpha \\gg 1$ .", "(The relative amplitude of the curves do not depend on $x_\\alpha $ .)", "If just a small fraction of the kinetic energy in $v_{\\rm bc}$ were thermalized, this heating would lead to large temperature fluctuations and a larger 21cm signal than has been anticipated in models that ignore $v_{\\rm bc}$ .", "We showed in Paper I that the gaseous dynamical friction timescale is shorter than the Hubble time for dark matter overdensities with masses greater than $10^4~{M_{\\odot }}$ – roughly halos massive enough to significantly perturb the gas density.", "The short dynamical friction time results in the streaming gas decelerating into the dark matter potential wells and in visually apparent shocks throughout the simulation volume (see Fig.", "4, in Paper I).", "In fact, by $z=20$ in the simulation, all gas with $\\delta _b > 2$ has lost its relative velocity owing to dynamical friction.", "The amount of energy dissipated in this deceleration is enough to significantly heat the cosmic gas (by a factor of 3, on average) if it were all thermalized.", "However, it is difficult to estimate from first principles the amount of energy that this process would actually thermalize: This heating may primarily impact overdense gas, which occupies a small fraction of the cosmic volume, and the low Mach number shocks sourced by $v_{\\rm bc}$ are inefficient entropy generators.For low Mach number shocks, the energy lost in deceleration may not necessarily go into heating the gas.", "The temperature increase across an adiabatic shock is $\\frac{T}{T_0} = 16^{-1} \\, (5 {\\cal M}^2 - 1)(3 {\\cal M}^{-2} +1) ~~\\overrightarrow{_{~as~{\\cal M}\\rightarrow 1^+~}}~~ {\\cal M}.$ More fundamental is the entropy jump, where we define entropy here as $S \\equiv p/\\rho ^{5/3}$ .", "In this case, $\\frac{S}{S_0} = 4^{-8/3} \\, (5 {\\cal M}^2 - 1)(3 {\\cal M}^{-2} +1)^{5/3} ~~\\overrightarrow{_{~as~{\\cal M}\\rightarrow 1^+~}} ~~\\frac{5 ({\\cal M}-1)^3}{6},$ which is zero at first and second order in ${\\cal M}- 1$ , where ${\\cal M}$ is the shock Mach number.", "For an ${\\cal M}= 2$ shock, $\\lbrace T/T_0, S/S_0\\rbrace $ equals $\\lbrace 2.1, 1.2\\rbrace $ and, for an ${\\cal M}= 4$ , it equals $\\lbrace 5.9, 2.6\\rbrace $ .", "Thus, the regions where ${\\cal M}_{\\rm bc}$ is largest may lead to the most heating (even though dynamical friction is less effective there).", "In addition, if the simulations fail to capture ${\\cal M}-1 = 0.1$ shocks, they are missing only $\\approx 10^{-3}$ jumps in entropy.", "Finite $v_{\\rm bc}$ suppresses the collapse of gas onto halos, which also impacts the gas temperature.", "If the suppression of structure leads to temperature fluctuations between regions with different $v_{\\rm bc}$ that have standard deviations of $\\sigma _T = 0.5\\%$ , $1.5\\%$ , or $5\\%$ , each would lead to a contribution to $P_{21}$ given by one of the $P_{v^2}$ curves in Figure REF .", "We can estimate the level of temperature fluctuations that result from the impact of $v_{\\rm bc}$ on structure formation.", "In the limit that $T_b^{21{\\rm cm}} \\propto \\rho /{T_{\\rm K}}$ (that applies when $x_\\alpha \\gg 1$ ) and pure adiabatic evolution such that ${T_{\\rm K}}\\propto (1+\\delta _b)^{2/3}$ , the average 21cm brightness temperature depends on $v_{\\rm bc}$ as $\\bar{T}_b^{21} (v_{\\rm bc}) &=& \\bar{T}_{b}^{21} \\,\\left\\langle \\left(1+ \\delta _b \\right)^{1/3} \\right\\rangle _{{\\rm fixed}~v_{\\rm bc}}, \\nonumber \\\\&\\approx & \\bar{T}_{b}^{21} \\left( 1 - \\frac{1}{9} \\, \\sigma _{\\delta _b}(v_{\\rm bc})^2 + ... \\right),$ where $\\sigma _\\delta (v_{\\rm bc})$ is the standard deviation in the gas density in a region with differential velocity $v_{\\rm bc}$ .", "Thus, the average temperature is slightly lower in regions with larger variance, at least when $\\sigma _{\\delta _b} \\ll 1$ so that higher order moments are not important.", "Computing $\\sigma _T = \\langle \\bar{T}_b^{21} (v_{\\rm bc})/ \\langle T_{b}^{21} \\rangle _v -1 \\rangle _v^{1/2}$ using linear theory yields $0.002$ at $z=20$ (a time when the linear theory standard deviation in $\\delta _b$ is $\\approx 0.5$ ).", "At $z=20$ and for $x_\\alpha \\gg 1$ , the imprint of $\\sigma _T = 0.002$ on the 21cm power spectrum would lie a factor of 6 below the $0.5\\%$ curve in Figure REF , and, thus, be quite small.", "Equation (REF ) neglects the impact of Compton heating, which results in deviations from pure adiabatic evolution.", "Compton heating changes the temperature of a region by $0.6~K~$ per Hubble time at $z\\approx 20$ for the cosmic free electron fraction of $2\\times 10^{-3}$ , although the heating rate scales as $(1+z)^4$ .", "Because the Compton heating rate is density independent (noting that the electron fraction is largely uniform) whereas the adiabatic cooling rate scales as $\\delta ^{2/3}$ , the fractional temperature in voids is more impacted by Compton heating than in filaments.", "This results in a $v_{\\rm bc}$ –dependence to the heating as larger $v_{\\rm bc}$ results in shallower void depths.", "Thus, shocks, structure formation and Compton heating result in $v_{\\rm bc}$ –dependent temperatures.", "Cosmological simulations are our best recourse for a more accurate estimate.", "We use the cosmological codes GADGET3 [64] and Enzo v2.1.1 [50] with initial conditions that include $v_{\\rm bc}$ self-consistently (in addition to other improvements; Paper I).", "GADGET solves the equations of fluid dynamics with the smooth particle hydrodynamics method, whereas Enzo is a grid code with adaptive mesh refinement (AMR).", "Both the GADGET and Enzo codes have been shown to conserve entropy at the part in 1000–level for the test case of the expansion of a homogeneous Universe [51].", "This level of entropy conservation is unusual for hydrodynamics codes, and it owes to the entropy-conserving formalism of GADGET and the 3rd–order accurate in space, 2nd–order in time Riemann solver employed by Enzo.", "Thus, both codes are well-motivated choices for tracking thermal effects in the early Universe, and their much different hydrodynamics solvers tests the robustness of the results.In addition, we have put the cosmological codes GADGET and Enzo through a battery of tests in order to confirm that our results regarding the temperature evolution of the Universe are robust.", "We find that the relative difference in the 21cm intensity–weighted temperature in the simulations is robust to: i) the frame of reference for the relative velocity of the baryons and dark matter on the grid at least for Galilean transforms with boost velocity comparable to $M_{\\rm bc}$ ii) the maximum time-step size, iii) the grid size, iv) the box size, v) the number of particles, vi) the number of AMR levels in Enzo, and vii) the chosen hydrodynamics solver in Enzo.", "Each curve in Figure REF plots the difference in $\\langle 1/{T_{\\rm K}}\\rangle _M^{-1}$ between both GADGET and Enzo simulations between a simulation with ${\\cal M}_{{\\rm bc}} = 0$ and one with ${\\cal M}_{{\\rm bc}} = 1.9$ , where $\\langle ... \\rangle _M$ signifies a mass average.", "We refer to this difference as $\\delta \\langle 1/{T_{\\rm K}}\\rangle _M^{-1}$ .", "Note that $\\bar{T}^{21}_{b} \\propto [1+ {T_{\\rm CMB}}\\, \\langle {T_{\\rm K}}^{-1} \\rangle _M]$ when $x_\\alpha \\, {T_{\\rm CMB}}/{T_{\\rm K}}\\gg 1$ such that $\\langle {T_{\\rm K}}^{-1} \\rangle _M^{-1}$ is the gas temperature–weighting relevant to 21cm observations during the period of interest (when this signal appears in absorption).", "All of our simulations find that the average temperature is larger in the simulations with ${\\cal M}_{{\\rm bc}} = 0$ than in those with larger ${\\cal M}_{{\\rm bc}}$ .", "(Note that on the scale of our simulations, $\\le 1~{\\rm Mpc}/h$ , this differential velocity is a uniform wind with a single Mach number, allowing us to refer to simulations by their value of ${\\cal M}_{{\\rm bc}}$ .)", "The direction of the temperature difference indicates that the suppression of structure formation and the impact of Compton heat- ing are the dominant processes that shape the temperature rather than shock heating.", "The dot-dashed curve shows $\\delta \\langle 1/{T_{\\rm K}}\\rangle _M^{-1}$ in Enzo between a simulation with and one without Compton heating.", "Compton heating contributes more than half of this temperature difference.", "Figure: Fractional difference between the 21cm intensity–weighted temperature in simulations with ℳ bc =0{\\cal M}_{{\\rm bc}} = 0 and in those with ℳ bc =1.9{\\cal M}_{{\\rm bc}} = 1.9 (corresponding to v bc =3v_{\\rm bc}= 3 km s -1 ^{-1} at z=100z=100).", "Note that the temperature in the simulations with ℳ bc =0{\\cal M}_{{\\rm bc}} = 0 is higher, and the fractional temperature difference is similar in Enzo to this difference in GADGET.", "The horizontal line shows the difference that leads to the signal given by the lowermost P v 2 P_{v^2} curve in Figure .", "The Enzo simulations here are uni-grid, but we find negligible differences when compared with the Enzo AMR calculations.With both Enzo and GADGET, the temperatures are reasonably converged in resolution and in box size between the simulations shown in Figure REF .This is not the case if we run GADGET in the standard setting rather than with the gravitational softening equal to the gas softening as done here, where we find the temperature differences are twice as large owing to artificial particle coupling.", "We also find that Enzo is converged in $\\delta \\langle 1/{T_{\\rm K}}\\rangle _M^{-1}$ between both the adaptive-mesh refinement (AMR) and uni-grid simulations.", "We find that the temperature differences are roughly linear in $v_{{\\rm bc}}$ from comparing with also the ${\\cal M}_{{\\rm bc}} = 3.8$ simulations such that we find $\\sigma _T \\approx 0.4 \\, \\delta \\langle 1/{T_{\\rm K}}\\rangle _M^{-1}$ .", "This allows us to draw a horizontal line in Figure REF that corresponds to the lowest amplitude curve for the $P_{v^2}$ contribution to $P_{21}$ in Figure REF .", "For $v_{{\\rm bc}}$ –sourced temperature fluctuations to contribute a comparable fraction of the 21cm power as the density sourced intensity fluctuations, $\\sigma _T \\gtrsim 0.015$ must be satisfied (Fig.", "REF ).", "Thus, the simulations yield temperature differences that are a factor of a few too small for the $P_{v^2}$ contribution to be an ${\\cal O} (1)$ contribution to $P_{21}$ ." ], [ "star formation–sourced fluctuations", "While we found that the coupling of non-astrophysical processes to $v_{\\rm bc}$ is insufficient to significantly enhance the fluctuations in the 21cm background, the local amount of star formation can also couple to $v_{\\rm bc}$ and modulate associated radiation backgrounds, which in turn affect $P_{21}$ .", "We attempt to model this coupling here.", "We consider two disparate models for which halos at $z\\sim 10-20$ dominated the SFR: Molecular Cooling Halos: In this case, stars are primarily formed in minihalos with $10^{2.5} \\lesssim T_{\\rm vir} \\lesssim 10^4~K$ .", "These halos cool by exciting molecular hydrogen transitions.", "We use a simplistic parametrization to quantify how $v_{\\rm bc}$ could impact these halos.", "In particular, in addition to including the impact of $v_{\\rm bc}$ on the halo mass function, we assume that all halos with higher circular velocities than $V_{\\rm cool} (z)$ are able to retain their gas and form stars proportional to their mass.", "As in [20], we use $V_{\\rm cool} (z) = \\lbrace V_{\\rm cool, 0}^2 + \\left[\\alpha \\, v_{\\rm bc}(z) \\right]^2\\rbrace ^{1/2}.$ Previous numerical simulations find $V_{\\rm cool,0} = 3.7 ~{\\rm km ~s}^{-1}$ – the approximate threshold to cool via $H_2$ when $F_{\\rm LW, 21} \\ll 1$ and in the absence of baryonic streaming – and $\\alpha = 4.0$ (e.g., [20], who used fits to the numerical results of [65] and [27]).", "We leave $\\alpha $ as a free parameter that is calibrated with our simulations, and we use $V_{\\rm cool,0} = 3.7 ~{\\rm km ~s}^{-1}$ as well as $V_{\\rm cool,0} = 7.4~{\\rm km ~s}^{-1}$ .", "The latter choice roughly corresponds to the minimum circular velocity that can cool for a Lyman-Werner background with $x_\\alpha = 1$ (eqn.", "REF evaluated with the intensity from eqn.", "REF ).", "For reference, $3.7 ~{\\rm km ~s}^{-1}$ and $7.4~{\\rm km ~s}^{-1}$ correspond to $4\\times 10^5~{M_{\\odot }}$ and $3\\times 10^6~{M_{\\odot }}$ halos at $z=20$ .", "Atomic Cooling Halos: In this case, halos with $T_{\\rm vir} > 10^4~$ K dominate the total star formation rate (e.g., [25], [22]).", "It is unlikely that $v_{\\rm bc}$ significantly impacts the mass threshold that cools and forms stars in this case because (1) atomic transitions are a robust coolant that depends more weakly on the density of the gas and (2) the circular velocity of these halos is much larger than $v_{\\rm bc}$ .", "The impact of $v_{\\rm bc}$ on star formation in these halos should derive primarily from its effect on these halos' mass function.", "However, Figure REF illustrates that $v_{\\rm bc}$ has a small impact on the matter power spectrum at scales relevant to atomic cooling halos, so a large $v_{\\rm bc}$ -sourced contribution to $P_{21}$ is not anticipated.", "For both the molecular and atomic cooling cases, our calculations assume that star formation is proportional to collapsed mass above the minimum mass threshold." ], [ "numerical estimates for the star formation suppression", "Ultimately, our aim in this section is to estimate the contribution to the 21cm power spectrum that traces $P_{v^2}$ from star formation–sourced radiation backgrounds.", "We use our cosmological simulations to motivate how the star formation rate is suppressed.", "We first investigate $v_{\\rm bc}$ 's impact on the halo mass function in numerical simulations relative to what one would predict for its impact on the Sheth-Tormen halo mass function (which can be calculated directly from the linear theory matter power spectrum).", "Each panel in Figure REF shows halo mass functions calculated from our two largest GADGET simulations ($1~{\\rm Mpc}/h$ , $2\\times 768^3$ particle, one with ${\\cal M}_{\\rm bc}=0$ and the other with ${\\cal M}_{\\rm bc}=1.9$ ).In Paper I we showed that particle coupling can impact the GADGET simulations when ${\\cal M}_{\\rm bc}=0$ and this particle coupling is alleviated in the simulations with ${\\cal M}_{\\rm bc}>0$ .", "These mass functions were calculated using the friends-of-friends algorithm with a linking length of $0.2$ .", "For comparison, we also show the semi-analytic mass function using the Sheth-Tormen theory.", "We include finite box size effects in the Sheth-Tormen mass function as described in the ensuing footnote in order to compare on an equal footing with the simulation mass functions.To account for finite box size effects in Figures REF and REF , we multiply the Sheth-Tormen mass function by $n_{\\rm PS}(m_h | \\sqrt{\\sigma _{m_h}^2 - \\sigma _{l_{\\rm box}/2}^2})/ n_{\\rm PS}(m_h | \\sigma _m)$ , where $n_{\\rm PS} (m_h, \\sigma _X)$ is the Press-Schechter mass function at mass $m_h$ given $\\sigma _X$ , the RMS density contrast in a sphere of radius X.", "This prescription was motivated in [4].", "Both sets of curves are calculated with ${\\cal M}_{\\rm bc}$ equal to 0 and $1.9$ , and both simulations use the same random numbers to generate the density and velocity fields.", "While the simulated mass function is slightly below the Sheth-Tormen mass function, particularly at $z=25$ , the suppression of the amplitude of the mass function is comparable to that predicted by the semi-analytic mass function calculation.", "We will use this semi-analytic mass function model in subsequent calculations.", "Figure: Impact of v bc v_{\\rm bc} on mass that can cool within 1 Hubble time as a function of halo mass, where each marker shows the gas mass that meets our criterion within a virial radius from a halo centroid.", "The blue circles are the {1 Mpc /h,2×768 3 particle }\\lbrace 1~{\\rm Mpc}/h, 2\\times 768^3~{\\rm particle}\\rbrace simulation with ℳ bc =0{\\cal M}_{\\rm bc}= 0, and the red stars are the same but for ℳ bc =1.9{\\cal M}_{\\rm bc}= 1.9.", "The thin curves with the corresponding color show the mean in both simulations.", "The dashed curve is the cosmic baryon fraction times the halo mass, and the short vertical lines illustrate the cutoff mass in our simple model for ℳ bc =0{\\cal M}_{\\rm bc}= 0 and ℳ bc =1.9{\\cal M}_{\\rm bc}= 1.9 and the specified α\\alpha .", "Crudely, the impact of ℳ bc {\\cal M}_{\\rm bc} is to shift the minimum halo mass with α≈4\\alpha \\approx 4 in equation (), but ℳ bc {\\cal M}_{\\rm bc} also suppresses star formation in more massive systems and in a stochastic manner.Figure: Same as Figure but for the amount of gas that can cool in 0.10.1 Hubble times.The other parameter we aim to calibrate from the simulations is $\\alpha $ .", "This parameter regulates the minimum mass of a halo that is able to cool by molecular transitions.", "Figures REF and REF show the mass in gas within a virial radius of a halo of the specified dark matter mass that can cool by molecular hydrogen transitions in $1.0$ and $0.1$ Hubble times, respectively, at $z=20$ .", "The points show the individual halos in the $1~{\\rm Mpc}/h$ , $2\\times 768^3~$ particle simulations, and the curves are the average amount of mass in gas that can cool in these simulations.", "To calculate the cooling time, we use the formula in [67] under the crude approximation that $\\log (1+ N_{\\rm rec}) =1$ to calculate the amount of molecular hydrogen, where $N_{\\rm rec}$ is the number of recombinations.", "This equality approximately holds for gas at the virial density of halos at $z\\sim 20$ , and it allows us to calculate the cooling rate from a single simulation snapshot rather than following the density evolution of a fluid element.Most other studies of the impact of $v_{\\rm bc}$ have followed the collapse of gas parcels to much higher densities as a proxy for star formation [42], [65], [27].", "In our opinion, it is not necessarily a disadvantage to use our cooling criteria on as a proxy for star formation rather than following the cooling and condensing gas to much higher densities: Firstly, simulations that follow gas to much higher densities do not all agree on the character of star formation in the first halos (e.g., [26]).", "In addition, feedback processes either from stellar HII regions within the halo [2], [75] or the cosmological Lyman-Werner background [29], [40] drastically increase the complexity of modeling star formation at the Cosmic Dawn.", "However, we have run the same simulations in Enzo with molecular hydrogen cooling and AMR, and found that our simple estimates did roughly reproduce the amount of gas that cooled to much higher densities.", "The amount of mass that can cool trends to zero at halo masses less than $\\approx 1-4\\times 10^5~{M_{\\odot }}$ (Fig.s REF and REF ).", "Such a threshold was anticipated from more detailed first star calculations (e.g., [40]), and larger ${\\cal M}_{\\rm bc}$ shift this turn-over mass to higher values.", "The vertical lines show how much this mass would shift under our parametrization for the impact on molecular hydrogen cooling halos for $V_{\\rm cool, 0} = 3.7~$ km s$^{-1}$ and the specified $\\alpha $ .", "The shift in the critical mass that can form stars in the simulations most closely approximates the shift seen in the $\\alpha =4$ case.", "However, the $V_{\\rm cool, 0}-\\alpha $ parametrization does not explain the halo-to-halo stochasticity in the simulations.", "There are far more halos in the ${\\cal M}_{\\rm bc}=1.9$ simulation that have zero gas under our cooling criteria than in the ${\\cal M}_{\\rm bc}=0$ simulation.", "A small component of this suppression results from the halos being slightly less massive in the ${\\cal M}_{\\rm bc}=1.9$ case (see Fig.", "REF ), but most originates from $v_{\\rm bc}$ 's suppression of gas accretion onto these halos.", "In Paper I, we showed that one source of this variation from halo to halo results from the orientation of filamentary flows onto a halo.", "Figure: Ratio of the amount of gas that can cool within the specified period and form stars in simulations with ℳ bc =0{\\cal M}_{\\rm bc}= 0 and ℳ bc =1.9{\\cal M}_{\\rm bc}= 1.9.", "Other than the one Enzo curve, all others are calculated from simulations run with GADGET.", "Also shown for comparison are semi-analytic calculations for this ratio using the molecular hydrogen cooling model for the specified α\\alpha for a box size of 1 Mpc /h1~{\\rm Mpc}/h (thin dot-dashed curve) and 0.5 Mpc /h0.5~{\\rm Mpc}/h (thin double dotted curve).", "The simulations are closest to the model for α≈4-6\\alpha \\approx 4-6.Figure REF shows the ratio of the total gas that can cool in the simulation boxes with ${\\cal M}_{\\rm bc}= 1.9$ to their counterpart with ${\\cal M}_{\\rm bc}= 0$ , under the same cooling criteria and using three sets of the GADGET simulations and one set of Enzo simulations.", "Also shown for comparison are semi-analytic calculations for this ratio in the cases with $V_{\\rm cool,0} =3.7~$ km s$^{-1}$ and with $\\alpha =2$ , $\\alpha =4$ or $\\alpha = 6$ .", "The simulation curves are most consistent with the $\\alpha \\approx 4-6$ curves, but note that the simulations become less reliable at the highest $z$ owing to the rareness of these halos.", "We have also looked at the suppression of ${\\cal M}_{\\rm bc}= 3.8$ and reached a similar constraint on $\\alpha $ .", "In conclusion, the factor by which star formation is suppressed is on average similar for both the $1~H^{-1}$ and $0.1~H^{-1}$ cases in Figures REF , REF , and $\\ref {fig:SFRhist}$ .", "We note that the $0.1$ Hubble times criterion is likely most appropriate given the exponential growth with time in the number of collapsing halos.", "Also, we find a similar suppression in our simulations with Enzo as with GADGET.", "We conclude that $\\alpha = 4-6$ is the best match to the suppression in the SFR under this simple parametrization.", "While much of our discussion focussed on the $V_{\\rm cool,0} =3.7~$ km s$^{-1}$ , it seems natural to assume $\\alpha $ is fixed as different physics changes $V_{\\rm cool,0}$ , such as the presence of a Lyman-Werner background.", "We showed that this was the case if we changed the cooling time threshold (compare Fig.s REF and REF ).", "The ensuing discussion will further motivate this assumption." ], [ "explanation for SFR suppression", "We have found that the simulations prefer $\\alpha \\approx 4-6$ in equation (REF ), which parameterizes the minimum star forming halo.", "Let us attempt to understand this result with two toy models.", "An estimate for the overdensity of gas in a virialized minihalo when $v_{\\rm bc}= 0$ and ignoring cooling is $\\delta _b(x) \\approx \\left( \\frac{6 \\,T_{\\rm vir}}{5 \\, T_{\\rm K}^{\\rm ad}} \\right)^{3/2},$ where $ T_{\\rm K}^{\\rm ad}$ is the temperature of the IGM at virialization under the idealization of a purely adiabatic collapse and hydrostatic equilibrium [67].Equation (REF ) is starting to break down at interesting $T_{\\rm vir}$ : When $T_{\\rm vir} >260$ , equation (REF ) yields $\\delta _b > 200$ at $z=20$ (i.e., the densities it returns are higher than the virial density).", "In the case of finite $v_{\\rm bc}$ and where its energy thermalizes in shocks during the collapse, this expression is altered such that $T_{\\rm IGM, ad} \\rightarrow T_{\\rm IGM, ad} (1 +5 {\\cal M}_{\\rm bc}^2/9)$ , resulting in $\\delta _b \\propto (1 + 5{\\cal M}_{\\rm bc}^2/9)^{-3/2}$ .", "There is some threshold virial temperature, $T_{\\rm vir, *}$ , that obtains high enough $\\delta _b$ and $T_{\\rm vir}$ to cool within a Hubble time.", "The molecular-hydrogen cooling time is proportional to $\\exp [512 {\\rm ~K}/T]/n_{H_2}$ .", "At a crude level, we can ignore the factor $\\exp [512 {\\rm ~K}/T]$ because $V_{\\rm cool, 0} = 3.7~{\\rm km~s}^{-1}$ corresponds to an $800~$ K halo at $z=20$ so that this factor is changing by less than the factor of 2 above this velocity scale.", "With this approximation, $T_{\\rm vir, *}$ (and also $V_{\\rm cool}^2$ ) scales as $\\approx (1 + 5{\\cal M}_{\\rm bc}^2/9)$ in this model, since $n_{H_2} \\sim \\delta _b$ .", "This scaling for $V_{\\rm cool}$ results in a modulation comparable to $\\alpha =6$ at $z=20$ in the parameterization in equation (REF ) and with $\\alpha $ decreasing with increasing redshift.", "(Including the factor $\\exp [512 {\\rm ~K}/T]$ suppresses the effective $\\alpha $ somewhat.)", "An even simpler (but related) model uses that velocities scale as $(1+\\delta )^{1/3}$ in an adiabatically collapsing region, at least in the absence of dissipative processes.", "Thus, at halo densities $\\delta \\sim 200$ , the streaming velocity accelerates and becomes $\\approx 6 \\, v_{\\rm bc}$ , where $v_{\\rm bc}$ is the velocity at the cosmic mean density.", "The total effective pressure at a halo virial radius scales as $\\sim \\lbrace V_{\\rm cool, 0}^2 + \\left[6 \\, v_{\\rm bc}(z) \\right]^2/3\\rbrace $ , which has a similar form to the parameterization for $V_{\\rm cool}$ given in equation (REF ) and suggests $\\alpha \\approx 4$ .", "If the second term is larger than the first, the ram pressure is larger than the thermal pressure (potential depth).", "One can also understand this scaling as the requirement that the circular velocity of the halo must be larger than the local streaming velocity ($\\sim 6 \\, v_{\\rm bc}$ ) for the gas to be focussed into the potential well, where it can shock and cool." ], [ "Ultraviolet pumping", "Now that we have calibrated our model, finding $\\alpha \\approx 4-6$ , we can use it to make predictions for the 21cm signal.", "We first focus on 21cm inhomogeneities resulting form ultraviolet radiation from the first stars as this is likely to occur earlier than other star formation–sourced inhomogeneities (as argued in Section REF ).", "Figure REF shows the $P_{v^2}$ and $P_{\\delta }$ contributions to the 21cm power spectrum, where $\\Delta T_b(k)^2 \\equiv k^3 P_{21}/ (2 \\pi ^2)$ , for different models of how $v_{\\rm bc}$ impacts star formation assuming {$z=20$ , $\\bar{x}_\\alpha = 1$ } in the top panel and {$z=30$ , $\\bar{x}_\\alpha = 0.1$ } in the bottom panel.", "These curves assume that the X-ray heating and ionizations from these stars has yet to impart significant fluctuations in the 21cm background – that the only fluctuation sources are density and $x_\\alpha $ .", "Each curve can be rescaled to a different $\\bar{x}_\\alpha $ by noting that this parameter only changes the amplitude of $P_{21}$ via the factor $\\bar{T}_b^2 (1+ \\bar{x}_\\alpha )^{-2}$ .", "(This is not the case at $k \\gtrsim 0.1~{\\rm Mpc}^{-1}$ for the $P_\\delta $ contribution, where the fluctuations begin to be sourced directly by $\\delta _b$ rather than $J_\\alpha $ as at smaller wavenumbers.)", "Our calculations also assume that the sources' specific emissivity scales as $\\epsilon \\propto \\nu ^\\alpha $ with $\\alpha = 0$ , $\\epsilon (z) = \\exp [- (\\tau _{\\rm fcoll} H)^{-1} \\, \\Delta z/(1+z)]$ , noting that $\\tau _{\\rm fcoll}^{-1} = d \\log f_{\\rm coll}(m_h)/dt$ , $dz/(1+z) = H\\,dt$ , and $n_{\\rm max} = 20$ (see §REF ).We have ignored that that Lyman-Werner background can suppress star formation in regions that produce that are producing Ly$\\alpha $ .", "[32] found that such fluctuations are unlikely to have a large impact on $P_{21}$ .", "It would decrease the amplitude of both the $P_{v^2}$ and $P_\\delta $ –coupled 21cm fluctuations arising from star formation.", "In addition, we have not included the effects of lensing and peculiar velocities on $\\delta _\\alpha $ and $\\delta _T$ .", "These effects enter at $1/b$ [5] and alter the contribution proportional to $\\delta _b$ at the $\\sim 10\\%$ level (but do not quantitatively effect our results).", "Both effects do not alter the component that traces $P_{v^2}$ .", "We assume the same parametrization for $\\epsilon (z)$ when considering X-ray heating in §REF .", "The thick curves in Figure REF are the $P_{v^2}$ contribution to $P_{21}$ and the thin are this but for $P_\\delta $ .", "The molecular cooling halo cases with $\\alpha =4$ are given by the dashed curves ($V_{\\rm cool, 0} > 3.7~{\\rm km~s}^{-1}$ ) and dotted curves ($V_{\\rm cool, 0} > 7.4~{\\rm km~s}^{-1}$ ).", "The same calculations but with $\\alpha =6$ shift the $P_{v^2}$ curves in the minihalo–sourced models upward by a factor of $\\approx 2$ .", "The atomic cooling halos case is given by the solid curves, and it only appears in the top panel since atomic cooling halos would not exist in sufficient abundance to pump the 21cm line at $z=30$ .", "In all cases, the relative impact of $v_{\\rm bc}$ becomes smaller with time (decreasing redshift) because (1) $P_\\delta $ is growing as $(1+z)^{-2}$ , (2) halos at all mass scales are becoming less rare such that the impact on the growth of modes is less accentuated, and (3) $v_{\\rm bc}$ and, hence, its role as an effective pressure for a halo with potential depth $\\sim V_{\\rm cool, 0}^2$ is becoming smaller with time.", "Thus, we find in both considered minihalo cases that a significant component of the power is sourced by $P_{v^2}$ at $k\\sim 10^{-2}~{\\rm Mpc}^{-1}$ (Fig.", "REF ).", "This is consistent with the findings of [18], although our model for the impact of $v_{\\rm bc}$ on star formation is more conservative.", "Unfortunately, the power is suppressed at larger wavenumbers by the free streaming of ultraviolet photons, and 21cm instruments are most able to observe $k \\gtrsim 0.1~{\\rm Mpc}^{-1}$ for reasons detailed in §REF .", "Thus, we conclude that $P_{v^2}$ does not contribute significantly to $P_{21}$ on observable scales at $z=20$ .", "For the $z=30$ case in Figure REF , the $P_{v^2}$ contribution is beginning to become comparable to the contribution from $P_\\delta $ at $k\\approx 0.1~{\\rm Mpc}^{-1}$ , roughly the largest observable scale.", "However, observing the signal at $z=30$ is a more challenging venture.", "We can also quantify how much both (1) $v_{\\rm bc}$ 's effect on the suppression of the growth of dark matter structure and (2) its impact on disrupting gas accretion onto minihalos (and modulating the minimum mass that can form stars) contribute to the inhomogeneities $v_{\\rm bc}$ sources in the 21cm power spectrum.", "We find that effect (2) tends to dominate in our models, sourcing $\\approx 80\\%$ of the $P_{v^2}$ –coupled signal at $z=20$ in both minihalo models.", "This percentile also holds for our ensuing estimates in §REF ." ], [ "X-ray heating", "We argued in Section REF that Ly$\\alpha $ pumping with $\\bar{x}_\\alpha \\sim 1$ is likely to have occurred before X-rays significantly heated the IGM, but, admittedly, there is much uncertainty in the high-redshift production of X-rays.", "However, the fluctuations from X-ray heating are less damped than those from the Ly$\\alpha $ background because of the shorter mean free path of the X-ray photons that are responsible for such heating.", "The amount of damping is small on all scales we plot for the spectral index of the background that is assumed ($-1$ in intensity per unit frequency), and we found that this result was generic over a range of spectral indices.", "Figure REF shows predictions for $P_{21}$ when the fluctuations are primarily driven by $z\\sim 20$ X-ray backgrounds.", "For simplicity, we assume $x_\\alpha \\gg 1$ so that fluctuations in $x_\\alpha $ are zero and ${T_{\\rm K}}= (T_{\\rm ad} + {T_{\\rm CMB}})/2$ .", "The amplitude of our curves can be rescaled to other $x_\\alpha $ and ${T_{\\rm K}}$ .", "Values of $x_\\alpha \\gg 1$ would imply large Lyman-Werner backgrounds that would suppress star formation in the environments we consider.", "However, note that changing $x_\\alpha = 2$ to $x_\\alpha =\\infty $ has little impact on our predictions.", "In addition, we assumed that the critical velocity that can form stars is determined by equation (REF ) with $\\alpha =4$ for the minihalo curves in this figure.", "If we instead used $\\alpha =6$ (which is also consistent with the modulation observed in our simulations), this results in a factor of $\\approx 2$ increase in the normalization of these curves.", "The fluctuations are larger on observable scales in the case in which X-ray heating drives the fluctuations compared to the case where the fluctuations originate from the ultraviolet background.", "The $P_{v^2}$ contribution in both of the minihalo models in Figure REF may be detectable with an MWA-like instrument.", "However, Figure REF illustrates that only in the optimistic model with $V_{\\rm cool, 0} =3.7~$ km s$^{-1}$ is the $P_{v^2}$ contribution comparable to the $P_\\delta $ contribution and then only at $k < 0.1~{\\rm Mpc}^{-1}$ .", "Thus, even in this optimistic case in which X–rays dominate the fluctuations, the signal is not dramatically enhanced.", "Thus, the contribution of $P_{v^2}$ to the 21cm signal is largest when X-rays from minihalos reheated the Universe.", "However, in models in which atomic cooling halos dominate the $z\\sim 20$ X-ray background, the contribution to the 21cm power spectrum that is sourced by $v_{\\rm bc}$ is likely to be insignificant." ], [ "Conclusions", "This study presented semi-analytic calculations, which were calibrated with large cosmological simulations, aimed at understanding the early Universe and its 21cm signatures.", "We focused on $z\\sim 20$ , an epoch when fluctuations in the diffuse 21cm background are not necessarily more difficult to detect than these fluctuations from any other cosmic era (e.g., [57]).", "A detection of this signal would provide a window into when the first stars formed and into a time before the IGM had been reheated by astrophysical sources.", "We focused on the question of whether the dark matter–baryon supersonic differential velocity, $v_{\\rm bc}$ [68], is likely to enhance the $z\\sim 20$ 21cm signal.", "In [49], we showed that typical values for $v_{\\rm bc}$ have a dramatic impact on the morphology of structures on scales of $10-100~$ comoving kpc.", "Whether $v_{\\rm bc}$ impacted the 21cm signal in an observable manner boils down to whether $v_{\\rm bc}$ also affected the $\\sim 10\\,$ comoving Mpc modes to which interferometric 21cm efforts are anticipated to be sensitive.", "In fact, we showed that if just $\\sim 1\\%$ fractional fluctuations in the gas temperature were correlated with the large-scale $v_{\\rm bc}$ flows, this coupling could lead to a significant new component to the signal.", "We investigated three sources of $v_{\\rm bc}$ –coupled 21cm fluctuations relevant to the epoch after the first stars turned on but prior to reionization: (1) heating from structure formation and its associated shocks, (2) ultraviolet pumping of the 21cm line from the first stars' emissions, and (3) heating by the X-ray background produced by the first supernovae, X-ray binaries, and miniquasars.We did not consider the impact of $v_{\\rm bc}$ on the signal from the reionization epoch as the consensus is that this epoch was driven by star formation in the atomic cooling halos that are too massive to be impacted by $v_{\\rm bc}$ (but see [18] and [8]).", "$v_{\\rm bc}$ would also impact the clumpiness of gas and, thus, the number of recombinations during reionization (Joanne Cohn, private conversation).", "This suppression would delay reionization in regions with smaller $v_{\\rm bc}$ .", "However, any X-ray preheating prior to reionization would act to eliminate the clumpiness on scales impacted by $v_{\\rm bc}$ , and it is thought that $\\sim 10^7~{M_{\\odot }}$ minihalos – which are only moderately impacted by $v_{\\rm bc}$ – are likely to dominate the number of absorptions from minihalos in all reionization scenarios [34], [43].", "Nevertheless, an investigation of this coupling mechanism would be interesting.", "We found that the first of these sources, $v_{\\rm bc}$ 's impact on structure formation and shocking, did not contribute a significant level of large-scale 21cm fluctuations.", "In particular, we found that simulations both with and without $v_{\\rm bc}$ did not yield large enough differences in the average gas temperatures to result in fluctuations that were comparable to the fluctuations in the 21cm signal from density fluctuations.", "We estimated that shocking contributes $\\lesssim 10\\%$ of the power at all wavenumbers.", "This finding held true despite the fact that most of the overdense gas was decelerated into the potential wells of the dark matter via dynamical friction in our simulations, losing its relative velocity by $z\\sim 20$ .", "This process leads to supersonic wakes and shocking throughout the cosmic volume, but not to significant heating.", "The other mechanism by which $v_{\\rm bc}$ could imprint new fluctuations in the 21cm background involves the spatial modulation by $v_{\\rm bc}$ of the formation of the first stars and their associated radiation backgrounds.", "We investigated with a suite of cosmological simulations whether it is plausible that $v_{\\rm bc}$ modulates early star formation.", "At $z=20$ , we found $\\approx 3$ times more mass in gas that could cool in $1.0$ (or $0.1$ ) Hubble times and form stars in our simulations with ${\\cal M}_{\\rm bc}=0$ than in those with ${\\cal M}_{\\rm bc}= 1.9$ .", "Surprisingly, even the amount of dense gas in some of our most massive simulated halos ($10^6-10^7~{M_{\\odot }}$ ) could be suppressed at the order–unity level for some halos in the simulations with ${\\cal M}_{\\rm bc}= 1.9$ .", "We provided simplistic analytic estimates that reproduced the average amount of suppression in star-forming gas that was found in the simulations.", "These estimates revealed why the impact is significant on halos with circular velocities of $\\approx 10 \\,v_{\\rm bc}$ , in contrast to previous estimates.", "In detail, the amount of suppression in the simulations varied significantly from halo to halo at fixed halo mass and depended on, for example, the orientation of filamentary accretion flows with respect to the baryonic wind (see Paper I).", "The fluctuations in the 21cm background from inhomogeneous ultraviolet pumping are likely to be present at higher redshifts than fluctuations from other stellar-mediated radiation backgrounds.", "Unfortunately, the long mean free path of ultraviolet photons to redshift into the Ly$\\alpha $ resonance dampens the fluctuations from $v_{{\\rm bc}}$ on observable scales.", "We found that the $v_{\\rm bc}$ –dependence of ultraviolet pumping was a significant contribution to the 21cm anisotropy at $k\\sim 0.01~$ Mpc$^{-1}$ , although not as large as in the models of [18] in which the impact of $v_{\\rm bc}(z)$ on the star formation rate was rather extreme.", "However, we found that $v_{\\rm bc}$ could not source a significant component of the anisotropy on the largest observable scales ($k\\gtrsim 0.1~$ Mpc$^{-1}$ ) at $z=20$ .", "At $z=30$ (a redshift from which the diffuse 21cm background would be more difficult to observe), we found that the $v_{\\rm bc}$ contribution at $k\\sim 0.1~$ Mpc$^{-1}$ could be more significant.", "We concluded that $v_{\\rm bc}$ is most likely to leave an ${\\cal O}(1)$ imprint on the 21cm signal if X-ray heating from star-forming minihalos drives fluctuations in the intergalactic gas temperature.", "Such X-ray production at high redshifts is very uncertain, and it is by no means guaranteed that X-ray reheating was sourced primarily by minihalos rather than by more massive halos that could have cooled atomically.", "In fact, most previous models for the 21cm signal assumed the latter (e.g.", "[25], [22]).", "X-ray reheating by minihalos would be more likely if minihalos produced more X-rays per unit star formation rate at $z=20$ than $z=0$ galaxies (such that X-ray reheating occurred before the Lyman-Werner background quenched star formation in minihalos).", "Previous studies have made arguments for why this could have been the case [48], [46].", "In the cases we considered in which $> 3\\times 10^5~{M_{\\odot }}$ and $> 3\\times 10^6~{M_{\\odot }}$ halos formed stars in proportion to their mass, we found that the 21cm anisotropy from $v_{\\rm bc}$ 's modulation of the X-ray background could be comparable to the 21cm signal sourced by density fluctuations at $z\\sim 20$ .", "Most of this anisotropy was sourced by the impact of $v_{\\rm bc}$ on the gas accreted by these halos and not from the suppression of the halo mass function (the effect considered in [68]).", "The acoustic oscillations in this $v_{\\rm bc}$ –sourced anisotropy make this finding particularly interesting as it results in a more distinctive signature for 21cm observatories to target – a signature which would indicate that the Universe was reheated by star-forming minihalos.", "Finally, we also investigated whether structure formation–initiated shocks reheated the Universe in Appendix A, which would reduce the absolute brightness temperature of the 21cm signal from $z\\sim 20$ , times when this line is anticipated to appear in absorption.", "The one previous study that investigated this signal with cosmological simulations found that such shocks would dramatically suppress the amount of absorption [25].", "The recently-funded LEDA and recently-proposed DARE 21cm observatories aim to detect this 21cm absorption trough in the sky-averaged 21cm signal, and the sensitivity of these efforts scales with the depth of the absorption trough.", "Fortunately, we found that shock heating only suppresses the 21cm absorption at a fractional level of $\\lesssim 20\\%$ at $z>10$ .", "We would especially like to thank Dusan Keres and Mike Kuhlen for their help with GADGET and Enzo.", "We thank Rennan Barkana, Gianni Bernardi, Joanne Cohn, Lincoln Greenhill, Smadar Naoz, and Martin White for useful discussions.", "We thank Volker Springel for GADGET3.", "Computations described in this work were performed using the Enzo code, developed by the Laboratory for Computational Astrophysics at the University of California in San Diego (http://lca.ucsd.edu), and with the yt analysis software [69].", "MM and RO are supported by the National Aeronautics and Space Administration through Einstein Postdoctoral Fellowship Award Number PF9-00065 (MM) and PF0-110078 (RO) issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060.", "This research was supported in part by the National Science Foundation through TeraGrid resources provided by the San Diego Supercomputing Center (SDSC) [14] and through award number AST/1106059." ], [ "A. When does shock heating result in the intergalactic temperature departing from adiabatic evolution?", "Here we quantify the importance of shock heating in determining the temperature history of the gas during the cosmic Dark Ages.", "It is plausible that prior to X-ray reheating the Universe shock heated to significantly higher temperatures than if it had simply cooled adiabatically with the expansion of the Universe.", "This heating would be analogous to what cosmological simulations find at $z\\lesssim 1$ , where $\\sim 50\\%$ of the simulated mass shock heated to $\\gtrsim 10^6~$ K owing to $100~$ km s$^{-1}$ convergent flows [15].", "However, at $z\\sim 20$ , a $0.3~$ km s$^{-1}$ flow was supersonic, so weak flows could have sourced shocks and heated the gas.", "The middle and bottom panels in Figure REF quantify how much heating occurs in our simulations.", "The top panel shows $\\langle {T_{\\rm K}}^{-1} \\rangle _M^{-1}$ , where $\\langle ... \\rangle _M$ signifies a mass average, and the bottom panel shows the deviation of $\\langle {T_{\\rm K}}^{-1} \\rangle _M^{-1}$ from adiabatic evolution.", "Note that $\\bar{T}^{21}_{b} \\propto [1+ {T_{\\rm CMB}}\\, \\langle {T_{\\rm K}}^{-1} \\rangle _M]$ for $x_\\alpha \\, {T_{\\rm CMB}}/{T_{\\rm K}}\\gg 1$ .", "Thus, $\\langle {T_{\\rm K}}^{-1} \\rangle _M^{-1}$ is the gas temperature-weighting most relevant to 21cm observations (after ultraviolet pumping becomes effective).", "Both GADGET and Enzo have almost identical residuals when compared at the same resolution and box size.", "Both codes predict $\\approx 10-15\\%$ deviations above adiabatic evolution by $z= 15$ .", "About half of this deviation owes to Compton heating rather than shocking.", "(The “no Compton” curve does not include such heating.)", "We have also run several adaptive simulations with Enzo that result in much higher resolution in overdense gas and find good convergence with the temperature evolution in these uni-grid Enzo calculations.", "We conclude that structure formation shocks will not qualitatively change the 21cm signal, even in scenarios where the IGM is not impacted by astrophysics at $z\\approx 10$ .", "Two prior studies had looked at the importance of shock heating in setting the temperature of the early Universe.", "[24] estimated the fraction of mass that shock heated above the CMB to be $f_{\\rm sh} \\sim \\lbrace 0.1\\%,~ 3\\%,~ 25\\%\\rbrace $ at $z = \\lbrace 30,~ 20, ~10\\rbrace $ based on the amount of mass that was at turnaround in potential wells that were massive enough (i.e., had large enough characteristic velocities) to heat the IGM above the CMB temperature.", "However, shocks could occur even prior to turnaround, as only $\\sim 0.1-1~$ km s$^{-1}$ flows can be supersonic and shock.", "For example, as shells cross in voids, this crossing will generate shocks in the gas (e.g., [7]; although, shell crossing in spherical top-hat voids only occurs at a linear overdensity of $-2.7$ ).", "In support of the possibility that shocks do not just occur at or after turnaround, the study of [25] concluded using cosmological simulations that structure formation shocks are more prominent than the simple estimates in [24] suggest, creating order unity differences even at the highest redshift [25] considered, $z=17$ .", "(Although, the simulations in [25] did not resolve the Jeans' scale.)", "The [24] semi-analytic estimate is more consistent with the heating seen in our simulations." ], [ "B. sensitivity to 21cm signal", "The principle difficulty with detecting high-redshift 21cm radiation is that the sky is much brighter than the 21cm signal owing to foreground emission, the dominant foreground being synchrotron from the Galaxy.", "Foreground emission not only must be subtracted off to isolate the 21cm signal, but also is likely to be the dominant source of statistical noise in a measurement of $P_{21}$ .", "The anticipated spectral smoothness should allow all significant extraterrestrial foregrounds to be separated from the 21cm signal.", "However, since the brightness temperature of the sky scales as $T_{\\rm sky} \\approx 240 {\\rm \\, K \\,}\\left( \\nu /150 ~{\\rm MHz}\\right)^{-2.55}$ [61], it is thought that detecting the 21cm signal will be an even more difficult task as the targeted redshift increases (the observed frequency decreases; e.g., [44]).", "The challenge with statistically detecting the 21cm power spectrum at wavenumber ${k}$ scales proportionally to $T_{\\rm sky}^2/ \\Delta \\tilde{T}_{b, 21}^2({k})$ .", "To quantify the detectability of the signal in our models, we consider two hypothetical interferometer designs in this paper.", "Our first design is motivated by the design of the MWA instrument, which is targeting the 21cm signal from $z\\sim 10$ .", "Specifically, for $z=20$ we assume an instrument that consists of a 40 meter core of closely packed antennae with filling fraction $0.5$ and with an $r^{-2}$ baseline distribution to larger radii.", "We assume each dipole contributes $\\lambda ^2/4$ in collecting area and that the dipoles have spacing $\\lambda /2$ .", "In addition, we contract or dilate this $z=20$ array in proportion to the targeted wavelength when we consider the signal from $z=15$ or $z=30$ in order to optimize the instrument to the specified redshift.", "MWA's cost is $\\sim 20~$ million US dollars for 500 antennae.", "The cost of our analogous lower frequency instrument would be comparable.", "We also consider in the body of this paper a next generation array with 10 times the number of antennae elements and, for $z=20$ , a $160~$ meter core prior to an $r^{-2}$ falloff, which we denote as “MWA–like $\\times $  10”.", "The sensitivity of these two hypothetical instruments to the signal is discussed in Section REF .", "See [44] for additional details regarding our sensitivity calculations." ] ]

[ [ "Common zeros of inward vector fields on surfaces" ], [ "Abstract A vector field X on a manifold M with possibly nonempty boundary is inward if it generates a unique local semiflow $\\Phi^X$.", "A compact relatively open set K in the zero set of X is a block.", "The Poincar\\'e-Hopf index is generalized to an index for blocks that may meet the boundary.", "A block with nonzero index is essential.", "Let X, Y be inward $C^1$ vector fields on surface M such that $[X,Y]\\wedge X=0$ and let K be an essential block of zeros for X.", "Among the main results are that Y has a zero in K if X and $Y$ are analytic, or Y is $C^2$ and $\\Phi^Y$ preserves area.", "Applications are made to actions of Lie algebras and groups." ], [ "Introduction", "Let $M$ be an $n$ -dimensional manifold with boundary $\\partial M$ and $X$ a vector field on $M$ , whose value at $p$ is denoted by $X_p$ .", "The zero set of $X$ is ${\\mathsf {Z} ( X)}:= \\lbrace p\\in M\\colon \\,X_p=0\\rbrace .$ The set of common zeros for a set ${\\mathfrak {s}}$ of vector fields is $\\textstyle {\\mathsf {Z} ( {\\mathfrak {s}})}:=\\bigcap _{X\\in {\\mathfrak {s}}}{\\mathsf {Z} ( X)}.$ A block for $X$ is a compact, relatively open subset $K\\subset {\\mathsf {Z} ( X)}$ .", "This means $K$ lies in a precompact open set $U\\subset M$ whose topological boundary $\\mathsf {bd}(U)$ contains no zeros of $X$ .", "We say that $U$ is isolating for $(X,K)$ , and for $X$ when $K:={\\mathsf {Z} ( X)}\\cap U$ .", "When $M$ is compact, ${\\mathsf {Z} ( X)}$ is a block for $X$ with $M$ as an isolating neighborhood.", "Definition 1.1 If $p\\in M{\\tt \\}\\partial M$ is an isolated zero of $X$ , the index of $X$ at $p$, denoted by $\\mathsf {i}_p X$ , is the degree of the map of the unit $(n-1)$ -sphere ${\\bf S}^{n-1}\\rightarrow {\\bf S}^{n-1}, \\quad z\\mapsto \\frac{\\hat{X}(z)}{\\Vert \\hat{X} (z)\\Vert },$ where $\\hat{X}$ is the representative of $X$ in an arbitrary coordinate system centered at $p$ .", "When $U$ is isolating for $(X, K)$ and disjoint from ${\\mathsf {Z} ( X)} \\cap \\partial M$ , the Poincaré-Hopf index of $X$ at $K$ is $\\mathsf {i}^{\\rm PH}_K (X) :=\\sum _p\\mathsf {i}_pY, \\quad (p\\in {\\mathsf {Z} ( Y)} \\cap U)$ where $Y$ is any vector field on $M$ such that ${\\mathsf {Z} ( Y)} \\cap \\overline{U}$ is finite, and there is a homotopy of vector fields $\\left\\lbrace X^t\\right\\rbrace _{0\\le t\\le 1}$ from $X^0=X$ to $X^1=Y$ such that $\\bigcup _{t}]{\\mathsf {Z} ( X^t)} \\cap U$ is compact.The Poincaré-Hopf index goes back to Poincaré [28] and Hopf [15].", "It is usually defined only when $M$ is compact and $K={\\mathsf {Z} ( X)}$ .", "The more general definition above is adapted from Bonatti [3].", "The block $K$ is essential for $X$ if  $\\mathsf {i}^{\\rm PH}_K (X)\\ne 0$ .", "Christian Bonatti [3] proved the following remarkable result: Theorem 1.2 (Bonatti) Assume ${\\mathsf {dim}\\,}M\\le 4$ and $\\partial M=\\varnothing $ .", "If $X,Y$ are analytic vector fields on $M$ such that $[X, Y]= 0$ , then ${\\mathsf {Z} ( Y)}$ meets every essential block of zeros for $X$ .Bonatti assumes ${\\mathsf {dim}\\,}M = 3$ or 4, but the conclusion for ${\\mathsf {dim}\\,}M\\le 2$ follows easily: If ${\\mathsf {dim}\\,}M=2$ , identify $M$ with $M\\times \\lbrace 0\\rbrace \\subset M\\times {\\mathbb {R}}$ and apply Bonatti's theorem to the vector fields $\\left(X, x\\frac{\\partial ~}{\\partial x}\\right)$ and $\\left(Y, x\\frac{\\partial ~}{\\partial x}\\right)$ on $M\\times {\\mathbb {R}}$ .", "For ${\\mathsf {dim}\\,}M=1$ there is a simple direct proof.", "Our main results, Theorems REF and REF , reach similar conclusions for surfaces $M$ which may have nonsmooth boundaries, and certain pairs of vector fields that generate local semiflows on $M$ , including cases where the fields are not analytic and do not not commute.", "Applications are made to actions of Lie algebras and Lie groups.", "Next we define terms (postponing some details), state the main theorems and apply them to Lie actions.", "After sections on dynamics and index functions, the main results are proved in Section ." ], [ "Terminology", "${\\mathbb {Z}}$ denotes the integers, ${\\mathbb {N}}_{+}$ the positive integers, ${\\mathbb {R}}$ the real numbers, and ${\\mathbb {R}}_+$ the closed half line $[0,\\infty )$ .", "Maps are continuous, and manifolds are real, smooth and metrizable, unless otherwise noted.", "The set of fixed points of a map $f$ is $\\operatorname{\\mathsf {Fix}}(f)$ .", "The following assumptions are always in force: Hypothesis 1.3 $\\tilde{M}$ is an analytic $n$ -manifold with empty boundary.", "$M\\subset \\tilde{M}$ is a connected topological $n$ -manifold.The only role of $\\tilde{M}$ is to permit a simple definition of smooth maps on $M$ .", "Its global topology is never used, and in any discussion $\\tilde{M}$ can be replaced by any smaller open neighborhood of $M$ .", "If $n>2$ then $M$ might not be smoothable, as shown by a construction due to Kirby [17]: Let $P$ be a nonsmoothable closed 4-manifold (Freedman [7]).", "Let $D\\subset P$ be a compact 4-disk.", "Then $M:=P\\setminus \\operatorname{\\mathsf {Int}}D$ is not smoothable, for otherwise $\\partial M$ would be diffeomorphic to ${\\bf S}^{3}$ and $P$ could be smoothed by gluing $$ to $M$ .", "Define $\\tilde{M}$ as the connected sum of $P$ with a nontrivial ${\\bf S}^{2}$ -bundle over ${\\bf S}^{2}$ .", "Then $\\tilde{M}$ is smoothable and contains $M$ (compare Friedl et al.", "[8]).", "We call $M$ an analytic manifold when $\\partial M$ is an analytic submanifold of $\\tilde{M}$ .", "The tangent vector bundle of $\\tilde{M}$ is $T\\tilde{M}$ , whose fibre over $p$ is the vector space $T_p\\tilde{M}$ .", "The restriction of $T\\tilde{M}$ to a subset $S\\subset \\tilde{M}$ is the vector bundle $T_S\\tilde{M}$ .", "We set $TM:=T_M\\tilde{M}$ .", "A map $f$ sending a set $S\\subset M$ into a smooth manifold $N$ is called $C^r$ if it extends to a map $\\tilde{f}$ , defined on an open subset of $\\tilde{M}$ , that is $C^r$ in the usual sense.", "Here $r\\in {\\mathbb {N}}_{+}\\cup \\lbrace \\infty , \\omega \\rbrace $ , and $C^\\omega $ means analytic.", "If $f$ is $C^1$ and $S$ has dense interior in $M$ , the tangent map $T\\tilde{f}\\colon \\,T\\tilde{M}\\rightarrow T N$ restricts to a bundle map $Tf\\colon \\,T_SM\\rightarrow TN$ determined by $f$ .", "A vector field on $S$ is a section $X\\colon \\,S\\rightarrow T_S M$ , whose value at $p$ is denoted by $X_p$ .", "The set of these vector fields is a linear space $S)$ .", "The linear subspaces $r (S)$ and ${\\mathsf {L}}(M)$ , comprising $C^r$ and locally Lipschitz fields respectively, are given the compact-open topology (uniform convergence on compact sets).", "$X$ and $Y$ always denote vector fields on $M$ .", "When $X$ is $C^r$ , $\\tilde{X}$ denotes an extension of $X$ to a $C^r$ vector field on an open set $W\\subset \\tilde{M}$ .", "The Lie bracket of $X, Y \\in 1 (M)$ is the restriction to $M$ of $[\\tilde{X}, \\tilde{Y}]$ .", "This operation makes $\\omega (M)$ and $\\infty (M)$ into Lie algebras.", "$X\\wedge Y$ denotes the tensor field of exterior 2-vectors $p\\mapsto X_p\\wedge Y_p \\in \\Lambda ^2(T_pM)$ .", "Evidently $X\\wedge Y=0$ iff $X_p$ and $Y_p$ are linearly dependent at all $p\\in M$ ." ], [ "Inward vector fields", "A tangent vector to $M$ at $p$ is inward if it is the tangent at $p$ to a smooth curve in $M$ through $p$ .", "The set of inward vectors at $p$ is $T^{\\mathsf {in}}_pM$ .", "A vector field $X$ is inward if $X(M)\\subset T^{\\mathsf {in}} (M)$ , and there is a unique local semiflow $\\Phi ^X:=\\left\\lbrace \\Phi ^X_t\\right\\rbrace _{t\\in {\\mathbb {R}}_+}$ on $M$ whose trajectories are the maximally defined solutions to the initial value problems $ \\frac{dy}{dt} = X(y), \\quad y(0)=p, \\qquad p\\in M, \\quad t \\ge 0.$ The set of inward vector fields is ${\\mathsf {in}} (M)$ .", "When $\\partial M$ is a $C^1$ submanifold of $\\tilde{M}$ , it can be shown that $X$ is inward iff $X (M)\\subset T^{\\mathsf {in}} (M)$ .", "Define $r_{\\mathsf {in}} (M):= {\\mathsf {in}} (M)\\cap r (M), \\qquad {\\mathsf {L}}_{\\mathsf {in}}(M):= {\\mathsf {in}} (M)\\cap {\\mathsf {L}}(M).$ Proposition REF shows these sets are convex cones in $M)$ ." ], [ "The vector field index and essential blocks of zeros", "Let $K$ be a block of zeros for $X\\in {\\mathsf {in}} (M)$ , and $U\\subset M$ an isolating neighborhood for $(X, K)$ .", "The vector field index $\\mathsf {i}_K (X):=\\mathsf {i} (X, U)\\in {\\mathbb {Z}}$ is defined in Section as the fixed point index of the map $\\Phi ^X_t|U\\colon \\,U\\rightarrow M$ , for any $t >0$ so small that the compact set $\\overline{U}$ lies in the domain of $\\Phi ^X_t$ .", "The block $K$ is essential (for $X$ ) when $\\mathsf {i}_K (X)\\ne 0$ .", "A version of the Poincaré-Hopf theorem implies $K$ is essential if it is an attractor for $\\Phi ^X$ and has nonzero Euler characteristic $\\chi (K)$ ." ], [ "Statement of results", "In the next two theorems, besides Hypothesis REF we assume: Hypothesis 1.4   $M$ and $\\tilde{M}$ are surfaces, $X$ and $Y$ are $C^1$ inward vector fields on $M$ , $K\\subset M$ is an essential block of zeros for $X$ , $[X, Y]\\wedge X=0$ .", "The last condition has the following dynamical significance (Proposition REF ): $\\Phi ^Y$ permutes integral curves of $X$ .", "This implies: if $q=\\Phi ^Y_t (p)$ then $ X_q=\\lambda \\cdot T\\Phi ^Y_t(X_p)$ for some $ \\lambda >0$ , ${\\mathsf {Z} ( X)}$ is positively invariant under $\\Phi ^Y$ .", "(See Definition REF .)", "A cycle for $Y$ , or a $Y$ -cycle, is a periodic orbit of $\\Phi ^Y$ that is not a fixed point.", "Theorem 1.5 Assume Hypothesis REF.", "Each of the following conditions implies ${\\mathsf {Z} ( Y)}\\cap K\\ne \\varnothing $ : (a) $X$ and $Y$ are analytic.", "(b) Every neighborhood of $K$ contains an open neighborhood whose boundary is a nonempty union of finitely many $Y$ -cycles.", "When $[X, Y]=0$ this extends Bonatti's Theorem to surfaces with nonempty boundaries.", "The case $[X, Y]= cX, \\, c\\in {\\mathbb {R}}$ yields applications to actions of Lie algebras and Lie groups.", "Example 1.6 In his pioneering paper [18], E. Lima constructs vector fields $X, Y$ on the closed disk $$ , tangent to $\\partial $ and generating unique flows, such that $[X, Y]= X$ and ${\\mathsf {Z} ( X)}\\cap {\\mathsf {Z} ( Y)}=\\varnothing $ (see Remark REF ).", "Such fields can be $C^\\infty $ (M. Belliart & I. Liousse [2], F.-J Turiel [33]).", "The unique block of zeros for $X$ is ${\\mathsf {Z} ( X)}=\\partial $ , which is essential because $\\chi ()\\ne \\varnothing $ , but ${\\mathsf {Z} ( Y)}$ is a point in the interior of $$ .", "This shows that the conclusion of Theorem REF (a) can fail when $X$ and $Y$ are not analytic.", "The flows of $X$ and $Y$ generate an effective, fixed-point free action by a connected, solvable nonabelian Lie group.", "A local semiflow on a surface $M$ preserves area if it preserves a Borel measure on $M$ that is positive and finite on nonempty precompact sets.", "Theorem 1.7 Assume Hypothesis REF.", "If $\\Phi ^Y$ preserves area, each of the following conditions implies ${\\mathsf {Z} ( Y)} \\cap K\\ne \\varnothing $ : (i) $K$ contains a $Y$ -cycle, (ii) $Y$ is $C^2$ , (iii) $K$ has a planar neighborhood in $M$ .", "Definition 1.8 For $X\\in \\omega _{\\mathsf {in}} (M)$ define $\\mathsf {W}(X):=\\lbrace Y\\in \\omega _{\\mathsf {in}} (M) \\colon \\,[X, Y]\\wedge X=0\\rbrace ,$ which is the set of inward analytic vector fields on $M$ whose local semiflows permute integral curves of $X$ .", "Propositions REF and REF imply $\\mathsf {W}(X)$ is a convex cone that is closed under Lie brackets, and a subalgebra of $\\omega (M)$ if $M$ is an analytic manifold without boundary.", "Theorem 1.9 Assume Hypothesis REF holds.", "If $X$ is analytic and $\\partial M$ is an analytic subset of $\\tilde{M}$ , then $ {\\mathsf {Z} ( \\mathsf {W} (X))}\\cap K\\ne \\varnothing $ ." ], [ "Actions of Lie algebras and Lie groups", "Theorem 1.10 Let $M$ be an analytic surface and $\\mathfrak {g}$ a Lie algebra (perhaps infinite dimensional) of analytic vector fields on $M$ that are tangent to $\\partial M$ .", "Assume $X\\in \\mathfrak {g}$ spans a nontrivial ideal.", "Then: (a) ${\\mathsf {Z} ( \\mathfrak {g})}$ meets every essential block of zeros for $X$ .", "(b) If $M$ is compact and $\\chi (M)\\ne 0$ , then ${\\mathsf {Z} ( \\mathfrak {g})}\\ne \\varnothing $ .", "Note that $\\mathfrak {g}$ has 1-dimensional ideal if its center is nontrivial, or $\\mathfrak {g}$ is finite dimensional and supersoluble (Jacobson [16]).", "A finite dimensional solvable Lie algebra of vector fields on a surface has derived length $\\le 3$ (Epstein & Thurston [6]).", "Plante [26] points out that $\\omega ({\\mathbb {R}}^{2})$ contains infinite-dimensional subalgebras, such as the Lie algebra of quadratic vector fields.", "An action of a group $G$ on a manifold $M$ is a homomorphism $\\alpha \\colon \\,g\\rightarrow g^\\alpha $ from $G$ to the homeomorphism group of $P$ , such that the corresponding evaluation map $\\mathsf {ev_\\alpha }\\colon \\,G\\times P\\rightarrow P, \\quad (g,p)\\mapsto g^\\alpha (p)$ is continuous.", "When $\\mathsf {ev_\\alpha }$ is analytic, $\\alpha $ is an analytic action.", "Theorem 1.11 Assume $M$ is a compact analytic surface and $G$ is a connected Lie group having a one-dimensional normal subgroup.", "If $\\chi (M)\\ne 0$ , every effective analytic action of $G$ on $M$ has a fixed point.", "For supersoluble $G$ this is due to Hirsch & Weinstein [14]." ], [ "Background on group actions", "The literature on actions of connected Lie groups $G$ include the following notable results: Proposition 1.12 If $G$ is solvable (respectively, nilpotent) and acts effectively on an $n$ -dimensional manifold, the derived length of $G$ is $\\le n+1$ (respectively, $\\le n$ ) (Epstein & Thurston [6]).", "In the next two propositions $M$ denotes a compact connected surface.", "Proposition 1.13 Assume $G$ acts on $M$ without fixed points.", "(i) If $G$ is nilpotent, $\\chi (M)\\ne 0$ (Plante [25]).", "(ii) If the action is analytic, $\\chi (M)\\ge 0$ (Turiel [33], Hirsch [11]).", "Proposition 1.14 Let ${\\rm Aff}_+ ({\\mathbb {R}}^{m})$ denote the group of orientation-preserving affine homeomorphisms of ${\\mathbb {R}}^{m}$ .", "(a) If $\\chi (M) < 0$ and $G$ acts effectively on $M$ without fixed points, then $G$ has a quotient isomorphic to ${\\rm Aff}_+ ({\\mathbb {R}}^{1})$ (Belliart [1]).", "(b) ${\\rm Aff}_+({\\mathbb {R}}^{1})$ has effective fixed-point free actions on $M$ (Plante [25]).", "(c) ${\\rm Aff}_+ ({\\mathbb {R}}^{2})$ has effective analytic actions on $M$ (Turiel [33]).", "For related results see the references above, also Belliart [1], Hirsch [12], [13], Molino & Turiel [21], [22], Plante [27], Thurston [31], Turiel [32].", "Transitive effective surface actions are classified in Mostow's thesis [23], with a useful summary in Belliart [1]." ], [ "Dynamics", "Let $\\Psi :=\\lbrace \\Psi _t\\rbrace _{t\\in \\mathsf {T}}$ denote a local flow ($\\mathsf {T}={\\mathbb {R}})$ or a local semiflow ($\\mathsf {T}={\\mathbb {R}}_+)$ on a topological space $S$ .", "Each $\\Psi _t$ is a homeomorphism from an open set ${\\mathcal {D}}(\\Psi _t)\\subset S$ onto a set ${\\mathcal {R}} (\\Psi _t)\\subset S$ , such that: $\\Psi _t (p)$ is continuous in $(t,p)$ , $\\Psi _0$ is the identity map of $S$ , if $0 \\le |s| \\le |t|$ and $|st|\\ge 0$ then ${\\mathcal {D}}(\\Psi ^s)\\supset {\\mathcal {D}} (\\Psi ^t)$ , $ \\Psi _t (\\Psi _s (p)) = \\Psi _{t+s} (p)$ .", "We adopt the convention that notation of the form “$\\Psi _t (x)$ ” presumes $x\\in {\\mathcal {D}} (\\Psi _t)$ .", "Definition 2.1 A set $L\\subset S$ is positively invariant under $\\Psi $ provided $\\Psi _t$ maps $L\\cap {\\mathcal {D}}(\\Psi _t)$ into $L$ for all $t\\ge 0$ , and invariant when $L\\subset {\\mathcal {D}} (\\Psi _t)\\cap {\\mathcal {R}} (\\Psi _t)$ for all $t \\in \\mathsf {T}$ .", "When $\\Psi $ is generated by a vector field $Y$ we use the analogous terms “positively $Y$ -invariant” and “$Y$ -invariant.” Let $M, \\tilde{M}$ be as in Hypothesis REF .", "When $\\Psi $ is a local semiflow on $M$ , the theorem on invariance of domain shows that ${\\mathcal {R}} (\\Psi _t)$ is open in $M$ when $\\Phi $ is a local flow, and also when ${\\mathcal {D}} (\\Psi _t)\\cap \\partial M=\\varnothing $ .", "This implies $M\\setminus \\partial M$ is positively invariant under every local semiflow on $M$ .", "Proposition 2.2 Assume $X, Y\\in {\\mathsf {in}} (M)$ and $[X, Y]\\wedge X=0$ .", "(a) If $\\Phi ^Y_t (p)=q$ then $T_p\\Phi ^Y_t \\colon \\,X_p\\mapsto cX_q,\\, c >0$ .", "(b) $\\Phi ^Y_t$ sends integral curves of $X$ to integral curves of $X$ .", "(c) ${\\mathsf {Z} ( X)}$ is $Y$ -invariant.", "Proof Let $\\tilde{X}$ and $\\tilde{Y}\\in 1 (\\tilde{M})$ be extensions of $X$ and $Y$ , respectively.", "It suffices to prove:" ], [ "(*)", "If $p\\in M,\\ t\\ge 0$ and $p(t):= \\Phi _t^{\\tilde{X}} (p)$ , then the linear map $T_p\\Phi _t^{ \\tilde{Y}}\\colon \\,T_p \\tilde{M}\\rightarrow T_{p(t)}\\tilde{M}$ sends $X_p$ to a positive scalar multiple of $X_{p(t)}$ .", "By continuity it suffices to prove this when $Y_p\\ne 0$ , $X_p\\ne 0$ and $|t|$ sufficiently small.", "Working in flowbox coordinates $(u_j)$ for $\\tilde{Y}$ in a neighborhood of $p$ , we assume $\\tilde{M}$ is an open set in $ {\\mathbb {R}}^{n}$ ,   $\\tilde{Y}=\\frac{\\partial ~}{\\partial u_1}$ , and $\\tilde{X}$ has no zeros.", "Because $[X, Y]\\wedge X=0$ , there is a unique continuous map $f\\colon \\,M\\rightarrow {\\mathbb {R}}$ such that $[X, Y]=fX$ .", "Since $\\tilde{Y}$ is a constant vector field, the vector-valued function $t\\mapsto X_{p(t)}$ satisfies $\\frac{dX_{p(t)}}{dt} = -f(p(t))\\cdot X_{p(t)},$ whose solution is $X_{p (t)}= e^{-\\int _0^t f(s)ds}\\cdot X_{p(0)}.$ This implies (*).", "The following fact is somewhat surprising because $T^{\\mathsf {in}}_pM$ need not be convex in $T_pM$ : Proposition 2.3 ${\\mathsf {L}}_{\\mathsf {in}}(M)$ is a convex cone in $M)$ .", "Proof As ${\\mathsf {L}}(M)$ is a convex cone in $M)$ , it suffices to show that ${\\mathsf {L}}_{\\mathsf {in}} (M)$ is closed under addition.", "Let $X, Y\\in {\\mathsf {L}}(M)$ .", "We need to prove: $ \\mbox{\\em If $p\\in \\partial M$ there exists $\\epsilon >0$ such that $0\\le t\\le \\epsilon \\Rightarrow \\Phi _t^{ X + Y} (p)\\in M$.", "}$ This is easily reduced to a local result, hence we assume $M$ is relatively open in the closed halfplane ${\\mathbb {R}}\\times [0,\\infty )$ and $X, Y$ are Lipschitz vector rields on $M$ .", "Let $\\tilde{X}, \\tilde{Y}$ be extensions of $X, Y$ to Lipschitz vector fields on an open neighborhood $\\tilde{M}\\subset {\\mathbb {R}}^{2}$ of $M$ ( Johnson et al.", "[19]).", "Denote the local flows of $\\tilde{X}, \\, \\tilde{Y}, \\, \\tilde{X}+ \\tilde{Y}$ respectively by $\\lbrace f_t\\rbrace ,\\lbrace g_t\\rbrace , \\lbrace h_t\\rbrace , \\ (t\\in {\\mathbb {R}})$ .", "We use a special case of Nelson [24]: Proposition 2.4 For every $p\\in \\tilde{M}$ there exists $\\epsilon >0$ and a neighborhood $W\\subset \\tilde{M}$ of $p$ such that $h_t (x)=\\lim _{k\\rightarrow \\infty }\\left(f_{t/k}\\circ g_{t/k}\\right)^k (x)$ uniformly for $x\\in W$ and $|t|<\\epsilon $ .", "Because $X$ and $Y$ are inward, $M$ is positively invariant under the local semiflows $\\lbrace f_t\\rbrace _{t\\ge 0}$ and $\\lbrace g_t\\rbrace _{t\\ge 0}$ .", "Therefore $0\\le t\\le \\epsilon \\Rightarrow \\left(f_{t/k}\\circ g_{t/k}\\right)^k \\in M,\\qquad (k\\in {\\mathbb {N}}_{+}).$ As $M$ is relatively closed in $\\tilde{M}$ , Proposition REF implies $h_t (x)\\in M$ for $0\\le t\\le \\epsilon $ , which yields (REF ).", "Examination of the proof yields: Corollary 2.5 If $L$ is a closed subset of a smooth manifold $N$ , the set of locally Lipschitz vector fields on $N$ for which $L$ is positively invariant is a convex cone.", "Question 2.6 Is ${\\mathsf {in}}(M)$ is a convex cone in $M)$ ?" ], [ "Index functions", "We review properties of the fixed point index $I (f)$ defined by the late Professor Albrecht Dold ([4], [5]).", "Using it we define an equilibrium index $I_K (\\Phi )$ for local semiflows, and a vector field index $\\mathsf {i}_K (X)$ for inward vector fields." ], [ "Dold's Hypothesis:", " $V$ is an open set in a topological space $S$ .", "$f\\colon \\,V \\rightarrow S$ is a continuous map with compact fixed point set  $\\operatorname{\\mathsf {Fix}}(f)\\subset V$ .", "$V$ is a Euclidean neighborhood retract (ENR).This means $V$ is homeomorphic to a retract of an open subset of some Euclidean space.", "Polyhedra and connected metrizable manifolds are ENRs.", "On the class of maps satisfying these conditions, Dold constructs an integer-valued fixed point index denoted here by $I (f)$ , uniquely characterized by the following five properties (see [5]): (FP1) $I(f)=I(f|V_0)$   if $V_0\\subset V$ is an open neighborhood of $\\operatorname{\\mathsf {Fix}}(f)$ .", "(FP2) $I(f)={\\left\\lbrace \\begin{array}{ll}& 0 \\ \\mbox{if $\\operatorname{\\mathsf {Fix}}(f) =\\varnothing $,}\\\\& 1 \\ \\mbox{if $f$ is constant.}\\end{array}\\right.", "}$ (FP3) $I(f)=\\sum _{i=1}^m I(f| V_i)$   if $V$ is the disjoint union of $m$ open sets $V_i$ .", "(FP4) $I (f\\times g)=I (f)\\cdot I (g)$ .", "(FP5) $I(f_0)=I(f_1)$   if there is a homotopy $f_t\\colon \\,V\\rightarrow S,\\, (0\\le t \\le 1)$   such that $\\bigcup _t\\operatorname{\\mathsf {Fix}}(f_t)$ is compact.", "These correspond to (5.5.11) — (5.5.15) in [5].", "In addition: (FP6) If $f$ is $C^1$ and $\\operatorname{\\mathsf {Fix}}(f)$ is an isolated fixed point $p$ , then $I(f)= (-1)^\\nu $ where $\\nu $ is the number of eigenvalues $\\lambda $ of $f$ such that $\\lambda >1$ , ignoring multiplicities  ([5]).", "(FP7) If $S$ is an ENR and $f\\colon \\,S\\rightarrow S$ is homotopic to the identity map, then $I(f)= \\chi (S).$ See Dold [5].", "Lemma 3.1 If  $g$   is sufficiently close to $f$ in the compact open topology, then $\\operatorname{\\mathsf {Fix}}(g)$ is compact and $I(g)=I(f)$ .", "Proof We can assume $\\rho \\colon \\,W\\rightarrow V$ is a retraction, where $W\\subset {\\mathbb {R}}^{m}$ is an open set containing $V$ .", "For $g$ sufficiently close to $f$ the following hold: $W$ contains the line segment (or point) spanned by $\\lbrace f(p), g(p)\\rbrace $ for every $p\\in V$ , and the maps $f_t\\colon \\,(t,p)\\mapsto \\rho ((1-t)f(p) + t g(p)),\\qquad (0\\le t\\le 1,\\quad p\\in V)$ constitute a homotopy in $V$ from $f_0=f$ to $f_1=g$ with $\\bigcup _t\\operatorname{\\mathsf {Fix}}(f_t)$ is compact.", "Therefore the conclusion follows from (FP5)." ], [ "The equilibrium index for local semiflows", "Let $\\Phi :=\\lbrace \\Phi _t\\rbrace _{t\\ge 0}$ be a local semiflow in a topological space ${\\mathcal {C}}$ , with equilibrium set ${\\mathcal {E}} (\\Phi ):={\\textstyle \\bigcap }_{t\\ge 0}\\operatorname{\\mathsf {Fix}}(\\Phi _t).$ $K\\subset {\\mathcal {E}} (\\Phi )$ is a block if $K$ is compact and has an open, precompact ENR neighborhood $V\\subset {\\mathcal {C}}$ such that $\\overline{V}\\cap {\\mathcal {E}} (\\Phi )\\subset V$ .", "Such a $V$ is an isolating neighborhood for $K$ .", "With these assumptions on $\\Phi $ and $V$ we have: Lemma 3.2 There exists $\\tau >0$ such that the following hold when $0 <t \\le \\tau $ : (a) $\\operatorname{\\mathsf {Fix}}(\\Phi _t)\\cap V$ is compact, (b) $I (\\Phi _t|V)=I (\\Phi _\\tau |V)$ .", "Proof If (a) fails, there are convergent sequences $\\lbrace t_k\\rbrace $ in $[0,\\infty )$ and $\\lbrace p_k\\rbrace $ in $V$ such that $t_k\\searrow 0, \\quad p_k\\in \\operatorname{\\mathsf {Fix}}(\\Phi _{t_k})\\cap V, \\quad p_k\\rightarrow q\\in \\mathsf {bd} (V).$ Joint continuity of $(t,x)\\mapsto \\Phi _t (x)$ yields the contradiction $q\\in {\\mathcal {E}} (\\Phi ) \\cap \\mathsf {bd} (V)$ .", "Assertion (b) is a consequence of (a) and (FP5).", "It follows that the fixed point index $I(\\Phi _\\tau |V)$ depends only on $\\Phi $ and $K$ , and is the same for all isolating neighborhoods $V$ of $K$ .", "Definition 3.3 Let $\\tau >0$ be as in Lemma REF (b).", "We call $I(\\Phi _\\tau |V)$ the equilibrium index of $\\Phi $ in $V$ , and at $K$ , denoted by $\\mathsf {i}(\\Phi ,V)$ and $\\mathsf {i}_K (\\Phi )$ ." ], [ "The vector field index for inward vector fields", "In the rest of this section the manifolds $\\tilde{M}$ and $M\\subset \\tilde{M}$ are as in Hypothesis REF , $K$ is a block of zeros for $X$ for $X\\in {\\mathsf {in}} (M)$ , and $U$ an isolating neighborhood for $(X, K)$ .", "Then $K$ is also a block of equilibria for the local semiflow $\\Phi ^X$ , and the equilibrium index $\\mathsf {i} (\\Phi ^X,U)$ is defined in Definition REF .", "Definition 3.4 The vector field index of $X$ in $U$ (and  at $K$) is $\\begin{split}\\mathsf {i} (X, U)=\\mathsf {i}_K (X) & := \\mathsf {i} (\\Phi ^X, U).\\end{split}$ $K$ is essential (for $X$ ) if $\\mathsf {i}_K (X)\\ne 0$ , and inessential otherwise.", "Two vector fields $X_j\\in {\\mathsf {in}} (M_j), \\,j=1,2$ have isomorphic germs at $K_j\\subset M_j$ provided there are open neighborhoods $U_j\\subset M_j$ of $X_j$ and a homeomorphism $U_1\\approx U_2$ conjugating $\\Phi ^{X_1}|U_1$ to $\\Phi ^{X_2}|U_2$ .", "Proposition 3.5 The vector field index has the following properties: (VF1) $i_K (X)=\\mathsf {i}_{K^{\\prime }} (X^{\\prime })$   if $K^{\\prime }$ is a block of zeros for $X^{\\prime }\\in {\\mathsf {in}} (M^{\\prime })$ and the germs of $X$ at $K$ and $X^{\\prime }$ at $K^{\\prime }$ are isomomorphic.", "(VF2) If $\\mathsf {i} (X, U)\\ne 0$ then ${\\mathsf {Z} ( X)}\\cap U\\ne \\varnothing $ .", "(VF3) $\\mathsf {i} (X, U)=\\sum _{j=1}^m \\mathsf {i} (X, U_j)$ provided $U$ is the union of disjoint open sets $U_1,\\dots ,U_m$ .", "(VF4) If $Y$ is sufficiently close to $X$ in ${\\mathsf {in}}(M)$ , then $U$ is isolating for $Y$ and $\\mathsf {i} (X, U)=\\mathsf {i}(Y, U)$ .", "(VF5) $\\mathsf {i}_K (X)$ equals the Poincaré-Hopf index $\\mathsf {i}^{\\rm PH}_K (X)$ provided $K\\cap \\partial M=\\varnothing $ .", "Proof   (VF1): A consequence of (FP1).", "(VF2): Follows from (FP2).", "(VF3): Follows from (FP3).", "(VF4): Use Lemma REF .", "(VF5): Since $X$ can be approximated by locally $C^\\infty $ vector fields transverse to the zero section, using compactness of $\\overline{U}$ and (VF4) we assume ${\\mathsf {Z} ( X)} \\cap U$ is finite set of hyperbolic equilibria.", "By (FP3) we assume ${\\mathsf {Z} ( X)} \\cap U$ is a hyperbolic equilibrium $p$ .", "In this case the index of $X$ at $p$ is $(-1)^\\nu $ where $\\nu $ is the number of positive eigenvalues of $dX_p$ (ignoring multiplicity).", "The conclusion follows from (FP6) and Definitions REF , REF .", "Proposition 3.6 If $M$ is compact,  $\\mathsf {i} (X, M)= \\chi (M)$ .", "Proof Follows from (FP7).", "Proposition 3.7 Assume $X, Y\\in {\\mathsf {in}} (M)$ and $U\\subset M$ is isolating for $X$ .", "Then $U$ is isolating for $Y$ , and $\\mathsf {i} (X, U) =\\mathsf {i}(Y, U),$ provided one of the following holds: (i) $Y|\\mathsf {bd}(U)$ is sufficiently close to $X|\\mathsf {bd} (U)$ , (ii) $Y|\\mathsf {bd}(U)$ is nonsingularly homotopic to $X|\\mathsf {bd}(U)$ .", "Proof Both (i) and (ii) imply $U$ is isolating for $Y$ .", "Consider the homotopy $Z^t:= (1-t)X + tY,\\qquad (0\\le t\\le 1).$ When (i) holds each vector field $Z^t|{\\mathsf {bd} (U)}$ is nonsingular, implying (ii).", "In addition, $Z^t$ is inward by Proposition REF , and Lemma REF yields $\\tau >0$ such that $0<t\\le \\tau \\Rightarrow \\mathsf {i} (X, U)=I(\\Phi ^X_t|U), \\quad \\mathsf {i} (Y,U)=I(\\Phi ^Y_t|U).$ By Lemma REF , each $t\\in [0,1]$ has an open neighborhood $J_t\\subset [0,1]$ such that $s\\in J_t\\Rightarrow I (\\Phi ^{X^s}_\\tau |U) =I (\\Phi ^{X^t}_\\tau |U).$ Covering $[0,1]$ with sets $J_{t_1},\\dots , J_{t_\\nu }$ and inducting on $\\nu \\in {\\mathbb {N}}_{+}$ shows that $I (\\Phi ^X_\\tau |U) = I (\\Phi ^Y_\\tau |U),$ which by Definition REF implies the conclusion." ], [ "The index as an obstruction", "The following algebraic calculation of the index is included for completeness, but not used.", "Assume $M$ is oriented and $U$ is an isolating neighborhood for a block $K\\subset {\\mathsf {Z} ( X)}$ .", "Let $V\\subset U$ be a compact smooth $n$ -manifold with the orientation induced from $M$ , such that $K\\subset V{\\tt \\} \\partial V$ .", "The primary obstruction to extending $X|\\partial V$ to a nonsingular section of $TV$ is the relative Euler class ${\\bf e}_{(X, V)}\\in H^n (V, \\partial V)$ Let ${\\bf v}\\in H_n (V, \\partial V)$ be the homology class corresponding to the induced orientation of $V$ .", "Denote by $H^n (V, \\partial V) \\times H_n (V,\\partial V)\\rightarrow {\\mathbb {Z}},\\quad ({\\bf c}, {\\bf u}) \\rightarrow \\langle {\\bf c}, {\\bf u}\\rangle ,$ the Kronecker Index pairing, induced by evaluating cocycles on cycles.", "Unwinding definitions leads to: Proposition 3.8 With $M, X, K, V$ are as above, $\\mathsf {i}_K (X)= \\langle {\\bf v}, {\\bf e}_{(X, V)} \\rangle .$ When $M$ is nonorientable the same formula holds provided the coefficients for $H^n (V, \\partial V)$ and $ H_n (V,\\partial V)$ are twisted by the orientation sheaf of $V$ ." ], [ "Stability of essential blocks ", "An immediate consequence of Propositions REF (i) and property (VF2) of REF is: Corollary 3.9 If a block $K$ is essential for $X$ , and $Y\\in {\\mathsf {in}} (M)$ is sufficiently close to $X$ , then every neighborhood of $K$ contains an essential block for $Y$ .", "Thus essential blocks are stable under perturbations of the vector field.", "It is easy to see that a block is stable if it contains a stable block.", "For example, the block $\\lbrace -1, 1\\rbrace $ for $X= (x^2-1)\\frac{\\partial }{\\partial x}$ on ${\\mathbb {R}}$ is stable, but inessential.", "But the following result (not used) means that a block can be perturbed away iff every subblock is inessential: Proposition 3.10 Assume $\\partial M$ is a smooth submanifold of $\\tilde{M}$ and every block for $X$ in $U$ is inessential.", "Then $ X=\\lim _{k\\rightarrow \\infty }X^k $ where $X=\\lim _{k\\rightarrow \\infty }X^k, \\quad X^k\\in {\\mathsf {L}}_{\\mathsf {in}} (M), \\quad {\\mathsf {Z} ( X^k)}\\cap U =\\varnothing , \\qquad (k\\in {\\mathbb {N}}_{+})$ and $X^k$ coincides with $X$ outside $U$ .", "Proof Fix a Riemannian metric on $M$ .", "For every $\\epsilon >0$ choose an isolating neighborhood $W:=W(\\epsilon )\\subset U$ of $K$ having only finitely many components, and such that $\\Vert X_p\\Vert <\\epsilon , \\qquad (p\\in W).$ Thus $X (W)$ lies in the bundle $T^\\epsilon W$ whose fibre over $p$ is the open disk of radius $\\epsilon $ in $T_p W$ .", "Smoothness of $\\partial M$ enables an approximation $Y^\\epsilon \\in {\\mathsf {L}}_{\\mathsf {in}}$ to $X$ such that $Y^\\epsilon (W)\\subset T^\\epsilon W$ and ${\\mathsf {Z} ( Y^\\epsilon )}\\cap \\overline{U}$ is finite.", "By Proposition REF (ii) and the hypothesisw we choose the approximation close enough so that for each component $W_j$ of $W$ : $\\mathsf {i} (Y^\\epsilon , W_j) =0.$ Standard deformation techniques (compare Hirsch [10]) permit pairwise cancellation in each $W_j$ of the zeros of $Y^\\epsilon $ , without changing $Y^\\epsilon $ near $\\mathsf {bd} (W_j)$ .", "This yields a vector field $X^{\\epsilon }\\in {\\mathsf {L}}_{\\mathsf {in}} (M)$ coinciding with $X$ in a neighborhood of $M{\\tt \\} W$ and nonsingular in $W$ , and such that $\\Vert X^\\epsilon _p - X_p\\Vert < 2\\epsilon , \\qquad (p\\in M)$ The sequence $\\lbrace X^{1/k}\\rbrace _{k\\in {\\mathbb {N}}_{+}}$ has the required properties.", "Plante [27] discusses index functions for abelian Lie algebras of vector fields on closed surfaces." ], [ "Proof of Theorem ", "We recall the hypothesis: $\\tilde{M}$ is an analytic surface with empty boundary, $M\\subset \\tilde{M}$ is a connected topological surface embedded in $\\tilde{M}$ .", "$X$ and $Y$ are inward $C^1$ vector fields on $M$ .", "$[X, Y]\\wedge X=0$ .", "$K$ is an essential block of zeros for $X$ .", "Definition 4.1 Let $A, B$ be vector fields on a set $S\\subset \\tilde{M}$ .", "The dependency set of $A$ and $B$ is $\\mathsf {D} (A, B):=\\lbrace p\\in S\\colon \\,A_p\\wedge B_p=0\\rbrace $ Evidently $\\mathsf {D} (X, Y)= \\mathsf {D} (\\tilde{X}, \\tilde{Y})\\cap M.$ Proposition REF implies $ \\mathsf {D} (\\tilde{X}, \\tilde{Y})$ is $\\tilde{Y}$ -invariant and $\\mathsf {D} (X, Y)$ is positively $Y$ -invariant.", "Case (a): $X$ and $Y$ are analytic.", "Then $\\mathsf {D} (\\tilde{X}, \\tilde{Y})$ and its subset ${\\mathsf {Z} ( \\tilde{X})}$ are analytic sets in $\\tilde{M}$ , hence $\\tilde{M}$ is a simplicial complex with subcomplexes $\\mathsf {D} (\\tilde{X}, \\tilde{Y})$ and ${\\mathsf {Z} ( \\tilde{X})}$ by S. Łojasiewicz's triangulation theorem [20].", "We assume ${\\mathsf {dim}\\,}{\\mathsf {Z} ( \\tilde{Y})} < 2$ , as otherwise $Y=0$ and the conclusion is trivial.", "We also assume every component of $K$ has dimension 1 because isolated points of $K$ lie in ${\\mathsf {Z} ( Y)}$ and $K=M$ by analyticity if some component of $K$ is 2-dimensional, and either of these conditions imply the conclusion.", "Thus $\\Psi ^{\\tilde{Y}}$ restricts to a semiflow on the 1-dimensional complex $ D (\\tilde{X}, \\tilde{Y})$ having ${\\mathsf {Z} ( X)}$ and $\\mathsf {D} (X, Y)$ as positively invariant subcomplexes.", "Let $J\\subset K$ be any component.", "$J$ is a compact, connected, triangulable space of dimension $\\le 1$ which is positively $Y$ -invariant.", "From the topology of $J$ we see that ${\\mathsf {Z} ( Y)}$ meets $J$ and therefore $K$ , unless $ \\mbox{\\em $J$ is a Jordan curve on which $\\Phi ^Y$ actstransitively.", "}$ Henceforth (REF ) is assumed.", "Let $L\\subset \\mathsf {D} (X, Y)$ be the component containing $J$ .", "The set $Q:=J\\cap \\overline{L\\setminus J}$ is positively $Y$ -invariant, whence (REF ) implies $Q=J$ or $Q=\\varnothing $ .", "Therefore one of the following holds: (D1) $J\\subset \\operatorname{\\mathsf {Int}}_M\\mathsf {D} (X, Y)$ , (D2) $J$ is a component of $\\mathsf {D} (X, Y)$ .", "Assume (D1) and suppose per contra that ${\\mathsf {Z} ( Y)}\\cap K=\\varnothing $ .", "Then $ \\mathsf {D} (X, Y)$ contains the compact closure of an open set $U$ that is isolating for $(X,K)$ .", "We choose $U$ so that each component $C$ of the topological boundary $\\mathsf {bd}(U)$ is a Jordan curve or a compact arc.", "It suffices by Proposition REF (ii) to prove for each $C$ : $ \\mbox{\\em the vector fields $X|C$ and $Y|C$ are nonsingularly homotopic.", "}$ Since this holds when $C$ is an arc, we assume $C$ is a Jordan curve.", "Fix a Riemannian metric on $M$ and define $\\hat{X}_p=\\frac{1}{\\Vert X_p\\Vert }X_p, \\quad \\hat{Y}_p=\\frac{1}{\\Vert Y_p\\Vert }Y_p,\\qquad (p\\in C).$ These unit vector fields are nonsingularly homotopic to $X|C$ and $Y|C$ respectively, and the assumption $C\\subset \\mathsf {D} (X, Y)$ implies $\\hat{X}=\\hat{Y}$ or $\\hat{X}= -\\hat{Y}$ .", "In the first case there is nothing more to prove.", "In the second case $\\hat{X}$ and $\\hat{Y}$ are antipodal sections of the unit circle bundle $\\eta $ associated to $T_CM$ .", "As the identity and antipodal maps of the circle are homotopic through rotations, (REF ) is proved.", "Now assume (D2).", "There is an isolating neighborhood $U$ for $X$ such that $ U\\cap \\mathsf {D} (X, Y) = K.$ If $0<\\epsilon <1$ the field $X^\\epsilon := (1-\\epsilon X)+\\epsilon Y$ belongs to ${\\mathsf {L}}_{\\mathsf {in}}(M)$ (Proposition REF ), and has a zero $p\\in U$ provided $\\epsilon $ is sufficiently small (Proposition REF ).", "In that case $X_p$ and $Y_p$ are linearly dependent, therefore $p\\in K$ by (REF ), whence $Y_p=0$ .", "Case (b): Every neighborhood of $K$ contains an open neighborhood $W$ whose boundary consist of finitely many $Y$ -cycles.", "It suffices to prove that ${\\mathsf {Z} ( Y)}\\cap \\overline{W} \\ne \\varnothing $ if $W$ is isolating for $(X, K)$ .", "Given such a $W$ , let $C$ be a component of $\\mathsf {bd}(W)$ .", "By Proposition REF (a), $X_p$ and $Y_p$ are linearly dependent at all points of $C$ , or at no point of $C$ .", "In the first case $X|C$ and $Y|C$ are nonsingularly homotopic, as in the proof of (REF ).", "In the second case they are nonsingularly homotopic by the restriction to $C$ of the path of vector fields $(1-t)X+tY$ , $0\\le t\\le 1$ .", "It follows that $X|\\mathsf {bd}(W)$ and $Y|\\mathsf {bd}(W)$ are nonsingularly homotopic.", "Now Proposition REF (ii) implies $\\mathsf {i}(Y, W)=\\mathsf {i} (X, W),$ which is nonzero because $K$ is essential for $X$ .", "Hence either ${\\mathsf {Z} ( Y)}$ meets $\\mathsf {bd} (W)$ , or $W$ is isolating for $(Y, K)$ and ${\\mathsf {Z} ( Y)}\\cap W\\ne \\varnothing $ .", "This shows that ${\\mathsf {Z} ( Y)}$ meets $\\overline{W}$ .", "Remark 4.2 It is interesting to see where the proof Theorem REF breaks down in Lima's counterexample to a nonanalytic version (see Example REF ).", "Lima starts from the planar vector fields $X^1:= \\frac{\\partial }{\\partial x},\\qquad Y^1:=x \\frac{\\partial }{\\partial x} + y \\frac{\\partial }{\\partial y}$ satisfying $[X^1, Y^1]= X^1$ and transfers them to the open disk by an analytic diffeomormophism $f\\colon \\,{\\mathbb {R}}^{2} \\approx \\operatorname{\\mathsf {Int}}$ .", "This is done in such a way that the push-forwards of $X^1$ and $Y^1$ extend to continuous vector fields $X, Y$ on $M:=$ satisfying $[X, Y]= Y$ , with ${\\mathsf {Z} ( X)}=K= \\partial $ and $\\mathsf {i}_K (X) =1$ , while ${\\mathsf {Z} ( Y)}$ is a singleton in the interior of $$ .", "This can be done so that $X$ and $Y$ are $C^\\infty $ (see [2]) and therefore generate unique local semiflows.", "The dependency set $\\mathsf {D} (X, Y)$ is $R\\cup \\partial $ , where $R$ is the $\\Phi ^X$ -orbit of $z$ , a topological line in $\\operatorname{\\mathsf {Int}}\\, $ that spirals toward the boundary in both direction.", "$\\mathsf {D} (X, Y)$ is not triangulable because it is connected but not path connected.", "It is easily seen that neither (D1) nor (D2) holds." ], [ "Proof of Theorem ", "Here $K$ is essential for $X$ and $\\Phi ^Y$ preserves area.", "Suppose per contra $ {\\mathsf {Z} ( Y)}\\cap K=\\varnothing .$ We can assume $K$ contains a $Y$ -cycle $\\gamma $ , for (REF ) implies every minimal set for $\\Phi ^Y$ in $K$ is a cycle: This follows from the Poincaré-Bendixson theorem (Hartman [9]) when $K$ has a planar neighborhood, and from the Schwartz-Sacksteder Theorem [30], [29] when $Y$ is $C^2$ .", "Let $J\\subset M$ be a half-open arc with endpoint $p\\in \\gamma $ and otherwise topologically transverse to $Y$ orbits (Whitney [35]).", "For any sufficiently small half-open subarc $J_0\\subset J$ with endpoint $p$ , there is a first-return Poincaré map $f\\colon \\,J_0\\hookrightarrow J$ obtained by following trajectories.", "By the area-preserving hypothesis and Fubini's Theorem, $f$ is the identity map of $J_0$ .", "Therefore $\\gamma $ has a neighborhood $U\\subset M$ , homeomorphic to a cylinder or a Möbius band, filled with $Y$ -cycles.", "Theorem REF (b) implies $K$ is inessential for $X$ , contradicting to the hypothesis." ], [ "Proof of Theorem ", "$K$ is an essential block for $X\\in \\omega _{\\mathsf {in}}(M)$ and $\\partial M$ is an analytic set in $\\tilde{M}$ .", "We can assume $\\mathsf {W}(X)\\ne \\varnothing $ (see Definition REF ).", "Our goal is to prove ${\\mathsf {Z} ( \\mathsf {W}(X))}\\cap K\\ne \\varnothing .$ The main step is to show that the set ${\\mathcal {P}} (K):=\\lbrace \\mathfrak {t}\\subset \\mathsf {W} (X)\\colon \\,{\\mathsf {Z} ( \\mathfrak {t})}\\cap K\\ne \\varnothing \\rbrace $ is inductively ordered by inclusion.", "Note that ${\\mathcal {P}} (K)$ is nonempty because it contains the singleton $\\lbrace Y\\rbrace $ .", "In fact Theorem REF states: $ Y\\in \\mathsf {W} (X)\\Rightarrow \\lbrace Y\\rbrace \\in {\\mathcal {P}} (K).$ We rely on a consequence of Proposition REF : $ \\mbox{\\em $K$ is positively invariant under $\\mathsf {W} (X)$}$ The assumption on $\\partial M$ implies $M$ is semianalytic as a subset of $\\tilde{M}$ , and this implies $K$ is also semianalytic.", "Therefore $K$ , being compact, has only finitely many components and one of them is essential for $X$ by Proposition REF , (VF3).", "Therefore we can assume $K$ is a connected semianalytic set of $\\tilde{M}$ .", "Now $K$ is the intersection of $M$ with the component of ${\\mathsf {Z} ( \\tilde{)}}X$ that contains $K$ , which is an analytic set.", "This implies ${\\mathsf {dim}\\,}K \\le 1$ , and we assume ${\\mathsf {dim}\\,}K=1$ , as otherwise $K$ is finite and contained in ${\\mathsf {Z} ( \\mathsf {W} (X))}$ by (REF ).", "The set $K_{sing}\\subset K$ where $K$ is not locally an analytic 1-manifold is finite and positively invariant under $\\mathsf {W}(X)$ As this implies $K_{sing}\\subset {\\mathsf {Z} ( \\mathsf {W} (X))}\\cap K$ , we can assume $K_{sing}=\\varnothing $ , which under current assumptions means: $\\mbox{\\em $K$ is an analytic submanifold of $\\tilde{M}$diffeomorphic to a circle.", "}$ We can also assume: (K) $K\\subset L$ if $K\\cap L\\ne \\varnothing $ and $L$ is positively $\\mathsf {W} (X)$ -invariant semianalytic set in $\\tilde{M}$ .", "For if $Y\\in \\mathsf {W} (S)$ then $K\\cap L$ , being nonempty, finite and positively invariant under every $Y$ , is necessarily contained in ${\\mathsf {Z} ( Y)}$ .", "From (K) we infer $ \\mathfrak {t}\\in {\\mathcal {P}} (K) \\Rightarrow K\\subset {\\mathsf {Z} ( \\mathfrak {t})}.$ Consequently ${\\mathcal {P}} (K)$ is inductively ordered by inclusion.", "By Zorn's lemma there is a maximal element $\\mathfrak {m}\\in {\\mathcal {P}} (K)$ , and (REF ) implies $ K\\subset {\\mathsf {Z} ( \\mathfrak {m})}.$ To prove every $Y\\in \\mathfrak {g}$ lies in $\\mathfrak {m}$ , let $\\mathfrak {n}_Y\\subset \\mathfrak {g}$ be the smallest ideal containing $Y$ and $\\mathfrak {m}$ .", "Theorem REF implies ${\\mathsf {Z} ( Y)}\\cap K\\ne \\varnothing $ , whence ${\\mathsf {Z} ( \\mathfrak {n}_Y)}\\cap K= {\\mathsf {Z} ( Y)}\\cap {\\mathsf {Z} ( \\mathfrak {m})}\\cap K \\ne \\varnothing $ by (REF ).", "Property (K) shows that $\\mathfrak {n}_Y\\in {\\mathcal {H}}$ , so $\\mathfrak {n}_Y=\\mathfrak {m}$ by maximality of $\\mathfrak {m}$ ." ], [ "Proof of Theorem ", "The theorem states: Let $M$ be an analytic surface and $\\mathfrak {g}$ a Lie algebra of analytic vector fields on $M$ that are tangent to $\\partial M$ .", "If $X\\in \\mathfrak {g}$ spans a one-dimensional ideal, then: (a) ${\\mathsf {Z} ( \\mathfrak {g})}$ meets every essential block $K$ of zeros for $X$ , (b) if $M$ is compact and $\\chi (M)\\ne 0$ then ${\\mathsf {Z} ( \\mathfrak {g})}\\ne \\varnothing $ .", "The hypotheses imply $\\mathfrak {g}\\subset \\mathsf {W} (X)$ , because if $Y\\in \\mathfrak {g}$ then $[X, Y]=cX, c\\in {\\mathbb {R}}$ .", "Therefore (a) follows from Theorem REF .", "Conclusion (b) is a consequence, because its assumptions imply the block ${\\mathsf {Z} ( X)}$ is essential for $X$ (Proposition REF )." ], [ "Proof of Theorem ", "An effective analytic action $\\alpha $ of $G$ on $M$ induces an isomorphism $\\phi $ mapping the Lie algebra $\\mathfrak {g}_0$ of $G$ isomorphically onto a subalgebra $\\mathfrak {g}\\subset \\omega (M)$ .", "Let $X^0\\in \\mathfrak {g}_0$ span the Lie algebra of a one-dimensional normal subgroup of $G$ .", "Then $\\phi (X^0)$ spans a 1-dimensional ideal in $\\mathfrak {g}$ , hence Theorem REF implies ${\\mathsf {Z} ( \\mathfrak {g})}\\ne \\varnothing $ .", "The conclusion follows because ${\\mathsf {Z} ( \\mathfrak {g})}=\\operatorname{\\mathsf {Fix}}(\\alpha (G))$ ." ] ]

[ [ "Tipi: A TPTP-based theory development environment emphasizing proof\n analysis" ], [ "Abstract In some theory development tasks, a problem is satisfactorily solved once it is shown that a theorem (conjecture) is derivable from the background theory (premises).", "Depending on one's motivations, the details of the derivation of the conjecture from the premises may or may not be important.", "In some contexts, though, one wants more from theory development than simply derivability of the target theorems from the background theory.", "One may want to know which premises of the background theory were used in the course of a proof output by an automated theorem prover (when a proof is available), whether they are all, in suitable senses, necessary (and why), whether alternative proofs can be found, and so forth.", "The problem, then, is to support proof analysis in theory development; the tool described in this paper, Tipi, aims to provide precisely that." ], [ "Introduction", "A characteristic feature of theorem proving problems arising in theory development is that we often do not know which premises of our background theory are needed for a proof until we find one.", "If we are working in a stable background theory in which the axioms are fixed, we naturally include all premises of the background theory because it a safe estimate (perhaps an overestimate) of what is needed to solve the problem.", "We may add lemmas on top of the background theory to help a theorem prover find a solution or to make our theory more comprehensible.", "Since computer-assisted theorem proving is beset on all sides by intractability, any path through a formal theory development task is constantly threatened by limitations both practical (time, memory, patience, willpower) and theoretical (undecidability of first-order validity).", "Finding even one solution (proof, model, etc.)", "is often no small feat, so declaring victory once the first solution is found is thus quite understandable and may be all that is wanted.", "In some theory development tasks, though, we want to learn more about our problem beyond its solvability.", "This paper announces Tipi, a tool that helps us to go beyond mere solvability of a reasoning problem by providing support for answering such questions as: What premises of the problem were used in the solution?", "Do other automated reasoning systems derive the conclusion from the same premises?", "Are my premises consistent?", "Do they admit unintended models?", "What premises are truly needed for the conclusion?", "Can we find multiple sets of such premises?", "Is there a a “minimal” theory that derives the conclusion?", "Are my axioms independent of one another?", "Let us loosely call the investigation of these and related questions proof analysis.", "Tipi is useful for theory exploration both in the context of discovery and in the context of justification.", "In the context of discovery, one crafts lemmas, adds or deletes axioms, changes existing axioms, modifies the problem statement, etc., with the aim of eventually showing that the theory is adequate for one's purposes (it derives a certain conjecture, is satisfiable, is unsatisfiable, etc.).", "In the context of discovery, the set of one's axioms is in flux, and one needs tools to help ensure that the development is not veering too far off course into the unexpected countersatisfiability, admitting “nonsense” models, being inconsistent, etc.", "In the context of justification, after the initial work is done and a solution is found, one wants to know more about the relationship between the theory and the conjecture than simply that the latter is derivable from the former.", "What is the proof like?", "Are there other proofs?", "Tipi is designed to facilitate answering questions such as these.", "The theorem provers and model finders that make up Tipi include E, Vampire, Prover9, Mace4, and Paradox.", "The system is extensible; adding support for new automated reasoning systems is straightforward because we rely only on the SZS ontology to make judgments about theorem proving problems.", "Tipi uses a variety of automated reasoning technology to carry out its analysis.", "It uses theorem provers and model finders and is based on the TPTP syntax for expressing reasoning problems [7] and the SZS problem status ontology [6] thereby can flexibly use of a variety of automated reasoning tools that support this syntax.", "Going beyond solvability and demanding more of our solutions is obviously not a new idea.", "Our interests complement those of Wos and collaborators, who are also often interested not simply in derivability, but in finding proofs that have certain valuable properties, such as being optimal in various senses; see [11], [10].", "Tipi emphasizes proof analysis at the level of sets of premises, whereas one could be interested in more fine grained information such as the number of symbols employed in a proof, whether short proofs are available, whether a theory is axiomatized by a single formula, etc.", "Such analysis tends to involve rather expert knowledge of particular problems and low-level tweaking of proof procedures.", "Tipi uses automated reasoning technology essentially always in “automatic mode”.", "The philosophical background of Tipi is a classic problem in the philosophy of logic known as the proof identity problem: When are two proofs the same?", "Standard approaches to the proof identity problem work with natural deduction derivations or category theory.", "One well-known proposal is to identify “proof” with a natural deduction derivation, define a class of conversion operations on natural deduction derivations, and declare that two proofs are the same if one can be converted to the other.", "See [3] for a discussion of this approach to the proof identity problem.", "The inspiration for Tipi is to take on the proof identity problem with the assistance of modern automated reasoning tools.", "From this perspective, the TPTP library [7] can be seen as a useful resource on which to carry out experiments about “practical” proof identity.", "TPTP problems typically don't contain proofs in the usual sense of the term, but they do contain hints of proofs in the sense that they specify axioms and perhaps some intermediate lemmas.", "One does not need to share the philosophical background (or even care about it) to start using Tipi, which in any case was designed to facilitate TPTP-based theory development.", "Terminology In the following we sometimes equivocate on the term theory, understanding it sometimes in its mathematical sense as a set of formulas closed under logical consequence (and so is always infinite and has infinitely many axiomatizations), and sometimes in its practice sense, represented as a TPTP problem, which always has finitely many axioms.", "“TPTP theory” simply means an arbitrary (first-order) TPTP problem.", "Of course, from a TPTP theory $T$ we can obtain a theory in the mathematical sense of the term by simply reading the formulas of $T$ as logical formulas and closing $T$ under logical consequence.", "From an arbitrary first-order theory in the mathematical sense of the term obviously one cannot extract a unique finite axiomatization and, worse, many theories of interest are not even finitely axiomatizable.", "Still, we may at times, for precision, need to understand “theory” in its mathematical sense, even though of course we shall always work with finite TPTP theories (problems).", "Convention Some TPTP problems do not have a conjecture formula.", "Indeed, some TPTP problems are not theorem proving problems per se but are better understood as model finding problems (e.g., the intended SZS status is Satisfiable).", "For expository convenience we shall restrict ourselves to TPTP problems whose intended interpretation is that a set of premises entails a single conclusion.", "The structure of this paper is as follows.", "Section  describes some simple tools provided by Tipi to facilitate theory development.", "Section  discusses the problem of determining which premises are needed.", "Section  discusses two algorithms, one syntactic and the other semantic, for determining needed needed premises.", "Section  concentrates on independent sets of axioms.", "Section  discusses some simple model analysis tools provided by Tipi.", "Section  gives a sense of the experience so far with using Tipi on real-world proof analysis tasks.", "Section  says where can obtain Tipi and briefly discusses its implementation.", "Section  concludes and suggests further directions for proof analysis and dependencies." ], [ "Syntax analysis", "When designing TPTP theories, one needs to be careful about the precise language (signature) that one employs.", "An all-too-familiar problem is typos: one intends to write connected_to but writes conected_to instead.", "One quick check that can help catch this kind of error is to look for unique occurrences of constants, functions, or predicates.", "A unique occurrence of a relation or function symbol is a sign (though by no means a necessary or sufficient condition) that the theory is likely to be trivially inadequate to the intended reasoning task because it will fail to be (un)satisfiable, or fail to derive the conjecture.", "Detecting such hapax legomena early in the theory development process can prevent wasted time “debugging” TPTP theories that appear to be adequate but which are actually flawed." ], [ "Needed premises", "Once it is known that a conjecture $c$ is derivable from a background theory $T$ , one may want to know about the class of proofs that witness this derivability relation.", "Depending on which automated theorem prover (ATP) is used, there may not even be a derivation of $c$ from $T$ , but only the judgment that $c$ is a theorem of $T$ .", "If one does have a derivation (e.g., a resolution proof) $d$ , one can push the investigation further: Which premises of $T$ occur in $d$ ?", "Are all the premises occurring in $d$ needed?", "Various notions of “needed” are available.", "For lack of space we cannot give a complete survey of this interesting concept; see [1] for a more thorough discussion of the notion of “dependency” in the context of interactive theorem provers.", "One can distinguish whether a formula is needed for a derivation, or for a conclusion.", "In a Hilbert style calculus, the sequence $d$ of formulas $\\langle C, A, A \\rightarrow B, B \\rangle $ is a derivation of $D$ from the premises $X = \\lbrace C,A,A \\rightarrow B\\rbrace $ .", "All axioms of $X$ do appear in $d$ , so it is reasonable to assert that all of $X$ is needed for $d$ .", "But are all premises needed for the conclusion $B$ of $d$ ?", "The formula $C$ is not used as the premise of any application of a rule of inference (here, modus ponens is the only rule of inference).", "Thus one can simply delete the first term of $d$ and obtain a derivation $d^{\\prime }$ of $B$ from $X - \\lbrace C\\rbrace $ .", "In a plain resolution calculus, a derivation of the empty clause from a set $\\mathcal {C}$ of clauses can have unused premises in the same sense as there can be unused premises of a Hilbert-style derivation.", "Still, there can be “irrelevant” literals in clauses of $\\mathcal {C}$ whose deletion from $\\mathcal {C}$ and from a refutation $d$ of $\\mathcal {C}$ yields a more focused proof.", "Intuitively, any premise that is needed for a conclusion is also needed for any derivation of the conclusion (assuming sensible notions of soundness and completeness of the calculi in which derivations are carried out).", "However, a premise that is needed for a derivation need not be needed for its conclusion.", "Clearly, multiple proofs of a conclusion are often available, employing different sets of premises.", "In the ATP context, we may even find that, if we keep trimming unused premises from a theory $T$ that derives a conjecture $c$ until no more trimming is possible (so that every premise is needed by the ATP to derive the conjecture), there may still be proper subsets of “minimal” premises that suffice.", "The examples below in Section  illustrate this." ], [ "Reproving and minimal subtheories", "Given an ATP $A$ , a background theory $T$ and a conjecture $c$ , assume that $T$ does derive $c$ and is witnessed by an $A$ -derivation $d$ .", "Define $T_{0} \\mathrel {\\mathop :}=\\lbrace \\varphi \\in T \\colon \\text{$\\varphi $ occurs in $d$} \\rbrace $ as the set of premises of $T$ occurring in $d$ .", "Do we need all of $T$ to derive $c$ ?", "If $T_{0}$ is a proper subset of $T$ , then the answer is evidently “no”.", "One simple method to investigate the question of which premises of $T$ are needed to derive $c$ is to simply repeat the invocation of $A$ using successively weaker subtheories of $T$ .", "Given $T_{k}$ and an $A$ -derivation $d_{k}$ of $c$ from $T_{k}$ , define $T_{k+1} \\mathrel {\\mathop :}=\\lbrace \\varphi \\in T_{k} \\colon \\text{$\\varphi $ occurs in $d_{k}$} \\rbrace $ We are then after the fixed point of the sequence $T_{0} \\supset T_{1} \\supset T_{2} \\supset \\dots $ We can view this discussion as the definition of a new proof procedure: Definition 1 (Syntactic reproving) Given a background theory $T$ , an ATP $A$ , a conjecture $c$ , use $A$ to derive $c$ from $A$ .", "If this succeeds, extract the premises of $T$ that were used by $A$ to derive $c$ ; call this set $T^{\\prime }$ .", "If $T^{\\prime } = T$ , then stop and return $d$ .", "If $T^{\\prime }$ is a proper subset of $T$ , then let $T \\mathrel {\\mathop :}=T^{\\prime }$ and repeat.", "We call this proof procedure “syntactic” simply because we view the task of a proof finder as a syntactic one.", "The name is not ideal because an ATP may use manifestly semantic methods in the course of deriving $c$ from $T$ .", "The definition of syntactic reproving requires of $A$ only that we can compute from a successful search for a derivation, which premises were used; we do not require a derivation from $A$ , though in practice various ATPs do in fact emit derivations and from these we simply extract used premises.", "If $A$ is a complete ATP, then we can find a fixed point, provided we have unlimited resources; the existence of a fixed point follows from the assumption that $T_{0} \\vdash c$ and the fact that $T_{0}$ is finite.There may even be multiple $A$ -minimal subtheories of $T$ that derive $c$ , but the proof procedure under discussion will find only one of them.", "Of course, we do place restrictions on our proof searches, so we often cannot determine that a proper subset of $T$ suffices to derive $c$ , even if there is such a subset.", "It can happen that the syntactic reprove procedure applied with an ATP $A$ , a theory $T$ , and a conjecture $c$ , terminates with a subtheory $T^{\\prime }$ of $T$ even though there is a proper subset $T^{\\prime \\prime }$ of $T$ that also suffices and, further, $A$ can verify that $T^{\\prime \\prime }$ suffices.", "The syntactic reprove procedure does not guarantee that the solution it finds is truly minimal.", "Some other proof procedure, then, is needed.", "Definition 2 (Semantic reproving) Given a background theory $T$ and ATPs $A$ and $B$ , a conjecture $c$ , use $A$ to derive $c$ from $A$ .", "If this succeeds, extract the premises of $T$ that were used by $A$ to derive $c$ ; call this set $T^{\\prime }$ .", "Define $T^{*} \\mathrel {\\mathop :}=\\lbrace \\varphi \\in T^{\\prime } \\colon {T^{\\prime } - \\lbrace \\varphi \\rbrace } \\lnot \\vdash {c} \\rbrace $ Now use $A$ to check whether ${T^{*}} \\vdash {c}$ .", "If this succeeds, return $T^{*}$ .", "The semantic reprove procedure takes two ATPs $A$ and $B$ as parameters.", "$A$ is used for checking derivability, whereas $B$ is used to check underivability.", "This proof procedure is called “semantic” because the task of constructing $T^{*}$ is carried out in Tipi using a model finder (e.g., Paradox or Mace4), which solves the problem of showing that ${X} \\lnot \\vdash {\\varphi }$ by producing a model of $X \\cup \\lbrace \\lnot \\varphi \\rbrace $ .", "As with “syntactic” in “syntactic reprove”, the “semantic” in “semantic reprove” is not ideal because any ATP that can decide underivability judgments would work; whether $B$ uses syntactic or semantic methods (or a combination thereof) to arrive at its solution is immaterial.", "Indeed, in principle, for $B$ a theorem prover could be used.", "Even though Vampire and E are typically used to determine derivability, because of the properties of their search procedures, they can be used for determining underivability, though establishing underivability is not necessarily their strong suit and often a model finder can give an answer more efficiently to the problem of whether ${X} \\vdash {\\varphi }$ ." ], [ "Independence", "In proof analysis a natural question is whether, in a set $X$ of axioms, there is an axiom $\\varphi $ that depends on the others in the sense that $\\varphi $ can be derived from $X - \\lbrace \\varphi \\rbrace $ .", "Definition 3 (Independent set of formulas) A set $X$ of formulas is independent if for every formula $\\varphi $ in $X$ it is not the case that ${X - \\lbrace \\varphi \\rbrace } \\vdash {\\varphi }$ .", "Tipi provides a proof procedure for testing whether a set of formulas is independent.", "The algorithm for testing this is straightforward: given a finite set $X = \\lbrace \\varphi _{1}, \\dots ,\\varphi _{n} \\rbrace $ of axioms whose independence we need to test, test successively whether there is any $\\varphi = \\varphi _{k}$ in $X$ such that for some $j_{1} \\ne k$ , we have ${\\lbrace x_{j_{1}}\\rbrace } \\vdash {\\varphi }$ whether there is any $\\varphi = \\varphi _{k}$ in $X$ such that for some $j_{1}, j_{2} \\ne k$ we have ${\\lbrace x_{j_{1}}, x_{j_{2}} \\rbrace } \\vdash {\\varphi }$ , etc., for increasingly larger $k$ (the upper bound is of course $n - 1$ ).", "On the assumption that most set of axiom that arise in practice are not independent, Tipi employs a “fail fast” heuristic: if $X$ is not independent, then we can likely find, for some axiom $\\varphi $ in $X$ , a small subset $X^{*}$ suffices to prove $\\varphi $ .", "Other algorithms for testing independence are conceivable.", "It could be that the naive algorithm that is immediately suggested by the definition—enumerate the axioms, checking for each one whether it is derivable from the others—may be the best approach.", "Experience shows this is indeed an efficient algorithm if one really does have an independent set (obviously the iterative “fail fast” algorithm sketched requires $n(n - 1) = O(n^{2})$ calls to an ATP for a set of axioms of size $n$ , whereas the obvious algorithm just makes $n$ calls).", "A model finder can be used to facilitate this: if $X - \\lbrace \\varphi \\rbrace \\cup \\lbrace \\lnot \\varphi \\rbrace $ is satisfiable, then $\\varphi $ is independent of $X$ .", "If one is dealing with large sets of axioms, testing independence becomes prohibitively expensive, so one could employ a randomized algorithm: randomly choose an axiom $\\varphi $ and a proper subset $T^{\\prime }$ of $T$ that does not contain $\\varphi $ and test whether $T^{\\prime }$ proves $\\varphi $ .", "Tipi implements all these algorithms for checking independence.", "A typical application of independence checking first invokes one of the minimization algorithms described in Section .", "If there is a proper subset $T^{\\prime }$ of $T$ that suffices to derive $c$ , then the independence of the full theory $T$ is probably less interesting (and in any event requires more work to determine) than the independence of the sharper set $T^{\\prime }$ .", "Checking independence is related to the semantic reprove algorithm described in Section .", "If we are dealing with a theory that has a conjecture formula, then the two notions are not congruent, because the property of independence holds for a set of formulas without regard to whether they are coming from a theory that has a conjecture formula.", "If we are dealing with a theory $T$ without a conjecture whose intended SZS status is Unsatisfiable, i.e., the theory should be shown to be unsatisfiable, then an axiom $\\varphi $ of a theory $T$ gets included in the the set $T^{*}$ of Definition REF of semantic reproving if $T - {\\varphi }$ is satisfiable.", "$T$ is “semantically minimal” when no proper subtheory of $T$ is unsatisfiable, i.e., every proper subtheory of $T$ is satisfiable.", "Independence and semantic minimality thus coincide in the setting of theories without a conjecture formula with intended SZS status Unsatisfiable." ], [ "Model analysis", "When developing formal theories, one's axioms, lemmas, and conjecture are typically in flux.", "One may request the assistance of an automated reasoning system to check simply whether one's premises are consistent.", "One might go further an ask whether, if one is dealing with a TPTP theory that has a conjecture, there the theory is satisfiable when the conjecture is taken as simply another axiom.", "An “acid test” for whether one is proceeding down the right path at all is whether one's problem is countersatisfiable.", "Tipi provides tools for facilitating this kind of analysis.", "A single command is available that can check, given a theory whether the theory without the conjecture has a model whether the axioms of the theory together with the conjecture (if present) has a model whether the axioms of the theory together with the negation of the conjecture has a model.", "The second consistency check is useful to verify that one's whole problem (axioms together with the conjecture, considered as just another axiom) is sensible.", "It can happen that the axioms of a problem have very simple models, but adding the conjecture makes the models somewhat more complicated.", "If a set of axioms has a finite model but we cannot determine reasonably quickly that the axioms together with the conjecture have a finite model, then we can take such results as a sign that the conjecture may not be derivable from the axioms.", "(Of course, it is possible that that the set of axioms is finitely satisfiable but the set containing axioms and the conjecture is finitely unsatisfiable.", "One can use tools such as Infinox [2] to complement Tipi in such scenarios.)" ], [ "Experience", "Tipi has so far been used successfully to analyze a variety of theories occurring in diverse TPTP theory development tasks.", "It has proved quite useful for theory development tasks in computational metaphysics [4], which was the initial impetus for Tipi.", "To get a sense of how one can apply Tipi, we now consider several applications of Tipi to problems coming from the large TPTP library of automated reasoning problems.", "In these examples we use E as our theorem prover and Paradox as our model finder.", "Example 1 (GRA008+1) A problem in a first-order theorySee GRA001+0.", "about graphs has 17 premises.", "Syntactic reprove brings this down to 12.", "Progressing further with semantic repoving, we find that 8 of the 12 are needed (in the sense that for each of them, their deletion, while keeping the others, leads to countersatisfiability).", "Moreover, none of the other 4 is individually needed (the conjecture is still derivable from the 4 theories one obtains by deleting the 4).", "It turns out that there are two minimal theories that suffice to derive the conjecture; see Table REF .", "Table: Two minimal subtheories of GRA008+1.Example 2 (PUZ001+1) Pelletier's Dreadbury Mansion puzzle [5] asks: “Who killed Aunt Agatha?” Someone who lives in Dreadbury Mansion killed Aunt Agatha.", "Agatha, the butler, and Charles live in Dreadbury Mansion, and are the only people who live therein.", "A killer always hates his victim, and is never richer than his victim.", "Charles hates no one that Aunt Agatha hates.", "Agatha hates everyone except the butler.", "The butler hates everyone not richer than Aunt Agatha.", "The butler hates everyone Aunt Agatha hates.", "No one hates everyone.", "Agatha is not the butler.", "Among the 13 first-order sentences in the formalization are three lives(agatha), lives(butler), lives(charles) that turn out to be deletable, which the reader may find amusing since we are dealing with a murder mystery.", "Each of the 10 other premises turn out to be needed (their deletion leads to countersatisfiability), so there is a unique minimal subtheory of the original theory that suffices to solve the mystery (which is that Agatha killed herself).", "Note that the premise that Agatha is not the butler (an inference that would perhaps be licensed on pragmatic grounds if it were missing from the text) is needed, which perhaps explains why the puzzle explicitly states it.", "Example 3 (REL002+1) A problem about relation algebra is to show that $\\top $ is a right unit for the join operation ($+$ ): $\\forall x (x \\vee \\top = \\top ) .$ There are 13 premisesSee REL001+0.. Syntactic reprove with Vampire shows that 7 axioms can be cut, whereas syntactic reprove with E finds 6.", "The sets of syntactically minimal premises of E and Vampire are, interestingly, not comparable (neither is a subset of the other).", "Semantic reprove with the 10 distinct axioms used by either Vampire or E shows, surprisingly, that 2 are needed whereas each of the other 8 is separately eliminable.", "Of the 256 combinations of these 8 premises, we find two minima; see Table REF .", "Table: Minimal subtheories of REL002+1.With enough caution, Tipi can be used somewhat in the large-theory context, where there are “large” numbers of axioms (at least several dozen, sometimes many more).", "Although it is quite hopeless, in the large-theory context, to test all possible combinations of premises in the hope of discovering all minimal theories, one can, sometimes use syntactic reproving to weed out large classes of subsets.", "With these filtered premises, semantic reproving can be used to find minima using a more tractable number of combinations of premises.", "Example 4 (TOP024+1) Urban's mapping [9] of the Mizar Mathematical Library, with its rich language for interactively developing mathematics, into pure first-order theorem proving problems, is a rich vein of theorem proving problems.", "Many of them are quite challenging owing to the large number of axioms and the inherent difficulty of reasoning in advanced pure mathematics.", "Here the problem is to prove that every maximal $T_{0}$ subset of a topological space $T$ is dense.", "Of the 68 available premises, 9 are found through an initial syntactic reproving run using E and Vampire.", "Of these 9, 3 are (separately) not needed, whereas the other 6 are needed.", "Of the 8 combinations of these 3 premises, we find two minima; see Table REF .", "Table: Two minimal subtheories of TOP024+1." ], [ "Availability", "Tipi is available at https://github.com/jessealama/tipi At present Tipi relies on the GetSymbols, TPTP2X, and TPTP4X tools, which are part of the TPTP World distribution [8].", "These are used to parse TPTP theory files; a standalone parser for the TPTP language is planned, which would eliminate the dependency on these additional tools." ], [ "Conclusion and future work", "At the moment Tipi supports a handful of theorem provers and model finders.", "Supporting further systems is desirable; any automated reasoning system that supports the SZS ontology could, in principle, be added.", "Tipi supports, at the moment, only first-order logic, and so covers only a part of the space of all TPTP theories.", "There seems to be no inherent obstacle to extending Tipi to support higher-order theories as well.", "More systematic investigation for alternative proofs of a theorem could be carried out using Prover9's clause weight mechanism.", "One could have an alternative approach to the problem of generating multiple alternative proofs to the simple approach taken by Tipi.", "When working with models of a theory under development that makes true some rather unusual or unexpected formulas, it can sometimes be difficult to pinpoint the difficulty with the theory that allows it to have such unusual models.", "One has to infer, by looking at the raw presentation of the model, what the strange properties are.", "We would like to implement a smarter, more interactive diagnosis of “broken” theories.", "The problem of finding minimal subtheories sufficient to derive a conjecture, checking independence of sets of axioms, etc., clearly requires much more effort than simply deriving the conjecture.", "Tipi thus understandably can take a lot of time to answer some of the questions put to it.", "Some of this inefficiency seems unavoidable, but it is reasonable to expect that further experience with Tipi could lead to new insights into the problem of finding theory minima, determining independence, etc.", "The proof procedures defined by Tipi naturally suggest extensions to the SZS ontology [6].", "One can imagine SZS statuses such as IndependentAxioms: The set of axioms is independent.", "DependentAxioms: The set of axioms is dependent.", "MinimalPremises: No proper subset of the axioms suffices to derive the conjecture.", "NonMinimalPremises: A proper subset of the axioms suffices to derive the conjecture.", "UniqueMinimum: There is a unique subset $S$ of the axioms such that $S$ derives the conjecture and every proper subset of $S$ fails to derive the conjecture.", "MultipleIncomparableMinima: There are at least two proper subsets $S_{1}$ and $S_{2}$ of the axioms suffices to derive the conjecture, with neither $S_{1} \\subseteq S_{2}$ nor $S_{2} \\subseteq S_{1}$ .", "Tipi itself already can be seen as supporting these (currently unofficial) SZS statuses.", "One could even annotate the statistics for many problems in the TPTP library by listing the number of possible solutions (minimal subtheories of the original theory) they admit, or the number of premises that are actually needed." ] ]

[ [ "Information Transmission using the Nonlinear Fourier Transform, Part II:\n Numerical Methods" ], [ "Abstract In this paper, numerical methods are suggested to compute the discrete and the continuous spectrum of a signal with respect to the Zakharov-Shabat system, a Lax operator underlying numerous integrable communication channels including the nonlinear Schr\\\"odinger channel, modeling pulse propagation in optical fibers.", "These methods are subsequently tested and their ability to estimate the spectrum are compared against each other.", "These methods are used to compute the spectrum of various signals commonly used in the optical fiber communications.", "It is found that the layer-peeling and the spectral methods are suitable schemes to estimate the nonlinear spectra with good accuracy.", "To illustrate the structure of the spectrum, the locus of the eigenvalues is determined under amplitude and phase modulation in a number of examples.", "It is observed that in some cases, as signal parameters vary, eigenvalues collide and change their course of motion.", "The real axis is typically the place from which new eigenvalues originate or are absorbed into after traveling a trajectory in the complex plane." ], [ "Introduction", "The nonlinear Fourier transform (NFT) of a signal $q(t)$ is a pair of functions: the continuous spectrum $\\hat{q}(\\lambda )$ , $\\lambda \\in \\mathbb {R}$ , and the discrete spectrum $\\tilde{q}(\\lambda _j)$ , $\\Im \\lambda _j>0$ , $j=1,\\ldots ,\\mathsf {N}$ .", "The NFT arises in the study of integrable waveform channels as defined in Part I [1].", "In such channels, signals propagate (in a potentially complicated manner) according to a given integrable evolution equation, whereas the nonlinear Fourier transform of the signal propagates according to a (simple) multiplication operator.", "In [Part I], we proposed nonlinear frequency-division multiplexing (NFDM), a scheme that uses the nonlinear Fourier transform for data communication over integrable channels.", "NFDM extends traditional orthogonal frequency division multiplexing (OFDM) to channels generatable by a Lax pair.", "An example is the optical fiber channel, where signal propagation is modelled by the (integrable) nonlinear Schrödinger (NLS) equation.", "In general, the channel input-output relations in the NFT domain are (see [Part I]) rCl Y()=H()X() +Z(), Y(j)=H(j)X(j)+Z(j), where $\\hat{X}(\\lambda )$ and $\\tilde{X}(\\lambda _j)$ are continuous and discrete spectra at the input of the channel, $\\hat{Y}(\\lambda )$ and $\\tilde{Y}(\\lambda _j)$ are spectra at the output of the channel, and $\\hat{Z}(\\lambda )$ and $\\tilde{Z}(\\lambda _j)$ represent noise.", "The channel filter $H(\\lambda )$ for the NLS equation is given by $H(\\lambda )=\\exp (-4j\\lambda ^2 z)$ .", "NFDM is able to deal directly with nonlinearity and dispersion, without the need for additional compensation at the transmitter or receiver.", "In this scheme, information is encoded in the nonlinear spectrum at the channel input, and the corresponding time-domain signal is transmitted.", "At the receiver, the NFT of the received signal is computed, and the resulting spectra $\\hat{Y}(\\lambda )$ and $\\tilde{Y}(\\lambda _j)$ are subsequently used to recover the transmitted information.", "Similar to the ordinary Fourier transform, while the NFT can be computed analytically in a few cases, in general, numerical methods are required.", "Such methods must be robust, reliable and fast enough to be implemented in real time at the receiver.", "In this paper, we suggest and evaluate the performance of a number of numerical algorithms for computing the forward NFT of a given signal.", "Using these algorithms, we then perform extensive numerical simulations to understand the behavior of the nonlinear spectrum for various pulse shapes and parameters commonly used in data communications.", "We are aware of no published work presenting the NFT of various signals numerically, for many pulse shapes and parameters.", "Such work is necessary to clarify the structure of the nonlinear spectrum and help in its understanding.", "In part, this has been due to the fact that the NFT has largely remained a theoretical artifice, and practical implementation of the NFT as an applied tool has not yet been pursued in engineering.", "We review the relevant literature in Section  and suggest new schemes for the numerical evaluation of the NFT.", "Although these methods are general and work for the AKNS system [2], for the purpose of illustration, we specialize the AKNS system to the Zakharov-Shabat system.", "All these methods are put to test in cases where analytical formulae exist and are compared with one another in Section .", "Only some of these methods will be chosen for the subsequent numerical simulations; these are the layer-peeling method, Ablowitz-Ladik integrable discretization, and the spectral matrix eigenvalue scheme.", "These methods are used in the next sections to numerically compute the nonlinear Fourier transform of a variety of practical pulse shapes encountered in the data communications." ], [ "The Nonlinear Fourier Transform", "Details of the nonlinear Fourier transform can be found in [Part I].", "Here we briefly recall a few essential ingredients required in the numerical computation of the forward transform.", "As noted earlier, we illustrate numerical methods in the context of the Zakharov-Shabat system, which is a Lax operator for the nonlinear Schrödinger equation.", "For later use, we recall that the slowly-varying complex envelope $q(t,z)$ of a narrow-band small-amplitude signal propagating in a dispersive weakly-nonlinear medium, such as an optical fiber, satisfies the cubic nonlinear Schrödinger equation.", "By proper scaling, the equation can be normalized to the following dimensionless form in $1+1$ dimensions: rCl jqz = qtt+ 2|q|2 q.", "Here $t$ denotes retarded time, and $z$ is distance.", "The NFT for an integrable evolution equation starts by finding a Lax pair of operators $L$ and $M$ such that the evolution equation arises as the compatibility condition $L_z=[M,L]=ML-LM$ .", "For the NLS equation, we may take operator $L$ as rCl L=j t -q(t,z) -q*(t,z) t .", "(The corresponding $M$ operator can be found in [Part I].)", "The NFT is defined via the spectral analysis of the $L$ operator, given in this paper by ().", "The spectrum of $L$ is found by solving the eigenproblem $Lv=\\lambda v$ , where $\\lambda $ is an eigenvalue of $L$ and $v$ is its associated eigenvector.", "It can be shown that the operator $L$ in () has the isospectral flow property, i.e., its spectrum is invariant even as $q$ evolves according to the NLS equation.", "The eigenproblem $Lv = \\lambda v$ can be simplified to rCl vt= -j q(t) -q*(t) jv.", "Note that the $z$ -dependence of $q$ is suppressed in () (and throughout this paper), as this variable comes into play only in the propagation of the signal, not in the definition and computation of the NFT.", "Assumption 1 Throughout this paper we assume that (a) $q\\in L^1({\\mathbb {R}})$ , and (b) $q(t)$ is supported in the finite interval $[T_1,T_2]$ .", "$\\Box $ The set of eigenvectors $v$ associated with eigenvalue $\\lambda $ in () is a two-dimensional subspace $E_\\lambda $ of the continuously differentiable functions.", "We define the adjoint of a vector $v=[v_1(t),v_2(t)]^T$ as $\\tilde{v} = [v_2^*(t), -v_1^*(t)]^T$ .", "If $v(t,\\lambda ^*)$ is an element of $E_{\\lambda ^*}$ , then $\\tilde{v}(t,\\lambda ^*)$ is an element of $E_{\\lambda }$ .", "It can be shown that any pair of eigenvectors $v(t,\\lambda )$ and $\\tilde{v}(t,\\lambda ^*)$ form a basis for $E_\\lambda $ [Part I].", "Using Assumption REF (b), we can select an eigenvector $v^1(t,\\lambda )$ to be a solution of () with the boundary condition $v^1(T_2,\\lambda )=\\begin{pmatrix}0\\\\1\\end{pmatrix}e^{j\\lambda T_2}.$ The basis eigenvectors $v^1$ and $\\tilde{v}^1$ are called canonical eigenvectors.", "Having identified a basis for the subspace $E_\\lambda $ , we can project any other eigenvector $v^2$ on this basis according to rCl v2(t,)=a()v1(t,)+b()v1(t,).", "Following Assumption REF (b), a particular choice for $v^2$ is made by solving the system rCl vt= -j q(t) -q*(t) jv,   v(T1,)= 1 0e-jT1, in which we dropped the superscript 2 in $v^2$ for convenience.", "By solving () in the interval $[T_1, T_2]$ for a given $\\lambda $ and obtaining $v(T_2,\\lambda )$ , the nonlinear Fourier coefficients $a(\\lambda )$ and $b(\\lambda )$ can be obtained by considering () at $t=T_2$ .", "The resulting coefficients obtained in this manner are rCl a()=v1(T2)ejT2, b()=v2(T2)e-jT2.", "The NFT of a signal $q(t)$ consists of a continuous spectral function defined on the real axis $\\lambda \\in \\mathbb {R}$ rCl q()=b()a(),   R, and a discrete spectral function defined on the upper half complex plane $\\mathbb {C}^+=\\lbrace \\lambda : \\Im (\\lambda )>0\\rbrace $ rCl q(j)=b(j)da() /d|=j, j=1,,N, where $\\lambda _j$ are eigenvalues and correspond to the (isolated) zeros of $a(\\lambda )$ in $\\mathbb {C}^+$ , i.e.", "$a(\\lambda _j)=0$ .", "From the discussions made, in order to compute the nonlinear spectrum of $q(t)$ , the system of differential equations () needs to be solved in the interval $[T_1, T_2]$ .", "Except for special cases, () needs to be solved numerically.", "Numerical methods for the calculation of the forward nonlinear Fourier transform are divided into two classes in this article: Methods which estimate the continuous spectrum by directly integrating the Zakharov-Shabat system; see Section .", "Methods which find the (discrete) eigenvalues.", "Two approaches are suggested in this paper for this purpose.", "Similar to the continuous spectrum estimation, we can integrate the Zakharov-Shabat system numerically and obtain $a(\\lambda )$ .", "To find zeros of $a(\\lambda )$ , the scheme is often supplemented with a search method to locate eigenvalues in the upper half complex plane.", "One can also discretize and rewrite the Zakharov-Shabat system in the interval $[T_1, T_2]$ as a (large) matrix eigenvalue problem; see Section .", "We begin by discussing methods which estimate the continuous spectrum." ], [ "Numerical Methods for Computing the Continuous Spectrum", "In this section, we assume that $\\lambda \\in \\mathbb {R}$ is given and provide algorithms for calculating the nonlinear Fourier coefficients $a(\\lambda )$ and $b(\\lambda )$ .", "The continuous spectral function is then easily computed as the ratio $\\hat{q}(\\lambda )=b(\\lambda )/a(\\lambda )$ .", "This process can be repeated to compute the spectral amplitudes for any desired finite set of continuous frequencies $\\lambda $ ." ], [ "Forward and Central Discretizations", "The most obvious method to attempt to solve () is the first-order Euler method or one of its variations [3].", "Recall that the signal $q(t)$ is supported in the finite time interval $[T_1,T_2]$ , and partition this interval uniformly according to the mesh $T_1<T_1+\\epsilon <\\cdots <T_1+N\\epsilon =T_2$ with size $N$ , i.e., with $\\epsilon = (T_2-T_1)/N$ .", "Let $q[k]\\stackrel{\\Delta }{=}q(k\\epsilon )$ and let rCl P[k] = -j q[k] -q*[k] j.", "Integrating both sides of () from $k\\epsilon $ to $(k+1)\\epsilon $ and assuming that the right hand side is constant over this interval, we get rCl v[k+1] = v[k] + P[k] v[k] ,   k=0,...,N, v[0] = 1 0 e-jT1.", "Equation (REF ) is iterated from $k=0$ to $k=N$ to find $v[N]$ .", "The resulting vector is subsequently substituted in () to obtain $a(\\lambda )$ and $b(\\lambda )$ .", "We have implemented the Euler method for the calculation of the nonlinear Fourier transform of a number of pulse shapes.", "Unfortunately, the one-step Euler method does not produce satisfactory results for affordable small step sizes $\\epsilon $ .", "One can improve upon the basic Euler method by considering the central difference iteration [3], rCl v[k+1]=v[k-1]+2P[k]v[k].", "This makes the discretization second-order, i.e., the error $v(T_1+k\\epsilon )-v[k]$ is of order $\\mathcal {O}(\\epsilon ^2)$ .", "Here an additional initial condition is required too, which can be obtained, e.g., by performing one step of the regular forward difference (REF )." ], [ "Fourth-order Runge-Kutta Method", "One can also employ higher-order integration schemes such as the Runge-Kutta methods.", "Improved results are obtained using the fourth-order Runge-Kutta method [4], [5].", "However it takes significant time to estimate the spectrum using such higher order numerical methods in real-time.", "Since the method, with its typical parameters, is quite slow and does not outperform some of the schemes suggested in the following sections, we do not elaborate on this method here; see [3] for details.", "However, for comparison purposes we will include this scheme in our numerical simulations given in Section ." ], [ "Layer-peeling Method", "In Section IV.", "C of [Part I], we have calculated the nonlinear spectra of a rectangular pulse.", "One can approximate $q(t)$ as a piece-wise constant signal and use the layer-peeling property of the nonlinear Fourier transform to estimate the spectrum of any given signal.", "Let $a[k]$ and $b[k]$ be the nonlinear Fourier coefficients of $q(t)$ in the interval $[T_1, k\\epsilon )$ , and $x[k]$ and $y[k]$ coefficients in the small (rectangular) region $[k\\epsilon ,k\\epsilon +\\epsilon )$ .", "The iterations of the layer-peeling method read rCl (a[k+1],b[k+1]) = (a[k],b[k])(x[k],y[k]), (a[0],b[0]) = (1,0), where the $\\circ $ operation is defined as in [1] rCl a[k+1]=a[k]x[k]-b[k]y[k], b[k+1]=a[k]y[k]+b[k]x[k], in which rCl x[k]=((D)-jD(D))ej(t[k]-t[k-1]), y[k]= -qk*D(D)e-j(t[k]+t[k-1]), and $\\bar{x}[k](\\lambda )=x^*[k](\\lambda ^*)$ , $\\bar{y}[k](\\lambda )=y^*[k](\\lambda ^*)$ , $D=\\sqrt{\\lambda ^2+|q[k]|^2}$ .", "The desired coefficients are obtained as $a:=a[N]$ and $b:=b[N]$ .", "Note that the exponential factors in $x[k]$ and $y[k]$ enter (REF ) in a telescopic manner.", "As a result, for the numerical implementation, it is faster to drop these factors and just scale the resulting $a[N]$ and $b[N]$ coefficients by $\\exp (j\\lambda (T_2-T_1))$ and $\\exp (-j\\lambda (T_2+T_1))$ , respectively.", "This, however, reduces the accuracy as it involves the product of large and small numbers.", "We are motivated by [6] in which the layer-peeling identity (REF ) is mentioned as a property of the nonlinear Fourier transform.", "An equivalent presentation of this method is given in [7] as well.", "Note further that a different numerical method, but with the same name (layer-peeling), exists in geophysics and fiber Bragg design [8]; however this method is not related to the problem considered here.", "We shall see in Section  that the layer-peeling method gives remarkably accurate results in estimating the nonlinear Fourier transform." ], [ "Crank-Nicolson Method", "In the Crank-Nicolson method, the derivative of the evolution parameter is approximated by a finite-difference approximation, e.g., forward discretization, and other functions are discretized by taking their average over the end points of the discretization interval: rCl v[k+1]-v[k]=12(P[k] v[k]+P[k+1] v[k+1]), where $P[k]$ is defined in (REF ).", "This implicit iteration can be made explicit rCl v[k+1]=(I-2P[k+1])-1(I+2P[k] )v[k], k=0,...,N with initial condition (REF ).", "As we will see, this simple scheme too gives good results in estimating the nonlinear spectrum." ], [ "The Ablowitz-Ladik Discretization", "Ablowitz-Ladik discretization is an integrable discretization of the NLS equation in time domain [9].", "In this section, we suggest using the Lax pairs of the Ablowitz-Ladik discretization of the NLS equation for solving the Zakharov-Shabat eigenproblem in the spectral domain.", "Discretization sometimes breaks symmetries, making the discrete version of an integrable equation no longer integrable.", "A consequence of symmetry-breaking is that quantities which are conserved in the continuous model may no longer be invariant in the discretized equation.", "A completely integrable Hamiltonian system with an infinite number of conserved quantities might have a discretized version with no, or few, conserved quantities.", "The discrete equation therefore does not quite mimic the essential features of the original equation if the step size is not small enough.", "However, for some integrable equations, discretizations exist which are themselves completely integrable Hamiltonian systems, i.e., they possess an infinite number of conserved quantities and are linearizable by a Lax pair, and therefore are solvable by the nonlinear Fourier transform.", "Such developments exist for the NLS and Korteweg-de Vries (KdV) equations.", "For the NLS equation, the integrable discrete version was introduced by Ablowitz and Ladik [9].", "To illustrate the general idea, let us replace $1\\pm j\\lambda \\epsilon $ for small $\\epsilon $ with $e^{\\pm j\\lambda \\epsilon }$ in the forward discretization method (REF ) (the opposite of what is usually done in practice).", "Let $z=e^{-j\\lambda \\epsilon }$ represent the discrete eigenvalue, $Q[k]=q[k]\\epsilon $ , and rCl R[k]= z Q[k] -Q*[k] z-1 .", "The Ablowitz-Ladik iteration is rCl v[k+1]=R[k]v[k],   v[0]= 1 0 e-jT1.", "Under the $z$ transformation, the upper half complex plane in $\\lambda $ domain is mapped to the exterior of the unit circle in the $z$ domain.", "The continuous spectrum therefore lies on the unit circle $|z|=1$ , while the discrete spectrum lies outside of the unit circle $|z|>1$ .", "One can rewrite the $R$ -equation (REF ) in the eigenvalue form $Lv[k]=zv[k]$ , with the following $L$ operator rCl L= Z -Q[k] -Q*[k-1] [k-1]Z-1 , where $\\alpha [k]=1+|Q[k]|^2$ , and $\\mathcal {Z}$ is the shift operator, i.e., $\\mathcal {Z}^{-1}x[k]=x[k-1]$ , $k\\in \\mathbb {Z}$ .", "To the first order in $\\epsilon $ , $\\alpha [k]\\approx 1$ and (REF ) can be simplified to rCl L= Z -Q[k] -Q*[k] Z-1 .", "Given the $L$ operator (REF ), one can consider the $M$ operator of the continuous NLS equation and modify its elements such that the compatibility equation $L_z=[M,L]$ represents a discretized version of the NLS equation.", "It is not hard to verify that after doing so we are led to an $M$ operator resulting in the following discrete integrable NLS equation rCl jdq[k]dz=q[k+1]-2q[k]+q[k-1]2 +|q[k]|2(q[k+1]+q[k-1]).", "Here the space derivative remains intact and the signal $q[k]$ is discretized in time, in such a way that the nonlinearity is somehow averaged among three time samples.", "In the continuum limit $\\epsilon \\rightarrow 0$ , (REF ) approaches the continuous NLS equation and its merits lie in the fact that it is integrable for any $\\epsilon $ , not just in the limit $\\epsilon \\rightarrow 0$ .", "For example, soliton pulses can be observed in this model for any $\\epsilon $ .", "The equation has its own infinite number constants of motion, approaching integrals of motion in the continuum limit.", "The operator $M$ which leads to (REF ), and the details of the nonlinear Fourier transform for (REF ) can be found in [9], [10].", "We conclude that the Ablowitz-Ladik discretization can be used not only as a means to discretize the NLS equation in the time domain, but also as a means to solve the continuous-time Zakharov-Shabat system in the spectral domain.", "This is a non-finite-difference discretization, capable of dealing with oscillations $\\exp (\\pm j\\lambda t)$ in the Zakharov-Shabat system, which greatly enhances the accuracy of the one-step finite-difference methods.", "Following the Tao and Thiele's approach [6] and [4], we can also normalize the $R[K]$ matrix rCl v[k+1]=11+|Q[k]|2 z Q[k] -Q[k]* z-1 v[k].", "The scale factor does not change the spectrum significantly, since it is cancelled out in the ratios $\\hat{q}=b/a$ and $\\tilde{q}=b/a^\\prime $ , and also its effects are second order in $\\epsilon $ .", "However, numerically, normalization may help in reducing the numerical error.", "In subsequent sections, we refer to (REF ) as the Ablowitz-Ladik method (AL1) and to (REF ) as the modified Ablowitz-Ladik method (AL2)." ], [ "Methods for Calculating the Discrete Spectrum", "In order to compute the discrete spectrum, the zeros of $a(\\lambda )$ in the upper half complex plane must be found.", "One way to visualize this is to assume a two-dimensional mesh in $\\mathbb {C}^+$ and determine $a$ at all mesh points.", "Discrete eigenvalues are then easily identified by looking at the graph of $|a(\\lambda )|$ ; in many cases they correspond to deep and narrow “wells” corresponding to the zeros of the magnitude of $a$ .", "As noted earlier, two types of methods are suggested to calculate the point spectrum.", "One can use the integration-based algorithms mentioned in Section  which calculate nonlinear Fourier coefficients, and search for eigenvalues using a root finding method, such as the Newton-Raphson method.", "Such methods require good initial points and one needs to be careful about convergence [3].", "It is also possible to rewrite the spectral problem for an operator as a (large) matrix eigenvalue problem.", "The point spectrum of the operator can be found in this way too." ], [ "Search Methods", "To calculate the discrete spectral amplitude $\\tilde{q}=b/a^\\prime $ , we require $\\mathrm {d}a/\\mathrm {d}\\lambda $ as well.", "As we will show, information about the derivative of $a$ can be updated recursively along with the information about $a$ , without resorting to approximate numerical differentiation.", "Recall that the nonlinear Fourier coefficient $a(\\lambda )$ is given by () rCl a()=v1[N]ejt[N].", "Taking the derivative with respect to $\\lambda $ , we obtain rCl da()d=((v1[N])+j T2v1[N]) ejt[N].", "We can update the derivative information $\\mathrm {d}v/\\mathrm {d}\\lambda $ along with $v$ .", "In methods of Section , the transformation of eigenvectors from $t[k]$ to $t[k+1]$ can be generally represented as rCl v[k+1]=A[k]v[k], for some suitable one-step update matrix $A_k$ (which varies from method to method).", "Differentiating with respect to $\\lambda $ and augmenting $v$ with $v^\\prime =\\mathrm {d}v/\\mathrm {d}\\lambda $ , we get the iterations rCl v[k+1]=A[k]v[k], v[k+1]=A[k]v[k]+A[k]v[k], with initial conditions rCl v[0] = 1 0 e-jt[0],    v[0]= -jt[0] 0 e-jt[0].", "The derivative matrix $A^\\prime $ depends on the method used.", "For the forward discretization scheme: rCl A=M1= -j 0 0 j .", "For the Crank-Nicolson scheme: $A^\\prime =\\frac{1}{2}\\left(I+M_2\\right)^{-1} \\left(I+\\left(I+M_2\\right)^{-1}\\left(I-M_2\\right) \\right)M_1,$ where rCl M2= 12j -12q[k] 12 q*[k] -12j.", "For the Ablowitz-Ladik method: rCl A= -jz 0 0 jz-1 .", "The desired coefficients are obtained at $k=N$ as follows rCl a()=v1[N]ejt[N], b()=v2[N]e-jt[N], a()=(v1[N]+jt[N]v1[N])ejt[N].", "Similarly, the layer-peeling iteration can be augmented to update $a^\\prime (\\lambda )$ as well: rCl a[k+1]=a[k]x[k]-b[k]y[k], b[k+1]=a[k]y[k]+b[k]x[k], a[k+1]=a[k]x[k]+a[k]x[k]-(b[k]y[k]+b[k]y[k]), b[k+1]=a[k]y[k]+a[k]y[k]+b[k]x[k]+b[k]x[k], a[0]=1,   b[0]=a[0]=b[0]=0, where rCl x[k]=(D)-jD(D), y[k]= -qk*D(D), x[k]=-j2D2(D)-(j+D-j2D3)(D), y[k]= -qk*D3(D(D)-(D)).", "The expressions for $\\bar{x}^\\prime [k]$ and $\\bar{y}^\\prime [k]$ are similar, with $j$ replaced with $-j$ and $q[k]$ replaced with $-q^*[k]$ .", "With the derivative information being available, the Newton-Raphson method is a good scheme to search for the location of the (discrete) eigenvalues.", "The iteration for the complex-valued Newton-Raphson scheme is rCl k+1=k-ka(k)a(k), where $\\alpha _k$ is some step size modifier; usually $\\alpha _k=1$ .", "The iteration stops if $\\lambda _k$ is almost stationary, i.e., if $|\\alpha \\frac{a}{a^\\prime }|<\\delta $ for a small $\\delta $ .", "In practice, the quadratic convergence of the scheme is often very fast and occurs in just a few iterations.", "In data communications, since noise is usually small, the points in the transmitted constellation can (repeatedly) serve as the initial conditions for (REF ).", "In this case, convergence is usually achieved in a couple of iterations.", "For an unknown signal, random initial conditions are chosen.", "In either case, one or more sequence of Newton iterations have to be performed for any single eigenvalue.", "To make sure that all of the eigenvalues are found, we can check the trace formula for $n=1,2,3$ [Part I].", "The trace formula is a time frequency identity relating the hierarchy of infinitely many conserved quantities to the spectral components.", "In general, the trace formula represents a time domain conserved quantity as the sum of discrete and continuous spectral terms: rCl E(k)=Edisc(k)+Econt(k), where rCl Edisc(k)=4ki=1N(ik), Econt(k)= 1-k-1(1+|q()|2)d, and where $E^{(k)}$ are time domain conserved quantities (functionals of the signal).", "The first few conserved quantities are energy rCl E(1)=-|q(t,z)|2dt, momentum rCl E(2)=12j -q(t)dq(t)dtdt, and Hamiltonian rCl E(3)=1(2j)2 -(|q(t)|4-|dq(t)dt|2)dt.", "For $n=1$ , the trace formula is a kind of Parseval's identity (Plancherel's theorem), representing the total energy of the signal in time as the sum of the energy of the discrete and continuous spectral functions.", "When satisfied, the Parseval's identity ensures that all of the signal energy has been accounted for.", "We first calculate the continuous spectrum and its “energy terms” for a sufficiently fine mesh on the real $\\lambda $ axis.", "The energy difference $E_{\\textnormal {error}}=\\Vert E^{(k)}-E_{\\textnormal {cont}}^{(k)}\\Vert $ gives an estimate on the number of missing eigenvalues.", "When a new eigenvalue $\\lambda $ is found, this error is updated as $E_{\\textnormal {error}}:=\\Vert E^{(k)}-E_{\\textnormal {cont}}^{(k)}-\\frac{4}{k}\\Im \\lambda ^k\\Vert $ .", "The process is repeated until $E_{\\textnormal {error}}$ is less than a small prescribed tolerance value.", "To summarize, given the signal $q(t)$ , its nonlinear Fourier transform can be computed based on Algorithm 1.", "Numerical Nonlinear Spectrum Estimation Sample the signal at a sufficiently small step size $\\epsilon $ .", "Fix a sufficiently fine mesh $M$ on the real $\\lambda $ axis.", "each $\\lambda \\in M$ Iterate (REF ) from $k=0$ to $k=N$ to obtain $v[N]$ .", "Compute the continuous spectral amplitude rCl ()=v2[N]v1[N]e-2jt[N].", "Initialize the error $e=\\Vert E-E_{\\textnormal {cont}}\\Vert $ .", "$|e|>\\epsilon _1$ Choose $\\lambda _0\\in \\mathcal {D}$ randomly, where $\\mathcal {D}$ is a prescribed region in $\\mathbb {C}^+$ .", "Set $i=0$ ; Iterate (REF )-(REF ) from $k=0$ to $k=N$ to obtain $v[N]$ and $v^\\prime [N]$ , and perform a Newton-Raphson update rCl i+1=i-,   =i v1[N]v1[N]+jt[N]v1[N] If $\\lambda _{i+1}\\notin \\mathcal {D}$ choose $\\lambda _0\\in \\mathcal {D}$ randomly.", "Set $i:=-1$ .", "Set $i:=i+1$ .", "$|\\Delta \\lambda |<\\epsilon _2$ and $i>0$ $\\lambda _i$ is an eigenvalue and the associated spectral amplitude is rCl q(i)=v2[N]v1[N]+jt[N]v1[N]e-2ji t[N].", "Update $e:=\\Vert E-E_{\\textnormal {cont}}-E_{\\textnormal {disc}}\\Vert $ , where $E_{\\textnormal {disc}}=[4\\Im \\lambda _i,2\\Im \\lambda ^2_i,\\frac{4}{3}\\Im \\lambda ^3_i]$ ." ], [ "Discrete Spectrum as a Matrix Eigenvalue Problem", "The methods mentioned in Section REF find the discrete spectrum by searching for eigenvalues in the upper half complex plane.", "Sometimes it is desirable to have all eigenvalues at once, which can be done by solving a matrix eigenvalue problem [5].", "These schemes obviously estimate only (discrete) eigenvalues and do not give information on the rest of the spectrum.", "Since the matrix eigenvalue problem can be solved quickly for small-sized problems, it might take less computational effort to compute the discrete spectrum in this way.", "In addition, one does not already need the continuous spectrum to estimate the size of the discrete spectrum.", "On the other hand, for large matrices that arise when a large number of signal samples are used, the matrix eigenproblem (which is usually not Hermitian) is slow and it may be better to find the discrete spectrum using the search-based methods.", "The matrix-based methods also have the disadvantages that they can generate a large number of spurious eigenvalues, and one may not be able to restrict the algorithm for finding eigenvalues of a matrix to a certain region of the complex plane.", "Below we rewrite some of the methods mentioned in the Section  as a regular matrix eigenvalue problem, for the computation of the discrete spectrum." ], [ "Central-difference Eigenproblem", "The matrix eigenvalue problem can be formulated in the time domain or the frequency domain [5].", "Consider the Zakharov-Shabat system in the form $Lv=\\lambda v$ rCl j t -q -q* -t v =v.", "In the time domain, one can replace the time derivative $\\partial /\\partial t$ by the central finite-difference matrix rCl D=12 0 1 0 -1 -1 0 1 0 0 0 -1 0 1 1 0 0 -1 0 , and expand (REF ) as rCl j D -diag(q[k]) -diag(q*[k]) -D v[k] =v[k].", "The point spectrum is contained in the eigenvalues of the matrix in the left hand side of (REF ).", "Eigenvalues of a real symmetric or Hermitian matrix can be found relatively efficiently, owing to the existence of a complete orthonormal basis and the stability of the eigenvalues.", "In this case, a sequence of unitary similarity transformations $A_{k+1}=PA_kP^T$ can be designed, using, for instance, the QR factorization, the Householder transformation, etc., to obtain the eigenvalues rather efficiently [11].", "Unfortunately, most of the useful statements about computations using Hermitian matrices cannot usually be generalized to non-Hermitian matrices.", "As a result, the eigenvalues of a non-Hermitian matrix (corresponding to a non-self-adjoint operator) are markedly difficult to calculate [11], [12].", "Running a general-purpose eigenvalue calculation routine on (REF ) is probably not the most efficient way to get eigenvalues.", "Next we make suggestions to simplify the non-Hermitian eigenproblem (REF ) by exploiting its structure.", "The diagonal matrix in the lower left corner of (REF ) can be made zero by applying elementary row operations and using the entries of the $D$ matrix.", "Since elementary row operations, as in Gauss-Jordan elimination, generally change the eigenvalues, the corresponding column operations are also applied to induce a similarity transformation.", "In this way, an upper-Hessenberg matrix is obtained in $\\mathcal {O}(N)$ operations, as compared to a sequence of Householder transformations with $\\mathcal {O}(N^3)$ operations and $\\mathcal {O}(N^2)$ memory registers.", "The eigenvalues of the resulting complex upper-Hessenberg matrix can subsequently be found using QR iterations." ], [ "Ablowitz-Ladik Eigenproblem", "We can also rewrite the Ablowitz-Ladik discretization as a matrix eigenvalue problem.", "Using the $L$ operator (REF ), we obtain rCl v1[k+1]-Q[k]v2[k]=zv1[k], -Q*[k-1]v1[k]+[k-1] v2[k-1]=zv2[k], which consequently takes the form rCl U1 -diag(Q[k]) -diag(Q*[k-1]) UT2 v =z v, in which rCl U1= 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 , U2= 0 [1] 0 0 0 0 [2] 0 0 0 0 0 [N-1] [N] 0 0 0 0 , and $\\textnormal {diag}(Q[k])=\\textnormal {diag}\\left(Q[0],\\cdots ,Q[N]\\right)$ , $\\textnormal {diag}(Q^*[k-1])=\\textnormal {diag}\\left(Q^*[N],Q^*[0] \\cdots ,Q^*[N-1]\\right)$ .", "Note that all shifting operations are cyclic, so that all vector indices $k$ remain in the interval $0\\le k\\le N$ .", "It is not very different to consider the more simplified $L$ operator (REF ) instead of (REF ).", "This corresponds to the above discretization with $U_2:=U_1$ and $-Q^*[k-1]$ replaced with $-Q^*[k]$ .", "Similarly, one can rewrite the normalized Ablowitz-Ladik iteration as a matrix eigenvalue problem.", "This corresponds to (REF ) where instead of having $\\alpha [k-1]$ , $v_1[k+1]$ is multiplied by $\\alpha [k]$ in the first equation, i.e., $U_1$ and $U_2$ are interchanged." ], [ "Spectral Method", "In the frequency domain, one can approximate derivatives with the help of the Fourier transform.", "Let us assume that rCl v(t)=k=-M2M2k kej2ktT,   q(t)=k=-M2M2k ej2ktT, where $T=T_2-T_1$ .", "Then the Zakharov-Shabat system is rCl -k2kT-jm=-M2M2k-mm=k, -jm=-M2M2-k+m*m+k2kT=k.", "Thus we obtain rCl -H - = j , where $\\alpha =[\\alpha _{-\\frac{M}{2}},\\ldots ,\\alpha _{\\frac{M}{2}}]^T$ , $\\beta =[\\beta _{-\\frac{M}{2}},\\ldots , \\beta _{\\frac{M}{2}}]^T$ , $\\Omega =-\\frac{2\\pi }{T}\\textnormal {diag}(-\\frac{M}{2},\\ldots ,\\frac{M}{2})$ and rCl =-j ( cccccccc 0 -1 -M2 0 0 0 1 0-1 -M2 0 0 M2 M2-1 2c -M2+1 -M2 0 M2 M2-1 -M2+1 0 0 M2 M2-1 0 -1 0 0 0 M2 M2-1 0 ).", "The point spectrum is thus found by looking at the eigenvalues of the matrix rCl A= -H -.", "The numerical methods discussed in this paper are first-order matrix iterations and therefore the running time of all of them is $\\mathcal {O}(N)$ multiplications and additions per eigenvalue.", "This corresponds to a complexity of $\\mathcal {O}(N^2)$ operations for the calculation of the continuous spectrum on a mesh with $N$ eigenvalues.", "The exact number of operations depends on the details of the implementation and the memory requirement of the method.", "All iterative methods thus take about the same time asymptotically, albeit with different coefficients.", "An important observation is that, while the Fast Fourier Transform (FFT) takes $\\mathcal {O} (N\\log _2 N)$ operations to calculate the spectral amplitudes of a vector with length $N$ at $N$ equispaced frequencies, the complexity of the methods described in this paper to compute the continuous spectrum are $\\mathcal {O} (N^2)$ .", "Similarly, it takes $\\mathcal {O} (\\mathsf {N}N)$ operations to calculate the discrete spectrum.", "In other words, so far we do not exploit the potentially repetitive operations in our computations.", "It is evident from () that as $T_2\\rightarrow \\infty $ , $v_1[k]$ should grow as $\\sim \\exp (-j\\lambda T_2)$ so that $a(\\lambda )$ is a finite complex number.", "The canonical eigenvector $v[k;T_1,T_2]$ thus has an unbounded component as $T_2\\rightarrow \\infty $ (i.e., $\\Vert v[k]\\Vert \\rightarrow \\infty $ ).", "One can, however, normalize $v_1$ and $v_2$ according to rCl u1=v1ejt u2=v2e-jt, and transform () to rCl ut= 0 q(t)e2jt -q*(t)e-2jt 0 u,   u(T1,)= 1 0.", "The desired coefficients are simply $a(\\lambda )=u_1(T_2)$ and $b(\\lambda )=u_2(T_2)$ .", "Consequently, if one is interested in obtaining eigenvectors $v[k]$ in addition to the coefficients $a(\\lambda )$ and $b(\\lambda )$ , the discretization of the normalized system (REF ) has better stability properties: rCl (a[k+1], b[k+1])= 1 Q[k]z-2k -Q*[k]z2k 1 (a[k], b[k]) (a[0], b[0])=(1,0).", "The nonlinear Fourier coefficients are obtained as $a:=a[N]$ and $b:=b[N]$ .", "The discrete nonlinear Fourier transform mentioned in [6] is thus the forward discretization of the normalized Zakharov-Shabat system (REF ).", "We are interested in the convergence of $v[k]$ (or $a(\\lambda )$ and $b(\\lambda )$ ) as a function of $N$ for fixed values of $T_1$ and $T_2$ .", "That is to say, we require that the error $e=\\Vert v(k\\epsilon )-v[k]\\Vert \\rightarrow 0$ as $N\\rightarrow \\infty $ (for fixed $T_1$ and $T_2$ ).", "The (global) error in all methods described in this paper is at least $\\mathcal {O}(\\epsilon )$ , and therefore all these methods are convergent.", "Some of these methods are, however, not stable.", "This is essentially because the Zakharov-Shabat system in its original form has unbounded solutions, i.e., $\\Vert v(t)\\Vert \\rightarrow \\infty $ as $t\\rightarrow \\infty $ .", "Errors can potentially be amplified by the unstable system poles.", "One should be cautious about the normalized system (REF ) as well.", "For example, forward discretization of the normalized system (REF ) gives a first-order iteration $x[k+1]=A[k]x[k]$ .", "The eigenvalues of the matrix $A[k]$ in this method are rCl s1,2=1j|q[k]|.", "It follows that the forward discretization of (REF ) gives rise to eigenvalues outside of the unit disk, $|s|>1$ .", "As a result, first-order discretization of (REF ) are also unstable.", "In cases where $|s|>1$ , we can consider normalizing the iterations by dividing $A[k]$ by $\\sqrt{\\det A[k]}$ (in the case of (REF ), dividing the right-hand side by $\\sqrt{1+|Q[k]|^2}$ ).", "The resulting iteration has eigenvalues inside the unit disk.", "For $\\epsilon \\ll 1$ the effect is only second-order in $\\epsilon $ , however it helps in managing the numerical error if larger values of $\\epsilon $ are chosen.", "An issue pertinent to numerical methods is chaos.", "Chaos and numerical instability of finite-difference discretizations has been observed in [13] for the sine-Gordon equation, which is also integrable and shares a number of basic properties with the NLS equation.", "In [14], the authors conclude that the standard discretizations of the cubic nonlinear Schrödinger equation may lead to spurious numerical behavior.", "This instability is deeply related to the homoclinic orbits of the NLS equation, i.e., it occurs if the initial signal $q(t,z=0)$ is chosen to be close to the homoclinic orbit of the equation.", "It disappears only if the step size is made sufficiently small, which can be smaller than what is desired in practice.", "It is shown in [14] that the Ablowitz-Ladik discretization of the NLS equation (in time) has the desirable property that chaos and numerical instability, which are sometimes present in finite-difference discretizations of the NLS equation, do not appear at all.", "Though these results are for the original time-domain equations, the issue can occur in the spectral eigenvalue problem () as well, if the signal $q[k]$ is close to a certain family of functions (related to $\\sin \\omega t$ and $\\cos \\omega t$ ).", "Therefore, among discretizations studied, the Ablowitz-Ladik discretization of the Zakharov-Shabat system is immune to chaos and its resulting numerical instability.", "This is particularly important in the presence of amplifier noise, where chaos can be more problematic." ], [ "Testing and Comparing the Numerical Methods", "In this section, we test and compare the ability of the suggested numerical schemes to estimate the nonlinear Fourier transform (with respect to the Zakharov-Shabat system) of various signals.", "Numerical results are compared against analytical formulae, in a few cases where such expressions exist.", "Our aim is to compare the speed and the precision of these schemes for various pulse shapes in order to determine which ones are best suited for subsequent simulation studies.", "To derive the analytical formulae, recall that the continuous spectral function can be written as $\\hat{q}(\\lambda )=\\lim \\limits _{t\\rightarrow \\infty }y(t,\\lambda )$ , in which $y(t,\\lambda )$ satisfies [1] rCl dy(t,)dt+q(t)e2jty2(t,)+q*(t)e-2jt=0, y(-,)=0.", "Similarly, one can solve the second-order differential equation rCl d2z(t,)dt2-(2j+qtq)dz(t,)dt+|q|2z(t,) =0 , z(-,)=1,   dz(-,)dt=0, and obtain $a(\\lambda )=\\lim \\limits _{t\\rightarrow \\infty }z(t,\\lambda )$ .", "The zeros of $a(\\lambda )$ form the discrete spectrum.", "In the following, the discrete spectrum is found and compared using the following matrix-based schemes: central difference method; spectral method; Ablowitz-Ladik (AL) discretization (AL1); AL discretization with normalization (AL2).", "In the matrix-based schemes, the entire point spectrum is found at once by solving a matrix eigenvalue problem.", "The complete spectrum is found using search-based methods: forward discretization method; fourth-order Runge-Kutta scheme; layer-peeling methods; Crank-Nicolson method; AL discretization; AL discretization with normalization.", "In search-based methods, the Newton method is used together with the trace formula to find both discrete and continuous spectra.", "Each of the following pulses is sampled uniformly using a total of $n$ samples in a time window containing at least $99.99\\%$ of the pulse energy." ], [ "Satsuma-Yajima pulses", "One signal with known spectrum is the Satsuma-Yajima pulse [15] $q(t)=A\\textnormal {sech}(t).$ Solving the initial value problem () (or ()) analytically, the following continuous spectral function is obtained [15] rCl q()=-(-j+12+A)(-j+12-A)2(-j+12) (A)().", "The discrete spectrum is the set of zeros of $a(\\lambda )$ i.e., poles of $\\hat{q}(\\lambda )$ (when analytically extended in $\\mathbb {C}^+$ ).", "Recalling that $\\Gamma (x)$ has no zeros and is unbounded for $x=0,-1,-2,\\ldots $ , it follows that the discrete spectrum consists of $\\mathsf {N}=\\lfloor A+\\frac{1}{2}-\\epsilon \\rfloor $ eigenvalues, located at $(A-\\frac{1}{2})j,(A-\\frac{3}{2})j, \\ldots $ .", "In the special case in which $A$ is an integer, $A=\\mathsf {N}$ , the Satsuma-Yajima pulse is a pure $\\mathsf {N}$ -soliton with $\\mathsf {N}$ eigenvalues, and the continuous spectral function is zero.", "Figs.", "REF , REF and REF give the numerical results for $A=2.7$ , $N=2^{10}$ .", "Fig.", "REF shows that, in this example, the spectral and central difference methods produce good results among the matrix-based methods in estimating the discrete eigenvalues $\\lambda =0.2j, 1.2j, 2.2j$ .", "All methods generate a large number of spurious eigenvalues along the real axis.", "This behavior might be viewed as a tendency of the algorithms to generate the continuous spectrum too.", "However the spurious eigenvalues do not disappear completely even when the continuous spectrum is absent (when $A$ is an integer); only their range becomes more limited.", "The spurious eigenvalues across the real axis can easily be filtered, since their imaginary part has negligible amplitude.", "The AL methods, with and without normalization, produce the same eigenvalues plus another vertical line of spurious eigenvalues having a large negative real part.", "Normalization in the AL scheme does not make a significant difference in this example.", "Figure: Discrete spectrum of the Satsuma-Yajima pulse with A=2.7A=2.7 using(a) central difference method,(b) spectral method,(c) Ablowitz-Ladik scheme,(d) modified Ablowitz-Ladik scheme.Fig.", "REF shows the accuracy of the various matrix-based methods in estimating the smallest and largest eigenvalues of $q=2.7\\textnormal {sech}(t)$ in terms of the number of the sample points $N$ .", "As the number of sample points $N$ is decreased, the spectral and central difference methods maintain reasonable precisions, while the accuracy of the AL schemes quickly deteriorates.", "One can check that in these cases, the error in the approximation $e^{j\\lambda \\Delta t}\\sim 1+j\\lambda \\Delta t$ becomes large (since $\\Delta t\\gg 1$ ).", "The spectral method is generally more accurate than the other matrix-based methods.", "The AL discretizations seem to perform well as long as $\\lambda \\Delta t\\ll 1$ , i.e., when estimating eigenvalues with small size or when $N\\ge 200$ .", "The AL discretization eventually breaks down at about $N=50$ as the analogy between the continuous and discrete NLS equation is no longer justified at such low resolutions, whereas other schemes continue to track the eigenvalues to some accuracy.", "In other words, what the AL methods find at such small values of $N$ is the spectrum of the discrete soliton-bearing NLS equation, which is not a feature of finite-difference discretizations.", "(In fact, it is essential for this algorithm to deviate from the finite-difference discretizations as $N$ is reduced, to produce appropriate solitons with few samples.)", "The running time of all matrix eigenvalue methods is about the same.", "Figure: Error in estimating (a) the smallest eigenvalueand (b) the largest eigenvalue of theSatsuma-Yajima pulse q(t)=2.7sech(t)q(t)=2.7\\textnormal {sech}(t) as a function of the number of sample points NN usingmatrix eigenvalue methods.The Ablowitz-Ladik method 1 is the method of Section with nonormalization, and the Ablowitz-Ladik method 2 is the same scheme with normalization.Search-based methods can be used to estimate the point spectrum as well.", "Here we use the Newton method with random initial points to locate eigenvalues in $\\mathbb {C}^+$ .", "Naturally, we limit ourselves to a rectangular region in the complex plan, slightly above the real axis to avoid potential spurious eigenvalues.", "Since the number of eigenvalues is not known a priori, the continuous spectrum is found first so as to give an estimate of the energy of the discrete spectrum.", "It is essential that the continuous spectrum is estimated accurately so that a good estimate of the energy of the discrete spectrum can be obtained.", "Once this energy is known, and a suitable (rectangular) search region in $\\mathbb {C}^+$ is chosen, the Newton method is often able to locate all of the discrete eigenvalues using just a few iterations.", "Fig.", "REF shows the accuracy of the searched-based methods in estimating the largest eigenvalue of the signal $q(t)=2.7\\textnormal {sech}(t)$ .", "The Runge-Kutta, layer-peeling and Crank-Nicolson methods have about the same accuracy, followed closely by forward discretization.", "Since this is the largest eigenvalue, the AL schemes are not quite as accurate.", "As noted above, comparison at smaller values of $N$ is not illustrative, as the AL estimate quickly deviates from $\\lambda _{\\textnormal {max}}$ of the continuous signal.", "The Runge-Kutta method, at the accuracies shown in the above graphs, is of course very slow, and is not a practical method to implement.", "The running time for the other schemes is approximately the same.", "Search-based methods take an order of magnitude more time than matrix-based methods when $N$ is small.", "These methods fail when $N$ becomes too small ($N<200$ ), since the large error in estimating the energy terms of the continuous spectrum negatively influences the stopping criteria and consequently degrades the Newton increments.", "For large $N$ , on the other hand, the QR factorization, which takes $\\mathcal {O}(N^3)$ operations in calculating the eigenvalues of a matrix, becomes quite slow and restricts the use of matrix-based methods.", "Figure: Error in estimating the largest eigenvalue of Satsuma-Yajima pulse q(t)=2.7sech(t)q(t)=2.7\\textnormal {sech}(t) as a function of the number of sample points NN using search-based methods.The same conclusions are observed for various choices of real or complex parameter $A$ .", "As $|A|$ is increased, as before, the spectral and finite-difference schemes produce the correct eigenvalues, and the AL methods generate the same eigenvalues plus an additional vertical strip of spurious eigenvalues.", "The range of the spurious eigenvalues across the real axis remains about the same.", "As the phase of $A$ is increased, the true (non-spurious) eigenvalues remain the same in all methods (as expected analytically), while some of the vertical spurious eigenvalues in the AL schemes move from left to right or vice versa.", "The spectral and finite-difference schemes are relatively immune to these additional spurious eigenvalues.", "Normalization of the AL method sometimes produces slightly fewer spurious eigenvalues across the real axis, as can be seen in Figs.", "REF (c)-(d)." ], [ "Rectangular pulse", "Consider the rectangular pulse rCl q(t)= {ll A t[T1,T2] 0 otherwise ..", "It can be shown that the continuous spectrum is given by [1] rCl q()=A*je-2jt(1-Dj(D (T2-T1))), where $D=\\sqrt{\\lambda ^2+|A|^2}$ .", "To calculate the discrete spectrum, the equation () is reduced to a simple constant coefficient second-order ODE rCl d2 zdt2-2jdzdt+|A|2z=0,   z(T1)=1, z(T1)=0.", "It is easy to verify that the eigenvalues are the solutions of rCl e2j(T2-T1)2+|A|2=+2+|A|2-2+|A|2.", "Following the causality and the layer-peeling property of the NFT, one can generalize the above result to piece-wise constant pulses.", "This is the basis of the layer-peeling method of Section REF .", "Fig.", "REF (a)-(b) show the results of numerically computing the discrete spectrum of a rectangular pulse with parameters $A=2$ , $T_2=-T_1=1$ .", "The exact eigenvalue is found to be $\\lambda =1.5713j$ , by numerically finding the roots of (REF ) using the Newton-Raphson method.", "No other eigenvalue is found under a large number of random initial conditions.", "All methods generate the desired eigenvalue together with a large number of spurious eigenvalues across the real axis.", "The central-difference scheme visibly generates fewer spurious eigenvalues.", "The Ablowitz-Ladik schemes produce two more eigenvalues with a large negative real part.", "Figure: Discrete spectrum of the rectangular pulse () with A=2A=2, T 2 =-T 1 =1T_2=-T_1=1 using (a) Fourier method,(b) central difference method,(c) Ablowitz-Ladik scheme, (d) modified Ablowitz-Ladik scheme.Fig.", "REF compares the precision of various methods in estimating the nonlinear spectrum of the rectangular pulse with $A=2$ , $T_2=-T_1=1$ .", "The modified AL scheme performed the same as the basic AL scheme, and hence we do not include the modified AL scheme in the graphs.", "Figure: Error in estimating the largest eigenvalue of the rectangular pulse q(t)=2rect(t)q(t)=2\\textnormal {rect}(t) as a function of the number of sample points NN using (a) matrix-based methods and (b) search-based methods.Convergence to the discrete spectral amplitudes generally occurs much more slowly than convergence to the eigenvalues.", "Fig.", "REF (a) shows the precision of various methods in estimating the discrete spectral amplitude of the rectangular wave with $A=2$ , $T_2=-T_1=1$ .", "It can be seen that convergence does not occur until $N>1000$ .", "Fig.", "REF (b) shows the continuous spectrum for the same function.", "All methods produced essentially the same continuous spectrum, except for some very slight variations near zero frequency.", "Figure: (a) Convergence of the discrete spectral amplitude for the rectangular pulse q(t)=2rect(t)q(t)=2\\textnormal {rect}(t) as a function of the number of sample points NN.", "Factor -j-j is not shown in the figure.", "(b) Continuous spectrum.As $|A|$ is increased, more eigenvalues appear on the imaginary axis.", "The distance between these eigenvalues becomes smaller as $|\\lambda |$ is increased.", "All methods produce similar results, with the Ablowitz-Ladik methods reproducing the purely imaginary eigenvalues at spurious locations with large real part.", "Phase addition has no influence on any of these methods, as expected analytically." ], [ "$N$ -soliton pulses", "We consider an 4-soliton pulse with discrete spectrum rCl q(-1+0.25j)=1,    q(1+0.25j)=-j, q(-1+0.5j)=-1,   q(1+0.5j)=j The 4-soliton is generated by solving the Riemann-Hilbert linear system of equations with zero continuous spectrum [1] and can be seen in Fig.", "REF (a).", "Figs.", "REF (a)-(d) show the discrete spectrum of the signal using various matrix-based methods.", "The relative accuracy of these schemes in estimating the eigenvalue $\\lambda =1+0.5j$ is shown in Fig.", "REF (b).", "A very similar graph is obtained for other eigenvalues.", "Iterative methods fare similarly and their performance is shown in Fig.", "REF (b).", "Figure: (a) Amplitude profile of a 4-soliton pulse with spectrum ().", "(b) Error in estimating the eigenvalue λ=1+0.5j\\lambda =1+0.5j.Figure: Discrete spectrum of 𝖭\\mathsf {N}-soliton pulse withspectrum (): (a) Fourier method,(b) central difference method, (c) Ablowitz-Ladik scheme, (d) modified Ablowitz-Ladik scheme.The convergence of the discrete spectral amplitudes $\\tilde{q}(\\lambda _j)$ is not quite satisfactory.", "Discrete spectral amplitudes associated with eigenvalues with small $|\\Im (\\lambda _j)|$ can be obtained with reasonable accuracy, although the convergence of $\\tilde{q}(\\lambda _j)$ is slower than the convergence of the eigenvalues themselves.", "On the other hand, discrete spectral amplitudes associated with eigenvalues with large $|\\Im (\\lambda )|$ are extremely sensitive to the location of eigenvalues and even slight changes in eigenvalues lead to radically different estimates for the spectral amplitudes.", "In fact, as the energy of the pulse is increased by having eigenvalues with large $|\\Im (\\lambda )|$ , the Riemann-Hilbert system becomes ill-conditioned.", "Therefore the discrete spectral amplitudes cannot generally be obtained using the methods discussed in this paper.", "It is illustrative to see the surface of $|a(\\lambda )|$ in Fig.", "REF .", "The eigenvalues sometimes correspond to deep and narrow wells in the surface of $|a(\\lambda )|$ , and sometimes they correspond to flat minima.", "In cases that they correspond to narrow wells, the derivative $a^\\prime (\\lambda )$ is sensitive to the location of eigenvalues, leading to sensitivities in $\\tilde{q}(\\lambda _j)$ .", "It is also clear from (REF ) that $a^{\\prime }(\\lambda )$ is proportional to $\\lambda ^2$ and thus is sensitive to $\\lambda $ .", "Note that $\\tilde{q}(\\lambda _j)$ do not appear in the trace formula, and in particular they do not contribute to the signal energy.", "This part of the NFT controls the time center of the pulse and influences the signal phase too.", "Due to dependency on the time center of the signal and the fact that time center can hardly be used for digital transmission, the values of $\\tilde{q}(\\lambda _j)$ appear to be numerically chaotic and cannot carry much information.", "For this reason, we do not discuss these quantities in detail.", "Figure: (a) Error in estimating the eigenvalue λ=-1+0.25j\\lambda =-1+0.25j ina 4-soliton using search-based methods.", "(b)Error in estimating the discrete spectral amplitude |q ˜|=1|\\tilde{q}|=1." ], [ "Nonlinear Fourier Transform of Pulses in Data Communications", "In this section, we use the numerical methods discussed in Section  to compute the nonlinear Fourier transform of signals typically used in optical fiber transmission.", "The emphasis is on sinc functions as they constitute signal degrees of freedom, but we also consider raised-cosine functions, $\\textnormal {sech}$ signals, and Gaussian pulses.", "In particular, we study the effect of the amplitude and phase modulation on the structure of the nonlinear spectra.", "We will also discuss the spectrum of wavetrains formed by sinc functions.", "Since the layer-peeling and the spectral methods give accurate results in estimating the nonlinear spectra, they are chosen for subsequent simulations." ], [ "Amplitude and Phase Modulation of Sinc Functions", "Fig.", "REF shows the spectrum of $y(t)=A\\textnormal {sinc}(2t)$ under the amplitude modulation.", "It can be seen that a sinc function is all dispersive as $A$ is increased from zero, until about $A=\\pi $ ($||q||_{L_1}=1.2752\\pi $ ) where a new eigenvalue emerges from the origin.", "Starting from $A=0$ , the continuous spectrum is a rectangle, resembling the ordinary Fourier transform $-\\mathcal {F}(q^*(t))(2\\lambda )$ .", "As $A$ is increased, the continuous spectral function is narrowed until $A=\\pi $ , where it looks like a delta function and its energy starts to deviate from the energy of the time domain signal.", "As $A>\\pi $ is further increased, the dominant eigenvalue on the $j\\omega $ axis moves up until $A=1.27\\pi $ , where $\\lambda _1=1.4234j$ and a new pair of eigenvalues emerges, starting from $\\lambda _{23}=\\pm 3.2+0.05j$ .", "When the newly created eigenvalues are not pronounced enough, for instance in this example when transiting from $A=\\pi $ to $A=1.27\\pi $ , numerical algorithms have difficulties in determining whether these small emerging eigenvalues are part of the spectrum or not.", "Here it appears that for $\\pi <A<1.27\\pi $ there is just one dominant purely imaginary eigenvalue moving upward.", "At $A=1.27\\pi $ , $\\lambda _{23}$ emerge and move up in the complex plane as $A$ is increased.", "An important observation is that the sinc function appears to have not only purely imaginary eigenvalues, but also a pair of symmetric eigenvalues with nonzero real part emerging at high values of $A$ ; see Fig.", "REF (b).", "This means that, for example, a sinc function (viewed in the time domain) contains a stationary “central component” plus two small “side components” which travel to the left and right if the sinc function is subject to the NLS flow.", "The locus of the eigenvalues of the function $A\\textnormal {sinc}(2t)$ as a result of variations in $A$ is given in Fig.", "REF .", "Figure: Locus of eigenvalues of the sinc function under amplitude modulation: (a) A=0A=0 to A=5A=5, (b) A=0A=0 to A=20A=20.It follows that the sinc function is a simple example of a real symmetric pulse whose eigenvalues are not necessarily purely imaginary, as conjectured for a long time.", "However if $q(t)$ is real, non-negative, and “single-lobe”, then there are exactly $\\mathsf {N}=\\lfloor \\frac{1}{2}+\\frac{||q||_{L_1}}{\\pi }-\\epsilon \\rfloor $ eigenvalues, all purely imaginary [16].", "Under phase modulation, in the form of adding a constant phase term to the signal, the eigenvalues and the magnitude of the continuous spectrum remained unchanged.", "Vertical shift in the phase of the continuous spectrum as a result of phase modulation can be seen in Fig.", "REF .", "Figure: Phase of the continuous spectrum of a sinc function when:(a) A=4A=4 (b) A=4jA=4j.We may also examine the effect of time-dependent phase changes.", "The effect of linear chirp, of the form $\\exp (j\\omega t)$ , is shown in picture Fig.", "REF .", "Linear chirp results in just a shift of the discrete and continuous spectrum to the left or the right, depending on the sign of the chirp.", "Figure: (a) Amplitude of the continuous spectrum with nocarrier.", "(b) Amplitude of the continuous spectrum with carrier frequencyω=5\\omega =5.", "The phase graph is also shifted similarly with noother change (Δλ=2.5\\Delta \\lambda =2.5).", "(c) Locus of the eigenvalues of a sinc function with amplitude A=8A=8 as the carrier frequencyexp(-jωt)\\exp (-j\\omega t) varies.It is interesting to observe the effect of a quadratic chirp.", "The locus of eigenvalues that result due to changes in the quadratic phase $q\\exp (j\\omega t^2)$ has been studied in [16] for Gaussian pulses.", "In our sinc function example, in the case that there is one discrete eigenvalue in the chirp-free case (such as when $A=4$ ), increasing $\\omega $ will move the eigenvalue on the $j\\omega $ axis upward, but then the eigenvalue move down again and is absorbed in the real axis.", "Fig.", "REF (c) shows the moment before the eigenvalue is absorbed into the real axis.", "Note that the eigenvalues off the $j\\omega $ axis are considered to be spurious; their number increases as the number of sample points is increased.", "A more interesting behavior is observed when $A=12$ .", "Here, there are two eigenvalues on the $j\\omega $ axis: $\\lambda _1\\approx 10.0484j$ , $\\lambda _2=5.5515j$ , together with $\\lambda _{3,4}=\\pm 3.1315 +0.7462j$ (Fig REF (d)).", "As $\\omega $ is increased, $\\lambda _1$ and $\\lambda _2$ move down and a fifth eigenvalue $\\lambda _5$ emerges from the real axis and moves upwards on the $j\\omega $ axis.", "Eventually, at about $\\omega =41.41$ , $\\lambda _2$ and $\\lambda _5$ “collide” and move out of the $j\\omega $ axis to the left and right.", "If $\\omega $ is further increased, $\\lambda _2$ and $\\lambda _5$ are absorbed into the real axis; see Fig.", "REF .", "Figure: Eigenvalues of Ae -jωt 2 sinc(2t)Ae^{-j\\omega t^2}\\textnormal {sinc}(2t): (a) locus of eigenvaluesfor A=4A=4 and ω=0.5\\omega =0.5 to ω=50\\omega =50, (b) locus of eigenvalues for A=12A=12and ω=0\\omega =0 to ω=50\\omega =50, (c) eigenvalues for A=4A=4 and ω=15\\omega =15, (d)eigenvalues for A=12A=12 and ω=0.50\\omega =0.50, (e)eigenvalues for A=12A=12 andω=41.39\\omega =41.39 just before collision, (f) eigenvalues for A=12A=12 andω=41.43\\omega =41.43 after collision.Figure: (a)Nonlinear spectral broadening as a result of quadratic phase modulation Ae jωt 2 sinc(2t)Ae^{j\\omega t^2}\\textnormal {sinc}(2t) with A=1A=1 and ω=0\\omega =0, 10 and 30.", "(b) Phase of the continuous spectrum when A=1A=1 and ω=10\\omega =10.", "(c) Phase of the continuous spectrum when A=1A=1 and ω=30\\omega =30.Collision of eigenvalues also occurs with time dilation.", "Signal $q(t)=\\textnormal {sinc}(at)$ has 3 eigenvalues on the $j\\omega $ axis for $a=0.1$ , plus two small eigenvalues on two sides of the $j\\omega $ axis (Fig REF (b)).", "As $a$ is increased, the smaller eigenvalue on the $j\\omega $ axis comes down and a new eigenvalue is generated at the origin, moving upward.", "These two eigenvalues collide at $0.12j$ ($a=0.1330$ ) and are diverted to the first and second quadrant, and eventually absorbed in the real axis at about $\\Re \\lambda =\\pm 0.32$ ($\\omega =0.1990$ ), Fig.", "REF (c)-(d)).", "As $a$ is decreased, more eigenvalues appear on the $j\\omega $ axis and fewer on the real axis (Fig.", "REF (e)-(f)).", "Note that the eigenvalues are not necessarily on the $j\\omega $ axis.", "For example, the signal $y=\\textnormal {sinc}(0.1370 t)$ clearly has eigenvalues $\\lambda _1=0.8684j$ , $\\lambda _2=0.5797j$ , $\\lambda _{3,4}=\\pm 0.1055 + 0.1210i$ .", "Fig.", "REF shows the nonlinear spectrum of a sinc pulse under a quadratic chirp modulation, given by $Ae^{j\\omega t^2}\\textnormal {sinc}(2t)$ , is broadened as $\\omega $ varies.", "Figure: Locus of eigenvalues of sinc(at)\\textnormal {sinc}(at) as the bandwidthvaries (a) from a=0.1a=0.1 to a=0.6a=0.6.", "(Eigenvalues with small ℑλ\\Im \\lambda are not shown here.)", "(b) Eigenvalues for a=0.1a=0.1.", "(c) Eigenvalues before collision and (d) after collision.", "(e) Eigenvalues for a=0.06a=0.06 before collision and (f) for a=0.065a=0.065 after collision.The effect of time dilation on the continuous spectrum can be seen in Fig .REF .", "It can be observed that increasing bandwidth $a$ , will increase the continuous range of real nonlinear frequencies, leading to bandwidth expansion.", "Figure: Bandwidth expansion in sinc(at)\\textnormal {sinc}(at) for (a) a=0.06a=0.06, a=0.1a=0.1, a=0.3a=0.3 (b) a=1a=1, a=2a=2, a=3a=3." ], [ "Sinc wavetrains", "The nonlinear spectrum of a wavetrain can take on a complicated form, just like its ordinary Fourier transform counterpart.", "Eigenvalues of a two-symbol train, for instance, depend on the amplitude and phase of the two signals, and their separation distance.", "We first analyze the case in which there are only two sinc functions located at the fixed Nyquist distance from each other, i.e., $y(t)=a_1\\textnormal {sinc}(2t+\\frac{1}{2})+a_2\\textnormal {sinc}(2t-\\frac{1}{2})$ .", "For $a_1=a_2=2$ the spectrum consists of a single eigenvalue $\\lambda =0.3676j$ and a number of spurious eigenvalues as shown in Fig.", "REF (a).", "As the phase of $a_2$ is increased from $\\theta =0$ to $\\theta =\\pi $ , the eigenvalues moves off the $j\\omega $ axis to the left and a new eigenvalue emerges from the real axis in the first quadrant.", "Eigenvalues at $\\theta =\\pi $ are $\\pm 1.3908 + 0.3287i$ .", "The resulting locus of eigenvalues is shown in Fig.", "REF (b).", "Figs REF (c)-(d) depict similar graphs when both $a_1$ and $a_2$ change.", "Figure: Discrete spectrum of y(t)=a 1 sinc(2t-1 2)+a 2 sinc(2t+1 2)y(t)=a_1\\textnormal {sinc}(2t-\\frac{1}{2})+a_2\\textnormal {sinc}(2t+\\frac{1}{2}) for (a) a 1 =a 2 a_1=a_2, (b) a 1 =2a_1=2, a 2 =2e jθ a_2=2e^{j\\theta } for -π<θ≤π-\\pi <\\theta \\le \\pi , (c) a 1 =2a_1=2, 0≤a 2 ≤60\\le a_2 \\le 6, (d) a 1 =4ja_1=4j , 0≤a 2 ≤60 \\le a_2 \\le 6.Next we study the locus of the discrete spectrum as a function of pulse separation for fixed amplitudes.", "If the amplitude of the sinc functions is increased sufficiently, eigenvalues appear off the real axis and form a locus as the distance between pulses varies.", "Fig.", "REF (a) shows the locus of eigenvalues of $y(t)=4\\textnormal {sinc}(2t+\\tau )+4\\textnormal {sinc}(2t-\\tau )$ as $\\tau $ changes between zero to 5.", "At $\\tau =0$ , eigenvalues are $\\lambda _1=6j$ , $\\lambda _{2,3}=\\pm 2.3618 + 0.6476i$ and $\\lambda _{4,5}=\\pm 3.2429 + 0.0815i$ .", "As the distance between pulses is increased, $\\lambda _1$ rapidly decreases, and at about $\\tau =0.25$ , where $\\lambda _1=4.3j$ , the eigenvalues with non-zero real parts are absorbed into the real axis at $\\Re \\lambda =-3$ .", "As $\\tau $ is further increased, $\\lambda _1$ decreases further, until $\\tau =0.4$ where $\\lambda _1=2.4j$ and two new eigenvalues emerge at locations $\\Re \\lambda =\\pm 3.12$ going up and towards the $j\\omega $ axis.", "These eigenvalues return, before reaching the $j\\omega $ axis, to be absorbed into the real axis, while new eigenvalues are generated again from the real axis.", "At $\\tau =0.7$ eigenvalues are $\\lambda =2j,\\:\\pm 1+j$ .", "At some point, the newly created eigenvalues are not absorbed into the real axis, but they reach the $j\\omega $ axis and collide.", "A collision occurs, for instance, at $\\tau =1.05$ .", "One of these eigenvalues goes down to be absorbed into the origin, and the other one, interestingly, goes up to be united with the maximum eigenvalue on the $j\\omega $ axis (i.e., to create one eigenvalue with multiplicity two).", "Increasing the distance further does not change the location of this eigenvalue, which from now is fixed at $\\lambda =1.4j$ , but just changes the pattern of lower level eigenvalues.", "The collision does not occur when the amplitudes of the signals are smaller; see Fig.", "REF (b) for the locus of the eigenvalues when the amplitude of the two sinc functions is 2.", "Figure: (a) The locus of the discrete spectrum of y(t)=4sinc(2t+τ)+4sinc(2t-τ)y(t)=4\\textnormal {sinc}(2t+\\tau )+4\\textnormal {sinc}(2t-\\tau ) as a function of 0≤τ≤50\\le \\tau \\le 5.", "(b) The locus of the discrete spectrum of y(t)=2sinc(2t+τ)+2sinc(2t-τ)y(t)=2\\textnormal {sinc}(2t+\\tau )+2\\textnormal {sinc}(2t-\\tau ) as a function of 0≤τ≤50\\le \\tau \\le 5.For wavetrains with a larger number of signals, the number of eigenvalues increases proportionally.", "We generate these wavetrains randomly and examine the region to which the spectrum is confined.", "Fig.", "REF shows the locus of the discrete spectrum of all sinc wavetrains with 16 signals.", "All 16 signal degrees of freedom in the bandlimited signal are modulated here.", "The effect of the bandwidth constraint in the nonlinear spectral domain can seen in this picture.", "Figure: Effect of the bandwidth constraint on the location of the eigenvalues of a sinc wavetrain containing 16 pulses having random amplitudes." ], [ "Preservation of the spectrum of the NLS equation", "It is crucial to ensure that the spectrum found by the numerical methods, such as those discussed in the previous sections, is in fact correct.", "While it proved difficult to do so consistently and efficiently, there are various tests to increase one's confidence in the truth of the output of the numerical methods.", "Taking the inverse nonlinear Fourier transform in the continuous-time domain and comparing the resulting function in time with the original signal is generally quite cumbersome and not always feasible.", "One quick test is to examine a time frequency identity, such as the trace formula for $n=1,2,3,\\ldots $ as used in this paper.", "The first few conserved quantities in this identity can be written explicitly.", "One should allow higher tolerance values in the trace formula for large $n$ , as the discrete terms in this identity involve $\\lambda ^n$ and thus are increasingly more sensitive to the eigenvalues.", "Another test is to subject the signal to the flow of an integrable equation, such as the NLS equation, and check that the discrete spectrum is preserved and the spectral amplitudes are scaled appropriately according to that equation.", "In this section, we let the signal propagate according to the NLS equation and compare the spectra at $z=0$ and $z={\\mathcal {L}}$ for various ${\\mathcal {L}}$ .", "Figure: Propagation of pulses alongan optical fiber in the time domain (left),in the nonlinear Fourier transform domain (middle), and showing thesurface of |a(λ)||a(\\lambda )| (right).", "The pulses are(a) Gaussian pulse,(b) Satsuma-Yajima pulse,(c) raised-cosine pulse,(d) sinc pulse.The zeros of |a(λ)||a(\\lambda )| correspondto eigenvalues in ℂ + \\mathbb {C}^+.Fig.", "REF shows examples of the spectra of a number of pulses at $z=0$ and $z={\\mathcal {L}}$ evolving according to the NLS equation ().", "The distances mentioned in the graphs in $\\rm km$ correspond to a standard optical fiber with parameters in Table REF .", "Table: Fiber ParametersNote that in all these examples discrete the spectrum is completely preserved, and the continuous spectral amplitudes undergo a phase change properly.", "Compared to Gaussian and raised-cosine examples, whose nonlinear Fourier transform can be found easily, the discrete spectrum of sinc functions is much more challenging to find.", "This is because the non-dominant eigenvalues off the $j\\omega $ axis have small imaginary parts for typical parameters and are not sufficiently distinguished.", "They also have large real parts, increasing the search region.", "Sinc functions are thus not the best examples to illustrate the application of the NFT in optical fibers.", "We studied these ideal pulses primarily because of their fundamental utility in digital communications." ], [ "Conclusions", "In this paper, we have suggested and compared a variety of numerical methods for the computation of the nonlinear Fourier transform of a signal defined on the entire real line.", "A straightforward finite-difference discretization, such as the forward discretization, does not often produce satisfactory results.", "Among the methods studied in this paper, the layer-peeling and spectral methods gave accurate results in estimating the continuous and discrete spectrum over a wide class of examples.", "Given a waveform without having prior knowledge of the location of the discrete eigenvalues, we suggest the use of matrix-based methods to compute the discrete spectrum.", "If, on the other hand, the location of the eigenvalues is known approximately (as in data-communication problems, where the eigenvalues are chosen at the transmitter from a finite set) a search-based method is recommended.", "Although the eigenvalues and the continuous spectral function can be calculated with great accuracy, the discrete spectral amplitudes are quite sensitive to the location of the eigenvalues, even in the absence of noise.", "These discrete amplitudes control the time center of the pulse, and are therefore sensitive to timing jitter.", "For data communication purposes it follows that, whereas the presence or absence of the eigenvalue itself may allow for robust information transmission, encoding information in the time center of the pulse, i.e., in the discrete spectral amplitudes, is unlikely to be viable.", "Using these numerical methods, we studied the influence of various signal parameters on the nonlinear Fourier transform of a number of pulses commonly used in data communications.", "We found, for example, that the spectrum of an isolated normalized sinc function with amplitude $A$ is purely continuous for the $A<\\pi $ .", "However, as the pulse amplitude is increased, dominant eigenvalues appear on the $j\\omega $ axis, together with pairs of symmetric eigenvalues having nonzero real part.", "In general, amplitude variations result in variations in the location of the eigenvalues and the shape of the continuous spectrum.", "Eigenvalues follow particular trajectories in the complex plane.", "Phase variations, on the other hand, influence only the phase of the spectrum, not the location of the eigenvalues.", "One important observation, which may be beneficial for the design of data communication systems, is that the nonlinear spectrum of bandlimited pulses appears to be confined to a vertical strip in the complex plane with a width proportion to the signal bandwidth.", "This paper has only scratched the surface of a potentially rich research area.", "The development of efficient and robust numerical techniques suitable for various engineering applications of the nonlinear Fourier transform will require significant additional effort.", "A problem of particular interest is the development of a “fast” nonlinear Fourier transform method that would be the analog of the FFT." ] ]

[ [ "Nuclear-Modification Factor for Open-Heavy-Flavor Production at Forward\n Rapidity in Cu+Cu Collisions at sqrt(s_NN)=200 GeV" ], [ "Abstract Background: Heavy-flavor production in p+p collisions tests perturbative-quantum-chromodynamics (pQCD) calculations.", "Modification of heavy-flavor production in heavy-ion collisions relative to binary-collision scaling from p+p results, quantified with the nuclear-modification factor (R_AA), provides information on both cold- and hot-nuclear-matter effects.", "Purpose: Determine transverse-momentum, pt, spectra and the corresponding R_AA for muons from heavy-flavor mesons decay in p+p and Cu+Cu collisions at sqrt(s_NN)=200 GeV and y=1.65.", "Method: Results are obtained using the semi-leptonic decay of heavy-flavor mesons into negative muons.", "The PHENIX muon-arm spectrometers measure the p_T spectra of inclusive muon candidates.", "Backgrounds, primarily due to light hadrons, are determined with a Monte-Carlo calculation using a set of input hadron distributions tuned to match measured-hadron distributions in the same detector and statistically subtracted.", "Results: The charm-production cross section in p+p collisions at sqrt{s}=200 GeV, integrated over pt and in the rapidity range 1.4<y<1.9 is found to be dsigma_ccbar/dy = 0.139 +/- 0.029 (stat) ^{+0.051}_{-0.058} (syst) mb.", "This result is consistent with calculations and with expectations based on the corresponding midrapidity charm-production cross section measured earlier by PHENIX.", "The R_AA for heavy-flavor muons in Cu+Cu collisions is measured in three centrality intervals for 1<pt<4 GeV/c.", "Suppression relative to binary-collision scaling (R_AA<1) increases with centrality.", "Conclusions: Within experimental and theoretical uncertainties, the measured heavy-flavor yield in p+p collisions is consistent with state-of-the-art pQCD calculations.", "Suppression in central Cu+Cu collisions suggests the presence of significant cold-nuclear-matter effects and final-state energy loss." ], [ "Introduction", "Understanding the energy loss mechanism for partons moving through the hot dense partonic matter produced in heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) is a key priority in the field of heavy-ion collision physics [1], [2].", "Production of heavy quarks in heavy-ion collisions can serve as an important tool for better understanding properties of the dense matter created in such collisions.", "In particular, because of their large mass, heavy quarks are almost exclusively produced in the early stages of heavy-ion collisions and can therefore serve as a probe of the subsequently created medium.", "The large mass scale associated with the production of heavy quarks also allows one to test perturbative Quantum Chromodynamics (pQCD) based theoretical models describing high energy collisions.", "Recent measurements of heavy-quark production in heavy-ion collisions [3], [4], [5] exhibit a suppression, which is larger than expected and not easily reconciled with early theoretical predictions [6], [7].", "In these calculations the dominant energy loss mechanism for fast partons is gluon bremsstrahlung [8], [9].", "In this context, it was predicted that heavy quarks would lose less energy than light quarks due to the so-called dead-cone effect [10].", "The disagreement between this prediction and experimental results led to a consideration of alternative in-medium parton energy loss mechanisms, assumed earlier to have a small effect on heavy quarks compared to radiative energy loss.", "In particular, it was suggested that heavy quarks can lose a significant amount of their energy through elastic collisions with in-medium partons (collisional energy loss mechanism) [11], [12], [13], especially in the intermediate transverse momentum range (${p_T}\\approx 3--8$  GeV/$c$ ) in which most of the RHIC open heavy flavor measurements are performed.", "Additional mechanisms for in-medium energy loss for heavy quarks have also been suggested [14], [15].", "Despite recent progress, still needed is a universal theoretical framework describing precisely the production of heavy quarks and their subsequent interactions with the partonic medium created in heavy-ion collisions.", "Also needed are accurate measurements of heavy-quark production in heavy-ion collisions, which are critical to test and constrain the theoretical predictions.", "Hidden-heavy-flavor (${\\rm {J}/\\psi }$ ) production has also been extensively measured in heavy-ion collisions [16], [17].", "The production of ${\\rm {J}/\\psi }$ mesons is expected to be affected by the formation of a quark-gluon plasma due to the interplay of several competing mechanisms, including suppression due to a color screening mechanism similar to the Debye screening in QED [18] and enhancement due to the coalescence of uncorrelated ${c\\overline{c}}$ pairs from the hot medium [19], [20], [21].", "The magnitude of such an enhancement depends strongly on the production cross section of open-heavy flavor in heavy-ion collisions, measurements of which are therefore essential to the interpretation of heavy quarkonia results.", "A well-established observable for quantifying medium effects in heavy-ion collisions is the nuclear-modification factor, ${R_{\\rm AA}}$ : ${R_{\\rm AA}}=\\frac{1}{{N_{\\rm coll}}}\\frac{\\sigma _{\\rm AA}}{\\sigma _{pp}},$ where $\\sigma _{\\rm AA}$ and $\\sigma _{pp}$ are the invariant cross sections for a given process in ${{\\rm A}+{\\rm A}}$ collisions and ${p+p}$ collisions, respectively, and ${N_{\\rm coll}}$ is the average number of nucleon-nucleon collisions in the ${{\\rm A}+{\\rm A}}$ collision, evaluated using a simple geometrical description of the $\\rm A$ nucleus [22].", "For processes that are sufficiently hard (characterized by large energy transfer), ${R_{\\rm AA}}$ is expected to be equal to unity in the absence of nuclear effects.", "A value smaller (larger) than unity indicates suppression (enhancement) of the observed yield in ${{\\rm A}+{\\rm A}}$ collisions relative to expectations based on ${p+p}$ collision results and binary-collision scaling.", "Open-heavy-flavor production has been measured by the PHENIX experiment at midrapidity ($|\\eta |<0.35$ ) [3].", "This paper presents the measurement of open-heavy-flavor production at forward rapidity ($1.4<|\\eta |<1.9$ ) in ${{\\rm Cu}+{\\rm Cu}}$ and ${p+p}$ collisions, and the resulting ${R_{\\rm AA}}$ , using negatively-charged muons from the semi-leptonic decay of open-heavy-flavor mesons.", "The paper is organized as follows: Section  presents a short overview of the PHENIX detector subsystems relevant to these measurements followed by a description of the data sets and track selection criteria.", "Section  presents a detailed description of the methodology for measuring the invariant cross section in ${p+p}$ collisions and ${R_{\\rm AA}}$ in ${{\\rm Cu}+{\\rm Cu}}$ collisions for muons from heavy-flavor-meson decays.", "Results are presented in Section  and compared to existing measurements as well as theoretical predictions in Section .", "The PHENIX experiment is equipped with two muon spectrometers [23], shown in Fig.", "REF , that provide pion rejection at the level of $2.5\\times 10^{-4}$ in the pseudorapidity range $-1.2<\\eta <-2.2$ (south muon arm) and $1.2<\\eta <2.4$ (north muon arm) over the full azimuth.", "Each muon arm is located behind a thick copper and iron absorber and comprises three stations of cathode strip chambers (the Muon Tracker, or MuTr), surrounded by a radial magnetic field, and five \"gaps\" (numbered 0–4) consisting of a plane of steel absorber and a plane of Iarrocci tubes (the Muon Identifier, or MuID).", "The MuTr measures the momentum of charged particles by tracking their motion in the surrounding magnetic field.", "Matching the momentum of the particles reconstructed in the MuTr to the penetration depth of the particle in the MuID (that is, the last MuID gap a given particle reaches) is the primary tool used to identify muons with respect to the residual hadronic background.", "Measured muons must penetrate 8 to 11 interaction lengths in total to reach the last gap of the MuID.", "This corresponds to a reduction of the muon longitudinal momentum (along the beam axis) of $\\delta p_z=2.3 (2.45)$  GeV/$c$ in the south (north) muon arm.", "The MuID is also used in the online data acquisition to trigger on collisions that contain one or more muon candidates.", "Also used in this analysis are the Beam-Beam Counters (BBC) [24], which comprise two arrays of 64 quartz Čerenkov detectors that surround the beam, one on each side of the interaction point.", "The BBCs measure charged particles produced during the collision in the pseudorapidity range $3<|\\eta |<3.9$ and determine the collision's start-time, vertex longitudinal position, and centrality (in ${{\\rm Cu}+{\\rm Cu}}$ collisions).", "The BBCs also provide the minimum bias trigger.", "Figure: (color online)Side view of the PHENIX muon detectors (2005)." ], [ "Data Sets", "Two data sets, recorded in 2005, are used in this analysis: ${p+p}$ collisions and ${{\\rm Cu}+{\\rm Cu}}$ collisions at a center of mass energy per nucleon-nucleon collision of $\\sqrt{s_{\\rm NN}}=200$  GeV.", "The ${p+p}$ data used for this analysis have been recorded using two muon enriched triggers, in coincidence with the Minimum Bias (MB) trigger, which requires at least one hit in each of the BBCs and covers approximately 55% of the total ${p+p}$ inelastic cross section.", "These two muon triggers rely on the information recorded in the MuID.", "The first (Deep) trigger requires one or more muon candidates to reach the last plane of the MuID (Gap4), whereas the second, less strict, (Shallow) trigger requires one or more muon candidates to reach at least the third MuID gap (Gap2).", "The integrated luminosity sampled with these triggers and used for this analysis is $44.3$  nb$^{-1}$ (48.7 nb$^{-1}$ ) for the south (north) muon arm.", "All ${{\\rm Cu}+{\\rm Cu}}$ data used for this analysis have been recorded using the Minimum Bias trigger described above.", "For ${{\\rm Cu}+{\\rm Cu}}$ collisions, this trigger covers approximately 94 % of the total inelastic cross section.", "The integrated luminosity sampled with this trigger and used for this analysis is 0.13 nb$^{-1}$ , using a total ${{\\rm Cu}+{\\rm Cu}}$ inelastic cross section seen by the minimum bias trigger $\\sigma _{{{\\rm Cu}+{\\rm Cu}}}^{\\rm inel}=2.91$  b." ], [ "Centrality Determination", "The centrality of each ${{\\rm Cu}+{\\rm Cu}}$ collision is determined by the number of hits in the BBCs.", "Three centrality bins are used for this analysis: $0--20$ %, $20--40$ % and $40--94$ %, where $0--20$ % represents the most central 20% of the collisions.", "For a given centrality, the average number of nucleon-nucleon collisions (${N_{\\rm coll}}$ ) and the average number of participating nucleons (${N_{\\rm part}}$ ) are estimated using a Glauber calculation [22] coupled to a model of the BBC response.", "Values of ${N_{\\rm coll}}$ and ${N_{\\rm part}}$ for the three centrality bins defined above are listed in Table REF .", "To ensure that the centrality categories are well defined, collisions are required to be within $\\pm 30$  cm of the center of the PHENIX detector along the beam axis.", "Table: Centrality characterizationvariables for Cu + Cu {{\\rm Cu}+{\\rm Cu}} collisions." ], [ "Track Selection", "This section outlines the track-based selection variables.", "$\\bf {z_{\\rm BBC}}$ The event vertex longitudinal position is measured by the BBC detector.", "For low-momentum tracks ($p_T <2$  GeV/$c$ ) reconstructed in north (south) muon arm we demand ${z_{\\rm BBC}}> 0$ (${z_{\\rm BBC}}< 0$ ).", "This arm-dependent cut improves the signal to background ratio because light hadrons produced during the collision have a probability to decay into a muon that increases with their distance from the front muon arm absorber, whereas muons from short-lived heavy-flavor hadrons have a yield that is independent of ${z_{\\rm BBC}}$ (see also Section REF ).", "${\\bf z_{\\rm \\bf fit}}$ The vertex longitudinal position of a track evaluated using a fit of the track position and momentum measured in the MuTr and extrapolated backward through the front absorber towards the interaction point, together with the BBC vertex measurement.", "${{N_{\\rm \\bf hits}^{\\rm \\bf MuTr}}}$ The total number of track hits in the three MuTr stations.", "A given track can have up to 16 MuTr hits.", "${{N_{\\rm \\bf hits}^{\\rm \\bf MuID}}}$ The total number of track hits in the five MuID gaps.", "A given track can have up to 2 hits in each gap (10 in total).", "${{\\bf r_{\\rm \\bf ref}}}$ The distance to the beam axis of the track, as reconstructed in the MuID only, when extrapolated (backward) to $z=0$ (illustrated in Fig.", "REF ).", "Figure: (color online)Schematic representation of track selection variables DG0and DDG0.Road Slope The slope of the track, as reconstructed in the MuID only, measured at MuID Gap0: $\\sqrt{(dx/dz)^{2} + (dy/dz)^{2} }$ .", "A cut applied to this variable eliminates combinatorial background generated in the high hit-density region closest to the beam pipe.", "DG0 The distance between the track positions calculated in the MuTr and in the MuID, evaluated at the MuID Gap0 $z$ position (see Fig.", "REF ).", "DDG0 The difference between the track angles calculated in the MuTr and in the MuID, evaluated at the MuID Gap0 $z$ position (see Fig.", "REF ).", "${{\\mathbf {\\delta }z}}$ The difference between the event vertex longitudinal position reconstructed by the BBC (${z_{\\rm BBC}}$ ) and the track longitudinal position provided by the track reconstruction algorithm: $\\delta z=|{z_{\\rm BBC}}- {z_{\\rm fit}}|$ .", "${{\\mathbf {p}\\delta \\theta }}$ the effective scattering angle of the track in the front absorber, $\\delta \\theta $ , scaled by the average of the momentum measured at the vertex and at MuTr Station 1: $p$ = $(p_{\\rm vtx} + p_{\\rm st1})/2$ , where $\\delta \\theta $ is given by: $\\delta \\theta = \\cos ^{-1}\\left(\\frac{\\overrightarrow{p}_{\\rm vtx} \\cdot \\overrightarrow{p}_{\\rm st1} }{{p}_{\\rm vtx}.", "{p}_{\\rm st1} } \\right).$ where $\\overrightarrow{p}_{\\rm st1}$ is the momentum vector measured at Station 1 and $\\overrightarrow{p}_{\\rm vtx}$ is the momentum vector at the vertex.", "For a given track, $\\delta \\theta $ essentially measures the track deflection in the front absorber due mostly to multiple scattering and radiative energy loss, but also to the magnetic field upstream of station 1.", "This deflection is expected to be inversely proportional to the track total momentum.", "Scaling the scattering angle $\\delta \\theta $ by the track momentum therefore ensures that the ${p\\delta \\theta }$ distribution is approximately Gaussian with a constant width for all ${p_T}$ bins.", "Cut values applied to these variables are, in some cases, ${p_T}$ -, species- and/or centrality-dependent.", "Within a given ${p_T}$ , species and centrality bin, the same cut values are applied to both Monte Carlo simulations and real data.", "Even after all cuts are applied to select good quality muon candidates, there remains a small contamination of misreconstructed tracks caused by: Accidental combinations of hits in the muon tracker that do not correspond to a real particle.", "Tracks arising from interactions between the beam and residual gas in the beam pipe or between the beam and beamline components.", "These misreconstructed tracks, later denoted ${N_F}$ , are not completely reproduced by experimental simulations and must be estimated and properly subtracted from the inclusive muon sample to evaluate the amount of muons from heavy-flavor decay.", "The method by which ${N_F}$ is estimated is based on the distributions of the ${p\\delta \\theta }$ and $\\delta z$ variables and is described in more detail in Section REF .", "Note: positive muons are not used in this analysis due to a poorer signal/background ratio resulting from the fact that both anti-protons and negative kaons are more strongly suppressed by the MuTr front absorbers than their positive counterparts.", "The rapidity interval used for this measurement is smaller than the rapidity coverage of the PHENIX muon spectrometers ($1.2<|\\eta |<2.2$ ) to reduce uncertainties in the acceptance calculation." ], [ "Overview", "The methodology used to measure heavy-flavor muon (i.e., muons from heavy-flavor meson decay) production in ${p+p}$ and ${{\\rm Cu}+{\\rm Cu}}$ collisions is described in this section.", "This analysis is a refinement of techniques originally developed in [25], [26], [27].", "For both ${p+p}$ and ${{\\rm Cu}+{\\rm Cu}}$ collisions the double differential heavy flavor muon invariant yield is defined by: $\\frac{d^{2}N^{\\mu }}{2 \\pi {p_T}d{p_T}d\\eta }= \\frac{1}{2\\pi {p_T}\\Delta {p_T}\\Delta \\eta }\\frac{{N_I}-{N_C}-{N_F}}{N_{\\rm evt}\\epsilon _{\\rm BBC}^{c\\overline{c}\\rightarrow \\mu }A\\epsilon }$ where ${N_I}$ is the total number of muon candidates in the bin, consisting of the tracks that reach the last gap of the MuID (Gap4) and pass all track selection criteria; ${N_F}$ is the estimated number of misreconstructed tracks that pass the track selection cuts accidentally (Section REF ); ${N_C}$ is the number of tracks corresponding to the irreducible hadronic background, as determined using a hadron cocktail approach (Section REF ); $N_{\\rm evt}$ is the number of events, $A\\epsilon $ is the detector acceptance and efficiency correction (Section REF ), and $\\epsilon _{\\rm BBC}^{c\\overline{c}\\rightarrow \\mu }$ is the BBC trigger efficiency for events in which a heavy-flavor muon at forward rapidity is present.", "This efficiency amounts to 79% (100%) in ${p+p}$ (${{\\rm Cu}+{\\rm Cu}}$ ) collisions.", "The ${p+p}$ and ${{\\rm Cu}+{\\rm Cu}}$ invariant yields determined with Eq.", "REF can be used directly to determine the heavy-flavor muon ${R_{\\rm AA}}$ (Eq.", "REF ).", "However, in order to minimize the systematic uncertainty associated with the estimate of the hadronic background by canceling the part of this uncertainty that is correlated between the ${p+p}$ and the ${{\\rm Cu}+{\\rm Cu}}$ analyses, ${R_{\\rm AA}}$ is calculated separately for a given $i^{th}$ version of the Monte-Carlo simulation of hadron cocktail used in the estimate of ${N_C}$ : $R^{i}_{\\rm AA} = \\frac{1}{{N_{\\rm coll}}}\\left(\\frac{d^2N_{{{\\rm Cu}+{\\rm Cu}}}/d{p_T}d\\eta }{d^2N_{{p+p}}/d{p_T}d\\eta }\\right)^{i}$ The final value for ${R_{\\rm AA}}$ is then determined by taking the mean of the values obtained for the different cocktails, each weighted by its ability to reproduce measured data, as discussed in Section REF ." ], [ "Contamination from Misreconstructed Tracks", "${N_F}$ , the number of misreconstructed tracks that accidentally pass all track quality cuts, is estimated using the ${p\\delta \\theta }$ distribution inside and outside of the ${\\delta z}$ cut defined in Section REF .", "These two distributions are shown in the top panel of Fig.", "REF .", "The distribution inside the ${\\delta z}$ cut (black squares) shows two contributions: a peak at ${p\\delta \\theta }=0.05$  rad$\\cdot $ GeV/$c$ , corresponding to the expected multiple scattering of muons in the front absorber, and a tail out to large values of ${p\\delta \\theta }$ .", "In the distribution outside the ${\\delta z}$ cut (red triangles), the signal peak has disappeared, and only the tail remains.", "Note that the tail extends below the ${p\\delta \\theta }$ cut; this is the ${N_F}$ contribution.", "Using the fact that the shape of this tail appears to be the same on both sides of the ${\\delta z}$ cut, one can estimate ${N_F}$ using: ${N_F}= \\alpha {N_F}^{\\prime }$ where ${N_F}^{\\prime }$ is the number of tracks with ${p\\delta \\theta }< {p\\delta \\theta }_{\\rm max}$ but ${\\delta z}> {\\delta z}_{\\rm max}$ , and $\\alpha $ normalizes the tails of the two distributions above the ${p\\delta \\theta }$ cut: $\\alpha =\\frac{N({p\\delta \\theta }>{p\\delta \\theta }_{\\rm max},{\\delta z}<{\\delta z}_{\\rm max})}{N({p\\delta \\theta }>{p\\delta \\theta }_{\\rm max},{\\delta z}>{\\delta z}_{\\rm max})}$ The bottom panel of Fig.", "REF shows the ${p\\delta \\theta }$ distribution inside the ${\\delta z}$ cut (black squares, identical to the corresponding distribution in the top panel) and the distribution outside the ${\\delta z}$ cut (red triangles from the top panel) after scaling by $\\alpha $ (Eq.", "REF ).", "Using Equations REF and REF , it is found that ${N_F}$ amounts to less than $1\\%$ of the inclusive muon sample in the lowest ${p_T}$ bin ($1<{p_T}<1.5$  GeV/$c$ ) and increases with ${p_T}$ up to about $5\\%$ for the highest ${p_T}$ bins.", "Uncertainties on these estimations are negligible in the final results." ], [ "Hadron Cocktail", "Charged pions and kaons are the largest source of particles in the PHENIX muon arms.", "Other species ($p$ , $\\bar{p}$ , $K^{0}_{s}$ , $K^{0}_{L}$ ) have small but nonzero contributions.", "Altogether, these light hadrons constitute the main background source for the measurement of muons from heavy-flavor meson decay.", "One can define three contributions to this background, depending on how the particles enter the muon spectrometer: Decay muons - light hadrons that decay into muons before reaching the first absorber material.", "Since these particles enter the spectrometer as muons, a fraction of them also penetrate all the absorber layers of the MuID and enter the pool of inclusive muon candidates.", "Punch-through hadrons - hadrons produced at the collision vertex that do not decay, but penetrate all MuID absorber layers, thus also being (incorrectly) identified as muons.", "Decay-in-MuTr - hadrons produced at the collision vertex that penetrate the muon arm front absorber and decay into a muon inside the MuTr tracking volume, with the decay muon then passing through the rest of the MuTr and the MuID.", "Most such particles are simply not reconstructed because of the decay angle between the primary hadron and the decay muon.", "However, some can be reconstructed, usually with an incorrect momentum assigned to the track.", "Due to the exponential $p_T$ distribution, even a small number of such tracks can form a significant background at high $p_T$ , but for the $p_T$ range in this analysis this contribution is small.", "While decay muons can not be distinguished from punch-through hadrons and heavy-flavor muons on an event-by-event basis, their production exhibits a strong vertex dependence, as illustrated in Fig.", "REF .", "This feature plays a key role in constraining heavy-flavor background (Section REF ).", "Figure: (color online)Vertex zz distribution of muon candidates reconstructed in north(z>0z>0) MuID Gap4, relative to the event vertex zz distribution(black circles).", "The vertex zz dependencies of the variouscontributions to the inclusive muon spectra are representedschematically as colored boxes.A series of Monte Carlo simulations (“hadron cocktail packages\") are used to estimate the overall background due to light hadron sources.", "The construction of a given hadronic cocktail package involves the following steps: Generate a primary hadron sample based on parameterized ${p_T}$ and $y$ distributions (Section REF ).", "Propagate these hadrons through the muon spectrometer using the complete geant3 [28] PHENIX simulation.", "Each hadron cocktail package uses one of the two hadron shower codes provided by geant3: g-fluka or gheisha with a scaled value of the hadron-Iron interaction cross section (Section REF ).", "For the ${{\\rm Cu}+{\\rm Cu}}$ analysis the simulated hadrons are then embedded in real events in order to account for deterioration of the reconstructed track quality due to high hit multiplicity.", "Reconstruct the resulting particles using the same reconstruction code and track quality cuts used in the real data analysis.", "(Section REF ).", "Tune (that is, re-weight) the input ${p_T}$ distributions (from step 1) to match hadron distributions measured in the muon arm (Section REF )." ], [ "Input Particle Distributions", "Particle distributions required as input to the hadron cocktail have not been measured over the required $y$ and ${p_T}$ range at RHIC energies.", "We therefore use a combination of data from PHENIX, BRAHMS and STAR, together with Next-to-Leading Order (NLO) pQCD calculations to derive realistic parameterizations of these distributions.", "An exact match to actual distributions is not necessary since the input distributions are re-weighted to match measured hadron distributions before being used to generate estimates of ${N_C}$ (Section REF ).", "We start with the $\\pi ^0$ spectrum in ${p+p}$ collisions at $y=0$ measured by PHENIX [29].", "This is extrapolated to $y= 1.65$ in two steps.", "First, an overall scale factor is obtained from a Gaussian parameterization of the charged pion $dN/dy$ distribution measured by BRAHMS [30].", "Next, the ${p_T}$ shape is softened using a parameterization of the ratio of unidentified hadron ${p_T}$ spectra measured by BRAHMS at $\\eta =0$ and $\\eta = 1.65$  [31], [32]: $dN/d{p_T}(\\pi ^{\\pm },y=1.65) =dN/d{p_T}(\\pi ^{0},y=0) \\times \\exp (-\\frac{1}{2}(1.65/2.25)^{2}) \\times (1- (0.1\\cdot {p_T}[{\\rm GeV}/c] -1) )$ Next we extrapolate this spectrum over the range $1.0 \\le y \\le 2.4$ using a series of Next-to-Leading Order (NLO) calculations [33] to obtain the ratio $dN/d{p_T}(\\pi ^{\\pm },y)/dN/d{p_T}(\\pi ^{\\pm },y=1.65)$ .", "Figure REF shows a comparison of the hadron cocktail input for charged pions compared to charged-pion distributions at $y=0$ and $y=2.95$ .", "Spectra for other hadron species in the cocktail are obtained by multiplying the parameterized pion spectra by parameterizations of measured values of hadron-to-pion ratios, as a function of ${p_T}$ .", "With 8–11 interaction length of material prior to MuID Gap4, approximately 4000 hadrons must be simulated to obtain a single hadron reconstructed as a muon.", "Given this level of rejection, it is very CPU intensive to generate a sufficient sample of high ${p_T}$ hadrons using realistic ${p_T}$ spectra.", "A standard technique is to throw particles with a flat ${p_T}$ spectrum and then weight them with a realistic distribution.", "However, interactions in the absorber in front of the MuTr and decays in the MuTr volume can both result in particles being reconstructed with incorrect momentum.", "Due to the steeply falling nature of the ${p_T}$ spectrum, tracks with low momentum and incorrectly reconstructed with a higher momentum can have a significant contribution at high ${p_T}$ , with respect to properly reconstructed tracks.", "As a compromise designed to ensure statistically robust samples of both tracks with initial high ${p_T}$ and with misreconstructed high ${p_T}$ , we multiply the realistic ${p_T}$ distributions by ${p_T}^2$ to form the simulation input ${p_T}$ distributions, and re-weight the output of the simulation by $1/{p_T}^2$ to recover the initial distribution.", "Figure: (color online)Pion cross sections as a function of p T {p_T} used as initial hadroncocktail input, for several rapidity intervals in [1.0,2.2][1.0,2.2] (bluelines) compared to a fit to the PHENIX π 0 \\pi ^{0} data at y=0y=0(black line, open black circles) and BRAHMSπ - \\pi ^{-} data at y=2.95y=2.95 (open blackcircles).The particles in the primary hadron sample used as input to each hadron cocktail package are generated as follows: The particle type and rapidity are chosen based on $dN/dy$ values obtained by integrating the unweighted ${p_T}$ distributions described above.", "The particle's transverse momentum is chosen within the range $0.8\\le {p_T}\\le 8$  GeV/$c$ using the ${p_T}^2$ -weighted ${p_T}$ distributions described above.", "Since the muon spectrometer acceptance shows little dependence on the vertex $z$ position, the particle's $z$ origin is chosen from a flat distribution over the range $-35\\le z\\le 35$  cm.", "The particle's azimuthal angle, $\\phi $ , is chosen from a flat distribution over $2\\pi $ ." ], [ "Hadron Cocktail Packages", "Modeling hadron propagation through thick material is known to be difficult and neither hadron shower code available in geant3 (g-fluka and gheisha) is able to reproduce measured data in the PHENIX muon arms.", "The approach we have chosen to circumvent this issue is to produce a range of background estimates using a set of hadron cocktails (referred to as packages), each of which uses one of the geant hadron shower codes and a different, modified, value of the hadron-Iron interaction cross section.", "The set of background estimates are then combined in a weighted fashion to extract central values for production yields, ${R_{\\rm AA}}$ , and the contribution to the systematic uncertainty on these quantities due to the uncertainty in hadron propagation.", "Using the default hadron-ion cross section, fluka simulations produce more muon candidates than gheisha simulations, therefore the fluka cross sections are increased relative to the default and the gheisha cross sections are decreased.", "The cross section modifications are referred to in terms of percentage, so that a 6% increase is referred to as 106%.", "Five packages are used in this analysis: fluka105 (or fl105), fl106, fl107, gheisha91 (or gh91) and gh92." ], [ "Tuning the Hadron Cocktail Packages", "To tune and validate a given hadron-cocktail package we can compare its output to three measured hadron distributions: The ${p_T}$ distribution of tracks that stop in MuID Gap2 (counting from 0), with $p_z$ larger than a given minimum value.", "The ${p_T}$ distribution of tracks that stop in MuID Gap3 (counting from 0), with $p_z$ larger than a given minimum value.", "The vertex $z$ distribution of reconstructed tracks, normalized to the collision-vertex distribution.", "Particles that stop in MuID Gap2 or Gap3 are those tracks for which no hit is found in the downstream gaps (Gap3 and/or Gap4).", "Figure REF shows the longitudinal-momentum ($p_{z}$ ) distribution of tracks stopping in MuID Gap3 obtained using a given hadronic cocktail.", "Decay muons are characterized by a sharp peak, corresponding to electromagnetic energy loss in the absorber material.", "Note that the same peak would be obtained for muons from heavy-flavor decay.", "In contrast, hadrons are characterized by a broad shoulder that extends to much larger values of $p_z$ .", "For $p_z>p_z^{\\rm min}$ (with $p_z^{\\rm min}\\approx 3$  GeV/$c$ in this example) one obtains a clean hadron sample.", "The hadron-input ${p_T}$ distributions can then be tuned so that a good match between the number of stopped hadrons in the simulation and in real data is achieved in each ${p_T}$ bin.", "Figure: (color online)Simulated p z p_{z} distributions for particles that stop in MuIDGap3: (Black squares) all particles; (red triangles) stoppedhadrons; (blue circles) decay muons.Figure REF shows, for two muon-${p_T}$ ranges, comparisons for real data and hadron-cocktail simulations of the $z$ -vertex distributions of $dN_\\mu /dz_{\\rm BBC}$ tracks, which (a) are reconstructed in the north muon arm (located at positive $z$ ), (b) reach the MuID Gap4, and (c) are normalized by the event vertex distribution $dN_{\\rm evt}/dz_{\\rm BBC}$ .", "The approximately linear dependence on $z_{\\rm BBC}$ is entirely due to the contribution of muons from light hadrons decaying before the muon-tracker front absorber.", "Muons from short-lived heavy-flavor hadrons have no measurable dependence on $z_{\\rm BBC}$ and their contribution to the real-data sample is the source of the vertical offset between the hadron cocktail and the real-data distributions.", "Therefore, the hadron-cocktail package can be tuned by matching the slopes of these two distributions in each ${p_T}$ bin.", "The quality of this match is quantified by: $\\chi _{\\rm Gap4}^{2}(p_{T})=\\sum _{i=1}^{N_{\\rm bins}}\\frac{(\\Delta N_{i} - \\overline{\\Delta N})^{2}}{\\sigma _{i}^{2}+\\sigma _{mean}^{2}}$ where $N_{\\rm bins}$ is the number of $z_{\\rm BBC}$ bins; $\\Delta N_{i}=dN_{I}/dz_{\\rm BBC}-dN_{C}/dz_{\\rm BBC}$ is the difference between the data and simulation for the $i^{th}$ $z_{\\rm BBC}$ bin; $\\overline{\\Delta N}$ is the average difference over the entire $z_{\\rm BBC}$ range; $\\sigma _{i}$ and $\\sigma _{mean}$ are the statistical uncertainties of $\\Delta N_{i}$ and $\\overline{\\Delta N}$ , respectively.", "Figure: (color online)Vertex zz distribution of tracks reconstructed in North (z>0z>0)MuID Gap4, for two transverse momentum bins.", "The real data (blackclosed circles) are compared to a given hadron-cocktail package(blue open diamonds).", "The offset between data and the hadroncocktail is the contribution from heavy-flavor decays.Figure: (color online)Relative dispersion between the N C {N_C} yields obtained withthe five hadron cocktails for the p+p{p+p} analysis.", "Each hadroncocktail package is compared to the mean of the five packages forthe north (top panel) and south (bottom panel) muon arm.Tuning of each hadron-cocktail package is achieved by iteratively selecting a set of ${p_T}$ -dependent weights (applied to each track's thrown ${p_T}$ ) that simultaneously optimizes the agreement between data and simulation for the three distributions described above.", "Applying these weights to those simulated hadron tracks that reach MuID Gap4 determines the corresponding hadron contribution to the inclusive muon yield (${N_C}$ , Eq.", "REF ).", "Figure REF shows the relative dispersion between ${N_C}$ values obtained for the five different hadron cocktail packages used for the ${p+p}$ analysis, as a function of ${p_T}$ .", "For both muon arms, the largest differences exist between the gheisha and fluka cocktail packages for ${p_T}<2$  GeV/$c$ , with a spread of about 20%.", "For ${p_T}>3$  GeV/$c$ , most of the dispersion between the packages is due to increased statistical uncertainty in the data yields used to tune the hadron cocktail." ], [ "Systematic Uncertainties Associated with Individual\nHadron Cocktail Packages", "There are two systematic uncertainties associated with the implementation of a given hadron cocktail package: $\\mathbf {\\sigma }_{\\rm \\bf SystPack}$ the uncertainty associated with the implementation of the hadron cocktail packages.", "It is comprised of two components: the uncertainty on the hadron cocktail input distributions and the so called MuID Gap3 to Gap4 matching uncertainty.", "The uncertainty on the hadron cocktail input distributions amounts up to 20% and is correlated between the two arms.", "The uncertainty on the MuID Gap3 to Gap4 matching corresponds to tracks, in either real data or simulations, that get assigned an incorrect penetration depth, due to accidental addition of extra hits in the next MuID gap, or on the contrary, to detection inefficiencies.", "This uncertainty is evaluated using simulations.", "It is arm independent and amounts to 10%.", "These two contributions are uncorrelated and added in quadrature.", "$\\mathbf {\\sigma }_{\\rm \\bf PackMismatch}$ the uncertainty that characterizes, as a function of ${p_T}$ , the ability of a given hadron cocktail package to reproduce the measured distributions described in the previous section.", "To evaluate this uncertainty the cocktail is tuned three times, each time matching one of the three measured hadron distributions perfectly.", "The dispersion between the resulting background yields ${N_C}$ obtained with these three different tunings, along with the central value for ${N_C}$ obtained using the simultaneous tuning described above, is assigned to $\\sigma _{\\rm PackMismatch}$ .", "A different value is calculated for each muon arm, each ${p_T}$ (and centrality) bin, and each of the five hadron cocktail packages.", "Mathematical details of the calculation are outlined in Section REF .", "Since the optimization is arm independent, this uncertainty is uncorrelated between the two muon arms.", "The magnitude of this uncertainty varies from 10 to 20% depending on the muon arm and the ${p_T}$ bin.", "In addition to the hadronic background, other background sources include: muons from heavy-flavor-resonance leptonic decay (e.g.", "${\\chi _c}$ , ${\\rm {J}/\\psi }$ , ${\\psi ^{\\prime }}$ and the $\\Upsilon $ family); muons from Drell-Yan; muons from light vector meson decay ($\\rho $ , $\\phi $ and $\\omega $ ).", "These three sources contribute significantly less to the inclusive yields than the backgrounds from light hadrons.", "Monte Carlo simulations performed in the same manner as in [5] show that their contribution to the final heavy-flavor muon ${p_T}$ spectrum is less than 5% in the ${p_T}$ range used for this analysis and they have negligible impact with respect to the other sources of systematic uncertainties." ], [ "Acceptance and Efficiency Corrections", "Acceptance and efficiency corrections, $A\\epsilon $ , enter in the denominator of invariant yield measurements (Eq.", "REF ).", "They are evaluated using simulated prompt single muons, propagated through the detector using the PHENIX geant3 simulation and reconstructed with the same analysis code and the same track quality cuts as for the real data analysis.", "These corrections account for the detector's geometrical acceptance and inefficiencies (for example, due to tripped high voltage channels or dead front-end electronic channels).", "They also account for the muon triggers, reconstruction code and analysis cut inefficiencies.", "A reference run, representative of a given data taking period, is chosen to define the detector's response to particles passing through it.", "This includes notably the list of inactive high-voltage and electronic channels.", "Remaining run-to-run variations with respect to this reference run are small due to the overall stability of the detector's performance, and are included in the systematic uncertainties ($\\sigma _{\\rm run to run} = 2\\%$ ).", "A comparison between the hit distributions in the MuTr and the MuID obtained for the reference run in real data and simulations is used to assign an additional systematic error on our ability to reproduce the real detector's response in the simulations.", "Areas with unacceptable discrepancies are removed from both the simulations and the real data using fiducial cuts.", "Remaining discrepancies are accounted for with a 8% systematic uncertainty for the MuTr and 4.5% for the MuID.", "The hit multiplicity in the MuTr for ${{\\rm Cu}+{\\rm Cu}}$ collisions is much higher than for ${p+p}$ collisions and for the single muon simulations.", "To account for deterioration of the reconstruction efficiency in presence of such high multiplicity events, simulated single muon events are embedded into real data ${{\\rm Cu}+{\\rm Cu}}$ events before running the reconstruction and evaluating the $A\\epsilon $ correction.", "Another systematic uncertainty, $\\sigma _{\\rm p-scale}$ , is assigned to a possible systematic bias between the particle's reconstructed momentum and its real momentum.", "This uncertainty is estimated by comparing the measured ${\\rm {J}/\\psi }$ invariant mass (using the dimuon invariant mass distribution) and its Particles Data Group (PDG) value.", "This uncertainty amounts to $\\sim 1.5$ %.", "Table REF summarizes the acceptance and efficiency related uncertainties, which sum quadratically to 9.5%.", "Table: Uncertainties in the acceptance and efficiencycorrections.", "Individual components are added in quadrature toobtain the total value of σ Aϵ \\sigma _{A\\epsilon }." ], [ "Systematic Uncertainties", "This section summarizes systematic uncertainties associated with this analysis, most of which have been described in previous sections: Systematic uncertainties associated with individual hadron cocktail packages, $\\sigma _{\\rm SystPack}$ and $\\sigma _{\\rm PackMismatch}$ (Section REF ); Systematic uncertainty resulting from the dispersion of the results obtained with the different hadron cocktail packages (introduced in Section REF , mathematical details in Section REF ); Systematic uncertainty on the acceptance and efficiency correction factors, $\\sigma _{A\\epsilon }$ (Section REF and Table REF ); These systematic uncertainties are calculated independently for each arm, ${p_T}$ bin and centrality bin.", "The first three uncertainties listed above (first two items) are related to the hadronic background estimate and are combined to form a $\\sigma _{\\rm model}$ systematic uncertainty, following a method described in Section REF .", "For invariant cross section measurements (in ${p+p}$ collisions) and measurements of ${R_{\\rm AA}}$ one must add to the uncertainties above the systematic uncertainty on the ${p+p}$ inelastic cross section seen by the minimum bias trigger, $\\sigma ^{pp}_{\\rm BBC} = 9.6$ %.", "For ${R_{\\rm AA}}$ measurements, one must also add the systematic uncertainty on the mean number of binary collisions (${N_{\\rm coll}}$ ) in each centrality bin, as provided by the Glauber calculation used to determine this quantity.", "Table REF summarizes the systematic uncertainties in this analysis.", "Table: Uncertainties in the single muon analysis.", "The individualcomponents contribute to the final uncertainty as discussed inSection ." ], [ "Determination\nof the Central Value for Heavy-Flavor-Muon Production Yields and\n${R_{\\rm AA}}$", "This section details the procedure used to combine the results from multiple hadron cocktail packages to obtain the central values for the ${p_T}$ spectra and ${R_{\\rm AA}}$ and to propagate associated systematic uncertainties.", "This discussion includes the definition of $\\sigma _{\\rm PackMismatch}$ and $\\sigma _{\\rm Model}$ .", "Throughout this section the variable $Q$ is used to represent either the invariant yield or ${R_{\\rm AA}}$ , for a given ${p_T}$ and centrality bin; the procedure is the same for both, except where noted explicitly.", "For each ${p_T}$ bin $i$ , hadron cocktail package $j$ , and package tuning $k$ , we calculate the value $Q_{i,j,k}$ where: $k$ =1 is the optimal tuning that best matches all three hadron distributions simultaneously (see Section REF ); $k$ =2 is the tuning that best reproduces the ${p_T}$ distribution of particles stopping in MuID Gap2; $k$ =3 is the tuning that best reproduces the ${p_T}$ distribution of particles stopping in MuID Gap3; and $k$ =4 represents the tuning that best reproduces the vertex $z$ distribution of particles reaching MuID Gap4.", "The tuning $k=1$ is used for the central value whereas the other tunings are used to establish the systematic uncertainty for a single hadron cocktail package due to its inability to completely describe measured hadron distributions.", "The package mismatch contribution to the uncertainty on the measurement $Q_{i,j,k}$ is estimated by the standard deviation between the four tunings, $k$ : $\\sigma ^{2}_{{\\rm PackMismatch},i,j}=\\frac{1}{4}\\sum ^{4}_{k=1}(Q_{i,j,k} - \\langle Q_{i,j,k}\\rangle )^{2}$ For each ${p_T}$ bin $i$ and package $j$ , the associated total uncertainty $\\sigma _{i,j}$ is computed: $\\begin{array}{rcl}\\sigma ^{2}_{i,j}&=&\\sigma ^{2}_{{\\rm StatData},i} + \\sigma ^{2}_{{\\rm StatPack},i,j} \\\\&+&\\sigma ^{2}_{{\\rm SystPack},i}+\\sigma ^{2}_{{\\rm PackMismatch},i,j}\\\\&+&\\sigma ^{2}_{A\\epsilon ,i},\\end{array}$ where the first two contributions, $\\sigma ^{2}_{{\\rm StatData},i}$ and $\\sigma ^{2}_{{\\rm StatPack},i}$ are the statistical uncertainties on the data and on the simulation and all other terms have already been introduced in previous sections.", "Using $\\sigma _{i,j}$ from step 3 we calculate the weighted mean of the $Q_{i,j}$ values obtained for the optimal tuning ($k=1$ ) of the different packages, $j$ , in each ${p_T}$ bin, $i$ : $ \\nonumber \\langle Q_{i}\\rangle =\\sum _{j=1}^{5}w_{i,j}Q_{i,j,k=1}$ where $w_{i,j} \\equiv \\frac{1/\\sigma ^{2}_{i,j}}{\\displaystyle \\sum _{j=1}^{5} 1/\\sigma _{i,j}^{2}}.$ The total uncertainty on the final measurement is the variance of the weighted mean: $\\begin{array}{rcl}{\\rm Var}(\\langle Q_{i}\\rangle )&=&\\displaystyle \\sum ^{5}_{j=1} w^{2}_{i,j}\\sigma _{i,j}^{2} \\\\&+&2\\displaystyle \\sum ^{5}_{j<m}w_{i,j}w_{i,m}\\sigma ^{2}_{{\\rm common},i}\\end{array}$ where $\\sigma _{{\\rm common},i}$ is the part of the total uncertainty that is correlated between different packages: $\\begin{array}{rcl}\\sigma ^{2}_{{\\rm common},i} &\\equiv &\\sigma ^{2}_{{\\rm StatData},i} \\\\&+&\\sigma ^{2}_{{\\rm SystPack},i}+\\sigma ^{2}_{A\\epsilon ,i}\\end{array}$ For convenience, the total uncertainty ${\\rm Var}(\\langle Q_{i}\\rangle )^{1/2}$ is split into a statistical uncertainty, a model-related systematic uncertainty and an acceptance and efficiency correction related systematic uncertainty: ${\\rm Var}(\\langle Q_{i}\\rangle ) = \\sigma _{{\\rm StatCombined},i}^2 + \\sigma _{{\\rm model}, i}^2 + \\sigma _{A\\epsilon ,i}^2$ with: $\\sigma _{{\\rm StatCombined},i}^2=\\sigma ^{2}_{{\\rm StatData},i} + \\frac{1}{5}\\sum _{j=1}^5\\sigma ^{2}_{{\\rm StatPack},i,j}$ and (by construction): $\\sigma _{{\\rm model},i}^2\\equiv {\\rm Var}(\\langle Q_{i}\\rangle ) - \\sigma _{{\\rm StatCombined},i}^2 - \\sigma _{A\\epsilon ,i}^2$ so that the final measurement, in a given muon arm, is written: $\\langle Q_{i}\\rangle \\pm \\sigma _{{\\rm StatCombined},i}\\pm \\sigma _{{\\rm model},i}\\pm \\sigma _{A\\epsilon ,i}$ The independent North and South measurements are combined using: $\\langle Q_{i}\\rangle =\\sum ^{2}_{j=1} w_{i,j} Q_{i,j},$ where $i$ is the index of the ${p_T}$ bin, $j$ the arm index and $w_{i,j}$ a weight calculated in the same manner as in Eq.", "REF , using the following total uncertainty on the measurement $Q_{i,j}$ : $\\sigma _{i,j}^2=\\sigma _{{\\rm StatCombined},i,j}^2 + \\sigma _{{\\rm model}, i,j}^2+\\sigma _{A\\epsilon ,i,j}^2,$ which is identical to the expression of Eq.", "REF , but explicitly includes the arm index, $j$ .", "The total uncertainty on the arm-averaged $Q_i$ value is calculated in a manner similar to Eq.", "REF : $\\begin{array}{rcl}{\\rm Var}(\\langle Q_{i}\\rangle )&=&\\displaystyle \\sum ^{2}_{j=1}w^{2}_{i,j}\\sigma _{i,j}^{2} \\\\&+&2\\displaystyle \\sum ^{2}_{j<m}w_{i,j}w_{i,m} \\sigma ^{2}_{{\\rm arm\\;common},i}\\end{array}$ where $\\sigma ^{2}_{{\\rm arm\\;common},i}$ is the systematic uncertainty common to both muon arms due to uncertainty on cocktail input.", "For convenience, this uncertainty is again split into a statistical contribution $\\sigma _{{\\rm StatCombined},i}$ and a systematic contribution $\\sigma _{{\\rm SystCombined},i}$ defined by: $\\sigma ^{2}_{{\\rm SystCombined},i}\\equiv {\\rm Var}( \\langle Q_{i}\\rangle )-\\sigma ^{2}_{{\\rm StatCombined},i}$ so that the final, arm-averaged, measurement of $Q_i$ is written: $\\langle Q_i\\rangle \\pm \\sigma _{{\\rm SystCombined},i}\\pm \\sigma _{{\\rm StatCombined},i}$ As already noted in Section REF , for invariant cross section measurements (in ${p+p}$ collisions) and ${R_{\\rm AA}}$ measurements one must add the systematic uncertainty on the ${p+p}$ inelastic cross section seen by the minimum bias trigger, $\\sigma ^{pp}_{\\rm BBC}$ in quadrature to the uncertainties above.", "For ${R_{\\rm AA}}$ measurements one must also add the systematic uncertainty on the mean number of binary collisions ${N_{\\rm coll}}$ in each centrality bin." ], [ "Heavy-Flavor Muon ${p_T}$ Distributions in {{formula:c41498ef-7e5a-4224-89cb-66099f681ca8}} and\n{{formula:be8caaa2-103f-4185-a5ea-140ab6790d9e}} Collisions", "Figure REF shows the production cross section of negatively charged muons from decays of open-heavy-flavor mesons as a function of ${p_T}$ .", "Vertical bars correspond to statistical uncertainties.", "Boxes correspond to the systematic uncertainties calculated following the steps described in Section REF .", "The measurements from both muon arms have been combined to reduce the overall uncertainty.", "Measured values for each ${p_T}$ bin are listed in the Appendix (Table REF ).", "Figure: (color online)Invariant production yields of negative muons fromheavy-flavor-mesons decay as a function p T {p_T} inp+pp+p collisions at s=200\\sqrt{s}=200 GeV (open squares) and in Cu + Cu {{\\rm Cu}+{\\rm Cu}} collisions for three different centrality intervals(40--9440--94%, 20--4020--40% and 0--200--20%), scaled by powers of ten forclarity (filled circles).", "The solid line associated to each set ofpoints corresponds to a fit to the p+p{p+p} invariant yielddistribution described in the text, scaled by the appropriatenumber of binary collisions N coll {N_{\\rm coll}} when comparing to the Cu + Cu {{\\rm Cu}+{\\rm Cu}}measurements.Figure REF shows the invariant yield of negative muons from heavy-flavor mesons decay for all analyzed ${{\\rm Cu}+{\\rm Cu}}$ centrality classes, compared to the invariant yield measured in ${p+p}$ collisions.", "The solid lines correspond to a fit to the ${p+p}$ data using the function $A(1+({p_T}/B)^{2})^{-4.2}$, similar to the one used in [35], scaled by the average number of binary collisions ${N_{\\rm coll}}$ for each ${{\\rm Cu}+{\\rm Cu}}$ centrality bin.", "For peripheral ($40--94$ % centrality) and midcentral ($20--40$ % centrality) ${{\\rm Cu}+{\\rm Cu}}$ collisions, a reasonable agreement is observed between the measurement and the scaled fit to the ${p+p}$ data, whereas for central collisions ($0--20$ %), a systematic difference is visible for high ${p_T}$ muons (${p_T}\\ge 2$ GeV/$c$ ), and the measurements are below the scaled ${p+p}$ fit, indicating a suppression of the heavy-flavor yields with respect to binary collision scaling, which is best quantified by measuring ${R_{\\rm AA}}$ (see Section REF )." ], [ "Charm Cross Section, $\\left.d\\sigma _{c\\bar{c}}/dy \\right|_{\\langle y\\rangle =1.65}$ in {{formula:2f55e561-1edd-4c48-bc39-b72045eac526}} Collisions", "The $p+p$ heavy-flavor muon ${p_T}$ distribution is used to estimate the charm differential production cross section, $d\\sigma _{c\\bar{c}}/dy$ at forward rapidity ($\\langle y\\rangle =1.65$ ), as described in detail in reference [27].", "The muon ${p_T}$ spectrum measured in ${p+p}$ collisions spans from ${p_T}=1$ to 7 GeV/$c$ .", "Estimation of the full charm charm cross section requires a theoretical calculation in order to extrapolate the measurement down to ${p_T}=0$ GeV/$c$ .", "A set of fixed-order-plus-next-to-leading-log (FONLL) [36], [37] calculations have been used in this analysis.", "The charm production cross section $d\\sigma _{c\\bar{c}}/dy$ is derived from the heavy-flavor muon cross section using: $d\\sigma _{c\\bar{c}}/dy =\\frac{1}{BR(c \\rightarrow \\mu )} \\cdot \\frac{1}{C_{l/D}}\\cdot \\frac{d\\sigma _{\\mu ^-}}{dy}$ where $BR(c \\rightarrow D)$ is the total muon branching ratio of charm and is fixed to 0.103 in FONLL; $C_{l/D}$ is a kinematic correction factor, also provided by the FONLL calculation, which accounts for the difference in rapidity distributions between leptons and $D$ mesons; $d\\sigma _{\\mu ^-}/dy$ is the total cross section for negative muons from heavy-flavor mesons decay, integrated over ${p_T}$ and estimated by extrapolating our measurement down to ${p_T}=0$  GeV/$c$ using the FONLL calculation." ], [ "Extrapolation of the Data for ${p_T}<1.0$ GeV/{{formula:ca851728-3b62-4a14-b014-13232cc3bee9}}", "Low ${p_T}$ muons dominate the integrated heavy-flavor muon cross section due to the power-law like behavior of the ${p_T}$ distribution (Fig.", "REF ): according to the central value of the FONLL calculation, the integrated charm cross section for ${p_T}^{\\mu }>1$ GeV/$c$ represents about 6% of the total.", "Additionally, the contribution of bottom quark decay to the heavy-flavor muon ${p_T}$ distribution becomes increasingly relevant for ${p_T}>4$ GeV/$c$ , but has a negligible contribution to the integral and is ignored hereafter.", "The measured spectral shape matches the calculated shape.", "Therefore, extrapolation of the measured heavy-flavor muon ${p_T}$ spectra down to ${p_T}=0$  GeV/$c$ using FONLL is given by: $\\left.d\\sigma _{c\\bar{c}}/dy \\right|^{\\rm PHENIX}= \\left.d\\sigma _{c\\bar{c}}/dy \\right|^{\\rm FONLL}\\alpha ^{\\rm FONLL}$ where $\\alpha ^{\\rm FONLL}$ is a constant determined by fitting the central values of the FONLL ${p_T}$ distribution to the data for ${p_T}>1$  GeV/$c$ .", "It amounts to 3.75, and is used in determining the central point for PHENIX muons shown in Fig.", "REF ." ], [ "Systematic Uncertainties on $\\left.d\\sigma _{c\\bar{c}}/dy\\right|_{\\langle y\\rangle =1.65}$", "The total systematic uncertainty assigned to $\\left.d\\sigma _{c\\bar{c}}/dy\\right|_{\\langle y\\rangle =1.65}$ is a combination of experimental and theoretical uncertainties, added in quadrature.", "The experimental systematic uncertainty on the integral above ${p_T}> 1$ ,GeV/$c$ is determined by the appropriate quadrature sum of the systematic uncertainties on the individual ${p_T}$ points.", "This uncertainty is up/down symmetric and is equal to 32 %.", "The theoretical uncertainty for $\\left.d\\sigma _{c\\bar{c}}/dy\\right|_{\\langle y\\rangle =1.65}$ originates from the FONLL uncertainties.", "The variation in the FONLL calculation are determined by variation of the factorization scale, $\\mu _{F}$ , the renormalization scale, $\\mu _{R}$ , and the charm quark mass.", "Other contributions, such as fragmentation and parton distribution functions are smaller and neglected in this analysis.", "The FONLL upper and lower bounds obtained by varying the scales and the charm quark mass are treated as approximations for a one standard deviation systematic uncertainty.", "The ratio of the measured ${p_T}$ distributions for ${p_T}>1$  GeV/$c$ to the upper and lower FONLL bounds are fit independently to determine the corresponding two normalization factors.", "The difference between these two normalization factors is then used as a theoretical uncertainty.", "This uncertainty is asymmetric and amounts to $^{+29}_{-37}$  %.", "These FONLL systematic uncertainties are consistent with those of a previous study [27], which examined the different ${p_T}$ distributions obtained by varying the FONLL parameters, $1.3< M_{c}$ [GeV/$c$ ]$< 1.7$ , $0.5 <\\mu _R/m_T < 2$ , $0.5 < \\mu _F/m_T < 2$ , with $m_{T}$ representing transverse mass.", "The different predicted theoretical ${p_T}$ ranged within an envelope of $\\pm $ 35% relative to the central spectrum." ], [ "Integrated Charm Production Cross Section at $\\langle y\\rangle =1.65$ in {{formula:a1824eda-cc25-4d60-8e7f-097774637909}} collisions", "The integrated charm production cross section at forward rapidity ($\\langle y\\rangle =1.65$ ) obtained with this method is: $\\left.d\\sigma _{{c\\overline{c}}}/dy\\right|_{\\langle y\\rangle =1.65} = 0.139\\pm 0.029\\ {\\rm (stat)~}^{+0.051}_{-0.058}\\ {\\rm (syst)}$ This measurement is shown in Fig.", "REF , together with the measurement performed by PHENIX at midrapidity [38], as well as the FONLL calculation and its uncertainty band, calculated as discussed in the previous section.", "The full circle, located at $y=-1.65$ , corresponds to the combined measurement performed in both muon arms.", "The open circle, located at $y=1.65$ , corresponds to its mirror image.", "Figure: (color online)cc ¯c\\overline{c} production cross section as a function of rapidityin p+pp+p collisions, measured using semileptonic decay to electrons(closed square) and to muons (closed circle).Figure REF shows $R_{\\rm AA}({p_T})$ for muons from heavy-flavor decay in ${{\\rm Cu}+{\\rm Cu}}$ collisions as a function of muon ${p_T}$ for three centrality classes ($40--94$ %, $20--40$ % and $0--20$ %).", "As was the case for invariant yields and cross sections, the two independent measurements obtained with each muon arm are statistically combined, following the method discussed in Section REF .", "Vertical bars correspond to the statistical uncertainties; boxes centered on the data points correspond to point-to-point correlated uncertainties and the vertical gray band centered on unity corresponds to the uncertainty on ${N_{\\rm coll}}$ , as listed in Table REF .", "Also shown in the bottom panel of Fig.", "REF is a theoretical calculation from [39], [40], discussed in Section .", "The measured values for each ${p_T}$ bin and each centrality class are listed in the Appendix (Table REF )." ], [ "Discussion and Conclusions", "The measurement of open-heavy-flavor muon production in ${p+p}$ collisions at $\\sqrt{s}=200$  GeV reported in this paper is a significant improvement over the previous PHENIX published result [25].", "The transverse momentum range of the present measurement is extended to ${p_T}=7$ GeV/$c$ (compared to ${p_T}=3$ GeV/$c$ in the previous analysis).", "The differential production cross section is integrated over ${p_T}$ to calculate a production cross section at forward rapidity of $d\\sigma _{c\\bar{c}}/dy|_{\\langle y\\rangle = 1.65} = 0.139\\pm 0.029\\ {\\rm (stat)~}^{+0.051}_{-0.058}\\ {\\rm (syst)}$  mb.", "This cross section is compatible with a FONLL calculation within experimental and theoretical uncertainties.", "It is also compatible with expectations based on the corresponding midrapidity charm production cross section measured by PHENIX.", "Muons from heavy-flavor decay have also been measured in ${{\\rm Cu}+{\\rm Cu}}$ collisions at $\\sqrt{s_{NN}} = 200$  GeV/$c$ , in the same rapidity and momentum range.", "This allows determination of the heavy-flavor-muon ${R_{\\rm AA}}$ as a function of ${p_T}$ in three centrality classes, $40--94$ %, $20--40$ % and $0--20$ %.", "As shown in Fig.", "REF , no suppression is observed across most of the transverse momentum range for muon yields measured in peripheral ($40--94$ %) and midcentral ($20--40$ %) ${{\\rm Cu}+{\\rm Cu}}$ collisions compared to ${N_{\\rm coll}}$ -scaled ${p+p}$ collisions.", "On the contrary, open heavy flavor production is significantly suppressed for central ${{\\rm Cu}+{\\rm Cu}}$ collisions ($0--20$ %), with the largest effect observed for ${p_T}>2$  GeV/$c$ .", "Interestingly, as demonstrated in Fig.", "REF , the level of suppression for these higher ${p_T}$ heavy-flavor muons (the last red point on right) is comparable to the level of suppression observed for high ${p_T}$ nonphotonic electrons measured at midrapidity in the most central ${\\rm Au}+{\\rm Au}$ collisions (the last blue point on right).", "One expects the Bjorken energy density of the matter produced in the midrapidity region in the most central ${\\rm Au}+{\\rm Au}$ collisions to be at least twice as large as that of the matter produced in the forward rapidity region in most central ${{\\rm Cu}+{\\rm Cu}}$ collisions [41], [26].", "Therefore the large suppression observed in ${{\\rm Cu}+{\\rm Cu}}$ collisions suggests significant (cold) nuclear effects at forward rapidity in addition to effects due to strongly interacting partonic matter.", "As shown in the bottom panel of Fig.", "REF , the suppression of open-heavy-flavor muon production for central ${{\\rm Cu}+{\\rm Cu}}$ collisions is consistent with a recent theoretical calculation performed at the same rapidity ($y=1.65$ ) for ${p_T}>2.5$  GeV/$c$  [40], [39].", "This calculation includes effects of heavy-quark energy loss (both elastic and inelastic) and in-medium heavy meson dissociation.", "Additionally, the calculation accounts for cold nuclear matter effects relevant for open heavy flavor production [42], namely shadowing (nuclear modification of the parton distribution functions of the nucleon) and initial state energy loss due to multiple scattering of incoming partons before they interact to form the ${c\\overline{c}}$ pair.", "New PHENIX inner silicon vertex detectors will greatly improve heavy flavor production measurements and allow separation of charm and bottom contributions [43], [44]." ], [ "ACKNOWLEDGMENTS", "We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions.", "We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (P. R. China), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat à l'Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Ministry of Industry, Science and Tekhnologies, Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), Hungarian National Science Fund, OTKA (Hungary), Department of Atomic Energy (India), Israel Science Foundation (Israel), National Research Foundation and WCU program of the Ministry Education Science and Technology (Korea), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and the Wallenberg Foundation (Sweden), the U.S.", "Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the US-Hungarian NSF-OTKA-MTA, and the US-Israel Binational Science Foundation.", "Table REF gives the differential invariant cross section of muons from heavy-flavor decay in $\\sqrt{s} = 200$  GeV ${p+p}$  collisions and corresponds to Fig.", "REF .", "Table REF gives ${R_{\\rm AA}}$ of muons from heavy-flavor mesons decay for the different centrality classes of ${\\sqrt{s_{\\rm NN}}}=200$  GeV ${{\\rm Cu}+{\\rm Cu}}$  collisions and corresponds to Fig.", "REF .", "Table: Nuclear-modification factor, R AA {R_{\\rm AA}}, of negative muonsfrom heavy-flavor mesons decay as a function of p T {p_T} forthe specified centrality classes of Cu + Cu {{\\rm Cu}+{\\rm Cu}} collisions ats NN =200{\\sqrt{s_{\\rm NN}}}=200 GeV." ] ]

[ [ "Using the J1-J2 Quantum Spin Chain as an Adiabatic Quantum Data Bus" ], [ "Abstract This paper investigates numerically a phenomenon which can be used to transport a single q-bit down a J1-J2 Heisenberg spin chain using a quantum adiabatic process.", "The motivation for investigating such processes comes from the idea that this method of transport could potentially be used as a means of sending data to various parts of a quantum computer made of artificial spins, and that this method could take advantage of the easily prepared ground state at the so called Majumdar-Ghosh point.", "We examine several annealing protocols for this process and find similar result for all of them.", "The annealing process works well up to a critical frustration threshold." ], [ "Introduction", "The ability to send data from one part of a computer to another accurately and quickly is an essential feature in virtually any design.", "The use of artificial spin clusters in quantum computing has been of growing interest.", "There is an implementation which has been demonstrated using superconducting flux q-bits[1], [2], [3], [4], [5].", "This paper demonstrates an effective and scalable way of sending arbitrary q-bit states along a spin chain with Heisenberg type coupling using quantum annealing.", "Assuming one could implement a Hamiltonian which follows this model, for example using the methods proposed in [6] using coupled cavities, this system design could be used for a data bus which transports quantum states to different sections of a quantum computer system.", "For instance, the protocols discussed in this paper could potentially be used to move states from memory to a system of quantum gates in an implementation of the circuit model.", "There has already been significant work done on the subject of quantum data buses using spin chains, [7], [8], [9], [10].", "However these works differ significantly from the method proposed in this paper in that the encoded q-bit is not transmitted through a degenerate ground state manifold, but through excitations of the Hamiltonian.", "This paper investigates a method of using q-bits as an intermediate bus for the transfer of quantum information.", "This method can be compared to another method which is that of pulses [11], where a Hamiltonian is applied to a system for a period of time to perform a given operation.", "In the case of information transfer this operation is usually a swap.", "Unlike the method of using pulses, this method of using q-bits does not require precise timing to insure that the correct operation is performed.", "The method of using a spin chain Hamiltonian as a data bus also means that one does not need to either be able to address any pair of q-bits in the system or perform multiple operations to transfer an arbitrary q-bit.", "The pulse method does have the advantage that every intermediate spin can be used as quantum memory.", "However this is at the cost of the increased complexity of using dynamic quantum evolution in excited states, and the requirement of precise timing.", "The adiabatic quantum bus method also has the advantage that, as in any adiabatic quantum process, only the lowest energy parts of Hamiltonian need to be faithfully realised by the implementation method.", "For example, a Hamiltonian which actually has an infinite number of excited states on each “spin”, but where only the low energy states which act like a spin $\\frac{1}{2}$ Heisenberg system, contribute to the ground state would be perfectly acceptable to use as an adiabatic quantum bus without modification.", "But the higher energy states may cause issues using a method such as pulses.", "This general feature of adiabatic quantum processes such as the one illustrated in this paper makes them more versatile than their non-adiabatic counterparts.", "The effect we will examine exploits the SU(2) symmetry of the Heisenberg Hamiltonian and uses the ground state degeneracy created by this symmetry in a chain with an odd number of spins.", "It has already been demonstrated [12] that disturbances can be sent an unlimited distance along such chains because of their degenerate ground state.", "This paper goes a step further and actually demonstrates how a specific state can be transported across the chain using a quantum annealing protocol.", "Further investigation will also be provided into application of this method to systems such as the XYZ spin chain which only have a $\\mathbb {Z}_{2}$ symmetry.", "The model we consider is the J1-J2 Heisenberg spin chain with open boundaries, $H=\\sum _{n=1}^{N-1}J_{1}^{n}\\vec{\\sigma }_{n}\\cdot \\vec{\\sigma }_{n+1}+\\sum _{n=1}^{N-2}J_{2}^{n}\\vec{\\sigma }_{n}\\cdot \\vec{\\sigma }_{n+2}.$ This model has SU(2) symmetry, which is expressed by the Hamiltonian being block diagonal, such that there are N+1 blocks each with $\\binom{N}{k}$ states.", "Each block represents all of the states with a given number, k, of up spins.", "If the number of spins in the model is odd, then the additional symmetry under a flip of the spins in the z direction, i.e.", "$\\sigma ^{z}\\rightarrow -\\sigma ^{z}$ implies that all states of the Hamiltonian have a twofold energy degeneracy.", "In the anti-ferromagnetic case, ( $J_{1},\\: J_{2}>0$ ) the ground state manifold consists of one state from the $\\textrm {k=floor(}\\frac{N}{2}\\textrm {)}$ and one from the $\\textrm {k=ceil(}\\frac{N}{2}\\textrm {)}$ sector.", "A simple example of this would be taking a system with 5 spins, the ground state would be twofold degenerate and would span the k=2 and k=3 sectors.", "One can now consider an initial Hamiltonian of the form of Eq.", "REF where the couplings are the ones given by $J_{1}^{n} & = & {\\left\\lbrace \\begin{array}{ll}J_{1}^{n,init} & n<N-1\\\\0 & n=N-1\\end{array}\\right.", "},\\\\J_{2}^{n} & = & {\\left\\lbrace \\begin{array}{ll}J_{2}^{n,init} & n<N-2\\\\0 & n=N-2\\end{array}\\right.", "}.$ The general condition on $J_{1}^{n,init}$ and $J_{2}^{n,init}$ is that the coupling is predominantly anti-ferromagnetic everywhere and that each spin is coupled to the others by at least one non zero J.", "For simplicity this paper considers only $J_{1}^{n,init}=1$ and $J_{2}^{n,init}=J_{2}^{init}$ .", "This ground state manifold consists of the tensor product of the (unique) ground state of the chain of length N-1 with the Nth spin in an arbitrary state, a state in this manifold is of the from given by $|\\Psi ^{init} \\rangle =|\\Psi _{0}^{N-1} \\rangle \\times |\\psi ^{init} \\rangle ,$ where $|\\Psi _{0}^{N-1} \\rangle $ is the ground state of the spin chain of length N-1 and $|\\psi ^{init} \\rangle $ is an arbitrary single spin state.", "One can now consider the same Hamiltonian, but with $n\\rightarrow (N-n)+1$ .", "This Hamiltonian also has the form of Eq.", "REF , but with couplings $J_{1}^{n} & = & {\\left\\lbrace \\begin{array}{ll}J_{1}^{n,final} & n>1\\\\0 & n=1\\end{array}\\right.", "},\\\\J_{2}^{n} & = & {\\left\\lbrace \\begin{array}{ll}J_{2}^{n,final} & n>2\\\\0 & n=2\\end{array}\\right.", "}.$ The general condition on $J_{1}^{n,final}$ and $J_{2}^{n,final}$ is that the coupling is predominantly anti-ferromagnetic everywhere and that each spin is coupled to the others by at least one non-zero J.", "For simplicity this paper considers only $J_{1}^{n,final}=1$ and $J_{2}^{n,final}=J_{2}^{final}$ .", "A state in the ground state manifold is now given by $|\\Psi ^{final} \\rangle =|\\psi ^{final} \\rangle \\times |\\Psi _{0}^{N-1} \\rangle ,$ where $|\\psi ^{final} \\rangle $ is an arbitrary single spin state.", "One can now consider a quantum annealing process with described by $\\textrm {\\textrm {H}(t;}\\tau \\textrm {)}=\\textrm {\\textrm {\\textrm {A}(t;}\\tau \\textrm {)} H}^{\\textrm {init}}+\\textrm {\\textrm {\\textrm {B}(t;}\\tau \\textrm {)} H}^{\\textrm {final}},$ where $\\textrm {H}^{\\textrm {init}}$ is REF with the conditions given in REF and and $\\textrm {H}^{\\textrm {final}}$ is REF with the conditions given in REF and .", "Also A and B follow the conditions $\\textrm {\\textrm {A}(}t\\le 0\\textrm {;}\\tau \\textrm {)} & = & 1,\\\\\\textrm {\\textrm {B}(}t\\le 0\\textrm {;}\\tau \\textrm {)} & = & 0,\\\\\\textrm {\\textrm {A}(}t\\ge \\tau \\textrm {;}\\tau \\textrm {)} & = & 0,\\\\\\textrm {\\textrm {B}(}t\\ge \\tau \\textrm {;}\\tau \\textrm {)} & = & 1.$ For all values of A and B the SU(2) symmetry is preserved.", "Therefore the Hamiltonian remains block diagonal at all times.", "The symmetry of the Hamiltonian under $\\sigma ^{z}\\rightarrow -\\sigma ^{z}$ is also preserved at all times.", "This implies that the ground-state degeneracy (as well as the twofold degeneracy of all states) is preserved.", "The block diagonal structure implies that there will be no exchange of amplitude between spin sectors during the annealing process, while the degeneracy implies that no relative phase can be acquired between the states in the $\\textrm {k=floor(}\\frac{N}{2}\\textrm {)}$ and the $\\textrm {k=ceil(}\\frac{N}{2}\\textrm {)}$ sector.", "From the combination of these two conditions one can see that as long as one anneals slowly enough with $\\textrm {\\textrm {H}(t;}\\tau \\textrm {)}$ Technically one must give the additional condition that there is no true crossing within the spin sectors on the annealing path.", "one can start with a state of the form given in Eq.", "REF and reach a final state in the form Eq.", "REF where $|\\psi ^{fin} \\rangle =\\exp (\\imath \\varphi )|\\psi ^{init} \\rangle $ , and $\\varphi $ is an irrelevant phase.", "One specific example of such an annealing protocol to transport a spin is given in Fig.", "REF .", "Figure: Cartoon representation of a process where a spin is joined to thechain, then the spin on the opposite end is removed.", "Note that thisis only one specific example of many possible processes for transportinga q-bit.The use of the J1-J2 Heisenberg chain for transport by quantum annealing has several advantages.", "First the model with uniform coupling is gapped for $\\frac{J_{2}}{J_{1}}\\gtrsim 0.25$ [13].", "This suggests that within the adiabatic evolution process, at least locally, the system should behave as a gapped system in this regime, as long as global effects such as odd length frustration do not cause problems.", "It is important to note that even the largest system size considered here is far from the thermodynamic limit.", "One should note, however, that given the connectivity schemes of adiabatic quantum chips already in existence [5], one may only need to transport a q-bit state a few spins to get it to any part of the system.", "Further evidence of favorable scaling comes from [12] which demonstrates that disturbances can travel an unlimited distance in the presence of a degenerate ground state, even in a gapped system.", "Furthermore, [12] suggests that these disturbances can carry entanglement, polarization, and quantum information.", "The transport by annealing given here is a specific example of how this effect can be taken advantage of.", "Another advantage of the use of the J1-J2 Heisenberg Hamiltonian is the existence of the so called Majumdar-Ghosh point [14] ($\\frac{J_{2}}{J_{1}}=0.5$ ).", "At this point the ground state (with an even number of spins) has the simple form of a matrix product of singlets.", "Due to this fact the system should be relatively easy to prepare.", "The system is also gapped at the Majumdar-Ghosh point, making the Majumdar-Ghosh Heisenberg Hamiltonian, an ideal system for transport by quantum annealing and the ideal candidate for building an adiabatic quantum data bus.", "Although this paper focuses on the J1-J2 Heisenberg model, it should be noted that this same annealing scheme should work with any pattern of coupling in the intermediate spins (i.e.", "J1-J2-J3)At least this should work for small systems.", "In the continuum limit many of these systems may become gapless, so that quantum annealing cannot be effectively performed.", "Also one may be able to construct certain pathological cases with paths which pass through true crossings.. One would also expect this scheme to work in models where the SU(2) symmetry is broken but there is a remaining $\\mathbb {Z}_{2}$ symmetry such as the XYZ or XY model, again with arbitrary patterns of coupling.", "Note however that this mehhod will not work in the Ising model, because although there is a $\\mathbb {Z}_{2}$ symmetry, the Hamiltonian lacks terms to exchange q-bits between sites because it is diagonal in the computational basis." ], [ "Proof of Principle", "None of the arguments so far have given much illumination to the difficulty or ease of annealing within the sector.", "While we have discussed that transport of a q-bit state is possible in principle by annealing, we have not yet shown that the annealing process is fast enough to be practical.", "For this we turn to numerics.", "For the purposes of this paper we will consider the annealing time, $\\tau $ , required to reach a given fixed fidelity, $F(\\tau )$ , with the true final ground state, $F(\\tau )=|\\left\\langle \\Psi ^{fin} \\mid _{0}^{\\tau }dt\\, H(t,\\tau ) \\mid \\Psi ^{init} \\right\\rangle |.$ The J1-J2 Heisenberg model is not an analytically solved model, at least for finite values of $J_{2}$ , so numerical methods must be used in this calculation.", "One can first consider one part of the annealing process, in which a single spin is joined to a even length J1-J2 spin chain, using both $J_{1}$ and $J_{2}$ couplings which are linearly increased to equal values of those used in the rest of the chain Note that this Hamiltonian (and all other annealing Hamiltonians in this paper) can be rewritten in the form of REF .", "However it is much more compact not to write the unchanging parts of the Hamiltonian twice., $H(t,\\tau )=\\sum _{n=1}^{N-2}J_{1}\\vec{\\sigma }_{n}\\cdot \\vec{\\sigma }_{n+1}+\\sum _{n=1}^{N-3}J_{2}\\vec{\\sigma }_{n}\\cdot \\vec{\\sigma }_{n+2}+\\lambda (t,\\tau )(J_{1}\\vec{\\sigma }_{N-1}\\cdot \\vec{\\sigma }_{N}+J_{2}\\vec{\\sigma }_{N-2}\\cdot \\vec{\\sigma }_{N}),$ $\\lambda (t,\\tau )={\\left\\lbrace \\begin{array}{ll}0 & t\\le 0\\\\\\frac{t}{\\tau } & 0<t<\\tau \\\\1 & t\\ge \\tau \\end{array}\\right.", "}.$ Figure: Coupling constant λ(t,τ)\\lambda (t,\\tau ) from Eq.", "and Eq.", "versus t τ\\frac{t}{\\tau } .Figure: Annealing time required to reach a 90% fidelity with the true groundstate within one of the two largest spin sectors of the Hamiltonianvs.", "J 2 J_{2}, with J 1 J_{1} set to unity.", "One can see that for largervalues of J 2 J_{2} the annealing time behaves unpredictably.", "The annealingtime also scales poorly with system size close to the Majumdar-Ghoshpoint.As shown in Fig.", "REF , the annealing time required becomes large and highly sensitive to small variations for larger values of $J_{2}$ .", "Also the behavior seems to get worse in this regime as system size is increased, and is poor at the Majumdar-Ghosh point At least for fixed coupling, the case of dynamically changing coupling will be considered later..", "Figure: Scaling of annealing time to achieve 90%final ground state fidelity (in units of inverse Hamiltonian energy)versus length of chain on a log-log plot.As a further demonstration of the scaling with annealing time versus $J_{2}$ , one can plot the annealing time versus system size, as we have done in Fig.", "REF .", "This figure shows polynomial or even sub polynomial scaling for small values of $J_{2}$ , but than shows strongly non-monotonic behavior for stronger coupling.", "It is important to note however that even the longest chain length considered here is probably far from the infinite system limit, and this data may not be trustworthy for making predictions for scaling as the chain length approaches the infinite system limit.", "Figure: Plots of gap for joining a single spin to aneven length J1-J2 Heisenberg spin chain.", "For density plots lightercolors indicate larger gap.", "a) gap versus λ\\lambda in Eq.", "and J 2 J_{2} for 15 total spins d) Gap versus J 2 J_{2} with λ=1\\lambda =1By examining the gap one can hope to gain insight into the underlying cause of the behaviour of annealing time curves.", "As Figs.", "REF (a) and (b) show, the behavior of the annealing time curves is reflected by the presence of what appear to be true crossings Strictly speaking nothing in this paper has demonstrated them to be true crossings, they could just be close avoided crossings, it does not matter for the purposes of this paper.", "for the odd length spin chain with uniform coupling.", "Fig.", "REF (b) shows the gap for an odd length spin chain and seems to confirm the presence of points with very small gap with uniform coupling for $J_{2}$ above 0.5.", "Figs.", "REF and REF together show that, at least at the length scales considered here, there are good annealing paths for joining a single spin to an even length chain.", "However, the simplest method of taking advantage of the simple ground-state wavefunction at the Majumdar-Ghosh point is not optimal.", "Fortunately there are many other possible options to take advantage of the easily prepared ground state and hopefully avoid the regions of small gap found here." ], [ "Dynamically Tuning J2", "One method to avoid regions of small gap while still taking advantage of the Majumda-Ghosh point would be to start at the Majumdar-Ghosh point and then dynamically reduce the value of $J_{2}$ during the annealing process, a simple way of doing this would be to use the Hamiltonian in Eq.", "REF .", "$H(t,\\tau )=\\sum _{n=1}^{N-2}J_{1}\\vec{\\sigma }_{n}\\cdot \\vec{\\sigma }_{n+1}+\\sum _{n=1}^{N-3}J_{2}(t,\\tau )\\vec{\\sigma }_{n}\\cdot \\vec{\\sigma }_{n+2}+\\lambda (t,\\tau )(J_{1}\\vec{\\sigma }_{N-1}\\cdot \\vec{\\sigma }_{N}+J_{2}(t,\\tau )\\vec{\\sigma }_{N-2}\\cdot \\vec{\\sigma }_{N}),$ $\\lambda (t,\\tau )={\\left\\lbrace \\begin{array}{ll}0 & t\\le 0\\\\\\frac{t}{\\tau } & 0<t<\\tau \\\\1 & t\\ge \\tau \\end{array}\\right.", "},$ $J_{2}(t,\\tau )={\\left\\lbrace \\begin{array}{ll}0.5 & t\\le 0\\\\0.5+\\frac{t}{\\tau }(J_{2f}-0.5) & 0<t<\\tau \\\\J_{2f} & t\\ge \\tau \\end{array}\\right.", "}.$ Figure: In this annealing protocol not only is a spin coupled to the chain,but J 2 J_{2}is also changed dynamically.Figure: Annealing time required to reach a 90% fidelitywith the true ground state within one of the two largest spin sectorsof the Hamiltonian with dynamical coupling starting at J 2 J_{2}=0.5and linearly changing to J 2f J_{2f} while also joining a spin to thechain, with J 1 J_{1} set to unity throughout the process.", "Notice thatthis figure is qualitatively and quantitatively very similar to Fig..Fig.", "REF shows that taking advantage of the easily prepared ground state at the Majumdar-Ghosh point does in fact work, and the curves in this figure are strikingly similar to those in Fig.", "REF .", "This similarity is to be expected because Fig.", "REF demonstrates that the gap is the smallest where the spin is completely joined.", "Hence this part of the process should dominate the annealing time.", "It is reasonable to argue that because the regions of phase space which are visited are the same in the uncoupling process as coupling, the behavior of the system during the uncoupling process is determined by the gaps shown in Fig.", "REF , and therefore the annealing times for the uncoupling process should be at least qualitatively similar to those given in Fig.", "REF .", "One advantage to the uncoupling process is that unlike the coupling process, the need is not as strong to end in an easily prepared state.", "The only reason one may have to want to end in the Majumdar-Ghosh point is as an error check.", "The spins in the chain can be measured after the end of the process to ensure that no error has occurred For example if two spins which should be in a singlet together ended up being measured to be facing in the same direction than the annealing process would have failed..", "Figure: Annealing time required to reach a 90%Fidelity with the true ground state for uncoupling process withinone of the two largest spin sectors of the Hamiltonian vs. J 2 J_{2}with J 1 J_{1} set to unity.", "One can see that this figure is very similarto Fig.", "as one would expect because it is simplythe time reversed version of that process.Fig.", "REF shows the time required to uncouple a spin from the chain, not surprisingly this figure looks very similar to Fig.", "REF which is the coupling process.", "Note that in this system the Hamiltonian is simply Eq.", "REF with $\\frac{t}{\\tau }\\rightarrow (1-\\frac{t}{\\tau })$ .", "Figure: Annealing time required to reach a 90%fidelity with the true ground state for uncoupling process withinone of the two largest spin sectors of the Hamiltonian vs. initialJ 2i J_{2i} with a final J 2 J_{2} at the Majumdar-Ghosh point with J 1 J_{1}set to unity.", "This figure is very similar to Fig.", "as one would expect, because it is simply the time reversed versionof that process.As expected, except for one curve where a numerical error made some points unable to plot one can see from Fig.", "REF that the uncoupling process also requires roughly the same time as the coupling process for dynamically tuned $J_{2}$ .", "Note that the Hamiltonian for this process is simply Eq.", "REF with $\\frac{t}{\\tau }\\rightarrow (1-\\frac{t}{\\tau })$ and $J_{2f}\\rightarrow J_{2i}$ ." ], [ "Simultaneous Uncoupling and Coupling", "Because many of the issues encountered with the coupling protocol seem to relate to odd-spin frustration, it may be reasonable to consider simultaneously coupling one q-bit to the chain while uncoupling the other.", "The Hamiltonian in this case is given in Eq.", "REF .", "$H(t,\\tau )=\\sum _{n=1}^{N-2}J_{1}\\vec{\\sigma }_{n}\\cdot \\vec{\\sigma }_{n+1}+$ $\\sum _{n=1}^{N-3}J_{2}(t,\\tau )\\vec{\\sigma }_{n}\\cdot \\vec{\\sigma }_{n+2}+\\lambda (t,\\tau )((J_{1}\\vec{\\sigma }_{N-1}\\cdot \\vec{\\sigma }_{N}+J_{2}\\vec{\\sigma }_{N-2}\\cdot \\vec{\\sigma }_{N})-(J_{1}\\vec{\\sigma }_{1}\\cdot \\vec{\\sigma }_{2}+J_{2}\\vec{\\sigma }_{1}\\cdot \\vec{\\sigma }_{3})),$ $\\lambda (t,\\tau )={\\left\\lbrace \\begin{array}{ll}0 & t\\le 0\\\\\\frac{t}{\\tau } & 0<t<\\tau \\\\1 & t\\ge \\tau \\end{array}\\right.", "}.$ Figure: Plots of gap for simultaneously joininga single spin to an even length J1-J2 Heisenberg spin chain and unjoininga spin from the other end.", "For density plots lighter colors indicatelarger gap.", "a) gap versus λ\\lambda from Eq.", "and J2 for 17 total spins b) Gap versus J 2 J_{2} with λ=0.5\\lambda =0.5.Fig.", "REF shows the gaps for various system sizes for the process where the couplings are turned on and off simultaneously.", "This process does not seem to avoid the area of low gap for $J_{2}\\gtrsim 0.5$ seen in Fig.", "REF .", "However by comparing Fig.", "REF d) and Fig.", "REF d) one can see that it appears that the process of simultaneous uncoupling and coupling is characterized by avoided crossings rather than true crossings This statement is based on the fact that the gap does not have a cusp when plotted on a log scale.", "Strictly speaking this just shows that there is not a true crossing at the line where the two couplings are equal..", "Figure: Annealing time required to reach a 90%Fidelity with the true ground state for combined coupling and uncouplingprocess within one of the two largest spin sectors of the Hamiltonianvs.", "J 2 J_{2} with J 1 J_{1} set to unity.Fig.", "REF shows the time required for annealing processes with for the combined coupling and uncoupling process, the results are consistent with what one would expect from looking at Fig.", "REF , and confirm that the annealing time also tends to be very long and vary a lot for larger values of $J_{2}$ ." ], [ "Requirements for use as an Adiabatic Quantum Bus", "It is now useful to consider a broader class of models that may be used as adiabatic quantum buses, as in general the full SU(2) symmetry of the Heisenberg Hamiltonian is not required.", "The requirements for a spin chain (or network) Hamiltonain to be usable as an adiabatic quantum bus are as follow: The ground state must be at least 2 fold degenerate, and the ground state manifold must be able to encode a q-bit.", "In this paper this is achieved by having at least a $\\mathbb {Z}_{2}$ symmetry, and an odd number of spins, but there may be other ways.", "The Hamiltonian (or at least the low energy states) must be predominantly anti-ferromagnetic in nature.", "This guarantees that the encoded q-bit will be excluded from the larger spin chain (or network) when a single q-bit is removed.", "The Hamiltonian must contain terms which perform exchanges between sites.", "This excludes models such as the Ising model which, although it has the required symmetry, cannot be used a quantum bus because its Hamiltonian is diagonal in the computational basis One must be able to slowly couple in a spin with an arbitrary state on one end of the chain (network) and also to slowly remove coupling on the other end.", "More control may improve performance, but is not necessary.", "Annealing paths in parameter space must not contain true crossings.", "This is a general requirement for adiabatic quantum computing." ], [ "XXZ and XYZ model", "As previously mentioned, the full SU(2) symmetry of the Heisenberg Hamiltonian is not required.", "The Hamiltonian must only have a $\\mathbb {Z}_{2}$ symmetry to encode and transport one q-bit of information.", "In this section we will briefly examine two other possibilities: the XXZ model, where the SU(2) symmetry is broken, but the block diagonal structure imparted by this symmetry remains, and the XYZ model where only the block diagonal structure of a $\\mathbb {Z}_{2}$ symmetry is present.", "Figure: Annealing time to reach 90% fidelity on using theadiabatic quantum bus protocol on an XXZ spin chain versus the ratioof X and Z coupling strengths note that Z/X=0 is an XX model whileZ/X=1 is a J1 Heisenberg spin chain.", "This data was obtained with joiningand disconnecting of spins occurring simultaneously.As one can see from Fig.", "REF , the XXZ model can be used as an adiabatic quantum data bus.", "There is a regime where this system outperforms the XXX Heisenberg model for Z/X between 1 and roughly 2.", "This is to be expected because adding additional coupling in the z direction may serve to open the gap between the the ground-state manifold and the next excited state.", "The increasing time as the z coupling is increased further can be explained because the system would behave like an Ising model in the limit of $\\frac{Z}{X}\\gg 1$ .", "One can further examine the behavior of an XYZ model as an adiabatic quantum spin bus.", "For this purpose we consider the quantum bus protocol performed on the following normalized XYZ Hamiltonian $H_{XYZ}(\\Delta ;N)=C_{\\Delta }\\sum _{i=1}^{N-1}\\sigma _{i}^{x}\\sigma _{i+1}^{x}+(1+\\Delta )\\sigma _{i}^{y}\\sigma _{i+1}^{y}+(1+2\\Delta )\\sigma _{i}^{z}\\sigma _{i+1}^{z},$ where the normalization is $C_{\\Delta }=\\frac{\\sqrt{3}}{\\sqrt{1+(1+\\Delta )^{2}+(1+2\\Delta )^{2}}}.$ One can now examine the performance of this Hamiltonian for different values of $\\Delta $ , noting that $H_{XYZ}(0;N)$ is simply the J1 Heisenberg spin chain of length N. As Fig.", "REF shows, a slight advantage can be gained by using an XYZ model rather than a simple Heisenberg chain.", "Fig.", "REF also seems to suggest that the benefit gained is relatively independent of chain length.", "Figure: Plot of fractional difference from annealingtime for an chain with small Δ\\Delta (Heisenberg chain).", "This datais for the adiabatic quantum bus protocol performed on a chain ofthe form eq.", "with spins being attached and removedsimultaneously." ], [ "Other Protocols", "So far we have only investigated a small subset of the possible annealing protocols which meet the criteria given in the introduction.", "For example the XY spin chain should also have and easily prepared ground state and may be easier to experimentally realize [1].", "One could also try to examine the case of dynamically tuning the y and or z direction coupling and starting out at the Majumdar-Ghosh point but using modified coupling in the y and z directions with an XYZ model to avoid low gap regions.", "One could also try to change the coupling scheme to avoid the low gap region, by either randomly or systematically modifying the coupling between intermediate spins, if this is done dynamically, one can still take advantage of the Majumdar-Ghosh point.", "This technique could also be used in conjunction with any of the ideas in the previous paragraph.", "This paper is intended only to provide proof of principle for this method and is by no means an exhaustive search of all possible protocols." ], [ "Conclusions", "We have demonstrated how a J1-J2 Heisenberg spin chain can be used to transport a q-bit state adiabatically.", "We have also shown that many extensions of this Hamiltonian; such as different coupling schemes or the XY or XYZ model which have only a $\\mathbb {Z}_{2}$ symmetry, will also be able to be used to transport a q-bit Assuming there is not a true crossing along the annealing path, the coupling must also be (at least predominately) anti-ferromagnetic so that the excess spin does not become trapped in the larger spin chain.. We have found that for values of high frustration, transport by quantum annealing does not work very well.", "We have also demonstrated that this does not prevent us from exploiting the easily prepared ground state at the Majumdar-Ghosh point.", "We have given some examples of possible annealing protocols in this paper, but have really only investigated a very small section of a vast space of possible protocols for transportation of quantum states by annealing." ], [ "Acknowledgements", "The numerical computations were carried out on the University of Southern California high performance supercomputer cluster.", "This research is partially supported by the ARO MURI grant W911NF-11-1-0268." ] ]

[ [ "The close T Tauri binary system V4046 Sgr: Rotationally modulated X-ray\n emission from accretion shocks" ], [ "Abstract We report initial results from a quasi-simultaneous X-ray/optical observing campaign targeting V4046 Sgr, a close, synchronous-rotating classical T Tauri star (CTTS) binary in which both components are actively accreting.", "V4046 Sgr is a strong X-ray source, with the X-rays mainly arising from high-density (n_e ~ 10^(11-12) cm^(-3)) plasma at temperatures of 3-4 MK.", "Our multiwavelength campaign aims to simultaneously constrain the properties of this X-ray emitting plasma, the large scale magnetic field, and the accretion geometry.", "In this paper, we present key results obtained via time-resolved X-ray grating spectra, gathered in a 360 ks XMM-Newton observation that covered 2.2 system rotations.", "We find that the emission lines produced by this high-density plasma display periodic flux variations with a measured period, 1.22+/-0.01 d, that is precisely half that of the binary star system (2.42 d).", "The observed rotational modulation can be explained assuming that the high-density plasma occupies small portions of the stellar surfaces, corotating with the stars, and that the high-density plasma is not azimuthally symmetrically distributed with respect to the rotational axis of each star.", "These results strongly support models in which high-density, X-ray-emitting CTTS plasma is material heated in accretion shocks, located at the base of accretion flows tied to the system by magnetic field lines." ], [ "Introduction", "In the context of star formation and evolution, understanding the physics of young low-mass stars is essential.", "Such stars possess strong magnetic fields that regulate the transfer of mass and angular momentum to and from the circumstellar disk, via accretion and outflow phenomena.", "Young low-mass stars are also intense sources of high-energy emission (UV and X-rays) that ionizes, heats, and photoevaporates material in the circumstellar disk, thus affecting its physical and chemical evolution and, eventually, the disk lifetime [15], [19].", "Low-mass pre-main sequence stars are classified as classical T Tauri stars (CTTS) when they still accrete mass from the circumstellar disk.", "They become weak-line T Tauri stars (WTTS) when the accretion process ends.", "Both CTTS and WTTS are bright in X-rays due to the presence of hot coronal plasmas, heated and confined by the intense stellar magnetic fields [17], [16], [38], [20].", "It was suggested that in CTTS also the accretion process, beside the coronal magnetic activity, can provide a further X-ray emission mechanism [48], [23], [35].", "Magnetospheric accretion models predict that in CTTS mass transfer from the inner disk onto the star occurs via accretion streams funneled by magnetic flux tubes [33], [26], [6], where material moves in a almost free fall with typical velocities of $\\sim 300-500\\,{\\rm km\\,s^{-1}}$ .", "The impact with the stellar atmosphere, usually involving small fractions of the stellar surface, generates shock fronts that heat the infalling material up to temperatures of a few MK, and therefore should yield significant emission in the soft X-ray band ($0.1-1$  keV).", "Numerical modeling predicts high $L_{\\rm X}$ ($\\sim 10^{30}\\,{\\rm erg\\,s^{-1}}$ ) even for low accretion rates ($10^{-10}\\,{\\rm M_{\\odot }\\,yr^{-1}}$ ), indicating that X-ray emission related to the accretion process can rival or exceed coronal emission [25], [41] at least in principle.", "Strong evidence of accretion-driven X-rays from CTTS has been provided by the observed high densities of the X-ray emitting plasma at $T\\sim 2-4$  MK [30], [43], [24], [3], [28], [39], [2].", "These densities, considering the typical accretion rates and surface filling factors, are compatible with predictions of shock-heated material, and are significantly higher than that of typical quiescent coronal plasmas at temperatures of a few MK [36], [45].", "Moreover [22] observed a soft X-ray excess in CTTS with respect to WTTS, compatible with the scenario of a further plasma component at a few MK produced by accretion.", "However other results are discrepant with predictions: the observed $L_{\\rm X}$ of the high-density cool-plasma component in CTTS is lower than that predicted from the accretion rate by more than a factor 10 [4], [9], leaving the coronal component the major contributor to the X-ray emission in CTTS; furthermore in the cool plasma of CTTS the density increases for increasing temperature, at odds with predictions based on a single accretion stream [8].", "Because of these apparent discrepancies different scenarios were proposed, suggesting that the high-density cool plasma in CTTS could be coronal plasma, confined into magnetic loops, that is somehow modified by the accretion process [21], [8], [14].", "In addition to containing plasma at a few MK, the shock region is known to be associated with material at $T\\sim 10^{4}$  K or more, significantly hotter than the surrounding unperturbed photosphere, as a consequence of the energy locally deposited by the accretion process.", "This photospheric hot spot produces excess emission in the UV and optical band, which is often rotationally modulated because of the very small filling factor of the accretion-shock region and because accretion streams are usually not symmetric with respect to the rotation axis [7], [27], [37].", "Therefore, if the observed high-density X-ray emitting plasma also originates in the accretion shock, then, its X-ray emission might display rotational modulation.", "Specifically plasma heated in the accretion-shock, observed in the X-rays, could display periodic variations in density, emission measure, average temperatures, absorption, and source optical depth, as a consequence of stellar rotation.", "First hints of accretion driven X-rays that vary because of the stellar rotation were provided by [2] for the star V2129 Oph.", "Understanding the origin of this high-density plasma is important, both for constraining the total amount of X-rays emitted in CTTS, and setting the energy balance of the accretion-shock region [42].", "Eventually, a definitive confirmation that this plasma component is material heated in the accretion shock, would make its X-ray radiation an insightful tool to probe the physical properties (i.e.", "density and velocity) of the accretion stream, and to measure the chemical composition of the inner disk material [13].", "To search for such X-ray modulation effects we planned and carried out X-ray monitoring of V4046 Sgr, a close binary CTTS system in which both components are actively accreting from a circumbinary disk (see § ).", "In this work we describe the first results from an XMM-Newton Large Program (LP) focused on V4046 Sgr, based on time-resolved high-resolution X-ray spectroscopy on timescales down to 1/10 of the system orbital period.", "To constrain the large-scale magnetic field and the accretion geometry, we also carried out a coordinated multi-wavelength campaign involving photometry, spectroscopy, and spectropolarimetry of V4046 Sgr.", "In §  we summarize the project focused on V4046 Sgr, whose properties are described in § .", "Details of the data processing and analysis are reported in § .", "The observing results are presented in § , and then discussed in § ." ], [ "The V4046 Sgr project", "The XMM-Newton observation of V4046 Sgr consists of a 360 ks exposure performed on 2009 September 15-19 (Obs-id: 0604860201, 0604860301, and 0604860401).", "This observation is part of a quasi-simultaneous multi-wavelength campaign (optical photometry with REM/ROSS, 2009 September 1-30; optical spectroscopy with TNG/SARG, 2009 September 10-17; optical spectropolarimetry with CFHT/ESPADONS, 2009 September 2-8), aimed at studying simultaneously the properties of coronal plasmas, stellar magnetic field structure, photospheric spots (both cool spots and hot spots), and the accretion process.", "Here we present the results obtained with the XMM-Newton/RGS specifically aimed at searching for rotational modulation in the accretion-driven X-rays.", "The results of the entire observing campaign are presented in a series of papers describing, among other results, the properties of the X-ray emitting plasma (A. Maggio et al., in preparation), maps of the large-scale magnetic field structure and accretion geometry as inferred from optical spectropolarimetry [11], variations in the accretion process over a range of timescales (G. G. Sacco et al., in preparation), detection and identification of a distant comoving WTTS system [31]." ], [ "V4046 Sgr properties", "V4046 Sgr is a close CTTS binary system composed of two solar-like mass stars [11], separated by $8.8\\,{\\rm R_{\\odot }}$ .", "The two components are synchronously rotating with a period of 2.42 d, in circularized orbits [44].", "V4046 Sgr is estimated to lie at a distance of 73 pc [47] and it is viewed with an inclination of $35^{\\circ }$ [44], [32], with the orbital axis of the binary likely aligned with the individual stellar rotation axes [11].", "At an age of $\\sim 10-15$  Myr [47], [11] and classified as a CTTS, V4046 Sgr is still surrounded by a dusty, molecule-rich circumbinary disk [40] from which both the components are actively accreting [44].", "A previous Chandra observation [24] showed that V4046 Sgr has a cool plasma component ($T\\approx 2-4$  MK) at high density ($n_{\\rm e}\\approx 0.3-1\\times 10^{12}\\,{\\rm cm^{-3}}$ ), interpreted as material heated in the accretion shock.", "At the time of the XMM-Newton observation, the spectroscopic optical monitoring demonstrated that both components were accreting with a constant rate of $5\\times 10^{-10}\\,{\\rm M_{\\odot }\\,yr^{-1}}$ [11].", "Both components displayed complex magnetic fields [11], significantly weaker than that of younger solar-like CTTS [12].", "These magnetic fields are not strong enough to disrupt local disks farther than $1\\,R_{\\star }$ above stellar surface, thus the formation of circumstellar stellar disks around each component, distinct from the circumbinary disk, may be possible [10].", "The accretion process, based on Ca2 IRT, did not show significant rotational modulation, suggesting that post shock material contributing to these lines is symmetrically distributed with respect to stellar poles.", "The optical monitoring campaign confirmed the orbital/rotational period ($2.42\\,{\\rm d}$ ), and determined the conjunction and quadrature epochs at the time of the XMM-Newton observationThe quadrature with primary receding occurred at 2455078.199 HJD.", "In this work we adopt the phase reference defined in [44].", "However our optical monitoring revealed a phase shift of 0.069 with respect to that ephemeris, with quadratures occurring at phases 0.93 and 0.43, and conjunctions at phases 0.18 and 0.68 [11]." ], [ "Observations", "The XMM-Newton observation of V4046 Sgr, composed of three observing segments of $\\sim 120$  ks each separated by gaps of $\\sim 50$  ks, covered 2.2 system rotations.", "X-ray emitting material heated in the accretion shock is expected to have temperatures of a few MK at most.", "Therefore to search for X-ray variability possibly produced in the accretion shock we analyzed the XMM-Newton/RGS spectra, that contain emission lines that specially probe the coolest plasma components.", "The RGS spectrograph, composed of two nominally identical gratings (RGS1 and RGS2), covers the $\\sim 2-38$  Å  wavelength range.", "The first order spectrum, embracing the $4-38$  Å  band, has a resolution FWHM of 0.06 Å, while the second order provides a resolution FWHM of 0.03 Å  in the $\\sim 2-19$  Å  range.", "We extracted RGS spectra using the standard rgsproc task.", "Data were filtered discarding time segments affected by high background count rates.", "The final net exposures of the three observing segments were of 115, 122, 120 ks, respectively.", "We then applied the rgscombine task to add the RGS1 and RGS2 spectra of the same order.", "Totally 34800 and 8200 net counts were registered in the first and second order RGS spectra, respectively.", "We analyzed the RGS spectra using the IDL package PINTofALE v2.0 [29] and the XSPEC v12.5 [5] software.", "We measured individual line fluxes by fitting simultaneously first and secondSecond order spectrum was used only for lines contained in its wavelength range.", "order RGS spectra.", "Fit procedure was performed in small wavelength intervals ($\\Delta \\lambda \\lesssim 1.0$  Å).", "The adopted best fit function takes into account the RGS line spread function (determined by the matrix response fuction), and the continuum contribution (determined by adding a constant to the line emission, and leaving this constant as a free parameter in the fit)." ], [ "Results", "The RGS spectra collected during the entire observation (see details in A. Maggio et al., in preparation) indicate that the main properties of the X-ray emitting plasma of V4046 Sgr are similar to those observed during the previous Chandra observation [24]: the plasma at $T\\sim 1-4$  MK has high density, $n_{\\rm e}\\sim 10^{11}-10^{12}\\,{\\rm cm^{-3}}$ , as determined by the $f/i$ line ratio of He-like triplets of N6, O7, and Ne9The measurements of the Ne9 triplet was performed by including in the fit the Fe19 line at 13.52 Å, that is anyhow weaker than the Ne9 lines.. c@c@r@$\\pm $ l@ r@$\\pm $ l@ r@$\\pm $ l@ r@$\\pm $ l@ r@$\\pm $ l@ r@$\\pm $ l@ r@$\\pm $ l@ r@$\\pm $ l@ r@$\\pm $ l@  0pt Observed line fluxes at different time intervals 2c Ne X 2c Ne IX 2c Ne IX 2c Ne IX 2c Fe XVII 2c O VIII 2c O VIII 2c O VII 2c N VII rot.", "2c 12.13 Å 2c 13.45 Å 2c 13.55 Å 2c 13.70 Å 2c 15.02 Å 2c 16.01 Å 2c 18.97 Å 2c 21.60 Å 2c 24.78 Å phase 2c 6.3 MK 2c 4.0 MK 2c 3.5 MK 2c 4.0 MK 2c 5.6 MK 2c 3.2 MK 2c 3.2 MK 2c 2.0 MK 2c 2.0 MK $F_{\\rm int1/seg1}$ 0.67 34.0 5.2 58.3 6.6 21.1 5.2 21.3 4.8 14.9 3.2 15.3 3.3 90.6 8.9 18.9 8.9 67.6 9.4 $F_{\\rm int2/seg1}$ 0.79 33.0 5.0 76.0 7.0 17.7 5.1 16.8 4.3 7.4 3.0 12.8 3.7 81.8 8.1 17.0 9.3 69.5 9.4 $F_{\\rm int3/seg1}$ 0.91 31.7 4.7 56.1 6.9 34.3 6.2 25.3 5.5 14.0 3.2 17.6 3.7 101.0 9.1 42.5 11.7 89.9 11.0 $F_{\\rm int4/seg1}$ 0.03 45.9 5.4 65.7 6.9 31.6 5.8 24.9 4.9 10.5 3.3 9.8 3.3 118.2 9.7 48.3 11.0 62.3 9.2 $F_{\\rm int5/seg1}$ 0.15 40.9 6.6 59.8 8.3 22.0 6.8 27.1 6.3 11.6 3.6 22.3 5.0 52.4 8.9 30.6 12.7 52.3 10.8 $F_{\\rm int1/seg2}$ 0.50 40.2 5.2 71.0 7.4 33.8 6.1 37.3 5.6 14.5 3.4 10.9 3.7 95.6 8.4 33.2 11.8 119.1 10.7 $F_{\\rm int2/seg2}$ 0.62 54.3 5.5 37.7 6.1 30.3 5.7 35.9 5.3 16.9 3.6 13.1 3.7 92.0 8.5 31.2 10.1 64.3 9.3 $F_{\\rm int3/seg2}$ 0.73 45.6 5.5 43.1 6.3 45.4 6.2 20.2 4.9 18.5 3.7 20.4 4.0 101.9 8.4 19.0 7.2 72.5 9.5 $F_{\\rm int4/seg2}$ 0.85 42.5 5.2 55.4 6.6 31.7 6.0 29.5 5.3 17.4 3.4 10.6 3.4 112.6 9.4 42.6 9.6 77.0 10.4 $F_{\\rm int5/seg2}$ 0.97 42.6 5.7 75.1 8.2 60.8 7.5 39.7 6.1 5.2 2.8 18.8 3.7 121.3 9.6 24.1 9.7 107.4 11.6 $F_{\\rm int1/seg3}$ 0.32 61.6 6.0 46.3 6.0 26.6 5.4 20.7 4.6 5.8 3.4 9.7 3.7 135.8 9.8 35.0 9.1 69.4 10.0 $F_{\\rm int2/seg3}$ 0.44 48.3 5.6 68.5 7.2 41.3 6.2 32.5 5.1 14.7 3.7 12.8 3.7 98.3 8.7 53.3 12.3 76.7 10.0 $F_{\\rm int3/seg3}$ 0.56 43.3 5.3 66.6 7.1 20.4 5.3 35.9 5.1 15.1 3.4 16.6 3.9 112.0 9.1 39.9 10.4 94.6 10.9 $F_{\\rm int4/seg3}$ 0.68 23.7 4.3 36.9 5.9 9.9 4.3 30.7 5.1 4.7 3.0 11.9 3.1 69.6 7.7 24.9 9.9 43.4 7.8 $F_{\\rm int5/seg3}$ 0.79 39.1 5.8 41.9 6.4 21.4 6.1 20.7 4.9 13.8 3.5 13.7 3.7 75.0 8.9 26.0 11.9 56.4 10.0  For each line, the Table head reports: ion, wavelength, and maximum formation temperature.", "$F_{\\rm int {\\it i}/seg {\\it j}}$ refers to the line fluxes measured in the $i-$ th interval of the $j-$ th observing segment.", "For each time interval, the listed phase corresponds to the central time of the bin.", "Line fluxes are in $10^{-6}\\,{\\rm ph\\,s^{-1}\\,cm^{-2}}$ .", "Errors correspond to 1$\\sigma $ .", "Figure: Total flux of the cool line set versus time.", "The set of cool lines is composed of: Ne9 triplet (13.45, 13.55, and 13.70 Å), O8 Lyα\\alpha and Lyβ\\beta (16.00 and 18.98 Å), O7 resonance line (21.60 Å), and N7 Lyα\\alpha (24.78 Å)).", "Horizontal error bars represent the time-bin width.", "Dotted line marks the best-fit sinusoidal function.", "Orbital/rotational phases are computed according to the ephemeris HJD =2446998.335+2.4213459E{\\rm HJD} = 2446998.335+2.4213459\\,E defined in .", "Vertical dashed lines (dark gray) indicate quadrature and conjunction epochs, with the corresponding schematic views of the system plotted above (white and gray circles represent the primary and secondary components, respectively).", "Time intervals adopted for extracting spectra corresponding to low and high phases are marked by the vertical bands (light blue and light red for the low and high phase, respectively).Figure: Total flux of the hot line set versus time.", "The set of hot lines is composed of: Ne10 Lyα\\alpha line at 12.13 Å  and Fe17 line at 15.02 Å.", "Dotted line marks a sinusoidal function with the same period, phase, and relative amplitude obtained from the best fit of the total flux of the cool line set.", "The hot lines do not show rotational modulation, unlike the cool lines, see Fig.", ", suggesting that their variability is associated with coronal plasma variability." ], [ "Time resolved RGS spectra", "To investigate variability on short timescales, we analyzed RGS spectra gathered in time intervals of $\\sim 25$  ks (i.e.", "bins of 0.12 in rotational phase).", "Totally nine lines have fluxes detected at 1$\\sigma $ level in all the time intervals.", "These lines, and their fluxes at different time intervals, are reported in Table .", "Significant variability on the explored timescales is observed for all the listed lines.", "To check for variations in the coolest plasma components we considered lines with peak formation temperature $T_{\\rm max}<5$  MK among the lines reported in Table .", "This sample of lines, named cool lines, is composed of: the Ne9 triplet (13.45, 13.55, and 13.70 Å), O8 Ly$\\beta $This is blended with an Fe18, that is however negligible because of the $EMD$ and abundances of the X-ray emitting plasma.", "and Ly$\\alpha $ (16.00 and 18.98 Å), O7 resonance line (21.60 Å), and N7 Ly$\\alpha $ (24.78 Å).", "Among the lines reported in Table , the Ne10 and Fe17 lines stay out of the cool line sample, because of their $T_{\\rm max}$ higher than 5 MK.", "Therefore their flux likely includes significant contributions from hot plasma.", "These two lines compose the hot line sample.", "ccccccc 0pt Best fit parameters Set name Line set Pa (d) Ab (%) $t_{\\rm max}$ c (d) ($\\chi ^2_{\\rm red,1}$ )d ($\\chi ^2_{\\rm red,2}$ )e cool lines Ne9+O8+O7+N7 $1.22\\pm 0.01$ $23\\pm 2$ $1.36\\pm 0.05$ 2.91 8.93 cool line subset O8+O7+N7 $1.21\\pm 0.02$ $22\\pm 3$ $1.35\\pm 0.06$ 2.31 5.50 cool line subset O7+N7 $1.22\\pm 0.03$ $30\\pm 5$ $1.30\\pm 0.05$ 0.65 2.89 cool line subset Ne9 $1.22\\pm 0.02$ $25\\pm 4$ $1.40\\pm 0.06$ 2.33 5.12 hot lines Ne10+Fe17 $1.12\\pm 0.03$ $14\\pm 5$ $1.61\\pm 0.10$ 3.63 3.46 aPeriod.", "bAmplitude.", "cFirst epoch of maximum flux after observation start, since HJD 2455089. dReduced $\\chi ^2$ obtained with a sinusoid as best-fit function (4 free parameters).", "eReduced $\\chi ^2$ obtained with a constant as best-fit function (1 free parameters).", "To maximize the $S/N$ of the coolest plasma emission we added the measured fluxes of the cool lines for each time interval.", "This total line flux, plotted in Fig.", "REF , is variable and the observed modulation is clearly linked to the stellar rotation: the flux is higher near phases 0.0 and 0.5, i.e.", "quadrature phases, and lower near phases 0.25 and 0.75, i.e.", "conjunction phases.", "To confirm this variability pattern we fitted these observed flux variations with a sinusoid plus a constant.", "We left all the best-fit function parameters (period, phase, amplitude, and the additive constant) free to vary.", "We obtained a best-fit period of $1.22\\pm 0.01$  d, and an amplitude of $23\\pm 2\\,\\%$ with respect to the mean value (Table REF ).", "The inferred period is exactly half the rotational period of the system.", "As guessed maximum and minimum phases occur approximately at quadrature and conjunction, respectively.", "To check whether this observed modulation is effectively linked to the cool plasma emission, and not to a given line emission, we performed the same fit by separately considering the total flux obtained from different and independent cool line subsets.", "In all the inspected cases (see Table REF ) we found the same periodic variability (period, phase, amplitude).", "We checked whether this modulation is present also in the emission of hotter plasma by applying the same fit procedure to the total flux of the hot lines, Ne10 and Fe17.", "Fit results are reported in Table REF , in this case the periodic modulation is not detected.", "The observed variability is instead likely dominated by hot (coronal) plasma.", "Figure REF shows a comparison between Ne10+Fe17 line variability with modulation observed for the cool lines.", "The detected X-ray rotational modulation is also not visible in the EPIC lightcurves (A. Maggio et al., in preparation), even considering only a soft band.", "The substantial continuum contribution mostly due to the highly variable hot plasma likely masks the rotationally modulated signal.", "Hence we conclude that the observed X-ray line flux modulation is due to the high-density, cool plasma component.", "To understand the nature of the observed variability we searched for variations in the average temperature by considering ratios of lines originating from the same element.", "All the inspected ratios display significant variability, but are not correlated among themselves, and are not related to the rotational phase.", "We also searched for variations in the plasma density, probed by the $f/i$ ratio of the Ne9 triplet.", "This line ratio is approximately constant ($f/i\\approx 1$ , indicating $n_{\\rm e}\\approx 10^{12}\\,{\\rm cm^{-3}}$ ) during the entire observation, except for a lower value measured during the third interval of the second segment ($f/i=0.45\\pm 0.13$ , corresponding to $n_{\\rm }=(5.2^{+2.0}_{-1.3})\\times 10^{12}\\,{\\rm cm^{-3}}$ ), and a higher value observed during the fourth interval of the third segment ($f/i=3^{+2.5}_{-1.1}$ , corresponding to $n_{\\rm }<4\\times 10^{11}\\,{\\rm cm^{-3}}$ ).", "These variations appear to be associated with episodic events, like clumpy accretion flows, and not with a rotational modulation effect.", "lccr@$\\pm $ lr@$\\pm $ l 0pt Observed line fluxes at different phases 1cIon 1c$\\lambda $ 1c$T_{\\rm max}$ 2c$F_{Low}$ 2c$F_{High}$ 1c(Å) 1c(MK) 2c$({\\rm 10^{-6}\\,ph\\,s^{-1}\\,cm^{-2}})$ 2c$({\\rm 10^{-6}\\,ph\\,s^{-1}\\,cm^{-2}})$ Ne X 10.24 6.3 5.9 1.3 4.1 1.2 Ne IX 11.55 4.0 4.1 1.6 11.9 2.1 Ne X 12.13 6.3 46.2 3.0 50.6 3.0 Fe XXI 12.28 10.0 4.3 1.6 3.7 1.8 Ne IX 13.45 4.0 72.2 4.0 75.3 3.9 Fe XIX 13.52 10.0 16.3 3.3 9.4 3.3 Ne IX 13.55 3.5 27.3 3.1 49.4 3.5 Ne IX 13.70 4.0 29.9 2.9 41.7 3.1 Fe XVIII 14.20 7.9 5.1 1.6 3.8 1.5 Fe XVII 15.02 5.6 13.6 1.8 14.3 1.9 O VIII 15.18 3.2 4.3 1.7 7.9 1.8 O VIII 16.01 3.2 19.5 2.1 16.2 2.0 Fe XVII 16.78 5.0 11.4 2.1 10.3 1.9 Fe XVII 17.05 5.0 12.7 2.4 14.1 2.4 Fe XVII 17.10 5.0 5.8 2.6 8.9 2.6 O VIII 18.97 3.2 90.5 4.7 113.9 4.8 N VII 20.91 2.2 11.0 3.1 7.0 2.9 O VII 21.60 2.0 24.8 4.0 46.0 6.1 O VII 21.81 2.0 17.3 5.9 24.4 6.3 O VII 22.10 2.0 12.2 3.9 10.3 3.3 N VII 24.78 2.0 72.0 5.4 106.6 6.0 N VI 28.79 1.6 25.0 4.5 17.9 4.5 N VI 29.08 1.3 12.7 4.1 7.5 3.9 C VI 33.73 1.6 25.3 5.2 35.6 5.7  Line fluxes measured during the high and low phases are listed.", "Flux errors correspond to 1$\\sigma $ .", "crr 0pt Line flux ratios at different phases 1cLine flux ratio 1c$R_{Low}$ 1c$R_{High}$ Ne9 (13.45 Å) / Ne9 (11.55 Å) $18^{+11}_{-5}$ $6.4^{+1.4}_{-1.0}$ O8 L$\\alpha $ (18.97 Å) / O8 Ly$\\beta $ (16.01 Å) $4.6^{+0.6}_{-0.5}$ $7.0^{+1.1}_{-0.8}$ N7 L$\\alpha $ (24.78 Å) / N7 Ly$\\beta $ (20.91 Å) $6.5^{+2.6}_{-1.5}$ $15^{+11}_{-5}$  Ratio errors correspond to 68% confidence level.", "Figure: RGS spectra corresponding to minimum (low) and maximum (high) phases, corresponding to exposure times of 84 and 94 ks respectively.", "For clarity reasons the two spectra are slightly smoothed, and the spectrum of the high-flux phase is shifted toward longer wavelengths by 0.10.1 Å." ], [ "RGS spectra at different phases", "The total flux of the cool lines from V4046 Sgr displayed variations in time linked to the stellar rotation.", "To investigate the differences of the X-ray emitting plasma corresponding to epochs of low and high fluxes of the cool lines, we added RGS data collected at the same phases with respect to the X-ray rotational modulation.", "We extracted two RGS spectra obtained by adding all the events registered during time intervals centered on maximum and minimum times, with duration of one fourth of the observed X-ray period (integration time intervals are shown in Fig.", "REF ).", "The two resulting low and high spectra, whose exposure times are 84 and 94 ks respectively, are shown in Fig.", "REF , while the measured line fluxes, detected at 1$\\sigma $ level in the two spectra, are listed in Table REF .", "We searched for differences in the low and high spectra to investigate how the emitting plasma properties vary between these two phases.", "The two spectra display significantly different photon flux ratios of N7, O8, Ne9 lines, as reported in Table REF .", "In principle, these line ratios may vary due to changes of absorption, plasma temperature, or source optical depth.", "In Fig.", "REF we plot the measured line ratios together with the values predicted in the optically thin regime for different temperatures and different hydrogen column densities, $N_{\\rm H}$ .", "Absorption can change line ratios because, on average, lines at longer wavelengths suffer larger attenuation for increasing $N_{\\rm H}$ .", "The two N7 lines considered here are an exception, because the absorption cross section of the interstellar medium has the oxygen K-shell edge [50] located between their wavelengths, making the longer wavelength line, the Ly$\\alpha $ (24.78 Å), slightly less absorbed than the Ly$\\beta $ (20.91 Å).", "The two lines however suffer very similar absorption, making the absorption effect of little relevance in the case of the N7 Ly$\\alpha $ /Ly$\\beta $ ratio (as can be seen from the upper panel of fig.", "REF , where the curves predicted for different $N_{\\rm H}$ are very similar).", "Therefore any change in this line ratio, as that observed, is hardly explained in terms of $N_{\\rm H}$ variability.", "Instead, an $N_{\\rm H}$ decrease from the low to the high state might explain the variation of the O8 line ratios, but an opposite $N_{\\rm H}$ variation should be invoked to justify the Ne9 variability (middle and lower panels of fig.", "REF ).", "All this findings indicate that the hydrogen column density toward the source appears to be unchanged, and that line ratio variability is produced by a different mechanism.", "This conclusion is supported by the similar fluxes between low and high spectra measured for the two lines at long wavelengths (N6 at 28.8 Å  and C6 at 33.7 Å), the most affected by absorption, and it is also confirmed by the full fledged analysis of the EPIC data presented in A. Maggio et al., in preparation, where $N_{\\rm H}$ is found to vary by only a factor 2 over the whole observation around a mean value of $3\\times 10^{20}\\,{\\rm cm^{-2}}$ (i.e.", "$\\log N_{\\rm H} = 20.5$ ).", "Figure: Lyα\\alpha to Lyβ\\beta photon flux ratio for the N7 and O8 H-like ions and 13.4513.45 Å  to 11.5511.55 Å  for the Ne9 He-like ion, vs plasma temperature.", "Horizontal bands indicate values measured during high (red) and low (blue) phases.", "Black, dark gray, and light gray lines represent predicted values for different absorptions N H N_{\\rm H}, with labels reporting the corresponding logN H \\log N_{\\rm H} value, and with thicker lines marking curve portions compatible with the observed ratios.The three explored ratios depend also on temperature, because of the different energy of the upper levels of the two electronic transitions considered in each ratio.", "In this respect, the three ratios do not vary consistently.", "In fact, a temperature decrease from the low state to the high state could explain the increasing N7 and O8 line ratios, but not the variation Ne9 lines.", "The derivation of the plasma model (see A. Maggio et al., in preparation) is beyond the scope of this work, but we anticipate here that the $EMD$ does not appear to vary enough between the two phases to justify the observed variations of the line ratios.", "Moreover the average plasma temperature ($\\log T \\sim 6.6-6.7$ ), together with the measured $N_{\\rm H}$ , indicates that, in some phases, line ratios are not compatible with the optically thin limit, irrespective of the nature of their variability.", "Optical depth effects can change line ratios because each line optical depth is directly proportional to the oscillator strength of the transition [1].", "Therefore, if optically thin emission does not apply, transitions with very different oscillator strength may suffer different attenuation/enhancement, with stronger effects occurring in lines with higher oscillator strengths.", "In the inspected ratios the lines with higher oscillator strength are the $Ly\\alpha $ line of N7 and O8, and the $13.45$  Å line of Ne9 [46].", "Non negligible optical depth is expected in strong X-ray resonance lines produced from shock-heated plasma in CTTS [4].", "The observed variable ratios might indicate that some lines are affected by a changing optical depth.", "Since the expected attenuation/enhancement with respect to optically thin emission depends on the source geometry and viewing angle, stellar rotation can produce periodic changes in the line opacity, and hence in the observed ratios.", "However, once again the three ratios do not vary in the same direction, with N7 and O8 ratios being higher in the low state, whereas the ratio of the slightly hotter Ne9 lines results higher in the high phase.", "Summarizing we stress the significant variations observed in line ratios between low and high phases.", "The origin of these variations remains unclear.", "If these variations were due to changes in plasma temperature or absorption, a coherent behavior would be expected for the three ratios, and this is not the case.", "Opacity effects instead can operate in a more complex way, provoking both line enhancements or reductions, depending on the source geometry and viewing angle.", "This hypothesis is therefore the most intriguing, especially considering that in some phases line ratios are discrepant from the value expected in the optically-thin limit." ], [ "Discussion", "The main result of the time resolved spectral analysis of the X-ray emitting plasma from V4046 Sgr (§ ) is that the high-density plasma component at $3-4$  MK is rotationally modulated with a period of half the system orbital period, with maximum and minimum phases occurring at quadrature and conjunction epochs, respectively.", "The observed X-ray rotational modulation indicates that this high-density plasma component is not symmetrically distributed with respect to the stellar rotational axes.", "We also found that strong emission lines from this plasma component provide some indications of non negligible optical depth effects, and that the periodic modulation appears to be associated with variations in the source optical depth, as evidenced by the significant variations in line ratios sensitive to optical depth observed between low and high phases.", "The strongest X-ray emission lines, produced by shock-heated material in CTTS, are expected to have non-negligible optical depth due to the high density and typical size of the post-shock region [4].", "Moreover the optical depth should vary if the viewing geometry of the post shock region changes.", "Hints of non-negligible optical depth observed in the strongest X-ray lines of V4046 Sgr indicate that the high-density plasma is mostly concentrated in a compact portion of the stellar surface, as predicted for the post-shock material.", "Moreover, the variability of the optical depth can be naturally explained with the changing viewing geometry of the volume occupied by the high-density plasma during stellar rotation.", "This scenario requires plasma confinement by the stellar magnetic fields.", "We observed an X-ray period of half the system orbital period, as already observed for accretion indicators [49], [34] and X-ray emission [18] from some CTTS.", "That could be explained, in the case of V4046 Sgr, by different scenarios.", "If the X-ray emitting plasma is located only in one of the two system components, then a period of half the rotational period is observed when there are two accretion-shock regions on the stellar surface at opposite longitudes, or there is only one accretion-shock region and the maximum X-ray flux is observed when the base of the accretion stream is viewed sideways [2].", "Considering the system symmetry and the accretion geometry previously suggested by [44], it is conceivable that both components possess similar amounts of high-density cool plasma.", "In this scenario the half period can be naturally explained assuming that the location on each stellar surface of this plasma, compact and not azimuthally symmetric with respect to each stellar rotation axis, is symmetric for $180^{\\circ }$ rotations with respect to the binary rotation axis.", "The simultaneous optical monitoring campaign indicated that the two components have similar accretion rates, validating the assumption that the two components possess similar amounts of high-density plasma.", "However, the optical accretion spots, probed by Ca2 IRT, did not show rotational modulation [11].", "Therefore accretion regions emitting Ca2 should be symmetrically distributed with respect to the stellar poles.", "This scenario, different from that obtained from the X-ray data, could be reconciled considering that X-rays are likely produced only by a fraction of the entire accretion-shock region [42].", "In conclusion, our XMM-Newton/RGS data of the V4046 Sgr close binary system have shown for the first time the rotational modulation of X-ray lines characteristic of a cool, high-density plasma corotating with the stars.", "This strongly support the accretion-driven X-ray emission scenario, in which the high-density cool plasma of CTTS is material heated in the accretion shock.", "It moreover suggests that the accretion flow is channeled by magnetic field lines anchored on the stars, along small magnetic tubes.", "This is consistent with the general framework of magnetic accretion, but brings new insights into the accretion mechanism in close binary systems of CTTS.", "This work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.", "C.A., A.M., and F.D.", "acknowledge financial contribution from the agreement ASI-INAF I/009/10/0." ] ]

[ [ "The processes $e^{+}e^{-} \\to \\pi\\pi(\\pi')$ in the extended NJL model" ], [ "Abstract The process $e^{+}e^{-} \\to \\pi^{+}\\pi^{-}$ is described in the framework of the extended NJL model.", "Intermediate vector mesons $\\rho^0(770)$, $\\omega(782)$ and \\rho'(1450)$ are taken into account.", "Our results are in satisfactory agreement with experimental data.", "The prediction for the process $e^{+}e^{-} \\to \\pi\\pi'(1300)$ is given.", "Here the main contribution is given by the diagram with intermediate $\\rho'(1450)$ meson." ], [ "Introduction", "Recently, the processes $e^{+}e^{-} \\rightarrow \\pi ^0 \\gamma ,\\,\\, \\pi ^{\\prime } \\gamma ,\\,\\, \\pi ^0 \\omega ,\\,\\, \\pi ^0 \\rho ^0\\,, \\pi ^0 \\pi ^0 \\gamma $ have been described in the framework of the extended NJL model [1], [2], [3], [4].", "Intermediate vector mesons in the ground and excited states were took into account.", "In all these works satisfactory agreement with experimental data was obtained without using any arbitrary parameters.", "In this work the same method is used for description of the processes $e^{+}e^{-} \\rightarrow \\pi \\pi (\\pi ^{\\prime })$ .", "Now this process is thoroughly investigated from both experimental (see [5] and other references in this work) and theoretical [6], [7], [8], [9], [10], [11] points of view.", "However, in all these works it is necessary to use the number of additional parameters.", "The extended NJL model allows us to describe these processes without attraction of additional parameters.", "Let us note that for the description of the processes $e^{+}e^{-} \\rightarrow \\pi ^{+}\\pi ^{-}$ with intermediate photon and vector mesons in the ground state it is sufficient to use the standard NJL model [12], [13], [14], [15], [17], [18], [19], [20], [16].", "On the other hand, for the description of the amplitude of this process with the intermediate $\\rho ^{\\prime }(1450)$ meson the extended NJL model [21], [22], [23], [24], [20] should be used." ], [ "Lagrangian and process amplitudes", "The amplitude of the process $e^{+}e^{-} \\rightarrow \\pi ^{+}\\pi ^{-}$ is described by the diagrams given on Figs.", "REF and REF .", "Figure: Contact interaction of the photon with a pion pair through the photon propagatorFigure: Interaction with intermediate vector mesonsFor the description of the first three diagrams with intermediate $\\gamma $ , $\\rho $ , and $\\omega $ mesons in the ground state we need the part of the standard NJL Lagrangian which describes interactions of photons, pions, and the ground states of vector mesons with quarks $&& \\Delta {\\mathcal {L}}_{1} = \\bar{q}\\biggl [i\\hat{\\partial }- m+ \\frac{e}{2}\\biggl (\\tau _3+\\frac{1}{3}I\\biggr )\\hat{A}+ig_{\\pi }\\gamma _5\\tau _{\\pm }\\pi ^{\\pm }+ \\frac{g_\\rho }{2}\\gamma _\\mu \\left(I\\hat{\\omega } + \\tau _3\\hat{\\rho }^0\\right) \\biggr ]q,$ where $\\bar{q}=(\\bar{u},\\bar{d})$ with $u$ and $d$ quark fields; $m=diag(m_u,m_d)$ , $m_u=280$  MeV is the constituent quark mass, $m_d - m_u \\approx 3.7$  MeV as will be shown below; $e$ is the electron charge; $A$ , $\\pi ^{\\pm }$ , $\\omega $ and $\\rho ^0$ are the photon, pion, $\\omega $ and $\\rho $ meson fields, respectively; $g_\\pi $ is the pion coupling constant, $g_\\pi =m_u/f_\\pi $ , where $f_\\pi =93$  MeV is the pion decay constant; $g_\\rho $ is the vector meson coupling constant, $g_\\rho \\approx 6.14$ corresponds to the standard relation $g_\\rho ^2/(4\\pi )\\approx 3$ ; $\\tau _{\\pm } = (\\tau _1 \\mp i\\tau _2)/\\sqrt{2}$ , $I=diag(1,1)$ and $\\tau _{1,2,3}$ are Pauli matrices.", "For description of the radial excited mesons interactions we use the extended version of the NJL Lagrangian [22], [23], [3]: $&& \\Delta {\\mathcal {L}}_2^{\\mathrm {int}} =\\bar{q}(k^{\\prime })\\biggl \\lbrace A_\\pi \\tau _{\\pm }\\gamma _5\\pi (p)- A_{\\pi ^{\\prime }} \\gamma _5\\tau _{\\pm }\\pi ^{\\prime }(p)+ A_{\\omega ,\\rho } \\left(\\tau _3{\\hat{\\rho }}(p)+I\\hat{\\omega }(p)\\right)- A_{\\rho ^{\\prime }} \\tau _3{\\hat{\\rho }^{\\prime }}(p)\\biggr \\rbrace q(k),\\\\ \\nonumber && p = k-k^{\\prime },\\\\ \\nonumber && A_\\pi = g_{\\pi _1}\\frac{\\sin (\\alpha +\\alpha _0)}{\\sin (2\\alpha _0)}+g_{\\pi _2}f({k^\\bot }^2)\\frac{\\sin (\\alpha -\\alpha _0)}{\\sin (2\\alpha _0)},\\nonumber \\\\&& A_{\\pi ^{\\prime }} = g_{\\pi _1}\\frac{\\cos (\\alpha +\\alpha _0)}{\\sin (2\\alpha _0)}+g_{\\pi _2}f({k^\\bot }^2)\\frac{\\cos (\\alpha -\\alpha _0)}{\\sin (2\\alpha _0)},\\nonumber \\\\&& A_{\\omega ,\\rho } = g_{\\rho _1}\\frac{\\sin (\\beta +\\beta _0)}{\\sin (2\\beta _0)}+g_{\\rho _2}f({k^\\bot }^2)\\frac{\\sin (\\beta -\\beta _0)}{\\sin (2\\beta _0)},\\nonumber \\\\ \\nonumber && A_{\\rho ^{\\prime }} = g_{\\rho _1}\\frac{\\cos (\\beta +\\beta _0)}{\\sin (2\\beta _0)}+g_{\\rho _2}f({k^\\bot }^2)\\frac{\\cos (\\beta -\\beta _0)}{\\sin (2\\beta _0)}.$ The radially-excited states were introduced in the NJL model with the help of the form factor in the quark-meson interaction: $f({k^\\bot }^2) &=& (1-d |{k^\\bot }^2|) \\Theta (\\Lambda ^2-|{k^\\bot }^2|),\\nonumber \\\\{k^\\bot } &=& k - \\frac{(kp)p}{p^2},\\qquad d = 1.788\\ {\\mathrm {GeV}}^{-2},$ where $k$ and $p$ are the quark and meson momenta, respectively.", "The cut-off parameter $\\Lambda =1.03$  GeV.", "The coupling constants $g_{\\rho _1}=g_\\rho $ and $g_{\\pi _1}=g_\\pi $ are the same as in the standard NJL version.", "The constants $g_{\\rho _2}=10.56$ and $g_{\\pi _2} = g_{\\rho _2}/\\sqrt{6}$ , and the mixing angles $\\alpha _0=58.39^\\circ $ , $\\alpha =58.70^\\circ $ , $\\beta _0=61.44^\\circ $ , and $\\beta =79.85^\\circ $ were defined in refs.", "[23], [3].", "The amplitude $e^{+}e^{-} \\rightarrow \\pi ^{+}\\pi ^{-}$ has the form $T = \\bar{e}\\gamma _\\mu e \\frac{4\\pi \\alpha _e}{s} \\left(B_{\\gamma \\rho } + B_\\omega + B_{\\rho ^{\\prime }}\\right) f_{a_1}(s) (p_{\\pi ^{+}}^\\mu - p_{\\pi ^{-}}^\\mu ) \\pi ^{+} \\pi ^{-}\\, ,$ where $\\alpha _e = e^2/4\\pi \\approx 1/137$ , $s = (p_{e^{+}} + p_{e^{-}})^2$ , and $f_{a_1}(p^2) = Z + (1 - Z) + \\left(\\frac{p^2 - m_\\pi ^2}{(g_\\rho F_\\pi )^2}\\right)\\left(1 - \\frac{1}{Z}\\right) = 1 + \\left(\\frac{p^2 - m_\\pi ^2}{(g_\\rho F_\\pi )^2}\\right)\\left(1 - \\frac{1}{Z}\\right)\\, ,$ where $Z = (1 - 6m_u^2/m_{a_1}^2)^{-1}$ is the additional renormalizing factor pion fields that appeared after the inclusion of $a_1$ – $\\pi $ transitions.", "This function describes the creation of pions at the ends of the triangle quark diagram with taking into account the possibility of creation of these pions through the intermediate axial-vector $a_1(1260)$ meson.", "The first term of this amplitude corresponds to the triangle diagram without $a_1$ – $\\pi $ transitions, the second term corresponds to diagram with $a_1$ – $\\pi $ transition on the one of the pion lines and the third term corresponds to the diagram with transitions on both pion linesLet us note that $f_{a_1}(p^2)$ has the value which is close to the factor introduced in [11]..", "Figure: Triangle diagrams with a 1 a_1 – π\\pi transitionsThe transition $\\gamma $ – $\\rho $ takes the form (see [14]) $\\frac{e}{g_\\rho }(g^{\\nu \\nu ^{\\prime }}q^2 - q^{\\nu } q^{\\nu ^{\\prime }})\\,.$ Thus, one can write $B_{\\rho \\gamma }$ contribution in the form $B_{\\gamma \\rho } = 1 + \\frac{s}{m_\\rho ^2 - s - i\\sqrt{s}\\Gamma _\\rho (s)} = \\frac{1 - i\\sqrt{s}\\Gamma _\\rho (s)/m_\\rho ^2}{m_\\rho ^2 - s - i\\sqrt{s}\\Gamma _\\rho (s)}m_\\rho ^2\\, .$ Let us note that this expression is close to vector meson dominance model.", "The term describing the transition $\\gamma $ – $\\omega $ is equal of the term $\\gamma $ – $\\rho $ multiplied on the factor 1/3 [14], [3].", "The $\\omega \\rightarrow \\pi \\pi $ process was described in [14] $C(m_\\rho ^2)\\omega _\\mu (p_{\\pi ^{+}}^\\mu - p_{\\pi ^{-}}^\\mu )\\,,$ where $C(s) = C_1(s) + C_2(s)$ .", "$C_1$ describes the amplitude of transition $\\omega \\rightarrow \\rho $ due to the difference of two quark loops.", "The first of them contains only $u$ quark and second contains only $d$ quark.", "Using the last experimental data for the decay $\\omega \\rightarrow \\pi \\pi $  [29] we obtain $m_d - m_u \\approx 3.7$ MeV.", "This difference allows us to describe not only the decay $\\omega \\rightarrow \\pi \\pi $ , but the mass difference of charge and neutral pion and kaon (see [14]) and obtain the interference $\\omega $ – $\\rho $ in process $e^{+}e^{-} \\rightarrow \\pi ^{+}\\pi ^{-}$ in good agreement with experimental data $C_1(s) = \\frac{8(\\pi \\alpha _\\rho )^{3/2}m_\\omega ^2}{3(m_\\omega ^2 - s - i\\sqrt{s}\\Gamma _\\rho (s))}\\frac{3}{(4\\pi )^2}\\log \\left(\\frac{m_d}{m_u}\\right)^2 \\, ,$ and $C_2$ describes the amplitude $\\omega \\rightarrow \\gamma \\rightarrow \\rho $ $C_2(s) = -\\sqrt{\\frac{\\pi }{\\alpha _\\rho }}\\frac{2\\alpha s}{3(m_\\omega ^2 - s - i\\sqrt{s}\\Gamma _\\rho (s))} \\, .$ Then for the part of the amplitude with intermediate $\\omega $ meson we get $B_\\omega = \\frac{C(s)}{3g_\\rho } \\frac{s}{m_\\omega ^2 - s - i\\sqrt{s}\\Gamma _\\omega (s)}\\, .$ The last part of the amplitude contains intermediate $\\rho ^{\\prime }(1450)$ meson.", "The transition $\\gamma $ – $\\rho ^{\\prime }$ has the form $C_{\\gamma \\rho ^{\\prime }} \\frac{e}{g_\\rho }(g^{\\nu \\nu ^{\\prime }}q^2 - q^{\\nu } q^{\\nu ^{\\prime }})\\, ,$ $C_{\\gamma \\rho ^{\\prime }} = -\\left(\\frac{\\cos (\\beta +\\beta _0)}{\\sin (2\\beta _0)} + \\Gamma \\frac{\\cos (\\beta - \\beta _0)}{\\sin {(2\\beta _0)}}\\right)\\, ,$ $\\Gamma = \\frac{I^f_2}{\\sqrt{I_2 I^{ff}_2}} = 0.54$ Figure: Comparison of experimental results for e + e - →π + π - e^{+}e^{-} \\rightarrow \\pi ^{+}\\pi ^{-} with the NJL predictionThe vertex $\\rho ^{\\prime } \\pi \\pi $ is proportional to $C_{\\rho ^{\\prime }\\pi \\pi } = -\\left(\\frac{\\cos (\\beta + \\beta _0)}{\\sin (2\\beta _0)}g_{\\rho _1} + \\frac{\\cos (\\beta - \\beta _0)}{\\sin (2\\beta _0)}\\frac{I^f_2}{I_2}g_{\\rho _2}\\right) = 1.68.$ Unfortunatly, our model can not describe relative phase between $\\rho (770)$ and $\\rho ^{\\prime }(1450)$ in $e^{+}e^{-} \\rightarrow \\pi \\pi (\\pi ^{\\prime })$ .", "Thus, we should get phase from $e^{+}e^{-}$ annihilation and $\\tau $ decays experiments: $B_{\\rho ^{\\prime }} \\rightarrow e^{i\\pi }B_{\\rho ^{\\prime }}$ Thus, the $\\rho ^{\\prime }$ meson contribution reads $B_{\\rho ^{\\prime }} = e^{i\\pi }\\frac{C_{\\gamma \\rho ^{\\prime }} C_{\\rho ^{\\prime }\\pi \\pi }}{g_\\rho }\\frac{s}{m_{\\rho ^{\\prime }}^2 - s - i\\sqrt{s}\\Gamma _{\\rho ^{\\prime }}(s)}\\, ,$ where the running width $\\Gamma _{\\rho ^{\\prime }}$ reads [3] $\\Gamma _{\\rho ^{\\prime }}(s) &=& \\Theta (2m_\\pi - \\sqrt{s}) \\Gamma _{\\rho ^{\\prime }\\rightarrow 2\\pi }\\\\ &+& \\Theta (\\sqrt{s} - 2m_\\pi )(\\Gamma _{\\rho ^{\\prime }\\rightarrow 2\\pi } + \\Gamma _{\\rho ^{\\prime }\\rightarrow \\omega \\pi }\\frac{\\sqrt{s}-2m_\\pi }{m_\\omega -m_\\pi })\\Theta (m_\\omega + m_\\pi - \\sqrt{s})\\nonumber \\\\ &+& \\Theta (m_{\\rho ^{\\prime }} - \\sqrt{s})\\Theta (\\sqrt{s} - m_\\omega - m_\\pi ) \\cdot \\nonumber \\\\ && \\left(\\Gamma _{\\rho ^{\\prime }\\rightarrow 2\\pi } + \\Gamma _{\\rho ^{\\prime }\\rightarrow \\omega \\pi }+ (\\Gamma _{\\rho ^{\\prime }} - \\Gamma _{\\rho ^{\\prime }\\rightarrow 2\\pi } -\\Gamma _{\\rho ^{\\prime }\\rightarrow \\omega \\pi })\\nonumber \\frac{\\sqrt{s}- m_\\omega - m_\\pi }{m_{\\rho ^{\\prime }} - m_\\omega - m_\\pi }\\right)\\nonumber \\\\ &+& \\Theta (\\sqrt{s} - m_{\\rho ^{\\prime }})\\Gamma _{\\rho ^{\\prime }}\\,.\\nonumber $ The values $\\Gamma (\\rho ^{\\prime }\\rightarrow 2\\pi )=22$  MeV and $\\Gamma (\\rho ^{\\prime }\\rightarrow \\omega \\pi ^0)=75$  MeV were calculated in [23].", "The $\\Gamma _{\\rho ^{\\prime }} = 340$ MeV was taken the value of lower boundary from PDG [29].", "For the total cross-section we get $\\sigma (s) = \\frac{\\alpha ^2 \\pi }{12s} f_{a_1}^2(s)(1 - 4m_\\pi ^2/s)^{3/2} \\left|B_{\\rho \\gamma } + B_\\omega + B_{\\rho ^{\\prime }}\\right|^2\\,.$ The total cross-section is defined by $\\rho $ и $\\omega $ mesons, the $\\rho ^{\\prime }$ meson contributes only to the differential cross-section.", "Let us note that for description of the $e^{+}e^{-} \\rightarrow \\pi ^{\\prime }\\pi $ not only the intermediate state with $\\rho ^{\\prime }(1450)$ but the intermediate state $\\rho ^{\\prime \\prime }(1700)$ can play the important role.", "However, here we take into account only rift radial excitation of $\\rho $ meson.", "Therefore, we can pretend only to quality description of this process.", "The corresponding total cross-section takes the form $\\sigma (s) = \\frac{\\alpha ^2 \\pi }{12s^2}\\Lambda ^{3/2}(s,m_{\\pi ^{\\prime }}^2,m_{\\pi }^2) \\left|B_{\\rho \\gamma }^{\\pi \\pi ^{\\prime }} + B_{\\rho ^{\\prime }}^{\\pi \\pi ^{\\prime }}\\right|^2\\,,$ $B_{\\rho \\gamma }^{\\pi \\pi ^{\\prime }} = \\frac{C_{\\rho \\pi \\pi ^{\\prime }}}{g_\\rho }\\left(1 + \\frac{s}{m_\\rho ^2 - s - im_\\rho \\Gamma _\\rho }\\right) = \\frac{C_{\\rho \\pi \\pi ^{\\prime }}}{g_\\rho }\\frac{1 - i\\Gamma _\\rho /m_\\rho }{m_\\rho ^2 - s - im_\\rho \\Gamma _\\rho }m_\\rho ^2,$ $B_{\\rho ^{\\prime }}^{\\pi \\pi ^{\\prime }} = e^{i\\pi }\\frac{C_{\\gamma \\rho ^{\\prime }} C_{\\rho ^{\\prime }\\pi \\pi ^{\\prime }}}{g_\\rho }\\frac{s}{m_{\\rho ^{\\prime }}^2 - s - im_{\\rho ^{\\prime }}\\Gamma _{\\rho ^{\\prime }}},$ where $\\Lambda (s,m_{\\pi ^{\\prime }}^2,m_{\\pi }^2) = (s - m_{\\pi ^{\\prime }}^2 - m_{\\pi ^{\\prime }}^2)^2 - 4m_{\\pi ^{\\prime }}^2m_{\\pi }^2$ , $m_{\\pi ^{\\prime }} = 1300$  MeV is mass of $\\pi ^{\\prime }$ meson [29], $C_{\\rho \\pi \\pi ^{\\prime }}$ and $C_{\\rho ^{\\prime }\\pi \\pi ^{\\prime }}$ is defined in a similar way as $C_{\\rho ^{\\prime }\\pi \\pi }$ with the use of the Lagrangian (REF ).", "Figure: NJL prediction for e + e - →ππ ' e^{+}e^{-} \\rightarrow \\pi \\pi ^{\\prime }.", "The solid line is the total cross-section, the dashed line is the ρ ' (1450)\\rho ^{\\prime }(1450) contribution onlyNow for the processes $e^{+}e^{-} \\rightarrow 4\\pi $ we have the following experimental data: $\\sigma (e^{+}e^{-}\\rightarrow \\pi ^{+}\\pi ^{-}\\pi ^{0}\\pi ^{0}) \\approx 10$ nb at energy $1.5$ GeV; $\\sigma (e^{+}e^{-}\\rightarrow \\pi ^{+}\\pi ^{-}\\pi ^{+}\\pi ^{-}) \\approx 30$ nb at energy $1.5$ GeV.", "Therefore, we can see that our result does not contradict these data and can give a noticeable contribution to these processes." ], [ "Conclusions", "Let us note than our version of the NJL model allows us to describe not only meson production in the $e^{+}e^{-}$ processes but branching of the decay of tau lepton into mesons.", "Indeed, in the works [26], [27], [28] decays $\\tau $ into $3\\pi \\nu $ , $\\pi \\gamma \\nu $ and $\\pi \\pi \\nu $ were described in satisfactory agreement with experimental data.", "The calculation of the last process $\\tau \\rightarrow \\pi \\pi \\nu $ is very close to process $e^{+}e^{-} \\rightarrow \\pi ^{+}\\pi ^{+}$ which was considered here.", "In [28] we obtained satisfactory agreement of both branching and differential width with experimental data.", "With the help of the method use here we can obtain also a qualitative prediction for branching of the process $\\tau \\rightarrow \\pi \\pi ^{\\prime }(1300)\\nu $ .", "This value approximately equals $0.2$ %, which does not contradict modern experimental data regarding the decays $\\tau \\rightarrow 4\\pi \\nu $ .", "This prediction can be useful result for future experimental measurement." ], [ "Acknowledgments", "We are grateful to E. A. Kuraev and A. B.", "Arbuzov for useful discussions.", "This work was supported by RFBR grant 10-02-01295-a." ] ]

[ [ "A Combinatorial Proof of the Enumeration of Alternating Permutations\n with Given Peak Set" ], [ "Abstract Using the correspondence between a cycle up-down permutation and a pair of matchings, we give a combinatorial proof of the enumeration of alternating permutations according to the given peak set." ], [ "Introduction", "Let ${\\mathfrak {S}}_n$ denote the symmetric group of all permutations of $[n]:= \\lbrace 1, 2, \\ldots , n\\rbrace $ .", "An alternating permutation on $[n]$ is defined to be a permutation ${\\sigma }={\\sigma }_1 {\\sigma }_2 \\cdots {\\sigma }_n$ $\\in {\\mathfrak {S}}_n$ satisfying $ {\\sigma }_1>{\\sigma }_2<{\\sigma }_3>{\\sigma }_4<\\cdots $ , etc., in an alternating way.", "Similarly, ${\\sigma }$ is reverse alternating if $ {\\sigma }_1<{\\sigma }_2>{\\sigma }_3<{\\sigma }_4>\\cdots $ , which is also referred as an up-down permutation.", "Denote by ${\\mathcal {E}}_n$ the set of alternating permutations on $[n]$ , and further let $E_n=|{\\mathcal {E}}_n|$ .", "Note that $E_n$ is called Euler number, and was shown by André [1], [2] to satisfy $\\sum _{n\\ge 0}E_n \\frac{x^n}{n!", "}=\\sec x+\\tan x.$ The reverse map ${\\sigma }\\mapsto {\\sigma }^{r}$ defined by ${\\sigma }^{r}_i={\\sigma }_{n+1-i}$ on ${\\mathfrak {S}}_n$ shows that $E_n$ is also the number of up-down permutations in ${\\mathfrak {S}}_n$ .", "Recently, Elizalde and Deutsch [4] introduced the concept of cycle up-down permutations.", "A cycle is said to be up-down if, when written in standard cycle form, say $(a_1,a_2, a_3,\\ldots )$ , one has $a_1<a_2> a_3<a_4>\\cdots $ , and a permutation ${\\sigma }$ is a cycle up-down permutation if it is a product of up-down cycles.", "They prove both bijectively and analytically that Proposition 1 ([4], Lemma 2.2) The number of cycle up-down permutations of $[2k]$ all of whose cycles are even is $E_{2k}$ .", "For out purpose, let us briefly recall the bijection $\\tau $ developed in [4] to prove the above proposition.", "Given ${\\sigma }={\\sigma }_1 {\\sigma }_2 \\cdots {\\sigma }_{2k}$ $\\in {\\mathcal {E}}_{2k}$ , let ${\\sigma }_{i_1} > {\\sigma }_{i_2} >\\cdots >{\\sigma }_{i_m}$ be its left to right minima, the corresponding cycle up-down permutation $\\tau ({\\sigma })$ with only even cycles is defined by $ \\tau ({\\sigma }) = ({\\sigma }_{i_1},\\ldots ,{\\sigma }_{i_2-1})({\\sigma }_{i_2},\\ldots ,{\\sigma }_{i_3-1}) \\cdots ({\\sigma }_{i_m}, \\ldots ,{\\sigma }_{2k}).$ The element ${\\sigma }_i(1\\le i \\le n)$ is called a peak if ${\\sigma }_{i-1}<{\\sigma }_i>{\\sigma }_{i+1}$ , where we set ${\\sigma }_0=0$ and ${\\sigma }_{n+1}=0$ , and the peak set of ${\\sigma }$ are the elements of peaks in ${\\sigma }$ .", "For other definitions of peaks see [5], [6].", "For $n=2k$ even, and for any sequence $2\\le i_1<i_2<\\cdots <i_k=n$ , let $S_k(i_1,i_2,\\ldots ,i_k)$ denote the set of permutations in ${\\mathcal {E}}_{2k}$ with peak set equals to $\\lbrace i_1,i_2,\\ldots ,i_k\\rbrace $ , and let $s_k(i_1,i_2,\\ldots ,i_k)=|S_k(i_1,i_2,\\ldots ,i_k)|$ .", "For $n=2k+1$ odd, and for any sequence $2\\le i_1<i_2<\\cdots <i_k<i_{k+1}=n$ , let $T_k(i_1,i_2,\\ldots ,i_{k+1})$ denote the set of permutations in ${\\mathcal {E}}_{2k+1}$ with peak set equals to $\\lbrace i_1,i_2,\\ldots ,i_{k+1}\\rbrace $ , and let $t_k(i_1,i_2,\\ldots ,i_{k+1})=|T_k(i_1,i_2,\\ldots ,i_{k+1})|$ .", "Using induction on $n$ , Strehl [7] derived the following enumeration formula for the numbers $s_k$ and $t_k$ .", "Theorem 2 ([7]) $s_k(i_1,i_2,\\ldots ,i_k)&= \\prod _{1\\le j \\le k-1}(i_j-2j+1)^2, \\\\ t_k(i_1,i_2,\\ldots ,i_{k+1})&=\\prod _{1\\le j \\le k}(i_j-2j+2)(i_j-2j+1).$ In this note we will give a combinatorial proof for the identities (REF ) and ()." ], [ "Combinatorial Proof of Theorem ", "In order to give a combinatorial proof of Theorem REF , we begin by deducing a formula for the number of matchings on $[2k]$ with a given closer set.", "Recall that a matching $\\pi $ of $[2k]$ is a partition of the set $[2k]$ with the property that each block has exactly two elements.", "It can be represented as a graph with vertices $1, 2, \\ldots , 2k$ drawn on a horizontal line in increasing order, where two vertices $i$ and $j$ are connected by an edge if and only if $\\lbrace i,j\\rbrace $ (with $i < j$ ) is a block, and we say that $i$ is the opener and $j$ is the closer of this edge.", "Since the graph is undirected, the edge can be denoted by $(i,j)$ or $(j,i)$ with no difference.", "The set of all the closers (resp.", "openers) of a matching is called its closer set (resp.", "opener set).", "It is known that the number of matchings on $[2k]$ equals $(2k-1)!", "!$ .", "If in addition, the $k$ closers among these $2k$ vertices are given, we have Lemma 3 The number of matchings on $[2k]$ with closer set $\\lbrace i_1<i_2<\\cdots <i_k\\rbrace $ equals $\\prod _{1\\le j \\le k-1}(i_j-2j+1).$ Proof.", "Once the closer set is given, the opener set is accordingly known, and a matching on $[2k]$ can be constructed step by step by determining the $k$ edges with these closers $\\lbrace i_1,i_2,\\cdots ,i_k\\rbrace $ from left to right.", "For the closer $i_1$ , there are $i_1-1$ openers before it with labels $1,2,\\ldots ,i_1-1$ , so there are $i_1-1$ choices for the first closer to form an edge.", "Generally, for $2\\le j \\le k-1$ , there are $j-1$ closers before the closer $i_j$ , thus $j-1$ openers have been chosen so far by these closers, so the $j$ -th closer has $i_j-1-2(j-1)$ openers to be chosen to generate a new edge.", "For the last closer $i_k$ , there is only one opener left to form an edge with it.", "By considering all the $k$ closers, we see that there are $\\prod _{1\\le j \\le k-1}(i_j-2j+1)$ possibilities to form a matching with this given closer set.", "As in [3], we can also represent a permutation of $[n]$ as a graph on $n$ vertices labeled $1, 2, \\ldots , n$ , with an edge from $i$ to $j$ if and only if ${\\sigma }_i = j$ .", "Explicitly, put the $n$ vertices on a horizontal line, ordered from left to right by increasing label and we draw an edge from $i$ to ${\\sigma }_i$ above the line if $i\\le {\\sigma }_i$ and under the line otherwise.", "Using this drawing, cycle up-down permutations having only even cycles correspond precisely to a pair of independent matchings whose vertices agree on openers and closers.", "By convention, we refer the matching with edges above the line as above matching and the matching with edges below the line as below matching.", "By this representation of permutations, we are now in the position to give a combinatorial proof of Theorem REF .", "Combinatorial Proof of (REF ).", "Given an alternating permutation ${\\sigma }\\in {\\mathcal {E}}_{2k}$ with peak set $\\lbrace i_1,i_2,\\ldots ,i_k\\rbrace $ , the reverse permutation ${\\sigma }^{r}$ is an up-down permutation on $[2k]$ , and ${\\sigma }^{\\prime }=\\tau ({\\sigma }^{r})$ is a cycle up-down permutation with only even cycles.", "Let ${\\mathcal {G}}({\\sigma }^{\\prime })$ be the corresponding graph of the permutation ${\\sigma }^{\\prime }$ , it is easy to check that the closer set of the pair of matchings is $\\lbrace i_1,i_2,\\ldots ,i_k\\rbrace $ .", "On the other hand, given a pair of matchings with closer set $\\lbrace i_1,i_2,\\ldots ,i_{k}\\rbrace $ , we can recover an alternating permutation on $[2k]$ with peak set $\\lbrace i_1,i_2,\\ldots ,i_{k}\\rbrace $ by reversing the above procedure.", "By Lemma REF , the number of above matchings and the number of below matchings with closer set $\\lbrace i_1,i_2,\\ldots ,i_{k}\\rbrace $ are both $\\prod _{1\\le j \\le k-1}(i_j-2j+1)$ , thus identity (REF ) follows.", "For example, if ${\\sigma }=5\\,3\\,8\\,1\\,4\\,2\\,7\\,6 \\in {\\mathcal {E}}_8$ , then ${\\sigma }^{r}=6\\,7\\,2\\,4\\,1\\,8\\,3\\,5$ and $\\tau ({\\sigma }^{r})=(6,7)\\,(2,4)\\,(1,8,3,5)$ , the graph ${\\mathcal {G}}(\\tau ({\\sigma }^{r}))$ is depicted in Figure REF .", "Figure: The graph of the cycle up-down permutation (6,7) (2,4) (1,8,3,5).Combinatorial Proof of ().", "Given an alternating permutation ${\\sigma }\\in {\\mathcal {E}}_{2k+1}$ with peak set $\\lbrace i_1,i_2,\\ldots ,i_{k+1}\\rbrace $ , let $\\overline{{\\sigma }}:={\\sigma }\\,0$ be a permutation on the set $\\lbrace 0,1,2,\\ldots 2k,2k+1\\rbrace $ obtained by appending a “0\" after the last element of ${\\sigma }$ .", "Since $\\overline{{\\sigma }}_{2k}={\\sigma }_{2k}<\\overline{{\\sigma }}_{2k+1}={\\sigma }_{2k+1}>0=\\overline{{\\sigma }}_{2k+2}$ , and $\\overline{{\\sigma }}_{i}={\\sigma }_i$ for $i\\le 2k+1$ , we can view $\\overline{{\\sigma }}$ as an alternating permutation of ${\\mathcal {E}}_{2k+2}$ and the reverse permutation $\\overline{{\\sigma }}^{r}$ is an up-down permutation on $[2k+2]$ with the first element being 0.", "From the position of 0, we see that 0 is the unique left to right mimimum of $\\overline{{\\sigma }}$ , thus $\\overline{{\\sigma }}^{\\prime }=\\tau (\\overline{{\\sigma }}^{r})$ is a cycle up-down permutation with only one even cycle.", "Let ${\\mathcal {G}}(\\overline{{\\sigma }}^{\\prime })$ be the corresponding graph of the permutation $\\overline{{\\sigma }}^{\\prime }$ , then the closer set of the pair of matchings is $\\lbrace i_1,i_2,\\ldots ,i_{k+1}\\rbrace $ .", "Since there are only one cycle in $\\overline{{\\sigma }}^{\\prime }$ , the above matching and the below matching are not independent now.", "It requires that in the graph ${\\mathcal {G}}(\\overline{{\\sigma }}^{\\prime })$ , the edges not containing the last closer $i_{k+1}$ can not form a closed circle.", "That is to say, for any closer $i_j, j\\le k$ , there exists no opener $a$ such that $(a,b_1),(b_1,b_2),\\cdots , (b_m,i_j)(i_j,a)$ are the edges of ${\\mathcal {G}}(\\overline{{\\sigma }}^{\\prime })$ .", "On the other hand, given a pair of matchings with closer set $\\lbrace i_1,i_2,\\ldots ,i_{k+1}\\rbrace $ and satisfy the above condition, we can also reverse the above procedure to get an alternating permutation on $[2k+1]$ with peak set $\\lbrace i_1,i_2,\\ldots ,i_{k+1}\\rbrace $ .", "By the same analysis as Lemma REF , the number of above matchings on $\\lbrace 0,1,2,\\ldots 2k,2k+1\\rbrace $ with closer set $\\lbrace i_1,i_2,\\ldots ,i_{k+1}\\rbrace $ equals $\\prod _{1\\le j \\le k}(i_j-2j+2)$ since there is an extra opener 0.", "After the above matching is determined, we next construct the edges of the below matching.", "For $1\\le j \\le k$ , there are $i_j-1-2(j-1)+1$ openers before the $j$ -th closer $i_j$ .", "Among these openers, the opener $a$ such that $(i_j,a),(a,b_1),(b_1,b_2),\\cdots , (b_m,i_j)$ are the edges of the graph have been constructed so far can not be chosen, otherwise there will exist at least two cycles in $\\overline{{\\sigma }}^{\\prime }$ .", "Hence the $j$ -th closer of the below matching has $i_j-2j+1$ choices for its opener to generate an edge.", "Combining the number of possible ways of constructing this pair of matchings with the same given closer set leads to the identity () immediately.", "Let us illustrate the argument with an example.", "For ${\\sigma }=8\\,6\\,7\\,3\\,4\\,1\\,9\\,2\\,5 \\in {\\mathcal {E}}_9$ , then $\\overline{{\\sigma }}=8\\,6\\,7\\,3\\,4\\,1\\,9\\,2\\,5\\,0\\in {\\mathcal {E}}_{10}$ , $\\overline{{\\sigma }}^{r}=0\\,5\\,2\\,9\\,1\\,4\\,3\\,7\\,6\\,8$ and $\\tau (\\overline{{\\sigma }}^{r})=(0,5,2,9,1,4,3,7,6,8)$ , see Figure REF for its corresponding graph.", "Figure: The graph of the cycle up-down permutation (0,5,2,9,1,4,3,7,6,8)." ] ]

[ [ "Suppressing nano-scale stick-slip motion by feedback" ], [ "Abstract When a micro cantilever with a nano-scale tip is manipulated on a substrate with atomic-scale roughness, the periodic lateral frictional force and stochastic fluctuations may induce stick-slip motion of the cantilever tip, which greatly decreases the precision of the nano manipulation.", "This unwanted motion cannot be reduced by open-loop control especially when there exist parameter uncertainties in the system model, and thus needs to introduce feedback control.", "However, real-time feedback cannot be realized by the existing virtual reality virtual feedback techniques based on the position sensing capacity of the atomic force microscopy (AFM).", "To solve this problem, we propose a new method to design real-time feedback control based on the force sensing approach to compensate for the disturbances and thus reduce the stick-slip motion of the cantilever tip.", "Theoretical analysis and numerical simulations show that the controlled motion of the cantilever tip tracks the desired trajectory with much higher precision.", "Further investigation shows that our proposal is robust under various parameter uncertainties.", "Our study opens up new perspectives of real-time nano manipulation." ], [ "Introduction", "One of the central problems in nano science and technology [1] is the realization of high-precision nano manipulation, e.g., pushing, pulling, rotating, rolling, and cutting nano-scale objects.", "In the widely applied AFM experiments [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], a large amount of frontier progresses have been achieved about nano manipulation.", "However, the theoretical analysis for such system is mainly focused on the static force analysis within a Newton-mechanical framework [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] that describes the macroscopic system.", "Recent experiments show that the force analysis in the nano scale is not the same as that at the macroscopic scale.", "Typically, nano-scale friction [14], [15] is logarithmically dependent on the velocity [16], [17], [18], which is quite different from the macroscopic sliding friction (the friction is independent on the velocity) and the traditional viscous-type friction (the friction is linearly proportional to the velocity).", "Additionally, the atomic scale periodic structure on the substrate has to be seriously considered in the friction analysis that is usually done with the continuous mechanics in macroscopic system [19], [20].", "The periodic lateral force induced by the atomic periodic structure on the surface of the substrate may lead to stick-slip motions [21], [22], [23] of the cantilever tip of the AFM, which deteriorates the precision of the nano manipulation.", "In the literature [19], [20], the cantilever of the AFM is usually modelled as a soft spring, and the effective nano friction force is described by a sinusoidal lateral force.", "Temperature-dependent white noises are introduced to represent the stochastic fluctuations in the lateral force induced by the thermal motion of the substrate atoms.", "Such a dynamical model predicts a motion transition of the cantilever tip of the AFM from the continuous sliding to a stick-slip mode by varying the ratio between the amplitude of the sinusoidal lateral force and the stiffness of the cantilever tip.", "To improve the precision in nano manipulation, many strategies [24], [25], [26], [27], [28], [29], [30] have been proposed to control the motions of the nano objects under nano frictions.", "For example, in the literature [24], the authors designed a non-Lipschitzion control function under which one can push the nano sample to asymptotically track the target velocity.", "This proposal is efficient and robust, but the natural fluctuation cannot be removed.", "To overcome this difficulty, more complex control function was designed based on the Lyapunov theory in Refs.", "[29], [30].", "However, the control designs in these proposals require the knowledge of the exact position of the sample during the nano manipulation.", "Such schemes are uneasy to be realized with the present experimental techniques, e.g., the haptic sensing and the virtual reality visual feedback techniques [9], [10], [31], for the inability of simultaneous position sensing and manipulation processing by the AFM.", "In fact, in these techniques, one has to stop the nano manipulation process and scan the surface of the substrate by the cantilever tip of the AFM to position the sample on the substrate, after which the next step of nano manipulation can go on.", "To solve this problem, in this paper, we propose a feedback control strategy based on the real-time signal sampled from the force sensor of the AFM, which can be conditionally done without pending the nano manipulation.", "The signal is used to estimate the position of the cantilever tip of the AFM, with which we can design feedback control to reduce the stick-slip motion of the cantilever tip and thus improve the manipulation precision.", "The paper is organized as follows.", "Section  describes the dynamical model, following which the stick-slip motions of the cantilever tip under open-loop control are presented in Sec. .", "Section  is devoted to the design of the real-time feedback control to reduce the stick-slip motion of the cantilever tip.", "Section  discusses the robustness of our method against various parameter uncertainties.", "Conclusions and forecast of the future work are given in Sec.", "." ], [ "Modelling of the one-dimensional nano manipulation\nsystem", "We first present the model used to describe the one-dimensional nano manipulations such as pushing a nano sample or etching the surface of the substrate to draw desired pattern.", "In this model, the cantilever of the AFM is taken as a spring with an effective stiffness $k_c$ .", "Thus, according to the Hooke's law, the lateral force imposed by the cantilever tip is: $F_p=k_c(u-x),$ where $x$ and $u$ are the relative positions of the cantilever tip with and without deformation on the platform of the nano manipulation, respectively (see Fig. ).", "In our method, $u$ is the control parameter to be designed.", "The mechanism of control is shown in Fig. (a).", "As shown in Fig.", "REF (a), in the designed nano manipulation system, the cantilever of the AFM is fixed, while the platform of the nano manipulation moves.", "In such a system, the motion of the platform of the nano manipulation is controlled by a piezoelectric transducer (PZT).", "By adjusting the voltage $V_{\\rm PZT}$ added on the PZT via a control circuit, the deformation of the PZT can be controlled, by which the relative position $u$ between the cantilever of the AFM and the platform is tunable.", "Typically, the functional relationship between the deformation of the PZT and the voltage $V_{\\rm PZT}$ shows hysteresis, creep, and structural vibration behaviors and thus is nonlinear.", "However, such a nonlinear characteristic response of PZT can be compensated by auxiliary control devices.", "Our recent theoretical and experimental study [32] shows that a linear functional relationship between $V_{\\rm PZT}$ and $u$ can be obtained by introducing optimal design of feedforward controller by the Prandtl-Ishlinskii model [33].", "Based on this study, in order to simplify our discussions, we do not go into details about the relationship between $V_{\\rm PZT}$ and $u$ , but simply take the deformation of the PZT represented by $u$ as the control parameter.", "Additionally, in the moving frame of the platform, the control system shown in Fig.", "(a) is equivalent to that shown in Fig.", "REF (b), in which the platform is fixed and the cantilever moves.", "Figure: (color online) Schematic diagrams of thecontrolled nano manipulation system: (a) the control system inexperiments in which the cantilever of the AFM is fixed and theplatform moves; (b) the equivalent control system in which theplatform is fixed while the cantilever moves.", "The two controlsystems (a) and (b) are equivalent if we fix the origin of thecoordinate with the moving platform.", "“PZT\" denotes thepiezoelectric transducer.", "The deformation of the PZT is controlledby the voltage V PZT V_{\\rm PZT} added on the PZT.", "Thus, the relativeposition uu can be controlled as what we want by a controlcircuit.When the cantilever tip moves on the substrate, the nano friction may occur on the contact area.", "In recent experiments [16], the average frictional force is observed to be logarithmically dependent on the velocity of the cantilever tip $v=dx/dt$ , i.e., $\\bar{F}_f=F_{f_0}+F_{f_1}\\ln \\left(\\frac{v}{v_1}\\right),$ where $F_{f_0}$ , $F_{f_1}$ , $v_1$ are constant parameters.", "There also exists periodic lateral force induced by the surface potential of the substrate, and stochastic noises induced by the thermal motions of the substrate atoms.", "For simplicity, we only consider the fundamental-frequency component of the periodic lateral force, and omit the correlation effects of the thermal motions of the substrate atoms.", "Additionally, to simplify our discussions, the origin of coordinate is chosen such that the periodic lateral force is zero at the origin.", "Thus, the resulting modification of the lateral force can be expressed as: $\\delta F_f=F_{f_2}\\sin \\left(2\\pi \\frac{x}{a}\\right)-\\xi (t),$ where $F_{f_2}$ is the amplitude of the periodic lateral force; $a$ is the lattice constant of the substrate; and $\\xi (t)$ is a white noise such that $E(\\xi (t))=0$ , $E(\\xi (t)\\xi (t^{\\prime }))=D_{\\xi }\\delta (t-t^{\\prime })$ , with $D_{\\xi }$ being the strength.", "$E\\left(\\cdot \\right)$ denotes the ensemble average over the white noise.", "From Eqs.", "(REF ), (), and (REF ), the equations of the system can be expressed in the Ito notation as: $dx & = & v dt,\\nonumber \\\\m dv & = & k_c(u-x)dt+\\left[-F_{f_0}-F_{f_1}\\ln \\left(\\frac{v}{v_1}\\right)\\right.\\nonumber \\\\&&\\left.-F_{f_2}\\sin \\left(2\\pi \\frac{x}{a}\\right)\\right]dt+\\sqrt{D_{\\xi }}dW,$ where $m$ is the effective mass of the cantilever tip; and $dW=W\\left(t+dt\\right)-W\\left(t\\right)$ is the increment of the Wiener process $W\\left(t\\right)=\\int _0^t \\xi \\left(\\tau \\right)d\\tau $ satisfying: $E(dW)=0,\\quad (dW)^2=dt.$" ], [ "Stick-slip motion under open-loop control", "In order to implement nano manipulation by the cantilever tip of the AFM with a constant velocity $v^*$ , i.e., to control the position of the cantilever tip such that $x=v^* t$ , a simple strategy is to set $u=v^* t$ .", "However, this would be ineffective due to the existence of the periodic lateral force and the stochastic fluctuations especially when the roughness of the substrate is relatively large compared with the stiffness of the cantilever of the AFM.", "In such case, existing studies [19], [20] have predicted stick-slip motions of the cantilever tip, which greatly reduce the precision of the nano manipulation.", "The lateral pushing force imposed by the cantilever tip oscillates under the stick-slip motion, which can be so large that the fragile sample and substrate are damaged, or the cantilever tip of the AFM slide over them.", "The drawbacks of the open-loop constant control can be seen from the following numerical examples.", "The system parameters are chosen as [19]: $&F_{f_0}=10\\,\\,{\\rm nN},\\quad F_{f_1}=1\\,\\,{\\rm nN},\\quad a=0.25\\,\\,{\\rm nm},&\\nonumber \\\\&v_1=1\\,\\,{\\rm nm/s},\\quad F_{f_2}=\\,\\,{\\rm either}\\,\\,0.25\\,\\,{\\rm nN}\\,\\,{\\rm or}\\,\\,15\\,\\,{\\rm nN},&\\nonumber \\\\&v^*=3\\,\\,{\\rm nm/s}, \\quad \\sqrt{D_{\\xi }}=0.1\\,\\,{\\rm nN\\cdot s^{-1/2}},\\nonumber \\\\&k_c=1\\,\\,{\\rm N/m},\\quad m=5\\times 10^{-11}\\,\\,{\\rm kg},&$ under which the simulation results of the position $x$ , velocity $v$ , and the lateral force $F_p$ imposed by the cantilever tip are shown in Fig. .", "Figure: (color online) Plots of (a) the positions,(b) the velocities of the cantilever tip, and (c) the pushingforces imposed by the cantilever tip.", "The red curves with plussigns denote the ideal trajectories for which the cantilever tipmoves with constant velocity v=v * =3v=v^*=3 nm/s; the green triangle(blue solid) curves represent the controlled trajectories underopen-loop control x=v * tx=v^* t, with the amplitudes of the periodiclateral force F f 2 F_{f_2} as 0.250.25 nN (15 nN).As shown in Fig.", ", when the amplitude of the periodic lateral force $F_{f_2}$ is relatively small such that $F_{f_2}/a$ is comparable with $k_c$ (the case with $F_{f_2}=0.25$ nN), which is the case when the cantilever tip moves on an incommensurate substrate [14], there exists no stick-slip motion and the cantilever tip moves with a constant velocity after a transient process (see the green triangle curves).", "However, when the amplitude of the periodic lateral force is relatively large such that $F_{f_2}/a\\gg k_c$ (the case with $F_{f_2}=15$ nN), which may be valid when the cantilever tip moves on a commensurate substrate, stick-slip motions can be observed (see the blue solid curves).", "Such stick-slip motions can be explained by the elastic instability predicted by the Frenkel-Kontorova model [14].", "In this case, the cantilever tip moves only when the pushing force $F_p$ exceeds the critical value, i.e., the peak value of the sawtoothlike trajectory shown in Fig. (c).", "It is also shown in Fig.", "(a) that no matter whether the stick-slip motion occurs or not, there exist tracking errors between the controlled trajectories (the green triangle curve and the blue solid curve) and the ideal trajectory (the red curve with plus signs), which need to be compensated to improve the precision of the nano manipulation." ], [ "Reduction of the stick-slip motion by real-time feedback control", "In this section, we introduce feedback control to reduce this unexpected stick-slip motion under open-loop control.", "The object of our method is to monitor the deformation of the cantilever and thus make the cantilever tip move with a given constant velocity.", "To acquire feedback signals, there are two available sensing methods supported by the AFM-based nano manipulation systems: position sensing and force sensing.", "Position sensing is one of the basic functions of the AFM, by which one can easily obtain the tomography of a surface with nano-scale roughness.", "However, real-time position sensing is unavailable during the nano manipulation.", "Another sensing capacity of the AFM is the force sensing, which can be done by detecting the deformation of the cantilever of the AFM.", "In contrast with position sensing, force sensing can be done during the nano manipulation, which makes the real-time feedback control possible.", "The main idea of our feedback control method is to estimate the position of the cantilever tip by force sensing to adjust the control parameter $u$ , and further control the motion of the tip.", "The control process thus can be divided into two steps, i.e., a filtering and estimation step and a feedback control step, which will be specified below." ], [ "Position estimation by force sensing", "By the force sensing capacity of the AFM, the pushing force measured can be expressed as: $F_p^m=k_c\\left(u-x\\right)+\\eta (t),$ where $\\eta \\left(t\\right)$ is a white noise induced by the measurement apparatus.", "$\\eta \\left(t\\right)$ satisfies $E\\left(\\eta \\left(t\\right)\\right)=0,\\quad E\\left(\\eta \\left(t\\right)\\eta \\left(t^{\\prime }\\right)\\right)=D_{\\eta }\\delta \\left(t-t^{\\prime }\\right),$ where $D_{\\eta }$ is the strength of the noise $\\eta \\left(t\\right)$ .", "To reduce the measurement-induced disturbance of $\\eta \\left(t\\right)$ in feedback control, we filter the measured signal $F_p^m$ by a low-pass filter over a time window $[t-T,t]$ .", "The output signal can be expressed as (see, e.g., Refs.", "[34], [35]): $\\hat{F}_p^m\\left(t\\right)=\\frac{1}{T}\\int _{t-T}^te^{-\\gamma _{ft}\\left(t-\\tau \\right)}F_p^m\\left(\\tau \\right)d\\tau ,$ where $\\gamma _{ft}$ is the damping rate of the low-pass filter.", "Then, the position of the tip $x$ can be estimated from $\\hat{F}_p^m$ by: $\\hat{x}\\left(t\\right)=u(t)-\\frac{1}{k_c}\\hat{F}_p^m\\left(t\\right).$ Under the filtering condition $\\gamma _{ft}\\ll 1/T,$ it can be verified that $x\\left(t\\right)-\\hat{x}\\left(t\\right)=e^{-\\gamma _{ft}t}\\left(x_0-\\hat{x}_0\\right),$ where $x_0$ and $\\hat{x}_0$ are the initial states of $x\\left(t\\right)$ and $\\hat{x}\\left(t\\right)$ respectively (see the derivation in Appendix ).", "Thus, the estimated position $\\hat{x}\\left(t\\right)$ exponentially converges to the position of the tip $x\\left(t\\right)$ with the convergence rate $\\gamma _{ft}$ , and thus $\\hat{x}\\left(t\\right)$ can be taken as a good estimation of $x\\left(t\\right)$ when $t\\gg 1/\\gamma _{ft}$ ." ], [ "Feedback control design based on the estimated position", "Based on the estimated position $\\hat{x}$ given in Eq.", "(REF ), we can design the position-based feedback control $u\\left(\\hat{x}\\right)$ to reduce the stick-slip motion of the cantilever tip, which is given as follows: $u\\left(\\hat{x}\\right)&=&\\frac{1}{k_c}\\left[k_cx^*+F_{f_0}+F_{f_1}\\ln \\left(\\frac{v^*}{v_1}\\right)+F_{f_2}\\sin \\left(2\\pi \\frac{\\hat{x}}{a}\\right)\\right.\\nonumber \\\\&&\\left.-k_x\\left(\\hat{x}-x^*\\right)-k_I\\int _0^t\\left(\\hat{x}\\left(\\tau \\right)-x^*\\right)d\\tau \\right],$ where $x^*=v^*t$ is the desired motion of the tip; and $k_x,\\,k_I>0$ are the control parameters to be determined.", "The term $k_c x^*+F_{f_0}+F_{f_1}{\\rm ln}\\left(v^*/v_1\\right)+F_{f_2}\\sin \\left(2\\pi \\hat{x}/a\\right)$ in the control given by Eq.", "(REF ) is introduced to compensate the mean and the periodic friction forces.", "The proportional and integral feedback control terms $k_x\\left(\\hat{x}-x^*\\right)$ and $k_I\\int _0^t\\left(\\hat{x}-x^*\\right)d\\tau $ are introduced to speed up the convergence of the system dynamics to the stationary motion and reduce the static error.", "Figure: (color online) Schematic diagram of thefeedback control loop, where r * =k c x * +F f 0 +F f 1 ln v * /v 1 r^*=k_c x^*+F_{f_0}+F_{f_1}{\\rm ln}\\left(v^*/v_1\\right) and e ^=x ^-x * \\hat{e}=\\hat{x}-x^*; “PZT\",“PEC\", “LPF\", and “SWG\" are the piezoelectric transducer, thephotonelectric convertor, the low-pass filter, and the sine-wavegenerator respectively.", "The feedback control circuit is dividedinto two parts, i.e., a filtering circuit and a control circuitThe designed feedback control uu in Eq.", "() can be used to find the voltage V PZT =V PZT uV_{\\rm PZT}=V_{\\rm PZT}\\left(u\\right) added on the PZT.The schematic diagram of the feedback control proposed is given in Fig. .", "The deformation of the cantilever tip of the AFM is detected by an optical refracting system, and then converted into electric signals by a photonelectric convertor.", "The output signal masked by the measurement noise $\\eta \\left(t\\right)$ is fed into a low-pass filter followed by a control circuit to generate the feedback control signal.", "The feedback electric signal is used to control the motion of the platform by adjusting the voltage $V_{\\rm PZT}$ added on the piezoelectric transducer connected to the platform.", "Under the filtering condition (REF ), the weak noise condition $D_{\\xi }\\ll 2F_{f_1}\\left(k_c+k_x\\right)a^2/v^*,\\,2F_{f_1}mv^*,$ and choosing the control parameters $k_I$ and $k_x$ such that $k_I<F_{f_1}\\left(k_c+k_x\\right)/mv^*,$ the controlled trajectory $x\\left(t\\right)$ tracks the desired trajectory $x^*=v^*t$ in average (see Appendix ), i.e., $\\lim _{t\\rightarrow \\infty }\\left(E\\left(x\\left(t\\right)\\right)-x^*\\right)=0,$ where $E\\left(\\cdot \\right)$ denotes the ensemble average of the stochastic signal.", "This result indicates that the stick-slip motion of the cantilever tip can be efficiently suppressed by the designed feedback control.", "To evaluate the magnitude of the stochastic fluctuation, we further estimate the variances of the position $x$ and velocity $v$ of the cantilever tip, which are defined by: $V_{x}=E\\left(\\left(x-E\\left(x\\right)\\right)^2\\right),\\quad V_{v}=E\\left(\\left(v-E(v)\\right)^2\\right).$ With additional calculations, the stationary values of the variances $V_{x}$ and $V_{v}$ can be approximately estimated as (see the analysis in Appendix ): $V_{x}^{\\infty }=\\frac{D_{\\xi }v^*}{2F_{f_1}\\left(k_c+k_x\\right)},\\quad V_{v}^{\\infty }=\\frac{D_{\\xi }v^*}{2F_{f_1}m}.$ The effectiveness of our proposal can be demonstrated via numerical examples.", "Given the system parameters $&F_{f_0}=10\\,\\,{\\rm nN},\\quad F_{f_1}=1\\,\\,{\\rm nN},\\quad a=0.25\\,\\,{\\rm nm},&\\nonumber \\\\&v_1=1\\,\\,{\\rm nm/s},\\quad F_{f_2}=15\\,\\,{\\rm nN},&\\nonumber \\\\&v^*=3\\,\\,{\\rm nm/s}, \\quad \\sqrt{D_{\\xi }}=0.1\\,\\,{\\rm nN\\cdot s^{1/2}},\\nonumber \\\\&k_c=1\\,\\,{\\rm N/m},\\quad m=5\\times 10^{-11}\\,\\,{\\rm kg},\\quad k_I=1\\,\\,{\\rm N/\\left(m\\cdot s\\right)},&\\nonumber \\\\&k_x=5\\,\\,{\\rm N/m},\\quad T=0.1\\,\\,{\\rm s},\\quad \\gamma _{ft}=1\\,\\,{\\rm s}^{-1},&$ we compare the motions of the cantilever tip driven by the feedback control given in Eq.", "(REF ) and by the open-loop control $u=v^* t$ .", "Each curve is obtained by averaging over 20 sample (stochastic) trajectories.", "As shown in Fig.", ", the stick-slip motion observed under open-loop control (the green triangle curves) is greatly reduced by the proposed feedback control (the blue solid curves).", "To compare the stochastic fluctuations with the mean trajectories, we calculate the square roots of the variances $\\sigma _{x}=V_{x}^{1/2}$ and $\\sigma _{v}=V_{v}^{1/2}$ .", "It can be obtained over 20 stochastic trajectories that $\\sigma _{x}^{\\infty }=0.06$ nm and $\\sigma _{v}^{\\infty }=0.26$ nm/s, which are quite close to the estimated values $\\sigma _{x}^{\\infty }=0.05$ nm, $\\sigma _{v}^{\\infty }=0.38$ nm/s given by Eq.", "() and are negligible compared with the average motion ($E\\left(x\\right)$ is about tens of nm, and $E\\left(v\\right)$ is about 3 nm/s).", "Figure: (color online) Plots of themotions of the cantilever tip with (a) the position xx, (b) thevelocity vv, and (c) the lateral force F p F_p imposed by thecantilever tip; (d) shows the trajectories in presence ofparameter uncertainties given in Eq. ().", "The red curves with plus signs are the idealtrajectories with v=v * =3v=v^*=3 nm/s; the green curves with trianglesigns represent the trajectories driven by the open-loop controlu=v * tu=v^* t; and the blue solid curves denote the mean trajectoriesunder the feedback control given in Eq. ().", "The mean trajectories in the plot are obtained over20 stochastic trajectories." ], [ "Robustness against parameter uncertainties", "The system parameters involved in Eq.", "(REF ) can be identified offline by, e.g., pre-designed nanofriction experiments.", "In practical experiments, we also need to consider the uncertainties in the system parameters which may deteriorate the performances of the nano manipulation.", "For example, the plastic deformation of the tip and the adhesion force between the tip and the substrate can lead to small deviation $\\Delta k_c$ of the stiffness $k_c$ (typically $\\Delta k_c/k_c<10\\%$ in the literature [36]).", "To reduce the effects of uncertainties, we add an integral control term $-k_I\\int _0^t\\left(\\hat{x}-x^*\\right)d\\tau $ in Eq.", "() to reduce the static error induced by the uncertainties.", "Denote the additive uncertainties in the system parameters $k_c$ , $F_{f_0}$ , $F_{f_1}$ , $F_{f_2}$ , $v_1$ , and $a$ by $\\Delta k_c$ , $\\Delta F_{f_0}$ , $\\Delta F_{f_1}$ , $\\Delta F_{f_2}$ , $\\Delta v_1$ , $\\Delta a$ , and assume that there exists a phase offset $\\Delta \\phi $ in the sine function in Eq. ().", "With the analysis given in Appendix , the static tracking error can be controlled to zero, i.e., $\\lim _{t\\rightarrow \\infty }\\left(E\\left(x\\right)-x^*\\right)=0$ , if the uncertainties of the system parameters are not too large to satisfy $&\\Delta _x=\\left|\\Delta k_c\\right|+2\\pi \\frac{F_{f_2}}{a^2}\\left|\\Delta a\\right|+\\frac{2\\pi }{a}\\left|\\Delta F_{f_2}\\right|<k_c+k_x,&\\nonumber \\\\&\\left|\\Delta F_{f_1}\\right|<F_{f_1},&$ and the gain of the integrator $k_I$ is chosen such that $k_I<\\left(F_{f_1}-\\left|\\Delta F_{f_1}\\right|\\right)\\left(k_c+k_x-\\Delta _x\\right)/mv^*.$ It can be seen that the proportional and integral feedback terms $k_x\\left(\\hat{x}-x^*\\right)$ , $k_I\\int _0^t\\left(\\hat{x}-x^*\\right)d\\tau $ in the control (REF ) both contribute to the robustness of our method: (i) the integral term is used to reduce the static error; and (ii) the proportional term is used to increase the robustness of our method about the parameter uncertainty (the parameter regime given in Eqs.", "() and (REF ) is enlarged when we increase the control parameter $k_x$ ).", "Given the parameter uncertainties: $&\\Delta F_{f_0}=1\\,\\,{\\rm nN},\\,\\,\\,\\Delta F_{f_1}=0.1\\,\\,{\\rm nN},\\,\\,\\,\\Delta a=0.01\\,\\,{\\rm nm},&\\nonumber \\\\&\\Delta F_{f_2}=0.1\\,\\,{\\rm nN},\\,\\,\\,\\Delta k_c=0.1\\,\\,{\\rm N/m},&$ Figure (d) shows that our method is still valid under the given parameter uncertainties (the controlled trajectory, i.e., the blue solid curve, matches very well with the ideal trajectory, i.e., the red curve with plus signs)." ], [ "Conclusion", "In summary, we propose a feedback control strategy to reduce the stick-slip motion of the cantilever tip in a AFM-based nano manipulation system.", "The feedback control is designed based on the position estimation of the cantilever tip obtained by the force sensing capacity of the AFM.", "Compared with open-loop control, our proposal can greatly reduce the stick-slip motion of the cantilever tip.", "Our method is robust against small uncertainties in the system parameters, e.g., the stiffness of the cantilever of the AFM, the lattice constant of the substrate, and the phase offset in the surface potential of the substrate.", "Future study will be focused on extending the method to more practical cases.", "For example, as shown in our discussions, our designed feedback control is valid only under small uncertainties.", "More robust design should be developed for large uncertainties.", "Since the uncertainties in the model of the system can be reduced to a DC plus periodic disturbance, internal model principle is a good choice of the control design.", "Additionally, our control design can be naturally extended to the two-dimensional nano manipulation systems.", "More interesting results, such as the suppression of the S-shaped motion of the cantilever tip, are hopeful to be observed for the two-dimensional case.", "J. Zhang would like to thank Prof. Y.-X.", "Liu for helpful discussions.", "J. Zhang and R. B. Wu are supported by the National Natural Science Foundation of China under Grant Nos.", "61174084, 61134008, 60904034.", "T. J. Tarn would also like to acknowledge partial support from the U. S. Army Research Office under Grant W911NF-04-1-0386." ], [ "Derivation of Eq. (", "From Eqs.", "(REF ) and (), we have $u-x&=&\\left(F_p^m-\\eta \\left(t\\right)\\right)/k_c,\\\\u-\\hat{x}&=&\\hat{F}_p^m/k_c.$ It can be calculated from Eqs.", "() and (REF ) that $&&\\frac{1}{T}\\int _{t-T}^te^{-\\gamma _{ft}\\left(t-\\tau \\right)}\\left(u\\left(\\tau \\right)-x\\left(\\tau \\right)\\right)d\\tau \\nonumber \\\\&=&\\frac{1}{k_c}\\left(\\hat{F}_p^m-\\frac{1}{T}\\int _{t-T}^te^{-\\gamma _{ft}\\left(t-\\tau \\right)}\\eta \\left(\\tau \\right) d\\tau \\right)\\nonumber \\\\&=&\\frac{1}{k_c}\\left(\\hat{F}_p^m-\\frac{1}{T}\\int _0^Te^{-\\gamma _{ft}\\tilde{\\tau }}\\eta \\left(t-\\tilde{\\tau }\\right) d\\tilde{\\tau }\\right)\\nonumber \\\\&\\approx &\\frac{1}{k_c}\\left(\\hat{F}_p^m-\\frac{1}{T}\\int _0^T\\eta \\left(t-\\tilde{\\tau }\\right) d\\tilde{\\tau }\\right)\\nonumber \\\\&=&\\frac{1}{k_c}\\left(\\hat{F}_p^m-\\frac{1}{T}\\int _{t-T}^t\\eta \\left(\\tau \\right)d\\tau \\right)\\nonumber \\\\&\\approx &\\frac{1}{k_c}\\hat{F}_p^m=u-\\hat{x}.$ Here, we have used the condition given in Eq.", "(), i.e., $T\\ll 1/\\gamma _{ft}$ , to obtain $e^{-\\gamma _{ft}t}\\approx 1$ when $t\\in \\left[0,T\\right]$ and the ergodic property [37] of the white noise $\\eta \\left(t\\right)$ to replace the time average of $\\eta \\left(t\\right)$ by its ensemble average, i.e., $\\frac{1}{T}\\int _{t-T}^t\\eta \\left(\\eta \\right)d\\tau \\approx E\\left(\\eta \\left(t\\right)\\right)=0.$ Let us set $h\\left(t\\right)&=&u\\left(t\\right)-x\\left(t\\right),\\\\\\hat{h}\\left(t\\right)&=&u\\left(t\\right)-\\hat{x}\\left(t\\right),$ then from Eq.", "(REF ), i.e., $\\hat{h}\\left(t\\right)=\\frac{1}{T}\\int _{t-T}^te^{-\\gamma _{ft}\\left(t-\\tau \\right)}h\\left(\\tau \\right)d\\tau ,$ we have $\\frac{d}{dt}\\hat{h}\\left(t\\right)=\\frac{1}{T}\\left(h\\left(t\\right)-e^{-\\gamma _{ft}T}h\\left(t-T\\right)\\right)-\\gamma _{ft}h\\left(t\\right).$ We assume that $g\\left(T\\right)=e^{\\gamma _{ft}T}h\\left(t+T\\right)$ , then we have $\\frac{1}{T}\\left(h\\left(t\\right)-e^{-\\gamma _{ft}T}h\\left(t-T\\right)\\right)=\\frac{1}{T}\\left(g\\left(0\\right)-g\\left(-T\\right)\\right),$ which can be replaced by $g^{\\prime }\\left(T\\right)\\left|_{T=0}\\right.=\\frac{d}{dt}h\\left(t\\right)+\\gamma _{ft}h\\left(t\\right)$ under the condition $T\\ll 1/\\gamma _{ft}$ .", "Thus, Eq.", "(REF ) can be reexpressed as: $\\frac{d}{dt}\\left(\\hat{h}-h\\right)=-\\gamma _{ft}\\left(\\hat{h}-h\\right).$ From the definition of $h$ and $\\hat{h}$ , we have $\\frac{d}{dt}\\left(\\hat{x}-x\\right)=-\\gamma _{ft}\\left(\\hat{x}-x\\right),$ which means that $\\hat{x}-x\\rightarrow 0$ when $t\\gg 1/\\gamma _{ft}$ , i.e., the estimated position of the cantilever tip $\\hat{x}$ tracks the actual position $x$ in the long time limit." ], [ "Derivation of Eqs. (", "By substituting the feedback control law given in Eq.", "(REF ) into Eq.", "(), we have $dx&=&v dt,\\nonumber \\\\mdv&=&\\left[-k_c\\left(x-x^*\\right)-k_x\\left(\\hat{x}-x^*\\right)\\right.\\nonumber \\\\&&+F_{f_2}\\sin \\left(2\\pi \\frac{\\hat{x}}{a}\\right)-F_{f_2}\\sin \\left(2\\pi \\frac{x}{a}\\right)\\nonumber \\\\&&\\left.-F_{f_1}\\left({\\rm ln}\\frac{v}{v_1}-{\\rm ln}\\frac{v^*}{v_1}\\right)-k_I q\\right]dt+\\sqrt{D_{\\xi }}dW,\\nonumber \\\\d\\delta \\hat{x}&=&-\\gamma _{ft}\\delta \\hat{x} dt,\\,\\,\\,dq=\\left(\\hat{x}-x^*\\right)dt,$ where $\\delta \\hat{x}=\\hat{x}-x$ and $q=\\int _0^t\\left(\\hat{x}-x^*\\right)d\\tau $ .", "By taking the ensemble average and denoting $\\delta \\bar{x}=E\\left(x\\right)-x^*$ , $\\delta \\bar{v}=E\\left(v\\right)-v^*$ , $\\delta \\bar{\\hat{x}}=E\\left(\\delta \\hat{x}\\right)$ , and $\\bar{q}=E\\left(q\\right)$ , we can expand the equation to the second-order quadratures to obtain: $\\delta \\dot{\\bar{x}}&=&\\delta \\bar{v},\\nonumber \\\\m\\delta \\dot{\\bar{v}}&=&-\\left(k_c+k_x\\right)\\delta \\bar{x}+F_{f_2}\\frac{2\\pi }{a}\\cos \\left(2\\pi \\frac{x^*}{a}\\right)\\delta \\bar{\\hat{x}}\\nonumber \\\\&&-k_x\\delta \\bar{\\hat{x}}-\\frac{F_{f_1}}{v^*}\\delta \\bar{v}+F_{f_2}\\frac{8\\pi ^3}{a^3}V_{x}\\cos \\left(2\\pi \\frac{x^*}{a}\\right)\\delta \\bar{\\hat{x}}\\nonumber \\\\&&+F_{f_1}\\frac{V_{v}}{2v^{*2}}+O\\left(\\frac{\\delta \\bar{x}}{a}\\frac{\\delta \\bar{\\hat{x}}}{a}\\right)+O\\left(\\frac{\\delta \\bar{v}^2}{v^{*2}}\\right)\\nonumber \\\\&&+O\\left(\\frac{V_{x}^2}{a^4}\\frac{\\delta \\bar{\\hat{x}}}{a}\\right)+O\\left(\\frac{V_{v}^2}{v^{*4}}\\right)-k_I\\bar{q},\\nonumber \\\\\\delta \\dot{\\bar{\\hat{x}}}&=&-\\gamma _{ft}\\delta \\bar{\\hat{x}},\\,\\,\\,\\dot{\\bar{q}}=\\delta \\bar{x}+\\delta \\bar{\\hat{x}},$ where $V_{x}=E\\left(x-E\\left(x\\right)\\right)^2$ and $V_{v}=E\\left(v-\\left(v\\right)\\right)^2$ are the variances of $x$ and $v$ ; and $O\\left(\\cdot \\right)$ denotes the higher-order terms.", "Since the nonlinear equation () is linearized near the origin $\\left(0,0,0,0\\right)^T$ in Eq.", "(REF ), we have introduced the Gaussian assumption to omit higher-order quadratures of $x$ and $v$ .", "We can further omit the terms related to $V_{x}/a^2$ and $V_{v}/v^{*2}$ in Eq. ().", "In fact, as shown in Eq.", "(), we have $V_{x}^{\\infty }\\ll a^2$ , $V_{v}^{\\infty }\\ll v^{*2}$ under the assumption ().", "Thus, we have $V_{x}\\ll a^2$ and $V_{v}\\ll v^{*2}$ for sufficiently long time.", "Since we are just interested in the stationary behaviors of the system dynamics, we can omit the terms related to $V_{x}/a^2$ and $V_{v}/v^{*\\,2}$ .", "With the above analysis, we can obtain the linearization equation of Eq.", "(REF ) in the neighborhood of $\\left(\\delta \\bar{x},\\delta \\bar{v},\\delta \\bar{\\hat{x}},\\bar{q}\\right)^T=\\left(0,0,0,0\\right)^T$ .", "It can be easily verified that the characteristic equation of the coefficient matrix of the linearization equation is $\\left(s+\\gamma _{ft}\\right)\\left(s^3+\\frac{F_{f_1}}{mv^*}s^2+\\frac{\\left(k_c+k_x\\right)}{m}s+\\frac{k_I}{m}\\right)=0.$ Since $k_I<F_{f_1}\\left(k_c+k_x\\right)/mv^*$ from Eq.", "(), the real parts of the eigenvalues of the above equation are all negative.", "It means that the linearization equation is exponentially stable, and thus the original nonlinear equation (REF ) is asymptotically stable at the origin, which leads to the fact that $\\lim _{t\\rightarrow \\infty }\\left(E\\left(x\\right)-x^*\\right)=0$ .", "To approximately estimate the stationary variances, we replace $\\hat{x}$ by $x$ , linearize Eq.", "(), and omit higher-order correlation terms.", "It can be verified that $V_{x}$ , $V_{v}$ , and $C_{x v}$ , i.e., the covariance between $x$ and $v$ , satisfy the following equation $\\left(\\begin{array}{c}\\dot{V}_{x} \\\\\\dot{V}_{v} \\\\\\dot{C}_{x v} \\\\\\end{array}\\right)&=&\\left(\\begin{array}{ccc}0 & 0 & 2 \\\\0 & -\\frac{2F_{f_1}}{mv^*} & -\\frac{2\\left(k_c+k_x\\right)}{m} \\\\-\\frac{k_c+k_x}{m} & 1 & -\\frac{F_{f_1}}{mv^*} \\\\\\end{array}\\right)\\nonumber \\\\&&\\left(\\begin{array}{c}V_{x} \\\\V_{v} \\\\C_{x v} \\\\\\end{array}\\right)+\\left(\\begin{array}{c}0 \\\\\\frac{D_{\\xi }}{m^2} \\\\0 \\\\\\end{array}\\right).$ To consider the stationary variances, the omission of the higher-order correlation terms to obtain Eq.", "() is reasonable, because $x-\\hat{x}\\rightarrow 0$ when $t\\rightarrow \\infty $ , and higher-order correlation terms are small compared with $V_x$ , $V_v$ , and $C_{xv}$ under the weak noise assumption ().", "From Eq.", "(), we can calculate the stationary variances of $x$ and $v$ as: $V_{x}^{\\infty }=\\frac{D_{\\xi }v^*}{2F_{f_1}\\left(k_c+k_x\\right)},\\quad V_{v}^{\\infty }=\\frac{D_{\\xi }v^*}{2F_{f_1}m}.$" ], [ "Robustness analysis of our method", "Let us replace the system parameters $k_c$ , $F_{f_0}$ , $F_{f_1}$ , $F_{f_2}$ , $v_1$ , and $a$ in Eq.", "(REF ) by $k_c+\\Delta k_c$ , $F_{f_0}+\\Delta F_{f_0}$ , $F_{f_1}+\\Delta F_{f_1}$ , $F_{f_2}+\\Delta F_{f_2}$ , $v_1+\\Delta v_1$ , and $a+\\Delta a$ , and consider the phase offset $\\Delta \\phi $ .", "We can expand the equation to the linear terms of the uncertainties $\\Delta k_c$ , $\\Delta F_{f_0}$ , $\\Delta F_{f_1}$ , $\\Delta F_{f_2}$ , $\\Delta v_1$ , $\\Delta \\phi $ , and $\\Delta a$ .", "By neglecting the higher-order nonlinear terms, we can obtain $\\delta \\dot{\\bar{x}}&=&\\delta \\bar{v},\\nonumber \\\\m\\delta \\dot{\\bar{v}}&=&-w_x\\delta \\bar{x}-w_v\\delta \\bar{v}-\\left(k_x-F_{f_2}\\frac{2\\pi }{a}\\cos \\frac{x^*}{a}\\right)\\delta \\bar{\\hat{x}}\\nonumber \\\\&&-k_I\\bar{q}+w_0,\\nonumber \\\\\\delta \\dot{\\bar{\\hat{x}}}&=&-\\gamma _{ft}\\delta \\bar{\\hat{x}},\\,\\,\\,\\dot{\\bar{q}}=\\delta \\bar{x}+\\delta \\bar{\\hat{x}},$ where $w_x&=&-\\frac{2\\pi }{a}\\cos \\left(2\\pi \\frac{x^*}{a}\\right)\\left(F_{f_2}\\frac{\\Delta a}{a}-\\Delta F_{f_2}\\right)\\\\&&+\\Delta k_c+k_c+k_x,\\\\w_v&=&\\left(F_{f_1}+\\Delta F_{f_1}\\right)/v^*,\\\\w_0&=&\\Delta F_{f_0}+{\\rm ln}\\left(\\frac{v^*}{v_1}\\right)\\Delta F_{f_1}+\\frac{F_{f_1}}{v_1}\\Delta v_1\\\\&&-\\sin \\left(2\\pi \\frac{x^*}{a}\\right)\\Delta F_{f_2}-F_{f_2}\\cos \\left(2\\pi \\frac{x^*}{a}\\right)\\Delta \\phi .$ The characteristic equation of the linear matrix of Eq.", "(REF ) can be expressed as: $\\left(s+\\gamma _{ft}\\right)\\left(s^3+w_v s^2+w_x s+k_I\\right)=0.$ It can be checked from Eqs.", "(REF ) and (REF ) that the real parts of the eigenvalues of the above equation are all negative.", "It means that there exists a stationary solution of the linearlization equation of Eq.", "(REF ), so does the original equation (REF ).", "Thus, we have $\\dot{\\bar{q}}=\\delta \\bar{x}+\\delta \\bar{\\hat{x}}\\rightarrow 0$ when $t\\rightarrow \\infty $ .", "Furthermore, from $\\delta \\dot{\\bar{\\hat{x}}}=-\\gamma _{ft}\\delta \\bar{\\hat{x}}$ , it can be verified that $\\delta \\bar{\\hat{x}}\\rightarrow 0$ .", "Thus, we have $\\lim _{t\\rightarrow \\infty }\\delta \\bar{x}=\\lim _{t\\rightarrow \\infty }\\left(E\\left(x\\right)-x^*\\right)=0$ ." ] ]

[ [ "Modeling a measurement-device-independent quantum key distribution\n system" ], [ "Abstract We present a detailed description of a widely applicable mathematical model for quantum key distribution (QKD) systems implementing the measurement-device-independent (MDI) protocol.", "The model is tested by comparing its predictions with data taken using a proof-of-principle, time-bin qubit-based QKD system in a secure laboratory environment (i.e.", "in a setting in which eavesdropping can be excluded).", "The good agreement between the predictions and the experimental data allows the model to be used to optimize mean photon numbers per attenuated laser pulse, which are used to encode quantum bits.", "This in turn allows optimization of secret key rates of existing MDI-QKD systems, identification of rate-limiting components, and projection of future performance.", "In addition, we also performed measurements over deployed fiber, showing that our system's performance is not affected by environment-induced perturbations." ], [ "Introduction ", "From the first proposal in 1984 to now, the field of quantum key distribution (QKD) has evolved significantly [1], [2].", "For instance, experimentally, systems delivering key at Mbps rates [3] as well as key distribution over more than 100 km [4], [5] have been reported.", "From a theoretical perspective, efforts aim at developing QKD protocols and security proofs with minimal assumptions about the devices used [6].", "Of particular practical importance are two recently developed protocols that do not require trusted single photon detectors (SPDs) [7], [8].", "One of these, the so-called measurement-device-independent QKD (MDI-QKD) protocol, has already been implemented experimentally [9], [10], [11], [12].", "Hence, it is foreseeable that it will play an important role in the future of QKD, and it is thus important to understand the interplay between experimental imperfections (which will always remain in real systems) and system performance to maximize the latter.", "In this work, we derive a widely applicable mathematical model describing systems that implement the MDI-QKD protocol.", "The model is based on facts about our [9], and other existing experimental setups [10], [11], [12], and takes into account carefully characterized imperfect state preparation, loss in the quantum channel, as well as limited detector efficiency and noise.", "It is tested by comparing its predictions with data taken with a proof-of-principle QKD system [9] employing time-bin qubits and implemented in a laboratory environment.", "Our model, which contains no free parameter, reproduces the experimental data within statistical uncertainties over three orders of magnitude of a relevant parameter.", "The excellent agreement allows optimizing central parameters that determine secret key rates, such as mean photon numbers used to encode qubits, and to identify rate-limiting components for future system improvement.", "In addition, we also find that the model accurately reproduces experimental data obtained over deployed fibers, showing that our system minimizes environment-induced perturbation to quantum key distribution in real-world settings.", "This paper is organized in the following way: In section  we detail some of the side-channel attacks (i.e.", "attacks exploiting incorrect assumptions about the working of QKD devices) proposed so far and review technological countermeasures.", "In section  we briefly describe the MDI-QKD protocol, which instead exploits fundamental quantum physical laws to render the most important of these attacks useless.", "Our model of MDI-QKD systems is presented in section .", "This section is followed by an in-depth account of experimental imperfections that affect MDI-QKD performance and a description of how we characterized them in our system (section ).", "Section  shows the results of the comparison between modelled and measured quantities, and section  details how to optimize the performance of our MDI-QKD system using the model.", "Finally, we conclude the article in section .", "Figure: Schematics for MDI-QKD.", "Charlie facilitates the key distribution between Alice and Bob without being able to learn the secret key." ], [ "Side-channel attacks ", "A healthy development of QKD requires investigating the vulnerabilities of QKD implementations in terms of potential side-channel attacks.", "Side-channels in QKD are channels over which information about the key may leak out unintentionally.", "One of the first QKD side-channel attacks proposed was the photon number splitting (PNS) attack [13] in which the eavesdropper, Eve, exploits the fact that attenuated laser pulses sometimes include more than one photon to obtain information about the key.", "This attack can be detected if the decoy state protocol [14], [15], [16] is implemented.", "In the decoy state protocol, Alice varies the mean photon number per pulse in order to allow her and Bob to distill the secret key only from information stemming from single photon emissions.", "More proposals of side-channel attacks followed, including the Trojan-horse attack [17], for which the countermeasure is an optical isolator [17], and the phase remapping attack [18], for which the countermeasure is phase randomization [18].", "Later on, attacks that took advantage of SPD vulnerabilities were also proposed and demonstrated [19], [20], [21], [22].", "For example, the time-shift attack [20] exploits a difference in the quantum efficiencies of the SPDs used in a QKD system.", "This attack can be prevented by actively selecting one of the two bases for the projection measurement, as well as by monitoring the temporal distribution of photon detections [20].", "Another example is the detector blinding attack [22] in which the eavesdropper uses high intensity pulses to modify the performance (i.e.", "blind) the SPDs.", "It can be detected by monitoring the intensity of light at the entrance of Bob's devices with a photodiode [22], [23], [24].", "Nevertheless, due to its power, the blinding attack is currently of particular concern.", "It is important to mention that open side-channels do not necessarily compromise the security of the final key if the information that Eve may have obtained through an attack is properly removed during privacy amplification.", "However, as technological fixes (as discussed above) or additional privacy amplification can only thwart known attacks, it is important to develop and implement protocols that use a minimum number of assumptions about the devices used to implement the protocol.", "An important example is the measurement-device-independent QKD protocol, which we will introduce in the next section." ], [ "The measurement-device-independent quantum key distribution protocol", "The MDI-QKD protocol is a time-reversed version of entanglement-based QKD.", "In this protocol, the users, Alice and Bob, are each connected to Charlie, a third party, through a quantum channel, e.g.", "optical fiber (see Fig.", "REF ).", "In the ideal version, the users have a source of single photons that they prepare randomly in one of the BB84 qubit states [25] $\\vert 0\\rangle $ , $\\vert 1\\rangle $ , $\\vert +\\rangle $ and $\\vert -\\rangle $ , where $\\vert \\pm \\rangle = 2^{-1/2}(\\vert 0\\rangle \\pm \\vert 1\\rangle )$ .", "The qubits are sent to Charlie where the SPDs are located.", "Charlie performs a partial Bell state measurement (BSM) through a 50/50 beam splitter and then announces the events for which the measurement resulted in a projection onto the $\\vert \\psi ^-\\rangle = 2^{-1/2} (\\vert 0\\rangle _A\\vert 1\\rangle _B - \\vert 1\\rangle _A\\vert 0\\rangle _B)$ state.", "Alice and Bob then publicly exchange information about the used bases (z, spanned by $\\vert 0\\rangle $ and $\\vert 1\\rangle $ , or x, spanned by $\\vert +\\rangle $ and $\\vert -\\rangle $ ).", "Associating quantum states with classical bits (e.g.", "$\\vert 0\\rangle ,\\vert -\\rangle \\equiv ~$ 0, and $\\vert 1\\rangle ,\\vert +\\rangle \\equiv $  1) and keeping only events in which Charlie found $\\vert \\psi ^-\\rangle $ and they picked the same basis, Alice and Bob now establish anti-correlated key strings.", "(Note that a projection of two photons onto $\\vert \\psi ^-\\rangle $ indicates that the two photons, if prepared in the same basis, must have been in orthogonal states.)", "Bob then flips all his bits, thereby converting the anti-correlated strings into correlated ones.", "Next, the so-called x-key is formed out of all key bits for which Alice and Bob prepared their photons in the x-basis; its error rate is used to bound the information an eavesdropper may have acquired during photon transmission.", "Furthermore, Alice and Bob form the z-key out of those bits for which both picked the z-basis.", "Finally, they perform error correction and privacy amplification[1], [2] to the z-key, which results in the secret key.", "The advantage of the MDI-QKD protocol over conventional prepare-and-measure or entangled photon-based QKD protocols is that, in the case of Charlie performing an ideal (partial) BSM as described above, detection events are uncorrelated with the final secret key bits.", "This is because a projection onto $\\vert \\psi ^-\\rangle $ only indicates that Alice and Bob sent orthogonal states, but does not reveal who sent which state.", "As a result, Charlie (or Eve) is unable to gain any information about the key from passively monitoring the detectors.", "Furthermore, a measurement that is different from the ideal BSM leads to an increased error rate and thus to a smaller, but still secret, key once privacy amplification has been applied.", "Notably, it does not matter wether the difference is due to experimental imperfections or to an eavesdropper (possibly Charlie himself) trying to gather information about the states that Alice and Bob sent by replacing or modifying the measurement apparatus.", "Hence, all detector side channels are closed in MDI-QKD.", "In the ideal scenario introduced above, Alice and Bob use single photon sources to generate qubits.", "However, it is possible to implement the protocol using light pulses attenuated to the single photon level.", "Indeed, as in prepare-and-measure QKD, randomly varying the mean photon number of photons per attenuated light pulse between a few different values (so-called decoy and signal states) allows making the protocol practical while protecting against a possible PNS attack [7], [26].", "The secret key rate is then given by [7]: $S=Q_{11}^z\\big (1-h_2(e_{11}^x)\\big ) - Q_{\\mu \\sigma }^z f h_2(e_{\\mu \\sigma }^z ),$ where $h_2$ is the binary entropy function, $f$ indicates the error correction efficiency, $Q$ indicates the gain (the probability of a projection onto $\\vert \\psi ^-\\rangle $ per emitted pair of pulses [27]) and $e$ indicates error rates (the ratio of erroneous to total projections onto $\\vert \\psi ^-\\rangle $ ).", "Furthermore, the superscripts, $x$ or $z$ , denote if gains or error rates are calculated for qubits prepared in the x- or the z-basis, respectively.", "Similarly, the subscripts, $\\mu $ and $\\sigma $ , show that the quantity under concern is calculated or measured for pulses with mean photon number $\\mu $ (sent by Alice) and $\\sigma $ (sent by Bob), respectively.", "Finally, the subscript 11 indicates quantities stemming from detection events for which the pulses emitted by Alice and Bob contain only one photon each.", "Note that $Q_{11}$ and $e_{11}$ cannot be measured; their values must be bounded using either a decoy state method, or employing qubit tagging [13].", "However, the latter yields smaller key rates and distances than the former.", "Shortly after the original proposal [7], a practical decoy state protocol for MDI-QKD was proposed [26].", "It requires Alice and Bob to randomly pick mean photon numbers between two decoy states and a signal state.", "One of the decoy states must have a mean photon number lower than the signal state, while the other one must be vacuum.", "A finite number of decoy states results in a lower bound for $Q_{11}^{x,z}$ and an upper bound for $e_{11}^x$ , which in turn gives a lower bound for the secret key rate in Eq.", "(REF ).", "We will elaborate more on decoy states in section REF ." ], [ "The model", "Our model takes into account imperfections present in a typical QKD system.", "Regarding the sources, located at Alice and Bob, we take into account imperfect preparation of the quantum state of each photon.", "Furthermore, we consider transmission loss of the links between Alice and Charlie, and Bob and Charlie.", "And finally, concerning the measurement apparatus at Charlie's, we consider imperfect projection measurement stemming from non-maximum quantum interference on Charlie's beam splitter, detector noise such as dark counts and afterpulsing, and limited detector efficiency.", "See also [28] for another model describing MDI-QKD performance, but with a more restrictive set of imperfections and not yet tested against actual experimental data.", "In the following paragraphs we present a detailed description of our model.", "It relies on the assumption of phase randomized laser pulses at Charlie's.", "While Alice and Bob generate coherent states in our proof-of-principle setup, this assumption is correct as the long fibres used to connect Alice and Bob with Charlie introduce random global phase variations (we will discuss the impact of the lack of phase randomization at Alice's and Bob's on the security of distributed keys in section ).", "We note that, in order to facilitate explanations, we have adopted the terminology of time-bin encoding.", "However, our model is general and can also be applied to MDI-QKD systems implementing other types of encoding [11]." ], [ "State preparation ", "In the MDI-QKD protocol, Alice and Bob derive key bits whenever Charlie announces a projection onto the $\\vert \\psi ^-\\rangle $ Bell state.", "We model the probability of a $\\vert \\psi ^-\\rangle $ projection for various quantum states of photons emitted by Alice and Bob as a function of the mean photon number per pulse ($\\mu $ and $\\sigma $ , respectively) and transmission coefficients of the fiber links ($t_A$ and $t_B$ , respectively).", "We consider photons in qubit states described by: $\\vert \\psi \\rangle =\\frac{1}{\\sqrt{1+2b^{x,z}}}\\left(\\sqrt{m^{x,z}+b^{x,z}}\\vert 0\\rangle + e^{i\\phi ^{x,z}}\\sqrt{1-m^{x,z}+b^{x,z}}\\vert 1\\rangle \\right)$ where $\\vert 0\\rangle $ and $\\vert 1\\rangle $ denote orthogonal modes (i.e.", "early and late temporal modes assuming time-bin qubits), respectively.", "Note that $\\vert \\psi \\rangle $ describes any pure state [29] and the presence of the $m^{x,z}$ and $b^{x,z}$ terms in Eq.", "(REF ), as opposed to using only one parameter, is motivated by the fact that they model different experimentally characterizable imperfections.", "In the ideal case, $m^{z}$ $\\in [0,1]$ for photon preparation in the z-basis (in this case, the value of $\\phi ^z$ is irrelevant), $m^{x}=\\frac{1}{2}$ and $\\phi ^{x}\\in [0,\\pi ]$ for the x-basis, and $b^{x,z}=0$ for both bases.", "Imperfect preparation of photon states is modelled by using non-ideal $m^{x,z}$ , $\\phi ^{x,z}$ and $b^{x,z}$ for Alice and Bob.", "The parameter $b^{x,z}$ is included to represent the background light emitted and modulated by an imperfect source.", "Furthermore, in principle, the various states generated by Alice and Bob could have differences in other degrees of freedom (i.e.", "polarization, spectral, spatial, temporal modes).", "This is not included in Eq.", "(REF ), but would be reflected in a reduced quality of the BSM, which will be discussed below." ], [ "Conditional probability for projections onto $\\vert \\psi ^-\\rangle $ ", "A projection onto $\\vert \\psi ^-\\rangle $ occurs if one of the SPDs after Charlie's 50/50 beam splitter signals a detection in an early time-bin (a narrow time interval centered on the arrival time of photons occupying an early temporal mode) and the other detector signals a detection in a late time-bin (a narrow time-interval centered on the arrival time of photons occupying a late temporal mode).", "Note that, in the following paragraphs, this is the desired detection pattern we search for when modeling possible interference cases or noise effects.", "Also, note that we assume that Charlie's two single-photon detectors have identical properties.", "A deviation from this approximation does not open a potential security loophole (in contrast to prepare-and-measure and entangled photon based QKD), as all detector side-channel attacks are removed in MDI-QKD.", "We build up the model by first considering the probabilities that particular outputs from the beam splitter (at Charlie's) will generate the detection pattern associated with a projection onto $\\vert \\psi ^-\\rangle $ .", "The outputs are characterized by the number of photons per output port as well as their joint quantum state.", "The probabilities for each of the possible outputs to occur can then be calculated based on the inputs to the beam splitter (characterized by the number of photons per input port and their quantum states, as defined in Eq.", "(REF )).", "Note that for the simple cases of inputs containing zero or one photon (summed over both input modes), we calculate the probabilities leading to the desired detection pattern directly, i.e.", "without going through the intermediate step of calculating outputs from the beam splitter.", "Finally, the probability for each input to occur is calculated based on the probability for Alice and Bob to send attenuated light pulses containing exactly $i$ photons, all in a state given by Eq.", "(REF ).", "The probability for a particular input to occur also depends on the transmissions of the quantum channels, $t_A$ and $t_B$ .", "We note that this model considers up to three photons incident on the beam splitter.", "This is sufficient as, in the case of heavily attenuated light pulses and lossy transmission, higher order terms do not contribute significantly to projections onto $\\vert \\psi ^-\\rangle $ .", "However, we limit the following description to two photons at most: the extension to three is lengthy but straightforward and follows the methodology presented for two photons." ], [ "Detector noise ", "Let us begin by considering the simplest case in which no photons are input into the beam splitter.", "In this case, detection events can only be caused by detector noise.", "We denote the probability that a detector indicates a spurious detection as $P_{n}$ .", "Detector noise stems from two effects: dark counts and afterpulsing [32].", "Dark counts represent the base level of noise in the absence of any light, and we denote the probability that a detector generates a dark count per time-bin as $P_{d}$ .", "Afterpulsing is an additional noise source produced by the detector as a result of prior detection events.", "The probability of afterpulsing depends on the total count rate, hence we denote the afterpulsing probability per time-bin as $P_a$ , which is a function of the mean photon number per pulse from Alice and Bob ($\\mu $ and $\\sigma $ ), the transmission of the channels ($t_A$ and $t_B$ ) and the efficiency of the detectors ($\\eta $ ) located at Charlie (see below for afterpulse characterization).", "The total probability of a noise count in a particular time-bin is thus $P_{n} = P_{d} + P_{a}$ .", "All together, we find the probability for generating the detection pattern associated with a projection onto the $\\vert \\psi ^-\\rangle $ -state, conditioned on having no photons at the input, specified by “in\", of the beam splitter, to be : $P(\\vert \\psi ^-\\rangle | \\mbox{0 photons, in}) = P(\\vert \\psi ^-\\rangle | \\mbox{0 photons, out})= 2P_{n}^2,\\\\$ Here and henceforward, we have ignored the multiplication factor (1-$P_{n}) \\sim 1$  [30], which indicates the probability that a noise event did not occur in the early time-bin (this is required in order to see a detection during the late time-bin assuming detectors with recovery time larger than the separation between the $\\vert 0\\rangle $ and $\\vert 1\\rangle $ temporal modes).", "Note that the probability conditioned on having no photons at the inputs of the beam splitter equals the one conditioned on having no photons at the outputs (specified in Eq.", "(REF ) by the conditional “out\")." ], [ "One-photon case ", "Next, we consider the case in which a single photon arrives at the beam splitter.", "To generate the detection pattern associated with $\\vert \\psi ^-\\rangle $ , either the photon must be detected and a noise event must occur in the other detector in the opposite time-bin, or, if the photon is not detected, two noise counts must occur as in Eq.", "(REF ).", "We find $P(\\vert \\psi ^-\\rangle | \\mbox{1 photon, in}) = \\eta P_{n} + (1-\\eta )P(\\vert \\psi ^-\\rangle | \\mbox{0 photons, out}),$ where $\\eta $ denotes the probability to detect a photon that occupies an early (late) temporal mode during an early (late) time-bin (we assume $\\eta $ to be the same for both detectors)." ], [ "Two-photon case ", "We now consider detection events stemming from two photons entering the beam splitter.", "The possible outputs can be broken down into three cases.", "In the first case, both photons exit the beam splitter in the same output port and are directed to the same detector.", "This yields only a single detection event, even if the photons are in different temporal modes (the latter is due to detector dead time.", "Note that as our model calculates detections in units of bits per gate, modeling a dead-time free detector is straightforward.).", "The probability for Charlie to declare a projection onto $\\vert \\psi ^-\\rangle $ is then $P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, 1 spatial mode, out}) =\\nonumber \\\\\\hspace{8.53581pt} (1-(1-\\eta )^2)P_{n} + (1-\\eta )^2P(\\vert \\psi ^-\\rangle | \\mbox{0 photons, out}).$ In the second case, the photons are directed towards different detectors and occupy the same temporal mode.", "Hence, to find detections in opposite time-bins in the two detectors, at least one photon must not be detected.", "This leads to $P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, 2 spatial modes, 1 temporal mode, out}) = \\nonumber \\\\\\hspace{8.53581pt} 2\\eta (1-\\eta )P_{n} + (1-\\eta )^2P(\\vert \\psi ^-\\rangle | \\mbox{0 photons, out}).$ In the final case, both photons occupy different spatial as well as temporal modes.", "In contrast to the previous case, a projection onto $\\vert \\psi ^-\\rangle $ can now also originate from the detection of both photons.", "This leads to $P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, 2 spatial modes, 2 temporal modes, out}) = \\nonumber \\\\\\hspace{8.53581pt} \\eta ^2 + 2\\eta (1-\\eta )P_{n} + (1-\\eta )^2P(\\vert \\psi ^-\\rangle | \\mbox{0 photons, out}).$ In order to find the probability for each of these three two-photon outputs to occur, we must examine two-photon inputs to the beam splitter.", "We note that it is possible for the two photons to be subject to a two-photon interference effect (known as photon bunching) when impinging on the beam splitter.", "As this quantum interference can lead to an entangled state between the output modes, the calculation must proceed with quantum mechanical operators.", "We consider three cases: two photons arrive at the same input of the beam splitter, one photon arrives at each input of the beam splitter and the two photons are distinguishable, and one photon arrives at each input of the beam splitter and the two photons are indistinguishable.", "For ease of analysis, we first introduce some notation: $p^{x,z}(0,0) & \\equiv & (m^{x,z}_1+b^{x,z}_1)(m^{x,z}_2+b^{x,z}_2) \\nonumber \\\\p^{x,z}(0,1) & \\equiv & (m^{x,z}_1+b^{x,z}_1)(1 - m^{x,z}_2+b^{x,z}_2)\\nonumber \\\\p^{x,z}(1,0) & \\equiv & (1-m^{x,z}_1+b^{x,z}_1)(m^{x,z}_2+b^{x,z}_2)\\nonumber \\\\p^{x,z}(1,1) & \\equiv & (1-m^{x,z}_1+b^{x,z}_1)(1-m^{x,z}_2+b^{x,z}_2)\\nonumber \\\\b^{x,z}_{norm} & \\equiv & 1+ 2b^{x,z}_1 + 2b^{x,z}_2 +4b^{x,z}_1b^{x,z}_2$ where $b^{x,z}_{1,2}$ and $m^{x,z}_{1,2}$ are the parameters introduced in Eq.", "(REF ); the subscripts label the photon (one or two) whose state is specified by the parameters.", "Furthermore, $p^{x,z}(i,j)$ is proportional to finding photon one before the beam-splitter in temporal mode $i$ and photon two in temporal mode $j$ , where $i,j \\in [0,1]$ .", "Finally, $b^{x,z}_{norm}$ is a normalization factor.", "First, considering the situation in which the two photons impinge from the same input on the beam splitter, one has the state $\\vert \\psi _{input}\\rangle = \\left( \\frac{1}{\\sqrt{1+2b^{x,z}}} \\left(\\sqrt{m^{x,z}+b^{x,z}} \\ \\hat{a}^\\dagger (0)+ e^{i\\phi ^{x,z}}\\sqrt{1-m^{x,z}+b^{x,z}} \\ \\hat{a}^\\dagger (1)\\right) \\right)^{\\otimes 2} \\vert vac\\rangle ,$ where $\\hat{a}^\\dagger (0)$ and $\\hat{a}^\\dagger (1)$ are the creation operators for a photon in the $\\vert 0\\rangle $ or $\\vert 1\\rangle $ state, respectively.", "Evolving this state through the standard unitary transformation for a lossless, 50/50 beam splitter, described by $\\hat{a}^\\dagger \\rightarrow (\\hat{c}^\\dagger + \\hat{d}^\\dagger )/\\sqrt{2}$ (where $\\hat{c}^\\dagger $ and $\\hat{d}^\\dagger $ are the two output modes of the beam splitter), one finds that with probability $1/2$ the two photons exit the beam splitter in the same output port (or spatial mode) and with probability $1/2$ in different ports.", "Furthermore, with probability $A = [p^{x,z}(0,0) + p^{x,z}(1,1)]/2b^{x,z}_{\\mbox{norm}}$ we find the photons in different spatial modes and in the same temporal mode, and with probability $B = [p^{x,z}(0,1) + p^{x,z}(1,0)]/2b^{x,z}_{\\mbox{norm}}$ we find the photons in different spatial and temporal modes.", "By symmetry, we find the same result if the two photons arrive from the other input mode of the beam splitter.", "Thus the probability that Charlie finds the desired detection pattern is: $P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, 1 spatial mode, in} ) = \\nonumber \\\\\\hspace{8.53581pt} \\frac{1}{2}P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, 1 spatial mode, out}) \\nonumber \\\\\\hspace{8.53581pt} + A \\times P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, 2 spatial modes, 1 temporal mode, out}) \\nonumber \\\\\\hspace{8.53581pt} + B \\times P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, 2 spatial modes, 2 temporal modes, out}).\\nonumber \\\\$ Second, consider the situation in which the two photons come from different inputs, and are completely distinguishable in some degree of freedom.", "This can be modelled by starting with the input state $\\vert \\psi _{input}\\rangle &= \\frac{1}{\\sqrt{1+2b_1^{x,z}}} \\left(\\sqrt{m_1^{x,z}+b_1^{x,z}} \\ \\hat{a}^\\dagger (0) + e^{i\\phi _1^{x,z}}\\sqrt{1-m_1^{x,z}+b_1^{x,z}} \\ \\hat{a}^\\dagger (1)\\right) \\nonumber \\\\&\\otimes \\frac{1}{\\sqrt{1+2b_2^{x,z}}} \\left(\\sqrt{m_2^{x,z}+b_2^{x,z}} \\ \\hat{b}^\\dagger (0) + e^{i\\phi _2^{x,z}}\\sqrt{1-m_2^{x,z}+b_2^{x,z}} \\ \\hat{b}^\\dagger (1)\\right) \\vert vac\\rangle , $ where $\\hat{b}^\\dagger $ is the creation operator for a photon in the second input mode of the beam splitter.", "One can then evolve the state with the beam splitter unitary described by $\\hat{a}^\\dagger \\rightarrow (\\hat{c}^\\dagger + \\hat{d}^\\dagger )/\\sqrt{2}$ (as before) and $\\hat{b}^\\dagger \\rightarrow (-\\hat{e}^\\dagger + \\hat{f}^\\dagger )\\sqrt{2}$ , where $\\hat{c}^\\dagger $ and $\\hat{e}^\\dagger $ correspond to the same spatial output mode but with distinguishability in another degree of freedom, and similarly for the other spatial output mode described by $\\hat{d}^\\dagger $ and $\\hat{f}^\\dagger $ .", "One finds the same result as for the previous case, described by Eq.", "(REF ): $P&(\\vert \\psi ^-\\rangle | \\mbox{2 photons, 2 spatial modes, non-interfering, in} )\\nonumber \\\\ &= P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, 1 spatial mode, in} ) \\nonumber \\\\&\\equiv P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, non-interfering, in} ).$ The definition reflects that there is no two-photon interference in both cases.", "Finally, consider the case in which the two photons impinge from different inputs are indistinguishable, and interfere on the beam splitter.", "This can be modelled by considering the same input state as in Eq.", "(REF ), but using a beam splitter unitary described by $\\hat{a}^\\dagger \\rightarrow (\\hat{c}^\\dagger + \\hat{d}^\\dagger )/\\sqrt{2}$ (as before) and $\\hat{b}^\\dagger \\rightarrow (-\\hat{c}^\\dagger + \\hat{d}^\\dagger )/\\sqrt{2}$ .", "In this case, the probabilities of finding the outputs from the beam splitter discussed in Eqs.", "(REF -REF ) depend on the difference between the phases $\\phi ^{x,z}_1$ and $\\phi ^{x,z}_2$ that specify the states of photons one and two, $\\Delta \\phi ^{x,z} \\equiv \\phi ^{x,z}_1 - \\phi ^{x,z}_2$ .", "Note that, due to the two-photon interference effect, finding the two photons in different spatial modes and the same temporal mode is impossible.", "We are thus left with the case of having two photons in the same output port (the same spatial mode), which occurs with probability $C = [p^{x,z}(0,0) + p^{x,z}(1,1) + 0.5(p^{x,z}(0,1) + p^{x,z}(1,0)) + \\sqrt{p^{x,z}(0,1)p^{x,z}(1,0)}\\cos (\\Delta \\phi ^{x,z})]/b^{x,z}_{norm}$ , and the case of having the photons in different temporal and spatial modes, which occurs with probability $D = [0.5(p^{x,z}(0,1) + p^{x,z}(1,0)) -\\sqrt{p^{x,z}(0,1)p^{x,z}(1,0)}\\cos (\\Delta \\phi ^{x,z})]/b^{x,z}_{norm} $ .", "This leads to $P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, interfering, in} ) = \\nonumber \\\\\\hspace{8.53581pt} C \\times P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, 1 spatial mode, out}) + \\nonumber \\\\\\hspace{8.53581pt} D \\times P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, 2 spatial modes, 2 temporal modes, out}).$" ], [ "Aggregate probability for projections onto $\\vert \\psi ^-\\rangle $ ", "Now that we have calculated the conditional probabilities of a detection pattern indicating $\\vert \\psi ^-\\rangle $ for various inputs to the beam splitter, let us consider with what probability each case occurs.", "This requires that we know the photon number distribution of the pulses arriving at Charlie's beam splitter from Alice and Bob, which can be computed based on the photon number distribution at the sources and the properties of the quantum channels.", "For the following discussion, we assume that the channels from Alice to Charlie, and from Bob to Charlie are characterized by the loss $t_A$ and $t_B$ , respectively, yielding pulses with number distribution $\\mathbb {D}$ and mean photon number, $\\mu t_A$ and $\\sigma t_B$ , respectively.", "This is equivalent to assuming that no PNS attack takes place, which was ensured by performing experiments with the entire setup (including the fiber transmission lines) inside a single laboratory in which no eavesdropping took place during the experiments.", "We limit our discussion to the cases with two or less photons at the input of the beam splitter (but recall that the actual calculation includes up to three photons).", "Hence, the cases we consider and their probabilities of occurrence, $P_O$ , are given by: 0 photons at the input from both sources: $P_O=\\mathbb {D}_0(\\mu t_A)\\mathbb {D}_0(\\sigma t_B)$ 1 photon at the input from Alice and 0 photons from Bob: $P_O=\\mathbb {D}_1(\\mu t_A)\\mathbb {D}_0(\\sigma t_B)$ 0 photons at the input from Alice and 1 photon from Bob: $P_O=\\mathbb {D}_0(\\mu t_A)\\mathbb {D}_1(\\sigma t_B)$ 2 photons at the input from Alice and 0 photons from Bob: $P_O=\\mathbb {D}_2(\\mu t_A)\\mathbb {D}_0(\\sigma t_B)$ 0 photons at the input from Alice and 2 photons from Bob: $P_O=\\mathbb {D}_0(\\mu t_A)\\mathbb {D}_2(\\sigma t_B)$ 1 photon at the input from both sources: $P_O=\\mathbb {D}_1(\\mu t_A)\\mathbb {D}_1(\\sigma t_B)$ where we denote the probability of having $i$ photons from a distribution $\\mathbb {D}$ with mean number $\\mu $ as $\\mathbb {D}_i(\\mu )$ .", "For each of these cases, we have already computed the probability that Charlie obtains the detection pattern associated with the $\\vert \\psi ^-\\rangle $ -state for arbitrary input states of the photons (as defined in Eq.", "(REF )).", "When zero or one photons arrive at the beam splitter, Eq.", "(REF ) and Eq.", "(REF ) are used, respectively.", "In the case in which two photons arrive from the same source, Eq.", "(REF ) is used.", "Finally, in the case in which one photon arrives from each source at the beam splitter, Eq.", "(REF ) would be used in the ideal case.", "However, perfect indistinguishability of the photons cannot be guaranteed in practice.", "We characterize the degree of indistinguishability by the visibility, $V$ , that we would observe in a closely-related Hong-Ou-Mandel (HOM) interference experiment [33] with single-photon inputs.", "Taking into account partial distinguishability, the probability of finding a detection pattern corresponding to the projection onto $\\vert \\psi ^-\\rangle $ is given by $P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, visibility $V$, in}) = \\nonumber \\\\\\hspace{8.53581pt} VP(\\vert \\psi ^-\\rangle | \\mbox{2 photons, interfering, in}) \\nonumber \\\\\\hspace{8.53581pt} + (1-V)P(\\vert \\psi ^-\\rangle | \\mbox{2 photons, non-interfering, in}).$ Equations REF -REF detail all possible causes for observing the detection pattern associated with a projection onto the $\\vert \\psi ^-\\rangle $ Bell state, if up to two photons at the beam splitter input are taken into account.", "We remind the reader that all calculations in the following sections take up to three photons at the input of the beam splitter into account.", "To calculate the gains, $Q_{\\mu \\sigma }^{x,z}$ , using these equations, we need only substitute in the correct values of $\\mu $ , $\\sigma $ , $t_A$ , $t_B$ , $m^{x,z}$ , $b^{x,z}$ , and $\\Delta \\phi ^{x,z}$ for the cases in which Alice and Bob both sent attenuated light pulses in the x-basis or z-basis, respectively.", "The error rates, $e_\\mu ^{x,z}$ , can then be computed by separating the projections onto $\\vert \\psi ^-\\rangle $ into those where Alice and Bob sent photons in different states (yielding correct key bits) and in the same state (yielding erroneous key bits).", "More precisely, the error rates, $e_{\\mu \\sigma }^{x,z}$ , are calculated as $e_{\\mu \\sigma }^{x,z}=p^{x,z}_{wrong}/(p^{x,z}_{correct}+p^{x,z}_{wrong}$ ) where $p^{x,z}_{wrong}$ ($p^{x,z}_{correct}$ ) denotes the probability for detections yielding an erroneous (correct) bit in the $x$ (or $z$ )-key." ], [ "Characterizing experimental imperfections", "The parameters used to model our system are derived from data established through independent measurements.", "To test our model, the characterization of experimental imperfections in our MDI-QKD implementation [9] is very technical at times.", "It can be broken down into time-resolved energy measurements at the single photon level (required to extract $\\mu $ , $\\sigma $ , $b^{x,z}$ and $m^{x,z}$ for Alice and Bob, as well as dark count and afterpulsing probabilities), measurements of phase (required to establish $\\phi ^{x,z}$ for Alice and Bob), and visibility measurements.", "In the following paragraphs we describe the procedures we followed to obtain these parameters from our system." ], [ "Our MDI-QKD implementation", "In our implementation of MDI-QKD [9] Alice's and Bob's setups are identical.", "Each setup consists of a CW laser with large coherence time, emitting at 1550nm wavelength.", "Time-bin qubits, encoded into single photon-level light pulses with Poissonian photon number statistics, are created through an attenuator, an intensity modulator and a phase modulator located in a temperature controlled box.", "More precisely, the intensity modulator is used to tailor pulse pairs out of the cw laser light, the phase modulator is used to change their relative phase, and the attenuator attenuates these pulses to the single-photon level.", "The two temporal modes defining each time-bin qubit are of 500 ps (FWHM) duration and are separated by 1.4 ns.", "Each source generates qubits at 2 MHz rate.", "We emphasize that our qubit generation procedure justifies the assumption of a pure state in Eq.", "(REF ).", "Indeed, all photons, including background photons due to light leaking through imperfect intensity modulators, have to be generated by the CW lasers whose coherence times exceeds the separation between the temporal modes $\\vert 0\\rangle $ and $\\vert 1\\rangle $  [31].", "Note that in all experiments reported to date [9], [10], [11], [12] background photons always add coherently to the modes describing qubits, making our pure-state description widely applicable.", "The time-bin qubits are sent to Charlie through an optical fiber link.", "The link consisted of spooled fiber (for the measurements in which Alice, Bob and Charlie were all located in the same laboratory) or deployed fiber (for the measurements in which the three parties were located in different locations within the city of Calgary).", "We remind the reader that all pulses arriving at Charlie's are phase randomized, due to the use of long fibers.", "Charlie performs a BSM on the qubits he receives using a 50/50 beamsplitter and two SPDs.", "See Figure REF .", "Note that, in order to perform a Bell state measurement the photons arriving to Charlie must be indistinguishable in all degrees of freedom: polarization, frequency, time and spatial mode.", "The indistinguishability of the photons is assessed through a Hong-Ou-Mandel interference measurement [33].", "As our system employs attenuated laser pulses, the maximum visibility we can obtain in this measurement is $V_{max}=50\\%$ (and not 100% as it would be with single photons) [34].", "In our implementation the visibility measurements resulted in $V=(47 \\pm 1)$ , irrespective of whether they were taken with spooled fiber inside the lab, or over deployed fiber.", "Figure: Time-bin qubits are created at Alice's and Bob's through a CW laser (LD), attenuator (ATT), and frequency shifter (FS) and temperature-controlled intensity (IM) and phase (PM) modulator.", "The projective measurements are done at Charlie's via a beam splitter (BS) and two single photon detectors (SPDs)." ], [ "Time-resolved energy measurements ", "First, we characterize the dark count probability per time-bin, $P_{d}$ , of the SPDs (InGaAs-avalanche photodiodes operated in gated Geiger mode [32]) by observing their count rates when the optical inputs are disconnected.", "We then send attenuated laser pulses so that they arrive just after the end of the 10 ns long gate that temporarily enables single photon detection.", "The observed change in the count rate is due to background light transmitted by the intensity modulators (whose extinction ratios are limited) and allows us to establish $b^{x,z}$ (per time-bin) for Alice and Bob.", "Next, we characterize the afterpulsing probability per time-bin, $P_{a}$ , by placing the pulses within the gate, and observing the change in count rate in the region of the gate prior to the arrival of the pulse.", "The afterpulsing model we use to assess $P_{a}$ from these measurements is described below.", "Once the background light and the sources of detector noise are characterized, the values of $m^{x,z}$ can be calculated by generating all required states and observing the count rates in the two time-bins corresponding to detecting photons generated in early and late temporal modes.", "Observe that $m^{z=1}$ for photons generated in state $\\vert 1\\rangle $ (the late temporal mode) is zero, since all counts in the early time-bin are attributed to one of the three sources of background described above.", "Furthermore, we observed that $m^{z=0}$ for photons generated in the $\\vert 0\\rangle $ state (the early temporal mode) is smaller than one due to electrical ringing in the signals driving the intensity modulators.", "Note that, in our implementation, the duration of a temporal mode exceeds the width of a time-bin, i.e.", "it is possible to detect photons outside a time-bin (see Figure REF for a schematical representation).", "Hence, it will be useful to also define the probability for detecting a photon arriving at any time during a detector gate; we will refer to this quantity as $\\eta _{gate}$ .The count rate per gate, after having subtracted the rates due to background and detector noise, together with the detection efficiency, $\\eta _{gate}$ ($\\eta _{gate}$ , as well as $\\eta $ , have been characterized previously based on the usual procedure [32]), allows calculating the mean number of photons per pulse from Alice or Bob ($\\mu $ or $\\sigma $ , respectively).", "The efficiency coefficient relevant for our model, $\\eta $ , is smaller than $\\eta _{gate}$ .", "Finally, we point out that the entire characterization described above was repeated for all experimental configurations investigated (the configurations are detailed in Table REF ).", "We found all parameters to be constant in $\\mu \\sigma t_At_B$ , with the obvious exception of the afterpulsing probability.", "Figure: Sketch (not to scale) of the probability density p(t) for a detection event to occur as a function of time within one gate.", "Detection events can arise from a photon within an optical pulse (depicted here as a pulse in the late temporal mode), or be due to optical background, a dark count, or afterpulsing.", "Also shown are the 400 ps wide time-bins.", "Within the early time-bin only optical background, dark counts and afterpulsing give rise to detection events in this case.", "Note that the width of the temporal mode exceeds the widths of the time-bins." ], [ "Phase measurements ", "To detail the assessment of the phase values $\\phi ^{x,z}$ determining the superposition of photons in early and late temporal modes, let us assume for the moment that the lasers at Alice's and Bob's emit light at the same frequency.", "First, we defined the phase of Bob's $\\vert +\\rangle $ state to be zero (this can always be done by appropriately defining the time difference between the two temporal modes $\\vert 0\\rangle $ and $\\vert 1\\rangle $ ).", "Next, to measure the phase describing any other state (generated by either Alice or Bob) with respect to Bob's $\\vert +\\rangle $ state, we sequentially send unattenuated laser pulses encoding the two states through a common reference interferometer.", "This reference interferometer featured a path-length difference equal to the time-difference between the two temporal modes defining AliceÕs and Bob's qubits.", "For the phase measurement of qubit states $\\vert +\\rangle $ and $\\vert -\\rangle $ (generate by Alice), and $\\vert -\\rangle $ generated by Bob), first, the phase of the interferometer was set such that Bob's $\\vert +\\rangle $ state generated equal intensities in each output of the interferometer (i.e.", "the interferometerÕs phase was set to $\\pi /4$ ).", "Thus, sending any of the other three states through the interferometer and comparing the output intensities, we can calculate the phase difference.", "We note that any frequency difference between Alice's and Bob's lasers results in an additional phase difference.", "Its upper bound for our maximum frequency difference of 10 MHz is denoted by $\\phi _{freq}$ ." ], [ "Measurements of afterpulsing ", "We now turn to the characterization of afterpulsing.", "After a detector click (or detection event, which includes photon detection, dark counts and afterpulsing), the probability of an afterpulse occuring due to that detection event decays exponentially with time.", "The SPDs are gated, with the afterpulse probability per gate being a discrete sampling of the exponential decay.", "This can be expressed using a geometric distribution: supposing a detection event occurred at gate $k=-1$ , the probability of an afterpulse occuring in gate $k$ is given by $P_k =\\alpha p(1-p)^k$ .", "Thus, if there are no other sources of detection events, the probability of an afterpulse occuring due to a detection event is given by $\\sum _{k=0}^\\infty \\alpha p(1-p)^k$ .", "Figure: Afterpulse probability per time-bin as a function of the average number of photons arriving at the detector per gate.In a realistic situation, the geometric distribution for the afterpulses will be cut off by other detection events, either stemming from photons, or dark counts.", "In addition, the SPDs have a deadtime after each detection event during which the detector is not gated until $k \\ge k_{dead}$ (note that time and the number of gates applied to the detector are proportional).", "The deadtime can simply be accounted for by starting the above summation at $k=k_{dead}$ rather than $k=0$ .", "However, for an afterpulse to occur during the $k^{{th}}$ gate following a particular detection event, no other detection events must have occured in prior gates.", "This leads to the following equation for the probability of an afterpulse per detection event: $P(\\mbox{a,det}) = \\sum _{k=k_{dead}}^\\infty \\left(\\gamma \\times \\upsilon \\times \\rho \\times P_k\\right)$ where: $\\hspace{8.53581pt}\\gamma = \\left(1-\\mu _\\mathrm {avg}(\\mu ,\\sigma ,t_A,t_B)\\eta _{gate}\\right)^{k-k_{dead}}\\nonumber \\\\\\nonumber \\\\\\hspace{8.53581pt} \\upsilon = (1 - P_{d,gate})^{k-k_{dead}}\\nonumber \\\\\\nonumber \\\\\\hspace{8.53581pt} \\rho = \\prod _{j=k_{dead}}^{k-1}1 - \\alpha p(1-p)^{j} \\nonumber \\\\\\nonumber \\\\\\hspace{8.53581pt} P_k = \\alpha p(1-p)^k$ and $P_{d,gate}$ denotes the detector dark count probability per gate (as opposed to per time-bin), and $\\mu _\\mathrm {avg}(\\mu ,\\sigma ,t_A,t_B)$ expresses the average number of photons present on the detector during each gate as follows: $\\mu _\\mathrm {avg}(\\mu ,\\sigma ,t_A,t_B) = \\frac{(\\mu +b_A)t_A + (\\sigma + b_B)t_B}{2},$ where $b_A$ and $b_B$ characterize the amount of background light per gate from Alice and Bob, respectively, and the factor of $\\frac{1}{2}$ comes from Charlie's beam splitter.", "The terms in the sum of Eq.", "(REF ) describe the probabilities of neither having an optical detection ($\\gamma $ ), either caused by a modulated pulse or background light, nor a detector dark count ($\\upsilon $ ) in any gate before and including gate $k$ , and not having an afterpulse in any gate before gate $k$ ($\\rho $ ), followed by an afterpulse in gate $k$ ($P_k$ ).", "Equation (REF ) takes into account that afterpulsing within each time-bin is influenced by all detections within each detector gate, and not only those happening within the time-bins that we post-select when acquiring experimental data.", "The afterpulse probability, $P_{a,gate}$ , for given $\\mu $ , $\\sigma $ , $t_A$ and $t_B$ can then be found by multiplying Eq.", "(REF ) by the total count rate $P_{a,gate} = \\left(\\mu _\\mathrm {avg}(\\mu ,\\sigma ,t_A,t_B)\\eta _{gate} + P_{d,gate} + P_{a,gate}\\right)P(\\mbox{a,det}).$ This equation expresses that afterpulsing can arise from prior afterpulsing, which explains the appearance of $P_{a,gate}$ on both sides of the equation.", "Equation (REF ) simplifies to $P_{a,gate} = \\frac{\\left(\\mu _\\mathrm {avg}(\\mu ,\\sigma ,t_A,t_B)\\eta _{gate} + P_{d,gate}\\right)P(\\mbox{a,det})}{1-P(\\mbox{a,det})}.$ Finally, to extract the afterpulsing probability per time-bin, $P_{a}(\\mu ,\\sigma ,t_A,t_B)$ , we note that we found that the distribution of afterpulsing across the gate to be the same as the distribution of dark counts across the gate.", "Hence, $P_{a}(\\mu ,\\sigma , t_A,t_B)= P_{a,gate}\\frac{P_{d}}{P_{d,gate}}.$ Fitting our afterpulse model to the measured afterpulse probabilities, we find $\\alpha =1.79 \\times 10^{-1}$ , $p= 2.90 \\times 10^{-2}$ , and $\\frac{P_{d}}{P_{d,gate}} =4.97 \\times 10^{-2}$ for $k_{dead} = 20$ .", "The fit, along with the measured values, is shown in Figure REF as a function of the average number of photons arriving at the detector per gate $\\mu _\\mathrm {avg}(\\mu ,\\sigma ,t_A,t_B)$ .", "A summary of all the values obtained through these measurements is shown in Table REF .", "Table: Experimentally established values for all parameters required to describe the generated quantum states, as defined in Eq.", "(), as well as two-photon interference parameters and detector properties." ], [ "Comparing modelled with actual performance", "To test our model, and to verify our ability to perform, in principle, QKD with deployed (real-world) fiber, we now compare the model's predictions with experimental data obtained using the QKD system characterized by the parameters listed in Table REF .", "We performed experiments in two configurations: inside the laboratory using spooled fiber (for four different distances between Alice and Bob ranging between 42 km and 103 km), and over deployed fiber (18 km).", "The first configuration allows testing the model, and the second configurations shines light on our system's capability to compensate for environment-induced perturbations, e.g.", "due to temperature fluctuations.", "For each test, three different mean photon numbers (0.1, 0.25 and 0.5) were used.", "All the configurations tested (as well as the specific parameters used in each test) and the results obtained are listed in Table REF .", "In Figure REF we show the simulated values for the error rates ($e^{z,x}$ ) and gains ($Q^{z,x}$ ) predicted by the model as a function of $\\mu \\sigma t_A t_B$ .", "The plot includes uncertainties from the measured parameters, leading to a range of values (bands) as opposed to single values.", "The figure also shows the experimental values of $e^{z,x}$ and $Q^{z,x}$ from our MDI-QKD system in both the laboratory environment and over deployed fiber.", "Considering the data taken inside the lab, the modelled values and the experimental results agree within experimental uncertainties over three orders of magnitude.", "This shows that the model is suitable for predicting error rates and gains.", "In turn, this allows us to optimize performance of our QKD systems in terms of secret key rate (see section ).", "In particular, the model allows optimizing the mean photon number per pulse that Alice and Bob use to encode signal and decoy states as a function of transmission loss, and identifying rate-limiting components.", "Furthermore, the measurement results over deployed fibre are also well described by the same model, indicating that this more-difficult measurement worked correctly.", "The increased difficult across real-world fiber arises due to the fact that BSMs require incoming photons to be indistinguishable in all degrees of freedom (i.e.", "arrive within their respective coherence times, with identical polarization, and with large spectral overlap).", "As we have shown in [9], time-varying properties of optical fibers in the outside environment (e.g.", "temperature dependent polarization and travel-time changes) can remove indistinguishability in less than a minute.", "Active stabilization of these properties is thus required to achieve functioning BSMs and, in fact, three such stabilization systems were deployed during the MDI-QKD measurements presented here (more details are contained in [9]).", "That our measurement results agree with the predicted values of the model demonstrates that the impact of environmental perturbations on the ability to perform Bell state measurements is negligible (which is the same conclusion drawn in [9]).", "Figure: Modelled and measured results.", "Figure a) shows the plot for the error rates in the zz-basis (green band) and in the xx-basis (blue band) as a function of the mean photon number per pulse sent by Alice (μ\\mu ) and Bob (σ\\sigma ) multiplied by the channel transmissions (t A t_A and t B t_B).", "Figure b) shows the plot of the gains as a function of μσt A t B \\mu \\sigma t_A t_B .", "The zz-basis is shown in green and the xx-basis is shown in blue.", "For both figures the results of the measurements done in the laboratory are shown with squares (blue or green) and the measurements done over deployed fiber are shown with diamond (red and purple).", "The difference in gains and error rates in the x- and the z-basis, respectively is due to the fact that, in the case in which one party sends a laser pulse containing more than one photon and the other party sends zero photons, projections onto the |ψ - 〉\\vert \\psi ^-\\rangle Bell state can only occur if both pulses encode qubits belonging to the x-basis.", "The Bell state projection cannot occur if both prepare qubits belonging to the z-basis (we ignore detector noise for the sake of this argument).", "This causes increased gain for the x-basis and, due to an error rate of 50% associated with these projections, also an increased error rate for the x-basis.Table: Measured error rates, e μσ x,z e_{\\mu \\sigma }^{x,z}, and gains, Q μσ x,z Q_{\\mu \\sigma }^{x,z}, for different mean photon numbers, μ\\mu and σ\\sigma (where μ=σ\\mu =\\sigma ), lengths of fiber connecting Alice and Charlie, and Charlie and Bob, ℓ A \\ell _{A} and ℓ B \\ell _{B}, respectively, and total transmission loss, ll.", "The last set of data details real-world measurements using deployed fiber.", "Uncertainties are calculated using Poissonian detection statistics." ], [ "Decoy-state analysis", "To calculate secret key rates for various system parameters, which allows optimizing these parameters, first, it is necessary to compute the gain, $Q_{11}^{z}$ , and the error rate, $e_{11}^{x}$ , that stem from events in which both sources emit a single photon.", "We consider the three-intensity decoy state method for the MDI-QKD protocol proposed in [26], which derives a lower bound for the secret key rate using lower bounds for $Q_{11}^{x,z}$ and an upper bound for $e_{11}^x$ .", "Note that we assume here that the the only effect of imperfectly generated qubit states on the secret key rate that we consider here is that it increases the error rates (further considerations require advancements to security proofs, which are under way [26], [35]) increases of error rates.", "We denote the signal, decoy, and vacuum intensities by $\\mu _s$ , $\\mu _d,$ and $\\mu _v$ , respectively, for Alice, and, similarly, as $\\sigma _s$ , $\\sigma _d$ , and $\\sigma _v$ for Bob.", "Note that $\\mu _v = \\sigma _v = 0$ by definition.", "This decoy analysis assumes that perfect vacuum intensities are achievable, which may not be correct in an experimental implementation.", "However, note that, first, intensity modulators with more than 50 dB extinction ratio exist, which allows obtaining almost zero vacuum intensity, and second, that a similar decoy state analysis with non-zero vacuum intensity values is possible as well [28].", "For the purpose of this analysis, we take both channels to have the same transmission coefficients (that is $t_A = t_B \\equiv t$ ), according to our experimental configuration, and Alice and Bob hence both select the same mean photon numbers for each of the three intensities (that is $\\mu _s = \\sigma _s \\equiv \\tau _s$ , $\\mu _d = \\sigma _d \\equiv \\tau _d$ , and $\\mu _v = \\sigma _v \\equiv \\tau _v$ ).", "Additionally, for compactness of notation, we omit the $\\mu $ and $\\sigma $ when describing the gains and error rates (e.g.", "we write $Q_{ss}^z$ to denote the gain in the z-basis when Alice and Bob both send photons using the signal intensity).", "Under these assumptions, the lower bound on $Q_{11}^{x,z}$ is given by $Q_{11}^{x,z} \\ge \\frac{ \\mathbb {D}_1(\\tau _s)\\mathbb {D}_2(\\tau _s)\\big (Q_{dd}^{x,z} - Q_0^{x,z}(\\tau _d)\\big ) - \\mathbb {D}_1(\\tau _d)\\mathbb {D}_2(\\tau _d)\\big (Q_{ss}^{x,z} - Q_0^{x,z}(\\tau _s)\\big ) }{\\mathbb {D}_1(\\tau _s)\\mathbb {D}_1(\\tau _d)\\big (\\mathbb {D}_1(\\tau _d)\\mathbb {D}_2(\\tau _s)-\\mathbb {D}_1(\\tau _s)\\mathbb {D}_2(\\tau _d)\\big )},$ where the various $\\mathbb {D}_i(\\tau )$ denote the probability that a pulse with photon number distribution $\\mathbb {D}$ and mean $\\tau $ contains exactly $i$ photons, and $Q_{0}^{x,z}(\\tau _d)$ and $Q_{0}^{x,z}(\\tau _s)$ are given by $Q_{0}^{x,z}(\\tau _d) & = & \\mathbb {D}_0(\\tau _d)Q_{vd}^{x,z} + \\mathbb {D}_0(\\tau _d)Q_{dv}^{x,z} - \\mathbb {D}_0(\\tau _d)^2Q_{vv}^{x,z},\\\\Q_{0}^{x,z}(\\tau _s) & = & \\mathbb {D}_0(\\tau _s)Q_{vs}^{x,z} + \\mathbb {D}_0(\\tau _s)Q_{sv}^{x,z} - \\mathbb {D}_0(\\tau _s)^2Q_{vv}^{x,z}.$ The error rate $e_{11}^x$ can then be computed as $e_{11}^x \\le \\frac{e_{dd}^xQ_{dd}^x - \\mathbb {D}_0(\\tau _d)e_{vd}^xQ_{vd}^x - \\mathbb {D}_0(\\tau _d)e_{dv}^xQ_{dv}^x + \\mathbb {D}_0(\\tau _d)^2e_{vv}^xQ_{vv}^x}{\\mathbb {D}_1(\\tau _d)^2Q_{11}^x},$ where the upper bound holds if a lower bound is used for $Q_{11}^x$ .", "Note that $Q_{11}^{x,z}$ , $Q_{0}^{x,z}(\\tau _d)$ , $Q_{0}^{x,z}(\\tau _s)$ and $e_{11}^x$ (Eqs.", "(REF -REF )) are uniquely determined through measurable gains and error rates." ], [ "Optimization of signal and decoy intensities", "For each set of experimental parameters (i.e.", "distribution function $\\mathbb {D}$ , channel transmissions and all parameters describing imperfect state preparation and measurement), the secret key rate (Eq.", "(REF )) can be maximized by properly selecting the intensities of the signal and decoy states ($\\tau _s$ and $\\tau _d$ , respectively).", "Here we consider its optimization as a function of the total transmission (or distance) between Alice and Bob.", "We make the assumptions that both the channel between Alice and Charlie and the channel between Bob and Charlie have the same transmission coefficient, $t$ , and that Alice and Bob use the same signal and decoy intensities.", "We considered values of $\\tau _d$ in the range $0.01 \\le \\tau _d < 0.99$ and values of $\\tau _s$ in the range $\\tau _d < \\tau _s \\le 1$ .", "An exhaustive search computing the secret key rate for an error correction efficiency $f=1.14$  [36] is performed from 2 km to 200 km total distance (assuming 0.2 dB/km loss), with increments of 0.01 photons per pulse for both $\\tau _s$ and $\\tau _d$ .", "For each point, the model described in section  is used to compute all the experimentally accessible quantities required to compute secret key rates using the three-intensity decoy state method summarized in Eqs.", "(REF -REF ).", "In our optimization, we found that, in all cases, $\\tau _d = 0.01$ is the optimal decoy intensity.", "We attribute this to the fact that $\\tau _d$ has a large impact on the tightness of the upper bound on $e_{11}^x$ in Eq.", "(REF ) (this is due to the fact that all errors in the cases in which both parties sent at least one photon, which increases with $\\tau _d$ , are attributed to the case in which both parties sent exactly one photon).", "Figure REF shows, as a function of total loss (or distance), the optimum values of the signal state intensity, $\\tau _s$ , and the corresponding secret key rate, $S$ , for decoy intensities of $\\tau _d \\in [0.01, 0.05, 0.1]$ , as well as for a perfect decoy state protocol (i.e.", "using values of $Q_{11}^z$ and $e_{11}^x$ computed from the model, as detailed in the preceeding section).", "Figure: a) Optimum signal state intensity, τ s \\tau _s, and b) corresponding secret key rate as a function of total loss in dB.", "The secondary axis shows distances assuming typical loss of 0.2 dB/km in optical fiber without splices.", "The optimum values for μ s \\mu _s for small loss have to be taken with caution as in this regime the model needs to be expanded to higher photon number terms." ], [ "Rate-limiting components", "Finally, we use our model to simulate the performance of the MDI-QKD protocol given improved components.", "We consider two straightforward modifications to the system: replacing the InGaAs single photon detectors (SPDs) with superconducting single photon detectors (SSPDs) [37], and improving the intensity modulation (IM).", "For various combinations of these improvements, the optimized signal intensities and secret key rates for $\\mu _{d}=0.05$ are shown in Figure REF .", "First, using state-of-the-art SSPDs in [37], the detection efficiency ($\\eta $ ) is improved from 14.5% to 93%, and the dark count probability ($P_{d}$ ) is reduced by nearly two orders of magnitude.", "Furthermore, the mechanisms leading to afterpulsing in InGaAs SPDs are not present in SSPDs (that is $P_{a} = 0$ ).", "This improvement results in a drastic increase in the secret key rate and maximum distance as both the probability of projection onto $\\vert \\psi ^-\\rangle $ and the signal-to-noise-ratio are improved significantly.", "Second, imperfections in the intensity modulation system used to create pulses in our implementation contribute significantly to the observed error rates, particularly in the z-basis.", "Using commercially-available, state-of-the-art intensity modulators [38] allow suppressing the background light (represented by $b^{x,z}$ in general quantum state given in Eq.", "(REF )) by an additional  10-20 dB, corresponding to an extinction ration of 40 dB.", "Furthermore, we considered improvements to the driving electronics that reduces ringing in our pulse generation by a factor of 5, bringing the values of $m^{x,z}$ in Eq.", "(REF ) closer to the ideal values.", "As seen in Figure REF , this provides a modest improvement to the secret key rate, both when applied to our existing implementation, and when applied in conjunction with the SSPDs.", "Note that in the case of improved detectors and intensity modulation system the optimized $\\tau _s$ for small loss (under 10 dB) is likely overestimated due to neglected higher-order terms.", "Figure: a) Optimum signal state intensity, τ s \\tau _s, and b) corresponding secret key rate as a function of total loss in dB.", "The secondary axis shows distances assuming typical loss of 0.2 dB/km in optical fiber without splices.", "The optimum values for μ s \\mu _s for small loss, are not shown as the model needs to be expanded to higher photon number terms in this regime." ], [ "Discussion and conclusion", "We have developed a widely applicable model for systems implementing the Measurement-Device-Independent QKD protocol.", "Our model is based on facts about the experimental setup and takes into account carefully characterized experimental imperfections in sources and measurement devices as well as transmission loss.", "It is evaluated against data taken with a real, time-bin qubit-based QKD system.", "The excellent agreement between observed values and predicted data confirms the model.", "In turn, this allows optimizing mean photon numbers for signal and decoy states and finding rate-limiting components for future improvements.", "We believe that our model, which is straightforward to generalize to other types of qubit encoding, as well as the detailed description of the characterization of experimental imperfections will be useful to improve QKD beyond its current state of the art.", "To finish, let us emphasize that tests of a model that describes the performance of a QKD system in terms of secret key rates has to happen in a setting in which eavesdropping can be excluded (i.e.", "within a secure lab and using spooled fibre) – otherwise, the measured data, which depends on the (unknown) type and amount of eavesdropping, may deviate from the predicted performance and no conclusion about the suitability of the model can be drawn.", "Interestingly, this implies that neither phase randomization, nor random selection of qubit states or intensities of attenuated laser pulses used to encode qubit states is necessary to test a model, as their presence (or absence) does not impact the measured data.", "However, it is obvious that these modulations are crucial to ensure the security of a key that is distributed through a hostile environment.", "We note that in this article, all effects of imperfections in the system on the measured quantities are still attributed to an eavesdropper, and accounted for in the calculation of the secret key rate as well in the optimization of system parameters." ], [ "Acknowledgments", "The authors thank E. Saglamyurek, V. Kiselyov and TeraXion for discussions and technical support, the University of Calgary's Infrastructure Services for providing access to the fiber link between the University's main campus and the Foothills campus, SAIT Polytechnic for providing laboratory space, and acknowledge funding by NSERC, QuantumWorks, General Dynamics Canada, iCORE (now part of Alberta Innovates Technology Futures), CFI, AAET and the Killam Trusts." ] ]

[ [ "Self-assembly of short DNA duplexes: from a coarse-grained model to\n experiments through a theoretical link" ], [ "Abstract Short blunt-ended DNA duplexes comprising 6 to 20 base pairs self-assemble into polydisperse semi-flexible chains due to hydrophobic stacking interactions between terminal base pairs.", "Above a critical concentration, which depends on temperature and duplex length, such chains order into liquid crystal phases.", "Here, we investigate the self-assembly of such double-helical duplexes with a combined numerical and theoretical approach.", "We simulate the bulk system employing the coarse-grained DNA model recently proposed by Ouldridge et al.", "[ J. Chem.", "Phys.", "134, 08501 (2011) ].", "Then we evaluate the input quantities for the theoretical framework directly from the DNA model.", "The resulting parameter-free theoretical predictions provide an accurate description of the simulation results in the isotropic phase.", "In addition, the theoretical isotropic-nematic phase boundaries are in line with experimental findings, providing a route to estimate the stacking free energy." ], [ "Introduction", "Self-assembly is the spontaneous formation through free energy minimization of reversible aggregates of basic building blocks.", "The size of the aggregating units, e.g.", "simple molecules, macromolecules or colloidal particles, can vary from a few angströms to microns, thus making self-assembly ubiquitous in nature and of interest in several fields, including material science, soft matter and biophysics [1], [2], [3], [4], [5].", "Through self-assembly it is possible to design new materials whose physical properties are controlled by tuning the interactions of the individual building blocks [6], [7], [8], [9].", "A relevant self-assembly process is the formation of filamentous aggregates (i.e.", "linear chains) induced by the anisotropy of attractive interactions.", "Examples are provided by micellar systems [10], [11], [12], formation of fibers and fibrils [13], [14], [15], [16], solutions of long duplex B-form DNA composed of $10^2$ to $10^6$ base pairs  [17], [18], [19], [20], filamentous viruses [21], [22], [23], [24], [25], chromonic liquid crystals [26] as well as inorganic nanoparticles [27].", "If linear aggregates possess sufficient rigidity, the system may exhibit liquid crystal (LC) phases (e.g.", "nematic or columnar) above a critical concentration.", "In the present study we focus on the self-assembly of short (i.e.", "6 to 20 base pairs) DNA duplexes (DNADs) [28], [29], [30] in which coaxial stacking interactions between the blunt ends of the DNADs favor their aggregation into weakly bonded chains.", "Such a reversible physical polymerization is enough to promote the mutual alignment of these chains and the formation of macroscopically orientationally ordered nematic LC phases.", "At present, stacking is understood in terms of hydrophobic forces acting between the flat hydrocarbon surfaces provided by the paired nucleobases at the duplex terminals, although the debate on the physical origin of such interaction is still active and lively [31], [32].", "In this respect, the self-assembly of DNA duplexes provides a suitable way to access and quantify hydrophobic coaxial stacking interactions.", "In order to extract quantitative informations from DNA-DNA coaxial stacking experiments, reliable computational models and theoretical frameworks are needed.", "Recent theoretical approaches have focused on the isotropic-nematic (I-N) transition in self-assembling systems  [33], [34], building on previous work on rigid and semi-flexible polymers [35], [36], [37], [38], [39], [40], [41], [42], [43].", "In a recent publication [44] we investigated the reversible physical polymerization and collective ordering of DNA duplexes by modeling them as super-quadrics with quasi-cylindrical shape [45] with two reactive sites [46], [47] on their bases.", "Then we presented a theoretical framework, built on Wertheim [48], [49], [50] and Onsager [51] theories, which is able to properly account for the association process.", "Here, we employ this theoretical framework to study the physical properties of a realistic coarse-grained model of DNA recently proposed by Ouldridge et al.", "[52], where nucleotides are modeled as rigid bodies interacting with site-site potentials.", "Following Ref.", "[44], we compute the inputs required by the theory, i.e.", "the stacking free energy and the DNAD excluded volume, for the Ouldridge et al.", "model [52].", "Subsequently we predict the polymerization extent in the isotropic phase as well as the isotropic-nematic phase boundaries.", "To validate the theoretical predictions, we perform large-scale molecular dynamics (MD) simulations in the NVT ensemble of a bulk system comprising 9600 nucleotides, a study made possible by the computational power of modern Graphical Processing Units (GPUs).", "The parameter-free theoretical predictions provide an accurate description of the simulation results in the isotropic phase, supporting the theoretical approach and its application in the comparison with experimental results.", "The article is organized as follows.", "Section  provides details of the coarse-grained model of DNADs, of the MD computer simulations and presents a summary of the theory.", "Section  describes the protocols implemented to evaluate the input parameters required by the theory via MC integrations for two DNADs.", "We also discuss some geometrical properties of the bonded dimer configurations.", "We then compare the theoretical predictions with simulation and experimental results.", "Finally, in Section  we discuss estimates for the stacking free energy and present our conclusions.", "We implement a coarse-grained model for DNA recently developed by Ouldridge et al.", "[53], [52].", "In such model, designed via a top-down approach, each nucleotide is described as a rigid body (see Figure REF ).", "The interaction forms and parameters are chosen so as to reproduce structural and thermodynamic properties of both single- (ssDNA) and double- (dsDNA) stranded molecules of DNA in B-form.", "All interactions between nucleotides are pairwise and, in the last version of the model [52], continuous and differentiable.", "Such feature makes the model convenient foe MD simulations.", "The interactions between nucleotides account for excluded volume, backbone connectivity, Watson-Crick hydrogen bonding, stacking, cross-stacking and coaxial-stacking.", "The interaction parameters have been adjusted in order to be consistent with experimental data [52], [54], [55].", "In particular, the stacking interaction strength and stiffness have been chosen to be consistent with the experimental results reported for 14-base oligomers by Holbrook et al. [55].", "Hydrogen-bonding and cross-stacking potentials were adjusted to give duplex and hairpin formation thermodynamics consistent with the SantaLucia parameterization of the nearest-neighbor model [54].", "Interaction stiffnesses were also further adjusted in order to provide correct mechanical properties of the model, such as the persistence length.", "The model does not have any sequence dependence apart from the Watson-Crick pairing, meaning that the strength of the interactions acting between nucleotides is to be considered as an average value.", "In addition, the model assumes conditions of high salt molarity ($0.5 \\,\\hbox{M}$ ).", "In this model, the coaxial-stacking interaction acts between any two non-bonded nucleotides and is responsible for the duplex-duplex bonding.", "It has been parametrized [56] to reproduce experimental data which quantify the stacking interactions by observing the difference in the relative mobility of a double strand where one of the two strands has been nicked with respect to intact DNA [57], [58] and by analyzing the melting temperatures of short duplexes adjacent to hairpins [59].", "To cope with the complexity of the model and the large number of nucleotides involved in bulk simulations, we employ a modified version of the CPU-GPU code used in a previous work [60], and extend it to support the force-fields [61].", "Harvesting the power of modern Graphical Processing Units (GPUs) results in a 30-fold speed-up.", "The CPU version of the code, as well as the Python library written to simplify generation of initial configurations and post-processing analysis, will soon be released as free software [62]." ], [ "Bulk simulations", "To compare numerical results with theory, we perform Brownian dynamics simulations of 400 dsDNA molecules made up by 24 nucleotides each, i.e.", "400 cylinder-like objects with an aspect ratio of $\\approx 2$ (see Figure REF  (d)).", "The integration time step has been chosen to be $0.003$ in simulation units which corresponds, if rescaled with the units of length, mass and energy used in the model, to approximately $1\\times 10^{-14}$ seconds.", "We study systems at three different temperatures, namely $T = 270\\,\\hbox{K}$ , $285\\,\\hbox{K}$ and $300\\,\\hbox{K}$ , and for different concentrations, ranging from $2 \\,\\hbox{mg/ml}$ to $241\\,\\hbox{mg/ml}$ .", "The $T=270\\,\\hbox{K}$ state point, despite being far from the experimentally accessed $T$ , is here investigated to test the theory in a region of the phase diagram where the degree of association is significant.", "To quantify the aggregation process we define two DNADs as bonded if their pair interaction energy is negative.", "Depending on temperature and concentration, we use $10^6 - 10^7$ MD steps for equilibration and $10^8 - 10^9$ MD steps for data generation on NVIDIA Tesla C2050 GPUs, equivalent to $1 - 10\\;\\mu s$ .", "Figure: Snapshots taken from simulations at T=300KT = 300\\,\\hbox{K}.", "At low concentrations (c=2mg/mlc=2 \\,\\hbox{mg/ml}, top) chain formation is negligible and the average chain length is approximately 1.", "As the concentration is increased (c=80mg/mlc=80 \\,\\hbox{mg/ml}, bottom), DNADs start to self-assemble into chains and the average chain length increases." ], [ "Theory", "We build on the theoretical framework previously developed to account for the linear aggregation and collective ordering of quasi-cylindrical particles [44].", "Here, we provide a discussion of how such a theory can be used to describe the reversible chaining and ordering of oligomeric DNADs at the level of detail adopted by the present model.", "According to Ref.", "[44], the free energy of a system of equilibrium polymers can be written as $\\frac{\\beta F}{V} &=& \\sum _{l=1}^{\\infty } \\nu (l) \\left\\lbrace \\ln \\left[ v_d \\nu (l) \\right] - 1 \\right\\rbrace +\\nonumber \\\\&+& \\frac{\\eta (\\phi )}{2} \\sum _{l=1 \\atop l^{\\prime }=1}^{\\infty } \\nu (l) \\nu (l^{\\prime }) v_{excl}(l, l^{\\prime })\\nonumber \\\\&-& \\beta \\Delta F_b \\sum _{l=1}^{\\infty } (l-1) \\nu (l)+ \\sum _{l=1}^{\\infty } \\nu (l) \\sigma _o(l)$ where $V$ is the volume of the system, $\\phi \\equiv v_d \\rho $ ($\\rho =N/V$ is the number density of monomers) is the packing fraction, $\\nu (l)$ is the number density of chains of length $l$ , normalized such that $\\sum _{l=1}^{\\infty } l\\, \\nu (l)= \\rho $ , $v_d$ is the volume of a monomer, $\\Delta F_b$ , as discussed in the subsection REF , is a parameter which depends on the free energy associated to a single bond and $v_{excl}(l,l^{\\prime })$ is the excluded volume of two chains of length $l$ and $l^{\\prime }$ .", "$\\eta (\\phi )$ is the Parsons-Lee factor [63] $\\eta (\\phi ) = \\frac{1}{4} \\frac{4-3\\phi }{(1-\\phi )^2}$ and $\\sigma _o(l) $[43] accounts for the orientational entropy that a chain of length $l$ loses in the nematic phase (including possible contribution due to its flexibility).", "The explicit form for $\\sigma _o(l) $ can be found in Ref.", "[44].", "The free energy functional (Eq.", "REF ) explicitly accounts for the polydispersity inherent in the equilibrium polymerization using a discrete chain length distribution and for the entropic and energetic contributions of each single bond through the parameter $\\Delta F_b$ ." ], [ "Isotropic phase", "In the isotropic phase $\\sigma _0=0$ and the excluded volume can be written as follows (see Appendix ): $v_{excl}(l,l^{\\prime }, X_0) = 2 B_I X_0^2 \\,l\\, l^{\\prime } + 2 v_d k_I \\frac{l+l^{\\prime }}{2}$ where the parameters $B_I$ and $k_I$ can be estimated via MC integrals of a system composed by only two monomers (see Appendix ) and $X_0$ is the aspect ratio of the monomers.", "We assume that the chain length distribution $\\nu (l)$ is exponential [44] with an average chain length $M$ $\\nu (l) = \\rho M^{-(l+1)} (M-1)^{l-1} $ where $M = \\frac{\\sum _{l=1}^{\\infty } l \\, \\nu (l)}{\\sum _{l=1}^{\\infty } \\nu (l)}.$ With this choice for $\\nu (l)$ the free energy in Eq.", "(REF ) becomes: $\\frac{\\beta F_{I}}{V} &=& -\\rho \\beta \\Delta F_b (1 - M^{-1}) +\\nonumber \\\\&+& \\eta (\\phi ) \\left[ B_I X_0^2 + \\frac{v_d k_I }{M} \\right]\\rho ^2 + \\nonumber \\\\&+& \\frac{\\rho }{M} \\left[ \\ln \\left( \\frac{v_d\\rho }{M} \\right)- 1\\right] + \\nonumber \\\\&+& \\rho \\frac{M-1}{M} \\ln (M-1) - \\rho \\ln M.$ Minimization of the free energy in Eq.", "(REF ) with respect to $M$ provides the following expression for the average chain length $M(\\phi )$ : $M = \\frac{1}{2} \\left( 1 + \\sqrt{1 + 4 \\phi e^{k_I \\phi \\eta (\\phi ) + \\beta \\Delta F_b} }\\right).$" ], [ "Nematic phase", "In the nematic phase the monomer orientational distribution function $f (\\theta )$ depends explicitly on the angle $\\theta $ between the particle and the nematic axis, i.e.", "the direction of average orientation of the DNAD, since the system is supposed to have azimuthal symmetry around such axis.", "We assume the form proposed by Onsager [51], i.e.", ": $f_{\\alpha }(\\theta ) = \\frac{\\alpha }{4 \\pi \\sinh \\alpha } \\cosh (\\alpha \\cos \\theta )$ where $\\alpha $ controls the width of the angular distribution.", "Its equilibrium value is obtained by minimizing the free energy with respect to $\\alpha $ .", "As discussed in Appendix , we assume the following form for the excluded volume in the nematic phase: $v_{excl}(l, l^{\\prime },X_0,\\alpha )= 2 B_N(\\alpha ) X_0^2 l\\, l^{\\prime } + 2 v_d k_N^{HC}(\\alpha ) \\frac{l+l^{\\prime }}{2}$ where the term $2 v_d k_N^{HC}(\\alpha )$ is the end-midsection contribution to the excluded volume of two hard cylinders (see Appendix ) and $B_N(\\alpha ) = \\frac{\\pi }{4} D^3 \\left( \\eta _1 + \\frac{\\eta _2}{\\alpha ^{1/2}} + \\frac{\\eta _3}{\\alpha } \\right).$ In Eq.", "(REF ), $D$ is the diameter of the monomer and $\\eta _k$ with $k=1,2,3$ are three parameters that we chose in order to reproduce the excluded volume calculated from MC calculations as discussed in Appendix .", "Inserting Eqs.", "(REF ) and (REF ) into Eq.", "(REF ) and assuming once more an exponential distribution for $\\nu (l)$ one obtains after some algebra: $\\frac{\\beta F_{N}}{V} &=&\\hat{\\sigma }_o - \\rho \\beta \\Delta F_b (1 - M^{-1}) +\\nonumber \\\\&+& \\eta (\\phi ) \\left[ B_N(\\alpha ) X_0^2 + \\frac{ v_d k_N^{HC}(\\alpha )}{M} \\right] \\rho ^2 +\\nonumber \\\\&+& \\frac{\\rho }{M} \\left( \\ln \\left[\\frac{v_d\\rho }{M}\\right]-1\\right)- \\rho \\ln M + \\nonumber \\\\&+& \\rho \\ln (M-1) \\frac{M-1}{M}$ where $\\hat{\\sigma }_o \\equiv \\sum _l \\sigma _o(l) \\nu (l)$ .", "The explicit calculation of the parameters $B_N$ and $k_N^{HC}$ is explained in Appendices   and .", "Assuming that the orientational entropy $\\hat{\\sigma }_o$ can be approximated with the expression valid for long chains [43], minimization with respect to $M$ results in $M = \\frac{1}{2} \\left( 1 + \\sqrt{1 + \\alpha \\phi e^{k_N(\\alpha ) \\phi \\eta (\\phi ) + \\beta \\Delta F_b} }\\right).$ while using the approximated expression for short chains [43], one obtains $M = \\frac{1}{2} \\left( 1 + \\sqrt{1 + 4 \\alpha \\phi e^{k_N(\\alpha ) \\phi \\eta (\\phi ) + \\beta \\Delta F_b - 1} }\\right).$ The equilibrium value of $\\alpha $ is thus determined by further minimizing the nematic free energy in Eq.", "(REF ), which has become only a function of $\\alpha $ .", "The parameter $\\alpha $ is related to the degree of orientational ordering in the nematic phase as expressed by the nematic order parameter $S$ as follows: $S(\\alpha ) = \\int (3 \\,\\cos ^2\\theta -1) f_{\\alpha }(\\theta ) \\pi \\,\\sin \\theta \\;d\\theta \\approx 1 - 3/\\alpha .$ Further refinements of the theory could be obtained by including a more accurate description of the orientational distribution $f_\\alpha (\\theta )$ in the proximity of the I-N phase transition, along the lines of Eqs.", "(40)-(42) of Ref. [44].", "For the sake of simplicity we have just presented the basic theoretical treatment.", "However, in the theoretical calculations in Sec.", "we will make use of the refined and more accurate free energy proposed in Ref.", "[44]." ], [ "Phase Coexistence", "The phase boundaries, at which the aggregates of DNAD are sufficiently long to induce macroscopic orientational ordering, are characterized by coexisting isotropic and nematic phases in which the volume fraction of DNADs are, respectively, $\\phi _N=v_d \\rho _{N}$ and $\\phi _I= v_d \\rho _{I}$ .", "The number densities $\\rho _I$ and $\\rho _N$ can be calculated by minimizing Eq.", "(REF ) with respect to $M_I$ and by minimizing Eq.", "(REF ) with respect to $M_N$ and $\\alpha $ .", "In addition, the two phases must be at equal pressure, i.e.", "$P_{I}=P_{N}$ , and chemical potential, i.e.", "$\\mu _{I}=\\mu _{N}$ .", "These conditions yield the following set of equations: $\\frac{\\partial }{\\partial M_{I}} F_{I} (\\rho _{I}, M_{I}) &=& 0\\nonumber \\\\\\frac{\\partial }{\\partial M_{N}} F_{N} (\\rho _{N}, M_{N}, \\alpha ) &=& 0\\nonumber \\\\\\frac{\\partial }{\\partial \\alpha } F_{N} (\\rho _{N}, M_{N},\\alpha ) &=& 0\\nonumber \\\\P_{I}(\\rho _{I}, M_{I}) &=& P_{N}(\\rho _{N},M_{N},\\alpha )\\nonumber \\\\\\mu _{I}(\\rho _{I}, M_{I}) &=& \\mu _{N}(\\rho _{N},M_{N},\\alpha )$" ], [ "Properties of the model", "To characterize structural and geometrical properties of the simulated DNADs monomers and aggregates we analyze conformations of duplexes extracted from large-scale GPU simulations (see Figure REF for some snapshots).", "In the following, the volume $v_d$ occupied by a single DNAD of length $X_0 D$ and double helix diameter $D$ ($D\\simeq 2 \\,\\hbox{nm}$ ) will be calculated as the volume of a cylinder with the same length and diameter, i.e.", "$v_d = \\pi X_0 D^3 / 4$ .", "When comparing numerical and experimental results with theoretical predictions we use the number of nucleotides $N_b$ in place of $X_0$ ($X_0 \\simeq 0.172 N_b$ ) and the concentration $c$ instead of the packing fraction $\\phi $ , which can be related to the former via: $\\phi = \\frac{0.172 D^3\\pi }{8 m_{N}}\\, c$ where $m_N=330 \\,\\hbox{Da}$ is the average mass of a nucleotide.", "Hence, in the following $c_I$ and $c_N$ will be used in place of $\\phi _I$ and $\\phi _N$ .", "First we calculate the dimensions (height $L$ and width $D$ ) of the DNADs for different $c$ and $T$ .", "We observe no concentration dependence on both quantities, while the variation in $T$ is negligible (of the order of $0.1\\%$ between DNADs of samples at $270\\,\\hbox{K}$ and $300\\,\\hbox{K}$ ).", "The effect of this small change does not affect substantially the value of the aspect ratio, which we consider constant ($X_0=2.06$ ) throughout this work.", "The geometrical properties of end-to-end bonded duplexes are not well-known since there are no experimental ways to probe such structures.", "In a very recent work, the interaction between duplex terminal base–pairs has been analyzed by means of large-scale full-atom simulations by Maffeo et al. [64].", "They found that blunt-ended duplexes (i.e.", "duplexes without dangling ends) have preferential binding conformations with different values of the azimuthal angle $\\gamma $ , defined as the angle between the projections onto the plane orthogonal to the axis of the double helix of the vectors connecting the O5' and O3' terminal base pairs.", "They report two preferential values for $\\gamma $ , namely $\\gamma = -20^\\circ $ and $\\gamma = 180^\\circ $ .", "Figure: Probability distributions for (a) the azimuthal angle γ\\gamma and (b) the end-to-end distance rr.In the present model the continuity of the helix under end-to-end interactions is intrinsic in the model and the azimuthal angle probability distribution is peaked around a single value $\\gamma _0 \\approx 40^\\circ $ (see Figure REF (a)).", "This is very close to the theoretical value $\\gamma \\approx 36^\\circ $ given by the pitch of the B-DNA double helix.", "The qualitative difference between the conformations of bonded DNADs found in this work and in Ref.", "[64] should be addressed in future studies describing the coaxial end-to-end interaction in a more proper way.", "In addition, we calculate the average distance $r$ between the centres of masses of the terminal base pairs.", "Figure REF (b) shows $P(r)$ , the probability distribution of $r$ .", "$P(r)$ is peaked at $0.39 \\,\\hbox{nm}$ , whereas Maffeo et al.", "[64] found an average distance of $r\\approx 0.5 \\,\\hbox{nm}$ .", "This difference can be understood in terms of the effect of the salt concentration which, being five times higher than the one used in Ref.", "[64], increases the electrostatic screening, thus effectively lowering the repulsion between DNA strands.", "The effect of the temperature is small, as lowering $T$ leads only to more peaked distributions for both $P(\\gamma )$ and $P(r)$ (and a very small shift towards smaller angles for $\\gamma $ ) but does not change the overall behavior." ], [ "Stacking free energy and excluded volume", "In this section we discuss the procedure employed to evaluate the input quantities required by the theory, namely $\\Delta F_b$ and $v_{excl}(l,l^{\\prime })$ .", "To this aim we perform a Monte Carlo integration over the degrees of freedom of two duplexes.", "$\\Delta F_b$ is defined as [44] $\\beta \\Delta F_b = \\ln \\left[2 \\frac{\\Delta (T)}{v_d}\\right] $ where [65] $\\Delta (T) = \\frac{1}{4}\\left\\langle \\int _{V_b} [ e^{-\\beta V({\\bf r}_{12},{\\Omega }_1,{\\Omega }_2)} - 1 ] \\,d{\\bf r}_{12} \\right\\rangle .$ Here ${\\bf r}_{12}$ is the vector joining the center of masses of particles 1 and 2, ${\\Omega }_i$ is the orientation of particle $i$ and $\\langle \\ldots \\rangle $ represents an average taken over all the possible orientations.", "$V_b$ is the bonding volume, defined here as the set of points where the interaction energy $V({\\bf r}_{12},{\\Omega }_1,{\\Omega }_2)$ between duplex 1 and duplex 2 is less than $k_BT$ .", "To numerically evaluate $\\Delta (T)$ we perform a MC integration using the following scheme: Produce an ensemble of 500 equilibrium configurations of a single duplex at temperature $T$ .", "Set the counter $N_{\\mathrm {tries}} = 0$ and the energy factor $F = 0$ .", "Choose randomly two configurations $i$ and $j$ from the generated ensemble.", "Insert a randomly oriented duplex $i$ in a random position in a cubic box of volume $V=1000$ $nm^{3}$ .", "Insert a second duplex $j$ in a random position and with a random orientation.", "Compute the interaction energy $V(i,j)$ between the two duplexes $i$ and $j$ and, if $V(i,j)< k_BT$ , update the energy factor, $F = F + \\left(e^{-\\beta V(i, j)} - 1\\right)$ .", "Increment $N_{\\mathrm {tries}}$ .", "Repeat from step 3, until $\\Delta (T) \\cong \\frac{1}{4} \\frac{V}{N_{\\rm tries}} F$ converges within a few per cent precision.", "The employed procedure to compute $v_{excl}(l,l^{\\prime })$ is fairly similar except that it is performed for duplexes with a various number of bases (i.e.", "with different $X_0$ ) and the quantity $F$ counts how many trials originate a pair configuration with $V(i,j)> k_BT$ (i.e.", "in step 4, $F = F+1$ ).", "In the nematic case, the orientations of the duplexes are extracted randomly from the Onsager distribution given by Eq.", "REF .", "With such procedure, $v_{excl}(l=1, l^{\\prime }=1,X_0) = \\frac{V}{N_{\\rm tries}} F$ We calculate $v_{excl}$ for 8 values of $\\alpha $ , ranging from 5 to 45 (see Appendix ).", "Since the $X_0$ and $l$ dependences of Eqs.", "(REF ) and (REF ) are the same and the $X_0$ dependence of the numerically calculated $v_{excl}$ on the shape of DNADs is negligible, the evaluation of the excluded volume as a function of $X_0$ provides the same information as the evaluation of $v_{excl}$ as a function of $l$ .", "Fig.", "REF shows $\\Delta (T)$ for all investigated $T$ in a $\\ln \\Delta $ vs $1/T$ plot.", "A linear dependence properly describes the data at the three $T$ .", "An alternative way to evaluate $\\Delta (T)$ is provided by the limit $\\rho \\rightarrow 0$ of Eq.", "(REF ).", "Indeed in the low density limit $M$ and $\\Delta (T)$ are related via the following relation: $\\Delta (T) = \\frac{M (1- M)}{2\\rho }.$ Therefore it is also possible to estimate $\\Delta (T)$ by extrapolating the low density data for $M$ at $T=270\\,\\hbox{K}$ , $285\\,\\hbox{K}$ and $300\\,\\hbox{K}$ .", "The results, also shown in Fig.", "REF , are in line with the ones obtained through MC calculations.", "The Arrhenius behavior of $\\Delta (T)$ suggests that bonding entropy and stacking energy are in first approximation $T$ independent.", "The coaxial stacking free energy $G_{ST}$ is related to $\\Delta (T)$ as follows $G_{ST}= - k_B T \\ln [ 2 \\rho \\Delta (T) ].$ Substituting the fit expression provided in Fig.", "REF for $\\Delta (T)$ results in a stacking free energy $G_{ST}^0=-0.086 \\,\\hbox{kcal/mol}$ at a standard concentration $1 \\,\\hbox{M}$ of DNADs and $T=293\\,\\hbox{K}$ comprising a bonding entropy of $-30.6$ $\\hbox{cal}/\\hbox{mol}\\,\\hbox{K}$ and a bonding energy of $-9.06 \\,\\hbox{kcal/mol}$ .", "Figure: Δ(T)\\Delta (T) calculated with the procedures described in Sec.", "for all investigated TT." ], [ "Isotropic phase: comparing simulation results with theoretical predictions ", "Fig.", "REF shows the concentration $c$ dependence of $M$ calculated from the MD simulation of the $N_b=12$ system.", "The average chain length increases progressively on increasing $c$ .", "The figure also shows the theoretical predictions calculated by minimizing the isotropic free energy in Eq.", "(REF ) with respect to $M$ using the previously discussed estimates for $\\Delta F_b$ and $v_{excl}$ .", "The theoretical results properly describe the MD simulation data up to concentrations around $200\\,\\hbox{mg/ml}$ , which corresponds to a volume fraction $\\phi \\approx 0.20$ .", "In Ref.", "[44] similar observations have been made and the discrepancy at moderate and high $\\phi $ has been attributed to the inaccuracy of the Parsons decoupling approximation.", "The $M$ values calculated using the excluded volume of two hard cylinders (HC) are also reported, to quantify the relevance of the actual shape of the DNA duplex.", "Indeed the HC predictions appreciably deviate from numerical data beyond 100 mg/ml.", "Figure: Average chain length MM in the isotropic phase at low concentration.", "Symbols are numericalresults and dashed lines are theoretical predictions.", "Dotted lines are theoretical predictions usingthe excluded volume of HCs v excl HC v_{excl}^{HC} (see Appendix )." ], [ "Phase Coexistence: Theoretical predictions", "A numerical evaluation of the phase coexistence between the isotropic and the nematic phases for the coarse-grained model adopted in this study is still impossible to obtain given the current computational power.", "We thus limit ourselves to the evaluation of the I-N phase coexistence via the theoretical approach discussed in Sec. .", "Fig.", "REF shows the theoretical phase diagram in the $c$ -$N_b$ plane for $T=270\\,\\hbox{K}$ and $300\\,\\hbox{K}$ .", "As expected, both $c_I$ and $c_N$ decrease on increasing $N_b$ , since the increase of the number of basis result in a larger aspect ratio.", "On decreasing $T$ , theory predicts a $10\\%$ decrease of $c_I$ and a similar decrease of $c_N$ , resulting in an overall shift of the I-N coexistence region toward lower $c$ values.", "This trend is related to the increase of the average chain length $M$ with increasing $\\beta \\Delta F_b$ (see Fig.", "REF ).", "Fig.", "REF also shows the phase boundaries calculated using the excluded volume of two hard cylinders.", "Assimilating DNADs to hard cylinders results in a 10–$15 \\%$ widening of the isotropic-nematic coexistence region.", "Figure: I-N phase diagram in the cc vs N b N_b plane for T=270KT=270 \\,\\hbox{K} (top) and 300K300 \\,\\hbox{K} (bottom).", "Dotted linesare theoretical phase boundaries calculated using the excluded volume of HCs v excl HC v_{excl}^{HC} (see Appendix )." ], [ "Comparison between theory and experiments", "The theoretical predictions concerning the isotropic-nematic coexisting concentrations can be compared to the experimental results reported in Refs.", "[28] and [30] for blunt-ended DNADs.", "Figure REF compares the experimentally determined nematic concentrations $c_N$ at coexistence with the values calculated from the present model for $T=293 \\,\\hbox{K}$ .", "Despite all the simplifying assumptions and despite the experimental uncertainty, the results provide a reasonable description of the $N_b$ dependence of $c_N$ .", "The experimental data refer to different base sequences and different salt concentrations.", "According to the authors $c_N$ is affected by an error of about $\\pm 50 \\,\\hbox{mg/ml}$ .", "In particular for the case $N_b=12$ the critical concentrations $c_N$ for distinct sequences shows that blunt-end duplexes of equal length but different sequences can display significantly different transition concentrations.", "Hence, for each duplex length, we consider the lowest transition concentration among the ones experimentally determined, since this corresponds to the sequence closest to the symmetric monomer in the model.", "Indeed the dependence of $c_N$ on the DNADs sequence is expected to be larger for the shortest sequences, i.e.", "$N_b < 12$ , for which DNAD bending could be significant [30].", "Unfortunately, quantitative experimental data on this bending effect are still lacking.", "In general it is possible that $c_N$ for $N_b < 12$ (for which a large number of sequences have been studied, see Fig.", "REF ), would be corrected to lower values if a larger number of sequences were explored.", "For more details on this phenomenon, we refer the reader to the discussions in Refs.", "[30], [66], [44].", "The overestimation of the phase boundaries for $N_b \\ge 12$ with respect to experimental results suggests that the DNA model of Ouldridge et al.", "[52] overestimates the coaxial stacking free energy.", "Such discrepancy can perhaps be attributed to the restricted number of microstates allowing for bonding states in the DNA model [52], [56], as discussed in Sec. .", "Indeed, allowing DNADs to form end-to-end bonds with more than one preferred azimuthal angle would increase the entropy of bonding, thus effectively lowering $G_{ST}$ .", "Allowing for both left- and right-handed binding conformations, a possibility supported by the results of Maffeo et al.", "[64], would double $\\Delta (T)$ in Eq.", "(REF ) and hence add an entropic contribution equal to $-k_B T\\ln (2)$ to $G_{ST}$ , which would result in a value $G_{ST}^0 - 0.403\\,\\hbox{kcal/mol} = -0.49 \\,\\hbox{kcal/mol}$ for $T=293K$ .", "Fig.", "REF also shows the theoretical prediction for such upgraded $G_{ST}$ value.", "In Fig.", "REF theoretical calculations of the I-N transition lines are shown for $G_{ST}=-0.4 \\,\\hbox{kcal/mol}$ and $G_{ST} =-2.4 \\,\\hbox{kcal/mol}$ at $T=293 \\,\\hbox{K}$ as the upper and lower boundaries of the grey band respectively.", "To calculate these critical lines we retain the excluded volume calculated in the subsection REF and, given the value of $G_{ST}$ , we evaluate $\\Delta F_b$ according to Eqs.", "(REF ) and (REF ) for $T=293 \\,\\hbox{K}$ and $\\rho $ corresponding to the standard $1 \\,\\hbox{M}$ concentration.", "The selected points with $N_b \\ge 12$ fall within the grey band shown in Fig.", "REF enabling us to provide an indirect estimate of $G_{ST}$ between $-0.4 \\,\\hbox{kcal/mol}$ and $-2.4 \\,\\hbox{kcal/mol}$ .", "For the points with $N < 12$ , where duplex bending might play a role, it would be valuable to have more experimental points corresponding to more straight sequences in order to validate the theoretical predictions.", "It is worth observing that for all DNAD lengths $N_b$ the electrostatics interactions are properly screened.", "For $N_b=20$ a concentration $1.2 \\,\\hbox{M}$ of NaCl has been added to the solution resulting in a Debye screening length $k_D^{-1} \\approx 0.23 \\,\\hbox{nm}$ .", "For all other lengths (i.e.", "$N_b \\le 18$ ) we note that at the lowest DNA concentration of $440 \\,\\hbox{mg/ml}$ corresponding to $N_b=14$ $k_D^{-1} \\approx 0.40 \\,\\hbox{nm}$ .", "Therefore the experimental $k_D^{-1}$ is always smaller than the excluded volume diameter for the backbone-backbone interaction of our coarse-grained model [52] ($ \\approx 0.6 \\,\\hbox{nm}$ ), thus enabling us to neglect electrostatic interactions.", "On the other hand, if electrostatics interactions are not properly screened the effective aspect ratio for such DNAD sequences would be smaller than the ones used in our theoretical treatment and this would result in a underestimate of $c_N$ .", "To account for this behavior one should at least have a reasonable estimate of the effective size of DNADs when electrostatics interactions are not fully screened.", "Moreover, the role of electrostatics interactions can be subtle and not completely accounted for by simply introducing an effective size of DNADs.", "A possible route to include electrostatics in our treatment can be found in Ref.", "[20] and it will be addressed in future studies.", "Figure: Critical nematic concentrations c N c_N as a function of the number of base pairs per duplex N b N_b for thepresent model, calculated theoretically at T=293KT= 293 \\,\\hbox{K} using the computed stacking free energy G ST 0 G_{ST}^0 (short dashed lines), G ST =-0.49kcal/molG_{ST}=-0.49 \\,\\hbox{kcal/mol} (long dashed lines), and for experiments  (circles and squares).", "Squares are c N c_N for different sequences at the same N b =12N_b = 12.The grey band has been built considering for G ST G_{ST} an upper bound of -0.4kcal/mol-0.4 \\,\\hbox{kcal/mol} and a lower bound of -2.4kcal/mol-2.4 \\,\\hbox{kcal/mol}." ], [ "Comparison with Onsager Theory", "The experimental average aggregation numbers are estimated in Refs.", "[13], [28] by mapping the self-assembled system onto an “equivalent” mono-disperse system of hard rods with an aspect ratio equal to $M X_0$ .", "In Ref.", "[44] it has been shown that the theoretically estimated isotropic-nematic coexistence lines for the case of polymerizing superquadric particles in the $M X_0-\\phi $ plane, parametrized by the stacking energy, are significantly different from the corresponding Onsager original predictions (as re-evaluated in Ref. [35]).", "In light of the relevance for interpreting the experimental data, we show in Figure REF the same curves for the DNA model investigated here.", "In this model, a clear re-entrant behavior of the transition lines in the $c-M X_0$ plane is observed.", "The re-entrant behavior occurs for values of the stacking free energy accessed at temperatures between $270 \\,\\hbox{K}$ and $330 \\,\\hbox{K}$ .", "We believe that the re-entrancy of the transition lines in the $c-M X_0$ plane is a peculiar mark of the system polydispersity resulting from the reversible self-assembling into chains, and as such it should be also observable experimentally.", "Figure: Isotropic-nematic coexistence lines in the average aspect ratio MX 0 M X_0 and concentration cc plane for two values of N b N_b, namely N b =12N_b=12 (top) and N b =20N_b = 20 (bottom).", "Solid lines indicate theoretical predictions, dashed lines indicate the Onsager original predictions, as re-evaluatedin Ref.", "for c I c_{I} and c N c_N.", "Symbols along the isotropic and nematic phase boundaries at coexistenceare joined by dotted lines, to indicate the change in concentration and average chain length at the transition." ], [ "Conclusions", "In this article, we have provided the first study of a bulk solution of blunt ended DNA duplexes undergoing reversible self-assembly into chains, promoted by stacking interactions.", "The simulation study, carried out at different concentrations and temperatures, provides a clear characterization of the $c$ and $T$ dependence of the average polymerization length $M$ and an indirect estimate of the stacking free energy.", "We have provided a theoretical description of the self-assembly process based on a theoretical framework recently developed in Ref. [44].", "The inputs required by the theory (the DNAD excluded volumes and the stacking free energy) have been numerically calculated for the present DNA model, allowing a parameter free comparison between the molecular dynamic results and the theoretical predictions.", "Such comparison has been limited to the isotropic phase, due to the difficulties to simulate the dense nematic phase under equilibrium conditions.", "The description of the isotropic phase is satisfactory: quantitative agreement between theory and simulations is achieved for concentrations up to $c\\approx 200$ mg/ml.", "The stacking free energy value that properly accounts for the polymerization process observed in the molecular dynamics simulations is $G^0_{ST}=-0.086\\,\\hbox{kcal/mol}$ at a standard concentration 1 M of DNADs and $T = 293~$ K comprising a bonding entropy of $-30.6 \\,\\hbox{cal/mol~K}$ and a bonding energy of $-9.06 \\,\\hbox{kcal/mol}$ .", "Theoretical predictions for the I-N transition have been compared with experimental results for several DNA lengths, ranging from 8 to 20 bases.", "For $N_b \\ge 12 $ the model predicts values for $c_N$ which are higher than experimental ones.", "This suggests that the DNA model employed overestimates $G_{ST}$ .", "In view of the recent results of Maffeo et al.", "[64], we speculate that the bonding entropy is underestimated, in agreement with the observation that the probability distribution of the azimuthal angle between two bonded DNADs, which is designed to be single-peaked, is too restraining.", "In this respect, the present study call for an improvement of the coarse-grained potential [52] in regard to the coaxial stacking interaction.", "The value of $G_{ST}$ can also be used as a fitting parameter in the theory for matching $c_N$ with the experimental results, retaining the excluded volume estimates calculated for the coarse-grained DNA model.", "Such procedure shows that values of the stacking free energy between $-0.4\\,\\hbox{kcal/mol}$ and $-2.4\\,\\hbox{kcal/mol}$ are compatible with the experimental location of the I-N transition line.", "In the work of Maffeo et al., the authors report a quite smaller value of $G_{ST}$ , namely $G^M_{ST}=-6.3\\,\\hbox{kcal/mol}$ , a value which was confirmed by the same authors by performing an investigation of the aggregation kinetic in a very lengthy all-atom simulation of DNAD with $N_b=10$ .", "If such $G_{ST}$ value is selected as input in our theoretical approach (maintaining the same excluded volume term), then one finds $c_N^M \\approx 250 $ mg/ml, a value significantly smaller than the experimental result ($c_N=650\\pm 50$ mg/ml).", "This casts some doubts on the effectiveness of the employed all-atom force-field to properly model coaxial stacking.", "Finally, our work draws attention to the errors affecting the estimate of the average chain length $M$ via a straightforward comparison of the nematic coexisting concentrations with analytic predictions based on the original Onsager theory for mono-disperse thin rods [28], [13].", "We have found that such approximation significantly underestimates $M$ at the I-N transition concentration $c_N$ .", "In addition, the theoretical approach predicts a re-entrant behavior of the transition lines in the $c$ -$M X_0$ plane, a distinct feature of the polydisperse nature of the equilibrium chains." ], [ "ACKNOWLEDGMENTS", "We thank Thomas Ouldridge, Flavio Romano and Teun Vissers for fruitful discussions.", "We acknowledge support from ERC-226207-PATCHYCOLLOIDS and ITN-234810-COMPLOIDS as well as from NVIDIA." ], [ "Excluded volume contributions", "Here we further discuss the calculation of the excluded volume term $v_{excl}(l,l^{\\prime })$ for the present model.", "Following Ref.", "[44], the excluded volume is assumed to be the following second order polynomial in $l$ and $l^{\\prime }$ : $&&v_{excl}[l,l^{\\prime }; f(\\theta )] = 2 \\int f(\\theta ) f(\\theta ^{\\prime }) D^3 \\left[ \\Psi _1(\\gamma ,X_0) + \\right.", "\\nonumber \\\\&&+\\frac{l + l^{\\prime }}{2} \\Psi _2(\\gamma ,X_0) X_0 +\\left.", "\\Psi _3(\\gamma ,X_0)X_0^2\\; l\\, l^{\\prime } \\right] \\nonumber \\\\&&d{\\Omega }\\, d{\\Omega }^{\\prime }$ where the functions $\\Psi _\\alpha $ , $\\alpha =1,2,3$ , describe the angular dependence of the excluded volume.", "The orientational probability $f(\\theta )$ is normalized such that $\\int f(\\theta ) d\\Omega = 1.$ The three contributions to the excluded volume in Eq.", "(REF ) come from end-end, end-midsection and midsection-midsection steric interactions [44] between two chains.", "In the isotropic phase the orientational distribution does not have any angular dependence, i.e.", "$f(\\theta )=1/4\\pi $ , and Eq.", "(REF ) reduces to the form $v_{excl}(l,l^{\\prime },X_0) &=& B_I X_0^2 \\,l\\, l^{\\prime } +k_I v_d \\frac{l+l^{\\prime }}{2} + A_I.$ The parameters $B_I$ , $k_I$ and $A_I$ appearing in Eq.", "(REF ) can be calculated via MC integration procedures as discussed in the subsection REF and in Ref. [44].", "We expect that these parameters do not depend on $X_0$ because each DNADs comprises $N_b$ stacked base pairs which are all identical with respect to excluded volume interactions (i.e.", "they all have the same shape).", "In particular, the calculated excluded volume of two DNADs is reported in Fig.", "REF for 5 different aspect ratios, together with the resulting values for the above parameters.", "Figure: Excluded volume in the isotropic phase together with analytic approximations.", "From the linearfit one has B I =0.959D 3 B_I= 0.959 D^3 and k I =3.084k_I=3.084, while we assume A I =0A_I=0Using the Onsager angular distribution $f_\\alpha (\\theta )$ in Eq.", "(REF ), the excluded volume in the nematic phase depends also on the parameter $\\alpha $ , i.e.", "the general form in Eq.", "(REF ) reduces to $v_{excl}(l,l^{\\prime },X_0,\\alpha ) &=& B_N(\\alpha ) X_0^2 \\,l\\, l^{\\prime } +k_N(\\alpha ) v_d \\frac{l+l^{\\prime }}{2} \\nonumber \\\\&+& A_N(\\alpha ).$ Assuming that $A_N(\\alpha )=0$ , $k_N(\\alpha )=k_N^{HC}(\\alpha )$ and $B_N(\\alpha )$ is given by Eq.", "(REF ), the three parameters $\\eta _k $ with $k=1, 2, 3$ have to be estimated.", "For $l=l^{\\prime }=1$ and several values of $\\alpha $ ($\\alpha =5\\dots 45$ in steps of 5) and $X_0$ we calculated numerically the nematic excluded volume for two DNADs.", "The results are shown in Fig.", "REF , where we plot $v_{excl}/v_d$ vs $X_0$ for various $\\alpha $ .", "The dashed lines shown in Fig.", "REF are obtained through a two-dimensional fit to numerical data for $v_{excl}(1,1,X_0,\\alpha )$ using Eq.", "(REF ) as fitting function." ], [ "Excluded volume of hard cylinders", "For two rigid chains of length $l$ and $l^{\\prime }$ which are composed of hard cylinders (HCs) of diameter $D$ and length $X_0 D$ , $v_{excl}(l,l^{\\prime })$ can be described by $v_{excl}^{HC}[l,l^{\\prime }; f(\\theta )] &=& \\int f(\\theta ) f(\\theta ^{\\prime }) D^3 \\left[ \\, \\frac{\\pi }{2} \\sin \\gamma + \\frac{\\pi }{2} X_0\\right.", "\\nonumber \\\\&& ( 1 + |\\cos \\gamma | + \\frac{4}{\\pi } E(\\sin \\gamma ) ) + \\nonumber \\\\&+& \\left.", "\\frac{l + l^{\\prime }}{2} + 2 X_0^2 \\sin \\gamma \\;\\; l\\,l^{\\prime }\\, \\right] d{\\Omega }\\, d{\\Omega }^{\\prime }$ where $\\cos \\gamma ={\\bf u}\\cdot {\\bf u}^{\\prime }$ , $\\bf u$ and $\\bf u^{\\prime }$ are the orientations of two HCs and $E(\\sin \\gamma )$ is the complete elliptical integral $E(\\sin \\gamma ) = \\frac{1}{4} \\int _0^{2\\pi } (1-\\sin ^2\\gamma \\sin ^2\\psi )^{1/2} d\\psi .$ The integrals in Eq.", "(REF ) can be calculated exactly in the isotropic phase, while in the nematic phase the calculation can be done analytically only for suitable choices of the angular distribution $f(\\theta )$ .", "Here we assume that the angular distribution is given by the Onsager function in Eq.", "(REF ).", "Using the Onsager orientational function the following approximate expressions for the coefficients $k_N(\\alpha )$ , $B_N(\\alpha )$ and $A_N(\\alpha )$ can be derived [43] $\\tilde{B}_N(\\alpha ) &=& D^3 (\\pi /4)\\rho _a(\\alpha ) \\nonumber \\\\\\tilde{k}_N(\\alpha ) &=& \\pi D^3\\frac{X_0}{v_d}\\left(1-\\frac{1}{\\alpha }\\right) \\nonumber \\\\\\tilde{A}_N(\\alpha ) &=& D^3\\left(\\pi /4\\right)^2\\rho _a(\\alpha )$ where $\\rho _a&=& 4 (\\pi \\alpha )^{-1/2}\\left(1 - \\frac{15}{16\\,\\alpha }+\\frac{105}{512\\,\\alpha ^2}\\right.+\\nonumber \\\\&& \\left.+\\frac{315}{8192\\,\\alpha ^3}\\right)$ We evaluate numerically the excluded volume in Eq.", "(REF ) for many values of $\\alpha $ and, building on the expressions in Eqs.", "(REF ), we perform a fit to this data using the following functions: $B_N^{HC} (\\alpha )&\\simeq & D^3 (\\pi /4) \\left( \\rho _a(\\alpha ) + \\frac{c_4}{\\alpha ^{9/2}}+\\frac{c_5}{\\alpha ^{11/2}}\\right)\\\\k_N^{HC}(\\alpha ) &=& 4 \\left(1-\\frac{1}{\\alpha }\\right)+ \\sum _{i=2}^{\\infty }\\frac{b_i}{\\alpha ^i}\\simeq \\frac{4}{\\pi } \\sum _{i=0}^4 \\frac{d_i}{\\alpha ^i}\\\\A_N^{HC}(\\alpha ) &\\simeq & D^3 (\\pi /4)^2 \\left( \\rho _a(\\alpha ) + \\frac{c_4}{\\alpha ^{9/2}}+\\frac{c_5}{\\alpha ^{11/2}}\\right)$ The coefficient values resulting from the fitting procedure are $c_4=1.2563$ , $c_5=-0.95535$ , $d_0=3.0846$ , $d_1=-4.0872$ , $d_2=9.0137$ , $d_3=-9.009$ and $d_4=3.3461$ ." ] ]

[ [ "Quantum Spin Holography with Surface State Electrons" ], [ "Abstract In a recent paper Moon and coworkers [C.R.", "Moon et al., Nature Nanotechnology 4, 167 (2009)] have shown that the single-atom limit for information storage density can be overcome by using the coherence of electrons in a two-dimensional electron gas to produce quantum holograms comprised of individually manipulated molecules projecting an electronic pattern onto a portion of a surface.", "We propose to further extend the concept by introducing quantum spin holography - a version of quantum holographic encoding allowing to store the information in two spin channels independently." ], [ "Accepted for publication in Applied Physics Letters (March 2012) Quantum Spin Holography with Surface State Electrons Oleg O. Brovko Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D06120 Halle, Germany [email protected] Valeri S. Stepanyuk Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D06120 Halle, Germany In a recent paper Moon and coworkers [C.R.", "Moon et al., Nature Nanotechnology 4, 167 (2009)] have shown that the single-atom limit for information storage density can be overcome by using the coherence of electrons in a two-dimensional electron gas to produce quantum holograms comprised of individually manipulated molecules projecting an electronic pattern onto a portion of a surface.", "We propose to further extend the concept by introducing quantum spin holography – a version of quantum holographic encoding allowing to store the information in two spin channels independently.", "75.75.-c,73.20.-r quantum, holography, spin, surface state, electron In the era of ever increasing information storage density the physical entity setting the limit to the progress has quickly been reduced to the size of a single atom.", "The concept of representing a single bit of information by a single atom or molecule has been conceived with the ability to manipulate individual atoms on a surface with the tip of a scanning tunneling microscope.", "[1] Two decades later it has also been shown that it is possible to store information in the state of a single free-standing atom trapped in an optical cavity [2] and that constructing truly atomic logic elements is not any more beyond the grasp of state-of-the-art experimental techniques [3], , And despite the fact that achieving information storage densities close to the single-atom limit in real-world devices is still far from being feasible, the search for an information carrier which would break the limit set by the finite spacing between single atoms in solid state systems has not been abandoned.", "The main requirement that such a carrier has to fulfill is that it's characteristic lateral dimensions should be smaller than those of a single atomic unit.", "In a recent paper Moon and coworkers have proposed that information can be stored in a fermionic state of a coherent two-dimensional electron gas.", "[6], They have dubbed the proposed concept quantum holographic encoding, drawing parallels between the free space optical waves and electron wavefuctions.", "They use atomic manipulation [1] to construct molecular holograms of carbon monoxide (CO) molecules on the (111) surface of a Cu crystal (Fig.", "REF ) hosting a two-dimensional quasi-free electron gas – a Shockley type surface state (SS).", "[8], , Figure: (color online) Principles of quantum holographic encoding in a two-dimensional electron gas.", "Inspired by Ref.", "Moon2009.Electron waves scatter at local potentials introduced by the molecules and interfere (owing to inherent long range coherence [11]) in a designated area of the surface (white area in Fig.", "REF ) to form an electron density of states (LDOS) pattern representing an information page.", "[6] This page can then be read out with a density sensitive scanning probe technique, such as the scanning tunneling microscopy (STM).", "There are several major advantages of such information encoding procedure: the information is projected onto a medium (i) free of lattice constraints and (ii) native to the system itself.", "This means that the size of a single bit is limited neither by the host surface lattice nor by the limitations of a readout technique (f.e.", "by lensing or collimation of an external readout beam [6]).", "Furthermore, owing to the parabolic dispersion of the SS, the interference patters formed by electrons with different energies are different.", "Using this property Moon et al.", "were able to project the hologram not only in two spatial degrees of freedom but also in the energy dimension (Fig.", "REF ), a concept very similar to optical volumetric holography.", "[12], [6] It is, however, conceivable, that in the same way as Moon and coworkers [6] have utilized the energy dispersion, other intrinsic properties of electrons can act as a new dimension for information storage, pushing further back the information density limit.", "In the present letter we propose to use the spin of electrons as such additional dimension.", "We show that if the molecules or atoms used in the construction of a molecular hologram have pronounced magnetic properties or the surface state is inherently spin-polarized the scattering of surface state electrons becomes spin-dependent, allowing one to store different information pages in different spin channels of a collinear uniaxial system.", "As an example we discuss Cu(111) surface decorated with bilayer cobalt (Co) islands capped with Cu thus obtaining a system with spin-polarized surface states.", "[13] With its help we demonstrate the possibility of simultaneously encoding different information pages with electrons of the same energy but opposite spins as sketched in Fig.", "REF .", "In order to have an effective means to treat the quantum holographic encoding theoretically we adopt the multiple scattering formalism, which has in the past proven to be a uniquely suitable tool for the description of surface state scattering by impurities.", "[14] In terms of complex amplitudes the process of scattering at a single potential modifies the amplitude of electrons as $a_{out}=a_{in}\\cdot \\frac{e^{2i\\eta }-1}{2i}\\;,$ where $a_{in}$ is the incoming amplitude and $\\eta $ is the complex phase shift.", "The latter can be decomposed into real and imaginary parts such that $e^{2i\\eta }=\\alpha \\exp (2i\\delta )$ , where the real part $\\delta $ describes the scattering phase and $\\alpha $ describes the attenuation of the amplitude due to various processes (e.g.", "adsorption or, in particular case of an atom on a surface, scattering into bulk states).", "Parameters $\\alpha $ and $\\delta $ are determined by the shape of the scattering potential and, in turn, uniquely define the process of scattering.", "The propagation of an electron across the surface can, to the first order, be considered as a free electron propagation [14]: $a=a_0 \\cdot \\sqrt{\\frac{2}{\\pi k \\Delta r}}\\cdot e^{ik\\Delta r}\\;,$ where $k$ is the wave length and $\\Delta r$ is the propagation distance.", "The local density of states (LDOS) at a certain point $r$ induced by a single atom located at $r_i$ is then determined by the interference of outgoing (in the spirit of the Huygens principle) and scattered amplitudes at that particular point.", "Considering the outgoing amplitude to be an order zero bessel function (being equal to 1 at the origin), the LDOS is then proportional to: $LDOS(r,k) \\mathrm {Re}[a] = \\mathrm {Re}\\left[\\frac{\\alpha e^{2i\\delta }-1}{2i}\\frac{e^{2ik(r_i-r)}}{r_i-r}\\right].$ The generalization to the multiple scattering case of $n$ potentials is straightforward [14]: $a_{MS}(r,k) = \\bf {a_{o}} [1-\\bf A]^{-1} \\bf {a_{s}}\\;,$ where $\\bf {a_{o}}$ and $\\bf {a_{s}}$ are vectors of length $n$ describing the propagation of the outgoing and scattered waves to and from scatterers $i=1..n$ , and $\\bf A$ is an $n\\times n$ matrix describing electron propagation between individual scatterers.", "The spatial LDOS distribution produced by an array of scatterers in a surface state is thus determined by 3 parameters: the $k$ -vector of electrons (tied to their energy by the dispersion relation) and the parameters $\\alpha $ and $\\delta $ describing the scattering potentials.", "Moon et al.", "have made use of the energy dispersion of the surface state ($k=k(E)$ ) to open the energy dimension for quantum holographic encoding.", "[6] We, however would like to utilize the fact, that some or all of the three above named parameters might be spin dependent.", "Two possibilities to achieve such spin-dependence spring to mind.", "One would be to use magnetic atoms or molecules for the construction of the hologram.", "The scattering potentials that they present for the surface state electrons shall then be different for electrons with differently oriented spins.", "In a uniaxial system one could say that $\\alpha =\\alpha (\\sigma )$ and $\\delta =\\delta (\\sigma )$ become dependent on the spin $\\sigma $ of the electron (either $\\uparrow $ or $\\downarrow $ in a uniaxial system), so that the same arrangement of scatterers would produce different LDOS patterns in different spin channels.", "In realization of such a scenario care should be taken to ensure the uniaxial character of the spins' alignment, their thermal stability and the absence of Kondo screening of the atomic or molecular moments.", "The first two and to some extent the third criterion could be satisfied, f.e., if one uses a substrate with substantial crystalline anisotropy and/or couples the spins of the adatoms or molecules constituting the hologram to an external magnetic field [15] or an underlying subsurface magnetic layer.", "[16], [17] To avoid the Kondo screening one could also concentrate on a system with a Kondo temperature below the experimental conditions.", "We, however, concentrate on another possibility of making electron scattering spin-dependent.", "Instead of using magnetic statterers, we switch to a system with an inherently spin-polarized surface state – Co nanoislands or multilayers on Cu(111).", "[8] This system is well studied both experimentally and theoretically [13], [8], [18] and provides us exactly with what we need – the spin-dependence of scattering parameters $\\alpha $ and $\\delta $ as well as the spin-dependence of the electron wave vector $k$ .", "However, Co nanoislands do themselves scatter the surface state electrons [8] and would thus slightly interfere with the encoding of the hologram.", "To avoid that, and at the same time protect the system from additional contamination and intermixing, the Cu surface with islands grown on top can be covered with a few additional layers of copper thus sealing the island within the surface and reducing their effect on the scattering of surface state electrons while still retaining the spin-polarized character of the surface state.", "Without limiting the generality we shall, in the following, concentrate on a particular example of such a system – bilayer cobalt nanoislands adsorbed on a Cu(111) surface and capped with two additional layers of copper [Cu/Co/Cu(111)].", "A sketch of the system is given in Fig.", "REF (a).", "We treat the system theoretically with a first-principles code based on Korringa-Kohn-Rostoker (KKR) Green's function method (in the framework of the density functional theory).", "This method is described in detail in numerous publications [19], [20], , and has been shown to be well suited for describing the kind of systems we are about to address.", "[8] Figure: (color) (a) Sketch of the system: bilayer Co island on Cu(111) covered with two monolayers of Cu.", "(b) Spectral density map of majority (left) and minority (right) electrons (along the K-Γ-K\\mathrm {K}-\\Gamma -\\mathrm {K} direction of the surface Brillouin zone) some 5Å5~\\mathrm {Å} above the system shown in (a).", "(c) Spin resolved Friedel oscillations at 0.6 eV 0.6~\\mathrm {eV} around a singe Cu adatom on Cu/Co/Cu(111).", "The data points represent the KKR calculation, the solid line is a fit of Eq.", "to the data (modified for minority electrons to account for a faster decay).The calculated spectral density maps of electrons some $5~\\mathrm {Å}$ above the surface of the system described above is presented in Fig.", "REF (b) for majority (left) and minority (right) electrons.", "It is apparent that the surface band structure of the system is spin-polarized.", "While majority electrons show a parabolic dispersion band (white dash-dotted line in the left panel) residing in the projected copper bulk band gap (black dashed lines), the minority electrons (right panel) form a dispersive band below the Fermi energy with a relatively high effective mass and have a number of additional dispersive features in the energy ranges of $0.2-0.6~\\mathrm {eV}$ (arising from the hybridization of surface $sp$ electrons with a $d$ -level of Co) and above $0.7~\\mathrm {eV}$ (the analogue of the majority parabolic surface state band pushed beyond the Fermi level).", "It is worth noting, that those bands cross the boundary of the projected bulk band gap, which in normal circumstances would lead to a formation of a fairly broad resonance.", "Here, however, the bands remain largely unbroadened, which means that the layered structure topologically protects the surface state from scattering into the bulk.", "This protection is, however, not complete as shall be seen later.", "Having at our disposal a spin-polarized surface state we can now attempt to encode a hologram using electrons of a certain energy but with different spin orientations to store different information pages.", "To do so, we need to extract the scattering parameters ($\\alpha $ , $\\delta $ and $k$ ).", "In the selection of the electron energy for holographic encoding we rely on two factors: high $k$ -vector to reduce the characteristic electron wavelength and thus maximize the potential information density and possibly small deviation from the Fermi energy in order to facilitate experimental realization.", "Based on that we select an energy of $0.6~\\mathrm {eV}$ above the Fermi energy, which corresponds to the upper end of the dispersive minority band in Fig.", "REF (b).", "Parameters $\\alpha $ , $\\delta $ and $k$ are then obtained for the given energy by fitting Eq.", "REF to a radial LDOS distribution (Friedel oscillations) produced by a single scatterer on a surface.", "Such a distribution for a single Cu adatom on Cu/Co/Cu(111) is given in Fig.", "REF (c) for majority (red circles) and minority (blue rectangles) electrons.", "It is apparent that the Friedel oscillations of majority and minority electrons are substantially different.", "Not only are they defined by different scattering parameters, but they also seem to have different decay ratios.", "While the majority curve can be well fitted with Eq.", "REF , yielding the red solid line, the minority curve seems to decay exponentially rather than follow the $1/r$ rule In fact, since the decay path cannot be unambiguously identified from our calculations, the correct analytic decay rate cannot be guaranteed.", "The exponential factor, however, yields much more accurate fitting results, than $1/r^n$ power law..", "This is the consequence of the upper part of the minority band (at $0.6~\\mathrm {eV}$ ), with which the scattered electrons can be associated, lying outside the projected copper bulk band gap.", "While the Co bilayer island and the capping Cu bilayer seem to largely prevent the surface state electrons from being scattered into the bulk, the probability of such a scattering is non-zero and thus the coherence length of surface state electrons is reduced causing the exponential, rather than $1/r$ , decay of Friedel oscillations.", "The blue solid curve represents a fit of the data to Eq.", "REF with an additional exponential factor added.", "Nevertheless ,the conditions for the spin-selective encoding are satisfied and we can assume that encoding different information pages with electrons of different spin character is possible.", "Following the notation of Moon and coworkers such encoding could be called the “quantum spin holography” (QSH).", "Figure: (color) Quantum spin hologram, encoding two information pages in different spin channels.", "Bit maps 5×75\\times 7 pixels in likeness of letters “S” (a) and “U” (b) encoded with majority and minority electrons, respectively.", "The insets show target (top) and resulting (bottom) information pages.", "On the right (c) the total LDOS distribution ρ(tot)=ρ(E F ,↑)+ρ(E F ,↓)\\rho (tot)=\\rho (E_F,\\uparrow )+\\rho (E_F,\\downarrow ) is shown.To prove the possibility of QSH we will try to encode the patterns used by Moon et al.", "in the “volumetric quantum holography” section of their paper [6] – namely $5\\times 7$ bit pages in likeness of letters “S” and “U”.", "The procedure of encoding the quantum spin hologram is virtually identical to the one described by Moon et al.", "for the volumetric quantum holography: one has to find a scatterer distribution producing the desired pattern in the target area of the surface (Fig.", "REF ).", "The pattern produced by a given atom distribution (atomic hologram) was calculated using the multiple scattering formalism as formulated in Eq.", "REF .", "The corresponding parameters $\\alpha $ , $\\delta $ and $k$ were obtained by fitting as described above.", "The quality of the pattern created by a particular distribution was assessed by calculating the standard Pearson correlation [23] between the calculated LDOS pattern [digitized on a dense real space mesh and averaged locally to obtain the $5\\times 7$ bit pattern, see lower insets in Figs.", "REF panels (a) and (b)] and the target bit map [upper insets in Fig.", "REF (a) and (b)].", "The correlation was then maximized over the whole set of possible atomic/molecular configurations to optimize the hologram.", "While theoretical annealing was effectively utilized by Moon and coworkers [6] to predict optimal molecular positions for volumetric quantum holography, we chose make use of a genetic algorithm technique – a search heuristic that mimics the process of natural evolution [24], [25] to enhance the theoretical annealing method.", "Genetic algorithms allow faster convergence and increase the chance of avoiding local minima in the optimization of the pattern-target correlation.", "In optimizing the atomic configuration of the molecular hologram, the following experimental necessity was taken into account: the atoms were prohibited to form dimers as those are virtually impossible to control with an STM tip.", "The procedure described above applied to an area of Cu/Co/Cu(111) surface approximately $10\\times 10~\\mathrm {nm}$ with a target area of $2\\times 3~\\mathrm {nm}$ yielded the holographic patterns shown in Fig.", "REF (a) for majority and in Fig.", "REF (b) for minority electrons at the Fermi level.", "The optimized positions of atoms forming the hologram can be deduced from the map of the total density of states in Fig.", "REF (c).", "It is apparent that while the total LDOS map [Fig.", "REF (c)] contains a seemingly random pattern of spots the majority and minority [Figs.", "REF (a) and REF (b)] LDOS patterns closely resemble (within the projection area in the center) the targeted information pages (upper insets).", "To quantify the information contained in the hologram we apply the procedure used by Moon et al.", "and dividing the projection area into a grid of $5\\times 7$ squares average the LDOS within each square.", "The obtained values can then be compared to a predefined threshold resulting in 35 bits of encoded information).", "The result of averaging of the holograms presented in panels (a) and (b) of Fig.", "REF is presented in the lower insets in the corresponding panels.", "In summary, we have demonstrated the feasibility of quantum spin holography, i.e.", "information can be stored in the local density of states patterns created independently by majority and minority electrons scattered at an ensemble of magnetic atoms.", "Introducing magnetism into the system thus theoretically allows one to double the information density as compared to the volumetric quantum holographic encoding.", "[6] We are also rather optimistic about the possibility of the experimental realization of the proposed concept using a spin-polarized scanning tunneling microscope for construction and readout of quantum spin holograms." ] ]

[ [ "Testing Booleanity and the Uncertainty Principle" ], [ "Abstract Let f:{-1,1}^n -> R be a real function on the hypercube, given by its discrete Fourier expansion, or, equivalently, represented as a multilinear polynomial.", "We say that it is Boolean if its image is in {-1,1}.", "We show that every function on the hypercube with a sparse Fourier expansion must either be Boolean or far from Boolean.", "In particular, we show that a multilinear polynomial with at most k terms must either be Boolean, or output values different than -1 or 1 for a fraction of at least 2/(k+2)^2 of its domain.", "It follows that given oracle access to f, together with the guarantee that its representation as a multilinear polynomial has at most k terms, one can test Booleanity using O(k^2) queries.", "We show an \\Omega(k) queries lower bound for this problem.", "Our proof crucially uses Hirschman's entropic version of Heisenberg's uncertainty principle." ], [ "Introduction", "Let $f$ be a function from $\\lbrace -1,1\\rbrace ^n$ to $\\mathbb {R}$ .", "Equivalently, one can consider functions on $\\lbrace 0,1\\rbrace ^n$ or $\\mathbb {Z}_2^n$ , as we do below.", "A natural way to represent such a function is as a multilinear polynomial.", "For example: $f(x_1,x_2,x_3) = x_1-2x_2x_3+3.5x_1x_2.$ This representation is called the Fourier expansion of $f$ and is extremely useful in many applications (cf., [19]).", "The coefficients of the Fourier expansion of $f$ are called the Fourier transform of $f$ .", "We denote the Fourier transform by $\\hat{f}$ , and think of it too as a function from $\\lbrace -1,1\\rbrace ^n$ to $\\mathbb {R}$ .", "We say that $f$ is Boolean if $f(x)=1$ or $f(x)=-1$ for all $x$ in its domain.", "An interesting question in the field of discrete Fourier analysis of Boolean functions is the following: what does the fact that $f$ is Boolean tell us about its Fourier transform $\\hat{f}$ ?", "Is there a simple characterization of functions that are the Fourier transform of Boolean functions?", "We propose the following observation that lies at the basis of our proofs: $f$ is Boolean if and only if the convolution (over $\\mathbb {Z}_2^n$ ) of $\\hat{f}$ with itself is equal to the delta function.", "This follows from the convolution theorem, as we show below in Proposition REF .", "Equipped with this characterization, we consider the question of determining whether or not $f$ is Boolean.", "In particular, we consider the case that we are given black box access to a function $f$ , together with the guarantee that its representation as a multilinear polynomial has at most $k$ terms, in which case we say that $f$ is $k$ -sparse.", "Sparse functions on the hypercube have been the subject of numerous studies (see, e.g., [18], [11], [15]).", "We show that $O(k^2)$ queries to $f$ suffice to answer this question correctly with high probability.", "This follows from the following combinatorial result: in Theorem REF we show that if $f$ is not Boolean then it is not Boolean for at least a $2/(k+2)^2$ fraction of its domain.", "More generally, we show that for any set $D \\subset \\mathbb {R}$ of size $d$ , either the image of $f$ is contained in $D$ , or else $f(x) \\notin D$ for at least a $d!/(k+d)^d$ fraction of the domain of $f$ .", "We prove an $\\Omega (k)$ lower bound for this problem.", "Booleanity testing bears resemblance to problems of property testing of functions on the hypercube (see, e.g., [3], [6], [7], [17]).", "See Section REF below for further discussion.", "Our proofs rely on the discrete version of Heisenberg's uncertainty principle.", "There have been very few applications of the discrete uncertainty principle in Computer Science, and in fact we are only familiar with one other such result, concerning circuit lower bounds [13].", "We expect that more applications can be found, in particular in cryptography.", "See Sections REF and REF below for further discussion.", "In the following Section REF we present our main results, and in Sections REF , REF , REF and REF we elaborate on the background and relation to other work, as well as propose a relaxation of our main claim.", "Section  contains formal definitions, and proofs appear in Section ." ], [ "Main results", "A function $f :\\lbrace -1,1\\rbrace ^n\\rightarrow \\mathbb {R}$ is $k$ -sparse if it can be represented as a multilinear polynomial with at most $k$ terms.", "Recall that we say that $f$ is Boolean if its image is contained in $\\lbrace -1,1\\rbrace $ .", "The following theorem is a combinatorial result, stating that a function with a sparse Fourier expansion is either Boolean or far from Boolean.", "Theorem 1.1 Every $k$ -sparse function $f$ is either Boolean, or satisfies ${\\mathbb {P}}_{x}\\left[{f(x) \\notin \\lbrace -1,1\\rbrace }\\right] \\ge \\frac{2}{(k+2)^2}$ where ${\\mathbb {P}}_{x}\\left[{\\cdot }\\right]$ denotes the uniform distribution over the domain of $f$ .", "We in fact prove a more general result: Theorem 1.2 Let $D \\subset \\mathbb {R}$ be a set with $d$ elements.", "Then, for any $k$ -sparse function $f$ , one of the following holds.", "Either ${\\mathbb {P}}_{x}\\left[{f(x) \\in D}\\right] = 1$ , or ${\\mathbb {P}}_{x}\\left[{f(x) \\notin D}\\right] \\ge \\frac{d!", "}{(k+d)^d}$ , where ${\\mathbb {P}}_{x}\\left[{\\cdot }\\right]$ denotes the uniform distribution over the domain of $f$ .", "That is, either $f$ 's image is in $D$ , or it is far from being in $D$ .", "In particular, for $D=\\lbrace -1,1\\rbrace $ (or $\\lbrace 0,1\\rbrace $ , or any other set of size two), this theorem reduces to Theorem  REF An immediate consequence of Theorem REF is the following result.", "Theorem 1.3 For every $\\epsilon >0$ there exists a randomized algorithm with query (and time) complexity $O(k^2\\log (1/\\epsilon ))$ that, given $k$ and oracle access to a $k$ -sparse function $f$ , returns true if $f$ is Boolean, and returns false with probability at least $1-\\epsilon $ if $f$ is not Boolean.", "This result can easily be extended to test whether the image of a function on the hypercube is contained in any finite set, using Theorem REF .", "We prove the following lower bound: Theorem 1.4 Let $A$ be a randomized algorithm that, given $k$ and oracle access to a $k$ -sparse function $f$ , returns true with probability at least $2/3$ if $f$ is Boolean, and returns false with probability at least $2/3$ if $f$ is not Boolean.", "Then $A$ has query complexity $\\Omega (k)$ ." ], [ "The Fourier transform of Boolean functions", "Let $f,g$ be functions from $\\mathbb {Z}_2^n$ to $\\mathbb {R}$ .", "Their convolution $f*g$ is also a function from $\\mathbb {Z}_2^n$ to $\\mathbb {R}$ defined by $[f*g](x) = \\sum _{y \\in \\mathbb {Z}_2^n}f(y)g(x+y),$ where the addition “$x+y$ ” is done using the group operation of $\\mathbb {Z}_2^n$ .", "Note that the convolution operator is both associative and distributive.", "An observation that lies at the basis of our proofs is a characterization of the Fourier transforms of Boolean functions: $\\hat{f} : \\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ is the Fourier transform of a Boolean function if its convolution with itself is equal to the delta function; that is, $\\hat{f} * \\hat{f} = \\delta $ (where $\\delta : \\mathbb {Z}_2^n \\rightarrow \\lbrace 0,1 \\rbrace $ is given by $\\delta (0)=1$ , and $\\delta (x)=0$ for every $x \\ne 0$ ).", "This is our Proposition REF ; it follows from the convolution theorem (see, e.g.,[14]).", "Equivalently, given a function $f$ on $\\mathbb {Z}_2^n$ , one can shift it by acting on it with $x \\in \\mathbb {Z}_2^n$ by $[xf](y) = f(x+y)$ .", "Hence the observation above can be stated as follows: If and only if a function is orthogonal to its shifted self, for all non-zero shifts in $\\mathbb {Z}_2^n$ , then it is the Fourier transform of a Boolean function." ], [ "The uncertainty principle", "A distribution over a discrete domain $S$ is often represented as a non-negative function $f: S \\rightarrow \\mathbb {R}^+$ which is normalized in $L_1$ , i.e., $\\sum _{x \\in S}f(x)=1$ .", "In Quantum Mechanics the state of a particle on a domain $S$ is represented by a complex function on $S$ , and the probability to find the particle in a particular $x \\in S$ is equal to $|f(x)|^2$ .", "Accordingly, $f$ is normalized in $L_2$ , so that $\\sum _{x\\in S}|f(x)|^2=1$ .", "Often, the domain $S$ is taken to be $\\mathbb {R}$ (or some power thereof).", "In this continuous case one represents the state of a particle by a function $f:\\mathbb {R}\\rightarrow such that $ x R|f(x)|2dx =1$, andthen $ |f(x)|2$ is the probability density function of thedistribution of the particle^{\\prime }s position.", "The Fourier transform of $ f$,denoted by $ f$, is then also normalized in $ L2$ (if one choosesthe Fourier transform operator to be unitary), and $ |f(x)|2$ isthe probability density function of the {\\em distribution of theparticle^{\\prime }s momentum}.$ The Heisenberg uncertainty principle states that the variance of a particle's position times the variance of its momentum is at least one - under an appropriate choice of units.", "Besides its physical significance, this is also a purely mathematical statement relating a function on $\\mathbb {R}$ to its Fourier transform.", "Hirschman [12] conjectured in 1957 a stronger entropic form, namely $H_e\\Big [f\\Big ]+H_{e}\\left[{\\hat{f}}\\right] \\ge 1-\\ln 2,$ where $H_{e}\\left[{f}\\right] = -\\int _{x \\in \\mathbb {R}} |f(x)|^2\\ln |f(x)|^2dx$ is the differential entropy of $f$ .", "This was proved nearly twenty years later by Beckner [1].", "When the domain $S$ is $\\mathbb {Z}_2^n$ (equivalently, $\\lbrace -1,1\\rbrace ^n$ ) then a similar inequality holds, but with a different constant.", "Let $f:\\mathbb {Z}_2^n\\rightarrow have Fourier transform $ f:Z2n .", "Then $H\\Bigg [\\frac{f}{||f||}\\Bigg ]+H\\left[{\\frac{\\hat{f}}{||\\hat{f}||}}\\right] \\ge n.$ where $H\\left[{f}\\right] = -\\sum _{x \\in \\mathbb {Z}_2^n}|f(x)|^2\\log _2|f(x)|^2$ , and $\\Vert f\\Vert = \\sqrt{\\sum _{x \\in \\mathbb {Z}_2^n}f(x)^2}$ .", "(For a further discussion on the foregoing inequality, see Section REF .)" ], [ "Relation to property testing", "We note that the problem of testing Booleanity is similar in structure to a property testing problem.", "Since its introduction in the seminal paper by Rubinfeld and Sudan [20], property testing has been studied extensively, both due to its theoretical importance, and the wide range of applications it has spanned (cf.", "[8], [9]).", "In particular, property testing of functions on the hypercube is an active area of research [3], [6], [7], [17].", "A typical formulation of property testing is as follows: Given a fixed property $P$ and an input $f$ , a property tester is an algorithm that distinguishes with high probability between the case that $f$ satisfies $P$ , and the case that $f$ is $\\epsilon $ -far from satisfying it, according to some notion of distance.", "The algorithm we present for testing Booleanity given oracle access is similar to a property testing algorithm.", "However, in our case there is no proximity parameter: we show that if a function is not Boolean then it must be far from Boolean, and can therefore be proved to not be Boolean by a small number of queries.", "This type of property testing algorithms have appeared in the context of the study of adaptive versus non-adaptive testers [10]." ], [ "Discussion and open questions", "In this paper we use a discrete entropy uncertainty principle to prove a combinatorial statement concerning functions on the hypercube.", "To the best of our knowledge, this is the first time this tool has been used in the context of theoretical computer science, outside of circuit lower bounds.", "We note that Theorem REF and Theorem REF are, in a sense, a dual to the Schwartz-Zippel lemma [22], [21]: both limit the number of roots of a polynomial, given that it is sparse.", "Given the usefulness of the Schwartz-Zippel lemma, we suspect that more combinatorial applications can be found for the discrete uncertainty principle.", "For example, Biham, Carmeli and Shamir [2] show that an RSA decipherer who uses hardware that has been maliciously altered can be vulnerable to an attack resulting in the revelation of the private key.", "The assumption is that the dechiperer is not able to discover that it is using faulty hardware, because the altered function returns a faulty output for only a very small number of inputs.", "The uncertainty principle shows that such malicious alteration is impossible to accomplish with succinctly represented functions: when the Fourier transform of a function is sparse then it is impossible to “hide” elements in its image.", "As for the scope of this study, many questions still remains open.", "In particular, there is a gap between the lower bound and the upper bound for testing Booleanity with oracle access; we are disinclined to guess which of the two is not tight.", "A natural extension of our results is to functions with a Fourier transform $\\hat{f}$ that is not restricted to having support of size $k$ , but rather having entropy $\\log k$ ; the latter is a natural relaxation of the former.", "Unfortunately, we have not been able to generalize our results given this constraint.", "However, another natural constraint which does yield a generalization is the requirement that the entropy of $\\hat{f}*\\hat{f}$ , the convolution of the Fourier transform with itself, is at most $2\\log k$ .", "See Proposition REF for why this is indeed natural.", "Two additional amendments are needed to be added for Theorem REF for it to be thus generalized.", "First, we require that $|f|^2=2^n$ .", "Next, recall that we call a function $f$ Boolean if $f^2=1$ .", "We likewise say that $f$ is $\\epsilon $ -close to being Boolean if $\\sqrt{\\frac{1}{2^n}\\sum _{x \\in \\mathbb {Z}_2^n}(f(x)^2-1)^2} \\le \\epsilon .$ This is simply the $L_2$ distance of $f^2$ from the constant function 1.", "In the following theorem we do not test for Booleanity, but for $\\epsilon $ -closeness to Booleanity.", "Theorem 1.5 Let $H\\left[{\\frac{\\hat{f}*\\hat{f}}{\\Vert \\hat{f}*\\hat{f}\\Vert }}\\right] \\le 2\\log k$ , and let $\\Vert f\\Vert ^2=2^n$ .", "Then $f$ is either $\\epsilon $ -close to Boolean, or satisfies ${\\mathbb {P}}_{x}\\left[{f(x) \\notin \\lbrace -1,1\\rbrace }\\right] = \\Omega \\left(\\frac{1}{k^{2(\\epsilon ^2+1)/\\epsilon ^2}}\\right)$ where ${\\mathbb {P}}_{x}\\left[{\\cdot }\\right]$ denotes the uniform distribution over the domain of $f$ .", "We prove this Theorem in Section REF ." ], [ "Definitions", "The following definitions are mostly standard.", "We deviate from common practice by considering both a function and its Fourier transform to be defined on the same domain, namely $\\mathbb {Z}_2^n$ .", "Some readers might find $\\lbrace 0,1\\rbrace ^n$ or $\\lbrace -1,1\\rbrace ^n$ a more familiar domain for a function, and likewise the power set of $[n]$ a more familiar domain for its Fourier transform.", "Denote $\\mathbb {Z}_2=\\mathbb {Z}/2\\mathbb {Z}$ .", "For $x,y \\in \\mathbb {Z}_2^n$ we denote by $x+y$ the sum using the $\\mathbb {Z}_2^n$ group operation.", "The equivalent operation in $\\lbrace -1,1\\rbrace ^n$ is pointwise multiplication (i.e., $xy = (x_1y_1, \\ldots ,x_ny_n)$ ).", "Let $f:\\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ .", "We denote its $L_2$ -norm by $\\Vert f\\Vert = \\sqrt{\\sum _{x \\in \\mathbb {Z}_2^n}f(x)^2},$ denote its support by $\\operatorname{supp}f = \\lbrace x \\in \\mathbb {Z}_2^n \\::\\: f(x) \\ne 0\\rbrace ,$ and denote its entropy by $H\\left[{f}\\right] = -\\sum _{x \\in \\mathbb {Z}_2^n}f(x)^2\\log f(x)^2,$ where logarithms are base two and $0\\log 0 = 0$ , by the usual convention in this case.", "We remark that for the simplicity of the presentation, we define norms and convolutions using summation rather than expectation.", "We call a function $f:\\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ Boolean if its image is in $\\lbrace -1,1\\rbrace $ , i.e., if $f(x) \\in \\lbrace -1,1\\rbrace $ for all $x \\in \\mathbb {Z}_2^n$ .", "Let $\\hat{f}:\\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ denote the discrete Fourier transform (also known as the Walsh-Fourier transform and Hadamard transform) of $f$ , or its representation as a multilinear polynomial: $\\hat{f}(x) = \\frac{1}{2^n}\\sum _{y \\in \\mathbb {Z}_2^n}f(y)\\chi _y(x),$ where the characters $\\chi _y$ are defined by $\\chi _y(x) ={\\left\\lbrace \\begin{array}{ll}-1 & \\sum _{i:y_i=1}x_i = 1\\\\1 & \\mbox{otherwise}\\end{array}\\right.", "}.$ Note that the sum $\\sum _{i:y_i=1}x_i$ is over $\\mathbb {Z}_2$ and that $x_i,y_i$ are (respectively) the $i$ 'th coordinate of $x$ and $y$ .", "It follows that the discrete Fourier expansion of $f$ is $f(x) = \\sum _{y \\in \\mathbb {Z}_2^n}\\hat{f}(y)\\chi _y(x).$ Note that this is a representation of $f$ as a multilinear polynomial.", "Hence $f:\\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ is $k$ -sparse if $|\\operatorname{supp}\\hat{f}| \\le k$ .", "We define $\\delta : \\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ by $\\delta (x) ={\\left\\lbrace \\begin{array}{ll}1& \\mbox{when } x = (0,\\ldots ,0)\\\\0& \\mbox{otherwise}\\end{array}\\right.", "}.$ If we denote by ${\\bf 1}(x): \\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ the constant function such that ${\\bf 1}(x)=1$ for all $x \\in \\mathbb {Z}_2^n$ , then it is easy to verify that $\\hat{\\bf 1}=\\delta .$ Given functions $f,g : \\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ , their convolution $f * g$ is also a function from $\\mathbb {Z}_2^n$ to $\\mathbb {R}$ , defined by $[f*g](x) = \\sum _{y \\in \\mathbb {Z}_2^n}f(y)g(x+y).$ We denote $f^{(2)} = f * f,$ and more generally $f^{(k)}$ is the convolution of $f$ with itself $k$ times.", "$f^{(0)}$ is taken to equal $\\delta $ , since $f * \\delta = f$ ." ], [ "The Fourier transform of Boolean functions", "The convolution theorem (see., e.g., [14]) for $\\mathbb {Z}_2^n$ states that, up to multiplication by a constant, the Fourier transform of the pointwise multiplication of two functions is equal to the convolution of their Fourier transforms, and that likewise the Fourier transform of a convolution is the product of the Fourier transforms (again up to a constant): $\\widehat{f \\cdot g} = \\hat{f} *\\hat{g},\\quad \\quad \\mbox{and}\\quad \\quad \\widehat{f * g} = 2^n\\hat{f}\\cdot \\hat{g}.$ The correctness of the constants can be verified by, for example, setting $f=g={\\bf 1}$ .", "The following proposition follows from Eqs.", "REF and REF .", "Proposition 3.1 $f :\\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ is Boolean iff $\\hat{f}*\\hat{f}=\\delta $ ." ], [ "The discrete uncertainty principle", "The discrete uncertainty principle for $\\mathbb {Z}_2^n$ is the following.", "It is a straightforward consequence of Theorem 23 in Dembo, Cover and Thomas [5]; we provide the proof for completeness, since it does not seem to have previously appeared in the literature.", "Theorem 3.2 For any non-zero function $f:\\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ (i.e., $\\Vert f\\Vert > 0$ ) it holds that $H\\Bigg [\\frac{f}{||f||}\\Bigg ]+H\\left[{\\frac{\\hat{f}}{||\\hat{f}||}}\\right]\\ge n.$ Let $U$ be a unitary $n$ by $n$ matrix such that $\\max _{ij}|u_{ij}|=M$ .", "Let $x \\in n$ be such that $\\Vert x\\Vert >0$ .", "Then Theorem 23 in Dembo, Cover and Thomas [5] states that $H\\left[{\\frac{x}{\\Vert x\\Vert }}\\right] + H\\left[{\\frac{Ux}{\\Vert Ux\\Vert }}\\right] \\ge 2 \\log (1/M),$ where for $x \\in n$ we define $H\\left[{x}\\right] = -\\sum _{i \\in [n]}|x_i|^2\\log |x_i|^2$ .", "Let $F$ be the matrix representing the Fourier transform operator on $\\mathbb {Z}_2^n$ .", "Note that by our definition in Eq.", "REF , the transform operator $F$ is not unitary.", "However, if we multiply it by $\\sqrt{2^n}$ (i.e., normalize the characters $\\chi _y$ ) then it becomes unitary.", "The normalized matrix elements (which are equal to the elements of the normalized characters $\\chi _y$ ), are all equal to $\\pm 1/\\sqrt{2^n}$ .", "Hence $M = 1/\\sqrt{2^n}$ , and $H\\left[{\\frac{f}{\\Vert f\\Vert }}\\right]+H\\left[{\\frac{Ff}{\\Vert Ff\\Vert }}\\right] \\ge 2 \\log (1/M) =n.$ A distribution supported on a set of size $k$ has entropy at most $\\log k$ , as can be shown by calculating its Kullback-Leibler divergence from the uniform distribution (see, e.g., [4]).", "Hence any distribution with entropy $\\log k$ has support of size at least $k$ .", "This fact, together with the discrete uncertainty principle, yields a proof of the following claim (see Matolcsi and Szucs [16] or O'Donnell [19] for an alternative proof of Eq.", "REF .)", "Claim 3.3 For any non-zero function $f:\\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ (i.e., $\\Vert f\\Vert > 0$ ) it holds that $|\\operatorname{supp}f| \\cdot |\\operatorname{supp}\\hat{f}| \\ge 2^n$ and $|\\operatorname{supp}f| \\cdot 2^{H\\left[{\\hat{f}/ \\Vert \\hat{f}\\Vert }\\right]} \\ge 2^n.$ By Theorem REF we have that $H\\left[{\\frac{f}{||f||}}\\right]+H\\left[{\\frac{\\hat{f}}{||\\hat{f}||}}\\right] \\ge n.$ Since $\\log |\\operatorname{supp}(f)| = \\log |\\operatorname{supp}(f/||f||)| \\ge H\\left[{f/||f||}\\right]$ then $|\\operatorname{supp}f |\\cdot 2^{H\\left[{\\hat{f}/ \\Vert \\hat{f}\\Vert }\\right]} \\ge 2^n$ and likewise $|\\operatorname{supp}f |\\cdot |\\operatorname{supp}\\hat{f} | \\ge 2^n.$ We note that for the proof of Theorem REF we rely on Claim REF , whereas for the more general Theorem REF , using Claim REF does not suffices and we must use (the stronger) Theorem REF ." ], [ "Testing Booleanity given oracle access", "We begin by proving the following standard proposition, which relates the support of functions $f$ and $g$ with the support of their convolution.", "Proposition 3.4 Let $g,f :\\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ .", "Then $\\operatorname{supp}(f*g) \\subseteq \\operatorname{supp}f + \\operatorname{supp}g.$ Here $\\operatorname{supp}f + \\operatorname{supp}g$ is the set of elements of $\\mathbb {Z}_2^n$ that can be written as the sum of an element in $\\operatorname{supp}f$ and an element in $\\operatorname{supp}g$ .", "Let $x \\in \\operatorname{supp}(f * g)$ .", "Then, from the definition of convolution, there exist $y$ and $z$ such that $f(y) \\ne 0$ , $g(z) \\ne 0$ and $x=y+z$ .", "Hence $x \\in \\operatorname{supp}f + \\operatorname{supp}g$ .", "We consider a $k$ -sparse function $f$ to which we are given oracle access.", "We are asked to determine if it is Boolean, or more generally if its image is in some small set $D$ .", "We here think of $k$ as being small - say polynomial in $n$ .", "We first prove the following combinatorial result: Theorem (REF ) Let $D \\subset \\mathbb {R}$ be a set with $d$ elements.", "Then for any $k$ -sparse $f$ one of the following holds.", "Either ${\\mathbb {P}}_{x}\\left[{f(x) \\in D}\\right] = 1$ , or ${\\mathbb {P}}_{x}\\left[{f(x) \\notin D}\\right] \\ge \\frac{d!", "}{(k+d)^d}$ , where ${\\mathbb {P}}_{x}\\left[{\\cdot }\\right]$ denotes the uniform distribution over the domain of $f$ .", "Let $D=\\lbrace y_1,\\ldots ,y_d\\rbrace $ .", "Denote $g = \\prod _{i=1}^d(f-y_i),$ so that $g(x) = 0$ iff $f(x) \\in D$ .", "Then $\\hat{g} &=\\left(\\hat{f}-y_1\\delta \\right)*\\cdots *\\left(\\hat{f}-y_d\\delta \\right)= \\hat{f}^{(d)}+a_{d-1}\\hat{f}^{(d-1)}+ \\cdots a_1\\hat{f}+a_0\\delta ,$ for some coefficients $a_0,\\ldots ,a_{d-1}$ .", "Therefore $\\operatorname{supp}\\hat{g} \\subseteq \\bigcup _{i=1}^d\\operatorname{supp}\\hat{f}^{(i)}\\cup \\lbrace 0\\rbrace .$ We show that $|\\operatorname{supp}\\hat{g}| \\le (k+d)^d/d!$ .", "Let $A = \\operatorname{supp}\\hat{f} \\cup \\lbrace 0\\rbrace $ .", "Then by Proposition REF $\\operatorname{supp}\\hat{f}^{(i)}$ is a subset of $iA = A+\\cdots +A$ , where the sum is taken $i$ times; this is the set of elements in $\\mathbb {Z}_2^n$ that can be written as a sum of $i$ elements of $A$ .", "Hence $\\operatorname{supp}\\hat{g} \\subseteq A \\cup 2A \\cup \\cdots \\cup dA.$ Since $0 \\in A$ , then for all $i \\le d$ we have that $iA \\subseteq dA$ .", "Hence $\\operatorname{supp}\\hat{g} \\subseteq dA.$ Therefore $\\operatorname{supp}\\hat{g}$ is a subset of the set of elements that can be written as the sum of at most $d$ elements of $A$ .", "This number is bounded by the number of ways to choose $d$ elements of $A$ with replacement, disregarding order.", "Hence $|\\operatorname{supp}\\hat{g}| \\le \\binom{|A|-1+d}{d} \\le \\frac{(k+d)^d}{d!", "},$ since $|A| \\le |\\operatorname{supp}\\hat{f}| + 1=k+1$ .", "Now, if $f(x) \\in D$ for all $x \\in \\mathbb {Z}_2^n$ , then clearly ${\\mathbb {P}}_{x}\\left[{f(x) \\in D}\\right] = 1$ .", "Otherwise, $g(x)$ is different than zero for some $x$ , and so $\\Vert g\\Vert > 0$ .", "Hence we can apply Claim REF and $|\\operatorname{supp}g| \\cdot |\\operatorname{supp}\\hat{g}| \\ge 2^n.$ By Eq.", "REF this implies that $|\\operatorname{supp}g| \\ge \\frac{2^nd!", "}{(k+d)^d}.$ Since the support of $g$ is precisely the set of $x \\in \\mathbb {Z}_2^n$ for which $f(x) \\notin D$ then it follows that ${\\mathbb {P}}_{x}\\left[{f(x) \\notin D}\\right] \\ge \\frac{d!", "}{(k+d)^d}.$ A consequence is that a function that is not Boolean (i.e., the case $D=\\lbrace -1,1\\rbrace $ ) is not Boolean over a fraction of at least $2/(k+2)^2$ of its domain.", "Theorem REF is a direct consequence of this result: assuming oracle access to $f$ (i.e., $O(1)$ time random sampling), the algorithm samples $f$ at random ${\\textstyle \\frac{1}{2}}(k+2)^2\\ln (1/\\epsilon )$ times, and therefore will discover an $x$ such that $f(x) \\notin \\lbrace -1,1\\rbrace $ with probability at least $1-\\epsilon $ - unless $f$ is Boolean.", "While we were not able to show a tight lower bound, we show that any algorithm would require at least $\\Omega (k)$ queries to perform this task (even when two-sided error is allowed).", "Theorem (REF ) Let $A$ be a randomized algorithm that, given $k$ and oracle access to a $k$ -sparse function $f$ , returns true with probability at least $2/3$ if $f$ is Boolean, and returns false with probability at least $2/3$ if $f$ is not Boolean.", "Then $A$ has query complexity $\\Omega (k)$ .", "[Proof of Theorem REF ] Let $A$ be an algorithm that is given oracle access to a function $f:\\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ , together with the guarantee that $\\operatorname{supp}\\hat{f}\\le k$ .", "When $f$ is Boolean then $A$ returns “true”.", "When $f$ is not Boolean then $f$ returns “false” with probability at least $2/3$ .", "We show that $A$ makes $\\Omega (k)$ queries to $f$ .", "Denote by $B_k$ the set of Boolean functions that depend only on the first $\\log k$ coordinates.", "Denote by $C_k$ the set of functions that likewise depend only on the first $\\log k$ coordinates, return values in $\\lbrace -1,1\\rbrace $ for some $k-1$ of the $k$ possible values of the first $\\log k$ coordinates, but otherwise return 2.", "Note that functions in both $B_k$ and $C_k$ have Fourier transforms of support of size at most $k$ .", "We prove the lower bound on the query complexity of the randomized algorithm by showing two distributions, a distribution of Boolean functions and a distribution of non-Boolean functions, which are indistinguishable to any algorithm that makes a small number of queries to the input.", "That is, we present two distributions: one for which the algorithm should return “false” (denoted by $\\mathcal {D}_0$ ) and another for which the algorithm should return “true” (denoted by $\\mathcal {D}_1$ ).", "We prove that any randomized algorithm which performs at most $o(k)$ queries would not be able to distinguish between the two distributions with non-negligible probability.", "This proves the claim.", "Let $\\mathcal {D}_1$ be the uniform distribution over $B_k$ , and let $\\mathcal {D}_0$ be the uniform distribution over $C_k$ .", "Observe that an arbitrary query to $f$ in either distribution would output a non-Boolean value with probability at most $1/k$ , independently of previous queries with different values of the first $\\log k$ coordinates.", "Therefore any algorithm that performs $o(k)$ queries would find an input for which $f(x)=2$ with probability $o(1)$ , and would therefore be unable to distinguish between $\\mathcal {D}_0$ and $\\mathcal {D}_1$ with noticeable probability." ], [ "Proof of Theorem ", "Recall the statement of Theorem REF .", "Theorem (REF ) Let $H\\left[{\\frac{\\hat{f}*\\hat{f}}{\\Vert \\hat{f}*\\hat{f}\\Vert }}\\right] \\le 2\\log k$ , and let $\\Vert f\\Vert ^2=2^n$ .", "Then $f$ is either $\\epsilon $ -close to Boolean, or satisfies ${\\mathbb {P}}_{x}\\left[{f(x) \\notin \\lbrace -1,1\\rbrace }\\right] = \\Omega \\left(\\frac{1}{k^{2(\\epsilon ^2+1)/\\epsilon ^2}}\\right)$ where ${\\mathbb {P}}_{x}\\left[{\\cdot }\\right]$ denotes the uniform distribution over the domain of $f$ .", "We begin by proving a preliminary proposition.", "Proposition 3.5 Let $X$ be a discrete random variable, and let $x_0$ be a value that $X$ takes with positive probability.", "Then $H(X|X \\ne x_0) \\le \\frac{H(X)}{{\\mathbb {P}}\\left[{X \\ne x_0}\\right]}.$ Let $A$ be the indicator of the event $X=x_0$ .", "Then $H(X) & \\ge H(X|A) \\\\ &= {\\mathbb {P}}\\left[{X=x_0}\\right]H(X|X=x_0) + {\\mathbb {P}}\\left[{X\\ne x_0}\\right]H(X|X\\ne x_0)\\\\ &= {\\mathbb {P}}\\left[{X\\ne x_0}\\right]H(X|X \\ne x_0),$ since $H(X|X=x_0)=0$ .", "[Proof of Theorem REF ] Assume that $f$ is $\\epsilon $ -far from being Boolean.", "Observe that $\\Vert \\hat{f}^{(2)}\\Vert ^2 = \\frac{1}{2^n}\\Vert f^2\\Vert ^2 =\\frac{1}{2^n}\\sum _{x \\in \\mathbb {Z}_2^n}f(x)^4 = \\frac{1}{2^n}\\sum _{x \\in \\mathbb {Z}_2^n}(f(x)^2-1)^2 + 1 \\ge 1+\\epsilon ^2,$ where the equality before last follows from the fact that $\\Vert f\\Vert ^2=2^n$ .", "Let $X$ be a $\\mathbb {Z}_2^n$ -valued random variable such that ${\\mathbb {P}}\\left[{X=x}\\right]=\\hat{f}^{(2)}(x)^2/\\Vert \\hat{f}^{(2)}\\Vert ^2$ .", "Since $f$ is normalized, then $\\hat{f}^{(2)}(0)=1$ .", "Furthermore, ${\\mathbb {P}}\\left[{X \\ne 0}\\right] = 1-{\\mathbb {P}}\\left[{X = 0}\\right] =1-\\frac{\\hat{f}^{(2)}(0)^2}{\\Vert \\hat{f}^{(2)}\\Vert ^2} \\ge \\frac{\\epsilon ^2}{\\epsilon ^2+1},$ since $\\hat{f}^{(2)}(0)=1$ , and by Eq.", "REF .", "Let $g=f^2-1$ .", "Then $\\hat{g} = \\hat{f}^{(2)}-\\delta $ , $\\hat{g}(0) =0$ , and ${\\mathbb {P}}\\left[{X=x|X \\ne 0}\\right]=\\hat{g}(x)^2/\\Vert \\hat{g}\\Vert ^2$ .", "Hence by Proposition REF it follows that $H\\left[{\\frac{\\hat{g}}{\\Vert \\hat{g}\\Vert }}\\right] \\le H\\left[{\\frac{\\hat{f}^{(2)}}{\\Vert \\hat{f}^{(2)}\\Vert ^2}}\\right] \\cdot \\frac{\\epsilon ^2+1}{\\epsilon ^2} \\le 2 \\frac{\\epsilon ^2+1}{\\epsilon ^2}\\log k,$ where the second inequality follows from the proposition hypothesis that $H\\left[{\\frac{\\hat{f}*\\hat{f}}{\\Vert \\hat{f}*\\hat{f}\\Vert }}\\right] \\le 2\\log k.$ By Claim REF it follows that $|\\operatorname{supp}g| \\cdot 2^{H\\left[{\\hat{g}/\\Vert \\hat{g}\\Vert }\\right]} \\ge 2^n.$ Hence $|\\operatorname{supp}(f^2-1)| \\cdot k^{2(\\epsilon ^2+1)/\\epsilon } \\ge 2^n$ , from which the proposition follows directly, since ${\\mathbb {P}}_{x}\\left[{f(x) \\notin \\lbrace -1,1\\rbrace }\\right] = \\frac{|\\operatorname{supp}(f^2-1)|}{2^n}.$" ], [ "Acknowledgments", "The authors would like to thank Elchanan Mossel for a helpful initial discussion of the problem and for suggesting the application to property testing.", "We would like to thank Adi Shamir for suggesting the relevance to cryptography, and we would like to thank Oded Goldreich for discussions regarding the relevance to property testing.", "Last, we would like to thank the anonymous referees for the helpful comments that allowed us to improve the presentation of the results." ] ]

[ [ "GPGPU for orbital function evaluation with a new updating scheme" ], [ "Abstract We accelerated an {\\it ab-initio} QMC electronic structure calculation by using GPGPU.", "The bottleneck of the calculation for extended solid systems is replaced by CUDA-GPGPU subroutine kernels which build up spline basis set expansions of electronic orbital functions at each Monte Carlo step.", "We achieved 30.8 times faster evaluation for the bottleneck, confirmed on the simulation of TiO$_2$ solid with 1,536 electrons.", "To achieve better performance in GPGPU we propose a new updating scheme for Monte Carlo sampling, quasi-simultaneous updating, which is in between the configuration-by-configuration updating and the widely-used particle-by-particle one.", "The energy deviation caused both by the single precision treatment and the new updating scheme is found to be within the accuracy required in the calculation, $\\sim 10^{-3}$ hartree per primitive cell." ], [ "Introduction", "General Purpose computing on Graphical Processing Units (GPGPU) [1], [2] has attracted recent interest in HPC (High Performance Computing) for accelerating calculations at a reasonable cost.", "Environments for developing GPGPU, such as CUDA (Compute Unified Device Architecture) [3], [4], have also contributed to the recent trend for using GPGPU for scientific applications with much increased portability [2].", "Electronic structure calculations [5] form one of the largest fields within HPC and there have been many attempts to accelerate such calculations using GPGPU [2].", "Electronic structure calculation using quantum Monte Carlo (QMC) methods can provide highly reliable estimates of material properties for a wide range of compounds [7], [10], [6].", "The very high computational demands are not so important because of the inherently high efficiency of massively parallel computational facilities for Monte Carlo computations [11].", "There have been several attempts to apply GPGPU to ab-initio QMC electronic structure calculations [13], [12].", "Previously we reported GPGPU acceleration of a QMC calculation for molecular systems, in which we achieved a speedup of more than a factor of 20 [12].", "The key idea was to replace only the bottleneck subroutines in the main code by the CUDA kernel running on the GPU.", "We emphasized that the replacement of the entire simulation code by its GPU version is not practical from the viewpoint of version administration [12].", "This becomes more serious for practical program packages with large number of users, as is common in ab-initio electronic structure simulations [12].", "It was challenging to achieve substantial acceleration using such a `partial replacement strategy', and it should give a speedup of at least more than a factor of ten to be advantageous to use multi-core processor technology.", "In Ref.", "[12] the main code written in Fortran90 (F90) was partially replaced by the GPU kernel, which were at the object code level.", "Users could switch back to the original CPU version of the subroutine if the GPU was not available.", "In the previous study GPGPU was applied to molecular simulations, although solid systems are the most attractive target for GPU-QMC electronic structure simulations [10] because of the vast CPU time required and the potential of QMC to achieve more reliable results than frameworks such as density functional theory (DFT).", "The bottleneck in the present work differs from that in our previous molecular simulation [12].", "In our previous work the bottleneck was the routine for computing the Hartree fields corresponding to the particle configuration [14].", "In the present work the bottleneck is the routine for evaluating the single particle orbitals at the required particle positions.", "We have achieved a speedup of more than a factor of 30 with GPGPU compared with the single core performance of the conventional CPU evaluation.", "This acceleration does not, in principle, harm the MPI (massively parallel interface) parallelization efficiency, which is essentially the same as in our previous work [12].", "The conventional MPI parallel evaluation [10] can be accelerated further by attaching a GPU to each node.", "In QMC calculations the electronic orbitals are calculated many times at different electronic positions.", "It is quite common in ab-initio electronic structure methods, including QMC, that one builds up orbital functions for given electronic positions.", "Our implementation achieved here would be useful also in self-consistent field (SCF) methods used in density functional theories (DFT) or molecular orbitals (MO) methods.", "MPI parallelization has successfully been used in QMC electronic structure calculations [7], [10], [6], obtaining $\\sim $ 99% parallel efficiency by distributing the huge number of configurations over the processing nodes.", "On each node the evaluation is usually sequential, though there have been several attempts to exploit further parallelization within the node using, for example, OpenMP [6].", "The evaluations performed on each node include updating an electronic configuration $\\vec{R}^{(\\alpha )} = (\\vec{r}_1^{(\\alpha )},\\cdots ,\\vec{r}_j^{(\\alpha )}, \\cdots ,\\vec{r}_N^{(\\alpha )})$ , and sampling with the updated configuration, where $\\alpha $ is the index for MC steps.", "There are two major types of updating scheme, the configuration-by-configuration scheme (simultaneous updating) and the particle-by-particle scheme (PbP, sequential updating).", "In the former, attempted trial $N$ -electron moves are generated to update a configuration, $\\vec{R}^{(\\alpha )} \\rightarrow \\vec{R}^{(\\alpha + 1)}$ , and then the new configuration is accepted or rejected.", "In the latter, a trial move of a single electron is attempted and accepted or rejected, $\\vec{r}_j^{(\\alpha )} \\rightarrow \\vec{r}_j^{(\\alpha +1)}$ , and the process is repeated $N$ times.", "Sequential updating is more efficient than simultaneous updating, and it is widely used in QMC electronic structure calculations [6].", "For hybrid parallelization, including GPGPU and OpenMP, one seeks further parallelization in the sequential evaluation within an MPI node.", "The GPGPU performs the accept/reject steps for each particle `simultaneously'.", "The ratio of the probabilities evaluated in the Metropolis accept/reject algorithm [7] differs both from those for simultaneous and sequential updating.", "In this sense our updating scheme can be viewed as `quasi-simultaneous updating' (Q.S.).", "This scheme is designed to obtain GPGPU acceleration by improving the sequential memory access (so called `coalescing'), and the concealment of memory latencies.", "We have confirmed that our new updating scheme does not change the results within the required statistical accuracy, namely the chemical accuracy.", "The paper is organized as follows.", "In §II we briefly summarize the VMC method (Variational Monte Carlo method).", "The evaluation of the orbitals represented in a spline basis set is the bottleneck in the computation, as described in this section.", "The benchmark systems used in the performance evaluation are also introduced.", "§III is devoted to a description of the GPU architecture.", "The structure of processors and memories in the GPU used in the present work is briefly explained.", "Other features, such as how we assign the number of threads and blocks for parallel processing, are discussed in §IV, in connection with the design and implementation of the quasi-updating scheme.", "The quasi-updating scheme is also introduced in this section.", "Several other possible implementations with different updating schemes or thread/block assignments are introduced here and their performances are compared.", "The results are summarized in §V, including comparisons of the energies, operation costs and data transfers, and the dependence of the performance on system size.", "In §VI we discuss the results, comparing with the ideal performance in terms of operations and memory access.", "We also discuss the possibility of using high-speed memory devices in the GPU and the relation to linear algebra packages.", "In ab-initio calculations the system is specified by a hermitian operator $\\hat{H}$ called the Hamiltonian [15].", "The operator includes information about the positions and charges of the ions, the number of electrons, and the form of the potential functions in the system.", "The fundamental equation at the electronic level is the many-body Schrödinger equation, which takes the form of a partial differential equation with $\\hat{H}$ acting on a multivariate function $\\Psi \\left(\\vec{r}_{1},\\cdots ,\\vec{r}_{N}\\right)$ , known as the many-body wave function, where $N$ denotes the number of electrons.", "The energy of the system, $E$ , is the eigenvalue of the partial differential equation.", "The equation has the variational functional [7] $E &=& \\frac{\\int \\Psi ^* \\hat{H} \\Psi \\, d{\\vec{r}_1}\\cdots d{\\vec{r}_N}}{\\int \\Psi ^* \\Psi \\,d{\\vec{r}_1}\\cdots d{\\vec{r}_N}}\\nonumber \\\\&=&\\frac{\\int |\\Psi |^2 \\cdot \\Psi ^{-1} \\hat{H} \\Psi \\, d{\\vec{r}_1}\\cdots d{\\vec{r}_N}}{\\int |\\Psi |^2 \\, d{\\vec{r}_1}\\cdots d{\\vec{r}_N}} ,$ which is minimized when the above integral is evaluated with $\\Psi $ being an exact solution of the eigen equation.", "For a trial $\\Psi $ the functional can be evaluated as an average of the local energy, $E_L \\left(\\vec{r}_1,\\cdots ,\\vec{r}_N\\right) = \\Psi ^{-1} \\hat{H} \\Psi $ over the probability density distribution $p(\\vec{r}_1,\\cdots ,\\vec{r}_N) = |\\Psi |^2/\\int |\\Psi |^2 \\, d{\\vec{r}_1}\\cdots d{\\vec{r}_N} \\ .$ In VMC the energy is evaluated by Monte Carlo integration using the Metropolis algorithm to generate configurations $\\left\\lbrace \\vec{R}^{(\\alpha )}\\right\\rbrace _{\\alpha =1}^{M}$ distributed according to the probability distribution $p(\\vec{r}_1,\\cdots ,\\vec{r}_N)=p(\\vec{R})$ , where $\\vec{R}$ denotes a configuration $(\\vec{r}_1,\\cdots ,\\vec{r}_N)$ , as $E = \\frac{1}{M}\\sum _{\\alpha =1}^{M} {E_L\\left(\\vec{R}^{(\\alpha )}\\right)} \\ ,$ with $M$ being typically of the order of millions.", "The trial function $\\Psi $ is improved by an optimization procedure so that the integral of Eq.", "(REF ) can be minimized numerically [8], [9].", "Since each $E_L \\left( \\vec{R}^{(\\alpha )} \\right)$ is evaluated independently, the summation over $\\alpha $ can be divided into sub-summations distributed over the processors by MPI with high efficiency [10].", "In this work GPGPU is used to accelerate the evaluation of each $E_L \\left( \\vec{R}^{(\\alpha )} \\right)$ , rather than parallelization over the suffix $\\alpha $ .", "We used the `CASINO' program package [6] for the VMC calculations.", "There are several possible forms of trial $\\Psi $ , and we chose to use the common Slater-Jastrow type wave function [10], [7], $\\Psi _{\\rm SJ} \\left( {\\vec{R}} \\right) = e^{J\\left( {\\vec{R}} \\right)}\\cdot \\Psi _{\\rm D} \\left( {\\vec{R}} \\right)\\ ,$ where $e^{J\\left( {\\vec{R}} \\right)}$ is known as the Jastrow factor [16], [17].", "$\\Psi _{\\rm D}$ is a Slater determinant [18] $\\Psi _{\\rm D} \\left( {\\vec{r}_1 , \\cdots ,\\vec{r}_N } \\right)= \\left| {\\begin{array}{*{20}c}{\\psi _1 \\left( {\\vec{r}_1 } \\right)} & \\cdots & {\\psi _N \\left( {\\vec{r}_1 } \\right)} \\\\\\vdots & \\ddots & \\vdots \\\\{\\psi _1 \\left( {\\vec{r}_N } \\right)} & \\cdots & {\\psi _N \\left( {\\vec{r}_N } \\right)} \\\\\\end{array}} \\right|\\ ,$ which is an anti-symmetrized product of one-particle orbital functions, $\\left\\lbrace \\psi _l \\left( {\\vec{r}} \\right)\\right\\rbrace _{l=1}^L$ .", "The number of independent orbitals, $L$ , can be reduced by using the symmetries of the system.", "The bottleneck of the whole simulation has been found to be the construction of the $\\left\\lbrace \\psi _l \\left( {\\vec{r}} \\right)\\right\\rbrace $ [6].", "In this study the computational power of the GPU is devoted to the bottleneck process, as described in the following subsection." ], [ "Orbital evaluation", "In each MPI process the following evaluations are performed sequentially: An attempted move, $\\vec{R}^{(\\alpha )} \\rightarrow \\vec{R}^{(\\alpha +1)}$ , is randomly generated, The updated probability $p(\\vec{R}^{(\\alpha + 1)})$ and the ratio $\\xi = p(\\vec{R}^{(\\alpha +1)})/p(\\vec{R}^{(\\alpha )})$ is evaluated, Based on the ratio $\\xi $ , the attempted move $\\vec{R}^{(\\alpha + 1)}$ is accepted or rejected, The local energy $E_L \\left( \\vec{R}^{(\\alpha + 1)}\\right)$ is evaluated.", "Each configuration is a set of electronic positions (we omit the spin coordinate for simplicity), $\\vec{R}^{(\\alpha )} = (\\vec{r}^{(\\alpha )}_1,\\vec{r}^{(\\alpha )}_2,\\cdots ,\\vec{r}^{(\\alpha )}_j,\\cdots ,\\vec{r}^{(\\alpha )}_N)$ .", "Following Eqs.", "(REF ), (REF ), and (REF ), one can reduce the evaluation of the ratio $\\xi = p(\\vec{R}^{(\\alpha +1)})/p(\\vec{R}^{(\\alpha )})$ to that of the orbital functions, $\\left\\lbrace \\psi _l \\left( {\\vec{r}^{(\\alpha +1)}_j}\\right)\\right\\rbrace $ .", "In practical QMC calculations for extended systems the orbital functions are expanded in a $B$ -spline basis set $\\left\\lbrace \\Theta _s \\left( {\\vec{r}} \\right)\\right\\rbrace $ [20], [19], [6] as $\\psi _l \\left( {\\vec{r}_j} \\right)= \\sum \\limits _{s = 1}^{4^3=64} {a_{ls}\\cdot \\Theta _s \\left( {\\vec{r}_j} \\right)} \\ .$ The index $s$ runs over the subset of the spatial sites within the unit cell of the periodic system.", "The spline basis functions, $\\left\\lbrace \\Theta _s \\left( {\\vec{r}} \\right)\\right\\rbrace $ , have non-zero values only at sites $s$ within the fourth nearest neighbor of the position $\\vec{r}$ along each direction.", "The total number of terms in the summation (REF ) is therefore 64 = 4$^3$ (four spatial points along each direction in the three dimensional space), as a subset of the whole lattice within the unit cell amounting to $S=50^3\\sim 60^3$ .", "The lattice is indexed by $\\left\\lbrace \\tilde{s}\\right\\rbrace _{\\tilde{s} = 1}^{S}$ , as depicted schematically in the two dimensional plane in Fig.", "REF .", "The subset $\\left\\lbrace s\\right\\rbrace _{s=1}^{4^3} \\subset \\left\\lbrace \\tilde{s}\\right\\rbrace _{\\tilde{s} = 1}^{S=50^3\\sim 60^3}$ is the spatial region where $\\left\\lbrace \\Theta _s \\left( {\\vec{r}} \\right)\\right\\rbrace $ has non-zero values, depending on the given $\\vec{r}$ .", "Since the indices introduced so far are complicated we summarize them in Table REF .", "Table: Conventions for indices used in this paper.The value of $\\Theta _s \\left( {\\vec{r}} \\right)$ at a $s$ -lattice site, $\\vec{R}_s = (X_s,Y_s,Z_s)$ , is given by the function depending on the distance between $\\vec{r}$ and $\\vec{R}_s$ as, $\\Theta _s \\left( {\\vec{r}} \\right)= \\varphi \\left( {\\frac{{x - X_s}}{b_x}} \\right)\\cdot \\varphi \\left( {\\frac{{y - Y_s}}{b_y}} \\right)\\cdot \\varphi \\left( {\\frac{{z - Z_s}}{b_z}} \\right) \\ ,$ where $\\varphi \\left( \\zeta \\right)$ is a second order polynomial in $\\zeta $ , and $(b_x,b_y,b_z)$ denote grid spacings for each direction.", "The coefficients in Eq.", "(REF ), $\\left\\lbrace {{{\\tilde{a}}_{l,\\tilde{s}}}}\\right\\rbrace _{\\tilde{s} = 1}^{{S\\sim 250,000}}$ , are precomputed and provided as an input file, stored in memory at the beginning of the simulation.", "For each MC step with a updated particle position $\\vec{r}_j$ , the subset $\\left\\lbrace {\\left\\lbrace {{a_{l,s\\left( {{{\\vec{r}}_j}} \\right)}}}\\right\\rbrace _{s = 1}^{64}} \\right\\rbrace _{j = 1}^N\\subset \\left\\lbrace {{{\\tilde{a}}_{l,\\tilde{s}}}} \\right\\rbrace _{\\tilde{s} = 1}^{{S}}$ is identified and used in the summation (REF ).", "Denoting $a_{l,s\\left( {\\vec{r}_j } \\right)} = a\\left[ {l,j_x ,j_y ,j_z }\\right]$ as an array, $s\\left( {\\vec{r}_j } \\right)=(j_x ,j_y ,j_z)$ forms a simply connected region in the three dimensional space but it does not allow sequential memory access in one dimensional address space, as it is interrupted with some stride due to the higher dimensions (see Fig.", "REF ).", "The orbital index $l$ , is, however, inherently one dimensional and we exploit this for sequential memory access, which is very important for GPGPU, as discussed in §IV.B.", "Figure: Data structure for the expansion coefficients of the orbitalfunctions in Eq.", "(), schematically depicted in twodimensional space.", "Black lattice points show the nearest site foreach given r → j \\vec{r}_j.", "In the shaded regions the basis functionsΘ s r → j \\left\\lbrace \\Theta _s \\left( {\\vec{r}_j} \\right)\\right\\rbrace in Eq.", "() have non-zero values.", "We have identified the `multiply and add' operations in Eq.", "(REF ), by which the orbital functions, $\\left\\lbrace \\psi _l \\left( {\\vec{r}_j} \\right)\\right\\rbrace _{l=1}^{L}$ , are evaluated as the bottleneck in the present QMC simulation [6].", "This operation appears at every MC step when the particle position is updated, $\\vec{r}_j^{(\\alpha )} \\rightarrow \\vec{r}_j^{(\\alpha +1)}$ .", "The number of operations for a single evaluation is proportional to the number of orbitals, $L$ , and hence to $N$ .", "In the present study we treat system sizes up to $L=384$ and $N$ =1,536." ], [ "Benchmark systems", "To investigate the dependence of the acceleration on system size, we prepared three different benchmark systems, as reported in Table REF .", "For each system the atomic cores are replaced by pseudopotentials [5] in Si-diamond ($N$ =216) and cubic TiO$_2$ ($N$ =648 and 1,536).", "The periodic boundary conditions for $(3\\times 3\\times 3)$ or $(4\\times 4\\times 4)$ arrays of unit cells form a simulation cell.", "More detailed specifications for each system are given in Ref.", "[21] for Si and Ref.", "[22] for TiO$_2$ .", "Table: Benchmark systems used in the present study.", "NN and LL denotethe number of electrons and orbitals for each system, respectively.", "Thetiming data and the acceleration factors achieved by the best coding aresummarized, see §V.A.", "The bottleneck routine (orbital evaluation) to be replaced by the GPU processing kernel occupies 20$\\sim $ 30% of the entire CPU time for TiO$_2$ ($N$ =1,536), as analyzed by a profiler (Intel VTune Amplifier [23]).", "This depends on the choice of the `dcorr' parameter in CASINO [6], [24], which specifies the interval between sampling; in order to reduce the correlation in the sampling, the local energy is evaluated every `dcorr' MC steps (an MC step corresponds to the update of a configuration).", "The ratio of the CPU time spent in the bottleneck is reduced from 39.5% to 22.0% by increasing `dcorr' from one to ten, as measured for a simulation with 10,000 MC steps.", "Typically `dcorr = 5' is chosen, for which the reduction becomes 27.5%.", "The reason for this dependence is that the orbital evaluation is called not only by the configuration updating but also by the local energy evaluation.", "Increasing `dcorr' means less frequent evaluation of the local energy and hence less frequent calls to the orbital evaluation.", "General descriptions of the architecture of a GPU can be found in the literature [2], [3], [4], and our previous paper [12] also provides such a description.", "This section provides the minimum amount of information required to understand the present work which was performed with the NVIDIA GeForce GTX 480 architecture.", "A GPU has hundreds of processing cores.", "The key points for the acceleration in the present work can be summarized as follows: (1) Parallelized tasks are distributed over many cores.", "The large number of processing cores of a GPU allows the whole task to be processed more rapidly than by a CPU.", "(2) The parallelized tasks are grouped into several sets (called `warps').", "The GPU processes each warp in order (`barrel processing'), skipping those still waiting for data load from memory.", "As there are usually hundreds of warps, barrel processing conceals the memory latency.", "In our previous work [12] the acceleration was achieved mainly by dividing a huge number of loops into several subsets and distributing them over GPU processor cores.", "The present work does not follow this strategy, and we do not divide the loop for the summation in Eq.", "(REF ).", "Instead a huge number of independent parallel tasks, $N\\times L =1,536 \\times 384 = 589,824$ , for the orbitals $\\left\\lbrace \\left\\lbrace \\psi _l \\left( {\\vec{r}^{\\prime }_j}\\right)\\right\\rbrace _{l=1}^{L}\\right\\rbrace _{j=1}^N$ are distributed over the GPU processing cores.", "Other key points for the present work include, (3) memory latency is much improved when the access occurs with sequential memory address (memory coalescing), and (4) a command set called a `Fused Multiply Add (FMA) which performs multiply and add operations within a clock cycle (two operations at once)." ], [ "Processors and performance", "GTX480 has 480 processor cores, each of which is termed a `streaming core' (SP) for AMD products while `cuda core' is used for NVIDIA products.", "In the present paper we use the term SP.", "The specs of the GTX480 are summarized in Table REF .", "As shown in Fig.", "REF every 32 SPs are grouped into a unit called a Streaming Multi-Processor (SM), the total number of which is hence fifteen.", "Each SP includes 32 bit scalar operators for floating point (FP32) and integer (Int32) data.", "These two operators can process the data independently within a clock cycle, giving a contribution to the ideal performance with 2 OP/cycle (two operations per clock cycle) for single precision operations.", "For double precision each 32 bit operator deals with 64 bit floating point (integer) data with two clock cycles, termed FP64 (Int64), giving a 1 OP/cycle contribution on average for double precision operations.", "There is another kind of operation unit called a `Special Function Unit' (SFU), devoted to evaluating hyper functions including exponential, logarithmic, and trigonometric functions.", "Each SM includes four SFUs in addition to the 32 SPs.", "A SFU performs four floating point operations per clock cycle, in parallel with other SM operations, which therefore contributes a further 4 OP/cycle to the ideal performance.", "Table: Spec of the NVIDIA GTX480 GPU architecture used in the present work.With a clock frequency of 1.401 GHz, the peak performance of GTX480 is hence evaluated as ${\\rm {1.401 GHz}} \\times 15{\\rm {SM}} \\times ({\\rm {32SP}} \\times {\\rm {2OP}}+ {\\rm {4SFU}} \\times {\\rm {4OP) = 1,681[GFlops]}} \\ .$ Note that, unlike the previous GTX275, multiply-and-add operations are not subject to a SFU in the present GTX480.", "For evaluating Eq.", "(REF ) there is hence no place for a SFU to be applied, giving an ideal performance to be compared with our achievement of ${\\rm {1.401 GHz}} \\times 15{\\rm {SM}} \\times ({\\rm {32SP}} \\times {\\rm {2OP)= 1,345 [GFlops] \\ ,}}$ by omitting the contribution from SFU.", "Though the present work concentrates on single precision GPU evaluation, the ideal double precision performance is estimated to be 672 [GFlops], which is half of that for single precision.", "GeForce GTX480, however, limits it to a quarter of this value, 168 [GFlops], by its driver, for some reason [12].", "These estimates are summarized in Table REF , compared with that of the CPU (Intel Core i7 920) used in the present work.", "Figure: A schematic picture of the structure of the GPU device used in thepresent work.Table: Estimated ideal performances of GTX480 to be compared with ourachievement.", "The peak performance of the Intel Core i7 CPU is also listedfor reference.", "Note that the `ideal' performance differs from the peakperformance of the device (see the text).", "Table REF summarizes the specification of a computational node used for the experiments.", "On each node an Intel Core i7 920 processor [25] and a GPU is mounted on a mother board.", "Hyper-Threading [26] in the Core i7 processor is turned off, and it is used purely as a four-core CPU.", "Compute Capability specifies the version of hardware level controlled by CUDA, above ver.1.3, which supports double precision operations.", "We used the Intel compiler version 12.0.0 for Fortran/C codes using options, `-O3' (optimizations including those for loop structures and memory accesses), `-no-prec-div' and `-no-prec-sqrt' (acceleration of division and square root operations with slightly less precision), `-funroll-loops' (unrolling of loops), `-no-fp-port' (no rounding for float operations), `-ip' (interprocedural optimizations across files), and `-complex-limited-range' (acceleration for complex variables).", "For CUDA we used the nvcc compiler with options `-O3' and `-arch=sm_13' (enabling double precision operations).", "Table: Setup of a computational node." ], [ "Memory architecture", "It is essential in GPGPU coding to design efficiently the parallelized tasks (termed `threads') to be grouped into subsets with several different classes.", "With the variety of memory devices provided in a GPU, see Table REF , each subset has a different `distance' from these devices.", "The performance of the GPGPU is critically affected by the choice of theses subsets because a good design can effectively reduce the memory latency.", "In the present study all of the threads are grouped into `blocks'.", "Threads within a block can share memory devices with fewer latencies.", "Each block is assigned to a SM by which the threads within the block are processed.", "The SM processes 32 threads at once, as in vector processing.", "A bunch of 32 threads is called a `warp'.", "When a warp accesses with sequential memory addresses, the latency is much reduced (called `coalescing').", "To conceal the memory latency, the scheduler and dispatcher for warps monitor which warps are immediately executable (namely which ones have already completed their memory loads).", "Then the scheduled warps are processed sequentially by the SM.", "A schematic picture is shown in Fig.", "REF .", "Figure: A schematic picture showing the relation between blocks, threads, andstreaming multiprocessors (SM).", "Table REF contains only those kinds of memory relevant to this study, and excludes the texture memory [4].", "Off-chip memories are located within the GPU board but not on the device chip.", "They have larger capacities and are accessible directly from CPU hosts but are in general slower.", "On-chip memories are complementary, namely with higher speed and lower capacity.", "Data required for GPU processing is transferred from the mother board to off-chip global memory, and is then loaded to on-chip shared memory, as usual.", "The capacity of the global memory ranges from 1 GB to 3 GB, depending on the product.", "As a trade-off against the large capacity it is about 100 times slower than on-chip memories.", "In GTX480, 768 KB of off-chip L2 cache is available to cover the low speed of the global memory.", "Another off-chip memory with high speed accessibility is the 64 KB constant memory.", "Via the constant caches located on every SM the constant memory can be accessed from all the threads with higher speed, although it is limited to read-only.", "As its name suggests, it is used to store constants referred to by threads.", "On-chip memories inside each SM include registers, shared memories, and L1 caches.", "In GTX480 there are 32,768 registers available for each SM.", "Registers are usually used to store the loop index variables defined within kernel codes, as in the present study.", "The 64 KB memory device on each SM can be shared by all the threads within a block with high speed access.", "The 64 KB capacity is divided into 48 KB and 16 KB parts which work as a shared memory and L1 cache, respectively.", "The user can specify which 48 or 16 KB region corresponds to the shared memory or L1 cache when the kernel code is compiled.", "The access to the global memory refers first to L1 cache and then L2 and finally to the off-chip global memory device when it fails to load from cache, which are called cache misses.", "Data loading from the global memory takes at least 200 cycles, and more usually 400 $\\sim $ 600 cycles.", "To conceal the latency, the GPU administrates all of the warps and monitors whether it is ready to be executed with the completion of data load.", "With sufficient warps one can ensure that the processors are almost always executing operations without waiting for data loads.", "To achieve better concealment it is essential to design the code so that it maintains a large number of warps.", "Since it depends on the specs of each architecture, such as the number of threads per warp, and the maximum possible number of threads per block etc., programming for better performance requires tuning for each GPU product.", "Table: Various kinds of memory in a GPU relevant to this study.R and W stand for readable and writable, respectively." ], [ "Coding", "Only the bottleneck routine for evaluating Eq.", "(REF ) is replaced by the CUDA kernel code executed on the GPU.", "The interface between the main code in F90 and the CUDA kernel is the same as that in our previous study [12]." ], [ "Quasi-simultaneous updating", "To construct appropriate parallelized degrees of freedom, we introduced a new scheme for the MC updating of configurations.", "Let us denote a configuration at MC step $\\alpha $ by $\\vec{R}^{(\\alpha )} =(\\vec{r}_1^{(\\alpha )},\\cdots ,\\vec{r}_j^{(\\alpha )}, \\cdots ,\\vec{r}_N^{(\\alpha )})$ , and consider the update of a particle position $\\vec{r}_j^{(\\alpha )}\\rightarrow \\vec{r}^{(\\alpha +1)}_j$ .", "In configuration-by-configuration updating (simultaneous updating), the accept/reject of the updating is evaluated based on the ratio of the probabilities in Eq.", "(REF ), $\\xi _{\\rm sim}= \\frac{{p\\left( {\\vec{r}^{(\\alpha +1)}_1 , \\cdots ,\\vec{r}^{(\\alpha +1)}_{j - 1} ,\\vec{r}^{(\\alpha +1)}_j ,\\vec{r}^{(\\alpha +1)}_{j + 1} ,\\cdots ,\\vec{r}^{(\\alpha +1)}_N } \\right)}}{{p\\left( {\\vec{r}_1^{(\\alpha )} ,\\cdots ,\\vec{r}_j^{(\\alpha )} ,\\cdots ,\\vec{r}_N^{(\\alpha )} } \\right)}} \\ ,$ while in particle-by-particle (PbP, sequential updating), $\\xi _{\\rm seq}^{\\left( j \\right)}= \\frac{{p\\left( {\\vec{r}^{(\\alpha +1)}_1 , \\cdots ,\\vec{r}^{(\\alpha +1)}_{j - 1} ,\\vec{r}^{(\\alpha +1)}_j ,\\vec{r}_{j + 1}^{(\\alpha )} ,\\cdots ,\\vec{r}_N^{(\\alpha )} } \\right)}}{{p\\left( {\\vec{r}^{(\\alpha +1)}_1 , \\cdots ,\\vec{r}^{(\\alpha +1)}_{j - 1} ,\\vec{r}^{(\\alpha )}_j ,\\vec{r}_{j + 1}^{(\\alpha )} ,\\cdots ,\\vec{r}_N^{(\\alpha )} } \\right)}} \\ .$ The index $j$ in $\\xi _{\\rm seq}^{\\left( j \\right)}$ means that the accept/reject step is made for each particle move, unlike in configuration-by-configuration updating.", "These two updating schemes give slightly different values for the statistical estimates because of the different paths of the evaluations, but they coincide with each other within the statistical errors.", "We introduce another updating scheme based on the ratio $\\xi _{\\rm q.sim}^{\\left( j \\right)}= \\frac{{p\\left( {\\vec{r}^{(\\alpha )}_1 , \\cdots ,\\vec{r}^{(\\alpha )}_{j - 1} ,\\vec{r}^{(\\alpha +1)}_j ,\\vec{r}_{j + 1}^{(\\alpha )} ,\\cdots ,\\vec{r}_N^{(\\alpha )} } \\right)}}{{p\\left( {\\vec{r}_1^{(\\alpha )} ,\\cdots ,\\vec{r}_j^{(\\alpha )} ,\\cdots ,\\vec{r}_N^{(\\alpha )} } \\right)}} \\ ,$ termed `quasi-simultaneous updating' (Q.S.).", "In this scheme the accept/reject evaluation for the $j$ th particle at step $(\\alpha + 1)$ is based on the previously fixed $\\alpha $ step configuration and on each particle position $j$ , which gives $N$ individual parallel tasks.", "Evaluating Eq.", "(REF ) reduces to computing the orbital functions with updated positions, $\\left\\lbrace \\left\\lbrace \\psi _l \\left( {\\vec{r}^{(\\alpha +1)}_j} \\right)\\right\\rbrace _{l=1}^{L}\\right\\rbrace _{j=1}^{N}$ , which requires $(N\\times L)$ independent evaluations.", "New trial moves at the $(\\alpha + 1)$ step $(\\vec{r}_1^{(\\alpha )},\\cdots ,\\vec{r}_j^{(\\alpha )},\\cdots ,\\vec{r}_N^{(\\alpha )})\\rightarrow (\\vec{r}_1^{(\\alpha +1)},\\cdots ,\\vec{r}_j^{(\\alpha +1)},\\cdots ,\\vec{r}_N^{(\\alpha +1)}) \\ ,$ are generated on a CPU and sent to a GPU and the orbital functions are evaluated (see Fig.", "REF ).", "For TiO$_2$ (4$\\times $ 4$\\times $ 4) this gives $N\\times L =1536 \\times 384 =589,824$ parallelized tasks.", "Note that the parallelized multiplicity $(N\\times L)$ scales as $N^2$ since the number of orbitals $L \\propto N$ .", "The concealment of memory latency is more efficient when the multiplicity increases, and hence we expect better performance for larger systems.", "Figure: Data transfer between a CPU and GPU in the present implementation.Trial moves for particle positions are generated and sent to a GPU.", "The GPUcomputes updated values of the orbitals to send back to the CPU.", "Theaccept/reject evaluation based on the orbital values are performed on theCPU." ], [ "Assigning blocks", "Each thread evaluating $\\psi _l \\left( {\\vec{r}^{(\\alpha +1)}_j} \\right)$ is labelled by $(j,l)$ .", "As mentioned in §II.B, the memory access for the coalescing should take $l$ as the sequential index for the data load of $a_{l,s\\left( {\\vec{r}_j } \\right)} = a\\left[ {l,j_x ,j_y,j_z } \\right]$ , as required for Eq.", "(REF ).", "We therefore assigned the threads sharing the same $j$ with sequentially varying $l=1\\sim L$ within a block for the coalescing.", "The number of orbitals, $L$ , cannot therefore exceed the maximum possible number of threads within a block, which is 1,024 for GTX480.", "The number $L$ can usually be reduced to $L<N$ using the symmetry of the system.", "It is then a natural choice to assign $L$ within a block rather than $N$ , which is always being larger than $L$ .", "For non-magnetic solid systems as in the present work, $L$ is reduced to $\\sim N/4$ by the symmetries, giving the limit $N \\lesssim $ 4,096 for GTX480.", "For magnetic systems, $L \\sim N/2$ , and hence $N \\lesssim $ 2,048, and for systems without time-reversal symmetry it becomes 1,024.", "This limitation is consistent with the maximum simulation size of contemporary QMC electronic structure calculations, which are able to treat at most a few thousand electrons in extended systems [21].", "Furthermore the limit of 1,024 in GTX480 is expected to double in future architectures In GTX275 used in our previous work [12] it was 512 for the limitation of threads within a block., making this issue less important.", "When evaluating Eq.", "(REF ), all the threads within a block refer to the same $\\Theta _s \\left( {\\vec{r}_j }\\right)$ with fixed $j$ .", "Once a warp loads $\\left\\lbrace \\Theta _s \\left( {\\vec{r}_j } \\right)\\right\\rbrace _{s=1}^{64}$ , these data are stored in L1 and L2 cache including their neighboring data.", "We can then expect effective cache hits for the data load.", "The operation of each thread, 64 terms multiply and add, easily fits the FMA of the GPU." ], [ "Other code prepared for comparisons", "We prepared several other versions of the code with different updating schemes and thread/block assignments, in order to compare the performance.", "The original CPU implementation provided by the CASINO distribution [6] is the particle-by-particle (PbP) algorithm, with double precision updating, which we refer to as [(0a) CPU/PbP].", "Another version, [(1) GPU/PbP], is the GPU version of (0a) but with single precision updating, which is useful for studying deviations between single and double precision computations.", "In this version the GPU kernel is called with a single fixed $j$ (PbP).", "Each term, $a_{ls}\\cdot \\Theta (\\vec{r}_j)$ in the summation of Eq.", "(REF ), labelled by $s$ , is calculated by each thread, the sum of which is obtained by the reduction operation [4] among all threads.", "The threads indexed by $s$ are grouped into those sharing the same $l$ and hence the blocks are labelled by the orbital index $l$ .", "Another reference is the version [(2) GPU/Q.S./non-coalescing], which uses quasi-simultaneous (Q.S.)", "updating, but the same thread/block assignment as the version (1), namely each thread calculates only the product $a_{ls}\\cdot \\Theta (\\vec{r}_j)$ .", "In this case the blocks are labelled by $(j,l)$ .", "Coalescing does not work in this version.", "The indices for threads and blocks are summarized in Table REF .", "Comparing (1) and (2) shows firstly how the performance in speed is improved by the simultaneous data transfers for $j = 1\\sim N$ in Q.S.", "compared to the sequential transfers in PbP.", "Secondly we can see how much the energy deviates due to using Q.S.", "updating instead of PbP.", "Our final implementation, described in §IV.B, is termed [(3) GPU/Q.S./coalescing].", "The comparison between (1) and (2) is within the single precision treatment.", "To compare the Q.S.", "scheme in single and double precision we also prepared [(0b) CPU/Q.S.", "], which is a double precision version of Q.S." ], [ "Acceleration performance", "Tables REF and REF summarize the acceleration factors and computational time taken for the bottleneck kernel part within a MC step.", "The results are evaluated for systems with $N$ = 216 (Si, $3\\times 3\\times 3$ ) and 1,536 (TiO$_2$ , $4\\times 4\\times 4$ ).", "A better performance was obtained using implementation (3) rather than (2) and (1), and with larger system sizes $N$ .", "Table: Comparison of acceleration factors for each implementation evaluatedfrom Table .", "`PbP' and `Q.S.'", "stand for theparticle-by-particle and quasi-simultaneous updating schemes,respectively.", "The `Index' column shows which indices in Eq.", "() are assigned to threads and blocks in the GPU.", "Forsystem sizes NN, see Table for a more detaileddescription of the systems.The improvement from (1) to (2) is attributed to the increased number of threads in Q.S.", "due to processing the particle indices $j=1\\sim N$ simultaneously.", "This also brings about improved efficiency in the data transfer between the CPU and GPU, which is simultaneous transfer in version (2) and repeated transfers for each $j$ in PbP version (1).", "The improvement from (2) to (3) arises because the number of operations performed on each thread is increased; multiply and add summation with 64 terms in (3) and just one multiplication in (2).", "The fact that memory coalescing only works in (3) also contributes to the improvement, for which a detailed analysis will be given in §VI.C.", "Table: Comparison of the computational times (ms) per Monte Carlo step ineach implementation.", "`PbP' and `Q.S.'", "stand for the particle-by-particleand quasi-simultaneous updating schemes, respectively.", "More informationabout the systems (of sizes NN) are given in Table ." ], [ "Deviation in energy values", "In GPGPU for scientific simulations it is important to consider whether the deviation in the results due to the single precision operations are within the required accuracy.", "For the present electronic structure simulation the deviation should be within $\\Delta E \\sim $ 0.001 [hartree] in the energy estimation, known as the chemical accuracy.", "Table REF shows the results from each implementation, the ground state energy of Si, $3\\times 3\\times 3$ ($N$ = 216), by 1,000,000 MC steps with sampling every 10 MC steps (`dcorr' = 10, see §II.C).", "Table: The ground state energy of Si, 3×3×33\\times 3\\times 3 (NN = 216), from1,000,000 MC steps, sampling at every 10 steps, evaluated by eachimplementation.", "`PbP' and `Q.S.'", "stand for the particle-by-particle andquasi-simultaneous updating schemes, respectively.The comparison between (0a) and (1) gives the deviation due to the change from double to single precision evaluation.", "The deviation is within the statistical error bars, but the agreement is poorer than single precision, which is expected to be correct to around six digits.", "To understand this we must remember that the energies given in Table REF are statistical estimates obtained from different accept/reject paths after 1,000,000 MC steps.", "When we look at the difference after a single MC step, agreement is confirmed within six digits, as shown in Table REF .", "Even such small deviations may give rise to different decisions along a accept/reject branch.", "Once a different decision occurs, the subsequent random walk takes different paths, giving different estimates which are outside of the six digits but within the statistical error bars.", "Table: Comparison between the energies after a MC stepevaluated in each implementation.", "`PbP' stands for theparticle-by-particle updating scheme.", "A comparison of (0a) and (0b) in Table REF gives the deviation due purely to the two different updating schemes.", "As expected it is confirmed that these energies agree to within the statistical error bars.", "The result of (2,3) includes both the deviation due to the changes in the updating scheme and the numerical precision.", "Comparing with the reference (0a) demonstrates that our updating scheme keeps the result within the required accuracy.", "In the present study, the updated orbital values from the GPGPU/single precision evaluations are used only to determine if updated particle positions are accepted or rejected.", "The energies reported in Table REF were calculated using CPU/double precision.", "The difference between single and double precisions alters the paths of the random walks and hence where the $\\vec{R}$ space is sampled.", "If the energies themselves were also evaluated using single precision it might introduce significant biases, but we have not investigated this here.", "As mentioned in §VI.d, there are further possibilities for accelerating the second largest bottleneck, which is the updating of the Slater determinants (or equivalents) using GPGPU.", "In this scheme the updated single precision value of the many-body wave function is used to evaluate the energies, and the errors in these energies should be considered carefully to make sure the results are still within chemical accuracy." ], [ "System size dependence of the performance", "The acceleration factors achieved by implementation (3) applied to $N=$ 216 (Si/$2 \\times 2 \\times 2$ ), 648 (TiO$_2$ /$3 \\times 3 \\times 3$ ), and 1,536 (TiO$_2$ /$4 \\times 4 \\times 4$ ) are summarized in Table REF .", "Figure REF also shows the acceleration factors and computational times (ms) taken for a MC step in (0a), (1), and (3).", "Figure: Computational time [ms] per MC step and correspondingacceleration factors for system sizes NN (number of electrons).`PbP' and `Q.S.'", "stand for the particle-by-particle andquasi-simultaneous updating schemes, respectively.As mentioned in §IV.A, the number of parallelized threads scales as $(N \\times L) \\sim N^2$ , so that the efficiency of the GPU improves for larger systems.", "Coalescing in implementation (3) also becomes more effective for larger $L$ .", "However there is a drawback for larger systems in the data transfer costs, especially for returning $\\left\\lbrace {\\left\\lbrace {{\\psi _l}\\left( {{{\\vec{r}}_i}} \\right)} \\right\\rbrace _{i =1}^N} \\right\\rbrace _{l = 1}^L$ from the GPU to CPU.", "As overall cancellation, the acceleration seems to scale as $\\sim N$ , rather than $ \\sim N^2$ (the number of threads).", "Table: Data transfer costs between the CPU and GPUfrom implementation (3) in Table .The dependence on the system size NN (number of electrons) is shown.Table REF summarizes the data transfer times using ver.", "(3), which show that the transfer cost increases with system size.", "The percentage of the total GPU time is, however, decreased because the number of GPU operations also increases.", "Another remarkable fact illustrated by Fig.", "REF is that the performance of ver.", "(1) is inferior up to $N$ = 648 but superior for larger $N$ = 1,536.", "The only possible reason for that is the concealment of memory latency, because in this implementation there is no coalescing and the operation load on each thread is simply a multiplication, although it has the largest number of warps among all the implementations." ], [ "Discussions", "To prevent redundancy we omit discussions of the evaluation of how much of the original CPU code is optimized, and of how we interpret the acceleration factors achieved in terms of the actual usefulness of GPGPU, as these were discussed in our previous report [12]." ], [ "Acceleration performance", "The ideal limit of the acceleration factor to be compared with our achievement of 30.7 is that of (1345/10.64)=126.4, where 1345 [GFlops] is the ideal limit for the GPU discussed in §III.A and 10.64 is the single core performance of the Intel Core i7 processor used here for implementation (0a).", "This limit might be achieved when the ratio of the cost of memory loads to that for operations approaches zero, although this is not possible in practice.", "This ratio corresponds to those shown in the last column of Table REF as in percentages.", "As shown in the table, the ratio is decreased for larger systems and hence we expect better performance.", "As another evaluation of our achievement, we estimated the FLOPS of our implementations applied to TiO$_{2}$ ($N$ =1,536) as listed in Table REF .", "The values are obtained from the number of operations required only for Eq.", "(REF ) [(2 operations) $\\times $ (2 components in complex numbers) per term] divided by the time taken for the GPU kernel execution.", "We did not take into account the operations required to identify which subset $\\left\\lbrace s\\right\\rbrace _{s=1}^{4^3} \\subset \\left\\lbrace \\tilde{s}\\right\\rbrace _{\\tilde{s} = 1}^{S=50^3\\sim 60^3}$ should be chosen to form the coefficients $\\left\\lbrace a_{ls}\\right\\rbrace $ .", "The actual FLOPS should therefore be larger than those given in the table.", "Table: Estimated FLOPS for each implementation and the ratios to the idealperformances.", "`PbP' and `Q.S.'", "stand for the particle-by-particle andquasi-simultaneous updating schemes, respectively.In this evaluation, our achievement of a factor 30.7 gives an efficiency of only 5.5% of the ideal performance.", "For reference, the CUBLAS (GPGPU of BLAS Level 3) is known to give 400 GFlops on the NVIDIA Tesla C2050, which is 38.8% of the peak performance [27].", "The reason for the lower percentage in our case is the smaller number of operations per thread, 64 terms multiply and add summations for (3), and just a single multiplication for (1) and (2).", "The amount does not depend on the system size because of the use of a localized spline basis set, although this is a disadvantage for GPGPU in the sense that the number of operations is smaller." ], [ "Performances in memory access", "Table REF summarizes the memory load performances of each implementation as measured by `Compute Visual Profiler' for CUDA [28].", "Table: Performance in global memory access for each implementation.`PbP' and `Q.S.'", "stand for the particle-by-particle and quasi-simultaneousupdating schemes, respectively.The increase in the number of 32 bit load from (1) to (2) is simply because of the increased number of threads due to the simultaneous processing with respect to the number of particles $N$ in Q.S.", "From the coalescing in (3) we see a remarkable reduction in the number of memory loads by a factor of $\\sim $ 27.", "Correspondingly the throughput becomes much closer to the peak value, 177.4 GB/s, compared with the poor achievements in (1) and (2) due to the random access of $\\left\\lbrace a_{js}\\right\\rbrace $ .", "The SM activity in Table REF is a measure of the efficiency of the concealment of the memory latency.", "This quantity is defined as the ratio of the number of cycles at which the operations started after the completion of memory loads to the total number of cycles taken for the kernel execution.", "With efficient concealing, at least one of the threads is always ready for execution at each cycle, and hence this quantity is expected to be close to 100%.", "Since the concealment becomes more effective with larger numbers of warps, the SM activity is increased from (1) to (2) by the increased number of threads.", "Though the number is decreased again in (3) by a factor of $\\sim $ 64, greatly accelerated memory accesses by the coalescing improve the SM activity to 100%." ], [ "Shared memory and read only memory", "In our previous work [12] we found that it was quite effective to exploit high-speed read-only memory.", "In the present work we have investigated further improvements by exploiting high-speed read-only memory but found no significant gains.", "This is summarized as follows: (i) in the present case the data required for the operation on each thread is large and does not fit into the high-speed memory devices, and (ii) even without explicit use of high-speed devices, the compiler automatically assigns them to L1 cache, and the explicit data transfer to shared memory by the user gives rather slower performance than automatic assignment.", "The data loads considered in the present case are $\\left\\lbrace {{{\\tilde{a}}_{l,\\tilde{s}}}} \\right\\rbrace _{\\tilde{s} = 1}^{{S \\sim 250,000}}$ , which is initially stored in the global memory.", "After choosing the subset $\\left\\lbrace \\left\\lbrace {a_{l,s} } \\right\\rbrace _{s =1}^{64}\\right\\rbrace _{l=1}^{L}$ from $\\left\\lbrace {{{\\tilde{a}}_{l,\\tilde{s}}}} \\right\\rbrace $ , our best implementation (3) loads them by coalescing access to the global memory.", "However, the access speed to the on-chip shared memory is around 100 times faster than that.", "With coalescing the maximum speed for the global memory access is expected being around 200 clock cycles, compared with 2 clock cycles for on-chip memories.. A block sharing a shared memory device has a common $j$ so the set $\\left\\lbrace \\left\\lbrace {a_{l,s} } \\right\\rbrace _{s = 1}^{64}\\right\\rbrace _{l=1}^{L}$ is shared by all the threads within the block.", "The 64 KB capacity of the device corresponds to 16,000 (= 64 KB/4B) elements of single precision data.", "It is therefore possible to accommodate $\\left\\lbrace {a_{l,s} } \\right\\rbrace $ when $L < 16,000/64 = 250$ , so that the total number of orbitals is less than 250.", "Table REF shows that Si ($3\\times 3\\times 3$ ) and TiO$_2$ ($3\\times 3\\times 3$ ) correspond to this case.", "Storing the data within the on-chip memories becomes advantageous when the data is referenced repeatedly by the warps.", "The number of repeated references is given in our case by the ratio of $L \\sim $ 250 to the warp size, 32, which is less than 10.", "Beyond this number of repeats the SM switches to another process with different $j$ and hence the corresponding new data for $\\left\\lbrace {a_{l,s} } \\right\\rbrace $ should be loaded again from the global memory.", "We have tried such an implementation using shared memory devices but we did not find any improvements over (3), possibly because of the small number of repeated references.", "Such an improvement would already be included implicitly in (3) by L1 cache acting on the shared device.", "The explicit usage of the device as the shared memory seems to reduce the performance.", "Another choice is to use constant memory.", "Unlike shared memory it is located off chip and blocks with different $j$ indices pick up their subset $\\left\\lbrace {a_{l,s} } \\right\\rbrace $ .", "The memory should therefore be able to provide the whole set of $\\left\\lbrace {{{\\tilde{a}}_{l,\\tilde{s}}}} \\right\\rbrace $ , which is far beyond the size of the constant memory and is therefore infeasible.", "Another data required for the GPU operation is $\\left\\lbrace \\left\\lbrace {\\Theta _s \\left( {\\vec{r}_j } \\right)} \\right\\rbrace _{j =1}^N \\right\\rbrace _{s=1}^{64}$ in Eq.", "(REF ), the total size of which can be accommodated within the constant memory for $N <$ 250.", "Our trial implementation using constant memory for $\\left\\lbrace \\Theta _s \\right\\rbrace $ actually gives a slight improvement in performance, by $\\sim $ 4%, but this is only applicable to smaller system sizes.", "The data $\\left\\lbrace {\\Theta _s \\left( {\\vec{r}_j^{(k)} }\\right)} \\right\\rbrace _{j = 1}^N $ is updated at every MC step and hence the life time in cache is short, giving only a slight improvement in performance." ], [ "Acceleration of Slater Determinant Updating", "The bottleneck operation of updating of orbital functions, which are computed by the GPU, gives a system size dependence of $O(N^2)$ [6].", "The corresponding updating of the many-body wave function, (REF ), actually takes $O(N^2)$ in the PbP implementation [7], [29], in spite of the fact that evaluating a determinant scales as $O(N^3)$ .", "In PbP, say $\\vec{r}_j\\rightarrow \\vec{r}^{\\prime }_j$ , only one column of the determinant including $\\vec{r}^{\\prime }_j$ is updated at each step.", "The updating of the determinant can hence be evaluated based on the Laplace expansion with respect to the column, and the other co-factors are unchanged.", "This algorithm, known as Sherman-Morrison updating [29] therefore requires only $O(N^2)$ operations [6].", "In our Q.S., the updated orbitals are stored in an array and are read sequentially to update the determinant using this algorithm.", "The cost of the updating of the many-body wave function amounts to 19.4% of the time compared to 39.5 % for the present bottleneck, the orbital updating, when $N$ =1,536, dcorr=1.", "The ratio of the former to the latter increases with larger dcorr.", "This operation mostly involves BLAS routines (ddot and daxpy) which could be replaced by CUBLAS in future work." ], [ "As a prototype of linear algebra problems", "The evaluation of Eq.", "(REF ) can be written in terms of linear algebra, $\\left( {\\begin{array}{*{20}{c}}{\\psi _1\\left( {{{\\vec{r}}_j}} \\right)}\\\\{\\psi _2\\left( {{{\\vec{r}}_j}} \\right)}\\\\\\vdots \\\\{\\psi _L\\left( {{{\\vec{r}}_j}} \\right)}\\end{array}} \\right)= A\\left( {{{\\vec{r}}_j}} \\right)\\cdot \\left( {\\begin{array}{*{20}{c}}{{\\Theta _1}\\left( {{{\\vec{r}}_j}} \\right)}\\\\{{\\Theta _2}\\left( {{{\\vec{r}}_j}} \\right)}\\\\\\vdots \\\\{{\\Theta _{64}}\\left( {{{\\vec{r}}_j}} \\right)}\\end{array}} \\right) \\ ,$ for which the matrix $A$ is defined by $A = \\left\\lbrace {{a_{ls}}} \\right\\rbrace = \\left\\lbrace {{a_{l,s\\left( {{{\\vec{r}}_j}} \\right)}}} \\right\\rbrace = A\\left( {{{\\vec{r}}_j}} \\right) \\ .$ If the matrix $A$ is constant, we can write Eq.", "(REF ) as a matrix multiplication: $\\left( {\\begin{array}{*{20}{c}}{\\psi _1^{}\\left( {{{\\vec{r}}_1}} \\right)}&{\\psi _1\\left( {{{\\vec{r}}_2}} \\right)}& \\cdots &{\\psi _1\\left( {{{\\vec{r}}_N}} \\right)}\\\\{\\psi _2^{}\\left( {{{\\vec{r}}_1}} \\right)}& \\ddots & & \\vdots \\\\\\vdots &{}&{}&{}\\\\{\\psi _L^{}\\left( {{{\\vec{r}}_1}} \\right)}& \\cdots &{}&{\\psi _L\\left( {{{\\vec{r}}_N}} \\right)}\\end{array}} \\right) = A\\cdot \\left( {\\begin{array}{*{20}{c}}{{\\Theta _1}\\left( {{{\\vec{r}}_1}} \\right)}&{{\\Theta _1}\\left( {{{\\vec{r}}_2}} \\right)}& \\cdots &{{\\Theta _1}\\left( {{{\\vec{r}}_N}} \\right)}\\\\{{\\Theta _2}\\left( {{{\\vec{r}}_1}} \\right)}& \\ddots &{}& \\vdots \\\\\\vdots &&&\\\\{{\\Theta _{64}}\\left( {{{\\vec{r}}_1}} \\right)}& \\cdots &{}&{{\\Theta _{64}}\\left( {{{\\vec{r}}_N}} \\right)}\\end{array}} \\right) \\ ,$ and using this we could increase the ratio of operations to data transfer, and hence the efficiency of GPGPU by a large amount.", "Fig.", "REF , however, reminds us that the matrix $A$ in Eq.", "(REF ) is randomly varying in its elements choice indexed by $\\vec{r}_j$ , but within the given constant elements $\\left\\lbrace {{{\\tilde{a}}_{l,\\tilde{s}}}} \\right\\rbrace $ , and hence we cannot reduce Eq.", "(REF ) into a unified form as Eq.", "(REF ).", "Since our calculation of Eq.", "(REF ) is linear algebra we considered whether it could be performed efficiently using CUBLAS.", "To use CUBLAS we have to construct $A\\left( {{{\\vec{r}}_j}} \\right)$ on the CPU at every MC step, transfer it to the GPU, and then call CUBLAS.", "In our case the randomly varying matrix $A$ is not large, however, the cost of constructing it cannot be compensated by the high performance of CUBLAS.", "Since the enlarged matrix $\\tilde{A} = \\left\\lbrace {{\\tilde{a}_{l \\tilde{s}}}} \\right\\rbrace _{\\tilde{s} = 1}^{{S}}$ is a constant matrix, only one data transfer to the GPU is required in principle, but this is not possible with CUBLAS which requires the operands to be transferred every time for the operations.", "In this sense, our achievement here corresponds to an effective GPGPU implementation of the algorithm for the prototype as explained above, Eq.", "(REF ) with a randomly chosen matrix." ], [ "Concluding Remarks", "We have applied GPGPU to the evaluation of the orbital functions in ab-initio Quantum Monte Carlo electronic structure calculations, which we identified as the computational bottleneck.", "For efficiency we propose a new updating scheme for generating trial moves for the walkers in the Monte Carlo sampling, which we call quasi-simultaneous updating.", "Using this scheme we achieved a speedup of more than a factor of 30 compared with using a single core CPU.", "The GPGPU implementation gives a deviation in energy from original CPU evaluation which is smaller that the required chemical accuracy.", "Though the effective performance in operations amounts to 74 GFlops, which is only 5.5% of the peak performance, the memory throughput reaches 153 GB/s, which is 86% of the peak value with almost perfect concealment of memory latency as shown by the SM activity.", "The implementation presented here is a prototypical problem of linear algebra with a sort of random matrix, processed by GPGPU." ], [ "Acknowledgments", "RM is grateful for financial support provided by a Grant in Aid for Scientific Research on Innovative Areas “Materials Design through Computics: Complex Correlation and Non-Equilibrium Dynamics” (Grant No.", "22104011), and “Optical Science of Dynamically Correlated Electrons” (Grant No.", "23104714) of the Ministry of Education, Culture, Sports, Science, and Technology (KAKENHI-MEXT/Japan).", "The authors would like to express our special thanks to Richard J.", "Needs and Kenta Hongo for their useful comments and kind supports for us." ] ]

[ [ "Binomial coefficient-harmonic sum identities associated to\n supercongruences" ], [ "Abstract We establish two binomial coefficient--generalized harmonic sum identities using the partial fraction decomposition method.", "These identities are a key ingredient in the proofs of numerous supercongruences.", "In particular, in other works of the author, they are used to establish modulo $p^k$ ($k>1$) congruences between truncated generalized hypergeometric series, and a function which extends Greene's hypergeometric function over finite fields to the $p$-adic setting.", "A specialization of one of these congruences is used to prove an outstanding conjecture of Rodriguez-Villegas which relates a truncated generalized hypergeometric series to the $p$-th Fourier coefficient of a particular modular form." ], [ "Introduction and Statement of Results", "This work was supported by the UCD Ad Astra Research Scholarship program.", "For non-negative integers $i$ and $n$ , we define the generalized harmonic sum, ${H}^{(i)}_{n}$ , by ${H}^{(i)}_{n}:= \\sum ^{n}_{j=1} \\frac{1}{j^i}$ and ${H}^{(i)}_{0}:=0$ .", "In [3] Chu proves the following binomial coefficient-generalized harmonic sum identity using the partial fraction decomposition method.", "If $n$ is a positive integer, then $\\sum _{k=1}^{n} {\\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr )}^2 {\\biggl ({\\genfrac{}{}{0.0pt}{}{n}{k}}\\biggr )}^2\\biggl [ 1+ 2k H_{n+k}^{(1)} + 2kH_{n-k}^{(1)} -4kH_k^{(1)} \\biggr ]=0.$ This identity had previously been established using the WZ method [1] and was used by Ahlgren and Ono in proving the Apéry number supercongruence [2].", "In [4], [5] the author establishes various supercongruences between truncated generalized hypergeometric series, and a function which extends Greene's hypergeometric function over finite fields to the $p$ -adic setting.", "Specifically, let $p$ be an odd prime and let $n \\in \\mathbb {Z}^{+}$ .", "For $1 \\le i \\le n+1$ , let $\\frac{m_i}{d_i} \\in \\mathbb {Q} \\cap \\mathbb {Z}_p$ such that $0<\\frac{m_i}{d_i}<1$ .", "Let $\\Gamma _p{\\left({\\cdot }\\right)}$ denote Morita's $p$ -adic gamma function, $\\left\\lfloor x \\right\\rfloor $ denote the greatest integer less than or equal to $x$ and $\\langle x \\rangle $ denote the fractional part of $x$ , i.e.", "$x- \\left\\lfloor x \\right\\rfloor $ .", "Then define ${_{n+1}G} \\left( \\tfrac{m_1}{d_1}, \\tfrac{m_2}{d_2}, \\cdots , \\tfrac{m_{n+1}}{d_{n+1}} \\right)_p\\\\:= \\frac{-1}{p-1} \\sum _{j=0}^{p-2}{\\left((-1)^j\\Gamma _p{\\bigl ({\\tfrac{j}{p-1}}\\bigr )}\\right)}^{n+1}\\prod _{i=1}^{n+1} \\frac{\\Gamma _p{\\bigl ({\\langle \\frac{m_i}{d_i}-\\frac{j}{p-1}\\rangle }\\bigr )}}{\\Gamma _p{\\bigl ({\\frac{m_i}{d_i}}\\bigr )}}(-p)^{-\\lfloor {\\frac{m_i}{d_i}-\\frac{j}{p-1}}\\rfloor }.$ Note that when $p \\equiv 1 \\pmod {d_i}$ this function recovers Greene's hypergeometric function over finite fields.", "For a complex number $a$ and a non-negative integer $n$ let ${\\left({a}\\right)}_{n}$ denote the rising factorial defined by ${\\left({a}\\right)}_{0}:=1 \\quad \\textup {and} \\quad {\\left({a}\\right)}_{n} := a(a+1)(a+2)\\cdots (a+n-1) \\textup { for } n>0.$ Then, for complex numbers $a_i$ , $b_j$ and $z$ , with none of the $b_j$ being negative integers or zero, we define the truncated generalized hypergeometric series ${{_rF_s} \\left[ \\begin{array}{ccccc} a_1, & a_2, & a_3, & \\cdots , & a_r \\vspace{3.61371pt}\\\\\\phantom{a_1} & b_1, & b_2, & \\cdots , & b_s \\end{array}\\Big | \\; z \\right]}_{m}:=\\sum ^{m}_{n=0}\\frac{{\\left({a_1}\\right)}_{n} {\\left({a_2}\\right)}_{n} {\\left({a_3}\\right)}_{n} \\cdots {\\left({a_r}\\right)}_{n}}{{\\left({b_1}\\right)}_{n} {\\left({b_2}\\right)}_{n} \\cdots {\\left({b_s}\\right)}_{n}}\\; \\frac{z^n}{{n!", "}}.$ An example of one the supercongruence results from [5] is the following theorem.", "Theorem 1.1 ([5] Thm.", "2.7) Let $r, d \\in \\mathbb {Z}$ such that $2 \\le r \\le d-2$ and $\\gcd (r,d)=1$ .", "Let $p$ be an odd prime such that $p\\equiv \\pm 1 \\pmod {d}$ or $p\\equiv \\pm r \\pmod {d}$ with $r^2 \\equiv \\pm 1 \\pmod {d}$ .", "If $s(p) := \\Gamma _p{\\left({\\tfrac{1}{d}}\\right)} \\Gamma _p{\\left({\\tfrac{r}{d}}\\right)}\\Gamma _p{\\left({\\tfrac{d-r}{d}}\\right)}\\Gamma _p{\\left({\\tfrac{d-1}{d}}\\right)}$ , then ${_{4}G} \\Bigl (\\tfrac{1}{d} , \\tfrac{r}{d}, 1-\\tfrac{r}{d} , 1-\\tfrac{1}{d}\\Bigr )_p&\\equiv {_{4}F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{d}, & \\frac{r}{d}, & 1-\\frac{r}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d_1}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}+s(p)\\hspace{1.0pt} p\\pmod {p^3}.$ A specialization of this congruence is used to prove an outstanding supercongruence conjecture of Rodriguez-Villegas, which relates a truncated generalized hypergeometric series to the $p$ -th Fourier coefficient of a particular modular form [4],[6].", "Similar results to Theorem REF exist for $_4G$ with other parameters, and also $_2G$ and $_3G$ .", "The main results of the current paper, Theorems REF and REF below, are two binomial coefficient–generalized harmonic sum identities which factor heavily into the proofs of all the $_4G$ congruences.", "Taking particular values for $n, m, l, c_1$ and $c_2$ in these identities allows the vanishing of certain terms in the proofs.", "Note that letting $m=n$ in Theorem REF recovers (REF ).", "Theorem 1.2 Let $m,n$ be positive integers with $m\\ge n$ .", "Then $\\sum _{k=0}^{n} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n}{k}}\\biggr )\\biggl [ 1+k \\left(H_{m+k}^{(1)} +H_{m-k}^{(1)} + H_{n+k}^{(1)}+ H_{n-k}^{(1)} -4H_k^{(1)}\\right) \\biggr ]\\\\[6pt]+\\sum _{k=n+1}^{m} (-1)^{k-n} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\Big / \\biggl ({\\genfrac{}{}{0.0pt}{}{k-1}{n}}\\biggr )=(-1)^{m+n}.$ Theorem 1.3 Let $l,m, n$ be positive integers with $l > m\\ge n\\ge \\frac{l}{2}$ and $c_1, c_2 \\in \\mathbb {Q}$ some constants.", "Then $\\sum _{k=0}^{n} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n}{k}}\\biggr )\\Biggl \\lbrace \\biggl [1+k \\Bigl (H_{m+k}^{(1)} +H_{m-k}^{(1)} + H_{n+k}^{(1)} + H_{n-k}^{(1)}-4H_k^{(1)} \\Bigr )\\biggr ]\\\\[5pt]\\cdot \\biggl [c_1\\left(H_{k+n}^{(1)} - H_{k+l-n-1}^{(1)}\\right) + c_2 \\Bigl (H_{k+m}^{(1)} - H_{k+l-m-1}^{(1)}\\Bigr )\\biggr ]-k\\biggl [c_1\\left(H_{k+n}^{(2)} - H_{k+l-n-1}^{(2)}\\right)\\\\[5pt]+ c_2 \\left(H_{k+m}^{(2)} - H_{k+l-m-1}^{(2)}\\right)\\biggr ] \\Biggr \\rbrace + \\sum _{k=n+1}^{m} (-1)^{k-n} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\Big / \\biggl ({\\genfrac{}{}{0.0pt}{}{k-1}{n}}\\biggr )\\\\[5pt]\\cdot \\biggl [c_1\\left(H_{k+n}^{(1)} - H_{k+l-n-1}^{(1)}\\right) + c_2 \\left(H_{k+m}^{(1)} - H_{k+l-m-1}^{(1)}\\right)\\biggr ] = 0.$ The remainder of this paper is spent proving Theorems REF and REF ." ], [ "Proofs", "We first develop two algebraic identities of which the binomial coefficient–harmonic sum identities are limiting cases.", "Theorem 2.1 Let $x$ be an indeterminate and let $m,n$ positive integers with $m\\ge n$ .", "Then $\\sum _{k=0}^{n} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n}{k}}\\biggr )\\\\\\cdot \\Biggl \\lbrace \\frac{-k}{(x+k)^2}+ \\frac{1+k \\left(H_{m+k}^{(1)} +H_{m-k}^{(1)} + H_{n+k}^{(1)} + H_{n-k}^{(1)} -4H_k^{(1)}\\right)}{x+k} \\Biggr \\rbrace \\\\[6pt] +\\sum _{k=n+1}^{m} \\frac{(-1)^{k-n}}{x+k} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\Big / \\biggl ({\\genfrac{}{}{0.0pt}{}{k-1}{n}}\\biggr )=\\frac{x {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{n+1} {\\left({x}\\right)}_{m+1}}.\\phantom{=(2.1)}$ Using partial fraction decomposition we can write $f(x):= \\frac{x {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{n+1} {\\left({x}\\right)}_{m+1}}= \\frac{A}{x} + \\sum _{k=1}^{n}\\biggl \\lbrace \\frac{B_k}{(x+k)^2} + \\frac{C_k}{x+k} \\biggr \\rbrace + \\sum _{k=n+1}^{m} \\frac{D_k}{x+k}$ for some $A, B_k, C_k$ and $D_k \\in \\mathbb {Q}$ .", "We now isolate these coefficients by taking various limits of $f(x)$ as follows.", "$A = \\lim _{x \\rightarrow 0} x f(x)= \\lim _{x \\rightarrow 0}\\frac{{\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({1+x}\\right)}_{n} {\\left({1+x}\\right)}_{m}}=1.$ For $1 \\le k \\le n$ , $B_k = \\lim _{x \\rightarrow -k} (x+k)^2 f(x)&=\\lim _{x \\rightarrow -k} \\frac{x {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{k}^2 {\\left({x+k+1}\\right)}_{n-k} {\\left({x+k+1}\\right)}_{m-k}}\\\\[12pt]&=\\frac{-k {\\left({k+1}\\right)}_{n} {\\left({k+1}\\right)}_{m}}{{\\left({-k}\\right)}_{k}^2 {\\left({1}\\right)}_{n-k} {\\left({1}\\right)}_{m-k}}\\\\[12pt]&=\\frac{-k {\\left({k+1}\\right)}_{n} {\\left({k+1}\\right)}_{m}}{(-1)^{2k} k!^2 (n-k)!", "(m-k)!", "}\\\\[12pt]&=-k \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n}{k}}\\biggr ),\\\\[-3pt]$ and, using L'Hôspital's rule, $C_k &= \\lim _{x \\rightarrow -k} \\frac{(x+k)^2 f(x) - B_k}{x+k}\\\\[12pt]&=\\lim _{x \\rightarrow -k} \\frac{d}{dx}\\Biggl [(x+k)^2 f(x)\\Biggr ]\\\\[12pt]&=\\lim _{x \\rightarrow -k} \\frac{d}{dx}\\left[\\frac{x {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{k}^2 {\\left({x+k+1}\\right)}_{n-k} {\\left({x+k+1}\\right)}_{m-k}}\\right]\\\\[12pt]&=\\lim _{x \\rightarrow -k} \\left\\lbrace \\left[\\frac{ {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{k}^2 {\\left({x+k+1}\\right)}_{n-k} {\\left({x+k+1}\\right)}_{m-k}}\\right]\\left[1-x \\left(\\sum _{s=1}^{n}(-x+s)^{-1}\\right.\\right.\\right.\\\\[6pt]& \\left.\\left.\\left.\\quad \\phantom{=} +\\sum _{s=1}^{m}(-x+s)^{-1} +\\sum _{s=1}^{n-k}(x+k+s)^{-1}+\\sum _{s=1}^{m-k}(x+k+s)^{-1}+2\\sum _{s=0}^{k-1}(x+s)^{-1}\\right)\\right]\\right\\rbrace \\\\[12pt]&=\\left[\\frac{ {\\left({1+k}\\right)}_{n} {\\left({1+k}\\right)}_{m}}{{\\left({-k}\\right)}_{k}^2 {\\left({1}\\right)}_{n-k} {\\left({1}\\right)}_{m-k}}\\right]\\left[1+k \\left(\\sum _{s=1}^{n}(k+s)^{-1}+\\sum _{s=1}^{m}(k+s)^{-1} +\\sum _{s=1}^{n-k}(s)^{-1}\\right.\\right.\\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\;\\:\\:\\left.\\left.+\\sum _{s=1}^{m-k}(s)^{-1}+2\\sum _{s=0}^{k-1}(-k+s)^{-1}\\right)\\right]\\\\[6pt]&= \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n}{k}}\\biggr ) \\biggl [1+k \\left(H_{m+k}^{(1)} +H_{m-k}^{(1)} + H_{n+k}^{(1)} + H_{n-k}^{(1)} -4H_k^{(1)}\\right)\\biggr ].$ Similarly, for $n+1 \\le k \\le m$ , $D_k = \\lim _{x \\rightarrow -k} (x+k) f(x)&=\\lim _{x \\rightarrow -k} \\frac{x {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{n+1} {\\left({x}\\right)}_{k} {\\left({x+k+1}\\right)}_{m-k}}\\\\[12pt]&=\\frac{-k {\\left({k+1}\\right)}_{n} {\\left({k+1}\\right)}_{m}}{{\\left({-k}\\right)}_{n+1} {\\left({-k}\\right)}_{k} {\\left({1}\\right)}_{m-k}}\\\\[12pt]&=(-1)^{k-n} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\Big / \\biggl ({\\genfrac{}{}{0.0pt}{}{k-1}{n}}\\biggr ).$ Theorem 2.2 Let $x$ be an indeterminate and let $l,m, n$ be positive integers with $l > m\\ge n\\ge \\frac{l}{2}$ and $c_1, c_2 \\in \\mathbb {Q}$ some constants.", "Then $\\sum _{k=0}^{n} \\frac{1}{x+k} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n}{k}}\\biggr )\\Biggl \\lbrace \\biggl [c_1\\left(H_{k+n}^{(1)} - H_{k+l-n-1}^{(1)}\\right) \\\\+ c_2 \\left(H_{k+m}^{(1)} - H_{k+l-m-1}^{(1)}\\right)\\biggr ]\\cdot \\biggl [ \\frac{x}{x+k} + k \\left(H_{m+k}^{(1)} +H_{m-k}^{(1)} + H_{n+k}^{(1)} + H_{n-k}^{(1)} -4H_k^{(1)}\\right) \\biggr ]\\\\[6pt]\\multicolumn{1}{l}{\\qquad - k \\biggl [c_1\\left(H_{k+n}^{(2)} - H_{k+l-n-1}^{(2)}\\right)+ c_2 \\left(H_{k+m}^{(2)} - H_{k+l-m-1}^{(2)}\\right)\\biggr ]\\Biggr \\rbrace } \\\\\\multicolumn{1}{l}{+\\sum _{k=n+1}^{m} \\frac{(-1)^{k-n}}{x+k} \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\Big / \\biggl ({\\genfrac{}{}{0.0pt}{}{k-1}{n}}\\biggr )}\\\\[6pt]\\multicolumn{1}{l}{\\qquad }\\cdot \\biggl [c_1\\left(H_{k+n}^{(1)} - H_{k+l-n-1}^{(1)}\\right)+ c_2 \\left(H_{k+m}^{(1)} - H_{k+l-m-1}^{(1)}\\right)\\biggr ]\\\\[6pt]=\\frac{x {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{n+1} {\\left({x}\\right)}_{m+1}}\\left[c_1 \\sum _{s=l-n}^{n} (-x+s)^{-1} + c_2 \\sum _{s=l-m}^{m} (-x+s)^{-1} \\right].$ Using partial fraction decomposition we can write $f(x):&=\\frac{x {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{n+1} {\\left({x}\\right)}_{m+1}}\\left[c_1 \\sum _{s=l-n}^{n} (-x+s)^{-1} + c_2 \\sum _{s=l-m}^{m} (-x+s)^{-1} \\right]\\\\[6pt]&=\\frac{A}{x} + \\sum _{k=1}^{n} \\biggl \\lbrace \\frac{B_k}{(x+k)^2} + \\frac{C_k}{x+k} \\biggr \\rbrace + \\sum _{k=n+1}^{m} \\frac{D_k}{x+k}$ for some $A, B_k, C_k$ and $D_k \\in \\mathbb {Q}$ .", "As in the proof of Theorem REF , we isolate the coefficients $A, B_k, C_k$ and $D_k$ by taking various limits of $f(x)$ .", "For brevity, we first let $T_{a}^{(r)}:= c_1 \\sum _{s=l-n}^{n} (a+s)^{-r} + c_2 \\sum _{s=l-m}^{m} (a+s)^{-r}$ and $U^{(r)} :=c_1\\left(H_{k+n}^{(r)} - H_{k+l-n-1}^{(r)}\\right)+ c_2 \\left(H_{k+m}^{(r)} - H_{k+l-m-1}^{(r)}\\right).$ Then we have $A = \\lim _{x \\rightarrow 0} x f(x)&= c_1 \\lim _{x \\rightarrow 0} \\sum _{s=l-n}^{n} \\frac{{\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({1+x}\\right)}_{n} {\\left({1+x}\\right)}_{m} (s-x)}+ c_2 \\lim _{x \\rightarrow 0} \\sum _{s=l-m}^{m} \\frac{{\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({1+x}\\right)}_{n} {\\left({1+x}\\right)}_{m} (s-x)}\\\\[7pt]&=c_1 \\sum _{s=l-n}^{n} s^{-1} +c_2 \\sum _{s=l-m}^{m} s^{-1}\\\\[7pt]&=c_1\\left(H_n^{(1)} - H_{l-n-1}^{(1)}\\right) + c_2 \\left(H_m^{(1)} - H_{l-m-1}^{(1)}\\right).$ For $1 \\le k \\le n$ , $B_k = \\lim _{x \\rightarrow -k} (x+k)^2 f(x)&=\\lim _{x \\rightarrow -k} \\frac{x {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{k}^2 {\\left({x+k+1}\\right)}_{n-k} {\\left({x+k+1}\\right)}_{m-k}}\\;T_{-x}^{(1)}\\\\[7pt]&=\\frac{-k {\\left({k+1}\\right)}_{n} {\\left({k+1}\\right)}_{m}}{{\\left({-k}\\right)}_{k}^2 {\\left({1}\\right)}_{n-k} {\\left({1}\\right)}_{m-k}}\\;T_{k}^{(1)}\\\\[7pt]&=-k \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n}{k}}\\biggr ) U^{(1)}$ and $C_k &=\\lim _{x \\rightarrow -k} \\frac{d}{dx}\\Biggl [(x+k)^2 f(x)\\Biggr ]\\\\[7pt]&=\\lim _{x \\rightarrow -k} \\frac{d}{dx}\\left[\\frac{x {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{k}^2 {\\left({x+k+1}\\right)}_{n-k} {\\left({x+k+1}\\right)}_{m-k}}\\;T_{-x}^{(1)} \\right] \\\\[7pt]&=\\lim _{x \\rightarrow -k} \\Biggl \\lbrace \\Biggl [\\frac{ {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{k}^2 {\\left({x+k+1}\\right)}_{n-k} {\\left({x+k+1}\\right)}_{m-k}}\\Biggr ]\\Biggl [\\;x\\; T_{-x}^{(2)} +\\;T_{-x}^{(1)} -x\\;T_{-x}^{(1)} \\\\[4pt]& \\phantom{=} \\qquad \\qquad \\cdot \\Biggl (\\sum _{s=1}^{n}(-x+s)^{-1}+\\sum _{s=1}^{m}(-x+s)^{-1}+\\sum _{s=1}^{n-k}(x+k+s)^{-1}\\\\[4pt] &\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad +\\sum _{s=1}^{m-k}(x+k+s)^{-1}+2\\sum _{s=0}^{k-1}(x+s)^{-1}\\Biggr )\\Biggr ] \\Biggr \\rbrace \\\\[7pt]&=\\left[\\frac{ {\\left({1+k}\\right)}_{n} {\\left({1+k}\\right)}_{m}}{{\\left({-k}\\right)}_{k}^2 {\\left({1}\\right)}_{n-k} {\\left({1}\\right)}_{m-k}}\\right]\\Biggl [-k T_{k}^{(2)}+ T_{k}^{(1)}\\left(1+k \\left(\\sum _{s=1}^{n}(k+s)^{-1}+\\sum _{s=1}^{m}(k+s)^{-1} \\right.\\right.\\\\[4pt]&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\left.\\left.+\\sum _{s=1}^{n-k}(s)^{-1} +\\sum _{s=1}^{m-k}(s)^{-1}+2\\sum _{s=0}^{k-1}(-k+s)^{-1}\\right)\\right)\\Biggr ]\\\\[7pt]&= \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n}{k}}\\biggr ) \\\\*[4pt]& \\qquad \\qquad \\cdot \\biggl [-k \\,U^{(2)} +\\bigg (1+k \\biggl (H_{m+k}^{(1)} +H_{m-k}^{(1)} + H_{n+k}^{(1)}+ H_{n-k}^{(1)} -4H_k^{(1)}\\biggr )\\biggr ) U^{(1)} \\biggr ].$ For $n+1 \\le k \\le m$ , $D_k = \\lim _{x \\rightarrow -k} (x+k) f(x)&=\\lim _{x \\rightarrow -k} \\frac{x {\\left({1-x}\\right)}_{n} {\\left({1-x}\\right)}_{m}}{{\\left({x}\\right)}_{n+1} {\\left({x}\\right)}_{k} {\\left({x+k+1}\\right)}_{m-k}}\\;T_{-x}^{(1)}\\\\[12pt]&=\\frac{-k {\\left({k+1}\\right)}_{n} {\\left({k+1}\\right)}_{m}}{{\\left({-k}\\right)}_{n+1} {\\left({-k}\\right)}_{k} {\\left({1}\\right)}_{m-k}}\\;T_{k}^{(1)}\\\\[12pt]&=(-1)^{k-n} \\, U^{(1)} \\, \\biggl ({\\genfrac{}{}{0.0pt}{}{m+k}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{m}{k}}\\biggr ) \\biggl ({\\genfrac{}{}{0.0pt}{}{n+k}{k}}\\biggr ) \\Big / \\biggl ({\\genfrac{}{}{0.0pt}{}{k-1}{n}}\\biggr ).$ Multiply both sides of (REF ) and (REF ) respectively by $x$ and take the limit as $x \\rightarrow \\infty $ ." ] ]

[ [ "Faster Walks in Graphs: A $\\tilde O(n^2)$ Time-Space Trade-off for\n Undirected s-t Connectivity" ], [ "Abstract In this paper, we make use of the Metropolis-type walks due to Nonaka et al.", "(2010) to provide a faster solution to the $S$-$T$-connectivity problem in undirected graphs (USTCON).", "As our main result, we propose a family of randomized algorithms for USTCON which achieves a time-space product of $S\\cdot T = \\tilde O(n^2)$ in graphs with $n$ nodes and $m$ edges (where the $\\tilde O$-notation disregards poly-logarithmic terms).", "This improves the previously best trade-off of $\\tilde O(n m)$, due to Feige (1995).", "Our algorithm consists in deploying several short Metropolis-type walks, starting from landmark nodes distributed using the scheme of Broder et al.", "(1994) on a modified input graph.", "In particular, we obtain an algorithm running in time $\\tilde O(n+m)$ which is, in general, more space-efficient than both BFS and DFS.", "We close the paper by showing how to fine-tune the Metropolis-type walk so as to match the performance parameters (e.g., average hitting time) of the unbiased random walk for any graph, while preserving a worst-case bound of $\\tilde O(n^2)$ on cover time." ], [ "Introduction", "In the undirected $S$ -$$ connectivity problem (USTCON), the input to the algorithm is an undirected graph $G=(V,E)$ with $n$ vertices and $m$ edges.", "Two of the vertices of the graph, $S, \\in V$ , are distinguished.", "The goal is to determine whether $S$ and $$ belong to the same connected component of $G$ .", "USTCON has a spectrum of applications in various areas of computer science, ranging from tasks of network discovery to computer-aided verification.", "The problem has also made its mark on complexity theory, most famously, playing a central part in the rise and eventual collapse of the complexity class SL.", "The time complexity of algorithms for USTCON depends on the amount of space available to the algorithm.", "Given $\\widetilde{\\Theta }(n)$ space, USTCON can be solved deterministically in time $O(m)$ by fast algorithms such as BFS or DFS.", "Given $\\Theta (\\log n)$ space, the problem can still be solved deterministically [21] in polynomial time.", "However, in this case the fastest known solutions are randomized.", "Aleliunas et al.", "[2] proposed a log-space algorithm with bounded error probability, which consists in running a random walk, starting from node $S$ for $O(nm)$ steps, and testing if node $$ has been reached.", "The study of the interplay between the space complexity $S$ and the time complexity $T$ of randomized algorithms for USTCON was initiated by Broder et al.", "[10].", "They observed that both BFS/DFS, and the random walk, admit the same time-space trade-off of $T = \\widetilde{O}(\\frac{mn}{S})$ , and investigated whether there exist algorithms with such a trade-off for an arbitrary choice of $S$ , $c\\cdot \\log n \\le S \\le n$ , where $c>0$ is some model-dependent constant.", "After a sequence of papers relying on the deployment of many short random walks, this question was eventually settled in the affirmative by Feige [14], who proposed a family of algorithms which achieve such a time-space trade-off in the whole of the considered range of space bounds.", "The main result of this paper is an improved time-space trade-off for USTCON.", "Since the cover time of the random walk is precisely $\\Theta (nm)$ for some graphs, any improvement with respect to Aleliunas et al.", "[2] or Feige [14] requires a refinement of the performed walk on graphs.", "Instead of the random walk, we make use of the Metropolis-Hastings walk on graphs, with weighting proposed by Nonaka et al.", "[20].", "This walk covers any undirected graph in $\\widetilde{O}(n^2)$ steps, but its transition probabilities rely on knowledge of the degrees of neighboring nodes at every step.", "We start the technical sections of this paper with an explicit implementation of the walk from [20] using the Metropolis sampling algorithm from [18].", "This yields a solution to USTCON in $\\widetilde{O}(n^2)$ time and logarithmic space.", "Our contribution lies in completing this quadratic time-space trade-off for larger bounds on the space complexity of the algorithm.", "The main technical difficulty concerns overcoming problems with short runs of the Metropolis-Hastings walk, which sometimes exhibits inferior behavior to the random walk in terms of the speed of discovering new nodes.", "For the entire range of space bounds ($c\\cdot \\log n \\le S \\le n$ ), we propose algorithms running in time $T = \\widetilde{O}(\\max \\lbrace \\frac{n^2}{S}, m\\rbrace )$ .", "In other words, we obtain $T = \\widetilde{O}(\\frac{n^2}{S})$ for $S \\le \\frac{n^2}{m}$ , and $T = \\widetilde{O}(m)$ for $S > \\frac{n^2}{m}$ .", "(Note that $T = \\Omega (m)$ is a lower bound on execution time for any algorithm for USTCON, regardless of the space bound.)", "In particular, we prove that USTCON can be solved in time $\\widetilde{O}(m)$ using space $O(\\frac{n^2}{m})$ , which is, in general, less than the space requirement of BFS/DFS.", "All of the considered algorithms for USTCON are randomized (in the Monte Carlo sense), with bounded probability of one-sided error.", "This means that the positive answer “connected” may only be reached by the algorithm when $S$ and $$ belong the same connected component of $G$ , whereas the negative answer “not connected” signifies that, with probability at least $2/3$ , $S$ and $$ belong to different components of $G$ ." ], [ "Related work", "Much of the work on undirected $S$ -$$ connectivity has focused around its role as the fundamental complete problem for the symmetric log-space complexity class (SL).", "A survey of other important problems identified as belonging to the class SL, such as simulating symmetric Turing machines, and testing if a graph is bipartite, is provided in AGCR.", "A major line of study concerned determining the minimum space required to solve USTCON deterministically.", "The bound on the required space was reduced, over several decades, from the $O(\\log ^2 n)$ bound given by Savitch's theorem [22], through $O(\\log ^{3/2} n)$ [23], and $O(\\log ^{4/3} n)$  [4].", "Finally, in 2004, Reingold's [21] new construction of universal graph exploration sequences provided the first log-space algorithm for USTCON, showing that SL=L.", "Befor Reingold's paper, Nisan [19] had shown a deterministic algorithm for USTCON running in polynomial time and $O(\\log ^2 n)$ space.", "Borodin et al.", "[9] proposed a log-space Las-Vegas type algorithm for USTCON (with no-error) which runs in expected polynomial time.", "When considering randomized algorithms with bounded one-sided error, the unbiased random walk was shown to solve USTCON in $O(\\log n)$ space and $\\widetilde{O}(mn)$ time by Aleliunas et al.", "[2].", "Several years later, Broder et al.", "[10] proposed a family of algorithms based on short random walks starting from landmark nodes.", "Relying on landmarks chosen on the set of nodes according to the stationary distribution of the walk, they achieved a time-space trade-off of $T= \\widetilde{O}(\\frac{m^2}{S})$ .", "Subsequent algorithms from the literature [5], [14] make use of different landmark distribution schemes.", "Barnes and Feige [5] achieve a trade-off of $T = \\widetilde{O}(\\frac{m^{1.5}n^{0.5}}{S})$ by using a mixed landmark distribution scheme, which places half of the landmarks according to the stationary distribution of the random walk, and half according to the uniform distribution on nodes.", "Feige [14] introduces the inverse distribution scheme, which likewise places half of the landmarks according to the stationary distribution of the random walk, and the other half according to the inverse of node degrees.", "He achieves a time-space trade-off of $T = \\widetilde{O}(\\frac{mn /\\delta }{S})$ in general, where $\\delta $ is the minimum degree of the graph.", "Thus, the trade-off of $T = \\widetilde{O}(\\frac{n^2}{S})$ is reached for the case of (nearly) regular graphs.", "Undirected $S$ -$$ connectivity is a special case of the more general reachability problem in directed graphs (STCON), which is a complete problem for the class NL.", "STCON can also be solved deterministically in $O(\\log ^2 n)$ space using Savitch's theorem [22].", "So far, it has resisted fast solutions in small space.", "This problem was extensively studied in different variants of a model of computation based on Jumping Automata on Graphs (JAG-s).", "The memory of a JAG is organized in the form of $P$ pebbles placed in the graph and $Q$ states of the automaton, with space defined as $S = P \\log n + \\log Q$ .", "Cook and Rackoff [12] show a way of solving STCON in the JAG model deterministically in $O(\\log ^2 n)$ space, and also prove an almost matching lower bound on space of $\\Omega (\\log ^2 n /\\log \\log n)$ .", "This lower bound is also known to apply to randomized JAG-s running in slightly super-polynomial time [8].", "Gopalan et al.", "[15] propose a family of algorithms for STCON based on short random walks, whose runtime increases from $O(n^{\\log n})$ to $O(n^n)$ as space decreases from $O(\\log ^2 n)$ to $O(\\log n)$ .", "Finally, we remark on recent developments in the area of graph exploration with biased random walks.", "Ikeda et al.", "[17] and Nonaka et al.", "[20] studied possible adjustments to the transition matrix of the walk based on the availability of local topological information (otherwise known as “look-ahead”).", "In general, the idea of these approaches is to increase the probability of transition to a node of lower degree.", "The former paper introduces a new type of walk, called the $\\beta $ -walk, whose transition matrices are biased so that transition from a node to its neighbor of degree $d$ is proportional to $d^{-\\beta }$ .", "Such a walk was shown to visit all nodes of the graph in $O(n^2 \\log n)$ steps in expectation for an optimal choice of parameter $\\beta = 1/2$ .", "Nonaka et al.", "[20] later used the key lemmas from this work to prove an analogous result for a walk with a modified transition matrix, which fits into the class of Metropolis-Hastings walks.", "This walk is the starting point for considerations in our paper.", "A somewhat different approach was adopted by Berenbrink et al.", "[7], who show that a random walk with the additional capability of marking one unvisited node in its neighborhood as visited can be used to speed up exploration." ], [ "Overview of the paper", "The organization of the technical parts of the paper is the following.", "In Section , we recall the definition of the Metropolis-Hastings walk and provide its efficient implementation using the Metropolis algorithm.", "In this way, given a representation of graph $G$ , each step of the walk can be simulated by a procedure running in $\\widetilde{O}(1)$ time and using $\\Theta (\\log n)$ bits of space.", "We subsequently identify the key properties of the unit-potential Metropolis-Hastings walk, denoted $RW(G_1)$ , which allow it to be used as a replacement for the (unbiased) random walk on $G$ , denoted $RW(G)$ , in algorithms solving USTCON.", "The major difference between these types of walks is that a short random walk $RW(G)$ has the desirable property of low edge-return rate, i.e., each edge of the graph is visited $O(\\sqrt{t})$ times in expectation during $t$ steps of the walk (for sufficiently small $t$ ).", "However, no analogous property hold for the Metropolis-Hastings walk.", "In fact, on some graphs (e.g., the glitter star defined in [20]), the Metropolis-Hastings walk $RW(G_1)$ , will in expectation discover only $O(1)$ edges during $t$ steps of the walk, visiting each of these edges $\\Omega (t)$ times (for any choice of $t\\le n$ ).", "We overcome this problem in two stages: In Section  we prove that in a graph of maximum degree $\\Delta $ , the Metropolis-Hastings walk $RW(G_1)$ begins to achieve a low node-return rate starting from a threshold length of $\\Delta ^2$ steps: a Metropolis-Hastings walk of length $t$ , $\\Delta ^2 < t < n^2$ , visits each node of the graph $O(\\sqrt{t})$ times in expectation.", "This property is formally stated as Lemma REF .", "In Section  we show how to obtain the trade-off $T = \\widetilde{O}(\\max \\lbrace \\frac{n^2}{S}, m\\rbrace )$ for an arbitrary choice of space bound $S$ .", "Our initial approach makes use of a modification of a technique introduced by Broder et al.", "[10].", "It consists in running $p \\approx S$ walks of length $t \\approx \\frac{n^2}{S^2}$ each, which originate from an appropriately chosen subset of $p$ nodes of the graph called landmarks.", "In our formulation, the walks used are Metropolis-Hastings walks (rather than random walks on $G$ ), and the set of landmarks is sampled uniformly on $V$ .", "By observing the visits of each of these walks to other landmarks from the set, it is possible to obtain information about paths connecting different landmarks.", "When the performed Metropolis-Hastings walks have a low node-return rate (i.e., when $t>\\Delta ^2$ ), the obtained information turns out to be w.h.p.", "sufficient to find an answer to $S$ -$$ connectivity with a low probability of error.", "Otherwise, when $t < \\Delta ^2$ , we modify the approach, performing a logical transformation of graph $G$ .", "We split each node of degree greater than $\\sqrt{t}$ , so that the maximum degree of the modified graph does not exceed $\\sqrt{t}$ .", "Then, all of the considerations are performed for this modified graph.", "In particular, the set of landmark nodes is chosen by uniform sampling on the set of nodes of this modified graph.", "The overhead associated with this transformation is just small enough for our algorithm to have the claimed time complexity of $T = \\widetilde{O}(\\max \\lbrace \\frac{n^2}{S}, m\\rbrace )$ .", "An implementation of the complete algorithm is provided in Appendix A.", "Finally, in the closing Section  we discuss the tightness of the obtained results.", "We also propose a modified weighting of the Metropolis-Hastings walk which performs faster than uniform-weighted Metropolis-Hastings for many classes of graphs, while still covering all the nodes of the graph in $\\widetilde{O}(n^2)$ time.", "This walk satisfies the property that its commute time between any pair of nodes (and consequently also the average hitting time) is asymptotically upper-bounded by the values of the respective parameters for the unbiased random walk.", "In particular, it covers all the nodes of the previously mentioned glitter star, in expected $\\widetilde{O}(n)$ steps." ], [ "Notation and model", "The input graph $G=(V,E)$ , with $|V|=n$ and $|E|=m$ , is simple and not necessarily connected.", "In order to simplify notation for complexity bounds, we assume $m = \\Omega (n)$ .", "The degree of a node $v\\in V$ is denoted by $\\deg (v)$ , the neighborhood of node $v$ by $\\Gamma (v)$ , and the closed neighborhood of $v$ by $\\Gamma ^+(v) = \\Gamma (v) \\cup \\lbrace v\\rbrace $ .", "The maximum degree of the graph is denoted by $\\Delta $ .", "The arc set $\\vec{E} \\subseteq V \\times V$ of undirected graph $G$ is understood as the set of arcs of all edges and self-loops of $G$ : $\\vec{E} = \\lbrace (v,u) : v\\in V, u \\in \\Gamma ^+(v)\\rbrace $ .", "An arc $(v,u)\\in \\vec{E}$ is sometimes denoted as $e_{vu}$ for compactness of notation.", "Note that the symbols $V$ , $E$ , $\\Delta $ , $n$ , $m$ always refer to the input graph $G$ .", "When considering a different graph $X$ , we will sometimes denote its vertex, edge, and arc sets by $V(X)$ , $E(X)$ , and $\\vec{E}(X)$ , respectively.", "Our algorithms are designed for the classical RAM model of computation.", "No special assumptions are made on the representation of graph $G$ , except that for any node $v\\in V$ , there should exist a local ordering on the set of its neighbors, given by the bijective function $PORT_v : \\Gamma (v) \\rightarrow \\lbrace 0,1,\\ldots ,\\deg (v)-1\\rbrace $ .", "Each of the following operations should be possible to implement in $\\widetilde{O}(1)$ time: computing $\\deg (v)$ , computing $PORT_v(u)$ for a node $u \\in \\Gamma (v)$ , and “traversing an edge” by computing $PORT^{-1}_v(i)$ , for port $i\\in \\lbrace 0,1,\\ldots ,\\deg (v)-1\\rbrace $ .", "An example of a permissible representation is a lexicographically sorted array of ordered pairs of identifiers of neighboring nodes $(u,v)$ , taken over $\\lbrace u,v\\rbrace \\in E$ .", "For most of the paper, we consider weighted reversible Markovian processes corresponding to a random walk $RW(X)$ on some weighted undirected graph $X$ with positive weights on arcs.", "The walk is located on the nodes of graph $X$ , and the next state of the walk is reached by following an arc incident to the current node, chosen with probability proportional to the weight of this arc.", "By a slight abuse of notation, we denote the transition matrix of the walk in the same way as the weighted graph.", "Most other notation follows that of Aldous and Fill [1].", "In particular, we consider the following random variables: $N_a (t)$ denotes the number of steps in the time interval $[0,t)$ during which the walk visits $a$ , where the symbol $a$ may represent a node, edge, or arc of the graph.", "$T_a$ denotes the first moment of time $t>0$ at which the walk first visits (or returns to) a node from $a$ , where the symbol $a$ may represent a subset of nodes or a single node of the graph.", "By writing $\\mathbb {E}_\\alpha Y$ and $\\mathrm {Pr}_\\alpha [E]$ , respectively, we mean the expectation of random variable $Y$ , and the probability of event $E$ occurring, taken over walks starting from probability distribution $\\alpha $ (which may be concentrated on a single node or arc).", "A walk starting from an arc is understood as one which starts from the head of the arc at time 0, and then moves to the tail of the arc at time 1.", "Given a weighted graph $X$ , we denote by $Com(i,j) \\equiv \\mathbb {E}_i T_j + \\mathbb {E}_j T_i$ the commute time between nodes $i,j \\in V(X)$ .", "Throughout the paper, we consider only walks representing reversible Markovian processes, corresponding to symmetric weightings of the graph: $w(e_{vu}) = w(e_{uv})$ , for all $(u,v)\\in \\vec{E}$ .", "In some of the proofs, we rely on the resistor network representation of reversible walks: for each edge $e=\\lbrace u,v\\rbrace \\in E(X)$ having weight $w(e)$ on each of its arc, a resistor with resistance $1/w(e)$ is placed between nodes $u$ and $v$ of the resistor network.", "The symbol $R(u,v)$ denotes the resistance of replacement between nodes $u$ and $v$ of the network.", "We recall that $Com(i,j) = R(i,j)\\sum _{e\\in \\vec{E}(X)}w(e)$ .", "[11]" ], [ "Preliminaries: The Metropolis-Hastings Walk on Graphs", "The Metropolis-Hastings walk with potential function $f : V \\rightarrow \\mathbb {R}^+$ is defined as a walk on the weighted graph $G_f = (V, E, w_f)$ with the following assignment of weights $w_f : \\vec{E} \\rightarrow \\mathbb {R}^+$ : $w_f(e_{vu}) = \\min \\left\\lbrace \\frac{f(v)}{\\deg (v)}, \\frac{f(u)}{\\deg (u)}\\right\\rbrace , \\textrm {\\quad for all } \\lbrace v,u\\rbrace \\in E.$ $w_f(e_{vv}) = f(v) - \\sum _{u\\in \\Gamma (v)} w_f(\\lbrace v,u\\rbrace ), \\textrm {\\quad for all } v \\in V,$ We recall that for a walk in state $v \\in V$ , the next state is chosen as $u \\in \\Gamma (v) \\cup \\lbrace v\\rbrace $ with probability proportional to the weight $w_f(e_{vu})$ .", "By a classical result due to Metropolis et al.", "[18], for a given representation of graph $G$ , a single step of the Metropolis-Hastings walk $RW(G_f)$ can be simulated in $\\widetilde{O}(1)$ time and space by means of the procedure shown in Algorithm .", "The algorithm takes advantage of the fact that $w_f(e_{vu}) / \\sum _{x\\in \\Gamma ^+(v)} w_f(e_{vx})\\le \\frac{1}{\\deg (v)}$ , for all $u \\in \\Gamma (v)$ .", "For a walk located at node $v$ , it samples a node $u\\in \\Gamma (v)$ with uniform probability $\\frac{1}{\\deg (v)}$ , and accepts $u$ as the new state with the appropriate probability.", "We remark that a step of $RW(G_f)$ can also be simulated by a log-space automaton which pushes a pebble along the arc $(v,u)$ .", "The pebble remains at $u$ if state $u$ is accepted, and otherwise reverts to $v$ by traversing the arc $(u,v)$ .", "Thus, one step of $RW(G_f)$ can be simulated by at most two moves of a pebble.", "0.5em function next_state ($v$ : node) {      $u \\leftarrow $ neighbor of $v$ in $G$ chosen uniformly at random;    // pick a new state      with probability $\\min \\lbrace \\frac{\\deg (v)}{\\deg (u)}\\frac{f(u)}{f(v)}, 1\\rbrace $ do return $u$ ; // accept: move to new state      return $v$ ;    // do not accept: keep current state } State transition function on $V$ for the walk $RW(G_f)$ .", "Definition 1 We denote by $G_1$ the weighted graph $G_f$ for the unit potential function $f(v) \\equiv 1$ .", "From the next two sections, we focus on the Metropolis-Hastings walk $RW(G_1)$ .", "We note that the weights on the edges of $G$ are now simply given by $w(e_{vu}) = \\min \\lbrace \\frac{1}{\\deg (v)}, \\frac{1}{\\deg (u)}\\rbrace $ .", "The bound on the time required by the Metropolis-Hastings walk to discover w.h.p.", "the entire connected component containing the starting node of the walk follows from the considerations of Nonaka et al.", "[20].", "(All omitted proofs are provided in the Appendix.)", "Lemma 1 ([20]) Let $i \\in V$ , let $H$ be the connected component of $G$ containing node $i$ , and let $n_H = |V(H)|$ .", "Then: a walk $RW(G_1)$ of length $12 n_H^2$ starting from $i$ covers an arbitrary node $j\\in V(H)$ with probability at least $\\frac{1}{2}$ .", "a walk $RW(G_1)$ of length $24 n_H^2 \\log n$ starting from $i$ covers all nodes from $V(H)$ with probability at least $1 - \\frac{1}{n}$ .", "By the above Lemma, a solution to USTCON, with probability $1-\\frac{1}{n}$ , is obtained by running the walk $RW(G_1)$ , starting from $S$ , for $24 n^2 \\log n$ steps.", "USTCON can therefore be solved in log-space by running Algorithm  in a loop for $O(n^2 \\log n)$ iterations.", "(We are unaware of any previous reference in the literature for this observation.)", "Corollary 1 There is a log-space algorithm for USTCON which runs in time $O(n^2\\log n)$ , with probability of one-sided error bounded by $\\frac{1}{n}$ .", "$\\Box $ For our purposes, we will need a more detailed analysis of the behavior of the Metropolis-Hastings walk.", "We start by recalling that the Metropolis-Hastings walk $RW(G_1)$ is a reversible Markovian process, since $w(e_{vu}) = w(e_{uv})$ for all arcs.", "Its stationary distribution is the uniform distribution $\\pi : V \\rightarrow \\mathbb {R}^+$ , with $\\pi (v)= \\frac{1}{n},$ for all $v\\in V$ .", "This allows us to show the following key lemma which captures the “low node-return rate” property of the Metropolis-Hastings walk, as highlighted in the Introduction.", "The first claim states that a Metropolis-Hastings walk starting within any subset of nodes $A\\subsetneq V$ is likely to leave it within $O(|A|^2)$ steps, while its second claim shows that a Metropolis-Hastings walk of length $t$ is likely to return to its starting node not more than $O(\\sqrt{t})$ times.", "However, both of the above statements hold only when considering walks of duration $\\Omega (\\Delta ^2)$ .", "Lemma 2 Suppose that $G$ is connected.", "Let $A \\subsetneq V$ , and let $i\\in A$ .", "For a weighted random walk $RW(G_1)$ starting from node $i$ : the expected time to reach a node from $V\\setminus A$ is bounded by: $\\mathbb {E}_i T_{V\\setminus A} < (|A|+1)(6|A|+2\\Delta ),$ the expected number of visits to node $i$ before any time $t$ , $0<t<6n^2$ , is bounded by: $\\mathbb {E}_i N_i (t) < 5 \\sqrt{t} + 2 \\Delta .$ The proof of the lemma follows by an analysis of resistances of replacement along shortest paths in the resistor network for the weighted graph $G_1$ ." ], [ "A time-space trade-off for USTCON", "The time-space tradoffs for USTCON proposed by Broder et al.", "[10] make use of a number of short random walks, originating from a subset of nodes of the graph called landmarks.", "Herein, we design an algorithm which replaces these random walks by Metropolis-Hastings walks.", "We start by a brief overview of the landmark-based approach.", "When considering an algorithm using space $S$ , the size of the set of landmarks is defined by a parameter $p = \\Theta (S/\\log n)$ .", "The algorithm first chooses a set of landmarks $L \\subseteq V$ consisting of $p+2$ nodes: node $S$ , node $$ , and $p$ nodes picked (in the case of our work) uniformly at random from $V$ .", "Then, a walk of suitably chosen length $t$ is released from each of the landmarks.", "Throughout this process, the algorithm maintains a disjoint-set data structure (also known as “Union-Find” [16]) on the set of landmarks, with each set corresponding to the landmarks identified as belonging to the same connected component of the graph.", "Initially, each landmark belongs to a separate set.", "Whenever a walk starting from one landmark hits some other landmark, the algorithm updates the data structure, merging the classes corresponding to these two landmarks.", "At the end of the process, if landmarks $S$ and $$ belong to the same class, then, with certainty, there exists an $S$ -$$ path in $G$ , and the answer to USTCON is positive.", "Otherwise, the algorithm returns a negative result, and, in the rest of this Section, we focus on proving that this result is correct w.h.p.", "The runtime of the algorithm of Broder et al.", "is determined by the time of running $p = \\widetilde{\\Theta }(S)$ random walks of length $t$ each, thus $T = \\widetilde{O}(tp) = \\widetilde{O}(\\frac{t p^2}{S})$ .", "To achieve the claimed trade-off of $T = \\widetilde{O}(\\frac{n^2}{S})$ , we will therefore need to use walks of length roughly $t \\approx \\frac{n^2}{p^2}$ ." ], [ "An initial approach", "We fix a value of the parameter $p = O(S)$ , describing the number of landmark nodes.", "The landmark-based algorithms are built around the premise that landmarks belonging to the same connected component of $G$ quickly discover each other with the help of the short walks they release.", "In particular, it is desirable that the set of landmarks in each connected component of $G$ has the property that for any partition of the set of landmarks into two subsets, a short walk originating from a landmark in one of these subsets is likely to reach some landmark from the other subset.", "Broder et al.", "[10] observe (cf.", "also [14] for a high-level exposition of the argument) that this property is satisfied if the considered set of landmarks is good, i.e., it fulfills the following two assumptions.", "Firstly, the set of short walks originating from all of the landmarks should be likely to jointly cover all the arcs of the graph.", "Secondly, a short walk originating from an arbitrary starting node of the graph should be likely to reach at least one landmark from the set.", "Most of the analysis and key lemmas in this subsection follow along the lines proposed by Broder et al.", "We confine ourselves to a summary of the approach, highlighting the subtle differences resulting from the use of Metropolis-Hastings walks.", "We start by re-setting the good landmark property in the context of Metropolis-Hastings walks $RW(G_1)$ of a specifically chosen length $\\tau $ .", "Property 1 Let $L \\subseteq V$ be the set of $p = |L|$ landmark nodes, let $H$ be a connected component of $G$ , and let $n_p = \\max \\lbrace \\gamma \\frac{n}{p} \\log n, \\Delta \\rbrace $ , where $\\gamma =60$ is a suitably chosen absolute constant (whose value follows from the proof of Lemma REF ).", "We say that the set of landmarks $L$ is good with respect to $H$ if the following properties hold: With probability at least $1 - \\frac{1}{n}$ , a set of $p$ walks $RW(G_1)$ of length $\\tau = n_p^2$ each, with one walk originating from each landmark from $L$ , covers an arbitrarily chosen arc of $H$ .", "With probability at least $1 - \\frac{1}{n}$ , a walk $RW(G_1)$ of length $\\tau = n_p^2$ , originating from an arbitrarily chosen node of $H$ , hits some landmark from $L$ .", "In the above property, the choice of the length $\\tau $ of the walk takes into account that walks of length $\\widetilde{O}(\\frac{n^2}{p^2})$ lead us to the sought time complexity of $\\widetilde{O}(\\frac{n^2}{p})$ for the algorithm.", "However, in order to ensure that a uniformly sampled landmark set is likely to be good, we will make use of the low node-return rate of the Metropolis-Hastings walk from Lemma REF , and thus we need to have $\\tau = \\Omega (\\Delta ^2)$ .", "We will now show that that Property REF holds w.h.p.", "for a set of landmarks, each of which is chosen according to the uniform distribution $\\pi $ on the set of nodes $V$ .", "To achieve this, we capture the “contribution” of a single Metropolis-Hastings walk to the probability of success of the events described in the Property.", "It turns out that a Metropolis-Hastings walk of the chosen length $\\tau $ , when starting from a landmark, has probability $\\Omega (1/p)$ of reaching an arbitrary arc of the graph.", "When starting from an arbitrary node from $V$ , such a walk has probability $\\Omega (1/p)$ of reaching any specific landmark.", "These claims are formulated in a slightly more general way as the two lemmas below.", "Their proofs take into account the low node-return rate property from Lemma REF $(ii)$ , and the properties of a walk starting from its stationary distribution $\\pi $ .", "Lemma 3 Suppose that $G$ is connected.", "For a weighted walk $RW(G_1)$ starting from a node chosen according to the uniform distribution $\\pi $ , the probability of traversing (a fixed) non-loop arc $e_{ij}$ before time $t$ , where $\\Delta ^2 \\le t < 6n^2$ , is: $\\mathrm {Pr}_{\\pi } [T_{e_{ij}} < t ] > 0.1 \\sqrt{t} / n.$ Lemma 4 Suppose that $G$ is connected.", "Let $j\\in V$ be picked according to the uniform distribution $\\pi $ .", "For a weighted walk $RW(G_1)$ starting from some node $i \\in V$ , the probability of reaching $j$ before time $t$ , where $\\Delta ^2 \\le t < 6n^2$ , is: $\\mathrm {Pr}_{i} [T_j < t ] > 0.1 \\sqrt{t} / n.$ After combining the above lemmas and applying some elementary arguments about unions of independent events, we finally obtain that Property REF is satisfied w.h.p.", "by landmarks uniformly chosen from $V$ , provided that the considered connected component is sufficiently large.", "Lemma 5 If a connected component $H \\subseteq G$ has $n_H \\ge n_p/6$ nodes, then a (multi)set of $p$ nodes, picked with uniform probability from $V$ , is a good set of landmarks with respect to $H$ with probability at least $1- \\frac{1}{2n}$ .", "The results of Broder et al.", "imply directly that if a set of landmarks is good with respect to a connected component $H$ , then all landmarks in $H$ can be identified as belonging to the same connected component by releasing a small number of walks from each landmark, and applying Union-Find type operations on a disjoint-set datastructure on the landmarks.", "Since the proof does not rely on any other assumptions beyond the properties of good landmarks, the result is directly applicable to our considerations of the Metropolis-Hastings walk.", "Lemma 6 ([10]) Let $L$ be a set of good landmarks with respect to connected component $H \\subseteq G$ .", "Then, a set of walks of length $n_p^2$ each, with $\\beta \\log n$ walks originating from each of the landmarks, with probability at least $1-\\frac{1}{2n}$ discovers that all landmarks located within $H$ belong to the same connected component.", "In the above, the absolute constant $\\beta $ can be chosen as $\\beta = 72$ .", "Our algorithm for USTCON is now obtained as follows.", "We pick a set of landmarks $L$ , consisting of $S$ , $$ , and $p$ nodes picked uniformly at random from $V$ , and then follow $\\beta \\log n$ Metropolis-Hastings walks from each landmark, updating the disjoint-set data structure.", "Finally, the algorithm decides whether $S$ and $$ are connected based on whether these two landmarks have been identified as belonging to the same connected component.", "The algorithm never provides a false-positive answer.", "The probability of identifying a pair of nodes $S$ and $$ from the same component $H\\subseteq G$ as not being connected, can be bounded using the following argument adapted from Broder et al.", "Let $H$ be the connected component of $G$ containing node $S$ .", "If $n_H \\ge n_p / 6$ , then by Lemma REF , the set $L$ is a set of good landmarks with respect to $H$ with probability at least $1-\\frac{1}{2n}$ (note that adding nodes $S$ and $$ to a good set of landmarks cannot make this set of landmarks a bad one).", "Conditioned on this, by Lemma REF , we obtain a correct answer to USTCON with probability $1-\\frac{1}{2n}$ .", "Thus, the algorithm works correctly with probability at least $1-\\frac{1}{n}$ .", "In the case when $n_H < n_p / 6$ , we consider only the walks originating from landmark $S$ .", "There are $\\beta \\log n$ such (independent) walks, each of length $n_p^2 > 36 n_H^2$ .", "It follows from Lemma 3, putting $i = S$ and $j = $ , that in this case, node $$ will be reached with probability at least $1 - \\frac{1}{n}$ .", "This completes the proof of correctness.", "Proposition 1 For all $p\\ge 1$ , there is an algorithm solving USTCON using space $S = \\widetilde{O}(p)$ and time $T=\\widetilde{O}(n_p^2 p)$ , where $n_p = \\max \\lbrace \\gamma \\frac{n}{p} \\log n, \\Delta \\rbrace $ , with probability of one-sided error bounded by $\\frac{1}{n}$ .$\\Box $ For the case when $p = \\widetilde{O}(\\frac{n}{\\Delta })$ , we have obtained the trade-off $T = \\widetilde{O}(\\frac{n^2}{S})$ .", "We now show how to obtain the claimed trade-off in the general case." ], [ "Removing the dependence on maximum degree", "We now remove the dependence of length of the used walks on the value of $\\Delta $ .", "We design a graph $G^*=(V^*,E^*)$ by subdividing the nodes of $G$ , so that each node from $V$ turns into a path of nodes in $G^*$ with maximum degree bounded by $D+2$ , where $D \\ge 1$ is an integer parameter, whose value is specified later.", "Formally, graph $G^*$ is defined as follows: For each node $v\\in V$ , $V^*$ contains $\\lceil \\frac{\\deg (v)}{D}\\rceil $ copies of $v$ , labeled $(v,0),(v,1),\\ldots , (v,\\lceil \\frac{\\deg (v)}{D}\\rceil -1)$ .", "Nodes $(u,i)$ and $(v,j)$ , $u\\ne v$ , are connected by an edge in $E^*$ if and only if $\\lbrace u,v\\rbrace \\in E$ , $iD \\le PORT_v(u) < (i+1)D$ , and $jD \\le PORT_u(v) < (j+1)D$ .", "Nodes $(u,i)$ and $(u,i+1)$ , for all $0 \\le i < \\lceil \\frac{\\deg (v)}{D}\\rceil -1$ , are connected by an edge of $E^*$ , with special port labels $`prev^{\\prime }$ and $`next^{\\prime }$ at its endpoints.", "Let $n^*$ and $\\Delta ^*$ be the number of nodes and the maximum degree of $G^*$ , respectively.", "We have $\\Delta ^* = D + 2$ , and the following bound on $n^*$ holds: $n^* = \\sum _{v \\in V} \\lceil \\frac{\\deg (v)}{D}\\rceil < n + \\sum _{v \\in V} \\frac{\\deg (v)}{D} = n + \\frac{2m}{D}.$ Solving USTCON on $G$ between nodes $S$ and $$ can be reduced to solving USTCON on $G^*$ between nodes $(S, 0)$ and $(, 0)$ , since the transformation of $G$ into $G^*$ does not affect connectivity.", "In order to apply the algorithm for USTCON to $G^*$ , rather than to $G$ , we introduce the following modifications: Landmarks need to be distributed following the uniform distribution on $V^*$ .", "This can be achieved by picking $p$ integers uniformly at random from the range $[1, n^*]$ , then enumerating all the nodes of $V^*$ in order, and associating the landmarks with the corresponding nodes from $V^*$ .", "This operation can be performed in $O(n^* + p\\log n^*)$ time, which is always $\\widetilde{O}(m)$ .", "The performed walks need to follow $RW(G^*_1)$ , rather than $RW(G_1)$ .", "A simulation of one step of the walk $RW(G^*_1)$ can be performed in $\\widetilde{O}(1)$ time.", "The duration of each of the performed walks is given as $n_p^{*2}$ , where: $n_p^* = \\max \\left\\lbrace \\gamma \\frac{n^*}{p} \\log n^*, \\Delta ^*\\right\\rbrace < \\max \\left\\lbrace \\gamma \\frac{n + 2m/D}{p} \\cdot 2\\log n, D+2\\right\\rbrace $ It follows that the time complexity of the entire algorithm is bounded by the $\\widetilde{O}(m)$ complexity of landmark distribution and the $\\widetilde{O}(n_p^{*2} p)$ complexity of simulating the Metropolis-Hastings walks on $G^*$ .", "Substituting the expression from (REF ) for $n_p^*$ , we have: $T = \\widetilde{O}(m + n_p^{*2} p) = \\widetilde{O}\\left(m + \\frac{n^2}{p} + \\frac{m^2}{D^2 p} + D^2 p\\right)$ Now, putting $D = \\lceil \\sqrt{m/p}\\rceil $ gives $D^2 p = \\Theta (m)$ , and we obtain the required time bound $T = \\widetilde{O}(m + \\frac{n^2}{p} + \\frac{m^2}{m} + m) = \\widetilde{O}(\\max \\lbrace \\frac{n^2}{p}, m\\rbrace )$ .", "Since the proposed solution can be implemented with a space bound of $S = \\widetilde{O}(p)$ , we have proven the main theorem of the paper.", "Theorem 1 For all $S \\ge c\\log n$ , where $c>0$ is some model-dependent constant, there is an algorithm solving USTCON using space $S$ and time $T=\\widetilde{O}(\\max \\lbrace \\frac{n^2}{S}, m\\rbrace )$ , with probability of one-sided error bounded by $\\frac{1}{n}$ .$\\Box $" ], [ "Tightness of the trade-off.", "For a space bound $S \\ge \\frac{n^2}{m}$ , we cannot hope for an algorithm with smaller run-time than $T=\\widetilde{O}(m)$ , achieved in Theorem REF .", "In fact, the lower bound of $T = \\Omega (m)$ holds for the RAM model under most reasonable representations of $G$ in the memory (cf.", "Proposition REF in Appendix B for a standard proof).", "For smaller values of $S$ , the optimality of the achieved trade-off $S\\cdot T = \\widetilde{O}(n^2)$ is open.", "For the extremal case of $S = O(\\log n)$ , the results of [6] imply that $T=\\widetilde{\\Omega }(n^2)$ for any deterministic algorithm using a jumping automaton (JAG) with at most one movable pebble.", "There is also little hope of improving the time complexity using randomized algorithms similar to the Metropolis-Hastings walk, since Nonaka et al.", "[20] showed that any walk, having a stationary distribution which is (almost) uniform on the nodes of the graph, has $\\Omega (n^2)$ cover time for some graphs.", "Even more strongly, one can ask whether there exists an algorithm for USTCON which runs in $\\widetilde{O}(1)$ space and $\\widetilde{O}(m)$ time.", "This appears unlikely in view of the negative result of Edmonds [13], who showed that a randomized JAG using $\\widetilde{O}(1)$ space and $o(\\log n / \\log \\log n)$ pebbles requires in expectation $n^{1+ \\Omega (1)/\\log \\log n}$ time to explore certain 3-regular graphs." ], [ "Fine-tuning the Metropolis-Hastings walk.", "In view of Lemma REF , the Metropolis-Hastings walk visits all the nodes of a graph within $\\widetilde{O}(n^2)$ steps.", "This is an improvement with respect to the bound of $O(nm)$ on the cover time of an unbiased random walk.", "Nevertheless, the Metropolis-Hastings walk may perform worse than the random walk for specific graph classes.", "A generic example of such a graph, called the glitter star, was defined by [20] as a tree on $n = 2l+1$ nodes, with one central node of degree $l$ connected to $l$ nodes of degree 2, which are in turn connected to $l$ leaves.", "On the glitter star, the cover time of the random walk is $\\Theta (n \\log n)$ , and the cover time of the Metropolis-Hastings walk is $\\Theta (n^2)$ .", "Below we propose a walk $RW(G_f)$ with a different potential function which combines some of the advantages of the random walk and the Metropolis-Hastings walk.", "Proposition 2 For a graph $G$ , let the node potential function $f : V \\rightarrow \\mathbb {R}^+$ be given as $f(u) = \\frac{\\deg (u)}{d} + 1$ , where $d = \\frac{2m}{n}$ is the average degree of the graph.", "Then, for any pair of nodes $u,v\\in V$ , the walk $RW(G_f)$ achieves a commute time of: $Com_{G_f}(u,v) = O(\\min \\lbrace Com_{G}(u,v),Com_{G_1}(u,v)\\rbrace ),$ where $Com_{G}(u,v)$ and $Com_{G_1}(u,v)$ denote the commute times for the random walk on $G$ and the Metropolis-Hastings walk, respectively.", "A step of the walk $RW(G_f)$ can be simulated using $\\widetilde{O}(1)$ space and time.", "The above Proposition implies that for any graph, the walk $RW(G_f)$ with $f(u) = \\frac{\\deg (u)}{d} + 1$ , is asymptotically not slower than the unbiased random walk in terms of parameters such as maximum hitting time and (arbitrarily weighted) average hitting time.", "At the same time, this walk preserves the upper bound of $\\widetilde{O}(n^2)$ on the cover time in the graph, making it an interesting alternative to the unbiased random walk in practical applications, e.g., for different random graph models.", "We remark that there exist different ways of combining the unbiased random walk and the Metropolis-Hastings walk.", "For example, one may consider an automaton which iteratively performs a phase of the walk $RW(G)$ , followed by a phase of the walk $RW(G_1)$ of the same length, doubling the lengths of both walks in each subsequent iteration.", "Such a walk visits all the nodes of the graph in expected time asymptotically equal to the cover time of the faster of the two walks.", "Appendix A: Implementation For the sake of completeness, below we provide the pseudocode of the algorithm for USTCON announced in Theorem REF .", "The implementation is self-contained, except for the following subroutines.", "The disjoint-set data structure is implemented by the procedures: $SET(x)$ which adds a new set containing only element $x$ to the data structure, $FIND(x)$ which returns (the identifier of) the set containing element $x$ , and $UNION(S_1, S_2)$ which replaces sets $S_1$ and $S_2$ by set $S_1\\cup S_2$ in the data structure.", "Each of these operations is performed in amortized $\\widetilde{O}(1)$ time.", "The routine $TRAVERSE\\_EDGE_v(port)$ , for a node $v\\in V$ , returns a pair $(u, inport)$ , such that $u \\in \\Gamma (v)$ with $PORT_v(u) = port$ , and $inport = PORT_u(v)$ .", "This routine can be performed in $\\widetilde{O}(1)$ time in the RAM model, as well as in most JAG-based models.", "We recall the values of the absolute constants: $\\gamma = 60$ and $\\beta = 72$ .", "// Solution to USTCON using $p$ auxiliary landmarks procedure test_connectivity ($S,$ : nodes from $V$ ) {      $D \\leftarrow \\lceil \\sqrt{m / p} \\rceil $ ;      $n^* \\leftarrow \\sum _{v \\in V} \\lceil \\frac{\\deg (v)}{D}\\rceil $ ;      $L \\leftarrow \\lbrace (S,0),(,0)\\rbrace $ ;      // Distribute $p$ landmarks uniformly on $V^*$      $\\ell \\leftarrow $ multi-set of $p$ integers chosen uniformly at random from the range $\\lbrace 1,2,\\ldots ,n^*\\rbrace $ ;      $i \\leftarrow 0$ ;      for $v\\in V$ do           for $j \\leftarrow 0,1,\\ldots , \\lceil \\frac{\\deg (v)}{D}\\rceil -1$ do {                $i \\leftarrow i+1$ ;                if $i \\in \\ell $ then $L \\leftarrow L \\cup \\lbrace (v, j)\\rbrace $ ;           }      for $l\\in L$ do $SET(l) \\leftarrow l$ ;      // From each landmark, run $\\beta \\log n^*$ Metropolis walks $RW(G^{*}_1)$ of length $n_p^{*2}$ each      repeat $\\beta \\log n^*$ times {           for $l\\in L$ do {                $s \\leftarrow l$ ;                repeat $\\lceil \\max \\lbrace \\gamma \\frac{n^*}{p} \\log n^*, D+2\\rbrace \\rceil ^{2}$ times {                     $s \\leftarrow $ next_state* ($s$ );                     $UNION (FIND(s), FIND(l))$ ;                }           }      }      if $FIND((S,0)) = FIND((,0))$ then return “connected”;      return “probably not connected”; } // Simulate one step of the walk $RW({G^{*}_1})$ from state $(v,i)\\in V^*$ function next_state* ($v$ : node, $v\\_i$ : integer) {      $v\\_deg^* \\leftarrow $ get_degree*$(v, v\\_i)$ ;      $port \\leftarrow $ get_random_port*$(v, v\\_i)$ ;      if $port = `prev^{\\prime }$ then {           $u \\leftarrow v$ ;           $u\\_i \\leftarrow v\\_i-1$ ;      } else if $port = `next^{\\prime }$ then {           $u \\leftarrow v$ ;           $u\\_i \\leftarrow v\\_i+1$ ;      } else { // $port \\in [left, right]$ is an integer corresponding to a port at $v$ in $G$           $(u, inport) \\leftarrow TRAVERSE\\_EDGE_{v} (port)$ ;           $u\\_i \\leftarrow \\lfloor \\frac{inport}{D}\\rfloor $ ;      }      $u\\_deg^* \\leftarrow $ get_degree*$(u, u\\_i)$ ;      with probability $\\min \\lbrace \\frac{v\\_deg^*}{u\\_deg^*}, 1\\rbrace $ do return $(u, u\\_i)$ ;      return $(v, v\\_i)$ ; } // Return the degree of $(v,i)$ in $G^*$ function get_degree* ($v$ : node, $i$ : integer) {      $left \\leftarrow i \\cdot D$ ;      $right \\leftarrow \\min \\lbrace (i+1)\\cdot D-1, \\deg (v)\\rbrace $ ;      $deg^* \\leftarrow right - left + 1$ ;      if $left >0$ then $deg^* \\leftarrow deg^* +1$ ;      if $right < \\deg (v)$ then $deg^* \\leftarrow deg^* +1$ ;      return $deg^*$ ; } // Return a port at node $(v,i)$ in $G^*$ chosen uniformly at random function get_random_port* ($v$ : node, $i$ : integer) {      $left \\leftarrow i \\cdot D$ ;      $right \\leftarrow \\min \\lbrace (i+1)\\cdot D-1, \\deg (v)\\rbrace $ ;      $deg^* \\leftarrow $ get_degree* $(v,i)$ ;      with probability $(right - left + 1) / deg^*$ do           return integer from range $[left, right]$ chosen uniformly at random;      $neighbors \\leftarrow \\emptyset $ ;      if $left >0$ then $neighbors \\leftarrow neighbors \\cup \\lbrace `prev^{\\prime }\\rbrace $ ;      if $right < \\deg (v)$ then $neighbors \\leftarrow neighbors \\cup \\lbrace `next^{\\prime }\\rbrace $ ;      return element of $neighbors$ chosen uniformly at random; } Appendix B: Auxiliary claims Lemma 7 For all $i,j\\in V$ , $\\mathbb {E}_i N_j(t) = \\mathbb {E}_j N_i(t).$ Let $X_u(\\tau )$ , $u\\in \\lbrace i,j\\rbrace $ , denote the random variable equal to 1 if a walk of length $\\tau $ is located at $u$ after $\\tau $ steps, and 0 otherwise.", "Since $RW(G_1)$ is a reversible Markovian process, by the properties of the $\\tau $ -th power of the transition matrix of the walk (cf.", "[1], Chapter 3.1), we have: $\\pi (i) \\mathbb {E}_i X_j(\\tau ) = \\pi (j) \\mathbb {E}_j X_i(\\tau ).$ Since $\\pi (i) = \\pi (j) = \\frac{1}{n}$ , it follows that $\\mathbb {E}_i X_j(\\tau ) = \\mathbb {E}_j X_i(\\tau )$ .", "Taking into account that $N_u(t) = \\sum _{\\tau =0}^{t-1} X_u(\\tau )$ , $u\\in \\lbrace i,j\\rbrace $ , by linearity of expectation we obtain the claim.", "Lemma 8 For any node $i\\in V$ : $\\mathbb {E}_\\pi N_i(t) = \\frac{t}{n}$ and for any arc $e_{ij}$ of $G$ corresponding to an edge $\\lbrace i,j\\rbrace \\in E$ : $\\mathbb {E}_\\pi N_{e_{ij}}(t) = \\frac{t}{n}\\min \\left\\lbrace \\frac{1}{\\deg (i)}, \\frac{1}{\\deg (j)}\\right\\rbrace .$ Follows directly from the stationary distribution of the Metropolis-Hastings walk on nodes and edges.", "Proposition 3 Any algorithm for USTCON requires time $\\Omega (m)$ .", "Consider a generic instance of USTCON defined as follows.", "Take two disjoint copies of some 2-edge-connected graph $H$ on $n/2$ nodes, with one distinguished node $x$ .", "The two copies of $H$ are assigned the subscripts 1 and 2, respectively.", "Now, as the considered instance of USTCON we use, with probability $1/2$ , the disconnected graph $H_1 \\cup H_2$ with $S=x_1$ and $=x_2$ .", "Otherwise, we pick an edge $\\lbrace u,v\\rbrace $ of $H$ uniformly at random, and use as the instance the connected graph $H_1 \\cup H_2 \\cup \\lbrace \\lbrace u_2,v_1\\rbrace ,\\lbrace u_1,v_2\\rbrace \\rbrace \\setminus \\lbrace \\lbrace u_1,v_1\\rbrace ,\\lbrace u_2,v_2\\rbrace \\rbrace $ , likewise with $S=x_1$ and $=x_2$ .", "Subject to a choice of node identifiers in the representations, the connected and disconnected instances differ on precisely 4 memory cells in the adjacency lists of the graph (for nodes $u_1$ , $u_2$ , $v_1$ , and $v_2$ ), and these cells, taken over the choices of edge $\\lbrace u,v\\rbrace $ , form a partition of the memory representation of the graph.", "Consequently, the expected number of memory reads for an algorithm deciding connectivity with probability $1/2 +p$ is lower-bounded by $p \\cdot m/4$ , and is thus $\\Omega (m)$ , within the range $n \\le m \\le n^2/8-O(n)$ .", "Appendix B: Proofs of technical lemmas Proof of Lemma  REF The interested reader may see this proof as an analogue of the discussion for short random walks in regular graphs, cf.", "Aldous and Fill, Chapter 6, Proposition 16.", "Claim $(i)$ : Consider a shortest path $P$ in graph $G$ from $i$ to a nearest vertex $j \\in V\\setminus A$ .", "Let $P = (i_0, i_1, \\ldots , i_{a}, j)$ , where $i_0=i$ , and $i_l \\in A$ , for $0\\le l \\le a$ .", "Let $G^\\circ $ be the subgraph of $G$ induced by nodes from set $A$ , their neighbors in $G$ , and node $j$ : $G^\\circ = G [A \\cup N(A) \\cup \\lbrace j\\rbrace ]$ .", "Since any random walk in $G$ which starts from $i$ and does not enter $V\\setminus A$ is confined to nodes and edges of graph $G^\\circ $ , we have the following relation between the walks $RW(G_1)$ and $RW(G^\\circ _1)$ : $\\mathbb {E}_i T_{V\\setminus A} \\le \\mathbb {E}_i T^\\circ _{V\\setminus A} = \\mathbb {E}_i T^\\circ _j < Com^\\circ (i,j) = R^\\circ (i,j)\\sum _{e \\in \\vec{E}(G^\\circ )} w(e),$ where the latter equality follows from the electrical network representation of random walks.", "The resistance $R^\\circ (i,j)$ is upper-bounded by the resistance of the series connection going through the nodes of path $P$ in $G$ : $R^\\circ (i,j) \\le \\tfrac{1}{w(e_{i_0 i_1})} + \\tfrac{1}{w(e_{i_1 i_2})} + \\ldots + \\tfrac{1}{w(e_{i_{a-1} i_a})} + \\tfrac{1}{w(e_{i_{a} j})} =$ $= \\max \\lbrace \\deg (i_0),\\deg (i_1)\\rbrace + \\ldots + \\max \\lbrace \\deg (i_{a-1}),\\deg (i_a)\\rbrace + \\max \\lbrace \\deg (i_{a}),\\deg (i_j)\\rbrace < 2\\sum _{l=0}^{a-1} \\deg (i_l) + 2\\Delta .$ Since the path $P_s = (i_0,i_1,\\ldots ,i_{a-1})$ is a shortest path in graph $G$ between nodes $i_0$ and $i_{a-1}$ , such that $P_s \\subseteq A$ and $\\Gamma (P_s) \\subseteq A$ , it follows that (cf.", "[1]): $\\sum _{l=0}^{a-1} \\deg (i_l) \\le 3 |A|,$ and: $R^\\circ (i,j) < 6|A| + 2\\Delta .$ Since the total weight of edges and self-loops of $G$ incident to a vertex in $V$ is equal to 1, we have: $\\sum _{e \\in \\vec{E}(G^\\circ )} w(e) \\le \\sum _{v\\in A} \\left( \\sum _{u \\in \\Gamma (v) \\cup \\lbrace v\\rbrace } w(e_{vu}) \\right) + \\sum _{u \\in \\Gamma (j) \\cup \\lbrace j\\rbrace } w(e_{ju}) \\le |A| + 1.$ Claim $(i)$ follows from inequalities (REF ), (REF ), and (REF ).", "Claim $(ii)$ : Suppose that $s = \\sqrt{6t} \\ge \\frac{t}{n}$ , and let: $A = \\lbrace j\\in V : \\mathbb {E}_i N_j(t) > s\\rbrace .$ Since the considered walk hits nodes from $V$ a total of (at most) $t$ times, we have $|A| < \\frac{t}{s} \\le n$ , and the considerations performed in the proof of Lemma REF $(i)$ can be applied for the above-defined set $A$ .", "First, we bound the expected number of returns to node $i$ for a walk starting at $i$ before reaching $V\\setminus A$ for the first time: $\\mathbb {E}_i N_i (T_{V\\setminus A}) = 1 + (1-{\\mathrm {Pr}_i[T_{V\\setminus A}<T_i]})\\cdot \\mathbb {E}_i N_i (T_{V\\setminus A}) \\Rightarrow \\mathbb {E}_i N_i (T_{V\\setminus A}) = \\frac{1}{\\mathrm {Pr}_i[T_{V\\setminus A}<T_i]}.$ Taking into account [1] (Chapter 3, eq.", "(28) and Corollary 11) and bound (REF ), we have: $\\mathbb {E}_i N_i (T_{V\\setminus A}) = \\frac{1}{\\mathrm {Pr}_i[T_{V\\setminus A} < T_i]} = \\pi (i) \\cdot R(i,j)\\cdot \\!\\!\\sum _{e \\in \\vec{E}(G)} w(e) \\;\\le \\; \\pi (i) \\cdot R^\\circ (i,j)\\cdot \\!\\!\\sum _{e \\in \\vec{E}(G)} w(e) <$ $< \\frac{1}{n}\\cdot (6|A|+2\\Delta ) n = 6|A|+2\\Delta < 6 \\frac{t}{s} + 2\\Delta .$ It follows from Lemma REF that the definition of set $A$ may be rewritten as: $A = \\lbrace j\\in V : \\mathbb {E}_j N_i(t) > s\\rbrace $ Thus, $V\\setminus A = \\lbrace j\\in V : \\mathbb {E}_j N_i(t) \\le s\\rbrace $ , which means that if a walk starting from $i$ reaches $V\\setminus A$ , it will return to $i$ at most $s$ times in expectation before time $t$ .", "So, using (REF ), we obtain the claim: $\\mathbb {E}_i N_i(t) \\le \\mathbb {E}_i N_i(T_{V\\setminus A}) + s < 6\\frac{t}{s} + s + 2\\Delta = 2\\sqrt{6t} + 2\\Delta < 5\\sqrt{t} + 2\\Delta .$ $\\Box $ Proof of Lemma  REF Fix an arbitrary arc $e_{ij}$ , with $\\lbrace i,j\\rbrace \\in E$ .", "We will bound the sought probability from the inequality: $\\mathbb {E}_{\\pi } N_{e_{ij}} (t) \\le \\mathrm {Pr}_{\\pi } [T_{e_{ij}} < t]\\ \\mathbb {E}_{e_{ij}} N_{e_{ij}} (t)\\Rightarrow \\mathrm {Pr}_{\\pi } [T_{e_{ij}} < t] \\ge \\frac{\\mathbb {E}_{\\pi } N_{e_{ij}} (t)}{\\mathbb {E}_{e_{ij}} N_{e_{ij}} (t)}.$ The expected number of traversals of $e_{ij}$ for a walk of even length starting from the stationary distribution on $V$ is given by equation (REF ).", "To bound the expectation from the denominator of (REF ), we note that by Lemma REF $(ii)$ , $\\mathbb {E}_i N_i (t) < 5 \\sqrt{t} + 2 \\Delta $ , and that arc $e_{ij}$ is chosen with probability $\\min \\left\\lbrace \\frac{1}{\\deg (i)}, \\frac{1}{\\deg (j)}\\right\\rbrace $ during each visit to $i$ : $\\mathbb {E}_i N_{e_{ij}} (t) < (5 \\sqrt{t} + 2\\Delta ) \\min \\left\\lbrace \\frac{1}{\\deg (i)}, \\frac{1}{\\deg (j)}\\right\\rbrace .$ Considering a walk starting from a traversal of arc $e_{ij}$ , we observe that after its traversal of $e_{ij}$ the walk must return to node $i$ before traversing $e_{ij}$ again: $\\mathbb {E}_{e_{ij}} N_{e_{ij}} (t) \\le 1 + \\mathbb {E}_{j} N_{e_{ij}} (t) < 1 + \\mathbb {E}_i N_{e_{ij}} (t) < 1 + (5 \\sqrt{t} + 2 \\Delta ) \\min \\left\\lbrace \\frac{1}{\\deg (i)}, \\frac{1}{\\deg (j)}\\right\\rbrace \\le $ $\\le (5 \\sqrt{t} + 3 \\Delta ) \\min \\left\\lbrace \\frac{1}{\\deg (i)}, \\frac{1}{\\deg (j)}\\right\\rbrace .$ By combining inequalities (REF ), (REF ), (REF ), and taking into account that $t > \\Delta ^2$ , we obtain the claim: $\\mathrm {Pr}_{\\pi } [T_{ e_{ij}} < t] > \\frac{t}{n (5\\sqrt{t} + 3 \\Delta )} > \\frac{t}{8n \\sqrt{t}} > 0.1 \\frac{\\sqrt{t}}{n}.$ $\\Box $ Proof of Lemma  REF Pick a node $j \\in V$ according to the uniform probability distribution $\\pi $ .", "We will bound the sought probability from the inequality: $\\mathbb {E}_{i} N_{j} (t) \\le \\mathrm {Pr}_{i} [T_{j} < t]\\ \\mathbb {E}_{j} N_{j} (t)\\Rightarrow \\mathrm {Pr}_{i} [T_{j} < t] \\ge \\frac{\\mathbb {E}_{i} N_{j} (t)}{\\mathbb {E}_{j} N_{j} (t)}.$ Taking into account Lemma REF and condition (REF ), and noting that $j$ is chosen according to the uniform distribution $\\pi $ on $V$ , we have: $\\mathbb {E}_{i} N_{j} (t) = \\mathbb {E}_{j} N_{i} (t) = \\mathbb {E}_{\\pi } N_{i} (t) = \\frac{t}{n}.$ The expectation from the denominator of (REF ) is bounded by Lemma REF $(ii)$ , $\\mathbb {E}_{j} N_{j} (t) < 5\\sqrt{t} + 2\\Delta $ .", "By combining the above relations, and taking into account that $t > \\Delta ^2$ , we obtain: $\\mathrm {Pr}_{i} [T_{j} < t] > \\frac{t}{n ( 5\\sqrt{t} + 2 \\Delta )} > \\frac{t}{7n \\sqrt{t}} > 0.1 \\frac{\\sqrt{t}}{n}.$ $\\Box $ Proof of Lemma  REF Fixing a connected component $H \\subseteq G$ with $n_H \\ge n_p/6$ , we introduce the following notation for a set of landmarks $L$ : let $L_H = L\\cap V(H)$ , let $X(L)$ denote the event that $|L_H| \\ge \\frac{1}{2} p \\frac{n_H}{n}$ , let $F_1(L)$ be the random variable over $L$ describing the maximum, over all non-loops arcs $e$ belonging to $H$ , of the probability that a set of $p$ random walks $RW(G_1)$ of length $\\tau = n_p^2$ each, with one random walk originating from each landmark from $L$ , does not cover arc $e$ .", "let $F_2(L)$ be the random variable over $L$ describing the maximum, over all nodes $u \\in V(H)$ , of the probability that a random walk $RW(G_1)$ of length $\\tau = n_p^2$ , originating from $u$ , does not hit any landmark of $L$ .", "Suppose that $L$ is a set of $p$ nodes picked according to the uniform distribution $\\pi ^p$ on $V^p$ .", "To prove the claim of the Lemma, we need to show the following bound: $\\mathrm {Pr}_{L\\sim \\pi ^p} [F_1 > n^{-1} \\wedge F_2 > n^{-1}] <\\frac{1}{2n}.$ We observe that each landmark from $L$ belongs to $V(H)$ with probability $n_H/n$ .", "Let $L_H = L\\cap V(H)$ .", "A w.h.p.", "lower bound on the size of $L_H$ follows from the Chernoff bound applied to $p$ Bernoulli trials with success probability $n_H/n$ : $\\mathrm {Pr}_{L\\sim \\pi ^p} [ X ] \\ \\ge 1 - e^{-\\frac{1}{8}p\\frac{n_H}{n}} \\ge 1 - e^{-\\frac{1}{48}p\\frac{n_p}{n}}\\ge 1 - e^{-\\frac{\\gamma }{48}\\log n} > 1 - \\frac{1}{4n},$ where we took into account that $n_p \\ge \\gamma \\frac{n}{p} \\log n$ , and that $\\gamma =60 > 48$ .", "In the following, we only need to show that, conditioned on the event $X(L)$ holding, $L$ is a good set of landmarks with probability $1 - \\frac{1}{4n}$ .", "Note that all the landmarks from $L_H$ are distributed uniformly at random on $V(H)$ , also when conditioned on $X(L)$ .", "To bound $F_1(L)$ , fix a non-loop arc $e$ of $H$ as the arc maximizing the failure probability in the definition of $F_1(L)$ .", "By applying Lemma REF to graph $H$ , the probability that a walk $RW(H_1)$ of length $\\tau =n_p^2$ , starting from the uniform distribution on $V(H)$ , does not cover arc $e$ , is at most $1 - \\frac{0.1 n_p}{n_H}$ .", "Thus, considering that: $|L_H| \\ge \\frac{1}{2} p \\frac{n_H}{n} = \\frac{n_H \\cdot 3 \\log n}{6 \\frac{n}{p}\\log n} \\ge \\frac{n_H \\cdot 3 \\log n}{0.1 n_p},$ the probability $F_{1,e}(L)$ that no random walk starting from a landmark hits arc $e$ is bounded by: $\\mathbb {E}_{L\\sim \\pi ^p} \\left[F_{1} \\ \\big |\\ X \\right] < \\left( 1 - \\frac{0.1 n_p}{n_H} \\right)^{\\frac{n_H}{0.1 n_p} 3 \\log n} < 2^{-3 \\log n} < n^{-3}.$ Likewise, to bound $F_2(L)$ , fix a node $u \\in V(H)$ maximizing the probability that a walk $RW(G_1)$ of length $\\tau = n_p^2$ , originating from $u$ , does not hit any landmark of $L$ .", "By Lemma REF , the probability that the considered walk of length $\\tau $ does not cover a node chosen according to the uniform distribution on $V(H)$ , is at most $1 - \\frac{0.1 \\sqrt{\\tau }}{n_H}$ .", "Thus, taking into account that $|L_H| > \\frac{n_H}{0.1 \\sqrt{\\tau }} 3 \\log n$ , the probability that the walk does not hit any landmark can once again be bounded as less than $n^{-3}$ : $\\mathbb {E}_{L\\sim \\pi ^p} \\left[F_2\\ \\big |\\ X \\right] < n^{-3}.$ It follows that: $\\mathbb {E}_{L\\sim \\pi ^p} \\left[F_1 + F_2\\ \\big |\\ X \\right] < 2n^{-3},$ and by the Markov bound: $\\mathrm {Pr}_{L\\sim \\pi ^p} [F_1 + F_2> n^{-1}\\ \\big |\\ X ] < \\frac{2}{n^2} < \\frac{1}{4n}.$ Now, inequalities (REF ) and (REF ) imply that inequality (REF ) holds, which completes the proof.", "$\\Box $ Proof of Proposition  REF We begin by observing that the unbiased random walk on $G$ can be described as a weighted Metropolis-Hastings walk $RW(G_{f_c})$ , where, for all $u\\in V$ , the potential function on nodes is given as $f_c(u) = c\\deg (u)$ , where $c>0$ is an arbitrarily chosen constant of proportionality ($w(e) = c$ for all edges).", "Now, looking at the electrical networks analogy, by identifying with each other the corresponding nodes of the electrical networks describing the walks $RW(G_{f_c})$ and $RW(G_1)$ , and leaving the edges of both these networks in parallel connection, we obtain a new network on $G$ with edge weights $w_f$ given by: $w_f(e) = w_{f_c}(e) + w(e),$ corresponding to the potential function on nodes: $f(u) = f_c(u) + 1 = c\\deg (u) + 1.$ It follows that the resistance of replacement of the network of $RW(G_f)$ for any two nodes $u,v\\in V$ can be bounded as: $R_{G_f}(u,v) \\le R_{G_{f_c}}(u,v) \\quad \\text{and} \\quad R_{G_f}(u,v) \\le R_{G_1}(u,v).$ Moreover, the following relations hold between resistances and commute times: $Com_{G_{f_c}}(u,v) = 2cm R_{G_{f_c}}(u,v)$ $Com_{G_1}(u,v) = n R_{G_1}(u,v)$ $Com_{G_{f}}(u,v) = (2cm + n) R_{G_f}(u,v)$ Fixing $c = \\frac{1}{d} = \\frac{n}{2m}$ , i.e., $2cm = n$ , we obtain from all of the above relations: $Com_{G_f}(u,v) = O(\\min \\lbrace Com_{G_{f_c}}(u,v),Com_{G_1}(u,v)\\rbrace ).$ $\\Box $" ], [ "Appendix A: Implementation", "For the sake of completeness, below we provide the pseudocode of the algorithm for USTCON announced in Theorem REF .", "The implementation is self-contained, except for the following subroutines.", "The disjoint-set data structure is implemented by the procedures: $SET(x)$ which adds a new set containing only element $x$ to the data structure, $FIND(x)$ which returns (the identifier of) the set containing element $x$ , and $UNION(S_1, S_2)$ which replaces sets $S_1$ and $S_2$ by set $S_1\\cup S_2$ in the data structure.", "Each of these operations is performed in amortized $\\widetilde{O}(1)$ time.", "The routine $TRAVERSE\\_EDGE_v(port)$ , for a node $v\\in V$ , returns a pair $(u, inport)$ , such that $u \\in \\Gamma (v)$ with $PORT_v(u) = port$ , and $inport = PORT_u(v)$ .", "This routine can be performed in $\\widetilde{O}(1)$ time in the RAM model, as well as in most JAG-based models.", "We recall the values of the absolute constants: $\\gamma = 60$ and $\\beta = 72$ .", "// Solution to USTCON using $p$ auxiliary landmarks procedure test_connectivity ($S,$ : nodes from $V$ ) {      $D \\leftarrow \\lceil \\sqrt{m / p} \\rceil $ ;      $n^* \\leftarrow \\sum _{v \\in V} \\lceil \\frac{\\deg (v)}{D}\\rceil $ ;      $L \\leftarrow \\lbrace (S,0),(,0)\\rbrace $ ;      // Distribute $p$ landmarks uniformly on $V^*$      $\\ell \\leftarrow $ multi-set of $p$ integers chosen uniformly at random from the range $\\lbrace 1,2,\\ldots ,n^*\\rbrace $ ;      $i \\leftarrow 0$ ;      for $v\\in V$ do           for $j \\leftarrow 0,1,\\ldots , \\lceil \\frac{\\deg (v)}{D}\\rceil -1$ do {                $i \\leftarrow i+1$ ;                if $i \\in \\ell $ then $L \\leftarrow L \\cup \\lbrace (v, j)\\rbrace $ ;           }      for $l\\in L$ do $SET(l) \\leftarrow l$ ;      // From each landmark, run $\\beta \\log n^*$ Metropolis walks $RW(G^{*}_1)$ of length $n_p^{*2}$ each      repeat $\\beta \\log n^*$ times {           for $l\\in L$ do {                $s \\leftarrow l$ ;                repeat $\\lceil \\max \\lbrace \\gamma \\frac{n^*}{p} \\log n^*, D+2\\rbrace \\rceil ^{2}$ times {                     $s \\leftarrow $ next_state* ($s$ );                     $UNION (FIND(s), FIND(l))$ ;                }           }      }      if $FIND((S,0)) = FIND((,0))$ then return “connected”;      return “probably not connected”; } // Simulate one step of the walk $RW({G^{*}_1})$ from state $(v,i)\\in V^*$ function next_state* ($v$ : node, $v\\_i$ : integer) {      $v\\_deg^* \\leftarrow $ get_degree*$(v, v\\_i)$ ;      $port \\leftarrow $ get_random_port*$(v, v\\_i)$ ;      if $port = `prev^{\\prime }$ then {           $u \\leftarrow v$ ;           $u\\_i \\leftarrow v\\_i-1$ ;      } else if $port = `next^{\\prime }$ then {           $u \\leftarrow v$ ;           $u\\_i \\leftarrow v\\_i+1$ ;      } else { // $port \\in [left, right]$ is an integer corresponding to a port at $v$ in $G$           $(u, inport) \\leftarrow TRAVERSE\\_EDGE_{v} (port)$ ;           $u\\_i \\leftarrow \\lfloor \\frac{inport}{D}\\rfloor $ ;      }      $u\\_deg^* \\leftarrow $ get_degree*$(u, u\\_i)$ ;      with probability $\\min \\lbrace \\frac{v\\_deg^*}{u\\_deg^*}, 1\\rbrace $ do return $(u, u\\_i)$ ;      return $(v, v\\_i)$ ; } // Return the degree of $(v,i)$ in $G^*$ function get_degree* ($v$ : node, $i$ : integer) {      $left \\leftarrow i \\cdot D$ ;      $right \\leftarrow \\min \\lbrace (i+1)\\cdot D-1, \\deg (v)\\rbrace $ ;      $deg^* \\leftarrow right - left + 1$ ;      if $left >0$ then $deg^* \\leftarrow deg^* +1$ ;      if $right < \\deg (v)$ then $deg^* \\leftarrow deg^* +1$ ;      return $deg^*$ ; } // Return a port at node $(v,i)$ in $G^*$ chosen uniformly at random function get_random_port* ($v$ : node, $i$ : integer) {      $left \\leftarrow i \\cdot D$ ;      $right \\leftarrow \\min \\lbrace (i+1)\\cdot D-1, \\deg (v)\\rbrace $ ;      $deg^* \\leftarrow $ get_degree* $(v,i)$ ;      with probability $(right - left + 1) / deg^*$ do           return integer from range $[left, right]$ chosen uniformly at random;      $neighbors \\leftarrow \\emptyset $ ;      if $left >0$ then $neighbors \\leftarrow neighbors \\cup \\lbrace `prev^{\\prime }\\rbrace $ ;      if $right < \\deg (v)$ then $neighbors \\leftarrow neighbors \\cup \\lbrace `next^{\\prime }\\rbrace $ ;      return element of $neighbors$ chosen uniformly at random; } Lemma 7 For all $i,j\\in V$ , $\\mathbb {E}_i N_j(t) = \\mathbb {E}_j N_i(t).$ Let $X_u(\\tau )$ , $u\\in \\lbrace i,j\\rbrace $ , denote the random variable equal to 1 if a walk of length $\\tau $ is located at $u$ after $\\tau $ steps, and 0 otherwise.", "Since $RW(G_1)$ is a reversible Markovian process, by the properties of the $\\tau $ -th power of the transition matrix of the walk (cf.", "[1], Chapter 3.1), we have: $\\pi (i) \\mathbb {E}_i X_j(\\tau ) = \\pi (j) \\mathbb {E}_j X_i(\\tau ).$ Since $\\pi (i) = \\pi (j) = \\frac{1}{n}$ , it follows that $\\mathbb {E}_i X_j(\\tau ) = \\mathbb {E}_j X_i(\\tau )$ .", "Taking into account that $N_u(t) = \\sum _{\\tau =0}^{t-1} X_u(\\tau )$ , $u\\in \\lbrace i,j\\rbrace $ , by linearity of expectation we obtain the claim.", "Lemma 8 For any node $i\\in V$ : $\\mathbb {E}_\\pi N_i(t) = \\frac{t}{n}$ and for any arc $e_{ij}$ of $G$ corresponding to an edge $\\lbrace i,j\\rbrace \\in E$ : $\\mathbb {E}_\\pi N_{e_{ij}}(t) = \\frac{t}{n}\\min \\left\\lbrace \\frac{1}{\\deg (i)}, \\frac{1}{\\deg (j)}\\right\\rbrace .$ Follows directly from the stationary distribution of the Metropolis-Hastings walk on nodes and edges.", "Proposition 3 Any algorithm for USTCON requires time $\\Omega (m)$ .", "Consider a generic instance of USTCON defined as follows.", "Take two disjoint copies of some 2-edge-connected graph $H$ on $n/2$ nodes, with one distinguished node $x$ .", "The two copies of $H$ are assigned the subscripts 1 and 2, respectively.", "Now, as the considered instance of USTCON we use, with probability $1/2$ , the disconnected graph $H_1 \\cup H_2$ with $S=x_1$ and $=x_2$ .", "Otherwise, we pick an edge $\\lbrace u,v\\rbrace $ of $H$ uniformly at random, and use as the instance the connected graph $H_1 \\cup H_2 \\cup \\lbrace \\lbrace u_2,v_1\\rbrace ,\\lbrace u_1,v_2\\rbrace \\rbrace \\setminus \\lbrace \\lbrace u_1,v_1\\rbrace ,\\lbrace u_2,v_2\\rbrace \\rbrace $ , likewise with $S=x_1$ and $=x_2$ .", "Subject to a choice of node identifiers in the representations, the connected and disconnected instances differ on precisely 4 memory cells in the adjacency lists of the graph (for nodes $u_1$ , $u_2$ , $v_1$ , and $v_2$ ), and these cells, taken over the choices of edge $\\lbrace u,v\\rbrace $ , form a partition of the memory representation of the graph.", "Consequently, the expected number of memory reads for an algorithm deciding connectivity with probability $1/2 +p$ is lower-bounded by $p \\cdot m/4$ , and is thus $\\Omega (m)$ , within the range $n \\le m \\le n^2/8-O(n)$ .", "The interested reader may see this proof as an analogue of the discussion for short random walks in regular graphs, cf.", "Aldous and Fill, Chapter 6, Proposition 16.", "Claim $(i)$ : Consider a shortest path $P$ in graph $G$ from $i$ to a nearest vertex $j \\in V\\setminus A$ .", "Let $P = (i_0, i_1, \\ldots , i_{a}, j)$ , where $i_0=i$ , and $i_l \\in A$ , for $0\\le l \\le a$ .", "Let $G^\\circ $ be the subgraph of $G$ induced by nodes from set $A$ , their neighbors in $G$ , and node $j$ : $G^\\circ = G [A \\cup N(A) \\cup \\lbrace j\\rbrace ]$ .", "Since any random walk in $G$ which starts from $i$ and does not enter $V\\setminus A$ is confined to nodes and edges of graph $G^\\circ $ , we have the following relation between the walks $RW(G_1)$ and $RW(G^\\circ _1)$ : $\\mathbb {E}_i T_{V\\setminus A} \\le \\mathbb {E}_i T^\\circ _{V\\setminus A} = \\mathbb {E}_i T^\\circ _j < Com^\\circ (i,j) = R^\\circ (i,j)\\sum _{e \\in \\vec{E}(G^\\circ )} w(e),$ where the latter equality follows from the electrical network representation of random walks.", "The resistance $R^\\circ (i,j)$ is upper-bounded by the resistance of the series connection going through the nodes of path $P$ in $G$ : $R^\\circ (i,j) \\le \\tfrac{1}{w(e_{i_0 i_1})} + \\tfrac{1}{w(e_{i_1 i_2})} + \\ldots + \\tfrac{1}{w(e_{i_{a-1} i_a})} + \\tfrac{1}{w(e_{i_{a} j})} =$ $= \\max \\lbrace \\deg (i_0),\\deg (i_1)\\rbrace + \\ldots + \\max \\lbrace \\deg (i_{a-1}),\\deg (i_a)\\rbrace + \\max \\lbrace \\deg (i_{a}),\\deg (i_j)\\rbrace < 2\\sum _{l=0}^{a-1} \\deg (i_l) + 2\\Delta .$ Since the path $P_s = (i_0,i_1,\\ldots ,i_{a-1})$ is a shortest path in graph $G$ between nodes $i_0$ and $i_{a-1}$ , such that $P_s \\subseteq A$ and $\\Gamma (P_s) \\subseteq A$ , it follows that (cf.", "[1]): $\\sum _{l=0}^{a-1} \\deg (i_l) \\le 3 |A|,$ and: $R^\\circ (i,j) < 6|A| + 2\\Delta .$ Since the total weight of edges and self-loops of $G$ incident to a vertex in $V$ is equal to 1, we have: $\\sum _{e \\in \\vec{E}(G^\\circ )} w(e) \\le \\sum _{v\\in A} \\left( \\sum _{u \\in \\Gamma (v) \\cup \\lbrace v\\rbrace } w(e_{vu}) \\right) + \\sum _{u \\in \\Gamma (j) \\cup \\lbrace j\\rbrace } w(e_{ju}) \\le |A| + 1.$ Claim $(i)$ follows from inequalities (REF ), (REF ), and (REF ).", "Claim $(ii)$ : Suppose that $s = \\sqrt{6t} \\ge \\frac{t}{n}$ , and let: $A = \\lbrace j\\in V : \\mathbb {E}_i N_j(t) > s\\rbrace .$ Since the considered walk hits nodes from $V$ a total of (at most) $t$ times, we have $|A| < \\frac{t}{s} \\le n$ , and the considerations performed in the proof of Lemma REF $(i)$ can be applied for the above-defined set $A$ .", "First, we bound the expected number of returns to node $i$ for a walk starting at $i$ before reaching $V\\setminus A$ for the first time: $\\mathbb {E}_i N_i (T_{V\\setminus A}) = 1 + (1-{\\mathrm {Pr}_i[T_{V\\setminus A}<T_i]})\\cdot \\mathbb {E}_i N_i (T_{V\\setminus A}) \\Rightarrow \\mathbb {E}_i N_i (T_{V\\setminus A}) = \\frac{1}{\\mathrm {Pr}_i[T_{V\\setminus A}<T_i]}.$ Taking into account [1] (Chapter 3, eq.", "(28) and Corollary 11) and bound (REF ), we have: $\\mathbb {E}_i N_i (T_{V\\setminus A}) = \\frac{1}{\\mathrm {Pr}_i[T_{V\\setminus A} < T_i]} = \\pi (i) \\cdot R(i,j)\\cdot \\!\\!\\sum _{e \\in \\vec{E}(G)} w(e) \\;\\le \\; \\pi (i) \\cdot R^\\circ (i,j)\\cdot \\!\\!\\sum _{e \\in \\vec{E}(G)} w(e) <$ $< \\frac{1}{n}\\cdot (6|A|+2\\Delta ) n = 6|A|+2\\Delta < 6 \\frac{t}{s} + 2\\Delta .$ It follows from Lemma REF that the definition of set $A$ may be rewritten as: $A = \\lbrace j\\in V : \\mathbb {E}_j N_i(t) > s\\rbrace $ Thus, $V\\setminus A = \\lbrace j\\in V : \\mathbb {E}_j N_i(t) \\le s\\rbrace $ , which means that if a walk starting from $i$ reaches $V\\setminus A$ , it will return to $i$ at most $s$ times in expectation before time $t$ .", "So, using (REF ), we obtain the claim: $\\mathbb {E}_i N_i(t) \\le \\mathbb {E}_i N_i(T_{V\\setminus A}) + s < 6\\frac{t}{s} + s + 2\\Delta = 2\\sqrt{6t} + 2\\Delta < 5\\sqrt{t} + 2\\Delta .$ $\\Box $ Fix an arbitrary arc $e_{ij}$ , with $\\lbrace i,j\\rbrace \\in E$ .", "We will bound the sought probability from the inequality: $\\mathbb {E}_{\\pi } N_{e_{ij}} (t) \\le \\mathrm {Pr}_{\\pi } [T_{e_{ij}} < t]\\ \\mathbb {E}_{e_{ij}} N_{e_{ij}} (t)\\Rightarrow \\mathrm {Pr}_{\\pi } [T_{e_{ij}} < t] \\ge \\frac{\\mathbb {E}_{\\pi } N_{e_{ij}} (t)}{\\mathbb {E}_{e_{ij}} N_{e_{ij}} (t)}.$ The expected number of traversals of $e_{ij}$ for a walk of even length starting from the stationary distribution on $V$ is given by equation (REF ).", "To bound the expectation from the denominator of (REF ), we note that by Lemma REF $(ii)$ , $\\mathbb {E}_i N_i (t) < 5 \\sqrt{t} + 2 \\Delta $ , and that arc $e_{ij}$ is chosen with probability $\\min \\left\\lbrace \\frac{1}{\\deg (i)}, \\frac{1}{\\deg (j)}\\right\\rbrace $ during each visit to $i$ : $\\mathbb {E}_i N_{e_{ij}} (t) < (5 \\sqrt{t} + 2\\Delta ) \\min \\left\\lbrace \\frac{1}{\\deg (i)}, \\frac{1}{\\deg (j)}\\right\\rbrace .$ Considering a walk starting from a traversal of arc $e_{ij}$ , we observe that after its traversal of $e_{ij}$ the walk must return to node $i$ before traversing $e_{ij}$ again: $\\mathbb {E}_{e_{ij}} N_{e_{ij}} (t) \\le 1 + \\mathbb {E}_{j} N_{e_{ij}} (t) < 1 + \\mathbb {E}_i N_{e_{ij}} (t) < 1 + (5 \\sqrt{t} + 2 \\Delta ) \\min \\left\\lbrace \\frac{1}{\\deg (i)}, \\frac{1}{\\deg (j)}\\right\\rbrace \\le $ $\\le (5 \\sqrt{t} + 3 \\Delta ) \\min \\left\\lbrace \\frac{1}{\\deg (i)}, \\frac{1}{\\deg (j)}\\right\\rbrace .$ By combining inequalities (REF ), (REF ), (REF ), and taking into account that $t > \\Delta ^2$ , we obtain the claim: $\\mathrm {Pr}_{\\pi } [T_{ e_{ij}} < t] > \\frac{t}{n (5\\sqrt{t} + 3 \\Delta )} > \\frac{t}{8n \\sqrt{t}} > 0.1 \\frac{\\sqrt{t}}{n}.$ $\\Box $ Pick a node $j \\in V$ according to the uniform probability distribution $\\pi $ .", "We will bound the sought probability from the inequality: $\\mathbb {E}_{i} N_{j} (t) \\le \\mathrm {Pr}_{i} [T_{j} < t]\\ \\mathbb {E}_{j} N_{j} (t)\\Rightarrow \\mathrm {Pr}_{i} [T_{j} < t] \\ge \\frac{\\mathbb {E}_{i} N_{j} (t)}{\\mathbb {E}_{j} N_{j} (t)}.$ Taking into account Lemma REF and condition (REF ), and noting that $j$ is chosen according to the uniform distribution $\\pi $ on $V$ , we have: $\\mathbb {E}_{i} N_{j} (t) = \\mathbb {E}_{j} N_{i} (t) = \\mathbb {E}_{\\pi } N_{i} (t) = \\frac{t}{n}.$ The expectation from the denominator of (REF ) is bounded by Lemma REF $(ii)$ , $\\mathbb {E}_{j} N_{j} (t) < 5\\sqrt{t} + 2\\Delta $ .", "By combining the above relations, and taking into account that $t > \\Delta ^2$ , we obtain: $\\mathrm {Pr}_{i} [T_{j} < t] > \\frac{t}{n ( 5\\sqrt{t} + 2 \\Delta )} > \\frac{t}{7n \\sqrt{t}} > 0.1 \\frac{\\sqrt{t}}{n}.$ $\\Box $ Fixing a connected component $H \\subseteq G$ with $n_H \\ge n_p/6$ , we introduce the following notation for a set of landmarks $L$ : let $L_H = L\\cap V(H)$ , let $X(L)$ denote the event that $|L_H| \\ge \\frac{1}{2} p \\frac{n_H}{n}$ , let $F_1(L)$ be the random variable over $L$ describing the maximum, over all non-loops arcs $e$ belonging to $H$ , of the probability that a set of $p$ random walks $RW(G_1)$ of length $\\tau = n_p^2$ each, with one random walk originating from each landmark from $L$ , does not cover arc $e$ .", "let $F_2(L)$ be the random variable over $L$ describing the maximum, over all nodes $u \\in V(H)$ , of the probability that a random walk $RW(G_1)$ of length $\\tau = n_p^2$ , originating from $u$ , does not hit any landmark of $L$ .", "Suppose that $L$ is a set of $p$ nodes picked according to the uniform distribution $\\pi ^p$ on $V^p$ .", "To prove the claim of the Lemma, we need to show the following bound: $\\mathrm {Pr}_{L\\sim \\pi ^p} [F_1 > n^{-1} \\wedge F_2 > n^{-1}] <\\frac{1}{2n}.$ We observe that each landmark from $L$ belongs to $V(H)$ with probability $n_H/n$ .", "Let $L_H = L\\cap V(H)$ .", "A w.h.p.", "lower bound on the size of $L_H$ follows from the Chernoff bound applied to $p$ Bernoulli trials with success probability $n_H/n$ : $\\mathrm {Pr}_{L\\sim \\pi ^p} [ X ] \\ \\ge 1 - e^{-\\frac{1}{8}p\\frac{n_H}{n}} \\ge 1 - e^{-\\frac{1}{48}p\\frac{n_p}{n}}\\ge 1 - e^{-\\frac{\\gamma }{48}\\log n} > 1 - \\frac{1}{4n},$ where we took into account that $n_p \\ge \\gamma \\frac{n}{p} \\log n$ , and that $\\gamma =60 > 48$ .", "In the following, we only need to show that, conditioned on the event $X(L)$ holding, $L$ is a good set of landmarks with probability $1 - \\frac{1}{4n}$ .", "Note that all the landmarks from $L_H$ are distributed uniformly at random on $V(H)$ , also when conditioned on $X(L)$ .", "To bound $F_1(L)$ , fix a non-loop arc $e$ of $H$ as the arc maximizing the failure probability in the definition of $F_1(L)$ .", "By applying Lemma REF to graph $H$ , the probability that a walk $RW(H_1)$ of length $\\tau =n_p^2$ , starting from the uniform distribution on $V(H)$ , does not cover arc $e$ , is at most $1 - \\frac{0.1 n_p}{n_H}$ .", "Thus, considering that: $|L_H| \\ge \\frac{1}{2} p \\frac{n_H}{n} = \\frac{n_H \\cdot 3 \\log n}{6 \\frac{n}{p}\\log n} \\ge \\frac{n_H \\cdot 3 \\log n}{0.1 n_p},$ the probability $F_{1,e}(L)$ that no random walk starting from a landmark hits arc $e$ is bounded by: $\\mathbb {E}_{L\\sim \\pi ^p} \\left[F_{1} \\ \\big |\\ X \\right] < \\left( 1 - \\frac{0.1 n_p}{n_H} \\right)^{\\frac{n_H}{0.1 n_p} 3 \\log n} < 2^{-3 \\log n} < n^{-3}.$ Likewise, to bound $F_2(L)$ , fix a node $u \\in V(H)$ maximizing the probability that a walk $RW(G_1)$ of length $\\tau = n_p^2$ , originating from $u$ , does not hit any landmark of $L$ .", "By Lemma REF , the probability that the considered walk of length $\\tau $ does not cover a node chosen according to the uniform distribution on $V(H)$ , is at most $1 - \\frac{0.1 \\sqrt{\\tau }}{n_H}$ .", "Thus, taking into account that $|L_H| > \\frac{n_H}{0.1 \\sqrt{\\tau }} 3 \\log n$ , the probability that the walk does not hit any landmark can once again be bounded as less than $n^{-3}$ : $\\mathbb {E}_{L\\sim \\pi ^p} \\left[F_2\\ \\big |\\ X \\right] < n^{-3}.$ It follows that: $\\mathbb {E}_{L\\sim \\pi ^p} \\left[F_1 + F_2\\ \\big |\\ X \\right] < 2n^{-3},$ and by the Markov bound: $\\mathrm {Pr}_{L\\sim \\pi ^p} [F_1 + F_2> n^{-1}\\ \\big |\\ X ] < \\frac{2}{n^2} < \\frac{1}{4n}.$ Now, inequalities (REF ) and (REF ) imply that inequality (REF ) holds, which completes the proof.", "$\\Box $ We begin by observing that the unbiased random walk on $G$ can be described as a weighted Metropolis-Hastings walk $RW(G_{f_c})$ , where, for all $u\\in V$ , the potential function on nodes is given as $f_c(u) = c\\deg (u)$ , where $c>0$ is an arbitrarily chosen constant of proportionality ($w(e) = c$ for all edges).", "Now, looking at the electrical networks analogy, by identifying with each other the corresponding nodes of the electrical networks describing the walks $RW(G_{f_c})$ and $RW(G_1)$ , and leaving the edges of both these networks in parallel connection, we obtain a new network on $G$ with edge weights $w_f$ given by: $w_f(e) = w_{f_c}(e) + w(e),$ corresponding to the potential function on nodes: $f(u) = f_c(u) + 1 = c\\deg (u) + 1.$ It follows that the resistance of replacement of the network of $RW(G_f)$ for any two nodes $u,v\\in V$ can be bounded as: $R_{G_f}(u,v) \\le R_{G_{f_c}}(u,v) \\quad \\text{and} \\quad R_{G_f}(u,v) \\le R_{G_1}(u,v).$ Moreover, the following relations hold between resistances and commute times: $Com_{G_{f_c}}(u,v) = 2cm R_{G_{f_c}}(u,v)$ $Com_{G_1}(u,v) = n R_{G_1}(u,v)$ $Com_{G_{f}}(u,v) = (2cm + n) R_{G_f}(u,v)$ Fixing $c = \\frac{1}{d} = \\frac{n}{2m}$ , i.e., $2cm = n$ , we obtain from all of the above relations: $Com_{G_f}(u,v) = O(\\min \\lbrace Com_{G_{f_c}}(u,v),Com_{G_1}(u,v)\\rbrace ).$ $\\Box $" ] ]

[ [ "Fermions as Global Correction: the QCD Case" ], [ "Abstract It is widely believed that the fermion determinant cannot be treated in global acceptance-rejection steps of gauge link configurations that differ in a large fraction of the links.", "However, for exact factorizations of the determinant that separate the ultraviolet from the infrared modes of the Dirac operator it is known that the latter show less variation under changes of the gauge field compared to the former.", "Using a factorization based on recursive domain decomposition allows for a hierarchical algorithm that starts with pure gauge updates of the links within the domains and ends after a number of filters with a global acceptance-rejection step.", "Ratios of determinants have to be treated stochastically and we construct techniques to reduce the noise.", "We find that the global acceptance rate is high on moderate lattice sizes and demonstrate the effectiveness of the hierarchical filter." ], [ "Introduction ", "The state of the art simulation algorithm for lattice QCD is the Hybrid Monte Carlo (HMC) [1], [2].", "As the continuum limit is approached, when the lattice spacing $a$ goes to zero, the simulation cost for a given observable scales typically as $a^{-(5+z)}$ .", "The dynamical critical exponent $z$ depends on the observable and is responsible for the critical slowing down of the simulations.", "Recently in [3] it was shown that $z(Q^2)=5$ for the topological charge $Q$ (the scaling might even be exponential in $1/a$ cf.", "[4]).", "This is a common problem for all present algorithms for gauge theories and the reason has been traced back to the fact that simulations on periodic lattices get stuck in topological sectors [5].", "In fact, on lattices with open boundary conditions $z(Q^2)=2$ is found in [6].", "Our original motivation was to look for an alternative algorithm which allows for larger steps in the space of gauge fields.", "In recent years new actions to simulate QCD on the lattice have been developed, in particular, based on smearing of the gauge links in the Dirac operator.", "In the case of Wilson fermions the stability of the HMC algorithm is influenced by the fluctuations of the smallest eigenvalues of the Wilson–Dirac operator [7].", "The results of [8] show evidence that smearing improves the stability.", "The HMC requires the computation of forces (i.e.", "derivatives of the Dirac operator with respect to the gauge links) and this can be very complicated or even impossible, like when HYP smearing [9] is used.", "A number of solutions exist, like using stout [10], nHYP [9] or HEX [11], [8] smearing or a differentiable approximation to the SU(3) projection for the smeared links [12], but flexibility in the choice of gauge and fermion actions is highly desirable and so the question arises, whether an alternative algorithm without force computations exists.", "In this article we study, motivated by a previous work in the Schwinger model [13], an algorithm based on global acceptance-rejection steps accounting for the fermion determinant in QCD with $N_{\\rm f}=2$ quark flavors.", "The basic idea is to make a gauge proposal which is accepted with a probability that depends on the ratio of fermion determinants on the “new” and “old” gauge configurations.", "Such an algorithm has already been used in QCD simulations with HYP-smeared link staggered fermions [14], [15], [16], with the fixed point action [17] and in [18].", "The problem with this type of algorithms is their scaling with the lattice volume $V$ .Unless otherwise specified, in this article we use lattice units (i.e.", "we set $a=1$ ) and $V$ is the number of lattice points.", "The cost of an exact determinant computation grows with $V^3$ and the acceptance to change a finite fraction of links decreases like $\\exp {(-V)}$ .", "In order to avoid the computation of exact determinants we use a stochastic estimation.", "This estimation can naively introduce a noise which grows like $\\exp {(V)}$ .", "In order to tackle these problems we construct a hierarchical filter of acceptance-rejection steps which successively filters the large fluctuations of the gauge proposal [19].", "Hierarchical acceptance-rejection steps based on approximations of the determinant with increasing accuracy were introduced and tested in [20].", "Here the filter relies on an exact factorization of the fermion determinant based on domain decomposition [21], which separates the short distance from the long distance scales of the lattice.", "A hierarchy of block acceptance-rejection steps was proposed in [22] but has never been tested.", "The article is organized as follows.", "In Section  we introduce the hierarchical filter of acceptance-rejection steps.", "Its construction based on domain decomposition is detailed in Section .", "The techniques we use for the stochastic estimation of determinant ratios are presented in Section .", "In particular we introduce an interpolation of the gauge fields which also allows to compute the exact (i.e.", "without stochastic noise) acceptance.", "Results for the latter and the effectiveness of the filter are shown in Section .", "Section  presents simulation results of $16^4$ and $32\\times 16^3$ lattices using a filter with three acceptance-rejection steps.", "A comparison with the HMC is made for observables like the plaquette or the topological charge.", "In the conclusions Section  we also discuss the scaling with the volume.", "Appendix contains the proof of detailed balance, Appendix describes the technique of relative gauge fixing used for the stochastic estimation and Appendix explains how the acceptance is enhanced by the use of additional parameters." ], [ "Hierarchy of acceptance steps ", "Let $P(s)$ be the desired distribution of the states $s$ of a system.", "Suppose a process that proposes a new state $s^{\\prime }$ with transition probability $T_0(s\\rightarrow s^{\\prime })$ and fulfills detailed balance with respect to $P_0(s)$ .", "A process with fixed point distribution $P(s)$ is then obtained by the combination of such a proposal with a subsequent Metropolis acceptance-rejection step [23] $\\begin{split}0) &\\quad \\text{Propose $s^{\\prime }$ according to $T_0(s\\rightarrow s^{\\prime })$}\\\\1) &\\quad P_{\\rm acc}(s\\rightarrow s^{\\prime }) = \\min \\left\\lbrace 1,\\frac{P_0(s)P(s^{\\prime })}{P(s)P_0(s^{\\prime })}\\right\\rbrace \\,.\\end{split}$ This hierarchy of a proposal step and an acceptance-rejection step can easily be generalized to an arbitrary number of acceptance-rejection steps.", "The result of the first acceptance-rejection step $1)$ is then interpreted as the proposal for a second acceptance-rejection step $2)$ and so on.", "If the target distribution $P(s)$ factorizes into $n+1$ parts $P(s)=P_0(s)\\,P_1(s)\\,P_2(s)\\dots P_{n}(s)\\,,$ the resulting hierarchical acceptance-rejection steps take the form $\\begin{split}0) &\\quad \\text{Propose $s^{\\prime }$ according to $T_0(s\\rightarrow s^{\\prime })$} \\\\1) &\\quad P_{\\rm acc}^{(1)}(s\\rightarrow s^{\\prime }) = \\min \\left\\lbrace 1,\\frac{P_1(s^{\\prime })}{P_1(s)}\\right\\rbrace \\\\2) &\\quad P_{\\rm acc}^{(2)}(s\\rightarrow s^{\\prime }) = \\min \\left\\lbrace 1,\\frac{P_2(s^{\\prime })}{P_2(s)}\\right\\rbrace \\\\...\\\\n) &\\quad P_{\\rm acc}^{(n)}(s\\rightarrow s^{\\prime }) = \\min \\left\\lbrace 1,\\frac{P_n(s^{\\prime })}{P_n(s)}\\right\\rbrace \\,.\\end{split}$ In the context of lattice QCD it is plausible to assume $P_i(s)\\propto \\exp (-S_i(s))$ with real actions $S_i$ and thus $\\frac{P_i(s^{\\prime })}{P_i(s)} = {\\rm e}^{-\\Delta _i(s,s^{\\prime })}\\,,$ where $\\Delta _i(s,s^{\\prime })=S_i(s^{\\prime })-S_i(s)$ .", "The average acceptance rate in step $i)$ is defined by $\\left\\langle P_{\\rm acc}^{(i)} \\right\\rangle _{s,s^{\\prime }} = \\sum _s P(s)\\sum _{s^{\\prime }} P_0(s^{\\prime })\\,P_1(s^{\\prime })\\dots P_{i-1}(s^{\\prime })\\,\\min \\left\\lbrace 1,{\\rm e}^{-\\Delta _i(s,s^{\\prime })}\\right\\rbrace \\,.$ It can be computed assuming a Gaussian distribution for $\\Delta _i(s,s^{\\prime })$ with variance $\\Sigma _i^2$ and the result is [13] (see also [24]) $\\left\\langle P_{\\rm acc}^{(i)} \\right\\rangle _{s,s^{\\prime }} = {\\rm erfc}\\left(\\sqrt{\\Sigma _i^2/8}\\right)\\,.$ The acceptance rates might be enhanced by parameterizing and tuning the factorization (REF ), see Appendix .", "Our goal is to simulate QCD with $N_{\\rm f}=2$ mass-degenerate fermions.", "After integration over the Grassmann fermion fields the states $s$ are defined by the gauge field $U$ and the target probability distribution is $P(U) = \\frac{\\left|\\det (D(U))\\right|^2{\\rm e}^{-S_g(U)}}{Z} \\,, $ where $S_g$ is the gauge action, $D$ is the lattice Dirac operator and $Z$ is the partition function $Z = \\int D[U] |\\det D(U)|^2 {\\rm e}^{-S_g(U)} \\,.$ The integration measure is $D[U] = \\prod _{x,\\mu } dU(x,\\mu )$ , where $dU(x,\\mu )$ is the SU(3) Haar measure for the link $U(x,\\mu )$ .", "A simple two-step algorithm would consist of some update of the gauge link configuration $U\\rightarrow U^\\prime $ , which fulfills detailed balance with respect to $P_0(U)\\propto \\exp (-S_g(U))$ , followed by an acceptance-rejection step with the fermion determinant ratio $P_{\\rm acc}^{(1)}(U\\rightarrow U^\\prime ) = \\min \\left\\lbrace 1,\\det \\frac{D(U^\\prime )^\\dagger D(U^\\prime )}{D(U)^\\dagger D(U)}\\right\\rbrace \\,.", "$ The proof of detailed balance can be found in Section REF .", "If the proposal changes only one link and the Dirac operator $D$ is ultra-local it is easy to show that the acceptance-rejection step requires only few inversionsFor example, in the case of the Wilson–Dirac operator 12 inversions are needed.. An ergodic algorithm is then obtained by sweeps through the lattice.", "Thus the cost of such an algorithm would scale with the lattice volume $V$ at least like $V^2$ [25] and it requires O($V$ ) inversions per sweep.", "If, on the other hand, a finite fraction $\\propto V$ of the links is updated for the proposal, the acceptance rate decreases exponentially with the volume.", "In order to see this we write the distribution $P_1$ as $P_1(U)\\propto \\exp (\\ln (\\det \\, D^\\dagger D))$ .", "The action difference $\\Delta _1(U,U^\\prime )=\\ln (\\det \\, D(U^\\prime )^\\dagger D(U^\\prime )) - \\ln (\\det \\, D^\\dagger (U) D(U))$ can be written as $\\Delta _1=-\\sum _i\\ln (\\lambda _i)$ in terms of the eigenvalues $\\lambda $ of the operator $M^\\dagger M$ with $M = D(U^\\prime )^{-1}D(U) \\,.", "$ If we assume a Gaussian distributionWe verified numerically that this assumption is valid to a good approximation.", "(after averaging over the gauge ensemble $U$ and the proposals $U^\\prime $ ) for the logarithms of the eigenvalues $\\hat{\\lambda }_i=\\ln (\\lambda _i)$ with mean zero and variance $\\sigma ^2_{\\hat{\\lambda }}$ , we can approximate $\\Sigma _1^2 \\approx \\bar{N}_1\\,\\sigma ^2_{\\hat{\\lambda }}/2 \\,, $ where $\\bar{N}_1$ is the number of eigenvalues $\\lambda \\ne 1$ .", "Typically $\\bar{N}_1\\propto V$ and this implies that $\\Sigma _1^2$ is proportional to the volume $V$ .", "The complementary error function in the formula for the acceptance (REF ) has the asymptotic expansion ${\\rm erfc}(x)\\sim \\exp (-x^2)/(x\\sqrt{\\pi })(1-1/(2x^2)+\\cdots )$ for $|x|\\gg 1$ which shows the exponential decrease with the volume.", "From the preceding discussion it is obvious that such two-step algorithms will not be efficient for large lattices.", "Indeed numerical experiments show that for lattices larger than $\\sim (0.2\\;\\mathrm {fm})^4$ (where all links are updated) the acceptance rate quickly becomes less than a percent.", "However, in the context of low mode reweighting the fluctuations of the determinant of $D_{\\text{low}}^\\dagger D_{\\text{low}}$ , where $D_{\\text{low}}$ is a restriction of $D$ to its low modes, are found to depend only mildly on the volume [26].", "The explanation for this observation might be the fact that the width of the distribution of the small eigenvalues of $\\sqrt{D^\\dagger D}$ decrease like $1/V$ [26] (the fluctuations of the eigenvalue gap go instead like $1/\\sqrt{V}$ [7]).", "Thus, given a factorization of the determinant that separates low (infrared IR) and high (ultraviolet UV) modes $\\det (D) = \\det (D_{\\text{UV}}) \\cdots \\det (D_{\\text{IR}})\\,,$ a hierarchy of acceptance steps can be constructed, where the large fluctuations of the UV modes go through a set of filters (acceptance-rejection steps) which are more and more dominated by the IR modes: Table: NO_CAPTIONThis hierarchy of modes may induce also a hierarchy of costs since it is the low modes that cause the most cost in lattice QCD.", "Furthermore the factorization should be exact and the terms simple to compute.", "Factorizations that realize these conditions are already used to speed-up the HMC algorithm, i.e., in the context of mass-preconditioning [27] and domain decomposition [21].", "Only the latter also allows for a decoupling of local updates and will be discussed in the following." ], [ "Domain decomposition ", "Domain decomposition was introduced in lattice QCD in [22] and in [21] the resulting factorization of the fermion determinant was used to separate short distance and long distance physics in the HMC algorithm.", "For definiteness we consider here the Wilson–Dirac operator $D(U)$ [28], which may include the clover term needed for O($a$ ) improvement [29], [30].", "But our algorithm is applicable to a more general class of Dirac operators, see below.", "Figure: Block decomposition of a 2-dimensional lattice.", "The blocks arecoloured like a checker board.", "Picture taken from .Suppose a decomposition $\\mathcal {C}$ of the lattice in non-overlapping blocks $b\\in \\mathcal {C}$ (cf. Fig.", "REF for a 2-dimensional visualization).", "The lattice sites are labeled such that the sites belonging to the first black block come first, then the second black block and after the last black block the first white block and so on.", "The Dirac operator can then be written as $D = \\left(\\begin{array}{cc} D_{\\rm bb} & D_{\\rm bw} \\\\D_{\\rm wb} & D_{\\rm ww} \\end{array}\\right)\\,,$ where $D_{\\rm bb}$ ($D_{\\rm ww}$ ) is a block-diagonal matrix with the black (white) block Dirac operators $D_b$ on the diagonal.", "The block Dirac operators $D_b$ fulfill Dirichlet boundary conditions and therefore are dominated by short distance physics (if the blocks are small enough).", "The matrices $D_{\\rm bw}$ and $D_{\\rm wb}$ contain the block interaction terms.", "The form (REF ) induces a factorization of the determinant $\\det (D) = \\prod _{b \\in \\mathcal {C}}\\det (D_b)\\det (\\hat{D})\\,, \\quad \\hat{D} = 1- D_{\\rm bb}^{-1}D_{\\rm bw}D_{\\rm ww}^{-1}D_{\\rm wb}\\,,$ where $\\hat{D}$ is the Schur complement of the decomposition (REF ) and contains block interactions, i.e.", "the long distance physics.", "A natural separation scale is given by the inverse block size $1/L_b$ .", "In the context of the domain decompositioned HMC the average force associated with the Schur complement is an order of magnitude smaller than the force associated with the block Dirac operators [21].", "This indicates that the fluctuations of the determinant of the Schur complement are smaller than that of the block determinants.", "Furthermore the factorization (REF ) can be iterated using a recursive domain decomposition $\\det (D_b) = \\prod _{b^{\\prime } \\in \\mathcal {C}_b}\\det (D_{b^{\\prime }})\\det (\\hat{D}_b)\\,.$ We note that the Schur complement $\\hat{D}_b$ fulfills Dirichlet boundary conditions.", "We have implemented the recursive domain decomposition in the freely available software package DD-HMC by M. Lüscher [31].", "In the case of one level of recursion the hierarchy of acceptance-rejection steps is given by $\\begin{split}1) &\\quad P_{\\rm acc}^{(1)} = \\min \\left\\lbrace 1,\\det \\frac{D_{b^{\\prime }}(U^\\prime )^\\dagger D_{b^{\\prime }}(U^\\prime )}{D_{b^{\\prime }}(U)^\\dagger D_{b^{\\prime }}(U)}\\right\\rbrace \\,,\\quad \\forall b, \\forall b^{\\prime }\\in \\mathcal {C}_b \\\\2) &\\quad P_{\\rm acc}^{(2)} = \\min \\left\\lbrace 1,\\det \\frac{\\hat{D}_b(U^\\prime )^\\dagger \\hat{D}_b(U^\\prime )}{\\hat{D}_b(U)^\\dagger \\hat{D}_b(U)}\\right\\rbrace \\,,\\quad \\forall b \\in \\mathcal {C} \\\\3) &\\quad P_{\\rm acc}^{(3)} = \\min \\left\\lbrace 1,\\det \\frac{\\hat{D}(U^\\prime )^\\dagger \\hat{D}(U^\\prime )}{\\hat{D}(U)^\\dagger \\hat{D}(U)}\\right\\rbrace \\,.\\end{split}$ At the beginning the set of links to be updated, the so called active links, is chosen such that the acceptance-rejection steps for the smallest blocks, $b^{\\prime }$ , at stage $1)$ in Eq.", "(REF ) decouple and can therefore be processed in parallel.", "In the case of Wilson fermions with or without clover term the active links are the links that have at most one endpoint on the boundary of a block (white points in Fig.", "REF ).", "In this case the block acceptance steps also decouple if the links in the Wilson–Dirac operator (but not in the clover term) are replaced by one level of HYP smearing [32].", "After the last and global acceptance-rejection step the gauge field is translated by a random vector, see Appendix C of [21].", "If the smallest blocks, $b^{\\prime }$ , at stage $1)$ in Eq.", "(REF ) consist of no more than $\\sim 6^4$ lattice points, the determinant ratios can be efficiently computed exactly by LU-decomposition [33].", "If the smallest blocks are larger, we compute their determinants by a factorization like in Eq.", "(REF ).", "The Schur complements at the stages $2)$ and $3)$ in Eq.", "(REF ) are usually too large for their determinant ratios to be computed exactly and have to be treated stochasticallyThe same applies to Schur complements arising from a factorization of the smallest blocks, if that is needed..", "The stochastic estimation of determinant ratios is the topic of the next section.", "Following this discussion we give to our algorithm the name of Partially Stochastic Multi-Step (PSMS) algorithm." ], [ "Stochastic techniques for determinant ratios ", "Since the numerical cost for the computation of exact determinants grows with the cube of the size of the matrix, determinants of Dirac operators for lattices larger than $6^4$ have to be estimated stochastically.", "In particular for our problem we have to estimate ratios of determinants of Schur complements, which arise from a domain decomposition and appear in the acceptance-rejection steps of Eq.", "(REF ).", "In Appendix we show that such stochastic acceptance-rejection steps fulfill detailed balance.", "In this section we describe in detail the techniques we use to reduce the associated stochastic noise.", "In Section REF we discuss the stochastic noise introduced when the determinant ratio in Eq.", "(REF ) (for generic Dirac operators $D$ ) is evaluated stochastically.", "The stochastic noise depends on the spectrum of generalized eigenvalues of the operators forming the ratio [13].", "In order to reduce it we apply techniques described in Section REF and Section REF .", "In Section REF we discuss a relative gauge fixing of the gauge field $U$ and $U^\\prime $ .", "This gauge fixing is applied for the construction of a gauge field interpolation, a new method which we present in Section REF .", "The gauge fields are linearly interpolated and this induces a factorization in terms of ratios of operators which can be made arbitrarily close as the number of interpolation steps increases.", "In particular, there exists the limit in which the exact ratio is obtained.", "In Section REF the properties of the Schur complement are reviewed.", "In this particular case the noise vector can be restricted to a subspace of the boundary points of the blocks.", "In Section REF we support the introduction of these techniques by numerical results." ], [ "Stochastic estimation of determinant ratios ", "We replace the determinants of ratios of Dirac operators in Eq.", "(REF ) by stochastic estimators $\\min \\left\\lbrace 1,\\det (M^\\dagger M)^{-1}\\right\\rbrace \\longrightarrow \\min \\left\\lbrace 1,{\\rm e}^{-|M\\eta |^2+|\\eta |^2}\\right\\rbrace \\,,$ where the ratio operator $M$ is defined in Eq.", "(REF ).", "In Eq.", "(REF ) $\\eta $ is a complex Gaussian noise vector that is updated before each acceptance-rejection step and $|\\eta |^2$ is its norm squared, see Section REF .", "The average over $\\eta $ of a function $f(\\eta )$ is defined by $\\left\\langle f(\\eta ) \\right\\rangle _\\eta = \\int D[\\eta ]\\,{\\rm e}^{-|\\eta |^2}f(\\eta ) \\,.", "$ The measure $D[\\eta ]$ is normalized such that $\\int D[\\eta ]\\,\\exp (-|\\eta |^2)=1$ .", "The algorithm satisfies detailed balance (the proof is given in Section REF ) and yields an acceptance rate that is bounded from above by the exact acceptance in Eq.", "(REF ) [13].", "There are other possible choices for the distribution of $\\eta $ than a Gaussian distribution.", "But because of the central limit theorem these other choices are equivalent to the Gaussian distribution in the large volume limit.", "The stochastic noise introduced in the acceptance-rejection step by Eq.", "(REF ) has the effect of replacing in Eq.", "(REF ) $\\Sigma ^2 \\longrightarrow \\sigma ^2 = \\Sigma ^2+\\left(\\sigma ^{\\rm stoch}\\right)^2\\,, $ where $\\left(\\sigma ^{\\rm stoch}\\right)^2 =\\left\\langle \\Delta ^2 \\right\\rangle _{U,U^\\prime ,\\eta } - (\\left\\langle \\Delta \\right\\rangle _{U,U^\\prime ,\\eta })^2 $ with $\\Delta = |M\\eta |^2 - |\\eta |^2$ .", "The average $\\left\\langle \\cdot \\right\\rangle _{U,U^\\prime ,\\eta }$ is taken over the gauge ensemble $U$ , the proposals $U^\\prime $ and the noise vectors $\\eta $ .", "For given $U$ and $U^\\prime $ Eq.", "(REF ) can be computed by performing the integrations over $\\eta $ in the basis of orthonormal eigenvectors of $M^\\dagger M$ with eigenvaluesThe eigenvalues $\\lambda $ of $M^\\dagger M$ are equivalent to the generalized eigenvalues of the problem $D(U)D(U)^\\dagger \\chi =\\lambda D(U^\\prime )D(U^\\prime )^\\dagger \\chi $ .", "$\\lambda _k$ , cf.", "[13].", "The result is $\\left(\\sigma ^{\\rm stoch}\\right)^2 = \\left\\langle \\sum _k (\\lambda _k-1)^2 \\right\\rangle _{U,U^\\prime } \\,.", "$ The eigenvalues $\\lambda =1$ do not contribute to the variance.", "If we denote by $h_1$ the full width at half maximum (FWHM) of the distribution of the eigenvalues $\\lambda _k$ and by $\\bar{N}_1$ the number of eigenvalues which are not one, we can approximate $\\left(\\sigma ^{\\rm stoch}\\right)^2 \\approx \\bar{N}_1\\,h_1^2 \\,.$ It becomes clear that the smaller $\\bar{N}_1$ and $h_1$ are, the larger the stochastic acceptance will be.", "Furthermore in [34] it was noted that the spectrum of $M^\\dagger M$ has to fulfill the condition $\\lambda >0.5$ , because otherwise the variance of the quantity under the minimum function in Eq.", "(REF ) is not defined." ], [ "Relative gauge fixing ", "In [13] (see also [17]) it was noticed that relative gauge fixing of the configuration $U$ and $U^\\prime $ reduces the stochastic noise in Eq.", "(REF ).", "Under a gauge transformation $g(x)\\in \\,{\\rm SU(3)}$ , the gauge links transform as $U(x,\\mu )\\rightarrow U^g(x,\\mu )=g(x)U(x,\\mu )g(x+\\hat{\\mu })^{-1}$ and the Dirac operator as $D(U^g)_{xy} = g(x) D(U)_{xy} g(y)^{-1} \\qquad \\mbox{(no sum over $x$ and $y$)}\\,,$ where we suppress the spin indices.", "Further we define a scalar product of two gauge fields as $(U,U^\\prime ) = \\frac{1}{12V}\\sum _{x,\\mu }{\\rm Re}\\,{\\rm Tr}\\,\\left\\lbrace 1-U(x,\\mu )^\\dagger U^\\prime (x,\\mu )\\right\\rbrace \\,.", "$ Relative gauge fixing is defined through the minimization $&& \\min _{g_1,g_2}\\, (U^{g_1},{U^\\prime }^{g_2}) =\\min _{g_1,g_2} \\frac{1}{12V}\\sum _{x,\\mu }{\\rm Re}\\,{\\rm Tr}\\,\\Big \\lbrace \\nonumber \\\\&& 1-g_1(x+\\hat{\\mu })U(x,\\mu )^\\dagger g_1(x)^{-1}g_2(x)U^\\prime (x,\\mu )g_2(x+\\hat{\\mu })^{-1}\\Big \\rbrace \\,.", "$ We determine $g_1$ and $g_2$ before the acceptance-rejection step Eq.", "(REF ), where we use $M = D({U^\\prime }^{g_2})^{-1}D(U^{g_1}) \\,.", "$ Relative gauge fixing does not change the exact acceptance rates in Eq.", "(REF ) but in general improves the stochastic acceptance rate in Eq.", "(REF ).", "In order to show detailed balance in the latter case, consider the reverse transition $U^\\prime \\rightarrow U$ , for which the minimization is $\\min _{\\tilde{g}_1,\\tilde{g}_2}\\, ({U^\\prime }^{\\tilde{g}_1},U^{\\tilde{g}_2})$ .", "As one can immediately see by taking the complex conjugate of Eq.", "(REF ) the result is given by $\\tilde{g}_1=g_2$ and $\\tilde{g}_2=g_1$ .", "This implies for the reverse transition $M\\rightarrow \\tilde{M} = D(U^{\\tilde{g}_2})^{-1}D({U^\\prime }^{\\tilde{g}_1}) = M^{-1} \\,,$ which is precisely the property needed to prove detailed balance [13].", "In the above procedure, the choice of $g_1$ and $g_2$ is not unique.", "In fact one can transform $g_1\\rightarrow g_1h$ and $g_2\\rightarrow g_2h$ by some other gauge transformation $h(x)$ and the minimization condition Eq.", "(REF ) is unchanged.", "Instead we chooseWe thank Ulli Wolff for suggesting this choice.", "$g_1 = g_2^{-1} = g \\,.", "$ The numerical procedure for the minimization Eq.", "(REF ) using Eq.", "(REF ) is described in Appendix .", "In the proposal $U\\rightarrow U^\\prime $ we only change active links in the blocks and we restrict the gauge transformations $g$ in Eq.", "(REF ) to the black points in Fig.", "REF .", "One reason for this is that the critical slowing down of such a local (i.e.", "restricted to the blocks) minimization is reduced compared to a global minimization over the entire lattice." ], [ "Gauge field interpolation ", "In order to ensure $\\lambda >0.5$ and bring the spectrum of $M^\\dagger M$ closer to one, one could employ the method of determinant breakup introduced in [20], [34].", "It uses the factorization $\\det (M^\\dagger M)= [\\det ((M^\\dagger M)^{1/N})]^N$ and in the stochastic acceptance-rejection step Eq.", "(REF ) each factor is then replaced by a stochastic estimator with an independent noise vector.", "The effect on the spectrum of $M^\\dagger M$ is to replace $\\lambda \\rightarrow \\lambda ^{1/N}$ .", "The gauge field interpolation which we propose in this article has a similar effect but avoids the computation of $1/N$ th roots of $M^\\dagger M$ .", "We introduce a sequence of intermediate fields $U_i,\\;i=0,\\ldots ,N$ which starts from the gauge field $U_0=U^{g}$ and ends with the gauge field $U_N={U^\\prime }^{g^{-1}}$ .", "$g$ is the gauge transformation in Eq.", "(REF ).", "The determinant of $M^\\dagger M$ can be factorized like $\\det (M^\\dagger M) = \\prod _{i=0}^{N-1} \\det (M_i^\\dagger M_i)\\,,$ where $M_i = D(U_{i+1})^{-1} D(U_i) \\,.$ The stochastic acceptance-rejection step in Eq.", "(REF ) is done by drawing one independent Gaussian noise vector $\\xi _i$ for each factor $\\min \\left\\lbrace 1,{\\rm e}^{\\sum _{i=0}^{N-1}-|M_i\\xi _i|^2+|\\xi _i|^2}\\right\\rbrace \\,.$ The cost is then one inversion for each factor.", "In order for the algorithm to fulfill detailed balance the intermediate gauge configurations have to be the same when doing the reverse change $U^\\prime \\rightarrow U$ .", "The proof of detailed is given in Section REF .", "The simplest way to construct such an interpolation is $U_i(x,\\mu ) = \\frac{N-i}{N} U^{g}(x,\\mu ) + \\frac{i}{N} {U^\\prime }^{g^{-1}}(x,\\mu ) \\,, \\quad i=0,1,\\cdots ,N-1 \\,,$ which interpolates linearly between $U_0=U^{g}$ and $U_N={U^\\prime }^{g^{-1}}$ .", "The interpolation has no physical meaning, only numerical efficiency counts.", "The intermediate fields are not SU(3) matrices, in the Dirac operator we use $U_i^{\\dagger }$ (and not $U_i^{-1}$ ) in order to preserve the $\\gamma _5$ Hermiticity of the Wilson–Dirac operator.", "Since $||U_i- U_{i+1}||\\propto 1/N$, $\\forall \\, i<N$ , we expect the eigenvalues $\\lambda _k^{(i)}$ of $M_i^\\dagger M_i$ to be $\\lambda _k^{(i)}=1+{\\rm O}(h_1/N)$ and so the FWHM of their distributionIn the case of the full Dirac operator, we find numerically that the smallest (largest) eigenvalue change with $N$ as $\\lambda ^{(i)}_{\\rm min}\\sim \\exp \\lbrace -b/N\\rbrace $ ($\\lambda ^{(i)}_{\\rm max}\\sim \\exp \\lbrace b^{\\prime }/N\\rbrace $ ) for positive constants $b$ ($b^{\\prime }$ ).", "This is the same behavior one obtains using the determinant breakup in $1/N$ th roots.", "can be approximated by $h_N\\approx h_1/N$ in terms of the FWHM $h_1$ of the eigenvalue distribution of $M^\\dagger M$ .", "The stochastic noise in the acceptance-rejection step is reduced to $\\left(\\sigma _N^{\\rm stoch}\\right)^2 \\approx N\\,\\bar{N}_1\\,h_N^2 \\approx \\bar{N}_1\\,\\frac{h_1^2}{N} $ as compared to Eq.", "(REF ).", "An important feature of this method is the limit $N\\rightarrow \\infty $, for which $\\sigma _N^{\\rm stoch}\\rightarrow 0$ and we recover the exact acceptance, cf.", "Eq.", "(REF )." ], [ "Schur Complement ", "The Schur complement in Eq.", "(REF ) is $\\hat{D}=1-Q$ with $Q=D_{\\rm bb}^{-1}D_{\\rm bw}D_{\\rm ww}^{-1}D_{\\rm wb}$ .", "Let us denote by $P$ the orthonormal projector to the space of the white points in the black blocks in Fig.", "REF .", "For the points which have only one nearest neighbor on a different block, $P$ projects to only two of the four spin components.", "The explicit definition of $P$ can be found in Appendix B of [21].", "It does not depend on the gauge field and it satisfies the properties $D_{\\rm wb}P=D_{\\rm wb}$ and $P^2=P$ which imply $\\det (1-Q) = \\det (1-PQ) \\,.", "$ This means that one can use $1-PQ$ instead of $\\hat{D}$ in Eq.", "(REF ) and therefore the noise $\\eta $ is defined only on the space invariant under $P$ .", "We also need to apply the inverse of the operator $1-PQ$ which is [21] $(1-PQ)^{-1} = 1-PD^{-1}D_{\\rm wb} \\,.$ Here $D_{\\rm wb}$ is meant to act on the total space of points (by padding with zeros).", "For a global lattice of sizes $L_\\mu $ in directions $\\mu =0,1,2,3$ and a domain decomposition into blocks of sizes $l_\\mu $ , the dimension of the space invariant under $P$ is ${\\rm dim}(P) =6\\prod _{\\mu =0}^3\\frac{L_\\mu }{l_\\mu }\\left(\\sum _{\\nu =0}^3\\frac{l_0l_1l_2l_3}{l_\\nu }-4\\sum _{\\nu =0}^3(l_\\nu -1) \\right) \\,.$ For the number $\\bar{N}_1$ in Eq.", "(REF ) we have $\\bar{N}_1\\le {\\rm dim}(P)$ .", "On a lattice with the same number of points $L$ in all directions, if we choose $l_\\mu =L/2$ (16 blocks) then ${\\rm dim}(P)\\approx 48L^3$ , to be compared to $V=12L^4$ if we were to consider the full Dirac operator.", "The reduction of $\\bar{N}_1$ alone turns out not to be sufficient to make stochastic acceptance-rejection steps like in Eq.", "(REF ), with the Schur complement ratio, efficient.", "Moreover the relative gauge fixing described in Section REF does not directly help in reducing the stochastic noise in this case.", "The reason is that the restriction of the gauge transformations to the black points in Fig.", "REF leaves the Schur complement invariant.", "This is why the gauge field interpolation is necessary to further reduce the noise.", "As we show in the next section relative gauge fixing has an impact on the interpolation." ], [ "Numerical results ", "The interpolated fields $U_i$ in Eq.", "(REF ) change if we apply first a relative gauge fixing of $U$ and $U^\\prime $ , which minimizes their distance in the sense of Eq.", "(REF ).", "In Fig.", "REF we show the behavior of the plaquette of the interpolated fields $U_i$ .", "In the computation of the plaquette, if $U$ denotes a link then the link in reversed direction is defined by $U^{\\dagger }$ (and not by $U^{-1}$ ).", "Without relative gauge fixing the intermediate configurations look like if they were thermalized configurations of a smaller $\\beta $ .", "The links become rougher.", "This is understandable if one imagines that the gauge configurations $U$ and $U^\\prime $ lay somewhere randomly in the configuration space.", "So the path will not go over configurations which are similar to the “thermalized” ones.", "With relative gauge fixing, the path of the interpolated links yields plaquette values which are approximately constant, cf.", "Fig.", "REF which also shows the two-sigma band of a thermalized ensemble.", "In Fig.", "REF we show the spectra of the Schur complement ratios $M_i^\\dagger M_i$ in Eq.", "(REF ).", "Since relative gauge fixing is applied to all links the spectrum is narrower and the requirement $\\lambda >0.5$ can be fulfilled for a relatively low value of interpolation steps $N$ .", "As expected from the behavior of the plaquette the width of the spectrum does not change significantly along the interpolation.", "Figure: The spectrum of the Schur complement ratioM i † M i M_i^\\dagger M_i defined in Eq.", "() is shown as a function of ii.The start and end fields U 0 U_0 and U N U_N (N=12N=12) are quenched4 4 4^4 lattices where all links are changed (and relative gauge fixed).The plain Wilson–Dirac operator with mass am 0 =0.56am_0=0.56 is used and the Schurcomplement is defined for a domain decomposition in 2 4 2^4 blocks.There are many possible ways to define alternative interpolations replacing Eq.", "(REF ).", "For example we could normalize the links by substituting $U_i(x,\\mu )\\rightarrow U_i(x,\\mu )\\det (U_i(x,\\mu ))^{-1/3}$ .", "It turns out that in this case the intermediate configurations look like if they were thermalized configurations at larger $\\beta $ .", "As a consequence the spectrum can develop negative eigenvalues for small quark masses.", "We note that a mass-shift towards larger masses can be generated by multiplying the links with a common factor $\\exp (\\alpha )$ , $\\alpha <0$ , in the Dirac operator.", "Effectively such a factor (albeit with a different value for each link) can be easily incorporated into Eq.", "(REF ) by multiplying the links $U_i(x,\\mu )$ with an appropriate power of their determinant $\\det (U_i(x,\\mu ))$ .", "If we do not normalize the links, a “mass shift” towards larger masses is automatically realized because $\\det (U_i(x,\\mu ))<1$ .", "But there is some room for improving the efficiency of the method.", "In the following we will use the simple interpolation given in Eq.", "(REF ).", "Finally we discuss what happens if the relative gauge fixing is extended to the entire lattice and is not restricted to the points inside the blocks.", "Links which are unchanged after the pure gauge update would change through a global minimization.", "This could introduce additional noise and indeed this is the case for the full Dirac operator but not for the Schur complement.", "The global minimization slightly improves the behavior of the interpolated fields in Eq.", "(REF ) but this effect is not large and the danger to run into negative eigenvalues as discussed above increases." ], [ "Volume dependence of the exact acceptance rate ", "We simulate QCD with $N_{\\rm f}=2$ flavors of mass-degenerate quarks.", "The action for the gauge field is the Wilson plaquette gauge action [28] $S_g=\\beta S_{\\rm w}(U) = \\frac{\\beta }{6}\\sum _p{\\rm Re}\\,{\\rm Tr}\\,\\lbrace 1-U(p)\\rbrace \\,, $ where $p$ runs over all oriented plaquettes (i.e., each plaquette is counted with two orientations).", "For the fermions we use the plain Wilson–Dirac operator [28] (without clover term and without smearing) with bare quark mass $m_0$ , whose action on a quark field $\\psi $ is given by $&&(D_{\\rm w}(U)+m_0)\\psi (x) = (4+m_0)\\psi (x) - \\nonumber \\\\&&\\sum _{\\mu =0}^3\\frac{1}{2}\\lbrace U(x,\\mu )(1-\\gamma _\\mu )\\psi (x+\\hat{\\mu }) +U(x-\\hat{\\mu },\\mu )^{\\dagger }(1+\\gamma _\\mu )\\psi (x-\\hat{\\mu })\\rbrace \\,.", "$ The hopping parameter is defined as $\\kappa =1/(2m_0+8)$ .", "In this section we simulate at parameters $\\beta =5.6$ and $\\kappa =0.15825$ .", "Theses values corresponds to a lattice spacing $a=0.0717(15)\\,{\\rm fm}$ [35] and a pseudoscalar mass $m_{\\rm PS}\\approx 404\\,{\\rm MeV}$ [36] (determined on a larger $32\\times 24^3$ lattice).", "Table: Optimal parameters for the 4-step PSMS algorithm (representative set) forplain Wilson fermions at β=5.6\\beta =5.6 and κ=0.15825\\kappa =0.15825.We implement a 4-step PSMS algorithm based on a domain decomposition with block size $4^4$ and on a hierarchy of three acceptance-rejection steps.", "Our code is based on the freely available software package DD-HMC by M. Lüscher [31].", "In order to enhance the acceptance rates we introduce parameters as explained in Appendix .", "In the first step we update the active links in the $4^4$ blocks, which amount to a fraction of about 9.4% of all links.", "The gauge proposal consists of 500 iterations of two Cabibbo-Marinari heat-bath [37] sweeps (with reversed sequence of gauge link updates and random choice of SU(2) subgroups) at the shifted coupling $\\beta _0^{(0)}=5.6918$ .", "The gauge proposal is then subjected to a first acceptance-rejection step containing a plaquette action $S_{\\rm w}^{\\rm HYP}$ like in Eq.", "(REF ) but where the plaquettes are constructed from HYP smeared links with the parameters of [32] (one level of smearing).", "We do one iteration of this step with 95% acceptance.", "The resulting proposal goes into a second acceptance-rejection step containing the action $S_b=\\sum _{b \\in \\mathcal {C}}2\\ln (\\det (D_b))$ of the block determinants (one iteration with 76% acceptance).", "We emphasize that the first and second acceptance-rejection (or filter) steps are done block-wise and can be therefore parallelized.", "Finally the gauge proposal which passed through the first two filter steps enters the global acceptance-rejection step with the Schur complement of the $4^4$ block decomposition.", "This is a stochastic acceptance-rejection step performed according to Eq.", "(REF ) using the interpolation with intermediate fields $U_i,\\;i=0,\\ldots ,N$ in Eq.", "(REF ).", "The optimal parameters can be tuned following the prescription given in Section REF .", "We note that they depend only mildly on the global lattice volume and a representative set is listed in Table REF .", "Figure: The variance of the stochastic estimator in the global stepfrom simulations of plain Wilson fermions at β=5.6\\beta =5.6, κ=0.15825\\kappa =0.15825with the 4-step PSMS algorithm.The left plot shows the exact variance Σ 3 2 \\Sigma ^2_3 (black circles)as a function of the lattice volume VV together with a linear fit (red line).The blue diamond is the result using a 5-step PSMS algorithm, see text.The right plot shows the volume dependence of the slope s(V)s(V) definedin Eq.", "() together with a linear fit.The global acceptance-rejection probability is $P_{\\rm acc}^{(3)} = \\min \\left\\lbrace 1,\\exp (-\\Delta _3)\\right\\rbrace $ , where (cf.", "Eq.", "(REF )) $\\Delta _3 = \\beta _3^{(0)}\\Delta S_{\\rm w}+\\beta _3^{(1)}\\Delta S_{\\rm w}^{\\rm HYP}+\\beta _3^{(2)}\\sum _{b \\in \\mathcal {C}}2\\Delta \\ln (\\det (D_b))+\\sum _{i=0}^{N-1} \\eta _i^\\dagger (M_i^\\dagger M_i-1) \\eta _i$ and $M_i$ is the ratio of Schur complements.", "On lattices with $V=8^4$ up to $V=16^4$ we measure, for different values of the gauge field interpolation steps $N$ in Eq.", "(REF ), the variance $\\sigma _3^2(V,N)=\\left\\langle (\\Delta _3-\\left\\langle \\Delta _3 \\right\\rangle )^2 \\right\\rangle _{U,U^\\prime ,\\eta } \\,.$ At fixed volume $V$ we extrapolate linearly in $1/N$ to zero, thus obtaining an estimate for the exact variance $\\Sigma ^2_3(V)$ as a function of the volume.", "The justification for this extrapolation is given by Eq.", "(REF ), which in this case means $\\sigma _3^2(V,N)=\\Sigma ^2_3(V)+\\left(\\sigma _{3,N}^{\\rm stoch}(V)\\right)^2 $ and by Eq.", "(REF ), which implies $\\left(\\sigma _{3,N}^{\\rm stoch}(V)\\right)^2\\approx \\frac{1}{N}\\,s(V) \\,,$ with the slope $s(V)$ is approximately given by $\\bar{N}_1\\,h_1^2$ .", "Here $\\bar{N}_1$ and $h_1$ refer to the Schur complement ratio.", "Note that $\\sigma _3^2(V,N)$ contains also contributions from parts of the action other than the Schur complement (cf.", "Eq.", "(REF )) but which do not depend on $N$ .", "The extrapolated exact variance $\\Sigma ^2_3(V)$ is shown in the left plot of Fig.", "REF as a function of $V$ .", "The data can be very well fitted by a straight line constrained to zero at zero volume (red line).", "The slopes $s(V)$ of the linear fits of $\\sigma _3^2(V,N)$ in $1/N$ are plotted against the volume $V$ in the right plot of Fig.", "REF .", "The data of the slope can be also well fitted by a straight line constrained to zero at zero volume (red line).", "Assuming that $\\bar{N}_1$ is equal to the dimension of the projector $P$ in Eq.", "(REF ) and taking into account that the block size is here constant and equal to $4^4$ , we deduce that $\\bar{N}_1\\propto V$ .", "Therefore our results for the slope means that the FWHM $h_1$ of the generalized eigenvalues of the Schur complements does not significantly depend on the volume.", "Figure: The exact global acceptance is plotted as a functionof the variance Σ 2 \\Sigma ^2 forsimulations of plain Wilson fermions at β=5.6\\beta =5.6, κ=0.15825\\kappa =0.15825with the 4-step PSMS algorithm (black circles).The point corresponding to the blue diamond is from a simulation of a16 4 16^4 lattice with a 5-step PSMS algorithm.Via (REF ) the exact acceptance rate can be determinedWe tested the (tacitly assumed) validity of the Gaussian model for finite values of $N$ .", "from the variance $\\Sigma ^2_3(V)$ .", "The exact acceptance rates as determined from the variances are plotted in Fig.", "REF (black circles) together with the result from the fit to $\\Sigma ^2_3(V)$ shown in the left plot of Fig.", "REF .", "The 4-step PSMS algorithm of this section shows a good acceptance for lattices up to $16^3\\times 8$ .", "This is the region where the error function can be approximated by a Taylor expansion with a linear leading term ${\\rm erfc}(x)=1-2x/\\sqrt{\\pi }+{\\rm O}(x^3)$ .", "Fig.", "REF shows that the acceptance rates, which one would obtain from the Schur complement alone (blue diamonds), are much smaller.", "The efficiency of the hierarchy of filters in enhancing the acceptance of the global step can be demonstrated by simulating the largest $16^4$ lattice using a 5-step PSMS algorithm.", "For this we use a recursive domain decomposition of the $16^4$ lattice in $8^4$ and $4^4$ blocks, cf.", "Eq.", "(REF ).", "The additional filter with respect to the 4-step PSMS algorithm is a stochastic acceptance-rejection step accounting for the Schur complements of the $4^4$ blocks within the $8^4$ blocks, cf. Eq.", "(REF ).", "The acceptance of the global step (accounting for the global Schur complement) is increased by this further filter step, cf.", "the blue diamond in Fig.", "REF .", "Using recursive domain decomposition to keep the largest block size at $L/2$ (where $V=L^4$ ), the volume dependence of $\\Sigma ^2$ in the global step is $V^q$ (dotted line in the left plot of Fig.", "REF ) with $q\\approx 0.9$ (determined on our available lattices $8^4$ and $16^4$ ).", "At large $V$ one expects the asymptotic behavior $q=3/4$ , cf.", "Section REF ." ], [ "Numerical tests of the algorithm ", "We present results of simulations of $N_{\\rm f}=2$ flavors of mass-degenerate plain Wilson fermions on a $16^4$ lattice at $\\beta =5.8$ and $\\kappa =0.15462$ .", "The clover coefficient is set to zero and the fermions have anti-periodic boundary conditions in time direction.", "The lattice spacing is estimated in [35] to be $0.0521(7)\\,{\\rm fm}$ and the pseudoscalar mass is $381\\,{\\rm MeV}$ [36] (determined on a larger $64\\times 32^3$ lattice).", "Table: Parameters for the 4-step PSMS algorithm for plain Wilson fermionsat β=5.8\\beta =5.8 and κ=0.15462\\kappa =0.15462.In the simulations in Section  our smallest blocks are $4^4$ and the gauge proposal changes the active links in these blocks.", "It turns out that larger blocks are better in terms of changing the topological charge and allow for higher global acceptances (at somewhat higher computational cost).", "That is why we change our setup in this section and use $8^4$ blocks as our smallest ones.", "The gauge proposal changes the active links in a $6^4$ hypercube inside each of the $8^4$ blocks, which amounts to updating $7.9$ % of all gauge links.", "We adopt a 4-step PSMS algorithm whose parameters and acceptances are summarized in Table REF .", "For each of the steps $i=0,1,2,3$ , $n_i$ is the number of iterations per step and $N_i$ is the number of gauge field interpolation steps (for stochastic estimates of Schur complement ratios).", "The gauge proposal consists of a number $n_0$ of iterations of symmetrized sweeps of Cabibbo-Marinari heat-bath [37] and over-relaxation [38], [39] updates.", "One iteration consists of one heat-bath (HB) sweep and $L/2$ over-relaxation (OR) sweeps followed by the reversed sequence of link updates (so in total one HB + $L/2$ OR + $L/2$ OR + one HB sweeps), where for each link we choose with probability $1/2$ one sequence of SU(2) subgroups and with probability $1/2$ the reversed sequence.", "The first acceptance-rejection step is a Metropolis step for a HYP plaquette gauge action which has to be subtracted in the successive filter steps.", "The determinant of the $8^4$ blocks is factorized by a domain decomposition in $4^4$ blocks.", "The second acceptance-rejection step accounts for the exact product of the $4^4$ block determinants times the determinant of the Schur complement of the decomposition of the $8^4$ blocks in $4^4$ blocks.", "The latter is treated stochastically.", "These two acceptance-rejection steps are performed independently for each $8^4$ block.", "The third stochastic acceptance-rejection step contains the global Schur complement of the decomposition of the $16^4$ lattice in $8^4$ blocks.", "Figure: Histogram distribution of the plaquette values fromsimulations of plain Wilson fermions andWilson plaquette action at β=5.8\\beta =5.8, κ=0.154620\\kappa =0.154620, 16 4 16^4 lattices.We compare results for the 4-step PSMS algorithm and for the HMC.In Fig.", "REF we show the histogram distribution of the plaquette value.", "We compare the results from 4 replica simulated using the 4-step PSMS algorithm (red bins) with the results from a long HMC simulation (white bins).", "The HMC simulation is done with the DD-HMC algorithm [21], [40] using $8^4$ blocks.", "The distributions agree perfectly.", "Figure: Histories of the topological charge QQ (upper plots) andhistograms of the deviations of the replicum means of Q 2 Q^2 from the totalmean divided by the replicum errors (lower plots).The left plots show the results of 4 replica simulated with the 4-step PSMSalgorithm.", "The right plots show the HMC results from 3 replica.In the upper two plots of Fig.", "REF the histories of the topological charge are shown.", "The topological charge is defined by $Q = \\frac{1}{16\\pi ^2}\\sum _{x,\\mu ,\\nu }F_{\\mu \\nu }(x)\\tilde{F}_{\\mu \\nu }(x)\\,,$ using a discretization of the field strength tensor $F_{\\mu \\nu }$ (see e.g.", "[30]) in which gauge links constructed from three levels of HYP smearing are used.", "We consider 4 replica simulated using the 4-step PSMS algorithm (left plot) and 3 replica simulated with the HMC (right plot).", "The horizontal dotted lines are determined from an ad hoc fit to the histogram of the topological charge using 3 Gaussian functions (one centered at zero and the other at values $\\pm m$ corresponding to the dotted lines).", "In order to compare the Monte Carlo histories of the two algorithms, we take the Monte Carlo units which correspond to a full change of the gauge configuration.", "To this end, on the x-axis of the history plots we take, for the PSMS algorithm, the number of global acceptance steps multiplied by the fraction $R$ of links changed and by the global acceptance while, for the HMC algorithm, we take the number of trajectories multiplied by the ratio $R$ of active links and by the acceptance.", "The right plot shows that the long HMC replicum was not able to really tunnel to a topological sector different than zero, while such a tunneling occurred at least once for all PSMS replica.", "Indeed we compared the distributions of the topological charge squared $Q^2$ for the PSMS replica and the long HMC replicum and found that they agree well around $Q^2=0$ but differ at larger values.", "Therefore we started two more HMC replica from configurations with topological charge different than zero (generated in the PSMS ensembles), which are also shown in Fig.", "REF .", "In one of these two additional replica we observe a clear tunneling from topological sector zero to nonzero.", "In the lower two plots of Fig.", "REF we show histograms of the deviations of the replicum means of $Q^2$ from the total mean divided by the replicum errors (the quantity in Eq.", "(30) of [41]; left plot, PSMS; right plot, HMC).", "The goodness of the replica distribution is measured by the probability (goodness-of-fit) of a constant fit to the replicum means.", "The goodness is 0.7 for the PSMS algorithm and 0.05 for the HMC.", "A value much below 0.1 is very unlikely.", "The expecation value $\\left\\langle Q^2 \\right\\rangle $ is $0.37(15)$ for the 4-step PSMS algorithm and $0.281(81)$ for the HMC algorithm.", "The errors are determined using the method of [41].", "From leading order chiral perturbation theory we expect $\\left\\langle Q^2 \\right\\rangle \\approx 0.19$ .", "We emphasize that Fig.", "REF is a comparison made at one lattice spacing only.", "The main problem is the scaling with the lattice spacing which we cannot address in the scope of this paper.", "Figure: The variance σ 3 2 (V,N)\\sigma _3^2(V,N) (left plot) in the globalstep for simulations of plain Wilson fermions andWilson plaquette action at β=5.8\\beta =5.8, κ=0.154620\\kappa =0.154620.Data for V=16 4 V=16^4 (circles) and V=32×16 3 V=32\\times 16^3 (diamonds) are very wellfitted as functions of 1/N1/N using a global linear fit (red lines).", "In theright plot we show the resulting acceptances.In Fig.", "REF we plot the variance $\\sigma _3^2(V,N)$ (left plot) and the acceptance (according to Eq.", "(REF ), right plot) in the global acceptance-rejection step (see Eq.", "(REF )) as a function of $1/N$ and $N$ respectively.", "Together with the data for the $16^4$ lattices we present data for $32\\times 16^3$ lattices.", "Motivated by the results of Section  we perform a global fit to the variances of the form $\\sigma _3^2(V,N) = V\\,\\left(a_1+\\frac{a_2}{N}\\right)$ with fit parameters $a_1$ and $a_2$ .", "The exact variance turns out to be $\\Sigma ^2_3=0.50(2)$ and $\\Sigma ^2_3=1.00(4)$ for the $16^4$ and $32\\times 16^3$ lattices respectively.", "This corresponds to acceptances $0.724(5)$ and $0.617(7)$ .", "A cost comparison of the simulations of the $16^4$ lattice can be performed by comparing the number of full inversions of the Wilson–Dirac operator needed to update all the links.", "Using the 4-step PSMS algorithm at optimal parameters, the global acceptance-rejection step has $N=96$ gauge field interpolation steps (each of which requires one inversion of the full operator for the inversion of the global Schur complement, see Eq.", "(REF )) and 64% acceptance.", "This means that we need $\\approx 23$ global steps or 2200 inversions to get a new gauge configuration.", "If we instead run the 4-step PSMS algorithm with $N=24$ and 42% global acceptance, one new gauge configuration is obtained after $\\approx 35$ global steps or 840 inversions.", "The DD-HMC needs only 120 inversions for one new gauge configuration.", "This naive cost comparison does not take into account effects of autocorrelation times, which are hard to estimate for observables like the topological charge." ], [ "Conclusions ", "We have developed and tested the PSMS algorithm for lattice QCD that consists of a hierarchical filter of acceptance-rejection steps.", "The hierarchy is based on an exact factorization of the fermion determinant.", "Although other factorization are possible, we here deploy (recursive) domain decomposition as it separates the determinant in a local (blocks) and global part (Schur complement).", "We were able to determine the exact global acceptance rates for volumes up to $(1.2\\; {\\rm fm})^4$ and demonstrate that the filter is successful in fighting the exponential decrease with the volume.", "The global acceptance-rejection step with the Schur complement remains expensive.", "We estimate a factor of ten in comparison with the HMC for the setup of Section .", "The expected scaling of the cost of the algorithm with the volume is $V\\;\\mbox{(inversion)}\\times V^{3/4}\\;(N)\\times 1/(\\mbox{acceptance})\\,.$ The first factor is due to the cost of one inversion of the Dirac operator and the second factor arises from the necessity to keep the stochastic noise low.", "A constant global acceptance is achieved for constant variance $\\Sigma ^2$ of the action differences that go into the global step, i.e., $\\sigma ^2_{\\hat{\\lambda }}\\propto 1/V$ is needed (cf.", "Eq.", "(REF )).", "Instead we find $\\sigma ^2_{\\hat{\\lambda }}\\sim {\\rm const}$ as $V$ is increased (cf.", "Fig.", "REF ).", "Previously the fluctuations of the small eigenvalues of $\\sqrt{D^\\dagger D}$ have been found to decrease like $1/V$ [26].", "We do not seem to see this behavior for $\\sigma ^2_{\\hat{\\lambda }}$ .", "The reason might be that our separation scale, given by the inverse block size $1/L_{\\rm b}$ , is too large.", "For the simulations at $\\beta =5.8$ ($\\beta =5.6$ ) with a block size of 8 this scale is approximately $500\\,{\\rm MeV}$ ($360\\,{\\rm MeV}$ ).", "At the moment the performance of the PSMS algorithm is worse than the one of the HMC algorithm, but the scaling of autocorrelation times of the topological charge with the lattice spacing has to be studied to make a definite conclusion.", "In Fig.", "REF we present evidence that the PSMS algorithm is more efficient in sampling the topological sectors compared to the HMC.", "It is still relevant to study alternatives to the HMC and there are prospects of using and improving the PSMS algorithm.", "One possibility is to apply reweighting for the Schur complement, cf.", "[42] where we demonstrate that reweighting factors for the Schur complement have a better scaling with the volume compared to the full operator.", "Improved gauge actions can be included in the hierarchy of acceptance steps and there is room for better choices of the gauge updates within the blocks.", "Also factorizations of the determinant other than domain-decomposition could be used.", "The techniques for the stochastic estimation of determinant ratios, which we introduced in this article for the acceptance-rejection steps, can be equally well applied to the case of reweighting, e.g., in the quark mass [42] or to account for electromagnetic effects.", "Acknowledgement.", "We thank Tony Kennedy for correspondence on the proof of detailed balance, Martin Lüscher for a clarification on the fluctuations of small eigenvalues of the Dirac operator, Rainer Sommer for discussions and Ulli Wolff for comments on the relative gauge fixing.", "The Monte Carlo simulations were carried out on the cluster Stromboli at the University of Wuppertal and we thank the University." ], [ "Exact acceptance-rejection steps ", "The simplest setup of our algorithm is to split up the gauge weight in Eq.", "(REF ) from the fermionic one.", "The idea is to propose a new gauge configuration $U^\\prime $ by a pure gauge updating algorithm and accept or reject it by a Metropolis step accounting for the fermionic weight.", "Let $T_0(U\\rightarrow U^\\prime )$ be the transition probability for the pure gauge proposal which has to satisfy detailed balance for the distribution (see below) $P_0(U) = \\frac{\\exp (-S_g(U))}{Z_0} \\,, $ where $Z_0$ is the partition function for the gauge action $S_g$ .", "The Metropolis acceptance-rejection step [23] consists of accepting or rejecting the proposal $U^\\prime $ with probability $P_{\\rm acc}(U,U^\\prime ) =\\min \\left\\lbrace 1,\\frac{P_0(U)P(U^\\prime )}{P(U)P_0(U^\\prime )} \\right\\rbrace \\,.$ The transition probability for this algorithm is $&&T(U\\rightarrow U^\\prime ) = T_0(U\\rightarrow U^\\prime )P_{\\rm acc}(U,U^\\prime )\\nonumber \\\\& & + \\delta (U-U^\\prime )\\left(1-\\int D[U^{\\prime \\prime }]\\, T_0(U\\rightarrow U^{\\prime \\prime })P_{\\rm acc}(U,U^{\\prime \\prime })\\right) \\,.", "$ In order for $T$ to satisfy detailed balance for the distribution $P$ in Eq.", "(REF ), $T_0$ has to satisfy detailed balance for the distribution $P_0$ in Eq.", "(REF ).", "If the gauge proposal is a sequence of gauge link updates, their order has to be symmetrized or chosen randomly [20]." ], [ "Stochastic acceptance-rejection steps ", "The exact calculation of the determinant ratio in Eq.", "(REF ) is numerically prohibitive.", "It can be replaced by a stochastic approximation that maintains detailed balance exactly [13].", "We follow closely Appendix A, in particular section A.5, of [22].", "The variables of the system (the gauge field) are enlarged by adding auxiliary stochastic variables, which are called pseudofermions and are only used in the stochastic acceptance-rejection step.", "The equilibrium probability distribution for the enlarged system of gauge field $U$ and pseudofermion $\\eta $ is $\\hat{P}(\\eta ,U) = \\frac{{\\rm e}^{-|D(U)^{-1}\\eta |^2}\\exp (-S_g(U))}{Z} \\,.$ The pseudofermion is a complex-valued field $\\eta $ with the measure $D[\\eta ] = \\prod _{x,\\alpha }\\frac{d{\\rm Re}\\,(\\eta _{x,\\alpha })d{\\rm Im}\\,(\\eta _{x,\\alpha })}{\\pi } \\,, $ where the index $\\alpha $ contains spin and color degrees of freedom.", "The norm squared of $\\eta $ is defined by the scalar product $(\\eta ,\\eta )$ : $|\\eta |^2 = (\\eta ,\\eta ) = \\sum _{x,\\alpha }\\eta _{x,\\alpha }^*\\eta _{x,\\alpha } \\,.$ The equilibrium distribution of the gauge field alone is recovered by integrating over the pseudofermion: $P(U) = \\int D[\\eta ]\\, \\hat{P}(\\eta ,U) \\,.", "$ We also define the conditional probability $\\hat{P}(\\eta |U) = \\frac{\\hat{P}(\\eta ,U)}{P(U)} =\\frac{{\\rm e}^{-|D(U)^{-1}\\eta |^2}}{|\\det D(U)|^2} $ to generate the pseudofermion field $\\eta $ given the gauge field $U$ .", "The algorithm to update the enlarged system consists of alternating two Markov steps.", "The first is a global heatbath step for updating the pseudofermion at given gauge field $U$ .", "A new pseudofermion $\\eta $ distributed according to $\\hat{P}(\\eta |U)$ in Eq.", "(REF ) is generated through $\\eta = D(U) \\xi \\,, $ where $\\xi $ is a Gaussian random pseudofermion generated with probability distribution $p(\\xi )=\\exp (-|\\xi |^2)$ The pseudofermion measure in Eq.", "(REF ) is normalized such that $\\int D[\\xi ] p(\\xi )=1$ ..", "The second step is a Metropolis step for the gauge field at given pseudofermion.", "A new gauge field $U^\\prime $ is proposed with transition probability $T_0(U\\rightarrow U^\\prime )$ , which satisfies detailed balance for the distribution $P_0(U)$ in Eq.", "(REF ).", "The proposal is followed by an acceptance-rejection step with probability $\\min \\left\\lbrace 1,\\frac{P_0(U)\\hat{P}(\\eta ,U^\\prime )}{\\hat{P}(\\eta ,U)P_0(U^\\prime )} \\right\\rbrace =\\min \\left\\lbrace 1,\\frac{{\\rm e}^{-|D(U^\\prime )|^2\\eta }}{{\\rm e}^{-|D(U)|^2\\eta }}\\right\\rbrace \\,.", "$ Both the heatbath and Metropolis steps separately fulfill detailed balance with respect to the combined probability distribution $\\hat{P}(\\eta ,U)$ in Eq.", "(REF ) [20].", "Therefore also their composition has the correct fixed point probability [43].", "We consider now a composite update step consisting of an heatbath update for the pseudofermion in Eq.", "(REF ) immediately followed by a Metropolis step for the gauge field in Eq.", "(REF ).", "If after this we forget the pseudofermion field, this can be viewed as an update for the gauge field alone with acceptance probabilityWe thank Tony Kennedy for clarifying this point in a correspondence.", "$P_{\\rm acc}(U,U^\\prime ) & = & \\int D[\\eta ]\\, \\hat{P}(\\eta |U) \\min \\left\\lbrace 1 ,\\frac{P_0(U) \\hat{P}(\\eta ,U^\\prime )}{\\hat{P}(\\eta ,U) P_0(U^\\prime )} \\right\\rbrace \\nonumber \\\\& = & \\int D[\\xi ]\\,{\\rm e}^{-|\\xi |^2}\\min \\left\\lbrace 1,{\\rm e}^{-|M\\xi |^2+|\\xi |^2}\\right\\rbrace \\,, $ where the ratio operator $M$ is defined in Eq.", "(REF ).", "The associated transition probability in Eq.", "(REF ), where now $P_{\\rm acc}(U,U^\\prime )$ is given by Eq.", "(REF ), satisfies detailed balance for the equilibrium probability $P(U)$ due to the property [22] $[P(U)/P_0(U)]\\,P_{\\rm acc}(U,U^\\prime ) = [P(U^\\prime )/P_0(U^\\prime )]\\,P_{\\rm acc}(U^\\prime ,U)\\,, $ or equivalently [13] $\\frac{P_{\\rm acc}(U,U^\\prime )}{P_{\\rm acc}(U^\\prime ,U)}=|\\det (M)|^{-2} \\,.$ In practice, the acceptance step Eq.", "(REF ) is done by drawing one Gaussian distributed pseudofermion $\\xi $ and accepting or rejecting depending on the argument under the $\\min $ function.", "We note that it is not possible to perform the average of the argument under the $\\min $ function over many pseudofermions, as this violates detailed balance.", "The acceptance probability in Eq.", "(REF ) was computed in [13] $P_{\\rm acc}(U,U^\\prime ) =\\sum _{i} \\; \\min (1,1/\\lambda _i)\\prod _{j\\ne i} \\; \\frac{\\lambda _i-1}{\\lambda _i-\\lambda _j}$ in terms of the eigenvalues $\\lambda _i$ of $M^\\dagger M$ .", "It is bounded by the exact (non-stochastic) acceptance probability in Eq.", "(REF ) [13] $P_{\\rm acc}(U,U^\\prime ) \\le \\min \\left\\lbrace 1,|\\det (M)|^{-2}\\right\\rbrace \\,.$ So far we discussed the case of a proposal followed by an acceptance-rejection steps.", "Eq.", "(REF ) can be generalized to an arbitrary number of acceptance-rejection steps as discussed in Section .", "The algorithm satisfies detailed balance and this is also true if (some of) the Metropolis acceptance-rejections steps are replaced by their stochastic counterpart Eq.", "(REF )." ], [ "Gauge field interpolation ", "In order to simplify a bit the notation we consider an algorithm like it is described in Section REF with one stochastic acceptance-rejection step.", "In practice we apply the gauge field interpolation method to acceptance-rejection steps involving the Schur complements (the global Schur complement $\\hat{D}$ as well as the Schur complement in the blocks $\\hat{D}_b$ when we use recursive domain decomposition, see Eq.", "(REF )).", "For the gauge proposal $U\\rightarrow U^\\prime $ we consider the gauge field interpolation $U_i$ as it is given in Eq.", "(REF ).", "For each of the transitions $U_i\\rightarrow U_{i+1}$ , $i=0,1,\\cdots ,N-1$ we introduce a pseudofermion field $\\eta _i$ .", "The equilibrium probability distribution for the enlarged system is $\\hat{P}(\\lbrace \\eta _j\\rbrace ,U,U^\\prime )=\\frac{{\\rm e}^{-|D(U^{g})^{-1}\\eta _0|^2} {\\rm e}^{-S_g(U)}}{Z}\\prod _{i=1}^{N-1} \\frac{{\\rm e}^{-|D(U_i)^{-1}\\eta _i|^2}}{|\\det (D(U_i))|^2}$ and depends now also on the proposed configuration $U^\\prime $ .", "Integrating over the pseudofermions gives $P(U) = \\int \\prod _{i=0}^{N-1} D[\\eta _i] \\, \\hat{P}(\\lbrace \\eta _j\\rbrace ,U,U^\\prime ) \\,.$ The conditional probability to generate the pseudofermions $\\lbrace \\eta _j\\rbrace $ given the proposal $U\\rightarrow U^\\prime $ is $\\hat{P}(\\lbrace \\eta _j\\rbrace |U,U^\\prime ) = \\frac{\\hat{P}(\\lbrace \\eta _j\\rbrace ,U,U^\\prime )}{P(U)} =\\prod _{i=0}^{N-1}\\frac{{\\rm e}^{-|D(U_i)^{-1}\\eta _{i}|^2}}{|\\det (D(U_i))|^2} \\,.$ We use the property $\\det (D(U))=\\det (D(U^{g}))$ .", "If we consider the reversed gauge proposal $U^\\prime \\rightarrow U$ (i.e.", "$U_0={U^\\prime }^{g^{-1}}$ and $U_N=U^{g}$ ), the intermediate configurations $U_i$ in Eq.", "(REF ) are the same but they are traversed in reversed order and therefore the pseudofermion $\\eta _i$ is associated with the transition $U_{i+1}\\rightarrow U_i$ .", "The probability distribution for the enlarged system is now $\\hat{P}(\\lbrace \\eta _j\\rbrace ,U^\\prime ,U)=\\frac{{\\rm e}^{-|D({U^\\prime }^{g^{-1}})^{-1}\\eta _{N-1}|^2} {\\rm e}^{-S_g(U^\\prime )}}{Z}\\prod _{i=0}^{N-2} \\frac{{\\rm e}^{-|D(U_{i+1})^{-1}\\eta _i|^2}}{|\\det (D(U_{i+1}))|^2}$ and the conditional probability to generate the pseudofermions $\\lbrace \\eta _j\\rbrace $ is $\\hat{P}(\\lbrace \\eta _j\\rbrace |U^\\prime ,U) = \\frac{\\hat{P}(\\lbrace \\eta _j\\rbrace ,U^\\prime ,U)}{P(U^\\prime )} =\\prod _{i=0}^{N-1}\\frac{{\\rm e}^{-|D(U_{i+1})^{-1}\\eta _{i}|^2}}{|\\det (D(U_{i+1}))|^2} \\,.$ The acceptance probability for the gauge proposal $U\\rightarrow U^\\prime $ is $P_{\\rm acc}(U,U^\\prime )&=&\\int \\prod _{i=0}^{N-1} D[\\eta _i] \\, \\hat{P}(\\lbrace \\eta _j\\rbrace |U,U^\\prime )\\min \\left\\lbrace 1,\\frac{P_0(U) \\hat{P}(\\lbrace \\eta _j\\rbrace ,U^\\prime ,U)}{\\hat{P}(\\lbrace \\eta _j\\rbrace ,U,U^\\prime ) P_0(U^\\prime )}\\right\\rbrace \\nonumber \\\\&=& \\int \\prod _{i=0}^{N-1} D[\\xi _i] \\, {\\rm e}^{-|\\xi _i|^2}\\min \\left\\lbrace 1,{\\rm e}^{\\sum _{j=0}^{N-1}-|M_j\\xi _j|^2+|\\xi _j|^2}\\right\\rbrace \\,,$ where $M_i = D(U_{i+1})^{-1} D(U_i) \\,.$ $P_{\\rm acc}(U,U^\\prime )$ in Eq.", "(REF ) fulfills the detailed balance condition Eq.", "(REF ) or equivalently $\\frac{P_{\\rm acc}(U,U^\\prime )}{P_{\\rm acc}(U^\\prime ,U)} = |\\det (M)|^{-2} \\,,\\quad M=D(U^\\prime )^{-1}D(U)\\,.$ In practice, the global acceptance step Eq.", "(REF ) is done by drawing $N$ Gaussian distributed pseudofermions $\\xi _i$ and accepting or rejecting depending on the argument of the $\\min $ function (i.e.", "we evaluate the sum in the exponent under the $\\min $ function)." ], [ "Relative gauge fixing ", "The relative gauge fixing of two gauge field configurations $U$ and $U^\\prime $ is done using a steepest descent scheme introduced by [44], [45] for gauge group SU(3).", "Using the condition Eq.", "(REF ) for the gauge transformation $g$ , the minimization condition in Eq.", "(REF ) can be written similarly to the case of the Landau gauge condition.", "Then one can apply the procedure of [45] to fix the relative gauge.", "At each point $x$ where the gauge transformation $g$ is defined we have to solve the condition $\\min _{g(x)} {\\rm Re}\\,{\\rm Tr}\\,\\left\\lbrace 1-(g(x)^\\dagger )^2\\cdot (W_{\\rm f}(x) + W_{\\rm b}(x) \\right\\rbrace \\,,$ where $W_{\\rm f}(x) & = & \\sum _{\\mu }U^\\prime (x,\\mu )g(x+\\hat{\\mu })^2U^\\dagger (x,\\mu ) \\,,\\\\W_{\\rm b}(x) & = & \\sum _{\\mu }{U^\\prime }^\\dagger (x-\\hat{\\mu },\\mu )g(x-\\hat{\\mu })^2U(x-\\hat{\\mu },\\mu ) \\,.$ Using the steepest descent method of [45] we get the minimizing transformation field $g(x)$ iteratively through $g(x)= \\exp \\left\\lbrace -\\frac{\\alpha }{2}\\left[\\Delta - \\Delta ^\\dagger -\\frac{1}{3} {\\rm Tr}\\,(\\Delta - \\Delta ^\\dagger )\\right]\\right\\rbrace $ with a scaling parameter $\\alpha $ and $\\Delta = W_{\\rm f}(x) + W_{\\rm b}(x) \\,.$ The minimum is reached when $\\theta (x)=0$ , where $\\theta (x) = {\\rm Tr}\\,\\left[\\Delta - \\Delta ^\\dagger -\\frac{1}{3} {\\rm Tr}\\,(\\Delta - \\Delta ^\\dagger ) \\right]^2\\,.", "$ We choose the value $\\alpha =0.15$ .", "If there is no convergence we reduce it in steps of $-0.01$ and reach values down to $\\alpha =0.10$ .", "For the SU(3) exponential function in Eq.", "(REF ) we use the matrix function described in Appendix A of [21].", "The numerical cost of the relative gauge fixing can be reduced in the case of a domain decomposition by defining $g(x)$ only inside the blocks where the active links are changed.", "Further it can be reduced by stopping the iteration when $\\theta (x)<10^{-3}$ , which we find good enough for the purpose of the gauge field interpolation discussed in Section REF ." ], [ "Parametrized acceptance-rejection steps ", "The general idea of the PSMS algorithm is to factorize the distribution in Eq.", "(REF ) in several pieces and introduce a recursive update procedure with a computational cost ordering.", "Naively speaking a gauge configuration is proposed by a pure gauge algorithm and the fermion determinant is treated in acceptance-rejection steps.", "It is easy to see that the plaquette gauge action and the determinant of the Dirac operator are strongly correlated.", "This correlation can be used to increase the acceptance [24], [14], [13] and also in the case of reweighting [46].", "This is an example of ultraviolet filtering [20], [47], [48]." ], [ "Optimization of the acceptance ", "We consider the hierarchy of acceptance-rejection steps $i=1,2,\\ldots ,n$ in Eq.", "(REF ).", "The factorization (REF ) is not unique and can be parametrized.", "In each acceptance-rejection step the action involved might be written as $S_i = \\sum _{j=0}^i\\beta _i^{(j)}S^{(j)} \\,,\\quad i=1,\\ldots ,n \\,, $ with real coefficients $\\beta _i^{(j)}$ and actions $S^{(j)}$ .", "The probability to accept the proposal for a new gauge field $U^\\prime $ starting from $U$ is $\\min \\left\\lbrace 1,\\exp (-\\Delta _i)\\right\\rbrace $ where $\\Delta _i = S_i(U^\\prime )-S_i(U) = \\sum _{j=0}^i\\beta _i^{(j)}\\Delta S^{(j)}\\,.$ For Gaussian distributed $\\Delta _i$ the acceptance rate is given by ${\\rm erfc}\\left(\\sqrt{\\Sigma _i^2/8}\\right)$ with $\\Sigma _i^2 = \\left\\langle (\\Delta _i-\\left\\langle \\Delta _i \\right\\rangle )^2 \\right\\rangle \\,.", "$ In the case of the factorization Eq.", "(REF ), $n=2$ and $S^{(0)}$ is the gauge action, $S^{(1)} = \\sum _{b \\in \\mathcal {C}}2\\ln (\\det (D_b))$ and $S^{(2)}=2\\ln (\\det (\\hat{D}))$ are the effective actions of the determinants of the blocks and of the Schur complement respectively.", "In stochastic acceptance-rejection steps, like we do for the Schur complement, we use $\\Delta S^{(2)} = \\eta ^\\dagger (M^\\dagger M-1) \\eta \\,,$ where $\\eta $ is a Gaussian noise vector and $M=\\hat{D}(U^\\prime )^{-1}\\hat{D}(U)$ .", "In such case the variance $\\Sigma _i^2$ in Eq.", "(REF ) is replaced by the sum of the exact variance and the stochastic variance according to Eq.", "(REF ).", "The variance in Eq.", "(REF ) can be written explicitly in terms of the coefficients as $\\Sigma _i^2 = \\sum _{j=0}^i\\left(\\beta _i^{(j)}\\right)^2C^{(jj)}+\\sum _{\\begin{array}{c}j,k=0\\\\j\\ne k\\end{array}}^{i}\\beta _i^{(j)}\\beta _i^{(k)}C^{(jk)} \\,,$ where $C^{(jk)}=\\left\\langle (\\Delta S^{(j)}-\\left\\langle \\Delta S^{(j)} \\right\\rangle ) (\\Delta S^{(k)}-\\left\\langle \\Delta S^{(k)} \\right\\rangle ) \\right\\rangle $ .", "We note that here $\\left\\langle \\cdot \\right\\rangle $ means an average over configurations $U$ in the dynamical ensemble and over gauge proposals $U^\\prime $ (and over noise $\\eta $ if applicable).", "At each step $i=1,\\ldots ,n$ the optimization of the parameters $\\beta _i^{(j)}$ is done by minimizing the variance $\\Sigma _i^2$ in Eq.", "(REF ).", "The idea is to use the correlation of low cost actions with the high cost action of the $i$ th step to increase the $i$ th-level acceptance rate.", "In order to get the right distribution after the last step the parameters of a specific action $S^{(j)}$ has to sum up to the target value $\\beta ^{(j)}=\\sum _i\\beta _i^{(j)}$ .", "This implies constraints on the parameter.", "For example the parameter of the plaquette action has to sum up to $\\beta ^{(0)} = \\beta $ .", "In principle, in order to solve for the parameters, we can start from the last step $i=n$ , solve for the parameters $\\beta _n^{(j)}$ and go to step $i-1$ .", "This provides an explicit solution scheme.", "At step $i$ , we solve the linear system of $i$ equations $2C^{(jj)}\\beta _i^{(j)} + \\sum _{\\begin{array}{c}k=0\\\\k\\ne j\\end{array}}^{i-1}C^{(jk)}\\beta _i^{(k)}=-C^{(ji)}\\beta _i^{(i)}\\,,\\quad j=0,\\ldots ,i-1\\,, $ to uniquely determine the values of the coefficients $\\beta _i^{(0)},\\ldots ,\\beta _i^{(i-1)}$ .", "The solutions of the steps $k>i$ and one constraint imply $\\beta _i^{(i)}=\\beta ^{(i)}-\\sum _{k=i+1}^n \\beta _k^{(i)}$ .", "We emphasize some properties of the parametrized acceptance-rejection steps.", "First of all this quite simple technique guarantees that the distribution of the pure gauge proposal has a good overlap with the dynamical distribution.", "Without parametrization the acceptance rate for lattices bigger than $4^4$ would be less than few %.", "The parametrization introduces a $\\beta $ -shift to higher $\\beta $ values in the pure gauge update, mainly reflecting the correlation with the determinants on the small blocks, see Table REF and Table REF .", "In general it is possible to introduce a new acceptance-rejection step $i$ by defining an auxiliary action with additional parameters.", "These parameters have to sum up to zero when considering all acceptance-rejection steps $k\\ge i$ .", "Their effect is to enhance the acceptance rate of these steps.", "For example we introduced a plaquette action, which uses HYP smeared links (one level of smearing) in order to better match the pure gauge update with the fermionic weight.", "This is particularly motivated for simulations with HYP smeared Wilson fermions but also helps for plain Wilson fermions, see Table REF and Table REF .", "We remark that it is not possible to introduce parameters for terms which are evaluated stochastically like Eq.", "(REF ).", "The effectiveness of the parametrization of acceptance-rejection steps is demonstrated in Fig.", "REF , where we compare the exact acceptances in the global step with optimal parameters (black circles) to the exact acceptances without parameters (blue diamonds)." ], [ "Tuning the optimal parameters ", "The parameters in the acceptance-rejection steps $i=1,2,\\ldots ,n-1$ are estimated from a simulation where the global step $i=n$ (the computationally most costly) is left out.", "Subsequently a full simulation is performed in order to determine the optimal parameters for the global step.", "Iterating further this procedure does not significantly change the values of the parameters." ] ]

[ [ "Coulomb potential and the paradoxes of PT-symmetrization" ], [ "Abstract Besides the standard quantum version of the Coulomb/Kepler problem, an alternative quantum model with not too dissimilar phenomenological (i.e., spectral and scattering) as well as mathematical (i.e., exact-solvability) properties may be formulated and solved.", "Several aspects of this model are described.", "The paper is made self-contained by explaining the underlying innovative quantization strategy which assigns an entirely new role to symmetries." ], [ "Abstract", "Besides the standard quantum version of the Coulomb/Kepler problem, an alternative quantum model with not too dissimilar phenomenological (i.e., spectral and scattering) as well as mathematical (i.e., exact-solvability) properties may be formulated and solved.", "Several aspects of this model are described.", "The paper is made self-contained by explaining the underlying innovative quantization strategy which assigns an entirely new role to symmetries." ], [ "Introduction", "The list of the traditional roles of symmetries in Quantum Theory has recently been enriched by the increasingly active studies of the role of the so called ${\\cal PT}-$ symmetries [1].", "Typically, one considers a Hamiltonian composed of a kinetic- and potential-energy terms, $H= -\\frac{\\ \\hbar ^2}{2m} \\,\\triangle +V(x)$ and replaces the current requirement $H = H^\\dagger $ of its Hermiticity in Hilbert space $L^2(\\mathbb {R}^d):={\\cal H}^{(friendly)}$ by the less trivial physical Hermiticity (or rather “crypto-Hermiticity” [2]) $H = H^\\ddagger $ which must be constructively, ad hoc defined in another Hilbert space ${\\cal H}^{(standard)}$ .", "In the language of symmetries this means that, firstly, the “usual” Hermiticity $H = H^\\dagger $ is reinterpreted as a formal time-reversal symmetry $H \\,{\\cal T} = {\\cal T}\\,H$ (with a suitable time-reversal operator ${\\cal T}$ ).", "In the second step one rejects the constraint (REF ) as “too formal” [3] and postulates, instead, the Hermiticity of $H=H^*$ in an auxiliary Krein space with a suitable indefinite pseudometric ${\\cal P}$ .", "In other words, in the language of symmetries one follows the recommendations of mathematicians [4], [5] and re-facilitates the mathematics by the replacement of Eq.", "(REF ) by the modified requirement called ${\\cal PT}$ symmetry, ${\\cal PT}\\,H=H\\,{\\cal PT}\\,.$ In the context of physics the recipe finds its most ambitious theoretical encouragement in the appeal of the concept of such a form of symmetry in relativistic quantum field theory [6], with ${\\cal P}$ representing the parity and with ${\\cal T}$ mimicking the time reversal in this implementation.", "Another explanation of the increasing popularity of the whole concept certainly lies in the amazing productivity of the ${\\cal PT}-$ symmetrizations of various quantum systems.", "In the popularization of such a trick the key role has been played by the serendipitious letter [7] in which Bender and Boettcher proposed and demonstrated that the ${\\cal PT}-$ symmetrization of a given local Schrödinger equation in single dimension (i.e., a transition from constraint (REF ) to constraint (REF ) at $d=1$ ) may represent an efficient theoretical tool and way towards finding new phenomenologically useful Hamiltonians with real spectra.", "They also recommended to achieve this simply by the replacement of the original Hermitian potential $V(x)$ by its non-Hermitian alternative $V({\\rm i}x)$ (thorough review offered by paper [3] may be recommended as an introductory reading).", "In our present paper we intend to return to the older application of this idea to the exactly solvable Coulomb/Kepler problem [8] and to its recent upgrades [9] – [11].", "We intend to review the related recent theoretical developments and to show how the incessant progress in the field applies to this particular but important example." ], [ "The ${\\cal PT}-$ symmetric version of the Coulomb problem", "In loc.", "cit., the standard quantum Coulomb/Kepler problem has been assigned a new, non-equivalent quantum system version which is formally represented by the ${\\cal PT}-$ symmetric Schrödinger equation $\\frac{\\ \\hbar ^2}{2m} \\,\\left[-\\frac{{\\rm d}^2}{{\\rm d}{x}^2}+ \\frac{L(L+1)}{{x}^2}\\right]\\, \\Psi ({x})+\\frac{{\\rm i}Z}{x} \\, \\Psi ({x}) = E \\,\\Psi ({x})\\,,\\ \\ \\ L>-\\frac{1}{2}\\,,\\ \\ \\ \\ 2L\\ \\notin \\mathbb {Z}\\,.$ We shall abbreviate here $2mE/\\hbar ^2=-k^2$ and emphasize that even in the presence of the imaginary unit, this equation remains solvable in terms of the well known confluent hypergeometric functions, $\\Psi _1(x)=C_1 \\Psi _1(x) +C_2 \\Psi _2(x)\\,,$ $\\Psi _1(x)=e^{-kx}x^{L+1}{_1}{F}{_1}(1+L+{{\\rm i}Z}/{(2k)},2L+2,2kx)\\,,$ $\\Psi _2(x)=e^{-kx}x^{-L}{_1}{F}{_1}(-L+{{\\rm i}Z}/{(2k)},-2L,2kx)\\,.$ The well known Coulomb – harmonic oscillator correspondence has been studied in [8].", "In the role of a formal postulate it helped us to fix the physical asymptotic boundary conditions which would remain, otherwise, ambiguous (Ref.", "[12] and previous loci citati should be consulted for all details).", "It has been shown [13] that all of the models of the class sampled by Eqs.", "(REF ) + (REF ) may be perceived, under certain conditions which we explained in Ref.", "[2], as fully compatible with the standard postulates of Quantum Mechanics.", "Our present paper will expose Eq.", "(REF ) as a special case of the broader class of Schrödinger equations which all exemplify an extension of quantum model-building strategies.", "In Ref.", "[2] we called this approach a “crypto-Hermitian” or “three-Hilbert-space” quantum mechanics.", "We shall emphasize here that the transition from the Hermitian to ${\\cal PT}-$ symmetric language is extremely productive while, at the same time, its multistep nature often leads to conceptual misunderstandings.", "While “teaching by example”, we shall try to clarify here some of the most blatant ones." ], [ "The abstract formalism", "Several compact review papers [3], [2], [14] may be recommended for reference.", "At the same time, an introductory explanation of the structure of Quantum Mechanics using the pseudometric in Krein space may be given, for our present purposes, a much shorter form.", "First of all the readers should be warned that in the majority of textbooks on Quantum Mechanics the meaning of the Dirac-ket symbols $|\\psi \\rangle \\in {\\cal V}$ is merely explained via examples.", "Most often one deals just with the most common quantum motion of a point particle inside a local potential well.", "Thus, it is assumed that there exists an operator $\\hat{Q}$ of the particle position with eigenvalues $q\\in \\mathbb {R}^d$ and eigenkets $|q\\rangle $ .", "The standard (often called Dirac's) Hermitian conjugation is then represented by the transposition plus complex conjugation of any ket vector, yielding the bra vector, ${\\cal T}^{(Dirac)}: |\\psi \\rangle \\ \\rightarrow \\ \\langle \\psi |\\,.$ In the basis $|q\\rangle $ one may then represent the ket of any pre-prepared state $|\\psi \\rangle $ by the overlap $\\psi (q) = \\langle q|\\psi \\rangle $ , i.e., by the square-integrable wave function of the coordinate $q$ .", "Our present purpose is not a criticism of this approach as such (interested readers may find such a criticism elsewhere [15]) but rather just of one of its consequences.", "In virtually all of the similar classes of examples, indeed, the vector space of states ${\\cal V}$ is simply assumed endowed with the most common inner product $\\langle \\psi _a|\\psi _b\\rangle = \\int _{-\\infty }^\\infty \\,\\psi _a^*(q)\\psi _b(q)\\ dq$ with, possibly, the integration replaced by the infinite or finite summation.", "Thus, we may (and usually do) set ${\\cal V}\\equiv L^2(\\mathbb {R})$ , etc.", "Without any real danger of misunderstanding we may speak here about the “friendly\" Hilbert space of states ${\\cal H}^{(F)} \\equiv {\\cal V}$ , calling the variable $q$ in Eq.", "(REF ) “the coordinate”.", "In parallel we usually perform a maximally convenient choice of the Hamiltonian $H$ based on the so called principle of correspondence which encourages us to split the Hamiltonian into the kinetic and potential energies, $H=T+V$ .", "Whenever the general interaction operator $V$ is represented, say, by a kernel $V(q,q^{\\prime })$ when acting upon the wave functions, this kernel is most often chosen as proportional to the Dirac's delta-function so that $V$ becomes an elementary multiplicative operator $V =V_{local}= V(q)$ .", "Similarly, the most popular and preferred form of the “kinetic energy\" $T$ is a differential operator, say, $T = T_{local}=-d^2/dq^2$ in single dimension and in the suitable units.", "The word of strong warning emerges when we perform a Fourier transformation in ${\\cal H}^{(F)} $ so that the variable $q$ becomes replaced by $p$ (= momentum).", "One should rather denote the latter, Fourier-image space by the slightly different symbol ${\\cal H}^{(P)}$ , therefore (with the superscript still abbreviating “physical\" [2]).", "Paradoxically, after the latter change of frame the kinetic operator $T_{local}$ becomes multiplicative while $V_{local}$ becomes strongly non-local in momenta.", "Nevertheless, all this does not modify the overall paradigm.", "A truly deep change of the paradigm only comes with the models where the necessity of the observability of the coordinate $q$ is abandoned completely.", "One may still start from the vector space of kets ${\\cal V}$ but it makes sense to endow it with another Hilbert-space structure, via the inner product defined by an integral over a complex path, $\\langle \\psi _a|\\psi _b\\rangle = \\int _{\\cal C}\\,\\psi _a^*(s)\\psi _b(s)\\ ds\\,.$ This is one of the most characteristic intermediate steps made in the so called ${\\cal PT}-$ symmetric quantum theories [3].", "The resulting loss of simplicity of the position operator $\\hat{Q}$ changes the physics of course.", "The key point is that we lose the one-to-one correspondence between the integration path ${\\cal C}$ and the spectrum $\\mathbb {R}$ of any coordinate-mimicking operator.", "The physics-independent optional variable $s$ becomes purely formal.", "In such a setting our choice of the physical observables must still obey the old quantization paradigm, which is just set in a modified context.", "The loss of the observability of the coordinate proves essential, anyhow.", "For illustration one might recall the pedagogically motivated paper [16] in which, in a slightly provocative demonstration of the abstract nature of quantum theory, the variable $s$ in Eq.", "(REF ) has been interpreted as an observable “time\" of a hypothetical “quantum clock\" system.", "Once we wish to understand our Coulombic Schrödinger eigenvalue problem (REF ), we must make one more step and generalize further the inner products (REF ).", "Such a second-step generalization of the inner product will certainly move us from the two Hilbert spaces ${\\cal H}^{(F)}$ and ${\\cal H}^{(P)}$ to the third one, viz., to the final and physical “standard” Hilbert space ${\\cal H}^{(S)}$ (this notation is taken from Ref. [2]).", "The introduction of the third Hilbert space forms the theoretical background of an amendment of the traditional quantum mechanics, the key nonstandard features of which can be seen in the admissibility of the complex potentials sampled by the power-law-anharmonic family $V(s) = -({\\rm i} s)^{2+\\delta }$ of Ref.", "[7] and generating the real and discrete bound-state spectra at any $\\delta >0$ (cf.", "the proofs in [17]); in the replacement of the usual real line of $s$ by a complex curve ${\\cal C}=\\mathbb {C}(\\mathbb {R})$ which may even be, in principle, living on a complicated multisheeted Riemann surface [12]; in the theoretical imperative of the construction of certain operator $\\Theta $ (see below); in the possibility of a systematic study of the discretizations and simplifications.", "In our present paper, the emphasis will be put on the last feature.", "At the stage of development where we did not yet explain the meaning and role of the operator $\\Theta $ (called Hilbert space metric) the theory remains incomplete.", "We already cannot rely upon a more or less safe guidance of quantization as offered by the principle of correspondence.", "Just a partial revitalization of such guidance is possible in the new context (cf., e.g., a nice example-based discussion of this point in Ref. [13]).", "This being said, the main theoretical obstacle lies in the vast ambiguity of the necessary appropriate generalization of the Hermitian conjugation as prescribed by Eq.", "(REF ).", "The general recipe (explained already in [18] or, more explicitly, in [2]) is $\\Theta -$ dependent and reads ${\\cal T}^{(general)}_\\Theta : |\\psi \\rangle \\ \\rightarrow \\ \\langle \\!\\langle \\psi |:=\\langle \\psi |\\Theta \\,.$ This means that using the language of wave functions $\\psi (s)$ with $s \\in {\\cal C}$ we must replace the most common single-integral definition (REF ) of the inner product in the original “friendly\" Hilbert space ${\\cal H}^{(F)}$ by the more sophisticated double-integral formula $\\langle \\!\\langle \\psi _a|\\psi _b\\rangle = \\int _{\\cal C}\\,\\int _{\\cal C}\\,\\psi _a^*(s)\\,\\Theta (s,s^{\\prime })\\,\\psi _b(s^{\\prime })\\ ds\\,ds^{\\prime }\\,.$ In terms of an integral-operator-kernel representation $\\Theta (s,s^{\\prime })$ of our abstract metric operator $\\Theta =\\Theta ^\\dagger >0$ this recipe defines the inner product which converts the same ket-vector space ${\\cal V}$ into the amended and final, metric-dependent and physics-representing Hilbert space ${\\cal H}^{(S)}$ of the very standard quantum theory (cf.", "[2] for more details)." ], [ "The upgraded formalism in applications", "In the attempted applications of all of the new ideas to the traditional benchmark models like Coulomb scattering one may make use of its traditional merits (like, e.g., exact solvability) as well as of the flexibility of the choice of the ${\\cal PT}-$ symmetric (i.e., complex and left-right symmetric) integration path, sampled in Ref.", "[10] as follows, $x(s)=x^{U}_{(\\varepsilon )}(s)\\,= \\,\\left\\lbrace \\begin{array}{ll}-{\\rm i}(s+\\frac{\\pi }{2}\\varepsilon )-\\varepsilon , & s \\in (-\\infty , -\\frac{\\pi }{2}\\varepsilon ),\\\\\\varepsilon e^{ {\\rm i}({s/\\varepsilon +3/2\\pi })},& s \\in ( -\\frac{\\pi }{2}\\varepsilon ,\\frac{\\pi }{2}\\varepsilon ),\\\\{\\rm i}(s-\\frac{\\pi }{2}\\varepsilon )+\\varepsilon \\,, & s \\in (\\frac{\\pi }{2}\\varepsilon , \\infty ).\\end{array}\\right.$ This leads to new results of course.", "Typically, in spite of the non-unitarity of the scattering (remember that the Coulomb potential is strictly local!)", "the bound-state energies still emerge from the poles of the scattering matrix [10]." ], [ "Discretizations", "The use of discretizations of the differential forms of Schrödinger operators may be, typically, Runge-Kutta-inspired.", "In practice, they are slowly becoming useful in solid-state physics [19], optics [20] and statistical physics [21].", "Less expectedly, the use of lattice models proved crucial for the unitarity of the scattering.", "It has been shown [22] that the theory of scattering by non-Hermitian obstacles may be made unitary and consistent via a certain selfconsistently prepared transition to non-local potentials.", "Unfortunately, it is not yet clear how such a requirement of selfconsistency could be realized in the continuous limit of the discrete models.", "Whenever we discretize the coordinates and replace the differential Hamiltonians by matrices with property $H \\ne H^\\dagger $ , the above-reviewed theory applies without changes.", "The usual Hilbert space becomes unphysical and it must be replaced by its unitarily non-equivalent correct alternative ${\\cal H}^{(S)}$ endowed with a sophisticated metric $\\Theta =\\Theta ^{(S)}\\ne I$ which defines the ad hoc inner product.", "In the discretized version of the theory, the integral kernel of the metric must merely be replaced by a matrix.", "Naturally, also the double integral (REF ) gets replaced by the double sum, $(\\psi ,\\phi )^{(S)}=\\sum _{j,k=1}^N\\,\\psi ^*_j\\,\\Theta _{j,k}^{(S)}\\,\\phi _k\\ $ in ${\\cal H}^{(S)}$ .", "In this setting, the imaginary choice of the Coulomb coupling may still be made compatible with the standard postulates of Quantum Theory, provided only that it still generates the real, i.e., potentially observable spectrum of the bound-state energies.", "Once we started our considerations from the imaginary Coulomb model defined along a continuous complex trajectory, we may expect that many of its properties will survive also the transition to its discrete descendants.", "For inspiration we may recall Ref.", "[10] where the bound-state energies were shown to coincide with the poles of transmission coefficients.", "Still, as long as the potential $V \\ne V^\\dagger $ is local, the unitarity of the scattering cannot be required [10], [22].", "At the same time, the unitarity of the time evolution of the system itself may be achieved.", "Indeed, although the Hamiltonian $H$ is non-Hermitian in ${\\cal H}^{(F)}$ , (abbreviated $H\\ne H^\\dagger $ ), it is Hermitian in ${\\cal H}^{(S)}$ .", "This feature is called cryptohermiticity, requiring $H=H^\\ddagger $ alias $ H^\\dagger \\Theta ^{(S)}= \\Theta ^{(S)}\\,H\\,.$ Here, the operator or matrix $\\Theta ^{(S)}$ is precisely the one which defines the physical inner product.", "This being said, the loss of easy constructions is a problem [23].", "Still, the discretization of the coordinates may be recommended as the recipe." ], [ "Interpretations", "The generic ${\\cal PT}-$ symmetric quantum model describes a closed system defined via a doublet of operators $ \\ {H,\\Theta }\\ $ or via a triplet of operators (adding a new observable $\\Lambda $ and having, typically, Hamiltonian $H \\ne H^\\dagger $ accompanied by a charge [3]), etc.", "In other words, the dynamical content of phenomenological quantum models is encoded in Hamiltonian $H$ and in metric $\\Theta $ .", "In this setting the metric $\\Theta $ guarantees the unitarity of time evolution in an ad hoc, “standard” Hilbert space [18], to be denoted by the symbol ${\\cal H}^{(S)}$ in what follows.", "In addition, one can also impose some other, phenomenologically motivated requirements like a short-range smearing of coordinates [24], etc.", "One of the remarkable features of such an upgrade of applications of quantum mechanics may be seen in the robust nature of its “first principles” which remain unchanged.", "Thus, its traditional probabilistic interpretation is not changed (notice that it practically did not change during the last cca eighty years!).", "In the language of textbooks one could speak just about the use of a non-unitary generalizations $\\Omega \\sim \\sqrt{\\Theta }$ of the most common Fourier transformations.", "Still, the new physics behind the trick may be nontrivial (in nuclear physics, for example, the mapping $\\Omega $ (called Dyson's [18]) was used to represent fermions as images of bosons).", "Among the most innovative consequences of the upgraded formulation of quantum models one notices, first of all, the existence and possibility of constructions of a horizon [25].", "Formally, this notion coincides with the set $\\partial {\\cal D}$ of the Kato's [26] exceptional points in the (real or complex) manifold ${\\cal D}$ of available free parameters (like coupling strengths, etc).", "The practical appeal of this notion may be based, e.g., on its tunability [27] and/or a new physics near instabilities and quantum catastrophes [28].", "As another emergent concept one should list fundamental length, i.e., a quantity $\\theta $ defined, in the simplified discrete models, as the number of diagonals in the metric which is required to possess a band-matrix form, $\\Theta ^{(S)}_{mn} =0 \\ {\\rm for} \\ |m-n|>\\theta $ .", "In this context one might mention the first papers devoted to the study of ${\\cal PT}-$ symmetric quantum graphs [29] in which one might search for a connection between the fragile parts of the spectrum and the topological characteristics of the underlying graph structure.", "Last but not least, it is necessary to emphasize the challenging character of a generic scenario with more observables, each of which may be responsible for its own part of the physical horizon, “invisible\" from the point of view of the other observables.", "In other words, a lot of work is still to be done before one could speak about a “classification\" of exceptional points (i.e., about a a sort of “quantum theory of catastrophes\") – the first attempts in this direction only dealt with the hardly realistic, too oversimplified and schematic quantum systems [30]." ], [ "\nCoulomb potential $V(x_j)={\\rm i}/x_j$", "The main weak point of the above-cited choice of the Coulomb potential may be identified not only with its strict locality (i.e., with the necessary loss of the unitarity of the scattering, cf.", "also the detailed study [23] in this respect) but also with the difficulties encountered during transition to any model which would not be exactly solvable.", "For this reason, our present use of the Runge-Kutta-inspired discretization will help also in the case of the Coulomb potential.", "As long as our present main ambition is the presentation of the upgraded formalism, we shall try to simplify many inessential mathematical aspects of our Coulomb/Kepler model.", "In parallel, we shall also try to treat this potential as a special case of a broader class of forces.", "For the sake of definitness and in a way insspired by Ref.", ", we shall pick up the class $V(x)={\\rm i}x^z$ with a real exponent which does not lie too far from its Coulombic value of $z=-1$ .", "For this purpose we must replace, first of all, the typical differential Schrödinger equation $-\\frac{d^2}{dx^2}\\,\\psi _n(x)+V(x)\\,\\psi (x) =E_n\\,\\psi _n(x)\\,\\ \\ \\ \\ \\ \\ \\ \\psi (\\pm \\Lambda )=0\\,,\\ \\ \\ \\ 1 \\ll \\Lambda \\le \\infty $ by its discrete version (i.e., approximation or analogue) $-\\frac{\\psi (x_{k-1})-2\\,\\psi (x_k)+\\psi (x_{k+1})}{h^2}+V(x_k)\\,\\psi (x_k)=E\\,\\psi (x_k)\\,.$ An equidistant grid of the Runge-Kutta points $x_k=-\\Lambda +k\\,h$ with $k = 0, 1, \\ldots ,N+1$ will be used.", "In this sense, also the standard general double-integral inner product will be replaced by the above-mentioned double sum, etc.", "Naturally, the discretization recipe also involves the change of the asymptotic boundary conditions, with $x_{N+1}=\\Lambda $ , $h=2\\Lambda /(N+1)$ and $\\psi (x_{0})=\\psi (x_{N+1})=0$ .", "In other words, the eigenvalue calculations become reduced to the mere diagonalizations of the $N$ by $N$ matrix Hamiltonians ${H}^{(N)}=\\left(\\begin{array}{ccccc}2+h^2V(x_{1})&-1&&&\\\\-1&2+h^2V(x_{2})&-1&&\\\\&-1& 2+h^2V(x_{3}) & \\ddots &\\\\& &\\ddots &\\ddots &-1 \\\\&&&-1&2+h^2V(x_{N})\\end{array}\\right)\\,.$ The insertion of any potential $V(x_j)={\\rm i}x_j^z$ will lead to the eigenvalues $\\varepsilon _j:=h^2E^{(N)}_j\\in (0,4)$ which must be computed numerically in general.", "In some applications the transition to the continuous limit $N=\\infty $ is made or, at worst, postponed till the end of the calculations.", "In the present methodical context we shall rather keep the dimension $N$ constant and, in fact, not too large.", "The insertion of formula $V(x_j)={\\rm i}/x_j$ in Eq.", "(REF ) with even $N=2K$ yields the sequence of the discrete ${\\cal PT}-$ symmetric Coulomb Hamiltonians $ H^{(2K)}(a)=$ $\\left[ \\begin{array}{cccccccc}2-i\\,a\\,/(2K-1)&-1&0&\\ldots &&&\\ldots &0\\\\-1&\\ddots &\\ddots &\\ddots &&&&\\vdots \\\\0&\\ddots &2-i\\,a\\,/3&-1&0&&&\\\\{\\medskip }\\vdots &\\ddots &-1&2-ia&-1&0&&\\\\{\\medskip }&&0&-1&2+ia&-1&\\ddots &\\vdots \\\\{\\medskip }&&&0&-1&2+i\\,a\\,/3&\\ddots &0\\\\\\vdots &&&&\\ddots &\\ddots &\\ddots &-1\\\\{\\medskip }0&\\ldots &&&\\ldots &0&-1&2+i\\,a\\,/(2K-1)\\end{array}\\right]\\,.$ In its first nontrivial example let us set $N=4$ , $H^{(4)}(a,z)=\\left[ \\begin{array}{cccc}2-i\\,a\\,3^z&-1&0&0\\\\{\\medskip }-1&2-ia&-1&0\\\\{\\medskip }0&-1&2+ia&-1\\\\ {\\medskip }0&0&-1&2+i\\,a\\,3^z\\end{array}\\right]\\,,\\ \\ \\ \\ z=-1\\,.$ We see that a natural generalization may be targeted not only at the growing dimensions $N > 4$ but also towards the small deviations of the exponent $z$ from its Coulombic value.", "Empirically, one can verify that in both of these directions, the spectral loci (i.e., eigenvalues $\\varepsilon ^{(N,z)}(a)$ ) remain topologically the same.", "More precisely, at a fixed $N$ , the topology of the Coulomb-potential pattern as sampled by Figs.", "REF - REF may be expected to survive all the negative exponents $z$ [11].", "At $N=4$ the model is exactly solvable at the Coulombic exponent $z=-1$ .", "The secular equation ${{\\it {E}}}^{4}-8\\,{{\\it {E}}}^{3}+ \\left( 21+{\\frac{10}{9}}\\,{a}^{2}\\right) {{\\it {E}}}^{2}+ \\left( -{\\frac{40}{9}}\\,{a}^{2}-20 \\right) {\\it {E}}+5+1/9\\,{a}^{4}+5\\,{a}^{2}=0$ generates the closed-form spectrum $\\varepsilon (a)=2 \\pm 1/6\\,\\sqrt{54-20\\,{a}^{2}\\pm 2\\,\\sqrt{405-720\\,{a}^{2}+64\\,{a}^{4}}}\\,,$ which is real iff $|a| \\le 3/4\\,\\sqrt{10-4\\,\\sqrt{5}} \\approx 0.7706147226$ .", "At $N=6$ the model is still exactly solvable at $z=-1$ , yielding the secular equation ${{\\it {E}}}^{6}-12\\,{{\\it {E}}}^{5}+ \\left( 55+{\\frac{259}{225}}\\,{a}^{ 2} \\right) {{\\it {E}}}^{4}+ \\left( -120-{\\frac{2072}{225}}\\,{a}^{2}\\right) {{\\it {E}}}^{3}+$ $+\\left( 126+{\\frac{5894}{225}}\\,{a}^{2}+{\\frac{7}{45}}\\,{a}^{4} \\right) {{\\it {E}}}^{2}+ \\left( -56-{\\frac{280 }{9}}\\,{a}^{2}-{\\frac{28}{45}}\\,{a}^{4} \\right) {\\it {E}}+$ $+7+14\\,{a}^{2 }+{\\frac{7}{9}}\\,{a}^{4}+{\\frac{1}{225}}\\,{a}^{6}=0$ which may be solved using Cardano formulae.", "Although the closed form of the spectrum becomes extremely clumsy in this representation, it decisively facilitates the graphical representation of the spectral loci which all appear topologically equivalent to the vertical array of circles.", "In particular, the survival of the exact solvability of the problem enables us to conclude that the whole $N=6$ spectrum remains real iff $|a| \\le 0.589586$ .", "These results indicate that the topological pattern remains generic and $N-$ independent.", "Such a conjecture is persuasively confirmed by the larger$-N$ graphical samples which are presented in Figs.", "REF and REF .", "Figure: The a-a-dependence of energiesε(a)\\varepsilon (a) at N=4N=4." ], [ "Graphical methods", "Numerical evaluation of the spectra is sampled in Figs.", "REF , REF and REF .", "These graphical constructions indicate that at any $N=2K$ the spectrum is real iff $a \\in \\left(-\\alpha ^{(2K)},\\alpha ^{(2K)}\\right)$ and fully complex for $a\\notin \\left(-\\beta ^{(2K)},\\beta ^{(2K)}\\right)$ .", "where $\\alpha ^{(2K)} $ is a quickly decreasing function of $K$ .", "The latter observation may be interpreted in two ways.", "For the finite lattices in which the numerical value of parameter $a$ is fixed, the reality of the spectrum is, undoubtedly, fragile.", "In the alternative approach in which our model serves just as a simulation of the (NB: exactly solvable!)", "differential-equation system, the definition of parameter $a$ is prescribed by the Runge-Kutta recipe (see above).", "For this reason, its numerical value decreases, with $N$ , much more quickly than $\\alpha ^{(N)}$ .", "This implies that in the latter setting the reality of the spectrum may be declared robust and guaranteed.", "Figure: The a-a-dependence of energiesε(a)\\varepsilon (a) at N=8N=8.Marginally, we may add that in the former scenario using small and fixed $N$ , the loss of the reality of the spectrum is caused by the confluence of the ground state with the first excited state and by their subsequent complexification.", "Due to the up-down symmetry of the spectrum, this instability is paralleled by the upper two states of course.", "Figure: The a-a-dependence of energiesε(a)\\varepsilon (a) at N=14N=14." ], [ "The $N=2$ metrics ", "The condition of hidden Hermiticity of any Hamiltonian with real spectrum is often called Dieudonné equation [2], $H^{(N)}=\\left[H^{(N)}\\right]^\\ddagger :=\\Theta ^{-1}\\left[H^{(N)}\\right]^\\dagger \\Theta \\,.$ Its structure is best illustrated at $N=2$ where the metrics form just the two-parametric family $\\Theta \\left[H^{(N)}(a) \\right]=\\Theta \\left[H^{(N)}(a) \\right]_{(k,m)}=\\left[ \\begin{array}{cc} k&km-ika\\\\{\\medskip }km+ika&k\\end{array} \\right]\\,,\\ \\ k,m \\in \\mathbb {R}\\,.$ With the condition $a \\in (-1,1)$ of the reality of energies and with the metric-positivity constraint $\\theta =\\theta _{\\pm }=k\\pm \\sqrt{{k^2m}^{2}+{k}^{2}{a}^{2}}>0\\,$ we may conclude that $k$ must be positive and larger than the square root.", "We may reparametrize $a=\\cos \\beta \\,\\sin \\gamma $ , $m=\\cos \\beta \\,\\cos \\gamma $ , $\\beta \\in (0,\\pi )$ and $\\gamma \\in (0,\\pi )$ and $-1<\\cos \\beta <1$ and get the final result $\\Theta =\\Theta \\left\\lbrace H^{(2)}[a(\\beta ,\\gamma )]\\right\\rbrace _{[k,m(\\beta ,\\gamma )]}=k\\cdot \\left[\\begin{array}{cc} 1&e^{-{\\rm i}\\gamma }\\cos \\beta \\\\{\\medskip }e^{{\\rm i}\\gamma }\\cos \\beta &1\\end{array} \\right]\\,.$ In a search for the other eligible observables with crypto-Hermiticity property $\\Lambda _{(\\beta ,\\gamma )}^\\dagger \\,\\Theta _{[k,m(\\beta ,\\gamma )]}=\\Theta _{[k,m(\\beta ,\\gamma )]}\\,\\Lambda _{(\\beta ,\\gamma )}\\,.$ the use of the ansatz $\\Lambda =\\left[\\begin{array}{cc}G+{\\rm i}g&B+{\\rm i}b\\\\{\\medskip }C+{\\rm i}c&D+{\\rm i}d\\end{array} \\right]\\,.$ leads to the four real constraints imposed upon eight free parameters.", "The family of observables is four-parametric, therefore.", "Three constraints define $B$ , $C$ and $G-D$ .", "The remaining one relates the sums $c_\\Sigma =b+c$ and $g_\\Sigma =g+d$ and leads to the unique solution $g_\\Sigma =0$ .", "We may conclude that from the input $m=m(\\beta ,\\gamma )$ and $a=a(\\beta ,\\gamma )$ one gets the class of admissible observables $\\Lambda =\\Lambda (D,b,c,g)=\\frac{1}{a}\\cdot \\left[ \\begin{array}{cc}D\\,a- b-c+i\\,g\\,{a}\\,,&g-b\\, m+i \\,b\\,a\\\\{\\medskip }{g+c\\,m}+i\\,c\\,a\\,,&D\\,a-i\\,g\\,a\\end{array} \\right]\\,.$ In particular, the initial Hamiltonian is reobtained at $D=2$ , $G=2$ , $b=c=0$ , $B=C=-1$ and $g= -a\\ ( =-d)$ .", "In the literature the concept of charge is considered particularly useful [3].", "Its essence lies, in the present model, in an additional auxiliary assumption $\\left[H\\right]^\\dagger {\\cal P}={\\cal P}\\,H\\,$ where ${\\cal P}$ is the operator of parity.", "Under this assumption a unique metric is sought such that a very specific metric called ${\\cal CPT}$ metric is prescribed by formula $\\Theta ^{({\\cal CPT})}={\\cal CP}$ where ${\\cal C}$ is called “charge\".", "Thus, at a given $N$ we may define the parity ${\\cal P}={\\cal P}^{(N)}$ which contains units along the secondary diagonal, i.e., $ {\\cal P}^{(N)}_{m,n}= 1$ iff $m+n=N+1$ while ${\\cal P}^{(N)}_{m,n}= 0$ otherwise.", "Incidentally one may note that ${\\cal P}^{(2)}$ = limit of $\\Theta \\left[H^{(2)}(a) \\right]$ such that $k^{({\\cal P})} \\rightarrow 0$ , $m^{({\\cal P})} \\rightarrow \\infty $ , $k^{({\\cal P})}m^{({\\cal P})} \\rightarrow 1$ .", "The key merit of the use of charge is that its use makes the metric unique, ${\\cal C}^{({\\cal CPT})}=k \\cdot \\left[\\begin{array}{cc}- {\\rm i} a& 1\\\\{\\medskip }1& {\\rm i} a\\end{array} \\right]\\,.$ Moreover, it also represents one of the special cases of observable $\\Lambda $ using $D=b=c=0$ and $g = - \\cos \\beta /\\sin \\beta =-\\sqrt{k^2-1}$ .", "Indeed, from $\\Theta ^{({\\cal CPT})}={\\cal CP}$ we have ${\\cal C}=\\Theta \\left[H^{(2)}(a) \\right]_{(k,m)}{\\cal P}^{(2)}=\\left[\\begin{array}{cc}u&v\\\\{\\medskip }y&z\\end{array} \\right]\\,$ yielding $v=y=k$ and $z=u^*=ke^{i\\gamma }\\cos \\beta $ .", "Then, condition ${\\cal C}^2=I$ requires that $\\gamma =\\gamma ^{({\\cal CPT})}= \\pi /2$ (i.e., $a=\\cos \\beta $ ).", "Thus, we have $\\beta =\\beta ^{({\\cal CPT})}$ such that $\\sin \\beta ^{({\\cal CPT})}=1/k$ .", "This proves the above statement." ], [ "The $N=4$ metrics ", "With the natural ansatz for $\\Theta \\left[H^{(4)}(a,z) \\right]_{(k,m,r,h)}=$ $=\\left[ \\begin{array}{cccc}k&m-ikw&W^*&Z^*\\\\{\\medskip }m+ikw&r&h-i\\left( kw+ra \\right)&W^*\\\\{\\medskip }W &h+i\\left( kw+ra \\right) &r&m-ikw\\\\{\\medskip } Z&W &m+ikw&k\\end{array} \\right]$ where $w=w(z,a) = {3}^{z}{a}\\ $ and $W=W(k,m,r)=-{w}^{2}k+r-k- kwa+i \\left( wm+ma \\right)\\,,$ $Z=Z(k,m,r,h)=m{a}^{2}-{w}^{2}m-m+h-i \\left( kw-ka-kw{a}^{2}-rw+ {w}^{3}k\\right)\\,,$ the problem of the determination of the domain of positivity of the metric starts to be merely tractable graphically.", "For the numerous practical purposes the metric is sought in a special form.", "One of the phenomenologically inspired options is the choice of the matrix with units along its main diagonal, $k=r=1$ .", "In addition, let us select $m=h=0$ and compute $W(1,0,1)=-{w}\\left(w+a \\right)$ , $Z(1,0,1,0)=i \\left( a+w{a}^{2}- {w}^{3} \\right)$ .", "Such a restricet construction leads to the following four closed-form eigenvalues of the metric $\\Theta \\left[ H^{(4)}(a,z)\\right]_{(1,0,1,0)}$ , viz, $\\theta _+^\\pm =1+ \\frac{1}{2}\\,\\left(w-{a}^{2}w+{w}^{3}\\right)\\pm \\frac{1}{2}\\,\\sqrt{\\triangle ^+}$ $\\theta _-^\\pm =1- \\frac{1}{2}\\,\\left(w-{a}^{2}w+{w}^{3}\\right)\\pm \\frac{1}{2}\\,\\sqrt{\\triangle ^-}$ $\\triangle ^\\pm ={w}^{6}+ \\left( 2-2\\,{a}^{2} \\right) {w}^{4}+ \\left( \\pm 8+4\\,a \\right) {w}^{3}+ \\left(5 \\pm 8\\,a+ 6\\,{a}^{2}+{a}^{4} \\right) {w}^{2}+$ $+\\left( 4\\,a+4 \\,{a}^{3} \\right) w+4\\,{a}^{2}\\,.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $ Some details and numerical results of its analysis may be found elsewhere [11]." ], [ "Discussion", "In a climax of our present discussion of the discrete ${\\cal PT}-$ symmetric Coulomb problem characterized by the purely imaginary coupling constant, let us now summarize the overall method via the following scheme $\\begin{array}{c}\\begin{array}{|c|}\\hline \\ {\\rm textbook \\ level\\ quantum\\ theory:}\\ \\\\{\\rm {} prohibitively\\ complicated}\\ {\\rm \\ Hamiltonian}\\ \\mathfrak {h}\\\\{\\rm generating \\ unitary\\ time\\ evolution}\\ \\\\\\ \\ \\ \\ {\\rm \\fbox{P:} \\ {{} physics\\ = \\ trivial} } \\ \\\\\\ \\ \\ \\ \\fbox{\\rm calculations\\ =\\ practically\\ impossible\\ } \\ \\\\\\hline \\end{array}\\\\\\stackrel{{} simplification}{}\\ \\ \\ \\ \\swarrow \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\searrow \\nwarrow \\ \\ \\ \\stackrel{{} unitary\\ equivalence}{}\\\\\\begin{array}{|c|}\\hline \\ {\\rm \\ state}\\ \\psi \\ {\\rm is\\ represented}\\\\\\ {\\rm in\\ the} \\ \\fbox{{} false\\ {\\rm Hilbert\\ space}}\\ \\\\\\ {\\rm \\fbox{F:} \\ calculations\\ = \\ {{} feasible}\\ }\\ \\\\\\ {\\rm physical\\ meaning \\ = \\ lost}\\ \\\\\\ \\ \\ \\ H = {\\rm non-Hermitian\\ } \\ \\\\\\hline \\end{array}\\stackrel{ {{} hermitization} }{ \\longrightarrow }\\begin{array}{|c|}\\hline \\ \\fbox{\\rm amended \\ inner\\ product}\\ \\\\\\ {\\rm {} standardized} {\\rm \\ representation } \\ \\\\\\ \\ {\\rm \\fbox{S:} \\ picture\\ }=\\ {\\rm {} synthesis\\ } \\ \\\\ \\ \\ {\\rm physics\\ = \\ reinstalled}\\ \\\\\\ \\ \\ \\ H = {\\rm Hermitian\\ } \\ \\\\\\hline \\end{array}\\\\\\\\\\end{array}$ which characterizes the “three-Hilbert-space\" pattern of quantization as described in Ref.", "[2] as a recipe in which the usual Schrödinger equation $H\\,|\\psi _n\\rangle = E_n\\,|\\psi _n\\rangle \\,$ finds the standard probabilistic interpretation even if the Hamiltonian matrix (with real spectrum) proves manifestly non-Hermitian.", "The specific feature of non-Hermitian matrices $H$ may be seen in their ability of having the reality of their spectra controlled by a parameter (for this purpose we used $a$ in our present models).", "In other words, one can simulate the abrupt loss of the stability of the time evolution of the system by a mere smooth change of this parameter.", "In other words, we may speak about a non-empty (quasi-)Hermiticity domain of parameters, with the qualitative changes of physics at its boundary, and with a guaranteed reality of the spectrum in its interior.", "In our present paper we emphasized that another important aspect of physics with real spectra but non-Hermitian matrices of observables lies in the necessity of a fine-tuning mediated by the Dieudonne equation.", "Typically, a given Hamiltonian $H$ must be assigned a Hermitizing metric $\\Theta $ .", "As long as we merely considered $N<\\infty $ , we could avoid any difficulties by simply solving the second, conjugate Schrödinger equation $\\langle \\!\\langle \\psi _m|\\,H=F_m\\,\\langle \\!\\langle \\psi _m|\\,$ which may be also written in the form $H^\\dagger \\,|\\psi _m\\rangle \\!\\rangle = F_m^*\\,|\\psi _m\\rangle \\!\\rangle \\,$ This enabled us to work with the solutions as forming a bicomplete and biorthogonal basis, $I = \\sum _{n=0}^{N-1}\\,|\\psi _n\\rangle \\,\\frac{1}{\\langle \\!\\langle \\psi _n|\\psi _n\\rangle }\\,\\langle \\!\\langle \\psi _n|\\,,\\ \\ \\ \\ \\ \\langle \\!\\langle \\psi _m|\\psi _n\\rangle = \\delta _{m,n}\\,\\langle \\!\\langle \\psi _n|\\psi _n\\rangle \\,.$ The main benefit may be then found in the closed formula $\\Theta = \\sum _{n=0}^{N-1}\\,|\\psi _n\\rangle \\!\\rangle \\,|\\kappa _n|^2\\,\\langle \\!\\langle \\psi _n|\\,$ which defines all of the eligible metrics.", "This, in its turn, specifies all the dynamics given by the operator doublet $(H({\\lambda }),\\Theta ({\\kappa }))$ ." ], [ "Acknowledgment", "Work supported by the GAČR grant Nr.", "P203/11/1433." ] ]

[ [ "Planck Intermediate Results II: Comparison of Sunyaev-Zeldovich\n measurements from Planck and from the Arcminute Microkelvin Imager for 11\n galaxy clusters" ], [ "Abstract A comparison is presented of Sunyaev-Zeldovich measurements for 11 galaxy clusters as obtained by Planck and by the ground-based interferometer, the Arcminute Microkelvin Imager.", "Assuming a universal spherically-symmetric Generalised Navarro, Frenk & White (GNFW) model for the cluster gas pressure profile, we jointly constrain the integrated Compton-Y parameter (Y_500) and the scale radius (theta_500) of each cluster.", "Our resulting constraints in the Y_500-theta_500 2D parameter space derived from the two instruments overlap significantly for eight of the clusters, although, overall, there is a tendency for AMI to find the Sunyaev-Zeldovich signal to be smaller in angular size and fainter than Planck.", "Significant discrepancies exist for the three remaining clusters in the sample, namely A1413, A1914, and the newly-discovered Planck cluster PLCKESZ G139.59+24.18.", "The robustness of the analysis of both the Planck and AMI data is demonstrated through the use of detailed simulations, which also discount confusion from residual point (radio) sources and from diffuse astrophysical foregrounds as possible explanations for the discrepancies found.", "For a subset of our cluster sample, we have investigated the dependence of our results on the assumed pressure profile by repeating the analysis adopting the best-fitting GNFW profile shape which best matches X-ray observations.", "Adopting the best-fitting profile shape from the X-ray data does not, in general, resolve the discrepancies found in this subset of five clusters.", "Though based on a small sample, our results suggest that the adopted GNFW model may not be sufficiently flexible to describe clusters universally." ], [ "Introduction", "Clusters of galaxies are the most massive gravitationally bound objects in the Universe and as such are critical tracers of the formation of large-scale structure.", "The size and formation history of massive clusters is such that the ratio of cluster gas mass to total mass is expected to be representative of the universal ratio, once the relatively small amount of baryonic matter in the cluster galaxies is taken into account (e.g., [98]).", "Moreover, the comoving number density of clusters as a function of mass and redshift is expected to be particularly sensitive to the cosmological parameters $\\sigma _8$ and $\\Omega _{\\rm m}$ (e.g., [17]).", "The Sunyaev–Zeldovich (SZ) effect (see [19]; [23] for reviews) produces secondary anisotropies in the cosmic microwave background (CMB) radiation through inverse-Compton scattering from the electrons in the hot intracluster gas (which also radiates via thermal Bremsstrahlung in the X-ray waveband) and the transfer of some of the energy of the electrons to the low-energy photons.", "Moreover, the surface brightness of an SZ signal does not depend on the redshift $z$ of the cluster.", "Hence an SZ-effect flux-density-limited survey can provide a complete catalogue of galaxy clusters above a limiting mass (e.g., [16], [45], [48], [90]).", "Analyses of observations of galaxy clusters via their SZ effect, X-ray emission or gravitational lensing are often based on some spherically-symmetric cluster model in which one assumes parameterised functional forms for the radial distribution of some cluster properties, such as electron density and temperature [91], [95], [94], [52], [37], [9], [6], [4], [7], electron pressure and density [59], [57], [13], [63], [67], or electron pressure and entropy [5], [1].", "The motivation for this paper is to augment SZ measurements obtained with Planck Planck (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states (in particular the lead countries France and Italy), with contributions form NASA (USA) and telescope reflectors provided by a collaboration between ESA and a scientific consortium led and funded by Denmark.", "for a sample of 11 galaxy clusters with refined higher-resolution SZ measurements obtained with the Arcminute Microkelvin Imager (AMI) interferometer.", "Such a combination is an interesting and potentially very powerful way to pin down the gas pressure profile of individual galaxy clusters as it relies on a single well-understood astrophysical effect.", "In addition, Planck and AMI SZ measurements exploit very different aspects of the SZ signature: Planck effectively uses its wide frequency coverage to identify the characteristic frequency spectrum of the SZ effect while AMI exploits its higher angular resolution to perform spatial filtering to identify SZ clusters and constrain their parameters.", "Combining measurements by these two instruments not only provides a powerful consistency check on both sets of observations, but may also break, or at least reduce, the observed parameter degeneracy between the derived SZ Compton-$Y$ parameter and the cluster angular size which often results due to the finite resolution of SZ telescopes [67].", "The paper is organised as follows.", "In Section , we outline how we selected our sample of 11 galaxy clusters for this comparison work.", "In Sections  and , we describe the Planck and AMI observations of our cluster sample, respectively.", "In Section  we present the pressure profile that we have used to model the clusters and constrain parameters.", "The analysis of the real data from both experiments is also described in Section .", "We follow this by validating our analysis methodology and investigating the effects of diffuse foreground emission on the Planck constraints with simulations in Section REF .", "Section REF presents a similar simulations-based investigation of the effects of residual point sources and analysis methodology on the constraints derived from the AMI interferometric data.", "With a view to explaining some of the discrepancies we observe, in Section , we investigate the possibility of relaxing the assumptions regarding the universal pressure profile adopted for our cluster sample.", "Here we also examine the consistency of the Planck and AMI SZ results with complementary constraints from high-quality X-ray observations for a subset of our cluster sample.", "We conclude with a discussion in Section ." ], [ "Selection of the cluster sample", "An original sample of 26 clusters was defined at the beginning of this study.", "24 of the clusters were identified as members of the sample by virtue of the fact that they were both present in the Planck Early Sunyaev–Zeldovich (ESZ) cluster catalogue [67], and had also already been observed and detected with AMI during the course of its normal observing programme.", "Note that these 24 clusters had been observed by AMI as part of differing scientific programmes and while each programme had a well-defined sample, the resulting set of clusters used in this paper does not constitute a well-defined or complete sample.", "To this sample of 24, two newly-discovered Planck clusters were added, for which AMI made follow-up observations.", "The complete list of the original cluster sample, their coordinates and redshifts is presented in Table REF .", "The sample was then screened to include only clusters that had (i) a firm SZ detection by AMI ($\\ge 3\\sigma $ before source subtraction and $\\ge 5\\sigma $ after source subtraction) and (ii) a benign environment in terms of radio point sources (we discard clusters with total integrated source flux densities greater than $5 \\, \\rm {mJy}$ within a radius of $3{^{\\scriptstyle \\prime }}$ or greater than $10 \\, \\rm {mJy}$ within a radius of 10 from the phase centre).", "This reduced the sample to 11 clusters spanning a wide range in redshift, $0.11 < z < 0.55$ .", "The clusters in the new sample are A2034, A1413, A990, A2409, A1914, A2218, A773, MACS J1149+2223, RXJ0748+5941, PLCKESZ G139.59+24.18, and PLCKESZ G121.11+57.01.", "The sample includes two cool-core clusters, A1413 and A2034 [88], [44], [95], [39] and two newly-discovered Planck clusters, PLCKESZ G139.59+24.18 and PLCKESZ G121.11+57.01 [67].", "The last two have been observered in the optical with the RTT-150 telescopehttp://hea.iki.rssi.ru/rtt150/en/ as part of the Planck follow-up programme.", "The resulting spectroscopic redshifts measured for the brightest cluster galaxies within these two clusters are given in Table REF .", "Table: Original sample of 26 clusters.", "References for clusterinformation are: (1) ; (2) ; (3) ; (4); (5) ; (6); (7) ; (8); and (9) Planck RTTfollow-up programme; see Section .", "The meaning of the notes in theright-most column are: (i) cluster has SZ detectionsmaller than 3σ3\\sigma before source subtraction and smaller than5σ5\\sigma after source subtraction; (ii-a) cluster has a totalintegrated source flux density greater than 5 mJy within a radius of3 ' 3{^{\\scriptstyle \\prime }}; (ii-b) cluster has a total integrated source flux densitygreater than 10 mJy within a radius of 10 ' 10{^{\\scriptstyle \\prime }}.", "“in sample”indicates that the cluster is included in the 11-cluster sampleanalysed in this paper." ], [ "Description of ", "Planck [93], [64] is the third generation space mission to measure the anisotropy of the cosmic microwave background (CMB).", "It observes the sky in nine frequency bands covering 30–857 GHz with high sensitivity and angular resolution from 31$^{\\scriptstyle \\prime }$ to 5$^{\\scriptstyle \\prime }$ .", "The Low Frequency Instrument (LFI; [54], [18], [56]) covers the 30, 44, and 70 GHz bands with amplifiers cooled to 20 K. The High Frequency Instrument (HFI; [51], [86]) covers the 100, 143, 217, 353, 545, and 857 GHz bands with bolometers cooled to 0.1 K. Polarisation is measured in all but the highest two bands [53], [89].", "A combination of radiative cooling and three mechanical coolers produces the temperatures needed for the detectors and optics [65].", "Two Data Processing Centers (DPCs) check and calibrate the data and make maps of the sky [87], [99].", "Planck's sensitivity, angular resolution, and frequency coverage make it a powerful instrument for Galactic and extragalactic astrophysics as well as cosmology.", "Early astrophysics results are given in Planck Collaboration VIII–XXVI 2011, based on data taken between 13 August 2009 and 7 June 2010.", "Intermediate astrophysics results are now being presented in a series of papers based on data taken between 13 August 2009 and 27 November 2010.", "We note that the Planck maps used for the analysis in this paper are not the same as those used in the Early Planck results papers.", "In particular, we stress that both the data and the analysis techniques employed for this study are not the same as was used to construct the Planck ESZ catalogue [67].", "A later version of the data has been used for the analysis here and we use different analysis techniques for reasons which will be explained in Section REF .", "However, as part of our suite of internal tests, we have repeated our analysis on the older version of the Planck data which was used to derive the ESZ catalogue and we find excellent agreement.", "In Fig.", "REF we present maps of the dimensionless Compton-$y$ parameter for each of the clusters as estimated from the Planck data.", "The $y$ -parameter is related to the observed brightness as a function of frequency ($\\nu $ ) by $\\frac{\\Delta T_{\\rm tSZ}}{T_{\\rm CMB}} (\\nu ) = y \\cdot g(\\nu ),$ where $\\Delta T_{\\rm tSZ}$ is the brightness fluctuation due to the thermal SZ (tSZ) effect and $T_{\\rm CMB}$ is the temperature of the CMB, which we take to be $2.7255\\pm 0.0006$ [38].", "The function, $g(\\nu )$ is the frequency dependence of the SZ effect [92].", "For the HFI channel frequencies ($100, 143, 217,353, 545$ and 857 GHz), $g(\\nu )$ takes the values ($-4.03, -2.78,0.19, 6.19, 14.47$ and $26.36$ $K_{\\rm CMB} / y$ ).", "The maps of Fig.", "REF were estimated from HFI channel data taken between 13 August 2009 and 27 November 2010, corresponding to slightly more than 2.5 full-sky scans.", "The measured noise levels on the Planck frequency channel maps are listed in Table REF (Planck HFI Core Team, in prep.).", "The tSZ signal reconstructions were performed using the MILCA method ([43] and references therein) on the six Planck all-sky maps from 100 GHz to 857 GHz.", "MILCA (Modified Internal Linear Combination Algorithm) is a component separation approach aimed at extracting a chosen component (here the tSZ signal) from a multi-channel set of input maps.", "It is based on the well known ILC approach (e.g., [34]), which searches for the linear combination of the input maps that minimises the variance of the final reconstructed map, while imposing spectral constraints.", "For our cluster SZ reconstructions, we applied MILCA using two constraints, the first one to preserve the tSZ signal and the second one to remove CMB contamination in the final tSZ $y$ -map.", "In addition, to compute the weights of the linear combination, we have used the extra degrees of freedom to minimise residuals from other components (two degrees) and from the noise (two degrees).", "The noise covariance matrix was estimated from jack-knife maps.", "The final $y$ -maps have an effective resolution of 10 arcmin.", "Note that, in general, the properties of the foreground emission depend on both the position on the sky and on the frequency of observation.", "We have therefore allowed the weights to vary as a function of both position and frequency.", "We have confirmed using simulations that such an approach maximises the signal-to-noise and minimises the bias in the extraction of the tSZ signal.", "We emphasise that the MILCA SZ reconstructions presented in Fig.", "REF are intended for visual examination and qualitative assessment of the cluster signals only.", "Our quantitative analysis of the Planck data, which we use to compare with the AMI results, is based mainly on the PowellSnakes [25] SZ extraction algorithm (see Section ).", "Table: Noise levels per N side =2048N_{\\rm side} = 2048 pixel for each Planck frequency band.Figure: Maps of the dimensionless Compton-yy parameter(equation ) as estimated from thePlanck observations using the MILCA algorithm.", "The maps have aneffective resolution of 10 arcmin.", "The clusters are ordered in terms ofincreasing redshift, from top left to bottom right.", "Each panel shows a 100 ' ×100 ' 100^\\prime \\times 100^\\prime region." ], [ "Description of AMI data", "AMI comprises two arrays, the Small Array (SA) and the Large Array (LA), located at the Mullard Radio Astronomy Observatory near Cambridge.", "The SA consists of ten 3.7-m diameter equatorially–mounted antennas, with a baseline range of $ \\simeq 5$ –20 m and synthesised beam (resolution) of around $3{^{\\scriptstyle \\prime }}$ .", "The LA consists of eight 13-m diameter antennas with a baseline range of $\\simeq 20$ –100 m and synthesised beam of around 30.", "Both arrays observe Stokes $I+Q$ in the band 13–18 GHz, each with system temperatures of about 25 K. Note that AMI defines Stokes $Q$ and $U$ with respect to celestial north.", "The backends are analogue Fourier transform spectrometers, from which the complex signals in each of eight channels of 750-MHz bandwidth are synthesised, and the signals in the adjacent channels are correlated at the $\\simeq $ 10 % level.", "Further details of the instrument are given in [8].", "SA pointed observations of our cluster sample were taken during 2007–2011.", "The observation lengths per cluster before any flagging of the data are presented in Table REF ; the noise levels on the SA maps reflect the actual observation time used.", "The SA observations were made with single pointings interspersed with a phase calibration source, while the LA observations were made in a 61+19-point raster mode configuration with $4{^{\\scriptstyle \\prime }}$ spacing.", "This consisted of 61 pointings arranged in a hexagonal grid, with grid points separated by $4{^{\\scriptstyle \\prime }}$ with further observations of the central 19 pointings designed to increase the sensitivity at the centre of the field.", "In this mode the integration time on the area $\\le 7.5{^{\\scriptstyle \\prime }}$ from the cluster centre is twice as long as the area $>7.5{^{\\scriptstyle \\prime }}$ away, so as to provide a better match to the primary beam sensitivity of the AMI SA.", "Phase calibrators were chosen from the Jodrell Bank VLA Astrometric Survey (JVAS, [61]) on the basis of proximity ($\\le 2^\\circ $ for the AMI LA, $\\le 8^\\circ $ for the AMI SA) and 15 GHz flux density ($\\ge 0.2$  Jy for the AMI LA, $\\ge 0.7$  Jy for the AMI SA).", "The JVAS is based on observations made with the VLA in “A\" configuration [26], [27], [28], [97].", "The reduction of the AMI data was performed using a dedicated software tool reduce.", "This is used to apply path-compensator and path-delay corrections, to flag interference, shadowing and hardware errors, to apply phase and amplitude calibrations and to Fourier transform the correlator data readout to synthesise the frequency channels, before outputting to disk.", "Flux calibration was performed using short observations of 3C48 and 3C286 near the beginning and end of each run.", "The assumed $I+Q$ flux densities for these sources in the AMI channels are listed in Table REF and are consistent with [15].", "As Baars et al.", "measure $I$ and AMI measures $I+Q$ , these flux densities include corrections for the polarisation of the sources.", "An amplitude correction is also made for the intervening air mass during the observation.", "Flux calibration is expected to be accurate to ${\\simeq }\\,3$ % for the AMI SA and ${\\simeq }\\,5$ % for the AMI LA.", "After phase calibration, the phase of both arrays over one hour is generally stable to $5^{\\circ }$ for channels 4–7, and to $10^{\\circ }$ for channels 3 and 8.", "(Channels 1 and 2 are generally not used for science analysis as they tend to suffer from interference problems.)", "Table: Assumed I+QI+Q flux densities of 3C286 and 3C48 over thecommonly-used AMI band, and the full width at half maximum of the LAprimary beam (approximate field of view, Θ LA \\Theta _{\\rm LA}) for eachchannel.Maps were made using the Astronomical Image Processing System (aips, [41]) from each channel of the AMI SA and LA; however here we present only the combined-channel maps of the SA and LA observations.", "The aips task imean was used on the LA individual maps to attach the map noise to the map header.", "imean fits a Gaussian to the histogram of the map pixels (ignoring extreme pixels that might be due to sources) and uses the standard deviation of the fitted Gaussian as a measure of the random noise in the data.", "The aips task flatn was then used to form a mosaiced image from the multiple pointings.", "Data from the pointings were primary beam corrected using parameters listed in Table REF and weighted accordingly when combined.", "The AMI SA combined-channel map noise and the LA map noise are given in Table REF .", "The raw $uv$ data for all good observations were concatenated together to make a visibility data file for each channel.", "All maps were made using natural $uv$ weighting and all images were cleaned to three times the thermal noise with a single clean box encompassing the entire map.", "The data were also binned into bins of width $40\\lambda $ .", "This reduced the size of the data to a manageable level without adversely affecting the subsequent inference of cluster properties.", "Fig.", "REF shows a typical example of the SA synthesised beam, in this case for the observations of A2218.", "Figure: SA synthesised beam for the A2218observations.", "Contours start at 6%6\\,\\% and increase linearly by 3%3\\,\\%per contour.", "Contours drawn as red dashed lines are negative.", "Thesynthesised beams for the other cluster observations are qualitativelysimilar.As contamination from radio sources at 16 GHz tends to be significant, removing or modelling this emission accurately can often be essential to recover SZ decrements from AMI maps.", "To address this issue, we use the AMI-developed source extraction software, sourcefind [96], [3] to determine the position, flux density and spectral index of the radio sources with flux density $\\ge 3.5\\sigma _{\\rm {LA}}$ on the cleaned LA continuum maps, where $\\sigma _{\\rm {LA}}$ is the LA thermal noise.", "Spectral indices were fit with a Markov Chain Monte Carlo (MCMC) method using LA maps for all six channels and assuming that source flux densities follow a power-law relation of $S\\propto \\nu ^{-\\alpha }$ for the AMI frequencies.", "These source parameter estimates are subsequently used as priors in our Bayesian analysis of the cluster SZ signals (Section REF ).", "Tables REF and REF summarise the observational details of our cluster sample and Figs.", "REF and REF present the maps of the AMI observations of these clusters before and after source subtraction, respectively.", "Once again, as with the reconstructed Planck maps presented in the previous section, the AMI maps presented in Figs.", "REF and REF are intended for visual examination and qualitative assessment of the cluster signals only.", "Our quantitative analysis of the AMI data is described later in Section REF .", "Table: Details of the AMI observations of our cluster sample.Table: Details of radio point sources and the thermal noise levelsfor the AMI observations of the cluster sample.", "Here σ SA \\sigma _{\\rm {SA}}and σ LA \\sigma _{\\rm {LA}} refer to the thermal noise levels reached inthe LA and SA maps, respectively.", "The S/N values are calculated bydividing the peak flux values by σ SA \\sigma _{\\rm {SA}}.Figure: AMI maps before source subtraction.", "The clusters are orderedas in Fig.", ", in terms of increasingredshift.", "Black solid lines represent positive contours and reddashed lines indicate negative contours.", "The contours increaselinearly from ±2σ SA \\pm 2\\sigma _{\\rm SA} to ±10σ SA \\pm 10\\sigma _{\\rm SA} whereσ SA \\sigma _{\\rm SA} is listed in Table  for eachcluster.", "Each map covers a region approximately 30 ' ×30 ' 30^\\prime \\times 30^\\prime and the resolution is around 3 ' 3^\\prime .Figure: AMI maps after subtraction of radio point sources.", "The solidblack lines represent positive contours and the dashed red linesindicate negative contours.", "The cluster ordering, contour levels andresolution are the same as inFig.", "." ], [ "Analysing the SZ signal", "The SZ surface brightness (the Compton-$y$ parameter, cf., equation REF ) is proportional to the line of sight integral of the electron pressure, $y=\\frac{\\sigma _{\\rm T}}{m_{\\rm e}c^2} \\int _{-\\infty }^{+\\infty }{P_{\\rm e}(r){\\rm d}l},$ where $P_{\\rm e}(r)$ is the electron pressure at radius $r$ , $\\sigma _{\\rm T}$ is the Thomson scattering cross-section, $m_{\\rm e}$ is the electron mass, $c$ is the speed of light and $dl$ is the line element along the line of sight.", "In this context, [59] analysed the pressure profiles of a series of simulated clusters [49] as well as a sample of relaxed real clusters presented in [95], [94].", "They found that the pressure profiles of all of these clusters could be described by a generalisation of the Navarro, Frenk, and White (NFW, [60]) model used to describe the dark matter halos of simulated clusters.", "Assuming spherical geometry, the GNFW pressure profile [59] reads $P_{\\rm e}(r) = P_{\\rm {0}}\\left(\\frac{r}{r_{\\rm s}}\\right)^{-\\gamma }\\left[1+\\left(\\frac{r}{r_{\\rm s}}\\right)^{\\alpha }\\right]^{\\,(\\gamma -\\beta )/\\alpha },$ where $P_{\\rm {0}}$ is the overall normalisation coefficient of the pressure profile and $r_{\\rm s}$ is the scale radius.", "It is common to define the latter in terms of $r_{\\rm 500}$ , the radius at which the mean density is 500 times the critical density at the cluster redshift, and to define the gas concentration parameter, $c_{\\rm 500}=r_{\\rm 500}/r_{\\rm s}$ .", "The parameters $(\\alpha , \\beta , \\gamma )$ describe the slopes of the pressure profile at $r\\simeq r_ {\\rm s}$ , $r>r_{\\rm s}$ , and $r \\ll r_{\\rm s}$ respectively.", "In order to retain consistency between the Planck and AMI analysis pipelines, we follow [13] (see also [67], [69]) and fix the values of the gas concentration parameter and the slopes to be $(c_{\\rm 500},\\alpha ,\\beta ,\\gamma ) =(1.156,1.0620, 5.4807, 0.3292)$ .", "These values describe the “universal pressure profile”, derived from XMM-Newton observations of the REXCESS cluster sample [20], and from three different sets of detailed numerical simulations by [22], [62], and [59], which take into account radiative cooling, star formation, and energy feedback from supernova explosions.", "In Section , we will relax these restrictions for a subset of our cluster sample and will include information from X-ray observations of individual clusters in our analysis.", "We note that the profile of equation (REF ) has recently been used to analyse SZ data from the South Pole Telescope (SPT, [63]) in addition to the Planck survey data [67], [69].", "The integral of the $y$ parameter over the solid angle $\\Omega $ subtended by the cluster is denoted by $Y_{\\rm SZ}$ , and is proportional to the volume integral of the gas pressure.", "It is thus a good indicator of the total thermal energy content of the cluster and its mass (e.g., [16]).", "The determination of the normalisation and the slope of the $Y_{\\rm SZ}-M$ relation has therefore been a major goal of studies of the SZ effect [30], [58], [50], [63], [13], [10], [67], [68], [69], [70], [71].", "In particular, [10] investigated the $Y_{\\rm SZ}-Y_ {\\rm X}$ scaling relation within a sample of 15 clusters observed by SPT, Chandra and XMM-Newton and found a slope of close to unity ($0.96 \\pm 0.18$ ).", "Similar studies were carried out by [70] using a sample of 62 nearby ($z < 0.5$ ) clusters observed by both Planck and by XMM-Newton.", "The results are consistent with predictions from X-ray studies [13], [10].", "These studies at low redshifts, where data are available from both X-ray and SZ observations of galaxy clusters, are crucial to calibrate the $Y_{\\rm SZ}-M$ relation, as such a relation can then be scaled and used to determine masses of SZ selected clusters at high redshifts in order to constrain cosmology.", "The integrated $y$ parameter ($Y_{\\rm SZ}$ ) adopting a spherical geometry $Y_{\\rm sph}$ , is given by $Y_{\\rm sph}(r)= \\frac{\\sigma _{\\rm T}}{m_{\\rm e}c^2}\\int _{0}^{r}{P_{\\rm e}(r^{\\prime })4 \\pi {r^{\\prime }}^{2}{\\rm d}r^{\\prime }}.$ Following [13], we consider the radius of $5r_{500}$ as the boundary of the cluster where the pressure profile flattens, and we use this boundary to define the total volume integrated SZ signal, $Y_{\\rm tot}$ .", "In the simplest case, where $\\alpha $ , $\\beta $ , $\\gamma $ , and $c_{500}$ in equation (REF ) have fixed values, our cluster model depends only on four parameters: $x_{\\rm c}$ and $y_{\\rm c}$ which define the projected cluster position on the sky and ${P_{\\rm {0}}}$ and $r_{\\rm s}$ in the pressure profile (equation REF ).", "In this paper, we define clusters in terms of the parameter set $\\mbox{$\\Theta $}_{\\rm c}\\equiv (x_{\\rm c}\\,,\\, y_{\\rm c}\\,, \\,\\theta _{\\rm s}=r_{\\rm s}/D_{\\rm A}\\, , \\,Y_{\\theta }=Y_{\\rm tot}/D^2_{\\rm A})$ , where $D_{\\rm A}$ is the angular-diameter distance to the cluster [67], and we determine the model parameter ${P_{\\rm {0}}}$ by evaluating equation (REF ) at $r = 5r_{500}$ .", "To calculate $D_{\\rm A}$ we assume a flat Universe with matter density $\\Omega _m = 0.27$ and Hubble constant $H_0 =70.4 \\, {\\rm km}\\,{\\rm s}^{-1} {\\rm Mpc}^{-1}$ [47].", "We adopt an exponential prior for $\\theta _{\\rm s}$ and a power-law prior for $Y_{\\theta }$ to analyse both the Planck and AMI data [25].", "The prior on $\\theta _{\\rm s}$ is $\\lambda e^{-\\lambda \\theta _{\\rm s}}$ for $1.3<\\theta _{\\rm s}<45$ and zero outside this range, with $\\lambda =0.2$ .", "The prior on $Y_{\\theta }$ is $Y^{-\\alpha }_{\\theta }$ for $5.0 \\times 10^{-4} \\, \\rm {arcmin}^2 < {\\it Y}_{\\theta } < 0.2 \\, \\rm {arcmin}^2$ and zero outside this range, with $\\alpha = 1.6$ .", "These priors have already been used in Planck detection and extraction algorithms to identify and characterise compact objects buried in a diffuse background [67].", "For the cluster position, however, in order to ensure that we are comparing integrated SZ fluxes centred on identical positions on the sky, we performed an initial analysis of the AMI data using a Gaussian prior centred on the cluster phase centre and with a standard deviation of 1$^{\\scriptstyle \\prime }$ in order to find the best-fitting cluster coordinates.", "We then fixed the cluster position to these best-fitting coordinates in the subsequent analysis of the Planck data and also in a subsequent re-analysis of the AMI data." ], [ "Analysis of ", "The analysis of the Planck data was performed using PowellSnakes (PwS), which is a Bayesian package for discrete object detection, photometry and astrometry, as described in [24], [25].", "PwS is part of the Planck HFI pipeline and is regularly used to produce catalogues of objects [66] and to measure and characterise the SZ signal [67].", "Note that we have chosen to use PwS as our primary SZ extraction algorithm for the Planck analysis in this study as PwS naturally returns the full posterior distribution in the $Y_{500} - \\theta _{500}$ 2D parameter space.", "It is thus naturally suited for combining with the AMI results to produce joint constraints.", "We will also present a comparison with results obtained using the Matched Multi-Filter algorithm (MMF3; [55]) which was the reference algorithm adopted for the production of the Planck ESZ catalogue [67].", "However, in its current implementation, the MMF3 algorithm does not produce any information on the correlation between the two cluster parameters of interest ($Y_{500}$ and $\\theta _{500}$ ) and so producing joint constraints as obtained from AMI and Planck via MMF3 is currently not possible.", "The analysis of the Planck data using the PwS algorithm proceeded as follows.", "For each cluster, flat patches ($14.7^\\circ \\times 14.7^\\circ $ ; $512 \\times 512$ pixels) were created using a gnomonic projection, centred on the targeted cluster, for each of the Planck HFI channels.", "By operating on such a large patch of sky enough statistics are collected in order to produce a smooth cross-channel covariance matrix.", "The position of each cluster was assumed to be known precisely (we adopted delta-function priors at the AMI-determined position) as described in Section .", "The data model for a single isolated cluster located in the centre of the patch is then described by ${\\bf d}({\\bf x}) = Y \\, {f} \\, \\Gamma (\\theta _{\\rm s},{\\bf x})+ {\\bf n}({\\bf x}),$ where ${\\bf x}$ is sky position, ${\\bf d}({\\bf x})$ is a vector containing the data, ${\\bf n}({\\bf x})$ is the background composed of instrumental noise plus all other astronomical components except the SZ signal, ${f}$ is a vector containing the SZ surface brightness at each frequency, $Y$ is the total integrated Comptonisation parameter, $\\theta _{\\rm s}$ is a parameter controlling the cluster radial scale and $\\Gamma (\\theta _{\\rm s},{\\bf x})$ is the convolution of the canonical GNFW model integrated along the line of sight with the Planck beam at that channel.", "It is assumed that the background is a realisation of a stationary Gaussian random field.", "A direct computation of the likelihood is very expensive.", "Therefore, PwS instead computes the likelihood ratio of two competing models describing the data: a cluster is present ($H_1$ ); or no cluster is present ($H_0$ ).", "Note that the latter hypothesis does not contain any parameters and therefore only multiplies the target likelihood in $H_1$ by a constant.", "The representation of the likelihood ratio in real space reads $\\ln \\left[\\frac{\\mathcal {L}_{H}({\\bf \\Theta })}{\\mathcal {L}_{H_0}({\\bf \\Theta })}\\right] &=& Y \\mathcal {F}^{-1}\\left[\\mathcal {P}_j({\\eta })\\widetilde{\\tau }(-{\\eta };\\theta _{\\rm s})\\right]_{{\\bf x}={\\bf 0}} - \\nonumber \\\\& &\\frac{1}{2}Y^2 \\sum _{{\\eta }}\\mathcal {Q}_{jj}({\\eta })|\\widetilde{\\tau }({\\eta };\\theta _{\\rm s})|^2,$ where ${\\eta }$ is the spatial frequency (the conjugate variable to ${\\bf x}$ ) and $\\mathcal {F}^{-1}[\\ldots ]_{\\bf x}$ denotes the inverse Fourier transform of the quantity in brackets, evaluated at the point ${\\bf x}$ .", "We have also defined the quantities $\\mathcal {P}_j({\\eta }) \\equiv \\widetilde{{\\bf d}}^t({\\eta }){\\bf \\mathcal {N}}^{-1}({\\eta }) {\\psi }({\\eta })$ and $\\mathcal {Q}_{ij}({\\eta }) \\equiv \\widetilde{{\\psi }}_i^t({\\eta }){\\bf \\mathcal {N}}^{-1}({\\eta }) {\\psi }_j({\\eta })$ , in which the vector $\\psi _i({\\eta })$ has the components $({\\psi }_{i})_\\nu = \\widetilde{B}_\\nu ({\\eta })({\\bf f}_{i})_\\nu $ , with $\\nu $ labeling frequency channels and $\\widetilde{B}_\\nu ({\\eta })$ is the beam transfer function.", "The quantity $\\widetilde{\\tau }(-{\\eta };\\theta _{\\rm s})$ is the Fourier transform of $\\Gamma (\\theta _{\\rm s},{\\bf x})$ and the matrix ${\\bf \\mathcal {N}}({\\eta })$ contains the generalised noise cross-power spectra.", "We refer the interested reader to [24], [25] for further technical details on the PwS algorithm.", "The cross-channel covariance matrix is computed directly from the pixel data, by averaging the Fourier modes in concentric annuli.", "This operation is only possible because of the assumed isotropy of the background.", "To reduce the contamination of the background by the signal itself, the estimation of the covariance matrix is performed iteratively.", "After an initial estimate, all detected clusters in the patch are subtracted from the data using their best fit values and the covariance matrix is re-estimated.", "To enforce our assumption of a single source in the centre of the patch, PwS removes from the data all other detections with SNRs higher than our target cluster to reduce possible contamination of the signal by power leakage from nearby objects.", "Bright point sources are masked or subtracted from the maps as part of a pre-processing routine run prior to the production of the flat patches.", "To construct the joint posterior distributions of $(Y, \\theta _{\\rm s})$ , we have used the set of priors as described in Section .", "To draw the posterior distribution manifold, PwS grids the parameter space using a uniformly spaced lattice of $(256 \\times 256)$ cells, appropriately chosen to enclose all posterior regions significantly different from zero.", "Since the LFI channels of Planck have relatively coarse resolution, the use of LFI bands in current implementations of the extraction algorithms results in beam dilution of the SZ signal and thus decreases the S/N for the detected clusters [67].", "This can potentially be improved in the future with modifications to the algorithms but for the purposes of the present study, we use only the HFI data." ], [ "Analysis of AMI data", "An interferometer like AMI operating at a frequency $\\nu $ measures samples from the complex visibility plane $\\tilde{I}_\\nu ({\\bf u})$ .", "These are given by a weighted Fourier transform of the surface brightness $I_\\nu ({\\bf x})$ , namely $\\tilde{I}_\\nu ({\\bf u})=\\int {A_\\nu ({\\bf x})I_\\nu ({\\bf x})\\exp (2\\pi i{\\bf u\\cdot x}){\\rm d}{\\bf x}},$ where ${\\bf x}$ is the position relative to the phase centre, $A_\\nu ({\\bf x})$ is the (power) primary beam of the antennas at observing frequency $\\nu $ (normalised to unity at its peak) and ${\\bf u}$ is the baseline vector in units of wavelength.", "In our model, the measured visibilities are defined as $V_\\nu ({\\bf u})=S_\\nu ({\\bf u}) + N_\\nu ({\\bf u}),$ where the signal component, $S_\\nu ({\\bf u})$ , contains the contributions from the SZ cluster and identified radio point sources, whereas the generalised noise part, $N_\\nu ({\\bf u})$ , contains contributions from a background of unsubtracted radio point sources, primary CMB anisotropies and instrumental noise.", "We assume a Gaussian distribution for the generalised noise.", "This then defines the likelihood function for the data $\\mathcal {L}(\\mbox{$\\Theta $})=\\frac{1}{Z_{\\rm N}}\\exp \\left(-\\frac{1}{2}\\chi ^2\\right),$ where $\\chi ^2$ is the standard statistic quantifying the misfit between the observed data $\\mbox{$D$}$ and the predicted data $\\mbox{$D$}^p(\\mbox{$\\Theta $})$ , $\\chi ^2=\\sum _{\\nu , \\nu ^{\\prime } }(\\mbox{$D$}_\\nu -\\mbox{$D$}_\\nu ^p)^T(\\mbox{$C$}_{\\nu , \\nu ^{\\prime }})^{-1}(\\mbox{$D$}_{\\nu ^{\\prime }}-\\mbox{$D$}_{\\nu ^{\\prime }}^p),$ where $\\nu $ and $\\nu ^{\\prime }$ are channel frequencies.", "Here $\\mbox{$C$}$ is the generalised noise covariance matrix $\\mbox{$C$}=\\mbox{$C$}^{\\rm {rec}}_{\\nu , \\nu ^{\\prime }} +\\mbox{$C$}^{\\rm {CMB}}_{\\nu , \\nu ^{\\prime }} +\\mbox{ $C$}^{\\rm {conf}}_{\\nu , \\nu ^{\\prime }}.$ The first term on the right hand side of equation (REF ) is a diagonal matrix with elements $\\sigma ^2_{\\nu ,i}\\,\\delta _{ij}\\delta _{\\nu \\nu ^{\\prime }}$ , where $\\sigma _{\\nu ,i}$ is the rms Johnson (receiver) noise on the $i$ th element of the data vector $\\bf D_{\\nu }$ at frequency $\\nu $ .", "The second term denotes the noise due to primordial CMB anisotropies and contains significant off-diagonal elements, both between visibility positions and between frequencies.", "This term can be calculated from a given primary CMB power spectrum $C^{\\rm {CMB}}_{l}(\\nu )$ following [42]; note that in intensity units the CMB power spectrum is a function of frequency.", "To calculate this term, we adopt the best-fitting CMB power spectrum to the WMAP 7-year data [47].", "The third term on the right hand side of equation (REF ) is the source confusion noise, which accounts for remaining unresolved radio sources with flux densities less than the flux limit ($S_{\\rm {lim}}$ ) of the AMI observations and which remain after high resolution observation and subtraction.", "We estimate this term assuming that sources are randomly distributed on the sky, in which case we can describe the source confusion noise with a power spectrum calculated as $C^{\\rm conf}_\\ell (\\nu ) = \\int _0^{S_{\\rm lim}} S^2 n_\\nu (S) \\, {\\rm d}S,$ where $n_\\nu (S) = dN_\\nu (>\\!S) / dS$ is the differential source count at frequency $\\nu $ as a function of flux density $S$ .", "We use the source counts as measured by the 10C survey [2] for our calculation.", "The limiting flux density for the integration ($S_{\\rm lim}$ ) is determined from the noise in the LA maps and is different for each cluster, but is typically in the range 0.2–0.5 mJy.", "The normalisation factor $Z_{\\rm N}$ in equation (REF ) is given by $Z_{\\rm N}=(2\\pi )^{(2N_{\\rm {vis}})/2}|\\mbox{$C$}|^{1/2},$ where $N_{vis}$ is the total number of visibilities.", "Further details on our Bayesian methodology, generalised noise model, likelihood function and resolved radio point-source models are given in [35] and [36], [37].", "Radio sources detected in the LA maps were modeled by four source parameters, $\\mbox{$\\Theta $}_{\\rm s}\\equiv (x_{\\rm s}\\, ,\\,y_{\\rm s}\\,,\\, S_0\\, ,\\,\\alpha )$ , where $x_{\\rm s}$ and $y_{\\rm s}$ refer to the right ascension and declination of radio sources, respectively, while $S_0$ and $\\alpha $ are the flux density and spectral index of the radio source at the central frequency, $\\nu _0$ .", "As mentioned in Section , this modelling is necessary because of source variability and some difficulty with inter-array calibration.", "Therefore, the properties of point sources detected at $>3.5\\,\\sigma _{\\mathrm {LA}}$ by the LA were used as priors when modelling the SA data.", "We used a delta-function prior on the position of the source since the resolution of the LA is around three times that of the SA.", "We used Gaussian priors on the source flux densities, with LA (integrated, where applicable) flux densities generating the peak of the prior, and the Gaussian $\\sigma $ s were set to 40 % of the source flux densities.", "Spectral index ($\\alpha $ ) priors were also set as Gaussians, with $\\sigma $ equal to the error on the spectral index fit.", "This is because for sources with high signal-to-noise ratio, the determination of the spectral index is dominated by the AMI frequency channel mean and the error on $\\alpha $ is Gaussian.", "For sources with low signal-to-noise ratio which just meet our continuum detection threshold, the spectral index probability distribution is dominated by the prior, which is determined from the 10C survey [2]." ], [ "Results", "Fig.", "REF presents the 2D marginalised posterior distributions in the $Y_{500}-\\theta _{500}$ plane and Table REF summarises the mean and the dispersion of these two parameters for each cluster, as estimated from the Planck and AMI data, respectively.", "Note that in Fig.", "REF the inner and outer contours show the areas enclosing $68\\,\\%$ and $95\\,\\%$ of the probability distributions.", "Estimates of $\\theta _{500}$ as derived from X-ray observations, and which were included in the Planck ESZ catalogue [67], are also indicated in the figures for comparison.", "Figure: Recovered Planck and AMI 2D posterior distributions inthe Y 500 -θ 500 Y_{500}-\\theta _{500} plane.", "Blue contourplots are the results from the AMI analysis and purple contoursshow the Planck results (specifically using the PwS method).", "Red arrows show the values ofθ 500 \\theta _{500} as determined from X-ray measurements of these clusterswhere available.", "The grey points with error bars show the MMF3Planck results.", "The inner and outer contours in each set indicatethe areas enclosing 68 % and 95 % of the probability distribution,while the MMF3 error-bars indicate the 1σ1\\sigma uncertainties.", "Wherethe recovered AMI and PwS Planck constraints are consistent, thejoint constraints are also indicated by the heavy black contours.", "Incases where the contours do not close at the lower ends of theparameter ranges, the corresponding constraints represent upper limitsonly.Table: Recovered mean and dispersion values for θ 500 \\theta _{500} andY 500 Y_{500} for the 11 clusters.", "Where consistency is found betweenthe Planck and AMI measurements, the joint constraints are alsogiven.", "The cluster redshift and signal-to-noise of the PwSdetections are also listed.Recall that in this figure the clusters are ordered in terms of increasing redshift.", "The constraints from both Planck and AMI demonstrate a strong cluster size–integrated Compton parameter ($\\theta _{500}-Y_{500}$ ) degeneracy/correlation in all cases.", "It is important to account for such effects when attempting to use the SZ signal to estimate cluster masses [30], [12].", "We also note that the Planck constraints appear to be weaker for high redshift clusters, which can generally be understood as a resolution effect – Planck's relatively poor resolution (e.g., as compared with AMI) means it has difficulty resolving and thus estimating the parameters of clusters with small angular extent – and high-redshift clusters are likely to be smaller in angular size.", "AMI's increased resolution, on the other hand, means that it can still constrain the sizes of these high-redshift, small-angular size clusters.", "For three clusters (A1413, A1914, and PLCKESZ G139.59+24.18), the AMI and Planck constraints are clearly discrepant.", "On the other hand there is significant overlap in the posterior distributions for the remaining eight clusters.", "However, taking our cluster sample as an ensemble, there is some evidence that the cluster parameter estimates derived from the AMI data are systematically lower than those derived from the Planck data (i.e., AMI is finding the clusters to be fainter and smaller in angular extent compared to what the Planck data indicate).", "In addition, the $Y_{500} - \\theta _{500}$ degeneracies are significantly different for the Planck and AMI constraints, with the degeneracies of the AMI constraints being generally steeper than the Planck ones.", "This arises because of the interplay between the angular size–redshift relation and the differing angular scales that AMI and Planck are sensitive to, as well as the very different observational techniques and frequencies used by the two instruments.", "In the cases where the Planck and AMI-derived constraints are compatible with one another, we also overplot the joint constraints obtained from multiplying the Planck and AMI posteriors.", "In many cases, the resulting joint constraints are far tighter than either analysis alone which is a direct result of the differing parameter degeneracies for the two instruments, as described above.", "The marginalised constraints from this combined analysis are also presented in Table REF .", "In Fig.", "REF , we have also over-plotted the constraints as obtained from the Planck data using the MMF3 algorithm.", "Comparing these results with the PwS Planck results, we see good agreement in most cases, although there may be a tendency for the MMF3 estimates to be systematically brighter and larger than the PwS results.", "However, it is clear that our broad conclusions regarding the general levels of agreement between the Planck and AMI results remain unchanged if we consider the MMF3 results in place of the PwS constraints.", "In Fig.", "REF , we plot the Planck determined integrated Compton-$Y$ parameter versus the Compton-$Y$ parameter as derived from the AMI data.", "Note that, for this correlation plot, we have fixed the cluster scale size to be that determined from X-ray observations (as indicated by the red arrows in Fig.", "REF ).", "(Three of the clusters have no reported X-ray size so only eight of the 11 clusters contribute to this correlation analysis.)", "The measured correlation coefficient is 0.79 and the best-fitting linear relationship has a slope of $1.18\\pm 0.07$ , again indicating that the Planck SZ fluxes appear to be systematically larger that the AMI derived fluxes.", "We have also repeated this analysis fixing the cluster size to both the Planck-determined size and the AMI-determined size.", "In both cases we see the same general trend, with the Planck Compton-$Y$ parameter being consistently larger than the AMI-derived value.", "Figure: Comparison of the integrated Compton-YYparameters obtained from the Planck and AMI fits, when the clustersize is fixed to that determined from X-ray observations.", "Theone-to-one relation is denoted with the dashed line.", "Thebest-fitting linear relation is plotted as the unbroken line.", "Theslope of this latter relation is 1.18±0.071.18\\pm 0.07 and thecorrelation coefficient is 0.790.79.", "Note that the same generalbehaviour (slope >1> 1) is also observed when we fix the clustersize to be that determined from either the Planck or the AMI SZobservations.In summary, our results suggest a systematic difference between the Planck and AMI measurements of the SZ signal coming from our cluster sample.", "Such a systematic difference could be an indicator of a shortcoming in some part of our analysis and could have important implications for performing cosmological studies with larger samples of SZ clusters.", "For example, the observed systematic could indicate that the way the clusters are being modeled in the analysis (e.g., the fixed GNFW profile adopted and/or the assumption of spherical symmetry) is not flexible enough to describe both the Planck and AMI results simultaneously.", "If this were to be the source of the discrepancy then such effects would need to be accounted for in future cosmological studies.", "However, before considering such an explanation, it is important to first consider if possible instrumental and/or astrophysical systematic effects could be responsible for the results we have found.", "We now turn to simulations to investigate the potential impact of such effects." ], [ "Simulations", "In order to test the SZ signal extraction techniques used and to investigate whether the systematic discrepancy observed in the real data is due to unaccounted-for astrophysical foregrounds, instrumental systematics or data-analysis induced biases, we have conducted detailed simulations of both the Planck and AMI experimental setups.", "For each of the 11 clusters in our sample, to create an input SZ signal for the simulations, we simulated a cluster SZ signal using the GNFW pressure profile (equation REF ), with input parameters based on the best-fitting $Y_{500}$ and $\\theta _{500}$ values from an analysis based on intermediate Planck maps, which are, in practice, close to the best-fitting Planck parameters quoted in Table REF ." ], [ "For Planck, the main worry in terms of astrophysical systematics is probably thermal emission from dust in the Galaxy.", "The Planck simulation ensemble comprised of CMB and noise realisations and a fixed foreground dust template, produced by re-scaling the Planck 857 GHz channel map to the other HFI frequencies and reconvolving so as to apply the appropriate beam for each channel.", "The dust template assumed a modified blackbody spectrum with emissivity $\\beta = 1.8$ and temperature $T=18$  K. The beams were assumed to be Gaussian with the appropriate mean FWHM for each channel as calculated by the FEBeCoP algorithm [87].", "The noise component of the simulations was generated using the Springtide destriping pipeline [14].", "This pipeline creates realisations of the noise in the nominal mission time-ordered data streams, compresses them to rings and destripes the rings to produce noise maps.", "It is assumed that the noise is uncorrelated between rings and that in each ring it is drawn from a power spectrum.", "For these simulations, the noise power spectrum used was the mean of the ring-by-ring spectra.", "In turn, these were determined by applying the noise estimation pipeline [87] to the exact same version and time-span of the Planck data that was used for the real SZ cluster analysis of the previous section.", "The simulations were then analysed using the PwS algorithm in exactly the same manner as was applied during the analysis of the real data.", "For each of the 11 clusters, ten simulations were run.", "The dust template based on the Planck 857 GHz map was the same for each of these ten simulations, but the CMB and noise realisations were different.", "The results of the simulations are shown in Fig.", "REF .", "In each panel, input parameters are indicated with a star and the recovered parameter constraints from the ten different simulations are indicated by ten sets of differently coloured contours.", "Comparing with the input parameters, the recovered constraints for each of our clusters are clearly distributed about the input model and there is no indication of any significant bias due to dust contamination, noise bias, the Planck beams or the PwS extraction technique employed.", "Figure: Recovery of SZ cluster parameters fromsimulated Planck observations (see Section for details).", "Each set of recovered parameter constraints(contours with different colours) represents a differentrealisation of the instrument noise and primordial CMBfluctuations and the star shows the input parametervalues.", "The inner and outer contours in each set indicate theareas enclosing 68 % and 95 % of the probability distribution.", "Anybias in the recovery of the input parameters averaged overrealisations is negligible compared to the random errors.", "In caseswhere the contours do not close at the lower ends of the parameterranges, the corresponding constraints represent upper limitsonly." ], [ "AMI simulations", "As mentioned in Section , contamination from radio point sources is a significant issue at AMI frequencies ($\\sim $ 16 GHz).", "Although the AMI LA observations are used to accurately find and model sources in the AMI SA observations, there is the possibility of contamination from source residuals if this modelling is not perfect.", "In a similar manner to the Planck simulations described in the previous subsection, we have investigated potential issues associated with either residual foreground radio sources or with the AMI data-analysis methodology and instrument response using simulations.", "The simulated input clusters were the same as used for the Planck simulations.", "To simulate the interferometric AMI observations, we used the in-house simulation package, profile [40] to create the mock visibilities.", "In addition to the cluster signal, the simulations included primordial CMB fluctuations and Gaussian noise, the amplitude of which was chosen to match that measured from the real observations.", "The simulation package also mimics the actual $uv$ coverage and synthesised beam of the real observations.", "The point sources in each cluster were simulated using the best-fitting values from the analysis of the real data.", "These simulated observations were then analysed in the exact same way as was used for the real data.", "Once again, for each of the 11 clusters, ten simulations were performed.", "Here, the point source environment was kept the same for these ten simulations but the CMB and noise realisations were again different.", "The results of the simulations are shown in Fig.", "REF .", "In each panel, input parameters are indicated with a star and the recovered parameter constraints from the ten different simulations are indicated by ten sets of differently coloured contours.", "As was the case with the Planck simulations, the recovered constraints for each of our clusters are clearly distributed about the input model.", "Once again, there is no indication of any significant bias due to residual point source contamination, noise bias, the AMI $uv$ coverage and resolution effects, or the extraction technique employed.", "Figure: Recovery of SZ cluster parameters from thesimulated AMI observations in the presence of residual pointsource contamination from imperfectly modeled radio sources foreach cluster in the sample.", "Also included in the simulations arethe AMI uvuv coverage, the instrument beams and realisations ofthe instrument noise and primordial CMB.", "The different sets ofcontours indicate different CMB and noise realisations and the starshows the input parameters used to generate the simulatedcluster.", "The inner and outer contours in each set indicate theareas enclosing 68 % and 95 % of the probabilitydistribution." ], [ "Adopting individual pressure profiles as measured from X-ray\nobservations", "The simulations presented in the previous section indicate that the discrepancies seen in the analysis of the real data cannot be easily explained by astrophysical contamination, instrumental effects, or any issues associated with the SZ signal extraction techniques.", "It is then interesting to ask whether the discrepancies observed might be associated with the way in which the clusters have been modeled using the universal pressure profile [13].", "For a number of clusters in the sample, we have high-quality determinations of the clusters' individual pressure profiles, as estimated from X-ray observations.", "Rather than adopting the [13] profile (which is essentially an average profile taken over many clusters), one might expect to achieve better consistency on a case-by-case basis if we use these individual best-fitting X-ray derived profiles in the SZ analysis.", "We have performed such a re-analysis for five clusters in our sample for which we have high-quality measured X-ray profiles.", "The clusters concerned are A1413, A1914, A2034, A2218, and A773.", "We fitted a GNFW pressure profile to the measured X-ray profiles and the results are presented in Table REF .", "We then re-analysed the Planck and AMI SZ data for these five clusters using the best fitting values of the profile shape parameters $(c_{\\rm 500},\\alpha ,\\beta ,\\gamma )$ as given in Table REF .", "The resulting constraints from this re-analysis are shown in Fig.", "REF .", "Table: Best-fitting GNFW shape and concentration parameters(cf.", "equation ) derived by fitting the parameterisedGNFW profile to the measured X-ray pressure profiles of fiveclusters in our 11 cluster sample.", "Note that β=5.49\\beta =5.49 is fixedand a prior of γ>0\\gamma > 0 is imposed.", "This latter constraint isenforced to avoid unphysical pressure gradients being allowed by theGNFW parameterisation.Figure: Constraints obtained from the re-analysis ofthe Planck and AMI observations for the five clusters for which high-qualityX-ray observations are available.", "These re-analyses adopted theGNFW shape parameters which best fit the X-ray data as given inTable .", "Comparison with the correspondingpanels in Fig.", "reveals no obvious improvement in thelevel of agreement between the Planck and AMI constraints.", "TheX-ray sizes indicated here are also derived from the GNFW fits tothe high-quality X-ray data.", "These are slightly different from theX-ray sizes plotted in Fig.", ", which were taken fromthe ESZ catalogue.Comparing with the corresponding constraints for these five clusters in our original analysis, we see that the updated constraints for A2034 have tightened significantly.", "This appears to be due to the fact that the previously used [13] GNFW profile was not a good match to this particular cluster's pressure profile, particularly in the central region of the cluster, where AMI is sensitive.", "The GNFW profile variant used to produce the updated constraints is a much better match to the measured X-ray profile and so the AMI data are better able to constrain the remaining cluster parameters.", "Apart from this single case, comparing with our original results, there does not appear to be a systematic improvement in the agreement between the Planck and AMI constraints when we move from the [13] profile to the best-fitting GNFW profile as measured from the individual X-ray observations.", "This, and similar reasoning based on an adaptation of these modified profiles to the other clusters in our sample, suggests that a more significant widening of the parameter space describing the cluster profiles will be required in order to simultaneously fit both the Planck and AMI SZ measurements for the entire cluster sample considered in this paper." ], [ "Conclusions", "We have studied the $Y_{500}-\\theta _{500}$ degeneracy from the SZ effect for a sample of 11 clusters ($0.11<z<0.55$ ) observed with both Planck and AMI.", "This is motivated by the fact that such a study can potentially break the well-known $Y$ -size degeneracy which commonly results from SZ experiments with limited resolution.", "Modelling the radial pressure distribution in each cluster using a universal GNFW profile, we have shown that there is significant overlap in the 2D posterior distributions for eight of the clusters.", "However, overall, AMI finds the SZ signal to be smaller in angular extent and fainter than Planck finds.", "The derived parameter degeneracies are significantly different for the two instruments.", "Hence, where the constraints from the two instruments are mutually consistent, their combination can be powerful in terms of reducing the parameter uncertainties.", "Significant discrepancies are found between the Planck and AMI parameter constraints for the remaining three clusters in our sample.", "We have investigated the origin of these discrepancies by carrying out a detailed analysis of a series of simulations assessing the potential impact of diffuse thermal emission from dust and residual contamination from imperfectly modeled radio point sources.", "Our simulations also include a number of systematic effects associated with the two instruments in addition to primordial CMB fluctuations and thermal noise.", "We find that the results of the simulations of both the Planck and AMI analyses are unbiased, confirming the accuracy of the two analysis pipelines and their corresponding methodologies.", "We have attempted to reconcile some of the discrepancies seen by re-analysing the Planck and AMI data adopting individual best-fitting pressure profiles, as measured from high-quality X-ray observations for five of the clusters in the sample.", "However, we do not observe a systematic improvement in the agreement between the Planck and AMI parameter constraints when we perform this re-analysis.", "We conclude that: either (i) there remain unaccounted for systematic effects in one or both of the data sets beyond what are included in our simulations; or (ii) a further expansion of the parameter space used to model the SZ cluster signal is required to simultaneously fit the Planck and AMI SZ data.", "Such further expansion of the model parameter space, which we leave for future studies, could potentially include using the Planck and AMI data in conjunction with X-ray observations to find a global fit for the GNFW shape parameters, going beyond the GNFW parameterisation to investigate other cluster profiles, and/or dropping the assumption of spherical symmetry for the SZ (and X-ray) emission.", "A description of the Planck Collaboration and a list of its members, indicating which technical or scientific activities they have been involved in, can be found at http://www.rssd.esa.int/Planck.", "The Planck Collaboration acknowledges the support of: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN and JA (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and DEISA (EU).", "The AMI telescope is supported by Cambridge University and the STFC.", "The AMI data analysis was carried out on the COSMOS UK National Supercomputer at DAMTP, University of Cambridge and the AMI Consortium thanks Andrey Kaliazin for computing assistance." ] ]

[ [ "On the lemniscate components containing no critical points of a\n polynomial except for its zeros" ], [ "Abstract We prove a sharp inequality for the modulus of the logarithmic derivative of a polynomial in the lemniscate components containing no critical points." ], [ " V.N.", "Dubinin ON THE LEMNISCATE COMPONENTS CONTAINING NO CRITICAL POINTS OF A POLYNOMIAL EXCEPT FOR ITS ZEROS Let $P$ be a complex polynomial of degree $n$ and let $E$ be a connected component of the set $\\lbrace z:|P(z)|\\le 1$ containing no critical points of P different from its zeros.", "We prove the inequality $|(z-a)P^{\\prime }(z)/P(z)|\\le n$ for all $z\\in E\\backslash \\lbrace a\\rbrace $ , where a is the zero of the polynomial $P$ lying in $E$ .", "Equality is attained for $P(z) = cz^n$ and any $z, c\\ne 0$ .", "Bibliography: 4 titles.", "Introduction Let $R$ be a rational function of degree $n$ represented as $R=P/Q$ , where $P$ and $Q$ are polynomials of degrees $n$ and not exceeding $n$ , respectively, which have no zeros in common.", "Assume that $R(0)=R^{\\prime }(0)\\ne 0$ .", "Sheil-Small [1, 10.3.2] posed a question on finding a neighborhood of the point $w=0$ in which one could distinguish a one-valued branch, $f(w)$ , of the inverse function $z=R^{-1}(w)$ , f(0)=0, satisfying the inequality ${\\rm Re}\\,\\frac{wf^{\\prime }(w)}{f(w)}\\ge \\frac{1}{n}$ The inequality obtained would be of interest in connection with the shape of the level curves $|R(z)|=const$ in the domain where the function $R$ is univalent.", "In the present note, a closely related problem for polynomials $P(Q\\equiv 1)$ is considered.", "More precisely, the following result is proved.", "Theorem.", "Let $P$ be a polynomial of degree not exceeding $n$ and let $E$ be a connected component of the lemniscate $|P(z)|\\le 1$ containing no critical points of the polynomial $P$ different from its zeros.", "Then, for any point $z\\in E\\backslash \\lbrace a\\rbrace $, $\\left|\\frac{(z-a)P^{\\prime }(z)}{P(z)}\\right|\\le n,\\qquad \\mathrm {(1)}$ where $a$ is the zero of the polynomial $P$ belonging to yhe component $E$ .", "Equality in (1) is attained for any point $z$ in the case where $P(z)=cz^n, c\\ne 0$ .", "If the component $E$ contains no critical points, then from (1) it follows that for the corresponding branch $f$ of the function inverse to the polynomial $P$ , the inequality $\\left|\\frac{wf^{\\prime }(w)}{f(w)-a}\\right|\\ge \\frac{1}{n}.$ holds in the disk $|\\omega |<1$ .", "This result is weaker than the Sheil-Small statement.", "However, contrary to [1], critical points in $E$ are allowed.", "The result obtained has the following geometric interpretation.", "Assume, for simplicity, that $a=0$ and let log ($\\cdot $ ) denote the one-valued branch of the logarithm mapping the plane slit along the real positive semiaxis onto a strip of width 2$\\pi $ .", "For any level curve $c(\\tau )(|P(z)|=\\tau <1)$ , the \"curve\" $\\gamma (\\tau )=\\log c(\\tau )$ connects the opposite sides of the strip mentioned; consequently,its length is no less than $2\\pi $ .", "On the other hand, the image $\\gamma (\\tau )$ under the map $\\log P(\\exp (\\cdot ))$ covers a vertical interval of length $2\\pi $ no more than $n$ times.", "Therefore, on the curve $\\gamma (\\tau )$ , there is a point $\\zeta $ at which the distortion coefficient satisfies the condition.", "$\\left|[\\log P(\\exp \\zeta )]^{\\prime }\\right|\\le n.$ This means that inequality (1) holds at a certain point $z=\\exp \\zeta $ of the level curve $c(\\tau )$ .", "The theorem of the present paper claims that this inequality holds at any point of the curve $c(\\tau )$ .", "The assumption that the set $E$ contains no critical points different from the polynomial zero is essential.", "For instance, for the polynomial $P(z)=z^3/2-3z^2/4$ , the lemniscate $|P(z)|\\le 1$ contains both critical points $z=0$ and $z=1$ , whence it is connected.", "The point $z=2$ belongs to this lemniscate, but $\\frac{2P^{\\prime }(2)}{P(2)}=6>3.$ Corollary.", "If, under the assumption of the theorem, the inequality $P(z)>0$ holds at point $z\\in E$ , then the polar derivative with respect to the point $a$ , $D_aP$ , satisfies the bound ${\\rm Re}\\,D_aP(z)={\\rm Re}\\,[nP(z)-(z-a)P^{\\prime }(z)]\\ge 0,\\qquad {\\rm Im}.$ The theorem will be proved in Sec.", "2.", "Ideologically, it comes back to the proof of Hayman's conjecture on coverings of vertical under a conformal mapping of the disk [2].", "§1.", "Auxiliary constructions and assertions Let $P$ be a polynomial of degree $n$ and let $E$ be a connected component of the lemniscate $|P(z)|\\le 1$ that contains no critical points of the polynomial $P$ other than its zeros (i.e., no points $\\zeta $ such that $P^{\\prime }(\\zeta )=0$ and $P(\\zeta )\\ne 0$ ).", "Let $a$ be the zero of $P$ lying in $E$ and let $z_0$ be a point of the component $E$ such that $P(z_0)>0$ .", "By $R$ denote the Riemann surface of the function $P^{-1}$ inverse to the polynomial $P$ .", "In what follws, we consider the function $P^{-1}$ as a one-valued function given on the surfase $R$ .", "Let $P:\\overline{\\mathbb {C}}_z\\rightarrow R$ be the function inverse to $P^{-1}$ in this sense.", "The projection of a point $W\\in R$ is defined as the point $P(P^{-1}(W))\\in \\overline{\\mathbb {C}}_w$ .", "Assume that the ray $\\lbrace w:{\\rm Im}\\,w=0,\\,0<{\\rm Re}\\,w<\\infty \\rbrace $ contains no critical values of the polynomial $P$ (i.e., no points $P(\\zeta )$ such that $P^{\\prime }(\\zeta )=0$ for a certain $\\zeta $ ).", "By $L$ denote the ray on the surface $R$ or, more exactly, the Jordan curve univalently lying over the above ray of the sphere $\\overline{\\mathbb {C}}_w$ and connecting the points $P(a)$ and $P(\\infty )$ .", "Let $T=\\lbrace t_k\\rbrace _{k=0}^m,\\;0=t_0<P(z_0)=t_1<\\ldots <t_{m-1}<t_m=\\infty $ , be a partition of the interval $0\\le t\\le \\infty $ containing all those values of $t$ in $1<t<\\infty $ at which the circle $\\gamma (t):=\\lbrace w:|w|=t\\rbrace $ contains at least one critical value $P(\\zeta )$ with $\\zeta \\in E$ .", "Finally, by $C(t)$ denote the closed Jordan curve on $R$ intersecting the ray $L$ and lying over the circle $\\gamma (t)$ whose orientation corresponds to the positive orientstion on the projection $\\gamma (t)$ , $0<t<\\infty ,\\;t\\notin T;\\;c(t)$ is the image of the curve $C(t)$ under the mapping $P^{-1}$ .", "Lemma 1.", "The argument increment $\\triangle _{ c(t)}\\arg P(z)$ is a nondecreasing function of $t$ on the set $\\lbrace t:0<t<\\infty ,\\,t\\notin T\\rbrace $ .", "Proof.", "Let $0<t^{\\prime }<t^{\\prime \\prime }<\\infty ,\\;t^{\\prime },t^{\\prime \\prime }\\notin T$ .", "The points $P(a)$ and $P(\\infty )$ are located on different sides of the curves $C(t^{\\prime })$ and $C(t^{\\prime \\prime })$ on the surfase $R$ .", "Therfore, the nonintersecting Jordan curves $ c(t^{\\prime })$ and $ c(t^{\\prime \\prime })$ separate the point $a$ from $\\infty $ on the sphere $\\overline{\\mathbb {C}}_z$ .", "Consequently, one of them lies in the interior of the other.", "Furthermore, as we move along the ray $L$ from the point $P(a)$ to the point $P(\\infty )$ , we first meet the curve $C(t^{\\prime })$ and then the curve $C(t^{\\prime \\prime })$ .", "This means thst the curve $c(t^{\\prime })$ lies in the interior of the urve $c(t^{\\prime \\prime })$ .", "Therefore, the number $N_{t^{\\prime }}$ of the zeros of the polynomial $P$ lying inside $c(t^{\\prime })$ does not exceed the number $N_{t^{\\prime \\prime }}$ of the zeros lying inside $c(t^{\\prime \\prime })$ (with account for their multiplicities).", "It remains to apply the argument principle: $2\\pi N_t=\\triangle _{ c(t)}\\arg P(z),\\qquad t=t^{\\prime },t^{\\prime \\prime }.$ The lemma is proved.", "The points $P(a)$ and $P(\\infty )$ lie on different sides of the curve $C(t)$ for any $t,0<t<\\infty , t\\notin T$ .", "This implies that for every $k=0,\\ldots ,m-1$ , the doubly-connected domain $G_k=\\bigcup \\limits _{t_k<t<t_{k+1}}C(t)$ also separates the points $P(a)$ and $P(\\infty )$ .", "At the same time, the curve $P(H),\\;H=\\lbrace z:z_0+(a-z_0)\\tau ,\\,1\\le \\tau \\le \\infty \\rbrace $ , connects these points.", "Therefore, for every $k=0,\\ldots , m-1$ , there is at least one Jordane arc $H_k$ , on the curve $P(H)$ that lies in the domain $G_k$ and connects its boundary components.", "Thus, in the above notation, the following assertion holds.", "Lemma 2.", "For any $k=0,\\ldots ,m-1$ the domain $G_k\\setminus H_k$ is simply connected.", "Below, we will need the notio of condenser capacity (e.g., see [3]).", "For sufficiently small positive $r$ and $\\rho $ on the sphere $\\overline{\\mathbb {C}}_z$ , consider the condensers $C(r)=(H,\\lbrace z:|z-z_0|\\le r\\rbrace )$        and $C(r,\\rho )=(H\\cup \\lbrace z:|z-a|\\le \\rho \\rbrace \\cup \\lbrace z:|z|\\ge 1/\\rho \\rbrace \\cup \\bigcup \\limits _{P^{\\prime }(\\zeta )=0}\\lbrace z:|z-\\zeta |\\le \\rho \\rbrace ,\\lbrace z:|z-1|\\le r\\rbrace ).$ Lemma 3.", "For a fixed $r,\\;0<r<|a-z_0|,$ the condensers capacities satialy the relation $\\lim \\limits _{\\rho \\rightarrow 0}{\\rm cap}\\,C(r,\\rho )={\\rm cap}\\,C(r).$ Proof.", "We make use of the continuity of the capacity and of the fact that the latter is invariant under addition of a finite number of points to points to the condenser's plates: $\\lim \\limits _{\\rho \\rightarrow 0}{\\rm cap}\\,C(r,\\rho )={\\rm cap}\\,\\left(H\\cup \\bigcup \\limits _{P^{\\prime }(\\zeta )=0}\\lbrace \\zeta \\rbrace ,\\lbrace z:|z-z_0|\\le r\\rbrace \\right)={\\rm cap}\\,C(r)$ (see Propositions 1.4 and 1.6 in [3]).", "This proves the lemma.", "Below, we introduce new notation and give some comments.", "$\\zeta =f_k(W)$ is the one-valued branch of the function $\\zeta =\\log (W/ P(z_0))$ that maps the domain $G_k\\setminus H_k$ conformally and univalently into the \"strip\" $\\Pi _k:=\\lbrace \\zeta :\\xi _{k}<{\\rm Re}\\,\\zeta <\\xi _{k+1}\\rbrace ,\\;k=0,\\ldots ,m-1$ .", "Here, $\\xi _k=\\log (t_k/P(z_0)),\\;k=0,1,\\ldots ,m$ .", "The choice of such a branch is feasible in view of Lemma 2.", "For $k=1$ and $k=m$ , $\\Pi _k$ is a half-plane.", "$u(z)$ is the potential function of the condenser $C(r,\\rho )$ , i.e., the resl-valued function continuous on $\\overline{\\mathbb {C}}_z$ , vanishing on the first plate of the condenser $C(r,\\rho )$ , equal to unity on its second plate, and harmonic in the complement to these plates: $v_k(\\zeta )=\\left\\lbrace \\begin{array}{ll} u\\left(P^{-1}(f_k^{-1}(\\zeta ))\\right),\\qquad &\\zeta \\in f_k(G_k\\setminus H_k),\\\\0,\\qquad & \\zeta \\in \\Pi _k\\setminus f_k(G_k\\setminus H_k),\\qquad k=0,\\ldots ,m-1\\end{array}\\right.$ On $\\partial \\Pi _k$ the function $v_k$ is defined by continuity.", "The function obtained in this way is also denoted by $v_k$ .", "As is not difficult to see, the function $v_k$ satisfies the Lipschitz condition in the strip $\\overline{\\Pi }_k,\\;k=0,\\ldots ,m-1$ , whereas the function $v_j$ is equal to unity on the set $f_j(P(\\lbrace z:|z-z_0|\\le r\\rbrace )\\cap G_j ),\\;j=0,1$ .", "$v_k^*(\\zeta )$ is the result of Steiner symmetrization of the function $v_k(\\zeta ),\\;\\zeta \\in \\overline{\\Pi }_k$ , with respect to the real axis (see [4]).", "Every function $v_k^*(\\zeta )$ is a Lipschitz function in $\\overline{\\Pi }_k$ and vanishes on the set $\\lbrace \\zeta \\in \\overline{\\Pi }_k:|{\\rm Im}\\,\\zeta |\\ge \\pi n\\rbrace ,\\;k=0,\\ldots ,m-1$ .", "Lemma 1 implies the following inequalities: $v_{k-1}^*(\\xi _k+i\\eta )\\le v_k^*(\\xi _k+i\\eta ),\\qquad -\\infty <\\eta <\\infty ,\\quad k=2,\\ldots ,m-1.\\qquad \\mathrm {(2)}$ $\\zeta =F(z)$ is the function that maps the unit disk $|z|<1$ , conformally and univalently, onto the strip $|{\\rm Im}\\,\\zeta |<\\pi n$ in such a way that $F(0)=0,\\;F^{\\prime }(0)>0$ .", "$\\widetilde{r}$ is the upper bound for all $r$ for which the set $F(\\lbrace z:|z|<r\\rbrace )\\cap \\lbrace \\zeta :{\\rm Re}\\,(-1)^j\\zeta <0\\rbrace $ belongs to the result of Steiner symmetrization with respect to the real axis of the set $f_j(P(\\lbrace z:|z-z_0|\\le r\\rbrace )\\cap G_j)$ for $j=0$ and $j=1$ .", "$v(\\zeta )$ is the potential function of the condenser $\\widetilde{C}(\\widetilde{r})=(\\overline{\\mathbb {C}}_{\\zeta }\\setminus \\lbrace \\zeta :|{\\rm Im}\\,\\zeta |<\\pi n\\rbrace ,F(\\lbrace z:|z|\\le \\widetilde{r}\\rbrace ))$ .", "It is readily seen that $\\begin{array}{lll}\\dfrac{\\partial v}{\\partial \\xi }=0\\quad &\\mbox{on the line}\\quad {\\rm Re}\\,\\zeta =0,\\\\\\;&\\;\\\\\\dfrac{\\partial v}{\\partial \\xi }\\le 0\\quad &\\mbox{on every line}\\quad {\\rm Re}\\,\\zeta =\\xi >0.\\end{array}\\qquad \\mathrm {(3)}$ The level curves of the potential function $v$ coincide with the level curves of the function $F$ (i.e., with the curves $|F^{-1}(\\zeta )|=\\textrm {const}$ ).", "Given a sufficiently smooth function $\\lambda $ on an open set $\\Omega \\subset \\mathbb {C}$ , denote $I(\\lambda ,\\Omega )=\\int \\limits _{\\Omega }|\\nabla \\lambda |^2d\\sigma .$ Lemma 4.", "The following inequality holds: $\\sum \\limits _{k=0}^{m-1} I(v_k^*,\\Pi _k)\\ge I(v,\\mathbb {C}).$ Proof.", "Set $G_k=\\lbrace \\zeta \\in \\Pi _k:|{\\rm Im}\\,\\zeta |<\\pi n\\rbrace ,\\;k=0,1,\\ldots ,m-1$ , and $l_k=\\lbrace \\zeta :{\\rm Re}\\,\\zeta =\\xi _k,\\;|{\\rm Im}\\,\\zeta |<\\pi n\\rbrace ,\\;k=2,\\ldots ,m-1$ .", "For every $k,\\;0\\le k\\le m-1$ , we have $I(v_k^*,\\Pi _k)=I(v_k^*,G_k)=I(v_k^*-v+v,G_k)=I(v_k^*-v,G_k)+I(v,G_k)+$ $+2\\iint \\limits _{G_k}\\left[\\frac{\\partial (v_k^*-v)}{\\partial \\xi }\\frac{\\partial v}{\\partial \\xi }+\\frac{\\partial (v_k^*-v)}{\\partial \\eta }\\frac{\\partial v}{\\partial \\eta }\\right]d\\xi d\\eta \\ge I(v,G_k)-2\\int \\limits _{\\partial G_k}(v_k^*-v)\\frac{\\partial v}{\\partial n}ds,$ where $\\partial /\\partial n$ means differentiation along the inward normal to the boundary of the domain $G_k$ (angle points are excluded).", "With account for relations (2) and (3), we derive $\\sum \\limits _{k=0}^{m-1}I(v_k^*,\\Pi _k)\\ge \\sum \\limits _{k=0}^{m-1}I(v,G_k)-2\\sum \\limits _{k=1}^{m-1}\\int \\limits _{\\partial G_k}(v_k^*-v)\\frac{\\partial v}{\\partial n}ds=$ $=\\sum \\limits _{k=0}^{m-1}I(v,G_k)-2\\sum \\limits _{k=2}^{m-1}\\int \\limits _{l_k}\\left[(v_{k-1}^*-v)\\left(-\\frac{\\partial v}{\\partial \\xi }\\right)+(v_k^*-v)\\frac{\\partial v}{\\partial \\xi }\\right] ds=$ $=\\sum \\limits _{k=0}^{m-1}I(v,G_k)+2\\sum \\limits _{k=2}^{m-1}\\int \\limits _{l_k}(v_{k-1}^*-v_k^*)\\frac{\\partial v}{\\partial \\xi }ds\\ge I(v,\\mathbb {C})$ This completes the proof.", "§2.", "Proof of the theorem It is sufficient to prove inequality (1) at an arbitrary point $z_0\\in E\\setminus \\lbrace a\\rbrace $ such that $P(z_0)>0$ under the assumption that the ray $\\lbrace w:{\\rm Im}\\,w=0,\\,0<{\\rm Re}\\,w<\\infty \\rbrace $ contains no critical points of the polynomial $P$ .", "We use the notation introduced in Sec.", "1.", "The chain of relations below stems from the conformal invariance of the Dirichlet integral, from the Pólya and Szegő theorem on function symmetrization (see [4]), and from Lemma 4: ${\\rm cap}\\,C(r,\\rho )=I(u,\\mathbb {C})\\ge \\sum \\limits _{k=0}^{m-1}I(v_k,\\Pi _k)\\ge \\sum \\limits _{k=0}^{m-1} I(v_k^*,\\Pi _k)\\ge $ $\\ge I(v,\\mathbb {C})={\\rm cap}\\,\\widetilde{C}(\\widetilde{r}).$ In view of Lemma 3, we ultimately obtain ${\\rm cap}\\,C(r)\\ge {\\rm cap}\\,\\widetilde{C}(\\widetilde{r}).\\qquad \\mathrm {(4)}$ In order to compute the asymptotics of the condenser capacity as $r\\rightarrow 0$ , we use known formulas (e.g., see [3, (1.6) and (1.8)]), in which $r(B,a)$ stands for the inner radius of the domain $B$ with respect to a point $a\\in B$ .", "As a result, we obtain ${\\rm cap}\\,C(r)=-\\frac{2\\pi }{\\log r}-\\frac{1}{2\\pi }(\\log r(\\mathbb {C}_z\\setminus H,z_0))\\left(\\frac{2\\pi }{\\log r}\\right)^2+o\\left(\\left(\\frac{1}{\\log r}\\right)^2\\right)=$ $=-\\frac{2\\pi }{\\log r}-2\\pi (\\log [4|a-z_0|])\\left(\\frac{1}{\\log r}\\right)^2+o\\left(\\left(\\frac{1}{\\log r}\\right)^2\\right),\\qquad r\\rightarrow 0.$ Further, the second plate of the condenser $\\widetilde{C}(\\widetilde{r})$ is an \"almost disk\" of radius $(r|P^{\\prime }(z_0)|/P(z_0))(1+o(1))$ as $r\\rightarrow 0$ .", "Consequently, ${\\rm cap}\\,\\widetilde{C}(\\widetilde{r})=-\\frac{2\\pi }{\\log (r|P^{\\prime }(z_0)|/P(z_0))}-$ $-2\\pi (\\log r(\\lbrace \\zeta :|{\\rm Im}\\,\\zeta |<\\pi n\\rbrace ,0))\\left(\\frac{1}{\\log (r|P^{\\prime }(z_0)|/P(z_0))}\\right)^2+$ $+o\\left(\\left(\\frac{1}{\\log r}\\right)^2\\right)=-\\frac{2\\pi }{\\log (r|P^{\\prime }(z_0)|/P(z_0))}-$ $-2\\pi (\\log (4n))\\left(\\frac{1}{\\log (r|P^{\\prime }(z_0)|/P(z_0))}\\right)^2+o\\left(\\left(\\frac{1}{\\log r}\\right)^2\\right)=$ $=-\\frac{2\\pi }{\\log r}\\left(1-\\frac{\\log |P^{\\prime }(z_0)/P(z_0)|}{\\log r}+o\\left(\\frac{1}{\\log r}\\right)\\right)-$ $-2\\pi (\\log (4n))\\left(\\frac{1}{\\log r}\\right)^2+o\\left(\\left(\\frac{1}{\\log r}\\right)^2\\right)=-\\frac{2\\pi }{\\log r}-$ $-2\\pi (\\log |4nP(z_0)/P^{\\prime }(z_0)|)\\left(\\frac{1}{\\log r}\\right)^2+o\\left(\\left(\\frac{1}{\\log r}\\right)^2\\right),\\qquad r\\rightarrow 0.$ Substituting the asymptotics obtained into inequality (4), we arrive at the inequality $|a-z_0|\\le |nP(z_0)/P^{\\prime }(z_0)|.$ Under the assumptions considered, the latter relation coincides with (1) $(z=z_0)$ .", "The case of equality is verified straightforwardly.", "The theorem is proved.", "Ëèòåðàòóðà T.Sheil-Small, Complex polynomials, Cambridge Univ.", "Press., Cambridge (2002).", "V.N.", "Dubinin, \"Coverings of vertical segments under a conformal mapping\", Mat.", "Zametki 28, No.", "1, 25-32 (1980).", "V.N.", "Dubinin, \"Symmetrization in the geometric theory of function of a complex variable\", Usp.", "Mat.", "Nauk, 49, No.", "1, 3-76 (1994).", "W.K.Hayman, Multivalent functions, Cambridge Univ.", "Press., Cambridge (1994).", "T.Sheil-Small, Complex polynomials, Cambridge Univ.", "Press., Cambridge (2002).", "V.N.", "Dubinin, \"Coverings of vertical segments under a conformal mapping\", Mat.", "Zametki 28, No.", "1, 25-32 (1980).", "V.N.", "Dubinin, \"Symmetrization in the geometric theory of function of a complex variable\", Usp.", "Mat.", "Nauk, 49, No.", "1, 3-76 (1994).", "W.K.Hayman, Multivalent functions, Cambridge Univ.", "Press., Cambridge (1994)." ] ]